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  • Open Access

Pulsar timing array analysis in a Legendre polynomial basis

Bruce Allen* and Arian L. von Blanckenburg†

Ken D. Olum‡

  • Institute of Cosmology, Department of Physics and Astronomy, Tufts University, Medford, Massachusetts 02155, USA

  • *Contact author: bruce.allen@aei.mpg.de
  • †Contact author: arian.von.blanckenburg@aei.mpg.de
  • ‡Contact author: kdo@cosmos.phy.tufts.edu
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Phys. Rev. D 113, 102001 – Published 4 May, 2026

DOI: https://doi.org/10.1103/2dft-4rjj

Abstract

We use Legendre polynomials, previously employed in this context by Lee et al., van Haasteren and Levin, and Pitrou and Cusin, to model signals in pulsar timing arrays. These replace the (Fourier mode) basis of trigonometric functions normally used for data analysis. The Legendre basis makes it simpler to incorporate pulsar modeling effects, which remove constant-, linear-, and quadratic-in-time terms from pulsar timing residuals. In the Legendre basis, this zeros the amplitudes of the first three Legendre polynomials. We use this basis to construct an optimal quadratic cross-correlation estimator μ^ of the Hellings and Downs correlation and compute its variance σμ^2 in the way described by Allen and Romano. Remarkably, if the gravitational-wave background and pulsar noise power spectra are (sums of) power laws in frequency, then in this basis one obtains analytic closed forms for many quantities of interest.

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References (22)

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Outline

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Phys. Rev. D 113, 102001– Published 4 May, 2026
  • Received 27 October 2025
  • Accepted 5 February 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Open access publication funded by the Max Planck Society.

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