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

TL;DR

This paper develops a halo-model framework to jointly analyze the angular power spectra of Planck tSZ maps and the WISE galaxy density field, including their cross-correlation. By fitting $C^{yy}_{\ell}$, $C^{gg}_{\ell}$, and $C^{g y}_{\ell}$ with a calibrated foreground model and a redshift-/multipole-dependent galaxy bias $b_g(z,\ell)=b_g^0(1+z)^\alpha(\ell/\ell_0)^\beta$, the authors constrain the tSZ mass bias $B$ and the galaxy bias parameters, obtaining $B=1.50\pm0.07\,({\rm stat})\pm0.34\,({\rm sys})$ and $1-b_H=0.67\pm0.03\,({\rm stat})\pm0.16\,({\rm sys})$. They demonstrate that the cross-correlation with WISE helps reveal the gas distribution in halos and provides a robust constraint on hydrostatic mass bias, with results consistent with prior cross-correlation analyses. The study also quantifies foreground contributions (CIB/IR/radio) and demonstrates a nontrivial redshift and scale dependence in galaxy bias, suggesting refinements for future HOD-like treatments. The findings have implications for interpreting cluster masses in cosmology and motivate future cross-correlation analyses with upcoming surveys (e.g., LSST, CMB-S4).

Abstract

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 $21.8\,σ$. We also measure the auto-correlation spectrum for the tSZ ($yy$) and the galaxy density field maps (gg) with a significance of $150\,σ$ and $88\,σ$, 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 $b_{\rm g}(z,\ell)=b_{\rm g}^0(1+z)^α(\ell/\ell_0)^β$, 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 $α=0.20^{+0.11}_{-0.07}\,(\textrm{stat}) \pm{0.10}\,(\textrm{sys})$ and $β=0.45{\pm 0.01}\,(\textrm{stat}) \pm{0.02}\,(\textrm{sys})$ 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.

Figures (9)

• Figure 1: WISE galaxy overdensity map, computed using Eq. (\ref{['eq:wisemap']}). The masked region, where the overdensity is null, includes the contribution of both the Galactic plane mask and the cuts applied to the WISE catalogue to remove pointings affected by Moon contamination.
• Figure 2: Histogram showing the normalised redshift distribution $p_{\rm s}(z)$ of WISE galaxies, taken from Fig. 4 in Yan2013. The fitting function is plotted as the dashed blue line.
• Figure 3: Power spectra for the gg (top), g$y$ (middle) and $yy$ (bottom) correlations. Points represent our measurements binned at the effective multipoles from Table \ref{['tab:spectra']}, with error bars computed as the square root of the diagonal terms in the corresponding covariance matrices. Lines represent the associated theoretical predictions computed using the best-fit parameters quoted in Table \ref{['tab:estimates']} (full covariance case), obtained as described in Sec. \ref{['sec:parest']}. For each case we show separately the contribution of the one-halo and the two-halo terms, and the contribution of the foregrounds affecting the correlation.
• Figure 4: The six independent blocks of the full correlation matrix defined in Eq. (\ref{['eq:correlation']}). The diagonal blocks show the standard correlations for the gg, $yy$ and g$y$ spectra; the off-diagonal blocks show the additional cross-correlations between different spectra. The correlation values are shown for pairs of the effective bandpower multipoles defined in Table \ref{['tab:spectra']}.
• Figure 5: Cross-correlation between the WISE galaxy overdensity map and each of the three Planck CIB maps at different frequencies. Because the spectra show a similar multipole dependence, we can take their average (red dashed line) to estimate the effect of CIB contaminations in the g$y$ cross-correlation, as it is made explicit in Eq. (\ref{['eq:cibcl']}).