Early dark energy, the Hubble-parameter tension, and the string axiverse
  Precise measurements of the cosmic microwave background (CMB) power spectrum
are in excellent agreement with the predictions of the standard $\Lambda$CDM
cosmological model. However, there is some tension between the value of the
Hubble parameter $H_0$ inferred from the CMB and that inferred from
observations of the Universe at lower redshifts, and the unusually small value
of the dark-energy density is a puzzling ingredient of the model. In this
paper, we explore a scenario with a new exotic energy density that behaves like
a cosmological constant at early times and then decays quickly at some critical
redshift $z_c$. An exotic energy density like this is motivated by some
string-axiverse-inspired scenarios for dark energy. By increasing the expansion
rate at early times, the very precisely determined angular scale of the sound
horizon at decoupling can be preserved with a larger Hubble constant. We find,
however, that the Planck temperature power spectrum tightly constrains the
magnitude of the early dark-energy density and thus any shift in the Hubble
constant obtained from the CMB. If the reionization optical depth is required
to be smaller than the Planck 2016 $2\sigma$ upper bound $\tau\lesssim 0.0774$,
then early dark energy allows a Hubble-parameter shift of at most 1.6
km~s$^{-1}$~Mpc$^{-1}$ (at $z_c\simeq 1585$), too small to fully alleviate the
Hubble-parameter tension. Only if $\tau$ is increased by more than $5\sigma$
can the CMB Hubble parameter be brought into agreement with that from local
measurements. In the process, we derive strong constraints to the contribution
of early dark energy at the time of recombination---it can never exceed
$\sim2\%$ of the radiation/matter density for $10 \lesssim z_c \lesssim 10^5$.
