Power spectrum in the presence of large-scale overdensity and tidal fields: breaking azimuthal symmetry

Abstract

We consider the power spectrum of a biased tracer observed in a finite volume in the presence of a large-scale overdensity and tidal fields. Expanding both the observed power spectrum and the source fields (linear power spectrum, scalar overdensity and tidal field tensor) in spherical harmonics, we explicitly confirm that each $(\ell,m)$ source generates just the corresponding $(\ell,m)$ modes of power spectrum in real space. In redshift space, each $(\ell,m)$ source additionally couples only to $(\ell+2n,m)$ modes of tracer power spectra. This generalizes the Kaiser formula for monopole, quadrupole and hexadecapole of the power spectrum to all $(\ell,m)$ modes generated to the second order in perturbation theory. This formalism can find applications in constraining the super-sample covariance and in the local power spectrum based bispectrum estimators. As an example application, we forecast the ability to measure these modes a survey with BOSS-like galaxy number densities.