Complete Treatment of Galaxy Two-Point Statistics: Gravitational Lensing Effects and Redshift-Space Distortions

Abstract

We present a coherent theoretical framework for computing gravitational lensing effects and redshift-space distortions in an inhomogeneous universe and investigate their impacts on galaxy two-point statistics. Adopting the linearized FRW metric, we derive the gravitational lensing and the generalized Sachs-Wolfe effects that include the weak lensing distortion, magnification, and time delay effects, and the redshift-space distortion, Sachs-Wolfe, and integrated Sachs-Wolfe effects, respectively. Based on this framework, we first compute their effects on observed source fluctuations, separating them as two physically distinct origins: the volume effect that involves the change of volume and is always present in galaxy two-point statistics, and the source effect that depends on the intrinsic properties of source populations. Then we identify several terms that are ignored in the standard method, and we compute the observed galaxy two-point statistics, an ensemble average of all the combinations of the intrinsic source fluctuations and the additional contributions from the gravitational lensing and the generalized Sachs-Wolfe effects. This unified treatment of galaxy two-point statistics clarifies the relation of the gravitational lensing and the generalized Sachs-Wolfe effects to the metric perturbations and the underlying matter fluctuations. For near future dark energy surveys, we compute additional contributions to the observed galaxy two-point statistics and analyze their impact on the anisotropic structure. Thorough theoretical modeling of galaxy two-point statistics would be not only necessary to analyze precision measurements from upcoming dark energy surveys, but also provide further discriminatory power in understanding the underlying physical mechanisms.