Gravitational waves of quasi-circular, inspiraling black hole binaries in an ultralight vector dark-matter environment

TL;DR

This work analyzes how an ultralight vector dark-matter environment perturbs the gravitational-wave signal from quasi-circular black-hole binaries. By modeling the vector field as a Proca field in coherent patches, the authors derive a conservative, time-varying perturbation to the binary’s binding energy that induces a dephasing in the GW phase, calculated through post-Newtonian methods and the stationary-phase approximation. A Fisher analysis for a four-year LISA mission shows that, for vector masses in $m_A\in(10^{-19},10^{-16})$ eV, the GW imprint is detectable if the local dark-matter density satisfies $\rho_A \gtrsim 10^{14}-10^{15}\,M_\odot\,\text{pc}^{-3}$ (or $\gtrsim 10^{16}$ GeV cm$^{-3}$), with the strongest constraints for asymmetric, lower-mass binaries that sweep through the resonance during observation. These findings indicate that future space-based GW observations could probe or constrain local vector-type ultralight dark matter in binary environments, providing a novel astrophysical handle on the properties and distribution of such DM candidates.

Abstract

The gravitational waves emitted by massive black hole binaries can be affected by a variety of environmental effects, which, if detected, could inform astrophysics and cosmology. We here study how gravitational waves emitted by black holes in quasi-circular orbits are affected by the presence of an ultra-light, vector-field, dark-matter environment that is minimally coupled to the binary. This dark-matter environment induces oscillatory gravitational potentials that perturb the orbit of the binary, leaving an imprint in the binary's binding energy, and thus, on the gravitational waves emitted. We here compute the effect of this environment on the gravitational-wave phase using the stationary-phase approximation within the post-Newtonian formalism. We then perform a Fisher analysis to estimate the detectability of this environmental effect with a four-year LISA observation, focusing on vector fields with ultra-light masses in the $(10^{-19}, 10^{-16}) \; \rm{eV}$ range. We conclude that the observation of such gravitational waves with space-borne interferometers, like LISA, could yield a measurement or constraint on local, vector dark-matter environments, provided the dark-matter density is larger than roughly $10^{14} \rm{M}_\odot/{\rm{pc}}^3$.

Figures (6)

• Figure 1: Cartoon illustration of the dark-matter environment effect on the orbit (left) and on the gravitational waves (right) due to an ultra-light vector field (green arrows).
• Figure 2: Diagram of the effective one-body problem. The origin of coordinates is placed at the center of mass of the binary, and the position of the reduced mass is described by the vector $\vec{r} = b\left[\cos(\theta)\, \hat{x} + \sin(\theta)\, \hat{y}\right]$. The green arrow points in the direction of the vector field, which is defined with the angles $\varphi$ and $\alpha$.
• Figure 3: Dephasing of the gravitational wave calculated as in Eq. (\ref{['psi_GR_integral_bis']}) under the stationary-phase approximation (black dashed line) and numerically integrated (red dots), for $m = 10^{-18}\rm{eV}$, $M=10^5 \rm{\rm{M}_{\odot}}$, $\eta=0.25$, $\varphi = 0$ and $\alpha = \pi/4$. The vertical dash-dot gray line corresponds to the stationary point $\pi f_{\rm{stp}} = \frac{8}{5} m_A$. For $f>f_{\rm{stp}}$ the integral averages to zero. Observe that the stationary-phase approximation is very accurate, with percentage errors $\ll 1\%$ in the range shown.
• Figure 4: LISA's sensitivity strain (black curve). The different colors indicate the gravitational wave characteristic strain for each binary mass during the inspiral, defined as $|h_c (f)|= 2 f |\mathcal{A}(f)|$. We consider symmetric-mass binaries at $D_L = 1 \rm{Gpc}$. The mass of each member of the binary is the one indicated in the label. The dashed lines indicate $f_{\rm{min}}$ for each system, calculated as the frequency four years before the ISCO is reached (corresponding to LISA's observation period). On the top axes we show $\omega_{GW} \equiv 2 \pi f$.
• Figure 5: $1\sigma$ accuracy (estimated with a Fisher analysis) to which $\rho_A$ can be measured, given a value of the vector mass in eV, for a binary with various total masses (shown with different colors). The vertical dash-dot lines indicates $f_{\min}$, corresponding to the frequency of the system four year before the ISCO. We assume $\eta = 0.25$ for the analysis. If a binary system is immersed in a dark-matter vector environment with a density larger than these values (shaded regions), then the gravitational waves emitted carry a dark-matter signature that is detectable with LISA for a 4-year observation.