Cosmology - Hubble & Density Parameter

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.

$$H^2(t) equiv left ( frac{dot{a}}{a} ight )^2 = frac{8pi G}{3} ho - frac{kc^2}{a^2} + frac{wedge}{3}$$

where, ∧ is a cosmological constant.

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

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

For a matter dominated universe, the density varies as −

$$frac{ ho_m}{ ho_{m,0}} = left ( frac{a_0}{a} ight )^3 Rightarrow ho_m = ho_{m,0}a^{-3}$$

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

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

Presently, we are pving in a matter dominated universe. Hence, considering $ ho ≡ ho_m$, we get −

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

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

$$ ho_wedge = frac{wedge}{8 pi G} Rightarrow wedge = 8pi G ho_wedge$$

From this, we get −

$$left ( frac{dot{a}}{a} ight )^2 = frac{8pi G}{3} ho_{m,0}a^{-3} + frac{8 pi G}{3} ho_wedge$$

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

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

From this, we get −

$$left ( frac{dot{a}}{a} ight )^2 = frac{H_0^2}{ ho_{c,0}} ho_{m,0}a^{-3} + frac{H_0^2}{ ho_{c,0}} ho_wedge$$

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

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

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

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

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

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

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

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 −

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

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

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

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 $ ho_c$. For any quantity $x$ the corresponding density parameter, $Omega_x$ can be expressed mathematically as −

$$Omega_x = frac{ ho_x}{ ho_c}$$

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.