Exploring compensated isocurvature perturbations with CMB spectral
  distortion anisotropies
  We develop a linear perturbation theory for the spectral $y$-distortions of
the cosmic microwave background (CMB). The $y$-distortions generated during the
recombination epoch are usually negligible because the energy transfer due to
the Compton scattering is strongly suppressed at that time, but they can be
significant if there is a considerable amount of compensated isocurvature
perturbation (CIP), which is not tightly constrained from the present CMB
observations. The linear $y$-distortions explicitly depend on the baryon
density fluctuations, therefore $y$ anisotropies can completely resolve the
degeneracy between the baryon isocurvature perturbations and the cold dark
matter ones. This novel method is free from lensing contaminations that can
affect the previous approach to the CIPs based on the nonlinear modulation of
the CMB anisotropies. We compute the cross correlation functions of the
$y$-distortions with the CMB temperature and the $E$ mode polarization
anisotropies. They are sensitive to the correlated CIPs parameterized by
$f'\equiv\mathcal P_{\rm CIP\zeta}/\mathcal P_{\zeta \zeta}$ with $\mathcal
P_{\zeta \zeta}$ and $\mathcal P_{\rm CIP\zeta}$ being the auto correlation of
the adiabatic perturbations and the cross correlation between them and the
CIPs. We investigate how well the $y$ anisotropies will constrain $f'$ in
future observations such as those provided by a PIXIE-like and a PRISM-like
survey, LiteBIRD and a cosmic variance limited (CVL) survey, taking into
account the degradation in constraining power due to the presence of Sunyaev
Zel'dovich effect from galaxy clusters. For example, our forecasts show that it
is possible to achieve an upper limit of $f'< 2 \times 10^{5}$ at 68% C.L. with
LiteBIRD, and $f'<2\times 10^{4}$ with CVL observations.
