Cosmology
• Large scale structure of the Universe
• Hot Big Bang Theory
• Concepts of General Relativity
• Geometry of Space/Time
• The Friedmann Model
• Dark Matter
• (Cosmological Constant)
The large scale structure of the Universe
Cosmology is an evolutionary science (at least in principle) which does not
allow controlled repetition of the system. (We cannot build a universe in a
laboratory). Analogy with archaeology, geology, paleo-biology.
 Age of the Universe: 15 billion years. Evidence from dynamics of
universe expansion (model) AND age of oldest stars.
 Size of the Universe: more complicated question.
Units in astronomy:
• Astronomical Unit AU = 150 millions km (Earth/Sun distance)
• Parsec = 3.26 light years (ly)
• Light Year = 9.46 x 10^15 m
Size of Solar System (Pluto’s orbit) : about 6 light hours.
Size of Milky Way: 10^5 ly x 10^3 ly
Galaxies: bunches of stars (in evolution), with typically 10^11 stars.
Galaxies agglomerate in clusters with size of a few Mpc (e.g. Local Group)
Galaxy Clusters agglomerate in Superclusters with size: 200 Mpc
Dominant interaction in the Universe: Gravitation
How distances are measured?
The “ Cosmic distance ladder “
 Parallax methods
 Main-sequence fitting (HR plot)
 Variable (Cepheid) stars
 Supernovae, cosmological methods
The Universe as seen by us is
strongly dishomogeneus and
anisotropic.
This statement holds true also on
the galactic scales (kpc distances)
….and remains true also on the
scale of galaxy clusters (Mpc
distances)
However, if seen from distances of 100 Mpc or more,
the universe gets homogeneus and isotropic. This is
homogeneity and isotropy at large scales!
The Hot Big Bang Model
 Model for the large scale structure and evolution of the Universe.
 Based on important experimental observations.
Cosmological Red Shift
Radiation is emitted from stars and other celestial bodies
This radiation has the same physical origin of the radiation we study in
terrestrial laboratories (e.g. atom absorption and emission).
Stellar evolution and many other branches of astrophysics are based on such
evidence. E.g. chemical composition of star surfaces are well known.
The radiation emitted by any source can be affected by the Doppler effect if
there is a relative motion between the source and the receiver
From a distant galaxy
'  
z

In laboratory
Red shift
1929: Hubble discovered the empirical relation
From the nonrelativistic Doppler formula:
A relation between the Galaxy velocity (away
from us) and its distance
Birth of Modern Cosmology!
H
z d
c
v
z
c
vH d
v  Hd
Since our position in the Universe is hardly a privileged one, galaxy superclusters
recede from each other with the cosmological Hubble law. Universe is expanding!
Two immediate consequences:
• In the far past all matter was lumped in very little space (the Big Bang)
• The timescale for this is roughly 1/H (assuming the expansion law was the same
all over, which is not really the case)
The Universe is expanding
into what ?
It is the space itself that is
expanding? Yes.
Are rulers expanding? No,
only gravitationally
independent systems
participate in the expansion!
The Hubble law is a linear expansion law which generates an homologous
expansion (it is the same as seen from every Galaxy)
The expansion looks the same as seen from A or from B
H = 70 ± 7 km/sec Mpc
Naïve expansion model (assuming H = const)
= Patch of size
100 Mpc
What we see in our Patch is consistent with isotropic and homogeneous
expansion plus the “Cosmological Principle” (no privileged place in Universe!)
2
r12  r1  r2
r23  r2  r3
1
1
3
r13  r1  r3
Homogeneous and isotropic
expansion: the shape of the
triangle must be preserved.
Therefore
r12 (t )  a(t ) r12 (t0 )
r13 (t )  a(t ) r13 (t0 )
r23 (t )  a(t ) r23 (t0 )
Seen from patch 1:
v12 (t ) 
dr12
a
 a r12 (t0 )  r12 (t )
dt
a
v13 (t ) 
dr13
a
 a r13 (t )  r13 (t )
dt
a
Seen from patch 2:
v21 (t ) 
dr21
a
 a r21 (t0 )  r21 (t )
dt
a
v23 (t ) 
dr23
a
 a r23 (t0 )  r23 (t )
dt
a
In any universe undertaking homogeneous and isotropic
expansion, the velocity/distance relation must have the form
Now we see that:
H (t ) 
a
a
a
v (t )  r (t )
a
a(t): scale parameter
Elements of a naïve thermal history of the Universe
Going backward in time means:
• No structures (No stars, galaxies…..) Only Matter and Radiation
• Higher densities and higher temperatures
Matter
e
p
Radiation
Eb 13.6 eV
When E(γ) > 13.6 eV radiation and matter are coupled. This took place at
cosmic time 400,000 yrs. Radiations is in equilibrium with atoms.
  H e p
e p  H 
(photodissociation)
(radiative recombination)
Before this era, let us imagine: nuclei, electrons and radiation, at some T.
Energy ~ kT. Electrons streaming freely at this point.
Then by going backward some more in time energy increases to:
kT  0.5 MeV
This took place at about T=10^10 K and cosmic time 100 sec
  e e
Electrons cannot free stream.
Nucleosynthesis already taking place at that time (from 1 sec to 300 sec).
Then by going backward some more in time energy increases to give a mean
energy 10 MeV. Therefore the reactions
n  e  p  e
n  e  p  e
became possible. These reactions mix p and n together making nucleosynthesis
impossible. This is T around 10^10 K (and cosmic time 0.1 sec).
To summarize, a timeline of important events:
• T>10^10 K, E>10 MeV, t<0.1 sec . Neutrons and protons kept into equilibrium by
weak interactions. Neutrinos and photons in equilibrium.
• t = 1 sec. No more p/n equilibrium. Beginning of nucleosyntesis. Neutrinos
decoupling from matter.
• T=10^9 K ,E =1 MeV, t= 100 sec. Positrons and electrons annihilate into photons
• t = 300 sec nucleosysnthesis finished because of low energy available and no
more free neutrons around  Low mass nuclei abundance fixed
• Protons, photons, electrons, neutrinos (decoupled)
• T=5000 K, E=10 eV, t=400,000 years. No more radiation,e,p equilibrium. Atoms
formation (hydrogen, helium). Photons decouple  CMB
Primordial Nucleosynthesis
Gamow, Alpher and Herman proposed
that in the very early Universe,
temperature was so hot as to allow fusion
of nuclei, the production of light elements
(up to Li), through a chain of reactions that
took place during the first 3 min after the
Big Bang.
The elemental abundances of light
elements predicted by the theory agree with
observations.
Y ~ 24% Helium mass abundance in the
Universe
Cosmic Microwave Background
Probably the most striking
evidence that something
like the Big Bang really
happened is the all
pervading Cosmic
Background predicted by
G. Gamow in 1948 and
discovered by Penzias and
Wilson in 1965.
This blackbody gamma radiation
originated in the hot early
Universe.
As the Universe expanded and cooled the
radiation cooled down.
CMB temperature fluctuations (COBE)
By way of summary, the 3 experimental evidences for Big Bang:
• Red
shift (Cosmic Expansion)
• Primordial Nucleosynthesis
• Cosmic Microwave Background
Key concepts of the Hot Big Bang Model:
• General Relativity as a theory of Gravitation
• (Inflation)
Concepts of General Relativity
Classical Physics concepts
Special Relativity concepts
Spacetime of Classical Physics and Special Relativity
 General Relativity: a theory of Gravitation in
agreement with the Equivalence Principle
Spacetime must be curved !!
Classical Physics
• Existence of Inertial Reference Frames (IRF)
• Relativity Principle (Hey man, physics gotta be the same in any IRF!)
• Invariance of length and time intervals
x '  x vt
t'  t
Special Relativity
• Existence of Inertial Reference Frames (IRF)
• Relativity Principle (Hey man, physics gotta be the same in any IRF!)
• Invariance of c
x '   ( x  vt )
t '   (t  vx / c 2 )
Gravitation, a peculiar force field
Electric field
F = qE
F = m(i)a
qE = m(i)a
a = E q/m(i)





Depending on
particle charge
 a=g
 One for all bodies
Gravity field
P = m(g) g
P = m(i) a
m(g)g = m(i)a
a = g m(g)/m(i)
If gravitation does not depend on the characteristics of a body then it can be
ascribed to spacetime. It is a spacetime property.
Equivalence between inertial mass and gravitational mass
Free fall in gravitational field (apple from a tree) cannot be
distinguished from acceleration (the rocket)
1. Free fall the same for every body  geometric theory of gravitation
2. Gravitation equivalent to non-inertial frames (EP)
Einstein replaced the idea of force with the idea of geometry. To him the space
through which objects move has an inherent shape to it and the objects are just
travelling along the straightest lines that are possible given this shape (J. Allday).
Understanding gravitation requires understanding spacetime geometry.
The concept of elementary interaction
Newton
Action at a distance
Faraday
Field concept
Maxwell
Quantum Fields
Gravity
(field quanta exchange)
(spacetime curvature)
Spacetime geometry
Geometry: study of the properties of space.
Euclidean geometry: based on postulates
- example: given an infinitely long line L and a point P, which is not on the line,
there is only one infinitely long line that can be drawn through P that is not
crossing L at any other point.
P
L
Some consequences:
• The angles in a triangle when added together sum up to 180°
• The circumference of a circle divided by its diameter is a fixed number : 
• In a right angled triangle the lengths of the sides are related by
(Pythagoras Theorem)
c 2  a 2  b2
Euclid geometry is a description of our common sense (= classical physics) threedimensional space
However there are spaces that do not obey Euclid axioms. Spaces having a nonEuclidean geometry. We will consider the (2-dimensional) example of the surface
of a sphere.
What are the “straight lines” on the sphere
surface? They are the great cirlces! (the
shortest path between two point is an element
of a great circle).
Now, suppose we choose A as a point and
we draw from B the parallel to A.
They meet at the North Pole!
(Euclid axiom does not hold)
Another consequence: the sum of
the angles of a triangle is higher
than 180°
With the example of a bidimensional space (the sphere surface) we have
shown the existence of non-Euclidean (Riemannian) spaces. In this case
parallel axiom does no hold true!
Einstein’s theory replaced gravity as a force with the notion that space can have a
different geometry from the Euclidean. It is a curved space.
The sphere surface is 2-d and is a curved space when seen from “outside” (3-d)
We live in a 4-d curved (by gravity) spacetime
Three kind of geometry
are in general possible
(depending on energy
content of Universe)
Newtonian, Minkowski, General Relativity
geometries
Newtonian physics spacetime.
Length of a rules is invariant
(as well as time interval dt)
Special Relativity spacetime:
the 4-interval is invariant
2
1 0 0   x 


s 2  0 1 0   y 2   x 2  y 2  z 2
0 0 1   z 2 
 1 0 0 0   c 2 t 2 
 0 1 0 0  
2 

x
  c 2 t 2   x 2   y 2   z 2

s 2  
 0 0 1 0   y 2 

 2 
0
0
0

1

  z 
g Matrix (spacetime metric)
General Relativity Spacetime: similar in structure to Special Relativity spacetime
but now the gravity field makes the metric spacetime dependent.
g  ( x)
Einstein Tensor
Energy-Momentum Tensor T
G  8 G T
Ricci Tensor
R  g  R
1 


G R  g R
2
Riemann Tensor

R  R
R , 
Ricci Scalar R
1
g ,   g ,  g  ,  g  , 

2
Spacetime curvature
f ( g  )  8 T
Gravitational potential
2  4 G 
(Classical Physics)
Momentum/Energy
Mass Density
A picture of the Universe expansion
can be drawn by using the surface of
the sphere analogy
The “center of the Universe” lies
outisde of the Universe
The Big Bang takes place
everywhere!
The evidence is recession of other
parts of the Universe from us
This space is closed (one can
go all the way around and get
back to the same place)
Our Universe 4-d expansion is in
analogy with this toy-model
In the surface of the sphere analogy, the
geometry is non-Euclidean (but locally
Euclidean) and the space has a positive
curvature.
Geometry locally Euclidean means
neglecting the curvature (or neglecting
gravity). It is Minkowski space.
Friedmann Models (Newtonian version)
Friedmann Models are models of the Universe as (large-scale) systems that
are governed by (General Relativity) physical laws.
Recipe:
- Use the General Relativity Law (means selecting the equations)
- Assume the Universe in Homogeneus and Isotropic (means selecting a metric)
- Assume an energy content for the universe (means selecting E-p tensor)
Result: Equations for the evolution of the Universe (Friedmann, LeMaitre)
Let us start with a Newtonian model
F 
G MS m
RS2 (t )
m
d 2 RS
GM S


d t2
RS2 (t )
M S , RS (t )
2
GM S
1  dRs 

U


2  dt 
RS (t )
4
MS 
 (t ) RS3 (t )
3
RS (t )  a(t ) rS
1 2 2 4
rS a 
G rS2  (t ) a 2 (t )  U
2
3
Expansion of a classical
distributed mass
2U 1
 a  8 G
 (t )  2 2
  
3
rS a (t )
a
2
Friedmann equation in Newtonian form
General form of the solutions
2
1  dRs  GM S
A

 
2  dt 
Rs
We can calculate A using the present-day values (H, R, density)
4 R03 0
1 2 2
A  H 0 R0  G
2
3
3
2


dR
G

R
3
H
8

8

 S
2
0 0
0

GR0  0 


 
dt
3
R
3
8

G




s
2
Qualitative comments
Going in the past the first term dominates (R was smaller)
Rs  0
There was a time when
dRs

dt
(the beginning of the Big Bang)
What is the future o’ the Universe?
Let us define:
3H 02
 c
8 G
 
0
c
Critical density
3
dR
G

R
8

8
 S
2
0 0


GR
0  0  c 


3 Rs
3
 dt 
2
If current density greater than critical density the second term is negative and then
there will exist a time in which
dRs
0
dt
The expansion will then stop and the Universe will collapse back to the initial state
If current density smaller than critical density the second term is positive and the
derivative will never get down to zero. Expansion will go on forever.
Dark Matter
This is a golden era for cosmology:
• Measurements of the CMB (and its anisotropies)
• Existence of Dark Matter
• Existence of Cosmological Constant
In this section we discuss evidence for Dark Matter
Popular wisdom: that the matter in the Universe is made of ordinary baryons
(the so called barionic matter). This matter has the property of emitting
radiation (being mostly concentrated in stars).
However, there seems to be more matter than the one which is visible.
The presence of Dark Matter is deduced by using its gravitational effect on luminous
matter. At different scales!
Galaxy scale
For instance, let us consider a spiral galaxy and plot the velocity of matter in the
galaxy as a function of the distance from the center:
Keplerian rotation: due to gravitational force
This can be explained by the
existence of a halo of matter
surrounding the galaxy
Galaxy Cluster scale
In clusters of Galaxies: a galaxy cluster is a group of galaxies held together by
their own gravity. However, when we measure the speed with which each galaxy
moves, it appears a lot more gravity is required to hold the cluster together than
can be explained by the stars we can see. Therefore, there must be a lot of dark
matter that we can’t see.
Cosmological scale
Gravitational lensing effect.
The light from a far away source is
deflected by Dark Matter: distortions
and multiple images (Einstein Rings)
Gravity Optics ! !
Some Einstein rings
observed by Hubble
Space Telescope
So, what is dark matter ?
• MACHO’s (Massive Compact Halo Objects): Black Holes, Planets, Dead Stars
• Non-baryonic Matter (particle physics explanation): WIMP (Weakly Interacting
Massive Particles) like neutralinos, neutrinos
Dark Matter is about 90% of the total matter in the Universe ! !
The Friedmann Model strikes back
Friedmann Models are models of the Universe as (large-scale) systems that
are governed by (General Relativity) physical laws.
Recipe:
- Use the General Relativity Law (means selecting the equations)
- Assume the Universe in Homogeneus and Isotropic (means selecting a metric)
- Assume an energy content for the universe (means selecting E-p tensor)
Result: Equations for the evolution of the Universe (Friedmann, LeMaitre)
Let us do the “full” model
2U 1
 a  8 G
 (t )  2 2
  
3
rS a (t )
a
2
Friedmann equation in
Newtonian form
Why the Newtonian Universe is not a good representation?
1. Not homogeneuos and isotropic
2. It is Euclidean
What is then the correct equation for the scale parameter?
Recipe:
- Use the General Relativity Law (means selecting the equations)
- Assume the Universe in Homogeneus and Isotropic (means selecting a metric)
- Assume an energy content for the universe (means selecting E-p tensor)
G  8 GT   g 
Einstein equation with the Cosmological
constant
2

dr
2
2
2
2 
ds 2  dt 2  a 2 (t ) 

r
d


sin

d




2
1

kr


a(t )
scale parameter
k
space curvature (1 open, 0 flat ,  1 closed )
T

 pg


 (   p) U U

Homogeneity and Isotropy
(Friedmann, Robertson,
Walker metric)
Scale parameter and
geometry of the Universe
Energy-momentum tensor of a
perfect fluid
The result of all this produces two changes (with respect to the Newtonian form) in
the scale parameter equation:
2U 1
 a  8 G


(
t
)

 
3
rS2 a 2 (t )
a
2
 (t )   (t )
energy density
2U
kc 2
 2
rS2
R0
geometrical
The cosmological constant
The cosmological constant is an extra term in Einstein’s equation of General
Relativity which physically represents the possibility that there is a density and
pressure associated with “empty” space. This term acts as a “negative” pressure.
After some simplification the result is:
a 2 8 G

k

 M  R    2
2
a
3
3 a0
Data indicates that the dynamics of the Universe is dominated by Dark Matter and
Cosmological Constant. And that the Universe is geometrically flat.
In other words the equation is effectively
a 2 8 G

k

  DarkMatter    2
2
a
3
3 a0
The energetics of the Universe is mostly determined by the Matter component
(30% of total energy, which is 95% Dark Matter) and the Cosmological Constant
(70% of total energy content)
Our Universe is located at about
   0.7
 DM  0.3
These slides at
http://pcgiammarchi.mi.infn.it
Two suggestions for further reading
 B. F. Schutz, A first course in general relativity, Cambridge University
press.
 B. Ryden, Introduction to cosmology, Addison Wesley