Weak Lensing Peak Counts in Cosmology

• Peak counts are local maxima in smoothed weak-lensing convergence maps that capture non-Gaussian aspects of cosmic structure.
• They employ diverse filtering strategies, including Gaussian and starlet filters, to isolate signals across multiple scales.
• Combining peak counts with traditional power spectrum analyses significantly tightens constraints on parameters like Ωm, σ8, and neutrino masses.

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Peak counts in weak gravitational lensing are the abundance of local maxima in smoothed convergence maps, or in closely related aperture-mass or filtered shear maps, binned by peak height or signal-to-noise. In operational terms, a peak is usually a pixel whose value exceeds that of its eight neighbors on a 2D grid, and the statistic is measured as a histogram in $\kappa$ or $\nu \equiv \kappa_s/\sigma_{\rm noise}$. Because the lensing field on arcminute scales is shaped by nonlinear structure formation, halo profiles, line-of-sight projections, and observational noise, peak counts are an explicitly non-Gaussian observable complementary to the power spectrum (Li et al., 2018).

1. Definition and basic formalism

In the weak-lensing context, the basic scalar field is the convergence $\kappa(\boldsymbol{\theta})$, related in one common formulation to the lensing potential by

$$\kappa(\boldsymbol{\theta}) = \frac{1}{2}\nabla^2\Psi(\boldsymbol{\theta}) .$$

Peak-count analyses first smooth the convergence field,

$$\kappa_s(\boldsymbol{\theta}) = \int d^2\theta' \, W(|\boldsymbol{\theta} - \boldsymbol{\theta}'|)\,\kappa(\boldsymbol{\theta}'),$$

typically with a Gaussian kernel, and then identify local maxima on the pixelized map. A common height variable is

$\nu \equiv \frac{\kappa_s}{\sigma_{\rm noise}},$

with $\sigma_{\rm noise}$ the rms of the smoothed noise field in the relevant tomographic bin or survey configuration (Li et al., 2018).

A closely related construction uses the aperture mass,

$$M_{\rm ap}(\boldsymbol{\theta}_0)=\int d^2\boldsymbol{\theta}\, Q(\theta)\, g_t(\boldsymbol{\theta},\boldsymbol{\theta}_0),$$

where $g_t$ is the tangential component of the reduced shear relative to $\boldsymbol{\theta}_0$. In that formulation, peaks are local maxima of the aperture-mass or SNR map rather than of $\nu \equiv \kappa_s/\sigma_{\rm noise}$0 itself (Bard et al., 2013).

The statistic is then the peak histogram, measured either directly in $\nu \equiv \kappa_s/\sigma_{\rm noise}$1 or in SNR. Different studies adopt different binning conventions, but the core observable is always the number density of local maxima as a function of height. This makes peak counts a one-point summary of the extrema of a smoothed projected-mass field, rather than of its Fourier amplitudes.

2. Physical origin of peaks and their interpretation

Peak height has a direct but nontrivial relation to the underlying matter distribution. High-S/N peaks are typically produced by single, massive halos, whereas medium peaks are predominantly due to a superposition of several smaller halos along the line of sight, and low peaks arise from subtle combinations of many low-mass structures and noise (Li et al., 2018). This stratification is central: peak counts are not simply cluster counts in another guise.

That point is reinforced by halo-only tests. A comparison between full ray-traced simulations and maps built from only particles associated with halos shows that halo-only contributions replicate peak counts qualitatively well, but halos do not explain all weak-lensing peaks. In particular, halo-only maps underpredict negative peaks, which are associated with local overdensities in large-scale underdense regions along the line of sight, and they miss contributions from other elements of the cosmic web outside and far away from dark matter halos (Sabyr et al., 2021).

The same logic extends beyond galaxy weak lensing. In CMB lensing, peaks are defined as local maxima of reconstructed convergence maps after smoothing, and they probe nonlinear structure along a lensing kernel that peaks at $\nu \equiv \kappa_s/\sigma_{\rm noise}$2. The observable is formally analogous, but its redshift weighting and noise properties differ from those of galaxy-lensing peak counts (Liu et al., 2016).

3. Cosmological information content

Peak counts were developed precisely because second-order statistics do not exhaust the information content of the nonlinear lensing field. In an LSST-like forecast for tomographic weak-lensing peak counts, combining peak counts with the traditional lensing power spectrum improves the constraint on the neutrino mass sum, $\nu \equiv \kappa_s/\sigma_{\rm noise}$3, and $\nu \equiv \kappa_s/\sigma_{\rm noise}$4 by 39%, 32%, and 60%, respectively, over that from the power spectrum alone (Li et al., 2018). In a later multiscale analysis based on MassiveNuS, peak counts perform better than the power spectrum on $\nu \equiv \kappa_s/\sigma_{\rm noise}$5 respectively by 63%, 40% and 72% when using a starlet filter and by 70%, 40% and 77% when using a multiscale Gaussian (Ajani et al., 2020).

Observational analyses show the same complementarity. In CFHTLenS, constraints from peak counts are comparable to those from the power spectrum, and somewhat tighter when different smoothing scales are combined. When the power spectrum and peak counts are combined, the area of the error “banana” in the $\nu \equiv \kappa_s/\sigma_{\rm noise}$6 plane reduces by a factor of $\nu \equiv \kappa_s/\sigma_{\rm noise}$7, compared to using the power spectrum alone; for a flat $\nu \equiv \kappa_s/\sigma_{\rm noise}$8 cold dark matter model, the combined constraint is

$\nu \equiv \kappa_s/\sigma_{\rm noise}$9

(Liu et al., 2014).

Peak counts have also been proposed as probes of primordial non-Gaussianity. For survey specifications similar to those of Euclid and without assuming knowledge of the lens and source redshifts, the peak functions of models with $\kappa(\boldsymbol{\theta})$0 differ by up to 15% from the Gaussian peak function at the high-mass end, and assuming the other cosmological parameters known, $\kappa(\boldsymbol{\theta})$1 can be measured with an error $\kappa(\boldsymbol{\theta})$2 (Marian et al., 2010).

A common misconception is that only the highest peaks matter. Multiple studies explicitly show otherwise: low and medium peaks carry information comparable to that of high peaks, but with different degeneracy directions in parameter space (Li et al., 2018). This suggests that the cosmological content of peak counts is distributed across the full peak-height distribution rather than concentrated solely in the rarest maxima.

4. Filtering strategies, forward models, and inference

Filtering is not a mere technicality; it determines which structures enter the statistic and how correlated the resulting scales are. Comparisons among Gaussian filtering, the starlet wavelet, aperture mass, and the nonlinear MRLens method show that starlet filtering outperforms the Gaussian kernel, and that including peak counts from different smoothing scales helps to lift parameter degeneracies. Peak counts from different smoothing scales with a compensated filter show very little cross-correlation, and adding information from different scales can therefore strongly enhance the available information. Measuring peak counts separately from different scales yields tighter constraints than using a combined peak histogram from a single map that includes multiscale information (Lin et al., 2016).

The multiscale neutrino analysis sharpens this point. Both multiscale starlet and multiscale Gaussian filters behave similarly in raw constraining power, but with the starlet filter the majority of the information in the data covariance matrix is encoded in the diagonal elements, which can be an advantage when inverting the matrix and speeding up the numerical implementation (Ajani et al., 2020).

On the modeling side, halo-based forward models were introduced to bypass the cost of full $\kappa(\boldsymbol{\theta})$3-body suites. One such framework samples halos from a mass function, assigns them NFW profiles, places those halos randomly on the field of view, and performs ray-tracing through these “fast simulations.” In its original comparison to $\kappa(\boldsymbol{\theta})$4-body runs, this approach was found to be in good agreement with full simulations; it also showed that the lensing signal dominates shape noise and Poisson noise for peaks with SNR between 4 and 6, and that counts from the same SNR range are sensitive to $\kappa(\boldsymbol{\theta})$5 and $\kappa(\boldsymbol{\theta})$6 (Lin et al., 2014).

Emulation-based inference has become standard. Gaussian-process interpolation is used in MassiveNuS-based analyses to emulate the cosmology dependence of power spectra and peak histograms across $\kappa(\boldsymbol{\theta})$7 space (Li et al., 2018). In the CFHTLenS analysis, a dedicated emulator interpolates both the power spectrum and the peak counts to an accuracy of $\kappa(\boldsymbol{\theta})$8 across 91 cosmological models in $\kappa(\boldsymbol{\theta})$9 (Liu et al., 2014).

5. Systematics and modeling limitations

The most prominent contemporary limitation is baryonic physics. Semi-analytic baryonic correction models were proposed as a reduced-cost alternative to full hydrodynamical simulations, but their adequacy depends on survey depth and peak height. In a comparison between IllustrisTNG and a baryon-corrected version of IllustrisTNG-Dark, peak counts in baryonic correction models are accurate at the percent level for peaks with $$\kappa(\boldsymbol{\theta}) = \frac{1}{2}\nabla^2\Psi(\boldsymbol{\theta}) .$$0, statistically indistinguishable from IllustrisTNG in most current and ongoing surveys, but insufficient for deep future surveys covering the largest solid angles, such as LSST and $$\kappa(\boldsymbol{\theta}) = \frac{1}{2}\nabla^2\Psi(\boldsymbol{\theta}) .$$1. These models match individual peaks accurately but underpredict the amplitude of the highest peaks (Lee et al., 2022).

A broader hydrodynamical assessment with FLAMINGO finds that realistic baryonic feedback prescriptions induce differences typically $$\kappa(\boldsymbol{\theta}) = \frac{1}{2}\nabla^2\Psi(\boldsymbol{\theta}) .$$2 for $$\kappa(\boldsymbol{\theta}) = \frac{1}{2}\nabla^2\Psi(\boldsymbol{\theta}) .$$3, smaller than the $$\kappa(\boldsymbol{\theta}) = \frac{1}{2}\nabla^2\Psi(\boldsymbol{\theta}) .$$4 shifts induced by reasonable cosmological-parameter variations over the same $$\kappa(\boldsymbol{\theta}) = \frac{1}{2}\nabla^2\Psi(\boldsymbol{\theta}) .$$5 range, but still large enough that they must be modeled carefully to obtain unbiased results. For the range of models investigated, the baryonic suppression is insensitive to changes in cosmology up to $$\kappa(\boldsymbol{\theta}) = \frac{1}{2}\nabla^2\Psi(\boldsymbol{\theta}) .$$6, while the higher-$$\kappa(\boldsymbol{\theta}) = \frac{1}{2}\nabla^2\Psi(\boldsymbol{\theta}) .$$7 regime is dominated by Poisson noise and cosmic variance (Broxterman et al., 2023).

Several observational systematics act directly on the map-making stage. Neglecting the shear-to-convergence inversion introduces a small but non-negligible bias, and when propagated to parameter estimation, constraint contours involving the dark-energy equation of state can differ by 2-sigma (Lin et al., 2017). Planar projections also matter: projected peak counts consistently overestimate the counts at low SNR thresholds and underestimate them at high SNR thresholds across the projections evaluated, although the difference is reduced when smoothing of the maps is increased (Vallis et al., 2019).

Selection effects in the source population provide another route to bias. Magnification and size bias modulate the effective source density in a way correlated with $$\kappa(\boldsymbol{\theta}) = \frac{1}{2}\nabla^2\Psi(\boldsymbol{\theta}) .$$8, and if not accounted for, they introduce systematic errors for cosmological measurements. At $$\kappa(\boldsymbol{\theta}) = \frac{1}{2}\nabla^2\Psi(\boldsymbol{\theta}) .$$9, expected in the I band for LSST, the inferred values of $$\kappa_s(\boldsymbol{\theta}) = \int d^2\theta' \, W(|\boldsymbol{\theta} - \boldsymbol{\theta}'|)\,\kappa(\boldsymbol{\theta}'),$$0, $$\kappa_s(\boldsymbol{\theta}) = \int d^2\theta' \, W(|\boldsymbol{\theta} - \boldsymbol{\theta}'|)\,\kappa(\boldsymbol{\theta}'),$$1 and $$\kappa_s(\boldsymbol{\theta}) = \int d^2\theta' \, W(|\boldsymbol{\theta} - \boldsymbol{\theta}'|)\,\kappa(\boldsymbol{\theta}'),$$2 are biased by many sigma; the parameters are also biased differently in the $$\kappa_s(\boldsymbol{\theta}) = \int d^2\theta' \, W(|\boldsymbol{\theta} - \boldsymbol{\theta}'|)\,\kappa(\boldsymbol{\theta}'),$$3 space when the power spectrum and peak counts are used, which suggests that combining the two can help mitigate the impact of magnification and size biases (Liu et al., 2013).

Not all instrumental effects are equally damaging. In an LSST forecast based on aperture-mass peaks, realistic galaxy shape measurement errors were found to have relatively little impact on the constraining power of shear peak counts (Bard et al., 2013). This suggests that the dominant theoretical uncertainties lie less in simple shape-measurement noise than in the nonlinear astrophysics and map-level systematics that reshape the peak distribution itself.

6. Domains of application and current scope

Peak counts now span several observational and theoretical regimes. In galaxy weak lensing they have been applied to CFHTLenS data and forecast for DES, KiDS, HSC, LSST, $$\kappa_s(\boldsymbol{\theta}) = \int d^2\theta' \, W(|\boldsymbol{\theta} - \boldsymbol{\theta}'|)\,\kappa(\boldsymbol{\theta}'),$$4, and $$\kappa_s(\boldsymbol{\theta}) = \int d^2\theta' \, W(|\boldsymbol{\theta} - \boldsymbol{\theta}'|)\,\kappa(\boldsymbol{\theta}'),$$5; in CMB lensing they have been studied as a higher-order complement to the reconstructed convergence power spectrum. For AdvACT, a CMB-lensing analysis forecasts a 6$$\kappa_s(\boldsymbol{\theta}) = \int d^2\theta' \, W(|\boldsymbol{\theta} - \boldsymbol{\theta}'|)\,\kappa(\boldsymbol{\theta}'),$$6 detection of peaks and finds that combining the power spectrum, PDF, and peak counts tightens cosmological constraints in the $$\kappa_s(\boldsymbol{\theta}) = \int d^2\theta' \, W(|\boldsymbol{\theta} - \boldsymbol{\theta}'|)\,\kappa(\boldsymbol{\theta}'),$$7 plane by $$\kappa_s(\boldsymbol{\theta}) = \int d^2\theta' \, W(|\boldsymbol{\theta} - \boldsymbol{\theta}'|)\,\kappa(\boldsymbol{\theta}'),$$8, compared to using the power spectrum alone (Liu et al., 2016).

The following examples summarize the range of uses already demonstrated in the literature.

Domain	Representative result	arXiv
CFHTLenS weak lensing	Combined power spectrum and peak counts reduce the $$\kappa_s(\boldsymbol{\theta}) = \int d^2\theta' \, W(|\boldsymbol{\theta} - \boldsymbol{\theta}'|)\,\kappa(\boldsymbol{\theta}'),$$9 error area by a factor of $\nu \equiv \frac{\kappa_s}{\sigma_{\rm noise}},$0	(Liu et al., 2014)
LSST-like neutrino forecasts	Peaks + power spectrum improve constraints on $\nu \equiv \frac{\kappa_s}{\sigma_{\rm noise}},$1, $\nu \equiv \frac{\kappa_s}{\sigma_{\rm noise}},$2, $\nu \equiv \frac{\kappa_s}{\sigma_{\rm noise}},$3 by 39%, 32%, 60%	(Li et al., 2018)
Multiscale neutrino forecasts	Multiscale peaks alone outperform the power spectrum by 63–77% depending on filter and parameter	(Ajani et al., 2020)
Baryonic modeling	Baryonic prescriptions alter peak counts by typically $\nu \equiv \frac{\kappa_s}{\sigma_{\rm noise}},$4 for $\nu \equiv \frac{\kappa_s}{\sigma_{\rm noise}},$5	(Broxterman et al., 2023)
CMB lensing	Peaks forecast at 6$\nu \equiv \frac{\kappa_s}{\sigma_{\rm noise}},$6 for AdvACT, with joint constraints tightened by $\nu \equiv \frac{\kappa_s}{\sigma_{\rm noise}},$7	(Liu et al., 2016)

For large-area future surveys, two methodological directions recur across the literature. First, multiscale and tomographic analyses are favored because they recover information that single-scale summaries miss (Ajani et al., 2020). Second, spherical treatments are increasingly important, since planar projections introduce SNR-dependent distortions that become non-negligible over large sky areas (Vallis et al., 2019). A plausible implication is that peak-count cosmology is moving toward end-to-end forward models in which baryons, survey geometry, shear inversion, tomography, and covariance structure are all treated at the same level of fidelity as the peak statistic itself.