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The Complementarity of Redshift-space Distortions and the Integrated Sachs-Wolfe Effect: A 3D Spherical Analysis

Assuming General Relativity is correct on large-scales, Redshift-Space Distortions (RSDs) and the Integrated Sachs-Wolfe effect (ISW) are both sensitive to the time derivative of the linear growth function. We investigate the extent to which these probes provide complementary or redundant information when they are combined to constrain the evolution of the linear velocity power spectrum, often quantified by the function $f(z)σ_8(z)$, where $f$ is the logarithmic derivative of $σ_8$ with respect to $(1+z)$. Using a spherical Fourier-Bessel (SFB) expansion for galaxy number counts and a spherical harmonic expansion for the CMB anisotropy, we compute the covariance matrices of the signals for a large galaxy redshift survey combined with a CMB survey like Planck. The SFB basis allows accurate ISW estimates by avoiding the plane-parallel approximation, and it retains RSD information that is otherwise lost when projecting angular clustering onto redshift shells. It also allows straightforward calculations of covariance with the CMB. We find that the correlation between the ISW and RSD signals are low since the probes are sensitive to different modes. For our default surveys, on large scales ($k<0.05 \Mpc/h$), the ISW can improve constraints on $fσ_8$ by more than 10% compared to using RSDs alone. In the future, when precision RSD measurements are available on smaller scales, the cosmological constraints from ISW measurements will not be competitive; however, they will remain a useful consistency test for possible systematic contamination and alternative models of gravity.