[Paper Review] Weak Lensing On the Celestial Sphere

Albert Stebbins|arXiv (Cornell University)|Sep 22, 1996

Galaxies: Formation, Evolution, Phenomena26 references8 citations

TL;DR

This paper develops a tensor spherical harmonic formalism to model weak gravitational lensing effects on the celestial sphere due to large-scale cosmic inhomogeneities. It computes the angular power spectrum of shear and shows detectable shear at levels ≤10⁻³ for nearby galaxies (z ≲ 0.2), with the highest signal-to-noise on large angular scales (θ ≳ 10°).

ABSTRACT

This paper details a description of the pattern of galaxy image distortion over the entire sky caused by the gravitational lensing which is the result of large scale inhomogeneities in our universe. We present a tensor spherical harmonic formalism to describe this pattern, giving many useful formulae. This is applied to density inhomogeneities, where we compute the angular power spectrum of the shear pattern, as well as the noise properties due to finite galaxy sampling and cosmic variance. We show that a detectable level of shear is present for very nearby galaxies, $z\simlt0.2$. For such a shallow sample much of the largest signal-to-noise comes from very large angular scales, $θ\simgt10^\circ$, although it is in the form of very small shear at a level $\simlt10^{-3}$.

Motivation & Objective

• To model the pattern of galaxy image distortion across the entire sky due to weak gravitational lensing from large-scale inhomogeneities.
• To develop a mathematical framework suitable for analyzing weak lensing on a curved, spherical sky rather than in flat-sky approximations.
• To compute the angular power spectrum of shear induced by density inhomogeneities in the universe.
• To quantify noise contributions from finite galaxy sampling and cosmic variance in weak lensing measurements.
• To assess the detectability of weak lensing shear in shallow surveys with redshift z ≲ 0.2.

Proposed method

• Adopts a tensor spherical harmonic formalism to describe the shear field on the celestial sphere, generalizing vector and scalar harmonics to tensor fields.
• Derives expressions for the shear power spectrum in terms of the power spectrum of the underlying density field.
• Uses the Limber approximation and spherical Bessel functions to relate the 3D density power spectrum to the 2D angular shear spectrum.
• Incorporates statistical noise models, including cosmic variance and shot noise from finite galaxy sampling.
• Applies the formalism to a redshift distribution of galaxies with z ≲ 0.2 to estimate observable shear amplitudes.
• Evaluates the signal-to-noise ratio across angular scales, emphasizing large-scale contributions.

Experimental results

Research questions

• RQ1 How can weak lensing shear on the celestial sphere be systematically described using a consistent harmonic formalism?
• RQ2 What is the angular power spectrum of shear induced by large-scale density inhomogeneities in the universe?
• RQ3 What level of shear is expected for galaxies at low redshift (z ≲ 0.2), and on which angular scales is it most prominent?
• RQ4 What are the dominant noise sources—cosmic variance and finite sampling—that limit detectability in weak lensing surveys?
• RQ5 Can weak lensing shear be detected in shallow surveys, and what is the optimal angular scale for such detection?

Key findings

• The tensor spherical harmonic formalism provides a rigorous and complete framework for modeling weak lensing shear on the full celestial sphere.
• For galaxies at redshift z ≲ 0.2, a detectable level of shear is present, with typical amplitudes ≤10⁻³.
• The largest signal-to-noise ratio in shallow surveys arises from very large angular scales, θ ≳ 10°, despite the small shear amplitudes.
• Cosmic variance and finite galaxy sampling contribute significant noise, with the former dominating on large scales.
• The shear angular power spectrum is sensitive to the underlying matter power spectrum and redshift distribution of source galaxies.
• The formalism enables precise computation of expected signal and noise, supporting future weak lensing cosmology surveys.