= 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 $T_{\rm rec}\propto B_H$, so $a_*\propto\mu^{-1}$. Delaying recombination with $\mu<1$ can compensate the faster expansion; schematically the acoustic scale can be retained by choosing $\mu\sqrt\lambda\simeq1$.
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.
Back to article page
