Probing the Anisotropy and Non-Gaussianity in the Redshift Space through the Conditional Moments of the First Derivative

Focusing on the redshift space observations with plane-parallel approximation and relying on the rotational dependency of the general definition of excursion sets, we introduce the so-called conditional moments of the first derivative (\(cmd\)) measures for the smoothed matter density field in three dimensions. We derive the perturbative expansion of \(cmd\) for the real space and redshift space where peculiar velocity disturbs the galaxies' observed locations. Our criteria can successfully recognize the contribution of linear Kaiser and Finger-of-God effects. Our results demonstrate that the \(cmd\) measure has significant sensitivity for pristine constraining the redshift space distortion parameter \(\beta=f/b\) and interestingly, the associated normalized quantity in the Gaussian linear Kaiser limit has only \(\beta\) dependency. Implementation of the synthetic anisotropic Gaussian field approves the consistency between the theoretical and numerical results. Including the first-order contribution of non-Gaussianity perturbatively in the \(cmd\) criterion implies that the N-body simulations for the Quijote suite in the redshift space have been mildly skewed with a higher value for the threshold greater than zero. The non-Gaussianity for the perpendicular direction to the line of sight in the redshift space for smoothing scales \(R\gtrsim 20\) Mpc h\(^{-1}\) is almost the same as the real space. In contrast, the non-Gaussianity along the line of sight direction in redshift space is magnified. The Fisher forecasts indicate an almost significant enhancement in constraining the cosmological parameters, \(\Omega_m\), \(\sigma_8\), and \(n_s\) when using \(cmd+cr\) jointly.