**1. Homework Statement**

Currently, the density of matter ##\rho_0^M## and of radiation

Estimate the ratio of the cosmic scale factors ##a_{eq}## (scale factor at equality) and ##a_0## (scale factor now)

Hence obtain a rough estimate of the age of the universe at the time of matter-radiation equality. (Assume ##t_0 = 10^{10}## years)

Currently, the density of matter ##\rho_0^M## and of radiation

**##\rho_0^R## have values of approximately ##10^{-29}## g/cm^3 and ##10^{-33}##**

**respectively.****g/cm^3**Estimate the ratio of the cosmic scale factors ##a_{eq}## (scale factor at equality) and ##a_0## (scale factor now)

Hence obtain a rough estimate of the age of the universe at the time of matter-radiation equality. (Assume ##t_0 = 10^{10}## years)

**2. Homework Equations**

##\rho = \rho_0 (\frac{a_0}{a})^{3(1+w)}## ??

##\rho = \rho_0 (\frac{a_0}{a})^{3(1+w)}## ??

**3. The Attempt at a Solution**

I may have found the ratio of scale factors, but I am almost certain it is incorrect, but whether it is or not I am a bit stuck on how to then get the estimate for the age of the universe .

I have done this so far:

I may have found the ratio of scale factors, but I am almost certain it is incorrect, but whether it is or not I am a bit stuck on how to then get the estimate for the age of the universe .

I have done this so far:

[tex]

\rho_{eq}^M = \rho_0^{M} (\frac{a_0}{a_{eq}})^{3} \\

\implies \frac{a_0}{a_{eq}}) = \sqrt[3]{(\frac{\rho_{eq}^R}{ \rho_0^{R}})} \\

\rho_{eq}^R = \rho_0^{R} (\frac{a_0}{a_{eq}})^{4} \\

\implies \frac{a_0}{a_{eq}}) = \sqrt[4]{(\frac{\rho_{eq}^R}{ \rho_0^{R}})} \\

\implies \sqrt[4]{\frac{\rho_{eq}^R}{ \rho_0^{R}}} = \sqrt[3]{\frac{\rho_{eq}^M}{ \rho_0^{M}}} \\

\therefore \frac{\sqrt[3]{\rho_0^M}}{\sqrt[4]{\rho_0^R}} = \frac{a_0}{a_{eq}} \\

\frac{\sqrt[3]{10^{-29}}}{\sqrt[4]{10^{-33}}} = 0.0383

[/tex]

**Any help/advice is much appreciated :)**
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