Chapter 3: THE FRIEDMANN MODELS

Chapter 3: THE FRIEDMANN MODELS
In this chapter we incorporate gravity and our knowledge of how the
density of the Universe changes as it expands to generate the family of
Friedmann models for R(τ). The dynamics and the curvature are both
determined by the average density, parameterized by Ω. These
models enable us to calculate the R 0 S k (ω) term in the Robertson-
Walker metric as a function of redshift.
The results in the previous chapter were all derived directly from the assumptions of
isotropy and homogeneity. To make further progress we need to determine both R(τ)
and k/A 2 and for this we need to have a theory of gravity and an equation of state to
describe the Universe. The construction, and solution, of Einstein's Field Equations
are beyond the scope of this course. We will therefore simply state the basic
dynamical equation that results from a rigourous treatment in General Relativity and
then seek to justify this in terms of firstly a purely Newtonian analysis and then a
simple-minded approach to General Relativity.
The following equation for R(τ) is the solution to the Field Equations and is called the
Friedmann Equation.
2
(3.1) R & 2 8πG kc R
2 Λ
= ρ − + R
2
3 A 3
2
Hre R(τ) is the scale factor and k/A 2 the curvature introduced in the previous chapter,
ρ is the density of the Universe. The Λ-term is the so-called cosmological constant,
originally omitted by Einstein but then introduced since it enabled him to find a static
solution for the Universe (thereby missing the considerable scientific coup of
predicting the non-static nature of the Universe). We'll see below that it can be viewed
as a curvature of the vacuum. After it was realized that the Universe is not static, it
became conventional to set Λ to zero. However, it has now reappeared in a new and
interesting guise.
3.1 An almost-completely Newtonian approach
Let us first see how far we can get with a purely Newtonian approach. It turns out
that if the Universe is made up of dust (i.e. a pressure-less material), a purely
Newtonian treatment of gravity produces the correct expressions for R(τ), but of
course fails to describe the curvature k/A 2 in the Friedmann equation.
Consider the classical Newtonian approach to calculating the motion of a galaxy A
located a physical distance x from us. Newton's two theorems concerning the

gravitational effects of uniformly distributed distributions of mass allow us to
consider only the mass enclosed within the sphere of radius x = R(τ)ω OA centered on
O whan calculating the gravitational attraction between O and A..
O
x
A
Thus if x is the distance OA, then we have
(3.2) &&x =− 4 π
G ρ x
3
Now, since the comoving distance to A from O, ω OA , is by definition constant during
the expansion, we can write:
x = Rω ⇒ x& = R& ω ⇒ x&& = R&&
ω
OA OA OA
We may thus rewrite our dynamical equation in terms of R(τ) alone, thus eliminating
reference to any particular galaxy or distance x.
(3.3) R && 4π
=− Gρ
R
3
So, the dynamics of our shell of material are the same as the dynamics of the whole
Universe. In a sense, this is then the justification for the Newtonian approach since we
can make our radius r arbitrarily small so that curvature and light travel time effects
are completely irrelevant.
To integrate this equation we need to know how ρ behaves as functions of R(τ), i.e.
we need to know the equation of state of the matter-energy in the Universe. For the
time being lets assume it is simple pressure-less matter (known as “dust”) that evolves
as ρ ∝ R -3 .
We adopt the convention of putting in square brackets quantities that are invariant
under the expansion. If we multiply both sides by & R and set [ρR 3 ] = constant, then we
have

(3.4)
&
&&& 4πG R
RR R
3
=− ρ
2
3 R
R&
2 8πG = R
3 1
ρ + 2ξ
3 R
R&
2 8πG = ρR
2
+ 2 ξ
3
Here, ξ is a constant of integration that, in this Newtonian analysis, corresponds to the
binding energy of the system: If ξ > 0, then we have solutions with
&R 2 → 2ξ as R→ ∞. R(τ) is proportional to τ and the Universe is unbound and
expands forever. If ξ < 0, we get R & = 0 at finite R so the Universe is bound and at
some point the expansion is halted and the Universe recollapses. If ξ = 0, then we
have the asymptotic limit R & → 0 as R→ ∞
Note that this has exactly the same form as the Relativistic Friedmann equation (3.1)
with Λ= 0 if we associate the ξ binding energy with the curvature term.
Let us now return to the equation of state and consider possibilities other than the
familiar one considered above, where we assumed that the density is dominated by
dust with ρ ∝ R -3 .
An obvious possibility is that the density of the Universe is dominated by
electromagnetic radiation, e.g. the cosmic background radiation field. As the
Universe expands, the density of this radiation field will drop as R -4 . We get a factor
of R -3 from the decreasing number density of photons, but an additional factor of R -1
from the fact that all the photons lose energy due to the redshift.
There are other more exotic possibilities. As we’ll see below, we could imagine a
scalar density field that does not change its density as the Universe expands, i.e. ρ =
constant (more than that – in fact we have evidence that this is the case!). We will
refer to this as a false vacuum density.
Other possibilities exist. A few years ago there was some interest in hypothetical
Universes that were dominated by the mass from “cosmic strings” – strange entities
that have a constant density per unit length regardless of how much they are stretched.
These give ρ ∝ R -2 . It is easy to see that putting these other ρ(R) dependencies into
our Newtonian equation (3.3) will not yield the right form of (3.4) upon integration
unless we introduce a small fudge. To see how to proceed, it is useful to consider the
ρ(R) dependence in thermodynamic terms.
We can assume that the expansion is adiabatic (i.e. that there is no inflow or outflow
of heat). The 1st Law of Thermodynamics gives us that the change in the internal
energy (ρV) equals the pdV work done by the system in expanding:
2
(3.5) c d( ρV)
=−pdV
⇒
dρ
=−
( ρ + pc)
2
3
dR
R
If the pressure can be neglected, then

(3.6)
dρ
dR
=−3
⇒ ρ ∝R
−
ρ R
3
This is thus nothing more than the conservation of mass that we had above for the
“dust” case.
If the Universe is dominated by radiation or other relativistic particles then in this case
p = ρc 2 /3, and
(3.7)
dρ
dR
=−3
⇒ ρ ∝R
4
−4
ρ R
3
This is precisely the R -4 dependence that we considered above, arising from the
decrease in photon number density and the loss of energy of photons.
A more exotic case is when we have “material” that exerts a negative pressure, p = −
ρc2. This then gives
d ρ = 0
i.e. the ρ = constant false vacuum case that we mentioned above.
It is trivial to convince yourself that when we we integrate the Newtonian && R equation
(3.3) we will after all get the correct Relativistic Friedmann equation if we introduce a
small fudge: In the Newtonian && R equation, we should use what is called the active
density ρ' defined as follows:
p
(3.8) ρ′ = ρ+
3
c 2
where ρ is the proper density of mass-energy in the Universe and p is the pressure. It
should be stressed that the presence of a pressure term has nothing to do with the
effects of pressure in classical dynamics, since it is pressure gradients that have
dynamcal effects, and there will be no pressure gradients in a homogeneous Universe.
For a radiation field, this has the effect of making this co-called active density twice
the real density. For a false vacuum, the active density becomes negative, so we
would anticipate a repulsive effect for gravity.
It is important to note that we needed to make this fudge of using the active density in
the Newtonian && R equation so as to reproduce the correct Friedmann equation for & R.
When we come to solve the Friedmann equation for different equations of state to get
R(τ), we use the normal density regardless of the pressure, but must, of course, use the
appropriate ρ(R) dependence.

3.2 A poor-man’s General Relativistic treatment
After exploring the Newtonian approach above, in this section we will look at a
simple-minded approach to General Relativity. It must be stressed that the solution of
Einstein's field equations involves the association of the curvature tensor of the metric
and the tensor describing the distribution of mass in the Universe and the tensor
calculus involved is beyond the scope of the course. In this section, we will look at a
highly simplified analysis (taken from Berry's book) in which we will consider two
dimensional surfaces in which the curvature may be represented by a single number.
The point of this is simply to see how dynamical information can come from an
analysis of the curvature of space time.
If we suppress the two angular dimensions, then the alternative form of the
Robertson-Walker metric, which has the Euclidean angular part of the metric (which
makes the suppression of the angular part easier) reduces to:
2
2 2 2 R ( τ)
(3.9) ds = c dτ
−
kr A dr
2 2
1 −
Thus the metric coefficients, g µν, are:
2
(3.10)
g
c
2
0
2
µν
= R ( τ )
0 −
2
1−
kr
A
2
Gauss's classic formula for calculating the curvature of a two-dimensional surface
from the metric coefficients is, in the case where g 12 = g 21 = 0:
(3.11)
1
K =
2g
g
⎪⎧
2
∂ g
⎨−
⎪⎩
∂r
2
∂ g
−
2
∂τ
1
+
2g
⎡∂g11
∂g
22 ⎛ ∂g
⎢ + ⎜
⎢⎣
∂τ ∂τ ⎝ ∂r
2
⎞ ⎤
⎟ ⎥ +
⎠ ⎥⎦
11 22
11
2
11 22
11
2
1
g
22
⎡∂g
⎢
⎢⎣
∂r
11
∂g
∂r
22
⎛ ∂g
22
+ ⎜
⎝ ∂τ
2
⎞
⎟
⎠
⎤⎪⎫
⎥⎬
⎥⎦
⎪⎭
If we evaluate this, we find, since all the derivatives of g 11 vanish:
(3.12) K
R
=− && 2
Rc
Knowing a little about GR, we can plausibly associate this curvature of the space-time
surface, which is a scalar quantity, with the mass density of the Universe, another
scalar, plus possibly a constant representing a curvature of the vacuum when the
density is zero. i.e. we can guess that:
R&&
(3.13) K =− = αρ +
β
2
Rc 2

This may be integrated in the usual way to produce:
(3.14) R & 2 c
2 R
2 R
2 c
2
= 2αρ − β − χ
Here χ will be a constant of integration representing the initial conditions.
You will see that this equation is the Friedmann equation if I associate α with
4πG/3c 2 , β, which we introduced as the curvature of the vacuum, with -Λ/3c 2 , and χ
with kc 2 /A 2 .
Only a full solution of the field equations gives the correct value for the constant χ
that gives, through A, the curvature of the constant τ surface.
3.3 Asymptotic solutions to the Friedmann equation
We have from above (3.3) the Friedmann equation. We have three terms on the right
hand side: (a) a density term; (b) a curvature term and (c) the Λ term.
&R
2
8πG kc Λ
= ρR
− + R
2
3 A 3
2 2
2
It is important to notice that a non-zero Λ term has exactly the same effect as setting ρ
= constant in the first, i.e. density, term. For this reason, a scalar field which produces
a false vacuum density that does not decrease with the expansion is often stilll
referred to as “Λ”, even though I would argue that the physical origin is slightly
different and that calling it Λ is slightly confusing. Whether to regard such a
component as a density component with weird behaviour (false vacuum) or as a nonzero
Λ will introduce a possible source of confusion in what follows. I will try to
consistently denote with a subscript different components of the density.
As an introduction, let us look first at the simplest asymptotic solutions to the
Friedmann equation that arise when the different terms on the right hand side
dominate.
3.3.1 Density term dominates with matter ρ ∝R -3
For a regular matter-dominated Universe with k = 0 and Λ = 0, we can solve the
Friedmann equation as follows

R&
2
3
2
0
8πG
=
3
RR&
=
R
R
R
3
2
=
3
[ ρR
]
R
8πG
3
⎛ 3 ⎞
= ⎜ H
0τ
⎟
⎝ 2 ⎠
3
[ ρR
]
⎛ 8πG
⎜ ρ
0R
⎝ 3
2
3
3
0
⎞
⎟τ
⎠
Thus the Universe expands with R proportional to τ 2/3 . The elapsed time, τ 0 , since the
time when R = 0, is:
2 1
(3.25) τ 0
=
3 H0
3.3.2 Density term dominates with radiation ρ ∝R -4
If the density of the Universe is dominated by radiation, or some other relativistic
species, then with energy density ε, then the basic Friedmann equation will be:
4
(3.37) R & G R
2 8π
ε kc
= −
2 2
3c
R A
We can thus solve the Friedmann equation as before:
2
2
(3.38)
G
RR& 8π
=
2
3c
εR
8πG
4
R = 4 εR
2
3c
τ
8πG
R=
R 4
0 2 ε
0
3c
τ
4
The Universe expands as τ 1/2 , and the age of the Universe is given by
(3.39) τ = 1
2H
3.3.3 Curvature term dominates
The completely undecelerated case with ρ = 0 (and Λ = 0) is trivial to solve:

(3.20)
R&
=
R
R
0
c
A
= H 0τ
with
k
= −1
This is a linear expansion as expected for an undecellerated Universe. The age is
simply the inverse of H.
1
(3.21) τ 0
= H
0
Note that the curvature in this solution, i.e. k = -1 and c/A = dR/dτ, are exactly as we
found in the Milne model in Chapter 2.
3.3.4 Λ-term dominates (or false vacuum density with constant ρ)
Let us first consider the non-zero Λ case. From the form of the Friedmann equation,
we will get:
(3.22)
2 1
R&
= ΛR
3
R&
Λ
=
R 3
R
R
0
⎛
= exp⎜
⎝
2
Λ ⎞
( τ −τ
) ⎟
0
3
⎠
Thus, for positive Λ, we get an accelerating, exponential expansion. Hubble's
parameter is, for once, actually a constant, and is given by:
Λ
(3.23) H = ⇔ R∝
3
exp( H τ)
Note that an identical result is obtained by simply inserting an equation of state such
that ρ = constant. If I denote this constant density by ρ Λ then we get:
(3.24)
H
=
8
Λ
πGρ
3
The accelerating effects under gravity of the weird ρ = constant material should not be
surprising when it is recalled that the “active density” in the original Newtonian && R
equation ρ’ = ρ+3p/c 2 is negative, since p = -rc 2 .
3.3.5 The R dependence of the different terms in the Freiedmann equation
It is instructive to see how the relative importance of the different terms of the
Friedmann equation will change as the Universe changes in size through R(τ).

(a)
Matter vs. Radiation in the density term
If the Universe contains both matter and radiation (as ours most certainly
does) then at some earlier time, radiation will come to dominate the density,
even if at later times, the matter dominates. The difference of one power of R
in the behaviour of ρ(R) means that radiation will dominate for R < R eq :
(3.25)
R eq
=
ρ
ρ
rad
matter
0
(b)
Matter vs. curvature
For regular pressure-less dust, the density term may be written [ρR 3 ]/R,
whereas the curvature term is independent of R. Thus as R increases, the
effects of curvature will become more important, as R. Conversely, as we
look back to smaller R, the effects of curvature will become less and less
important. Even if curvature dominates today, at some earlier epoch curvature
will have been negligible, but the relative importance of matter and curvature
will only change as a single power of R.
If the density is dominated by radiation, then the decline of the relative
importance of the curvature term will be more rapid, going as R 2 .
(c)
Matter vs. Λ or false vacuum energy
As with curvature, as we look back to smaller R for a Universe that today
contains both matter (or radiation) and false vacuum energy (or non-zero Λ),
we will find at smaller R that the latter becomes less important. However, the
transition to a matter- (or radiation-) dominated Universe is much faster than
was the case for curvature, going as R 3 (or R 4 for radiation).
3.4 General solutions: the cosmological density parameter Ω and the
deceleration parameter q
It is convenient to introduce a critical density, ρ c and a density parameter Ω, which is
simply the ratio of the actual density to that critical density:
(3.26)
2
3H
ρc
=
8πG
ρ
Ω =
ρ
c
Dividing by R 2 , we can rewrite the Friedmann equation to give a relationship between
the density parameter, the expansion rate and the curvature. Note that the density
here is the sum of all components; matter, radiation, even false vacuum. In what
follows, I will assume that the Universe may contain significant amounts of (pressure-

less) matter (which could be baryonic or non-baryonic), radiation (with pressure) and
false-vacuum (with negative pressure and ρ = constant). I will neglect any other even
more exotic components, including the possibility that the false-vacuum changes with
R. However, it should be clear how to include any other general term in the
discussion below.
Neglecting a Λ-term for the time being (i.e. putting false vacuum, if relevant, into the
density ρ tot to compute an Ω tot ), i..e.
ρ
Ω
tot
tot
= ρ + ρ + ρ
= Ω
m
m
r
+ Ω
r
Λ
+ Ω
Λ
we have
(3.27)
H
2
k
2
R A
( RA)
= H
2
2
=
2
⎛
= ⎜
⎝
Ω
H
c
2
tot
2
c
H
( Ω
2
⎞
⎟
⎠
2
kc
−
2
A R
tot
( Ω
2
−1)
k
−1)
tot
Thus the critical density is also the density of a Universe with zero spatial curvature.
As we’ll see later, (c/H) is a basic length scale in the Universe (it is obviously roughly
c times the age of the Universe τ) and the value of Ω tot tells us about the (physical)
radius of curvature of the Universe, RA, compared with c/H.
Note again that for ρ = 0, i.e. Ω tot = 0, and k = -1, equation (3.16) reduces to A -1 =
HR/c, which is exactly the expression that we derived in our Special Relativistic
analysis.
Note as an aside that if we had chosen to keep the Λ-term in the Friedmann equation
(i.e. to consider the density only to be made up of “normal” matter and radiation),
then the above equations become:
(3.28)
2
2 2 kc Λ
H = H Ω − +
2 2
A R 3
2
k H ⎛ Λ
= ( 1)
2 2 2
⎜ Ω − +
R A c ⎝ 3H
( RA)
2
⎛
= ⎜
⎝
c
H
2
⎞
⎟
⎠
k
( Ω −1+ Λ / 3H
2
⎞
⎟
⎠
2
)
The quantity Λ/3H 2 is often written as λ. The condition for a spatially flat Universe
with k = 0 is thus Ω tot = 1 if I include possible false vacuum energy in my
computation of ρ and hence Ω tot , or Ω + Λ/3H 2 = 1 if I do not and consider instead a
non-zero Λ. These are very important relations, worthy of there own line in the notes.

(3.29) Conditions for a flat Universe:
Ω
tot
= 1
( Ω + λ)
= 1
There is another way to look at Ω. In terms of regular matter and radiation where we
have a decelerating Universe, using the definition of H from the previous chapter we
can rewrite the Friedmann equation in its Newtonian form with the binding energy as:
(3.30) 2 ξ
H
2 (1 − Ω)
=
2
R
Clearly, if ρ < ρ c , i.e. Ω < 1, then ξ > 0 and the Universe is unbound and if ρ > ρ c, i.e .
Ω > 1 then ξ < 0. and the Universe will recollapse at some point in the future. Thus
the critical density is simply the density of a Universe of zero binding energy. The
concept of binding energy with a false vacuum component is not so well defined.
As noted in the previous lecture, we can also define a dimensionless deceleration
parameter, q,
RR
(3.31) q =− &&
R& 2
It is easy to see that the Newtonian R & equation, with the “correct” ρ’ = ρ + 3p/c 2 ,
reduces to:
4π
3
(3.32) R &
= − GR( ρ + 2ρ
− 2ρ
)
so that
Ω
m
(3.30) q = + Ω
r
− Ω
Λ
2
m
r
Remembering that a flat Universe has Ω tot = 1, the condition for a flat Universe
becomes:
3Ω
m
(3.31) q = + 2Ω
r
−1
2
The relationship between Ω and q is a test of the equation of state of the Universe.
Note that Ω 0 is a density measurement that can in principle be determined locally
(questions of homogeneity aside) whereas q 0 is a kinematic measurement that requires
us to observe sufficiently far away that we can measure the second derivative of R(τ).
3.5 General Solutions to the Friedmann equation
Λ

We may write the density of the Universe at any R in terms of the present density (i.e.
at R = R 0 ) which is parameterized by Ω 0 for the different components.
(3.32)
8πGρ
3H
2
0
⎛
⎜Ω
⎜
⎝
⎛ R ⎞
⎜
⎟
⎝ R0
⎠
−3
+ Ω
⎛ R ⎞
⎜
⎟
⎝ R0
⎠
−4
+ Ω
=
0, m
0, r
0, Λ
⎞
⎟
⎟
⎠
We can introduce a new time coordinate, η, called the conformal time though we
won’t encounter it much, such that
(3.33)
cdt
d η =
AR
so that, indicating differentiation w.r.t η with a prime, the Freidmann equation
becomes:
(3.34)
⎛ R′
⎞
⎜
⎟
⎝ R ⎠
2
8πGρ
2
= A R
2
c
0 3
2
⎛
⎜
⎝
R
R
0
⎞
⎟
⎠
2
⎛
− k
⎜
⎝
R
R
0
⎞
⎟
⎠
2
We can now substitute in the expression above for ρ(R)
⎛ R′
⎞
⎜
⎟
⎝ R0
⎠
2
=
H
c
2
0
2
⎛
⎜Ω
⎜
⎝
0, Λ
+ Ω
0, m
⎛
⎜
⎝
R
R
0
⎞
⎟
⎠
−3
+ Ω
0, r
⎛
⎜
⎝
R
R
0
⎞
⎟
⎠
−4
⎞
⎟ 2
A R
⎟
⎠
2
0
⎛
⎜
⎝
R
R
0
⎞
⎟
⎠
4
⎛
− k
⎜
⎝
R
R
0
⎞
⎟
⎠
2
and can clean this up with a = R/R 0 and substitute in the (3.27) relation:
k H
0
(3.35) = ( Ω0,
−1)
2 2 2 tot
R A c
and rearrange to get:
0
2
(3.36)
a′
2
=
k
( ) ( 2
4
Ω
)
0, r
+ Ω0,
ma
− ( Ω0,
tot
−1)
a + Ω0,
Λa
Ω −1
0
This is straightforward to solve provided that Ω 0,Λ = 0 (see 3.6 below).
Taking this equation and substituting in the relation for ρ(R) above (3.32) it is easy to
show that H(R) is given by:
2 2
−3
−4
−2
(3.37) H = H ( Ω + Ω a + Ω a − ( Ω −1)
a )
0
0, λ
0, m
0, r
0, tot
This tells us the value of Hubble’s parameter at all epochs.

Now, since from the Robertson-Walker metric, we have for the path of a photon that
(3.38)
cdR
Rd ω = cdτ
= =
R&
cdR
HR
We can calculate the rate at which a photon travels over comoving radial distance in
terms of the observable redshift, by noting that R=R 0 /(1+z). Thus
c
H
c
H
2
3
4
(3.39) R dω
= dz = [(1
− Ω )(1 + z)
+ Ω + Ω (1 + z)
+ Ω (1 z ] dz
0 0, tot
0, Λ 0, m
0, r
+ )
0
We would now in principle be able to calculate ω(z) and, knowing the curvature term
S k (ω) and k/A 2 , compute things like D(z), D L (z), D θ (z), and the incremental volume
element dV/dz, etc. These are all quantities that we need to know to make sense of the
distant Universe and/or test the cosmology or determine the cosmological parameters
from observation. Unfortunately, simple analytic solutions are only derivable for the
case of Ω 0,Λ = 0.
Finally, since we always have
2
kc
(3.40) Ω
tot
−1
=
2 2 2
H R A
we can calculate Ω tot at all R.
−0.5
(3.41)
Ω
tot
−1
=
−2
2
( 1− Ω + Ω (1 + z)
+ Ω (1 + z)
+ Ω (1 + z)
)
0, tot
0, Λ
Ω
0, tot
−1
0, m
0, r
An important point is that unless Ω 0,m ~ Ω 0,r ~ 0 (i.e. unless we are completely
dominated by a false vacuum at the present epoch) then at very high redshifts, all
Universes will have Ω tot ~ 1, regardless of their current Ω 0,m , Ω 0,r or Ω 0,Λ .
3.6 Case study: matter dominated Universes
The treatment of a matter dominated Universe of arbitrary density is relatively simple
analytically, allowing us to explore several topics addressed in the general case in the
previous section quite easily. Until very recently, we were fairly confident that this
also in fact described our own Universe. Within the last 2-3 years, there has emerged
evidence, that certainly deserves to be taken seriously, that there is a dominant false
vacuum term (or non-zero Λ). If correct, this makes the general solution for R(τ) and
k/A 2 much messier as discussed above. However, it is still worthwhile looking at the
matter-dominated case.
The rest of this extensive section deals with such a Universe (which may be
hypothetical). In this section Ω and Ω 0 will be taken be taken to be the current density
of matter, i.e. what we called Ω m above.

3.6.1 General solutions for R(τ)
Before looking at general solutions to the Friedmann equation, it is useful to look at
the asymptotic behaviour of models with general Ω. Setting Λ = 0, we may develop
the Friedmann equation, using the relations between curvature and Ω (equation 3.27)
and between R and z, as follows:
(3.42)
Thus
R&
R&
R
2
2
2
0
8π
GρR
kc
= −
3 R A
3 2
8π
Gρ0R0
= −( Ω −1)
H
3 R
2 R0
= H0
Ω0
−( Ω −1)
H
R
2
= H Ω ( 1+ z)
− Ω + 1
0
2
0 0
2
0 0
2
0 0
(3.43)
R&
R
2
2
= H ( 1 +Ω z)
2 0
0
0
Hence, if Ω 0 < 1, we can see that the Universe has had roughly constant & R (i.e. has
had undecelerated expansion) since the epoch corresponding to (1+z) ~ Ω 0
-1. At
earlier times, & R behaves as for an Ω = 1 Universe (though, note, with a different H
than the Ω 0 = 1 Universe that would have the same H 0 today).
We can see this also by looking at Ω(z), which may be calculated as follows: From
(3.43) we have
2 2 2
H = H ( 1+ z) ( 1+
Ω z)
0
0
Substituting this expression for H(z) into the definition of Ω and knowing ρ(R), we
get
Ω =
=
8πGρ
2 2
3H
( 1+ z) ( 1+
Ω z)
0
8πGρ
( 1+
z)
0
2
3H
( 1+
Ω z)
0
0
0
Thus
(3.44) Ω Ω ( 1 + z)
=
0
( 1 + Ω z)
0

Note that if the density was exactly critical at some point then it will be critical for all
later time (as is obvious from the identification of the critical density as that density
which produces the critically expanding Universe). Of more interest, if Ω 0 < 1, then
Ω was approximately unity for all z > Ω 0
-1 as we discussed above.
Thus, an Ω 0 < 1 Universe behaves to first order like a critical flat Universe for (1+z) >
Ω 0
-1 and then like an undecelerated Universe for (1+z) < Ω 0
-1. This approximation
can simplify many calculations, particularly those involving the growth of density
inhomogeneities in the Universe since the growth modes in Ω ~ 1 Universes and
those in Ω ~ 0 Universes are particularly easy to calculate.
One final interesting result is that, if Ω 0 > 1, then the radius of maximum expansion,
i.e. the point at which Η = 0, is given by the condition:
(3.45)
1+ Ω z = 0
R
max
R
0
0
max
Ω0
=
Ω −1
0
We gave above the solution for matter-dominated Ω = 1 (section 3.31). In the case of
a general Ω, parametric solutions may be obtained for R(τ).
It helps to define "natural" units for R and τ as follows in terms of a quantity M
τ 3
η =
2GM c
R 2
ζ =
2GM Ac
4π M = ρ R A
3
3 3
Then the basic Friedmann equation (3.1) becomes
2
c ⎛ dζ
⎞
⎜ ⎟
2
A ⎝ dη
⎠
⎛ dζ
⎞
⎜ ⎟
⎝ dη
⎠
2
2
1 c
=
ζ A
1
= − k
ζ
ζ dζ
= dη
1−
kζ
2
2
kc
−
A
We now introduce a secondary variable, u, such that
2
ζ = S ( u) ⇒ dζ
= 2S ( u) C ( u)
du
k
2
2
k
k

Here S k (u) and C k (u) are analogous to S k (ω) and are sin, sinh, cos, cosh etc.
depending on the value of k. Then
2
Sk
( u)
C
dη
= 2
1−
kS
= 2S
2
k
( u)
du
= k(1
− C
k
⎛ S
η = k⎜u
−
⎝
2
k
( u)
du
( u)
(2u))
du
k
k
(2u)
⎞
⎟
2 ⎠
Now making a final substitution of Θ = 2u, we get parametric solutions
k
η = ( Θ−Sk
( Θ))
2
k
ζ = ( −Ck
2 1 ( Θ))
So, we finally have for R and τ:
(3.46)
kGM
R =
2
Ac
kGM
τ =
3
c
{ 1−
C ( Θ)
}
k
{ Θ − S ( Θ)
}
k
Note that dτ = AR/c dΘ. The parameter Θ is (as noted above) called the "conformal
time" (although we will not make great use of it), and dω is proportional to dΘ for a
light ray.
3.6.2 Determining S k (ω) in the matter-dominated case
We saw in the previous chapter how the effective distance D enters into almost all the
relations between the intrinsic and observed properties of objects. Recall that D = R 0
S k (ω). Thus we need to know S k (ω) or rather S k (z) since the redshift z is the only real
observable related to distance.
We want to get ω(z), which we can get by integrating dω/dz, which in turn can be
obtained by recognizing that:
dω
dz
dω
dt
= × ×
dt dR
We had from equation (3.21):
dR
dz
(3.47)
R&
R
2
2
= H ( 1+Ω
z)
2 0
0
0

Now, using the definition of z,
R0 R0
R = ⇒ dR = −
( 1+
z) ( 1+
z)
2
dz
We get:
(3.48)
dz
H z z
dτ =− + 2
+
0
( 1 ) ( 1 Ω
0
This equation may be integrated to yield τ(z), the age of the Universe as a function of
z and Ω 0 .
Since light propagates along null-geodesics with ds = 0, we have from the metric that
light following a radial trajectory towards us has:
d c
(3.49) Rdω
=− cdτ
⇒ ω
dτ
=− R ( τ )
Thus we can get the change in comoving radial distance along a light ray with
redshift, z,
(3.50) R d ω c 1
0 =−
dz H ( 1+ z) 1+
Ω z
0 0
This expression can be integrated to give ω(z). Then, since we know the functional
form of S k (ω) (= A sin ω/A or whatever, see 2.7) and we know A in terms of Ω 0 or q 0 ,
we can find S k (ω) in terms of the observable z and q 0 .
The quantity R o S k (ω) which we identified in Chapter 2 as the effective distance D, is
often written in terms of a function Z q (z) as follows:
c
(3.51) D= R S
H Z z
0 k
( ω ) =
q
( )
0
Recall that at very low redshifts, d = cz/H o , so Z q (z), a function of q 0 , is a kind of
effective redshift which makes the expressions for D look familiar.
For the k = 0 Euclidean case ( Ω = 1), we can integrate (3.32) to give ω(z), and hence
since S k (ω) = ω, we have
(3.52)
0
2 −1/
2
c ⎡ 1 ⎤
D = R0ω
= ⎢ = 2(1 − (1 + )
1/ 2 ⎥
z
H
0 ⎣ ( 1+
z)
⎦
z
)
The actual derivation of Z q (z) in the general case is tedious and extremely messy.
However the final result comes out as:

1
(3.53) z)
= { q z + ( q −1)(
1+
2q
z −1}
Z q
(
2
0 0
0
q0
(1 + z)
This reduces in three simple cases to:
q
0
= 0
k
= −1
Z
q
z(1
+ 0.5z)
( z)
=
(1 + z)
(3.54)
q
q
0
0
= 0.5
= 1
k = 0
k = 1
⎛
Z
q
( z)
= 2⎜1
−
⎝
z
Z
q
( z)
=
1+
z
1 ⎞
⎟
1+
z ⎠
These expressions are very useful. The q 0 = 1 case is noteworthy primarily because it
produces a luminosity distance D L (see section 2.26) that is proportional to z. Thus the
magnitude-redshift relation (the classical Hubble diagram) is a straight line in this
cosmology.
It should be stressed that all of the foregoing relations apply only to a pressureless
matter-dominated Universe, like ours at the present epoch, because they were based
on the particular form of R(t) that is produced in such a model.
The various relations above are sufficient to derive a number of interesting quantities.
For instance, one often encounters the comoving volume element, dV c /dz. This
describes the incremental increase in comoving volume (i.e. in which galaxies have
constant number density assuming that their numbers are conserved) with redshift and
is required for instance when calculating the expected number of faint galaxies seen
within a survey of a given surface area on the sky since dN/dz is proportional to
dV c /dz.
Consider a cone of solid angle dΞ. If this projects to a physical area A on a sphere of
constant radius at a redshift z, then
(3.55)
dV
dz
dV
dz
c
c
2 dw
2
= A( 1+
z)
with A = dΞDθ
dz
⎛
=
⎜
⎝
c
H
0
⎞
⎟
⎠
3
Z
(1 + z)
2
q
( z)
dΞ
1+ Ω z
0
3.7 The interrelation between the curvature, the density and the expansion
rate
Remember that all solutions to the Friedmann equation must have the same
relationship (3.17) between the curvature and the expansion rate and the density:

k
( RA)
2
( Ω −1)
=
2
⎛ c ⎞
⎜ ⎟
⎝ H ⎠
with
Ω= 8 πG
ρ
3H
2
Here ρ is the density, be it that of normal matter density, radiation density or false
vacuum energy density, or the sum of all three if appropriate. Indeed, in the general
case we have:
k
( RA)
2
( Ωtot
=
⎛ c
⎜
⎝ H
−1)
2
⎞
⎟
⎠
with
Ω
tot
8πG
=
2
3H
∑
i
ρ
i
This inter-relationship, which can be written in several different ways, including that
of the original Friedmann equation (3.1), is the heart of the General Relativistic
Friedmann-type models.
Having been brought up since the cradle on Newtonian ideas, we tend to think of
densities and velocities as the key quantities to focus on, with gravity acting on the
density to decellerate the Universe and so on. We think intuitively of Newtonian
concepts such as escape speeds, binding energies, etc. For most of us, the curvature
is then added on as an uncomfortable afterthought. This Newtonian approach is a
useful and pragmatic way to approach questions such as the appearance of distant
objects and the physical evolution of the contents of the Universe. Indeed, as we saw,
we can derive perfectly correct expressions describing the expansion of R(τ).
However, in a fundamental way it is really the other way around. The one quantity
that can never change during the expansion of a homogeneous Universe is the
comoving curvature k/A 2 . It is the curvature that describes the Universe. Once the
curvature is defined, the density at any epoch follows from the expansion rate and
vice versa from (3.17). As the Universe expands, the equation of state tells us how the
density will change (e.g. as R -3 , R -4 and R 0 in the three cases above) and, as if by
magic (but not really of course), the expansion rate is decelerated or accelerated in the
ways that we calculated above to compensate.
In the evolution of the Universe, the effective equation of state (i.e. that of the
dominant density component) has changed and may change again. Each of these
changes brings a different form of R(τ) and of ρ(R) with smooth transitions in each so
as to preserve the correct relationship between them.
However, the topology of the Universe, represented in the homogeneous case by the
comoving curvature scalar k/A 2 , remains constant.

3.8 The Big Bang
We have now calculated R(τ) for three generic models for a pressure-less expanding
Universe. Because gravity acts on matter to decelerate it, it should not be surprising
that these three all have R = 0 at some finite time in the past. As noted above, any
curvature present today will be less important at earlier epochs, as Ω → 1. We would
expect radiation to become dynamically dominant at some earlier point in time, but it
too has R = 0 at some point in the past. Of course, R = 0 implies a singularity and it is
likely that some new physics must be introduced at such early times (as we’ll see
below) but a compact, rapidly expanding state for the Universe is a strong prediction
of the Friedmann models.
As noted above, even a non-zero false vacuum density today would, if constant, be
unimportant before some earlier epoch.
The idea that the Universe was once in an extremely compressed state is the
fundamental feature of the Friedmann models. This compressed but rapidly expanding
state is known as the Big Bang. The term was introduced, in a derisory way, by Fred
Hoyle during a radio broadcast.
Chapter 3: Key points
1. The correct Friedmann equation in & R (from GR) can in fact
be generated from the Newtonian && R equation if we set the density to
be the so-called "active density".
2. The dynamics of the Universe depend on the equation of state
of the matter-radiation in the Universe as this determines how the
density changes as the Universe expands. Our matter-dominated
Universe was once radiation dominated.
3. The density parameter Ω determines in fundamnetal way
both the dynamics and the curvature of the Universe.
4. For a given equation of state, the term in the Robertson-
Walker metric mat be written in terms of (c/H 0 )Z q (z), where q is a
deceleration parameter.
5. A false vacuum energy density is equivalent to a non-zero Λ
term in the Friedmann equation.