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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,

Equation 3 (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

Equation 4 (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

Equation 5 (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, H0. When needed we will use 9

Equation 6 (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, lambdaobs, of a feature in the spectrum that had wavelength lambdaem at emission satisfies

Equation 7 (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 lambda propto 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 lambdaz is the first-order Doppler shift in the wavelength lambda, 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 gtapprox 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

Equation 8 (8)

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

Equation 9 (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

Equation 10 (10)

It is conventional to rewrite this as

Equation 11 (11)

The first equation defines the function E(z) that is introduced for later use. The second equation assumes constant Lambda; 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 rhoM propto a-3 propto (1 + z)3. The second term represents radiation and relativistic matter, with pressure pR = rhoR / 3, whence rhoR propto (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 Omegai0 are the fractional contributions to the square of Hubble's constant, H02, that is, Omegai0(t) = 8pi G rhoi0 / (3 H02). At the present epoch, z = 0, the present value of dot{a} / a is H0, and the Omegai0 sum to unity (Eq. [1]).

In this notation, Eq. (8) is

Equation 12 (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 xµ of time and space. Neighboring events 1 and 2 at separation dxµ have invariant separation ds defined by the line element

Equation 13 (13)

The repeated indices are summed, and the metric tensor gµnu is a function of position in spacetime. If ds2 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

Equation 14 (14)

A freely falling, inertial, observer can choose locally Minkowskian coordinates, such that along the path of the observer gµnu = etaµnu and the first derivatives of gµnu vanish.

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

Equation 15 (15)

In the second expression, which assumes K > 0, the radial coordinate is r = K-1/2 sinhchi. 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 OmegaK0 in Eq. (11), which represents the constant of integration in Eq. (10), is related to the constant K:

Equation 16 (16)

where a0 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 OmegaK0, and OmegaK0 and the other density parameters Omegai0 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

Equation 17 (17)

The left side is a function of gµnu and its first two derivatives and represents the geometry of spacetime. The stress-energy tensor Tµnu 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

Equation 18 (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, rho(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 H0 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 H0 = 72 ± 8 km s-1 Mpc-1 (1 sigma uncertainty, Freedman et al., 2001) is in very good agreement with Eq. (6), as is the recent Tammann et al. (2001) summary value H0 = 60 ± 6 km s-1 Mpc-1 (approximate 1 sigma systematic uncertainty). This is an example of the striking change in the observational situation over the previous 5 years: the uncertainty in H0 has decreased by more than a factor of 3, making it one of the better measured cosmological parameters. Back.

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