Prospects and Limitations of PTAs Anisotropy Searches -- The Frequentist Case

Abstract

Recent findings from several Pulsar Timing Array (PTA) collaborations point to the existence of a Gravitational Wave Background (GWB) at nanohertz frequencies. A key next step towards characterizing this signal and identifying its origin is to map the sky distribution of its power. Several strategies have been proposed to reconstruct this distribution using PTA data. In this work, we compare these different strategies to determine which one is best suited to detect GWB anisotropies of different topologies. We find that, for both localized and large-scale anisotropies, reconstruction methods based on pixel and radiometer maps are the most promising. However, in both scenarios, even the optimistically large anisotropic signals discussed in this work remain challenging to detect with near-future PTA sensitivities. For example, we find that for a GWB hotspot contributing to $80\%$ of the GWB power in the second frequency bin, detection probabilities reach at most $\mathcal{O}(10\%)$ for a PTA with noise properties comparable with the ones of the upcoming IPTA third data release. Finally, we consider the fundamental limitations that cosmic variance poses to these kinds of searches by deriving the smallest deviations from isotropy that could be detected by an idealized PTA with no experimental or pulsar noise.

Figures (13)

• Figure 1: In the left (center) panels, we report the null distributions for the anisotropic SNR of the maps reconstructed using the square-root spherical harmonic (pixel) basis. In the right panels, we show the max radiometer SNR for maps reconstructed using the radiometer basis. The null distributions for the broadband analysis are reported in the top row, while the null distributions for the second bin of the per-frequency analysis are shown in the bottom row. In all the panels, the blue (red) histograms show the null distributions derived from a data set mimicking the noise properties of the NANOGrav 15-year (IPTA DR3) data set. The vertical dashed lines indicate the SNR value with a $p$-value=$3\times10^{-3}$ (for the per-frequency null distributions, this is a local significance, i.e. it does not take into account the look-elsewhere effect introduced by the fact that we have multiple frequency bins). In the per-frequency case, it is possible that some realizations of the GWB result in negative estimates for $A_{\rm GWB}$ in some frequency bins. Since this shows that there is no information to extract from these realizations, we include them as $\rm{SNR}=0$, resulting in small peaks separate from the bulk of the null distributions.
• Figure 2: Null distributions for the $C_\ell$ coefficients of maps reconstructed using the square root spherical harmonics decomposition, for both the broadband (left panel) and the second bin of the per-frequency reconstruction (right panel). The blue (red) violins give the null distribution for a NANOGrav 15-year (IPTA DR3) like data set. The dashed lines correspond to the $C_\ell$ values with a local $p$-value=$3\times10^{-3}$ (which translates to a $\sim3\sigma$ significance).
• Figure 3: Radiometer SNR threshold (corresponding to a global $p$-value=$3\times10^{-3}$) for both broadband maps (upper panels) and the second frequency bin of the per-frequency reconstruction (lower panels). The left columns show the upper limits for a data set with the same noise properties as the NANOGrav 15-year data set, while the right columns show those for a data set with the noise properties of the upcoming IPTA DR3 data set. The numbers outside of the sky map represent the declination angle in degrees, while the numbers on the horizontal axis give the value of the right ascension in hours (1 hour corresponds to 15$^\circ$ of sky rotation). Before plotting, these and all other sky maps that we will show in this paper have been upscaled to a resolution of $N_{\rm side}=64$ and smoothed with a Gaussian symmetric beam with a full width at half maximum of $5^\circ$.
• Figure 4: In this plot, for each map parametrization, we report the ROC curves for the best performing detection statistic (according to the results reported in Table \ref{['tab:summary']}) and for different GWB anisotropies: single GWB hotspot (upper panels), two GWB hotspots (middle panels), and GWB dipole (lower panels). Specifically, on the x-axis, we report the global true positive rate, while on the y-axis we report the false positive rate of these different detection methods; both for the broadband searches (solid lines) and the per-frequency searches (dashed lines). The left (right) panels show the results for a PTA data set with similar noise properties to the NANOGrav 15-year (IPTA DR3) data set. The black dashed line is the ROC curve for the null distribution. The lines that are further from the black dashed line indicate the best-performing search strategies.
• Figure 5: Average of reconstructed GWB maps (using a broadband search) for the single hotspots benchmark. Maps are rotated to place the hotspot at the map center before averaging. Top panels: Radiometer SNR maps. Middle panels: Pixel basis maps. Bottom panels: Square-root spherical harmonics basis maps. Left (right) panels use PTA data with noise similar to the NANOGrav 15-year (IPTA DR3) data set.