**A. The Friedmann-Lemaître model**

The standard world model is close to homogeneous and isotropic on large scales, and lumpy on small scales -- the effect of the mass concentrations in galaxies, stars, people, and all that. The length scale at the transition from nearly smooth to strongly clumpy is about 10 Mpc. We use here and throughout the standard astronomers' length unit,

(3) |

To be more definite, imagine many spheres of radius 10 Mpc are placed at random, and the mass within each is measured. At this radius the rms fluctuation in the set of values of masses is about equal to the mean value. On smaller scales the departures from homogeneity are progressively more nonlinear; on larger scales the density fluctuations are perturbations to the homogeneous model. From now on we mention these perturbations only when relevant for the cosmological tests.

The expansion of the universe means the distance *l* (*t*) between
two well-separated galaxies varies with world time, *t*, as

(4) |

where the expansion or scale factor, *a*(*t*), is independent
of the
choice of galaxies. It is an interesting exercise, for those who have not
already thought about it, to check that Eq. (4) is
required to preserve homogeneity and isotropy.
^{8}

The rate of change of the distance in Eq. (4) is the speed

(5) |

where the dot means the derivative with respect to world time *t*
and *H* is the time-dependent Hubble parameter. When *v* is small
compared to the speed of light this is Hubble's law.
The present value of *H* is
Hubble's constant, *H*_{0}. When needed we will use
^{9}

(6) |

at two standard deviations. The first equation defines the
dimensionless parameter *h*.

Another measure of the expansion follows by considering the
stretching of the wavelength of light received from a
distant galaxy. The observed wavelength,
_{obs}, of a
feature in the spectrum that had wavelength
_{em} at
emission satisfies

(7) |

where the expansion factor *a* is defined in Eq. (4)
and *z* is the redshift. That is, the wavelength of freely
traveling radiation stretches
in proportion to the factor by which the universe
expands. To understand this, imagine a large part of the universe
is enclosed in a cavity with perfectly reflecting walls. The
cavity expands with the general expansion, the widths
proportional to *a*(*t*). Electromagnetic radiation is a sum of
the normal modes that fit the cavity. At interesting wavelengths
the mode frequencies are much larger than the rate of expansion
of the universe, so adiabaticity says a photon
in a mode stays there, and its wavelength thus must vary as
*a*(*t*),
as stated in Eq. (7). The cavity
disturbs the long wavelength part of the radiation, but the
disturbance can be made exceedingly small by
choosing a large cavity.

Equation (7) defines the redshift *z*. The redshift is a
convenient label for epochs in the early universe, where *z*
exceeds unity. A good exercise for the student is to check
that when *z* is small Eq. (7) reduces to Hubble's law,
where *z* is the
first-order Doppler shift in the
wavelength , and
Hubble's parameter *H* is given by
Eq. (5). Thus Hubble's law may be written as
*cz* = *Hl* (where we have put in the speed of light).

These results follow from the symmetry of the cosmological model
and conventional local physics; we do not need general
relativity theory. When
*z* 1 we
need the relativistic theory
to compute the relations among the redshift and other
observables. An example is the relation between redshift and apparent
magnitude used in the supernova test. Other cosmological tests
check consistency among these relations, and this checks the
world model.

In general relativity the second time derivative of the expansion factor satisfies

(8) |

The gravitational constant is *G*. Here and throughout we choose units
to set the velocity of light to unity. The mean mass density,
(*t*),
and the pressure, *p*(*t*), counting all contributions including
dark energy, satisfy the local energy conservation law,

(9) |

The first term on the right-hand side represents the decrease of mass
density due to the expansion that more broadly disperses the matter.
The *pdV* work in the second term is a familiar local concept, and
meaningful in general relativity. But one should note that energy
does not have a general global meaning in this theory.

The first integral of Eqs. (8) and (9) is the Friedmann equation

(10) |

It is conventional to rewrite this as

(11) |

The first equation defines the function *E*(*z*) that is
introduced for later use. The second equation assumes constant
;
the time-dependent dark energy case is reviewed in
Secs. II.C and
III.E. The first term in the last part of
Eq. (11) represents non-relativistic matter with negligibly small
pressure; one sees from Eqs. (7) and (9) that the
mass density in this form varies with the expansion of the universe as
_{M}
*a*^{-3}
(1 +
*z*)^{3}. The second term represents radiation and
relativistic matter, with pressure *p*_{R} =
_{R} /
3, whence
_{R}
(1 +
*z*)^{4}. The third term is the effect of Einstein's
cosmological constant, or
a constant dark energy density. The last term, discussed in more detail
below, is the constant of integration in Eq. (10). The four
density parameters
_{i0} are
the fractional contributions to the square of
Hubble's constant, *H*_{0}^{2}, that is,
_{i0}(*t*)
= 8 *G*
_{i0} /
(3 *H*_{0}^{2}). At the present
epoch, *z* = 0, the present value of
/ *a* is
*H*_{0}, and the
_{i0} sum to
unity (Eq. [1]).

In this notation, Eq. (8) is

(12) |

The constant of integration in Eqs. (10)
and (11) is related to the geometry of
spatial sections at constant world time. Recall that in general relativity
events in spacetime are labeled by the four coordinates
*x*^{µ} of time and space. Neighboring events 1 and 2
at separation *dx*^{µ} have
invariant separation *ds* defined by the line element

(13) |

The repeated indices are summed, and the metric tensor
*g*_{µ}
is a function of position in spacetime. If *ds*^{2} is
positive then *ds*
is the proper (physical) time measured by an observer who moves
from event 1 to 2; if negative, | *ds*| is the proper distance
between events 1 and 2 measured by an observer who is moving so
the events are seen to be simultaneous.

In the flat spacetime of special relativity one can choose coordinates so the metric tensor has the Minkowskian form

(14) |

A freely falling, inertial, observer can choose locally Minkowskian
coordinates, such that along the path of the observer
*g*_{µ} =
_{µ} and the first derivatives of
*g*_{µ} vanish.

In the homogeneous world model we can choose coordinates so the metric tensor is of the form that results in the line element

(15) |

In the second expression, which assumes *K* > 0, the radial
coordinate is *r* = *K*^{-1/2}
sinh. The expansion factor
*a*(*t*) appears in Eq. (4). If *a* were constant
and the constant *K* vanished this would represent the flat spacetime
of special relativity in polar coordinates. The key point for now is
that _{K0} in
Eq. (11), which represents the constant of
integration in Eq. (10), is related to the constant *K*:

(16) |

where *a*_{0} is the present value of the expansion factor
*a*(*t*). Cosmological tests that are sensitive to the geometry of
space constrain the value of the parameter
_{K0},
and _{K0}
and the other density parameters
_{i0} in
Eq. (11) determine the expansion history of the universe.

It is useful for what follows to recall that the metric tensor in Eq. (15) satisfies Einstein's field equation, a differential equation we can write as

(17) |

The left side is a function of
*g*_{µ} and
its first two derivatives and represents the geometry of spacetime. The
stress-energy tensor
*T*_{µ}
represents the material
contents of the universe, including particles, radiation, fields,
and zero-point energies. An observer in a homogeneous and
isotropic universe, moving so the universe is observed to be
isotropic, would measure the stress-energy tensor to be

(18) |

This diagonal form is a consequence of the symmetry; the diagonal components define the pressure and energy density. With Eq. (18), the differential equation (17) yields the expansion rate equations (11) and (12).

^{8} We feel we have to comment on a few
details about Eq. (4) to avoid
contributing to debates that are more intense than
seem warranted. Think of the world time *t* as the
proper time kept by each of a dense set of observers, each moving
so all the others are isotropically moving away, and with the
times synchronized to a common energy density,
(*t*), in the
near homogeneous expanding universe. The distance
*l* (*t*) is the sum of the proper distances between neighboring
observers, all measured at time *t*, and along the shortest distance
between the two observers. The rate of increase of the distance,
*dl* / *dt*,
may exceed the velocity of light. This is no more problematic in
relativity theory than is the large speed at which the beam of a
flashlight on Earth may swing across the face of the Moon
(assuming an adequately tight beam). Space sections at fixed *t*
may be non-compact, and the total mass of a homogeneous universe
formally infinite. As far as is known this is not meaningful: we can only
assert that the universe is close to homogeneous and isotropic
over observable scales, and that what can be observed is a finite
number of baryons and photons.
Back.

^{9} The numerical values in Eq. (6) are
determined from an analysis of
almost all published measurements of *H*_{0} prior to mid 1999
(Gott et al., 2001).
They are a very reasonable summary of
the current situation. For instance, the Hubble Space Telescope
Key Project summary measurement value
*H*_{0} = 72 ± 8 km s^{-1}
Mpc^{-1} (1
uncertainty,
Freedman et al., 2001)
is in very good agreement with Eq. (6), as is the recent
Tammann et al. (2001)
summary value
*H*_{0} = 60 ± 6 km s^{-1} Mpc^{-1}
(approximate 1
systematic uncertainty). This is an example of
the striking change in the observational situation over the previous
5 years: the uncertainty in *H*_{0} has decreased by more
than a factor of
3, making it one of the better measured cosmological parameters.
Back.