Gravitational wave detection with photometric surveys

Gravitational wave (GW) detections have considerably enriched our understanding of the universe. To date, all GW events from individual sources have been found by interferometer-type detectors. In this paper, we study a GW detection technique based on astrometric solutions from photometric surveys and demonstrate that it offers a highly flexible frequency range that can uniquely complement existing detection methods. From repeated point-source astrometric measurements, periodic GW-induced deflections can be extracted and wave parameters inferred. We emphasize that this method can be applied widely to any photometric surveys relying on relative astrometric measurements, in addition to surveys designed to measure absolute astrometry, such as Gaia. We illustrate how high-cadence observations of the galactic bulge, such as offered by the Roman Space Telescope’s Exoplanet MicroLensing (EML) survey, have the potential to be a potent GW probe with a complementary frequency range to Gaia, pulsar timing arrays, and the Laser Interferometer Space Antenna. We calculate that the Roman EML survey is sensitive to GWs with frequencies ranging from $7.7\times {10}^{-8}\mathrm{Hz}$ to $5.6\times {10}^{-4}\mathrm{Hz}$, which opens up a unique GW observing window for supermassive black hole binaries and their waveform evolution. While the detection threshold assuming the currently expected performance proves too high for detecting individual GWs in light of the expected supermassive black hole binary population distribution, we show that binaries with chirp mass ${\mathcal{M}}_c>{10}^{8.3}{M}_{\odot }$ out to 100 Mpc can be detected if the telescope is able to achieve an astrometric accuracy of 0.11 mas. To confidently detect binaries with ${\mathcal{M}}_c>{10}^7{M}_{\odot }$ out to 50 Mpc, a factor of 100 sensitivity improvement is required. We propose several improvement strategies, including recovering the mean astrometric deflection and increasing astrometric accuracy, number of observed stars, field-of-view size, and observational cadence. We also discuss how other existing and planned photometric surveys could contribute to detecting GWs via astrometry.

Figure 1

The expected stellar astrometric deflections. Left: Orthographically projected $dn$ for a subset of stars observed by Gaia in the Northern Hemisphere onto the galactic plane (inspired by a very similar plot in [24]). The North Galactic pole is at the center, which is also the position of the GW source. Black arrows correspond to the real part of the waveform at GW phase $\varphi =0$ (plus polarization), and the red arrows correspond to that at $\varphi =\pi /4$ (cross polarization). The source is a ${10}^9{M}_{\odot }$ equal-mass binary black hole at 1 Mpc at $(l=90\mathrm{deg},b=90\mathrm{deg})$ in galactic coordinates, emitting GWs at ${10}^{-6}\mathrm{Hz}$. This inclination angle is set to $i=0$ (i.e., face-on) and the polarization angle is $\psi =0$. Right: Deflections within the Roman Space Telescope’s FoV during the EML survey. The lower panel shows the total deflection, and the upper panel shows the deflection after subtracting the mean, since the mean is expected to be absorbed in the pointing reconstruction; for further discussion see Sec. 4. Star coordinates are selected from the Gaia Data Release 2 catalog, with brightness $0<G<9$ [32, 33]. Density of stars reflects only a subset of the true stellar density in the catalog.

Figure 2

Mean maximal deflection due to GW sources at different Galactic coordinates. The FoV is fixed to point toward the Galactic center, i.e., $l=0\mathrm{deg},b=0\mathrm{deg}$. The magnitude is calculated as that of the orthographic projection of $dn$ in the FoV, averaged over all observed stars. The maximum occurs when the source position vector is perpendicular to the star position vectors, which is consistent with Eq. (5). Aside from its coordinates, the GW source at each position has the same properties as that in Fig. 1. In combination with Fig. 1, we observe that the quadrupolar deflection pattern does not show up when we consider signal magnitude only.

Figure 3

Strain sensitivity of various GW detectors and corresponding example signals. Note that the sensitivity for different detectors is not represented by the same quantity, in anticipation of the signal source types. The sensitivity for LISA and IPTA is represented by the dimensionless characteristic noise strain from the detector noise power spectral density, given by ${\tilde{h}}_n(f)=\sqrt{f{S}_n(f)}$. In the frequency range of IPTA, the yellow block shows the expected ${\tilde{h}}_c$ of the SMBHBB. In the frequency range of LISA, the blue block shows the characteristic strain, ${\tilde{h}}_c(f)\equiv 2f\tilde{h}(f)$, of a fast-evolving ${10}^6{M}_{\odot }$ binary at $D_L=25\mathrm{Gpc}$. The sensitivity of Gaia and Roman EML Survey is represented by the detectable instantaneous (time-domain) strain, $h$, of monochromatic GWs, assuming end-of-survey performance. For Roman EML Survey, the solid line shows the sensitivity under signal mean subtraction, and the dashed line shows the sensitivity if full astrometric deflections are detectable. The red and violet block show the time-domain wave amplitude of monochromatic GWs with chirp masses ${10}^{9.7}{M}_{\odot }$ and ${10}^7{M}_{\odot }$. The shown frequency range of these GWs are limited by the mission lifetime of Roman EML survey and $f_{\mathrm{ISCO}}$ at this chirp mass. Note that the example signals of Roman EML and LISA are illustrative; they do not reflect the GW source population expected from binary black hole formation theory. As is shown, the frequency band from roughly $5\times {10}^{-7}\mathrm{Hz}$ to $1\times {10}^{-5}\mathrm{Hz}$ is uniquely accessed by Roman EML survey.

Figure 4

Detection sensitivity of the Roman EML Survey in ${\log }_{10}{\mathcal{M}}_c-{\log }_{10}{D}_L$ space at multiple fixed frequency. Colors indicate detection thresholds at different confidence levels. Top row: sensitivity when signals are mean subtracted. Bottom row: Sensitivity when the full signal is observable. For the three columns, the GW frequencies are fixed to be $7.7\times {10}^{-8}\mathrm{Hz}$, $6.6\times {10}^{-6}\mathrm{Hz}$, and $5.6\times {10}^{-4}\mathrm{Hz}$, respectively. In all panels, GW sources that reach the ISCO at the specified frequency or lower are blocked out in gray.

Figure 5

Sensitivity of the Roman EML Survey to ${\log }_{10}{\mathcal{M}}_c$. The upper right corner (shaded gray) excludes massive systems that reach the ISCO at each frequency or lower. Detection sensitivity threshold is represented by the detectable chirp mass at a $2\sigma$ confidence level at various luminosity distances. Yellow and red blocks show detectable mass ranges at 1 Mpc and 10 Mpc, respectively, assuming the mean astrometric deflection is subtracted from the signal. The dashed line shows the detection threshold at 100 Mpc if the astrometric accuracy were to improve to 0.11 mas, equivalent to a factor of 10 improvement in the sensitivity. The solid line shows the sensitivity at 1 Gpc if the mean signal were observable, roughly comparable to a factor of 100 sensitivity improvement. See Sec. 4 for further discussions.