Gravitational waves from quasicircular extreme mass-ratio inspirals as probes of scalar-tensor theories

A stellar-mass compact object spiraling into a supermassive black hole, an extreme mass-ratio inspiral, is one of the targets for future space-based gravitational-wave detectors. Such inspirals offer a unique opportunity to learn about astrophysics and test general relativity in the strong field. Here we study whether scalar-tensor theories in asymptotically flat spacetimes can be further constrained with these inspirals. In the extreme mass-ratio limit, and assuming analyticity of the coupling functions entering the action, we show that all scalar-tensor theories universally reduce to massive or massless Brans-Dicke theory. We also show that in this limit, black holes do not emit dipolar radiation to all orders in post-Newtonian theory. For massless theories and quasicircular orbits, we calculate the scalar energy flux in the test-particle Teukolsky formalism to all orders in post-Newtonian theory and fit it to a high-order post-Newtonian expansion. We then derive the post-Newtonian corrections to the scalar-tensor modified, Fourier transform of the gravitational-wave response function and map it to the parametrized post-Einsteinian framework. With the Teukolsky flux at hand, we use the effective-one-body framework adapted to extreme mass-ratio inspirals to calculate the scalar-tensor modifications to the gravitational waveform. We find that such corrections are smaller than those induced in the early inspiral of comparable-mass binaries, leading to projected bounds on the Brans-Dicke coupling parameter that are worse than current Solar System ones. This is because Brans-Dicke theory modifies the weak field, leading to deviations in the energy flux that are largest at small velocities. For massive theories, superradiance can produce resonances in the scalar energy flux that can lead to quasicircular floating orbits outside the innermost stable circular orbit and that last until the supermassive black hole loses enough mass and spin-angular momentum. If such floating orbits occur in the frequency band of a LISA-like mission, they would lead to a large dephasing (typically $\sim {10}^6\mathrm{rads}$) that would prevent detection of such modified inspirals using general relativity templates. A detection that is consistent with general relativity would then rule out the presence of floating resonances at frequencies lower than the lowest inspiral frequency observed, allowing for the strongest constraints yet on massive scalar-tensor theories.

Figure 1

Parameter space of massive Brans-Dicke theory for a quasicircular EMRI with typical neutron star sensitivity $s_{\mathrm{SCO}}=0.188$, supermassive black hole spin $a_{\mathrm{MBH}}/M_{\mathrm{MBH}}=0.9$, and mass $M_{\mathrm{MBH}}={10}^5{M}_{\odot }$. The red dashed curve is the boundary of the region in $({\omega }_{\mathrm{BD}},{\mu }_s)$ space that separates floating and nonfloating resonances [cf. Eq. (69)]. The area below the black curve shows the region that is ruled out by Solar System experiments [38, 54]. The area between the green dot-dashed curves shows the region where a resonance would occur inside the classic-LISA sensitivity band [$({10}^{-4},{10}^{-2})\mathrm{Hz}$]. Because of a large dephasing introduced by floating resonances, any detection of an EMRI in a LISA-like mission would rule out the (open) region delimited by ABC. The dot-dashed blue line shows the values of $({\omega }_{\mathrm{BD}},{\mu }_s)$ below which nonfloating resonances lead to dephasings larger than 1 rad (see Sec. 7c). The ISCO frequency ${\Omega }_{\mathrm{ISCO}}$ is shown by a vertical dotted line. The curves ${\omega }_{\mathrm{BD}}^c$ and ${\omega }_{\mathrm{BD}}^{\mathrm{nf}}$ terminate at ${\mu }_s\sim {\Omega }_{\mathrm{orb}}\sim {\Omega }_{\mathrm{ISCO}}\sim 0.22/M_{\mathrm{MBH}}$.

Figure 2

PN expansion of the (${\nu }^2$-normalized) gravitational-wave energy flux (black solid curve) and the scalar energy flux carried out to infinity for Brans-Dicke theory with ${\omega }_{\mathrm{BD}}=4\times {10}^4$ (dashed red curve) and ${\omega }_{\mathrm{BD}}=1$ (dotted blue curve) as a function of velocity for a supermassive black hole with spin $q=0.99$ and a small compact object with $s_{\mathrm{SCO}}=0.188$. Observe that the scalar flux dominates over the GR one at sufficiently small velocities.

Figure 3

Scalar flux at infinity (black solid curve) and at the horizon (blue dashed curve) for a generic massless scalar-tensor theory, normalized by $(\alpha \nu {)}^2$ as functions of the velocity $v$ for ${\mu }_s=0$, $q_{\mathrm{MBH}}=0.99$ (left panel), $q_{\mathrm{MBH}}=0.9$ (middle panel), and $q_{\mathrm{MBH}}=0.6$ (right panel). For comparison, we show the PN formula (44) (green dot-dashed line) and its truncation at $-1\mathrm{PN}$ order (red dotted line).

Figure 4

Fractional difference between the fits, ${\mathcal{F}}_{\infty }^{s,\mathrm{massless}}$, with parameters from Table  and Teukolsky-based fluxes, ${\mathcal{F}}_{\mathrm{Teuk}}$, as a function of velocity for spin $q_{\mathrm{MBH}}=0.99$ and compared with the fractional difference with the PN flux (44). These curves are an upper bound of the fit error: the fractional difference for fit I, fit II, and for the PN formula decreases for smaller values of $q_{\mathrm{MBH}}$.

Figure 5

Scalar energy flux, normalized by $(\alpha \nu {)}^2$, carried to infinity and into the supermassive black hole horizon as a function of the velocity for ${\mu }_s{M}_{\mathrm{MBH}}=0.2$ and $q_{\mathrm{MBH}}=0.9$. Different curves are obtained by truncating the series in Eq. (46) at different values of $l$. Observe that below a certain velocity, $v\lesssim 0.4$, the contribution to infinity vanishes in accordance with Eq. (54). The resonance corresponding to $l=m=1$ is also shown.

Figure 6

Dephasing and normalized amplitude difference as a function of time in units of months for system F (left panel) and system R (right panel). The black and red curves are obtained by using the numerical fluxes constructed in Sec. 3 and their PN approximation in Eq. (44), respectively. We consider a Brans-Dicke parameter which saturates the Solar System bound, ${\omega }_{\mathrm{BD}}=4\times {10}^4$.

Figure 7

The ISCO frequency and angular velocity at the horizon as functions of the spin parameter for a Kerr black hole. In massive scalar-tensor theories, resonances occur for particles in circular orbits at ${\Omega }_{\mathrm{orb}}\sim {\mu }_s+\mathcal{O}({\mu }_s^2)$. The stability of the orbit implies ${\Omega }_{\mathrm{orb}}<{\Omega }_{\mathrm{ISCO}}$; i.e. the relevant region is below the blue line. For $q_{\mathrm{MBH}}\gtrsim 0.36$, these frequencies always satisfy the superradiant condition for $m=1$, ${\Omega }_{\mathrm{orb}}<{\Omega }_{+}$. For $q_{\mathrm{MBH}}\lesssim 0.36$, the resonant frequency may also be nonsuperradiant in the region limited by the blue line and the red dashed line. Note that resonances may occur even for Schwarzschild black holes.

Figure 8

The critical value ${\alpha }_c$ as a function of the scalar mass ${\mu }_s$, for several values of $q_{\mathrm{MBH}}$ and for $M_{\mathrm{MBH}}={10}^5{M}_{\odot }$ (lower curves) and $M_{\mathrm{MBH}}={10}^7{M}_{\odot }$ (upper curves). The floating orbit exists for those values of $\alpha$ above the curves and for $q\gtrsim 0.36$. Curves terminate when ${\mu }_s\sim {\Omega }_{\mathrm{ISCO}}$.

Figure 9

Dephasing in radians for systems F and R as a function of the floating time $T_{\mathrm{float},\mathrm{fin}}$, which can be mapped to ${\mu }_s$ via Eq. (121).