Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 73 / 4 / ii

Under the usual interpretation that the extra force simply slows otherwise adiabatic expansion, without changing reaction rates or injecting particles, the primordial hydrogen mass fraction would be larger. A smaller Hubble parameter makes cooling slower, delays the end of the deuterium bottleneck, and permits more free-neutron decay after freeze-out before nuclear assembly. If the reduced expansion also keeps weak conversions effective longer, the lower eventual cosmological weak freeze-out temperature further reduces the neutron-to-proton ratio. Both effects lower the primordial helium mass fraction and raise the remaining hydrogen mass fraction.

For fixed initial neutron fraction $x_{n,1}=n_n/(n_n+n_p)$, the post-freeze-out estimate is $x_n=x_{n,1}{e}^{-(t_N-t_1)/{\tau }_n}$, where $t_N$ is the onset of efficient nuclear assembly. Thus $Y_p\simeq 2x_n$ decreases when cooling delays $t_N$. The extra time is the time to reach the nuclear-assembly temperature, not an increase in the specified clock interval $300-1$ seconds. An unspecified force that also reheats the plasma, changes weak rates or causes recollapse would require additional information; the simple expansion-only interpretation is essential to this expectation.