Breaking the $\tau_{\rm T}-f$ degeneracy of the kinetic Sunyaev Zel'dovich cosmology spontaneously in redshift space

The 'optical depth-linear growth rate' ($\tau_{\rm T}-f$) degeneracy is a long-standing problem in the kinetic Sunyaev Zel'dovich (kSZ) cosmology. It can be spontaneously broken in redshift space, where the velocity field leaves its own distinct imprint on the galaxies' redshift space positions and provides valuable information of the linear growth rate. We validate this idea with the Fisher matrix and Monte Carlo Markov Chain techniques in this work, finding that the level of this degeneracy breaking is further enhanced on non-linear scales due to the non-linear evolution of the density and velocity fields. This result emphasizes the importance of the redshift space analysis of the kSZ effect and its potential as a powerful cosmological probe, especially on non-linear scales. As a by-product, we develop a non-linear model of the redshift space density-weighted pairwise kSZ power spectrum. The fitted $f$ and $\tau_{\rm T}$ values from this model are shown to be accurate within $1-2\sigma$ ranges of the fiducial ones when confronted to the mock galaxies mimicking a DESI+CMB-S4 survey combination, even on small scales of $k\sim 0.5h/{\rm Mpc}$.