Mapping gravitational-wave backgrounds of arbitrary polarisation using pulsar timing arrays

We extend our previous work on mapping gravitational-wave backgrounds using techniques borrowed from the analysis of cosmic microwave background data to backgrounds which have non-general-relativity (non-GR) polarisations. Our analysis and results are presented in the context of pulsar timing array observations, but the overarching methods are general, and can be easily applied to LIGO or eLISA observations using appropriately modified response functions. Analytic expressions for the pulsar timing response to gravitational waves with non-GR polarisation are given for each mode of a spin-weighted spherical-harmonic decomposition of the background, which permit the signal to be mapped across the sky to any desired resolution. We also derive the pulsar timing overlap reduction functions for the various non-GR polarisations, finding analytic forms for anisotropic backgrounds with scalar-transverse (“breathing”) and vector-longitudinal polarisations, and a semianalytic form for scalar-longitudinal backgrounds. Our results indicate that pulsar timing observations will be completely insensitive to scalar-transverse mode anisotropies in the polarisation amplitude beyond dipole, and anisotropies in the power beyond quadrupole. Analogous to our previous findings that pulsar timing observations lack sensitivity to tensor-curl modes for a transverse-traceless tensor background, we also find insensitivity to vector-curl modes for a vector-longitudinal background.

Figure 1

Plots of ${\mathrm{\Gamma }}_{lm}^{+}$ for $l=0$ [panel (a)], $l=1$ [panel (b)], $l=2$ [panel (c)] and $l=3$ [panel (d)], as a function of the angle between the two pulsars for an uncorrelated, anisotropic background. Note that the vertical scale is not the same in all plots, but has been adjusted to more clearly show the curves in each panel.

Figure 2

Plots of ${\mathrm{\Gamma }}_{lm}^B$ for $l=0$ [panel (a)], $l=1$ [panel (b)], $l=2$ [panel (c)] and $l=3$ [panel (d)], as a function of the angle between the two pulsars for an uncorrelated, anisotropic background. As mentioned in the text, the overlap functions are identically zero for $l\ge 3$ or $|m|\ge 2$. We note that, as in Fig. 1, the vertical scale is not the same in all plots, but has been adjusted to more clearly show the curves in each panel.

Figure 3

Plots of the real part (left column) and imaginary part (right column) of ${\mathrm{\Gamma }}_{lm}^L(f)$ for $l=0$ (first row), $l=1$ (second row), $l=2$ (third row), $l=3$ (bottom row) as a function of the angle between the two pulsars for an uncorrelated, anisotropic background. These were calculated using the semianalytic approximation described in the main text. For these plots we have chosen $y_1=100$ and $y_2=200$, where $y_I=2\pi f{L}_I/c$ and $L_I$ is the distance to pulsar $I$. As in previous figures, we note that the vertical scale is not the same in all plots, but has been adjusted to more clearly show the curves in each panel.

Figure 4

Fractional percentage difference between the values of the $l=0$, $m=0$ overlap reduction function ${\mathrm{\Gamma }}_{00}^L(f)$ calculated (i) semianalytically [i.e., using the analytic expression for $I_{lm}(y,x)$ derived in Appendix pp7, and doing the $x$ integration numerically], and (ii) doing the full $(\theta ,\varphi )$ sky integration numerically using a healpix [45] pixelization of the sky. The dotted curve is for $y_1=10$, $y_2=20$, the dashed curve is for $y_1=50$, $y_2=100$, and the solid curve is for $y_1=100$, $y_2=200$, where $y_I=2\pi f{L}_I/c$ and $L_I$ is the distance to pulsar $I$. Note that the percentage difference decreases as $y_1$ and $y_2$ increase. The vertical dashed grey lines at the left- and right-hand edges of the plot correspond to the minimum and maximum angular separation (0.95° and 174°, respectively) over all pairs of pulsars in the European Pulsar Timing Array.

Figure 5

Plots of ${\mathrm{\Gamma }}_{lm}^X$ for $l=0$ [panel (a)], $l=1$ [panel (b)], $l=2$ [panel (c)] and $l=3$ [panel (d)], as a function of the angle between the two pulsars for an uncorrelated, anisotropic background. As in previous figures, we note that the vertical scale is not the same in all plots, but has been adjusted to more clearly show the curves in each panel.

Figure 6

Maps of the real amplitude component of a scalar-transverse (breathing mode) background. (Left) A randomly generated scalar-transverse gravitational-wave sky, with structure up to and including $l=10$. (Right) The corresponding recovered sky computed by first forming the observed signal vector for an array of $N_p=50$ pulsars via $\mathbf{r}=\mathbf{R}\mathbf{h}$, where each element of the array response matrix $\mathbf{R}$ corresponds to the response of a particular pulsar to gravitational waves propagating in a given sky direction. We perform a noiseless map recovery by computing ${\mathbf{R}}^{+}\mathbf{r}$ (where ${\mathbf{R}}^{+}$ is the pseudoinverse of $\mathbf{R}$) which gives the map in the right panel. We note the lack of small-scale angular structure in the recovered map compared to the injected map.

Figure 7

(Left) A comparison of the angular power spectrum estimator, ${\widehat{C}}_l=\sum _{m=-l}^l|{\widehat{a}}_{lm}^B{|}^2/(2l+1)$, of the injected scalar-transverse sky map shown in the left-hand panel of Fig. 6, and the PTA-recovered map shown in the right-hand panel of the same figure. We see that PTAs will completely lack sensitivity to angular structure in a scalar-transverse gravitational-wave sky beyond the dipole level. This result is confirmed analytically in Eq. (61). (Right) We use the full Earth and pulsar-term response from Eq. (28) to investigate map recovery with finite $y$. The pulsar term will be highly oscillatory across the sky, so we expect some numerical fluctuations in our results. As $y$ is increased the PTA shows greater sensitivity to higher multipole moments in the gravitational-wave background. At $y=10$ the PTA is able to recover all modes in the injected map, although the nonzero sensitivity of the pulsar-term response at higher multipoles amplifies noise from the pixelation of the sky. For $y\gtrsim 20$ the Earth term behavior is recovered, and we observe a lack of sensitivity to modes beyond dipole. See text for further details.

Figure 8

Approximations to the overlap reduction functions for an isotropic, unpolarized, and uncorrelated stochastic background plotted as a function of the angle between a pair of pulsars. The approximations are obtained by summing products of the response functions over $l$ for different values of $l_{\max }$. Panel (a): transverse-tensor background. Panel (b): scalar-transverse (breathing) background. Panel (c): scalar-longitudinal background. Panel (d): vector-longitudinal background. We are working in the large-$y$ limit for all of these cases. For the scalar-longitudinal background, we have taken $y_1=100$ and $y_2=200$. The thick black line in each plot is the “full” expression for the overlap reduction function, corresponding to the limit $l_{\max }\to \infty$. (These limiting expressions equal $\sqrt{4\pi }/2$ times the $l=0$, $m=0$ component of the overlap reduction functions calculated in Sec. 3.) For the scalar-longitudinal case, the full expression was calculated numerically.