Age of universe derivation

Age of Universe Derivation in Terms of Scale Factor and Cosmological Constants

1. Friedmann Equation

The foundation is the Friedmann equation for a homogeneous, isotropic universe:

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

where:

• $H$ is the Hubble parameter
• $a(t)$ is the scale factor
• $\rho$ is the total energy density
• $k$ is the curvature parameter
• $\Lambda$ is the cosmological constant

2. Energy Density Evolution

The total energy density consists of different components:

$\rho ={\rho }_m+{\rho }_r+{\rho }_{\Lambda }$

Each component evolves as:

• Matter: ${\rho }_m={\rho }_{m,0}{a}^{-3}$
• Radiation: ${\rho }_r={\rho }_{r,0}{a}^{-4}$
• Dark energy (cosmological constant): ${\rho }_{\Lambda }={\rho }_{\Lambda ,0}=\frac{\Lambda }{8\pi G}$

3. Critical Density and Density Parameters

Define the critical density: ${\rho }_c=\frac{3H_0^2}{8\pi G}$

And density parameters: ${\Omega }_i=\frac{{\rho }_{i,0}}{{\rho }_c}$

The Friedmann equation becomes: $H^2=H_0^2\left[{\Omega }_m{a}^{-3}+{\Omega }_r{a}^{-4}+{\Omega }_{\Lambda }+{\Omega }_k{a}^{-2}\right]$

where ${\Omega }_k=-k/H_0^2$.

4. Time as Function of Scale Factor

From $H=\dot{a}/a$, we have: $dt=\frac{da}{aH}$

Substituting the Friedmann equation: $dt=\frac{da}{a{H}_0\sqrt{{\Omega }_m{a}^{-3}+{\Omega }_r{a}^{-4}+{\Omega }_{\Lambda }+{\Omega }_k{a}^{-2}}}$

5. Age of Universe Integration

The age of the universe is the time from $a=0$ to $a=1$:

$t_0={\int }_0^1\frac{da}{a{H}_0\sqrt{{\Omega }_m{a}^{-3}+{\Omega }_r{a}^{-4}+{\Omega }_{\Lambda }+{\Omega }_k{a}^{-2}}}$

Simplifying: $t_0=\frac{1}{H_0}{\int }_0^1\frac{da}{\sqrt{{\Omega }_m{a}^{-1}+{\Omega }_r{a}^{-2}+{\Omega }_{\Lambda }{a}^2+{\Omega }_k}}$

6. Special Cases

6.1 Matter-Dominated Universe (${\Omega }_m=1$, others = 0)

$t_0=\frac{1}{H_0}{\int }_0^1\frac{da}{\sqrt{a^{-1}}}=\frac{1}{H_0}{\int }_0^1{a}^{1/2}da=\frac{2}{3H_0}$

6.2 Lambda-Dominated Universe (${\Omega }_{\Lambda }=1$, others = 0)

$t_0=\frac{1}{H_0}{\int }_0^1\frac{da}{\sqrt{a^2}}=\frac{1}{H_0}{\int }_0^1\frac{da}{a}$

This diverges at $a=0$, indicating a universe with no beginning.

6.3 Flat Universe with Matter and Lambda (${\Omega }_k=0$)

$t_0=\frac{1}{H_0}{\int }_0^1\frac{da}{\sqrt{{\Omega }_m{a}^{-1}+{\Omega }_r{a}^{-2}+{\Omega }_{\Lambda }{a}^2}}$

7. General Solution with Hyperbolic Functions

For a flat universe (${\Omega }_k=0$) with matter and lambda:

$t_0=\frac{2}{3H_0\sqrt{{\Omega }_{\Lambda }}}\ln \left(\frac{1+\sqrt{{\Omega }_{\Lambda }}}{\sqrt{{\Omega }_m}}\right)$

8. Complete Expression

The most general expression for the age of the universe:

$t_0=\frac{1}{H_0}{\int }_0^1\frac{da}{\sqrt{{\Omega }_r{a}^{-2}+{\Omega }_m{a}^{-1}+{\Omega }_k+{\Omega }_{\Lambda }{a}^2}}$

This integral can be evaluated numerically for any set of cosmological parameters.

9. Current Observational Values

Using current best-fit values:

• $H_0\approx 67.4$ km/s/Mpc
• ${\Omega }_m\approx 0.315$
• ${\Omega }_{\Lambda }\approx 0.685$
• ${\Omega }_r\approx 9.2\times 10^{-5}$

The calculated age is approximately $t_0\approx 13.8$ billion years.

10. Key Relationships

• Age is inversely proportional to $H_0$
• Higher ${\Omega }_{\Lambda }$ increases the age
• Higher ${\Omega }_m$ decreases the age
• Radiation contribution is negligible for the age calculation