Probing dynamical gravity with the polarization of continuous gravitational waves

The direct detection of gravitational waves provides the opportunity to measure fundamental aspects of gravity which have never been directly probed before, including the polarization of gravitational waves. In the context of searches for continuous waves from known pulsars, we present novel methods to detect signals of any polarization content, measure the modes present and place upper limits on the amplitude of nontensorial components. This will allow us to obtain new model-independent, dynamical constraints on deviations from general relativity. We test this framework on multiple potential sources using simulated data from three advanced-era detectors at design sensitivity. We find that signals of any polarization will become detectable and distinguishable for characteristic strains $h\gtrsim 3\times {10}^{-27}\sqrt{1\mathrm{yr}/T}$, for an observation time $T$. We also find that our ability to detect nontensorial components depends only on the power present in those modes, irrespective of the strength of the tensorial strain.

Figure 1

Effect of different GW polarizations on a ring of free-falling test particles. Plus ($+$) and cross ($\times$) tensor modes (green); vector-x (x) and vector-y (y) modes (red); breathing (b) and longitudinal (l) scalar modes (black). In all of these diagrams, the wave propagates in the $z$ direction. This decomposition into polarizations was first proposed for generic metric theories in [7].

Figure 2

Angular response of a quadrupolar detector to each GW polarization. The radial distance represents the response of a single quadrupolar antenna to a unit-amplitude gravitational signal of a tensor (top), vector (middle), or scalar (bottom) polarization, i.e. $|F_p|$ for each polarization $p$ as given by Eqs. (7)–(11). The polar and azimuthal coordinates correspond to the source location with respect to the detector, which is to be imagined as placed with its vertex at the center of each plot and arms along the $x$ and $y$-axes. The response is plotted to scale, such that the black lines representing the detector arms have unit length in all plots. The response to breathing and longitudinal modes is identical, so we only display it once and label it “scalar”.

Figure 3

Computation of ${\mathcal{O}}_{\mathrm{N}}^{\mathrm{S}}$. First, the Bayes factor ${\mathcal{B}}_{\mathrm{N}}^m$ is obtained from the data $\mathbf{B}$ and corresponding priors $p(\theta |{\mathcal{H}}_m)$ for each model $m\in M$, by evaluating the integral of Eq. (40) using a nested sampling algorithm that samples over $\overrightarrow{\theta }$ (step indicated by integral sign); these values are then added and multiplied by $P({\mathcal{H}}_m)/P({\mathcal{H}}_{\mathrm{N}})$ to obtain ${\mathcal{O}}_{\mathrm{N}}^{\mathrm{S}}$, as in Eq. (43). (Note that here we have set $P({\mathcal{H}}_m)/P({\mathcal{H}}_{\mathrm{N}})=1/7$, as explained Sec. 4.) The computation of ${\mathcal{O}}_{\mathrm{GR}}^{\mathrm{nGR}}$ is analogous.

Figure 4

Model priors. Distribution of prior probability over subhypotheses for the construction of ${\mathcal{O}}_{\mathrm{N}}^{\mathrm{S}}$ (left) and ${\mathcal{O}}_{\mathrm{GR}}^{\mathrm{nGR}}$ (right), according to Eqs. (43) and (45), respectively. For ${\mathcal{O}}_{\mathrm{N}}^{\mathrm{S}}$, we assign equal weight to the ${\mathcal{H}}_{\mathrm{N}}$ (white) and ${\mathcal{H}}_{\mathrm{S}}$ (gray); as in Eq. (64), we make no a priori distinction between non-GR models (solid) and GR (hatched). For ${\mathcal{O}}_{\mathrm{GR}}^{\mathrm{nGR}}$, we set equal prior probability for ${\mathcal{H}}_{\mathrm{GR}}$ and ${\mathcal{H}}_{\mathrm{nGR}}$, distributing the prior equally among non-GR models, as in Eq. (66).

Figure 5

Signal vs noise log-odds background distributions for any-signal and GR hypotheses. Histograms of $\ln {\mathcal{O}}_{\mathrm{N}}^{\mathrm{S}}$ (black line, hatched) and $\ln {\mathcal{O}}_{\mathrm{N}}^{\mathrm{GR}}$ (gray line) over an ensemble of 1000 simulated noise instantiations corresponding to the Crab pulsar. For each instantiation, three time series of Gaussian noise were produced using the design noise spectra of H1, L1 and V1, as outlined in Sec. 4; the data are analyzed coherently across detectors. (Note that here $\ln {\mathcal{O}}_{\mathrm{N}}^{\mathrm{GR}}=\ln {\mathcal{B}}_{\mathrm{N}}^{\mathrm{GR}}$, since we assign equal weight to both models.)

Figure 6

Signal vs noise log-Bayes background distributions for all subhypotheses. Violin plots representing histograms of the log-Bayes of several models vs noise, computed over an ensemble of 1000 simulated noise instantiations each corresponding to H1, L1 and V1 design data prepared for the Crab pulsar; the data are analyzed coherently across detectors. The labels on the $x$ axis indicate which hypothesis is being compared against noise; the “GR” label indicates tensor modes parametrized by Eq. (29) with fixed $\psi$ and $\iota$. Black lines mark the range and median of each distribution. (The gray histogram in Fig. 5 corresponds to the leftmost distribution here.)

Figure 7

Expected sensitivity to GR and vector injections. Log-odds of any-signal (${\mathcal{H}}_{\mathrm{S}}$, black circles) and GR (${\mathcal{H}}_{\mathrm{GR}}$, gray triangles) vs noise (${\mathcal{H}}_{\mathrm{N}}$) hypotheses, as a function of injection amplitude, for signals corresponding to both GR (left) and the vector-only model from [21] (right). The any-signal odds is defined in Eq. (43). Each of the 500 points corresponds to a data instantiation (one time series for each detector: H1, L1 and V1) made up of Gaussian noise plus a simulated Crab-pulsar signal of the indicated strength. The injections were performed with random values of the nuisance phase parameters, and the data were analyzed coherently across detectors. A logarithmic scale is used for the $y$ axis, except for a linear stretch corresponding to the first decade.

Figure 8

Expected sensitivity to scalar-tensor injections. Log-odds of any-signal (${\mathcal{H}}_{\mathrm{S}}$, left), GR (${\mathcal{H}}_{\mathrm{GR}}$, center) and $\mathrm{GR}+\mathrm{s}$ (${\mathcal{H}}_{\mathrm{GR}+\mathrm{s}}$, right) hypotheses vs noise. The any-signal odds is defined in Eq. (43). Each plot was produced by analyzing 2500 instantiations of data (one time series for each detector: H1, L1 and V1) made up of Gaussian noise plus a simulated Crab-pulsar $\mathrm{GR}+\mathrm{s}$ signal of the indicated tensor ($x$ axis) and scalar ($y$ axis) amplitudes. The color of each hexagon represents the average value of the log-odds in that region of parameter space; color is normalized logarithmically, except for a linear stretch in the $(-1,1)$ range.

Figure 9

Signal log-odds vs GW frequency for noise-only data. Circles mark the mean of the distribution of $\ln {\mathcal{O}}_{\mathrm{N}}^{\mathrm{S}}$, as a function of the expected GW frequency for each pulsar in our set; vertical lines indicate one-sided standard deviations for each source. Each data point and corresponding bars summarize the shape of a distribution like Fig. 5 for each of the pulsars, but produced from only 100 runs per source. The effective noise amplitude spectral density $\sqrt{S_n}$ (ASD, red curve), corresponding to the harmonic mean of each detector PSD, is overlaid for comparison (scaling obtained from a linear regression).

Figure 10

Categorizing tensor and vector injections (${\mathcal{H}}_{\mathrm{nGR}}$ vs ${\mathcal{H}}_{\mathrm{GR}}$). Non-GR vs GR log-odds, as a function of effective injection amplitude, for both GR (left) and the vector-only model from [21] (right). Each of the 500 points corresponds to a data instantiation (one time series for each detector: H1, L1 and V1) made up of Gaussian noise plus a simulated Crab-pulsar signal of the indicated strength. The injections were performed with random values of the nuisance phase parameters, and the data were analyzed coherently across detectors. Note that, on the right, a logarithmic scale is used for the $y$ axis, except for a linear stretch corresponding to the first decade.

Figure 11

Categorizing scalar-tensor injections (${\mathcal{H}}_{\mathrm{nGR}}$ & ${\mathcal{H}}_{\mathrm{GR}+\mathrm{s}}$ vs ${\mathcal{H}}_{\mathrm{GR}}$). Log-odds comparing the non-GR and $\mathrm{GR}+\mathrm{s}$ hypotheses to GR. The non-GR odds is defined in Eq. (45). Each plot was produced by analyzing 2500 instantiations of data (one time series for each detector: H1, L1 and V1) made up of Gaussian noise plus a simulated Crab-pulsar $\mathrm{GR}+\mathrm{s}$ signal of the indicated tensor ($x$ axis) and scalar ($y$ axis) amplitudes. The color of each hexagon represents the average value of the log-odds in that region of parameter space; color is normalized logarithmically, except for a linear stretch in the $(-1,1)$ range.

Figure 12

Ensemble non-GR vs GR log-odds. Non-GR vs GR log-odds computed from data for multiple sources, vs the number of sources in the set. Each light-gray trace marks a possible progression of the ensemble log-odds as new sources are added; the red line corresponds to the best quadratic fit. For each pulsar, we chose an arbitrary data instantiation containing a GR (left) or $\mathrm{GR}+\mathrm{s}$ (right); GR signals are restricted to ${10}^{-27}<h_{\mathrm{t}}<{10}^{-26}$, while $\mathrm{GR}+\mathrm{s}$ signals also satisfy $0.3<h_{\mathrm{s}}/h_{\mathrm{t}}<1$. We compute the value of $\ln {\mathcal{O}}_{\mathrm{GR}}^{\mathrm{nGR}}$ for each signal in the set and combine them according to Eq. (54) to obtain the ensemble value plotted in the $y$ axis.

Figure 13

Expected Crab non-GR upper limits in absence of signal. Histogram of 95%-credible upper limits for the scalar (left) and vector (right) amplitudes, for a set of 1000 noise-only data sets, computed using priors uniform in the amplitude (black) or uniform in the logarithm of the amplitude (hatched gray); the differences between these two priors are discussed in detail in Appendix pp2. Each instantiation (one time series for each detector: H1, L1 and V1) is made up of simulated Gaussian noise with standard deviation given by the advanced design PSDs. Scalar and vector upper limits are produced using the $\mathrm{GR}+\mathrm{s}$ and $\mathrm{GR}+\mathrm{v}$ models respectively.

Figure 14

Expected Crab scalar and vector upper limits in presence of $\mathrm{GR}+\mathrm{s}$ and $\mathrm{GR}+\mathrm{v}$ signals. Shading represents the 95%-credible upper limit for the scalar ($h_{\mathrm{s}}^{95\%}$, left) and vector ($h_{\mathrm{v}}^{95\%}$, right) amplitudes, vs the amplitude of injected GR ($x$ axis) and corresponding non-GR ($y$ axis) components. Each plot was produced by analyzing 2500 instantiations of data (one time series for each detector: H1, L1 and V1) made up of Gaussian noise plus a simulated Crab-pulsar $\mathrm{GR}+\mathrm{s}$ (left) or $\mathrm{GR}+\mathrm{v}$ (right) signal with indicated strains. The color of each hexagon represents the average value of the upper limit in that region of parameter space.

Figure 15

Expected upper limits in absence of signal vs GW frequency. Circles mark the mean of the distribution of $h_{\mathrm{s}}^{95\%}$ (top), $h_{\mathrm{v}}^{95\%}$ (middle) and $h_{\mathrm{t}}^{95\%}$ (bottom), as a function of expected GW frequency for each pulsar in our set; vertical lines mark one-sided standard deviations for each source. Each datapoint and corresponding bars summarize the shape of a distribution like those of Fig. 13, but produced from 100 noise instantiations each. The scalar, vector and tensor upper limits were produced assuming st, vt and t models respectively. We use uniform priors in all amplitude parameters (see Fig. 13 and Appendix pp2).

Figure 16

Tensor and vector upper limits as a ratio of scalar upper limits. Histogram of tensor (top) and vector (bottom) upper limits divided by the scalar upper limit for each pulsar. The top plot shows ratios for $h_{\mathrm{t}}^{95\%}$ (black), $h_{+}^{95\%}$ (light gray, hatched), and $h_{\times }^{95\%}$ (dark gray); the bottom plot shows ratios for $h_{\mathrm{v}}^{95\%}$ (black), $h_{\mathrm{x}}^{95\%}$ (light gray, hatched), and $h_{\mathrm{x}}^{95\%}$ (dark gray). Vertical dashed lines mark the mean of each distribution.

Figure 17

Effect of changing polarization angle. Norm of the complex plus ($a_{+}^{\prime }$, solid line) and cross ($a_{\times }^{\prime }$, dashed line) weights after rotating the source by $\mathrm{\Delta }\psi$ in the plane of the sky, i.e. letting $\psi \to {\psi }^{\prime }=\psi +\mathrm{\Delta }\psi$; this transformation is expressed in Eqs. (a5) and (a6). In this case, we start from $a_{+}=1$, $a_{\times }=0$ and $\psi =0$.

Figure 18

Tensor posteriors in presence of signal. Posterior PDFs for parameters of ${\mathcal{H}}_{\mathrm{GR}}$ (left) and ${\mathcal{H}}_{\mathrm{t}}$ (right) with fixed (top) and unfixed (bottom) source orientation ($\psi$, $\iota$). Each panel consists of a corner plot displaying the two-dimensional posteriors for each pair of parameters as indicated by the $x$ and $y$ labels, with the diagonals showing a histogram of the one-dimensional PDF for each parameter [i.e. the one-dimensional PDF obtained after marginalization of the multidimensional posterior PDF all other quantities, as in Eq. (61)]. The data analyzed contain signals with parameters indicated by the red lines; note that $C_{22}=h_0/2$ is the quantity that was actually used to parametrize GR triaxial amplitudes in the code [33]. In the top row, $\cos \iota$ and $\psi$ are fully known, and their resolution in these plots is limited by binning only. These plots were produced using the corner.py package [49].

Figure 19

Effect of inclination. Posterior PDFs for parameters of ${\mathcal{H}}_{\mathrm{GR}}$ with unfixed source orientation ($\psi$, $\iota$). Each panel consists of a corner plot displaying the two-dimensional posteriors for each pair of parameters as indicated by the $x$ and $y$ labels, with the diagonals showing a histogram of the one-dimensional PDF for each parameter. The data sets analyzed contain signals with parameters indicated by the red lines; note that $C_{22}=h_0/2$ is the quantity that was actually used to parametrize GR triaxial amplitudes in the code [33]. On the left, the injected signal corresponds to a face-off source ($\cos \iota \approx -1$), making it difficult to constrain the polarization angle $\psi$; on the right, the injection has similar amplitude but corresponds to an edge-on source ($\cos \iota \approx 0$), making it easy to constrain $\psi$ [modulo $\pi /2$ due to the $2\mathrm{\Delta }\psi$ dependence of Eqs. (a3) and (a4)]. These plots were produced using the corner.py package [49].

Figure 20

Free-tensor vs GR. Natural logarithm of the Bayes factor comparing ${\mathcal{H}}_{\mathrm{t}}$ to ${\mathcal{H}}_{\mathrm{GR}}$, as a function of GR injection amplitude for fixed (left) and unfixed (right) source orientation. On the left, the analysis correctly gives preference to ${\mathcal{H}}_{\mathrm{GR}}$ for signals above the detection threshold; on the right, however, the analysis is unable to satisfactorily distinguish between ${\mathcal{H}}_{\mathrm{t}}$ and ${\mathcal{H}}_{\mathrm{GR}}$, due to the orientation degeneracies discussed in Appendix pp1.

Figure 21

Log-uniform prior and upper limits. For a one-dimensional random variable $x$, we show the probability densities corresponding to a uniform likelihood with upper cutoff $\log x_{\max }=-22$ (red) and log-uniform priors with different lower cutoffs ($\log x_{\min }=-25$ for box 1, $\log x_{\min }=-26$ for box 2 and $\log x_{\min }=-27$ for box 3). Vertical dashed lines mark areas of equal probability mass for each distribution. The combined effect of the likelihood and each of the prior distributions is to produce 95%-credible upper limits on $x$ with values shown in the legend. The value obtained using only the likelihood corresponds to that obtained with a uniform prior with a broad enough range. As expected from Eq. (b4), the upper limit is not very sensitive to the lower bound set by the prior.

Figure 22

Log-uniform vs flat amplitude priors. The logarithm of the GR vs noise Bayes factor is computed for 1000 instantiations of Crab pulsar noise. For the GR amplitude $h_0$, we apply priors uniform in the quantity (black) and uniform in the logarithm of the quantity (hatched gray). The flat prior causes one to more strongly favor the noise model, due to a larger implicit Occam’s penalty.

Figure 23

Numerical error in Bayes factor computation. The logarithm of the GR vs noise Bayes factor is computed 500 times for different values of injected GR signal amplitude. The noise realization is not varied between computations with the same injection strength, only the seed for the random number generator used by the nested sampling algorithm. The red bars mark one standard deviation around the mean.

Figure 24

Observed error and prediction. Error in the computation of the logarithm of the GR vs noise Bayes factor as a function of injected GR signal amplitude. The solid, black curve corresponds to measured standard deviations from the computation of $\ln {\mathcal{B}}_{\mathrm{N}}^{\mathrm{GR}}$ 500 times per injection strength (red bars in Fig. 23). The dashed, gray curve shows the theoretical prediction for the error in the logarithm of the evidence, Eq. (c1).