Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 310 / 3

Consider a species in thermal equilibrium, with energy density and pressure depending on temperature and with vanishing chemical potential, as in the supplied thermodynamic relation. Put $U=\rho (T)V$. The first law of thermodynamics givesSince $dP/dT=(\rho +P)/T$,The differential is therefore exact:After fixing the irrelevant additive entropy zero, the entropy density at zero chemical potential isMore precisely the first law leaves an additive constant in $S$; extensivity fixes that constant for the usual entropy-density normalization. A nonzero chemical potential would instead require $Ts=\rho +P-\mu n$, so the stated formula is not a universal identity for a decoupled massive species with conserved particle number.

For a reversible thermodynamic process that is an adiabatic process in a comoving volume, $\dot{V}=3HV$ and the cosmological perfect-fluid continuity equation gives $\dot{\rho }=-3H(\rho +P)$. Differentiating the extensive expression explicitly, or using the exact differential above, givesThis is cosmological entropy conservation for the closed equilibrium gas with no entropy-producing energy injection.

For radiation in cosmology, $P=\rho /3$, so $s=4\rho /(3T)$. Comparing $s=c_1{g}_{\star }{T}^3$ and $\rho =c_2{g}_{\star }{T}^4$ yields the coefficient ratioWhen all relativistic species share the same temperature and count, the familiar normalizations are $c_1=2{\pi }^2/45$ and $c_2={\pi }^2/30$. More generally the energy and entropy effective counts can differ; the common $g_{\star }$ here uses the assumptions stated in the question.

Let $T_{-}$ be the residual gas temperature immediately before decay. This is the $T_d$ that enters the target abundance formula, not the higher reheating temperature after decay. Before the decay, separately conserved relic number and gas entropy give the conserved cosmological relic abundance $Y=n_{\varphi }/s_{-}$. The cold nonrelativistic relic has negligible kinetic energy, henceAssume its density dominates over that of the residual gas and that curvature and dark energy are negligible in the Friedmann equation. Using the reduced Planck mass $M_{\mathrm{Pl}}^2=(8\pi G{)}^{-1}$ gives $3M_{\mathrm{Pl}}^2{H}_d^2\simeq {\rho }_{\varphi }(t_d^{-})$. Under the allowed sudden decay of a dominant nonrelativistic relic convention $H_d\simeq \Gamma =t_d^{-1}$, the temperature immediately before decay isThe coefficient inherits the approximation $H_d\simeq 1/t_d$: the result is a sudden-decay estimate, not an exact age relation for a matter-only universe. If $t_d$ were instead the exact cosmic age in an uninterrupted spatially flat matter era, $H_d=2/(3t_d)$ would replace the numerator $3M_{\mathrm{Pl}}^2$ by $4M_{\mathrm{Pl}}^2/3$.

The assumptions also require thermal decoupling in cosmology and negligible prior decay so that $Y$ remains conserved, a fixed relativistic count $g_{\star }$, and rapid thermalization after the energy release. In the same approximation, the reheating temperature $T_{+}$ follows from $c_2{g}_{\star }{T}_{+}^4\simeq 3M_{\mathrm{Pl}}^2/t_d^2$:Its absence of $Y$ at fixed decay rate makes clear why the two temperatures must be distinguished.

The instantaneous-decay approximation keeps the scale factor and physical volume fixed across the decay. Energy conservation and rapid thermalization giveMeanwhile the abundance relation gives $s_{-}={\rho }_{\varphi ,-}/(Ym)\simeq 3M_{\mathrm{Pl}}^2/(Ym{t}_d^2)$. Dividing yields the entropy-injection factorThere is a numerical error in the printed coefficient: part (c) omits the factor $1/3$. The denominator $s_{-}=3M_{\mathrm{Pl}}^2/(Ym{t}_d^2)$ supplies that factor, while the radiation energy contributes $3^{3/4}$ to the numerator. Thus the consistent coefficient is $c_1{3}^{-1/4}{c}_2^{-3/4}$, rather than $c_1{3}^{3/4}{c}_2^{-3/4}$. Both cannot follow from the stated part (b) using the same decay-time approximation.

Its increase is especially transparent without dropping the original radiation. Let $R={\rho }_{\varphi ,-}/{\rho }_{r,-}$. At unchanged $g_{\star }$, $T_{+}^4/T_{-}^4=1+R$, soRelic domination means $R\gg 1$, so the leading estimate is much larger than one. This is entropy production by decay of a dominant relic.

There is no conflict with part (a). The equilibrium gas before the decay was a separately conserved, reversibly expanding system. During decay it receives energy from the nonthermal relic, and the decay products thermalize irreversibly. Its cosmological perfect-fluid continuity equation now has an energy-injection source, so $d({\rho }_{r}V)+P_{r}dV$ does not vanish. The assumptions behind cosmological entropy conservation therefore fail during the event. Moreover the zero-chemical-potential equilibrium expression in part (a) cannot be imposed on the decoupled relic as though it were already part of that same equilibrium radiation bath. The Second law of thermodynamics permits, and here requires, the resulting entropy increase.