The Big Bang — Question 2

Question 2

Matter catches radiation as their densities dilute at different rates

Today let $\Omega_m=.30$ and $\Omega_r=9.0\times10^{-5}$, with radiation including all relativistic components. Use $\rho_m=\rho_{m0}a^{-3}$ and $\rho_r=\rho_{r0}a^{-4}$, with a0=1.

1. Find the scale factor and redshift of equality.

2. At z=9999, find the radiation-to-matter density ratio and identify the dominant component.

Question 2 - Solution

Matter catches radiation as their densities dilute at different rates

(a) Method and calculation

$$\rho_m=\rho_r\Rightarrow\boxed{a_{\mathrm{eq}}=\Omega_r/\Omega_m=3.0\times10^{-4}}.$$$$\boxed{1+z_{\mathrm{eq}}=3333.33,\quad z_{\mathrm{eq}}=3332.33}.$$

(b) Interpretation and further calculation

$$\rho_r/\rho_m=(\Omega_r/\Omega_m)(1+z).$$$$\boxed{(\rho_r/\rho_m)_{9999}=3:\ \text{radiation dominates}}.$$

Physical check

Equality redshift includes neutrinos when they are relativistic; using photons alone would give a different answer.