The Big Bang — Question 1

Question 1

A radiation-era clock links temperature to cosmic age

In a supplied radiation-dominated model with constant relativistic degrees of freedom, $t=1.0\ {\mathrm{s}}\,(1\ {\mathrm{MeV}}/k_BT)^2$ and $a\propto t^{1/2}$.

1. Find the ages when thermal energy kT is 2.0 MeV and .10 MeV.

2. Find the scale-factor and temperature ratios between those epochs and verify entropy-free cooling.

Question 1 - Solution

A radiation-era clock links temperature to cosmic age

(a) Method and calculation

$$\boxed{t_{2}=1/2^2=.25\ {\mathrm{s}}}.$$$$\boxed{t_{.10}=1/(.10)^2=100\ {\mathrm{s}}}.$$

(b) Interpretation and further calculation

$$\boxed{a_2/a_1=\sqrt{100/.25}=20}.$$$$\boxed{T_2/T_1=.10/2=.05=1/20,\quad aT=\text{constant}}.$$

Physical check

The normalization and constant-degree assumption are supplied approximations. Particle thresholds can change the temperature-time relation.