Detectability of dark matter density distribution via gravitational waves from binary black holes in the Galactic center

TL;DR

This work addresses how dark matter (DM) environments near the Galactic Center affect gravitational waves (GWs) from binary black holes (BBHs) in the nHz–μHz range detectable by pulsar timing arrays. It couples two DM density models—a generalized NFW (gNFW) profile without a spike and a spike-modified profile resulting from adiabatic compression—to orbital dynamics, incorporating DM dynamical friction, DM accretion, and GW reaction to predict GW waveform deviations. The main finding is that detectable DM-induced GW signatures after 30 years are expected only for steep inner slopes (e.g., γ=2) in the gNFW profile, and these signatures are amplified when a DM spike forms; GC-PTA could probe a broader portion of parameter space than SKA-PTA. Overall, the paper provides explicit observational criteria to constrain the inner DM density profile of the Galactic Center via GW observations, offering a complementary avenue to traditional dynamical and lensing constraints.

Abstract

The fundamental nature of dark matter (DM) remains unknown, with significant uncertainties in its density profile. DM environments surrounding massive binary black holes (BBHs) modify their orbital dynamics, thereby altering gravitational wave (GW) emissions. For BBH systems at the Galactic Center, dynamical friction induced by DM spikes could produce detectable deviations in GW spectra, potentially observable by future space-based detectors. To address the uncertainties in the Galactic Center's DM profile, we systematically examine two scenarios: the generalized Navarro-Frenk-White (gNFW) profile and its post-spike modification. We investigate the evolutionary effects of DM dynamical friction and accretion on the eccentricity and semi-latus rectum of secondary black holes (BHs) in elliptical orbits. By constructing orbital models with varying initial eccentricities across the mass-semi-latus rectum parameter space and utilizing 30 years of simulated pulsar timing array data from the Square Kilometer Array (SKA), we identify detectable parameter regimes of DM effects and employ these GW observational signatures to constrain different DM density profiles. Our analysis reveals that among gNFW profiles ($γ=2,1.5,1,0.5$), only $γ=2$ produces significant detectable signatures. The formation of DM spikes further enhances these observable waveform deviations for all gNFW slopes.

Figures (20)

• Figure 1: The DM density profiles for the gNFW halo in Eq. \ref{['eq::gnfw_halo']} are illustrated in the figure. The possible gNFW profile of DM in our MW is represented by a solid red line for $\gamma = 0.5$, a blue line for $\gamma = 1$, a yellow line for $\gamma = 1.5$, and a yellow line for $\gamma = 2$. At the center of the MW, there exists a SMBH with a mass of $4.26 \times 10^6 \, M_{\odot}$. As one approaches the GC, the DM distribution becomes increasingly dense for different indices. However, within the innermost stable circular orbit, the DM distribution is assumed to be zero.
• Figure 2: In the gNFW profile in Eq. \ref{['eq::gnfw_halo']}, the power-law indices $\gamma = \{0.5, 1, 1.5, 2\}$ correspond to the formation of spike power-law indices $\gamma_{\text{sp}} = \{16/7, 7/3, 2.4, 2.5\}$ in Eq. \ref{['p-spike']}. The relevant parameters are listed in Table \ref{['tab:2']}.
• Figure 3: The schematic diagram of orbital motion viewed in the fundamental reference frame.
• Figure 4: The variation of $\left|\left\langle\frac{de}{{\mathrm{d}}t}\right\rangle\right|$ with respect to the initial semi-latus rectum $p=p_0$ is analyzed, where we assume $e = 0.6$ and $m_2 = 1000\, M_{\odot}$. The solid black line represents the eccentricity change due to GWs, with $\left|\left\langle\frac{de}{{\mathrm{d}}t}\right\rangle_{\text{GW}}\right|$ in Eq. \ref{['pk-rr-ef']}. The solid line corresponds to the rate of eccentricity change $\left|\left\langle\frac{{\mathrm{d}}e}{{\mathrm{d}} t}\right\rangle_{\text{AC}}\right|$ in Eq. \ref{['de_dt_ac']} for the gNFW profile in Eq. \ref{['eq::gnfw_halo']}, and the dashed line represents the case for a DM spike in Eq. \ref{['p-spike_1']}.
• Figure 5: This figure presents the temporal evolution of eccentricity and semi-latus rectum $p$ for BBH systems embedded in a gNFW profile of DM distribution in Eq. \ref{['eq::gnfw_halo']} with slope $\gamma = \{0.5, 1, 1.5, 2\}$. The initial eccentricities $e_0$ are $\{0.3, 0.6, 0.9\}$ for each case. The top row shows the evolution of the semi-latus rectum $p$ with time, while the bottom row displays the corresponding eccentricity evolution. In all panels, the black curves represent the reference case without DM. For the plots of semi-latus rectum $p$ (top row), the horizontal axis shows time in years (yr), and the vertical axis is normalized in units of $Gm_1/c^2$. The initial semi-latus rectum is set to $p_0=1000\,Gm_1/c^2$, where $m_1 = 4.26 \times 10^6\,M_\odot$ is the primary BH mass and the secondary BH has a mass of $m_0=1000\,M_\odot$. The evolution of semi-latus rectum $p$ and eccentricity $e$ terminates when $r < r_{\mathrm{ISCO}}$.