= 2g 2 0 g0 ’

1

g00

00

which is easily shown to be zero by using the fact that g 0i = 0 for i = 1 2 3. Thus

the worldlines x are geodesics and hence can describe particles (observers)

moving only under the influence of gravity.

14.4 Homogeneity and isotropy of the universe

The metric (14.1) does not yet incorporate the property that space is homogeneous

and isotropic. Indeed this form of the metric can be used, with the help of a special

coordinate system obtained by singling out a particular fundamental observer,

to derive some general properties of the universe, without the assumptions of

homogeneity and isotropy, although we will not consider such cases here.

Let us now incorporate the postulates of homogeneity and isotropy. The former

demands that all points on a particular spacelike hypersurface are equivalent,

whereas the latter demands that all directions on the hypersurface are equiv-

alent for fundamental observers. The (squared) spatial separation on the same

359

14.5 The maximally symmetric 3-space

hypersurface t = constant of two nearby galaxies at coordinates x1 x2 x3 and

x1 + x1 x2 + x2 x3 + x3 is

= gij xi xj

2

d

If we consider the triangle formed by three nearby galaxies at some particular

time t, then isotropy requires that the triangle formed by these same galaxies at

some later time must be similar to the original triangle. Moreover, homogeneity

requires that the magnification factor must be independent of the position of the

triangle in the 3-space. It thus follows that time t can enter the gij only through

a common factor, so that the ratios of small distances are the same at all times.

Hence the metric must take the form

ds2 = c2 dt2 ’ S 2 t hij dxi dxj (14.4)

where S t is a time-dependent scale factor and the hij are functions of the

coordinates x1 x2 x3 only. We note that it is common practice to identify

fundamental observers loosely with individual galaxies (which are assumed to be

pointlike). However, since the magnification factor is independent of position, we

must neglect the small peculiar velocities of real individual galaxies.

14.5 The maximally symmetric 3-space

We clearly require the 3-space spanned by the spacelike coordinates x1 x2 x3

to be homogeneous and isotropic. This leads us to study the maximally symmet-

ric 3-space. In three dimensions, the curvature tensor Rijkl has, in general, six

independent components, each of which is a function of the coordinates. We

therefore need to specify six functions to define the intrinsic geometric properties

of a general three-dimensional space. Clearly, the more symmetrical the space,

the fewer the functions needed to specify its properties. A maximally symmetric

space is specified by just one number “ the curvature K, which is independent

of the coordinates. Such constant curvature spaces must clearly be homogeneous

and isotropic.

The curvature tensor of a maximally symmetric space must take a particularly

simple form. It must clearly depend on the constant K and on the metric tensor

gij . The simplest expression that satisfies the various symmetry properties and

identities of Rijkl and contains just K and the metric tensor is given by

Rijkl = K gik gjl ’ gil gjk (14.5)

360 The Friedmann“Robertson“Walker geometry

In fact, a maximally symmetric space is defined as one having a curvature tensor

of the form (14.5).

The Ricci tensor is given by

Rjk = g il Rijkl = Kg il gik gjl ’ gil gjk

=K k gjl ’ l gjk

l l

= K gjk ’ 3gjk = ’2Kgjk

The curvature scalar is thus given by

R = Rk = ’2K = ’6K

k

k k

As in our derivation of the general static isotropic metric in Section 9.1, the

metric of an isotropic 3-space must depend only on the rotational invariants

x · x ≡ r2 dx · dx x · dx

and in spherical polar coordinates r it must take the form

= C r x · dx 2 + D r dx · dx

2 2

d

= C r r 2 dr 2 + D r dr 2 + r 2 d + r 2 sin2 d

2 2

Following our analysis in Chapter 9, we can simplify this line element by redefin-

ing the radial coordinate r 2 = r 2 D r . Dropping the bars on the variables, the

¯

metric can thus be written as

= B r dr 2 + r 2 d + r 2 sin2 d

2 2 2

d

where B r is an arbitrary function of r.

We have met this line element before “ it is identical to the space part of the

general static isotropic metric. In Chapter 9, we showed that the only non-zero

connection coefficients are

r sin2

1 dB r r

= =’ =’

r r r

rr

2B r dr Br Br

1

= = = ’ sin cos = cot

r r

r

The Ricci tensor is given in terms of the connection coefficients by

Rij = ik ’ k + ’

k k l k l k

j ij ik lj ij lk

361

14.5 The maximally symmetric 3-space

and, after some algebra, we find that its non-zero components are

1 dB

Rrr = ’

rB dr

1 r dB

= ’1’ 2

R

2B dr

B

= R sin2

R

For our 3-space to be maximally symmetric, however, we must have

Rij = ’2Kgij

and so we require

1 dB

= 2KB r (14.6)

rB dr

r dB 1

1+ 2 ’ = 2Kr 2 (14.7)

2B dr B

Integrating (14.6) we immediately obtain

1

Br =

A ’ Kr 2

where A is a constant of integration. Substituting this expression into (14.7) then

gives

1 ’ A + Kr 2 = Kr 2

from which we see that A = 1. Thus, we have constructed the line element for

the maximally symmetric 3-space, which takes the form

dr 2

= + r 2d + r 2 sin2 d

2 2 2

(14.8)

d

1 ’ Kr 2

and has a curvature tensor specified by one number, K, the curvature of the space.

Notice also that this is exactly the same form as the metric for a 3-sphere

embedded in four-dimensional Euclidean space, which we discussed in Chapter 2.

The metric contains a ˜hidden symmetry™, since the origin of the radial coordinate

is completely arbitrary. We can choose any point in this space as our origin since

all points are equivalent. There is no centre in this space. We also note that,

on scales small compared with the spatial curvature, the line element (14.8) is

equivalent to that of a three-dimensional Euclidean space.

362 The Friedmann“Robertson“Walker geometry

14.6 The Friedmann“Robertson“Walker metric

Combining the our expression (14.8) for the maximally symmetric 3-space with

the line element (14.4), which incorporates the cosmological principle and Weyl™s

postulate, we obtain

dr 2

ds = c dt ’ S t + r2 d + sin2 d

2 2 2 2 2 2

(14.9)

1 ’ Kr 2

It is usual to write this line element in an alternative form in which the

arbitrariness in the magnitude of K is absorbed into the radial coordinate and the

scale factor. Assuming firstly that K = 0 we define the variable k = K/ K in

such a way that k = ±1 depending on whether K is positive or negative. If we

introduce the rescaled coordinate

r= K

¯ 1/2

r

then (14.9) becomes

S2 t d¯ 2

r

ds = c dt ’ + r2 d

¯ + sin2 d

2 2 2 2 2

1 ’ k¯ 2

K r

Finally, we define a rescaled scale function R t by

§

⎨ St if K = 0

R t = K 1/2

©

if K = 0

St

Then, dropping the bars on the radial coordinate, we obtain the standard form for

the Friedmann“Robertson“Walker (FRW) line element,

dr 2

ds = c dt ’ R t + r2 d + sin2 d

2 2 2 2 2 2

(14.10)

1 ’ kr 2

where k takes the values ’1, 0, or 1 depending on whether the spatial section

has negative, zero or positive curvature respectively. It is also clear that the

coordinates r appearing in the FRW metric are still comoving, i.e. the

worldline of a galaxy, ignoring any peculiar velocity, has fixed values of r .

14.7 Geometric properties of the FRW metric

The geometric properties of the homogeneous and isotropic 3-space corresponding

to the hypersurface t = constant depend upon whether k = ’1, 0 or 1. We now

consider each of these cases in turn.

363

14.7 Geometric properties of the FRW metric

Positive spatial curvature: k = 1

In the case k = 1, we see that the coefficient of dr in the FRW metric becomes

singular as r ’ 1. We therefore introduce a new radial coordinate , defined by

the relation

r = sin ’ dr = cos d = 1 ’ r 2 1/2

d

so that the spatial part of the FRW metric takes the form

= R2 d + sin2 + sin2 d

2 2 2 2

d d

where R is the value of the scale factor at the particular time t defining the

spacelike hypersurface of interest.

Some insight into this spatial metric may be gained by considering the 3-space

as embedded in a four-dimensional Euclidean space with coordinates w x y z ,

where

w = R cos

x = R sin sin cos

y = R sin sin sin

z = R sin cos

In fact we have already discussed exactly this embedding in Section 2.9. Such an

embedding is possible since one can write

= dw2 + dx2 + dy2 + dz2 = R2 d + sin2 + sin2 d

2 2 2 2

d d

where, from the transformation equations, we have the constraint

w2 + x2 + y2 + z2 = R2

This shows that our 3-space can be considered as a three-dimensional sphere in the

four-dimensional Euclidean space. This hypersurface is defined by the coordinate

ranges

0¤ ¤ 0¤ ¤ 0¤ ¤2

= constant are 2-spheres with surface area

The surfaces

2

A= = 4 R2 sin2

R sin d R sin sin d

=0 =0

and are the standard spherical polar coordinates of these 2-spheres. Thus, as

varies from 0 to , the area of the 2-spheres increases from zero to a maximum

value of 4 R2 at = /2, after which it decreases to zero at = . The proper

364 The Friedmann“Robertson“Walker geometry

radius of a 2-sphere is R , and so the surface area is smaller than that of a sphere

of radius R in Euclidean space.

The entire 3-space has a finite total volume given by

2

V= =2 23

R sin d R sin sin d

Rd R

=0 =0 =0

which is the reason why, in this case, R is often referred to as the ˜radius of the

universe™.

Zero spatial curvature: k = 0

In this case, if we set r = (to keep our notation consistent), the 3-space line

element is

= R2 d + + sin2 d

2 2 2 2 2

d d

which is simply the ordinary three-dimensional Euclidean space. As usual, under

the transformation

x = R sin cos y = R sin sin z = R cos

the line element becomes

= dx2 + dy2 + dz2

2

d

Negative spatial curvature: k = ’1

In this case, it is convenient to introduce a radial coordinate given by

r = sinh ’ dr = cosh d = 1 + r 2 1/2

d

so that the spatial part of the FRW metric becomes

= R2 d + sinh2 + sin2 d

2 2 2 2

d d

We cannot embed this 3-space in a four-dimensional Euclidean space, but it can

be embedded in a four-dimensional Minkowski space with coordinates w x y z

given by

w = R cosh

x = R sinh sin cos

y = R sinh sin sin

z = R sinh cos

In this case, we can write

= dw2 ’ dx2 ’ dy2 ’ dz2

2

d

365

14.8 Geodesics in the FRW metric

together with the constraint

w2 ’ x2 ’ y2 ’ z2 = R2

which shows that the 3-space can be represented as a three-dimensional hyper-

boloid in the four-dimensional Minkowski space. The hypersurface is defined by

the coordinate ranges

0¤ ¤ 0¤ ¤ 0¤ ¤2

= constant are 2-spheres with surface area

The 2-surfaces

A = 4 R2 sinh2

which increases indefinitely as increases. The proper radius of such a 2-sphere

is R , and so the surface area is larger than the corresponding result in Euclidean

space. The total volume of the space is infinite.

From the above discussion, we see that a convenient form for the FRW metric is

ds2 = c2 dt2 ’ R2 t d + S2 + sin2 d

2 2 2

(14.11)

d

where the function r = S is given by

§

if k = 1

⎪sin

⎨

= if k = 0 (14.12)

S

⎪

©

if k = ’1

sinh

Once again, it is clear that are comoving coordinates.

14.8 Geodesics in the FRW metric

In the comoving coordinate system(s) we have defined above, the galaxies have

fixed spatial coordinates (by construction; any peculiar velocities are ignored).

Thus the ˜cosmological fluid™ is at rest in the comoving frame we have chosen.

We now consider the motion of particles travelling with respect to this comoving

frame. In particular, we consider the geodesic motion of ˜free™ particles, i.e. those

experiencing only the ˜background™ gravitational field of the cosmological fluid

and no other forces. Examples of such particles might include a projectile shot out

of a galaxy or a photon travelling through intergalactic space. We could use the

˜Lagrangian™ procedure to calculate the geodesic equations for the FRW metric,

but instead we take advantage of the fact that the spatial part of the metric is

homogeneous and isotropic to arrive at the equations rather more quickly.

366 The Friedmann“Robertson“Walker geometry

It is convenient to express the FRW metric in the form (14.11) and write

x =t , so that

g00 = c2 g11 = ’R2 t g22 = ’R2 t S 2 g33 = ’R2 t S 2 sin2

The path of a particle is given by the geodesic equation

u+

™ u u =0

where u = x and the dot corresponds to differentation with respect to some

™

affine parameter. For our present purposes, however, it will be more useful to

rewrite the geodesic equation in the form

u=

™ 1

g uu

2

which shows, as expected, that if the metric is independent of a particular coor-

dinate x then u is conserved along the geodesic.

Let us suppose that the geodesic passes through some spatial point P. Since the

spatial part of the metric is spatially homogeneous and isotropic we can, without

loss of generality, take the spatial origin of the coordinate system, i.e. = 0, to

be at the point P. This simplifies the analysis considerably.

Consider first the -component u3 . Since the metric is independent of , we

have u3 = 0 so that u3 is constant along the geodesic. But

™

u3 = g33 u3 = ’R2 t S 2 sin2 u3

so that u3 = 0 at the point P where = 0. Thus u3 = 0 along the path and so also

we have u3 = ™ = 0. Hence, along the geodesic,

= constant

For the -component, we have

u2 =

™ 1

(14.13)

2g uu

2

The only component of g that depends on x2 = is g33 , but the contribution of

the corresponding term in (14.13) vanishes since u3 = 0. Thus u2 = 0 and so u2

™

is constant along the geodesic. Again

u2 = g22 u2 = ’R2 t S 2 u2

= 0 , and so u2 is zero along the geodesic, as is u2 , so that

which vanishes at P

= constant

For the r-component,

u1 =

™ 1

(14.14)

1g uu

2

367

14.9 The cosmological redshift

We have u2 = u3 = 0, while g00 and g11 are independent of . Thus, u1 = 0 so

™

that u1 is constant along the geodesic, so u1 = g11 u1 must be constant. Thus, we

have

R2 t ™ = constant (14.15)

Finally, u0 can be found from the appropriate normalisation condition,

u u = c2 for massive particles or u u = 0 for photons. Thus, we have

§

⎪1 + R t ™

2 2

⎪

⎨ for a massive particle

c2

™=2

t

⎪ R2 t ™ 2

⎪

© for a photon

c2

14.9 The cosmological redshift

We can use the results of the last section to derive the cosmological redshift.

Suppose that a photon is emitted at cosmic time tE by a comoving observer with

fixed spatial coordinates E E E and that the photon is received at time

tR by another observer at fixed comoving coordinates. We may take the latter

observer to be at the origin of our spatial coordinate system.

For a photon one can choose an affine parameter such that the 4-momentum

is p = x . From our above discussion, d = d = 0 along the photon geodesic,

™

or equivalently p2 = p3 = 0, and (14.15) shows that p1 is constant along the

geodesic. Since the photon momentum is null, we also require that g p p = 0,

which reduces to

1 1

p0 2 ’ 2 p1 2 = 0

c2 Rt

from which we find p0 = cp1 /R t .

In Appendix 9A we showed that, for an emitter and receiver with fixed spatial

coordinates, the frequency shift of the photon is given, in general, by

1/2

pR g00 E

=0

R

(14.16)

p0 E g00 R

E

For the FRW metric we have g00 = c2 , and so we find immediately that

R tR

E

1+z ≡ = (14.17)

R tE

R

Thus we see that if the scale factor R t is increasing with cosmic time, so that the

universe is expanding, then the photon is redshifted by an amount z. Conversely,

368 The Friedmann“Robertson“Walker geometry

if the universe were contracting then the photon would be blueshifted. Only if the

universe were static, so that R = constant, would there be no frequency shift.

In fact, we may also arrive at this result directly from the FRW metric. Since

ds = d = d = 0 along the photon path, from (14.11) we have, for an incoming

photon,

tR c dt E

= d

Rt 0

tE

Now, if the emitter sends a second light pulse at time tE + tE , which is received

at time tR + tR , then

tR + t R tR

c dt c dt

E

= d=

Rt Rt

tE + tE 0 tE

from which we see immediately that

tR + tR tE + t E

c dt c dt

=

Rt Rt

tR tE

Assuming that tE and tR are small, so that R t can be taken as constant in

both integrals, we have

tR tE

=

R tR R tE

Considering the pulses to be the successive wavecrests of an electromagnetic

wave, we again find that

tR R tR

E

1+z ≡ = =

tE R tE

R

14.10 The Hubble and deceleration parameters

In a common notation we shall write the present cosmic time, or epoch, as t0 . Thus

photons received today from distant galaxies are received at t0 . If the emitting

galaxy is nearby and emits a photon at cosmic time t, we can write t = t0 ’ t,

where t t0 . Thus, let us expand the scale factor R t as a power series about

the present epoch t0 to obtain

R t = R t0 ’ t0 ’ t

™ ¨

= R t0 ’ t0 ’ t R t0 + 2 t0 ’ t 2 R t0 ’ · · ·

1

= R t0 1 ’ t0 ’ t H t0 ’ 2 t0 ’ t 2 q t0 H 2 t0 ’ · · ·

1

(14.18)

369

14.10 The Hubble and deceleration parameters

where we have introduced the Hubble parameter H t and the deceleration param-

eter q t . These are given by

™

Rt

Ht ≡

Rt

(14.19)

¨

RtRt

q t ≡’

™

R2 t

where the dot corresponds to differentiation with respect to cosmic time t. It

should be noted that these definitions are valid at any cosmic time. The present-day

values of these parameters are usually denoted by H0 ≡ H t0 and q0 ≡ q t0 .

Using these definitions, we can write the redshift z in terms of the ˜look-back

time™ t ’ t0 as

R t0 ’1

z= ’ 1 = 1 ’ t0 ’ t H0 ’ 2 t0 ’ t 2 q0 H0 ’ · · · ’1

2

1

Rt

and, assuming that t0 ’ t t0 , we have

z = t0 ’ t H0 + t0 ’ t 1 + 2 q0 H0 + · · ·

2 2

1

(14.20)

Since it is the redshift that is an observable quantity, it is more useful to invert

the above power series to obtain the look-back time t0 ’ t in terms of z. Thus for

z 1 we have

’1 ’1

t0 ’ t = H0 z ’ H0 1 + 2 q0 z2 + · · ·

1

(14.21)

It is worth noting that, as one might expect in this approximation, the relations

(14.20) and (14.21) depend only on the present-day values H0 and q0 of the Hubble

and deceleration parameters and hence may be evaluated without knowledge of

the complete expansion history R t of the universe.

Using the Taylor expansion (14.18), we can also obtain an approximate expres-

sion for the -coordinate of the emitting galaxy, which is given by

t0 t0

c dt

cR’1 1 ’ t0 ’ t H0 ’ · · · ’1

= = dt

0

Rt

t t

Assuming once more that t0 ’ t t0 , we have

= cR’1 t0 ’ t + 2 t0 ’ t 2 H0 + · · ·

1

(14.22)

0

370 The Friedmann“Robertson“Walker geometry

We may now substitute for the look-back time t0 ’ t in this result using (14.21),

to obtain an expression for the -coordinate of the emitting galaxy in terms of its

redshift (assuming z 1), which reads

c

= z ’ 2 1 + q0 z2 + · · ·

1

(14.23)

R0 H0

Once again, in this approximation the results (14.22) and (14.23) only depend

on the present-day values H0 and q0 and may be evaluated without knowing the

expansion history of the universe.

From the FRW metric, we see that the proper distance d to the emitting galaxy3

at cosmic time t0 is d = R0 . Thus, for very nearby galaxies, d ≈ c t0 ’ t .

Moreover, from (14.20), in this case z ≈ t0 ’ t H0 . So, if we were to interpret

the cosmological redshift as a Doppler shift due to a recession velocity v of the

emitting galaxy, we would obtain

v = cz = H0 d (14.24)

which is approximately valid for small z. The galaxies will therefore appear to

recede from us with a recession speed proportional to their distance from us.

This is, of course, Hubble™s law, named after Edwin Hubble, who discovered

the expansion of the universe in 1929 by comparing redshifts with distance

measurements to nearby galaxies (derived from the period“luminosity relation of

Cepheid variables). His results suggested a linear recession law, as in (14.24).

This was an amazing result. It implies that the universe started off at high density

at some finite time in the past. You will notice from (14.24) that the Hubble

˜constant™ has the dimensions of inverse time. As we will see later, the quantity

1/H0 gives the age of the universe to within a factor of order unity. It is clear

that, in general, the Hubble parameter will vary with cosmic time t and hence

with redshift z. By combining the expressions (14.18), (14.19) and (14.21), we

can obtain an expression for how the Hubble parameter varies with z for small

redshift,

H z = H0 1 + 1 + q0 z ’ · · · (14.25)

So far, we have been considering the low-z limit. Having introduced the Hubble

parameter, however, we may use it to derive useful general expressions for the

3

In order to measure the proper distance d, one would in fact have to arrange for all the ˜civilisations™ along

the route to the galaxy to lay out measuring rods at the same cosmic time t0 . This could be synchronised by,

for example, requiring the temperature of the cosmic microwave background or the mean matter density of

the universe to have a given value. We will discuss more practical measures of distance shortly.

371

14.11 Distances in the FRW geometry

look-back time to an emitting galaxy, and for its -coordinate, as functions of the

redshift z of the received photon. In general, we have

R0 R0

™

dz = d 1 + z = d =’ R dt = ’ 1 + z H z dt

R R

which provides a very useful relation between an interval dz in redshift and the

corresponding interval dt in cosmic time. Thus, we can write the look-back time as

t0 z dz

t ’ t0 = dt = (14.26)

1+z H z

0

t

and the galaxy™s -coordinate is given by

t0 z

c dt c dz

= = (14.27)

Rt R0 Hz

0

t

It is clear, however, that in order to evaluate either of these integrals we must

know how H z varies with z, which requires knowledge of the evolution of the

scale factor R t .

14.11 Distances in the FRW geometry

Distance measures in an expanding universe can be confusing. For example, let us

consider the distance to some remote galaxy. The light received from the galaxy

was emitted when the universe was younger, because light travels at a finite

speed c. Evidently, as we look at more distant objects, we see them as they were

at an earlier time in the universe™s history when proper distances were smaller,

since the universe is expanding. What, therefore, do we mean by the ˜distance™ to

a galaxy? In fact, interpreting and calculating distances in an expanding universe

is straightforward, but one must be clear about what is meant by ˜distance™.

From the FRW metric

ds2 = c2 dt2 ’ R2 t d + S2 + sin2 d

2 2 2

d

we can define a number of different measures of distance. The parameter is

a comoving coordinate that is sometimes referred to as the coordinate distance,

whereas the proper distance to an object at some cosmic time t is d = R t , but

this cannot be measured in practice. Thus, we must look for alternative ways of

defining the distance to an object. The two most important operationally defined

distance measures are the luminosity distance and the angular diameter distance.

These distance measures form the basis for observational tests of the geometry of

the universe.

372 The Friedmann“Robertson“Walker geometry

Luminosity distance

In an ordinary static Euclidean universe, if a source of absolute luminosity L

(measured in W = J s’1 ) is at a distance d then the flux that we receive (measured

in W m’2 ) is F = L/ 4 d2 . Now suppose that we are actually in an expanding

FRW geometry, but we know that the source has a luminosity L and we observe

a flux F . The quantity

1/2

L

dL = (14.28)

4F

is called the luminosity distance of the source. This is an operational definition,

and we must now investigate how to express it in terms of the FRW metric.

Consider an emitting source E with a fixed comoving coordinate relative to

an observer O (note that, by symmetry, the emitter would assign the same value of

to the observer). We assume that the absolute luminosity of E as a function of

cosmic time is L t and that the photons it emits are detected by O at cosmic time

t0 . Clearly, the photons must have been emitted at an earlier time te . Assuming

the photons to have been emitted isotropically, the radiation will be spread evenly

over a sphere centred on E and passing through O (see Figure 14.2). The proper

area of this sphere is

A = 4 R2 t0 S 2

However, each photon received by O is redshifted in frequency, so that

e

=

0

1+z

O E

t = t0

t = te

Figure 14.2 Geometry associated with the definition of luminosity distance

(with one spatial dimension suppressed).

373

14.11 Distances in the FRW geometry

and, moreover, the arrival rate of the photons is also reduced by the same factor.

Thus, the observed flux at O is

1

L te

F t0 =

1+z

2 2

4 R0 S

The luminosity distance defined above is now evaluated as

dL = R0 S 1+z (14.29)

This is an important quantity, which can be used practically, but note that it

depends on the time history of the scale factor through the dependence on .

Angular diameter distance

Another important distance measure is based upon the notion of the existence of

some standard-length ˜rods™, whose angular diameter we can observe. Suppose

that a source has proper diameter . Then, in Euclidean space, if it were at a

= D/d. In an FRW geometry,

distance d it would subtend an angular diameter

we thus define the angular diameter distance to an object to be

dA = (14.30)

This is again an operational definition, and we now investigate how to express it

in terms of the FRW metric.

Suppose we have two radial null geodesics (light paths) meeting at the observer

at time t0 with an angular separation , having been emitted at time te from

a source of proper diameter at a fixed comoving coordinate (assuming,

for simplicity, that the spatial axes are oriented so that = constant along the

photon paths); see Figure 14.3. To obtain a clearer view of the specification of

the coordinates, we may look vertically down the worldline of O and define the

coordinates as in Figure 14.4. From the angular part of the FRW metric we have

= R te S

so that

R te R t0 S

dA = R te S = R t0 =

S

1+z

R t0

Thus the angular diameter distance is given by

R0 S

dA = (14.31)

1+z

374 The Friedmann“Robertson“Walker geometry

O E

t = t0

t = te

Figure 14.3 Geometry associated with the definition of angular diameter

distance (with one spatial dimension suppressed).

(te, χ, θ + ∆θ, φ)

∆θ

(t0, 0, 0, 0)

(te, χ, θ, φ)

Figure 14.4 Specification of the coordinates in the definition of angular diameter

distance.

This differs from the luminosity distance dL by a factor 1 + z 2 , emphasizing

again that ˜distance™ depends on definition. Again, because of the -dependence

we need to know the time history of the scale factor R t to evaluate dA .

14.12 Volumes and number densities in the FRW geometry

The interpretation of cosmological observations often requires one to determine

the volume of some three-dimensional region of the FRW geometry. Consider a

comoving cosmological observer, whom we may take to be at the origin = 0

of our comoving coordinate system. From the FRW metric

ds2 = c2 dt2 ’ R2 t d + S2 + sin2 d

2 2 2

d

375

14.12 Volumes and number densities in the FRW geometry

we see that, at cosmic time t0 , the proper volume of the region of space lying in

the infinitesmial coordinate range ’ + d and subtending an infinitesmial

solid angle d = sin d d at the observer is

dV0 = R0 d = R3 S 2

R2 S 2 d dd

0 0

For the interval ’ + d in the radial comoving coordinate there exists a

corresponding interval z ’ z + dz in the redshift of objects lying in this radial

range (and also a corresponding cosmic time interval t ’ t + dt within which the

light observed by O at t = t0 was emitted). We may therefore write the volume

element as

d

dV0 = R3 S 2 dz d

0

dz

From (14.27), however, we have

d c

=

dz R0 H z

and so

cR2 S 2 z

dV0 = 0 dz d

Hz

where we have made explicit that is also a function of z. This volume element

is illustrated in Figure 14.5. For an expanding universe, the proper volume of this

O dV0

t = t0, z = 0

χ

χ + dχ

t = t, z = z

t = t “ dt,

z = z + dz

Figure 14.5 Geometry associated with the definition of a proper volume element

dV0 at cosmic time t = t0 (with one spatial dimension suppressed).

376 The Friedmann“Robertson“Walker geometry

comoving region will be smaller at some earlier cosmic time t (which corresponds

to some redshift z). Indeed, using (14.17), we have

cR2 S 2 z

dV0

dV z = =03 (14.32)

dz d

1+z 1+z H z

3

The main use of the result (14.32) is in predicting the number of galaxies (of a

certain type) that one would expect to observe in a given area of sky and redshift

interval, and comparing that result with observations. Suppose, for example, that

the proper number density of galaxies of a certain type at a redshift z is given by

n z . Using (14.32), the total number dN of such objects in the redshift interval

z ’ z + dz and in a solid angle d is

cR2 S 2 z nz

dN = n z dV z = 0 (14.33)

dz d

1+z 3

Hz

The above expression has been arranged to make use of the fact that, if objects are

conserved (so that, once formed, galaxies are not later destroyed), we may write

n z / 1 + z 3 = n0 , where n0 is the present-day proper number density of such

objects; hence the resulting expression is simplified somewhat. As an illustration,

let us consider a population of galaxies which are formed instantaneously at a

redshift z = zf , which are not later destroyed and which have a present-day number

density n0 . From (14.33), the total number of such objects in the whole sky is

S2 z

zf

N =4 cn0 R2 dz

0

Hz

0

Clearly, in order to evaluate this integral one requires knowledge of the expansion

history R t of the universe.

14.13 The cosmological field equations

So far we have investigated only the geometric and kinematic consequences of the

FRW metric. The dynamics of the spacetime geometry is characterised entirely

by the scale factor R t . In order to determine the function R t , we must solve

the gravitational field equations in the presence of matter.

From Chapter 8 the gravitational field equations, in the presence of a non-zero

cosmological constant, are

’ 2g R+ g =’ T

1

R

where = 8 G/c4 . It is, however, more convenient to express the field equations

in the alternative form

=’ T ’ 2 Tg +g

1

(14.34)

R

377

14.13 The cosmological field equations

where T = T . In order to solve these equations, we clearly need a model for the

energy“momentum tensor of the matter that fills the universe. For simplicity, we

shall grossly idealise the universe and model the matter by a simple macroscopic

fluid, devoid of shear-viscous, bulk-viscous and heat-conductive properties. Thus

we assume a perfect fluid, which is characterised at each point by its proper density

and the pressure p in the instantaneous rest frame. The energy“momentum

tensor is given by

p

T = + 2 u u ’ pg (14.35)

c

Since we are seeking solutions for a homogeneous and isotropic universe, the

density and pressure p must be functions of cosmic time t alone.

We may perform the calculation in any coordinate system, but the algebra is

simplified slightly by adopting the comoving coordinates x = t r , in

which 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

Thus the covariant components g of the metric are

R2 t

g00 = c g11 = ’ g22 = ’R2 t r 2 g33 = ’R2 t r 2 sin2

2

1 ’ kr 2

Since the metric is diagonal, the contravariant components g are simply the

reciprocals of the covariant components.

The connection is given in terms of the metric by

= 2g + g’

1

g g

from which it is straightforward to show that the only non-zero coefficients are

™ ™ ™

= RR/ c 1 ’ kr 2 = RRr 2 /c = RRr 2 sin2

0 0 0

/c

11 22 33

™ ™

= cR/R = kr/ 1 ’ kr 2 = RRr 2 sin2

1 1 0

/c

01 11 33

= ’r 1 ’ kr 2 sin2

1

33

™

= cR/R = 1/r = sin cos

2 2 2

02 12 33

™

= cR/R = 1/r = cot

3 3 3

03 13 23

where the dots denote differentiation with respect to cosmic time t. We next

substitute these expressions for the connection coefficients into the expression for

the Ricci tensor,

= ’ + ’

R

378 The Friedmann“Robertson“Walker geometry

After some tedious but straightforward algebra, we find that the off-diagonal

components of the Ricci tensor are zero and the diagonal components are given by

¨

R00 = 3R/R

¨ ™

R11 = ’ RR + 2R2 + 2c2 k c’2 / 1 ’ kr 2

¨ ™

R22 = ’ RR + 2R2 + 2c2 k c’2 r 2

¨ ™

R33 = ’ RR + 2R2 + 2c2 k c’2 r 2 sin2

We must now turn our attention to the right-hand side of the field equa-

tions (14.34). In our comoving coordinate system t r , the 4-velocity of the

fluid is simply

=1000

u

which we can write as u = 0. Thus the covariant components of the

4-velocity are

u =g =g = c2 0

0

0

so we can write the energy“momentum tensor (14.35) as

= c2 + p c2 ’ pg

00

T

Moreover, since u u = c2 , contraction of the energy“momentum tensor gives

p2

T =T = + c ’p = c2 ’ 3p

c2

Hence we can write the terms on the right-hand side of the field equations (14.34)

that depend on the energy“momentum as

’ 2 Tg = c2 + p c2 ’2 c2 ’ p g

00

1 1

T

Including the cosmological-constant term, we find that the right-hand side of the

field equations (14.34) vanishes for = . The non-zero components read

’ T00 ’ 2 Tg00 + g00 = ’ 2 c2 + 3p c2 + c2

1 1

’ T11 ’ 2 Tg11 + g11 = ’ c2 ’ p + R2 / 1 ’ kr 2

1 1

2

’ T22 ’ 2 Tg22 + g22 = ’ c2 ’ p + R2 r 2

1 1

2

’ T33 ’ 2 Tg33 + g33 = ’ c2 ’ p + R2 r 2 sin2

1 1

2

Combining these expressions with those for the components of the Ricci tensor,

we see that the three spatial field equations are equivalent, which is essentially

379

14.14 Equation of motion for the cosmological fluid

due to the homogeneity and isotropy of the FRW metric. Thus the gravitational

field equations yield just the two independent equations,

¨

3R/R = ’ 2 c2 + 3p c2 + c2

1

¨ ™

RR + 2R2 + 2c2 k = c2 ’ p + c2 R2

1

2

¨ = 8 G/c4 , we

Eliminating R from the second equation and remembering that

finally arrive at the cosmological field equations

4G 3p

¨

R=’ + R + 1 c2 R

3

c2

3

(14.36)

8G 2 1 22

™

R2 = R + 3 c R ’ c2 k

3

These two differential equations determine the time evolution of the scale factor

R t and are known as the Friedmann“Lema®tre equations. In the case = 0 they

are often called simply the Friedmann equations. We will discuss the solutions

to these equations in various cases in Chapter 15.

14.14 Equation of motion for the cosmological fluid

For any particular model of the universe, the two cosmological field equa-

tions (14.36) are sufficient to determine R t . Nevertheless, we can derive one

further important equation (which is often useful in shortening calculations) from

the fact that energy“momentum conservation requires

=0

T

From our discussion of a perfect fluid in Chapter 8, we know that this requirement

leads to the relativistic equations of continuity and motion for the cosmological

fluid. These equations read

p

u + 2 u =0 (14.37)

c

p uu

+2u u= g ’ 2 (14.38)

p

c c

The second equation is easily shown to be satisfied identically, since both sides

equal zero. This confirms that the fluid particles (galaxies) follow geodesics,

which was to be expected since p is a function of t alone, and so there is no

pressure gradient to push them off geodesics. The continuity equation (14.37) can

be written

p

pu + + 2 u+ u =0

c

380 The Friedmann“Robertson“Walker geometry

is a function of t alone, and with u =

Remembering that 0, this reduces to

™

p 3R

™+ + =0 (14.39)

c2 R

which expresses energy conservation. This equation can in fact be derived directly

¨

from the field equations (14.36) by eliminating R. Thus, only two of the three

equations (14.36) and (14.39) are independent. One may use whichever two

equations are most convenient in any particular calculation.

Equation (14.39) can be simply rearranged into the useful alternative form

™

d R3 3pRR2

=’ (14.40)

c2

dt

Moreover, by transforming the derivative with respect to t to a derivative with

respect to R, one obtains a third useful form of the equation, namely

d R3 3pR2

=’ 2 (14.41)

dR c

Finally, we note that the density and pressure of a fluid are related by its

equation of state. In cosmology, it is usual to assume that (each component of)

the cosmological fluid has an equation of state of the form

p = w c2

where the equation-of-state parameter w is a constant (in the more exotic cosmo-

logical models one sometimes allows w to be a function of cosmic time t, but we

shall not consider such models here). The energy equation (14.41) can then be

written

d R3

= ’3w R2

dR

This equation has the immediate solution

∝ R’3 1+w (14.42)

which gives the evolution of the density as a function of the scale factor R t .

Note that in general c2 is the energy density of the fluid. In particular w = 0

for pressureless ˜dust™, w = 1 for radiation and w = ’1 for the vacuum (if the

3

cosmological constant = 0; see Section 8.7).

381

Exercises

14.15 Multiple-component cosmological fluid

Suppose that the cosmological fluid in fact consists of several distinct components

(for example, matter, radiation and the vacuum) that do not interact except through

their mutual gravitation. Let us suppose further that each component can be

modelled as a perfect fluid, as discussed above.

The energy“momentum tensor of a multiple-component fluid is given simply by

=

T T i

i

where i labels the various fluid components. Since each component is modelled

as a perfect fluid, we have

p

+ i u u ’ pi g

T=i i

c

p

= i i+ i u u ’ pg

ii

c

Thus, the multicomponent fluid can itself be modelled as a single perfect fluid with

= p=

and (14.43)

pi

i

i i

which can be substituted directly into our cosmological field equations (14.36).4

Moreover, since we are assuming that the fluid components are non-interacting,

conservation of energy and momentum requires that the condition

=0

T i

holds separately for each component. Then each fluid will obey an energy equation

of the form (14.39). Thus, if wi = pi / i c2 then the density of each fluid evolves

independently of the other components as

∝ R’3 1+wi (14.44)

i

Exercises

14.1 In an N -dimensional manifold, consider the tensor

Rijkl = K gik gjl ’ gil gjk

where K may be a function of position. Show that this tensor satisfies the symmetry

properties and the cyclic identity of the curvature tensor. Show that, in order to

satisfy the Bianchi identity, one requires K to be constant if N > 2.

4

Unfortunately, if the individual equation-of-state parameters wi are constants one cannot, in general, define

a single effective equation-of-state parameter w = p/ c2 that is also independent of cosmic time t.

382 The Friedmann“Robertson“Walker geometry

14.2 For a 3-space with a line element of the form

= B r dr 2 + r 2 d + r 2 sin2 d

2 2 2

d

show that the non-zero components of the Ricci tensor are

1 dB 1 r dB

Rrr = ’ R= ’1’ 2 = R sin2

R

2B dr

rB dr B

Hence show that if the 3-space is maximally symmetric then B r must take the

form

1

Br =

A ’ Kr 2

where A and K are constants.

14.3 In a four-dimensional Euclidean space with ˜Cartesian™ coordinates w x y z , a

3-sphere of radius R is defined by w2 + x2 + y2 + z2 = R2 . Show that the metric

on the surface of the 3-sphere can be written in the form

= R2 d + sin2 + sin2 d

2 2 2 2

d d

Show that the total volume of the 3-sphere is V = 2 2 R3 .

14.4 In a four-dimensional Minkowski space with ˜Cartesian™ coordinates w x y z ,

a 3-hyperboloid is defined by w2 ’ x2 ’ y2 + ’z2 = R2 . Show that the metric on

the surface of the 3-hyperboloid can be written in the form

= R2 d + sinh2 + sin2 d

2 2 2 2

d d

Show that the total volume of the 3-hyperboloid is infinite.

14.5 At cosmic time t1 , a massive particle is shot out into an expanding FRW universe

with velocity v1 relative to comoving cosmological observers. At a later cosmic

time t2 the particle has a velocity v2 with respect to comoving cosmological

observers. Show that, at any intermediate cosmic time t, the velocity of the particle

as measured by a comoving cosmological observer is

d

v t =R t

dt

Hence show that

v2 v2 R t1

=

v1 v1 R t2

where v = 1 ’ v2 /c2 ’1/2 and R t is the scale factor at cosmic time t. By

considering the particle momentum, show that as v1 ’ c the photon redshift

formula is recovered.

14.6 In the limit z 1, show that the look-back time for a galaxy with redshift z is

’1 ’1

t0 ’ t = H0 z ’ H0 1 + 2 q0 z2 + · · ·

1

383

Exercises

Show also that, in this limit, the variation of the Hubble parameter with redshift

is given by

H z = H0 1 + 1 + q0 z ’ · · ·

14.7 In a spatially flat FRW geometry, show that the luminosity and angular diameter

distances to an object of redshift z are given, in the limit z 1, by

c

dL = z + 2 1 ’ q0 z 2 + · · ·

1

H0

c

dA = z ’ 2 3 + q0 z 2 + · · ·

1

H0

Hence show that the angular diameter of a standard object can increase as z

increases. Do these results still hold in a spatially curved FRW geometry?

14.8 In the FRW geometry, show that the look-back time to a nearby object at proper

distance d is

d H d2

t0 ’ t = ’ 0 2 + · · ·

2c

c

Hence show that the redshift to the object is

H 0 d 1 + q0 H0 d 2

2