Varying fundamental constants principal component analysis: additional
  hints about the Hubble tension
  Varying fundamental constants (VFC) [e.g., the fine-structure constant,
$\alpha_{\rm EM}$] can arise in numerous extended cosmologies. Through their
effect on the decoupling of baryons and photons during last scattering and
reionisation, these models can be directly constrained using measurements of
the cosmic microwave background (CMB) temperature and polarization
anisotropies. Previous investigations focused mainly on time-independent
changes to the values of fundamental constants. Here we generalize to
time-dependent variations. Instead of directly studying various VFC
parameterizations, we perform a model-independent principal component analysis
(PCA), directly using an eigenmode decomposition of the varying constant during
recombination. After developing the formalism, we use Planck 2018 data to
obtain new VFC limits, showing that three independent VFC modes can be
constrained at present. No indications for significant departures from the
standard model are found with Planck data. Cosmic variance limited modes are
also compared and simple forecasts for The Simons Observatory are carried out,
showing that in the future improvements of the current constraints by a factor
of $\simeq 3$ can be anticipated.
  Our modes focus solely on VFC at redshifts $z\geq 300$. This implies that
they do not capture some of the degrees of freedom relating to the reionisation
era. This aspect provides important new insights into the possible origin of
the Hubble tension, hinting that indeed a combined modification of
recombination and reionisation physics could be at work. An extended PCA,
covering both recombination and reionisation simultaneously, could shed more
light on this question, as we emphasize here.
