[Paper Review] Lens distortion effects on CMB maps

Francis Bernardeau|arXiv (Cornell University)|Feb 18, 1998

Galaxies: Formation, Evolution, Phenomena13 citations

TL;DR

This paper investigates how weak gravitational lensing distorts Cosmic Microwave Background (CMB) temperature maps, showing that lensing significantly alters the probability distribution function (PDF) of ellipticity derived from local temperature curvature, inducing an excess of elongated structures. Analytical and numerical results confirm this effect, with signal-to-noise ratios of 3–6 for Planck-like surveys, enabling marginal detection of lensing imprints in CMB maps.

ABSTRACT

Weak lensing effects are known to introduce non-linear couplings in CMB temperature maps. In inflationary scenario, the primary CMB anisotropies are expected to form a 2D Gaussian map, for which, the probability distribution function of the ellipticity defined from the local temperature curvature matrix has a very specific shape. I show that lenses alter significantly the shape of this PDF, inducing an excess of elongated structures. The precise functional form is computed for both the field points and the temperature extrema. These analytical results are confirmed by numerical experiments on 10x10 square degree maps. These numerical results allow to investigate the effects of smoothing and to estimate the cosmic variance. For the best resolution and sky coverage of the Planck mission the signal to noise ratio for the statistical indicators presented here is about 3 to 6 depending on the cosmological models. A marginal detection should therefore be possible.

Motivation & Objective

• To understand how weak gravitational lensing modifies the statistical properties of CMB temperature maps.
• To quantify the impact of lensing on the probability distribution function (PDF) of ellipticity derived from local temperature curvature matrices.
• To assess the detectability of lensing-induced non-Gaussian features in CMB maps under realistic observational conditions.
• To evaluate the effects of smoothing and cosmic variance on statistical indicators of lensing distortion.
• To estimate the signal-to-noise ratio for detecting lensing effects in future CMB experiments like Planck.

Proposed method

• Analytically compute the functional form of the ellipticity PDF for both field points and temperature extrema under lensing effects.
• Use numerical simulations on 10×10 square degree CMB maps to validate analytical predictions.
• Model the primary CMB anisotropies as a 2D Gaussian random field before lensing.
• Apply weak lensing formalism to introduce non-linear couplings into the temperature field.
• Apply smoothing to simulate instrumental beam effects and assess its impact on statistical indicators.
• Estimate cosmic variance contributions using ensemble-averaged numerical experiments.

Experimental results

Research questions

• RQ1 How does weak lensing alter the shape of the ellipticity PDF derived from CMB temperature maps?
• RQ2 What is the functional form of the lensed ellipticity PDF at field points and temperature extrema?
• RQ3 To what extent do smoothing and sky coverage affect the detectability of lensing-induced non-Gaussianity?
• RQ4 What is the expected signal-to-noise ratio for detecting lensing effects in CMB maps from the Planck mission?
• RQ5 Can lensing imprints be statistically distinguished from primary CMB anisotropies using the ellipticity PDF?

Key findings

• Lensing induces a significant distortion in the ellipticity PDF, causing an excess of elongated structures compared to the unlensed Gaussian case.
• The analytical form of the lensed PDF for field points and temperature extrema is derived and confirmed by numerical simulations.
• Numerical experiments show that smoothing reduces the amplitude of the lensing signal but preserves its statistical detectability.
• Cosmic variance is quantified and found to be a limiting factor, but not prohibitive, for detection.
• For the best resolution and sky coverage expected from the Planck mission, the signal-to-noise ratio for the statistical indicators ranges from 3 to 6 across different cosmological models.
• A marginal detection of lensing effects in CMB maps is therefore feasible with Planck-like data.