Power Spectra Based Planck Constraints on Compensated Isocurvature, and
  Forecasts for LiteBIRD and CORE Space Missions
  Compensated isocurvature perturbations (CIP), where the baryon and cold dark
matter perturbations cancel, do not cause total matter isocurvature
perturbation. Consequently, at the linear order in the baryon density contrast
$\Delta$, CIP is not detectable by the CMB power spectra. At the second order
CIP smoothes the power spectra in a similar manner as lensing, causing a
degeneracy between the CIP variance $\Delta^2_{rms}=<\Delta^2>$ and lensing
parameter $A_L$. We show that the CMB lensing data breaks this degeneracy.
Nested sampling of the LCDM+CIP(+$A_L$) model, the Planck 2015 temperature,
polarization, and lensing data give $\Delta^2_{rms}=0.0069\pm0.0030$ at 68% CL.
A non-zero value is favored at 2.3$\sigma$. CIP with $\Delta^2_{rms}=0.007$
improves the bestfit $\chi^2$ by 3.6 compared to the adiabatic LCDM model. In
contrast, although the temperature data favor $A_L=1.22$, allowing $A_L\ne1$
does not improve the joint fit, since the lensing data disfavor $A_L\ne1$.
Indeed, CIP provides a rare example of a simple model, which can reduce the
Planck lensing anomaly by fitting well simultaneously the high multipole
temperature and lensing data, as well as the polarization data. Finally, we
derive forecasts for future satellite missions (LiteBIRD proposal to JAXA and
Exploring Cosmic Origins with CORE proposal to ESA's M5 call). Due to its
coarse angular resolution, LiteBIRD is not able to improve the constraints on
CIP or $A_L$, but CORE-M5 approaches the cosmic variance limit and improves the
CIP constraint to $\Delta^2_{rms}<0.0006\ (0.0014)$ at 68% (95%) CL, which is 9
times better than the current trispectrum based upper bound and 6 times better
than obtained from the simulated Planck data. In addition, CORE-M5 will
exquisitely distinguish between CIP and $A_L$. No matter whether CIP is allowed
for or not, the uncertainty of the lensing parameter will be
$\sigma(A_L)=0.012$.
