Constraining reionization with the first measurement of the
  cross-correlation between the CMB optical-depth fluctuations and the Compton
  y-map
  We propose a new reionization probe that uses cosmic microwave background
(CMB) observations; the cross-correlation between fluctuations in the CMB
optical depth which probes the integrated electron density, $\delta\tau$, and
the Compton $y$-map which probes the integrated electron pressure. This
cross-correlation is much less contaminated than the $y$-map power spectrum by
late-time cluster contributions. In addition, this cross-correlation can
constrain the temperature of ionized bubbles while the optical-depth
fluctuations and kinetic SZ effect can not. We measure this new observable
using a Planck $y$-map as well as a map of optical-depth fluctuations that we
reconstruct from Planck CMB temperature data. We use our measurements to derive
a first CMB-only upper limit on the temperature inside ionized bubbles, $T_{\rm
b}\lesssim 7.0\times10^5\,$K ($2\,\sigma$). We also present future forecasts,
assuming a fiducial model with characteristic reionization bubble size $R_{\rm
b}=5\,$Mpc and $T_{\rm b}=5\times10^4\,$K. The signal-to-noise ratio of the
fiducial cross-correlation using a signal dominated PICO-like $y$-map becomes
$\simeq7$ with CMB-S4 $\delta\tau$ and $\simeq13$ with CMB-HD $\delta\tau$. For
the fiducial model, we predict that the CMB-HD $-$ PICO cross-correlation
should achieve an accurate measurement of the reionization parameters; $T_{\rm
b}\simeq 49800^{+4500}_{-5100}\,$K and $R_{\rm b}\simeq
5.09^{+0.66}_{-0.79}\,$Mpc. Since the power spectrum of the electron density
fluctuations is constrained by the $\delta\tau$ auto spectrum, the temperature
constraints should be only weakly model-dependent on the details of the
electron distributions and should be statistically representative of the
temperature in ionized bubbles during reionization. This cross-correlation
could, therefore, become an important observable for future CMB experiments.
