Electromagnetic chirp of a compact binary black hole: A phase template for the gravitational wave inspiral

The gravitational waves (GWs) from a binary black hole (BBH) with masses ${10}^4\lesssim M\lesssim {10}^7{\mathrm{M}}_{\odot }$ can be detected with the Laser Interferometer Space Antenna (LISA) once their orbital frequency exceeds ${10}^{-4}–{10}^{-5}\mathrm{Hz}$. The binary separation at this stage is $a=O(100)R_{\mathrm{g}}$ (gravitational radius), and the orbital speed is $v/c=O(0.1)$. We argue that at this stage, the binary will be producing bright electromagnetic (EM) radiation via gas bound to the individual BHs. Both BHs will have their own photospheres in x-ray and possibly also in optical bands. Relativistic Doppler modulations and lensing effects will inevitably imprint periodic variability in the EM light curve, tracking the phase of the orbital motion, and serving as a template for the GW inspiral waveform. Advanced localization of the source by LISA weeks to months prior to merger will enable a measurement of this EM chirp by wide-field x-ray or optical instruments. A comparison of the phases of the GW and EM chirp signals will help break degeneracies between system parameters, and probe a fractional difference $\mathrm{\Delta }v$ in the propagation speed of photons and gravitons as low as $\mathrm{\Delta }v/c\approx {10}^{-17}$.

Figure 1

Tracks across the LISA band of binaries at $z=1$ with mass ratios of $q\equiv M_2/M_1=1/3$, and different primary masses $M_1$, as labeled. The break in the characteristic strain $h_c$ marks 5 years prior to merger. Along each track, the marks correspond to the times when (i) the binary enters the LISA band (vertical line), (ii) the sky localization of a typical binary reaches an accuracy of $10{\mathrm{deg}}^2$ (blue circle), and (iii) the tidal truncation radius of the circumprimary disk becomes smaller than $10R_{\mathrm{g}1}$ (red triangle). The dashed (magenta) curve shows LISA’s sensitivity, assuming a configuration with six links, 2 million km arm length, and a mission lifetime of five years [51]. Note that LISA is not planning to probe frequencies below ${10}^{-5}\mathrm{Hz}$, where the accelerometer noise is expected to rise steeply.

Figure 2

The time-domain GW chirp and the EM light curve for an $M_1=10^6{\mathrm{M}}_{\odot }$, $M_2=3\times 10^5{\mathrm{M}}_{\odot }$, circular, edge-on binary at $z=1$. For clarity, the EM light curve shows only the Doppler modulation and excludes lensing effects. The part of the signal shown in blue marks the $\approx 400$ cycles available for EM chirp measurement in x rays, and possibly in other bands. For this fiducial binary, the mean x-ray flux is $F_x=0.05L_{\text{Edd}}/4\pi d_L^2=2\times {10}^{-15}\mathrm{erg}{\mathrm{s}}^{-1}{\mathrm{cm}}^{-2}$.

Figure 3

Zooming in on the GW and EM chirp signals of the binary in Fig. 2 to show the few cycles around a look-back time of 32 days (when the binary is first localized on the sky to $10{\mathrm{deg}}^2$) and around 2.5 days (when the circumprimary disk is tidally stripped down to 10 $R_{\mathrm{g}1}$). The bottom panel assumes a tilt of the orbital plane 10° away from edge-on, to illustrate the asymmetric lensing of the primary and the secondary (the sharp peaks marked by the vertical dashed lines). Vertical dotted lines mark minima of the GW chirp, corresponding to alternative minima and maxima of the EM signal.