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a
1 x
aâˆ’1 +
a2 = H0 1 âˆ’
Ë™ â‡’ t=
2 2
0a dx
0
H0 1âˆ’ +
0 4
0x
0

This integral is a little more difficult than those considered earlier, but it can be
made tractable by the substitution y2 = x3 0 / 1âˆ’ 0 , which yields
âˆš3
0 / 1âˆ’
a
2 dy
0
H0 t =
1 Â± y2
3 00

where the plus sign in the integrand corresponds to the case 0 > 0 and the
minus sign to 0 < 0. This may now be integrated easily to give

âŽ§
âŽªsinhâˆ’1
âŽ¨ / 1âˆ’
a3 if >0
2 0 0 0
H0 t = (15.36)
âŽª âˆ’1
3 / 1âˆ’
a3 if <0
0 0 0 0
407
15.6 Analytical cosmological models

which may be inverted to give a t in each case. One can also obtain analytic
expressions for H t and m t (see Exercise 15.24) and thus for m t and
t.

The de Sitter model
The de Sitter model is a particular special case of a Lemaitre model defined
by the cosmological parameters m 0 = 0, r 0 = 0 and 0 = 1. This model
is therefore spatially flat (k = 0) but is not a true cosmological model in the
strictest sense, since it assumes that the matter and radiation densities are zero.
Nevertheless, it is interesting in its own right both for historical reasons and
because of its close connection with the theory of inflation (see Section 16.1).
For the de Sitter model, the cosmological field equation (15.13) reads
2
Ë™
a
= H0
2
a
which immediate tells us that the Hubble parameter H t is a constant and the
normalised scale factor increases exponentially as

a t = exp H0 t âˆ’ t0 = exp /3c t âˆ’ t0

where, in the second equality, we have expressed the solution in terms of the
cosmological constant . Thus, the de Sitter model has no big-bang singularity
at a finite time in the past.

Einsteinâ€™s static universe
All the cosmological models that we have constructed so far are evolving cosmolo-
gies. We know now, of course, that the universe is expanding and so there is no
conflict with the field equations. Nevertheless, it is interesting historically to look
at Einsteinâ€™s static model of the universe. Einstein derived his field equations
well before the discovery of the expansion of the universe and he was worried
that he could not find static cosmological solutions. He therefore introduced the
cosmological constant with the sole purpose of constructing static solutions.
For > 0, we seek a solution to the field equations in which the universe is
static, i.e. a = a = 0. In this case, the Hubble parameter H is zero always, and
Â¨Ë™
so the dimensional densities in (15.5) are formally infinite. It is more convenient
therefore to work with the field equations in their original forms (14.36). We see
immediately that we require

c2 k
4 G m0= c = 2 2
R0
408 Cosmological models

In fact, the first equality can be more succinctly written as m 0 = 2 0 . Since
is positive we thus require k = 1, and so the universe has positive spatial
curvature.
How well did Einsteinâ€™s static universe fit with cosmological observations of
the time? The mean matter density of the universe is still a matter of great debate,
but recent cosmological observations suggest that

â‰ˆ 3 Ã— 10âˆ’27 kg mâˆ’3
m0

In Einsteinâ€™s time, this value was estimated only to within about two orders
of magnitude. Nevertheless, adopting the above value of m 0 we find that the
scale factor is R0 â‰ˆ 2 Ã— 1026 m â‰ˆ 6000 Mpc, which is more than sufficient for the
closed spatial geometry to be large enough to encompass the observable universe.
Also = 1/R2 = 2 5 Ã— 10âˆ’53 mâˆ’2 , which is small enough to evade the limits on
0
â‰¤ 10âˆ’46 mâˆ’2 . Thus the Einstein static
from Solar System experiments
universe was not immediately and obviously wrong.
However, aside from the fact that the model disagreed with later observations
indicating an expanding universe, it has the theoretically undesirable feature of
being unstable. The cosmological constant must be fine-tuned to match the density
of the universe. Thus, if we add or subtract one proton from this universe, or
convert some matter into radiation, we will disturb the finely tuned balance
between gravity and the cosmological constant and the universe will begin to
expand or contract.

15.7 Look-back time and the age of the universe
Since the cosmological model may be fixed by specifying the values of the four
(present-day) cosmological parameters H0 , m 0 r 0 and 0 , it is possible to
use these quantities to determine other useful derived cosmological parameters.
In this section we consider the look-back time and the age of the universe.
In Chapter 14, we showed that if a comoving particle (galaxy) emitted a photon
at cosmic time t that is received by an observer at t = t0 then the â€˜look-back timeâ€™
t0 âˆ’ t is given as a function of the photonâ€™s redshift by
z dÂ¯z
t0 âˆ’ t = (15.37)
1+z H z
Â¯ Â¯
0

âˆ’1
From the cosmological field equation (15.13), on noting that a = R/R0 = 1+z
we obtain the useful result

H 2 z = H0 1+z 3 + 1+z 4 + 0+ 1+z
2 2
(15.38)
m0 r0 k0
409
15.7 Look-back time and the age of the universe

Thus, the look-back time to a comoving object with redshift z is given by

z
1 dÂ¯
z
t0 âˆ’ t =
H0 1+z
Â¯ 1+z 3 +
Â¯ 1+z 4 +
Â¯ 0+ 1+z
Â¯
0 2
m0 r0 k0

We note that the differential form of this relation is perhaps more useful since
one is often interested simply in the cosmic time interval dt corresponding to an
interval dz in redshift. In any case, a more convenient form of the integral for
evaluation is obtained by making the substitution x = z + 1 âˆ’1 , which yields

1
1 x dx
t0 âˆ’ t = (15.39)
H0 âˆ’1
m 0x + r 0+ x4 +
1+z x2
0 k0

Assuming r 0 = 0 (which is a reasonable approximation for our universe), in
Figure 15.7 we plot H0 t âˆ’ t0 , the look-back time in units of the Hubble time,
as a function of redshift for several values of m 0 and 0.
In any cosmological model with a big-bang origin, an extremely important
quantity is the age of the universe, i.e. the cosmic time interval between the point
when a t = 0 and the present epoch t = t0 . Since z â†’ at the big bang, we
may immediately obtain an expression for the age of the universe in such a model
1

(0.3, 0.7)

(0.3, 0)

(1, 0)
H0(t â€“ t0)

0.5
0

0.1 1 10
z

Figure 15.7 The variation in look-back time, in units of the Hubble time, as
a function of redshift z for several sets of values 0 as indicated,
m0
assuming that r 0 is negligible.
410 Cosmological models

Table 15.1 The age of the universe in Gyr for various
cosmological models (with r 0 = 0)

H0 in km sâˆ’1 Mpcâˆ’1

50 70 90
m0 0

1.0 0.0 13.1 9.3 7.2
0.3 0.0 15.8 11.3 8.8
0.3 0.7 18.9 13.5 10.5

by letting z â†’ in (15.39), so that the lower limit of the integral equals zero.
Since the resulting integral is dimensionless, we can write
1
t0 = f m0 r0 0
H0
where f is the value of the integral, which is typically a number of order unity.
The age of the universe is therefore the Hubble time multiplied by a number of
order unity. For general values of the density parameters m 0 , r 0 and 0,
it is not possible to perform the integral analytically and so one has to resort to
numerical integration. Table 15.1 lists the age of the universe t0 for the same
values of m 0 and 0 as considered in Figure 15.7. It is interesting to compare
these values with estimates of the ages of the oldest stars in globular clusters,
tstars â‰ˆ 11 5 Â± 1 3 Gyr
where the uncertainty is dominated by uncertainties in the theory of stellar evolu-
tion. Clearly, one requires t0 > tstars for a viable cosmology!
It is worth noting that, in our discussion of analytical cosmological models in
the previous section, we have already performed (a generalised version of) the
relevant integral required to calculate the corresponding age of the universe in each
case. Thus, for each model with a big-bang origin for which we have calculated an
analytical form for a t or t a , the corresponding age of the universe is obtained
simply by setting t = t0 and a = 1. For example, from (15.28), the age of an
Einsteinâ€“de-Sitter universe is simply t0 = 2/ 3H0 . Similarly, from (15.36), the
age of a spatially flat matter-only Lemaitre model with 0 > 0 is given by

2 tanhâˆ’1
2 âˆ’1 0
0
t0 = =
sinh
1âˆ’ 3H0
3H0 0
0 0

where, in the second equality, we have rewritten the result in a more useful form
involving 0 , using standard formulae for inverse hyperbolic trigonometric
functions.
411
15.8 The distanceâ€“redshift relation

15.8 The distanceâ€“redshift relation
We may also obtain a general expression for the comoving -coordinate of a
galaxy emitting a photon at time t that is received at time t0 with redshift z. This
is given by
Â¯
t0 c d t c z dÂ¯ z
= =
Â¯ Â¯
t Rt R0 0 H z
We may now subsitute for H z using the expression (15.38) derived in the
previous section. Thus the -coordinate of a comoving object with redshift z is
given by
z
c dÂ¯
z
z= (15.40)
R0 H0 1+z
Â¯ 3+ 1+z
Â¯ 4+
0+ 1+z
Â¯
0 2
m0 r0 k0

Once again, the differential form of this result is perhaps more useful, since one
is often interested in the comoving coordinate interval d corresponding to an
interval dz in redshift. As before, a simpler form for the integral is obtained by
making the substitution x = 1 + z âˆ’1 , which yields
Â¯

1
c dx
z= (15.41)
R0 H0 âˆ’1
m 0x + r 0+ x4 +
1+z x2
0 k0

From (14.29) and (14.31), the corresponding luminosity distance dL z and angu-
lar diameter distance dA z to the object are given by
R0
dL z = R0 1 + z S dA z =
and
z S z
1+z
where S is given by (14.12), whereas the proper distance to the object is simply
d z = R0 S z . It is useful to introduce the notation z = cE z / R0 H0 , so
that E z denotes the integral in (15.41). Using the expression (15.10) to obtain
k 0 , one can then write

âˆ’1/2 S =0
for
Ez
c k0 k0 k0
=
R0 S z
=0
H0 E z for k0

which allows simple direct evaluation of dL z and dA z in each case.
As was the case in the previous section, for general values of m 0 , r 0 and
0 it is not possible to perform the integral (15.41) analytically and so one
has to resort to numerical integration. Figure 15.8 shows plots of dimensionless
luminosity distance c/H0 âˆ’1 dL z (top panel) and dimensionless angular diame-
ter distance c/H0 âˆ’1 dA z (bottom panel) for various values of m 0 and 0,
assuming that r 0 is negligible; the solid, broken and dotted lines correspond to
412 Cosmological models

15
(0, 1) (0, 0) (0.3, 0)
(0.3, 0.7)

(1, 0)
10
(c/H0)â€“1dL(z)

5

(4, 0)
0

0 2 4 6 8 10
z
1

(0, 1)
(c/H0)â€“1dA(z)

(0, 0)
0.5

(0.3, 0)

(0.3, 0.7)
(1, 0)
(4, 0)
0

0 2 4 6 8 10
z

Figure 15.8 The variation in dimensionless luminosity distance (top panel) and
dimensionless angular diameter distance (bottom panel) as functions of redshift,
for different sets of values 0 as indicated, assuming that r 0 is
m0
negligible. The solid, broken and dotted lines correspond to spatially flat, open
and closed models respectively.

spatially flat, open and closed models respectively. In particular, it is worth noting
that, for the models with a non-zero matter density, the angular diameter distance
has a maximum at some finite value of the redshift z = zâˆ— . Thus, for a source of
= /dA declines with redshift for
fixed proper length , the angular diameter
z < zâˆ— , as one might naively expect, but then increases with redshift for z > zâˆ— .
A very-high-redshift galaxy (if such a thing existed) would therefore cast a large,
413
15.9 The volumeâ€“redshift relation

but dim, ghostly image on the sky. The physical reason for this is that the light
from a distant object was emitted when the universe was much younger than it is
now â€“ the object was close to us when the light was emitted. This, coupled with
gravitational focussing of the light rays by the intervening matter in the universe,
means that the galaxy looks big!
The integral (15.41) can, in fact, be evaluated analytically in some simple cases.
As an example, consider the Einsteinâ€“de-Sitter (EdS) model ( m 0 = 1, r 0 = 0,
0 = 0). In this case, we find that

1 2c
c dx âˆ’1/2
z= âˆš= 1âˆ’ 1+z
R0 H0 x R0 H0
âˆ’1
1+z

Thus, the luminosity distance in the EdS model is given as a function of z by
2c âˆ’1/2
dL z = 1+z 1âˆ’ 1+z
H0
and the angular diameter distance by
2c 1 âˆ’1/2
dA z = 1âˆ’ 1+z
H0 1 + z
Note that, in this case, dA z has a maximum at a redshift z = 5/4.
The relations between redshift and luminosity distance (angular diameter
distance), form the basis of observational tests of the geometry of the universe. All
one needs is a standard candle (for application of the luminosity-distanceâ€“redshift
relation) or a standard ruler (for application of the angular-diameter-distanceâ€“
redshift relation). Comparison with the predicted relations shown in Figure 15.8
can then fix the values of m 0 and 0 . Unfortunately, standard candles and
standard rulers are hard to find in the universe! Nevertheless, in recent years
there has been remarkable progress, using distant Type Ia supernovae as standard
candles and anisotropies in the cosmic microwave background radiation as a
standard ruler. The results of these observations suggest that we live in a spatially
flat universe with m 0 â‰ˆ 0 3 and 0 â‰ˆ 0 7.

15.9 The volumeâ€“redshift relation
In Section 14.10 we found that, at the present cosmic time t0 , the proper volume
of the region of space lying in the infinitesmial coordinate range â†’ + d and
subtending an infinitesmial solid angle d = sin d d at the observer is

cR2 S 2 z
dV0 = 0 (15.42)
dz d
Hz
414 Cosmological models

the corresponding volume of this region at a redshift z being given by dV z =
dV0 / 1 + z 3 . We may now express dV0 in terms of the cosmological parameters
H0 , m 0 , r 0 and 0 . Using the expressions (15.40), (15.38) and (15.10) for
z , H z and k respectively, we find immediately that

âˆ’1 âˆ’1 S 2 =0
3
for
cH0 Ez
k0 k0 k0
dV0 = (15.43)
k0=0
E2 z
hz for

where we have defined the new function

Hz
hz â‰¡ = 1+z 3 + 1+z 4 + 0+ 1+z 2
m0 r0 k0
H0

z
and E z â‰¡ 0 dÂ¯ /h z is the function defined in the previous section.
zÂ¯
For general values of m 0 , r 0 and 0 , one must once again resort to
numerical integration to obtain dV0 . In Figure 15.9, we plot the dimensionless
differential comoving volume element c/H0 âˆ’3 dV0 / dz d as a function of
r 0 = 0. In
redshift z for several values of m 0 and 0 , assuming that
0 = 03 07 ,
particular, we note that, in the currently favoured case m0
we may explore a large comoving volume by observing objects in the redshift
range z = 2â€“3.
1

(0.3, 0)

(0.3, 0.7)
(c/H0)â€“3 dV0 / (dz dâ„¦)

0.1

(1, 0)
0.01
10â€“3

0.1 1 10
z

Figure 15.9 The variation in the dimensionless differential comoving volume
element as a function of redshift z for several sets of values 0 as
m0
indicated, assuming that r 0 is negligible.
415
15.10 Evolution of the density parameters

15.10 Evolution of the density parameters
For the majority of our discussion so far, we have concentrated on exploring
cosmological models with properties determined by fixing the values of the
present-day densities m 0 , r 0 and 0 . From the definition (15.5), however, it
is clear that each density is, in general, a function of cosmic time t. It is therefore
of interest to investigate the evolution of these densities as the universe expands.
From (15.5) we have
Ë™
8G Ë™i= 8 G 2H
t= â‡’ Ë™i âˆ’ (15.44)
t
i i i
3H 2 t 3H 2 H
where the label i denotes â€˜mâ€™, â€˜râ€™ or â€˜ â€™ and the dots denote differentiation with
respect to cosmic time t. From the equation of motion (14.39) for a cosmological
fluid, however, we have
Ë™ i = âˆ’3 1 + wi H i

Ë™
where we have written H = R/R, and wi = pi / 2 is the equation-of-state
ic
parameter. Thus (15.44) becomes
Ë™
2H
Ë™i=âˆ’ i H 3 1 + wi + (15.45)
H2
where we have taken a factor of H outside the brackets for later convenience. We
Ë™
now need an expression for H, which is given by
2
Ë™ Â¨ Ë™ Â¨
d R R R R
Ë™
H= =âˆ’ = âˆ’ H2
dt R R R R
and so we may write
Ë™ Â¨
H RR
= âˆ’1 = âˆ’ q+1
Ë™
H2 R2
where q is the deceleration parameter. Substituting this result into (15.45) and
using the expression (15.14) for q, we finally obtain the neat relation
Ë™i= m +2 r âˆ’2 âˆ’ 1 âˆ’ 3wi
iH

Setting wi = 0, and âˆ’1 respectively for matter (dust), radiation and the vacuum,
1
3
we thus obtain
Ë™m= m âˆ’1 +2 r âˆ’2
mH

Ë™r= m +2 r âˆ’1 âˆ’2 (15.46)
rH

Ë™ = m +2 r âˆ’2 âˆ’1
H
416 Cosmological models

By dividing these equations by one another, we may remove the dependence on
the Hubble parameter H and the cosmic time t and hence obtain a set of coupled
first-order differential equations in the variables m , r and alone. Therefore,
given some general point in this parameter space, these equations define a unique
trajectory that passes through this point. As an illustration, let us consider the
case in which r = 0. Dividing the remaining two equations then gives

m âˆ’2 âˆ’1
d
=
m âˆ’1 âˆ’2
d m m

which defines a set of trajectories (or â€˜flow linesâ€™) in the -plane. This
m
equation also highlights the significance of the points (1, 0) and (0, 1) in this plane,
which act as â€˜attractorsâ€™ for the trajectories. This is illustrated in Figure 15.10,
which shows a set of trajectories for various cosmological models. Since any
general point in the plane defines a unique trajectory passing through that point, it
is convenient to specify each trajectory by the present-day values m 0 and 0
(although one could equally well use the values at any other cosmic time). In
the left-hand panel, we plot trajectories passing through m 0 = 0 3 and 0=
0=
01 02 1 1, and in the right-hand panel the trajectories pass through
0 7 and m 0 = 0 1 0 2 1 1. We see that the trajectories all start at (1, 0),
which is an unstable fixed point, and converge on (0, 1), which is a stable fixed
point.
1.5

1.5

â„¦m, 0 = 0.3 â„¦Î›, 0 = 0.7
â„¦Î›, 0 = 0.1, 0.2, . . . , 1.1 â„¦m, 0 = 0.1, 0.2, . . . , 1.1
1

1
â„¦Î›

â„¦Î›
0.5

0.5
0

0

0 0.5 1 1.5 0 0.5 1 1.5
â„¦m â„¦m

Figure 15.10 Evolution of the density parameters and for various
m
m 0 = 0 3 and 0=
cosmological models passing through the points
0 = 0 7 and m0=01 02
01 02 1 1 (left-hand panel) and 11
(right-hand panel).
417
15.11 Evolution of the spatial curvature

It is worth noting the profound effect of a non-zero cosmological constant on
the evolution of the density parameters. In the case = 0, any slight deviation
from m = 1 in the early universe results in a rapid evolution away from the point
(1, 0) along the m -axis, tending to (0, 0) for an open universe and to 0 for
a closed one. If > 0, however, the trajectory is â€˜refocussedâ€™ and tends to the
spatially flat de Sitter case (0, 1). Indeed, for a wide range of initial conditions,
by the time the matter density has reached m â‰ˆ 0 3 the universe is close to
spatially flat.

15.11 Evolution of the spatial curvature
We may investigate directly the behaviour of the spatial curvature as the universe
expands by determining the evolution of the curvature density parameter
c2 k
= 1âˆ’ mâˆ’ râˆ’ =âˆ’ 2 2 (15.47)
k
HR
Differentiating the final expression on the right-hand side with respect to cosmic
time, or combining the derivatives (15.46), one quickly finds that

Ë™k=2 = m +2 r âˆ’2 (15.48)
k Hq kH

= 0 then the quantity
where q is the deceleration parameter. We observe that if
in parentheses is always positive. Thus, in this case, if k differs slightly from
zero at some early cosmic time then the spatial curvature rapidly evolves away
from the spatially flat case. In particular, k â†’ 1 in the open case and k â†’ âˆ’
in the closed case. The presence of a positive cosmological constant, however,
changes this behaviour completely. In this case, at some finite cosmic time the
2 term in (15.48) will dominate the matter and radiation terms, with the result
that k is â€˜refocussedâ€™ back to k = 0.
We may in fact obtain an analytic expression for the spatial curvature as a
function of redshift z, in terms of the present-day values of the density parameters.
Substituting for c2 k from (15.47) evaluated at t = t0 , and noting that R0 /R = 1+z,
we obtain the useful general formula
2
H0 1 + z
z= k0
k
Hz
Using our expression (15.38) for H z then gives

k0
z= âˆ’2 +
k
1+z + 1+z 2 + 1+z
m0 r0 0 k0
418 Cosmological models

In particular, we see that (apart from models with only vacuum energy), even if
the present-day value k 0 differs greatly from zero, at very high redshift (i.e.
in the distant past) k z must have differed by only a tiny amount from zero.
Since today we measure the value k 0 to be (conservatively) in the range âˆ’0 5
to 0.5, this means that at very early epochs k must have been very finely tuned
to near zero. This tuning of the initial conditions of the expansion is called the
flatness problem and has no solution within standard cosmological models. From
our above discussion, however, the presence of a positive cosmological constant
goes some way to explaining why the universe is close to spatially flat at the
present epoch.

15.12 The particle horizon, event horizon and Hubble distance
Thus far, we have considered the evolution of the entire spatial part of the
FRW geometry. It is, however, interesting to consider the extent of the region
â€˜accessibleâ€™ (via light signals) to some comoving observer at a given cosmic
time t.

Particle horizon
Let us consider a comoving observer O situated (without loss of generality) at
= 0. Suppose further that a second comoving observer E has coordinate 1 and
emits a photon at cosmic time t1 , which reaches O at time t. Assuming light to
be the fastest possible signal, the only signals emitted at time t1 that O receives
by the time t are from radial coordinates < 1 .
The comoving coordinate 1 of the emitter E is determined by
Â¯
t dt
1=c (15.49)
Â¯
Rt
t1

If the integral on the right-hand side diverges as t1 â†’ 0 then 1 can be made as
large as we please by taking t1 sufficiently small. Thus, in this case, in principle
it is possible to receive signals emitted at sufficiently early epochs from any
comoving particle (such as a typical galaxy). If, however, the integral converges
as t1 â†’ 0 then 1 can never exceed a certain value for a given t. In this case our
vision of the universe is limited by a particle horizon. At any given cosmic time
t, the -coordinate of the particle horizon is given by

Â¯
t Rt
dt dR
t =c =c (15.50)
Ë™
Â¯
p
Rt RR
0 0

where in the second equality we have rewritten the expression as an integral over
R. The corresponding proper distance to the particle horizon is dp t = R t p t .
419
15.12 The particle horizon, event horizon and Hubble distance

Ë™
We see that expression (15.50) will be finite if RR âˆ¼ R with < 1, which is
Â¨
equivalent to the condition R < 0. Hence, any universe for which the expansion
has been continually decelerating up to the cosmic time t will have a finite particle
horizon at that time. Clearly, this includes all the Friedmann models that we
discussed earlier, but particle horizons also occur in other cosmological models,
for example in the spatially flat Lemaitre model with m 0 â‰ˆ 0 3 and 0â‰ˆ0 7
that seems to provide a reasonable description of our universe.
On differentiating (15.50) with respect to t, we have d p /dt = c/R t , which is
always greater than zero. Thus, the particle horizon of a comoving observer grows
as the cosmic time t increases, and so parts of the universe that were not in view
previously must gradually come into view. This does not mean, however, that a
galaxy that was not visible at one instant suddenly appears in the sky a moment
later! To understand this, we note that if the universe has a big-bang origin then
we have R t1 â†’ 0 as t1 â†’ 0, and so z â†’ . Thus, the particle horizon at any
given cosmic time is the surface of infinite redshift, beyond which we cannot see.
If the particle horizon grew to encompass a galaxy, the galaxy would therefore
appear at first with an infinite redshift, which would gradually reduce as more
cosmic time passed. Hence the galaxy would not simply â€˜popâ€™ into view.7
In fact, we can obtain explicit expressions for the particle horizon in some
cosmological models. For example, a matter-dominated model at early epochs
obeys R t /R0 = t/t0 2/3 , whereas a radiation-dominated model at early epochs
obeys R t /R0 = t/t0 1/2 . Substituting these expressions into (15.50) gives the
proper distance to the particle horizon at cosmic time t as
dp t = 3ct dp t = 2ct
These proper distances are larger than ct because the universe has expanded while
the photon has been travelling. Alternatively, if one has an analytic expression
for z for some cosmological model then the corresponding expression for p
may be obtained simply by letting z â†’ .
The existence of particle horizons for the common cosmological models illus-
trates the horizon problem, i.e. how do vastly separated regions display the
same physical characteristics (e.g. the nearly uniform temperature of the cosmic
microwave background) when, according to standard cosmological models, these
regions could never have been in causal contact? This problem, like the flatness
problem, is a serious challenge to standard cosmology that can only be resolved
by invoking the theory of inflation (see Chapter 16).

7
In practice, our view of the universe is not limited by our particle horizon but by the epoch of recombination,
which occurred at zrec â‰ˆ 1500 (long before the formation of any galaxies). Prior to this epoch, the universe
was ionised and photons were frequently scattered by the free electrons, whereas after this point electrons and
protons (and neutrons) combined to form atoms and the photons were able to propagate freely. This surface
of last scattering is therefore the effective limit of our observable universe.
420 Cosmological models

The horizon problem can be illustrated by a simple example. Consider a galaxy
at a proper distance of 109 light years away from us. Since the age of the universe
is âˆ¼ 1 5 Ã— 1010 years, there has been sufficient time to exchange about 15 light
signals with the galaxy. At earlier times, when the scale factor R was smaller,
everything was closer together and so we might have naively expected that this
would improve causal contact. In a continuously decelerating universe, however,
it makes the problem worse. At, for example, the epoch of recombination (when
the cosmic microwave background photons were emitted) the redshift z was
approximately 1000, so R trec /R0 â‰ˆ 10âˆ’3 and the proper distance to the â€˜galaxyâ€™
is 106 light years.8 If we assume, for simplicity, that after trec the expansion
followed a matter-dominated Einsteinâ€“de-Sitter universe, then
2/3
trec Rrec
= 10âˆ’3
=
t0 R0
and so trec = 1 5 Ã— 105 5 years. However, assuming that prior to trec the expansion
followed a radiation-dominated Einsteinâ€“de-Sitter model, the proper distance to
the particle (causal) horizon is 2ctrec = 3 Ã— 105 5 light years. Thus, by trec â€˜weâ€™
could not have exchanged even one light signal with the other â€˜galaxyâ€™.

Event horizon
Although our particle horizon grows as the cosmic time t increases, in some
cosmological models there could be events that we may never see (or, conversely,
never influence). Returning to our expression (15.49), we see that if the integral
on the right-hand side diverges as t â†’ (or the time at which R equals zero
again), then it will be possible to receive light signals from any event. However,
if the integral instead converges for large t then, for light signals emitted at t1 , we
will only ever receive those from events for which the -coordinate is less than

tmax dt
t1 = c
e
Rt
t1

where tmax is either infinity or the time of the big crunch (i.e. R tmax = 0). This
is called the event horizon. By symmetry, e t0 is the maximum -coordinate
that can be reached by a light signal sent by us today.

Hubble distance
From our discussion in Section 15.7, the elapsed cosmic time t since the big
bang is, in general, of the order H âˆ’1 t , which is known as the Hubble time and

8
In reality the galaxy would not yet have formed, but this does not affect the main point of the argument.
421
Exercises

provides a characteristic time scale for the expansion of the universe. In a similar
way, at a cosmic time t one can define the Hubble distance

dH t = cH âˆ’1 t

which provides a characteristic length scale for the universe. We may also define
the comoving Hubble distance
dH t c c
t= = = (15.51)
Ë™
H
Rt HtRt Rt
Ë™
where in the last equality we have used the fact that H = R/R. The above
expression simply gives the -coordinate corresponding to the Hubble distance.
The Hubble distance dH t corresponds to the typical length scale (at cosmic
time t) over which physical processes in the universe operate coherently. It is also
the length scale at which general-relativistic effects become important; indeed,
on length scales much less than dH t , Newtonian theory is often sufficient to
describe the effects of gravitation. From our discussion above, we further note
that the proper distance to the particle horizon for standard cosmological models
is typically
dp t âˆ¼ ct âˆ¼ cH âˆ’1 t
Thus, we see that the particle horizon in such cases is of the same order as the
Hubble distance. As a result, the Hubble distance is often described simply as the
â€˜horizonâ€™. It should be noted, however, that the particle horizon and the Hubble
distance are distinct quantities, which may differ by many orders of magnitude in
inflationary cosmologies, which we discuss in the next chapter. In particular, we
note that the particle horizon at time t depends on the entire expansion history of
the universe to that point, whereas the Hubble distance is defined instantaneously
at t. Moreover, once an object lies within an observerâ€™s particle horizon it remains
so. On the contrary, an object can be within an observerâ€™s Hubble distance at one
time, lie outside it at some later time and even come back within it at a still later
epoch.

Exercises
15.1 For blackbody radiation, the number density of photons with frequencies in the
+ d is given by
range
82
T d = 3 h /kT (E15.1)
n d
âˆ’1
ce
where T is the â€˜temperatureâ€™ of the radiation. By conserving the total number
of photons, show that the photon energy distribution of the cosmic microwave
422 Cosmological models

background (CMB) radiation retains its general blackbody form as the universe
expands. Show further that the total number density n of photons is
3
2 kB T
n T = 0 244
hc
Hence show that the present-day number density of CMB photons in the universe is
n0 â‰ˆ 4 Ã— 108 mâˆ’3 , and compare this with the present-day number density of protons.
How does this ratio vary with cosmic time?
x2
dx = 0 244 2 .
Hint: 0 x
e âˆ’1
15.2 Suppose that the present-day energy densities of radiation and matter (in the form
of dust) are r t0 c2 and m t0 c2 respectively. Show that the energy densities of
the two components were equal at a redshift zeq given by
t0
1 + zeq = m
t0
r

What assumptions underlie this result? Hence show that
3c2 m 0 H02
1 + zeq = 4
8 GaT0
where a is the reduced Stefanâ€“Boltzmann constant and T0 is the present-day temper-
ature of the cosmic microwave background. Show that for our universe zeq â‰ˆ 5000.
What was the temperature of the CMB radiation at this epoch?
15.3 Show that in the early, radiation-dominated, phase of the universe, the temperature
T of the radiation satisfies the equation
2
Ë™ 8 GaT 4
T
=
3
T

where the dot denotes differentiation with respect to the cosmic time t and a is the
reduced Stefanâ€“Boltzmann constant. Hence show that
1/4 âˆ’1/2
3c2 t
âˆ’1/2
T= â‰ˆ 1 5 Ã— 10 10
K
t
32 Ga s
and that the cosmic time at matterâ€“radiation equality is teq â‰ˆ 16 000 years.
15.4 The CMB radiation was emitted at the epoch of recombination at redshift zrec â‰ˆ
1500. Show that trec â‰ˆ 450 000 years.
15.5 Consider a cylindrical piston chamber of cross-sectional area A â€˜filledâ€™ with vacuum
energy. The piston is withdrawn a linear distance dx. Show that the energy created
by withdrawing the piston equals the work done by the vacuum, provided that

pvac = âˆ’ 2
vac c

Hence show that, in this case, the vacuum energy density is constant as the piston
is withdrawn.
423
Exercises

15.6 Show that the present-day value of the scale factor of the universe may be written as
1/2
c k
R0 =
H0 k0

What value does R0 take in a spatially flat universe?
15.7 Show that, for our universe to be spatially flat, the total density must be equivalent
to â‰ˆ 5 protons mâˆ’3 .
15.8 In the Newtonian cosmological model discussed in Exercise 14.14, show that the
total energy E of the test particle of mass m can be written as

E = 2m 1âˆ’ R2 H 2
1
m

and interpret this result physically.
15.9 Show that at all cosmic times the density parameters obey the relation

+ + + =1
m r k

15.10 In terms of the dimensionless density parameters, show that the two cosmological
field equations can be written in the forms
âˆ’3 âˆ’4 âˆ’2
H 2 = H0 + + 0+
2
m0a r0a k0a

q= +2 âˆ’2
1
m r
2

where H and q are the Hubble and deceleration parameters respectively, and
a = R/R0 is the normalised scale factor.
15.11 The conformal time variable is defined by d = c dt/R. Hence show that the
second cosmological field equation can be written as
2
da k
=âˆ’ m 0 a+ r0+ +
4 2
0a k0a
d k0

15.12 Show that the density parameter for matter, radiation or the vacuum varies with
the normalised scale factor as
2
H0
aâˆ’3 1+wi
= i0
i
H
where wi is the appropriate equation-of-state parameter.
15.13 Show that the condition for the a t -curve to have a turning point is

fa â‰¡ + 1âˆ’ m0âˆ’ a+ =0
3
0a 0 m0

In the case 0 > 0, show by evaluating the derivatives f a and f a that the
condition for f a to have a single positive root at a = aâˆ— is f aâˆ— = f aâˆ— = 0.
Show further that this root occurs at
1/3
m0
aâˆ— =
2 0
424 Cosmological models

Hence show that the values m 0 and 0 , along any dividing line in this plane
that separates those models with a turning point in the a t -curve from those
without, must satisfy

4 1âˆ’ m0âˆ’ + 27 =0
3 2
0 0
m0

15.14 Show that the substitution x = 1/3
4 reduces the final cubic equation
0/ m0
in Exercise 15.13 to
m 0 âˆ’1
3x
x3 âˆ’ + =0
4 4 m0
By using the standard formulae for the roots of a cubic, or otherwise, verify the
results (15.18â€“15.20).
Ë†
15.15 Show that, in terms of the variable t = H0 t âˆ’ t0 , the evolution of the normalised
scale factor obeys the equation
2
da âˆ’1 âˆ’2
= + + +1âˆ’ m0âˆ’ r0âˆ’
2
m 0a r 0a 0a
Ë† 0
dt
Show that, when one is integrating this equation numerically, an iterative algorithm
of the form
da
Ë†
an+1 â‰ˆ an + t
Ë†
dt n
Ë†
would not be able to propagate the solution through points for which da/dt = 0
15.16 For a k = âˆ’1 Friedmann model containing no matter or radiation, show that the
line element becomes

ds2 = c2 dt2 âˆ’ c2 t2 d + sinh2 + sin2 d
2 2 2
d

Show that this metric describes a Minkowski spacetime.
15.17 For a dust-only Friedmann model with m 0 > 1, show that

m0 m0
a= 1 âˆ’ cos t= âˆ’ sin
m 0 âˆ’1 m 0 âˆ’1
3/2
2 2H0
Hence show that the a t -curve has a maximum at

m0 m0
amax = tmax =
m 0 âˆ’1 m 0 âˆ’1
3/2
2H0
and that the age t0 of such a universe is given by
2 2 2
cosâˆ’1
m0
t0 = âˆ’1 âˆ’ m 0 âˆ’1
1/2
<
m 0 âˆ’1
3/2
2H0 3H0
m0 m0

15.18 For the Einsteinâ€“de-Sitter model, prove the following useful results:
2/3
2 1 1
t
at = Ht = = H0 1+z q0 = t=
3/2
m
6 Gt2
3t 2
t0
425
Exercises

= 1, show that
15.19 For a radiation-only Friedmann model with r0

1/2
1âˆ’ r0
1/2 1/2
a t = 2H0 1+
r0t H0 t
1/2
2 r0

Hence, for > 1, show that the a t -curve has a maximum at
r0

1/2
1/2
1
r0 r0
amax = tmax =
r 0 âˆ’1 r 0 âˆ’1
H0
and that the age t0 of such a universe is given by
1 1 1
t0 = <
1/2
r 0 +1
2H0
H0

15.20 For the spatially flat, radiation-only, Friedmann model, prove the following useful
results:
1/2
1 3
t
at = Ht = = H0 1 + z q0 = 1 t=
2
r
32 Gt2
2t
t0
15.21 For a spatially flat Friedmann model containing both matter and radiation, show
that
2 3/2
H0 t = m 0a + r 0 m 0a âˆ’ 2 r 0 + 2 r 0
1/2
2
3 m0
15.22 For a Lemaitre model containing no radiation, show that at the point of inflection
of the a t -curve the value of the normalised scale factor is
1/3
m0
aâˆ— =
2 0

and calculate aâˆ— for our universe. Show further that, in the vicinity of the point of
inflection, the scale factor obeys the equation

a2 â‰ˆ H0
Ë™ k 0 +3 0 aâˆ— + 3 a âˆ’ aâˆ—
2 2 2
0

and that this has the solution
âˆ’1/3 1/2
a t = a âˆ— + aâˆ— 1 + 1 t âˆ’ tâˆ—
2 1/2
1
sinh H0 3
k0 0 0
m0
3 4

15.23 For a spatially flat Lemaitre model containing no radiation, show that
âˆš3
0 / 1âˆ’
a
2 dy
0
H0 t =
1 Â± y2
3 0
0

Hence show that
1âˆ’ 1/3
sinh2/3 3
if >0
0 H0 t
0 0
at = 2
2/3 3
sin if <0
H0 t
0 0 0
2
426 Cosmological models

15.24 Show that, in general,
Â¨
R
Ë™
= H2 + H
R
Hence use the cosmological field equations to show that, for a spatially flat
Lemaitre model containing no radiation, the Hubble parameter and the matter
density satisfy the equations
Ë™
2H + 3H 2 = c2
3H 2 âˆ’ c2 = 8 G m

Assuming > 0 and requiring > 0, thus show that
m

c2 c2
3
Ht = coth t
3 2 3

c2 c2
3
t= cosech2 t
m
8G 2 3
and therefore find expressions for t and t . Show further that
m

2 tanhâˆ’1
t=
3H

Hint: a2 / a2 âˆ’ x2 dx = cothâˆ’1 x/a + constant, for x2 > a2 .
15.25 Show that for a physically reasonable perfect fluid (i.e. density > 0 and pressure
â‰¥ 0) there is no static isotropic homogeneous solution to Einsteinâ€™s equations with
= 0. Show that it is possible to obtain a static zero-pressure solution by the
introduction of a cosmological constant such that
c2 k
c =4 G m0= 2
2
R0
Show that this solution is unstable, however.
15.26 Show that the comoving -coordinate of a galaxy emitting a photon at time t that
is received at t0 is given by
1
c da
=
Ë™
R0 aa
âˆ’1
1+z

Ë™
Using the cosmological field equation (15.13) to substitute for a, show that
1
c dx
z=
m 0x + r0+
4+
R0 H 0 2
âˆ’1
0x k 0x
1+z

15.27 For a dust-only Friedmann model, show that the luminosityâ€“distance relation varies
with redshift as
2c
dL z = m 0z + m 0 âˆ’2 m 0z + 1 âˆ’ 1
H0 2 0
m

This result is known as Mattigâ€™s formula.
427
Exercises

15.28 For a Friedmann model dominated by a single source of energy density, show that
âˆ’1
i 0 âˆ’1
âˆ’1
z âˆ’1 =
1 + z 1+3wi
i

where wi is the equation-of-state parameter of the source. Use this result to
comment on the flatness problem.
15.29 For a general cosmological model, show that
Ë™i= +2 âˆ’2 âˆ’ 1 âˆ’ 3wi
iH m r

where i denotes matter, radiation or the vacuum.
15.30 By differentiating the definition k = âˆ’kc2 / R2 H 2 , show that
Ë™k=2 = +2 âˆ’2
k Hq kH m r

where q is the deceleration parameter.
15.31 Show that the particle horizon at cosmic time t is given
at
c dx
t=
p
m 0x + r0+
4+
R0 H 0 2
0x k 0x
0

15.32 Consider the cosmological line element

ds2 = c2 dt2 âˆ’ e2t/b dr 2 + r 2 d + r 2 sin2 d
2 2

Light signals from a galaxy at coordinate distance r are emitted at epoch t1 and
received by an observer at epoch t0 . Show that
r
= eâˆ’t1 /b âˆ’ eâˆ’t0 /b
bc
For a given r, show that there is a maximum epoch t1 and interpret this result
physically. Show that a light ray emitted by the observer asymptotically approaches
the coordinate r = bc but never reaches it.
16
Inflationary cosmology

In the last two sections of the previous chapter, we saw that standard cosmological
models suffer, in particular, from the flatness problem and the horizon problem.
To these problems, one might also add the â€˜expansion problemâ€™, which asks
simply why the universe is expanding at all. Although this appears as an initial
condition in cosmological models, one would hope to explain this phenomenon
with an underlying physical mechanism. In this chapter, we therefore augment
our discussion of cosmological models with a brief outline of the inflationary
scenario, which seeks to solve these problems (and others) and has, over the
past two decades, become a fundamental part of modern cosmological theory.1 In
particular, we will discuss the effect of inflation on the evolution of the universe
as a whole and also consider how inflation gives rise to perturbations in the
early universe that subsequently collapse under gravity to form all the structure
we observe in the universe today. Given the general algebraic complexity of
these topics (particularly the perturbation analysis), we will adopt the convention
throughout this chapter that

8 G=c=1

This choice of units makes many of the equations far less cluttered and amounts
only to a rescaling of the scalar field and its potential (see below), which we can
remove at the end if desired.

16.1 Definition of inflation
As noted in Section 15.12, the horizon problem is a direct consequence of the
deceleration in the expansion of the universe. Thus, a possible solution is to

1
For a detailed discussion of inflationary cosmology, see, for example, A. Liddle & D. Lyth, Cosmological
Inflation and Large-scale Structure, Cambridge University Press, 2000.

428
429
16.1 Definition of inflation

postulate an accelerating phase of expansion, prior to any decelerating phase. In
an accelerating phase, causal contact is better at earlier times and so remotely
separated parts of our present universe could have â€˜coordinatedâ€™ their physical
characteristics in the early universe. Such an accelerating phase is called a period
of inflation. Hence the basic definition of inflation is that

Â¨
R>0 (16.1)

In fact, we may recast this condition in an alternative manner that is physically
more meaningful by considering the comoving Hubble distance defined in (15.51),
namely H t = H âˆ’1 t /R t . The derivative with respect to t is given by

Â¨
H âˆ’1 1
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