In this chapter, we will discuss regarding the Density and Hubble parameters.

Hubble Parameter

The Hubble parameter is defined as follows −

$$H(t) equiv frac{da/dt}{a}$$

which measures how rapidly the scale factor changes. More generally, the evolution of the scale factor is determined by the Friedmann Equation.

$$H^2(t) equiv left ( frac{dot{a}}{a} right )^2 = frac{8pi G}{3}rho – frac{kc^2}{a^2} + frac{wedge}{3}$$

where, ∧ is a cosmological constant.

For a flat universe, k = 0, hence the Friedmann Equation becomes −

$$left ( frac{dot{a}}{a} right )^2 = frac{8pi G}{3}rho + frac{wedge}{3}$$

For a matter dominated universe, the density varies as −

$$frac{rho_m}{rho_{m,0}} = left ( frac{a_0}{a} right )^3 Rightarrow rho_m = rho_{m,0}a^{-3}$$

and, for a radiation dominated universe the density varies as −

$$frac{rho_{rad}}{rho_{rad,0}} = left ( frac{a_0}{a} right )^4 Rightarrow rho_{rad} = rho_{rad,0}a^{-4}$$

Presently, we are living in a matter dominated universe. Hence, considering $rho ≡ rho_m$, we get −

$$left ( frac{dot{a}}{a} right )^2 = frac{8pi G}{3}rho_{m,0}a^{-3} + frac{wedge}{3}$$

The cosmological constant and dark energy density are related as follows −

$$rho_wedge = frac{wedge}{8 pi G} Rightarrow wedge = 8pi Grho_wedge$$

From this, we get −

$$left ( frac{dot{a}}{a} right )^2 = frac{8pi G}{3}rho_{m,0}a^{-3} + frac{8 pi G}{3} rho_wedge$$

Also, the critical density and Hubble’s constant are related as follows −

$$rho_{c,0} = frac{3H_0^2}{8 pi G} Rightarrow frac{8pi G}{3} = frac{H_0^2}{rho_{c,0}}$$

From this, we get −

$$left ( frac{dot{a}}{a} right )^2 = frac{H_0^2}{rho_{c,0}}rho_{m,0}a^{-3} + frac{H_0^2}{rho_{c,0}}rho_wedge$$

$$left ( frac{dot{a}}{a} right )^2 = H_0^2Omega_{m,0}a^{-3} + H_0^2Omega_{wedge,0}$$

$$(dot{a})^2 = H_0^2Omega_{m,0}a^{-1} + H_0^2Omega_{wedge,0}a^2$$

$$left ( frac{dot{a}}{H_0} right )^2 = Omega_{m,0}frac{1}{a} + Omega_{wedge,0}a^2$$

$$left ( frac{dot{a}}{H_0} right )^2 = Omega_{m,0}(1+z) + Omega_{wedge,0}frac{1}{(1+z)^2}$$

$$left ( frac{dot{a}}{H_0} right)^2 (1+z)^2 = Omega_{m,0}(1+z)^3 + Omega_{wedge,0}$$

$$left ( frac{dot{a}}{H_0} right)^2 frac{1}{a^2} = Omega_{m,0}(1 + z)^3 + Omega_{wedge,0}$$

$$left ( frac{H(z)}{H_0} right )^2 = Omega_{m,0}(1+z)^3 + Omega_{wedge,0}$$

Here, $H(z)$ is the red shift dependent Hubble parameter. This can be modified to include the radiation density parameter $Omega_{rad}$ and the curvature density parameter $Omega_k$. The modified equation is −

$$left ( frac{H(z)}{H_0} right )^2 = Omega_{m,0}(1+z)^3 + Omega_{rad,0}(1+z)^4+Omega_{k,0}(1+z)^2+Omega_{wedge,0}$$

$$Or, : left ( frac{H(z)}{H_0} right)^2 = E(z)$$

$$Or, : H(z) = H_0E(z)^{frac{1}{2}}$$

where,

$$E(z) equiv Omega_{m,0}(1 + z)^3 + Omega_{rad,0}(1+z)^4 + Omega_{k,0}(1+z)^2+Omega_{wedge,0}$$

This shows that the Hubble parameter varies with time.

For the Einstein-de Sitter Universe, $Omega_m = 1, Omega_wedge = 0, k = 0$.

Putting these values in, we get −

$$H(z) = H_0(1+z)^{frac{3}{2}}$$

which shows the time evolution of the Hubble parameter for the Einstein-de Sitter universe.

Density Parameter

The density parameter, $Omega$, is defined as the ratio of the actual (or observed) density ρ to the critical density $rho_c$. For any quantity $x$ the corresponding density parameter, $Omega_x$ can be expressed mathematically as −

$$Omega_x = frac{rho_x}{rho_c}$$

For different quantities under consideration, we can define the following density parameters.

Where the symbols have their usual meanings.

Points to Remember

• The evolution of the scale factor is determined by the Friedmann Equation.

• H(z) is the red shift dependent Hubble parameter.

• The Hubble Parameter varies with time.

• The Density Parameter is defined as the ratio of the actual (or observed) density to the critical density.