Constraint on massive vector field with extreme-mass-ratio inspirals around a slowly rotating black hole

Abstract

We study the influence of a massive vector (Proca) field on the energy fluxes from extreme-mass-ratio inspirals (EMRIs) around a slowly rotating Kerr black hole. The secondary compact object, carrying a Proca hair, emits additional dipolar radiation that alters total energy flux relative to general relativity (GR). These modifications induce a secular drift in the orbital evolution of circular geodesic orbits, leading to measurable dephasing in the resulting EMRIs waveforms. By evaluating waveform mismatches between the Einstein-Proca framework and its GR counterpart, we show that the Laser Interferometer Space Antenna (LISA) can distinguish the signatures of a light Proca field when black hole rotation is included. Furthermore, using a Fisher information matrix analysis, we forecast LISA's capability to place stringent constraints on the Proca mass with EMRIs signal from slowly rotating Kerr black holes. For representative EMRIs configurations, we find that LISA can detect or constrain Proca masses down to $μ_v\sim 10^{-20}$eV, with typical fractional uncertainties at the level of tens percent, depending on the black-hole spin.

Figures (7)

• Figure 1: Interpolation errors of the gravitational and Proca energy fluxes, obtained by summing over multipole indices $(l,m)$ off a rectangular grid as functions of the Proca mass and MBH spin $(\mu,a)$, assuming a fixed vector charge $q=0.1$. The symbols $\dot{E}^{\mathcal{G}}_{\rm error}$ and $\dot{E}^{\mathcal{P}}_{\rm error}$ represent the discrepancies between the fluxes computed via the interpolation method and those obtained from perturbation theory. All fluxes are expressed in units of the mass ratio $m_p^2/M^2$. Black dashed contours indicate lines of constant flux error between the two methods.
• Figure 2: The Proca flux and the ratio between the gravitational and Proca fluxes as a function of the orbital radius $r_p/M$ are plotted for different values of MBH spin and Proca mass, which consider the cases of different spinning Kerr BH with $a\in \{0.0,0.0001,0.001,0.005,0.01,0.05,0.1,0.2\}$ in the left panels, and different Proca fields with a mass $\mu \in \{0.0,0.0001,0.0005,0.001,0.005,0.01,0.02\}$ and a fixed charge $q=0.1$ in the right panels. The subfigures are the zoom magnified pictures in the ranges $r_p/M\in[r_{\rm ISCO},12]$ of orbital radius. Note that these fluxes have the units of $m_p^2/M^2$.
• Figure 3: Comparison of the plus polarization $h_{+}(t)$ of EMRIs waveforms in the cases of GR and Einstein-Proca family with vector field masses $\mu = \{0,\,10^{-5},\,10^{-4},\,10^{-2}\}$ and fixed charge $q = 0.1$. The time-domain signals display three phases. Increasing $\mu$ leads to phase shifts and amplitude modulations relative to the GR case, highlighting the impact of the massive vector field on the waveform phase.
• Figure 4: The mismatches between the massless vector field or standard GR cases and the Proca field cases are plotted, where the parameters of the vector fields carried by the smaller objects are set as $( \{ \mu=0,q=0\}, \{\mu\neq0,q\neq0 \} )$ and $( \{ \mu=0,q=0.1\}, \{\mu\neq0,q=0.1 \} )$, and the other parameters are set as $\{r_p=12M, m_p=10 {\rm M}_\odot, M=10^6 {\rm M}_\odot\}$. Note that the horizontal black lines denote to the threshold value of mismatch for the top and bottom panels.
• Figure 5: Corner plot of the posterior probability distributions for EMRI source parameters: the component masses, MBH spin, initial orbital radius, initial orbital azimuthal phase, vector mass and charge are assumed as follows $(M = 10^6 M_\odot,\, m_p = 10 M_\odot,\, a = 0.001,\, r_p = 12M,\, \Phi_{\phi,0}=1.0,\,\mu = 0.02,\, q = 0.05)$. Other parameters are fixed as described in Sec. \ref{['result']}. The distributions are inferred from a two-year LISA observation. Vertical dashed lines indicate the $1\sigma$ credible intervals for each parameter, while the contours represent the 68%, 95%, and 99% confidence levels.