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 $R_{\rm b}=5,$Mpcand$T_{\rm b}=5\times10^4,$K. The signal-to-noise ratio of the fiducial cross-correlation using a signal dominated PICO-like $y$-map becomes $\simeq 7$ with CMB-S4 $\delta \tau$ and $\simeq 13$ 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; $T_{\rm b}\simeq 49800^{+4500}{-5100},$Kand$R{\rm b}\simeq 5.09^{+0.66}_{-0.79},$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.