Constraining alternative polarization states of gravitational waves from individual black hole binaries using pulsar timing arrays

Pulsar timing arrays are sensitive to gravitational wave perturbations produced by individual supermassive black hole binaries during their early inspiral phase. Modified gravity theories allow for the emission of gravitational dipole radiation, which is enhanced relative to the quadrupole contribution for low orbital velocities, making the early inspiral an ideal regime to test for the presence of modified gravity effects. Using a theory-agnostic description of modified gravity theories based on the parametrized post-Einsteinian framework, we explore the possibility of detecting deviations from general relativity using simulated pulsar timing array data, and provide forecasts for the constraints that can be achieved. We generalize the enterprise pulsar timing software to account for possible additional polarization states and modifications to the phase evolution, and study how accurately the parameters of simulated signals can be recovered. We find that while a pure dipole model can partially recover a pure quadrupole signal, there is little possibility for confusion when the full model with all polarization states is used. With no signal present, and using noise levels comparable to those seen in contemporary arrays, we produce forecasts for the upper limits that can be placed on the amplitudes of alternative polarization modes as a function of the sky location of the source.

Figure 1

GW antenna pattern sky maps. The color gradient indicates how sensitive the response is to a GW based on the pulsar’s sky position. The GW is originating from the green circle. From left to right, the columns correspond to values of $\mathrm{fL}=1$, 10, and 100, respectively. From top to bottom, the polarizations are the rms of TT/ST, VL, and SL, respectively.

Figure 2

Joint posteriors for the strain parameters of an all modes search. The histogram titles are the 50% quantiles with 1-sigma errors. The injected signal has all modes present and a total effective $\mathrm{SNR}\sim 20$. The priors for the strains are log-uniform. The blue lines are the injected values of the strains, and the dotted lines are the 95% quantiles.

Figure 3

Joint posteriors for the sky parameters of an all modes search. The histogram titles are the 50% quantiles with 1-sigma errors. The injected signal has all modes present and a total effective $\mathrm{SNR}\sim 20$. The priors for the parameters shown here are all uniform. The blue lines are the injected values of the parameters. The GW is originating from behind the GC.

Figure 4

Joint posterior for the radiative-loss coupling parameters of an all modes search. The histogram titles are the 50% quantiles with 1-sigma errors. The injected signal has all modes present and a total effective $\mathrm{SNR}\sim 20$. The priors for the parameters shown here are log-uniform. The blue lines are the injected values of the parameters.

Figure 5

Joint posteriors for an analysis using a pure ST model on data with a pure TT signal with ${\mathrm{SNR}}_{\mathrm{eff}}\sim 10$ at $f_{\mathrm{GW}}=1\times {10}^{-8}\mathrm{Hz}$. The histogram titles are the 50% quantiles with 1-sigma errors. The ST model is able to recover the TT signal, albeit with a biased ${\alpha }_D$ and amplitude. Identical analyses with longitudinal modes render biases in the sky localization. Two truth lines have been modified. Equation (16) shows that ${\mathrm{\Phi }}_0=\pi /4$ for a TT mode injection will be seen as ${\mathrm{\Phi }}_0=\pi /2$ for an ST mode that is in phase with that injection, and we have reflected that here. Also, since ${\log }_{10}{\alpha }_Q=8.5$ for this injection, we recast the truth value in terms of the correct units, ${\log }_{10}{\alpha }_Q{\omega }^{2/3}={\log }_{10}{\alpha }_D=3.5$, which is outside the posterior.

Figure 6

Joint posteriors of the sky location for an all modes upper limit search. The histogram titles are the 50% quantiles with 1-sigma errors. No signal is present in this injection. All parameter priors are uniform except for ${\alpha }_D$ and ${\alpha }_Q$, which are log-uniform. The sky location is restricted in a region around J1024-0719, whose position is indicated in blue lines. The enhanced response from longitudinal modes causes the posterior to peak away from the pulsar.

Figure 7

Joint posteriors of the strain upper limit strains in the presence of a pure GR signal with ${\mathrm{SNR}}_{\mathrm{eff}}\sim 10$ located at (0, $\pi$). The histogram titles are the 50% quantiles with 1-sigma errors. The dashed lines are the 95% quantiles.

Figure 8

Sky maps of an all modes upper limit search at fixed sky locations for GW frequency $f_{\mathrm{GW}}^{\mathrm{TT}}=1\times {10}^{-8}\mathrm{Hz}$. From left to right, top to bottom, the plots correspond to SL, ST, VL, and TT strains.