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Pulsar timing array analysis in a Legendre polynomial basis

Bruce Allen, Arian L. von Blanckenburg, Ken D. Olum

TL;DR

This paper develops a Legendre polynomial basis for pulsar timing array analyses, replacing the conventional Fourier basis to model timing residuals. By mapping the timing-model subtraction to the first three Legendre modes, it yields analytic, closed-form expressions for the timing-residual covariance under power-law GWB and pulsar-noise spectra, and it derives the optimal Hellings-Downs estimator directly in the Legendre framework. The work shows the estimator’s variance and SNR are basis-invariant and remains consistent when transitioning between timing-residual and redshift formulations, even with nonuniform observing times across pulsars. It also extends the formalism to nonstationary post-fit residuals through transfer functions and transmission functions, clarifying how low-frequency power is filtered by timing-model subtraction and enabling efficient, accurate PTA analyses in practical data sets.

Abstract

We use Legendre polynomials, previously employed in this context by Lee et al. [1], van Haasteren and Levin [2], and Pitrou and Cusin [3], to model signals in pulsar timing arrays (PTA). 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 zeroes the amplitudes of the the first three Legendre polynomials. We use this basis to construct an optimal quadratic cross-correlation estimator $\widehatμ$ of the Hellings and Downs (HD) correlation and compute its variance $σ^2_{\widehatμ}$ in the way described by Allen and Romano [4]. Remarkably, if the gravitational-wave background (GWB) 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.

Pulsar timing array analysis in a Legendre polynomial basis

TL;DR

This paper develops a Legendre polynomial basis for pulsar timing array analyses, replacing the conventional Fourier basis to model timing residuals. By mapping the timing-model subtraction to the first three Legendre modes, it yields analytic, closed-form expressions for the timing-residual covariance under power-law GWB and pulsar-noise spectra, and it derives the optimal Hellings-Downs estimator directly in the Legendre framework. The work shows the estimator’s variance and SNR are basis-invariant and remains consistent when transitioning between timing-residual and redshift formulations, even with nonuniform observing times across pulsars. It also extends the formalism to nonstationary post-fit residuals through transfer functions and transmission functions, clarifying how low-frequency power is filtered by timing-model subtraction and enabling efficient, accurate PTA analyses in practical data sets.

Abstract

We use Legendre polynomials, previously employed in this context by Lee et al. [1], van Haasteren and Levin [2], and Pitrou and Cusin [3], to model signals in pulsar timing arrays (PTA). 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 zeroes the amplitudes of the the first three Legendre polynomials. We use this basis to construct an optimal quadratic cross-correlation estimator of the Hellings and Downs (HD) correlation and compute its variance in the way described by Allen and Romano [4]. Remarkably, if the gravitational-wave background (GWB) 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.

Paper Structure

This paper contains 16 sections, 105 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Fourier Basis
  3. Legendre basis
  4. Effects of fitting a pulsar timing model
  5. Covariance between harmonic coefficients in the Legendre basis
  6. Evaluation of covariance matrix for power-law power spectra
  7. Relation between Fourier basis and Legendre basis
  8. The optimal HD estimator
  9. Variance sigmasq and SNR rhosq in the Legendre basis
  10. Differing pulsar observation times
  11. Transmission functions
  12. Conclusion
  13. Fitting of the "universal terms" in the Fourier basis
  14. Transmission function integral
  15. Redshifts
  16. ...and 1 more sections

Figures (4)

  • Figure 1: The Legendre polynomials $P_\mu(z)$ for $\mu=0,1,\cdots,4$.
  • Figure 2: The transmission function $R_a(f)$ shows low-frequency behavior identical to that of Hazboun:2019vhv, and pitrou-cusin:2024.
  • Figure 3: Comparison of ${\rm sinc}\bigl(\pi(f - f_j) T\bigr)$ (solid lines) with $V_j(f)$ (\ref{['e:Vj']}) (dashed lines) for $j = 0, \dots, 3$. The discrepancy decreases as $j$ and $f T$ increase.
  • Figure 4: Starting from timing residuals $\mathcal{T}_a$ in the top left corner, we always end up with the same $\check{\sigma}_{\hat{\mu}}$ and $\check{\rho}$. This is independent of the choice of Fourier or Legendre basis to evaluate $\check{\sigma}_{\hat{\mu}}$ and $\check{\rho}$.