Addressing leakage and mode suppression in angular power spectrum estimation for gravitational-wave backgrounds using pulsar timing arrays

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. Spherical harmonic expansions must be truncated at finite multipole l_max^rec, often set to l_max^N_pair$\equiv {\rm int}\left[\sqrt{\text{N_pair}}-1\right]$, where N_pair is the number of distinct pulsar pairs in an array. This choice is motivated by the counting argument that cross-correlations provide at most N_pair independent constraints. We obtain the multipole l_max^res corresponding to the maximum informative angular scale of a PTA. It is defined such that expansions to l_max^res (approximately) span the space of "observable skies" encoded in the N_pair eigenmaps of the Fisher information matrix, and therefore depends on the array configuration. We explicitly show that GWB power contained in multipoles l$\gtrsim$l_max^res do not significantly affect analyses that use expansions out to l_max^res, because the PTA response acts as a low-pass filter. In contrast, truncating at l_max^rec< l_max^res leads to leakage of small-scale angular power from l_max^rec<l$\leq$l_max^res. Even choosing l_max^rec=l_max^res, the standard frequentist estimator of the angular power spectrum C_l remains biased by the modes unobservable by the array. Although we can (partially) debias the standard estimator -- improving its agreement with an injected spectrum -- this reduction in bias comes at the expense of an increase in variance, particularly for poorly constrained modes with l$\gg$l_eff. We therefore recommend: (i) using l_max^res for PTA analyses involving spherical harmonic expansions, and (ii) using the debiased standard estimator for C_l recovery, but only out to multipoles l<l_eff ($\ll$l_max^res) corresponding to sufficiently constrained modes.

Figures (16)

• Figure 1: Sky location of 34 pulsars drawn from a statistically-uniform distribution, used for the main analyses in Sec. \ref{['s:demos']}.
• Figure 2: Plots showing mode suppression (left panel) and mode coupling (right panel) calculated using the Fisher matrix for the $N_{\rm psr}=34$, Stat-Unif configuration shown in Fig. \ref{['f:34pulsars']}.
• Figure 3: Eigenvalue spectrum of the Fisher matrix $\mathcal{F}_{ab,cd}$ in the PTR basis, expanded in terms of pixels for different choices of $N_{\rm side}$ (left panel); and expanded in terms of spherical harmonics for different choices of $\ell_{\rm max}^{\rm rec}$ (right panel). These plots correspond to the PTA configuration shown in Fig. \ref{['f:34pulsars']}. Insets in both panels show zoomed-in eigenvalues for large eigenmode indices, which correspond to the smallest eigenvalues. Note that an $N_{\rm side}=32$ pixel expansion (left panel) and an $\ell_{\rm max}^{\rm rec}=52$ SpH expansion (right panel) converges to the true eigenvalue spectrum, while an $N_{\rm side}=8$ pixel expansion and an $\ell_{\rm max}^{\rm rec}=22$ SpH expansion have not converged to the true spectrum.
• Figure 4: Left panel: Match between eigenvectors of the Fisher matrix $\mathcal{F}_{ab,cd}$ in the PTR basis, expanded in terms of spherical harmonics for different choices of $\ell_{\rm max}^{\rm rec}$. (The expansion of the eigenvectors of $\mathcal{F}_{ab,cd}$ in terms of pixels having $N_{\rm side}=32$ is used as reference.) The spherical harmonic expansion with $\ell_{\rm max}^{\rm rec}=52$ (blue circles) recovers most ($>90\%$) of the $N_{\rm pair}$ eigenvectors (vertical dashed), while $\ell_{\rm max}^{\rm rec}=22$ (orange triangles) captures only the lowest modes. Right panel: Fraction of eigenmodes with match in the range (0.95,1) normalized by $N_{\rm pair}$, as a function of $\ell_{\rm max}^{\rm rec}$. The vertical line marks $\ell_{\rm max}^{\rm res}=52$, the maximum informative angular scale for this configuration corresponding to fraction 0.9 (horizontal dashed grey line). Both of these plots correspond to the PTA configuration shown in Fig. \ref{['f:34pulsars']}.
• Figure 5: Demonstration of leakage-induced bias in the reconstructed angular power spectrum $\hat{C}_\ell$ for the PTA configuration shown in Fig. \ref{['f:34pulsars']}. As $\ell_{\rm max}^{\rm gwb}$ increases from 10 to 52 (left to right panels), a significant positive bias appears when truncating recovery at $\ell_{\rm max}^{\rm rec}=22$ (red solid), but not when using $\ell_{\rm max}^{\rm rec}=52$ (blue dashed). The reconstruction is compared with the true (theoretical) $C_\ell$ (blue dot-solid) and sampled-universe $\tilde{C}_\ell$ (orange dotted). The shaded region around the estimated $\hat{C}_\ell$ values denotes $\pm 1\sigma$ cosmic uncertainty.