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 $C^{\rm gg}_{\ell}$, $C^{yy}_{\ell}$ and $C^{{\rm g}y}_{\ell}$ to constrain the tSZ mass bias $B\equiv M_{500}/M^{\rm tSZ}_{500}$. We also fit for the galaxy bias, which is included with explicit redshift and multipole dependencies as $b_{\rm g}(z,\ell)=b_{\rm g}^0(1+z)^{\alpha}(\ell/\ell_0)^{\beta}$, with $\ell_0=117$. We obtain the constraints to be $B =1.50{\pm 0.07}\,(\textrm{stat}) \pm{0.34}\,(\textrm{sys})$, i.e. $1-b_{\rm H}=0.67\pm 0.03\,({\rm stat})\pm 0.16\,({\rm sys})$ (68\% confidence level) for the hydrostatic mass bias, and $b_{\rm g}^0=1.28^{+0.03}_{-0.04}\,(\textrm{stat}) \pm{0.11}\,(\textrm{sys})$, with $\alpha=0.20^{+0.11}_{-0.07}\,(\textrm{stat}) \pm{0.10}\,(\textrm{sys})$ and $\beta=0.45{\pm 0.01}\,(\textrm{stat}) \pm{0.02}\,(\textrm{sys})$ 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.