Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 310 / 1

In natural units, the equilibrium phase-space distribution function iswhere the minus sign gives the Bose-Einstein distribution for bosons and the plus sign gives the Fermi-Dirac distribution for fermions.

Solved by gpt-5.6-sol high.

Summing over the $g$ internal states, the number density, energy density, and isotropic pressure areandThe factor ${\mathbf{p}}^2/(3E)$ in the kinetic pressure of an isotropic gas is the angular average of one diagonal component of the momentum flux.

Solved by gpt-5.6-sol high.

For an ultrarelativistic particle, $E\simeq |\mathbf{p}|$. Setting $\mu =0$ and writing the three-dimensional momentum integral radially givesThe standard Bose integrals yieldFor fermions, the corresponding integrals differ by the familiar factorsThus the number density scales as $T^3$, while the energy density and pressure scale as $T^4$.

Solved by gpt-5.6-sol high.

The entropy density isFor the homogeneous cosmological plasma, adiabaticity gives the cosmological entropy conservation equationor equivalently $s{a}^3=\text{constant}$ in a comoving volume.

Solved by gpt-5.6-sol high.

After thermal decoupling in cosmology, the collisionless relativistic species has $T_{X}a=\text{constant}$. The still-coupled plasma separately conserves entropy, so $g_{\ast s}{T}_{\gamma }^3{a}^3$ remains constant. At decoupling $T_X=T_{\gamma }$, while at late times the coupled plasma contains only the two photon polarizations. Therefore the temperature of a decoupled relativistic relic isHere $g_{\ast }$ is understood as the effective entropy degrees of freedom in the plasma that remains coupled after $X$ decouples.

Solved by gpt-5.6-sol high.