Technical Report 2005–12
Ka Chun Yu
Lecture Notes for
Introduction to Cosmology
Denver – 2005
DMNS Technical Report 2005-12, 167 pages total
Lecture Notes for
Introduction to Cosmology
October, 2005
by
Dr. Ka Chun Yu
Curator of Space Science
Denver Museum of Nature & Science
Available at https://scientists.dmns.org/sites/kachunyu
Contents
1 Cosmology From Ancient to Modern Times
1.1 The Ancients . . . . . . . . . . . . . . . . .
1.2 European Thought Before the 20th Century
1.3 The Beginnings of Modern Science . . . . .
1.4 The Copernican Revolution . . . . . . . . .
1.5 Modern Cosmology . . . . . . . . . . . . . .
1.6 The Expanding Universe . . . . . . . . . . .
2 What Is In the Observable Universe?
2.1 The Extra-Galactic Zoo . . . . . . . .
2.2 Baryonic Composition of Galaxies . . .
2.3 Dark Matter Composition of Galaxies .
2.4 Galaxy Clusters and Superclusters . . .
2.5 The Cosmic Distance Ladder . . . . . .
2.5.1 Trigonometric Techniques . . .
2.5.2 Standard Candles . . . . . . . .
2.5.3 Cepheid Variables . . . . . . . .
2.5.4 Other Standard Candles . . . .
2.5.5 Redshifts and the Hubble Flow
2.6 Galaxy Cluster Mass . . . . . . . . . .
2.7 More on Dark Matter . . . . . . . . . .
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3 Theoretical Universes
3.1 The Curvature of Space-Time . . . . . . . . . . . . . .
3.2 The Distribution of Matter and Energy in the Universe
3.3 Modeling the Universe . . . . . . . . . . . . . . . . . .
3.3.1 Einstein’s Universe . . . . . . . . . . . . . . . .
3.3.2 The de Sitter Universe . . . . . . . . . . . . . .
3.3.3 The Friedmann-Robertson-Walker Universes . .
3.4 Cosmological Redshifts and the Hubble Constant . . .
3.5 The Critical Density . . . . . . . . . . . . . . . . . . .
3.6 The Age of the Universe . . . . . . . . . . . . . . . . .
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iv
CONTENTS
4 The
4.1
4.2
4.3
Big Bang
Cosmic Element Abundances . . . . . . . . . . . . . . . . . . . . . . .
The Cosmic Microwave Background . . . . . . . . . . . . . . . . . . .
The Steady-State Universe . . . . . . . . . . . . . . . . . . . . . . . .
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5 The
5.1
5.2
5.3
Accelerating Universe
Deceleration Parameter . . . . . . . . . . . . . . . . . . . . . . . . . .
Type Ia Supernovae . . . . . . . . . . . . . . . . . . . . . . . . . . . .
More on Dark Energy . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
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6 Anisotropies in the Cosmic Microwave Background
97
6.1 Analyzing the Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . 99
6.2 Origins of the Power Spectrum . . . . . . . . . . . . . . . . . . . . . 105
6.3 Analyzing the Anisotropies . . . . . . . . . . . . . . . . . . . . . . . . 110
7 Structure Formation in the Universe
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7.1 The Millenium Simulation . . . . . . . . . . . . . . . . . . . . . . . . 114
8 Inflation and the Early Universe
8.1 Problems With the Big Bang . . . . . . . . . . . . . . . . . . . . . .
8.2 Problems Solved? . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3 The Earliest Universe . . . . . . . . . . . . . . . . . . . . . . . . . . .
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9 Rampant Speculation
133
9.1 Chaotic and Eternal Inflation . . . . . . . . . . . . . . . . . . . . . . 133
9.2 Parallel Universes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
9.3 The End of the Universe . . . . . . . . . . . . . . . . . . . . . . . . . 139
A Notation and Astronomical Units
A.1 Scientific Notation . . . . . . . .
A.2 Scientific Units . . . . . . . . . .
A.3 Distances . . . . . . . . . . . . .
A.4 Magnitudes . . . . . . . . . . . .
A.5 Angular Measurements . . . . . .
A.6 Other Astronomical Units . . . .
Further Reading
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147
List of Figures
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
Thales, Anaxagoras, Aristotle & Ptolemy . . . . . . .
Nicolaus Copernicus, Giordano Bruno, & Tycho Brahe
Johannes Kepler, Galileo Galilei, & Sir Isaac Newton .
Immanuel Kant & Sir William Herschel . . . . . . . .
Harlow Shapley & Herbert Curtis . . . . . . . . . . .
Albert Einstein & Aleksandr Friedmann . . . . . . . .
Edwin Hubble . . . . . . . . . . . . . . . . . . . . .
Einstein at Mt. Wilson . . . . . . . . . . . . . . . . .
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2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13
2.14
2.15
2.16
2.17
2.18
2.19
2.20
2.21
2.22
2.23
2.24
Elliptical Galaxies . . . . . . . . . . . . . . . .
Unbarred Spiral Galaxies . . . . . . . . . . . . .
Barred Spiral Galaxies . . . . . . . . . . . . . .
Lenticular Galaxies . . . . . . . . . . . . . . . .
The Magellanic Clouds . . . . . . . . . . . . . .
More Irregular Galaxies . . . . . . . . . . . . .
The elliptical galaxy NGC 3923 . . . . . . . . .
Population synthesis Model . . . . . . . . . . .
Rotation Curve of a Rigid Body . . . . . . . . .
Rotation Curve of a Planetary System . . . . . .
Rotation Curve Based on Milky Way Gas Clouds
Rotation Curve of the Milky Way . . . . . . . .
Satellite Galaxies of the Milky Way . . . . . . .
The Local Group . . . . . . . . . . . . . . . . .
The Virgo Supercluster . . . . . . . . . . . . . .
The Virgo Cluster . . . . . . . . . . . . . . . .
The Coma Cluster . . . . . . . . . . . . . . . .
Neighboring Superclusters . . . . . . . . . . . .
2-Degree Field Galaxy Survey . . . . . . . . . .
The APM Survey of Galaxies . . . . . . . . . .
The Observable Universe . . . . . . . . . . . . .
Actual Size/Angular Size Relation . . . . . . . .
The Cosmic Distance Ladder . . . . . . . . . . .
Trigonometric Parallax . . . . . . . . . . . . . .
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vi
LIST OF FIGURES
2.25
2.26
2.27
2.28
2.29
2.30
2.31
Cepheid Variable Light Curves . . . . . . . . . .
Cepheid Variable Light Curves, Example II . . .
Cepheid Variable Light Curves, Example III . . .
Tully-Fisher Relationship . . . . . . . . . . . . .
Hubble Relation for Galaxies and Galaxy Clusters
Hydra A Galaxy Cluster . . . . . . . . . . . . .
Gravitational Lensing by Abell 2218 . . . . . . .
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3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
Einstein’s View of Space-Time . . . . . . . . .
The Curvature of Space-Time and Light . . . .
Pythagoras’ Theorem . . . . . . . . . . . . . .
Infinitesimal Pythagoras’ Theorem . . . . . . .
Curved Two-Dimensional Surfaces . . . . . . .
Flattened Circulars . . . . . . . . . . . . . . .
The Scale-Factor at Two Different Times . . .
Friedmann-Robertson-Walker Model Universes
Cosmological Redshift . . . . . . . . . . . . .
ΩΛ vs. Ωm . . . . . . . . . . . . . . . . . . .
Scale-Factor R Over Time t . . . . . . . . . .
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4.1
4.2
4.3
4.4
Light Element Abundances from Big Bang Nucleosynthesis
Seeing to the Edge of the Universe . . . . . . . . . . . . .
COBE’s View of the Cosmic Microwave Background . . . .
COBE Spectrum . . . . . . . . . . . . . . . . . . . . . .
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5.1
5.2
5.3
5.4
5.5
5.6
Evolution of Accelerating, Decelerating, and Constantly Expanding Universes
Hubble’s 1929 Velocity-Distance Relationship . . . . . . . . . . . . . . .
High-z vs. Distance Relationship for Galaxies . . . . . . . . . . . . . . .
Type Ia Supernovae Light Curve Shapes . . . . . . . . . . . . . . . . . .
Supernova Cosmology Project Hubble Diagram . . . . . . . . . . . . . .
Confidence Levels for Ωm and ΩΛ . . . . . . . . . . . . . . . . . . . . .
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
6.10
6.11
Blackbody Spectra . . . . . . . . . . . . . .
Maps of the Earth . . . . . . . . . . . . . .
The BOOMERanG Field . . . . . . . . . . .
The CMB from WMAP and COBE . . . . .
Sound Wave Description . . . . . . . . . . .
Water Waves in the Ocean . . . . . . . . . .
Sky Maps with Corresponding Power Spectra
CMB Power Spectrum Summary . . . . . . .
CMB Power Spectrum Data and Fit . . . . .
Origin of Acoustic Waves . . . . . . . . . . .
Origin of the Acoustic Peaks . . . . . . . . .
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108
109
LIST OF FIGURES
7.1
7.2
7.3
7.4
Millenium
Millenium
Millenium
Millenium
8.1
8.2
8.3
8.4
Expansion from Inflation . . . . . . .
The Inflaton Potential . . . . . . . .
Inflation Solves the Flatness Problem
Unification of Forces . . . . . . . . .
9.1
9.2
vii
Run Summary . . . . . . . .
Simulation at Large Scales .
Simulation at Medium Scales
Simulation at Small Scales .
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128
Chaotic Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Eternal Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
134
135
.
.
.
.
viii
LIST OF FIGURES
List of Tables
2.1
Population Model of M 31 Bulge: The derived mass and V-band luminosity
fraction of stars within each spectral type range. . . . . . . . . . . . . . .
24
6.1
Properties of the Universe . . . . . . . . . . . . . . . . . . . . . . . . .
111
8.1
State of the Universe Over Time . . . . . . . . . . . . . . . . . . . . . .
129
ix
Chapter 1
Cosmology From Ancient to
Modern Times
Cosmology is the branch of science concerned with the study of the entire Universe as
a whole. The fundamental questions that are asked by cosmologists include: “What
is the Universe made of?” “What is its structure?” “How did the Universe begin?”
“What is its eventual fate?” Long before we had the technology to enable us to firmly
answer them, these questions have interested philosophical and religious thinkers since
ancient times.
We are therefore very fortunate to be living in a time when we have the technology
and know-how to begin answering these questions. Cosmology is not a static, stodgy
science. Progresss in discoveries are coming quicker than ever before. Although we
will be covering much of what has been discovered from the early years of the 20th
century, we will also concentrate on major discoveries in the last ten years. We are
entering a period of “precision cosmology,” first coined by the American cosmologist
Michael Turner. Cosmological theories are being put to the test by observations.
Instead of theorists conjecturing wildly about the past, current and future states of
the Universe, they are being constrained by what observers are finding from the latest
telescopes and instrumentation. What is also exciting is that from multiple lines of
independent evidence, we are starting to piece together a very coherent, consistent
picture of the Universe.
For this course, I am assuming a basic understanding of astronomy as a prerequisite. If you have an amateur’s interest in the field, read the occasional issue of Sky &
Telescope or Astronomy at the dentist’s office, or took an Astro 101 course a decade or
two ago, then you are probably prepared. If this is your first exposure to astronomy
in the formal school or informal museum education sense, then there might be some
rough patches. Although there will be opportunities for defining terms, more often
than not, we will be plunging ahead assuming you know basic definitions like star,
galaxy, spectrum, and Doppler shift. If you feel rusty about what a light year or an
absolute magnitude is, you may want to check out the Appendix to get brief primers.
1
2
1.1
CHAPTER 1. COSMOLOGY FROM ANCIENT TO MODERN TIMES
The Ancients
Before we jump into the scientific details of what we know of the Universe today,
and how we know it, it is useful to remind ourselves of what human beings have
historically thought about the Universe. Every single culture on Earth has their
own unique creation myth. This fact suggests that humanity has an innate desire
to understand the origins of not just ourselves, but everything that we can perceive.
The scientific field of cosmology is merely a modern extension of this primitive need.
The Babylonians are as good a place to start as any other, when discussing
the history of cosmology. They were some of the first astronomers. They invented a
sexagesimal (base 60) numbering system that is reflected in our modern day usage of
seconds, minutes, and hours, and degrees. Babylonian astronomers kept careful logs
of the motions of the Moon and the planets in the sky in order to predict the future
using astrology. Their beliefs about the Universe are typical of pre-scientific peoples.
They believed in a cosmology where the Earth was at the center of the Universe,
bound below by water. The seven heavenly bodies that moved in the sky represented
deities, with each one moving in a progressively further sphere from the Earth. (In
order, they were the Moon, Mercury, Venus, the Sun, Mars, Jupiter, and Saturn.)
The fixed stars lay beyond Saturn, and beyond that was more water binding the outer
edge of the known Universe.
The Babylonian creation myth is similar also to that of most other cultures. The
gods were involved in the creation of the Universe, and making of form in the cosmos
from formlessness and chaos. The Babylonian tale of Genesis can be found in the
Enuma Elish (c. 1450 BCE, but evolving from much older Sumerian myths). Here an
epic primordial struggle occurs between the god Marduk, representing order, and the
dragon Tiamat, representing chaos. Marduk comes out victorious, and he divides up
the body of Tiamat, with half going to construct the heavens, and the other half to
build the Earth. Thus even early peoples recognized the need for explaining how the
structure of the Universe got the way it was, and the Babylonians were not alone in
suggesting that order was shaped from chaos. This need to understand our origins is
a quest that continues today in astronomy.
The Rig Vedas were Hindu texts that date back to 1000 BCE. Part of them
discussed the cyclical nature of the universe. The universe underwent a cycle of
rebirth followed by fiery destruction, as the result of the dance of Shiva. The length
of each cycle is a “day of Brahma” which lasts 4.32 billion years (which coincidentally
is roughly the age of our Earth and only a factor a few off from the actual age of the
universe). The cosmology has the Earth resting on groups of elephants, which stand
on a giant turtle, who in turn is supported by the divine cobra Shesha-nāga.
The Ancient Greeks: Although early Greek thought on the heavens mirrored
that of the Babylonians, with a reliance on gods and myths, by the 7th century
BCE, a new class of thinkers, relying in part on observations of the world around
them, began to use logic and reason to arrive at theories of the natural world and
1.1. THE ANCIENTS
3
of cosmology. Although the ancient Greek philosophers had a variety of ideas about
the nature of the universe—many of which we know to be incorrect—we can begin to
sense a thread of discovery and inquiry that will eventually lead to modern science.
ˆ Thales of Miletus (634–546 BCE) believed the Earth was a flat disk surrounded
by water.
ˆ Anaxagoras (ca. 500–ca. 428 BCE) believed the world was cylindrically shaped,
and we lived on the flat-topped surface. This world cylinder floats freely in space
on nothingness, with the fixed stars in a spherical shell that rotated about the
cylinder. The Moon shone as a result of reflected light from the Sun, and lunar
eclipses were the result of the Earth’s shadow falling on the Moon.
Figure 1.1: Thales, Anaxagoras, Aristotle & Ptolemy
Left to right: Thales, Anaxagoras, Aristotle, and Claudius Ptolemy.
ˆ Eudoxus of Cnidus (ca. 400–ca. 347 BCE) also had a geocentric model for the
Earth, but added in separate concentric spheres for each of the planets, the Sun,
and the Moon, to move in, with again the fixed stars located on an outermost
shell. Each of the shells for the seven heavenly bodies moved at different rates
to account for their apparent motions in the sky. To keep the model consistent
with observations of the planets’ motions, Eudoxus’ followers added more circles
to the mix—for instance, seven were needed for Mars. The complexity of this
system soon made his model unpopular.
ˆ Aristotle (384–322 BCE) refined the Eudoxus model, by adding more spheres
to make the model match the motions of the planets, especially that of the retrograde motions seen in the outermost planets. Aristotle believed that “nature
abhors a vacuum,” so he postulated a universe that was filled with crystalline
spheres moving about the Earth. Aristotle also believed that the universe was
eternal and unchanging. Outside of the fixed sphere of stars was “nothingness.”
ˆ Aristarchus (ca. 310–ca. 230 BCE) made a first crude determination of the
relative distance between the Moon and the Sun. His conclusion was that the
4
CHAPTER 1. COSMOLOGY FROM ANCIENT TO MODERN TIMES
Sun was 20× further, and the only reason they appeared to be of the same size
was that the Sun was also 20× larger in diameter. Aristarchus then wondered,
if the Sun was so much larger, does it really make sense for it to move around
the Earth? Would it make more sense for the Earth to circle it?
ˆ Claudius Ptolemy (ca. 100–ca. 170 CE) writing in Syntaxis (aka Almagest;
∼ 140 CE) took the basic ideas of Eudoxus’ and Aristotle’s cosmology, but had
the planets move in circular epicycles, the centers of which then moved around
the Earth on the deferent, an even bigger orbit. Ptolemy’s ideas gave the most
accurate explanations for the motion of the planets (as best as their positions
were known at the time). (Ptolemy’s and Aristotle’s ideas about the universe
and its laws of motion remained the dominant idea in Western thought until
the 15th century CE!)
1.2
European Thought Before the 20th Century
Nicolaus Copernicus (1473–1543) made a radical break from Ptolemaic thought
by proposing that the Earth was not at the center of the universe. In his De Revolutionibus Orbium Celestium, he believed a Sun-centered universe to be more elegant:
In no other way do we perceive the clear harmonious linkage between the
motions of the planets and the sizes of their orbs.
Although Copernicus made an immense leap by moving the displacing the Earth from
the center of the Universe, in other ways he was still stuck with the past. The heavens
were still moving via perfect circular motions. In order for his model to accurately
reflect the actual motions of the planets, Copernicus still had to use the motion of
smaller circles, known as an epicyclet, that orbited an offset circle.
However it cannot be understated how revolutionary was Copernicus’ De Revolutionibus. Displacing the Earth from the center of the Universe was an attack on
a worldview held by all serious medieval European thinkers, and one which had a
lineage that could be traced back more than two millenia. Copernicus was attacked
immediately by both Catholic Church officials and by leaders of the Protestant movement. However he died soon after the publication of his work and so was spared the
wrath of the authorities. Others who also had heretical views about the Universe
were not so lucky. Galileo Galilei (see below) was forced to recant by the Church,
and Giordano Bruno (also more below) was burned at the stake.
Thomas Digges (1546–1595), a leading English admirer of Copernicus, published
A Perfect Description of the Celestial Orbes, which re-stated Copernicus’ heliocentric
theory. However Digges went further by claiming that the universe is infinitely large,
and filled uniformly with stars. This is one of the first pre-modern statements of the
Cosmological Principle.
1.3. THE BEGINNINGS OF MODERN SCIENCE
5
Figure 1.2: Nicolaus Copernicus, Giordano Bruno, & Tycho Brahe
Left to right: Nicolaus Copernicus, Giordano Bruno, and Tycho Brahe.
Giordano Bruno (1548–1600) goes even further: not only are there an infinite
number of stars in the sky, but they are also suns with their own solar systems, and
orbited by planets filled with life. These and other heretical ideas (e.g., that all these
other life-forms, planets, and stars also had their own souls) resulted in him being
imprisoned, tortured, and finally burned at the stake by the Church.
Tycho Brahe (1546–1601) made and recorded very careful naked eye observations
of the planets, which revealed flaws in their positions as tabulated in the Ptolemaic
system. He played with a variety of both geocentric and heliocentric models.
Johannes Kepler (1571–1630) finally was able to topple the Ptolemaic system
by proposing that planets orbited the Sun in ellipses, and not circles. He proposed
his three laws of planetary motion. In 1610, Kepler also first pointed out that an
infinite universe with an infinite number of stars would be extremely bright and hot.
This issue was taken up again by Edmund Halley (1656–1742) in 1720 and Heinrich
Olbers (1758–1840) in 1823. Olbers suggested that the universe was filled with dust
that obscured light from the most distant stars. Only 20 years later, John Herschel
showed that this explanation would not work. The problem of Olber’s paradox would
not be resolved until the 20th century.
1.3
The Beginnings of Modern Science
Galileo Galilei (1564–1642) found observational evidence for heliocentric motion,
including the phases of Venus and the moons of Jupiter. When he pointed his telescope at the Milky Way, the dim, nebulous band in the sky revealed itself to contain
a myriad of faint stars that were otherwise invisible to the naked eye. He not only
supported a heliocentric view of the universe in his book Dialogue on the Two Great
World Systems, but his work on motion also attacked Aristotelian thought.
Sir Isaac Newton (1642–1727) discovered the mathematical laws of motion and
6
CHAPTER 1. COSMOLOGY FROM ANCIENT TO MODERN TIMES
Figure 1.3: Johannes Kepler, Galileo Galilei, & Sir Isaac Newton
Left to right: Johannes Kepler, Galileo Galilei, and Sir Isaac Newton.
gravitation that today bear his name. His Philosophiae Naturalis Principia Mathematica—or simply, the Principia—was the first book on theoretical physics, and
provided a framework for interpreting planetary motion. He was thus the first to
show that the laws of motion which applied in laboratory situations, could also apply
to the heavenly bodies.
Newton also wrote about his own view of a cosmology with a static universe in
1691: he claimed that the universe was infinite but contained a finite number of stars.
Self gravity would cause such a system to be unstable, so Newton believed (incorrectly) that the finite stars would be distributed infinitely far so that the gravitational
attraction of stars exterior to a certain radius would keep the stars interior to that
radius from collapsing.
The English astronomer Thomas Wright (1711–1786) published An Original
Theory or New Hypothesis of the Universe (1750), in which he proposed that the
Milky Way was a grouping of stars arranged in a thick disk, with the Sun near the
center. The stars moved in orbits similar to the planets around our Sun.
Immanuel Kant (1724–1804), the German philosopher, inspired by Wright, proposed that the Milky Way was just one of many “island universes” in an infinite
space. In his General Natural History and Theory of Heaven (1755), he writes of the
nebulous objects that had been observed by others (including Galileo!), and reflects
on what the true scale of the universe must be:
Because this kind of nebulous stars must undoubtedly be as far away from
us as the other fixed stars, not only would their size be astonishing (for
in this respect they would have to exceed by a factor of many thousands
the largest star), but the strangest point of all would be that with this
extraordinary size, made up of self-illuminating bodies and suns, these
stars should display the dimmest and weakest light.
1.4. THE COPERNICAN REVOLUTION
7
Figure 1.4: Immanuel Kant & Sir William Herschel
Immanuel Kant (left) and Sir William Herschel (right).
Sir William Herschel (1738–1822) and his son John (1792–1871) used a telescope, based on a design by Newton, to map the nearby stars well enough to conclude
that the Milky Way was a disk-shaped distribution of stars, and that the Sun was
near the center of this disk. He mapped some 250 diffuse nebulae, but thought they
were really gas clouds inside our own Milky Way. Others however took Kant’s view
that the nebulae were really distant galaxies. The German mathematician Johann
Heinrich Lambert (1728–1777) adopted this idea, plus he discarded heliocentrism,
believing the Sun to orbit the Milky Way like all of the other stars.
1.4
The Copernican Revolution
The Copernican Revolution radically changed the philosophical mindset of humanity’s
role in the Universe. It can be summarized by the statement that there are no
“special” observers. We have found ourselves to be less special, more marginalized
in the grand scope of things, and not necessarily occupying the center of anything,
never mind the Universe. Not only was the Earth displaced to be just a small body
orbiting the Sun, but the Sun itself was not unique. It was merely one star out of
billions orbiting by their mutual gravity in a giant stellar conglomeration known to
us today as the Milky Way Galaxy. And even in the last century, the Milky Way was
shown to be among hundreds of billions of other galaxies in the known, observable
Universe. The idea of this Copernican Principle has spread far beyond the realm
of the astronomy, since some philosophers of science have even associated Charles
Darwin (1809–1882) and his landmark work as being Copernican in spirit. From
Darwin’s On the Origin of Species (1859) and The Descent of Man (1871), humanity
was not specially created but could now be linked via common ancestors to the rest
of the animal kingdom. Even Sigmund Freud (1856–1939) was said to have removed
8
CHAPTER 1. COSMOLOGY FROM ANCIENT TO MODERN TIMES
the mind from its godly pedestal.
The Copernican Principle is merely a guideline to scientific thinking; it cannot be
proved in a mathematical sense and has no roots in any particular physical model
or theory. However like Occam’s Razor, it has been an extremely useful principle to
follow and has proven to be true from empirical observations. It is in the back of
many scientists’ minds as they consider what their research implies about the nature
of the Universe.
1.5
Modern Cosmology
The Copernican Principle takes its form in modern cosmology in the Cosmological
Principle: that the Universe is homogeneous and isotropic. Homogeneity any one
place is like any other. Isotropy implies that any direction one looks will be similar
to any other direction. There is nothing special about where we are, nor is there a
preferred direction. This may not be true on small scales, such as that of our Solar
System, or even that of our galaxy and nearby galaxy clusters. But on the scale of
galactic superclusters, hundreds of millions of light years across, the Universe does
appear to obey these two precepts quite well.
But we are getting slightly ahead of ourselves. Early in the 20th century, the
nature of our Milky Way and the nebulae was still unresolved. The location of the
Sun inside the Milky Way, and the nature of the diffuse nebulae remained the subjects
of heated debates.
Harlow Shapley (1885–1972), an American astronomer, observed globular clusters and the RR Lyrae variable stars in them. From their directions and distances,
he was able to show that they were placed in a spherical distribution not centered
on the Sun, but at a point nearly 50,000 light years away. (We know today that
Shapley over-estimated his distance by a factor of two.) The Copernican revolution
was almost complete: not only was the Earth not at the center of the universe, but
the Sun was far from the center of the Milky Way as well.
The American astronomer Vesto Slipher (1875–1969), working at Lowell Observatory, used spectroscopy to study the Doppler shift of spectral lines in the “spiral
nebulae,” thus establishing the rotation of these objects (1912–1920). Most of the
galaxies (as they are known today) in his sample, except for M31, the Andromeda
Galaxy, were found to be moving away from the Milky Way.
Albert Einstein (1879–1955) publishes his General Theory of Relativity in 1916,
which explains how matter causes space and time to be warped. The resulting force
of gravity can now be thought as the motion of objects moving in a warped spacetime. He realized that General Relativity could be used to explain the structure of
the entire universe. He assumed that the universe obeyed the Cosmological Principle:
it was infinite in size with the same average density of matter everywhere, with spacetime in the universe warped by the presence of matter within it. However his field
1.5. MODERN COSMOLOGY
9
Figure 1.5: Harlow Shapley & Herbert Curtis
Harlow Shapley (left) and Herbert Curtis (right).
equations:
T,
G = 8πT
predicted a Universe to be either expanding or contracting. This contradicted what
was known about the Universe at the time, and it was also against Einstein’s sensibilities. Einstein as a result added a term into his equations, the cosmological constant
to keep his model Universe from being dynamic:
T,
G + Λ = 8πT
Depending on its sign, a cosmological constant can keep a growing universe from
getting bigger, and it keep keep a shrinking universe from getting smaller. However to
keep the universe perfectly static, the cosmological constant has to balance out exactly
the other terms. Why it should have such an arbitrary value cannot be explained
from first principles, but is more of an ad hoc solution to fit the requirement of a
static universe. This is therefore not a very satisfactory solution for physicists.
Dutch astronomer Willem de Sitter (1872–1934) used Einstein’s General Relativity equations with a low (or zero) matter density but without the cosmological
constant to arrive at an expanding universe (1916–1917). His view was that the
cosmological constant:
. . . detracts from the symmetry and elegance of Einstein’s original theory,
one of whose chief attractions was that it explained so much without
introducing any new hypothesis or empirical constant.
Russian mathematician Aleksandr Friedmann (1888–1925) finds a solution to
Eintein’s equation with no cosmological constant (1920), but with any density of
matter. Depending on the matter density, his model universes either expanded forever
or expanded and collapsed in a manner that was periodic with time. His work was
dismissed by Einstein and generally ignored by other physicists.
10
CHAPTER 1. COSMOLOGY FROM ANCIENT TO MODERN TIMES
Figure 1.6: Albert Einstein & Aleksandr Friedmann
Albert Einstein (left) and Aleksandr Friedmann (right).
In 1920, Harlow Shapley and Herbert Curtis held a debate on the “Scale of
the Universe,” or really about the nature of the “spiral” nebulae. Shapley argued that
these were gas clouds inside our own Milky Way and that the universe consisted just
of our Milky Way. Curtis on the other hand argued that they were other galaxies
just like the Milky Way, but much further away. Although the debate laid open
the positions of the two sides, nothing was immediately resolved. (That same year,
Johannes Kapteyn was arguing that the Sun was in the center of a small Milky Way,
based on star counts.) It was only in the following decade that as Edwin Hubble
and other astronomers found novae and Cepheid variable stars in nearby galaxies,
that Curtis’ view was slowly adopted. (When a letter from Hubble describing the
period-luminosity relation for Cepheids in M 31 arrived at Shapley’s office, Shapley
held out the letter and said, “Here is the letter that destroyed my universe!”)
1.6
The Expanding Universe
Edwin Hubble (1889–1953) worked at Mt. Wilson Observatory, California in 1923–
1925, to systematically survey spiral galaxies, following up on Slipher’s work. In 1929
he published his observations showing that the galaxies around us appeared to be
receding away from us. This is similar to the Doppler shift in sound waves when
an ambulance passes by on the street, and its siren shifts in frequency depending on
whether it is moving toward or away from you. The Doppler shift of the galaxies that
1.6. THE EXPANDING UNIVERSE
11
Hubble found was particular, since it also depended on distance. Galaxies twice as
far away moved twice as fast from us; those three times as far traveled three times as
fast, and so on. These observations can be summarized by “Hubble’s Law:”
v = H◦ d,
which relates the velocity v of a galaxy to its distance d by a scale factor, H◦ , or the
Hubble constant. Hubble worked out a value for the eponymous constant to be
H◦ = 500 km s−1 Mpc−1 , nearly 10 times the current accepted value.
Figure 1.7: Edwin Hubble
Edwin Hubble.
One could be tempted to interpret this result as suggesting something special
about our own galaxy. But why would all of the other galaxies be speeding away
from us? Applying the Copernican principle, it makes more sense to suppose that all
the galaxies are speeding equally away from each other. This would give the results
observed: that more distant galaxies recede faster. The galaxies are not repelled
by each other; they are carried along in space as it expands. Note that although
the distances between galaxies (technically clusters of galaxies) increase with time,
neither the galaxies (nor the clusters to some degree) expand in size. The clusters
and galaxies are bound together by gravity locally, while globally, space expands.
By 1932, Einstein had come around to excepting the idea of an expanding Universe. When he went to Mt. Wilson to meet Hubble, he said the invention of the
cosmological constant was the “the biggest blunder of my life.” That same year,
he and de Sitter published a joint paper on their Einstein-de Sitter universe, an
expanding universe without a cosmological constant. (Although Einstein eventually
12
CHAPTER 1. COSMOLOGY FROM ANCIENT TO MODERN TIMES
Figure 1.8: Einstein at Mt. Wilson
Albert Einstein’s 1931 visit to Mt. Wilson Observatory. From left to right: Milton Humason,
Edwin Hubble, Charles St. John, Albert Michelson, Einstein, W.W. Campbell, and Walter
Adams.
rejected it, we shall see the cosmological constant, or Λ crop up again later in the
story.)
In 1927, the Belgian astronomer Georges Lemaı̂tre (1894–1966) independently
arrived at Friedmann’s solutions to Einstein’s equations, and realized they must correctly describe the universe, given Hubble’s recent discoveries. Lemaı̂tre was the first
person to realize that if the universe has been expanding, it must have had a beginning, which he called the “Primitive Atom.” This is the precursor to what is today
known as the “Big Bang.”
Chapter 2
What Is In the Observable
Universe?
2.1
The Extra-Galactic Zoo
From the resolution of the early 20th century debates about the nature of the diffuse
and spiral nebulae, we now know that our Sun resides in the Milky Way Galaxy,
a collection of about 200–300 billion stars bound together by each others’ collective
gravitational pull. Because we live inside the Milky Way, it is difficult to imagine
what the actual shape and size of our parent galaxy is. Today we know that it
belongs to a class of objects known as spiral galaxies (and more specifically, the
Milky Way is a barred spiral galaxy). A galaxy that might look similar to ours is M 31
(the Andromeda Galaxy) or M 51. A spiral galaxy is distinguished by a large flat,
disk of stars, with a central bulge at the center. The bulge tends to have smoothlydistributed stars that are older, and hence orange-ish in color. The disk, on the other
hand, has spiral arms that run through it pinwheel-like. The arms themselves are
clumpy, with brighter regions where star formation is taking place, as well as finer
dark lanes, which consist of enormous clouds of cold interstellar gas and dust. The
extent of the disk of our Milky Way is about 100,000 light years in diameter, and is
relatively thin, “only” about 1000 light years thick at the location of the Sun (which
is itself about 25,000 light years from the Galactic center). The thickness of the disk
is greatest toward the center and decreases outward.
The Milky Way Galaxy is just one galaxy among hundreds of billions of other
observable external galaxies. Even in the 1920s, it was clear that there was enough
variety in the appearance of galaxies that a morphological classification was devised.
The first scheme was introduced by Edwin Hubble and we still use his terminology
today. The four basic Hubble classes are elliptical, spiral, lenticular, and irregular
galaxies. The spiral and lenticular classes are divided additionally into barred and
unbarred varieties. Each Hubble class is also subdivided into Hubble types using
letters and numbers which refine the classifications further.
13
14
CHAPTER 2. WHAT IS IN THE OBSERVABLE UNIVERSE?
The Hubble class of elliptical galaxies are objects that look elliptical in shape (obviously!), with a bright core and a luminosity that drops off smoothly away from the
center, with little or no clumpiness. The Hubble types range from E0, for ellipticals
that look circular, to E7, for ellipticals that are highly elongated. Note that this
classification is made by the elliptical’s apparent shape. It has nothing to do with its
actual three-dimensional shape, since a long, cigar-shaped object might look circular
with the right perspective. Ellipticals are the most common type of galaxies, making
up more than 60% of all those that are observed. Ellipticals also come in a range
of sizes. The most prominent are the giant elliptical galaxies, which can be 3–4
times larger than the Milky Way. Most ellipticals belong to the faint dwarf elliptical class, containing only a few million stars. Dwarf galaxies may be undercounted
in galaxy surveys because they are more difficult to find than their brighter cousins.
The stars in elliptical galaxies tend to be old, with no significant star formation in
the last 10 billion years. This makes sense since there is little gas and dust left in
them.
Spiral galaxies are categorized by how tightly the spiral arms wrap around the
bulge, and whether the galaxy has a bar at the center of the spiral. The Hubble types
Sa, Sb, and Sc are for unbarred spirals, with Sa galaxies having the tightest arms and
largest bulges, and Sc galaxies having the loosest wound arms and smallest bulges.
Because of the relationship to bulge size, the exact Hubble type for a spiral galaxy can
be guessed even if the galaxy is nearly edge-on. Barred spirals have the inner parts
of the spiral arms meeting at the ends of a stellar bar. The barred spiral types are
SBa, SBb, and SBc, with SBa tightly wound, and SBc loosely wound. Barred spirals
are only evident if the galaxy is seen face-on. Spiral galaxies are often seen close
to edge-on, such as M 31, the Andromeda Galaxy. However many spiral galaxies,
including our own Milky Way, show evidence of a bar, and it may turn out that
all spirals have bars, with many too small to be seen. And because of the physical
and chemical similarities between barred and unbarred spirals, some extra-galactic
astronomers1 do not even bother differentiating between the two categories.
The disks in spirals have copious gas and dust allowing for ongoing star formation.
As a result, both young and old stars can be found mixed together in the disk. The
lack of gas in the bulge and halo means only older stars are found in those regions.
Lenticular galaxies are similar to spiral galaxies by having a disk and a bulge.
However their disks possess no spiral arms. The bulge may also be relatively large,
compared to the disk, and may contain a bar-like structure as well.
Spirals and lenticulars make up about 30% of all galaxies, with about 60% of each
class having noticeable bars.
Irregular galaxies do not have the clear symmetries or regular shapes found in the
previously described Hubble classes. Some have traces of spiral arms or hints of a
disk. These are the Type I irregulars, which have regions of vigorous star formation,
1
“Extra-galactic” here refers to outside our own Galaxy, i.e., other galaxies. “Galactic” astronomers study our own Milky Way.
2.1. THE EXTRA-GALACTIC ZOO
(a) M 89 (E0)
(c) M 32 (E2)
(e) M 59 (E5)
15
(b) M 105 (E1)
(d) M 49 (E4)
(f) M 110 (E6)
Figure 2.1: Elliptical Galaxies
Examples of a range of elliptical galaxies with different Hubble types. All of these images
were taken using telescopes at the National Optical Astronomy Observatories (NOAO), and
can be found at http://www.noao.edu/image gallery/.
16
CHAPTER 2. WHAT IS IN THE OBSERVABLE UNIVERSE?
(a) NGC 7217 (Sab)
(b) M 77 (Sb)
(c) M 99 (Sc)
Figure 2.2: Unbarred Spiral Galaxies
Examples
of
unbarred
spirals
taken
using
NOAO
telescopes
(from
http://www.noao.edu/image gallery/). The Sab type is intermediate between Sa
and Sb. The Hubble classification scheme places spirals with more tightly wrapped arms in
the Sa category, and galaxies with the loosest spiral arms in the Sc category.
2.1. THE EXTRA-GALACTIC ZOO
(a) NGC 4650(SBa)
17
(b) M 91 (SBb)
(c) NGC 1073 (SBc)
Figure 2.3: Barred Spiral Galaxies
Examples of barred spirals taken using NOAO telescopes (and can be downloaded from
http://www.noao.edu/image gallery). The SBa type is the most tightly wrapped barred
spiral, while the SBc type is the least tightly wrapped. Note that the spiral arms terminate
at the ends of the bar in these galaxies.
18
CHAPTER 2. WHAT IS IN THE OBSERVABLE UNIVERSE?
(a) M 85 (S0)
(b) NGC 936 (SB0)
Figure 2.4: Lenticular Galaxies
Examples
of
lenticular
galaxies
taken
using
NOAO
telescopes
(from
http://www.noao.edu/image gallery). Lenticulars can be subdivided into the unbarred (S0) and barred SB0 types.
2.1. THE EXTRA-GALACTIC ZOO
19
(a) Large Magellanic Cloud
(b) Small Magellanic Cloud
Figure 2.5: The Magellanic Clouds
The closest irregular galaxies to the Milky Way, taken with NOAO telescopes (from
http://www.noao.edu/image gallery/).
20
CHAPTER 2. WHAT IS IN THE OBSERVABLE UNIVERSE?
(a) IC 4182 (Irregular I)
(b) NGC 55 (Irregular I)
(c) NGC 3077 (Irregular II)
Figure 2.6: More Irregular Galaxies
2.2. BARYONIC COMPOSITION OF GALAXIES
21
with vast H ii regions and brilliant OB stars. The Magellanic Clouds are the two
closest examples of this sub-class. Type II irregulars are much more chaotic looking,
often containing odd arms, loops, or explosive filaments. These may be the result of
mergers or collisions between neighboring galaxies. Irregulars are the least common of
the traditional galaxy classes, making up about 15% of all observed galaxies. Finally
there is a small fraction of peculiar galaxies as well, that can not be made to easily
fit any of the Hubble categories.
2.2
Baryonic Composition of Galaxies
The fundamental building blocks of galaxies can be divided into dark matter and
baryonic matter. We will have more to say about dark matter later, but baryonic
matter is simply our familiar everyday matter that consists of protons, neutrons, and
electrons2 Based on spectral analyses of stars, nebulae, and galaxies, we can come to
the conclusion that hydrogen makes up roughly 75% of the normal baryonic matter
by mass in the Universe, with helium adding up to less than 25%, and the rest of the
chemical elements (the “metals”) no more than 1–2%.
A minor component of the baryonic mass in galaxies is in gas, and its mass can be
directly measured by observing in spectral emission lines associated with the atomic
and molecular species in the gas. Since hydrogen can be observed in its atomic
form (H i) via radio observations using the 21 cm line, the total atomic gas mass of
galaxies can be found by taking observed 21 cm emitted flux, infer from it the atomic
hydrogen gas mass, and then add an appropriate amount of helium to get the total
atomic gas mass. Molecular hydrogen (H2 ) locked in colder giant molecular clouds is
more difficult to observe, but can be studied by using submillimeter radio tracers like
carbon monoxide (CO). Using assumptions of the ratio of CO to H2 , one can infer
the total molecular gas mass. Finally extensive halos of ionized hydrogen (H ii) at
temperatures of a million degrees Kelvin or more have been found around many giant
elliptical galaxies, and the mass of such gas can be determined by X-ray observations.
Observations of gas in galaxies show that spiral galaxies have 5–15% of their mass
in mostly molecular gas, while a few percent exists as atomic hydrogen. Elliptical
galaxies have very little atomic hydrogen, but the ionized gas mass percentage in
giant ellipticals can be as much as that of the gas fraction in spiral galaxies. Smaller
ellipticals tend not to have ionized halos, and they appear to have much less total gas
than spiral galaxies, perhaps 1% of the total mass or less. Irregular galaxies have the
most gas on average: 15–25% of the total galaxy mass.
2
Protons and neutrons fall under the subatomic particle class of baryons, with masses mp =
1.673 × 10−27 kg and mn = 1.675 × 10−27 kg, respectively. Electrons are classified as leptons, but
have a mass of me = 9.109 × 10−31 kg, or roughly 1/1800th the mass of a proton or a neutron.
Therefore ordinary matter is made up almost entirely by mass of baryons, and hence is called
baryonic matter.
22
CHAPTER 2. WHAT IS IN THE OBSERVABLE UNIVERSE?
Figure 2.7: The elliptical galaxy NGC 3923
The optical image of the E4 elliptical galaxy NGC 3923 is shown in orange-red, while the
white contour lines show X-ray emission. The halo of X-ray emitting gas extends far beyond
the edge as defined by the visible stars.
2.2. BARYONIC COMPOSITION OF GALAXIES
23
Figure 2.8: Population synthesis Model
Spectra modeled from different stellar components in a galaxy. The spectra at the bottom of
the plot are the contributions by different star populations (e.g., “LMS” means Lower Main
Sequence, and “AGB” is asymptotic giant branch) assuming that the galaxy had a single
burst of star formation 5 billion years in the past, and the stars had metallicities similar to
the Sun. The spectrum at the top is the result of adding the bottom components and would
be compared to a galaxy’s observed spectrum. From A.J. Pickles, 1998, “A Stellar Spectral
Flux Library: 1150–25000 Å,” Publications of the Astronomical Society of the Pacific, 110,
pp. 863–878.
24
CHAPTER 2. WHAT IS IN THE OBSERVABLE UNIVERSE?
Table 2.1: Population Model of M 31 Bulge: The derived mass and V-band luminosity
fraction of stars within each spectral type range.
Stellar Type
Main Sequence
G0–G4
G5–K0
K1–K2
K3–K4
K5–K7
M0–M2
M3–M4
M5–M6
M7
Subgiants
G0–G4
G5–G9
K0–K1
K2
Giants
K3
K4–K5
M5–M6
Contribution to
Total Mass (%)
Contribution to
Luminosity (%)
0.77
0.76
0.40
0.78
1.12
0.73
10.3
4.6
69.4
11.56
3.10
2.29
3.07
1.24
0.27
1.09
0.15
1.74
0.35
0.26
0.13
0.12
11.88
8.79
6.74
26.57
0.03
0.01
0.003
12.23
5.98
1.32
Determining the stellar mass of galaxies is not as easy as simply measuring a single
spectral line from the gas. The luminous visible light emission of galaxies is from the
cumulative glow of all of its member stars. Each star will have a different spectrum
depending on its mass. The total light from a galaxy is therefore due to the total
numbers of stars of each spectral type in that galaxy. The method of population
synthesis looks at the spectrum of an entire galaxy. Using assumptions of what
fraction of stars are to be expected in each spectral type, a theoretical spectrum can
be constructed and compared with the actual observed spectrum. The numbers of
stars within each spectral class can be adjusted until the population model gives
a reasonable match to the observed spectrum. Then from the total luminosity of the
galaxy, total numbers of stars in each spectral class can be inferred, and by adding
up the mass associated with each spectral type, the galaxy’s total stellar mass can be
determined.
Table 2.1 shows the results of such a population model calculated for the Andromeda Galaxy, M 31. As to be expected, the orange-ish bulge of a spiral galaxy is
dominated by cool, low-mass red stars (the M-dwarfs) which contribute to more than
80% of the total bulge mass, but because they are so faint, only add to a few percent
of the total luminosity. What is visible in the bulges of spiral galaxies (including our
own Milky Way) are the red giant and subgiant stars. K giants are responsible for
2.3. DARK MATTER COMPOSITION OF GALAXIES
25
roughly one-fifth of the bulge’s total luminosity.
2.3
Dark Matter Composition of Galaxies
We can measure the speeds of objects orbiting in a rotating system and plot them
against their radial distance to give a rotation curve. For instance, a wheel spinning
at one revolution per second will give the rigid body rotation curve shown in Fig. 2.9.
The speed of each part of the wheel has to increase the further away it is from the
center. A point on the rim of the wheel has a much larger distance to rotate through
than a point near the axle. Note however that although the linear speeds of parts of
the wheel might be different, the angular speeds are all the same. Every portion of
the wheel takes an equal amount of time—one second—to sweep through 360◦ around
the axis.
For planets orbiting the Sun, the rotation curve shows differential rotation,
where each planet has different orbital and angular velocities. The drop-off of speed
with radial distance from the Sun shown in Fig. 2.10 is because each planet is orbiting
due to the dominating gravity from the Sun. Since the force of gravity falls off with
distance via the inverse square law (see p. 53), smaller angular and linear speeds are
necessary to keep Pluto in its orbit compared with Mercury.
Figure 2.9: Rotation Curve of a Rigid Body
Finally rotation curves for spiral galaxies can also be constructed by measuring
the Doppler velocities of stars within the disk. Such a rotation curve for the Milky
Way is shown in Fig. 2.11. The shape of the curve looks fundamentally different from
26
CHAPTER 2. WHAT IS IN THE OBSERVABLE UNIVERSE?
Figure 2.10: Rotation Curve of a Planetary System
the previous two examples because although the stars in the Galaxy are not rotating
like a rigid body, they also do not all orbit around the same common center of mass
as do the planets around the Sun. Instead a star orbits around the collective mass of
everything inside its orbit. The further a star is from the center of the Milky Way,
the more mass its orbit encloses.
The velocities of stars within much of the disk are consistent with the observed
luminous mass of the Galaxy. If the visible edge of the Galactic disk was indeed where
there was a sharp drop-off in mass, then one would expect the rotation curve for stars
beyond the edge to drop like that for planetary systems. However what is actually
observed are orbital velocities that stay constant with distance rather than falling off
(Fig. 2.12). This suggests that in addition to the visible matter in the outer reaches
of a spiral galaxy, there is also a dark matter halo, consisting of matter that has
not been observed in any (including non-visible) wavelengths. Although the visible
edge of the Milky Way’s disk is located about 50,000 light years from its center, dark
matter extends out at least to 65,000 light years. Depending on the assumptions used
in setting the boundaries of the Galaxy and the distributions of the dark matter, the
total mass of our Milky Way, including the dark matter, might be anywhere from
4–60 times the luminous mass from stars. That is, given the total luminous mass of
the Milky Way is 1011 M , then the luminous plus dark matter mass can be anywhere
from 4 × 1011 M to 6 × 1012 M .
2.3. DARK MATTER COMPOSITION OF GALAXIES
27
Figure 2.11: Rotation Curve Based on Milky Way Gas Clouds
The rotation curve of the Milky Way based on observations of CO and H i clouds. Plotted
along with each data point is the uncertainty of the observation as an associated “error
bar.” The solid and dashed lines are two different rotation curves that have been derived by
different authors. Note that the observations in the outer Galaxy are much noisier, because
observations of the outer Galaxy are much more difficult to make. From F. Combes, 1991,
“Distribution of CO in the Milky Way,” Annual Review of Astronomy & Astrophysics, 29,
195–237.
28
CHAPTER 2. WHAT IS IN THE OBSERVABLE UNIVERSE?
Figure 2.12: Rotation Curve of the Milky Way
A plot showing the orbital velocities of stars in the Milky Way (vertical axis) versus their
distance (horizontal axis). Based on Newton’s law of gravity, the more mass an object (like
a star) orbits around, the faster its orbital speed. From a census of the visible matter in our
Galaxy (stars plus gas and dust), we expect the velocities to follow the lower line. However
the actual observed velocities of the stars (including our Sun) follows the top line, suggesting
that there is more matter than can be observed. We therefore posit the existence of a dark
matter halo that must be driving the orbital velocities up.
2.4. GALAXY CLUSTERS AND SUPERCLUSTERS
2.4
29
Galaxy Clusters and Superclusters
Our own Milky Way is in a small cluster of galaxies, the Local Group (Fig. 2.14). This
is a gravitationally bound grouping of galaxies, within a volume of space 6 million light
years across. Each member of the group moves according to the net gravitational force
of the entire Local Group, and galaxies cannot escape unless they are ejected from a
collision or other close encounter. The two largest galaxies are spirals, the Milky Way
and M 31 (the Andromeda Galaxy). The next largest are the two Magellanic Clouds
and the small spiral M 33 (the Triangulum galaxy). The Milky Way and Andromeda
each have a small flock of smaller dwarf elliptical and irregular galaxies that orbit
nearby. A current census of the Local Group shows 30 members, but because dwarf
elliptical members are difficult to find, we may expect the membership to grow as our
telescopic instrumentation improves.
Figure 2.13: Satellite Galaxies of the Milky Way
The immediate vicinity of our Milky Way galaxy showing our satellite galaxy neighbors.
From An Atlas of The Universe, http://www.anzwers.org/free/universe/.
As clusters go, the Local Group is rather sparse and small. Other clusters are
30
CHAPTER 2. WHAT IS IN THE OBSERVABLE UNIVERSE?
Figure 2.14: The Local Group
Within 5 million light years of the Milky Way are a loose collection of galaxies that make
up the Local Group. The two massive galaxies, our own Galaxy and the Andromeda
Galaxy (M 31), dominate the cluster gravitationally. From An Atlas of The Universe,
http://www.anzwers.org/free/universe/.
2.4. GALAXY CLUSTERS AND SUPERCLUSTERS
31
Figure 2.15: The Virgo Supercluster
A plot showing galaxy clusters within 100 million light years of the Milky Way. This volume
of space is dominated gravitationally by the Virgo Supercluster. Although our Local Group
is speeding away from it currently, it will probably slow down, reverse course, and eventually
fall toward the local supercluster and merge with it in the far distant future. From An Atlas
of The Universe, http://www.anzwers.org/free/universe/.
32
CHAPTER 2. WHAT IS IN THE OBSERVABLE UNIVERSE?
Figure 2.16: The Virgo Cluster
The nearest supercluster is located in the constellation of Virgo. Although the center of the
Virgo cluster has a few giant ellipticals (such as M 87 and M 86), there are also many
spirals as well.
2.4. GALAXY CLUSTERS AND SUPERCLUSTERS
33
Figure 2.17: The Coma Cluster
A spherically shaped cluster of over 1000 galaxies, in the constellation of Coma Berenices,
and located about 5 times further away than the Virgo Cluster. The Coma Cluster is dominated by elliptical and lenticular (S0) galaxies. The central giant elliptical is NGC 4889.
34
CHAPTER 2. WHAT IS IN THE OBSERVABLE UNIVERSE?
Figure 2.18: Neighboring Superclusters
Expanding out to a volume 1 billion light years in radius shows not only additional superclusters of galaxies, but we see now that galaxies at such scales are distributed unevenly
in filaments and sheets, with vast voids in between. From An Atlas of The Universe,
http://www.anzwers.org/free/universe/.
2.4. GALAXY CLUSTERS AND SUPERCLUSTERS
35
Figure 2.19: 2-Degree Field Galaxy Survey
Part of the data from the 2-Degree Field Redshift Galaxy Survey, containing positions for
over 245,000 galaxies. In this image, the Milky Way is at the center of the two pie-shaped
wedges. Astronomers observed two narrow strips of sky stretching out in the southern hemisphere (the right wedge) and the northern hemisphere (the left wedge). The galaxies are
clearly distributed along giant filaments that surround enormous voids, like a mass of soap
bubbles. The distribution of galaxies decrease the further we are from the center in this
survey because the more distant galaxies are increasingly more difficult to observe.
36
CHAPTER 2. WHAT IS IN THE OBSERVABLE UNIVERSE?
Figure 2.20: The APM Survey of Galaxies
Instead of trying to determine the 3-dimensional spatial positions of galaxies over a narrow
strip of sky, this survey uses automated computer routines to look for galaxies in photographic plates, and determines their 2-dimensional location in the sky. About 4000 square
degrees (1/10th of the entire sky!) are surveyed, with roughly 2 million galaxies found. The
“holes” in the survey are regions containing bright stars that could not be scanned. The
large-scale distribution of galaxies can be discerned in this map as well.
2.4. GALAXY CLUSTERS AND SUPERCLUSTERS
37
Figure 2.21: The Observable Universe
Making one more jump shows the edge of the observable Universe, roughly 14 billion light
years in radius. This is an artist’s conception of the Universe, filled with vast voids 600 million light years across and superclusters of galaxies concentrated at the edges of these voids.
The Universe finally starts to look uniform and homogeneous at such a scale. From An
Atlas of The Universe, http://www.anzwers.org/free/universe/.
38
CHAPTER 2. WHAT IS IN THE OBSERVABLE UNIVERSE?
far richer, meaning not only do they have larger memberships (up to thousands of
observable galaxies), but they are also more densely packed. In fact the typical rich
cluster is comparable in diameter to the Local Group, but there are far more galaxies
at the cluster cores. Two well-known clusters are the Coma cluster and the Virgo
cluster. Each has over a thousand known members.3 The Coma cluster is about
250–300 million light years from the Milky Way, and is a spherically shaped cluster
consisting mainly of elliptical and lenticular galaxies. The Virgo cluster is closer, at
slightly more than 50 million light years, and is irregular in shape, with a mix of
spirals as well as ellipticals.
At larger scales, even larger superclusters of galaxies can be seen. The Local
Supercluster is centered on the Virgo cluster, includes the Local Group, and is
roughly 100 million light years across. The galaxy clusters that make up the supercluster are not gravitationally bound to each other, so the boundaries of a supercluster
depend on the cutoff number density that separates regions of high galaxy count versus low galaxy count. At larger scales of 800 million light years, other superclusters
are evident.
At the largest scales, superclusters appear to be organized in a vast network
of sheets and filaments that surround nearly-empty, great voids. The large-scale
structure of the Universe therefore appears to consist of chains of superclusters of
galaxies, arranged around voids, of order 600 million light years across, like the pores
inside sponges. The largest structures appear to have a scale of about 600 million
light years. At even larger scales, the Universe finally starts to appear uniform or
homogeneous—because any one 600 million light year sized region will look like any
other 600 million light year patch.
2.5
The Cosmic Distance Ladder
From §2.2–2.4, we have a good idea of the compositions and distributions of galaxies.
But how did astronomers ascertain the layout of galaxies and their positions, which
allow us to build maps like those in Figs. 2.13–2.18? Determining the direction of a
galaxy is easy enough. But how do we actually determine the distances to external
galaxies? Knowing distances is critical, because if we know the distance d to a galaxy,
then we can obtain the physical size of the galaxy L by the relation
L = d × θ,
where θ is the angular size that we measure for the galaxy in the sky in units of
radians.4 Once we know its true distance, we can calibrate the flux received from the
3
Remember that these are observed galaxies. Many galaxies, such as dwarf ellipticals, are below
the current observable threshold, and therefore would not be counted in surveys.
4
This relationship holds as long as the angular size θ is small, which is true for all observable
external galaxies.
2.5. THE COSMIC DISTANCE LADDER
39
galaxy by our instruments and determine its true light output. A distance measurement allows us to fix not only the size of the galaxy, but many other derived physical
quantities that depend on the size.
Figure 2.22: Actual Size/Angular Size Relation
It turns out that there are many different methods for determining distances,
and new methods are being devised and old methods are being revised constantly.
Each distinct methodology has its own advantages and disadvantages. When quoting
a distance using a particular technique, the careful astronomer is always aware of
the errors and uncertainties associated with that technique. It is important to keep
these errors in mind, since any one particular distance measurement method is good
over a limited range. A nearby distance measurement technique is used to calibrate
a second distance method that overlaps it slightly. This second technique is then
used to calibrate a third method that works on objects yet more distant, and so
on. Uncertainties associated with one method can propagate to other methods if the
former is used as the calibration.
2.5.1
Trigonometric Techniques
The most familiar of the distance measurement techniques is trigonometric parallax. This is the apparent shift of nearby astronomical objects against the background
of more distant ones as we change positions. For traditional parallax measurements,
the shift is due to the orbit of the Earth about the Sun (Fig. 2.24). The distance to
the star d is (for small angular shifts):
d = R × p,
(2.1)
where p, measured in radians, is the parallax angle or half of the total angular shift that
we see the foreground star make, and R is the Sun-Earth distance (or 1 astronomical
unit). This equation also shows us where the term parsec (“parallax-second”) arises.
An object is at a distance of 1 parsec if it has a parallax p = 100 :
1 parsec =
1 AU
= 3.0856 × 1018 cm
(2π/1.296 × 106 arcsec)
40
CHAPTER 2. WHAT IS IN THE OBSERVABLE UNIVERSE?
Figure 2.23: The Cosmic Distance Ladder
Some of the methods used to determine distances to astronomical objects. One technique
can be calibrated to another technique if both techniques overlap along the distance scale.
Figure 2.24: Trigonometric Parallax
A diagram showing the geometry of a nearby star with respect to the Sun and the Earth for
trigonometric parallax. The parallax angle p, distance to the star d, and Sun-Earth distance
R all follow Eq. 2.1 if angle p is small.
2.5. THE COSMIC DISTANCE LADDER
2.5.2
41
Standard Candles
The next important category of distance determination methods is assuming that
an object class has a known, fixed luminosity. If all the members of this class are
of the same brightness, then observed differences between one standard candle and
another can only be due to their relative distances. Isaac Newton was the first person
to suggest the use of standard candles. To estimate the distances to the stars, he
assumed that all stars were equally bright, and that they all have the same brightness
as the Sun. (This means that the brightest stars are simply the closest to us.) Since
the brightest stars are 100 billion times fainter than the Sun, using the inverse square
law, Newton worked out the closest star to be about D ≈ 1.5 parsecs away. This is
surprisingly close to the correct distances for the nearest stars—although not quite
correct for the brightest stars in the sky.
RR Lyrae stars are one category of stars that can be used as standard candles
within the Milky Way. They have a fixed ‘standard’ light output, so variations in
brightness can be mostly attributed to their different distances. (There are additional
effects that can add uncertainty to the results, but we will not go into the details here.)
2.5.3
Cepheid Variables
Cepheid variables are perhaps the most important distance indicators to be discovered, and were used to settle the debate over the spiral nebulae in the 1920s. Cepheids
are giant or supergiant stars whose light output changes regularly over a period ranging from days to hundreds of days. They are named after δ Cephei (“delta Cephei”),
the first member of this class to be described. They are intrinsically bright, so they
can be observed at great distances, including in nearby external galaxies. Their luminosity can vary by a factor of 10 or more, and when this luminosity variation is
plotted over time, a light curve results.
In 1907, Henrietta Leavitt (1868–1921) was studying variable stars located in the
Small Magellanic Cloud (SMC), one of the closest satellite galaxies to the Milky
Way. She found a correlation between the period of the light curve (the amount of
time it took for the star to reach successive peaks in brightness) and the average
apparent magnitude of the star. Since all the stars in the SMC are roughly the same
distance from the Sun, it followed that Cepheid variables have a period-luminosity
relationship. The greater the average luminosity of a star, the faster it varied in
brightness.
Another astronomer, Ejnar Hertzsprung (1873–1967) realized that the stars Leavitt was investigating were Cepheid variables. Although they now had a potential
standard candle for measuring distances, there was no direct way to calibrate their
method. None of the Cepheids within the Milky Way were close enough to have their
distances directly measured via trigonometric parallax. Hertzsprung had to resort
to a series of more complicated secondary methods to obtain a calibration distance
for nearby Cepheids. This along with an underestimate of the amount of starlight
42
CHAPTER 2. WHAT IS IN THE OBSERVABLE UNIVERSE?
reddening by the interstellar medium resulted in a distance measurement to the SMC
that was quite incorrect.
However since then, many new recalibrations have been done to the Cepheid
variable technique (including direct trigonometric measurements of nearby Cepheids).
It is thought that distances obtained via this method today have uncertainties of no
more than ±15%.
Cepheids were identified by Edwin Hubble in spiral nebulae, which along with
other evidence, now directly showed them to be giant collections of stars like our own
Milky Way. Starting in 1924, his measurement of light curves from Cepheids in the
galaxies M 31, M 33, and NGC 6822, placed these nebulae clearly outside our Milky
Way galaxy. The debate over the nature of the nebulae and the size of the Universe
was over, and the new field of extra-galactic astronomy had begun.
In the past decade, the Hubble Space Telescope has been used to study Cepheids
in galaxies up to 30 million parsecs (or 100 million light years) from the Milky Way.
Although this is not a great distance compared to the size of the observable Universe, it is large enough to allow for the calibration of additional distance scales that
otherwise cannot be calibrated.
2.5.4
Other Standard Candles
Other important standard candles include Type Ia supernovae. These are supernovae that occur in binary star systems where gas from one star overflows onto a
companion white dwarf. The white dwarf has a mass that is just below the maximum mass for white dwarfs. The smaller star accretes gas until it reaches the critical
Chandrasekhar limit of 1.4 M , at which point the heavier elements in the white
dwarf undergo a fast nuclear reaction, resulting in a supernova explosion. Since the
thermonuclear reactions are thought to occur right after the white dwarf reaches
the Chandrasekhar limit, the luminosity of all Type Ia supernova should be close to
constant. (We will have more to say in § 5.2 about recent observations of Type Ia
supernovae that show a surprising result about the expansion of the Universe.)
An entire galaxy by itself can also be used as a standard candle. In 1977, Brent
Tully and Richard Fisher discovered that the luminosity of spiral galaxies is correlated
with their maximum rotation speed, as measured by the width of the 21 cm line of
H i. This correlation can be explained in the most simplest terms as saying that
the more mass a spiral galaxy has, the more luminous it will be from the increased
number of stars. But one might also expect the faster the rotation rate for all of the
stars and gas in its disk, because of the increased gravitational potential. The TullyFisher relation was calibrated using ten nearby spiral galaxies, whose distances
could be determined by measuring Cepheid variables. In practice, uncertainties arise
from the orientation of the observed galaxy (which affects the Doppler measurements
of the hydrogen gas) and absorption and scattering of light along the line-of-sight
(which affects the luminosity measurement). A similar but different correlation, the
2.5. THE COSMIC DISTANCE LADDER
43
Figure 2.25: Cepheid Variable Light Curves
Cepheid variable light curves from a paper by Ejnar Hertzsprung (1928, Bulletin of the Astronomical Institute of the Netherlands, 4, pp. 164–171). Observations taken over multiple
periods are plotted together to show how the light output varies over time.
44
CHAPTER 2. WHAT IS IN THE OBSERVABLE UNIVERSE?
Figure 2.26: Cepheid Variable Light Curves, Example II
Cepheid variable light curves from a paper by Edwin Hubble (1925, Astrophysical Journal,
62, pp. 409–433). Light curves of two of the Cepheid variables from NGC 6822 are shown,
with periods of 21.06 and 37.45 days.
2.5. THE COSMIC DISTANCE LADDER
45
Figure 2.27: Cepheid Variable Light Curves, Example III
Light curves from the team that studied Cepheid variables in M 100 in the Virgo cluster
(Freedman et al., 1996, Astrophysical Journal, 464, pp. 568–599). Using the data from 52
observed Cepheids, the distance to M 100 (and hence the Virgo cluster) was determined to
be 16.1 ± 1.3 Mpc.
46
CHAPTER 2. WHAT IS IN THE OBSERVABLE UNIVERSE?
Faber-Jackson relation has been found for elliptical galaxies.
Figure 2.28: Tully-Fisher Relationship
The Tully-Fisher relationship between absolute magnitude and width of the H i line for ten
nearby spiral galaxies (left), which was then extrapolated to spiral galaxies in the more
distant Virgo cluster (right). From R.B. Tully and J.R. Fisher, 1977, “A New Method of
Determining Distances to Galaxies,” Astronomy & Astrophysics, 54, pp. 661–673.
2.5.5
Redshifts and the Hubble Flow
Hubble discovered the expansion of the Universe, as described in §1.6. What he and
other astronomers measure to get this result are the spectral lines of gases in distant
galaxies. The lines have associated “rest” wavelengths. When a source is moving
away or toward the observer, the spectral lines are respectively, red-shifted and
blue-shifted. The wavelengths of the lines actually change, and even Hubble’s early
work showed that the majority of the 24 galaxies he studied had red-shifted lines, or
were moving away from the Milky Way.
The redshift is given by a single parameter, z, which is defined as:
z=
λobs − λem
λobs
=
− 1,
λem
λem
(2.2)
where λobs is the observed wavelength of the line, and λem is the original emitted
wavelength. Assuming a linear relationship between the redshift and distance, we
2.5. THE COSMIC DISTANCE LADDER
47
get:
H◦
d,
(2.3)
c
where c is the speed of light (2.9979 × 108 m sec−1 ) and H◦ is the Hubble constant.
z=
Figure 2.29: Hubble Relation for Galaxies and Galaxy Clusters
A plot showing the relationship of redshift versus distance, using several independent techniques. The straight and dotted lines in the top plot show fits using three different Hubble
constants. The bottom plot shows the best fit result of H◦ = 72 km s−1 Mpc−1 . From
W.L. Freedman et al., 2001, “Final Results from the Hubble Space Telescope Key Project
to Measure the Hubble Constant,” Astrophysical Journal, 553, pp. 47–72.
Rewriting Eq. 2.3 as cz = H◦ d, and assuming that cz is the same as the velocity (which is correct when the redshift is small), we can use Eq. 2.3 to give us a
relationship between velocity and distance:
v = H◦ d.
(2.4)
48
CHAPTER 2. WHAT IS IN THE OBSERVABLE UNIVERSE?
This equation states that the recessional speed of a galaxy is exactly proportional to
its distance. This is what would be expected from a uniformly expanding Universe.
Thus if the Hubble constant, H◦ , was known accurately, one can use this equation to
determine the distance to a galaxy. This is the primary reason why an accurate determination of H◦ has long been a goal of astronomers. However as we see in Fig. 2.29,
even for galaxies with distances determined via standard candle techniques, there is
enough “noise” in the data to not quite match the smooth Hubble flow predicted by
Eq. 2.4. Some is due to uncertainties and unaccounted-for errors in measuring the
distances. However other uncertainty is due to intrinsic velocity differences that a
galaxy might have. If a galaxy was part of a large, rich cluster and was gravitationally interacting with the other cluster members, its orbital motions could result in
its recessional velocity to be quite different than the overall recessional speed of the
cluster.
2.6
Galaxy Cluster Mass
The mass of clusters of galaxies can be estimated in a number of ways. The first gives
us a luminous mass by simply adding up all of the light from the stars, gas, and dust
in the galaxies. This is what was described in § 2.2 but applied to all of the galaxies
within a cluster.
Another method is via the virial theorem. This theorem simply states that
the distribution of velocities of a group of gravitationally bound objects will depend
on their total collective mass. The greater the overall cluster mass, the faster the
orbital speeds of the individual galaxies. When astronomers started applying the
virial theorem to galaxy clusters, they found a very surprising result: the virial cluster
mass was many times higher than the luminous mass. In fact, ordinary luminous
matter in galaxies could account for only 20%–30% of the total cluster mass obtained
from the virial method.
Another mass-finding method was to measure hot X-ray emitting intracluster
gas that filled the space between galaxies. This is 10–100 million K gas, which
is extremely tenuous—several orders of magnitude less dense than the gas in the
Solar corona. To determine the mass of the entire cluster, we need the concept
of hydrostatic equilibrium. In galaxy clusters, a balance is assumed between the
pressure of the intracluster medium against the gravitational combined force of the gas
and galaxies. The measured temperatures and pressures of the intracluster medium
again implies that the total gravitational field is higher than the gravity expected
from the luminous matter.
Finally gravitational lensing has been used to derive the masses of galaxy clusters. Einstein’s General Theory of Relativity predicts that the path that light takes
can be bent if it passes close by a massive object. Images of background galaxies
are distorted because of the gravity of the foreground cluster, and the amount of the
distortion can be used to infer the total cluster mass. Again the results are congruent
2.7. MORE ON DARK MATTER
49
Figure 2.30: Hydra A Galaxy Cluster
A look at the Hydra A galaxy cluster in the optical (left) and in X-rays (right). The X-ray
image shows an enormous bubble of extremely hot coronal gas that fills the inter-galactic
medium in this cluster.
with the virial and X-ray gas methods.
We come to the conclusion based on three completely independent techniques
that there is hidden matter that is gravitationally influencing the galaxies within the
galactic clusters. Depending on the estimate used, the total luminous mass of galaxies
in a cluster is 10% or less, with the intracluster gas making up another 10–25%. The
dark matter contributes 70–90% of the total cluster mass. Thus based on studies of
individual galaxies as well whole groups of galaxies in clusters, it appears that the
majority of matter in the universe is not in any form that we can observe except by
its gravitational effects!
2.7
More on Dark Matter
What is the nature of dark matter? This is still under considerable debate, since
we do not know enough about it. One conjecture suggests that the dark matter
halo around the Milky Way could consist of ordinary matter in the form of brown
dwarfs, star-like objects too small to start hydrogen fusion in their cores. These
objects would be less than 80 times the mass of Jupiter, and would be too cool to
radiate very much radiation—so little in fact that they would escape detection at
both optical and infrared wavelengths. Another possibility is a population of stellar
remnants, the end products of stellar evolution such as white dwarfs, neutron stars,
50
CHAPTER 2. WHAT IS IN THE OBSERVABLE UNIVERSE?
Figure 2.31: Gravitational Lensing by Abell 2218
The rich galaxy cluster Abell 2218, as viewed by the Hubble Space Telescope (top). The
arcs that are centered around the bright elliptical just to the left of center are actually
distorted images of a background galaxy. The diagram (bottom) shows how light from the
background galaxy is bent by the gravity from the foreground cluster, arriving at our eyes
(or our instrumentation) as circular arcs centered on the cluster.
2.7. MORE ON DARK MATTER
51
and black holes. The white dwarfs and neutron stars would have stopped glowing
after a few billion years. If no matter was available to be accreted, a black hole
would be invisible. Attempts have been made to survey the halo for small, compact
objects via gravitational lensing, and such a population has been found. However the
numbers of these objects are only about 20% of what is necessary to account for all
of the dark matter halo mass.
In addition to this relatively cold, baryonic dark matter, physicists and astronomers
have also suggested that non-baryonic matter could make up the bulk of the dark
matter. This is the so-called hot dark matter component, because such particles
would move at velocities very close to the speed of light. The neutrino has been mentioned as a candidate for such a particle. Although neutrinos were thought to have
zero rest mass, and would therefore travel at the speed of light, physicists realized
that there was no fundamental reason why they should be massless. Therefore if
neutrinos did have a very small mass, they could provide some fraction of the dark
matter mass by sheer numbers. Recent experiments involving neutrinos from the Sun
showed that they have a mass about 5 million times less massive than the electron.
This however is not enough; the neutrinos would constitute on order only 1% of all
dark matter.
Physicists have also suggested WIMPs, the weakly interacting massive particles as a dark matter candidate. The “weak” in the name refers to one of the four
fundamental forces in the Universe that affects such particles.5 According to proposed
supersymmetry theories of particle physics, relationships between fundamental particles and the forces of nature implies the existence of new classes of undiscovered
particles which interact only by gravity and the weak force. One such postulated
particle, the neutralino, has a mass 20–1000 times that of a proton. It is expected
that if supersymmetry is correct, many neutralinos would have been created in the
early universe. Currently there are efforts (unsuccessful so far) to generate artificially
neutralinos in particle accelerators, as well as to detect any cosmic neutralinos that
might be wandering through the Solar System from the Galactic halo.
5
The other three forces are: the strong force which mediates interactions between protons
and neutrons in atomic nuclei, and keeps them bound together; the electromagnetic force which
controls interactions between charged particles; and gravity. See more on this in § 8.3.
52
CHAPTER 2. WHAT IS IN THE OBSERVABLE UNIVERSE?
Chapter 3
Theoretical Universes
To understand how the Universe got to its current state, we need a basic understanding of some of the basic physical laws that are most important to its evolution.
The first of these are Einstein’s Special Theory of Relativity and General Theory of
Relativity. In deriving these theories, Einstein showed how space and time could be
radically re-thought.
Before Einstein’s theories, space and time merely contained matter and energy.
Every particle of matter and each bit of energy could be located within a single
position in space and at a single moment of time. Furthermore, space and time were
passive: they made up the stage in which matter and energy interacted, but played
no part otherwise in the Universe. Einstein showed that the individual dimensions of
space and time were really components of a single space-time, and it was impossible
to disentangle space completely from time. Furthermore matter and energy affected
the geometry of this four-dimensional space-time, and the shape of space-time could
affect matter and energy in return. This radical restructuring of the fabric of reality
also showed how gravity became perhaps the most important force in shaping the
evolution of the Universe.
According to Newton, gravity was a force between masses, which acted instantly
across intervening space. The Sun exerted a gravitational force on the Earth, and
the Earth also exerted a far weaker gravitational force (because of its much smaller
mass) on the Sun. Newton’s law of universal gravitation can be described by:
F =G
m1 m2
,
r2
(3.1)
where the force F between two bodies is given by the product of the Gravitational
constant G and the masses of the two bodies, m1 and m2 , divided by the square of
the distance r between the two. Newton’s laws of gravity and motion were highly
successful for more than two centuries in explaining the motions of Solar System
bodies. However Newton could not explain the origin of gravity; his laws could only
describe accurately the motions.
53
54
CHAPTER 3. THEORETICAL UNIVERSES
Einstein was able to go further than Newton. Instead of describing gravitation as
a force between two bodies, Einstein showed that an object with mass geometrically
distorts space-time. This imposed curvature on space-time can then affect the motions of matter and energy through space-time. The Sun therefore curves space-time
in such a way that affects the Earth’s motion. The net effect is that the Earth appears
to move under the force of gravity.
Figure 3.1: Einstein’s View of Space-Time
Space-time, as viewed by Einstein, is a four-dimensional fabric that is distorted by the massenergy and momentum of matter and radiation within it. This warpage or curvature further
tells how matter moves through space-time.
However Einstein also showed from his Special Theory of Relativity that matter
and energy were equivalent and convertible so that one could turn one into the other.
The maximum amount of energy that can result from a 100% perfect conversion from
some lump of matter is given by perhaps the most famous equation in all of science:
E = mc2 .
(3.2)
This equivalence suggests that matter is not solely responsible for distorting spacetime. In fact the distribution of all energy and momentum throughout a four-
55
dimensional volume of space-time will have an effect on the local curvature of that
volume. Since the motion due to gravitational forces can be described as being caused
by the curvature of space-time, then gravity is due not only to matter, but also energy
and momentum as well. The energy and momentum of matter and radiation cause
space-time to warp, and the warped space-time tells how the matter and radiation
move through space-time.
Figure 3.2: The Curvature of Space-Time and Light
One additional consequence of Einstein’s General Theory of Relativity is that the curvature
of space-time can affect the path of light rays.
Einstein published his General Theory of Relativity in 1916. The prediction that
energy in the form of electromagnetic radiation would be affected by the curvature
of space-time was tested only a few years later. Observations of stars during a total
eclipse of the Sun in 1919 showed them to be exactly where General Relativity predicted they would be. This validation of Einstein’s theory made him internationally
famous.
56
3.1
CHAPTER 3. THEORETICAL UNIVERSES
The Curvature of Space-Time
The curvature of space-time from Einstein’s General Theory of Relativity is one of
the more difficult concepts to understand in modern cosmology for non-specialists.
One can work by analogy and first ask ourselves, what does it mean when there is
no curvature? In our everyday world, we are most familiar with Euclidean geometry.
This is the geometry that we learned in high school: parallel lines will go off to infinity
without ever crossing; triangles have interior angles that add up to 180◦ . Pythagoras’
theorem which relates the lengths of the sides of a right triangle also holds:
c 2 = a2 + b 2 ,
(3.3)
where c is the length of the hypotenuse of the right triangle, and a and b are the
lengths of the other two sides. One can generalize the Pythagorean theorem to threedimensions as well:
c 2 = a2 + b 2 + c 2 ,
(3.4)
For a mathematical geometrician, these two equations are all that is required to completely describe the surface. Such flat geometry is suitable for architects, surveyors,
and most everything that we deal with in our everyday life.
Figure 3.3: Pythagoras’ Theorem
Pythagoras’ Theorem, which is a relationship between the length of the hypotenuse c and the
lengths of the other two sides a and b of a right triangle.
However this is not the only geometry that is possible. Although the flat Euclidean
geometry is familiar to us, it is not a completely accurate description of our world.
The Earth is not flat, but spherical. But because the Earth is so large relative to
the scales and distances that we are used to normally, a small patch of its surface
will look extremely flat to us. But on a larger scale, expanding to a size where the
3.1. THE CURVATURE OF SPACE-TIME
57
curvature of the Earth becomes important, our geometry is now curved—in fact it is
positively curved. Here lines that start off parallel eventually meet. A circle with
radius r will have a circumference C < 2πr—for a flat geometry, recall that circles
have circumferences C = 2πr. The interior angles of a triangle add up to be a value
greater than 180◦ . (In fact it is possible to draw a triangle on a sphere with each
angle being a right angle of 90◦ !)
In the 19th century, mathematicians Karl Friedrich Gauss (1777–1855) and Bernhard Riemann (1826–1866) generalized geometry so that it was not restricted to just
flat surfaces where Euclidean rules applied. To derive rules for these other geometries,
Gauss assumed as a starting axiom that the Pythagorean theorem, Eqs. 3.3 and 3.4,
would be true for any geometry as long as the triangle you studied was small enough.
So even for a highly curved, non-flat surface, one could shrink down and look at a
tiny, local part of that surface that would look flat. A triangle drawn on that would
still follow the Pythagorean theorem. This triangle would have sides of length dx, dy,
and ds (where the d prefix comes from calculus and is used to denote variables that
are very small), and they would be related by:
(ds)2 = (dx)2 + (dy)2 .
(3.5)
Again in a three-dimensional space, we have the equivalent equation
(ds)2 = (dx)2 + (dy)2 + (dz)2 .
(3.6)
Eq. 3.5 would apply to a flat sheet that extends infinitely in all directions. If however
you were on the surface of the Earth which only looks locally flat, then one could also
write an equation similar to 3.5, but only more complicated to completely describe
the curved geometry.
Similarly one can use Eq. 3.6 to completely describe a three-dimensional space
that is flat. But if that three-dimensional space is curved, then you would need a
variant of Eq. 3.6 that was similarly much more complicated. Never mind how you
can imagine a three-dimensional space that is curved—it’s not possible so don’t even
try! Just realize that it can be mathematically described.
Actually one can understand this by looking at a two-dimensional analogy. Imagine a two-dimensional universe filled with 2D creatures. If their universe was flat, then
they would appear to us—3D beings—as living on an infinite flat sheet. However if
their universe was actually positively curved, then we might see their universe as
curved like a sphere. However it would be impossible for the 2D creatures to visualize
how their 2D universe could be positively curved, although they would be able to use
mathematics to describe it.
In addition to a positively curved sphere, there can also be negatively curved
surface as well. A negatively curved two-dimensional surface is usually represented
by a saddle shape. The rules for geometry are now the opposite of what they were in
the positively curved case. Parallel lines always diverge. Circles have circumferences
C > 2πr. Triangles have interior angles which add up to less than 180◦ .
58
CHAPTER 3. THEORETICAL UNIVERSES
Figure 3.4: Infinitesimal Pythagoras’ Theorem
An infinitesimal version of Pythagoras’ Theorem, which states that for an arbitrarily small
size, any two- or three-dimensional triangle will follow a version of the Pythagorean theorem.
When constructing his Theory of Relativity, Einstein took this previous work
on generalizing geometry and instead of just dealing with three-dimensional space,
he expanded it to four-dimensional space-time. Instead of measuring the physical
distance of points in three-dimensional spaces, Einstein’s theory dealt with the spacetime separation of events. Einstein’s version of Eq. 3.6 for flat space-time is:
(ds)2 = (dx)2 + (dy)2 + (dz)2 − c2 (dt)2 ,
(3.7)
where c is the speed of light and dt is the time separation of the two events. If spacetime was curved in either the positive or negative direction, then there would be a
corresponding equation that would also be much more complicated.
3.2
The Distribution of Matter and Energy in the
Universe
We mentioned back in § 1.5 the Cosmological Principle, which states that the
Universe is homogeneous and isotropic. Note that both properties are important,
since one can envision a distribution of matter that is homogeneous (with the same
density) everywhere, but does not look isotropic (does not look the same in every
direction), and vice versa.
The Cosmological Principle was not obvious to astronomers, even after Hubble
was able to measure the distance to the nearby galaxies. At that time, the universe
3.2. THE DISTRIBUTION OF MATTER AND ENERGY IN THE UNIVERSE 59
Figure 3.5: Curved Two-Dimensional Surfaces
Some curved two-dimensional surfaces seen in three-dimensional space. The behavior of
parallel lines, and the geometric relationships for circles and triangles are Euclidean for
the flat surface. However parallel lines converge in the positively-curved surface, while they
diverge in the negatively-curved surface. The interior angles are either, respectively, greater
than and less than 180◦ , and the circumferences of circles are less than and greater than
2πr.
60
CHAPTER 3. THEORETICAL UNIVERSES
Figure 3.6: Flattened Circulars
If we cut out the circles from the three different geometries shown in Fig. 3.5, and then
attempted to flatten them, the results are shown above. The flat geometry circle (left),
by definition, “stays” flat. The circle cut from the positively-curved sphere (center) has
a circumference smaller than the zero-curvature circle, and when flattened out, will have
“gaps.” The circle from the negatively-curved surface (right) has a circumference larger
than the flat circle, and when flattened, will have sections that overlap each other.
appeared mostly empty, with an occasional galaxy as an island of stars in the sea
of space. This lumpiness certainly does not look smoothly distributed as would be
required by homogeneity. We now have surveys (Figs. 2.19 and 2.20) that show the
Universe to look uniform and isotropic at large enough scales. Early in the 20th
century, it was a leap of faith to assume that this was true given the state of the
observational data at the time.
However cosmologists working with Einstein’s General Theory of Relativity found
themselves facing an extremely complicated set of equations. The theoretical cosmologists had to make whatever simplifications they could, in order to get tractable
solutions from Einstein’s equations. They therefore modeled the Universe as completely filled by a uniform gas or fluid. (And at large enough scales, superclusters
of galaxies will look uniform enough to be described by this “uniform fluid.”) From
these simple model assumptions, a cosmologist can now describe properties of the
gas by its density ρ (Greek letter ‘rho’). Since the fluid is uniform everywhere, the
pressure and density is the same everywhere. However if the Universe is expanding
or collapsing, the density will change with time t, which we can express as ρ(t).
Given such a simplified model, early theoretical cosmologists could now use Einstein’s equations to predict the behavior of the Universe over time. Einstein’s field
equations are represented by:
T,
G + Λ = 8πT
(3.8)
where as we noted back in § 1.5 that Λ represents the cosmological constant. We can
now also state that G is a mathematical construct called a tensor that describes the
curvature of space, and T is another tensor that describes the distribution of energy
and momentum in space.
3.3. MODELING THE UNIVERSE
61
However Eq. 3.8 is far more complicated than Newton’s universal law of gravity
(Eq. 3.1). Using Newton’s equation is difficult enough, since you have to apply the
effect of gravity of all of particles in your system on all of the other particles. This
can be quite a burden—though not impossible using computers—to calculate for realistic descriptions of physical systems. But this is simple compared with Einstein’s
equation, which is actually not one, but a family of ten equations that have to be all
solved simultaneously. Furthermore as noted before, matter and energy are equivalent, so both matter and energy create gravitational fields. But a gravitational field
is also a form of energy, so that also provides an additional warpage to space. This
non-linearity of the equations means that there are not that many instances where
one can come up with exact solutions to the field equations!
3.3
Modeling the Universe
By using the cosmological principle and Einstein’s field equation, early cosmologists
were able to describe different scenarios for the evolution of the Universe. If the
cosmological principle applies, then all of space is uniformly filled by matter with the
same density. Since the curvature of space is caused by the matter and energy within
it, then the same curvature exists at every point in space in these models, which
vastly simplifies the possible models. Before we describe the three main general types
of models, it is useful to look at the very first model universe, which was constructed
by Einstein himself.
3.3.1
Einstein’s Universe
When Einstein developed his General Theory of Relativity in 1916, the expansion
of the Universe had not been discovered yet. When he applied his equations to
the universe as a whole, Einstein found a model universe that was unstable to either
collapse or expansion. Remember from p. 9 that Einstein introduced the cosmological
constant to keep his Universe static, neither expanding nor contracting.
Although we know today that Einstein’s original model is not an accurate description of the Universe, it is useful to look at it carefully to discern its geometric
properties. This static universe is finite in size. But if you travel off in any direction
in a straight line, you will not hit a wall or an edge to the universe; you will however
eventually arrive back at your starting point. Similarly if you shine a laser beam off in
a direction in space, given enough time, the laser light will arrive back at its starting
position.
How can a straight path curve back on itself? This is due to the curvature of
space. A uniformly homogeneous universe will have the same curvature at every
point in space. The cosmological constant can provide a positive curvature to every
point in space so that even for a beam of light moving in a straight line, it will
eventually curve back on itself and arrive back at its starting point.
62
3.3.2
CHAPTER 3. THEORETICAL UNIVERSES
The de Sitter Universe
The Dutch physicist Willem de Sitter (1880–1933) was the second person to apply
Einstein’s equations to create a model of the universe. Unlike Einstein, he did not have
as a requirement a static universe. He also assumed that the amount of matter-energy
in the universe was negligible compared to the effect from the cosmological constant.
As a result, his universe was infinite in extent and expanded not only forever, but
accelerated over time, so that the rate of expansion increased exponentially. What
little matter that was in de Sitter’s universe was carried along in the expansion.
De Sitter realized that such an expansion could result in redshifts being observed
from the matter that was expanding away. He however never pressed this point,
so de Sitter is not considered as having “discovered” the expansion of the Universe
(which would have to wait for Hubble’s pioneering work).
3.3.3
The Friedmann-Robertson-Walker Universes
From the description of Einstein’s universe in § 3.3.1, curvature is the most important
property of space-time when describing model universes. The curvature parameter
is usually denoted by k, and its values corresponding to zero, positive, and negative
curvatures are k = 0, k = +1, and k = −1, respectively.
Later work by the Russian mathematician Aleksandr Friedmann, and the American physicists Howard P. Robertson and Arthur G. Walker showed that a generalized set of model universes for any value of k could be constructed from Einstein’s
equations. These Friedmann-Robertson-Walker (or FRW) models describe the
infinitesimal separation of two events in space-time by the equation:
(ds)2 = [R(t)]2
2
2
2
2
2
2 [(dx) + (dy) + (dz) ] − c (dt)
kr2
1+ 4
(3.9)
where k is the curvature parameter, r is the distance of the two events from the origin
of the coordinate system, and R(t) is a scale-factor that is time-dependent (hence the
“(t)”). For descriptions of either finite or infinite universes, it is almost meaningless
to talk about the “radius” of a universe. However one can define a coordinate system
that is attached to space-time, and can expand or contract with it. This co-moving
coordinate system will change its scale, and the distances between matter attached
to the coordinate system, by the scale factor R(t).
To determine the evolution of a model universe, the cosmic scale-factor R(t) must
be determined by solving something known as Friedmann’s equation:
Ṙ2 =
8πG 2 Λ 2
ρR + R − k c2 .
3
3
(3.10)
The solution will depend on the value of k, the density ρ, and the cosmological
constant Λ. Where this equation comes from and its solution is beyond the scope
3.3. MODELING THE UNIVERSE
63
Figure 3.7: The Scale-Factor at Two Different Times
The expansion of space as seen by a coordinate system that is attached to space-time. Two
galaxies attached to this co-moving coordinate system will have their separation vary by the
scale-factor R(t). Note that the stars in the galaxies themselves are dominated by their
mutual gravity within each galaxy, so they do not expand with space-time; the galaxies stay
the same size at the two times.
64
CHAPTER 3. THEORETICAL UNIVERSES
Figure 3.8: Friedmann-Robertson-Walker Model Universes
Friedmann-Robertson-Walker universes catalogued by the values of the curvature parameter
k and the cosmological constant Λ, showing the evolution of the scale-factor R over time t,
showing some form of expansion and/or contraction with time. The only exception is the
static universe, with Λ = ΛE where ΛE is Einstein’s value of the cosmological constant. .
3.3. MODELING THE UNIVERSE
65
of this course but the results are shown in Fig. 3.8, which show how the scale-factor
changes over time.
Again it is important to re-emphasize that only the positive curvature, k = +1
universes are finite. All the other k = 0, −1 universes are infinite in extent. In
any case, R(t) is not the radius or curvature of a universe, which is meaningless.
It is instead a scale-factor that tells you how a coordinate system attached to the
expanding or contracting space-time changes over time.
Note that in most of the model universes, running time back to t = 0 results in
R = 0 as well. This means that all of these universes have a beginning in a big
bang. Some universes return back to a state where the scale-factor is 0, R = 0, but
at a future time. These universes suffer a big crunch. Other universes continue
expanding forever without ever contracting again.
Λ < 0 Universes
For universes with negative Λ, the cosmological constant causes all the universes,
regardless of curvature to stop expanding and collapse back on themselves in big
crunches.
Λ = 0 Universes
Ignoring the cosmological constant or letting it be very small means effectively Λ = 0.
Until recently, it was thought that such models best described the Universe. If so, then
there are some general statements that we can make about the Universe depending
on what the curvature:
1. k = 00: The simplest case is the flat universe with zero curvature. But if
there is matter and energy in the universe, then why do they not warp space into
a non-flat curvature? It turns out that the curvature from the matter-energy
in the universe can be negated if the universe is also expanding. The energy
from the expansion of the universe will create a curvature that can cancel the
curvature from the matter and energy that is already in the universe. Note
that the expansion has to balance out the matter-energy density in the universe
exactly for the universe to stay flat. Hence this solution is also known as the
critical case. Since gravity from the mass-energy is negated and cannot slow
or stop the expansion, a flat universe will expand forever.
A flat universe also has several other properties. A pair of light beams that
start out parallel will stay parallel forever.
Also a flat universe is infinite in extent. You can travel in any direction forever
without coming back to a part of the universe that you have visited before.
2. k = +1
+1: If the matter-energy density dominates, then the universe has a positive curvature and is closed. The gravity of the mass-energy warps space-time
66
CHAPTER 3. THEORETICAL UNIVERSES
into a analog of the surface of a sphere. For a flat universe, the expansion will
eventually slow and the universe will collapse back on itself.
If you send out two laser beams that start off parallel, the beams will eventually
converge and cross in a closed universe.
Finally a closed universe is also finite in size. If you travel indefinitely in one
direction, you will not hit a wall or run into the edge of the universe, but you
will eventually arrive back at your starting point.
3. k = −1
−1: If the universe has negative curvature—for instance, because the
expansion dominates over the positive curvature from the matter-energy—then
the universe is open, and the universe will expand forever.
Light beams that start off parallel will diverge.
An open universe, like the flat universe, is also infinite in extent.
Λ > 0 Universes
We first consider the k = +1 models together:
1. Λ = ΛE : For the positive cosmological constant universes, let us consider the
case Λ = ΛE , where ΛE is Einstein’s value of cosmological constant that is
required to keep the universe static. The plot therefore shows a straight line for
all time t: the universe is neither expanding nor contracting, and is infinitely
old. However there are two additional FRW solutions. One is a universe that
started off with a big bang at t = 0, but has slowly increased in size until
the R approaches the scale-factor specified by the Einstein model. The other
possibility is the universe starts out with R as specified by Einstein, but it very
gradually expands. Universes with Λ = ΛE and flat or negative curvatures
cannot be static. As shown in the figure, they will expand forever.
2. 0 < Λ < ΛE : If the cosmological constant is greater than zero but less than ΛE ,
then for k = +1, there are two possible behaviors. One is the familiar big bang
at time zero followed by a big crunch (the bottom line in the plot). The second
line in the plot however does not have a definite t = 0. The universe contracts
from some infinite time in the past, but rebounds before it reaches R = 0, and
starts expanding again.
3. Λ > ΛE : Finally for the case where the cosmological constant is greater than
for Einstein’s static universe, the k = +1 scenario is known as the Lemaı̂tre
model, advocated by the Belgian cosmologist and priest Georges Lemaı̂tre
(1894–1966). After the initial big bang, the Lemaı̂tre universe’s scale-factor
evolves to the “flat” portion of the graph where it stays virtually static and
doesn’t grow very much. This was popular for a time in the 1930s when it was
3.4. COSMOLOGICAL REDSHIFTS AND THE HUBBLE CONSTANT
67
thought that the early phase after the big bang could be the time when the
chemical elements were created, while the “coasting” phase was when stars and
galaxies had time to form.
The k = 0, Λ > 0 model is right now thought to be the best description we have for
our Universe. We will see in later chapters evidence for why this is so. This universe
model starts off with a big bang and continues to expand, accelerating forever. There
is a slight slowdown, but not as long as the Lemaı̂tre model’s “coasting” phase.
Finally the k = −1, Λ > 0 model results in an infinitely large universe, that again
expands forever, with the expansion accelerating over time.
3.4
Cosmological Redshifts and the Hubble Constant
Recall that the redshift z of a galaxy can be related to its distance d and the Hubble
constant H◦ :
H◦
d,
(3.11)
z=
c
where c is the speed of light. In addition, the redshift z can be found by a formula
relating the observed and emitted wavelengths of a spectral line from the galaxy:
z=
λobs
− 1.
λem
(3.12)
We saw in the discussion of the FRW universes that their description of an expanding
universe implied an expansion of space-time. This can be measured by the scalefactor R(t). The stretching of space-time over time will cause light waves traveling
through space to stretch as well, resulting in a redshift. Thus the redshifts we observe
in distant galaxies is not due to them moving through space, but due to the galaxies
being carried along by space as it expands! Eq. 3.12 shows a cosmological redshift,
and not a Doppler redshift.
(Galaxies do have peculiar motions, or velocities through space. This can be
caused by gravitational interactions due to neighboring galaxies in a cluster. For
instance the Andromeda Galaxy and the Milky Way are falling toward each other
within the Local Group of galaxies. These peculiar motions add to the scatter to the
measured velocities of galaxies in Hubble relationship diagrams like in Fig. 2.29.)
Furthermore, the Hubble constant can be related to the scale factor R(t) by:
H(t) =
Ṙ(t)
R(t)
(3.13)
where H(t) is a time-dependent version of the Hubble constant and Ṙ(t) represents the
rate of change of the scale-factor over time—you can think of it relating to the velocity
68
CHAPTER 3. THEORETICAL UNIVERSES
Figure 3.9: Cosmological Redshift
The expansion of the Universe, shown by the increasing scale-factor R(t), results in a cosmological redshift. Light traveling from one galaxy is “stretched” by the space that it travels
through, so that it is red-shifted—relative to the original emitted wavelength of the light—
when it is observed at the second galaxy.
3.5. THE CRITICAL DENSITY
69
of the expansion. This means that the Hubble’s constant is not really constant. As
the Universe grows and expands, its value changes. The present day value of the
Hubble constant, at time t◦ after the Big Bang, is given by:
H(t◦ ) = H◦ =
Ṙ(t◦ )
.
R(t◦ )
(3.14)
The Hubble constant that we measure today is therefore the fractional rate of change
of the scale-factor R(t) at the present time t◦ .
3.5
The Critical Density
Recall that for different FRW universes, and depending on the value of Λ and the
curvature of the universe, the expansion could be slowing to a halt (after which time
the universe will collapse), or accelerating (which results in an infinite, open universe).
The critical universe which has k = 0 and Λ = 0 is particularly important, because
it is the borderline case between the closed and open universes.
For any FRW universe, the average density of matter at any time t is given by
ρ(t). In the critical universe, this average density is ρcrit (t). When astronomers first
tried to determine whether our Universe will expand forever or halt and collapse, they
try to measure all of the observable matter, and calculate a density of our Universe.
When comparing the actual density that they measure to the critical density, they
take ratio of the two, represent it by the Greek letter Ω (“Omega”), and call it the
density parameter:
Ω(t◦ ) =
actual density of Universe
ρ(t◦ )
=
.
critical density for a flat universe
ρcrit (t◦ )
If the density in the Universe matches the critical density, then Ω = 1. If the Universe
over-dense which will eventually lead to a big crunch, then Ω > 1. And if there is
less actual density than the critical density, then Ω < 1.
We can also define a density parameter for just the matter (which includes the
baryonic and the dark matter):
Ωm (t) =
ρm (t)
.
ρcrit (t)
(3.15)
If the density of matter at some time in the Universe was one-third of the critical
density, then Ωm (t) = 1/3; if it was one-half of the critical density, then Ωm (t) = 1/2..
Notice that in Friedmann’s equation (Eq. 3.10), the cosmological constant plays a
similar role as the density ρ. This suggests that we can also define a density parameter
for Λ as well:
ρΛ
ΩΛ (t) =
.
(3.16)
ρcrit (t)
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CHAPTER 3. THEORETICAL UNIVERSES
The density ρΛ can be thought of as the energy density of vacuum or of space. Unlike
the matter density which decreases as space expands, the energy density of vacuum
stays constant. But if space expanded and increased in volume, then the total amount
of vacuum energy will increase since there is more vacuum which leads to a greater
total vacuum energy!
As we will see later, there is growing observational evidence that the Universe does
contain a cosmological constant-like energy. Since there are many hypotheses that
have been proposed to explain it, with the cosmological constant being only one of
the possibilities, it has been generically called dark energy. You will therefore often
see in the literature references to ΩΛ as the density parameter for dark energy, not for
the cosmological constant (although Λ did originally refer only to the cosmological
constant).
The density parameters Ωm and ΩΛ can be related directly to the curvature parameter k:


< 1, k = −1;
Ωm + ΩΛ = 1, k = 0;
(3.17)


> 1, k = +1.
Therefore whether the Universe is closed, flat, or open can be determined by measuring Ωm and ΩΛ . All the possible different scenarios for different combinations of
Ωm and ΩΛ is given in Fig. 3.10. The dashed line represents all possibilities where
ΩΛ + Ωm = 1, and k = 0. Above the line is where k is positive; below the line has a
negative k.
So what are the values of ΩΛ and Ωm ? The best measurements imply that ΩΛ +
Ωm = 1, while,
ΩΛ ≈ 0.7,
Ωm ≈ 0.3.
(3.18)
Thus the Universe is flat, and will continue to expand and accelerate in its expansion
forever.
3.6
The Age of the Universe
A critical universe with curvature k = 0 and cosmological constant Λ = 0 can be
shown to have a very simple relationship between the Hubble constant H(t) and the
age of the universe t:
Hcritical
universe (t)
=
2
3tcritical
.
(3.19)
universe
If we lived in such a universe (which we probably do not, since all recent observations
point to Λ 6= 0), then we would be able to determine how old the universe is by
3.6. THE AGE OF THE UNIVERSE
71
Figure 3.10: ΩΛ vs. Ωm
A plot of showing the possibility space of ΩΛ versus Ωm . Depending on the values of these
two parameters, the Friedmann-Robertson-Walker universe will either expand forever (light
blue and yellow regions) or collapse back on itself (light purple region); decelerate while
expanding (light blue region), or keep accelerating forever (yellow region). The red line
represents all flat universes, where k = 0 and ΩΛ + Ωm = 1, with the region to the right of
it representing all universes with positive curvature, k > 0, and the region to the left of it
all universes with negative curvature, k < 0. Our Universe which appears to have ΩΛ ≈ 0.7
and Ωm ≈ 0.3 is represented by the red dot.
72
CHAPTER 3. THEORETICAL UNIVERSES
inverting the above equation. This gives us the Hubble time:
t◦ =
2
,
3H◦
(3.20)
where we measure the value of the Hubble constant at the present time t◦ . If we lived
in a critical universe, and using the best measured value of H◦ we have today, then
our Universe would be about 14 billion years old.
Figure 3.11: Scale-Factor R Over Time t
How the scale-factor R evolves over time t for four different Friedmann-Robertson-Walker
universes. All four model universes have the same Hubble constant at time t◦ . The oldest
universe has a non-zero cosmological constant, ΩΛ > 0.
Other relationships between the Hubble constant and the age of the universe can
be found for the other FRW models. However the forms of the equations are much
more complicated than 3.20. How the scale-factor evolves with time t for several
different FRW cases is plotted in Fig. 3.11. The present time is represented by t0 .
The Hubble constant is the same for each model universe, but the total elapsed time
from the big bang to the present ends up being different.
The closed universe with k = +1 and Λ = 0 has the shortest age, while the critical
universe with k = 0 and Λ = 0 is older, and the open universe with k = −1 and Λ = 0
3.6. THE AGE OF THE UNIVERSE
73
is even older still. The oldest is the model with Λ > 0 and k = 0, which is a universe
that accelerates as it expands.
Cosmologists have calculated possible ages for a range of different cosmological
parameters. Given our best measurements of Ωm and ΩΛ (Eqs 3.19), the age of our
Universe is just shy of one Hubble time, or t◦ = 13.7 billion years old.
74
CHAPTER 3. THEORETICAL UNIVERSES
Chapter 4
The Big Bang
If the Universe is expanding, then if we ran the movie of the Universe backward, the
volume of space-time would decrease the further back we went in time. The density of
matter and radiation would increase, and so would the temperatures and pressures.
From basic physics cosmologists can determine what the physical conditions must
have been like in the early Universe, while from knowledge of nuclear and particle
physics, they can also determine what sort of processes must have occurred in the
past.
4.1
Cosmic Element Abundances
In the first few minutes after the Big Bang, the temperature and pressure conditions
were such that there were only free protons (p), neutrons (n), electrons (e− ), positrons
(e+ ), neutrinos (ν), anti-neutrinos, (ν̄), and electromagnetic radiation in the form of
photons (γ). In fact, above a temperature of 10 billion K, the photon field would be
so energetic that it would spontaneously create electron-positron pairs which would
annihilate to form photons again:
γ + γ e− + e+
(4.1)
Within this sea of radiation, electrons, and positrons, the following reactions were
reversible (and hence the arrows pointing left and right):
e− + p n + ν
ν̄ + p n + e+
n p + e− + ν̄,
(4.2)
Thus as long as the temperature was hot enough, protons were being converted into
neutrons, but there also existed a process which converted neutrons back into protons.
As the temperature dropped below 30 billion K, about 2 seconds after the Big
Bang, primordial nucleosynthesis became possible: the creation of atomic nuclei
75
76
CHAPTER 4. THE BIG BANG
in the aftermath of the Big Bang. The important difference between this and the
stellar nucleosynthesis that is still occurring today inside stars is that the conditions
after the Big Bang were constantly changing. As the Universe grew larger after the
Big Bang, the pressures and temperatures dropped. The intense initial temperatures
meant however the nuclear reactions occurred at rates much higher than that found
inside stars.
The first important nuclear reaction created deuterium or 21 H:
p + n 21 H + γ.
Note that this reaction is again reversible, so that sufficiently energetic photon can destroy the deuterium nucleus. However once the temperature of the Universe dropped
below 1 billion K, the average photon energies dropped to the point where quantities
of deuterium could build up. At about t ≈ 0.7 seconds after the Big Bang, the ratio
of neutrons to protons “freezes out” at:
n
= 0.22.
(4.3)
p
Free neutrons have a decay half-life of time of tN = 10.5 minutes, so the number of
neutrons will decrease a little bit more. However after the temperature drops well
below 1 billion K, or about 170 seconds after the Big Bang, the photons are no longer
energetic enough to destroy deuterium nuclei. The electron-positron pairs all end up
annihilating, and the neutron-proton ratio ends up at:
n
≈ 0.14.
(4.4)
p
At this point the deuterium abundance has grown large enough for the deuterons
to produce helium, via the sequence of reactions:
d+d
d+d
t+d
3
2 He + d
↔ t+p
↔ 32 He + n
↔ 32 He + n
↔ 42 He + p,
(4.5)
where t is tritium or 31 H.
Essentially most of the neutrons wind up in Helium-4 (42 He). Based on the final
number of neutrons after they freeze out (Eq. 4.4), knowing that the number of 42 He
nuclei is half the number of neutrons, and ignoring any other atomic nuclei (other
than Hydrogen), then the mass fraction of 42 He is:
4n4 He
nN
4(nn /2)
2(n/p)
=
=
nn + np
1 + (n/p)
' 0.25,
x4 He ≈
(4.6)
4.2. THE COSMIC MICROWAVE BACKGROUND
77
where nn is the number of neutrons, np is the number of protons, nN = nn + np is
the total number of baryons, and n4 He is the number of 42 He nuclei.
A small amount of Lithium-7 (73 Li) was also created before the temperature dropped
too much:
4
7
2 He + t → 3 Li + γ
3
4
7
2 He + 2 He → 3 Li + γ.
(4.7)
After about 1000 seconds (17 minutes) after the Big Bang, the temperature was below
500 million K and all element creation ended.
There are no known processes in the present Universe that can create substantial
amounts of deuterium; stars can only destroy deuterium as a fuel source in reactions
at their cores. All of the deuterium today is the result of these primordial reactions.
As a result, there are many studies underway today to determine the exact amount
of present day deuterium in order to understand the conditions of the early Universe.
Observations to measure the abundance of deuterium, usually expressed as the D/H
ratio, has been made for Solar System objects and in the local interstellar medium
(ISM). For instance, the best Solar System measurement comes from the atmosphere
of Jupiter, with D/H ' 1–4 × 10−5 , which is similar to values from the local ISM.
More sophisticated modeling of the nucleosynthesis period in the early Universe
showed that the total amount of 42 He created should be about 24% (still not too far off
from our simplistic result in Eq. 4.6. This agrees remarkably well with observations
of not just interstellar gas not polluted by heavy element enrichment from stars, but
with the rough proportion of observed helium abundance everywhere in the Universe.
That the Big Bang (and not any other model) can predict elemental abundances so
close to what is actually observed is one of its underlying strengths as a theory.
4.2
The Cosmic Microwave Background
During the next 300,000–400,000 years, the temperature continued to drop in the
Universe as it expanded. The composition of the Universe consisted of photons,
neutrinos, protons (which would become hydrogen nuclei), helium nuclei, a smattering
of deuterium and lithium nuclei, electrons, and dark matter particles (whatever they
might be!). The photons collided with and scattered between the free electrons,
sharing energy. The electrons also collided and shared energy with the baryons, the
protons and atomic nuclei. All of the protons, nuclei, electrons, and photons could
therefore be described as having the same temperature.
However once conditions cooled to 4500 K, the era of recombination started.
Electrons started combining with atomic nuclei and free protons to form neutral
atoms. By the time the temperature had dropped to 3000 K, most of the electrons
were bound up in atoms. There were not enough free electrons for the photons to
scatter off of, and the Universe became transparent. The photons could now stream
78
CHAPTER 4. THE BIG BANG
Figure 4.1: Light Element Abundances from Big Bang Nucleosynthesis
The abundances of the light elements as a function of the baryon density Ωb which is plotted
at the top of the graph. Plotted along the bottom is the related parameter η, which represents
the number of baryons per photon. The smooth curves show the predicted number of nuclei
of each light element per nucleus of Hydrogen (H) due to Big Bang nucleosynthesis. The
blue and orange boxes show the range of observed abundances for deuterium and 73 Li. The
vertical stripe shows the range of Ωb that is consistent with the observations. From Walker,
T.P., Steigman, G., Schramm, D.N., Olive, K.A., & Kang, H.-S., 1991, “Primordial Nucleosynthesis Redux,” Astrophysical Journal, 376, pp. 51–69.
4.2. THE COSMIC MICROWAVE BACKGROUND
79
freely, instead of interacting with the electrons and nuclei. Because the photons could
no longer easily share their energy with other particles, they became decoupled from
the matter. The matter and photons now evolved separately from each other.
One consequence of the cosmological expansion of the Universe is the cosmological redshift of photons, where their wavelengths are stretched out by the expansion
of space-time (see § 3.4). Because the photons were originally coupled to the dense
hot matter soup (containing protons, electrons, and heavier nuclei), they retained a
blackbody spectrum. A blackbody spectrum originates whenever a source perfectly
absorbs and perfectly re-emits radiation, which aptly describes the early Universe
since the photons are continually absorbed and re-emitted by the matter. Another
term for this radiation is thermal radiation, since the perfect absorption and emission brings the source and the radiation into thermal equilibrium. Blackbody spectra
have a unique shape as we shall see below.
After decoupling, the blackbody spectrum continued to red-shift to lower temperatures and longer wavelengths. From a temperature of 3000–4500 K at a time
300,000–400,000 years after the Big Bang, the radiation has cooled from the infrared
to the microwave portion of the electromagnetic spectrum today. Today this radiation
makes up the cosmic microwave background (CMB), with a temperature just a
few degrees above absolute zero (equivalent to its emission mostly in the microwaves).
The CMB radiation consists of photons coming from the last-scattering surface. These photons last encountered another matter particle when the Universe was
opaque, just before recombination. Immediately after recombination, they were able
to free-stream through space to reach our detectors. Although this last-scattering
surface appears as a spherical shell (like the celestial sphere) that surrounds us, this
does not mean we are at the center of the Universe. A useful analogy is if you go out
walking on a foggy day and there is a visibility of 50 feet: light from objects does
not reach you from beyond a distance of 50 feet. Instead the photons are scattered
by the suspended water droplets that make up the fog. As you walk around this
“foggy universe,” it will feel as if the universe has a radius of 50 feet, defined by
how far you can see. Another observer taking a separate walk outside will see a
different last-scattering surface. Each observer will have her own observable universe
that is slightly different than any other observer’s. Thus there is a limit to how far
we can observe in the Universe! The Universe might be much larger than our visible
horizon as defined by the CMB, but there is no way for us to detect it.
The CMB is the third observational triumph that supports the Big Bang theory.
It was predicted by theoretical calculations in the 1940s, and was first detected in
1965 by Arno Penzias and Robert Wilson of Bell Laboratories. They were using
a horn-shaped microwave antenna to study microwave emission from the Earth’s
atmosphere, in order to identify sources of interference for satellite communications.
However Penzias and Wilson discovered a uniform background source of noise that did
not vary by the time of day, which suggested that it did not originate from the Earth.
It was highly uniform across the sky, suggesting it was also not galactic in nature
80
CHAPTER 4. THE BIG BANG
Figure 4.2: Seeing to the Edge of the Universe
Although we are looking backward in time when we peer out into the deepest reaches of the
Universe (since light has a finite velocity), this does not mean that we can see all the way to
the Big Bang. Photons before the time of recombination were scattered like light in a deep
fog. We can only detect photons that were created after the period of recombination (or light
coming from objects closer than the fog limit.)
4.3. THE STEADY-STATE UNIVERSE
81
either. Furthermore, most matter in the nearby Universe is transparent to CMB
photons, so it was unlikely the radiation was being emitted by nearby matter. They
checked their instrumentation repeatedly, removed pigeons that had been nesting in
their telescope, and finally came to the conclusion that their signal was real.
Throughout the 1970s and 1980s, ground- and balloon-based measurements were
able to confirm that the CMB had a blackbody-like spectrum. In 1989, the Cosmic
Microwave Background Explorer (COBE) was launched into space and it was able to
make detailed, unprecedented measurements of the characteristics of this emission.
Since the changes associated with the CMB are on the order of 10−5 (1 in 100,000)
of the mean temperature, larger effects such as foreground emission from the Milky
Way and other gross effects have to be subtracted out. The largest departure from
uniformity in the background is the dipole pattern, which is a Doppler blue-shift
towards one direction in the sky, and an equal and opposite red-shift in the other
direction. This is consistent with the idea that the Local Group of galaxies is moving
with respect to the CMB at 627 ± 22 km s−1 towards the direction with galactic
coordinates [l, b] = [276◦ ± 3◦ , 30◦ ± 3◦ ].
Not only did it confirm the uniform and isotropic nature of the emission, but
it found the blackbody spectrum to have a temperature of 2.725 ± 0.002 K. The
blackbody spectrum that it measured spectroscopically was perhaps the most perfect
example of a blackbody ever seen. No other emission in nature or from laboratories
on Earth have spectra that matches so closely to that of a theoretical blackbody
spectrum.
4.3
The Steady-State Universe
Although the Friedmann universe models were around since the 1920s, it took many
years for the Big Bang theory of the origins of the Universe to become accepted.
The evidence for an expanding Universe was discovered by Hubble in 1929, and
one immediate consequence of these observations was a universe that had a definite
beginning, which was philosophically unappealing to some cosmologists. As a result
an alternative model known as Steady State was devised.
The Steady State model had its most important advocate in cosmologist Sir Fred
Hoyle (1915–2001), as well as other astronomers such as Thomas Gold (1920–2004),
Hermann Bondi (1919–2005), and Jayant V. Narlikar (1938–). The basic idea behind
the Steady State universe was a re-working of the Cosmological Principle known as the
Perfect Cosmological Principle. In addition to being homogeneous and isotropic
over all space, the universe was also unchanging over time.
Since the Universe appears to be expanding, then the density of matter per cubic
volume would decrease over time. How did the Steady State model explain this?
Theorists postulated that matter must be continuously being created to balance out
the decrease in density as the universe expanded. This was due to a hypothetical
C-field, which results in a continuous creation of new matter. Such a process
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CHAPTER 4. THE BIG BANG
Figure 4.3: COBE’s View of the Cosmic Microwave Background
CMB fluctuations as seen by the COBE DMR. The top picture shows the dipole anisotropy
as a result of the Local Group’s motion with respect to the CMB. The second picture shows
mostly emission from warm dust in the plane of the Milky Way, after the dipole component
has been subtracted out. The last picture shows the data after the galactic emission has been
removed, showing fluctuations on the order of ∼ 10−5 of the 2.725 K background.
4.3. THE STEADY-STATE UNIVERSE
83
Figure 4.4: COBE Spectrum
The spectrum of the cosmic microwave background as measured by the COBE spacecraft.
The dots are the actual measurements. The error bars of the measurements are shown
400 times larger than their actual size; normally they would be undetectable in this image!
Finally the smooth curve is a theoretical blackbody spectrum at a temperature of 2.725 K,
which the observations show a nearly perfect agreement with.
84
CHAPTER 4. THE BIG BANG
has never observed in a laboratory. However since the matter creation rate required
was only about one atom of Hydrogen per cubic meter of space over the age of the
universe, this postulated effect could not be absolutely ruled out.
The discovery of the CMB in 1965 was perhaps the strongest piece of evidence
that killed the Steady State model, since it had no explanation for such a highly
uniform blackbody emission. From the Perfect Cosmological Principle, the Steady
State model also predicted that the Universe should look the same no matter how
deeply in space (and hence how far back in time since light has a finite speed) one
observed. The discovery of quasars in 1966 provided more evidence that directly
contradicted Steady State predictions, since these ultra-luminous compact objects
did not have any counterparts in the nearby Universe.
Chapter 5
The Accelerating Universe
After Hubble’s discovery of the expansion of the Universe, the cosmological constant
Λ was left out of Einstein’s Field Equation when describing our Universe. It felt
somewhat ad hoc to include it in, since its value had to be finely tuned to balance out
the rest of the matter-energy in the Universe to keep the Universe absolutely static.
For decades, cosmologists assumed Λ = 0 mostly as deference to Occam’s Razor.
There was no evidence that a cosmological constant was needed, so it was left out of
serious models of the Universe. Cosmologists took it for granted that the Universe
was gradually slowing due to the gravity of the matter-energy that it contained. The
only question was whether the expansion would continue forever or eventually slow,
stop, and reverse itself. However starting in the mid-1990s, work would be done that
would completely change this view.
5.1
Deceleration Parameter
The Hubble constant as we saw in § 3.4 can be written as,
H◦ =
Ṙ(t◦ )
,
R(t◦ )
(5.1)
where t◦ is the present time, R is the scale-factor, and Ṙ is the rate of change in
the scale factor. The Hubble constant can be thought of as a indicator of the speed
of expansion of the Universe. However as the above equation shows, it is also not
necessarily constant since it can depend on time; its value now at time t◦ might not be
what it is in the past, or what it will be in the future. The Hubble constant H◦ should
therefore really be thought of the present-day value of the Hubble parameter H(t):
H◦ = H(t◦ ).
(5.2)
One way to think of how the current value of the Hubble parameter relates to
the expansion of the Universe is to look at Fig. 5.1 which shows the evolution of the
85
86
CHAPTER 5. THE ACCELERATING UNIVERSE
scale-factor R(t) for an accelerating and a decelerating universe. In both universes,
alien cosmologists measure a value for the Hubble constant H◦ at time t◦ , which
happens to be the same for both universes. However this number only reflects the
rate of expansion of at time t◦ . In the decelerating universe, the rate of expansion
was greater in the past, while in the accelerating universe, the rate of expansion will
be greater in the future.
Figure 5.1: Evolution of Accelerating, Decelerating, and Constantly Expanding Universes
Evolution of an accelerating (green), a decelerating (green), and a constantly expanding
(gray) universe, with observers at time t◦ who measure the same number for the present
value of the Hubble parameter, H◦ . If the universe was neither accelerating nor decelerating,
but had a constant value for the expansion, then this expansion can be simply extrapolated
back in time along the gray line based on the current value of the Hubble constant to obtain
the age of the universe. The age of the accelerating universe is older than this number,
while for a decelerating universe, it is younger.
If a universe had constant rate of expansion (neither accelerating nor decelerating),
it would be represented by the straight gray line where the scale-factor R(t) increases
smoothly and at a steady rate. The time for the big bang start of the universe would
5.1. DECELERATION PARAMETER
87
be indicated at R(t) = 0, or where the gray line intersects the horizontal time axis.
An accelerating universe would be older than the constantly expanding universe, since
in the past it was expanding at a slower rate than indicated by the current value of
H◦ . A decelerating universe was expanding faster in the past—it had a larger value
for the Hubble parameter—and so is younger than either of the other two universes.
In all three of these universes, the value of the Hubble constant H◦ tells you
about the rate of expansion at the current time t◦ . As long as galaxies observed
are relatively nearby, they will follow the expansion of space-time given by H◦ . The
redshift z, distance d, and Hubble constant H◦ are therefore related simply by
z=
H◦
d,
c
(5.3)
d=
cz
.
H◦
(5.4)
or re-writing to give the distance:
In our universe, this linear relationship between the redshift and distance works up to
about z = 0.1–0.2. For instance, Hubble’s 1929 figure showing the measured distances
and velocities of galaxies (Fig. 5.2) can be fit by a straight line using an equation of
the same form as Eq. 5.4. Similarly data from measuring galaxies at distances several
hundred times further can be plotted as shown in Fig. 2.29 where the farthest galaxies,
receding at velocities 2–3 × 104 km s−1 , have redshifts of z <
∼ 0.1. In this case, the
data is also fit well by a straight line of the form of Eq. 5.4.
However if we observe more distant galaxies (which are galaxies existing at a much
earlier time), then a different value of H might be at work, since the universe is likely
to be expanding at a faster or slower rate. A version of Eq. 5.4 that includes a current
value of the deceleration parameter q◦ can be given as:
i
cz h
1
d=
1 + (1 − q◦ )z .
(5.5)
H◦
2
The measured value of the deceleration parameter q◦ is merely the current-day value of
the time-variable q(t). Its relationship to the scale-factor for a Friedmann-RobertsonWalker universe can be shown to be:
q(t) = −
R(t)
R̈(t),
[Ṙ(t)]2
(5.6)
where R̈(t) (“R double dot at time t”) is a measure of the acceleration or deceleration
of the scale-factor. Since there is a minus sign (−) in Eq. 5.6, a negative deceleration
parameter means the universe is accelerating, while a positive value for q means it is
decelerating.
For many decades, cosmologists (notably Allan Sandage) have focused their efforts
on determining precisely the value of q◦ . The deceleration parameter is linked to
88
CHAPTER 5. THE ACCELERATING UNIVERSE
Figure 5.2: Hubble’s 1929 Velocity-Distance Relationship
The Hubble diagram from Edwin Hubble’s 1929 paper, “A Relation Between Distance and
Radial Velocity Among Extra-Galactic Nebulae” (Proceedings of the National Academy of
Sciences, 15, pp. 168–173). The solid dots represent the galaxies which Hubble was able to
determine distances individually, while the solid line shows the Hubble relationship between
velocity and distance fit with these points. The circles and the broken line are for data where
the galaxies have been combined into groups.
5.1. DECELERATION PARAMETER
89
the ultimate fate of the Universe, whether it will continue to expand, or halt and
recollapse. However to determine q◦ , much more distant, high redshift galaxies must
be observed and their distances need to be measured to see the deviation from the
simple Hubble relation. Nearby galaxies will show a simple, straight relationship
between distance and z (see Fig. 5.3) that is based on the current value of the Hubble
parameter, H◦ .
Figure 5.3: High-z vs. Distance Relationship for Galaxies
Plotted are the redshift-distance relationships for a number of different universes with different values of the deceleration parameter, q◦ . For nearby galaxies with z <
∼ 0.1, all the plotted
lines are straight, with the slope determined by Eq. 5.4. At higher redshifts, the effects of q◦
start to kick in via Eq. 5.5.
Attempts to measure q◦ began in the late 1940s. George Abell focused on very
rich galaxy clusters that he was discovering—the so-called “Abell clusters” that were
named after him. He could not observe individual stars in these distant clusters
to identify Cepheid variables, so he assumed that statistically, the 10th brightest in
each galaxy had roughly the same intrinsic luminosity. These were then assumed to
be standard candles, which Abell used to determine approximate distances to the
clusters.
Work following along these lines led to a range of values for q◦ appearing in the
astronomical literature, with no clear agreement. Later work showed that there was
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CHAPTER 5. THE ACCELERATING UNIVERSE
substantial evolution among galaxies. Galaxies in clusters further away are observed
when they are younger, but also possibly with a very different luminosity than they
would have today. These evolutionary effects made it impossible to use whole galaxies
as standard candles in this fashion.
5.2
Type Ia Supernovae
As mentioned in § 2.5.4, Type Ia supernovae have been proposed as standard candles
since they are all the result of white dwarf stars accreting gas in binary systems, and
have reached the Chandrasekhar mass limit of 1.4 M . Above this limit, they explode
in a supernova as they turn themselves into neutron stars. The peak luminosities of
these Type Ia supernovae are extremely bright, about 109 L (or 1 billion solar
luminosities), making them useful for extragalactic distance determination.
However with enough observations of nearby Type Ia supernovae, it was clear that
Type Ia supernovae do not all reach the same maximum luminosity, but varied by
about 35%. The light curves, how the supernovae’s brightnesses vary with time,
also had slightly different shapes. The brighter supernovae tended to have broader
light curves, meaning it took longer for them to decrease from peak brightness (see
Fig. 5.4. The fainter supernovae, in contrast, decayed faster. There seemed then to
be a direct correlation between the peak brightness of a Type Ia supernova and the
width of its light curve. Thus by carefully measuring the fall-off in intensity over
several weeks in the light curve shape, one could determine the absolute magnitude
of the supernova peak luminosity.
Once the light curve shape technique was perfected, two teams set out to search for
and measure the light curves from distant, high-z supernovae. The Supernova Cosmology Project (SCP) and the High-z Supernova Search (HZSNS) both approached
the project in similar ways. You can’t predict when a particular galaxy will have
a star that goes supernova, so you look at lots of galaxies at once, hoping to catch
supernovae in at least one of them. For instance the SCP team made images at 50–
100 pointings in the sky in a single night at the Cerro Tololo 4 m telescope in Chile.
Each pointing typically had 100 galaxies. Three weeks later, they re-imaged these
fields and the later plates were compared to the earlier ones. Sophisticated computer
algorithms were used to identify individual supernovae candidates among the tens of
thousands of galaxies observed. Typically they would find 5–20 potential supernovae
for each night of observation. These candidate supernovae were followed up by additional observations at a variety of ground-based observatories, with the Hubble Space
Telescope used for the faintest, high-redshift candidates.
Spectroscopic observations were made to determine the redshift of the supernova
and its host galaxy, and also to confirm the object was indeed a Type Ia supernova.
Photometric observations—careful measurements of the brightness of the source—
were made over several weeks in order to construct the light curve. The results from
the two independent teams, announced in 1998, stunned the astrophysical community.
5.2. TYPE IA SUPERNOVAE
91
Figure 5.4: Type Ia Supernovae Light Curve Shapes
Plotted are a sample of light curves from Type Ia supernovae in nearby galaxies whose
distances were determined via other means, with their absolute magnitude plotted versus the
time of observation in units of days. Note that the supernova increases sharply in brightness
over the course of a few weeks before reaching its peak brightness (marked as time “0” in
the plot). The plot further shows that the brighter the peak intensity of the supernova, the
broader the shape of its light curve, and the longer for it to decay in brightness. From A. G.
Riess, W. H. Press, & R. P. Kirshner, “Using Type Ia Supernova Light Curve Shapes to
Measure the Hubble Constant,” Astrophysical Journal, 438, pp. L17–L20.
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CHAPTER 5. THE ACCELERATING UNIVERSE
The supernovae had smaller redshifts than would be expected for their brightnesses,
given calibrations with nearby Type Ia supernovae and Hubble’s law. This implied
that the Universe was accelerating in its expansion, something not predicted by the
Λ = 0 Friedmann-Robertson-Walker models.
Figure 5.5: Supernova Cosmology Project Hubble Diagram
A Hubble diagram constructed with 42 high-redshift supernovae from the Supernova Cosmology Project plotted as red points. Data from an earlier survey at low-redshift is plotted
in yellow. Plotted also are a number of lines for the redshift-magnitude relationship for
Λ = 0 (black lines) and non-zero Λ (dashed blue lines). From S. Perlmutter et al., 1998,
“Discovery of a Supernova Explosion at Half the Age of the Universe,” Nature, 391, 51.
This and other corroborating work after 1998 suggests a cosmological constant
(or something like it) is an important component of the Universe. However instead
of referring to it merely as a cosmological constant, cosmologists commonly give this
factor the generic name of dark energy, since Einstein’s cosmological constant is only
one of a number of proposed models for explaining the origins of the acceleration. We
will therefore refer to the symbol Λ interchangeably as dark energy or the cosmological
constant.
From the SCP and HZSNS data, estimates can be made for the value of the density
5.3. MORE ON DARK ENERGY
93
parameters ΩΛ and Ωm . (This was how Fig. 3.10 was generated.) The SCP results
can be used to construct the plot in Fig. 5.6. Plotted are ellipses showing confidence
levels, statistical measures of the uncertainty of the locations of the values for Ωm and
ΩΛ . For instance there is a 99% chance that the true values for Ωm and ΩΛ lie within
the dotted line ellipse. The smaller ellipses more tightly constrain the possible values
of the density parameters, but there is less certainty in these results. This (and plots
like this using combined data from both teams) show that our Universe appears to
be flat, meaning
ΩΛ + Ωm = 1,
(5.7)
while the total contributions from dark energy and matter to the density of the
Universe is given by
ΩΛ ≈ 0.7
Ωm ≈ 0.3.
5.3
(5.8)
More on Dark Energy
If we recall that measurements of matter and dark matter from § 1.4 that the amount
of normal, baryonic matter is outnumbered by the amount of dark matter. In fact
the baryon density parameter is only:
Ωb ≈ 0.02−0.04.
(5.9)
That is, the observed number of baryons is only 2–4% of the critical density necessary
for a flat universe.
From several independent methods of accounting for dark matter (some are recounted in § 2.6, we find roughly 10 times more dark matter than ordinary baryonic
matter:
Ωd ≈ 0.20−0.30.
(5.10)
Eqs. 5.9 and 5.10 are consistent with Ωm ≈ 0.3 in Eq. 5.9.
The discovery of dark energy makes a fascinating denouement to the Copernican
Revolution originally discussed in § 1.4. From the time of the Renaissance and the
beginnings of modern science, we have seen the Copernican principle periodically
manifest itself in our knowledge. The Earth was displaced from the center of the
Solar System and became just another planet that orbited the Sun. Our sense of the
size of the Universe also increase over time. By the early 20th century, the Sun was
further removed from the center of our collective system of stars that made up the
Milky Way Galaxy. Our own Galaxy soon became just another member of a small,
insignificant cluster in an odd corner of a Universe filled with other galaxies. And as
a final insult, we have learned that the baryonic matter that we are composed of is
94
CHAPTER 5. THE ACCELERATING UNIVERSE
Figure 5.6: Confidence Levels for Ωm and ΩΛ
A diagram of Ωm vs. ΩΛ , showing the different universes that result from a range of values
for the two parameters. Plotted are confidence contours that show the Universe is very
nearly flat, Ωm + ΩΛ = 1, while Ωm = 0.3 and ΩΛ = 0.7.
5.3. MORE ON DARK ENERGY
95
only a few percent at most of all matter-energy, and most of the Universe is made of
a bizarre dark energy that we are only beginning to understand!
So what is dark energy? One class of energies that might explain it is vacuum
energy, the energy of empty space devoid of traditional forms of matter-energy.
Although energy is absolutely conserved in classical physics, this is not the case in
quantum mechanics, where energy can appear and disappear out of nowhere, spontaneously and unpredictably. Thus even “empty” space is never empty, but is filled
with virtual particles, pairs of a particle and its anti-particle, appearing out of
nowhere and destroying themselves and disappearing before they can be detected.
The amount of energy ∆E available to create these virtual particles and the time ∆t
that they exist follow Heisenberg’s Uncertainty Principle, one form of which is:
h
,
(5.11)
2π
where h is the Planck constant, h = 6.626 × 10−27 erg sec. Even when there is
not enough energy to create a particle pair, it is “borrowed” for a very brief amount
of time, and after the particles annihilate, the energy “debt” is paid back. The
Uncertainty Principle implies that the more massive the virtual particles, the shorter
lived they will be. Such virtual particles filling space can have a calculable effect on
the energy levels of atoms, and this has been predicted as well as observed as the
Lamb shift. Another phenomenon, the Casimir effect whereby two plates hanging
close together in a vacuum are attracted toward each other, can also be explained by
virtual particles.
Physicists can calculate from first principles what the energy density of the quantum vacuum energy must be. They add up all the contributions from virtual particles
that could arise based on energies from Eq. 5.11. There is still uncertainty what cutoff one should make in the maximum virtual particle energies. Clearly one cannot
add up virtual particles with near-infinitely large energies. Whatever maximum particle energy they decide to pick, the resulting answer is much, much larger than what
is observed for the dark energy density. For example, using one typical maximum
cut-off gives a vacuum energy density of 10119 times bigger than observed—that is,
∆E∆t ∼
>
100, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000,
000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000,
000, 000, 000, 000, 000, 000, 000, 000, 000, 000!
This discrepancy between the theoretical value for the vacuum energy density and
the observed dark energy density has been described by physicist Steven Weinberg as
“the worst failure of an order-of-magnitude-estimate in the history of science.” There
must be some additional restrictions on the maximum energy (or perhaps other terms
that cancel out the largest energies) that is not clear to the theoreticians working on
this problem presently. Dark energy may yet turn out to be quantum mechanical
vacuum energy, but a clear theoretical case for it does not yet exist.
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CHAPTER 5. THE ACCELERATING UNIVERSE
If dark energy cannot be explained quantum mechanically right now, can it still
be Einstein’s cosmological constant? If it is, then dark energy would be true to its
name: it is a constant for any particular volume of space at any time. If your volume
of space doubles in size because of the expansion of space-time, then the amount
of cosmological constant present will also be doubled. However some theoretical
cosmologists think that if the cosmological constant had been present in its current
form from the start of the Big Bang, then the evolution of structure in the Universe
would not lead to what we see today. The cosmological constant would have to be
far weaker to allow for the formation of galaxies and stars that we see.
As a result, many other forms of dark energy have been proposed, including a class
called quintessence. Quintessence is named after the fifth element of the Ancient
Greeks (after earth, water, air, and fire), and which was thought to hold the stars
and planets in place in the sky in their old cosmogonies. The difference between
quintessence and Einstein’s cosmological constant is that quintessence is allowed to
be changing, over time and space.
There are many different models for quintessence, and they all arise to some extent
from particle physics. One idea is that it is a scalar field. A scalar field defines a
strength of the field at every point in space, but does not specify a preferred direction
for the field. Einstein’s cosmological constant can be thought of as a special case of a
scalar field, where the strength of the field is the same everywhere. In more general
scalar fields, the strength of the field can vary in time and position. Quintessence
is thought to vary in strength over time. One scenario has it increasing in strength
until it reaches the constant value that it still has today.
Chapter 6
Anisotropies in the Cosmic
Microwave Background
As first discussed in § 4.2, a highly uniform radiation field pervades the Universe. This
cosmic microwave background (CMB) radiation originates when the Universe first became transparent, about 300,000–400,000 years after the Big Bang. Before this time,
the radiation was strongly coupled to the matter via collisions with free electrons. In
fact, the radiation dominated over the matter, with roughly 109 (1 billion) photons
for every baryon. The constant collision, absorption, and re-emission of photons with
the matter particles meant the radiation retained a blackbody spectrum, which can be
characterized by a single temperature. Fig. 6.1 show several example blackbody spectra that would be expected from objects with temperatures 3000–6000 K. Notice that
as the temperature decreases, two things happen: (1) the flux density decreases—less
total radiation is emitted, and (2) the peak of the emission moves to longer (redder)
wavelengths. The peak of the blackbody spectrum is in fact given by Wien’s Law,
which in SI units is:
5.1 × 10−3
,
(6.1)
λpeak =
T
where the wavelength λpeak is in meters and the temperature T is in Kelvin. From
Fig. 4.4 the peak of the COBE spectrum can be measured accurately to be at a
wavelength of 1.87 mm. From Wien’s Law, the implied temperature is T = 5.1 ×
10−3 /λpeak , where λpeak = 1.87 mm = 1.87 × 10−3 m or T ≈ 2.73 K.
As also noted before, this is the current temperature of the radiation. When it
was emitted at the time of recombination at z = 1100, the Universe was 1100 times
smaller. The expansion of space-time has resulted also expanded the CMB radiation
by this factor. Since Wien’s Law is linear, a 1100 factor decrease in wavelength of
the radiation implies a temperature at recombination with a temperature 1100 times
The specific way in which the CMB is treated as an acoustic phenomenon in this
chapter follows the method developed by Mark Whittle.
See his excellent website
http://www.astro.virginia.edu/ dmw8f/ for more information.
97
98CHAPTER 6. ANISOTROPIES IN THE COSMIC MICROWAVE BACKGROUND
Figure 6.1: Blackbody Spectra
Examples of blackbody spectra with temperatures from 3000 K to 6000 K. As a blackbody
emitter’s temperature drops, its spectrum reduces in overall intensity, and its peak shifts
redward to longer wavelengths.
larger than the current value, or T ∼ 3000 K.
The temperature is highly uniform, after foreground emission and the dipole
anistropy has been subtracted out. However as seen in the bottom image in Fig. 4.3,
the CMB is not completely smooth: it has tiny temperatures on the order of 1:100,000
of the mean temperature. These fluctuations further appear at different sizes in the
sky in the COBE maps. They therefore have different angular scales. COBE’s
detectors had a resolution of 7◦ , meaning this was the smallest angular-size of an
object that it could detect. Fluctuations larger than this scale could be seen, but not
anything smaller.
There have been many follow-up attempts to measure finer scale fluctuations
than could be observed by COBE. Many telescopic experiments were planned and
executed from the ground, while a couple were planned for space. One ground-based
experiment was BOOMERanG (“Balloon Observations Of Millemetric Extragalactic
Radiation and Geophysics”), a balloon-borne telescope that floated above Antarctica,
and which observed a 40◦ × 25◦ patch of sky for 10 days.
However the highest resolution maps to date are being provided by the Wilkinson Microwave Anisotropy Probe (WMAP), which was launched in June 2001. The
angular resolution has been improved by almost a factor of ten over BOOMERanG,
resulting in structures that are resolved down to 0.25◦ . The first detailed maps from
WMAP were released in 2003, but the satellite continues to gather more data.
6.1. ANALYZING THE FLUCTUATIONS
99
Figure 6.2: Maps of the Earth
A high resolution map of the Earth’s surface (top) compared with a lower resolution version
where the finest detail that can be resolved is only 7◦ (bottom). The bottom map has
comparable resolution as the COBE maps of the CMB.
6.1
Analyzing the Fluctuations
For cosmologists, the locations of a specific warmer patch versus the location of
another cooler patch is not that important. What will reveal information about
the Universe as a whole is a statistical study of all of the fluctuations at once. This
can give us a global view of what is occurring in the Universe, rather than a local view
of the temperature variation in any one patch of sky. The method that cosmologists
have used to analyze the CMB is to study not just the temperature fluctuations,
but their wavelengths or angular size, and to see how strongly each sized fluctuation
contributes to the overall CMB.
The Universe before recombination should have been a hot plasma of protons,
neutrons, and electrons, mixed into a sea of photons, neutrinos, and dark matter
particles. Back then as now, the dark matter should dominate over the ordinary
baryonic matter. However the dark matter does not interact with the radiation field,
but interacts only gravitationally with the baryonic matter. But since the baryonic
matter is coupled tightly with the photons (to form a photon-baryon fluid), the
dark matter will be interacting indirectly with the radiation.
Non-uniformities in the dark matter means that the dark matter particles will
100CHAPTER 6. ANISOTROPIES IN THE COSMIC MICROWAVE BACKGROUND
Figure 6.3: The BOOMERanG Field
A view of the CMB as seen by the BOOMERanG experiment. It did not observe the whole
sky, but only a limited patch. Its 1◦ resolution meant that it could detect structures seven
times smaller than COBE. Note that the maximum range in temperature variations in
the CMB is ±300 µK, about ten thousand times smaller than the uniform temperature
of 2.725 K. From P. de Bernardis et al., 2000, “A Flat Universe from High-Resolution
Maps of the Cosmic Microwave Background Radiation,” Nature, 404, p. 955
6.1. ANALYZING THE FLUCTUATIONS
Figure 6.4: The CMB from WMAP and COBE
101
102CHAPTER 6. ANISOTROPIES IN THE COSMIC MICROWAVE BACKGROUND
Figure 6.5: Sound Wave Description
An example of a sound wave in air, which causes a variation in the pressure. Regions of
high air pressure are where the air molecules are compressed together, while low-pressure
regions are places where the air molecules are spread apart. The height or amplitudes of the
pressure waves give you the loudness of the sound.
6.1. ANALYZING THE FLUCTUATIONS
103
pool together in slightly higher density halos.1 The photon-baryon fluid will have a
tendency to flow toward these halos by the action of gravity. However the collapsing
photon-baryon fluid will resist being compressed by its internal pressure: it will tend
to “bounce back.” The photon-baryon fluid rebounding from falling into the dark
matter halos will oscillate, which create variations in density and pressure that spread
out as acoustic or sound waves. Thus the chaotic primordial stew in the aftermath
of the Big Bang can be viewed as being criss-crossed by sound waves.
Sound waves are compression waves that increase and decrease the pressures of
the medium that it travels through (Fig. 6.5). The amplitude of a wave is the height
of the peaks and the depth of the troughs as seen in a traditional line plot representation (Fig. 6.5a). In a sound wave, this amplitude difference is a difference in pressure
between the regions compressed and rarefied by the wave (Fig. 6.5b). In the compressed zones of a sound wave that is passing through the air, the air molecules are
squeezed closer together, while in the rarefaction zones, the air molecules are spread
further apart. The wavelength is the distance between successive peaks (or troughs)
within the wave.
The wavelengths of these sound waves depend on the time after the Big Bang that
the oscillations that generated them begin. At time t, any disturbances in pressure in
the photon-baryon fluid are not felt at distances greater than the distance ct, since the
maximum speed by which any signals can propagate is fixed by the speed of light, c.
Early enough in the Universe’s history, ct is smaller than the size of the dark matter
halos, so no fluctuations will originate from the halos. However given enough time
as the Universe expands, ct will be greater than the size of the halos and the sound
waves can begin to propagate from the halos. A useful point to remember is that the
largest wavelength of the waves will be roughly ct.
Early in the Universe, many waves with a large range of wavelengths and amplitudes can exist, just like the waves on the surface of a choppy sea (Fig. 6.6).
Looking at the CMB is therefore like looking at the multitude of ripples that are on
the surface of a wind-whipped ocean. The wavelengths measured so far by COBE
and its successor telescopes and experiments are in the range 20,000–200,000 light
years. The variations in pressure are about 1:10,000, which correspond to 110 dB
(decibels). Because this period in the Universe’s history is radiation-dominated, the
wave speeds are very high, about 0.5–0.6c. The number of waves passing per second
gives the pitch or sound frequency. Since these waves are so enormous and it would
take 40,000 years or more for a single wave to pass by, their frequencies are extremely
low, around f ≈ 10−12 –10−14 Hz.
To study the ripples in the CMB, cosmologists construct an angular power spectrum. This is a plot showing the contributions of all of the different wavelengths to
the CMB as seen in the entire sky. Using special mathematical techniques, they
extract out the amplitude of each wave with a wavelength corresponding to some
1
These slightly denser regions will be the seeds of future superclusters of galaxies, while the giant
voids between superclusters will come from the regions between the halos.
104CHAPTER 6. ANISOTROPIES IN THE COSMIC MICROWAVE BACKGROUND
Figure 6.6: Water Waves in the Ocean
Waves of many different wavelengths and amplitudes can be seen in the ocean.
Figure 6.7: Sky Maps with Corresponding Power Spectra
Three simple sky maps showing cooler (red) and hotter (blue) temperature patches, and
their equivalent power spectra. Each CMB has structures with only a single scale, which
have (from top to bottom) λ ≈ 180◦ , 7◦ , 0.25◦ . Based on C. H. Lineweaver, 1997, “Gold in
the Doppler Hills: Cosmological Parameters in the Microwave Background,” Proceedings
of the International School in Astrophysics: From Quantum Fluctuations to Cosmological
Structures, held in Casablanca, Morocco, December 1–10, 1996.
6.2. ORIGINS OF THE POWER SPECTRUM
105
angular size on the sky. The minimum size structure in the COBE maps for instance
are waves that correspond to structures 7◦ across. To make a more direct correspondence with sound waves, instead of talking about the wavelength of the fluctuations,
we refer to their angular frequency or multipole number l, which is roughly
l≈
180◦
,
λ
(6.2)
where λ is the wavelength. Therefore the minimum COBE-visible structure of 7◦
corresponds to l ∼ 25, while 1◦ -sized patches are equivalent to l ∼ 180. Thus small
waves have large l, while enormous waves show up at tiny l. Fig. 6.7 shows three
examples of very simple power spectra. The largest wave possible is one that fills
up the entire sky with the peak of the wave in hemisphere, and the trough in the
opposite hemisphere. The wavelength is λ = 180◦ , so l = 1. If this was the only wave
component in the CMB, then its power spectrum would show a single peak at l = 1.
If the CMB contained only fluctuations that were 7◦ across, then the sky might look
like the middle row in the figure, with a power spectrum spike at l ∼ 25. Finally
if there CMB was filled with structure 0.25◦ across, then the power spectrum would
have a single peak at l ∼ 700.
Fig. 6.8 shows a power spectrum from the CMB that summarizes the results
from seven different experiments, including BOOMERanG and WMAP. Plotted are
fluctuations with l in the range 2–1800. The data points show not only the amplitude
of each fluctuation, but also the corresponding error bars, which gives you a rough
idea of the uncertainties associated with each experiment.
Fig. 6.9 shows the result of averaging together all of the available data from
early 2003. The smooth curve shows the best fit from a cosmological model based
on Friedmann-Robertson-Walker universes that vary a large number of cosmological
parameters.
6.2
Origins of the Power Spectrum
What do these temperature fluctuations mean and what do they tell us? First remember that all of what we see of the CMB dates back to the last-scattering surface,
at the time the Universe became transparent. The radiation was being emitted and
absorbed through a matter soup that was filled with the ripples of overlapping acoustic waves. These ripples cause slight differences in the densities of the fluid, and they
result in the anisotropies that we measure in the CMB today. As soon as the Universe became transparent at recombination, the photons no longer interacted with
the matter. The pressure that gave support to the sound waves disappeared, and
the acoustic waves stop oscillating. The radiation then is a frozen signature of the
acoustic oscillations right before recombination.
The CMB at different values of the multipole number l reflect different physical
processes at work. For large angular scales of several degrees or more, with multiple
106CHAPTER 6. ANISOTROPIES IN THE COSMIC MICROWAVE BACKGROUND
Figure 6.8: CMB Power Spectrum Summary
A summary of CMB measurements from seven different experiments as of January 2004.
From Max Tegmark’s website http://space.mit.edu/home/tegmark.
6.2. ORIGINS OF THE POWER SPECTRUM
107
Figure 6.9: CMB Power Spectrum Data and Fit
A CMB power spectrum constructed from all of the available measurements as of early
2003. The specific shape of the power spectrum can give constraints on many cosmological
parameters simultaneously. Such a model is plotted as the black line, with ΩΛ = 0.743,
cold dark matter density parameter ΩCDM = 0.213, Ωbaryon = 0.0436, and Hubble constant
H = 72 km s−1 Mpc−1 .
108CHAPTER 6. ANISOTROPIES IN THE COSMIC MICROWAVE BACKGROUND
numbers l <
∼ 50, variations in the CMB are caused by the Sachs-Wolfe effect. A
region with a higher density in the last-scattering surface implied a slightly overdensity, which had a deeper gravitational potential well. General Relativity predicts
that photons become gravitationally red-shifted if they are forced to climb out of a
potential well (and conversely become blue-shifted if they fall in). Additionally, General Relativity also predicts a time dilation effect when they climb out of a potential
well. The slightly lower temperature patches in the CMB are therefore locations that
are over-dense compared to the average, and these lead to the “plateau” in the CMB
angular power spectrum at low l-numbers.
At l-numbers in the range ∼ 50–1000, the power spectrum reflects the acoustic
waves at the time of decoupling. Some pockets of the photon-baryon fluid are reaching
their maximum state of compression as they oscillate in the dark matter halos. These
compressions heat the photon-baryon fluid, causing the photons leaving these pockets
to be at a higher temperature than photons elsewhere. The size of these anisotropies
will be similar to the wavelength of the waves, or hence, will be at most ctdec in size,
where tdec = 380, 000 years is the time of recombination. Thus the largest fluctuations
should be no more than 380,000 light years in size. In a flat universe, where k = 0,
these largest fluctuations are thought to be about 1◦ in size, and correspond to the
peak at l = 220.
Figure 6.10: Origin of Acoustic Waves
Dark matter halos create potential wells, whose gravity pulls on the photon-baryon fluid,
causing them to fall into the wells. As the fluid collects and compresses inside the well, the
increasing pressure causes it to rebound out of the well. These oscillations are the origins of
the acoustic waves which pervade the early Universe. Regions where the gas is compressed
are heated to a higher temperature, and therefore appear brighter.
Another way to think of this is to think of these fluctuations as due to the photonbaryon fluid having a chance to compress once before it freezes out at recombination.
There are other fluctuations that are half as large; their wavelengths are 1/2 of the
6.2. ORIGINS OF THE POWER SPECTRUM
109
Figure 6.11: Origin of the Acoustic Peaks
The acoustic peaks in the CMB originate from the oscillation of the photon-baryon fluid
in the dark matter halos. The odd (1st, 3rd, 5th, . . .) peaks occur for fluctuations that
are in a state maximum compression when they are frozen out during recombination. The
even peaks (2nd, 4th, 6th, . . .) are for fluctuations that are at an extreme rarefaction during
recombination. Oscillations that are either in the midst of infall or rebound have gas motions
that Doppler blur the radiation, resulting in power between the peaks. Based on a diagram
from C. H. Lineweaver, 2004, “Inflation and the Cosmic Microwave Background,” submitted
to World Scientific.
110CHAPTER 6. ANISOTROPIES IN THE COSMIC MICROWAVE BACKGROUND
l = 220 mode, but as a result oscillates twice as fast. Then this anisotropy will have
a chance to compress and rarefy before combination. Finally for a mode that has
a wavelength 1/3 of the l = 220 mode, it can compress, rarefy, and compress itself
again. These three modes show up as the first, second, and third peaks in the CMB
power spectrum. The second, third, and higher number peaks can be thought of as
higher order harmonics of the first, fundamental peak.
For very small angular scales, with l >
∼ 1000, the acoustic waves are reduced by
an effect called Silk damping. The wavelengths of the oscillations are so short,
that they are comparable to the distance that photons travel during recombination.
Recombination does not happen instantly; during this period photons can bounce
collide with the charged electrons and baryons. During this random walk, photons can
scatter across the wavelength of the fluctuation, and the warmer and cooler photons
can mix and average out. The acoustic peaks therefore end up getting smeared out.
6.3
Analyzing the Anisotropies
Virtually every single cosmological parameter can affect the shape of the CMB power
spectrum. Cosmologists can therefore create models of the universe with different
values for H◦ , Ωm , ΩΛ , Ωb , and many others we have not discussed. Power spectra
for each of these universes can be constructed and compared with the observed power
spectrum. In fact, the location of the first peak, at l = 220, is strong evidence that the
Universe is very close to being spatially flat. The total energy density of the Universe
(from matter as well as dark energy) affects its geometry, and its curvature. For
k = +1 curvatures, the Universe bends light from the CMB like a convex lens, which
magnify the angular size of the patches. Conversely, k = −1 curvature results in
reducing the apparent sizes of patches. The CMB is therefore a completely separate
line of evidence, independent of the Type Ia supernovae measurements, that give
support to a curvature parameter of k = 0.
The amount of baryons can be determined from the ratio of the odd numbered
acoustic peaks compared to the even numbered peaks. A higher baryon fraction
means more mass in the dark matter halo potential wells. The photon-baryon gas
can therefore compress more due to gravity, before the pressure pushes the fluid back
out of the well. Since the odd numbered peaks are associated with the fluid falling
into the potential wells, and the even peaks with the fluid rebounding out of the
wells, then more baryons mean the odd peaks grow in size, while the even peaks are
suppressed. The implied baryon density of Ωb ≈ 0.02, which agrees with deuterium
measurements of quasars at high-z, and Big Bang nucleosynthesis theory.
The overall shape of the power spectrum, the locations and relative heights of
the acoustic peaks, can give quite a bit of information about the parameters that
define the Universe. By using all of the datasets available (from more than a dozen
different CMB experiments), one can build a model that best matches (or “fits”) the
data. This Concordance Model is constantly being refined, as newer and better
6.3. ANALYZING THE ANISOTROPIES
111
measurements are added. However the parameters derived from it are a remarkable
set of numbers, some of which have been pursued by cosmologists for much of the last
century. A measurement like the age of the Universe has been debated by thinkers for
millenia, and it is a testament to our scientific and technical prowess that we know it
to such a high degree of accuracy today.
Table 6.1: Properties of the Universe
Property
Age of Universe
Total density
Dark energy density
Matter density
Baryon density
Hubble constant
CMB Temperature
Baryon-to-photon ratio
Baryon-to-matter ratio
Redshift, recombination
Thickness of recombination
Time of first stars
Time of CMB, recombination
Symbol
t
Ωtot
ΩΛ
Ωm
Ωb
H◦
Tcmb
η
Ωb /Ωm
zdec
∆zdec
tr
tcmb
Value
13.7 billion years
1.02
0.73
0.27
0.044
71 km s−1 Mpc−1
2.725
6.1 × 10−10
0.17
1089
195
180 Myr
379 kyr
Uncertainty
±0.3
±0.02
±0.04
±0.04
±0.004
±4
±0.002
+0.3×10−10
−0.2×10−10
±0.01
±1
±2
+220 Myr
−80 Myr
+3 kyr
−2 kyr
112CHAPTER 6. ANISOTROPIES IN THE COSMIC MICROWAVE BACKGROUND
Chapter 7
Structure Formation in the
Universe
Galaxy and structure formation in the Universe after the Big Bang is thought to
have proceeded from the non-uniformity “seeds” provided by the dark matter halos
discussed in the previous chapter. These localized regions of density enhancements
attracted additional matter because of its gravity, which continued to increase size of
these localized regions of density. The dark matter is important because left to itself,
seeds containing baryonic matter only would take too long to grow and amplify into
the structures we see today; some models show it should take over 40 billion years
for gravity to coalesce structure from only baryonic matter. Therefore not only is the
existence of dark matter inferred from observations of galaxies and galaxy clusters,
but they appear necessary for baryonic matter to settle into large-scale structure
within the lifetime of the Universe. Dark matter also behaves differently than the
matter. Its only interaction is via gravity so it does not interact with the photons, like
the baryons. Dark matter merely clumps up and forms the potential wells depicted
in Fig. 6.10. And because dark matter is anywhere from 5–10 times more abundant
than ordinary baryonic matter, it is primarily responsible for structure formation in
the early Universe.
Two general categories of dark matter have been proposed. The first, hot dark
matter (HDM), is now somewhat out of favor. The “hot” in its name refer to the
subatomic particles which are thought to travel near the speed of light. The neutrino
was the favored candidate for HDM, but recent experiments have shown that its
mass is too insignificant to make up but a small component of the dark matter.
Because of its speeds close to the speed of light, HDM tends to wash out the small
scale fluctuations. Cosmologists who create theoretical simulations with HDM see
extremely large-scale structures forming first, which then break apart into smaller
objects, in an hierarchical scenario which gives top-down structure formation.
However in order to produce galaxy-sized objects in the present-day Universe, HDM
models tend to also create much more large scale structure than is currently observed.
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CHAPTER 7. STRUCTURE FORMATION IN THE UNIVERSE
The other class of models involve cold dark matter (CDM), which are typically
massive, slow moving particles such as the WIMPs. Computer simulations show
that a CDM-dominated universe tends to form small structures first, typically a
million solar masses. These then collect together to hierarchically build up to galaxysized (and larger) structures. This is then a bottom-up scenario. The timeline of
structure formation would therefore have stars forming first, which then would gather
into globular cluster-sized structures, which would themselves gravitationally coalesce
into galaxy-sized objects.
The most difficult part of CDM simulations is modeling the stars. Once stars
turn on, the physics in the simulation will grow much more complex. Massive stars
radiate copious amounts of UV radiation, expel matter out in powerful winds, and
die in supernova explosions. All three of these processes can dramatically change the
environment that they are in. Winds and supernovae, for instance, disrupt or rip
gas from proto-galaxies. Furthermore the first stars had no heavy elements, so they
represent a class of objects which we have no examples of today.
Whether a clump of matter actually collapses or remains stable depends on several
parameters. For a given gas cloud of a certain size, mass, and temperature, the cloud
can have enough internal pressure to be stable against collapse, or its gravity can
overwhelm this pressure. The mass of a cloud right at the boundary between these
two scenarios is called the Jeans mass. A cloud of a given size that is as massive as
a Jeans mass will collapse. A cloud with less mass than its Jeans mass will have an
internal pressure that can counteract the collapse.
In the early Universe, we must also worry about the expansion. Although gravity
is pulling matter into regions with increasing density, the expansion is decreasing the
overall density. About 10 million years after the Big Bang, the lumpiness of the matter
is about 50% that of the lumpiness of the dark matter wells. After 100 million years
after the Big Bang, the density concentrations are now twice the average density of the
surrounding average density. At this point, gravity has overwhelmed the expansion
of space-time, and the clumps start to free-fall and collapse. The largest galactic
superclusters that we see in the Universe are thought to result from the clumps that
give the 1st and 2nd peaks in the CMB angular power spectrum.
7.1
The Millenium Simulation
An example of a numerical simulation of structure formation comes from the Virgo
consortium, an international team of astrophysicists (Figs. 7.1–7.4). Using a cluster
of 512 computer processors located at the Max Planck Institute for Astrophysics in
Garching, Germany, the team ran the “Millenium Simulation,” which lasted 28 days,
and outputted 25 Terabytes of data. The model that was run is an N-body simulation,
because it simulates the changing positions of a large number of N particles; in this
case N ≈ 109 . Having large numbers of particles in your simulation is key: using
a small number means you can only simulate the largest galaxies. Supermassive
7.1. THE MILLENIUM SIMULATION
115
black holes which have masses a million times the mass of the Sun (or larger) could
effectively be invisible. Increasing the number of particles means each particle in the
simulation can represent a smaller mass, and you will be able to follow the formation
of less massive objects, like small galaxies.
Although the growth of density fluctuations can be derived analytically by mathematics for awhile, numerical simulations are necessary after the fluctuations begin
their collapse into the large-scale structure found in galaxy superclusters today. Such
simulations represent the matter in the Universe as a large number of discrete particles, each of which react to the collective gravity of all of the other particles. The
matter particles further can interact with radiation. The Millenium simulation followed 10 billion particles of ordinary matter and dark matter located in a volume
with a present width of 700 Mpc, from a redshift of z = 127 to the present, z = 0.
The simulation was able to follow the evolutionary histories for 20 million galaxies,
some as small as that of the Small Magellanic Cloud.
Additional cosmological parameters—such as the size of the Hubble constant, the
amount of dark matter relative to the baryonic matter, the size of the cosmological
constant, the size of the fluctuations expected from analyses of the CMB, etc.—are
input into the model as initial conditions. The computer model is then allowed to
evolve, with the interactions of all of the matter following realistic physics. The
simulation was then used to follow formation and growth of black holes (which lay at
the centers of most galaxies), quasars, and galaxies over the Hubble lifetime. Through
trial and error, the input parameters were adjusted and the model was repeatedly
run until it gave results for nearby, low-redshift structure that is similar to what is
observed in our local Universe.
Quasars and other active galactic nuclei are thought to be powered by central
black hole engines. Supermassive black holes have been discovered in the centers of
many galaxies, and they may be found in all galaxies. The formation of these enormous black holes (which can be millions of times the size of “ordinary” black holes
that originate from the collapse of a massive star) is therefore thought by many extragalactic astronomers to occur in the early Universe, concurrent with the hierarchical
formation of structure.
The Sloan Digital Sky Survey (SDSS) has made the surprising discovery of some
extremely bright quasars up to a redshift z = 6.43. If the distances are correct, then
the luminosity of these quasars implies that the central black hole engines have a
mass of 109 M , which is a thousand times larger than the mass of the central black
hole in our Milky Way. Cosmologists were unsure whether such super-supermassive
black holes could form, but the Millenium Simulation does show a number of such
objects coalescing. These eventually evolve into the most massive elliptical galaxies
in the simulation.
The goal of follow-ups to the Millenium Simulation and its ilk is that as they
successfully model large-scale and smaller-scale structure in their simulations (the
Millenium Simulation can resolve objects down to the Small Magellanic Cloud in
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CHAPTER 7. STRUCTURE FORMATION IN THE UNIVERSE
Figure 7.1: Millenium Run Summary
A poster summarizing the results of the Millenium simulation run of the VIRGO Consortium. More than 10 billion particles were used to simulate the formation of galaxies starting
with initial conditions after the Big Bang. A volume of space corresponding to a current
cubic volume about 700 Mpc (2 billion light years) across was followed from soon after recombination to the present. The resolution of the simulation was varied so that formation
of objects with a size and mass-scale of the Milky Way could be followed toward the end
of the simulation. More than 20 million artificial galaxies were formed in the simulation,
which had a data output of more than 25 Terabytes.
7.1. THE MILLENIUM SIMULATION
117
Figure 7.2: Millenium Simulation at Large Scales
Output from the Millenium Simulation at four different times. Upper left: (z = 18.3,
t = 0.1 billion years after the Big Bang) first inhomogeneities which are the seeds of galaxy
clusters grow and are visible. Upper right: (z = 5.7, t = 9.0 billion years) gravity concentrates matter into clusters, while also creating the large voids in-between the clusters. Lower
left: (z = 1.4, t = 12.7 billion years) mergers create supermassive black holes and larger
galaxies. Lower right: (z = 0, t = 13.7 billion years) At the present time, structures can be
found at all scales, from clusters to galaxies to stars. From V. Springel, 2005, “Simulating
the Joint Evolution of Quasars, Galaxies and Their Large-Scale Distribution,” Nature, 435,
pp. 629–636.
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CHAPTER 7. STRUCTURE FORMATION IN THE UNIVERSE
Figure 7.3: Millenium Simulation at Medium Scales
The same as Fig. 7.2 but at a resolution showing four times the detail.
7.1. THE MILLENIUM SIMULATION
Figure 7.4: Millenium Simulation at Small Scales
The same as Fig. 7.2 but at a resolution showing sixteen times the detail.
119
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CHAPTER 7. STRUCTURE FORMATION IN THE UNIVERSE
size), cosmologists and astronomers will be able to “fine-tune” the cosmological parameters. Some input parameters are well known, such as the Hubble constant (with
an uncertainty of 10%). Others are not as well known, such as the behavior of dark
energy at early times. By tweaking the parameter behavior—for instance by changing
the value of dark energy at different times—the cosmologists hope to find a simulation
that can be compared with the results from future, deep astronomical surveys, and
at the same time, deduce something about the nature of dark energy.
Chapter 8
Inflation and the Early Universe
The Big Bang theory is well established with predictions that are matched by observations. The three broad categories of evidence are the expanding universe, the
light element abundances, and the existence of a highly uniform cosmic microwave
background. However despite this set of consistent evidence, there are several issues
that remain unresolved with the basic model. The inflation model has therefore been
proposed to fix these problems. Inflation is added on so that it sets the initial conditions for the Big Bang. There is yet any slamdunk evidence to prove inflation is right
beyond reasonable doubt, unlike the evidence for the Big Bang.
8.1
Problems With the Big Bang
The Expansion of the Universe: This may sound like a weird reason to start
off a list of “Problems With the Big Bang.” However physicists do not like
their theories to have any arbitrariness. One of these is the expansion of the
Universe. Why did the Universe start expanding to create a Big Bang? The
answer, “Because it just did,” is not very satisfactory.
The Horizon Problem: Recall that in the cosmic microwave background is extremely uniform over the entire sky, about 1 part in 105 , meaning there is less
than a 0.0001◦ K difference in temperature from one part of the sky to another.
However at the time of last-scattering, when the photons were coupled to the
matter, the closest neighboring photons that could exchange energy with each
other at the speed of light is equivalent to only 2◦ on the sky today. This is
the horizon problem: how can parts of the cosmic microwave background 180◦
apart exchange energy to have the same temperature today? It could have been
sheer coincidence that all these disconnected regions had such close temperatures, but that seems very unlikely. Somehow these very widely separated parts
of the Universe today must have been far closer together to be able to swap energy and reach thermal equilibrium, then is suggested by the normal Big Bang
121
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CHAPTER 8. INFLATION AND THE EARLY UNIVERSE
model.
The Flatness Problem: As discussed in Chapter 3 are three possible end-states
to an expanding Friedmann-Robertson-Walker universe, and it depends on the
density of the universe. The universe might be over-dense, which would eventually cause it to contract toward a Big Crunch. Such a closed universe has a
curvature parameter k = +1. The universe might have far less mass in it to
stop or even slow down the expansion, which would mean faster expansion over
time. This is open universe has k = −1. Finally the third scenario is the critical
case where expansion will also continue forever, although more slowly than in
the open scenario. This flat universe has k = 0 and also Ω = 1..
As you may recall from Fig. 5.6 and § 6.3, it seemed that Ω was very close to
1, if not in fact equal to 1. However this presents a problem: if Ω is nearly 1
today, it would have to be even closer to 1 in the past. This is because if Ω
was slightly different than 1 in the past, this difference would have grown over
time. For instance, if Ω = 0.90 today, then when the Universe was 1/30th its
present age, Ω = 0.99. However when the Universe was only 1 sec old, Ω must
have been even closer to 1: 1.000000000001. At even earlier times, Ω would
have been even closer still to 1. Even if today Ω = 0.3, then this implies that a
mere 10−43 sec after the Big Bang, Ω couldn’t have deviated from exactly 1 by
more than one part in 1060 ! That is, it could not be larger than:
1.0000000000000000000000000000000000000000000000000000000000001,
or smaller than:
0.9999999999999999999999999999999999999999999999999999999999999.
It is awfully fortunate for us for Ω to be so close to 1, when it could be much
smaller or many times as large. How did the Universe come to be so perfectly
tuned? One might say it just happened this way, but that again is not a very
satisfactory answer to cosmologists.
The Hidden Relics Problem: According to Grand Unified Theories of particle
physics, the conditions in the Big Bang were expected to create a slew of unusual objects, including magnetic monopoles, gravitinos, and other exotic
subatomic particles. These are massive particles (up to 1016 times that of a
proton!) that do not decay. In fact there should be so many of these particles,
that they would dominate all other matter by a ridiculous amount, say a factor
of 1012 . Magnetic monopoles should therefore be easily found in the present
day Universe.
Also as the Universe cooled in the aftermath of the Big Bang, space-time is
expected to “settle” into its current lower-temperature state. This is due to
8.2. PROBLEMS SOLVED?
123
symmetry breaking when the strong force splits off from the electroweak
force at the end of the Grand Unified era (see § 8.3). However there are expected to be slight timing differences when this symmetry breaking occurs,
which results in volumes of space-time that are separated by “topological defects.” These interfaces in space-time can be “cosmic strings” (one-dimensional)
or “domain walls” (two-dimensional). However no topological defects have ever
been detected either.
One last problem with our model of the Universe is that there appears to be an
imbalance between matter and antimatter. Matter will annihilate perfectly with its
antimatter counterpart (electrons with positrons, protons with anti-protons, neutrons
with anti-neutrons, etc.) with high energy photons as a result. Similarly the reverse
reaction can occur: photons can spontaneously create matter-antimatter pairs, such
as an electron and a positron. One would there expect the early Big Bang to produce
equal amounts of matter and antimatter in the Universe. However by all accounts,
there is very little antimatter. The Universe is mostly matter.
8.2
Problems Solved?
One answer that would solve all of these problems is inflation, first proposed by
physicist Alan Guth in 1980. This hypothesis suggests that the universe went through
a period very early in its history where it expanded at an exponential rate, about
t = 10−37 sec after the Big Bang. An exponential growth can be characterized
by its doubling time, meaning the amount of time that it takes for something to
grow to twice its original size. Guth showed that the doubling time for inflation
was t = 10−37 sec and that there could be 100 doublings or more. Inflation theory
therefore predicted the size of the Universe to increase by a factor of as little as 1030
to as much as 1050 or more. Inflation did not continue forever; it lasted only from
t = 10−36 sec to 10−34 sec after the Big Bang. Afterward the Universe continued at
a more leisurely pace of expansion similar to what we see today.
This is an extraordinary amount of expansion within a very short time: in 10−34 sec,
a photon moving at the speed of light would have time to traverse a distance one hundred billionth the diameter of a single proton. Our current observable Universe at
the end of inflation was about 10 cm across, or about the size of a grapefruit. However this is a small part of a much, much larger Universe. (It may even be infinitely
large—see § 9.2.) This much larger Universe is not something we can observe since
there has not been enough time for light to travel from outside our observable volume
to reach us.
This hyper-expansion is thought to originate from properties of the vacuum. We
normally think of vacuum as the absence of any matter and energy, and so would
not think to attribute any properties to it. However from quantum mechanics, we
know that the vacuum is not empty but filled with virtual particles (§ 5.3). Also from
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CHAPTER 8. INFLATION AND THE EARLY UNIVERSE
Figure 8.1: Expansion from Inflation
The inflation hypothesis supposes that the Universe underwent an enormous expansion about
10−35 sec after the Big Bang. According to some versions of the model, it might have gotten
larger by a factor of 1050 times. After lasting between 10−36 and 10−34 sec, the inflationary
period (cyan) ended, and the Universe continued to expand at a much slower rate up through
the present day.
8.2. PROBLEMS SOLVED?
125
quantum mechanics, one can make predictions about the behavior of the vacuum.
One of these is that at extremely high temperatures, such as that which you find
immediately after the Big Bang, the vacuum can change states. This is akin to a
property of water, which normally freezes and turns to ice during a phase transition
at its critical temperature of 0◦ C. However it is possible to cool undisturbed water
below 0◦ C. This supercool water remains liquid, but any slight disturbance will cause
the water to quickly turn to ice.
The vacuum in the early Universe reached its critical, unstable state when it
cooled below 1027 K. This version of the vacuum is known as the false vacuum.
The phase transition that would drive inflation is thought to come as the result of
new hypothetical particles that exist during the Grand Unified Theory (GUT) era,
when the strong force merges with the electroweak force. (Gravity is still separate,
and does not unify with the other forces until times t < 10−43 sec; see § 8.3.) These
particles are thought to be described by scalar fields, which is exactly the type of
force necessary to create a vacuum-driven expansion.
A postulated inflaton particle, which gives rise to inflation, has a quantum field
φ that describes it, which varies with temperature. The energy of this field can
be described by a “Mexican hat” potential, with a peak at φ = 0 and the energy
increasing with greater values of the field φ. At high enough temperatures, the
location of the inflaton is at a minimum at φ = 0 of a “Mexican hat” potential
where the false vacuum is located (Fig. 8.2). While the inflaton is in the φ = 0 state,
inflation occurred.
However the location of the false vacuum is unstable: just as a marble located at
the peak of a Mexican hat tends to roll down the side, given just a slight nudge, the
inflaton can drop down in energy with a nudge from a quantum fluctuation. Such
a fluctuation (similar to those that create virtual particles) causes a pocket of the
false vacuum to decay into a true vacuum. Very quickly (within 10−34 sec) the
true vacuum would fill up the Universe. During the transition from a false vacuum
to a true vacuum, an enormous amount of energy is released which forms particleanti-particle pairs. Thus according to the inflation model, the vast majority of the
particles in the Universe were created as a result of inflation.
This enormous exponential expansion solved many of the problems we mentioned.
The Universe started expanding because it was impelled to do so by inflation. The
horizon problem is no longer an issue: The regions now separated by great distances
in the present epoch were actually once much closer together —so close that they
allowed these regions to reach the same temperature within a light travel time. It
was inflation that spread out space-time by such a vast amount that such regions
could never be in contact with each other again.
Inflation solves the flatness problem by expanding the space-time geometry of
the Universe. Even if the Universe had not started out being flat, the vast rate of
expansion stretched the curvature of Universe until it appeared flat. Therefore it does
not matter if Ω started as 1, 10, 100, 106 , or 10−6 . As long as there is enough inflation,
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CHAPTER 8. INFLATION AND THE EARLY UNIVERSE
Figure 8.2: The Inflaton Potential
A plot showing the energy density from the inflaton. At high enough temperatures, the
inflaton can settle at phi = 0 which corresponds with the Universe being filled with a false
vacuum. During this time, space expands exponentially. However a quantum fluctuation
can knock the potential down toward the true vacuum. As it settles to a zero energy density,
particle-antiparticle pairs arise during the inflationary transition to a true vacuum.
8.2. PROBLEMS SOLVED?
127
Ω would have evolved to 1. A similar analogy is that a small balloon might have a
noticeably curved geometry, with lumpiness or even wrinkles. However if you were
to blow up the balloon until it was the size of the Earth, the surface of the balloon
would appear to be extremely flat, even in the parts that were wrinkled. (If inflation
were to occur for our balloon, it would blow up to a size about 1024 or 1,000,000,000,000,000,000,000,000 times larger than the current size of the observable Universe.
The surface of such a balloon would indeed look extremely flat!)
Figure 8.3: Inflation Solves the Flatness Problem
Even if the initial Universe had an extremely curved space-time geometry, the inflationary
period blew up the Universe to such a great extent that today, it appears flat to us. Similarly
if a sphere the size of a proton were to expand until it was the size of the Earth, its surface
to us would look extremely flat.
The problem of magnetic monopoles and other exotic objects is also solved by
such an enormous expansion. The original pool of magnetic monopoles would have
expanded along with the rest of the Universe. The expansion is so great that they
would be so spread out, that there would likely not be a single magnetic monopole
left within our observable Universe.
Inflationary theory does not quite solve the imbalance problem between matter
and antimatter. However Grand Unified Theories involving the unification of the
strong nuclear force with the electromagnetic and weak forces (see the next section)
suggests that at high enough energies, matter could have formed with a slight excess
over antimatter. As the Universe cooled after this very early period to the time
t = 10−5 sec, all of the protons annihilated with the anti-protons to create gamma
ray photons. For every billion (109 ) proton-anti-proton pairs, there was an extra
proton that did not annihilate. These residual protons would eventually become all
the matter that we see today. The photons that came out of the annihilations became
the radiation bath that expanded with the Universe and turned into the radiation
background discussed back in §4.2.
128
8.3
CHAPTER 8. INFLATION AND THE EARLY UNIVERSE
The Earliest Universe
As we have seen in Chapters 4 and 6, the temperatures and conditions in the early
Universe around the time of recombination was very different than what they are
today. Baryonic matter only existed in a plasma state, and it was tightly coupled to
the radiation. However this period is still several hundred thousand years after the
Big Bang. What were the conditions like even earlier?
Figure 8.4: Unification of Forces
The evolution of the four fundamental forces with time as the Universe cools off after the
Big Bang.
In our current low-temperature Universe, there are four fundamental forces in
nature, which in order of decreasing strength are the strong nuclear force, the
weak nuclear force, the electromagnetic force, and gravity. The strong force is
responsible for binding atomic nuclei together by overcoming the repulsive force from
the positively charged protons. The weak force appears in certain subatomic particle
interactions, including radioactive decay. The electromagnetic force manifests itself
in forces due to electrical charges and magnetic fields. Although gravity is by far the
8.3. THE EARLIEST UNIVERSE
129
weakest of the four, it is more significant than either the strong or weak force, because it operates at a distance (instead of at subatomic length-scales). And although
electromagnetism also operates at long distances, gravity is the dominant force in the
Universe, since charge imbalances appear to be small, so that electromagnetic forces
tend to cancel each other out.
Particle physics experiments show that the strength of interactions vary with the
interaction energy. At higher temperatures and higher energies, the electromagnetic
and the weak nuclear forces are found to be closer and closer to each other in strength.
16
At T >
∼ 10 K, the strengths of the two interactions are expected to be the same,
and the electromagnetic and weak forces can therefore be thought of as variations
of a single electroweak force, via a theory that was first proposed by physicists
Abdus Salam and Steven Weinberg. Electroweak theory was verified in 1983 with the
discovery of the W + , W − , and Z 0 bosons in particle accelerators.
Because of this success with the electroweak theory, physicists are now pursuing
unification theories that will include the other two forces. The drive for unification
is in part philosophical: the belief that the Universe can be described by a minimal
set of particles and physical processes. The next goal for physicists is the Grand
Unified Theory (or GUT), which combines the strong and the electroweak forces
at T ∼ 1028 K. Inflation is thought to originate from the end of the GUT period, as
the strong and electroweak forces split. The current matter-antimatter imbalance is
also thought to be the result of processes from this period.
The final unification with gravity is believed not to occur until temperatures exceed T ∼ 1032 K. This is associated with a time t ∼ 10−43 sec (roughly the Planck
time—see below) after the Big Bang, during which quantum mechanical and gravitational effects become equally important. Stephen Hawking and Roger Penrose have
shown that at time t = 0, the field equations of General Relativity break down, and
singularities, or infinities appear. Analogous to how electromagnetic theory breaks
down at high enough energies, it is thought that with sufficiently strong gravitational
fields or extremely small length scales, gravity must be described with a quantized
model to avoid these singularities.
Table 8.1: State of the Universe Over Time
Temperature
(K)
∼ 1032
Energy
∼ 1019 GeV
Time
t
−43
∼ 10
sec
∼ 1026
∼ 1014 GeV
∼ 10−33 sec
Event
Planck Era, unification of all forces:
supergravity? supersymmetry? strings?
M-brane theory?
End of Grand Unification of the strong and
electroweak force; origin of matter-antimatter
asymmetry; creation of magnetic monopoles;
era of inflation
Continued on next page
130
CHAPTER 8. INFLATION AND THE EARLY UNIVERSE
Continued from previous page
Temperature
Energy
Time
(K)
t
15
−12
∼ 10
∼ 1 TeV ∼ 10
sec
1013
1 GeV
∼ 10−5 sec
3 × 1010
3 MeV
∼ 0.1 sec
1010
109
∼ 1 MeV
∼ 90 keV
∼ 1 sec
∼ 3 min
105
10 eV
∼ 104 yr
104
1 eV
∼ 105 yr
Event
End of electroweak unification; the
electromagnetic and weak nuclear forces split
Quark/hadron transition: quarks are confined into
baryons and mesons
The weak interactions that interconvert n and p
become unimportant, so that unequal numbers of
n and p freeze out
Neutrinos decouple; e− -e+ pairs annihilate
Neutron decay becomes important;
nucleosynthesis starts at this time and ends
∼ 30 min later
Domination of matter energy over radiation energy
densities
Formation of atoms; decoupling of matter and
radiation
However creating a theory for quantum gravity has proved to be extraordinarily
difficult. Einstein himself believed his theory to be incomplete, and spent the latter
half of his life unsuccessfully trying to unify it with quantum mechanics. Although
we do not have a theory of quantum gravity, we can make guesses as to what it
must involve. Since General Relativity is a theory of space-time, then space and time
must be quantized in quantum gravity. Thus although it appears to be continuous at
the macroscopic scales that we are familiar with, at sufficiently small scales and short
enough periods of time, space-time itself should appear lumpy. These length and timescales are the Planck length, 1.62 × 10−19 cm, and the Planck time, 5.31 × 10−44 sec.
A Planck mass can also be defined, which is the mass of a single particle that can
only be properly described by quantum gravity, 1.22 × 10−19 GeV.
A hypothetical Theory of Everything (TOE) would finally combine quantum
gravity with the theory for all particles and the rest of the force interactions. TOE
should summarize all of the particles and interactions with a single equation, or
perhaps a small set of equations—perhaps sufficiently short enough to appear as a
design on a t-shirt. Current attempts to create TOEs have been much publicized by
science writers and scientists and include categories with names like supersymmetry,
supergravity, string theory, branes, and M-theory.
Like GUTs and quantum gravity, there is no clear consensus on which class of
TOE could be correct. The high energies involved for GUTs make testing any of
these theories well beyond the practical possibility of even future particle accelerators. However microscopic conditions in the early Universe are linked to TOEs,
quantum gravity, and GUTs. Inflation would have multiplied these conditions by
such an enormous factor, as to make them potentially visible. For instance, quantum
8.3. THE EARLIEST UNIVERSE
131
fluctuations in the smooth energy-density of the early Universe can be enormously
magnified by inflation to create the seeds for the large-scale structure that we observe
today. Similarly structures in the cosmic microwave background that are greater than
the current horizon size would give you an idea of quantum fluctuations that must
have occurred in the pre-inflation Universe. Studying the CMB may therefore reveal
to us important clues not only to the earliest phases of our Universe, but to TOEs as
well.
132
CHAPTER 8. INFLATION AND THE EARLY UNIVERSE
Chapter 9
Chaotic Inflation, Parallel
Universes, and Other Rampant
Speculation
9.1
Chaotic and Eternal Inflation
Despite all its apparent successes, inflation is in no way as robust a model as the
Big Bang theory. In fact, there is not one inflationary hypothesis but a class of
many competing models. Alan Guth’s original model is not even under consideration
any more because of problems that were discovered with it. In fact, the plot of the
inflation potential in Fig. 8.2 is actually a revision of Guth’s original model called
new inflation.
Another variation of inflation is Andre Linde’s chaotic inflation. Instead of
the Mexican hat potential, the inflaton field is described by a general class of curves
that does not necessarily have a false vacuum state (Fig. 9.1. It is assumed that
chaotic conditions in the early Universe produce fluctuating patches of space where
the inflaton field φ was at some arbitrary high value somewhere on the potential.
Inflation occurs as the inflaton field rolls down the potential hill. If the initial value
of the inflaton field is high enough, then there will be sufficient inflation to expand
the Universe by the requisite amount to solve the Big Bang problems.
A variant of new inflation, eternal inflation (or “stochastic” inflation) was proposed in 1983 by Paul J. Steinhardt and Alexander Vilenkin. They found that as the
false vacuum decays, it never completely disappears; some volume of the false vacuum
always remains since it also grows exponentially once inflation starts. As a volume
of space grows via inflation and eventually turns into a true vacuum, the remaining
pockets of false vacuum also grows with time. These other false vacuums also have
the potential to go through an inflationary phase, decay into true vacuums, and then
end up as new post-big bang universes.
Fig. 9.2 shows how this would work. The initial state is a false vacuum, a portion
133
134
CHAPTER 9. RAMPANT SPECULATION
Figure 9.1: Chaotic Inflation
A plot showing the energy density from a chaotic inflaton scenario.
9.1. CHAOTIC AND ETERNAL INFLATION
135
Figure 9.2: Eternal Inflation
Eternal inflation occurs as volumes of false vacuum grow in size with time, and inflate into
new universes with true vacuums. Note that the false vacuums and universes are drawn
to show their relative positions next to each other, and not to scale since they would be
exponentially growing. Also the universes in this scenario would be expanding with three
dimensions of space, not one-dimensionally as shown.
136
CHAPTER 9. RAMPANT SPECULATION
of which goes through inflationary growth and evolves into a true vacuum, resulting
in a Big Bang and a universe similar to our observable universe. However at the same
time that the true vacuum universe is expanding, the space in the two remaining
regions of false vacuum has also expanded, so those two regions are now the same
size as the starting region of false vacuum. If we follow these regions for another
period of time, part of the false vacuum can also evolve into a local universe with a
true vacuum. This process continues forever with each false vacuum growing in size,
generating new universes, and growing in size again, ad infinitum.
What results are an infinite number of bubble universes or pocket universes.
None of the universes are accessible to any of the other universes, since inflation
will have expanded them into space-time volumes that are far beyond each of their
observable horizons. One consequence of eternal inflation is that the infinite web of
expanding universes are fractal in nature, with an infinite number of pocket universes
growing with different scale factors, while false vacuum pockets keep subdividing to
spawn new universes.
One possibility for eternal inflation is that there may not be a first universe or
an original false vacuum. One can imagine Fig. 9.2 extending forever upward as well
as forever downward so that there is no t = 0. In the plot, we have drawn only the
results of just one patch of false vacuum at one point in time. If eternal inflation
continues infinitely into the past, then this false vacuum may share a common false
vacuum ancestor with an infinite other false vacuums and pocket universes.
As we have seen from these two examples, inflation is still a model very much in a
state of flux. Even the exact mechanism for initiating inflation is not well understood.
Although inflation solves many fundamental problems of the Big Bang, this feature
does not guarantee that inflation actually occurred. The very ad hoc nature of the
many variants of inflation—people often come up with new variations to explain
different observations—may even argue against the idea as a whole. However many
physicists and cosmologists would argue that the concept of inflation has solved far
more problems than it introduces, and some version of it is here to stay.
9.2
Parallel Universes
There appears to many good arguments for some form of inflation to exist, even
though we do not have a solid theory for inflation. What are some possible consequences if we assumed that inflation is true? Max Tegmark has done just this and
he has come up with the prediction of parallel universes.1 These parallel universes
are similar to those that are prevalent in science fiction literature and sci-fi shows
on TV and the movies. In such a parallel universe, there is another version of you
(and indistinguishable from you in every way) sitting and reading this sentence, liv1
See M. Tegmark, “Parallel Universes,” May 2003, Scientific American for a layperson’s review.
A more technical version of this paper can be found at http://arxiv.org/abs/astro-ph/0302131.
9.2. PARALLEL UNIVERSES
137
ing in a world that is exactly the same as our world. This Parallel Earth orbits the
Parallel Sun with the same set of Solar System objects. Parallel universes may also
diverge from our own Universe, whether slightly or by a lot. Again jumping off from
the science fiction literature, there might be universes where JFK was not shot or
Hitler won World War II, or where Elvis is still alive (or perhaps this latter one is
our Universe).
Recall that the furthest that we can observe in the Universe is about 13.7 billion
light years, which is to the edge of the cosmic microwave background, at which point
the Universe becomes opaque. This Hubble volume is the furthest that we can see
now, but this has changed with time. When the Universe was half its current size,
our Hubble volume was then 7 billion light years in radius, because that was as far
light could travel to reach us from the beginning of the Universe. The expansion
of space-time is also constantly carrying galaxies out of our Hubble volume. If we
observe light from a galaxy whose distance we measure to be 10 billion light years
away, then that light was emitted 10 billion years ago. Today however, that galaxy is
not only 10 billion years older, but the expansion of space-time has carried it along
so that right now, it is 30 billion light years away. If we wanted to view it as it looks
today, it would not be possible since there would be no time for light being emitted
now to ever reach us. The expansion of the Universe is moving those galaxies outside
of our Hubble volume. An accelerating expansion makes it even worse: depending on
the exact amount of acceleration, we would lose sight of more galaxies faster.
There is also evidence that our Universe is much larger than our observable Hubble volume. The curvature of space as measured by the CMB appears to be very
flat (§ 6.3). Even taking the upper limit of the curvature parameter based on the
uncertainties in the measurements, the Universe will contain at least 1000 other Hubble volumes. The flatter the Universe, the more Hubble volumes will exist outside
our own Hubble volume. Completely flat Friedmann-Robertson-Walker universes are
infinitely large. If space is infinite in size now, then it was infinite to start with.
Inflation itself adds another complication, since it predicts the Universe to be
much, much larger than the observable Hubble volume. Some cosmologists, like
Jaume Garriga and Alexander Vilenkin, even argue that inflation can cause a Universe
that was originally finite in size to grow to infinite size.
Our Hubble volume however contains only a finite amount of space, with a finite
number of particles—estimated to be about 1090 . In an infinitely large universe, there
will be an infinite number of Hubble volumes. In classical physics, it turns out there
are an infinite number of ways to arrange a finite number of particles. But in quantum
mechanics, there will be only so many different ways you can arrange 1090 particles
within a single Hubble volume. Just from statistical arguments, one would expect
to run eventually into a repeat of that particle arrangement after a traveling a finite
distance away.
How far away would we find a repeat particle arrangement? Using basic quantum
mechanical arguments about how to arrange particles, Tegmark estimates that the
138
CHAPTER 9. RAMPANT SPECULATION
29
closest identical copy of any one of us is about 1010 meters away. For larger arrangements of particles, the probability is lower for an exact duplicate, so you have to go
even further to find a duplicate. For a sphere about 100 light years across (meaning
a volume that exactly matches our Sun, the Solar System, and nearby stellar neigh91
borhood), a duplicate will be found on average about 1010 meters away. Finally to
find a Hubble volume that is identical to ours, statistically speaking, one has to travel
115
about 1010 meters.
These are ridiculously huge numbers (although still finite and less than infinity).
29
Even the smallest distance, 1010 meters, is so large that it is difficult to imagine,
and is impossible to write out.2 These numbers are therefore far, far beyond anything
that is currently or will be observable.3
Tegmark calls such a set of parallel universes “Level I” multiverse. His “Level II”
multiverse involves elements of eternal inflation theory.4 Recall that extensions of
inflation like eternal inflation predict that universes continue to sprout from the false
vacuum (Fig. 9.2). Taken to one logical extreme, there will be an infinite number
of these bubble universes, for eternal inflation forward in time. If eternal inflation
continues backward in time as well, then there will be infinitely still more pocket
universes. Each of these bubble universes will also be infinitely big if the view of the
Level I multiverse is correct.
A post-inflation universe that evolves into a big bang however may have different
sets of physical constants. The value of the Planck constant, the electron-proton mass
ratio, the ratio of the electron charge to its mass, the strength of the weak force, etc.,
do not necessarily have to be locked to the values found in our Universe. Their values
could be different depending on the types of symmetry breaking that occur as the
post-inflation bubble cools. The Level II multiverse will therefore have much more
variation in it than just a rearrangement of particles, when compared to our Universe.
The number of space dimensions could be different, as well as the number of quark
families, or any of the other numerical constants in the “Standard Model” of particle
physics.
Thus while the Level I multiverse involves all possible arrangements of particles
to create parallel universes, the Level II multiverse is a super-set of that, involving
all possible values of physical parameters.
2
29
That is, 1010 is 1 followed by 1029 zeroes. To give you an idea of how many zeroes this is,
there are slightly less than 1029 protons in the human body.
3
That is unless dark energy oscillates between a repulsive and an attractive state in the far future,
which has been suggested by some cosmologists. If it moves out of the repulsive realm, dark energy
will act to reinforce gravity, causing the expansion of space-time to slow, eventually stop, and then
reverse. As the Universe grows smaller in size once unobservable regions of the Universe will come
into the observable Hubble volume.
4
If you guessed that Tegmark has come up with higher level multiverses, you are correct. We
will not have time to go through them here but you can learn more about them in his articles on
the subject.
9.3. THE END OF THE UNIVERSE
9.3
139
The End of the Universe
Finally what can we predict about the true “end” of the Universe given its accelerating
expansion and what we know about modern physics and astrophysics? A Universe
that not only expands but accelerates its expansion means that there will never be
a Big Crunch. The expansion will never slow, stop, and reverse. It will keep going
forever and faster, with distant galaxies receding farther away. The overall density
and temperature of the Universe will continue to drop. As it does so, we can imagine
four different periods that the Universe will go through.
1. The Stellar Era: This is the era that we are in now. Gas is locked up in stars,
and then expelled back into space when a star dies. However this process cannot
go on forever since eventually all the useful hydrogen will be used up. More and
more mass is locked up in white dwarfs, neutron stars, and black holes, until
there is not enough free gas to create new stars. The longest length of time that
a star can theoretically live is about 1014 years (or 100 million million years)
for objects with about one-tenth the mass of the Sun. We currently live in an
epoch 13.7 billion years after the Big Bang. After a length of time 10,000 times
longer than the current age of the Universe, all of the gas will be used up and
the Age of Stars will be at an end.
2. The Degenerate Era is reached once all matter is locked up in white dwarfs
and neutron stars. Here the word “degenerate” describes the quantum mechanical state of the electrons and protons in these compact objects. Over time
white dwarfs and neutron stars will cool off by radiation until they are the same
temperature as the cosmic microwave background (which itself will have cooled
off until it is just a fraction of a degree Kelvin above absolute zero). Galaxies
also gradually dissipate through a process called two-body relaxation, so that
the stars “evaporate” over time. What is left then is a Universe filled with free
floating black dwarfs and black holes. However even the black dwarfs themselves break down. According to Grand Unified Theories in particle physics,
the proton, one of the most stable of elementary particles, is expected to decay
and break down into a positron and a meson in 1032 years. Thus all of the
black dwarfs, planets, and whatever ordinary matter is left will have undergone
proton decay in 1037 years.
3. The Black Hole Era: Once all the protons have broken down, the only objects
left in the Universe with any appreciable mass are the black holes. However
even black holes themselves break down and evaporate through the process of
Hawking radiation emission. They will eventually disappear completely over
time as they are replaced by the pool of electrons, positrons, and photons that
are released. Black hole decay through Hawking radiation is an extremely slow
process, with the largest super massive black holes from the centers of galaxies
140
CHAPTER 9. RAMPANT SPECULATION
taking the disproportionate longest. We can estimate that it will be about
10100 years before all of the black holes evaporate.
4. The Dark Era: After 10100 years, and after the disappearance of black holes,
the Universe is left with nothing but a sea of photons and neutrinos, whose
wavelengths get longer as the Universe continues to expand. We are at a point
of maximum entropy and chaos. This photon and neutrino sea is so uniform,
without sources of energy or any sinks, so that it is not possible to do work
to create ordered structures. Eventually the expansion of the Universe will
continue to the point where each individual remaining particle is expanded
outside of the light horizon of every other particle. That is, if we were to examine
one photon, there would be no other observable photon or other particle within
the observable Universe. They are too widely separated for one to reach another
even traveling at the speed of light.
The photon energy will continue to decrease until it reaches the lowest possible
value of that of the quantum vacuum state. Not only is the Universe now
completely dark and alone except for a solitary particle at the bare minimum
energy level, but we have reached the End of Time as well. There is nothing
to distinguish one moment from the next. This is truly the End of the Universe.
Appendix A
Notation and Astronomical Units
A.1
Scientific Notation
Because science often deals with extremely large or incredibly small numbers, a shorthand notation for writing such figures has been developed. First we write an exponent as a superscript after a number, which signifies the number of factors of that
number to be multiplied together. Thus, an exponent of 2 over a 10 means two tens
multiplied together. Here is a list of exponents up to 8:
10 = 10
100 = 10 × 10
1000 = 10 × 10 × 10
10, 000 = 10 × 10 × 10 × 10
100, 000 = 10 × 10 × 10 × 10 × 10
1, 000, 000 = 10 × 10 × 10 × 10 × 10 × 10
10, 000, 000 = 10 × 10 × 10 × 10 × 10 × 10 × 10
100, 000, 000 = 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10
=
=
=
=
=
=
=
=
101
102
103
104
105
106
107
108 .
A simple way to remember what number is represented by the shortened scientific
notation form is to use the exponent as the total number of zeroes in the “long”
version of the number. “One million” of 106 will have 6 zeroes, meaning written out,
it will be “1,000,000.”
We can also define negative exponents, based on the number of factors of 10 that
141
142
APPENDIX A. NOTATION AND ASTRONOMICAL UNITS
have been divided:
0.1
0.01
0.001
0.0001
0.00001
0.000001
0.0000001
0.00000001
=
=
=
=
=
=
=
=
10−1
10−2
10−3
10−4
10−5
10−6
10−7
10−8 .
Comparing the two sequences, it also makes sense to define:
1 = 100 .
We can now use this system to express very large numbers, such as the length of a
light year (the distance light travels in one year):
1 lightyear = 9, 460, 000, 000, 000 kilometers = 9.46 × 1012 kilometers.
Very small numbers can also be expressed such as the density of gas out in interstellar
space between the stars:
0.000000000000000000000002 grams per cubic centimeter =
2 × 10−24 grams per cubic centimeter
A.2
Scientific Units
We will use almost all metric—or “SI” for Système Internationale—units in this
course. For length and distance measurements, we will use the meter (m) and its
variants:
1m =
=
=
=
39.37 inches
= 100 cm = 102 cm (centimeters)
= 1, 000 mm = 103 mm (millimeters)
= 1, 000, 000, 000 nm = 109 nm (nanometers).
For mass, we refer to the gram (g) and kilogram (kg):
1 kg = 2.20 pounds
= 1000 g = 103 g.
A.3. DISTANCES
143
The SI unit for temperature is the degree Kelvin (K). It is similar to the Celsius or
centigrade degree, so that 1 K = 1◦ C But while 0◦ C is set to the freezing point of
water, 0 K is defined to be at absolute zero, when all thermal motion stops, and
therefore the coldest temperature possible. To convert from degrees Kelvin to degrees
Celsius:
Temp(◦ C) = Temp(K) − 273.
To do the opposite and go from Celsius to the Kelvin scale:
Temp(K) = Temp(◦ C) + 273.
According to these formulae, the boiling point of water is then 100◦ C = 373 K.
A.3
Distances
Since the space sciences deal with the vast distances in the universe, a number
of length measurements have appeared in the astronomical sciences that are used
nowhere else in science. Although they have nothing to do with SI units, their use is
so widespread that it is unlikely they will go away in our lifetimes or perhaps ever.
The first is the astronomical unit or AU, and this is defined to be the distance
(technically the semi-major axis distance) between the Earth and the Sun. It is
defined to be:
1 A.U. = 149, 597.892 km = 1.49597892 × 1013 cm.
You will find it used whenever distances in the Solar System are referred to. You may
even see this unit in measurements of other solar systems, or proto-solar systems:
a planet orbiting 6 AUs from its parent star, or an accretion disk 60,000 AUs in
diameter around a Sun-like star.
A far larger unit of measuring distance is the light year, the distance that light,
moving at 299,790 km s−1 , covers in a single Earth year. This is a unit that is
appropriate for describing the distances between stars, and is:
1 ly = 9.4605 × 1012 km = 6.324 × 104 AU.
Another unit similar to the light year is the parsec. It derives from the measurement
of distances to stars using the technique of trigonometric parallax discussed in § 2.5.1.
A parsec (abbreviated as pc) is a little over 3 light years:
1 pc = 3.0856 × 1013 km = 2.0626 × 105 AU = 3.2616 ly.
What about distances far larger than a parsec or a light year? As you have seen in
§ 2.4, the distances between galaxies and clusters of galaxies can be many millions of
light years. Instead of coming up with a new unit for this size scale, astronomers use
144
APPENDIX A. NOTATION AND ASTRONOMICAL UNITS
the SI method of attaching prefixes to existing units. Just as we can scale up a meter
by 1000 times and call it a “kilometer,” a distance 1000 times a parsec is a kiloparsec
or kpc. A kiloparsec or kilo-light year is appropriate for describing distances from one
end of a galaxy to the other.
For distances between clusters of galaxies, we must resort to the “Mega-” prefix,
where a Megaparsec (or Mpc) is 1 million = 1,000,000 parsecs. Figs. 2.15 and 2.18
could have been re-labeled using Mpc or Mly units. Finally for the scales of voids
and filamentary superclusters in the large-scale structure in the observable Universe
(such as Fig. 2.21, we can go to the even larger prefix of Gigaparsecs (Gpc), which is
1 billion parsecs.
A.4
Magnitudes
Traditionally, the brightness of astrophysical objects have been expressed not as
fluxes, but as magnitudes. This is a logarithmic scale, with the apparent magnitude
proportional to the natural logarithm of the incoming radiation flux, or m ∝ log f .
If two objects have observed fluxes f1 and f2 , then
m2 − m1 = 2.5 log(f1 /f2 ).
(A.1)
The factor of 2.5 means that a difference in flux of a factor of 100 corresponds to 5
magnitudes.
The absolute magnitude M is defined as the magnitude a source would have if it
were at a standard distance of 10 parsecs. The absolute magnitude therefore gives a
measure of a star’s luminosity. The relation between apparent and absolute magnitude
is:
m − M = 5 log(D/10),
(A.2)
where D is the distance in parsecs. The absolute magnitude of the Sun is 4.72, while
its apparent magnitude is m = −26.85. The difference between the two, m − M , is
called the distance modulus.
A.5
Angular Measurements
Since astronomers are looking at and measuring the size of objects in the sky, it
became necessary to use angular measures of size. Most familiar is to divide a
circle into 360 equal-sized units, the degree (◦ ). For finer measures, there is the
arcminute (0 ), which is 1/60th of a degree, and the arcsecond (00 ), which is 1/60th
of an arcminute.
The relationships between these three units are:
1◦ = 600 = 360000 .
(A.3)
A.6. OTHER ASTRONOMICAL UNITS
145
Many familiar Earth-bound units are based on degrees, arcminutes, and arcseconds. For instance, the latitude and longitude of Denver is given as (39◦ 450 North,
105◦ 00 West), where the starting points for latitude is the equator (0◦ ), and the starting point for longitude is the Greenwich meridian. Celestial coordinates are given
in right ascension (or R.A.) and declination (or Dec.). Declination is measured
from the celestial equator to the celestial poles, while R.A. is measured along the
celestial equator with the vernal equinox as the starting point. However to make
things more confusing to the novice,1 instead of degrees, arcminutes, and arcseconds
for right ascension, astronomers have introduced the units of hours (h ), minutes
(m ), and seconds (s ). These are each, respectively, 15 times larger than degrees,
arcminutes, and arcseconds. Thus,
1h
1h
1m
1s
=
=
=
=
60m = 60s ,
15◦ ,
150 ,
1500 .
(A.4)
In mathematics, it is often simpler to use an angular unit called the radian (rad),
and you will often see this refered to in the astronomical literature. Its relationship
with more familiar units is shown in the following,
π rad =
1 rad =
=
=
180◦ ,
57.2957795◦
3437.746771 arcmin
2.062648062 × 105 arcsec.
(A.5)
Finally just as one can proceed from measurements of lengths to measurements of
areas when we add an orthogonal dimension, we can go from angular measures of
length to angular measures of area. The standard unit is the steradian (sr), and it
is scaled in such a way that a single spherical surface (such as the celestial sphere of
the sky) is 4π sr in size. Here is how a steradian relates to square degrees, square
arcminutes, and square arcseconds:
1 sr = (180◦ /π)2 = 32400/π 2 deg2
= 1.1664 × 108 /π 2 arcmin2
= 4.19904 × 1011 /π 2 arcsec2 .
A.6
(A.6)
Other Astronomical Units
One final note about astronomical units concerns units that are based on our Sun.
When looking at the energy output of other stars, it is often useful to compare them
1
Although citing R.A. in this fashion does save effort when performing ground-based observations.
146
APPENDIX A. NOTATION AND ASTRONOMICAL UNITS
to that of the Sun. Therefore, you might read or hear that a particular star has 12 L ,
or 12 times the luminosity of the Sun. (The symbol refers to the Sun.) Similarly
one can also use the mass of the Sun, as a unit: you can say a neutron mass has a
mass of 2.5 M . Or you can claim a supermassive black hole has a radius of 10 R ,
or 10 times the Solar radius.
Just for reference, the solar mass, luminosity, and radius can be written in traditional SI units as:
1 M = 1.989 × 1030 kg,
1 L = 3.826 × 1026 Joules/sec,
1 R = 6.9598 × 105 km.
(A.7)
Further Reading
The best part of the Internet is that—except for your connection charges—it’s free.
The worst part is that . . . it’s free. The fact that anyone can put up a webpage means
that there is so much content out there, that it can be difficult to sort out the useful
from the useless. Here is a minor attempt at finding and collecting together a list of
web resources that have some relevance to what was covered in class.
John Baez’s General Relativity Tutorial:
A somewhat advanced,
but very nice tutorial. Not for the mathematically weak-hearted
(http://math.ucr.edu/home/baez/einstein/).
G. Bothun’s Physics 410/510 Class:
Class notes from a Modern Observational Cosmology course at the University of Oregon
(http://zebu.uoregon.edu/1997/phys410.html).
Cambridge Cosmology Public Home Page: Contains info on the Big
Bang, galaxies, the CMB, inflation, cosmic strings, and other topics
(http://www.damtp.cam.ac.uk/user/gr/public/cos home.html).
Sean M. Carroll’s Lecture Notes on General Relativity: Another advanced set of class notes on the General Theory of Relativity, and the basis of
a textbook written by the Carroll. Note that the files are in PostScript format
(http://pancake.uchicago.edu/∼carroll/notes/).
Einstein Online:
Terrific website covering topics that range from the
elementary—like relativity—all the way to cosmology, black holes, extradimensional space, holographic universes, and more. Recommended if you
are interested in the some of the latest ideas that physicists are pursuing
(http://www.einstein-online.info/en/index.html).
Einstein Year 2005: All about Einstein, put together to celebrate the 100th
anniversary of his “miracle year” when he wrote five ground-breaking papers on physics (two of which involved the Special Theory of Relativity)
(http://www.einsteinyear.org/).
147
148
FURTHER READING
John Gribbin’s Cosmology for Beginners: An essay by the popular
science writer covering the Big Bang, inflation, dark matter, and more
(http://www.lifesci.sussex.ac.uk/home/John Gribbin/cosmo.htm).
Hubble’s 1929 Paper: A copy of Edwin Hubble’s original 1929 paper,
“A Relation Between Distance and Radial velocity Among Extra-Galactic
Nebulae,” which announced his discovery of the expansion of the Universe
(http://antwrp.gsfc.nasa.gov/diamond jubilee/d 1996/hub 1929.html).
Wayne Hu’s Physics of the Microwave Background Anisotropies: Very detailed, but understandable webpages on the CMB and how cosmologists are studying it (http://background.uchicago.edu/).
Dick McCray’s Cosmology Lecture Notes:
Lecture notes
a
University
of
Colorado-Boulder
undergraduate
astronomy
(http://super.colorado.edu/∼astr1020/lesson12.html).
from
class
Cosmos in a Computer: A large collection of computer animations and simulations from the National Center for Supercomputing Applications, at the
University of Illinois, Urbana-Champaign, on the evolution of the Universe
(http://archive.ncsa.uiuc.edu/Cyberia/Cosmos/CosmosCompHome.html).
Cosmology: A Research Briefing: Practically a book, this “briefing” from the
National Research Council is slightly out of date since it was published in 1995,
but is still a good source of information on just about every important topic in observational cosmology (http://www.nap.edu/readingroom/books/cosmology/).
The Particle Adventure:
Website funded by the National Science
Foundation and the Department of Energy containing introductory material to everything you might want to know about particle physics
(http://particleadventure.org/particleadventure/index.html).
The Official String Theory Web Site: I don’t know what makes this the “official”
site, but it’s a useful compendium of information on string theory and how it relates
to cosmology (http://superstringtheory.com/index.html).
Superstrings! Another string theory site with a tutorial and list of references
(http://www.sukidog.com/jpierre/strings/).
Max Tegmark’s Home Page:
Links
(http://space.mit.edu/home/tegmark/).
to
his
research
and
papers
VIRGO Consortium’s Millenium Simulation Project: Contains links
to movies and images from the N -body simulation discussed in § 7.1
(http://www.mpa-garching.mpg.de/galform/virgo/millennium/index.shtml).
FURTHER READING
149
Ned Wright’s Cosmology Tutorial: One of the best online resources for
all things cosmological and relativistic. Includes both tutorials and FAQs
(http://www.astro.ucla.edu/∼wright/cosmolog.htm).
Martin White’s Cosmology Reading List: Links to dozens of other useful sites
(http://cfa-www.harvard.edu/∼mwhite/readinglist.html).
Cosmology: The Study of the Universe: An educational site from the WMAP
folks, covering the theoretical and observational aspects of the Big Bang and the
Universe (http://map.gsfc.nasa.gov/m uni.html)
150
FURTHER READING
The following is a list of books and magazines for learning more about topics of this
course. Most have popular books written for the interested layperson, at the same
level as articles appearing in Scientific American or Discover magazines. The few that
are textbooks with some mathematics are noted as such.
Fred C. Adams & Greg Laughlin, The Five Ages of the Universe: Inside the
Physics of Eternity, 1999, Free Press. [A history of the Universe starting from
the Big Bang and ending at a staggering 10100 years later. Good if you are really
intrigued by The End of the Universe section from class.]
Kristy Ferguson, Measuring the Universe: Our historic Quest to Chart the Horizons
of Space and Time, 1999, Walker. [A chronicle of the attempts from the last 2000
years to measure the size of the Universe. It was also written late enough to
cover in its last chapter the Type Ia supernovae results.]
Timothy Ferris, The Whole Shebang: A State-of-the-Universe(s) Report, 1997, Simon & Schuster. [A nice introduction to cosmology and the Universe, although now
a little dated.]
George Gamow & Russell Stannard, The New World of Mr. Tompkins, 1999,
Cambridge University Press. [An update of a classic book for the layperson,
with explanations of the Special Theory of Relativity, quantum mechanics, and
the structure of the atom.]
Brian Greene, The Elegant Universe: Superstrings, Hidden Dimensions, and the
Quest for the Ultimate Theory, 2000, Vintage.
Brian Greene, The Fabric of the Cosmos: Space, Time, and the Texture of Reality,
2004, Knopf.
Alan H. Guth, The Inflationary Universe: The Quest for a New Theory of Cosmic
Origins, 1998, Perseus Books Group.
Edward R. Harrison Cosmology: The Science of the Universe, 2nd Edition, 2000,
Cambridge University Press. [Big and expensive, but a real cosmology textbook
but for the non-technical reader, with minimal mathematics.]
John E. Hawley & Katherine A. Holcomb Foundations of Modern Cosmology,
2nd Edition, 1997, Oxford University Press. [A textbook for an introductory
undergraduate level course, but not very math heavy. Summaries of the chapters
are online at http://astsun.astro.virginia.edu/∼jh8h/Foundations/]
Robert P. Kirshner The Extravagant Universe: Exploding Stars, Dark Energy,
and the Accelerating Cosmos, 2002, Princeton University Press. [A whole book
devoted to the evidence for dark energy, starting from Einstein’s first work on
the topic, up to the latest supernovae research of today.]
FURTHER READING
151
Lawrence Krauss, Quintessence: The Mystery of Missing Mass in the Universe, 2000,
Basic Books. [An update of his earlier book, The Fifth Essence.]
Mario Livio, The Accelerating Universe: Infinite Expansion, the Cosmological Constant, and the Beauty of the Cosmos, 2000, Wiley.
Malcolm S. Longair, Our Evolving Universe, 2nd Edition, 1996, Cambridge University
Press.
Laurence Marschall, The Supernova Story, 1994, Princeton University Press.
Martin J. Rees, Before the Beginning: Our Universe and Others, 1997, AddisonWesley Longman.
Martin J. Rees, Just Six Numbers: The Deep Forces that Shape the Universe, 1999,
Basic Books, NY.
Martin J. Rees, Our Cosmic Habitat, 2001, Princeton Univ. Press, 2001.
Vera C. Rubin, Bright Galaxies, Dark Matters, 1997, American Institute of Physics.
The editors
2002.
at
Scientific
American,
The
Once
and
Future
Cosmos,
[A reprint of the September 2002 issue of the magazine containing
more than a dozen articles by the field’s leading researchers on the
recent revolution in cosmology. You can order it through SciAm, e.g.,
http://www.sciam.com/special/toc.cfm?issueid=6&sc=rt nav list.]
Edwin F. Taylor & John A. Wheeler Spacetime Physics, 2nd Edition, 1997, W. H.
Freeman. [A very nice introductory---at the undergraduate science level---text
on the Special Theory of Relativity, including most of the famous ‘‘paradoxes’’
and other ‘‘weirdness.’’ The mathematics are all algebra-level.]
Kip Thorne, Black Holes & Time Warps: Einstein’s Outrageous Legacy, 1995,
W. W. Norton & Company. [An excellent mid-level introduction to General
Relativity, black holes, evidence for them in the Universe, and their many
strange properties. Excellent coverage of a topic that was just touched upon in
class!]
Neil DeGrasse Tyson & Donald Goldsmith, Origins: Fourteen Billion Years of Cosmic
Evolution, 2004, W. W. Norton & Company.
William H. Waller & Paul W. Hodge, Galaxies and the Cosmic Frontier, 2003,
Harvard University Press. [Covers all aspects of galaxies, from morphologies
to composition to formation and evolution. The subject of early chapters
concentrate on the Milky Way, and then gradually move focus to the Magellanic
Clouds, the Local Group, and so on until we get to the superclusters and large
scale structures in the Universe.]
152
FURTHER READING
Index
21 cm line, Hydrogen, 21, 42
Cepheid variables, 10, 41–42
Chandrasekhar limit, 90
chaotic inflation, 133–134
clusters, see galaxy clusters
Coma Cluster, 33, 38
Concordance Model, 110–111
continuous creation, 81
Copernican Principle, 7, 8, 11, 93–95
Copernican Revolution, 7–8
Copernicus, Nicolaus, 4, 5
cosmic distance ladder, 38–48
cosmic microwave background (CMB),
77–81, 84, 97–111, 121–122, 139
fluctuations, 99–111
Cosmic Microwave Background Explorer
(COBE), 81, 98–99
cosmological constant, 9, 11, 12, 60–62,
65, 66, 69, 70, 85, 92–96
negative, 65
positive, 66–67
zero, 65–66
Cosmological Principle, 4, 8, 58, 61
critical density, see density, critical
Curtis, Herbert, 9, 10
curvature, 54, 55
negative, 57, 58, 62, 66
positive, 57, 58, 61, 62, 65, 66
zero, 57, 58, 65, 66
curvature of the Earth, 57
curvature parameter k, 62, 110, 122
Abell, George, 89
Abell clusters, 89
acoustic waves, 99–103, 105–110
age of universe, see universe, age of
Almagest, 4
Ancient Greeks, the, 2–4
Anaxagoras, 3
Aristarchus, 3, 4
Aristotle, 3, 4
Eudoxus, 3, 4
Ptolemy, Claudius, 3, 4
Thales, 3
Andromeda Galaxy, see M 31
angular power spectrum, 103–111
anti-neutrinos, 75
antimatter, 123, 127
An Original Theory or New Hypothesis
of the Universe, 6
A Perfect Description of the Celestial
Orbes, 4
Babylonians, the, 2
baryonic matter, 21
Big Bang, the, 65–67, 69, 72, 75–84
Big Crunch, The, 65, 66, 69
blackbody spectrum, 79–81, 97–98
black holes
supermassive, 115
Bondi, Hermann, 81
BOOMERanG, 98, 100, 105
Brahe, Tycho, 5
Bruno, Giordano, 5
dark energy, 70, 92–96, 120
dark matter, 21, 26, 49–51, 93, 99–103,
113–115
cold, 113–114
C-field, 81
4
INDEX
halo, 26, 49, 51, 99–103, 108, 113,
114
hot, 113
in galaxies, 25–26
Day of Brahma, 2
deceleration parameter q◦ , 85–90
decoupling, 79
deferent, 4
density
critical, 69–70
density parameter
Ω, 69, 122, 125–127
baryonic matter Ωb , 93, 110
dark energy ΩΛ , 92–93
matter Ωm , 92–93
deuterium, 76–77
De Revolutionibus Orbium Celestium,
4
de Sitter, Willem, 9, 11, 62
Dialogue on the Two Great World Systems, 5
Digges, Thomas, 4
5
Friedmann, Aleksandr, 9, 10
Friedmann-Robertson-Walker universe,
62–67, 87, 122, 137
Einstein’s Field Equation, 9, 12, 60–62
Einstein, Albert, 8–12, 53–55, 58, 61,
62, 66
Einstein-de Sitter universe, 11
electromagnetic force, 127–130
electron, 123
electrons, 21, 51, 75–79, 97, 110, 123,
138, 139
electroweak force, 123, 125, 129
elliptical galaxy, see galaxies, elliptical
Enuma Elish, 2
epicycles, 4
epicyclet, 4
eternal inflation, 133–136, 138
galaxies, 13–38
barred, 13, 14, 17, 18
disk, 13, 14, 25, 42
elliptical, 13–15, 21, 22, 29, 32, 33,
38, 46, 50
elliptical, dwarf, 14, 29
elliptical, giant, 14, 21, 33
Hubble class, 13, 14
Hubble type, 13–15
irregular, 14, 19–21, 29
lenticular, 13, 14, 18, 33, 38
peculiar motions, 67
spiral, 13, 14, 16, 17, 21, 24–26, 41,
42, 46
unbarred, 13, 14, 16, 18
galaxy clusters, 29–38
Hydra A, 49
mass of, 48–49
Galileo Galilei, 5, 6
Garriga, Jaume, 137
Gauss, Karl Friedrich, 57
General Natural History and Theory of
Heaven, 6
General Theory of Relativity, 53, 55,
56, 60, 61, 129, 130
globular clusters, 8
Gold, Thomas, 81
Grand Unified Theory (GUT), 125, 129,
139
gravitational lensing, 48–50
gravity, force of, 6, 8, 25, 28, 53–54, 60–
61, 128, 129
Guth, Alan, 123
Faber-Jackson Relationship, 46
field equation, see Einstein’s Field Equation
Flatness Problem, The, 122, 125–127
Friedmann’s equation, 62
Halley, Edmund, 5
Hawking radiation, 139–140
Heisenberg’s Uncertainty Principle, 95
Helium, 76–77
Herschel, John, 5
6
Herschel, Sir William, 7
Hertzsprung, Ejnar, 41, 43
Hidden Relics Problem, The, 122–123,
127
High-z Supernova Search, 90
homogeneity, 8, 58, 60, 61
horizon, 79
Horizon Problem, The, 121–122, 125
Hoyle, Sir Fred, 81
Hubble class, see galaxies, Hubble class
Hubble type, see galaxies, Hubble type
Hubble’s Law, 11, 47–48, 67
Hubble, Edwin, 10–13, 42, 44, 46
Hubble constant H◦ , 11, 47, 48, 67–70,
85–87, 115, 120
Hubble parameter H(t), 85–87
Hubble Space Telescope, 42, 50
Hubble time, 72
Hubble volume, 137–138
inflation, 121–127, 129–131, 133–138
inflaton, 125–126, 133
irregular galaxy, see galaxies, irregular
isotropy, 8, 58, 60
Jeans mass, 114
Kant, Immanuel, 6, 7
Kepler, Johannes, 5, 6
Lambert, Johann Heinrich, 7
Large Magellanic Cloud (LMC), 19, 21,
29
last-scattering surface, 79, 105, 108
Leavitt, Henrietta, 41
Lemaı̂tre, Georges, 12, 66
lenticular galaxy, see galaxies, lenticular
light curve, 90
light element abundances, 75–77
Lithium, 77–78
Local Group, 29–31, 38
magnetic monopoles, 122, 127, 129
INDEX
Marduk, 2
matter-energy equivalence, 54–55
Milky Way, The, 5–8, 10, 13, 14, 24–31,
35, 38, 41, 42, 46, 49, 67
multipole number, 105
multiverse, see universe, Level I and
Level II
M 31, The Andromeda Galaxy, 8, 10,
13, 14, 24, 29, 30, 42, 67
N-body simulation, 114
Narlikar, Jayant V., 81
negative curvature, see curvature, negative
neutralino, 51
neutrinos, 51, 75, 77, 113, 140
neutron-proton ratio, 76–77
neutrons, 21, 75–77, 123
Newton’s law of universal gravity, 53,
61
Newton, Sir Isaac, 5, 6
nucleosynthesis, see primordial nucleosynthesis
Olber’s paradox, 5
Olbers, Heinrich, 5
parallax, see trigonometric parallax
peculiar motions, see galaxies, peculiar
motions
Penzias, Arno, 79–81
Perfect Cosmological Principle, 81–84
period-luminosity relationship, 10, 41
phase transition, 125
Philosophiae Naturalis Principia Mathematica, 6
photon-baryon fluid, 99–103, 108–110
photons, 75–79, 97, 99, 105, 108–110,
121, 123, 127, 139, 140
Planck constant h, 95
Planck length, 130
Planck mass, 130
Planck time, 129, 130
INDEX
population synthesis model, 21–25
positive curvature, see curvature, positive
positrons, 75–76, 123, 139
primordial nucleosynthesis, 75–77
proton decay, 139
protons, 21, 75–77, 79, 123, 139
Pythagorean theorem, 56, 57
quantum gravity, 129–131
quasars, 84, 115
quintessence, 96
recombination, 77–79, 97–98, 105, 108–
110
redshift, 46–48, 67, 87
cosmological, 62, 67–69, 79
gravitational, 108
reversible reactions, 75–76
Riemann, Bernhard, 57
Rig Vedas, The, 2
rotation curve, 25–28
RR Lyrae stars, 8, 41
Salam, Abdus, 129
Sandage, Allan, 87
scalar field, 96
scale-factor, 62, 65–67, 69, 72, 85–87
sexagesimal numbering system, 2
Shapley, Harlow, 8–10
Shesha-nāga, 2
Silk damping, 110
Slipher, Vesto, 8
Small Magellanic Cloud (SMC), 19, 21,
29, 41, 42
sound waves, see acoustic waves
space-time, 53–55, 62, 65, 122, 123, 125,
130
curvature, 55–58, 60–61, 127
Special Theory of Relativity, 53, 54
spiral galaxy, see galaxies, spiral
spiral arms, 13, 14, 16, 17
standard candles, 41–46
7
Steady State model, the, 81–84
Steinhardt, Paul J., 133
stellar nucleosynthesis, 76
strong force, 123, 127–129
structure formation, 113–120
bottom-up, 114
top-down, 113
Supernova Cosmology Project, 90
symmetry breaking, 123, 138
Syntaxis, 4
Tegmark, Mark, 136
Theory of Everything (TOE), 130–131
thermal radiation, see blackbody spectrum
Tiamat, 2
topological defects, 123
trigonometric parallax, 39, 41
tritium, 76–77
Tully-Fisher Relationship, 42–46
Turner, Michael, 1
Type Ia supernovae, 42, 90–93
universe
accelerating, 85–92
age of, 70–73
bubble, 136, 138
collapsing, 9, 60, 61, 65, 66
critical, 65
decelerating, 85–89
de Sitter, 62
expanding, 9–12, 42, 46, 48, 60–62,
65–67
flat, 65, 110, 122, 137
Level I, 138
Level II, 138
negatively curved, 66, 122
parallel, 136–138
pocket, 136, 138
positively curved, 65–66, 122
static, 6, 9, 61, 62, 66
vacuum
8
false, 125, 133–136, 138
true, 123–125
vacuum energy, 95
Vilenkin, Alexander, 133, 137
Virgo Supercluster, 31–33, 38, 45
virial theorem, 48
virtual particles, 95, 123
Weakly Interacting Massive Particles (WIMPs),
51, 114
weak force, 51, 127–130
Weinberg, Steven, 95, 129
Wien’s Law, 97
Wilkinson Microwave Anisotropy Probe
(WMAP), 98, 105
Wilson, Robert, 79–81
Wright, Thomas, 6
X-ray intracluster gas, 48–49
zero curvature, see curvature, zero
INDEX