Which Fundamental Constants for CMB and BAO?
  We use the three-scale framework of Hu et al. to show how the Cosmic
Microwave Background anisotropy spectrum depends on the fundamental constants.
As expected, the spectrum depends only on \emph{dimensionless} combinations of
the constants, and we emphasize the points that make this generally true for
cosmological observables. Our analysis suggests that the CMB spectrum shape is
mostly determined by $\alpha^2m_e/m_p$ and the proton-CDM-particle mass ratio,
$m_p/\mchi$. The distance to the last-scattering surface depends on
$Gm_p\mchi/\hbar c$, so published CMB observational limits on time variations
of the constants implicitly assume the time-independence of this quantity, as
well as assuming a flat-\lcdm~cosmological model. On the other hand,
low-redshift BAO, $H_0$ and baryon-mass-fraction measurements can be combined
with the \emph{shape} of the CMB spectrum to give information that is largely
independent of these assumptions. In particular we show that the
pre-recombination values of $G\mchi^2/\hbar c$, $m_p/\mchi$ and
$\alpha^2m_e/m_p$ are equal to their present values at a precision of
$\sim15\%$.
