Lec. 11
III-3 Cosmology
(Main Ref.: Lecture notes; FK p. 721; Sec. 171, 17-8, 19-6, 22-4, 23-4 & 8, 25-1 through 7)
1
III-3a. Introduction (Ref.: Lecture notes; FK p.
721, Sec. 25-1& 2)
(i) Steady Universe vs Evolutionary Universe:
Static,
Steady Universe; = static, infinite expanse of universe,
with no evolution – the universe is infinitely old and lasts
forever, with no major change of structure.
Evolutionary Universe; = non-static universe, with evolution –
the universe changes with time  leads to the big bang
cosmology!
Note: Einstein’s general theory of relativity gives NO solution for
the static universe – which means the universe is either
expanding or it will collapse. However, in those days (when
he derived the equations for general relativity first time),
before Hubble’s discovery of expansion of the universe, the
universe was considered to be static. To solve this problem,
Einstein introduced , cosmological constant, to make his
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solution for the collapsing universe to become static.
When Einstein learned the observational discovery by
Hubble that the universe is indeed expanding, he said that 
is the greatest mistake of his life and took it out, i.e., set  = 0
since he got expanding solutions. However, as we will
explain later in Section III-3c we do now need , after all.
Einstein’s mistake is NOT that  = 0, but that it must be much
larger than he predicted! See later sections.
(ii) Expanding Universe:
Hubble’s discovery, Hubble Law, established that the universe is
expanding.
*Hubble Law:
vc = H0 d,
Eqn(III-15)
where vc is receding velocity due to cosmological redshift;
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d is distance, and H0 is Hubble’s constant, and
d = vc / H0 = z c / H0.
Eqn(III-16)
where z = cosmological redshift.
Hubble Law is covered already. See Sec. II-9 (iii).
Note: Most advocates of the steady state cosmology gave it up
since Hubble’s discovery, although some still fight for it, e.g.,
Arp, G. Burbidge. However, one of the strongest advocates, Sir
Fred Hoyle, died in August 2000.
*Cosmological Principle: = Assumption that the universe is
homogeneous and isotropic in large enough scale, > 100 Mpc
(e.g., beyond the scale of largest clusters of galaxies) – well
supported by observations. Adopted by Einstein when he
developed his general theory of relativity.
4
The redshifts that we see from distant galaxies are caused by this
expansion, not by the motions of galaxies through space
The redshift of a distant galaxy is a measure of the scale of the
universe at the time the galaxy emitted its light
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Fig. III-73: Cosmological Redshift
III-3b. Big Bang Cosmology and Microwave
Background Radiation (Ref.: Lecture notes; FK Sec.
25-3 through 6)
(i) Hot Big Bang Theory: (Ref.: Lecture notes; FK Sec. 25-3
and 5)
The expanding universe emerged from a cataclysmic
event called the Big Bang
•
The universe began as an infinitely dense cosmic
singularity which began its expansion in the event
called the Big Bang, which can be described as the
beginning of time
•
During the first 10–43 second after the Big Bang, the
universe was too dense to be described by the
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known laws of physics
If the universe has been expanding, then, if we go back,
the universe will get denser, until the point in time
when the density gets infinite – the state called
singularity, when something, e.g., like an `explosion’,
may have happened
This singularity with density  = , then, takes place at
~ 15 billion years ago. G. Burbidge, who still believes
the steady state cosmology, coined this `singularity’
state, BIG BANG, which was adopted by the
community. This singularity is more like inside of a
black hole, where the distinction between space and
time disappears. So, this is more like the beginning
of the universe as we know it. We do not know what
things were before and at the big bang, because
physics as we know does not apply there!
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. *Planck Time tP: However, not only before and at the moment of
the big bang, but also after the big bang but before Planck
Time tP, classical mechanics breaks down. Planck Time is
defined as:
tP = square root of (G h / c5 ),
Eqn (III-17)
where G = gravitational constant, h = Planck constant, and c = velocity of light.
During time t < tP, classical mechanics breaks down, and so
quantum mechanics must be used. Since it is near singularity
(i.e., infinite density, like the center of a black hole), general
relativity must be used. The combination of the two, called
`Quantum Gravity’ is a very difficult field. Although some
excellent ambitious physicists are attacking quantum gravity,
the problems are not fully solved yet. Therefore, we do not
know what went on during 0 < t < tP, although some
speculate that singularity may be avoided at t = 0 if we adopt
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quantum gravity correctly.
*Why Hot Big Bang? - Helium Problem: We now know that
elements heavier than He (Z > 2) were produced through
nucleosynthesis during the presupervona stages of massive
stars. However, typical composition of ordinary stars is about
74% Hydrogen, 25% Helium, and 1% heavier elements
(`metals’, i.e., elements with Z > 2 ). There is no way to produce
25% helium by nucleosynthesis within stars. Earlier we noted
that high temperature of T > T☉c (core temperature of the sun)
= 1.6 x 107 K is required for hydrogen burning to convert H to
He. Here comes  need for Hot Big Bang!
°Idea: The universe was very hot right after the big bang. As it
expands, it cools down. In the earliest stage, radiation
(photons) and particles (first quarks but then protons and
electrons) coexist. When T > ~ 107 K in the hot universe, Hburning converts H to He. When ~ 25% were converted to He,
the universe has cooled down to T < ~ 107 K, and H-burning9
stops. In this way the He problem is solved.
• The observable universe extends about 14 billion
light-years in every direction from the Earth
Fig. III-74: The Observable Universe
We cannot see objects beyond this distance because light from
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these objects has not had enough time to reach us
*Age of the Universe To:
We can roughly estimate age of the universe by assuming the
universe has been expanding uniformly at the constant rate (ie.,
Hubble Constant Ho = constant with time). Then, To = d / v
(where d = current separation between two galaxies and v =
velocity with which they are separating). Then, from Hubble
Law, Eqn(III-15), we get:
T0 = 1 / H0.
Eqn (III-18)
See class notes for derivation
EX 62:
If Ho = 70 km/sec-Mpc, what is To?
Ans.
To = 14 billion years.
See class notes for details.
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Note: The oldest observed stars are in globular clusters, which are
~ 12 – 16 billion years old. The numbers roughly agree, and
so o.k. (Remember that the time after the big bang before first
stars are created must be very short compared with the age of
the current universe.)
Note: Later, we will learn that Ho has not been constant with time
(see Section III-3c), but this rough estimate is still o.k. – tells
roughly the age of the universe – the uncertainty involved
(mostly due to the uncertainty in distances which causes
uncertainty in the Ho value) is larger.
(ii) Evolution of the Universe: (Ref.: Lecture notes; FK Sec.
25-5)
After the hot big bang, the universe expands and cools down.
What will Happen as time goes on?
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Radiation Dominated Era and Matter Dominated Era:
*The universe is made of matter
and energy. Through Einstein’s
E = m c2 , Eqn (III-19)
energy E and mass m are similar –
They can be converted from one
to the other.
*Radiation
density R:
Define radiation energy ER and
the equivalent mass for radiation
mR, for convenience. Then, mass
density for radiation
R = mR/volume will become:
Fig. III-44: Evolution of Density
Fig. III-75:The Evolution of Density
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 R = 4  T / c3 ,
Eqn (III-20a) (Blackbody)
where  = Stefan Boltzman constant, c = velocity of light, and T is the
temperature of microwave background radiation today, which is
T = 2.725 K (see next Section (iii)). (Derivation of Eqn(III-20a) skipped.)
Then, Eqn(III-20a) gives:
R = 4.6 X 10–31 kg/m3,
Eqn (III-20b), now.
*Total matter density m: = total mass of all matter in the
universe/volume today, including all matter emitting radiation
(e.g., stars, galaxies, nebulae, etc.), and `invisible’ dark matter.
Then, observation gives:
m = (2 – 4) X 10–27 kg/m3, Eqn (III-21), now.
That means, today the universe is matter dominated. However,
that is not so earlier, because radiation density declines more
steeply with time, than matter density does (see Fig. III-75). 14
Then, we find m = R, at time tM, where: tM = 2500 years (after
the big bang); when zM = 25,000; M = 40 nm; and TM = 75000
K (comes in UV), where zM = redshift, M = wavelength of em
radiation (photons), and TM = temperature, at time tM. The
period BEFORE tM; t < tM; is called Radiation Dominated
Era.The period AFTER tM; t > tM; is called Matter Dominated
Era. See class notes for further details.
Recombination Era:
After tM, as the universe keeps cooling down, another critical
time tR is reached when temperature TR is so low that protons
and electrons combine to make Hydrogen atom. Here, tR = 3 x
105 years, with redshift zR = 1000, and TR = 3000 K. The period
AFTER tR is called Recombination Era.
Note: blackbody radiation photons at 3000K is visible red light.
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Note: Before tR, both protons and electrons
are free – matter is called ionized `plasma’
(charged particles). Then, photons hit the
particles and scatter – random walk (like inside
the sun). That means the matter is NOT
transparent, but opaque.
This hot opaque matter with photons is
sometimes called `primodial fire ball’.
After tR, however, when electrons and protons
are combined and become atoms, photons do
Fig. III-76: Evolution of
Radiation Temperature
NOT interact with H atoms – they travel freely, meaning the matter becomes
transparent to photons. Since photons and matter do not interact, they no
longer have the same temperature. Until tR, photons and matter have the
same temperature. (See Fig. III-76.)
(See class notes for further details.)
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Fig. III-77a:The Era of Recombination (before)
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Fig. III-77b:The Era of Recombination (after)
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Formation of Stars and Galaxies:
Right after tR, stars and galaxies are formed out of density
fluctuations at tR (see Section (iii) below). Which are formed first
is still controversial.
See class notes for the details.
(iii) Cosmic Microwave Background Radiation - CBR:
(Ref.: Lecture notes; FK Sec. 25-4)
The microwave radiation that fills all space is evidence of a hot
Big Bang.
The background radiation was hotter and more intense in the past
•The cosmic microwave background radiation, corresponding to
radiation from a blackbody at a temperature of nearly 3 K, is the
greatly redshifted remnant of the hot universe as it existed about
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380,000 years after the Big Bang
•During the first 380,000 years of the universe, radiation and
matter formed an opaque plasma called the primordial fireball
•We can never see photons from
t < tR because matter is opaque,
but we should be able to see
photons emitted at tR because
matter is transparent for t > tR.
These photons should have
reached us here today. However,
due to the expansion of the universe,
the wavelength should have
stretched to the microwave region.
Fig. III-78a: The Spectrum of CBR 20
(i)
(ii)
Fig. III-78b:
(i) The Bell Labs Horn Antenna
(ii) COBE
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★Peebles and Dicke of Princeton predicted that if the hot big
bang theory is correct, we should detect this microwave
background radiation. Indeed, Penzias and Wilson of Bell Lab
discovered this CBR, for which they received Nobel Prize. From
Big Bang Theory we can calculate the wavelength we should
detect today, and then from Wien’s Law for blackbody radiation we
can calculate the temperature of CBR. The results are:
 = 10 m – 1cm; T = 2.725 K. Eqn(III-22)
Note: Fig. III-78a shows that the data points (from COBE Satellite,
1989) fit blackbody spectrum of this temperature perfectly!
CBR is found to be isotropic and homogeneous (when the earth’s
motion is subtracted) – hence proving perfectly Einstein’s
Cosmological Principle!
Note: Discovery of CBR is, together with discovery of Hubble Law,
regarded as the strongest proof to support the hot big bang 22
theory.
Small Temperature/Density Fluctuations:
Although COBE was not sensitive enough to detect them, later balloon experiments
(BOOMERANG; MAXIMA (1998)) found small inhomogeneity, small temperature
and density fluctuations T~ 10-4 K (i.e., hot spots) and  - important as the origin
of stars and galaxies. See Section III-3c(ii).
III-3c. Dark Matter, Dark Energy, and Fate of
the Universe (Main Ref.: Lecture notes; FK Sec. 17-1, 17-8, 196, 22-4, 23-4 & 8, 25-6 & 7)
(i) Dark Matter: (Ref.: Lecture notes; FK
Sec. 22-4, 23-8)
Dark Matter (sometimes called `missing mass’) = all matter not observable in
all wavelengths of em radiation (i.e. light/photons).
Candidates:
MACHO(= massive compact halo objects) – e.g., brown dwarfs, black holes;
WIMPS (= weakly interacting massive particles) – e.g., neutrinos, heavy
neutrinos, axions.
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Detections: can estimate density from dynamics – e.g.,
galaxy rotation curves (motion of stars and gas around the
center of a galaxy) – See Fig. III-79.
Note: Boundary of a galaxy including
dark matter much larger than the
visible galaxy boundary. Also,
clusters of galaxies are filled with dark
matter – can tell from gravitational
lensing of distant quasars by clusters
FK
Fig. III-79: The Rotation Curves
of Four Spiral Galaxies
of galaxies in the foreground
Without dark matter
Conclusion: Dark matter ~90% of all matter!
See class notes for the details.
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(ii) Matter and Energy Content and Shape of the
Universe: (Ref.: Lecture notes; FK Sec. 25-6)
•The shape of the universe indicates its matter and energy content
•The curvature of the universe as a whole depends on how the
combined average mass density ρ0 compares to a critical density
ρc
•Definitions:
Combined average mass density of the universe 0: = the
sum of the average mass densities of matter, radiation, and any
other form of energy.
•Critical Density C : = critical density which is = the combined
average mass density of the flat universe
 the universe is flat when C = 0 .
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• Note:
C = 3 H02 / ( 8  G ) = 9.2 x 10–27 kg/m3. Eqn(III-23)
where H0 is Hubble Constant, G is Gravitational Constant.
• Density Parameter 0 : defined as:
0 = 0 / C .
Eqn(III-24)
Curvature of Space:
Density parameter tells whether the space is spherical (positive
curvature) and the universe is closed; space is flat (zero
curvature) and the universe is flat; or space is hyperbolic
(negative curvature) and the universe is open.
See Fig. III-80 and Table III-4.
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See class notes for the details.
Fig. III-80:The Geometry of the Universe
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What is the curvature of our space?
•Ans:
It is nearly FLAT,
from CBR observation, i.e.:
0 ~ 1.
Eqn(III-25)
• Can find it by measuring the size of the fine structure in CBR.
*Big bang theory predicts that the true angular size of the
structure r0 should be ~ 1o.
Table III-4 The Geometry and Average Density of the Universe
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Fig. III-49 (= FK Fig. 28-16) : CBR and the Curvature of Space
Fig. III-49 (= FK Fig. 28-16) : CBR
and the Curvature of Space
Fig. III-49 (= FK Fig. 28-16) : CBR
and the Curvature of Space
Fig. III-81:CBR and the Curvature of Space
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Observations of temperature variations in the cosmic microwave
background indicate that the universe is flat or nearly so, with a
combined average mass density equal to the critical density
Fig. III-82:
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Fig. III-83: WMAP
31
• If the measured size r > r0 the universe is closed.
If the measured size r < r0 the universe is open.
If the measured size r = r0 the universe is flat.
Result:
r ~ 1o ~ r0
Eqn(III-26)
See Fig. III-81.
See class notes for the details.
(iii) Dark Energy: (Ref.: Lecture notes; FK
Sec. 25-7)
Including dark matter, we find that the total matter density
parameter
m = m / C = 0.2 – 0.4 (*).
(*) Obtained by dividing Eqn(III-21) by Eqn(III-23).
Eqn(III-27)
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Note: m includes all visible mass (galaxies, stars, planets, etc)
+ Dark Matter.
However, CBR observation gives:
0 ~ 1.
Remember: C = 9.2 x 10–27 kg/m3,
and
Eqn(III-28)
Eqn(III-23) now
m = (2 – 4) X 10–27 kg/m3,
Eqn (III-21) now
R = 4.6 X 10–31 kg/m3
Eqn (III-20b) now.
So, m >> R now  m >> R. now.
So, 0 = m +  + R ~ m + 
Eqn(III-29) now,
which means:
0 = m +  + R ~ m + 
where  is dark energy density, and
Eqn(III-30) now,
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 =  / C = dark energy density parameter.
Eqn(III-31)
From Eqns(III-27), (III-28), and (III-30), we conclude that:
 = 0. 6 – 0.8 = 60 – 80%!
Eqn(III-32)
Note:  can be due to Einstein’s , cosmological constant,
which is more like `Antigravity’
Conclusion: A majority (60 – 80%) of the total ingredient of the
universe is dark energy! Out of the rest (total matter combined),
most (~ 90%) are dark matter!
(iv) Fate of the Universe ( Ref.: Lecture notes; FK
Sec. 17-1, 17-
8, 19-6, 22-4, 23-4, 25-7)
Distance Measurment – Summary only: (Ref.: Lecture notes; FK
Sec.17-1, 17-8, 19-6, 23-4)
How to measure distance?
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Summary: The closest can be measured by the direct parallax
method (FK Sec. 17-1), and the next closest by spectroscopic
parallax with the help of the HR Diagram (FK Sec. 17-8).
Method with Standard Candles: To measure distance to
objects further away, such as galaxies, we adopt Method with a
`standard candle’.
*Standard Candle in a Galaxy: = an object (e.g., a star) in the
galaxy whose absolute luminosity (i.e., absolute magnitude) is
known by some other means.
•Then, by measuring the apparent magnitude and with the known absolute
magnitude, can find distance (see FK Sec. 17-3). Distance to closer galaxies
can be found by using Population II RR Lyrae and Population I Cepheid
variables (see FK Sec. 19-6) as the standard candles. For those further away
Tully-Fisher method is used (see FK Sec.23-4). Those furthest away can be
found by adopting Type Ia supernovae as the standard candle.
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See Fig. III-84 for the summary.
Fig. III-84 :The Distance Ladder
Future of the Universe: (Ref.: Lecture notes; FK Sec.22-4, 25-7)
•Earlier, we assumed that Hubble constant H0 is constant with
time. We noted that it is essentially constant from FK Fig. 23-17.
However, the data used for this figure are all near-by galaxies,
which means that H0, and hence the rate of expansion of the
universe, has been constant in recent years
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How about the distant past? To
answer that question, we must
(a)
measure the distance to very
remote galaxies and quasars.
That has been done, most recently,
by using Type Ia supernovae as
standard candles.
• Note: on the distance d vs receding
velocity vC diagram, slow expansion
means steeper line than the rapid
(b)
expansion, and hence, if the
expansion of the universe has
Fig. III-85: Cosmological Expansion
speeded up, the d vs vc curve will look like the blue curve (i.e., steeper earlier)
in Fig. III-85, while it will look like the green curve (i.e., less steep earlier)
if the expansion has slowed down.
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Observations of distant supernovae reveal that we live in
an accelerating universe
Observations of galaxy
Clusters suggest that
the average density of
matter in the universe is
about 0.27 of the critical
density. The The remaining
contribution to the average
density is called dark energy
Measurements of Type Ia
supernovae in distant galaxies Fig. III-86: Varying Rates of Cosmic Expansion
38
show that the expansion of the universe is speeding up
Fig. III-87: Very Distant Supernovae
39
The most recent data from Type Ia supernova observations are shown in Fig.
III-88.
Black curve (upper curve) – flat universe with dark energy, the expansion
speeding up, with
0 = 1 = m + ; m = 0.24,  = 0.76.
Eqn(III-33)
Fits the data the best.
Blue curve refers to the flat universe with 0 = m = 1, no dark energy
This may be due to the presence of dark energy in the form of a cosmological
constant, which provides a pressure that pushes the universe outward
The conclusion: Comparison of the data and theoretical models
tells that the universe is not only expanding, but also that the
expansion is speeding up!
Note:The best fits to various observational constraints (see Fig.
III-89) gives: 0 = 1.02 (+-) 0.02, m = 0.27 (+-) 0.04,
 = 0.73 (+-) 0.04.
Eqn(III-34)
40
See class notes for the details.
Fig. III-88:The Hubble Diagram for Distant Supernovae41
Fig. III-89 : Limits on the Nature of the Universe
42
What is Dark Energy?: Non-zero  means we need dark
energy, and its contribution to the content of the universe is ~
76%, according to the supernova data. What is this dark energy?
One promising candidate is Einstein’s antigravity expressed by
the cosmological constant , which is constant.
Then, since both radiation and matter density decrease with time, while dark
energy density is constant, the dark energy now dominates the universe,
although earlier it was not important (see Fig. III-75). So, Einstein’s  is now
needed, after all!
Einstein’s only mistake is that he predicted it to be much smaller, than what we
now require from supernova observational data.
Note: there are other speculations about the nature of the dark
energy, but none of them, so far, got beyond speculation.
*Fate of the Universe:
Study Fig. III-90 and class notes.
Conclusion: the future universe will be dark and cold!
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Fig. III-90(a):Earlier Evolution of the Universe
44
Fig. III-90(b):Future Evolution of the Universe
45