Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 310 / 2

Chemical equilibrium for $e^{-}+p\leftrightarrow H+\gamma$ gives ${\mu }_e+{\mu }_p={\mu }_H$, since ${\mu }_{\gamma }=0$. Insert the supplied nonrelativistic equilibrium densities. Neglecting $m_e/m_p$ in the translational reduced mass and using the stated degeneracy convention givesCharge neutrality gives $n_p=n_e$, soThis is the hydrogen Saha ionization equation.

Since $n_e=n_p=X_e{n}_b$ and $n_H=(1-X_e)n_b$,Using $n_b=\eta n_{\gamma }$ and $n_{\gamma }=2\zeta (3)T^3/{\pi }^2$ yieldsCosmological recombination occurs far below $B_H$ because the baryon-to-photon ratio is only about $10^{-9}$. There are roughly a billion photons per baryon, so the high-energy tail of the blackbody distribution continues to photoionize hydrogen until the exponential Boltzmann factor overcomes this enormous entropy factor.

Neglecting baryon loading, the Sachs-Wolfe combination obeys a harmonic-oscillator equation with sound speed $1/\sqrt{3}$. Adiabatic initial conditions give a nonzero initial displacement and negligible initial velocity, henceProjection maps $k$ approximately to $\ell /{\chi }_{\ast }$, and the angular power is quadratic in the transfer function, producing the observed approximate ${\cos }^2(k{r}_s)$ sequence of acoustic peaks. A ${\sin }^2$ phase would instead indicate vanishing initial displacement and nonzero initial velocity, characteristic of an isocurvature rather than adiabatic primordial mode.

Multiplying every pre-recombination density by $\lambda$ changes $H$ by $\sqrt{\lambda }$ and therefore shrinks the sound horizon at fixed recombination epoch by ${\lambda }^{-1/2}$. Changing $B_H$ shifts the recombination temperature approximately as $T_{\mathrm{rec}}\propto B_H$, so $a_{\ast }\propto {\mu }^{-1}$. Delaying recombination with $\mu <1$ can compensate the faster expansion; schematically the acoustic scale can be retained by choosing $\mu \sqrt{\lambda }\simeq 1$.

This can preserve the peak positions approximately because post-recombination distances are unchanged. It cannot make the entire spectrum exactly identical: the photon-diffusion scale, visibility-function width, early integrated Sachs-Wolfe effect and relative peak heights scale differently. Thus a tuned pair $(\lambda ,\mu )$ creates an approximate degeneracy, with residual observables breaking it.