Physics & Astronomy Faculty Publications
Document Type
Article
Publication Date
9-24-2026
Abstract
Mapping gravitational-wave background (GWB) anisotropy with pulsar timing arrays (PTAs) is affected by harmonic space mode suppression and mode coupling arising from an array’s nonuniform sky response. Due to computational limitations, one must truncate spherical harmonic expansions at a finite multipole for recovery, ℓrec max, often chosen to equal ℓ𝑁pair max ≡int[√𝑁pair−1], where 𝑁pair ≡𝑁psr(𝑁psr −1)/2 is the number of distinct pulsar pairs in an array comprised of 𝑁psr pulsars. This choice is motivated by the counting argument that cross-correlations provide at most 𝑁pair independent pieces of information for a single frequency bin. Here, we explicitly obtain the multipole value (denoted ℓres max) that approximates the maximum informative angular scale of an array of pulsars. This value is defined by the requirement that a spherical harmonic expansion out to ℓres max approximately span the space of “observable skies” extracted by the array, which is encoded in the 𝑁pair eigenmaps of the Fisher information matrix. The value of ℓres max depends on specifics of the PTA configuration. We also explicitly show that GWB power contained in multipoles ℓ ≳ℓres max do not significantly affect analyses that use expansions out to ℓres max, due to the mode suppression (i.e., low-pass filtering) induced by the PTA response to the GWB. However, truncating spherical harmonic expansions at ℓrec max < ℓres max leads to leakage of small-scale angular power from multipoles ℓrec max < ℓ ≤ℓres max. Nonetheless, even if we use ℓres max for recovery, the standard frequentist estimator of the angular power spectrum components 𝐶ℓ is still biased due to modes that are not observable by the array. Although we can partially debias the standard estimator—improving its agreement with an injected angular power spectrum—this reduction in bias comes at the expense of an increase in variance, with especially large variances arising for poorly constrained modes, ℓ ≫ℓeff. In summary, the results of our analyses strongly advocate for: (i) using ℓres max for PTA analyses involving spherical harmonic expansions, and (ii) using the debiased standard estimator for 𝐶ℓ recovery, but only out to multipoles ℓ < ℓeff( ≪ℓres max) corresponding to sufficiently constrained modes.
Recommended Citation
Agarwal, Deepali, Joseph D. Romano, Yacine Ali-Haïmoud, and Tristan L. Smith. "Addressing leakage and mode suppression in angular power spectrum estimation for gravitational-wave backgrounds using pulsar timing arrays." Physical Review D 114, no. 6 (2026): 063048. https://doi.org/10.1103/cm48-4k18
Publication Title
Physical Review D
DOI
10.1103/cm48-4k18
Comments
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