PHYS 787:COSMOLOGY
Winter 2005
Mon & Fri 11:30-12:50 PM
Main Link Rooms (W: EIT 2053, G: MacN 101)
WWW: />Instructor: Mike Hudson
Office: Physics 252 (ext 2212)
Textbook: The primary textbook is Structure Formation in the Universe,
T. Padmanabhan, 1993, Camb. Univ. Press.
Other useful references are listed on the P787 WWW references page
Prerequisites: None. Some knowledge of General Relativity is advanta-
geous but is not required.
Syllabus:
1. Observational Overview
2. Homogeneous Universe
(a) Metric; Redshift
(b) Dynamics
(c) Distance; Ages; Volumes
3. Hot Big Bang
(a) Thermodynamics
(b) Recombination
(c) Nucleosynthesis
4. Structure Formation
(a) Linear Perturbation Theory
(b) Statistics of LSS
1
(c) Nonlinear models
5. Galaxies and Galaxy Formation
6. Cosmic Microwave Background Fluctuations
7. Gravitational Lensing
8. Inflation
Grading:

Assignments 50%
Term Paper & Seminar 50%
The course WWW page:
/>will always have the most up-to-date information.
2
Preamble
About this course
This course aims to give a broad review of modern cosmology. The emphasis is
on physical cosmology, i.e. its content, the physical processes in the expanding
Universe and the formation of structure from the horizon down to the scale of
galaxies. I will focus on the current paradigm, the Big Bang model and structure
formation in a Universe dominated by dark matter and dark energy.
A deep knowledge of General Relativity is not necessary, although a familiarity
with GR will make the course more palatable. Likewise a basic understanding of
astrophysical processes and some knowledge of basic particle physics are helpful.
In an effort to be broad some depth has necessarily been sacrificed, but I hope that
enough background and reference pointers have been provided for the interested
student to delve deeper on their own.
1 INTRODUCTION TO COSMOLOGY 1
1 Introduction to Cosmology
1.1 A Very Brief History
Early Cosmological Models
I will skip the full treatment of early cosmological models — which would cover the Ptolemaic
model and the Copernican revolution via Tycho and Kepler — except to note that the “Copernican
Principle”, i.e. that we do not live in a special place in the Universe, has proved to be influential.
Newton’s cosmology was infinite. Time and space were absolute and independent of the matter
in the Universe. Newton’s 1692 Letter to Richard Bentley:
It seems to me, that if the matter of our sun and planets, and all the matter of the universe,
were evenly scattered through all the heavens, and every particle had an innate gravity towards
all the rest, and the whole space throughout which this matter was scattered, was finite, the

matter on the outside of this would by its gravity tend towards all the matter on the inside,
and by consequence fall down into the middle of the whole space, and there compose one
great spherical mass. But, if the matter were evenly disposed throughout an infinite space, it
could never convene into one mass, but some of it would convene into one mass and some into
another, so as to make an infinite number of great masses, scattered great distances from one
to another throughout all that infinite space. And thus might the sun and fixed stars be formed,
supposing the matter were of a lucid nature.
Problems with Newton’s Universe:
• Stability
• Olber’s paradox - an infinite universe would produce an infinite amount of light at our posi-
tion, so ”why is the night sky dark?”
Einstein’s Static Model In 1917, before discovery of cosmological redshifts, Einstein proposed
a closed universe with a spherical geometry which was finite in extent, centreless and edgeless. In
order to make this model static, Einstein introduced into GR a small repulsive force known as the
cosmological constant.
Einstein believed in a static Universe – to the extent that he was willing to add an extra parameter
to his theory. Why? (Later he referred to the cosmological constant as his “greatest blunder”).
1 INTRODUCTION TO COSMOLOGY 2
Shortly afterward de Sitter discovered an expanding but empty solution of Einstein’s equations -
motion without matter. Friedmann (1922) found solutions with both expansion and matter, which
Lemaitre (1927) independently rediscovered.
Why was the Universe assumed to be homogeneous?
Early Extra-galactic Cosmography
At the beginning of the twentieth century, it was generally accepted that our galaxy was disk-
shaped and isolated. But what were the spiral “nebulae” like M31 (Andromeda) - were they inside
or outside the Milky Way? Immanuel Kant had speculated that they were other “island” universes.
In 1912, Slipher measured spectra from the nebulae, showing that many were Doppler-shifted. By
1924, 41 nebulae had been measured, and 36 of these were found to be receding.
In 1929, Hubble measured the distances to “nebulae”. He measured Cepheid stars in nearby galax-
ies such as M31 and then measured the relative distances between M31 and more distant galaxies

by assuming that brightest stars were standard candles.
Combining these with the known velocities (corrected to the velocity frame of the Milky Way), he
obtained the plot shown in Fig. 1.1.
Figure 1.1: Hubble’s plot of velocity versus distance
Fitting a straight line,
v = H
0
r , (1.1)
1 INTRODUCTION TO COSMOLOGY 3
Hubble found H
0
= 500 km/s/Mpc, a value about 7 times too large
1
The outstanding feature, however, is the possibility that the velocity-distance re-
lation may represent the de Sitter effect, and hence that numerical data may be intro-
duced into discussions of the general curvature of space.
(Hubble 1929)
1.2 Review of Observational Cosmology
1.2.1 Preliminary Definitions
Ω denotes a density divided by the critical density needed to close the Universe,
ρ
crit
=
3H
2
8πG
(1.2)
Subscripts m, b, r, v denote the densities of matter, baryonic matter, radiation and vacuum. No
subscript indicates the total density
2

Subscript 0 denotes the present-day value of a parameter, e.g. H
0
is the present-day value of the
Hubble constant.
Units In this section we will use “astronomer” units.
1 Megaparsec (Mpc) = 3.26 × 10
6
light years = 3.1 × 10
22
m
1 year = 3.16 × 10
7
s
1 Solar Mass (M) = 1.99 × 10
30
kg
1.2.2 Expansion of the Universe
Fig. 1.2 shows a modern Hubble diagram using Type Ia supernovae as distance indicators. Note
the deviations from linearity at large z, we will return to this later. Supernovae in all directions in
1
Hubble made two errors. First, Hubble assumed that the variable stars he observed in nearby galaxies (Cepheids)
were the same as a different class of variable stars (W Virginis) in our galaxy. Second, what Hubble thought were
bright stars in other galaxies were actually collections of bright stars. These errors were not discovered until the 1950s.
2
This convention is quite recent (and still by no means universal). In many sources Ω implicitly refers to matter.
The contribution from the vacuum is often denoted Ω
Λ
, Λ, or λ depending on how it is normalized.
1 INTRODUCTION TO COSMOLOGY 4
Calan/Tololo

(Hamuy et al, 
A.J. 1996)
Supernova
Cosmology
Project
effective m
B
(0.5,0.5)

(0, 0)
( 1, 0 ) (1, 0)
(1.5,–0.5)
(2, 0)
(Ω
Μ,
Ω
Λ
) = 
( 0, 1 )
Flat
Λ = 0
redshift z
14
16
18
20
22
24
26
0.02 0.05 0.1 0.2 0.5 1.00.02 0.05 0.1 0.2 0.5 1.0



Perlmutter, et al. (1998)
FAINTER
(Farther)
(Further back in time)
MORE REDSHIFT
(More total expansion of universe 
since the supernova explosion)
In flat universe: Ω
M
= 0.28 [± 0.085 statistical] [± 0.05 systematic]
Prob. of fit to Λ = 0 universe: 1%

Figure 1.2: Hubble diagram for Type Ia Supernovae (Perlmutter et al.)
the sky fit the curve: the expansion is indeed isotropic.
The Hubble Space Telescope Key Project measured the flux of Cepheid stars in nearby galaxies to
allow a calibration of the distance scale and hence the Hubble constant
3
h = H
0
/(100 km/s) = 0.72 ± 0.08 (1.3)
3
In fact, the Hubble constant is neither constant in space n– because of peculiar velocities – nor in time, so it would
be better called the Hubble parameter.
1 INTRODUCTION TO COSMOLOGY 5
from (Freedman et al. 2001).
1.2.3 Isotropy and Homogeneity of the Universe
The Universe is observed to be isotropic on very large scales. Fig. ?? plots a sample of distant
galaxies on the sky: clustering is evident on small angular scales but on the largest scales the

distribution looks smooth.
Figure 1.3: This picture covers a region of sky about 100 degrees by 50 degrees around the South Galactic
Pole. The intensities of each pixel are scaled to the number of galaxies in each pixel, with blue, green and
red for bright, medium and faint galaxies (1-mag slices centred on B magnitude 18, 19 and 20). The many
small dark ‘holes’ are excluded areas around bright stars, globular clusters etc. (From the APM survey.)
By obtaining redshifts of galaxies and using Hubble’s law, we can plot the distribution of galaxies
in 3D, as in Fig. 1.4. On the largest scales, the distribution of galaxies is homogeneous. On small
scales (1 − 10 Mpc), mass is clumped in galaxies and clusters of galaxies. On intermediate scales
(10 − 100 Mpc), clusters are grouped into superclusters and are connected by walls and filaments.
1.2.4 Cosmic Microwave Background (CMB)
Gamow predicted relic radiation from a primeval fireball in 1948. Penzias & Wilson (Bell Labs
Engineers) discovered the CMB in the radio in the 1960s.
The spectrum of the CMB is a perfect black body with a temperature of 2.728 ± 0.004K.
1 INTRODUCTION TO COSMOLOGY 6
Figure 1.4: The 2-Degree-Field Galaxy Redshift Survey
1 INTRODUCTION TO COSMOLOGY 7
Figure 1.5: Three false color images of the sky as seen at microwave frequencies. The orientation of the
maps are such that the plane of the Milky Way runs horizontally across the center of each image. The top
figure shows the temperature of the microwave sky in a scale in which blue is 0 K and red is 4. Note that
the temperature appears completely uniform on this scale. The middle image is the same map displayed in
a scale such that blue corresponds to 2.721 Kelvin and red is 2.729 Kelvin. The ”yin-yang” pattern is the
dipole anisotropy that results from the motion of the Sun relative to the rest frame of the cosmic microwave
background. The bottom figure shows the microwave sky after the dipole anisotropy has been subtracted
from the map. This removal eliminates most of the fluctuations in the map: the ones that remain are thirty
times smaller. On this map, the hot regions, shown in red, are 0.0002 Kelvin hotter than the cold regions,
shown in blue. The band across the centre is emission from our Galaxy.
1 INTRODUCTION TO COSMOLOGY 8
Fluctuations in the CMB
Dipole Term The CMB has a dipole anisotropy with an amplitude 3.358mK (known since 70s).
This is interpreted as a Doppler effect due to the motion of the Sun with respect to the “surface

of last scattering”. The corresponding velocity is 369.0 ± 2.5 km/s (Lineweaver et al. 1996, ApJ,
470, 38).
When we correct for the motion of the Sun wrt to the barycentre of the Local Group (LG), the LG
is moving at 600 km/s wrt to the CMB.
What is causing this motion?
Intrinsic Fluctuations
The intrinsic fluctuations are at the level of ∆T /T ∼ ×10
−5
. On large scales, these are due
primarily to (relative) gravitational redshifting (blueshifting) of photons as they escape potential
wells (hills) at the time of last scattering. On smaller scales, acoustic oscillations of the photon-
baryon fluid come into play and lead to the recently observed “acoustic peak(s)”.
1.2.5 Baryonic Universe
Abundance of Light Elements Big-bang nucleosynthesis allows us a prediction of the abun-
dances of helium, deuterium and lithium with only one free parameter (the baryon-to-photon ratio,
or equivalently the density of baryons).
The current best observational measurements of the primordial abundances of these elements (par-
ticularly deuterium) suggests
Ω
b
h
2
= 0.020 ± 0.002 (1.4)
(Burles, Nollett & Turner 01).
Baryon Budget Only a small fraction of the baryons in the Universe are in stars. The bulk of the
baryons are likely to be in between galaxies.
Note that the density in stars is quite negligible compared to the total baryonic density. This implies
that there is baryonic dark matter, primarily in groups/intergalactic medium.
1 INTRODUCTION TO COSMOLOGY 9
Figure 1.6: CMB temperature fluctuations as a function of multipole number l, showing data from many

recent experiments. The curve shows a cosmological model which fits the data. From Wayne Hu’sCMB
experiment page.
1.2.6 Evidence for Dark Matter
As noted above, the total density of baryonic matter is small: Ω
b
∼ 0.04 if h ∼ 0.7. Various lines
of evidence suggest that there is significantly more dark matter in the Universe. Some of these are:
Dynamics of Galaxies in Clusters As early as the thirties, Zwicky (1937) applied the virial
theorem to the orbits of galaxies in clusters and argued that there was evidence for the presence of
dark matter. The temperature of X-ray emitting plasma in clusters leads to the same conclusion.
1 INTRODUCTION TO COSMOLOGY 10
Figure 1.7: Predicted abundances (by mass) of
4
He, D,
3
He and Li
Rotation Curves of Spiral Galaxies How large are the DM haloes of galaxies?
Fig. 1.9 shows the rotational velocity of stars in the Milky Way. It is approximately constant over
the range of radii which have been measured. A similar behaviour is seen in other large spiral
galaxies. Beyond the edge of the light distribution, one expects the rotation velocity to fall as
r
−1/2
. However at large radii, well beyond the edge of the disk, the rotation curve is observed to
be approximately constant. This implies M(< r) ∝ r. This believed to be due to a dark matter
“halo”.
1 INTRODUCTION TO COSMOLOGY 11
Figure 1.8: The Baryon “Budget” (Fukugita, Hogan & Peebles, 1998 ApJ, 503,518)
Gravitational Lensing Strong gravitational lensing leads to the formation of multiple images
and giant arcs.
Weak gravitational lensing uses the small distortions of background galaxies to map the dark mat-

ter distribution. Because galaxies are not intrinsically round, there is considerable noise in this
method.
Microlensing is strong gravitational lensing where the multiple images are not resolved. This is
typically thecase when the stars are the lenses. Has been used to place constraints on the abundance
of dark compact objects in the halo of our Galaxy.
Cosmic Flows: Deviations from uniform Hubble expansion Deviations from homogeneity
lead to deviations from uniform expansion. The latter can be used to measure the former, provided
the scales are sufficiently large.
Observations of peculiar velocities of individual galaxies are noisy, so large numbers need to be
observed to obtain reliable results.
A comparison between mass and galaxies yields
Ω ≈ 0.3 (1.5)
1 INTRODUCTION TO COSMOLOGY 12
Figure 1.9: The rotation curve of the Milky Way Galaxy from a compilation of data. (From Fich &
Tremaine 1991)
Figure 1.10: Light and mass in the cluster Cl 0024+17
if fluctuations in the mass follow fluctuations in the light
1 INTRODUCTION TO COSMOLOGY 13
Figure 1.11: Density fluctuation fields of POTENT mass (left-hand column) vs. IRAS galaxies (middle
column), both smoothed G12. Contour spacing is 0.2 in δ; the heavier contour is δ = 0; solid contours mark
δ > 0 and dashed contours δ < 0. The density is also indicated by shading. Also drawn is the difference
field in units of the error, where the contour spacing is unity (right-hand column). The maps are drawn
out to a radius of 80h
−1
Mpc, and the very thick contour marks the boundaries of the R
e
= 40h
−1
Mpc
comparison volume. From Sigad et al., 1998, ApJ, 495, 516.

1.2.7 Ages of the oldest stars
Globular clusters are compact balls of stars orbiting in the halo of the Milky Way and other galax-
ies. The stars appear to be co-eval, and are thus easier to date by fitting to models of stellar
evolution. Allowing for 0.8 Gyr from the formation of the Universe to the formation of these stars,
Krauss and Chaboyer(2002) find that the age of the Universe > 11.2 Gyr at 95% CL, with a best
fit age of 13.4 Gyr.
5.1 shows the Hertzsprung-Russell diagram for the globular cluster M5. The three solid lines are
the model predictions for 10, 12 & 14 Gyr.
1.2.8 Galaxies
Luminosity function and mass functions The galaxy luminosity function (i.e. the probability
distribution function for galaxy luminosities), φ, is a power law at low luminosities and has an
exponential break at high luminosities
φ ∝ exp(−L/L
∗
)

L
L
∗

α
(1.6)
where the characteristic luminosity, L
∗
∼ 2.7 × 10
10
L

and α ∼ −1. This form is known as a
Schechter function.

The predicted mass function of virialized (dark-matter dominated) objects has a similar behaviour,
but breaks at larger masses corresponding to rich clusters.
1 INTRODUCTION TO COSMOLOGY 14
Figure 1.12: Isochrone fit to M92.The best fit age is 15±1 Gyr. From Harris et al 1997
What physics causes the break in the galaxy spectrum?
Morphologies There are two different types of galaxies: spirals are dominated by a rotationally-
supported, gas and dust rich disk of young stars; ellipticals are gas-poor and are dominated by
old stars on random orbits (“pressure supported”). However, many spirals have a small elliptical-
like bulge and many ellipticals have a weak disk. Ellipticals are found mainly in rich clusters of
galaxies, whereas spirals are found in low density regions.
How is galaxies related to mass on different scales?
1 INTRODUCTION TO COSMOLOGY 15
1.3 The “Standard Model”
Our Universe is expanding 1930s 99%
from a hot big bang 1950-60s 99%
in which the light elements were synthesized. 1960s 99%
There was a period of inflation 1980s 50%
which led to a “flat” universe today. 1990s 90%
Structure was seeded by Gaussian irregularities 1980-90s 75%
which are the relics of quantum fluctuations, 1980s 80%
and grew by gravity. 1970s 95%
The dominant matter is “cold dark matter” 1980s 80%
but Λ (or “quintessence”) is dynamically important. late 1990s 67%
(Adapted from Peebles Sci Am Jan 01)
The fundamental paradigm is the expansion of Universe governed by gravity and by the equation
of state of its constituents and the growth of structure within the Universe driven by gravity but
counteracted by pressure.
There are a number of parameters which characterize the standard model. At present few of these
are predicted a priori, and so must be fixed by observation. These include:
• The present-day expansion rate, h or H

0
[1 parameter].
• The present-day densities (Ω) of matter, baryonic matter, radiation and of the vacuum [4
parameters].
• The characteristics of the baryonic matter are described by the standard model of particle
physics. One could add the mass of the dark matter particle(s) [1 parameter?] and its inter-
actions. In the simplest standard model, the DM particle interacts only weakly.
• The equation of state of the dark energy [1 parameter?].
• The amplitude of primordial fluctuations, characterized by A and n, where the latter param-
eter describes how these fluctuations behave as a function of scale [2 parameters]
4
. To these
we might also add a gravitational wave background [2 more parameters]. Certain models of
inflation make specific predictions for combinations of the latter parameters.
4
Of course it is possible that the primordial fluctuation spectrum is not described by a simple power law as a
function of scale
1 INTRODUCTION TO COSMOLOGY 16
The simplest viable model contains only a few parameters that need to be measured (h, A, Ω
m
,
Ω
b
). A more general model might have as many as a dozen parameters. Occam says that we should
prefer the former unless we have evidence to think otherwise.
Some observations which might have disproved the Standard Model
• A non-blackbody spectrum for the CMB.
• CMB fluctuations ∆T/T  10
−5
.

• CMB fluctuations without the “acoustic peaks”
• A star with helium abundance Y  23%.
• Non-Gaussian galaxy or CMB power spectra.
• Direct evidence of violations of GR.
Status of the cosmological parameters (2002)
CMB data alone gives: Ω
Λ
+ Ω
m
= 1.04 ± 0.04. The Universe appears to be very close to flat!
The current best-fitting parameters from CMB anisotropy experiments plus the 2dF galaxy survey
give:
Ω
Λ
0.72±.09
h
2
Ω
m
0.138±.013
h
2
Ω
b
0.023±.003
n
s
0.99±.06
h 0.73±.11
(From Wang et al. 2002).

For the best fitting Hubble constant, the above yields: Ω
m
= 0.28, Ω
b
= 0.04.
Also note that Ω
b
/Ω
m
= 0.17 ± 0.027. A universe consisting only of baryons is ruled out at > 6σ.
Dark matter must be mostly non-baryonic.
• Supernovae give: Ω
m
= 0.28 in a flat Universe. Also in agreement with assessments from
weak lensing and large-scale flows. Errors remain large, though.
• Big-bang nucleosynthesis gives: Ω
b
h
2
= 0.02 in agreement with above.
1 INTRODUCTION TO COSMOLOGY 17
• Hubble constant is: 0.72 ± 0.08 in agreement with the above.
• These numbers yield an age of the Universe of 12.2 Gyr, in agreement with the globular
cluster ages.
Some outstanding issues
• The Very Early Universe
– What physics drove inflation?
∗ scalar field(s)?
∗ collisions in the brane world?
– How did the “Bang” begin?

∗ quantum gravity
• The Dark Sector
– What is the dark energy?
∗ how much is there?
∗ what is its equation of state?
∗ is it related to inflation? to dark matter?
– What is the dark matter?
∗ how much is there?
∗ what are the mass and interactions of the particle?
• Formation of Baryonic Objects
– When did the Universe re-ionize (when was “First Light”)?
– Were the first luminous objects stars or galaxies?
– When did galaxies assemble?
– Why are there two types of galaxies: spirals and ellipticals?
– Is there a connection between galaxy formation and active galactic nuclei (AGN) [mas-
sive black holes]
2 FRIEDMANN-ROBERTSON-WALKER COSMOLOGICAL MODEL 18
2 Friedmann-Robertson-Walker Cosmological Model
2.1 Cosmological Principle
Cosmological models are based on
• the observation that the Universe appears isotropic and;
• the assumption that observers on other galaxies also see an isotropic Universe
This cosmological principle is a “Copernican” assumption that the Earth (and the Milky Way
galaxy ) are not at a “special” location. (For example that we are not at the exact center of an
explosion).
Isotropy demands that on the surface of sphere whose radius is at a given “distance”, the local
matter and radiation densities, the local expansion rate, as well as the redshift of light and the
ticking rate of clocks must be independent of direction.
If we require all observers also find that the Universe is isotropic, this places strong restrictions on
the metric as we will see below. In particular once we have isotropy around all points, the Universe

must be homogeneous in the above properties as well.
2.2 The Metric
In General Relativity (GR), space-time is described by a metric ds
2
= c
2
dt
2
− dl
2
. This is com-
pactly written ds
2
= g
αβ
dx
α
dx
β
where Greek indices run from 0 to 3. Index 0 denotes time (so
dx
0
could also be written dt) and the indices 1 to 3 indicate the spatial dimensions.
Light travels along paths with ds = 0. Other particles follow geodesics, which can be thought of
as shortest paths in space time.
To gain some intuition about the nature of the metric, first consider 2-d surfaces at a fixed time.
Clearly one possible metric which satisfies the assumptions of isotropy and homogeneity is the
“x-y” plane, i.e. dl
2
= dx

2
+ dy
2
.
Another is the 2-d surface of a 3-d sphere (a “2-sphere”).
x
2
+ y
2
+ z
2
= a
2
(2.1)
In the 3-d space in which the 2-sphere is embedded, dl
2
= dx
2
+ dy
2
+ dz
2
. If we consider only
2 FRIEDMANN-ROBERTSON-WALKER COSMOLOGICAL MODEL 19
distances along the surface of the sphere, we can substitute for dz using equation (2.1) to get
dl
2
= dx
2
+ dy

2
+
(x dx + y dy)
2
a
2
− x
2
− y
2
(2.2)
With new coords defined by
x = r

cos(φ); y = r

sin(φ) (2.3)
we find
dl
2
=
a
2
dr
2
a
2
− r
2
+ r

2
d(φ)
2
(2.4)
Now rescaling r by defining r = r

/a gives
dl
2
= a
2

dr
2
1 − r
2
+ r
2
d(φ)
2

(2.5)
Note that this sphere is finite (with area 4πa
2
) and is unbounded. r = 0 is at the pole and r = 1 is
at the equator.
If instead we choose the usual polar coords θ, φ to be our 2-d coordinate system, ie.
x = a sin(θ) cos(φ); y = a sin(θ) sin(φ); z = cos(θ) (2.6)
then in terms of these coordinates
dl

2
= a
2
(dθ
2
+ sin(θ)
2
d(φ)
2
) (2.7)
Note that in addition to the flat and 2-sphere metrics, there is also a homogeneous hyperbolic
surface (with negative curvature). This has metric a plus sign in the denominator of equation (2.5)
and has a sinh(θ) in place of the sin(θ) in the above equation.
2.3 Robertson-Walker metric
The 2-d arguments can be extended to a 3-sphere embedded in a fictitious 4th dimension.
Instead let us re-derive the form of the RW metric a different way, following Gunn (1978). Isotropy
and the cosmological principle require that the spatial part of the RW metric has the following form
a
2
(t)

b
2
(r)dr
2
+ f
2
(r)dψ
2


(2.8)
2 FRIEDMANN-ROBERTSON-WALKER COSMOLOGICAL MODEL 20
where dψ
2
= dθ
2
+ sin
2
θdφ
2
. This last step is necessary so that distances in the plane of the sky
are the same in both the θ and φ directions.
Because of isotropy, we cannot have a different expansion rate in the r and ψ directions, so the
a(t) term has to be the same for both.
Similarly b, f and a cannot be functions of θ and φ.
Now let us choose a new radial coordinate χ so that
b(r)dr = dχ . (2.9)
This yields
a
2
(t)

dχ
2
+ S
2
(χ)dψ
2

(2.10)

P
S
QO
y
h
x
r
r
γ γ
lθ
Figure 2.1: Gunn’s “isosceles” triangle
We will now work out S(χ) using an argument from Gunn (1978). S(χ) is related to the “angular
diameter distance”
5
; it is a function that allows us to map a given angle dψ into a proper length.
For Euclidean space, S(χ) = χ, but as we saw above it need not be so.
Consider the three comoving observers O, P and Q in Fig. 2.1. Suppose P is slightly displaced
from the ray OQ so that γ and θ are small angles. Note that by isotropy the two γ’s must be equal.
Then
l = γS(2χ) = θS(χ) (2.11)
so
θ = γS(2χ)/S(χ) (2.12)
The distance y is
y = γS(χ + x) = γS(χ − x) + θS(x) (2.13)
5
The angular diameter distance in an expanding Universe is a little more complicated than simply S(χ). Here,
however, we are consider angles along the spatial surface at fixed cosmic time t.
2 FRIEDMANN-ROBERTSON-WALKER COSMOLOGICAL MODEL 21
Figure 2.2: In order to estimate the proper size of an object which subtends a given angle, we need the
geometry of space-time

differentiating with respect to x and setting x = 0 yields
2γS

(χ) = θS

(0) (2.14)
but S

(0) = 1 since S → χ for small χ, so substituting for θ gives
2S(χ)S

(χ) = S(2χ) (2.15)
Clearly S(χ) = χ is a solution of the above equation. More generally, this can be solved by series
expansion and it can be shown that sin(χ) and sinh(χ) are also solutions.
We can define
S
K
(χ) =





sin χ (K = 1)
sinh χ (K = −1)
χ (K = 0)
(2.16)
where the constant K is called the curvature constant. Note also that if A is a constant, AS
K
(χ/A)

is also a solution to equation (2.15). However, usually it is easier to absorb A into the definition of
a(t).
For example, K = +1 is positive curvature and the metric is a three-dimensional generalization of
the two-dimensional surface of sphere. This metric has a finite volume, in the same way that the
surface of a sphere has a finite area. This model is often called “closed”.
2 FRIEDMANN-ROBERTSON-WALKER COSMOLOGICAL MODEL 22
Similarly K = −1 is a generalization of a hyperbolic or saddle-shaped geometry. It is infinite and
“open”.
K = 0 gives a “flat” geometry, i.e. one which makes the comoving coordinates Euclidean at all χ,
not just at small χ.
So the Robertson-Walker metric is
ds
2
= c
2
dt
2
− a
2
(t)

dχ
2
+ S
2
K
(χ)dψ
2

(2.17)

An alternative form of the RW metric can be obtained if we change radial coordinates r = S
K
(χ).
ds
2
= c
2
dt
2
− a
2
(t)

dr
2
1 − Kr
2
+ r
2
dψ
2

(2.18)
Note that both of these forms look like Minkowski space for small r (or small χ). Both forms
appear in the literature, (and usually both use the notation r).
Comoving Coordinates
There is a special class of observers — those who a fixed value of r, θ and φ.
• It can be shown that such observers follow geodesics and are therefore “freely falling”.
• Note also that the proper time for all comoving observers is the same and is identical to the
cosmic variable time, t.

• These observers have no velocity with respect to the local matter
Such observers are labeled fundamental or comoving observers and the coordinates r are comoving
coordinates.
2.4 Kinematics of the Expansion
Hubble’s Law
At small distances, Hubble’s law should hold. Consider two nearby comoving galaxies, one at
r = 0 and another at a small comoving distance ∆r. They are separated by a proper radial distance