Relativistic signatures of scalar dark matter in extreme-mass-ratio inspirals

Author(s)

Keijzer, Robrecht, Maenaut, Simon, Inchauspé, Henri, Hertog, Thomas

Abstract

We study gravitational wave emission by circular extreme-mass-ratio systems in a spherically symmetric scalar environment. Previous studies have focused on the impact of scalar radiation channels, revealing a rich structure of resonances, sharp features and floating orbits. Through the backreaction of the cloud on the metric, corrections to the gravitational sector come in at the same order. We develop the computational methods, and provide a characterization of this new, fully relativistic cloud signature. Remarkably, corrections to the polar sector can dominate all other dissipative corrections. We identify scalar field masses $Mμ\lesssim 0.12$ as the regime where polar can overtake axial and scalar channels at small separation. For small $Mμ$, vacuum dephasing is dominated mostly by conservative and polar cloud corrections, with scalar radiation acting as only a minor correction. At large $Mμ$, both terms terms are shown to be highly non-negligible. Our results therefore motivate including these relativistic signatures in beyond-vacuum EMRI templates.

Figures

Caption

Cloud corrections to the vacuum flux for each of the terms in Eq. \eqref{flux term Eq}, with $M\mu=0.08$. At small (large) radii, axial and polar terms give a negative (positive) correction. The `Polar, no $\phi^+/\phi^-$' curve is computed with $\phi^+=\phi^-=0$. The fluxes at $r_p=10M$ are indicated as a triangle and cross, and are shown in Fig. \ref{fig: mass scaling} as well.

Caption

Scalar and polar flux corrections as a function of $M\mu$, at a fixed radius $r_p=10M$. Down-pointing triangles in red show a negative flux correction (redshift), up-pointing arrows in blue show a positive correction (blueshift/mass shift). Scalar fluxes are always positive for a spherical cloud \cite{BritoMain}. At small mass $M\mu$, we find a good quadratic fit for the polar term.

Caption

Relative difference between the $l\!=\!m\!=\!2$ infinity energy flux with cloud coupling and in vacuum (solid line), for clouds with low compactness. Dotted lines correspond to the relative difference with the redshifted flux.

Caption

Number of cycles dephasing $\Delta \mathcal{N}$ with the vacuum waveform, for $M_1=10^6M_\odot$, $M_2=10^1M_\odot$, $r_0 = 10M$, $M\mu=0.08$, $M_c=0.1M$. When including conservative corrections, we slightly reduce the radius $r_0 \approx 9.995M$ to make the initial frequencies match. The total number of cycles is $\sim 4\cdot10^5$ after one year.

Caption

Extrapolation of higher order terms. Unphysical $\mathcal{O}(M_c/M)$ terms are significant, but we have good control over the error (linear). Red dots show the $l=m=2$ flux recovered for each mass $M_c/M$, dark blue dots show the standard evaluation points $M_c=(0.001,0.002,0.003)M$. Through extrapolation, we can accurately recover the $\Delta\dot{E}/\dot{E}_{\text{vac}}$ correction (light blue dot).

Caption

Limit of close separation, low compactness, axial sector. Solid (dotted) lines correspond to the relative difference with the vacuum (redshifted) flux.

Caption

Limit of large radius, high compactness ($M\mu=0.3$). The expected limits for each of the harmonics are recovered, indicated as grey dotted lines (Eqs. \eqref{lim1}, \eqref{lim2}).

Caption

Number of cycles dephasing $\Delta \mathcal{N}$ with the vacuum waveform, for $M_1=10^6M_\odot$, $M_2=10^1M_\odot$, $r_0 = 10M$, $M\mu=0.2$, $M_c=0.1M$. When including conservative corrections, we slightly increase the radius $r_0 \approx 10.005M$ to make the initial frequencies match. The total number of cycles is $\sim 4\cdot10^5$ after one year.