Boson Stars Hosting Black Holes

TL;DR

This work studies boson stars formed from ultralight dark matter (ULDM) hosting a central black hole (BH) by solving the non-relativistic Gross-Pitaevskii-Poisson system, taking into account BH-induced boundary conditions. It reveals that a central BH enhances the boson-star core density while shrinking its size, with stability dependent on the sign of the self-interaction $\lambda$; attractive interactions introduce bounds on the maximum BS mass that depend on the BH mass. The authors introduce a Gaussian analytic ansatz to capture the BH effect and derive a mass–radius relation that agrees with full numerics in key regimes. They also assess gravitational-wave dephasing from an inspiraling secondary BH, showing ULDM environmental effects can leave measurable imprints in LISA observations and enabling forecasts of constraints on ULDM mass $m$ and self-coupling $\lambda$ via Fisher analysis.

Abstract

We study a system of a self-gravitating condensate, a boson star, formed from scalar ultra-light dark matter (ULDM), with a black hole hosted at its center. We numerically solve the equations of hydrostatic equilibrium in the non-relativistic limit, consistently incorporating the gravitational potential of the black hole, to obtain all possible configurations of this BS-BH system for different boson star masses, interaction types, and black hole masses. We also propose an analytic expression for the density profile and compare it with the numerical results, finding good agreement for attractive interactions and for a finite range of mass ratios between the black hole and boson star. Finally, considering the inspiral of this BS-BH system with a second, smaller black hole, we study the dephasing of gravitational waves due to the presence of the ULDM environment. A Fisher matrix analysis reveals the regions of parameter space of the ULDM mass and self-coupling that future gravitational-wave observatories such as LISA can probe.

Figures (12)

• Figure 1: Examples of the solutions to the differential equation \ref{['eq:red_hydro2']}, where $y$ is the scaled radius. The quantity $\bar{\chi}$ corresponds to the scaled interaction strength defined from \ref{['eq:chi-bar']}. The non-zero value of $f_1$ also evidently modifies the slope of the profiles. All of these colored lines meet at $y = 0$ as they share the same boundary conditions: $f(0) = 1$ and $f'(0) = f_1$.
• Figure 2: Examples of density profiles with fixed DM mass $m =5\times10^{-17}$ eV. The dotted line indicates a gravitating boson star $(\lambda = 0)$ with no central black hole, of size $\approx 2 \times 10^{-5}$ pc, resulting in a boson star of mass $\sim 1.5\times 10^5\,M_{\odot}$. The dashed and solid lines indicate that the boson star hosts a central black hole of mass $M_{\rm BH} = 10^5\,M_{\odot}$, with the boson star mass given by $M = M_{\rm BH}/\kappa$. Red and blue lines indicate attractive and repulsive interactions, respectively. The central black hole reduces the size of the boson star and increases the density of the boson star. Repulsive interactions $(\lambda >0)$ reduce the density, whereas attractive interactions $(\lambda < 0)$ enhance it. Inset: The same density profiles but in log-log scale.
• Figure 3: The central density (top) normalized to $\rho_a\equiv Gm^4/|a|^2$ and radius containing $99\%$ of the mass (bottom) of boson stars hosting central black holes, normalized to $R_a \equiv \sqrt{|a|/(Gm^3)}$. We show scenarios with attractive (left column) and repulsive (right column) interactions in the plane of the mass of the boson star, normalized by $M_a \equiv (Gm|a|)^{-1/2}$ and the ratio $\kappa$, for fixed magnitude of the interaction strength. For attractive-type interactions $(a < 0)$, there exists a maximum allowed mass $M_{\rm max}$, for fixed $\kappa$, beyond which the system becomes unstable. No such restriction arises for repulsive-type interactions, implying the full parameter space is allowed.
• Figure 4: Allowed parameter space in the plane of the scattering length, normalized to $a_M \equiv (GM^2m)^{-1}$, for a boson star with self-couplings, of fixed mass $M$, hosting a central black hole. There exists a minimum scattering length $a_{\rm min}$, below which the solutions are unstable. Increasing the black hole mass increases this $a_{\rm min}$. Top: The central density normalized to units of $\rho_M \equiv G^3 M^4m^6$. Bottom: The radius containing $99\%$ of the boson star mass, normalized to units of $R_M\equiv (GM\,m^2)^{-1}$.
• Figure 5: The potential in \ref{['eq:potential']} as a function of the radius $R$ with all quantities entering rendered dimensionless through appropriate normalizations, $E_a \equiv Gm/R_a^2$ and $R_a \equiv \sqrt{|a|/(Gm^3)}$. Top: For repulsive interactions, the potential is always bounded from below, with a single minimum, corresponding to a single stable configuration. Increasing $\kappa$ shifts the location of the minimum, but the system remains stable. Bottom: For attractive interactions, without the central black hole, there exists a maximum mass $M_{\rm max}$ above which $V(R)$ is unbounded (solid red line), see main text for details. Below $M_{\rm max}$ (dotted black line), the potential exhibits a local maximum (corresponding to a smaller $R$) and a local minimum (corresponding to a larger $R$), which are associated with an unstable and a stable configuration, respectively. Introducing a black hole lowers the $M_{\rm max}$, and if $\kappa$ is large enough, i.e., for a large $M_{\rm BH}$, the potential is again unbounded.