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1.2.1 Friedmann-Robertson-Walker Universes

For a homogeneous and isotropic fluid of density rho and pressure p in a homogeneous universe with curvature k and cosmological constant Lambda, Einstein's system of partial differential equations reduces to the two ordinary differential equations labeled in Table 1.2 as FRW E(00) and E(ii), for the diagonal time and spatial components (see, e.g., Rindler 1977, Section 9.9). Dividing E(00) by H02, and subtracting E(00) from E(ii) puts these equations into more familiar forms. Dividing the latter by 2E(00) and evaluating all expressions at the present epoch then gives the familiar expression for the deceleration parameter q0 in terms of Omega0 and OmegaLambda.

Multiplying E(00) by a3, differentiating with respect to a, and comparing with E(ii) gives the equation of continuity. Given an equation of state p = p (rho), this equation can be integrated to determine rho (a); then E(00) can be integrated to determine a (t).

Consider, for example, the case of vanishing pressure p = 0, which is presumably an excellent approximation for the present universe since the contribution of radiation and massless neutrinos (both having p = rho c2 / 3) to the mass-energy density is at the present epoch much less than that of nonrelativistic matter (for which p is negligible). The continuity equation reduces to (4 pi / 3)rho a3 = M = constant, and E(00) yields Friedmann's equation

Equation 1.2 (1.2)

This gives an expression for the age of the universe t0 which can be integrated in general in terms of elliptic functions, and for Lambda = 0 or k = 0 in terms of elementary functions (cf. standard textbooks, e.g. Peebles 1993, Section 13, and Felton & Isaacman 1986).

Figure 1.1

Figure 1.1. (a) Evolution of the scale factor a (t) plotted vs. the time after the present (t - t0 in units of Hubble time tH ident H0-1 = 9.78 h-1 Gyr for three different cosmologies: Einstein-de Sitter (Omega0 = 1,OmegaLambda = 0 dotted curve), negative curvature (Omega0 = 0.3, OmegaLambda = 0: dashed curve), and low-Omega0 flat (Omega0 = 0.3, OmegaLambda = 0.7: solid curve). (b) Age of the universe today t0 in units of Hubble time tH as a function of Omega0 for Lambda = 0 (dashed curve) and flat Omega0 + OmegaLambda = 1 (solid curve) cosmologies.

Figure 1.1 (a) plots the evolution of the scale factor a for three interesting examples: (Omega0, OmegaLambda = (1,0), (0.3,0), and (0.3,0.7). Figure 1.1 (b) shows how t / tH depends on Omega0 both for Lambda = 0 (dashed) and OmegaLambda = 1 - Omega0 (solid). Notice that for Lambda = 0, t0 / tH is somewhat greater for Omega0 = 0.3 (0.81) than for Omega = 1 (2/3), while for Omega0 = 1-OmegaLambda = 0.3 it is substantially greater: t0 / tH = 0.96. In the latter case, the competition between the attraction of the matter and the repulsion of space by space represented by the cosmological constant results in a slowing of the expansion at a ~ 0.5; the cosmological constant subsequently dominates, resulting in an accelerated expansion (negative deceleration q0 = -0.55 at the present epoch), corresponding to an inflationary universe. In addition to increasing t0, this behavior has observational implications that we will explore in Section 1.3.3.

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