Hubble & Density Parameter


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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.

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

where, is a cosmological constant.

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

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

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 −

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

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

$$\rho_\wedge = \frac{\wedge}{8 \pi G} \Rightarrow \wedge = 8\pi G\rho_\wedge$$

From this, we get −

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

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

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

From this, we get −

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

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

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

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

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

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

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

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

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.

S.No. Quantity Density Parameter
1 Baryons

$\Omega_b = \frac{\rho_b}{\rho_c}$

2 Matter(Baryonic + Dark)

$\Omega_m = \frac{\rho_m}{\rho_c}$

3 Dark Energy

$\Omega_\wedge = \frac{\rho_\wedge}{\rho_c}$

4 Radiation

$\Omega_{rad} = \frac{\rho_{rad}}{\rho_c}$

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.



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