How well can modified gravitational wave propagation be constrained with strong lensing?

Strong gravitational lensing produces multiple images of a gravitational wave (GW) signal, which can be observed by detectors as time-separated copies of the same event. It has been shown that under favorable circumstances, by combining information from a quadruply lensed GW with electromagnetic observations of lensed galaxies, it is possible to identify the host galaxy of a binary black hole coalescence. Comparing the luminosity distance obtained through electromagnetic means with the effective luminosity distance inferred from the lensed GW signal would then enable us to constrain alternative theories of gravity that allow for modified GW propagation. Here we analyze models including large extra spatial dimensions, a running Planck mass, and a model that captures propagation effects occurring in a variety of alternative theories to general relativity. We consider a plausible population of lenses and binary black holes and use Bayesian inference on simulated GW signals as seen in current detectors at design sensitivity, to arrive at a realistic assessment of the bounds that could be placed. We find that, due to the fact that the sources of lensed events will typically be at much larger redshifts, this method can improve over bounds from GW170817 and its electromagnetic counterpart by a factor of $\sim 5$ to $\mathcal{O}({10}^2)$, depending on the alternative gravity model.

Figure 1

The effect on the frequency domain GW signal in each of the modified propagation models assuming different amounts of deviations from GR denoted by different colors. In these examples, the GW source is assumed to be at $\sim 5\mathrm{Gpc}$, and the rest of the source parameters are similar to those of GW150914 [1]. For the running Planck mass model, the deviation is absolute since one has $c_M=0$ in GR. For large extra dimensions and $\mathrm{\Xi }$ parametrization, we consider percentage deviation in the parameters $D$ and ${\mathrm{\Xi }}_0$, taking the fiducial values to be $D=4$ and $\mathrm{\Xi }=1$, respectively. For the $\mathrm{\Xi }$ parametrization, we arbitrarily choose $n=1$ here, though in our subsequent analyses it will be a free parameter.

Figure 2

The fractional difference $\mathrm{\Delta }\equiv |(D_L^{\mathrm{GW}}-D_L^{\mathrm{EM}})/D_L^{\mathrm{EM}}|$ (color) between $D_L^{\mathrm{GW}}$ and $D_L^{\mathrm{EM}}$ as a function of source redshift (horizontal axis) and deviation parameter (vertical axis). The $\delta D/D$ (top panel) and $\delta {\mathrm{\Xi }}_0/{\mathrm{\Xi }}_0$ (middle) refer to changes in respectively $D$ and ${\mathrm{\Xi }}_0$ relative to their fiducial values $D=4$ and ${\mathrm{\Xi }}_0=1$ (with $n=1$ for the latter case), whereas for $c_M$ (bottom panel) we use the value of the parameter itself. In the light (dark) regions the impact of the deviation parameter on the relation between $D_L^{\mathrm{GW}}$ and $D_L^{\mathrm{EM}}$ is larger (smaller). At the redshift of GW170817 (green vertical line), for the $\mathrm{\Xi }$ parametrization and varying Planck mass, $\mathrm{\Delta }$ is smaller than at high redshifts, already suggesting that strong lensing measurements, which access the high-redshift regime, are likely to lead to better constraints on these deviation parameters. On the other hand, the effect of extra dimensions is less sensitive to redshift, and measurements of $D$ are not expected to improve as much as for the other two cases.

Figure 3

90% confidence intervals for measurements of $\delta D/D$, $\delta {\mathrm{\Xi }}_0/{\mathrm{\Xi }}_0$ (defined as in Fig. 2) and $c_M$. The dots refer to results from quadruply lensed events, whose source redshifts can be read off from the horizontal axis; in each case the colors indicate the combined SNRs from the four images. The triangle indicates bounds from GW170817. Even lensed events with combined SNR similar to that of GW170817 (which was $\simeq 32.4$) yield considerably better constraints on deviation parameters, again underscoring the benefit of being able to access the high-redshift regime.