Mapping gravitational-wave backgrounds using methods from CMB analysis: Application to pulsar timing arrays

We describe an alternative approach to the analysis of gravitational-wave backgrounds, based on the formalism used to characterize the polarization of the cosmic microwave background. In contrast to standard analyses, this approach makes no assumptions about the nature of the background and so has the potential to reveal much more about the physical processes that generated it. An arbitrary background can be decomposed into modes whose angular dependence on the sky is given by gradients and curls of spherical harmonics. We derive the pulsar timing overlap reduction functions for the individual modes, which are given by simple combinations of spherical harmonics evaluated at the pulsar locations. We show how these can be used to recover the components of an arbitrary background, giving explicit results for both isotropic and anisotropic uncorrelated backgrounds. We also find that the response of a pulsar timing array to curl modes is identically zero, so half of the gravitational-wave sky will never be observed using pulsar timing, no matter how many pulsars are included in the array. An isotropic, unpolarized and uncorrelated background can be accurately represented using only three modes, and so a search of this type will be only slightly more complicated than the standard cross-correlation search using the Hellings and Downs overlap reduction function. However, by measuring the components of individual modes of the background and checking for consistency with isotropy, this approach has the potential to reveal much more information. Each individual mode on its own describes a background that is correlated between different points on the sky. A measurement of the components that indicates the presence of correlations in the background on large angular scales would suggest startling new physics.

Figure 1

Mollweide projections of the frequency-independent response functions $F^{+}(\widehat{k})$, $F^{\times }(\widehat{k})$, for a pulsar located in direction $(\theta ,\varphi )=(50°,60°)$, indicated on the plots by a white star. The imaginary parts of both response functions are identically zero and are not shown above.

Figure 2

Mollweide projections of the real and imaginary parts of the Fourier transform of the redshift response $\tilde{z}(f)$ to a $+$-polarized point source located in direction $({\theta }_0,{\varphi }_0)=(50°,60°)$. The sky location of the pulsar is variable. The imaginary part of the response is identically zero, indicated by the solid green color of the second plot.

Figure 3

Mollweide projections of the real and imaginary parts of $h_{+}(\widehat{k})$, $h_{\times }(\widehat{k})$ for a simulated gravitational-wave background (top two rows) and the corresponding pulsar response map (bottom row).

Figure 4

Bayesian evidence for models of the overlap reduction function truncated at varying $l_{\text{max}}$. To produce this plot, we injected an isotropic, unpolarized and uncorrelated gravitational-wave background with a white-noise spectrum into a set of realistic-format pulsar TOAs. Testing truncated expansions of the form in Eq. (95), and recovering the Bayesian evidence, we find that an expansion up to and including $l=4$ is sufficient to recover the Hellings and Downs curve.

Figure 5

In (a) we explicitly show the envelope of overlap reduction functions corresponding to expansion coefficients lying in the 95% credible interval of the Bayesian analysis of an isotropic, unpolarized and uncorrelated gravitational-wave background. With an expansion up to and including $l=4$, our reconstruction is sufficiently consistent with the injected Hellings and Downs curve. In (b) we demonstrate that our numerical analysis is consistent with the analytic hypothesis that $C_l=\text{const}$, $\forall l\ge 2$.

Figure 6

Theoretical correlation maps ($|⟨h_{+}(\widehat{k})h_{+}^{*}({\widehat{k}}^{\prime }){⟩}_k|$) showing the degree to which Fourier modes along the $z$ axis and elsewhere on the sky are correlated. An uncorrelated gravitational-wave background should have a delta function in the sky location for the quadratic expectation value of Fourier modes [see Eq. (32)]. As we include higher multipoles in the expansion of Eq. (95), we converge toward this behavior.

Figure 7

Correlation maps [displaying $|⟨h_{+}(\widehat{k})h_{+}^{*}({\widehat{k}}^{\prime }){⟩}_k|$] constructed from the maximum–a posteriori $C_l$ values of our Bayesian analysis of a pulsar TOA data set containing an injected isotropic, unpolarized and uncorrelated gravitational-wave background. We see that the truncated expansion of Eq. (95) which gives the highest Bayesian evidence (corresponding to $l_{\text{max}}=4$) adequately recovers the restricted sky correlation that is characteristic of an uncorrelated gravitational-wave background.

Figure 8

Plots of ${\overline{\mathrm{\Gamma }}}_{12,lm}$ for $l=0,1,\dots ,5$ as a function of the angle between the two pulsars for an anisotropic, unpolarized and uncorrelated background.

Figure 9

Mollweide projections of $\text{Re}(h_{+})$ for the simulated gravitational-wave background. Panel (a) shows the total simulated background (grad+curl components); panel (b) shows the gradient component; and panel (c) shows the curl component. Sky maps of $\text{Im}(h_{+})$, $\text{Re}(h_{\times })$, and $\text{Im}(h_{\times })$ are similar.

Figure 10

Mollweide projections of $\text{Re}(h_{+})$ for the grad component of the simulated background (first column), the maximum-likelihood recovered sky maps (second column), and residual sky maps (third column) for PTAs containing different numbers of pulsars. The residual sky maps are the difference between the grad component of the simulated background and the recovered maps. Maps of $\text{Im}(h_{+})$, $\text{Re}(h_{\times })$, and $\text{Im}(h_{\times })$ are similar. The rows correspond to PTAs containing $N=1$, 5, 10, 20, 50, and 100 pulsars, respectively. The pulsar locations are shown as white stars. We used a variable color scale in the recovered maps to better show the angular structure in the small-$N$ maps, which would not have been visible if we had used the same (fixed) color scale used to make the simulated background and residual maps.

Figure 11

Mollweide projections of $\text{Re}(h_{+})$ for the different components of the simulated background (a)–(c), the maximum-likelihood recovered map for a PTA with $N=100$ pulsars (e), and the corresponding residual maps for the grad component (d) and the total simulated background (f). Sky maps of $\text{Im}(h_{+})$, $\text{Re}(h_{\times })$, and $\text{Im}(h_{\times })$ are similar. Note that the maximum-likelihood recovered map most closely resembles the gradient component of the simulated background, since a PTA is insensitive to the curl modes of a gravitational-wave background. The residual map for the grad component (d) is cleaner than the residual map for the total simulated background (f), which has angular structure that resembles the curl component of the background.

Figure 12

The average correlation between pairs of pulsars (and errors), as a function of pulsar angular separation, for an unpolarized, $l_{\text{max}}=2$ correlated gravitational-wave background (light-grey region) and for an isotropic, unpolarized and uncorrelated gravitational-wave background (dark-grey region). The averages were computed over 100 realizations of the backgrounds. Also shown are the analytic correlation functions expected in the two cases: the former being proportional to the Legendre polynomial $P_2(\cos \zeta )$, and the latter being the familiar Hellings and Downs curve.

Figure 13

A model-independent Bayesian reconstruction of the correlation between pairs of pulsars, as a function of pulsar angular separation, for a data set with an injected unpolarized, $l_{\text{max}}=2$ correlated gravitational-wave background. The background spectrum is white, with the signal injected into 10 pulsars which are observed fortnightly over a time span of 5 years. The correlation is parametrized at 13 distinct separations, and a cubic-spline interpolation used to compute the correlation at all other angular separations. The grey region shows the envelope of splines within the 95% credible interval of the full recovered posterior distribution, while the solid black curve is the expected analytic correlation function. The dashed black line indicates the largest angular separation between pulsars in our chosen array. As expected, beyond this line our reconstruction completely loses sensitivity.

Figure 14

Mollweide projections of the real and imaginary parts of $h_{+}$ and $h_{\times }$ for (a) a grad-only statistically isotropic correlated background with $C_l=1$ out to $l_{\text{max}}=10$ (top four plots), and (b) an anisotropic uncorrelated background that has the same power distribution as the statistically isotropic background but uncorrelated phase (bottom four plots). Note the pixel-to-pixel variation in the maps for the uncorrelated background.

Figure 15

Mollweide projection of the gravitational-wave power on the sky, $|h_{+}{|}^2+|h_{\times }{|}^2$, for both the statistically isotropic correlated background and the anisotropic uncorrelated background shown in Fig. 14.

Figure 16

Polarization ellipses for a patch of sky (centered at 0° latitude and 0° longitude) for (a) the statistically isotropic correlated background, and (b) the anisotropic uncorrelated background shown in Fig. 14. The polarization ellipses are superimposed on the (common) map of gravitational-wave power shown in Fig. 15. The crosses indicate the direction and principal axes of the $h_{+}$, $h_{\times }$ polarization ellipses. Linear polarization in a particular direction is represented by a line, and circular polarization by a cross with equal-length axes. Note that the polarization ellipses vary smoothly over the sky for the statistically isotropic correlated background, as compared to that for the anisotropic uncorrelated background, which have randomly oriented phase angles.