Cross Correlation of the Extragalactic Gamma-ray Background with Thermal
  Sunyaev-Zel'dovich Effect in the Cosmic Microwave Background
  Cosmic rays in galaxy clusters are unique probes of energetic processes
operating with large-scale structures in the Universe. Precise measurements of
cosmic rays in galaxy clusters are essential for improving our understanding of
non-thermal components in the intracluster-medium (ICM) as well as the accuracy
of cluster mass estimates in cosmological analyses. In this paper, we perform a
cross-correlation analysis with the extragalactic gamma-ray background and the
thermal Sunyaev-Zel'dovich (tSZ) effect in the cosmic microwave background. The
expected cross-correlation signal would contain rich information about the
cosmic-ray-induced gamma-ray emission in the most massive galaxy clusters at
$z\sim0.1-0.2$. We analyze the gamma-ray background map with 8 years of data
taken by the Large Area Telescope onboard Fermi satellite and the publicly
available tSZ map by Planck. We confirm that the measured cross-correlation is
consistent with a null detection, and thus it enables us to put the tightest
constraint on the acceleration efficiency of cosmic ray protons at shocks in
and around galaxy clusters. We find the acceleration efficiency must be below
5\% with a $2\sigma$ confidence level when the hydrostatic mass bias of
clusters is assumed to be 30\%, and our result is not significantly affected by
the assumed value of the hydrostatic mass bias. Our constraint implies that the
non-thermal cosmic-ray pressure in the ICM can introduce only a $\le 3\%$ level
of the hydrostatic mass bias, highlighting that cosmic rays alone do not
account for the mass bias inferred by the Planck analyses. Finally, we discuss
future detectability prospects of cosmic-ray-induced gamma rays from the
Perseus cluster for the Cherenkov Telescope Array.
