Abstract
We cross-correlate maps of the thermal Sunyaev-Zeldovich (tSZ) Compton-$y$
parameter published by Planck with the projected distribution of galaxies in a
set of low-redshift tomographic bins. We use the nearly full-sky 2MASS
Photometric Redshift and WISE $\times$ SuperCOSMOS public catalogues, covering
the redshift range $złesssim0.4$. Our measurements allow us to place
constraints on the redshift dependence of the mass-observable relation for tSZ
cluster count analyses in terms of the so-called 'hydrostatic mass bias'
parameter $1-b_H$. These results can also be interpreted as measurements
of the bias-weighted average gas pressure $bP_e\rangle$ as a function
of redshift, a quantity that can be related to the thermodynamics of gas inside
haloes and used to constrain energy injection processes. We measure $1-b_\rm
H$ with $\sim13\%$ precision in 6 equispaced redshift bins, and find no
evidence for a redshift-dependent mass bias parameter, in agreement with
previous analyses. Our mean value of $1-b_H = 0.59\pm0.03$ is also in
good agreement with the one estimated by the joint analysis of Planck cluster
counts and CMB anisotropies. Our measurements of $bP_e\rangle$, at the
level of $\sim10\%$ in each bin, are the most stringent constraints on the
redshift dependence of this parameter to date, and agree well both with
previous measurements and with theoretical expectations from shock-heating
models.
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