--- title: Marked Angular Power Spectra url: https://www.emergentmind.com/topics/marked-angular-power-spectra type: topic --- # Marked Angular Power Spectra 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 \(C_\ell^{\kappa\kappa}\) [2507.12315]. 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 [2006.10055]. ## 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 \(C_{AB}(l)\) is defined by the harmonic expansion of the projected fields and can be written as an integral over the 3D cross-power spectrum \(P_{AB}(k)\) and spherical Bessel transforms of the kernels [2304.13064]. In the notation used for the 21 cm signal, an angular statistic with frequency dependence takes the form \[ C_\ell(\nu_1,\nu_2)=\langle a_{\ell m}(\nu_1)a_{\ell m}^*(\nu_2)\rangle, \] which reduces to the ordinary angular auto-spectrum when \(\nu_1=\nu_2\) [2510.13752]. 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 \[ \Delta(\kappa)=m(\kappa_\theta)\,\kappa, \] with \(\kappa_\theta\) obtained by Gaussian smoothing of the convergence field on angular scale \(\theta\) [2507.12315]. The marked observables are then \[ C_\ell^{\kappa\kappa},\qquad C_\ell^{\kappa\Delta},\qquad C_\ell^{\Delta\Delta}. \] 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 \[ \rho_M(\vec x)=m(\vec x)\,n(\vec x), \] with a mark chosen as a local function of the smoothed overdensity, \[ m(\vec x)=\left(1+\frac{\delta_R(\vec x)}{1+\delta_s}\right)^{-p}. \] The marked overdensity \(\delta_M\) is then used to form a power spectrum \(M(k)\) [2006.10055]. 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, \[ m_{\mathcal B}=\frac{\kappa_\theta}{\sigma(\kappa_\theta)}. \] For the simplified marked field \(\Delta(\vec x)\equiv \kappa(\vec x)\kappa_\theta(\vec x)\), the cross-spectrum \[ C_\ell^{\kappa\Delta} = \int \frac{d^2L}{(2\pi)^2}\, W(L\theta)\,B_\kappa(\ell,L,|\boldsymbol{\ell}-\boldsymbol{L}|), \] behaves like a projected bispectrum contribution, while \[ C_\ell^{\Delta\Delta} \] contains both a Gaussian disconnected term and a connected trispectrum term [2507.12315]. In this sense, the marked angular spectra are still two-point observables, but they are not merely repackaged versions of the ordinary \(C_\ell\). 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 \(\delta(\vec x)\delta_R(\vec x)\), \(\delta_R^2(\vec x)\), and \(\delta_R^3(\vec x)\), so the marked one-loop power spectrum contains kernels that couple the unsmoothed field to its smoothed environment [2006.10055]. The paper shows that the linear marked spectrum, \[ M_L(\vec k)=\frac{1}{\bar m^2}\left[1-C_1W(kR)\right]^2P_L(\vec k), \] 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 [2006.10055]. 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-\(\ell\) cut is imposed, the marked spectra still receive information from smaller physical scales [2507.12315]. 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 [2507.12315] 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” \(\kappa\), and marks are built from smoothing scales \(\theta=2',\,4',\,10'\). | Mark | Definition | Stated role | |---|---|---| | \(\mathcal A\) | Gaussian-process interpolation mark | Anti-correlates low-to-medium overdensities and underdensities | | \(\mathcal B\) | \(m_{\mathcal B}=\kappa_\theta/\sigma(\kappa_\theta)\) | Up-weights extreme overdense and underdense regions | | \(\mathcal C\) | \(m_{\mathcal C}=(1+f(x))^{-p}\) with \(p=0.5,\ \delta=0.02,\ b=0.1\) | Preferentially up-weights underdense regions and is constant for sufficiently positive \(\kappa\) | The Gaussian-process mark \(\mathcal A\) is adapted from Cowell et al. (2024) and is defined by interpolation between four nodes using a squared-exponential kernel, \[ k_{\rm SE}(x,x')=\sigma^2\exp\!\left(-\frac{(x-x')^2}{2\ell^2}\right), \] with node positions recalibrated to the minimum and maximum of the smoothed field in each map [2507.12315]. The smoothed-field mark \(\mathcal B\) 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 \(\mathcal C\) uses \[ x=\frac{\kappa_\theta}{\sigma(\kappa_\theta)(1+b)}, \] with a “safety” function that becomes \((1-\delta)\tanh(10x)\) below \(x=-0.999\), preventing pathological behavior for sufficiently negative smoothed convergence [2507.12315]. 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 [2507.12315]. 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-\(C_\ell\) estimator implemented in NaMaster, using 14 logarithmic bins over \(80<\ell<6500\), while the baseline cosmological analysis imposes \(\ell<1500\) [2507.12315]. The data vector is \[ \mathbf D=\Big\{ C_\ell^{\kappa\kappa},\ C_\ell^{\kappa,\Delta_{\mathcal X,\theta}},\ C_\ell^{\Delta_{\mathcal X,\theta}\Delta_{\mathcal X,\theta}} \Big\}, \] with \(\mathcal X\in\{\mathcal A,\mathcal B,\mathcal C\}\) and \(\theta\in\{2',4',10'\}\), and excludes cross-correlations between different marks, different smoothing scales, and different tomographic bins [2507.12315]. 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 \(N_s=2268\) and \(N_d=380\), MOPED compression reduces the summary statistics to two numbers, and posterior sampling is performed with Cobaya under a Gaussian likelihood [2507.12315]. 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 [2304.13064]. 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 \(C_\ell\) values for cross-correlations [2304.13064; 1701.03592]. 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 [2507.12315]. The analysis uses six fields covering a total area of \(136.9\ \mathrm{deg}^2\), source redshifts \(0.3