Distinguishing dark matter halos with Extreme mass ratio inspirals

TL;DR

We address distinguishing three DM halo models around SMBHs by analyzing EMRIs in static, spherically symmetric spacetimes and their gravitational waves. To this end, we model a BH of mass $M_ ext{BH}$ embedded in a halo of mass $M$ with density profiles $ ho_ ext{DM}(r)$ for Hernquist, NFW, and Burkert halos, and compute EMRI orbital dynamics and GW signals using a metric with functions $A(r)$ and $m(r)$, parameterizing orbits by the semi-latus rectum $p$ and eccentricity $e$ and evaluating observables $P$ and $ abla\phi$ via $dt/d\chi$ and $d\phi/d\chi$. Gravitational radiation, Bondi–Hoyle accretion, and dynamical friction are incorporated to obtain the secular evolution of $p(t)$ and $e(t)$, with numerical integration complemented by analytic leading-order corrections in $M/a_0$. Gravitational waveforms are generated with the Numerical Kludge method and analyzed through orbital-cycle differences and waveform mismatches against LISA noise, yielding halo-differentiation thresholds such as $M/a_0>10^{-5}$ for Hernquist and $M/a_0>10^{-3}$ for Burkert/NFW halos, along with inter-halo distinctions; the results demonstrate EMRIs as feasible probes of DM halo structure in the strong-field regime.

Abstract

Using the static, spherically symmetric metric for a black hole (BH) immersed in dark matter (DM) halo characterized by Hernquist, Burkert, and Navarro-Frenk-White (NFW) density distributions, we calculate the orbital periods and precessions, along with the evolution of the semi-latus rectum and eccentricity for extreme mass ratio inspirals (EMRIs) surrounded by DM halos. For the Hernquist model, we find that the gravitational force exerted by the central BH is decreased by DM halos, while DM halos put additional gravitational force on the SCO. The presence of both Burkert-type and NFW-type DM halos enhances the gravitational force acting on the SCO, resulting in a decrease in the period $P$, with the decrease depending on $M/a_0^2$; additionally, we find that the reduction in orbital precession due to DM halos is influenced by $M/a_0^2$. The presence of DM halos leads to a slower evolution of EMRIs within Hernquist-type halos, while it accelerates evolution for EMRIs in Burkert-type and NFW-type halos; furthermore, it slows the decrease of eccentricity across all three types of DM halos. By calculating the number of orbital cycles and the gravitational waveform mismatches among these three types of DM halos, as well as between scenarios with and without DM halos, we find that DM halos can be detected when $M/a_0>10^{-5}$, $M/a_0>10^{-3}$, and $M/a_0>10^{-3}$ for Hernquist-type, NFW-type, and Burkert-type DM halos, respectively. Additionally, we can distinguish between NFW-type and Burkert-type DM halos when $M/a_0> 10^{-3}$; NFW-type and Hernquist-type DM halos, as well as Burkert-type and Hernquist-type DM halos, can be distinguished when $M/a_0> 10^{-5}$.

Figures (9)

• Figure 1: The density distributios of the DM halos and the mass function, "BUR", "HQ" and "NFW" represent the results calculated with Burkert-type, Hernquist-type, and NFW-type DM halos, respectively. The mass of central SMBH is taken as $M_\text{BH}=10^6M_\odot$, the length scale and the mass of DM halos are chosen as $a_0=100M$ and $M=100M_\text{BH}$, respectively.
• Figure 2: The results of the orbital periods and precessions for EMRIs in galaxies with different DM halos models. The symbols "BUR", "HQ", "NFW" and "No DM" represent the results for the Burkert, Hernquist, NFW density distributions, as well as the case without DM halos, respectively. The mass of central SMBH $M_\text{BH}$ is $10^6M_\odot$, and the eccentricity $e=0.6$. The parameters $(a_0,M)$ are chosen as $(100M,100M_\text{BH})$, $(100M,1000M_\text{BH})$, and $(1000M,10M_\text{BH})$ in the top, middle and bottom panels, respectively.
• Figure 3: The corrections to the orbital periods and precessions due to DM halos. The symbols "BUR", "HQ", and "NFW" represent the results for the Burkert, Hernquist, and NFW density distributions. The mass $M_\text{BH}$ of central SMBH is $10^6M_\odot$, and the eccentricity $e=0.6$. The parameters $(a_0,M)$ are chosen as $(100M,100M_\text{BH})$, $(100M,1000M_\text{BH})$, and $(1000M,10M_\text{BH})$ in the top, middle, and bottom panels, respectively.
• Figure 4: The energy and angular momentum fluxes for EMRIs in galaxies with three specific types of DM halos and without DM halos. The mass of central SMBH is $M_\text{BH}=10^6M_\odot$, the mass of the small BH is $\mu=10 M_\odot$, and the eccentricity is $e=0.6$. The parameters $(a_0,M)$ are chosen as $(100M,100M_\text{BH})$, $(100M,1000M_\text{BH})$, and $(1000M,10M_\text{BH})$ in the top, middle, and bottom panels, respectively.
• Figure 5: The evolution of $p(t)$ with three types of the DM halos and without DM halos for one year before the last stable orbit. The mass of central SMBH is $M_\text{BH}=10^6M_\odot$ and the mass of the small BH is $\mu=10 M_\odot$. The eccentricities at the last stable orbit are $e=0.05$ and $e=0.1$. The parameters $(a_0,M)$ are chosen as $(100M,100M_\text{BH})$, $(100M,1000M_\text{BH})$, and $(1000M,10M_\text{BH})$ in the top, middle, and bottom panels, respectively.