Forecasting Angular Cross Correlations Between Diffuse X-ray Emission and the Thermal Sunyaev-Zel'dovich Effect

X-ray emission and the thermal Sunyaev-Zel'dovich distortion to the Cosmic Microwave Background are two important handles on the gas content of the Universe. The cross-correlation between these effects eliminates noise bias and reduces observational systematic effects. Using analytic models for the cluster profile, we develop a halo model formalism to study this cross-correlation and apply it to forecast the signal-to-noise of upcoming measurements from eROSITA and the Simons Observatory. In the soft X-ray band (0.5--2 keV), we forecast a signal-to-noise of 174 for the cross-power spectrum. Over a wide range of the scales, the X-rays will be signal-dominated, and so sample variance is important. In particular, non-Gaussian (4-point) contributions to the errors highlight the utility of masking massive clusters. Masking clusters down to $10^{14}\ M_\odot$ increases the signal-to-noise of the cross-spectrum to 201. We perform a Fisher Analysis on the fitting coefficients of the Battaglia et al. gas profiles and on cosmological parameters. We find that the cross spectrum is most sensitive to the overall scale of the profiles of pressure and electron density, as well as cosmological parameters $\sigma_8$ and $H_0$, but that the large number of parameters form a degenerate set, which makes extracting the information more challenging. Our modeling framework is flexible, and in the future, we can easily extend it to forecast the spatial cross-correlations of surveys of X-ray lines available to high-energy-resolution microcalorimetry, to studies of the Warm-Hot Intergalactic Medium, and other effects.