Schutz B. A first course in general relativity

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363 12.4 Physical cosmology: the evolution of the universe we observe

distribution. Simulations show that only if the random velocities of dark matter particles

were small could the small irregularities grow in size fast enough to trap baryonic gas and

make it form galaxies. The matter had, therefore, to be cold, and we call this model of

galaxy formation the cold dark matter model. The standard cosmological model is called

CDM: cold dark matter with a cosmological constant.

The parameters of our cosmology – the cosmological constant, matter fraction, and so

on – leave their imprint on these ﬂuctuations. The ﬂuctuations occur on all length-scales,

but they do not have the same size on different scales. The angular spectrum of ﬂuctuations

contains a rich amount of information about the cosmological parameters, and it is here that

we ﬁnd the constraints shown in Fig. 12.4.

Dark matter and galaxy formation: the universe

after decoupling

Going forward from the time of decoupling, physicists have simulated the evolution of

galaxies and clusters of galaxies from the initial perturbations. The density perturbations

in the dark matter grow slowly, as they have been doing since before decoupling. But before

decoupling, the baryonic matter could not respond very much to them, because it remained

in equilibrium with the photons. Once the baryonic matter took the form of neutral atoms,

it could begin to fall into the gravitational wells created by the dark matter.

Unlike the dark matter, the baryonic matter had the ability to concentrate itself at the bot-

toms of these wells, so that the density irregularities of the baryonic matter soon became

stronger than those of the dark matter. The reason for this is that the baryonic matter was

charged: as the atoms fell into the potential wells, they collided with one another and colli-

sionally excited their electrons into higher energy levels. The density was low enough that

the electrons then decayed back to the ground state by radiating away the excess energy.

Now that decoupling had happened, this was a one-way street, a way of extracting energy

from the baryonic gas and allowing it to clump inside the dark-matter wells. Astronomers

call this process cooling, even though the net effect of radiating energy away is to make

the baryonic matter hotter as it falls deeper into the potential wells!

The dark matter itself is not charged, so it cannot form such strong contrast. It forms

extended ‘halos’ around galaxies today, as we shall see below. Extensive numerical sim-

ulations using supercomputers show that the clumps of baryonic gas eventually began to

condense into basic building-block clumps of a million solar masses or so, and then these

began to merge together to form galaxies. So although images of the universe seem to show

a lot of well-separated galaxies, the fact is that most of the objects we see were formed from

many hundreds or more of mergers. Mergers are still going on: astronomers have discov-

ered a fragment of several million solar masses that is currently being integrated into our

own Milky Way galaxy, on the other side of the center from our location. The unusual

star cluster Omega Centauri seems to be the core of such a mini-galaxy that was absorbed

by the Milky Way long ago. As mentioned in § 11.4, astronomers have found a massive

black hole in its center. And the Magellanic Clouds, easily visible in the sky in the southern

hemisphere, may be on their way to merging into the Milky Way.

364 Cosmology

In 2012 the European Space Agency plans to launch the astrometry satellite GAIA,

which will measure the positions and proper motions of a billion stars to unprecedented

accuracy. One of the many goals of GAIA will be to map the velocity ﬁeld of the Milky

Way, because astronomers expect that ancient mergers will still be apparent as ‘streams’ of

stars moving differently from others.

Sometime during this hierarchy of merging structures, the density of the gas got high

enough for the ﬁrst generation of stars to form. These are called Population III stars,

and they were unlike anything we see today. Since the gas from which they formed was

composed only of hydrogen and helium, with none of the heavier elements that were

made by this generation of stars and incorporated into the next, these stars were much

more massive. With masses between 100 and 1000 M



, they became very hot, evolved

quickly, generated heavier elements, blew much of their outer layers and much of the

new elements away with strong stellar winds, and then very likely left behind a large

population of black holes. The ultraviolet light emitted by these stars seems to have

re-ionized much of the hydrogen in the universe, which had been neutral since decou-

pling. All of this happened between redshifts of 10 and 20, the epoch of re-ionization.

This was the epoch of ﬁrst light for galaxies, the ﬁrst time that the expanding universe

would have looked optically a bit like it does today, if there had been anyone there to

observe it!

After re-ionization and the generation of heavier elements by the ﬁrst stars, the continued

expansion of the universe led galaxies to be more and more isolated from each other, and

they became nurseries for one generation of new stars after another, each with a bit more of

the heavier elements. Our Sun, whose age is about 5 billion years, was formed at a redshift

between 0.3 and 0.4.

One of the puzzles in this scenario of hierarchical structure formation is the appearance

of massive black holes in the centers of apparently all galaxies. Astrophysicists do not yet

know whether these formed directly by the collapse of huge gas clouds as the baryonic mat-

ter was accumulating in the potential wells, or if they arose later by the growth and merger

of intermediate-mass black holes left behind by Population III stars. The fact that galaxy

evolution is dominated by mergers suggests that the LISA gravitational wave observatory

(see the previous chapter) will have an abundance of black-hole mergers to study.

Although physicists do not know what kinds of particles (or indeed, more massive struc-

tures like black holes) might make up the dark matter, and although the dark matter emits

no electromagnetic radiation, it is possible to make indirect observations of it. One way it

shows itself is in the rotation curves of spiral galaxies: in the outer regions of spirals the

orbital speeds of gas and stars are much greater than could be accounted for by the grav-

itational pull of the visible stars. Indeed, observations suggest that in most spiral galaxies

the total mass of the galaxy inside a given radius continues to increase linearly with radius

even well outside the visible limits of the galaxy. This is the footprint of the dark matter

density concentration within which the galaxy formed. The other way of getting indirect

evidence for dark matter is through gravitational lensing. Astronomers have many images

like Fig. 11.8, as we discussed in the previous chapter. Both of these methods involve

essentially using gravity to ‘weigh’ the dark matter.

365 12.4 Physical cosmology: the evolution of the universe we observe

The early universe: fundamental physics meets cosmology

If we go back in time from the moment when the radiation and matter densities were equal,

then we are in a radiation-dominated universe. Eventually, when we are only about about

200s away from the Big Bang, the temperature rises to about 50 keV, the mass differ-

ence between a neutron and a proton. This is the temperature at which nuclear reactions

among protons and neutrons come into equilibrium with each other. Above this energy

all the baryons were free. As the universe cooled through this temperature, some heavier

elements were formed: mainly

He, but also small amounts of

He, Li, B, and traces of

other light elements. All the lithium and helium we see in the universe today was formed

at this time: processes inside stars tend to destroy light elements, not make them. The

ﬁnal abundances of these elements is very sensitive to the rate at which the universe was

expanding at this time. From extensive computations of the reaction networks, astrophysi-

cists have been able to show that the universe contains no signiﬁcant amounts of light or

massless particles other than photons and the three types of neutrinos that are known from

particle-physics experiments. If there were others, their self-gravity would have slowed

the universe more strongly, which means that to match the Hubble expansion today, a

universe with such extra particles would have had to have been expanding faster than

the nucleosynthesis computations allow. If there are extra particles, they must have an

energy density today that is signiﬁcantly less than that of the photons, which have 

∼

−5

. Gravitational waves from the Big Bang must, therefore, satisfy this nucleosynthesis

bound.

Notice that we are already within 200 s of the Big Bang in this discussion, and still we

are in the domain of well-understood physics. At about 1 s, the temperature was around 500

keV, which is the mass of the electron. In this plasma, therefore, there was an abundance

of electrons and positrons, constantly annihilating against one another and being created

again by photons. Much earlier than this the rest mass of the electrons is negligible, so

the number and energy density of photons and of electrons and positrons was similar. As

the universe expanded through this 500 keV temperature and cooled, the electrons and

positrons continued to annihilate, but no more were produced. After a few seconds, there

were apparently essentially no positrons, and there was about one electron for every 10

photons. This ratio of 10

is called the speciﬁc entropy of the universe, a measure of its

disorder.

Why were there any electrons left at all after this annihilation phase? Why, in other

words, was there any matter left over to build into planets and people? Extensive observa-

tional programs, coupled to numerical simulations, have convincingly established that there

is no ‘missing’ antimatter hidden somewhere, no anti-stars or anti-galaxies: signiﬁcant

amounts of antimatter just do not exist any more. Clearly, during the equilibrium plasma

phase, electrons and positrons were not produced in equal numbers. The same must also

have happened at a much earlier time, when protons and antiprotons were in equilibrium

with the photon gas, when the temperature was above a few hundred MeV (only 10 μsafter

the Big Bang): something must have favored protons over antiprotons in the same ratio as

for electrons over positrons, so that the overall plasma remained charge-neutral. This is one

366 Cosmology

of the central mysteries of particle physics. Something in the fundamental laws of physics

gave a slight preference to electrons. Nature has a mattter-antimatter asymmetry.

At times earlier than 10

−5

s, there are no protons or neutrons, just a plasma of quarks and

gluons, the fundamental building blocks of baryons. According to particle theory, quarks

are ‘conﬁned’, so that we never see a free one detached from a baryon. But at high enough

temperatures and densities, the protons overlap so much that the quarks can stay conﬁned

and still behave like free particles.

We can even push our perspective another step higher in temperature, to around 10

TeV, which is the frontier for current accelerators. Physics is not well understood at these

energies, and the Large Hadron Collider at CERN in Geneva will soon (end of 2008) start

doing experiments to look for the Higgs particle and to ﬁnd evidence for supersymmetry.

Both of these are theoretical constructs designed to solve deep theoretical problems in

fundamental physics. In particular, supersymmetry has the advantage of making it easier

within particle physics theories to predict a value of the cosmological constant in the range

of what we observe. If supersymmetry is found, it will encourage the idea that the dark

energy can be accommodated within the current framework of fundamental physics theory.

At 10 TeV, we are just 10

−14

s after the Big Bang. Although physics is poorly known, it is

unlikely that anything happened at this point that will challenge the picture presented here

of what happened later.

In this scheme there is one important thing that is missing: we have mentioned no mech-

anism in any of this physics for generating the density irregularities in the dark matter that

led to galaxy formation. We observe them in the microwave background, and we know

that they are needed in order to trigger all the processes that eventually led to our own

evolution. But even at 10

−14

s, they are simply an initial condition: they have to be there,

at a much smaller amplitude than we see them in the microwave background because the

irregularities grow as the universe expands, but they must be there. And known physics

has no explanation for how they got there. The exciting answer to this problem lies in the

scenario of inﬂation, which we come back to below.

But the density perturbations are not the only feature of the early universe that is not

explained. Right from the start we have assumed homogeneity and isotropy, based on

observations. We have had to accept a small amount of inhomogeneity in the density dis-

tribution at the time of decoupling, but that was inevitable: our assumption of homogeneity

does not hold on small scales, where galaxies and planets form. On the large scale, we need

to ask, why is the universe so smooth? This is particularly difﬁcult to explain because in

the standard Big Bang, there is no physical process that could work to smooth things out.

This needs some explanation.

Consider the primordial abundance of helium. It was ﬁxed when the universe was only

ﬁve minutes old. When we look with our telescopes in opposite directions on the sky, we

can see distant quasars and galaxies that appear to have the same element abundances as

we do, and yet they are so far away from each other that they could not have been in

communication: they are outside each other’s particle horizon in the standard cosmologi-

cal model. One way to ‘explain’ this is simply to postulate that the initial conditions for

the Big Bang were the same everywhere, even in causally disconnected regions. But it

would be more satisfying physically if some process could be found that enabled these

367 12.4 Physical cosmology: the evolution of the universe we observe

regions to communicate with each other at a very early time, even though they appear to be

disconnected. Again, inﬂation offers such a mechanism. Inﬂation. The basic idea of inﬂa-

tion (Starobinsky 1980, Guth 1981, Linde 1982) is that, at a very early time like 10

−35

the universe was dominated by a large positive cosmological constant, much larger than

we have today, but one that was only temporary: it turned on at some point and then turned

off again, for reasons we will discuss below. But during the time when the universe was

dominated by this constant, the matter and curvature were unimportant, and the universe

expanded according to the simple law

πρ



⇒

= 1/τ



, (12.64)

which is an exponential law with a growth time





8πρ





1/2

. (12.65)

If this exponential expansion lasted 20 or 30 e-foldings, then a region of very small size

could have been inﬂated into the the size of a patch that would be big enough to become

the entire observable universe today. The idea is that, before inﬂation, this small region

had been smoothed out by some physical process, which was possible because it was small

enough to do this even in the time available. Then inﬂation set in and expanded it into the

initial data for our universe.

This would explain the homogeneity of what we see: everything did indeed come from

the same patch. And inﬂation also explains the ﬂuctuation in the cosmic microwave back-

ground. Here we have to go into more detail about the mechanism for inﬂation. Attempts

to compute the cosmological constant today focus on the vacuum energy of quantum

ﬁelds, which we used in order to explain the Hawking radiation in the previous chapter.

The vacuum energy is attractive for this purpose because the vacuum must be invari-

ant under Lorentz transformations: there should not be any preferred observer for empty

space in quantum theory. This means that any stress-energy tensor associated with the vac-

uum must be Lorentz invariant. Now, the only Lorentz-invariant symmetric tensor ﬁeld of

type





is the metric tensor itself, so any vacuum-energy explanation of dark energy will

automatically produce something like a cosmological constant, proportional to the metric

tensor.

In some models of the behavior of the physical interactions at very high energies, beyond

the TeV scale, it is postulated that there is a phase transition in which the nature of the

vacuum changes, and a large amount of vacuum energy is released in the form of a cosmo-

logical constant, powering inﬂation. But this is a dynamical process, which sets in when

the phase change occurs and then stops when the energy is converted into the real energy

that eventually becomes the particles and photons in our universe. So for a limited time,

the universe inﬂates rapidly. Now, at the beginning there are the usual vacuum ﬂuctuations,

and the remarkable thing is that the exponential inﬂation ampliﬁes these ﬂuctuations in

much the same way as a nonlinear oscillator can pump up its oscillation amplitude. When

inﬂation ﬁnishes, what were small density perturbations on the quantum scale have become

much larger, classical perturbations.

368 Cosmology

When physicists perform computations with this model, it does very well. The amplitude

of the ﬂuctuations is reasonable, their spectrum matches that which is inferred from the

cosmic microwave background, and the physical assumptions are consistent with modern

views of uniﬁcation among the various interactions of fundamental physics. Inﬂation will

remain a ‘model’ and not a ‘theory’ until either a full theory unifying the nuclear forces

is found or until some key observation reveals the ﬁelds and potential that are postulated

within the model. However, it is a powerful and convincing paradigm, and it is currently

the principal framework within which physicists address the deepest questions about the

early universe.

Beyond general relativity

Inﬂation goes beyond standard physics, making assumptions about the way that the laws

governing the nuclear interactions among particles behave at the very high energies that

obtained in the early universe. But it does not modify gravity: it works within classical

general relativity. Nevertheless, as we have remarked before, the classical theory must

eventually be replaced by a quantum description of gravitation, and the search for this

theory is a major activity in theoretical physics today.

Although no consistent theory has yet emerged, the search has produced a number of

exciting ideas that offer the possibility of new kinds of observations, new kinds of expla-

nations. One approach, called loop quantum gravity, directly attacks the problem of how to

quantize spacetime, ignoring at ﬁrst the other forces in spacetime, like electromagnetism.

On a ﬁne scale, presumably the Planck scale, it postulates that the manifold nature of space-

time breaks down, and the smaller-scale structure is one of nested, tangled loops. There are

a number of variants on this approach, with different structures, but the common idea is

that spacetime is a coarse-grained average over something that has a much richer topology.

These ideas come from the mathematics in a natural way. A recent triumph of loop quan-

tum gravity is to show that the Big Bang may not have been singular after all, that going

backwards in time the universe is able to pass through the Big Bang and become a classical

collapsing universe on the other side (Bojowald 2005).

Even more active, in terms of the number of physicists working in it, is the string-theory

approach to quantum gravity. Here the aim is to unify all the interactions, including gravity,

so the theory includes the nuclear and electromagnetic interactions from the start. String

theory seems to be consistent, in the sense of not having to do artiﬁcial things to get rid

of inﬁnite energies, only in 11 spacetime dimensions. We live in just four of these, so

physicists are beginning to ask questions about the remaining ones.

The ﬁrst assumption was that they never got big: that attached to each point is a Planck-

sized seven-sphere offering the possibility of exiting from our four-dimensional universe

only to things that are smaller than the Planck length. This would not be easy to observe.

But it is also possible that some of these extra dimensions are big, and our four-dimensional

universe is simply a four-surface in this ﬁve- or more-dimensional surrounding. This sur-

face has come to be called a brane, from the word ‘membrane’. String theory on branes

has a special property: electromagnetism and the nuclear forces are conﬁned to our brane,

369 12.5 Further reading

but gravitation can act in the extra dimensions too. This would lead to a modiﬁcation of

the inverse-square-law of Newtonian gravity on short distances, on scales comparable to

a relevant length-scale in the surrounding space. All we can say is that this scale must be

smaller than about a millimeter, from experiments on the inverse square law. But there

are many decades between a millimeter and the Planck scale, and new physics might be

waiting to be discovered anywhere in between (Maartens 2004).

The new physics could take many different forms. Some kind of collision with another

brane might have triggered the Big Bang. A nearby extra brane might have a parallel world

of stars and galaxies, interacting with us only through gravity: shadow matter. There might

be extra amounts of gravitational radiation, due either to shadow matter or to unusual

brane-related initial conditions at the Big Bang.

Although these ideas sound like science ﬁction, they are ﬁrmly grounded in model the-

ories, which are deliberate over-simpliﬁcations of the full equations of string theory, and

which involve deliberate choices of the values of certain constants in order to get these

strange effects. They should be treated as neither predictions nor idle speculation, but

rather as harbingers of the kind of revolutionary physics that a full quantum theory of

gravity might bring us. Experimental hints, from high-precision physics, or observational

results, perhaps from gravitational waves or from cosmology, might at any time provide

key clues that could point the way to the right theory.

12.5 F urt h er re ad i ng

The literature on cosmology is vast. In the body of the chapter I have given the principal

references to original results, so I list here some recommended books on the subject.

Standard cosmology is treated in great detail in Weinberg (1972). Cosmological models

in general relativity become somewhat more complex when the assumption of isotropy

is dropped, but they retain the same overall features: the Big Bang, open vs. closed.

See Ryan and Shepley (1975). A well-balanced introduction to cosmology is Heidmann

(1980).

Early discussions on physical cosmology that remain classics include Peebles (1980),

Liang and Sachs (1980), and Balian et al. (1980). More modern is Liddle (2003).

An important current research area is into inhomogeneous cosmologies. See MacCal-

lum (1979). Another subject closely allied to theoretical cosmology is singularity theory:

Geroch and Horowitz (1979), Tipler et al. (1980). See also the stimulating article by

Penrose (1979) on time asymmetry in cosmology.

For greater depth on physical cosmology, see the excellent text by Mukhanov (2005).

For a different point of view on ‘why’ the universe has the properties it does, see the book

by Barrow and Tipler (1986) on the anthropic principle.

For popular-level cosmology articles written by research scientists, see the Einstein

Online website: http://www.einstein-online.info/en/.

370 Cosmology

12.6 E xe rc is e s

1 Use the metric of a two-sphere to prove the statement associated with Fig. 12.1, that

the rate of increase of the distance between any two points as the sphere expands (as

measured on the sphere!) is proportional to the distance between them.

2 The astronomer’s distance unit, the parsec, is deﬁned to be the distance from the Sun

to a star whose parallax is exactly one second of arc. (The parallax of a star is half the

maximum change in its angular position as measured from Earth as Earth orbits the

Sun.) Given that the radius of Earth’s orbit is 1 AU = 10

m, calculate the length of

one parsec.

3 Newtonian cosmology.

(a) Apply Newton’s law of gravity to the study of cosmology by showing that the gen-

eral solution of ∇

 = 4πρ for ρ = const. is a quadratic polynomial in Cartesian

coordintes, but is not necessarily isotropic.

(b) Show that if the universe consists of a region where ρ = const., outside of which

there is vacuum, then, if the boundary is not spherical, the ﬁeld will not be isotropic:

the ﬁeld will show signiﬁcant deviations from sphericity throughout the interior,

even at the center.

determine the shape of the boundary, even if the boundary is far outside our particle

horizon.

4 Show that if h

) = f (t

, t

) for all i and j in Eq. (12.3), then distances between

galaxies would increase anisotropically: the Hubble law would have to be written as

= H

(12.66)

for a matrix H

not proportional to the identity.

5 Show that if galaxies are assumed to move along the lines x

= const., and if we see

the local universe as homogeneous, then g

in Eq. (12.5) must vanish.

6 (a) Prove the statement leading to Eq. (12.8), that we can deduce G

of our three-

spaces by setting  to zero in Eqs. (10.15)–(10.17).

(b) Derive Eq. (12.9).

7 Show that the metric, Eq. (12.7), is not locally ﬂat at r = 0 unless A = 0inEq.(12.11).

8 (a) Find the coordinate transformation leading to Eq. (12.19).

(b) Show that the intrinsic geometry of a hyperbola t

− x

− y

− z

= const. > 0

in Minkowski spacetime is identical with that of Eq. (12.19) in appropriate

coordinates.

universe is homogeneous and isotropic.

9 (a) Show that a photon which propagates on a radial null geodesic of the metric,

Eq. (12.13), has energy −p

inversely proportional to R(t).

(b) Show from this that a photon emitted at time t

and received at time t

by observers

at rest in the cosmological reference frame is redshifted by

l + z = R(t

)/R(t

). (12.67)

371 12.6 Exercises

10 Show from Eq. (12.24) that the relationship between velocity and cosmological redshift

for a nearby object (small light-travel-time to us) is z = v, as we would expect for an

object with a recessional velocity v.

11 (a) Prove Eq. (12.29) and deduce Eq. (12.30) from it.

(b) Fill in the indicated steps leading to Eq. (12.31).

12 Derive Eq. (12.42)fromEq.(12.31). Derive Eq. (12.44) from Eq. (12.40).

13 Astronomers usually do not speak in terms of intrinsic luminosity and ﬂux. Rather, they

use absolute and apparent magnitude. The (bolometric) apparent magnitude of a star is

deﬁned by its ﬂux F relative to a standard ﬂux F

m =−2.5 log

(F/F

) (12.68)

where F

= 3 × 10

−8

−2

−1

is roughly the ﬂux of visible light at Earth from the

brightest stars in the night sky. The absolute magnitude is deﬁned as the apparent

magnitude the object would have at a distance of 10 pc:

M =−2.5 log

[L/4π(10 pc)

]. (12.69)

Using Eq. (12.42), with Eq. (12.27), rewrite Eq. (12.34) in astronomer’s language as:

m − M = 5log

(z/10 pc H

) + 1.09(1 −q

)z. (12.70)

Astronomers call this the redshift-magnitude relation.

14 (a) For the Robertson–Walker metric Eq. (12.13), compute all the Christoffel symbols



αβ

. In particular show that the nonvanishing ones are:



, 

1 − kr



θθ

=−r(1 −kr

), 

φφ

=−r(1 − kr

)sin

θ, (12.71)



rθ

= 

rφ

, 

φφ

= sin θ cos θ , 

θφ

= cot θ .

(b) Using these Christoffel symbols, show that the time-component of the divergence

of the stress-energy tensor of the cosmological ﬂuid is

0α

;α

=˙ρ + 3(ρ +p)

. (12.72)

, derive Eq. (12.46).

15 Show from Eq. (12.49) that if the radiation has a black-body spectrum of temperature

T, then T is inversely proportional to R.

16 Use the Christoffel symbols computed in Exer. 14 above to derive Eq. (12.50).

17 Use Eq. (12.46) and the time-derivative of Eq. (12.54) to derive Eq. (12.55)for

R.Make

sure you use the fact that p



=−ρ



18 In this chapter we saw that the negative pressure (tension) of the cosmological con-

stant is responsible for accelerating the universe. But is this a contradiction to ordinary

physics? Does a tension pull inward, not push outward? Resolve this apparent contra-

diction by showing that the net pressure force on any local part of the universe is zero.

Refer to the discussion at the end of § 4.6.

372 Cosmology

19 Assuming the universe to be matter-dominated and to have zero cosmological constant,

show that at times early enough for one to be able to neglect k in Eq. (12.54), the scale

factor evolves with time as R(t) ∝ t

2/3

20 Assume that the universe is matter dominated and ﬁnd the value of ρ



that permits the

universe to be static.

(a) Because the universe is matter-dominated at the present time, we can take ρ

(t) =

/R(t)]

where the subscript ‘0’ refers to the static solution we are looking

for. Differentiate the ‘energy’ equation Eq. (12.54) with respect to time to ﬁnd the

dynamical equation governing a matter-dominated universe:

R =

πρ



R −

πρ

−2

. (12.73)

Set this to zero to ﬁnd the solution



For Einstein’s static solution, the cosmological constant energy density has to be

half of the matter energy density.

(b) Put our expression for ρ

into the right-hand-side of Eq. (12.54) to get an energy-

like expression which has a derivative that has to vanish for a static solution. Verify

that the above condition on ρ



does indeed make the ﬁrst derivative vanish.

R and show that, at the static solution, it is positive. This means that the ’potential’

is a minimum and Einstein’s static solution is stable.

21 Explore the possible futures and histories of an expanding cosmology with nega-

tive cosmological constant. You may wish to do this graphically, by drawing ﬁgures

analogous to Fig. 11.1. See also Fig. 12.4.

22 (Parts of this exercise are suitable only for students who can program a computer.)

Construct a more realistic equation of state for the universe as follows.

(a) Assume that, today, the matter density is ρ

= m ×10

−27

kg m

−3

(where m is of

order 1) and that the cosmic radiation has black-body temperature 2.7 K. Find the

ratio ε = ρ

/ρ

, where ρ

is the energy density of the radiation. Find the number

of photons per baryon, ∼ εm

/kT.

(b) Find the general form of the energy-conservation equation, T

0μ

,μ

= 0, in terms of

ε(t) and m(t).

the present, assuming

R/R = 75 km s

−1

Mpc

−1

today, and assuming there is no

exchange of energy between matter and radiation. Do the integration for m = 0.3,

1.0, and 3.0. Stop the integration when the radiation temperature reaches E

/26.7 k,

where E

is the ionization energy of hydrogen (13.6 eV). This is roughly the tem-

perature at which there are enough photons to ionize all the hydrogen: there is

roughly a fraction 2 ×10

−9

photons above energy E

when kT = E

/26.7, and this

is roughly the fraction needed to give one such photon per H atom. For each m,

what is the value of R(t)/R

at that time, where R

is the present scale factor?

Explain this result. What is the value of t at this epoch?