Marked Angular Power Spectra

• Marked angular power spectra are angular two-point statistics computed from a field multiplied by a nonlinear mark that encodes local environmental information.
• They compress higher-order clustering information, effectively incorporating bispectrum and trispectrum contributions into a two-point framework.
• Practical implementations in weak-lensing surveys, such as the HSC analysis, have demonstrated significant improvements in cosmological parameter constraints.

Searching arXiv for recent and foundational papers on marked angular power spectra and closely related marked power spectrum methodology. arxiv_search query: "marked angular power spectra weak lensing marked convergence fields 2025" arxiv_search query: "\"marked angular power spectra\" OR \"marked power spectrum\" cosmology projection" arxiv_search query: "\"marked\" weak lensing power spectrum convergence" arxiv_search query: "\"What does the Marked Power Spectrum Measure\" OR marked statistics neutrino mass" Marked angular power spectra are angular two-point statistics formed not from an original projected field alone, but from a marked field in which the field is multiplied by a nonlinear weight constructed from its local, typically smoothed, environment. In the weak-lensing implementation applied to Subaru Hyper Suprime-Cam Year 1 data, the marked convergence field is defined as $\Delta(\kappa)=m(\kappa_\theta)\,\kappa$, where $m(\kappa_\theta)$ is a mark built from a Gaussian-smoothed convergence map $\kappa_\theta$; the resulting observables are the marked auto-spectrum $C_\ell^{\Delta\Delta}$, the original–marked cross-spectrum $C_\ell^{\kappa\Delta}$, and their combination with the ordinary convergence power spectrum $C_\ell^{\kappa\kappa}$ (Cowell et al., 16 Jul 2025). The rationale is that the nonlinear mark couples the field to its local environment, so the two-point spectra of the marked field can encode higher-order information while remaining accessible to standard angular power-spectrum estimators. The perturbative interpretation developed for the 3D marked power spectrum identifies the underlying mechanism as mode coupling induced by the mark, which transfers bispectrum- and trispectrum-like information into a power-spectrum observable (Philcox et al., 2020).

1. Definition and formal construction

For projected random fields $A(\mathbf{x})$ and $B(\mathbf{x})$ observed on the sky through radial kernels $F_A(r)$ and $F_B(r)$, the angular cross-power spectrum $m(\kappa_\theta)$0 is defined by the harmonic expansion of the projected fields and can be written as an integral over the 3D cross-power spectrum $m(\kappa_\theta)$1 and spherical Bessel transforms of the kernels (Feldbrugge, 2023). In the notation used for the 21 cm signal, an angular statistic with frequency dependence takes the form

$m(\kappa_\theta)$2

which reduces to the ordinary angular auto-spectrum when $m(\kappa_\theta)$3 (Kunze, 15 Oct 2025). These formulations emphasize that an angular power spectrum is fundamentally a harmonic-space summary of projected correlations.

Marked angular power spectra modify this construction at the field level. In the HSC weak-lensing analysis, the marked field is

$m(\kappa_\theta)$4

with $m(\kappa_\theta)$5 obtained by Gaussian smoothing of the convergence field on angular scale $m(\kappa_\theta)$6 (Cowell et al., 16 Jul 2025). The marked observables are then

$m(\kappa_\theta)$7

This preserves the formal simplicity of a two-point angular analysis while altering what the field itself represents.

The projected marked construction is closely related to the 3D marked power spectrum. There, the marked density field is written as

$m(\kappa_\theta)$8

with a mark chosen as a local function of the smoothed overdensity,

$m(\kappa_\theta)$9

The marked overdensity $\kappa_\theta$0 is then used to form a power spectrum $\kappa_\theta$1 (Philcox et al., 2020). The projected weak-lensing case is the direct angular analogue of this logic: a local environmental weight is applied before harmonic-space two-point estimation.

2. How higher-order information enters

The principal scientific interest of marked angular power spectra is that they are intended to compress information that would otherwise reside in higher-order correlators. In the HSC analysis, this is shown explicitly for the simplest mark,

$\kappa_\theta$2

For the simplified marked field $\kappa_\theta$3, the cross-spectrum

$\kappa_\theta$4

behaves like a projected bispectrum contribution, while

$\kappa_\theta$5

contains both a Gaussian disconnected term and a connected trispectrum term (Cowell et al., 16 Jul 2025). In this sense, the marked angular spectra are still two-point observables, but they are not merely repackaged versions of the ordinary $\kappa_\theta$6.

The perturbative analysis of the 3D marked power spectrum clarifies why this occurs. Expanding the mark in powers of the smoothed density produces cross-terms such as $\kappa_\theta$7, $\kappa_\theta$8, and $\kappa_\theta$9, so the marked one-loop power spectrum contains kernels that couple the unsmoothed field to its smoothed environment (Philcox et al., 2020). The paper shows that the linear marked spectrum,

$C_\ell^{\Delta\Delta}$0

is only the ordinary linear spectrum multiplied by a cosmology-independent window prefactor, implying that the extra constraining power does not arise at linear order. The additional information appears only once nonlinearities are included.

For projected statistics, the same paper argues that the same physical mechanism should operate after line-of-sight projection: scale mixing induced by the local mark should allow nonlinear, non-Gaussian structure to enter an angular two-point statistic (Philcox et al., 2020). This suggests that a marked angular power spectrum should be interpreted as a compressed observable carrying bispectrum- and trispectrum-like information into the two-point sector, not as a simple rescaling of an ordinary angular spectrum.

A further consequence is that marked spectra exhibit explicit scale mixing. In the weak-lensing case, the smoothing kernel implies that even if a high-$C_\ell^{\Delta\Delta}$1 cut is imposed, the marked spectra still receive information from smaller physical scales (Cowell et al., 16 Jul 2025). Part of the information gain can therefore reflect the way the smoothing kernel transfers small-scale structure into the measured statistic.

3. Mark functions and environmental weighting

The weak-lensing application (Cowell et al., 16 Jul 2025) uses three mark functions, each designed to weight different environments in the smoothed convergence field. The 1 arcmin-smoothed map is treated as the “original” $C_\ell^{\Delta\Delta}$2, and marks are built from smoothing scales $C_\ell^{\Delta\Delta}$3.

Mark	Definition	Stated role
$C_\ell^{\Delta\Delta}$4	Gaussian-process interpolation mark	Anti-correlates low-to-medium overdensities and underdensities
$C_\ell^{\Delta\Delta}$5	$C_\ell^{\Delta\Delta}$6	Up-weights extreme overdense and underdense regions
$C_\ell^{\Delta\Delta}$7	$C_\ell^{\Delta\Delta}$8 with $C_\ell^{\Delta\Delta}$9	Preferentially up-weights underdense regions and is constant for sufficiently positive $C_\ell^{\kappa\Delta}$0

The Gaussian-process mark $C_\ell^{\kappa\Delta}$1 is adapted from Cowell et al. (2024) and is defined by interpolation between four nodes using a squared-exponential kernel,

$C_\ell^{\kappa\Delta}$2

with node positions recalibrated to the minimum and maximum of the smoothed field in each map (Cowell et al., 16 Jul 2025). The smoothed-field mark $C_\ell^{\kappa\Delta}$3 is the analytically simplest and is the one for which the weak-lensing paper writes explicit bispectrum and trispectrum relations. The modified White-style mark $C_\ell^{\kappa\Delta}$4 uses

$C_\ell^{\kappa\Delta}$5

with a “safety” function that becomes $C_\ell^{\kappa\Delta}$6 below $C_\ell^{\kappa\Delta}$7, preventing pathological behavior for sufficiently negative smoothed convergence (Cowell et al., 16 Jul 2025).

The use of multiple marks is not a cosmetic choice. The HSC analysis reports that different marks probe different density environments and produce complementary degeneracy directions, which is one reason the combined three-mark analysis outperforms any single-mark case (Cowell et al., 16 Jul 2025). This suggests that mark design is part of the observable definition itself, not merely a tuning detail.

4. Measurement pipelines and theoretical modeling

The practical appeal of marked angular power spectra is that they can be estimated with existing angular-spectrum machinery. In the HSC-Y1 analysis, convergence maps are reconstructed from the shear catalog with Kaiser–Squires inversion, with inpainting before inversion to suppress E/B leakage from the mask. Power spectra are then measured with the flat-sky pseudo-$C_\ell^{\kappa\Delta}$8 estimator implemented in NaMaster, using 14 logarithmic bins over $C_\ell^{\kappa\Delta}$9, while the baseline cosmological analysis imposes $C_\ell^{\kappa\kappa}$0 (Cowell et al., 16 Jul 2025). The data vector is

$C_\ell^{\kappa\kappa}$1

with $C_\ell^{\kappa\kappa}$2 and $C_\ell^{\kappa\kappa}$3, and excludes cross-correlations between different marks, different smoothing scales, and different tomographic bins (Cowell et al., 16 Jul 2025).

The same analysis relies on simulations tailored to the HSC-Y1 footprint, mask geometry, source distribution, lensing weights, redshift distribution, and multiplicative bias. A Gaussian-process emulator is trained on cosmology-varied simulations, the inverse covariance is corrected with the Anderson–Hartlap factor using $C_\ell^{\kappa\kappa}$4 and $C_\ell^{\kappa\kappa}$5, MOPED compression reduces the summary statistics to two numbers, and posterior sampling is performed with Cobaya under a Gaussian likelihood (Cowell et al., 16 Jul 2025). This illustrates a general feature of marked angular analyses: although the estimator is a two-point statistic, calibration and inference still require survey-specific non-Gaussian simulations.

On the theory side, exact angular power-spectrum computation is already nontrivial even without marks. A projected angular spectrum requires integrals over spherical Bessel transforms, and the Limber approximation becomes inaccurate for narrow redshift bins, small overlap, and low multipoles (Feldbrugge, 2023). The complex-analysis method based on contour deformation and Picard–Lefschetz theory rewrites the oscillatory transform into an exact deformed integral with exponentially decaying integrand, while Angpow provides a separate fast exact pipeline based on Chebyshev expansions and Clenshaw–Curtis quadrature and emphasizes that Limber gives wrong $C_\ell^{\kappa\kappa}$6 values for cross-correlations (Feldbrugge, 2023, Campagne et al., 2017). A plausible implication is that any first-principles model of marked angular cross-spectra for narrow projected kernels will face the same non-Limber challenges, with the added complication that the mark itself introduces smoothing-induced mode coupling.

5. First weak-lensing constraints

The first direct cosmological application of marked angular power spectra uses Subaru Hyper Suprime-Cam Survey First-Year data (Cowell et al., 16 Jul 2025). The analysis uses six fields covering a total area of $C_\ell^{\kappa\kappa}$7, source redshifts $C_\ell^{\kappa\kappa}$8, and an effective galaxy number density of $C_\ell^{\kappa\kappa}$9 after masking. Although the source catalog is initially divided into four tomographic bins, the highest-redshift bin is omitted in the final analysis because of photo-$A(\mathbf{x})$0 calibration concerns, leaving three tomographic bins in the baseline setup (Cowell et al., 16 Jul 2025).

The parameter target is

$A(\mathbf{x})$1

The headline result is that combining multiple types of marked auto- and cross-spectra improves the $A(\mathbf{x})$2 constraint by $A(\mathbf{x})$3 relative to the standard convergence power spectrum alone at the same scale cuts (Cowell et al., 16 Jul 2025).

Observable set	Constraint on $A(\mathbf{x})$4	Note
Standard power spectrum only	$A(\mathbf{x})$5	Baseline $A(\mathbf{x})$6
Marked spectra only	$A(\mathbf{x})$7	Uses marked auto- and cross-spectra
Standard + marked	$A(\mathbf{x})$8	Adopted final HSC-Y1 result

The same analysis reports $A(\mathbf{x})$9, while noting that $B(\mathbf{x})$0 is less robust because of emulator limitations and prior domination (Cowell et al., 16 Jul 2025). Mark $B(\mathbf{x})$1 yields the strongest single-mark improvement, about $B(\mathbf{x})$2 relative to the baseline two-point result, but the three-mark combination performs best. The paper also states that the marked auto- and cross-spectra together can be nearly as constraining as the standard power spectrum alone even without $B(\mathbf{x})$3 (Cowell et al., 16 Jul 2025).

Systematics control is central to the claimed gain. The HSC paper explicitly tests baryonic effects, intrinsic alignment, photometric redshifts, and multiplicative shear bias. At the baseline cut $B(\mathbf{x})$4, the induced $B(\mathbf{x})$5 shifts are kept roughly within

$B(\mathbf{x})$6

which motivates the adopted scale cut (Cowell et al., 16 Jul 2025). The reported shifts include $B(\mathbf{x})$7 from baryons, up to $B(\mathbf{x})$8 from the tested NLA intrinsic-alignment amplitudes, about $B(\mathbf{x})$9 for FRANKEN-Z photo-$F_A(r)$0 calibration at the baseline cut, and up to $F_A(r)$1 for $F_A(r)$2 multiplicative shear calibration shifts (Cowell et al., 16 Jul 2025).

6. Interpretation, limitations, and relation to the broader $F_A(r)$3 framework

A recurrent misconception is that marked angular power spectra are simply ordinary angular spectra evaluated after an arbitrary reweighting. The existing theory and data argue against that reading. In 3D, the marked spectrum only becomes informationally distinct from the ordinary power spectrum once nonlinear terms are included; at linear order it is just the linear power spectrum multiplied by a window prefactor (Philcox et al., 2020). In projection, the weak-lensing calculation shows explicitly that the marked cross- and auto-spectra depend on bispectrum- and trispectrum-like terms (Cowell et al., 16 Jul 2025). Marked angular spectra are therefore best viewed as nonlinear compressed observables.

A second misconception is that their constraining gain should be attributed entirely to connected higher-order statistics. The HSC analysis cautions that the mark-induced smoothing causes scale mixing, so part of the gain may come from small-scale Gaussian information leakage through the smoothing kernel, not solely from intrinsic higher-order connected statistics (Cowell et al., 16 Jul 2025). This does not negate the utility of the statistic, but it does affect interpretation and the design of scale cuts.

Theoretical control is also limited. The perturbative treatment of the 3D marked spectrum becomes less controlled at low redshift and small smoothing scale $F_A(r)$4, and the paper states that the theory becomes non-perturbative at redshift zero for small smoothing scales, with important contributions from higher-order terms (Philcox et al., 2020). For projected marked statistics, this suggests that smoothing-scale choice is a central theoretical parameter, not just an analysis convenience.

Covariance estimation can be delicate in sparse-map applications. For ordinary angular power spectra estimated from finite counts, the exact variance depends not only on $F_A(r)$5 but also on a composite spectrum, an open bispectrum, and a disjoint trispectrum; neglecting these higher-order terms can produce a spurious detection of power (Campbell, 2014). This suggests analogous caution if marked angular power spectra are extended to sparse point-process data such as gamma rays, neutrinos, or other event maps.

Marked angular power spectra sit within a much broader family of angular-spectrum methods. Ordinary angular power spectra already serve as standard summaries of the CMB $F_A(r)$6, $F_A(r)$7, and $F_A(r)$8 spectra (Aghamousa et al., 2013), photometric galaxy clustering $F_A(r)$9 (Thomas et al., 2010), 21 cm multifrequency angular power spectra $F_B(r)$0 (Kunze, 15 Oct 2025), and anisotropic stochastic gravitational-wave backgrounds $F_B(r)$1 (Agarwal et al., 2023). Marked angular power spectra are a nonlinear extension of this established harmonic framework: they retain the observable form of an angular two-point statistic, but alter the field being projected and thereby alter the information content. The current literature establishes this program most concretely for weak lensing (Cowell et al., 16 Jul 2025), while the perturbative marked-power-spectrum literature provides the main physical blueprint for understanding why the construction works (Philcox et al., 2020).