Bias from small-scale leakage in Pulsar Timing Array maps

Abstract

Pulsar Timing Array experiments are rapidly approaching the era of gravitational wave background anisotropy detection. The timing residuals of each pulsar are an integrated measure of the gravitational-wave power across all angular scales. However, due to the limited number of monitored pulsars, current analyses are only able to reconstruct the angular structure of the background at large scales. We show analytically that this mismatch between the integrated all-sky signal and the truncated reconstruction introduces a previously unaccounted source of systematic bias in anisotropic background map reconstruction. The source of this systematic error, that we call ''small-scale leakage'', is the intrinsic presence of unaccounted gravitational wave power at scales smaller than the reconstructed scales. This unmodeled power leaks into large-scale modes, artificially increasing the recovered value of the inferred angular power spectrum by at least one order of magnitude in a wide range of scales. Importantly, this effect is fundamentally independent of the geometry of the pulsar configuration, the anisotropy reconstruction method, the use of different regularization schemes, and the presence of pulsar noise. As the quality of pulsar timing array experiments improves, a robust understanding of small-scale leakage will become paramount for reliable detection and characterization of the gravitational wave background. Thus, the theoretical formalism developed here will be essential to estimate the magnitude of this systematic uncertainty in anisotropy searches.

Figures (11)

• Figure 1: Left panels: fractional contribution of large-scale (blue lines) and small-scale (red lines) correlated residuals for four randomly displaced pulsars as a function of the reconstruction multipole $\ell_{\max}^{\rm rec}$. For $\ell_{\max}^{\rm rec}=0$ (monopole only), the entire signal originates from small scales. As $\ell_{\max}^{\rm rec}$ increases, progressively more power shifts into the large-scale component. The sum of both components, i.e., the total correlated residual, remains constant when varying $\ell_{\max}^{\rm rec}$. Right panel: absolute value of the small- to large-scale correlated residuals ratio for all pulsar pairs in the I34 configuration. The black line represents the median of the absolute value of the ratio.
• Figure 2: Left panels: individual realization of a true GWB with power up to $\ell_{\max}^{\rm GWB}=250$ (top panel), its large-scale component with power up to $\ell_{\max}^{\rm rec}=22$ (central panel), and reconstructed large-scale background, including small-scale leakage, up to $\ell_{\max}^{\rm rec}=22$ (bottom panel). Right panel: envelopes of true (black), reconstructed (red), and leakage (blue) correlated residual angular power spectra for $10^3$ realizations of the GWB and the I34 pulsar geometry. Solid lines indicate the median value of angular power spectra.
• Figure 3: Value of the mode-mixing function $M_{\ell \ell'}$ from equation \ref{['eq:cl_leakage_fitting']} in terms of large-scale $\ell$ and small-scale $\ell'$ multipoles. This matrix quantifies how the unmodeled scales $\ell'$ bias de the reconstructed $C_\ell$ (see equation \ref{['eq:mode_mixing_pure']}. In both panels we show how low large-scale multipoles are almost unaffected by small-scale leakage, however the situation radically changes for multipoles close to $\ell_{\max}^{\rm rec}$, where the order of magnitude of the mode-mixing function is around order unity.
• Figure 4: Envelopes of true (black), reconstructed (red), leakage (blue) and regularization (orange) $C_\ell$ for $10^3$ realizations of the GWB and the NG34 (top panels) and NG68 (bottom panels) pulsar geometry. Different panels show the effect of different values of the ridge regularization parameter, indicated on the top left corner. Solid lines indicate the median value of angular power spectrum. Although a mild regularization appears to successfully suppress the small-scale leakage effect, more aggressive choices suppress the reconstructed signal below its true value. Since the true value of GWB angular power spectrum is not known a priori, it is not possible to fine-tune the regularization process to filter out the spurious effect.
• Figure 5: Same of figure \ref{['fig:regularization_leakage_interplay']}, but for different amounts of injected pulsar red noise. Although in both cases the injected noise does not significantly affect the reconstruction of the signal, we observe as it increases the variance of the bands.