Weak Lensing and Cosmology

TL;DR

The paper develops a cosmology-dependent framework for weak gravitational lensing by relating the angular distortion power spectrum P_ψ(ω) to the 3D power spectrum of potential fluctuations, generalizing Limber's equation to FRW geometries. A central result is P_ψ(ω) = ω^4 ∫ dz [g^2(z)/sinh^6 z] P_Φ(ω/ sinh z; z), enabling predictions across open, flat, and Λ-dominated universes. It explores multiple P(k) models (δ-function, power-law, and Peacock's empirical spectrum), analyzes normalization choices, and assesses feasibility and survey design, showing that the growth of distortion with source redshift is a powerful cosmological discriminator, while the absolute amplitude depends on normalisation. The study also compares large-scale structure lensing with cluster lensing, highlighting that Λ models reduce Σ_crit at high zl and that mass estimates from lensing are only weakly cosmology dependent in many regimes. Overall, the work demonstrates that high-precision weak-lensing measurements, particularly with redshift information for source galaxies, can constrain cosmological models and complement cluster and dynamical probes.

Abstract

We explore the dependence of weak lensing phenomena on the background cosmology. We first generalise the relation between $P_ψ(ω)$, the angular power spectrum of the distortion, and the power spectrum of density fluctuations to non-flat cosmologies. We then compute $P_ψ$ for various illustrative models. A useful cosmological discriminator is the growth of $P_ψ$ with source redshift which is much stronger in low matter density models, and especially in $Λ$-dominated models. With even crude redshift information (say from broad band colours) it should be possible to constrain the cosmological world model. The amplitude of $P_ψ(ω)$ is also quite sensitive to the cosmology, but requires a reliable external normalisation for the mass fluctuations. If one normalises to galaxy clustering, with $M/L$ fixed by small-scale galaxy dynamics, then low density models predict a much stronger distortion. If, however, one normalises to large-scale bulk-flows, the predicted distortion for sources at redshifts $Z_s \sim 1-3$ is rather insensitive to the background cosmology. The signals predicted here can be detected at a very high level of significance with a photometric survey covering say 10 square degrees, but sparse sampling is needed to avoid large sampling variance and we discuss the factors influencing the design of an optimum survey. Turning to weak lensing by clusters we find that for high lens redshifts ($Z_l\simeq1$) the critical density is substantially reduced in $Λ$ models, but that the ratio of the shear or convergence to the velocity dispersions or X-ray temperature of clusters is only very weakly dependent on the cosmology.