Weak Gravitational Lensing

Understanding Weak Gravitational Lensing

Weak gravitational lensing is the coherent distortion of galaxy images by the large-scale distribution of dark matter. As light from distant galaxies travels through the universe, it is deflected by the gravitational potential of intervening matter, inducing correlations in galaxy shapes called cosmic shear. By measuring these tiny distortions (shear γ ~ 1%), we can map the dark matter distribution and constrain cosmological parameters, particularly the amplitude of matter fluctuations σ₈ and the matter density Ω_m. Tomographic lensing—dividing source galaxies into redshift bins—adds 3D information, breaking degeneracies and probing the evolution of structure.

What You're Seeing

Shear Power Spectrum C_ℓ(γ)

The shear power spectrum C_ℓ(γ) describes the angular correlation of galaxy shear at multipole ℓ (where ℓ ~ 180°/θ). It's computed by projecting the 3D matter power spectrum P(k,z) along the line of sight, weighted by the lensing efficiency q(z). The Limber approximation gives C_ℓ(γ) = ∫ dχ q²(χ) P(k=ℓ/χ, z(χ)) / χ². Each tomographic bin (i,j) has its own auto-spectrum (i=j) and cross-spectrum (i≠j), forming a matrix of observables that constrains both the amplitude and growth of structure. Higher ℓ probes smaller scales; the peak around ℓ ~ 1000-3000 comes from the nonlinear regime where dark matter halos dominate.

Lensing Kernel q(z)

The lensing kernel q(z) = (3/2) Ω_m H₀² (1+z) χ(z) ∫ dz' n(z') (χ'-χ)/χ' describes the efficiency of lensing at comoving distance χ. It's the product of the mass distribution along the line of sight and the geometric weight (χ'-χ)/χ', which peaks when the lens is halfway between us and the source. For a single source redshift bin with narrow distribution n(z), q(z) rises from zero, peaks around z ~ z_source/2, and drops at higher z. Multiple bins create overlapping kernels—cross-correlations between bins i and j measure the same dark matter structures at different effective distances, constraining the growth factor D(z).

Tomographic Constraints

Tomography dramatically improves cosmological constraints by breaking the degeneracy between σ₈ and Ω_m. Without redshift information, lensing measures only the combination S₈ = σ₈(Ω_m/0.3)^0.5. With multiple bins, we constrain both σ₈ (via the overall amplitude) and Ω_m (via the redshift evolution of q(z) and the shape of cross-correlations). This also tests the growth of structure: in ΛCDM, D(a) ∝ a at late times, but modified gravity theories predict different growth rates f = d ln D / d ln a. Current surveys like DES, HSC, and KiDS use ~5 tomographic bins from z=0.2 to z=1.5, measuring C_ℓ(γ) to percent-level precision.

The Physics of Weak Lensing

Convergence and Shear

Weak lensing induces two observable effects: convergence κ (isotropic magnification) and shear γ (anisotropic distortion). Both derive from the lensing potential ψ = ∫ Φ dχ, where Φ is the gravitational potential. In the weak-field limit, κ = ∇²ψ/2 and γ = (∂²ψ - ∂²ψ)/2 (complex notation). For matter perturbations δ(k,z), the potential is Φ(k) = -(3/2) Ω_m H₀² δ(k) / (k² a). Convergence and shear are related by κ = γ in Fourier space (for linear theory), so measuring the shear two-point function ⟨γ*γ⟩ directly probes the matter power spectrum.

Limber Approximation

The Limber approximation simplifies the projection from 3D power P(k,z) to 2D angular power C_ℓ by assuming that modes perpendicular to the line of sight dominate (valid for ℓ ≫ 1). This gives C_ℓ(γ) = ∫ dχ/χ² q²(χ) P(k=ℓ/χ, z(χ)), where the wavenumber k = ℓ/χ relates angular scale (ℓ) to physical scale at distance χ. The χ² factor comes from the solid angle; q²(χ) weights the matter distribution by lensing efficiency. For cross-spectra between bins i and j, replace q²(χ) with q_i(χ) q_j(χ). The Limber approximation breaks down at ℓ < 10, but is accurate for ℓ > 100 where most lensing signal lies.

Source Redshift Distribution n(z)

The source galaxy redshift distribution n(z) is critical for lensing cosmology. Real surveys measure n(z) from photometric redshifts (broad-band colors), which have uncertainties σ_z ~ 0.05(1+z). This toy uses a Gaussian model for each bin: n(z) ∝ exp(-(z-z_med)²/(2σ²)). Tomographic bins divide the full n(z) into slices—typically 3-5 bins with median redshifts from z~0.3 to z~1.5. Bin overlap creates cross-correlations; wider bins reduce shot noise but lose redshift resolution. Systematic errors in n(z) (e.g., catastrophic outliers) can bias cosmology, so surveys invest heavily in spectroscopic calibration.

Things to Try

1. Planck Cosmology: Use the Planck preset (Ω_m = 0.315, σ₈ = 0.81, h = 0.674). With 3 bins centered at z ~ 0.6, 0.9, 1.2, observe the family of C_ℓ curves. Auto-correlations (bin with itself) are strongest; cross-correlations (different bins) are weaker but probe overlapping redshift ranges. The peak around ℓ ~ 2000 comes from scales entering the nonlinear regime (k ~ 0.1-1 h/Mpc).
2. σ₈ Sensitivity: Increase σ₈ from 0.8 to 1.0 and watch C_ℓ rise by ~50% (since C_ℓ ∝ σ₈²). Lower σ₈ to 0.65 and the signal drops dramatically. This quadratic sensitivity makes lensing a powerful probe of σ₈. Current tension exists between CMB (σ₈ = 0.81 ± 0.01) and lensing surveys (σ₈ = 0.76 ± 0.02), possibly indicating new physics or systematics.
3. Tomographic Bins: Vary the number of bins from 1 to 5. With 1 bin (all sources), you see a single C_ℓ curve. With 5 bins, you get 15 spectra (5 auto + 10 cross). More bins improve constraints on Ω_m and growth, but shot noise increases (fewer galaxies per bin). Real surveys optimize this trade-off: DES uses 4 bins, Euclid will use ~10.
4. Lensing Kernel: Switch to "Lensing Kernel q(z)" view. For sources at z ~ 0.8, the kernel peaks around z ~ 0.4 (halfway to the source). Increase z_median to 1.5—the kernel extends to higher z and becomes broader, integrating more structure along the line of sight. Cross-correlations between nearby bins have similar kernels (high correlation); distant bins have less overlap (lower correlation).
5. Matter Density: Decrease Ω_m from 0.3 to 0.15. The lensing kernel drops (less matter to lens) and C_ℓ decreases. The peak also shifts: lower Ω_m means slower growth, so structure is less evolved. Increase Ω_m to 0.45—kernels rise and C_ℓ increases, especially at small scales where nonlinear effects are stronger.
6. Cross-Correlations: Enable "Show Cross-Correlations" to see spectra between different bins. Cross-spectra are always lower than auto-spectra (correlation coefficient r < 1). The ratio measures how much dark matter structure is shared between redshift slices. If bins are adjacent (e.g., z=0.6 and z=0.8), cross-correlation is strong. If bins are widely separated (z=0.4 and z=1.4), cross-correlation is weak—different cosmic volumes.

Why This Matters

Weak gravitational lensing is one of the most powerful probes of dark matter and dark energy:

• Dark Matter Mapping: Lensing is the only direct probe of all matter (dark + baryonic). Unlike galaxy clustering, which traces luminous matter, lensing measures total mass via gravity. Mass maps from DES and HSC reveal cosmic web filaments, voids, and massive halos, testing cold vs warm dark matter and self-interactions.
• σ₈ Tension: Current lensing surveys find σ₈ ~ 0.76, significantly lower than Planck CMB (σ₈ ~ 0.81). This 2-3σ tension could indicate new physics (e.g., decaying dark matter, modified gravity) or systematic errors (shear calibration, intrinsic alignments). Resolving this is a top priority for Stage-IV surveys like LSST and Euclid.
• Modified Gravity: In f(R) gravity or DGP models, the effective Newton's constant G_eff differs from GR, changing the lensing potential. Combining lensing (which measures Σ = Φ+Ψ) with redshift-space distortions (which measure Ψ) tests the consistency relation Σ/Ψ = 1 from GR. Deviations would rule out dark energy as cosmological constant.
• Neutrino Mass: Massive neutrinos suppress small-scale power, reducing C_ℓ at ℓ > 1000. Current constraints from lensing + CMB give Σm_ν < 0.12 eV (95% CL). Future surveys will reach σ(Σm_ν) ~ 0.02 eV, detecting the minimum mass from oscillations or ruling out inverted hierarchy.
• Dark Energy Equation of State: Tomographic lensing measures the growth rate f(z) and expansion H(z) via cross-correlations. This constrains the dark energy equation of state w(z). Combined with supernovae and BAO, lensing breaks the w₀-w_a degeneracy, testing whether dark energy is constant or evolving.
• Intrinsic Alignments: Galaxy shapes have intrinsic correlations from tidal torques during formation. This "intrinsic alignment" (IA) contaminates the lensing signal, mimicking extra shear. Tomography helps separate IA (which correlates nearby galaxies in redshift) from lensing (which correlates foreground and background). Modeling IA is critical for precision cosmology.
• Cluster Mass Calibration: Weak lensing provides unbiased mass estimates for galaxy clusters, calibrating scaling relations needed for cluster cosmology. This breaks the degeneracy between σ₈ and cluster selection function, enabling tight constraints from cluster counts.

Weak lensing tomography is entering a golden age. Current surveys (DES, KiDS, HSC) cover thousands of square degrees with ~10⁸ galaxies, measuring C_ℓ to ~5% precision. Next-generation experiments (LSST, Euclid, Roman) will image billions of galaxies over half the sky, reaching sub-percent systematics. Combined with CMB lensing (from SPT, ACT, Simons Observatory), 21cm surveys, and gravitational waves, tomographic shear will map the entire growth history of structure from z=0 to z=6, testing General Relativity on cosmic scales and potentially revealing new forces in the dark sector.