Contemporary cosmological models are based on the idea that the
universe is pretty much the same everywhere - a stance sometimes
known as the **Copernican principle**. On the face of it, such
a claim seems preposterous; the center of the sun, for example,
bears little resemblance to the desolate cold of interstellar
space. But we take the Copernican principle to only apply on
the very largest scales, where local variations in density are
averaged over. Its validity on such scales is
manifested in a number of different observations, such as number counts
of galaxies and observations of diffuse X-ray and -ray
backgrounds, but is most clear in the 3° microwave background
radiation. Although we now know that the microwave background is not
perfectly smooth (and nobody ever expected that it was), the
deviations from regularity are on the order of 10^{-5} or less,
certainly an adequate basis for an approximate description of
spacetime on large scales.

The Copernican principle is related to two more mathematically
precise properties that a manifold might have: isotropy and homogeneity.
**Isotropy** applies at some specific point in the space, and
states that the space looks the same no matter what direction you
look in. More formally, a manifold *M* is isotropic around a point
*p* if, for any two vectors *V* and *W* in
*T*_{p}*M*, there is an
isometry of *M* such that the pushforward of *W* under the
isometry is parallel with *V* (not pushed forward).
It is isotropy which is indicated by the observations of the
microwave background.

**Homogeneity** is the statement that the metric is the same
throughout the space. In other words, given any two points *p* and
*q* in *M*, there is an isometry which takes *p* into
*q*. Note that there is no necessary relationship between homogeneity
and isotropy; a manifold can be homogeneous but nowhere isotropic
(such as
× *S*^{2} in the usual metric), or it
can be isotropic
around a point without being homogeneous (such as a cone, which is
isotropic around its vertex but certainly not homogeneous). On the
other hand, if a space is isotropic *everywhere* then it is
homogeneous. (Likewise if it is isotropic around one point and
also homogeneous, it will be isotropic around every point.)
Since there is ample observational evidence for
isotropy, and the Copernican principle would have us believe that we
are not the center of the universe and therefore observers elsewhere
should also observe isotropy, we will henceforth assume both
homogeneity and isotropy.

There is one catch. When we look at distant galaxies, they appear to be receding from us; the universe is apparently not static, but changing with time. Therefore we begin construction of cosmological models with the idea that the universe is homogeneous and isotropic in space, but not in time. In general relativity this translates into the statement that the universe can be foliated into spacelike slices such that each slice is homogeneous and isotropic. We therefore consider our spacetime to be × , where represents the time direction and is a homogeneous and isotropic three-manifold. The usefulness of homogeneity and isotropy is that they imply that must be a maximally symmetric space. (Think of isotropy as invariance under rotations, and homogeneity as invariance under translations. Then homogeneity and isotropy together imply that a space has its maximum possible number of Killing vectors.) Therefore we can take our metric to be of the form

(8.1) |

Here *t* is the timelike coordinate, and
(*u*^{1}, *u*^{2}, *u*^{3}) are the
coordinates on ;
is the maximally symmetric
metric on . This formula is a special case of (7.2), which we
used to derive the Schwarzschild metric, except we have scaled *t*
such that *g*_{tt} = - 1. The function *a*(*t*)
is known as the
**scale factor**, and it tells us "how big" the spacelike
slice is at the moment *t*. The coordinates used here,
in which the metric is free of cross terms
*dt* *du*^{i} and the
spacelike components are proportional to a single function of *t*, are
known as **comoving coordinates**, and an observer who stays at constant
*u*^{i} is also called "comoving". Only a comoving
observer will
think that the universe looks isotropic; in fact on Earth we are
not quite comoving, and as a result we see a dipole anisotropy in
the cosmic microwave background as a result of the conventional
Doppler effect.

Our interest is therefore in maximally symmetric Euclidean three-metrics . We know that maximally symmetric metrics obey

(8.2) |

where *k* is some constant, and we put a superscript ^{(3)} on
the Riemann tensor to remind us that it is associated with the
three-metric
, not the metric of the entire spacetime.
The Ricci tensor is then

(8.3) |

If the space is to be maximally symmetric, then it will certainly be spherically symmetric. We already know something about spherically symmetric spaces from our exploration of the Schwarzschild solution; the metric can be put in the form

(8.4) |

The components of the Ricci tensor for such a metric can be obtained from (7.16), the Ricci tensor for a spherically symmetric spacetime, by setting = 0 and = 0, which gives

(8.5) |

We set these proportional to the metric using (8.3), and can solve
for (*r*):

(8.6) |

This gives us the following metric on spacetime:

(8.7) |

This is the **Robertson-Walker metric**. We have not yet
made use of Einstein's equations; those will determine the behavior
of the scale factor *a*(*t*).

Note that the substitutions

(8.8) |

leave (8.7) invariant. Therefore the only relevant parameter
is *k*/| *k*|, and there are three cases of interest: *k*
= - 1,
*k* = 0, and *k* = + 1. The *k* = - 1 case corresponds
to constant
negative curvature on , and is called **open**; the
*k* = 0 case corresponds to no curvature on , and is called
**flat**; the *k* = + 1 case corresponds to positive curvature
on ,
and is called **closed**.

Let us examine each of these possibilities. For the flat case
*k* = 0 the metric on is

(8.9) |

which is simply flat Euclidean space. Globally, it could describe
or a more complicated manifold, such as the
three-torus *S*^{1} × *S*^{1} ×
*S*^{1}. For the closed case *k* = + 1 we can define
*r* = sin to write the metric on as

(8.10) |

which is the metric of a three-sphere. In this case the only
possible global structure is actually the three-sphere (except for
the non-orientable manifold P^{3}). Finally in the open *k* = - 1
case we can set
*r* = sinh to obtain

(8.11) |

This is the metric for a three-dimensional space of constant negative curvature; it is hard to visualize, but think of the saddle example we spoke of in Section Three. Globally such a space could extend forever (which is the origin of the word "open"), but it could also describe a non-simply-connected compact space (so "open" is really not the most accurate description).

With the metric in hand, we can set about computing the connection
coefficients and curvature tensor. Setting
*da*/*dt*,
the Christoffel symbols are given by

(8.12) |

The nonzero components of the Ricci tensor are

(8.13) |

and the Ricci scalar is then

(8.14) |

The universe is not empty, so we are not interested in vacuum solutions to Einstein's equations. We will choose to model the matter and energy in the universe by a perfect fluid. We discussed perfect fluids in Section One, where they were defined as fluids which are isotropic in their rest frame. The energy-momentum tensor for a perfect fluid can be written

(8.15) |

where and *p* are the energy density and pressure
(respectively)
as measured in the rest frame, and *U*^{} is the four-velocity of
the fluid. It is clear that, if a fluid which is isotropic in some
frame leads to a metric which is isotropic in some frame, the two
frames will coincide; that is, the fluid will be at rest in comoving
coordinates. The four-velocity is then

(8.16) |

and the energy-momentum tensor is

(8.17) |

With one index raised this takes the more convenient form

(8.18) |

Note that the trace is given by

(8.19) |

Before plugging in to Einstein's equations, it is educational to consider the zero component of the conservation of energy equation:

(8.20) |

To make progress it is necessary to choose an **equation of
state**, a relationship between and *p*. Essentially all of
the perfect fluids relevant to cosmology obey the simple equation
of state

(8.21) |

where *w* is a constant independent of time. The conservation
of energy equation becomes

(8.22) |

which can be integrated to obtain

(8.23) |

The two most popular examples of cosmological fluids are
known as **dust** and **radiation**. Dust is collisionless,
nonrelativistic matter, which obeys *w* = 0. Examples include
ordinary stars and galaxies, for which the pressure is
negligible in comparison with the energy density. Dust is also
known as "matter", and universes whose energy density is mostly
due to dust are
known as **matter-dominated**. The energy density in matter
falls off as

(8.24) |

This is simply interpreted
as the decrease in the number density of particles as the universe
expands. (For dust the energy density is dominated by the rest
energy, which is proportional to the number density.) "Radiation"
may be used to describe either actual electromagnetic radiation, or
massive particles moving at relative velocities sufficiently close to
the speed of light that they become indistinguishable from photons (at
least as far as their equation of state is concerned).
Although radiation is a perfect fluid and thus has an energy-momentum
tensor given by (8.15), we also know that
*T*_{} can be expressed in
terms of the field strength as

(8.25) |

The trace of this is given by

(8.26) |

But this must also equal (8.19), so the equation of state is

(8.27) |

A universe in which most of the energy density is in the form of
radiation is known as **radiation-dominated**. The energy
density in radiation falls off as

(8.28) |

Thus, the energy density in radiation falls off slightly faster
than that in matter; this is because the number density of photons
decreases in the same way as the number density of nonrelativistic
particles, but individual photons also lose energy as *a*^{-1}
as they redshift, as we will see later. (Likewise, massive but
relativistic particles will lose energy as they "slow down" in
comoving coordinates.) We believe that today the
energy density of the universe is dominated by matter, with
/ 10^{6}. However, in the past
the universe was much smaller, and the energy density in radiation
would have dominated at very early times.

There is one other form of energy-momentum that is sometimes considered, namely that of the vacuum itself. Introducing energy into the vacuum is equivalent to introducing a cosmological constant. Einstein's equations with a cosmological constant are

(8.29) |

which is clearly the same form as the equations with no cosmological constant but an energy-momentum tensor for the vacuum,

(8.30) |

This has the form of a perfect fluid with

(8.31) |

We therefore have *w* = - 1, and
the energy density is independent of *a*, which is what
we would expect for the energy density of the vacuum. Since the
energy density in matter and radiation decreases as the universe
expands, if there is a nonzero vacuum energy it tends to win out over
the long term (as long as the universe doesn't start contracting).
If this happens, we say that the universe becomes **
vacuum-dominated**.

We now turn to Einstein's equations. Recall that they can be written in the form (4.45):

(8.32) |

The = 00 equation is

(8.33) |

and the
= *ij* equations give

(8.34) |

(There is only one distinct equation from
= *ij*, due to
isotropy.) We can use (8.33) to eliminate second derivatives in
(8.34), and do a little cleaning up to obtain

(8.35) |

and

(8.36) |

Together these are known as the **Friedmann equations**,
and metrics of the form (8.7) which obey these equations define
Friedmann-Robertson-Walker (FRW) universes.

There is a bunch of terminology which is associated with the
cosmological parameters, and we will just introduce the basics
here. The rate of expansion is characterized by the **Hubble
parameter**,

(8.37) |

The value of the Hubble parameter at the present epoch is the
Hubble constant, *H*_{0}. There is currently a great deal of
controversy about what its actual value is, with measurements
falling in the range of 40 to 90 km/sec/Mpc. ("Mpc" stands for
"megaparsec", which is
3 × 10^{24} cm.) Note that we
have to divide by *a* to get a measurable quantity, since
the overall scale of *a* is irrelevant. There is also the
**deceleration parameter**,

(8.38) |

which measures the rate of change of the rate of expansion.

Another useful quantity is the **density parameter**,

(8.39) |

where the **critical density** is defined by

(8.40) |

This quantity (which will generally change with time) is called the "critical" density because the Friedmann equation (8.36) can be written

(8.41) |

The sign of *k* is therefore determined by whether is
greater than, equal to, or less than one. We have

The density parameter, then, tells us which of the three Robertson-Walker geometries describes our universe. Determining it observationally is an area of intense investigation.

It is possible to solve the Friedmann equations exactly in
various simple cases, but it is often more useful to know
the qualitative behavior of various possibilities. Let us for
the moment set = 0, and consider the behavior of universes
filled with fluids of positive energy ( > 0) and nonnegative
pressure (*p* 0). Then by (8.35) we must have < 0.
Since we know from observations of distant galaxies that
the universe is expanding ( > 0),
this means that the universe is "decelerating." This is what
we should expect, since the gravitational attraction of the matter
in the universe works against the expansion. The fact that
the universe can only decelerate means that it must have been
expanding even faster in the past; if we trace the evolution
backwards in time, we necessarily reach a singularity at
*a* = 0. Notice that if were exactly zero, *a*(*t*)
would be a straight line, and the age of the universe would be
*H*_{0}^{-1}. Since is actually negative, the universe
must be somewhat younger than that.

This singularity at *a* = 0 is the **Big Bang**.
It represents the creation of the universe from a singular state,
not explosion of matter into a pre-existing spacetime. It might be
hoped that the perfect symmetry of our FRW universes was responsible
for this singularity, but in fact it's not true; the singularity
theorems predict that any universe with > 0 and *p* 0 must
have begun at a singularity. Of course
the energy density becomes arbitrarily high as
*a* 0,
and we don't expect classical general relativity to be an
accurate description of nature in this regime; hopefully a
consistent theory of quantum gravity will be able to fix things up.

The future evolution is different for different values of *k*.
For the open and flat cases, *k* 0, (8.36) implies

(8.42) |

The right hand side is *strictly* positive (since we are
assuming > 0), so never passes through zero. Since
we know that today > 0, it must be positive for all time.
Thus, the open and flat universes expand forever - they are
temporally as well as spatially open. (Please keep
in mind what assumptions go into this - namely, that there
is a nonzero positive energy density. Negative energy density
universes do not have to expand forever, even if they are "open".)

How fast do these universes keep expanding? Consider the
quantity *a*^{3} (which is constant in matter-dominated
universes). By the conservation of energy equation (8.20) we have

(8.43) |

The right hand side is either zero or negative; therefore

(8.44) |

This implies in turn that *a*^{2} must go to zero in an
ever-expanding universe, where
*a* . Thus (8.42)
tells us that

(8.45) |

(Remember that this is true for *k* 0.) Thus, for *k* = - 1
the expansion approaches the limiting value
1,
while for *k* = 0 the universe keeps expanding, but more and more
slowly.

For the closed universes (*k* = + 1), (8.36) becomes

(8.46) |

The argument that
*a*^{2} 0 as
*a*
still applies; but in that case (8.46) would become negative, which
can't happen. Therefore the universe does not expand indefinitely;
*a* possesses an upper bound
*a*_{max}. As *a* approaches
*a*_{max}, (8.35) implies

(8.47) |

Thus is finite and negative at this point, so *a*
reaches
*a*_{max} and starts decreasing, whereupon (since
< 0)
it will inevitably continue to contract to zero - the Big Crunch.
Thus, the closed universes (again, under our assumptions of
positive and nonnegative *p*) are closed in time as well
as space.

We will now list some of the exact solutions corresponding to
only one type of energy density.
For dust-only universes (*p* = 0), it is convenient to define
a **development angle** (*t*), rather than using *t* as
a parameter directly. The solutions are then, for open
universes,

(8.48) |

for flat universes,

(8.49) |

and for closed universes,

(8.50) |

where we have defined

(8.51) |

For universes filled with nothing but radiation,
*p* = ,
we have once again open universes,

(8.52) |

flat universes,

(8.53) |

and closed universes,

(8.54) |

where this time we defined

(8.55) |

You can check for yourselves that these exact solutions have the properties we argued would hold in general.

For universes which are empty save for the cosmological constant,
either or *p* will be negative, in violation of the
assumptions we used earlier to derive the general behavior of
*a*(*t*). In this case the connection between open/closed and
expands forever/recollapses is lost. We begin by considering
< 0. In this case
is negative, and from (8.41) this can only happen if *k* = - 1.
The solution in this case is

(8.56) |

There is also an open (*k* = - 1) solution for > 0, given by

(8.57) |

A flat vacuum-dominated universe must have > 0, and the solution is

(8.58) |

while the closed universe must also have > 0, and satisfies

(8.59) |

These solutions are a little misleading. In fact the three
solutions for > 0 - (8.57), (8.58), and (8.59) -
all represent the same spacetime, just in different coordinates.
This spacetime, known as **de Sitter space**, is actually
maximally symmetric as a spacetime. (See Hawking and Ellis for
details.) The < 0 solution (8.56) is
also maximally symmetric, and is known as **anti-de Sitter space**.

It is clear that we would like to observationally determine a
number of quantities to decide which of the FRW models
corresponds to our universe. Obviously we would like to determine
*H*_{0}, since that is related to the age of the universe.
(For a
purely matter-dominated, *k* = 0 universe, (8.49) implies that the
age is 2 / (3*H*_{0}). Other possibilities would predict
similar relations.) We would also like to know , which determines
*k* through (8.41). Given the definition (8.39) of ,
this means we want to know both *H*_{0} and . Unfortunately
both quantities are hard to measure accurately, especially .
But notice that the deceleration parameter *q* can be related
to using (8.35):

(8.60) |

Therefore, if we think we know what *w* is (*i.e.*, what kind
of stuff the universe is made of), we can determine by
measuring *q*. (Unfortunately we are not completely confident that
we know *w*, and *q* is itself hard to measure. But people are
trying.)

To understand how these quantities might conceivably be measured,
let's consider geodesic motion in an FRW universe. There are a
number of spacelike Killing vectors, but no timelike Killing vector
to give us a notion of conserved energy. There is, however, a
Killing tensor. If
*U*^{} = (1, 0, 0, 0) is the four-velocity of
comoving observers, then the tensor

(8.61) |

satisfies
*K*_{)} = 0 (as you can check), and is
therefore a Killing tensor. This means that if a particle has
four-velocity
*V*^{} = *dx*^{}/*d*, the quantity

(8.62) |

will be a constant along geodesics. Let's think about this, first
for massive particles. Then we will have
*V*_{}*V*^{} = - 1, or

(8.63) |

where
||^{2} =
*g*_{ij}*V*^{i}*V*^{j}. So
(8.61) implies

(8.64) |

The particle therefore "slows down" with respect to the comoving coordinates as the universe expands. In fact this is an actual slowing down, in the sense that a gas of particles with initially high relative velocities will cool down as the universe expands.

A similar thing happens to null geodesics. In this case
*V*_{}*V*^{} = 0, and (8.62) implies

(8.65) |

But the frequency of the photon as measured by a comoving
observer is
= - *U*_{}*V*^{}. The frequency of the photon
emitted with frequency will therefore be observed with
a lower frequency as the universe expands:

(8.66) |

Cosmologists like to speak of this in terms of the **redshift**
*z* between the two events, defined by the fractional change in
wavelength:

(8.67) |

Notice that this redshift is not the same as the conventional Doppler effect; it is the expansion of space, not the relative velocities of the observer and emitter, which leads to the redshift.

The redshift is something we can measure; we know the rest-frame
wavelengths of various spectral lines in the radiation from
distant galaxies, so we can tell how much their wavelengths have
changed along the path from time *t*_{1} when they were
emitted to
time *t*_{0} when they were observed. We therefore know the
ratio of the scale factors at these two times. But we don't know
the times themselves; the photons are not clever enough to tell
us how much coordinate time has elapsed on their journey. We have
to work harder to extract this information.

Roughly speaking, since a photon moves at the speed of light its
travel time should simply be its distance. But what is the
"distance" of a far away galaxy in an expanding universe?
The comoving distance is not especially useful, since it is not
measurable, and furthermore because the galaxies need not be
comoving in general. Instead we can define the **luminosity
distance** as

(8.68) |

where *L* is the absolute luminosity of the source and *F* is
the flux measured by the observer (the energy per unit time per
unit area of some detector). The definition comes from the
fact that in flat space, for a source at distance *d* the flux
over the luminosity is just one over the area of a sphere centered
around the source,
*F*/*L* = 1/*A*(*d* )= 1/4*d*^{2}. In an FRW universe,
however, the flux will be diluted. Conservation of photons
tells us that the total number of photons emitted by
the source will eventually pass through a sphere at comoving
distance *r* from the emitter. Such a sphere is at a physical
distance *d* = *a*_{0}*r*, where
*a*_{0} is the scale factor when the photons
are observed. But the flux is diluted by two additional effects:
the individual photons redshift by a factor (1 + *z*), and the photons
hit the sphere less frequently, since two photons emitted a time
*t* apart will be measured at a time
(1 + *z*)*t* apart.
Therefore we will have

(8.69) |

or

(8.70) |

The luminosity distance *d*_{L} is something we might hope to
measure, since there are some astrophysical sources whose
absolute luminosities are known ("standard candles"). But *r*
is not observable, so we have to remove that from our equation.
On a null geodesic (chosen to be radial for convenience) we have

(8.71) |

or

(8.72) |

For galaxies not too far away, we can expand the scale factor in a Taylor series about its present value:

(8.73) |

We can then expand both sides of (8.72) to find

(8.74) |

Now remembering (8.67), the expansion (8.73) is the same as

(8.75) |

For small
*H*_{0}(*t*_{1} - *t*_{0}) this
can be inverted to yield

(8.76) |

Substituting this back again into (8.74) gives

(8.77) |

Finally, using this in (8.70) yields **Hubble's Law**:

(8.78) |

Therefore, measurement of the luminosity distances and redshifts
of a sufficient number of galaxies allows us to determine
*H*_{0}
and *q*_{0}, and therefore takes us a long way to deciding
what kind
of FRW universe we live in. The observations themselves are
extremely difficult, and the values of these parameters in the
real world are still hotly contested. Over the next decade or so
a variety of new strategies and more precise application of
old strategies could very well answer these questions once and for all.