Marked Convergence Fields: Weak Lensing & Beyond

• Marked convergence fields are nonlinear reweightings of convergence maps using mark functions derived from a more heavily smoothed field, capturing key non-Gaussian features.
• They enhance weak-lensing analyses by combining Gaussian and non-Gaussian information, leading to up to a 43% improvement in S8 constraints over standard two-point spectra.
• The method extends to related frameworks such as marked random trees and extremal processes, connecting geometric transformations with asymptotic convergence in diverse applications.

Searching arXiv for papers on marked convergence fields and related weak-lensing marked statistics. Marked convergence fields are most concretely defined in weak gravitational lensing as nonlinearly reweighted convergence maps, obtained by multiplying a convergence field by a mark that depends on a smoothed version of that same field. In the notation used for Subaru Hyper Suprime-Cam Year 1, the marked field is

$$\Delta(\kappa)(\boldsymbol{\theta}) \equiv m\!\left(\kappa_\theta(\boldsymbol{\theta})\right)\,\kappa(\boldsymbol{\theta}),$$

where $\kappa$ is the $1'$-smoothed convergence field, $\kappa_\theta$ is a more heavily smoothed version, and $m$ is a nonlinear mark function (Cowell et al., 16 Jul 2025). A broader mathematical reading of the term links this construction to marked random fields on random trees, marked point processes of exceedances, and extremal fields governed by derivative-martingale-like limits; in each case, a base geometry or field is coupled to marks, and one studies convergence of the resulting marked object under conditioning, scaling, or asymptotic limits (Boulal et al., 23 Sep 2025, Ding et al., 2015, Cipriani et al., 2015).

1. Weak-lensing convergence as the underlying field

On the sphere, the weak-lensing convergence field is a scalar spin-0 field

$$\kappa(\hat{\mathbf{n}})\equiv {}_0\kappa(\theta,\phi),$$

encoding the projected line-of-sight matter fluctuations. In the formalism used for spherical mass mapping, convergence and shear derive from the lensing potential $\phi(r,\hat{\mathbf{n}})$, itself a line-of-sight projection of the Newtonian potential $\Phi$. The spherical relations are

$${}_0\kappa(r,\hat{\mathbf{n}})=\tfrac14(\eth\bar\eth+\bar\eth\eth)\,{}_0\phi(r,\hat{\mathbf{n}}), \qquad {}_2\gamma(r,\hat{\mathbf{n}})=\tfrac12\eth\eth\,{}_0\phi(r,\hat{\mathbf{n}}),$$

and in harmonic space

$${}_2\gamma_{\ell m} = {}_2F_\ell\,{}_0\kappa_{\ell m}, \qquad {}_2F_\ell = \frac{-1}{\ell(\ell+1)}\sqrt{\frac{(\ell+2)!}{(\ell-2)!}}.$$

This diagonal $\kappa$0-space kernel is the basis of spherical Kaiser–Squires inversion, which reconstructs $\kappa$1 from observed shear coefficients by

$\kappa$2

The same paper emphasizes that the full-sky formulation matters for wide surveys: for sky coverages typical of Euclid-, LSST-, and WFIRST-like surveys, projection-induced errors from planar approximations can be of order tens of percent and can exceed $\kappa$3 in some cases, while stereographic projection is the most effective planar option among those tested (Wallis et al., 2017).

This geometric baseline is essential for marked analyses. Marks act on $\kappa$4, but the fidelity of the marked field depends first on the fidelity of the convergence reconstruction itself. In particular, the spherical formalism avoids projection artefacts that would otherwise contaminate amplitude-based, environment-based, or E/B-based marks. The same source also notes that reduced shear,

$\kappa$5

makes the inversion nonlinear, and an iterative fixed-point scheme is used before any downstream manipulation of $\kappa$6 (Wallis et al., 2017).

2. Construction of marked convergence fields in weak lensing

The HSC-Y1 marked-field analysis starts from convergence maps reconstructed from galaxy shear catalogues by Kaiser–Squires inversion with inpainting to control E/B leakage, producing $\kappa$7-smoothed maps in four tomographic bins. From each such map, a more heavily smoothed field $\kappa$8 is constructed with a Gaussian filter of scale $\kappa$9, and the marked field is then defined by $1'$0. The central idea is to encode higher-order, non-Gaussian information into modified two-point statistics while retaining the computational structure of standard angular power-spectrum estimators (Cowell et al., 16 Jul 2025).

Three mark functions are used in that analysis.

Mark	Definition	Qualitative effect
$1'$1	GP-defined $1'$2	Smooth, non-polynomial, sign-flipping behaviour
$1'$3	$1'$4	Up-weights both strongly positive and strongly negative environments
$1'$5	$1'$6, with $1'$7, $1'$8, $1'$9, $\kappa_\theta$0	Up-weights underdense, void-like, or trough-like regions

Mark $\kappa_\theta$1 is a Gaussian-process–shaped mark inherited from earlier optimization work; its node positions are redefined for each map and smoothing scale to be equally spaced between the minimum and maximum $\kappa_\theta$2 values. Mark $\kappa_\theta$3 is analytically transparent and directly connects marked spectra to higher-order correlators. Mark $\kappa_\theta$4 is a modified White-type underdensity mark with a safety function

$\kappa_\theta$5

introduced to avoid singularities (Cowell et al., 16 Jul 2025).

The marked angular spectra are then defined in the same flat-sky pseudo-$\kappa_\theta$6 framework as the ordinary convergence spectrum. Besides $\kappa_\theta$7, the analysis measures the cross-spectra $\kappa_\theta$8 and auto-spectra $\kappa_\theta$9. For mark $m$0, the connection to higher-order structure is explicit: $m$1 so $m$2 is a weighted bispectrum integral, while $m$3 contains both disconnected Gaussian four-point terms and connected trispectrum terms (Cowell et al., 16 Jul 2025).

3. Statistical content and interpretation

The interpretive rationale of marked convergence fields is that $m$4 is strongly non-Gaussian on small scales, so standard two-point spectra do not exhaust the available cosmological information. Marked spectra repackage part of this information into objects that are still estimated with pseudo-$m$5 methods and standard covariance machinery. In the Gaussian limit, no genuine gain is expected: a nonlinear transformation would only reparameterize information already present in $m$6. The observed utility of marked fields therefore depends on non-Gaussianity, scale mixing, and the environmental selectivity of the chosen mark (Cowell et al., 16 Jul 2025).

A common simplification is to treat all gains as purely higher-order. The HSC-Y1 analysis shows a more nuanced picture. For mark $m$7, $m$8 is bispectrum-like and $m$9 is trispectrum-like, but the auto-spectrum also contains a Gaussian convolution term involving products $$\kappa(\hat{\mathbf{n}})\equiv {}_0\kappa(\theta,\phi),$$0. The paper therefore concludes that the improved constraints arise from a mixture of genuine non-Gaussian information and Gaussian small-scale power leaking into lower multipoles through the mark construction. This also means that scale cuts for marked spectra do not have the same interpretation as scale cuts for ordinary two-point spectra: removing $$\kappa(\hat{\mathbf{n}})\equiv {}_0\kappa(\theta,\phi),$$1 in $$\kappa(\hat{\mathbf{n}})\equiv {}_0\kappa(\theta,\phi),$$2 does not eliminate sensitivity to underlying convergence modes well above that range (Cowell et al., 16 Jul 2025).

This suggests that marked convergence fields occupy an intermediate methodological position. They are more interpretable and less combinatorially costly than full bispectrum or trispectrum estimation, yet they are more structurally informative than ordinary two-point functions. That balance is one reason they are presented as a practical route for extracting non-Gaussian information from Stage-III and Stage-IV weak-lensing surveys (Cowell et al., 16 Jul 2025).

4. HSC-Y1 implementation, constraints, and systematic control

The first data application uses Subaru Hyper Suprime-Cam Survey First-Year weak-lensing maps. The analysis works in a flat-sky approximation for each of six HSC fields, bins multipoles into 14 logarithmic bins over $$\kappa(\hat{\mathbf{n}})\equiv {}_0\kappa(\theta,\phi),$$3, and forms a full marked data vector from $$\kappa(\hat{\mathbf{n}})\equiv {}_0\kappa(\theta,\phi),$$4, $$\kappa(\hat{\mathbf{n}})\equiv {}_0\kappa(\theta,\phi),$$5, and $$\kappa(\hat{\mathbf{n}})\equiv {}_0\kappa(\theta,\phi),$$6 across three marks, three smoothing scales, and three tomographic source bins. The covariance is estimated from 2268 pseudo-independent HSC-Y1–like realizations derived from 108 full-sky $$\kappa(\hat{\mathbf{n}})\equiv {}_0\kappa(\theta,\phi),$$7-body ray-tracing simulations; the data vector length is 1064 before cuts and 380 after cuts, and the inverse covariance is corrected by the Hartlap factor. Cosmology dependence is interpolated by a Gaussian-process emulator trained on 100 cosmologies with 50 quasi-independent realizations each, varying $$\kappa(\hat{\mathbf{n}})\equiv {}_0\kappa(\theta,\phi),$$8 and $$\kappa(\hat{\mathbf{n}})\equiv {}_0\kappa(\theta,\phi),$$9 while holding other parameters fixed (Cowell et al., 16 Jul 2025).

The baseline analysis includes all three marks $\phi(r,\hat{\mathbf{n}})$0, all three smoothing scales $\phi(r,\hat{\mathbf{n}})$1, the marked auto- and cross-spectra, and the standard spectrum, with the highest redshift bin dropped and a conservative cut $\phi(r,\hat{\mathbf{n}})$2. The resulting HSC-Y1 constraints are

$\phi(r,\hat{\mathbf{n}})$3

from $\phi(r,\hat{\mathbf{n}})$4 alone,

$\phi(r,\hat{\mathbf{n}})$5

from the marked spectra alone, and

$\phi(r,\hat{\mathbf{n}})$6

from the combination. The abstract summarizes this as an $\phi(r,\hat{\mathbf{n}})$7 improvement in constraints on $\phi(r,\hat{\mathbf{n}})$8 relative to standard two-point power spectra (Cowell et al., 16 Jul 2025).

The paper also resolves the gain by mark type. No single mark matches the full three-mark combination. Mark $\phi(r,\hat{\mathbf{n}})$9 gives the strongest single-mark improvement in $\Phi$0, at about $\Phi$1 relative to $\Phi$2; mark $\Phi$3 is especially powerful for $\Phi$4 within the adopted prior; and mark $\Phi$5 yields degeneracy directions nearly orthogonal to those of $\Phi$6, making their combination particularly useful. Correlation matrices show strong self-correlations within a given mark and scale, strong cross-scale correlations for mark $\Phi$7, and very weak correlations between different marks, which the paper interprets as evidence of complementarity (Cowell et al., 16 Jul 2025).

Systematic tests determine the $\Phi$8 choice by imposing $\Phi$9. Within that regime, baryonic contamination from $${}_0\kappa(r,\hat{\mathbf{n}})=\tfrac14(\eth\bar\eth+\bar\eth\eth)\,{}_0\phi(r,\hat{\mathbf{n}}), \qquad {}_2\gamma(r,\hat{\mathbf{n}})=\tfrac12\eth\eth\,{}_0\phi(r,\hat{\mathbf{n}}),$$0TNG induces $${}_0\kappa(r,\hat{\mathbf{n}})=\tfrac14(\eth\bar\eth+\bar\eth\eth)\,{}_0\phi(r,\hat{\mathbf{n}}), \qquad {}_2\gamma(r,\hat{\mathbf{n}})=\tfrac12\eth\eth\,{}_0\phi(r,\hat{\mathbf{n}}),$$1; intrinsic-alignment injections in the NLA model produce shifts up to $${}_0\kappa(r,\hat{\mathbf{n}})=\tfrac14(\eth\bar\eth+\bar\eth\eth)\,{}_0\phi(r,\hat{\mathbf{n}}), \qquad {}_2\gamma(r,\hat{\mathbf{n}})=\tfrac12\eth\eth\,{}_0\phi(r,\hat{\mathbf{n}}),$$2; alternative photometric-redshift calibrations yield shifts of about $${}_0\kappa(r,\hat{\mathbf{n}})=\tfrac14(\eth\bar\eth+\bar\eth\eth)\,{}_0\phi(r,\hat{\mathbf{n}}), \qquad {}_2\gamma(r,\hat{\mathbf{n}})=\tfrac12\eth\eth\,{}_0\phi(r,\hat{\mathbf{n}}),$$3 for Mizuki and $${}_0\kappa(r,\hat{\mathbf{n}})=\tfrac14(\eth\bar\eth+\bar\eth\eth)\,{}_0\phi(r,\hat{\mathbf{n}}), \qquad {}_2\gamma(r,\hat{\mathbf{n}})=\tfrac12\eth\eth\,{}_0\phi(r,\hat{\mathbf{n}}),$$4 for FRANKEN-Z at $${}_0\kappa(r,\hat{\mathbf{n}})=\tfrac14(\eth\bar\eth+\bar\eth\eth)\,{}_0\phi(r,\hat{\mathbf{n}}), \qquad {}_2\gamma(r,\hat{\mathbf{n}})=\tfrac12\eth\eth\,{}_0\phi(r,\hat{\mathbf{n}}),$$5; and multiplicative shear-bias offsets of $${}_0\kappa(r,\hat{\mathbf{n}})=\tfrac14(\eth\bar\eth+\bar\eth\eth)\,{}_0\phi(r,\hat{\mathbf{n}}), \qquad {}_2\gamma(r,\hat{\mathbf{n}})=\tfrac12\eth\eth\,{}_0\phi(r,\hat{\mathbf{n}}),$$6 generate shifts up to $${}_0\kappa(r,\hat{\mathbf{n}})=\tfrac14(\eth\bar\eth+\bar\eth\eth)\,{}_0\phi(r,\hat{\mathbf{n}}), \qquad {}_2\gamma(r,\hat{\mathbf{n}})=\tfrac12\eth\eth\,{}_0\phi(r,\hat{\mathbf{n}}),$$7 (Cowell et al., 16 Jul 2025). These results support the baseline marked analysis at current precision, while also showing that marked spectra can respond to systematics in a more scale-mixed way than ordinary two-point functions.

5. Marked convergence on random trees

In probability theory, a closely related usage concerns marked Galton–Watson trees conditioned on having many marks. A marked Galton–Watson tree $${}_0\kappa(r,\hat{\mathbf{n}})=\tfrac14(\eth\bar\eth+\bar\eth\eth)\,{}_0\phi(r,\hat{\mathbf{n}}), \qquad {}_2\gamma(r,\hat{\mathbf{n}})=\tfrac12\eth\eth\,{}_0\phi(r,\hat{\mathbf{n}}),$$8 consists of a Galton–Watson tree with offspring distribution $${}_0\kappa(r,\hat{\mathbf{n}})=\tfrac14(\eth\bar\eth+\bar\eth\eth)\,{}_0\phi(r,\hat{\mathbf{n}}), \qquad {}_2\gamma(r,\hat{\mathbf{n}})=\tfrac12\eth\eth\,{}_0\phi(r,\hat{\mathbf{n}}),$$9 and independent vertex marks $${}_2\gamma_{\ell m} = {}_2F_\ell\,{}_0\kappa_{\ell m}, \qquad {}_2F_\ell = \frac{-1}{\ell(\ell+1)}\sqrt{\frac{(\ell+2)!}{(\ell-2)!}}.$$0, assigned conditionally on the tree with degree-dependent probabilities $${}_2\gamma_{\ell m} = {}_2F_\ell\,{}_0\kappa_{\ell m}, \qquad {}_2F_\ell = \frac{-1}{\ell(\ell+1)}\sqrt{\frac{(\ell+2)!}{(\ell-2)!}}.$$1. The total number of marked vertices is

$${}_2\gamma_{\ell m} = {}_2F_\ell\,{}_0\kappa_{\ell m}, \qquad {}_2F_\ell = \frac{-1}{\ell(\ell+1)}\sqrt{\frac{(\ell+2)!}{(\ell-2)!}}.$$2

The paper studies the local limit of

$${}_2\gamma_{\ell m} = {}_2F_\ell\,{}_0\kappa_{\ell m}, \qquad {}_2F_\ell = \frac{-1}{\ell(\ell+1)}\sqrt{\frac{(\ell+2)!}{(\ell-2)!}}.$$3

as $${}_2\gamma_{\ell m} = {}_2F_\ell\,{}_0\kappa_{\ell m}, \qquad {}_2F_\ell = \frac{-1}{\ell(\ell+1)}\sqrt{\frac{(\ell+2)!}{(\ell-2)!}}.$$4, distinguishing critical, generic subcritical, and non-generic subcritical regimes (Boulal et al., 23 Sep 2025).

A key structural device is a $${}_2\gamma_{\ell m} = {}_2F_\ell\,{}_0\kappa_{\ell m}, \qquad {}_2F_\ell = \frac{-1}{\ell(\ell+1)}\sqrt{\frac{(\ell+2)!}{(\ell-2)!}}.$$5-tilt of both reproduction and marking: $${}_2\gamma_{\ell m} = {}_2F_\ell\,{}_0\kappa_{\ell m}, \qquad {}_2F_\ell = \frac{-1}{\ell(\ell+1)}\sqrt{\frac{(\ell+2)!}{(\ell-2)!}}.$$6 with

$${}_2\gamma_{\ell m} = {}_2F_\ell\,{}_0\kappa_{\ell m}, \qquad {}_2F_\ell = \frac{-1}{\ell(\ell+1)}\sqrt{\frac{(\ell+2)!}{(\ell-2)!}}.$$7

For any $${}_2\gamma_{\ell m} = {}_2F_\ell\,{}_0\kappa_{\ell m}, \qquad {}_2F_\ell = \frac{-1}{\ell(\ell+1)}\sqrt{\frac{(\ell+2)!}{(\ell-2)!}}.$$8 in the admissible set $${}_2\gamma_{\ell m} = {}_2F_\ell\,{}_0\kappa_{\ell m}, \qquad {}_2F_\ell = \frac{-1}{\ell(\ell+1)}\sqrt{\frac{(\ell+2)!}{(\ell-2)!}}.$$9, the conditional law given $\kappa$00 is unchanged: $\kappa$01 This change of measure allows the conditioned marked process to be related to critical local-limit theory (Boulal et al., 23 Sep 2025).

The resulting limiting objects have a sharp dichotomy. In the critical case, and in the generic subcritical case after tilting to the unique $\kappa$02 such that $\kappa$03, the conditioned tree converges locally to a marked Kesten’s tree, an infinite random marked tree with a unique infinite spine. In the non-generic subcritical case, under the stated heavy-tail assumptions

$\kappa$04

with $\kappa$05, the local limit is a marked condensation tree with exactly one vertex of infinite outdegree and no infinite spine. The paper calls this the marked condensation phenomenon (Boulal et al., 23 Sep 2025).

This framework clarifies an important distinction: conditioning on the sum of marks is not equivalent to conditioning on tree size. The mark function enters both through the tilt $\kappa$06 in the generic regime and through the boundary behavior that decides whether the generic critical tilt exists. The paper presents this as a precise instance of a marked convergence field on a random tree, where the global mass of marks determines whether the local geometry converges to an infinite-spine object or to a condensation-point object (Boulal et al., 23 Sep 2025).

6. Extremal fields, marked point processes, and related convergence notions

Related literatures use the same ingredients—marks, fields, and convergence—in different asymptotic settings. For logarithmically correlated Gaussian fields on $\kappa$07, the centered maximum $\kappa$08 converges under assumptions (A.0)–(A.3), and the limiting law of $\kappa$09 is a randomly shifted Gumbel: $\kappa$10 The random shift is governed by the limit of a derivative-martingale-like variable

$\kappa$11

Although that paper proves only scalar maximum convergence, it explicitly interprets the result as a precursor to a derivative-martingale measure and a Poisson cluster extremal process, i.e. a marked extremal field in which cluster decorations would serve as marks (Ding et al., 2015).

A discrete extreme-value analogue appears for marked point processes of exceedances. For i.i.d. geometric or bivariate Marshall–Olkin geometric variables, exceedance processes are defined by random measures such as

$\kappa$12

and are approximated by Poisson random measures with explicit rates via the Stein–Chen method. In the univariate geometric case,

$\kappa$13

and in the bivariate Marshall–Olkin setting the limiting intensity has both an absolutely continuous off-diagonal part and a singular diagonal component. Here the “field” is a random configuration of marked exceedances in mark space, and convergence is quantified in total variation and Wasserstein-type metrics (Cipriani et al., 2015).

A different but conceptually adjacent framework concerns convergence of sub-$\kappa$14-fields on a fixed probability space. Weak convergence, strong convergence, Hausdorff-metric convergence, almost-sure convergence, set-theoretic convergence, and monotone convergence are all studied, and all preserve unconditional independence and are invariant under passage to an equivalent probability measure. For $\kappa$15, operator-norm convergence of conditional expectation operators $\kappa$16 is equivalent to Hausdorff convergence of $\kappa$17-fields. This literature does not define marked convergence fields in the weak-lensing sense, but it supplies a rigorous theory of convergence for information structures that underlies many conditioning arguments in marked random systems (Vidmar, 2016).

These adjacent formulations show that the phrase can point to several mathematically distinct objects. In weak lensing it denotes an explicitly constructed nonlinear transform of $\kappa$18; in random-tree theory it describes conditioned marked branching geometries; in extremal theory it suggests limiting point fields or cluster processes with marks; and in $\kappa$19-field theory it connects to convergence of the information generated by such objects. A plausible implication is that the common thread is not a single universal definition, but a recurring architecture: a base field or geometry, an attached mark structure, and an asymptotic regime in which the coupled object converges to a nontrivial limit.