Intermediate mass-ratio inspirals in a dense dark-matter environment: Effects of the initial dark-matter distribution

TL;DR

This work addresses how dense dark-matter spikes around intermediate-mass black hole binaries imprint gravitational-wave dephasing and how the formation history shapes the initial DM distribution. It develops a coupled binary–DM evolution framework with two cutoff prescriptions: an angular-momentum cutoff and a position-space cutoff, and treats the DM with a phase-space distribution function subject to dynamical-friction and secondary-accretion feedback. They find that using a physically motivated angular-momentum cutoff lowers DM density near the center and reduces dephasing compared with a position-space cutoff, with the largest differences at more extreme mass ratios; second-generation mergers show further depletion and reduced environmental effects. The results underscore the importance of the initial DM distribution for LISA measurements and point to extensions including eccentric orbits, Kerr spin, kicks, and additional dynamical processes.

Abstract

Recent work has shown the possibility of detecting dense dark-matter distributions surrounding intermediate or extreme mass-ratio inspirals through gravitational waves using LISA. Modeling these systems requires evolving the coupled dynamics of the binary and the dark matter. This also requires setting reasonable initial conditions for the dark-matter distribution, which itself relies upon understanding the formation history of these systems. In this paper, we investigate how two aspects of these systems' formation histories shape the dark-matter distribution: accretion onto the primary and prior merger events. We model accretion by introducing a minimum allowed angular momentum of dark-matter particles, which removes such particles that would have been accreted by the primary. When simulating an inspiral within such a distribution, we find a smaller dephasing of the gravitational-wave signal from a vacuum binary as compared to an inspiral without such a cutoff, particularly for more extreme mass-ratios. We also simulate an inspiral which takes place within a dark-matter distribution that remains after a prior merger. We find that the decrease in dephasing from vacuum binaries when compared to the prior inspiral is most significant for less extreme mass-ratios. Nevertheless, the environmental effects from the dark matter for these different cases of initial data are still expected to be measurable by future space-based detectors.

Figures (5)

• Figure 1: Dark-matter density with different angular-momentum cutoffs. In both panels, we show the density calculated from the distribution function given by Eq. \ref{['eq:static_f']}. We consider here a primary mass $m_1 = 10^5 \, \mathrm M_\odot$, $\rho_\mathrm{sp} = 200 \, \mathrm M_\odot/\mathrm{pc}^{3}$ and $\gamma_\mathrm{sp} = 7/3$. We plot a dotted, black, vertical line at $r_{\mathcal{J}} = 4\,R_\mathrm{s} = 8Gm_1/c^2$ for reference. Left: The analytic power law density ($j_\mathrm{min}= 0$) in dotted black, the fully Newtonian result ($j_\mathrm{min}= 4$) in dashed light-gray, the fully relativistic result in dash-dotted orange, and the modified Newtonian result ($j_\mathrm{min}=\sqrt 8$) in solid blue. Right: The latter two densities, on linear scale on the vertical axis. The figure is discussed in more detail in the text of Sec. \ref{['sec:case_cutform']}.
• Figure 2: Cycles of dephasing versus gravitational-wave frequency with a zero and nonzero angular-momentum cutoff. We show the GW dephasing relative to a vacuum system for an inspiral in an initial dark-matter distribution characterized by $\rho_\mathrm{sp} = 200~\mathrm M_\odot \, \mathrm{pc}^{-3}$ and $\gamma_\mathrm{sp} = 7/3$ (and a primary mass $m_1 = 10^5~\mathrm M_\odot$). The dotted black line is the dephasing using the power-law density profile without enforcing the cutoff in angular momentum, while the solid blue is the dephasing against vacuum using the density with a nonzero angular-momentum cutoff. A vertical, black, dotted line is plotted at the frequency four years from ISCO, $r_\mathrm{2,4y}$, which was computed using the prescription in Sec. \ref{['sec:cut_result_dephase']} (there is further discussion of this figure there, as well).
• Figure 3: Dark-matter density at three times during an inspiral. In all the times depicted in both panels, the density has a cutoff in the angular momentum ($j_\mathrm{min} = \sqrt{8}$) so that it vanishes at $r=2R_\mathrm{s}$. We chose $\rho_\mathrm{sp} = 200\,\mathrm M_\odot \, \mathrm{pc}^{-3}$ and $\gamma_\mathrm{sp} = 7/3$ for the dark-matter density in both panels and $m_1 = 10^3\,\mathrm M_\odot$ (left), or $m_1 = 10^5\,\mathrm M_\odot$ (right), for the primary mass. The dotted black line is the initial density profile (when $r_2 = 3 r_\mathrm{2,4y}$, but before the evolution begins), the dash-dotted orange curve is the density when $r_2 = r_\mathrm{2,4y}$, and the solid blue is the density when the secondary reaches the ISCO. Further discussion of the figure is given in Sec. \ref{['sec:cut_result_density']}.
• Figure 4: Cycles of dephasing versus gravitational-wave frequency for different initial dark-matter distributions and angular-momentum cutoffs. The dark-matter parameters ($\rho_\mathrm{sp} = 200\,\mathrm M_\odot/\mathrm{pc}^{3}$ and $\gamma_\mathrm{sp} = 7/3$) are the same as in Fig. \ref{['fig:dephase_PL1_100K']} for both panels and the primary mass is chosen to be $m_1 = 10^3\,\mathrm M_\odot$ (left) and $m_1 = 10^5\,\mathrm M_\odot$ (right). The dotted, black lines are the GW dephasing between vacuum binaries and those in a density profile without enforcing a cut to angular momentum ($j_\mathrm{min} = 0$), while the solid blue and dash-dotted orange correspond to the GW dephasing from vacuum for systems with a nonzero angular-momentum cutoff ($j_\mathrm{min} = \sqrt{8}$) for first- and second-generation inspirals, respectively. The vertical, black, dotted lines are plotted at the frequency for which the IMRI will merger in four years. More detailed discussion of the results is given in the text of Sec. \ref{['sec:successive_dephasing']}.
• Figure 5: Dark-matter density at three stages of the inspiral for a second-generation inspiral. The dark matter and binary parameters are the same as in the Fig. \ref{['fig:density_PL1_1K']}. We show the density of dark matter versus radius for several orbital separations during the inspiral of a second-generation merger. The dotted black line shows the initial density profile when $r_2 = 3 r_\mathrm{2,4y}$, the dash-dotted orange the density for $r_2 = r_\mathrm{2,4y}$, and the solid blue is the density for $r_2=r_\mathrm{ISCO}$. The light-gray dashed line is the initial density from the first-generation inspiral, which is given for comparison. More detailed discussion about the figure appears in the text of Sec. \ref{['subsec:2gdensity']}.