The electromagnetic chirp of a compact binary black hole: a phase template for the gravitational wave inspiral

TL;DR

This paper proposes that electromagnetic signals from gas around binary black holes can serve as phase templates for gravitational wave signals, enabling improved parameter estimation and tests of fundamental physics with LISA.

Contribution

It introduces the concept of an electromagnetic chirp as a phase template for GW inspiral signals from supermassive binary black holes.

Findings

Electromagnetic signals can track the orbital phase of binary black holes.

Phase comparison can measure differences in photon and graviton speeds as low as 10^-17.

Electromagnetic observations can improve localization and parameter estimation of GW sources.

Abstract

The gravitational waves (GWs) from a binary black hole (BBH) with masses between 10^4 and 10^7 Msun can be detected with the Laser Interferometer Space Antenna (LISA) once their orbital frequency exceeds 10^-4 - 10^-5 Hz. The binary separation at this stage is approximately a=100 R_g (gravitational radius), and the orbital speed is of order v/c=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 difference Delta v in the propagation speed of photons and gravitons as low as Delta v/c = O(10^-17).

Tables

Table 1. Table 1: For binaries with different redshift, primary and secondary mass ( M 1 subscript 𝑀 1 M_{1} and M 2 subscript 𝑀 2 M_{2} ; columns 1-2) the table shows the binary separation ( a 𝑎 a ), time-to-merger ( t m subscript 𝑡 𝑚 t_{m} ), GW frequency ( f min subscript 𝑓 min f_{\rm min} ), and EM variability amplitude ( Δ ​ F / F 0 = ( 3 + α ) ​ v | | / c \Delta F/F_{0}=(3+\alpha)v_{||}/c with α = − 1 𝛼 1 \alpha=-1 ) when the binary is first localized to 10 deg 2 (assumed to coincide with S/N=50; cols. 3-6); the time-to-merger ( t m subscript 𝑡 𝑚 t_{m} ), GW frequency ( f max (f_{\rm max} ), EM variability amplitude ( Δ ​ F / F 0 Δ 𝐹 subscript 𝐹 0 \Delta F/F_{0} ), S/N, and # of cycles accumulated since localization when the circumprimary disk is tidally truncated at 10 ​ R g1 10 subscript 𝑅 g1 10R_{\rm g1} (cols. 7-11); and a constant velocity difference ( Δ ​ v / c Δ 𝑣 𝑐 \Delta v/c ) or graviton mass ( m g subscript 𝑚 𝑔 m_{g} ) and equivalent Compton wavelength ( λ g subscript 𝜆 𝑔 \lambda_{g} ) which could be measured to S/N = 10 absent 10 =10 (cols. 12-14).

		at localization				at tidal truncation					shift
$M_1$	$M_2$	$a$	$t_m$	$f_{\min }$	$\Delta F/F_0$	$t_m$	$f_{\max }$	$\Delta F/F_0$	S/N	$N_{\mathrm{cyc}}$	$\Delta v/c$	$\log m_g$	$\log {\lambda }_g$
$\left[M_{\odot }\right]$	$\left[M_{\odot }\right]$	[$R_g$]	[day]	[Hz]	%	[day]	[Hz]	%	-	-	[log]	[eV/$c^2$]	[km]
z=1
${10}^3$	${10}^3$	190	23	-2.2	15	3min	-0.7	46	64	19670	$-17.5$	$-24.0$	15.1
${10}^4$	${10}^4$	99	17	-2.8	20	0.5hr	-1.7	46	530	3895	$-18.2$	$-25.1$	16.2
${10}^5$	${10}^5$	69	39	-3.6	25	0.2	-2.7		436	1510	$-17.3$	$-25.8$	16.9
${10}^6$	${10}^6$	35	26	-4.1	34	2.1	-3.7	46	214	232	$-16.1$	$-26.3$	17.4
$3\times {10}^6$	$3\times {10}^6$	24	18	-4.4	41	6.1	-4.2	46	87	58	$-15.2$	$-26.3$	17.4
${10}^7$	${10}^7$	16	12	-4.6	50	20	-4.7		–	–	–	–	–
${10}^5$	$3\times {10}^4$	68	34	-3.4	37	0.3	-2.5	69	387	2044	$-17.4$	$-25.5$	16.6
${10}^6$	$3\times {10}^5$	38	32	-4.0	50	2.5	-3.5	69	195	387	$-16.1$	$-26.1$	17.2
${10}^7$	$3\times {10}^6$	18	15	-4.5	74	25	-4.5	69	–	–	–	–	–
z=2
${10}^5$	${10}^5$	41	7.8	-3.4	31	0.3	-2.9	46	177	385	$-17.1$	$-25.5$	16.6
${10}^6$	${10}^6$	22	6.6	-4.0	42	3.1	-3.9	46	60	37	$-15.7$	$-25.8$	16.9
$3\times {10}^6$	$3\times {10}^6$	16	4.7	-4.2	51	9.2	-4.3	46	–	–	–	–	–
${10}^5$	$3\times {10}^4$	40	5.9	-3.2	49	0.4	-2.7	69	145	456	$-17.2$	$-25.2$	16.3
${10}^6$	$3\times {10}^5$	24	7.7	-3.8	63	3.7	-3.5	69	56	57	$-15.8$	$-25.7$	16.8

Equations

$$\frac{R_{\lambda }}{R_g}=68{\left(\frac{{\lambda }_{\mathrm{obs}}}{0.55\mu \mathrm{m}}\frac{3}{1+z}\right)}^{4/3}{\left(\frac{M}{10^6{\mathrm{M}}_{\odot }}\right)}^{-1/3}{\left(\frac{\dot{M}}{{\dot{M}}_{\mathrm{Edd}}}\right)}^{1/3}.$$

$$f_{\mathrm{GW}}=1.4\times 10^{-4}\mathrm{Hz}{\left(\frac{M}{10^6{\mathrm{M}}_{\odot }}\right)}^{-1}{\left(\frac{a}{60R_{\mathrm{g}}}\right)}^{-3/2}.$$

$$t_{\mathrm{GW}}=0.17q_s^{-1}\mathrm{yr}\left(\frac{M}{10^6{\mathrm{M}}_{\odot }}\right){\left(\frac{a}{60R_{\mathrm{g}}}\right)}^4,$$

$${\rho }^2(\delta h)=2\times 4{\int }_{f_{\min }}^{f_{\max }}df\frac{|\delta h(f){|}^2}{S_n^2(f)}$$

$${\rho }^2(\delta h)$$

$$\approx$$

Full text

I Introduction

Advanced LIGO has detected gravitational waves (GWs) from three stellar binary black hole (BBH) mergers, GW150914, GW151226 and LVT151012 Abbott et al. (2016a, b, c) and measured the GW inspiral waveform over the last $O(10)$ orbits. While no electromagnetic (EM) counterparts have been identified for these events, such counterparts could exist for stellar-mass BBHs embedded in dense gas, such as disks in active galactic nuclei (AGN) Bellovary et al. (2016); Bartos et al. (2017); Stone et al. (2017).

The space-based GW detector LISA Amaro-Seoane et al. (2017) will be sensitive to supermassive BBHs in the range $10^{4}-10^{7}{\rm M}_{\odot}$ and is expected to find dozens of merger events out to high redshifts ($z\sim 10$). These BBHs are produced in mergers of galaxies, which deliver the supermassive BHs, along with gas in the merging nuclei, to the center of the merger-remnant galaxy. The resulting compact BBHs are likely embedded in a gaseous environment, producing bright EM emission.

Identifying a GW source in EM bands would have considerable payoffs for cosmology and astrophysics (e.g. Phinney, 2010; Bloom et al., 2010). Examples for cosmology include measurement of the Hubble constant (Schutz, 1986), and probing gravity theories by comparing the luminosity-distance relation inferred from photons and gravitons (Deffayet and Menou, 2007). Measuring the differences in the arrival times of photons and gravitons from the same cosmological source would be an independent test of the massive character of gravity and of possible violations of Lorentz invariance in the gravity sector (Kocsis et al., 2008).

In this paper, we focus on the last possibility. While previous work has suggested the idea of comparing the arrival times of photons and gravitons (e.g. Csáki et al., 2001; Chung et al., 2002; Kocsis et al., 2008), a crucial limitation has remained: in order to measure a difference in their propagation speeds, one needs to know when the photons vs. gravitons were emitted by the source. In general, this would require modeling the astrophysics of the source. Here we argue that for a compact massive BBH in the LISA band, such modeling is not required: as long as the EM photospheres envelope individual BHs (rather than the binary as a whole), the relativistic Doppler modulation from the orbital motion of the binary will inevitably imprint the analog of the GW chirp on the EM light-curve D’Orazio et al. (2015)111Ref.HayasakiLoeb2016 considered a similar Doppler modulation that would arise following the tidal disruption of a star by a massive BBH at somewhat larger separations, but did not consider simultaneous direct measurement of GWs.. A measurement of the EM light-curve will determine the phase of the binary’s orbit. Comparing the EM and GW chirp signals will help break degeneracies between system parameters, and will measure the relative propagation speed $\Delta v$ of photons vs gravitons. Observing the same source with LISA and a wide-field X-ray (and possibly optical) instrument could probe a fractional difference as low as $\Delta v/c\approx 10^{-17}$.

II Periodic electromagnetic emission

EM emission from binaries – Recent hydrodynamical simulations have clarified the dynamics of a gaseous accretion disk around a BBH. Although the binary torques the inner disk and creates a low-density cavity, roughly twice the size of the binary separation, gas flows into this cavity through narrow streams Artymowicz and Lubow (1996). Simulations in which the BHs are in the computational domain have furthermore shown that the individual BHs are fueled efficiently via their own “minidisks” (Cuadra et al., 2009; Roedig et al., 2011; Nixon et al., 2011; Roedig et al., 2012; Gold et al., 2014; Farris et al., 2014; D’Orazio et al., 2016). This allows merging massive BHs to remain as bright as a typical quasar, nearly all the way to their merger (Farris et al., 2015a).

EM emission from the BH components – The gas accreting onto each BH forms a minidisk extending out to its tidal truncation radius. In units of their respective gravitational radii, these are $R_{1}/R_{\rm g1}\approx 0.27q^{-0.3}(1+q)(a/R_{\rm g})$ and $R_{2}/R_{\rm g2}\approx 0.27q^{-0.7}(1+q)(a/R_{\rm g})$ for the primary and secondary BH, respectively (Paczynski, 1977; Artymowicz and Lubow, 1994; Roedig et al., 2014), where $a$ is the binary’s semi-major axis. Simulations resolving the minidisks have found good agreement with these sizes (Farris et al., 2014). The nature of the emission produced by accretion via each minidisk should resemble those from quasars.

In standard quasar accretion disk models, the EM emission is radially stratified, with high-energy radiation produced closer in. In a steady, Keplerian, optically thick disk around a BH of mass $M$, the thermal emission $\sigma_{\rm B}T^{4}$ at radius $R$ balances the local dissipation $3GM\dot{M}/8\pi R^{3}$, where $T$ is the effective temperature at radius $R$, $\dot{M}$ the accretion rate, and $G$ and $\sigma_{\rm B}$ are Newton’s constant and the Stefan-Boltzmann constant (e.g. Frank et al., 2002).

Integrating the black-body emission over the face of a disk with a temperature profile of $T\propto R^{-3/4}$, the surface brightness profile $I_{\lambda}(R)\propto[\exp((R/R_{\lambda})^{3/4})-1]^{-1}$ has the scale radius at the observed wavelength $\lambda_{\rm obs}$.

[TABLE]

Here $\dot{M}_{\rm Edd}\equiv L_{\rm Edd}/c^{2}$ is the accretion rate corresponding to the Eddington luminosity $L_{\rm Edd}$, and we adopted a radiative efficiency of $\epsilon=15\%$ and a quasar luminosity of $L=\epsilon\dot{M}c^{2}=0.15L_{\rm Edd}$. The fiducial wavelength of 0.55$\mu$m corresponds to the optical $V$ band; in far UV bands ($\sim 0.15\mu$m) the disk would be $\sim 5$ times more compact ($R_{\lambda}\sim 12R_{\rm g}$). Note that $\approx 20\%$ of the total flux at $\lambda_{\rm obs}$ arises from inside the scale radius.

The X-ray emission observed from quasars is thought to arise from a combination of thermal disk emission and a hot corona. The size of the emitting region is limited by the broad observed FeK$\alpha$ line width and from variability measurements to be \mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt\sim}\hss}\raise 1.0pt\hbox{<}}10R_{\rm g} Reynolds and Nowak (2003); Miniutti and Fabian (2004). The X-ray emission from a handful of quasars has been directly constrained via microlensing, with half-light radii consistent with this compact size (e.g. Dai et al., 2010; Jiménez-Vicente et al., 2015; Chartas et al., 2016; Guerras et al., 2017). These X-ray-emitting regions fit around the individual BHs until the last stages of the merger, when the binary separation drops below \mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt\sim}\hss}\raise 1.0pt\hbox{<}}20R_{\rm g}.

Microlensing observations in the optical band have found sizes consistent with eq. (1). However, the observed luminosities are significantly below the value predicted by the above simple model, and imply $\sim 3$ smaller disks (e.g. Morgan et al. (2007)). One resolution of this discrepancy is that the optical/UV radiation emerging from the inner disk is scattered at larger radii. For the compact binaries considered in this paper, the individual minidisks are truncated at $(10-30)R_{\rm g}$. Reprocessing may then occur only by the narrow streams of gas inside the circumbinary cavity, with a small covering factor, and by the circumbinary disk farther out. We therefore consider the possibility below that the optical/UV emission is generated at radii $\sim 3$ times smaller than eq. (1).

Periodic relativistic modulation – The minidisks are fueled at a rate that varies periodically, on timescales comparable to the binary period (e.g. Hayasaki et al., 2007; MacFadyen and Milosavljević, 2008; Cuadra et al., 2009; Roedig et al., 2011; Noble et al., 2012; Shi et al., 2012; D’Orazio et al., 2013; Farris et al., 2014; Shi and Krolik, 2015; D’Orazio et al., 2016), which may introduce a corresponding periodic variation in the thermal EM emission Farris et al. (2015b). However, for compact binaries at relativistic separations (\beta\equiv v/c\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt\sim}\hss}\raise 1.0pt\hbox{>}}0.1, where $v$ is the orbital velocity and $c$ the speed of light), strong periodic variability is inevitably caused by relativistic Doppler modulation, and is expected to dominate the variability for mass ratios q\equiv M_{2}/M_{1}\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt\sim}\hss}\raise 1.0pt\hbox{<}}0.05 (D’Orazio et al., 2016). The Doppler effect results from the orbital motion, and exists irrespective of the details of the emission.

Photons emitted by gas bound to individual BHs suffer a Doppler shift in frequency $D=[\Gamma(1-\beta_{||})]^{-1}$, where $\Gamma=(1-\beta^{2})^{-1/2}$ is the Lorentz factor, $\beta_{||}=\beta\cos\phi\sin i$ is the component of the velocity along the line of sight, with $i$ and $\phi$ the orbital inclination and phase. The apparent flux $F_{\nu}$ at a fixed observed frequency $\nu$ is modified from the flux of a stationary source $F_{\nu}^{\rm 0}$ to $F_{\nu}=D^{3}F^{\rm 0}_{D^{-1}\nu}=D^{3-\alpha}F^{\rm 0}_{\nu}$. The last step assumes an intrinsic power-law spectrum $F^{\rm 0}_{\nu}\propto\nu^{\alpha}$. To first order in $v/c$, this causes a sinusoidal modulation of the apparent flux along a (circular) orbit, by a fractional amplitude $\Delta F_{\nu}/F_{\nu}=\pm(3-\alpha)(v/c)\sin i$ 222For a circular orbit; non-sinusoidal modulations from eccentric orbits are easily incorporated into the analysis..

This Doppler modulation is a plausible explanation for the remarkably sinusoidal optical and UV light-curves of the bright $z=0.3$ quasar PG 1302-102 D’Orazio et al. (2015), originally discovered by (Graham et al., 2015). This quasar, with a 5.2 yr observed period ($a\sim 0.01~{}{\rm pc}\sim 300~{}{\rm R_{g}}$), is a mildly relativistic binary BH candidate with $v/c\sim 0.07$. For more compact binaries, additional general relativistic effects from time-delays and lensing will be important. While we do not consider these effects here, we note that they will enhance the periodic brightness modulations, while tracking the phase of the orbital motion Schnittman et al. (2017).

As long as the BHs have their own photospheres, a periodic brightness modulation will be inevitable. The only exception would be the rare case of either (i) a face-on binary, or (ii) a widely separated equal-mass binary, inclined to the line of sight, for which the lensing effects are negligible, and the blue-shift/brightening of one BH cancels the red-shift/dimming of the other.

III Electromagnetic chirp for binaries in the LISA band

LISA binaries – The key feature of the Doppler-induced periodic variability is that its phase tracks the orbit of the binary. For a massive BBH whose inspiral is detected by LISA, this phase can be compared directly with the evolving phase of the GW chirp signal. The GW frequency (twice the orbital frequency) is given by

[TABLE]

The inspiral stage of a binary can be observed by LISA once its observed frequency enters the LISA band ($f_{\rm obs}=(1+z)^{-1}f_{\rm GW}\approx 10^{-4}-10^{-5}~{}{\rm Hz}$ for $M=10^{4}-10^{7}{\rm M_{\odot}}$). A compact binary will inspiral and merge, due to GW emission, on a timescale (to leading post-Newtonian quadrupole order; Peters (1964)) of

[TABLE]

where $q_{s}\equiv 4q/(1+q)^{2}$ is the symmetric mass ratio. Binaries can spend up to several years in the LISA band, and execute $\approx 1,900q_{s}^{-1}(a/60R_{\rm g})^{5/2}$ cycles, over which the GW chirp is measured.

The evolution across the LISA band of equal-mass binaries with several different masses (all at $z=1$) is illustrated in Figure 1. The characteristic strain $h_{c}$, defined as the product of the sky- and polarization-averaged Fourier strain amplitude $h(f)$ and the square root of the number of cycles ($n_{\rm cycle}\equiv\min[f^{2}/\dot{f},f\tau]$, where $\tau=5$ years is the assumed LISA mission lifetime) is shown, along with the spectral noise density $S_{\rm n}(f)$. We adopted the LISA configuration with six links, 2 million km arm length, and $\tau=5$yr (N2A2M5L6) from ref. Klein et al. (2016).

Along each track we mark, from left to right, three characteristic times. (i) First, equal-mass binaries with $(10^{7},10^{6},10^{5})~{}{\rm M_{\odot}}$ enter the LISA band (defined as the time when the S/N reaches $h_{c}/S_{\rm n}=1$) when their separation is (55, 110, 230) $R_{\rm g}$, respectively (vertical marks). (ii) As the binary evolves and accumulates S/N, its parameters can be estimated to increasingly better accuracy. With the configuration we adopted, LISA can localize a typical $10^{6}{\rm M}_{\odot}$ binary on the sky to an accuracy of 10 deg2 between a $\sim$week and a few months prior to the merger, with localization degrading rapidly with redshift, to $\sim$days at $z=3$ (Kocsis et al., 2008; Lang and Hughes, 2008; McWilliams et al., 2011). The blue circle along each curve denotes the time when the localization error drops to 10 deg2 for a typical binary. For simplicity, we assume that this occurs when the accumulated S/N=50; this gives good agreement with the advance localization accuracies found previously in (Kocsis et al., 2008) and Lang and Hughes (2008), who adopted a similar noise density (the accuracies vary by an order of magnitude, depending on the orientation of the orbital plane, spin magnitude and direction, and sky position). The advance localization is important for beginning an EM monitoring campaign, and occurs at separations of (16, 35, 69) $R_{\rm g}$. (iii) Finally, the red triangles denote the time when the tidal truncation radius of the circumprimary disk becomes smaller than $10R_{\rm g1}$. This ensures that the X-ray emitting regions of the disk have not yet been tidally stripped, and occurs (20 days, 2.1 day, 5 hrs) prior to merger. Note that the typical $10^{7}~{}{\rm M_{\odot}}$ binary is localized too late, but the $(10^{6},10^{5})~{}{\rm M_{\odot}}$ binaries are localized well before this tidal stripping; they accumulate a total S/N=(214, 436) and complete (232, 1510) cycles between localization and tidal stripping. We list the evolution of binaries with a range of masses at $z=1$ and 2 in Table 1.

Measuring the EM chirp – The advance localization will allow pointing a wide-field telescope at the LISA error box and obtain a densely sampled EM light-curve, covering hundreds of cycles. We consider both X-ray and optical telescopes, but note that because of the tidal truncation discussed above, it is unclear if optical emission will still be present for a binary in the LISA band.

In the optical, for a $10^{6}{\rm M_{\odot}}$ BH at $0.5<z<3$, shining at the Eddington limit, the detection of 10% variability will require a sensitivity corresponding to 22-27 magnitudes Kocsis et al. (2008). As an example, the Large Synoptic Survey Telescope (LSST; 333see www.lsst.org) has a field of view (FOV) of 10 deg2, which can cover the typical 2D sky localization error box from LISA, a month prior to merger, in a single pointing. LSST can reach a sensitivity of 27 mag (corresponding to a binary at $z=3$) in an integration time of less than 1hr (see Table 1 in Kocsis et al. (2008)). This is shorter than the orbital period and allows constructing a light-curve.

In the X-rays, as an example, the Wide Field Imager instrument on the proposed Athena mission 444see www.cosmos.esa.int/web/athena has a field of view of 0.5 deg2 and an effective area of $\sim 1$m2 at 1keV. Tiling the full LISA error box a month before merger would require $20$ pointings; alternatively, fewer cycles could be measured once the sky localization improves (an accuracy of 0.5 deg2 will be available several days prior to merger (Kocsis et al., 2008; Lang and Hughes, 2008; McWilliams et al., 2011)). The Lynx mission currently being developed 555See wwwastro.msfc.nasa.gov/lynx has a similar capability. For a $10^{6}{\rm M_{\odot}}$ BH at $z=1$, with an X-ray luminosity of $L_{X}=0.05L_{\rm Edd}$, Athena would collect $\approx$10 photons in a 103 sec integration, i.e. would obtain a flux measurement to $\sim 30$% accuracy throughout the 10 deg2 area in $\sim$5 hrs. Continued tiling for a week would yield a measurement of $\sim$30 X-ray cycles throughout the field.

Once the periodic EM counterpart has been identified, the rest of a month-long monitoring campaign can focus on this source, and can be done by many other, smaller-FOV telescopes. We conclude that obtaining a well sampled EM light-curve should be feasible for binaries at $z=1-2$ with masses {\rm few}\times 10^{3}{\rm M_{\odot}}\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt\sim}\hss}\raise 1.0pt\hbox{<}}M\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt\sim}\hss}\raise 1.0pt\hbox{<}}{\rm few}\times 10^{6}{\rm M_{\odot}}. Less massive binaries will have insufficient S/N, while more massive binaries enter the LISA band when their separation is already too compact for stable X-ray emission around the individual BHs (see Fig. 1 and Tab. 1).

IV Comparing the electromagnetic and gravitational wave chirp signals

Illustration – Figure 2 illustrates the comparison between the EM and GW chirp signals. The figure shows the strain from an edge-on ($\cos i=0$), circular, $M_{1}=10^{6}{\rm M_{\odot}}$, $M_{2}=3\times 10^{5}{\rm M_{\odot}}$ binary at $z=1$, with the chirp $df/dt$ computed from the lowest-order PN (quadrupole) term. The corresponding EM light-curve is shown assuming a spectral slope $\alpha=-1$, modulated by relativistic Doppler boost at order $v/c$. We have furthermore assumed that the secondary BH out-accretes the primary, and is twice as luminous, based on the simulation results in  Farris et al. (2014). Note that if the spectral slope $\alpha$ remains constant, the EM Doppler modulation amplitude ($v/c\propto f^{1/3}$) increases less steeply with frequency than the GW strain amplitude ($\propto f^{2/3}$). In principle, a few spectra, taken during the inspiral, could determine any evolution of $\alpha$). The figure high-lights in blue the 387 cycles available for EM chirp measurement in X-rays, and possibly in other bands (see Table 1).

Figure 3 zooms 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 deg2) and around 2.5 days (when the circumprimary disk is tidally stripped down to 10 $R_{\rm g1}$). In this figure, we have included the effect of gravitational lensing of whichever BH is behind the other, using the standard formula for lensing magnification to order $v/c$, $\mu=(x^{2}+2)/[x\sqrt{x^{2}+4}]$, where $x\equiv\theta/\theta_{E}$ is the angular distance between the two BHs on the sky, $\theta=\sqrt{(a\sin\theta)^{2}+(a\cos\theta\cos i)^{2}}d_{A}^{-1}$, measured in units of the Einstein radius of the BH in the front, $\theta_{E}\approx 2(R_{\rm gi}a)^{1/2}d_{A}^{-1}$. Note that the effect is asymmetric; to show this asymmetry, we have assumed a small 10∘ tilt of the orbital plane away from edge-on ($i=80$ deg).

Figure 3 shows several features of the correspondance between the time-domain GW chirp and the EM light-curve. First, in this example of a nearly edge-on binary, the GW is linearly polarized ($h_{+}$ only, $h_{\times}\approx 0$). The maxima of the GW strain correspond to the phase when the binary’s axis is aligned with the line of sight. The Doppler modulation $\Delta F/F=(3+\alpha)v_{||}/c$ at this phase vanishes, however, this also coincides with the maximum lensing magnification. The Einstein radius for the primary is larger, causing a more pronounced brightening when the secondary is being lensed (marked by the left vertical dashed line), but the lensing of the primary is still visible (vertical dashed line on the right). The minima of the GW strain alternate to sample the maxima and minima of the underlying Doppler modulation of the EM light-curve (shown the three vertical dotted lines).

Clearly, the shape of the EM light-curve will depend on inclination and mass ratio (see also 666D.J. D’Orazio and R. Di Stefano, in preparation). Larger inclinations would eliminate the lensing effect, but leave the phase of the EM chirp the same. On the other hand, the GW strain would acquire a cross-component ($h_{\times}\neq 0$) which is $\pi/2$ out of phase with the plus ($h_{+}$) component. Thus, comparing the EM and GW phases amounts to an independent measurement of the orbital inclination. Note that in our examples, the binary separation is \mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt\sim}\hss}\raise 1.0pt\hbox{>}}20R_{\rm g}, but for smaller separations, additional general relativistic effects, due to time-delay, and higher-order lensing, will become important.

Phasing accuracy – The signal-to-noise ratio $\rho$ for measuring a deviation in the GW waveform is given by

[TABLE]

where $f_{\rm min}$ is the frequency when the observation begins (i.e. when the localization accuracy drops below 10 deg2) and $f_{\rm max}$ is the frequency when the observation ends (i.e. when the tidal truncation radius of the primary BH drops below $10R_{\rm g1}$). The extra factor of two on the right-hand side is because of the assumed 3-arm configuration, equivalent to two independent interferometers. Because the EM chirp signal can provide a template for the phase evolution, we assume that these phases are precisely known, and evaluate the S/N for detecting a pure phase shift in the GW waveform, $\delta h=h(f)(1-\exp[i\Delta\psi(f)])$. Here $h(f)=\mathcal{A}f^{-7/6}\exp[i\psi(f)]$ is the frequency-domain inspiral waveform in the stationary-phase approximation, $\mathcal{A}=\pi^{-2/3}30^{-1/2}\mathcal{M}^{5/6}d_{L}^{-1}$ is the amplitude in the leading Newtonian order, $\mathcal{M}\equiv(M_{1}M_{2})^{3/5}/(M_{1}+M_{2})^{1/5}$ is the chirp mass, and $d_{L}$ is the luminosity distance.

As a simple case, we consider a shift of the time-domain waveform by a constant $\Delta t$, $\delta h=h(f)(1-\exp[2\pi if\Delta t])$. This depicts a constant propagation speed difference $\Delta v/c=c\Delta t/d_{L}$ of photons and gravitons, and yields

[TABLE]

where in the last step we assumed $2\pi f\Delta t\ll 1$. Fig. 1 shows that most of the signal-to-noise for M\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt\sim}\hss}\raise 1.0pt\hbox{>}}10^{4}~{}{\rm M_{\odot}} is contributed near $f_{\rm max}$, at the highest frequency at which there is still X-ray emission. To a good approximation, a detectable shift corresponds to $\approx 10/\rho$ rad, for a source with overall signal-to-ratio $\rho$ (Will, 2006; Kocsis et al., 2011). Evaluating eq. (5) numerically, Table 1 lists the shift $\Delta v/c$ for each binary that would produce $\delta\rho=10$. We find values as low as $\Delta v/c\approx 10^{-17}$.

If the velocity difference is due to gravitons with a nonzero mass $m_{g}$ in a Lorentz-invariant theory (e.g. Hassan and Rosen, 2012), then it will depend on frequency according to $\gamma m_{g}c^{2}=h_{p}f$, where $h_{p}$ is Planck’s constant, $f$ is the GW frequency, and $\gamma$ the Lorentz factor for the graviton. If $m_{g}c^{2}\ll h_{p}f$, the graviton speed differs from $c$ by $\Delta v/c=1/2(m_{g}c^{2}/h_{p})^{2}f^{-2}\propto f^{-2}$; this translates to a phase drift in the GW waveform $\Delta\psi(f)\propto f^{-1}$ Will (1998). In Table 1, we lists the graviton mass $m_{g}$for each binary that would produce $\delta\rho=10$, computed with this phase drift, using eqs. 3.8 & 3.9 in Will (1998). We also list the equivalent Compton wavelength ($\lambda_{g}=h_{p}/m_{g}c$).

Although Lorentz invariance for photons has been tested to high accuracy, it could be violated in the gravity sector, especially over cosmological scales. This occurs, e.g. in brane-world scenarios with extra dimensions (e.g. Chung et al., 2002; Csáki et al., 2001), or in vector-tensor scenarios (e.g. Jacobson and Mattingly, 2004); Lorentz-violating massive gravitons have also been proposed as candidates for cold dark matter (e.g, Dubovsky et al., 2005). See refs. Berti et al. (2005); Will (2006) for in-depth discussions. Following the EM and GW chirp in tandem as they evolve over a decade in frequency would explicitly test whether any measured phase drift is consistent with the expectation from $\gamma m_{g}c^{2}=hf$.

In principle, one should be able to tighten the above limits, by doing a direct cross-correlation analysis, or matched filtering, between the GW data stream and the light-curve of the EM counterpart(s). We also note that in case there are several variable quasars in the initial LISA error box, then this analysis will also help identify the correct counterpart to begin with (i.e. the object with period and phase closely matching those of the GWs).

Plasma effect for photons – The EM and GW signals can be offset from one another due to plasma effects slowing down the photons. The plasma frequency of the intergalactic medium is proportional to the density of free electrons, $n_{e}\approx 10^{-7}(1+z)^{3}{\rm cm}^{-3}$, yielding $\nu_{p}=(n_{e}e^{2}/\pi m_{e})^{1/2}=3~{}{\rm Hz}(1+z)^{3/2}$, and the corresponding index of refraction $n=\left(1-\frac{\nu_{p}^{2}}{\nu^{2}}\right)^{1/2}\approx 1-10^{-34}\left(\frac{E}{1{\rm keV}}\right)^{-2}$. This shows that plasma effects for photons can become comparable to the $\Delta v/c$ measurement limit for radio waves with \lambda\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt\sim}\hss}\raise 1.0pt\hbox{>}}10cm. This would become relevant in searching for periodic Doppler modulation of radio emission (e.g. from jets around the individual BHs, as proposed in Kulkarni and Loeb (2016)).

V Discussion and conclusions

Constraining the graviton mass by comparing the GW waveforms and the EM light-curves has been proposed previously in the context of compact white dwarf binaries (e.g. Cutler et al., 2003), and the possibility of an analogous measurement with massive BBHs was suggested by Kocsis et al. (2008). Here we followed up on the latter suggestion, and showed that the EM chirp signal, necessary to perform this experiment, will inevitably be produced by the relativistic Doppler modulations of the quasar-like emission from gas bound to the individual BHs. We also demonstrated that the EM chirp could be promptly (in a $\sim$week) identified by a wide-field X-ray telescope, a $\sim$month prior to merger, and would subsequently be available for monitoring for 2-3 weeks by other, smaller-field-of-view telescopes.

The EM chirp will serve as a template for the GW inspiral, allowing a novel test for theories with massive gravity or extra spatial dimensions. The constraints we find (Tab.1) are $\sim 4$ orders of magnitude tighter than the current upper limit, $1.2\times 10^{-22}~{}{\rm eV}/c^{2}$, from the LIGO waveforms Abbott et al. (2016a). They are comparable to the sensitivity that will be available from the phasing of GWs in LISA observations alone Will (2006). Compared to the latter forecasts, we have used only a restricted frequency-range, but have assumed that the EM template breaks degeneracies between the phase drift and other system parameters. Additionally, the EM chirp template should be useful in breaking parameter-degeneracies in the GW waveforms. As an example, the sky position is correlated with the orbital plane orientation; this degeneracy is broken by spin precession Lang and Hughes (2008) and by higher harmonics McWilliams et al. (2011), but only near the end of the inspiral, when these effects are large.

As the inspiral proceeds from a binary separation of $\sim 60R_{\rm g}$ to $20R_{\rm g}$, the minidisks surrounding each BH will gradually lose their outer annuli due to tidal stripping. Following the evolution of the luminosity and spectrum of the source, during this “peeling off” stage in the last few weeks of the inspiral, will provide a novel tomographic probe of the radial structure of the accretion disks.

We have conservatively focused here on the regime when the BHs are separated by \mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt\sim}\hss}\raise 1.0pt\hbox{>}}20R_{\rm g}, but the binary could possibly remain bright past this stage, and exhibit periodic fluctuations even during its last $\sim 10$ orbits (e.g. Bode et al., 2010). However, it is unclear whether the Doppler modulations, which arises from shocked gas near the binary but bound to individual BHs, can be as unambiguously related to the binary’s orbital phase in this case as at the larger separations we discussed.

We have focused on the X-ray emission, but binaries may be bright at radio and gamma-ray energies (e.g. via Bremsstrahlung emission and/or via jets Bode et al. (2010); Kulkarni and Loeb (2016)) all the way up to the merger. It may then be possible to discover the gamma-ray chirp signal without advance localization with an all-sky monitor. Barring this possibility, advance localization will be necessary for this experiment to work, and LISA would need to broadcast its data in real time.

While the gas accreting onto the BHs from a circumbinary disk can create bright quasar-like emission, it is unlikely to have an impact on the orbital evolution of the binary. When the binary is in the LISA band, and inspirals on time-scale of years, the gas torques are expected to be 4-5 orders of magnitude weaker Haiman et al. (2009); Kocsis et al. (2011); Tang et al. (2017).

Future work will need to develop the method proposed here, by taking into account effects we have neglected, such as the general relativistic modulations during the binary’s orbit, using higher-order GW waveforms, as well as addressing the degeneracies between the full set of system parameters. The flux from quasars is known to vary stochastically, including quasi-periodic modulations in the X-rays bands. In the case of binaries, significant X-ray emission can arise from shocks in accretion streams and the circumbinary disk, which may be modulated periodically, on time-scales different from those of the binary’s orbit Roedig et al. (2014); Farris et al. (2015b). The uncertainty to which the periodic Doppler modulations can be measured, in the presence of this variability, as well as realistic measurement errors of the EM light-curves, will need to be assessed in future work, as well.

Acknowledgements.

I thank Alessandra Buonanno, Csaba Csáki, Daniel Chung, Kohei Inayoshi, Feryal Ozel, Lorenzo Sironi, and Luigi Stella for useful discussions, and Daniel D’Orazio, Bence Kocsis and Geoffrey Ryan for useful comments on a draft of this manuscript. I also gratefully acknowledge support by a Simons Fellowship in Theoretical Physics (ZH) and by NASA grant NNX15AB19G.