Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 310 / 3

Take a homogeneous comoving cell of physical volume $V_c=a^3{V}_0$ and total entropy $S=s{V}_c$. At vanishing chemical potential, the reversible first law of thermodynamics readsFor an extensive equilibrium system, the thermodynamic Euler relation is $E=TS-P{V}_c+\mu N$. At $\mu =0$ this givesThe supplied pressure identity verifies consistency directly: $dP=(\rho +P)dT/T$ impliesThe thermodynamic Euler relation fixes the additive normalization that a differential identity alone would leave undetermined.

For adiabatic cosmological expansion, the cosmological perfect-fluid continuity equation gives $\dot{\rho }+3H(\rho +P)=0$, while ${\dot{V}}_c=3H{V}_c$. HenceThis proves cosmological entropy conservation in the stated perfect-fluid approximation. Kinetic equilibrium alone does not guarantee adiabatic evolution: entropy injection, dissipative stresses or out-of-equilibrium reactions would require additional source terms. Those effects are implicitly excluded in this calculation.

When the number-changing reaction is in chemical equilibrium, its chemical balance condition is $2{\mu }_{\varphi }=3{\mu }_{\varphi }$. Therefore zero chemical potential from number-changing reactions gives $\boxed{{\mu }_{\varphi }=0}$. Elastic scattering maintains kinetic equilibrium but imposes no independent condition on ${\mu }_{\varphi }$. The conclusion requires the number-changing rate to remain large compared with the Hubble parameter; merely listing an allowed reaction does not guarantee this after freeze-out.

Before thermal decoupling in cosmology, the sectors have a common temperature $T$. For relativistic species at that temperature their entropy densities aresoThese are the effective number of relativistic degrees of freedom for entropy. The paper's $g_{\ast }$ agrees with $g_{\ast s}$ for relativistic components at a common temperature. For one real scalar field, $g_{\ast s}^{\varphi }=1$. At temperatures where all Standard Model species are relativistic, both counts are constant, and so is $\xi$ before decoupling; thresholds can change this equilibrium ratio if they occur while the sectors remain coupled.

After decoupling, adiabatic isolation gives separately conserved visible and dark entropy: $s_{\mathrm{SM}}{a}^3$ and $s_{\varphi }{a}^3$ are individually constant. Thus $\boxed{\xi (T<T_d)=\xi (T_d)}$ even when either sector's entropy degrees of freedom change. Changes in particle content then alter their temperature ratio rather than the ratio of their comoving entropies. This assumes no late inter-sector transfer or entropy injection.

Use the dark-sector temperature in every dimensionless integration variable: $x=m_{\varphi }/T_{\varphi }$ and $y=p/T_{\varphi }$. Assume a single real scalar field, $g=1$, and fast number-changing reactions keeping ${\mu }_{\varphi }=0$. For $x\gg 1$, the populated momenta satisfy $y=O(\sqrt{x})$, soThis is the nonrelativistic Maxwell-Boltzmann distribution limit of the Bose-Einstein distribution. The given Gaussian Gamma function integral givesSince ${\rho }_{\varphi }=m_{\varphi }n+3n{T}_{\varphi }/2+\cdots$ and $P_{\varphi }=n{T}_{\varphi }$, the entropy density at zero chemical potential is $s_{\varphi }=n(x+5/2+\cdots )$. Its leading nonrelativistic entropy at zero chemical potential is thereforeAn internal multiplicity $g$ would multiply this expression; the absence of $g$ in the printed result assumes a real one-state scalar.

Now use the constant separately conserved visible and dark entropy ratio and the relativistic visible-bath entropy law, $s_{\mathrm{SM}}=(2{\pi }^2/45)g_{\ast s}^{\mathrm{SM}}{T}_{\gamma }^3=\xi s_{\varphi }$. Since $T_{\varphi }=m_{\varphi }/x$,Thus the cannibal-sector temperature ratio hasThe numerical coefficient is approximately $0.525$. The visible-bath temperature is the photon temperature while that bath is internally thermalized; after distinct visible species acquire different temperatures, its total entropy must instead use the appropriate temperature-weighted $g_{\ast s}^{\mathrm{SM}}$.

The displayed relativistic scalar entropy formula must not be continued into this regime with $g_{\ast s}^{\varphi }=1$ held constant. If it is used as an effective definition at arbitrary temperature, thenand $T_{\gamma }/T_{\varphi }=(\xi g_{\ast s}^{\varphi }/g_{\ast s}^{\mathrm{SM}}{)}^{1/3}$ gives the same result. This specifies how the relativistic expression can legitimately be combined with the nonrelativistic one.

During the cannibal phase of a decoupled sector, cosmological entropy conservation and the leading result of part (c) giveDifferentiating and using $T_{\varphi }=m_{\varphi }/x$ yields logarithmic cooling during cannibalism:Here $a_{\ast }$ absorbs the entropy normalization, and the logarithmic expression is a large-$x$ asymptotic relation. With fixed visible entropy degrees of freedom the photon bath has $T_{\gamma }\propto a^{-1}$. Thus for $x\gg 1$ the scalar particles cool much more slowly than the photons and become relatively hotter.

The physical reason is that $3\varphi \to 2\varphi$ converts rest-mass energy into kinetic energy of the remaining particles. Elastic scattering distributes this energy, offsetting much of the cooling caused by expansion. The total comoving entropy remains constant; the number density does not merely dilute as $a^{-3}$. This explanation applies only while number-changing chemical equilibrium persists. If those reactions freeze out, an isolated number-conserving nonrelativistic gas instead has $T_{\varphi }\propto a^{-2}$ under adiabatic expansion, which is faster cooling than the photons. The paper supplies no rates, so that transition time cannot be determined.