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We propose a new reionization probe that uses cosmic microwave background(CMB) observations; the cross-correlation between fluctuations in the CMBoptical depth which probes the integrated electron density, $\delta\tau$, andthe Compton $y$-map which probes the integrated electron pressure. Thiscross-correlation is much less contaminated than the $y$-map power spectrum bylate-time cluster contributions. In addition, this cross-correlation canconstrain the temperature of ionized bubbles while the optical-depthfluctuations and kinetic SZ effect can not. We measure this new observableusing a Planck $y$-map as well as a map of optical-depth fluctuations that wereconstruct from Planck CMB temperature data. We use our measurements to derivea first CMB-only upper limit on the temperature inside ionized bubbles, $T_{\rmb}\lesssim 7.0\times10^5\,$K ($2\,\sigma$). We also present future forecasts,assuming a fiducial model with characteristic reionization bubble size $R_{\rmb}=5\,$Mpc and $T_{\rm b}=5\times10^4\,$K. The signal-to-noise ratio of thefiducial cross-correlation using a signal dominated PICO-like $y$-map becomes$\simeq7$ with CMB-S4 $\delta\tau$ and $\simeq13$ with CMB-HD $\delta\tau$. Forthe fiducial model, we predict that the CMB-HD $-$ PICO cross-correlationshould achieve an accurate measurement of the reionization parameters; $T_{\rmb}\simeq 49800^{+4500}_{-5100}\,$K and $R_{\rm b}\simeq5.09^{+0.66}_{-0.79}\,$Mpc. Since the power spectrum of the electron densityfluctuations is constrained by the $\delta\tau$ auto spectrum, the temperatureconstraints should be only weakly model-dependent on the details of theelectron distributions and should be statistically representative of thetemperature in ionized bubbles during reionization. This cross-correlationcould, therefore, become an important observable for future CMB experiments.

Constraining reionization with the first measurement of the cross-correlation between the CMB optical-depth fluctuations and the Compton y-map