- Cosmology Tutorial
- Cosmology - Home
- The Expanding Universe
- Cepheid Variables
- Redshift and Recessional Velocity
- Redshift Vs. Kinematic Doppler Shift
- Cosmological Metric & Expansion
- Robertson-Walker Metric
- Hubble Parameter & Scale Factor
- Friedmann Equation & World Models
- Fluid Equation
- Matter Dominated Universe
- Radiation Dominated Universe
- The Dark Energy
- Spiral Galaxy Rotation Curves
- Velocity Dispersion Measurements of Galaxies
- Hubble & Density Parameter
- Age of The Universe
- Angular Diameter Distance
- Luminosity Distance
- Type 1A Supernovae
- Cosmic Microwave Background
- CMB - Temperature at Decoupling
- Anisotropy of CMB Radiation & Cobe
- Modelling the CMB Anisotropies
- Horizon Length at the Surface of Last Scattering
- Extrasolar Planet Detection
- Radial Velocity Method
- Transit Method
- Exoplanet Properties
- Cosmology Useful Resources
- Cosmology - Quick Guide
- Cosmology - Useful Resources
- Cosmology - Discussion
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.
$$H^2(t) \equiv \left ( \frac{\dot{a}}{a} \right )^2 = \frac{8\pi G}{3}\rho - \frac{kc^2}{a^2} + \frac{\wedge}{3}$$
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.