Friedmann Equations

The dependence of the scale of the universe, $a(t)$, on time is given by the Friedmann equations for the first and second derivatives of $a$. In physical units the first Friedmann equation in flat spacetime is [1,3,4]

$$\left(\frac{\dot{a}}{a}\right)^2 = \frac{\Lambda}{3} + \frac{8\pi G \rho_m(t)}{3} + \frac{8\pi G \rho_r(t)}{3}$$ (21)

$\Lambda$ is the cosmological constant which is presently interpreted as representing the role of dark energy in the dynamics of the universe. Mass and electromagnetic radiation determine the gravitational energy. $\rho_m(t)$ is the time-dependent mass density, and $\rho_r(t) = u_r/c^2$ is the time-dependent mass equivalent to the radiation energy density $u_r$. The left hand side of this equation gives the rate of expansion of the universe relative to its scale at that moment.

The ratio $\dot{a}/a$ has dimensions of $1/t$. At the present epoch ($t_0$) it is the Hubble Constant $H_0$

$$H_0 = \left[\frac{\dot{a}}{a}\right]_{t=t_0}$$ (22)

which is found by measuring the redshift $z$ through Eqs. 5 or 18. If $\dot{a}$ were always constant and equal to its present value, then the age of the universe would be $t_0$ such that

$$\frac{da}{dt} = a_0 H_0$$ (23)

$$\int_0^{a_0} da = a_0 H_0 \int_0^{t_0}dt$$ (24)

$$a_0 = a_0 H_0 t_0$$ (25)

$$t_0 = H_0^{-1}$$ (26)

Mass density simply scales with the volume of the universe

$$\rho_m(t) = \rho_m(t_0) \left( \frac{a_0}{a} \right)^3$$ (27)

but the contribution of radiation density to gravitation also includes the redshift of the photons

$$\rho_r(t) = \rho_r(t_0) \left( \frac{a_0}{a} \right)^4$$ (28)

It is useful to measure time in units of $t_0 = H_0^{-1}$, and distance in units of $a_0 = c t_0$. To that end, Eq. 21 becomes

$$\left(\frac{\dot{a}}{a}\right)^2 = H_0^2 \left(\frac{\Lambda... ... \pi G}{3 H_0^2} \rho_r \left(\frac{a_0}{a}\right)^4 \right)$$ (29)

A simpler form that shows the time scale explicitly is

$$\left(\frac{\dot{a}}{a}\right)^2 = \left(\frac{1}{t_0}\right... ...frac{\Omega_m}{(a/a_0)^3} + \frac{\Omega_r}{(a/a_0)^4}\right)$$ (30)

where

$$\Omega_\Lambda=\frac{\Lambda}{3 H_0^2}$$ (31)

$$\Omega_r=\frac{\rho_r(t_0)}{\rho_c}$$ (32)

and

$$\Omega_m = \frac{\rho_m(t_0)}{\rho_c}$$ (33)

A Newtonian universe with critical density $\rho_c$ given by

$$\rho_c=\frac{3 H_0^2}{8 \pi G}$$ (34)

would stop expanding after an infinite time. The parameters in Eq. 30 determine the past and future of the universe, but they are measured at $t_0$ in the universe today. Gott et al. [1] give values for them based on WMAP measurements of the cosmic background radiation, and an assumption of a flat universe:

$$H_0 = 71 \;\; \mathrm{km}/\mathrm{sec} \mathrm{Mpc}$$ (35)

$$\Omega_{\Lambda} = 0.73$$ (36)

$$\Omega_r = 8.35 \times 10^{-5}$$ (37)

$$\Omega_m = 0.27 - \Omega_r$$ (38)

$$a_0 = c t_0 = c/H_0$$ (39)