Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 62 / 3 / v

The small baryon-to-photon ratio suppresses nuclear equilibrium abundances through the factor ${\eta }^{A-1}$. Cooling eventually makes the binding-energy exponential favourable, but the deuterium bottleneck delays substantial nuclear burning until well after cosmological weak freeze-out. In ordinary Big Bang nucleosynthesis, nearly every neutron surviving that delay ends up in helium-4. This makes the final primordial helium mass fraction chiefly a neutron budget rather than an unrestricted equilibrium abundance.

Let $r_F$ be the neutron-to-proton ratio at freeze-out, let $\Delta t$ be the delay until effective nuclear burning, and let ${\tau }_n$ be the neutron mean lifetime. The initial fraction of baryons which are neutrons is $x_{n,F}=r_F/(1+r_F)$. Because neutron decay converts a neutron into a proton, total baryon number remains constant, while the neutron fraction becomes $x_{n,B}=x_{n,F}{e}^{-\Delta t/{\tau }_n}$. Each helium nucleus uses two neutrons and four baryons. Hence neutron-limited helium synthesis givesEquivalently, $Y_4=2r_B/(1+r_B)$ with $r_B=x_{n,B}/(1-x_{n,B})$. This accounts for the extra protons produced by neutron decay; multiplying $r_F$ alone by the survival factor would miss their contribution. The following responses refer to ordinary neutron-limited burning, without exotic particle injection or a change to the nuclear network.

Extra relativistic neutrino species increase the effective number of relativistic energy degrees of freedom. At fixed temperature the Hubble parameter rises as $g_{\ast }^{1/2}$, while the standard conversion rate is unchanged. Consequently cosmological weak freeze-out moves to higher temperature, $T_F\propto g_{\ast }^{1/6}$, and the neutron-to-proton ratio $r_F=e^{-Q/T_F}$ increases. The initial neutron fraction $r_F/(1+r_F)$ therefore increases as well.

The quicker expansion also reduces the cosmic time required to cool to the deuterium bottleneck temperature: during radiation domination, $t\propto g_{\ast }^{-1/2}{T}^{-2}$ while the thermal count is roughly constant. Fewer neutrons decay before burning. Both factors in the neutron budget increase, giving a larger final helium-4 mass fraction. This is the primordial helium response to extra relativistic species, under the assumption that the additional species contribute mainly to expansion rather than independently changing the conversion rates.

A longer neutron half-life means a larger mean lifetime ${\tau }_n={\tau }_{1/2}/\ln 2$. Holding the initial neutron-to-proton ratio and the nuclear-burning delay fixed, the derivative of the neutron survival factor is positive:Thus fewer neutrons disappear during the deuterium bottleneck, and the final helium-4 mass fraction increases.

There can be a second effect if the increased lifetime represents a uniform reduction of the relevant weak interaction strength. With fixed masses and other couplings, neutron decay and the conversion reactions both scale as $G_F^2$. Thus ${\tau }_n\propto G_F^{-2}$, whereas $T_F\propto G_F^{-2/3}\propto {\tau }_n^{1/3}$. Weaker conversion then freezes out earlier and increases $r_F$, reinforcing the survival effect. The survival argument alone already establishes the requested neutron half-life effect on primordial helium; linking lifetime to all conversion rates additionally requires this microscopic assumption.

A larger baryon-to-photon ratio favours bound nuclei at a fixed temperature through the equilibrium factor ${\eta }^{A-1}$. In particular it helps deuterium survive photodissociation sooner: there are fewer energetic photons per baryon, and nuclear reaction rates per baryon also rise. A schematic deuterium bottleneck condition is $\eta e^{B_D/T_B}\sim \mathrm{constant}$, suppressing slowly varying powers of temperature. It givesThus higher $\eta$ permits burning at a higher temperature. During radiation domination that means earlier cosmic time, $t_B\propto T_B^{-2}$, leaving less time for free-neutron decay after freeze-out. More neutrons enter helium, so the final helium-4 mass fraction increases, ordinarily rather weakly. This is the baryon-density effect on primordial helium.

The equilibrium helium factor ${\eta }^3$ indicates the direction of the change but is not a prediction that the final helium abundance grows without bound as ${\eta }^3$. Once almost all surviving neutrons are captured, the final primordial helium mass fraction is limited by their supply, as the shared neutron budget shows.