Gravitational Waves from Quasicircular Extreme Mass-Ratio Inspirals as Probes of Scalar-Tensor Theories

TL;DR

This work assesses how EMRIs can constrain scalar-tensor theories by combining a Teukolsky-based, relativistic treatment of scalar and gravitational radiation with an effective-one-body EMRI framework. It shows that in the EMRI limit generic ST theories collapse to massless or massive Brans-Dicke theory, with BHs emitting no dipolar radiation, and finds massless ST effects yield dephasings that do not improve BD bounds beyond Solar System tests. For massive ST theories, resonant scalar flux near certain orbital frequencies can produce floating orbits, leading to dramatic waveform dephasings; thus, a GR-consistent detection would impose tight constraints on the scalar mass $\\mu_s$ and coupling $\\alpha$ (or $\\omega_{BD}$), potentially surpassing current bounds by orders of magnitude. Overall, EMRIs are powerful probes of massive scalar-tensor modifications in the strong-field regime, while massless cases remain weaker than existing Solar System constraints, highlighting the importance of targeting resonant phenomena in future tests of gravity.

Abstract

A stellar-mass compact object spiraling into a supermassive black hole, an extreme-mass-ratio inspiral (EMRI), is one of the targets of future gravitational-wave detectors and it offers a unique opportunity to test General Relativity (GR) in the strong-field. We study whether generic scalar-tensor (ST) theories can be further constrained with EMRIs. We show that in the EMRI limit, all such theories universally reduce to massive or massless Brans-Dicke theory and that black holes do not emit dipolar radiation to all orders in post-Newtonian (PN) theory. For massless theories, we calculate the scalar energy flux in the Teukolsky formalism to all orders in PN theory and fit it to a high-order PN expansion. We derive the PN ST corrections to the Fourier transform of the gravitational wave response and map it to the parameterized post-Einsteinian framework. We use the effective-one-body framework adapted to EMRIs to calculate the ST 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 coupling that are worse than current Solar System ones. Brans-Dicke theory modifies the weak-field, with deviations in the energy flux that are largest at small velocities. For massive theories, superradiance can lead to resonances in the scalar energy flux that can lead to 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 LISA, they would lead to a large dephasing (~1e6 rads), preventing detection with GR templates. A detection that is consistent with GR would then rule out floating resonances at frequencies lower than the lowest observed frequency, allowing for the strongest constraints yet on massive ST theories.

Figures (9)

• Figure 1: Parameter space of massive Brans-Dicke theory for a quasi-circular EMRI with typical neutron star sensitivity $s_{\hbox{\tiny SCO}}=0.188$, supermassive black hole spin $a_{\hbox{\tiny MBH}}/M_{{\hbox{\tiny MBH}}}=0.9$ and mass $M_{\hbox{\tiny MBH}}=10^5 M_\odot$. The red, dashed curve is the boundary of the region in $(\omega_{\hbox{\tiny BD}},\mu_{s})$ space that separates floating and non-floating resonances (cf. Eq. \ref{['omegaBD_crit_Einstein']}). The area below the black curve shows the region that is ruled out by Solar System experiments Perivolaropoulos:2009akAlsing:2011er. The area between the green, dot-dashed curves shows the region where a resonance would occur inside the classic LISA sensitivity band [$(10^{-4},10^{-2})$ Hz]. Due to 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 $(\omega_{\hbox{\tiny BD}},\mu_{s})$ below which non-floating resonances lead to dephasings larger than $1$ rad (see Sec. \ref{['sec:resonant-effect-on-GW']}). The ISCO frequency, $\Omega_{\hbox{\tiny ISCO}}$, is shown by a vertical dotted line. The curves $\omega_{\hbox{\tiny BD}}^{c}$ and $\omega_{{\hbox{\tiny BD}}}^{\rm nf}$ terminate at $\mu_s\sim\Omega_{\hbox{\tiny orb}}\sim\Omega_{\hbox{\tiny ISCO}}\sim0.22/M_{\hbox{\tiny MBH}}$.
• 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 $\omega_{{\hbox{\tiny BD}}} = 4 \times 10^{4}$ (dashed red curve) and $\omega_{{\hbox{\tiny BD}}} = 1$ (dotted blue curve) as a function of velocity for a supermassive black hole with spin $q = 0.99$ and a small compact object with $s_{{\hbox{\tiny SCO}}} = 0.188$. 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 $\mu_s=0$, $q_{{\hbox{\tiny MBH}}}=0.99$ (left panel), $q_{{\hbox{\tiny MBH}}}=0.9$ (middle panel) and $q_{{\hbox{\tiny MBH}}}=0.6$ (right panel). For comparison, we show the PN formula \ref{['dipole-massless']} (green dot-dashed line) and its truncation at $-1$PN order (red dotted line).
• Figure 4: Fractional difference between the fits, ${\cal{F}}^{s,{\hbox{\tiny massless}}}_{\infty}$, with parameters from Table \ref{['tab:fit']} and Teukolsky-based fluxes , ${\cal{F}}_{{\hbox{\tiny Teuk}}}$, as a function of velocity for spin $q_{\hbox{\tiny MBH}}=0.99$ and compared with the fractional difference with the PN flux \ref{['dipole-massless']}. 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 $q_{\hbox{\tiny MBH}}$.
• 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 $\mu_s M_{\hbox{\tiny MBH}}=0.2$ and $q_{\hbox{\tiny MBH}}=0.9$. Different curves are obtained by truncating the series in Eq. \ref{['series']} at different values of $l$. Observe that below a certain velocity, $v\lesssim0.4$, the contribution to infinity vanishes in accordance with Eq. \ref{['dipole']}. The resonance corresponding to $l=m=1$ is also shown.