Cross Correlation between the Thermal Sunyaev-Zel'dovich Effect and Projected Galaxy Density Field

We present a joint analysis of the power spectra of the Planck Compton $y$-parameter map and the projected galaxy density field using the Wide Field Infrared Survey Explorer (WISE) all-sky survey. We detect the statistical correlation between WISE and Planck data (g$y$) with a significance of $21.8\sigma$. We also measure the auto-correlation spectrum for the tSZ ($yy$) and the galaxy density field maps (gg) with a significance of $150\sigma$ and $88\sigma$, respectively. We then construct a halo model and use the measured correlations $C_{\mathrm{\ell }}^{\mathrm{g}\mathrm{g}}$, $C_{\mathrm{\ell }}^{yy}$ and $C_{\mathrm{\ell }}^{\mathrm{g}y}$ to constrain the tSZ mass bias $B\equiv M_{500}\mathrm{/}{M}_{500}^{\mathrm{t}\mathrm{S}\mathrm{Z}}$. We also fit for the galaxy bias, which is included with explicit redshift and multipole dependencies as $b_{\mathrm{g}}(z,\mathrm{\ell })=b_{\mathrm{g}}^0(1+z{)}^{\alpha }(\mathrm{\ell }\mathrm{/}{\mathrm{\ell }}_0{)}^{\beta }$, with ${\mathrm{\ell }}_0=117$. We obtain the constraints to be $B=1.50\pm 0.07(\text{stat})\pm 0.34(\text{sys})$, i.e. $1-b_{\mathrm{H}}=0.67\pm 0.03(\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t})\pm 0.16(\mathrm{s}\mathrm{y}\mathrm{s})$ (68% confidence level) for the hydrostatic mass bias, and $b_{\mathrm{g}}^0=1.28_{-0.04}^{+0.03}(\text{stat})\pm 0.11(\text{sys})$, with $\alpha =0.20_{-0.07}^{+0.11}(\text{stat})\pm 0.10(\text{sys})$ and $\beta =0.45\pm 0.01(\text{stat})\pm 0.02(\text{sys})$ for the galaxy bias. Incoming data sets from future CMB and galaxy surveys (e.g. Rubin Observatory) will allow probing the large-scale gas distribution in more detail.