  The fundamental constants at recombination can differ from their present-day
values due to degeneracies in cosmological parameters, raising the possibility
of yet-undiscovered physics coupled directly to the Standard Model. We study
the cosmology of theories in which a new, hyperlight scalar field modulates the
electron mass and fine-structure constant at early times. We find new
degeneracies in cosmologies that pair early recombination with a new
contribution to the matter density arising at late times, whose predictions can
be simultaneously consistent with CMB and low-redshift distance measurements.
Such "late dark matter" already exists in the Standard Model in the form of
massive neutrinos, but is necessarily realized by the scalar responsible for
shifting the early-time fundamental constants. After detailing the physical
effects of varying constants and hyperlight scalar fields on cosmology, we find
that variations of the electron mass and fine structure constant are
constrained at the percent and permille level, respectively, and a hyperlight
scalar in the mass range $10^{-32}~\mathrm{eV} \lesssim m_\phi \lesssim
10^{-28}~\mathrm{eV}$ can impact what variations are allowed while composing up
to a percent of the present dark matter density. We comment on the potential
for models with a varying electron mass to reconcile determinations of the
Hubble constant from cosmological observations and distance-ladder methods, and
we show that parameter inference varies significantly between recent baryon
acoustic oscillation and type Ia supernova datasets.
