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

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.