A Summing Up — Question 5

Question 5

Extra relativistic species shift a weak freeze-out temperature

Assume weak rate $\Gamma\propto T^5$ and expansion rate $H\propto\sqrt{g_*}T^2$. All coefficients are fixed. Effective relativistic degrees of freedom rise from 10.75 to 14.25.

1. Derive how freeze-out temperature depends on gstar and find its fractional increase.

2. Using $n/p=e^{-\Delta/T_f}$, derive the sign of the neutron-ratio change and explain the direction of the approximate helium response.

Question 5 - Solution

Extra relativistic species shift a weak freeze-out temperature

(a) Method and calculation

$$T_f^5\propto\sqrt{g_*}T_f^2\Rightarrow\boxed{T_f\propto g_*^{1/6}}.$$$$\boxed{T_f^\prime/T_f=(14.25/10.75)^{1/6}\simeq1.0481}.$$

(b) Interpretation and further calculation

$$\frac{d\ln(n/p)}{dT_f}=\Delta/T_f^2>0.$$$$Y\simeq\frac{2r}{1+r},\quad\frac{dY}{dr}=\frac2{(1+r)^2}>0\Rightarrow\boxed{Y\text{ increases}}.$$

Physical check

The sign estimate holds other inputs fixed and omits changes in elapsed neutron-decay time. Precision abundances require coupled expansion and nuclear kinetics.