Angular DM Power Spectrum

• Angular DM power spectrum is a statistical measure quantifying dark matter clustering and anisotropy via spherical harmonic decomposition.
• Analytical models and simulation frameworks, including harmonic transforms and quadratic estimators, enable precise extraction of angular fluctuations.
• Extracted features from the spectrum inform dark matter microphysics, substructure, and secondary effects, guiding experimental and observational studies.

The angular dark matter (DM) power spectrum is a fundamental statistical tool quantifying the clustering and anisotropy of dark matter or dark matter–induced signals on the sky, typically as a function of angular multipole moment $\ell$. It serves as a bridge between theoretical models of DM distribution and observational or simulation data, offering a direct means of extracting information about the structure, interactions, and physical properties of dark matter via its angular correlations. Analyses of the angular DM power spectrum are foundational in studies ranging from cosmic microwave background (CMB) anisotropies and cosmic shear to unresolved gamma-ray backgrounds and galaxy clustering, as well as direct connections to experimental searches for DM substructure and secondary effects such as lensing.

1. Formal Definition and Statistical Foundation

The angular DM power spectrum, denoted $C_\ell$, describes the variance of the spherical harmonic coefficients $a_{\ell m}$ in the decomposition of a field $I(\mathbf{n})$ defined on the celestial sphere: $$I(\mathbf{n}) = \sum_{\ell=0}^{\infty} \sum_{m=-\ell}^{\ell} a_{\ell m} Y_{\ell m}(\mathbf{n}),\qquad C_\ell = \frac{1}{2\ell+1} \sum_{m=-\ell}^{\ell} \langle |a_{\ell m}|^2 \rangle$$ where $Y_{\ell m}$ are spherical harmonics and $\langle \cdot \rangle$ denotes a statistical ensemble average. For dark matter studies, $I(\mathbf{n})$ can represent projected DM density, matter-induced secondary signals (such as gamma rays from annihilation/decay), galaxy number counts tracing halos, or lensing convergence fields. The $C_\ell$ quantifies the mean-squared fluctuation at angular scales $\theta\sim\pi/\ell$, with large $\ell$ representing small angular separations.

2. Analytical Models, Simulation Frameworks, and Key Features

Multiple theoretical and computational methodologies are employed to compute or estimate the angular DM power spectrum across a range of contexts.

CMB and Cosmic String Scenarios: For cosmic (super-)strings, the contribution to the CMB temperature angular power spectrum from DM is dominated by Poisson-distributed, straight, and uniformly moving segments. The “segment formalism” computes $C_\ell$ by superposing the Kaiser–Stebbins temperature jumps from independent segments. The analytic $C_\ell$ exhibits a plateau at large angular scales ($\ell\lesssim\ell_{\mathrm{co}}$)—corresponding to the typical angular size of correlated DM structures—and transitions to a power-law decay $C_\ell\propto\ell^{-1}$ at small angular scales ($\ell\gg \ell_{\mathrm{co}}$) (Yamauchi et al., 2010). The position of this break is sensitive to the intercommuting probability, $P$, with lower $P$ (as in cosmic superstrings) pushing $\ell_{\mathrm{co}}\propto P^{-1/2}$ to higher $\ell$ and amplifying the spectrum due to increased segment density ($\rho\propto P^{-3/2}$).

Halo Model and Large-Scale Structure: In galaxy redshift surveys (Ando et al., 2017, Hayes et al., 2012), the angular power spectrum is modeled via the combination of one-halo (intra-halo clustering) and two-halo (halo–halo correlations) terms using a Halo Occupation Distribution (HOD) prescription. The 2D projected $C_\ell$ is related to the 3D power spectrum $P(k,z)$ via Limber's approximation: $$C_\ell = \int_0^\infty \frac{dz}{c\,H(z)} \frac{W^2(z)}{r^2(z)} P\!\left(k = \frac{\ell+1/2}{r(z)},z\right)$$ where $W(z)$ includes the redshift distribution, galaxy bias, and growth factor. For small angular scales ($\ell \gg 30$), the spectrum is frequently dominated by galaxies in nearby clusters, encoding the abundance and spatial profile of DM halos.

Non-Gaussian and Stochastic Effects: For finite event counts or sparse data—such as in high-energy gamma-ray or cosmic ray maps—the estimator

$$\widehat{C}_{\ell,N} = 4\pi \left(\frac{1}{N(N-1)} \sum_{i\neq j} P_\ell(\mathbf{n}_i\cdot\mathbf{n}_j) - \delta_{\ell,0}\right)$$

is unbiased but its variance contains higher-order terms (composite spectrum, open bispectrum, disjoint trispectrum) (Campbell, 2014), which are significant for low counts or non-Gaussian sky distributions and can lead to large uncertainties at high $\ell$. For cosmic ray anisotropies, simulations show $C_\ell$ can be modeled by a broken power law with a spectral break (typically $\ell_{\mathrm{break}}\approx4$), and flattening at large $\ell$—features closely related to magnetic turbulence scales and substructure (Bian et al., 12 Oct 2024).

3. Extraction Techniques and Statistical Inference

Direct Estimation: For uniform sky coverage, $C_\ell$ can be directly extracted via fast spherical harmonic transforms or, in the flat sky approximation, via 2D FFTs with appropriate scaling. Incomplete or masked data, uneven noise, or sparse events require unbiased or variance-minimizing estimators (e.g., quadratic estimators with KL-compression (Hayes et al., 2012), cross-power spectra between bands to suppress uncorrelated noise (Chiang et al., 2011), and Bayesian Gibbs sampling for high-dimensional interferometric data (Sutter et al., 2011)).

Distribution-Free and Goodness-of-Fit Methods: For model selection or hypothesis testing, a recent distribution-free approach (Algeri et al., 22 Apr 2025) constructs a test statistic from decorrelated residuals of the angular power spectrum, forming a partial sum process whose limiting distribution is Gaussian, regardless of the underlying distribution of $C_\ell$ estimators. This bypasses the need for distributional assumptions or Monte Carlo calibration and can be applied universally to test models of the angular DM power spectrum.

Approach	Estimator Principle	Key Application Context
Harmonic transform (FFT/HEALPix)	Direct inversion	Full-sky or high-resolution CMB/LSS
Quadratic estimator + KL	Covariance trace	Partial sky, pixelized galaxy maps
Gibbs sampler	Bayesian joint sampling	Interferometric, uv-plane coverage
Event-pair (finite counts)	Legendre polynomials sum	Diffuse gamma-ray, neutrino maps
Distribution-free residuals	Empirical process theory	Model testing, goodness-of-fit

4. Influence of Dark Matter Properties and Physical Mechanisms

The detailed features of the angular DM power spectrum encode information about fundamental DM physics:

• Secondary Cosmic Effects: DM-photon interactions damp small-scale CMB anisotropy, shifting or suppressing both the temperature and B-mode ($E/B$ polarization) spectra at high $\ell$; current Planck data constrain the DM–photon elastic scattering cross-section to $$\sigma_{DM-\gamma} < 8 \times 10^{-31} (m_{DM}/\mathrm{GeV})$$ cm$^2$ (for constant cross-section) (Wilkinson et al., 2013). These effects appear as scale-dependent amplitude reductions and altered peak positions.
• Clustering and Substructure: The one-halo component, subhalo abundance, and spatial profile are imprinted in excess small-scale power (large $\ell$) in galaxy and gamma-ray angular power spectra (Ando et al., 2017, Campbell, 2014). If DM annihilation or decay is spatially correlated with substructure, non-Gaussianity in the spatial distribution introduces corrections to expected $C_\ell$ variance beyond Poisson noise, possibly affecting detectability thresholds.
• Non-Gaussian Statistics: Significant higher-order moments (open bispectrum or trispectrum) alter the uncertainty estimates of angular power spectrum measurements; positive open bispectrum, for instance, increases variance, potentially masking faint DM-induced anisotropy signals (Campbell, 2014).

5. Algorithmic, Computational, and Experimental Considerations

• Computational Efficiency: Handling large datasets (e.g., from spectroscopic surveys) motivates hybrid binning strategies and block-diagonal covariance modeling, as in the optimized two-tier tomographic analysis (Faggioli et al., 2020), allowing efficient exploitation of both radial and angular information without prohibitive covariance matrix inversion.
• Compressiveness and Recovery: Compressive measurement architectures—using sub-Nyquist spatial and temporal sampling—enable estimation of angular power spectra even when the number of independent sources or modes exceeds the number of sensors, provided system matrices are of full rank (Ariananda et al., 2014).
• Foreground and Noise Mitigation: Radio interferometric power spectrum estimators (e.g., Tapered Gridded Estimator (Choudhuri et al., 2017)) deploy spatial windowing and calibrated noise bias removal to suppress contamination from imperfectly subtracted foregrounds, a critical challenge in recovering the low-brightness DM-induced signals.

6. Observational Implications and Future Prospects

The angular DM power spectrum serves as an essential bridge from theory to observation:

• CMB, Weak Lensing, and Large Scale Structure: Precision measurement of $C_\ell$ in the CMB and lensing maps restricts plausible DM microphysics (e.g., elastic scattering, suppression by warm or fuzzy DM), while the extracted features from galaxy or galaxy-clustering autocorrelations map the DM distribution's coevolution with baryons.
• Gamma-ray and Cosmic Ray Anisotropy: The angular power spectrum of unresolved gamma rays and cosmic rays can potentially detect signatures of DM subhalos and distinguish them from astrophysical backgrounds (Bian et al., 12 Oct 2024, Campbell, 2014). Distinctive scale-dependent features, such as plateau-to-power-law transitions or spectral breaks, can be used to infer substructure profiles or magnetic turbulence properties.
• Model Testing and Systematics Control: Modern distribution-free statistical techniques (Algeri et al., 22 Apr 2025) increase the robustness of hypothesis testing for DM-induced anisotropy models, mitigating dependency on estimator distribution assumptions and reducing computational overhead in massive surveys.

The angular DM power spectrum thus remains an indispensable tool in theoretical, experimental, and observational cosmology, functioning both as a sensitive probe of fundamental DM physics and a rigorous statistical measure for contemporary multi-messenger astrophysical analyses.