. 16
( 24)


z= + +···
14.9 The observed flux in the frequency range received from some distant
1 2
comoving object is given by
Fobs fobs d
1 2

where fobs v is the observed flux density (in W m’2 Hz’1 ) as a function of
frequency. If fem is the emitted (or intrinsic) flux density of the object, show
= em

where z is the redshift of the object. If fem over a wide range of frequencies,
show that
= Kz Fem
Fobs 1 2 1 2

where the K-correction is given by Kz = 1 + z ’1 .
14.10 The observed surface brightness obs of an extended object observed in the
frequency range 1 2 is defined as the observed flux per unit solid angle. Thus,
for a (small) circular object subtending an angular diameter we have
obs =
1 2
384 The Friedmann“Robertson“Walker geometry

where Fobs is defined in Exercise 14.9. Show that can be written as
1 2 obs

4 Lem Kz
= 1 2
obs 2 4

where is the physical (projected) diameter of the object, Lem 1 2 is the
intrinsic luminosity of the object in the frequency range 1 2 and Kz is the
Note: The above result is independent of cosmological parameters. Moreover,
setting aside the K-correction, the 1 + z ’4 -dependence means that the surface
brightness of extended objects drops very rapidly with redshift, making the detec-
tion of high-z objects difficult.
14.11 A commonly used distance measure in cosmology is the proper-motion distance
dM defined by
dM =

where v is the proper transverse velocity of (some part of) the object, which is
assumed known from astrophysics, and ™ is the corresponding observed angular
velocity. Show that
d M = 1 + z dA =

where dA and dL are the angular-diameter distance and the luminosity distance to
the object respectively.
14.12 A certain population of galaxies undergoes a short ultra-luminous phase at redshift
z = z— that lasts for a proper time interval t. After this phase, such galaxies are
neither created or destroyed. If z— 1, show that for a spatially flat universe the
total number of such galaxies in the sky that are in this phase is given by

4 c 3 n0 t
N= z— + 2 1 ’ q 0 z 2 + · · ·


where n0 is the present-day proper number density of these galaxies.
14.13 In the comoving coordinates x = t r , the FRW metric takes the form

dr 2
ds = c dt ’ R t + r2 d + sin2 d
2 2 2 2 2 2
1 ’ kr 2

Using the ˜Lagrangian™ method, or otherwise, calculate the corresponding connec-
tion coefficients . Hence calculate the non-zero elements of the Ricci
tensor R .
14.14 In Newtonian cosmology, the universe is modelled as an infinite gas of density t
that is expanding in such a way that the relative recessional velocity of any two

gas particles is v t = H t R t , where H t = R t /R t and R t is the separation

of the particles at time t. Use Gauss™ law to determine the force on a particle of
mass m on the edge of an arbitrary spherical region, and hence show that
R=’ R
By considering the total energy E of the above particle, show further that
8G 2

R2 = R ’ c2 k
where the constant k = ’2E/ mc2 . Compare these Newtonian cosmological field
equations with their general-relativistic counterparts.
14.15 Show that the relativistic equation of motion for the cosmological fluid is satisfied
identically and that the relativistic equation of continuity takes the form

p 3R
™+ + =0
c2 R
Show further that this equation may also be written in the forms

d R3 3pRR2 d R3 3pR2
=’ =’ 2
dt dR c
14.16 Use the cosmological field equations directly to derive the relativistic equation of
continuity for the cosmological fluid given in Exercise 14.15.
14.17 Consider a spherical comoving volume of the cosmological fluid whose surface is
defined by = constant. As the universe expands show that, for the infinitesimal
time interval t ’ t + dt, the conservation of energy requires that

c2 V = c2 +d V + dV + p dV

where c2 and p are the energy density and pressure of the fluid respectively.
Hence show that
= ’3 1 + w
dR R
where w = p/ c2 and R t is the scale factor of the universe. Show that this
equation has the solution ∝ R’3 1+w .
Cosmological models

In the previous chapter, we considered the geometric and kinematic properties of
the Friedmann“Robertson“Walker (FRW) metric and derived the cosmological
field equations for the scale factor R t . In this chapter, we will use the cosmo-
logical field equations to determine the behaviour of the scale factor as a function
of cosmic time in various cosmological models.

15.1 Components of the cosmological fluid
In a general cosmological model, the universe is assumed to contain both matter
and radiation. In addition, the cosmological constant is generally assumed to be
non-zero. As discussed in Section 8.7, the modern interpretation of is in terms
of the energy density of the vacuum, which may also be modelled as a perfect
fluid (with a peculiar equation of state). Thus, one usually adopts the viewpoint
that the cosmological fluid consists of three components, namely matter, radiation
and the vacuum, each with a different equation of state. The total equivalent mass
density is simply the sum of the individual contributions,

t= t+ t+ (15.1)
m r

where t is the cosmic time and we have adopted the commonly used cosmological
notation for the equivalent mass densities of matter, radiation and the vacuum
respectively. Moreover, we shall assume that these three components are non-
interacting (see Section 14.13); although matter and radiation did interact in the
early universe, this is a reasonable approximation for most of its history.
As mentioned in Section 14.12, each component of the cosmological fluid is
modelled as a perfect fluid with an equation of state of the form

pi = wi i c2

15.1 Components of the cosmological fluid

where the equation-of-state parameter wi is a constant (and i labels the compo-
nent). In particular wi = 0 for pressureless ˜dust™, wi = 1 for radiation and wi = ’1
for the vacuum. In general, if wi is a constant then, requiring that the weak energy
condition is satisfied (see Exercise 8.8) and that the local sound speed dp/d 1/2
is less than c, one finds that wi must lie in the range ’1 ¤ wi ¤ 1. We see that
this is indeed the case for dust, radiation and the vacuum. We now discuss each
of these components in turn and conclude with a description of their relative
contributions to the total density as the universe evolves.

In general, matter in the universe may come in several different forms. In addi-
tion to the normal baryonic matter of everyday experience (such as protons and
neutrons), the universe may well contain more exotic forms of matter consisting
of fundamental particles that lie beyond the ˜Standard Model™ of particle physics.
Indeed, observations of the large-scale structure in the universe suggest that most
of the matter is in the form of non-baryonic dark matter, which interacts electro-
magnetically only very weakly (and is hence invisible or ˜dark™). Moreover, dark
matter may itself come in different forms, such as cold dark matter (CDM) and
hot dark matter (HDM), the naming of which is connected to whether the typical
energy of the particles is non-relativistic or relativistic. We shall not pursue this
very interesting subject any further here1 but merely note that the total matter
density (at any particular cosmic time t) may be expressed as the sum of the
baryonic and dark matter contributions,

t= t+ t
m b dm

In the following discussion, we will not differentiate between different types of
matter, since it is only the total matter density that determines how the scale factor
R t evolves with cosmic time t. We shall also make the common assumption that
the matter particles (in whatever form) have a thermal energy that is much less
than their rest mass energy, and so the matter can be considered to be pressureless,
i.e. dust. In this case the equation of state parameter is simply w = 0. Thus, from
(14.44), if the matter has a present-day proper density of m t0 ≡ m 0 , its density
at some other cosmic time t is given by

t= z= 1+z 3
m m0 m m0

For a full discussion, see (for example) T. Padmanabhan, Structure Formation in the Universe, Cambridge
University Press, 1993.
388 Cosmological models

where in the second expression we have used (14.17) to write the density in terms
of the redshift z. These expressions concur with our expectation for the behaviour
of the space density of dust particles in an expanding universe.

The term radiation naturally includes photons, but also other species with very
small or zero rest masses, such that they move relativistically today. An example
of the latter is neutrinos (which may in fact have a small non-zero rest mass). The
total equivalent mass density of radiation in the universe at some cosmic time t
may then be written as the sum of the photon and neutrino contributions:
t= t+ t

Once again, we will not differentiate between different types of radiation in our
subsequent discussion, since it is only the total energy density that determines
the behaviour of the scale factor. For radiation, in general, we have w = 1 . Thus,
from (14.44), if the total radiation in the universe has a present-day energy density
of r 0 c2 then, at other cosmic times,

t= z= 1+z 4
r r0 r r0

In this case, the variation in the space density of photons (for example) again goes
as 1 + z 3 , but there is an additional factor 1 + z resulting from the cosmological
redshift of each photon.
It is worth noting that, to a very good approximation, the dominant contribution
to the radiation energy density of the universe is due to the photons of the
cosmic microwave background (CMB). This radiation is (to a very high degree
of accuracy) uniformly distributed throughout the universe and has a blackbody
form. For blackbody radiation, the number density of photons with frequencies
+ d is given by
in the range
T d = 3 h /kT (15.2)
n d
where T is the ˜temperature™ of the radiation. Since the energy per unit frequency
is simply u T = n T h , the total equivalent mass density of the radiation is
aT 4
T= 2 ud = 2
c c

where a = 4 2 kB / 60 3 c3 is the reduced Stefan“Boltzmann constant. Obser-

vations show that the CMB is characterised by a present-day temperature
15.1 Components of the cosmological fluid

T0 = 2 726 K, which corresponds to a total present-day number density n 0 ≈
4 — 108 m’3 . It is easily shown that the CMB photon energy distribution retains
its general blackbody form as the universe expands. Thus, at any given cosmic
time t, the temperature of the CMB radiation in the universe is given by

T t = T0 T z = T0 1 + z
or (15.3)

from which we see that the universe must have not only been denser in the past,
but also ˜hotter™.

As mentioned above the vacuum can be modelled as a perfect fluid having
an equation of state p = ’ c2 , so that the fluid has a negative pressure. This
corresponds to an equation of state parameter w = ’1. Thus, from (14.44), we
see that at any cosmic time t, we have

= 0=
Thus, the energy density of the vacuum always has the same constant value.

Relative contributions of the components
On combining the above results, we find that the variation in the total equivalent
mass density (15.1) may be written as
3 4
R0 R0
t= + + (15.4)
m0 r0 0
Rt Rt
From this expression, we see that the relative contributions of matter, radiation
and the vacuum to the total density vary as the universe evolves. The details
clearly depend on the relative values of m 0 , r 0 and 0 . Typically, however,
one would expect radiation to dominate the total density when R t is small. As
the universe expands, the radiation energy density dies away the most quickly
and matter becomes the dominant component. Finally, if the universe continues
to expand then the matter density also dies away and the vacuum ultimately
dominates the energy density. We conclude by noting that cosmologists often
define the normalised scale parameter
at ≡
in terms of which the above results are more compactly written, since a0 = 1 by
definition. We shall make use of this parameter further in subsequent sections.
390 Cosmological models

15.2 Cosmological parameters
In our very simplified model of the universe discussed above, its entire history
is determined by only a handful of cosmological parameters. In particular, if one
specifies the values of the equivalent mass densities m t— , r t— and at
some particular cosmic time t— then the value of each density, and hence the total
density, is determined at all other cosmic times t. Thus, specifying these quantities
is sufficient to determine the scale factor R t at all cosmic times. It is most
natural to take t— to be the present-day cosmic time t0 , and so the cosmological
model is entirely fixed by specifying the three quantities

m0 r0 0

It is, however, both convenient and common practice in cosmology to work
instead in terms of alternative dimensionless quantities, usually called density
parameters or simply densities, which are defined by

t≡ (15.5)
i i
3H 2 t

where H t is the Hubble parameter and the label i denotes ˜m™, ˜r™ or ˜ ™. It
is worth noting that t is, in general, a function of cosmic time t (unlike
, which is a constant). In terms of these new dimensionless parameters, the
cosmological model may thus be fixed by specifying the values of the four
present-day quantities
H0 m0 r0 0

A major goal of observational cosmology is therefore to determine these quantities
for our universe. Significant advances in the last decade mean that cosmologists
now know these values to an accuracy of just a few per cent.2 We simply note
here that

H0 ≈ 70 km s’1 Mpc’1 ≈ 5 — 10’5
≈03 ≈07
m0 r0 0

where the units of H0 are those most commonly used in cosmology, in which
1 Mpc ≡ 106 parsecs ≈ 3 09 — 1022 m; in SI units, H0 ≈ 2 27 — 10’18 s’1 . Perhaps
most astonishing is that the present-day energy density of the universe is domi-
nated by the vacuum!

How these observational advances have been achieved is discussed in, for example, J. Peacock, Cosmological
Physics, Cambridge University Press, 1999 or P. Coles & F. Lucchin, Cosmology: The Origin and Evolution
of Cosmic Structure (2nd edition), Wiley, 2002.
15.2 Cosmological parameters

We also note, for completeness, that cosmologists define further analogous
dimensionless density parameters for the individual contributions to the matter
and the radiation. For example, b , dm and are commonly used to denote
the dimensionless density of baryons, dark matter and neutrinos respectively. For
our universe, cosmological observations suggest the present-day values

≈ 0 05 ≈ 0 25 ≈0 (15.8)
b0 dm 0 0

noting, in particular, that only around one-sixth of the matter density is in the form
of the familiar baryonic matter. Moreover, the majority of the baryonic matter
seems not to reside in ordinary (hydrogen-burning) stars; the contribution of such
stars is only — ≈ 0 008. The values of the individual quantities (15.8) affect the
astrophysical process occurring in the universe and have a profound influence on,
for example, the formation of structure. For determining the overall expansion
history of the universe, however, only the quantities (15.6) need be specified.
The reason for defining the densities (15.5) becomes clear when we rewrite
the second of the cosmological field equations (14.36) in terms of them. Dividing

this equation through by R2 and noting that H = R/R, we obtain

c2 k
1= m+ r+ ’ 22 (15.9)
where, for notational simplicity, we have dropped the explicit time dependence of
the variables. Indeed, it is also common practice to define the curvature density

c2 k
t =’ 2 (15.10)
H t R2 t

so that, at all cosmic times t, we have the elegant relation

m+ r+ + =1 (15.11)

It should be noted that, in cosmological models with positive spatial curvature
k = 1 , the parameter k is negative. Moreover, if the cosmological constant
is negative then so too is the vacuum density parameter . This behaviour
should be contrasted with that of m and r , which are always positive.
From (15.9), we see that the values of m , r and determine the spatial
curvature of the universe in a simple fashion. We have three cases:

m+ r+ <1 ” negative spatial curvature k = ’1 ” ˜open™
m+ r+ =1 ” zero spatial curvature k = 0 ” ˜flat™
m+ r+ >1 ” positive spatial curvature k = 1 ” ˜closed™
392 Cosmological models

The above relations are valid at any cosmic time t but are most often applied to
the present day, t = t0 . In particular, it is also clear from (15.9) that, although
the density parameters m r and are all, in general, functions of cosmic
time t, their sum cannot change sign. Thus, the universe cannot evolve from one
form of the FRW geometry to another. We note that cosmologists often add to
the plethora of density parameters by also defining the total density parameter

≡ m+ r+ = 1’ (15.12)

which is related to the total equivalent mass density (15.1) by = 8 G / 3H 2 .
From (15.7), we see that for our universe 0 ≈ 1 or equivalently k 0 ≈ 0, and
it is therefore close to being spatially flat k = 0 .
Finally, it is worth noting that, for any cosmological model to be spatially
= 1 and it is common to describe the corresponding total
flat, one requires
equivalent mass density as the critical density, which is given by

3H 2
crit ≡
Hence, for any given value of the Hubble parameter, this expression gives the total
equivalent mass density required for the universe to be spatially flat. Since recent
cosmological observations suggest that our universe is indeed close to spatially
flat and, (15.7), that H0 ≈ 70 km s’1 Mpc’1 , one finds that the present-day total
equivalent mass density in our universe is
≈ 9 2 — 10’27 kg m’3
crit 0 =
As mentioned above, it is thought that only around 30 per cent of this equivalent
mass density is in the form of matter and only around 5 per cent in the form of
baryonic matter. Nevertheless, it is worth noting that crit 0 ≈ 5 5 protons m’3 ,
and so the critical density turns out to be extremely low by laboratory standards.3

15.3 The cosmological field equations
Since the cosmological model can be fixed by specifying the values of the quan-
tities listed in (15.6), it is worthwhile rewriting the cosmological field equations
(14.36) in terms of these parameters. Let us begin with the second field equation.

Recalling that H = R/R, this may be written
c2 k
H= ’2
3 R
The fact that this is a number of order unity is an accident of our choice of units!
15.4 General dynamical behaviour of the universe

where the label i includes matter, radiation and the vacuum. From (15.4), (15.5)
and (15.10), we therefore find

’3 ’4 ’2
H 2 = H0 + + 0+
m 0a r 0a k 0a

where we have written the result in terms of the dimensionless scale parameter
a = R/R0 . It should be remembered that k 0 = 1 ’ m 0 ’ r 0 ’ 0 and may
be considered merely as a convenient shorthand. It is also worth noting that, since
a = R/R0 = 1 + z ’1 , equation (15.13) immediately yields an expression for the
Hubble parameter H z as a function of redshift z.
We now turn to the first cosmological field equation in (14.36). Multiplying
™ ™
this equation through by R/R2 and again noting that H = R/R, we have
¨ 4G
=’ 1 + 3wi
™ i
3H 2 i
where the label i once more includes matter, radiation and the vacuum. The left-
hand side is equal to minus the deceleration parameter q defined in (14.19). Thus,
substituting the appropriate value of wi for each component and using (15.5), one
finds the neat relation

q= m +2 r ’2

If desired, one can easily write this equation explicitly in terms of the present-day
values of the density parameters by using the result (15.13) and the relation
a’3 1+wi
= i0
which holds generally for matter, radiation and the vacuum.

15.4 General dynamical behaviour of the universe
The cosmological field equations (15.13) and (15.14) allow us to determine the
general dynamical behaviour and the spatial geometry of the universe for any
given set of values for the parameters m 0 , r 0 and 0 . The observations
(15.7) suggest that the present-day value of the radiation density r 0 is signifi-
cantly smaller than the matter and vacuum densities. It is therefore a reasonable
approximation to neglect r 0 and parameterise a universe like our own in terms of
just m 0 and 0 (and H0 , which is irrelevant for our discussion in this section).
Figure 15.1 presents a summary of the properties of FRW universes dominated
by matter and vacuum energy (known as Lemaitre models) as a function of
394 Cosmological models


b an

ac g

expand forever




0 0.5 1 1.5 2

Figure 15.1 Properties of FRW universes dominated by matter and vacuum
energy, as a function of the present-day density parameters m 0 and 0 . The
circle indicates the region of the parameter space that is consistent with recent
cosmological observations.

position in the 0 parameter space. The dividing lines between the
various regions may be determined from the field equations (15.13, 15.14) and the
relation (15.11). In particular, the ˜open“closed™ line comes directly from (15.11)
evaluated at the present epoch, which gives the condition

= 1’
0 m0

Similarly, the ˜accelerating“decelerating™ line is obtained immediately by setting
q0 = 0 in (15.14) for t = t0 , which gives

= 1
0 m0

The ˜expand-forever“recollapse™ line and the ˜big-bang“no-big-bang™ line require
a little more work, as we now discuss.
In fact, both these lines are determined from the expression (15.13) for the
Hubble parameter. In particular, the condition for the graph of R t , or equivalently
15.4 General dynamical behaviour of the universe

of a t , to have a turning point at some cosmic time t = t— is simply that H t— = 0.
Setting k 0 = 1 ’ m 0 ’ 0 in (15.13), we find that, after rearranging, this
condition corresponds to

fa ≡ 0a + 1’ m 0’ a+ =0
0 m0

This is a cubic equation for the value(s) of the scale factor a = a— at which a t
has a turning point. We are not, in fact, interested in the particular value(s) a = a—
that solve (15.15) but only in whether a (real) solution exists in the region a ≥ 0
(which is the only physically meaningful regime).
For the case 0 < 0 we may deduce immediately from (15.14) that the
universe must have started with a ˜big bang™, at which a = 0, and must eventually
recollapse in a ˜big crunch™ as a ’ 0 once more. In (15.14), a negative value of
means that the deceleration parameter q is always positive. Thus a is always
negative, and hence the a t graph must be convex for all values of t. Since at

the present epoch a t0 > 0 (because we observe redshifts, not blueshifts), this
means that a t must have equalled zero at some point in the past, which it is
usual to take as t = 0;4 similar reasoning may be used to deduce that the universe
must eventually recollapse, although a little more care is required in this case. As
the universe expands, the vacuum energy eventually dominates and so we need
only consider the -term on the right-hand side of (15.14), which will not tend
to zero as the scale factor increases. Thus, a cannot tend to zero and so a ’ 0 at
some finite cosmic time in the future.
In our further analysis, we now need only consider the case in which
in (15.15), but this still requires some care. Let us first consider the case for
0 = 0. Immediately, we see that equation (15.15) then has the single
solution a— = m 0 / m 0 ’ 1 , which is negative in the range 0 ¤ m 0 ¤ 1,
indicating that there is no (physically meaningful) turning point. Therefore, over
0 = 0. We
this range, the ˜expand-forever“recollapse™ line is simply given by
must now address the far more complicated case for which 0 > 0. In this case
f a ’ ± as a ’ ± . Moreover f 0 = m 0 , which is positive. Thus, for
f a to have a positive root, it must have a turning point in the region a > 0.
On evaluating the derivatives f a and f a with respect to a, it is clear that,
in the limiting case of interest, f a must have the general form illustrated in
Figure 15.2. Thus, we require f a— = f a— = 0, which quickly yields
a— = (15.16)
2 0

0 = 0, provided that the universe contains even an
In fact, this reasoning is still valid in the case
infinitesimal amount of matter (or radiation). Thus all cosmological models with ¤ 0 have a big-bang
origin at some finite cosmic time in the past.
396 Cosmological models

f (a)


Figure 15.2 The limiting form of the cubic f a defined in (15.15) for > 0.

On substituting this expression back into (15.15) one then obtains a separate cubic
equation for 0 , given by

4 1’ m 0’ + 27 =0
3 2
0 0

By introducing the variable x = 1/3 ,
0/ 4 m 0 this equation quickly
reduces to
m 0 ’1
x3 ’ + =0
4 4 m0

which is amenable to analysis using the standard formulae for finding the roots
of a cubic. In particular, rewriting the resulting roots in terms of 0 , one finds
the following three cases:

• 0< ¤ 2 , one positive root at

1’ m 0
m 0 cosh (15.18)
3 m0

• ¤ 1, one positive root at
< m0

1’ m 0
=4 3
m 0 cos (15.19)
3 m0

• > 1, two positive roots, the larger given by (15.19) and the smaller by

1’ m 0
1 4
=4 +
m 0 cos (15.20)
3 3
15.5 Evolution of the scale factor

Moreover, from (15.16), one easily finds that a— < 1 for (15.18, 15.19), whereas
a— > 1 for (15.20). Since the universe is expanding, a— < 1 corresponds to a
turning point in the past (i.e. no big bang), whereas a— > 1 corresponds to a
turning point in the future (i.e. recollapse).
The resulting lines, plotted in Figure 15.1, show some interesting features. In
0 = 0 there is a direct correspondence between
particular, we note that when
the geometry of the universe and its eventual fate. In this case, open universes
expand forever, whereas closed universes recollapse. This correspondence no
longer holds in the presence of a non-zero cosmological constant, in which case
any combination of spatial geometry and eventual fate is possible. It is also
worth noting that the region of the 0 -plane consistent with recent
cosmological observations is centred on the spatially flat model (0.3,0.7) and
excludes the possibility of a zero cosmological constant at high significance.
These observations also show the expansion of the universe to be accelerating.
They also require the universe to have started at a big bang at some finite cosmic
time in the past and to expand forever in the future.

15.5 Evolution of the scale factor
So far, we have considered only the limiting behaviour of the (normalised)
scale factor a t for different values of the cosmological parameters; this was
summarised in Figure 15.1. We now discuss how to find the form of the a t -
curve at all cosmic times, for a given set of (present-day) cosmological parameter
values. This behaviour is entirely determined by the cosmological field equation
(15.13). Remembering that H = a/a, this may be written as

da ’1 ’2
= H0 + + 0a + 1 ’ m 0’ r 0’
2 2
m 0a r 0a 0
Instead of working directly in terms of the cosmic time t, it is more convenient
to introduce the new dimensionless variable
t = H0 t ’ t0 (15.22)
which measures cosmic time relative to the present epoch in units of the ˜Hubble
time™ H0 . In terms of this new variable, (15.21) becomes

da ’1 ’2
= + + +1’ m 0’ r 0’
m 0a r 0a 0a
ˆ 0

There exist some special cases, where m 0 , r 0 and 0 take on particular
simple values, for which equation (15.23) can be solved analytically; we will
398 Cosmological models

discuss some of these cosmological models in Section 15.6. In general, however, a
numerical solution is necessary. Starting at the point t = 0 (the present epoch), for
which a0 = 1, the normalised scale factor at time step n + 1 can be approximated
by the Taylor expansion
d2 a
ˆ ˆ
an+1 ≈ an + t+ 2
ˆ ˆ
dt 2
dt n n

ˆ ˆ ˆ
where t is the (small) step size in t. The coefficient of t is given by (15.23),
and the coefficient of t 2 may be obtained by differentiating (15.23). The latter
is important since, without the t 2 term, equation (15.24) would not carry the
integration correctly through a value of a for which da/dt is small or zero.
Figure 15.3 shows the variation in the normalised scale factor a t as a function
of t for different values of 0 as indicated, assuming that r 0 is
negligible, as it is for our universe. In the top panel m 0 + 0 = 1 in each case,
so each universe has a flat spatial geometry k = 0 . The solid line corresponds
to the case (0.3, 0.7), which is preferred by recent cosmological observations. An
interesting cosmological ˜coincidence™ for this model is that the present epoch,
ˆ ˆ
t = 0, corresponds almost exactly to the point of inflection on the a t curve.
A second such ˜coincidence™ is that the age of the universe in this model (i.e.
the time since the big bang) is very close to one Hubble time.5 The broken-and-
dotted line in the top panel of the figure corresponds to the case (0, 1), which is
known as the de Sitter model and will be discussed further in Section 15.6. For
the moment, we simply note that this model has no big-bang origin (although
a ’ 0 as t ’ ’ ) and will expand forever. The broken line in the top panel
corresponds to the case (1, 0), which is known as the Einstein“de-Sitter model and
will also be discussed in Section 15.6. As we see from the figure, this model does
have a big-bang origin. It is also on the borderline between expanding forever
and recollapsing; it will in fact expand forever, but a ’ 0 as t ’ .

In the bottom panel of Figure 15.3 we have m 0 + 0 = 1 in each case,
and so each universe is spatially curved; in particular the case (0.3, 0) is open
and the cases (0.3, 2) and (4, 0) are closed. We see that the case (0.3, 2) has no
big-bang origin, and is, in fact, what is known as a bounce model, where the
universe collapses from large values of the scale factor and ˜bounces™ at some
finite minimum value of a, after which it re-expands forever. Conversely, the case
(4, 0) corresponds to a cosmological model with a big-bang origin that expands
to some finite maximum value of a before recollapsing to a big crunch.
Before going on to discuss cosmological models that admit an analytic solution
for a t , it is worth discussing the general case in the limit a ’ 0. Whether

Whether such coincidences have some deeper significance is the subject of current cosmological research.
15.5 Evolution of the scale factor

a(t )

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

“2 “1 0 1 2
H0(t “ t0)

(0.3, 2)
a(t )

(0.3, 0) (4, 0)

“2 “1 0 1 2
H0(t “ t0)

Figure 15.3 The variation in the normalised scale factor as a function of the
dimensionless variable H0 t ’ t0 for different values of 0 as indi-
cated, assuming that r 0 is negligible. Top panel: m 0 + 0 = 1 in each
case, so the universes have a flat spatial geometry k = 0 . Bottom panel:
m 0+ 0 = 1 in each case, so the universes are spatially curved; in particular,
the case (0.3, 0) is open and the cases (0.3, 2) and (4, 0) are closed.

considering the big bang or the big crunch, in this limit we can assume that the
energy density of the universe is dominated by a one kind of source (which one
will depend on the particular cosmological model under consideration). In this
case, (15.23) can be written
da ’ 1+3wi
= + (15.25)
i 0a
ˆ k0
where the label i denotes the dominant form of the energy density as a ’ 0
and wi is the corresponding equation-of-state parameter. Moreover, if we restrict
400 Cosmological models

our attention to the (realistic) case, in which i denotes either dust wi = 0 or
radiation wi = 1 , then the first term on the right-hand side of (15.25) dominates
as a ’ 0, so we can neglect the curvature density k 0 . In this case, (15.25) can
be immediately integrated to give

2/ 3 1+wi
ˆ ˆˆ
a t =± 1 + wi t ’ t— 2/ 3 1+wi

ˆ ˆ
where t— is the value of t at which a = 0 and the plus and minus signs correspond
to the big bang and big crunch respectively. From (15.25), we also note that
ˆ ˆ ˆ
da/dt ’ as a ’ 0. Thus, we conclude that the a t -graph meets the t-axis at
right angles.

15.6 Analytical cosmological models
Although in the general case the evolution of the (normalised) scale factor a t
must be determined numerically, there exist a number of special cases, corre-
sponding to particular values of the cosmological parameters m 0 , r 0 and
0 , for which equation (15.23) can be solved analytically. We now discuss
some of these analytical cosmological models, all of which have inherited special
names that are widely used in cosmology. In this section, we will work in terms
of the cosmic time t directly, rather than the dimensionless variable t defined
in (15.22).

The Friedmann models
Cosmological models with a zero cosmological constant (and, strictly, a non-zero
matter or radiation density) are known as the Friedmann models. As noted in
Section 15.4, all Friedmann models have a big-bang origin at a finite cosmic time
in the past. Moreover, it is possible to place a strict upper limit on the age of the
universe in such models. Since the a t -curve is everywhere convex, it is clear
from Figure 15.4 that it crosses the t-axis at a time that is closer to the present
time t = t0 than the time at which the tangent to the point t0 a0 reaches the
t-axis (note that a0 = 1). Clearly, the point where the tangent meets the t-axis is
the point at which a t would have been zero for a = constant and a = 0. The
™ ¨
time elapsed from that point to the present epoch is simply a t0 /a t0 = H0 .

Thus, in Friedmann models, the age of the universe must be less than the Hubble
t0 < H0
15.6 Analytical cosmological models


t = t0 “ H0 t=0 t = t0

Figure 15.4 Diagram to illustrate that, for all Friedmann models, the age of the
universe is less than the Hubble time 1/H0 .

The behaviour of a t near the big-bang origin is given by (15.26) and is
independent of the curvature density parameter k 0 = 1’ m 0 ’ r 0 (and hence
of the sign of k). The future evolution, however, depends crucially on this constant.
From (15.21) we can distinguish three possible histories, depending on the value
of k 0 :

” open k = ’1 ” a ’ non-zero constant as a ’

k0=0 ” flat k = 0 ” a ’ 0 as a ’

” closed k = 1 ” a = 0 at some finite value amax


Thus, we see the main feature of Friedmann models, namely, that the dynamics of
the universe is directly linked to its geometry. The three cases above are illustrated
in Figure 15.5. We shall now find explicit analytical solutions for a t in the
special cases of a dust-only and a radiation-only Friedmann model. We will also
obtain an analytic form for t as a function of a for the case of a spatially flat
k = 0 Friedmann model containing both matter and radiation.

=0 =0 In this case (15.21) becomes
Dust-only Friedmann models 0 r0

1 x
™ +1’ ’ t=
2 2
H0 m 0a dx
m 0 + 1’
H0 x
0 m0


= 1,
which may be integrated straightforwardly in each of the three cases m0
m 0 > 1 and m 0 < 1 respectively, as follows.
402 Cosmological models

„¦k, 0 > 0 (open)

„¦k, 0 = 0 (flat)

„¦k, 0 < 0 (closed)

big bang big crunch

Figure 15.5 Schematic illustration of the evolution of the normalised scale factor
a t in closed, open and spatially flat Friedmann models.

• For = 1 (k = 0) the solution is immediate, and we find that

at = 3

This particular case is known as the Einstein“de-Sitter (or EdS) model.
• For m 0 > 1 k = 1 the integral (15.27) can be evaluated by substituting x =
m 0 ’ 1 sin
/2 , where is known as the development angle and varies
m 0/
over the range 0 . One then obtains

m0 m0
a= 1 ’ cos t= ’ sin
m 0 ’1 m 0 ’1
2 2H0

which shows that the graph of a t is a cycloid.
• For m 0 < 1 k = ’1 the integral (15.27) can be evaluated by substituting x =
m 0 / 1 ’ m 0 sinh
/2 , and one obtains

m0 m0
a= cosh ’ 1 t= sinh ’
2 1’ 2H0 1 ’ 3/2
m0 m0

t= and H t =
In each case, one may also obtain expressions for m 0a

a/a, and hence for m t .
15.6 Analytical cosmological models

=0 =0 In this case (15.21)
Radiation-only Friedmann models 0 m0
1 x
a2 = H 0
™ +1’ ’ t=
r 0a dx
r 0+ 1’
0 x2


= 1,
which may again be integrated straightforwardly for > 1 and
r0 r0
r 0 < 1 respectively.

• For = 1 (k = 0) the solution is again immediate, and we find that

a t = 2H0 t 1/2

• For r 0 < 1 k = ’1 and > 1 k = 1 the integral (15.29) can be evaluated
by inspection to give

1/2 1/2 r0
a t = 2H0 1+
r0t H0 t
2 r0

t= and H t =
In each case, one may again obtain expressions for m 0a

a/a, and hence for r t .

= 0, m 0+ =1 In this case (15.21)
Spatially flat Friedmann models 0 r0
1 x
’1 ’2
™ + ’ t=
2 2
H0 m 0a r 0a dx
m 0x +
H0 0 r0


which may be straightforwardly integrating by substituting y = m 0x + to

2 3/2
H0 t = m 0a + m 0a ’ 2 +2
r0 r0 r0
3 m0

Unfortunately, this expression cannot be easily inverted to give a t . Nevertheless,
it is simple to show that the above expression becomes 2 a3/2 for a matter-only
model and 2 a for a radiation-only model, and therefore agrees with our earlier
404 Cosmological models

The Lemaitre models
The Lemaitre models are a generalisation of the Friedmann models in which the
cosmological constant is non-zero. In particular, we will focus here on matter-
only models ( r 0 = 0), although our discussion is easily modified for radiation-
only models, and can be extended to include models containing both matter and
radiation. The general dynamical properties of Lemaitre models with r 0 = 0
were discussed in detail in Section 15.4, with particular focus on their limiting
behaviour. We concentrate here on determining the generic form of the a t -curve
for models of this type that have a big-bang origin and will expand forever.
We begin by considering the general case of arbitrary spatial curvature and then
specialise to the spatially flat case. A model of the latter sort appears to provide
a reasonable description of our own universe, if one neglects its radiation energy

=0 In this
Matter-only Lemaitre models with arbitrary spatial curvature r0
case the cosmological field equation (15.13) reads

a2 = H 0
™ + +
2 2
m 0a 0a k0

where k 0 = 1 ’ m 0 ’ 0 . Obtaining explicit formulae giving, for example,
the scale factor as a function of time is in general quite complicated, since the
integrals turn out to involve elliptic functions,6 which are unfamiliar to most
physicists these days. Nevertheless, we see that for small a the first term on the
right-hand side dominates and the equation is easily integrated. Thus, after starting
from a big-bang origin at t = 0, the a t -curve at first increases as

at = 3
for small t
2 H0 m 0t

which agrees with our earlier result (15.26). As the universe expands, however,
the matter energy density decreases and the vacuum energy eventually dominates.
Thus, for large t (and hence large a), the second term on the right-hand side of
(15.31) dominates. Once again the equation is then easily integrated to give

a t ∝ exp H0 for large t

See e.g. M. Abramowitz & I. A. Stegun, Handbook of Mathematical Physics, Dover, 1972.
15.6 Analytical cosmological models

From the above limiting behaviour at small and large t, it is clear that the
universe must, at some point, make a transition from a decelerating to an accel-
erating phase. This occurs when a = 0, at which point the a t -curve has a point
of inflection. Differentiating (15.31), we find that
a = 2 H0 2
¨ 12 0a ’ (15.32)
m 0a

From this result, we may verify immediately that at early cosmic times (when a
is small) we have a < 0, and so the expansion is decelerating. As the universe
expands, the deceleration gradually decreases until a changes sign, after which the
expansion accelerates ever more rapidly. We see that the value of the normalised
scale factor at which the point of inflection (a = 0) occurs is given by
a— = (15.33)
2 0

It is, in fact, possible to obtain an approximate analytic expression for the
normalised scale factor a t in the vicinity of the point of inflection. To do this,
we must first obtain an approximate form for the cosmological field equation
(15.31) in the vicinity of this point. Denoting the cosmic time at the point of

inflection by t— , we may perform separate Taylor expansions of a and a2 about
t = t— to obtain
a ≈ a— + a— t ’ t—
™ a2 ≈ a2 + a— a — t ’ t— 2
™ ™— ™

where, for notational convenience, we have written a— ≡ a t— , a— ≡ a t— , etc.
™ ™
Using the first expression to subtitute for t ’ t— 2 in the second, we obtain
a — a ’ a— 2
a ≈ a— +
™ ™
2 2

Differentiating (15.32) one easily obtains an expression for a . Then, substituting
(15.33) into the resulting expression, and into (15.31), one finds that (15.34)
a2 ≈ H0
™ k 0 +3 0 a— + 3 a ’ a—
2 2 2

This equation can now be integrated analytically and has the solution

’1/3 1/2
a t = a— + a— 1 + t ’ t—
2 1/2
1 1
sinh H0 3
k0 4 0 0


An interesting property of this type of model is that in the case of positive
spatial curvature (k = 1), for which k 0 < 0, there is a ˜coasting period™ in the
406 Cosmological models


Coasting period


Figure 15.6 The behaviour of a t in the Lemaitre model with k = 1. For k = 0
or k = ’1, there is no extended coasting period.

vicinity of the point where a = 0, during which the value of a t remains almost
equal to a— (see Figure 15.6). It is easily seen from (15.35) that, by setting the
’1/3 sufficiently close to ’1, one can
value of the quantity 1 k 0 4 1 2
0 m0
make the coasting period arbitrarily long. Indeed, in the limiting case, it is easy
to show that one requires that m 0 and 0 should satisfy (15.17).

r0=0 m 0+ 0=1 In this
Spatially flat matter-only Lemaitre models
case one can give an explicit formula for the scale factor. Moreover, even if the
universe turns out not to be exactly spatially flat, recent cosmological observations
show that it is close enough to flatness for the formulae involved to act as a
reasonable first approximation and so it is worthwhile to have them available.
In the spatially flat case, the cosmological field equation (15.13) may be written


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