Weak lensing and cosmic shear are powerful tools in observational cosmology. They allow us to map the invisible dark matter distribution by studying tiny distortions in galaxy shapes caused by gravitational lensing.

These techniques require precise measurements of galaxy shapes and sophisticated statistical analysis. By studying weak lensing across large cosmic volumes, we can constrain key cosmological parameters and test theories of gravity on the largest scales.

Weak Lensing Fundamentals

Gravitational Lensing and Weak Lensing

• Gravitational lensing occurs when massive objects bend light from distant sources
• Strong lensing produces multiple images or arcs of background galaxies
• Weak lensing causes subtle distortions in galaxy shapes, typically too small to detect individually
• Weak lensing effects accumulate statistically over large samples of galaxies
• Cosmic shear refers to coherent weak lensing distortions caused by large-scale structure

Shape Measurement Techniques

• Shape measurement quantifies the ellipticity and orientation of galaxy images
• Moment-based methods calculate quadrupole moments of galaxy light distribution
• Model-fitting approaches fit parametric models (Sérsic profiles) to observed galaxy images
• Machine learning techniques train on simulated galaxy images to predict intrinsic shapes
• Shape measurement challenges include accounting for point spread function and pixel noise

Sources of Systematic Error

• Atmospheric turbulence blurs galaxy images, complicating shape measurements
• Telescope optics introduce aberrations that must be carefully calibrated
• Detector effects like charge transfer inefficiency can mimic weak lensing signals
• Selection biases in galaxy samples can lead to spurious lensing signals
• Intrinsic alignments of galaxies can contaminate cosmic shear measurements

Weak Lensing Statistics

Shear Correlation Functions

• Shear correlation functions measure coherent galaxy shape distortions as a function of separation
• Two-point correlation function $\xi_\pm(\theta)$ quantifies tangential and cross components of shear
• Higher-order statistics like three-point functions probe non-Gaussian features of mass distribution
• Correlation functions averaged in annular bins around lens galaxies measure galaxy-galaxy lensing
• Theoretical models predict shear correlations for different cosmological parameters

Convergence Power Spectrum

• Convergence power spectrum $P_\kappa(l)$ describes statistical properties of lensing mass distribution
• Fourier transform of shear correlation function yields convergence power spectrum
• $P_\kappa(l)$ directly related to matter power spectrum $P_\delta(k)$ weighted by lensing efficiency
• Shape of convergence power spectrum sensitive to cosmological parameters ($\Omega_m$, $\sigma_8$)
• Baryonic effects modify small-scale convergence power, requiring careful modeling

Tomographic Weak Lensing Analysis

• Tomographic weak lensing divides source galaxies into redshift bins
• Cross-correlations between redshift bins probe 3D matter distribution
• Improves constraints on dark energy equation of state and modified gravity models
• Requires accurate photometric redshift estimates for large galaxy samples
• Self-calibration techniques use clustering information to constrain redshift distributions

Advanced Weak Lensing Techniques

Intrinsic Alignments and Mitigation Strategies

• Intrinsic alignments arise from correlations between galaxy shapes and large-scale structure
• Can mimic or contaminate cosmic shear signal, especially for nearby galaxy pairs
• Linear alignment model describes alignments of elliptical galaxies with large-scale potential
• Nonlinear alignment model incorporates additional tidal torquing effects
• Mitigation strategies include joint modeling of intrinsic alignments and cosmic shear
• Nulling techniques remove alignment-sensitive modes from weak lensing analysis

Mass Mapping and Peak Statistics

• Mass mapping reconstructs projected mass distribution from observed shear field
• Kaiser-Squires inversion relates shear to convergence in Fourier space
• Wiener filtering improves mass map signal-to-noise by suppressing noise
• Peak statistics count local maxima in convergence maps as function of height
• Abundance of high peaks sensitive to cosmological parameters and structure formation
• Machine learning techniques (convolutional neural networks) extract cosmological information from mass maps