On the Bondi accretion of a self-interacting complex scalar field

TL;DR

This work analyzes the relativistic Bondi accretion of a self-interacting complex scalar field with a global U(1) symmetry onto a Schwarzschild black hole in the test-field limit, and compares it to the corresponding $P(X)$ effective theory. The authors develop a gradient-expansion EFT that reduces to a $P(X)$ theory at leading order, then systematically incorporate gradient corrections and solve the full modulus profile equation to determine the accretion rate. They find that finite-gradient effects systematically lower the accretion rate compared with the pure $P(X)$ (perfect-fluid) case, with the suppression growing as the gradient parameter $\xi^2$ decreases and as the mass parameter $m^2$ becomes more tachyonic; in the limit $\xi^2\to\infty$ one recovers the $P(X)$ result and the accretion rate, $\dot{M} = 4\pi r_S^2 (\dot{\varphi}_0^4/\lambda) \beta^2$, is set by the horizon flux $\beta$. The analysis also reveals how gradient corrections modify the stress-energy content, producing heat flux and anisotropic stress that vanish in the $P(X)$ limit, and alter the equation-of-state parameter $w = p/\epsilon$ near the horizon. Together, these results show that black-hole accretion can, in principle, distinguish a UV-complete complex scalar from its $P(X)$ EFT, with implications for dark matter and dark-energy models where such scalars play a role, especially for small black holes or subdominant scalar components where finite-gradient effects are enhanced.

Abstract

Scalar fields with a global U(1) symmetry often appear in cosmology and astrophysics. We study the spherically-symmetric, stationary accretion of such a classical field onto a Schwarzschild black hole in the test-field approximation. Thus, we consider the relativistic Bondi accretion beyond a simplified perfect-fluid setup. We focus on the complex scalar field with canonical kinetic term and with a generic quartic potential which either preserves the U(1) symmetry or exhibits spontaneous symmetry breaking. It is well known that in the lowest order in gradient expansion the dynamics of such a scalar field is well approximated by a perfect superfluid; we demonstrate that going beyond this approximation systematically reduces the accretion rate with respect to the perfect fluid case. Hence, black holes can provide a way to distinguish a perfect fluid from its ultraviolet completion in form of the complex scalar field.

Figures (24)

• Figure 1: Effective potential landscape with red curves solving the algebraic $P(X)$ profile equation (\ref{['P(X) profile equation']}): left: sub-critical case where curves connect boundaries, but not the correct boundary points; middle: critical case where two curves touch tangentially connecting the correct boundary conditions; right: super-critical case where curves do not connect two boundaries.
• Figure 2: Phase diagrams of solutions of the $P(X)$ profile equation (\ref{['P(X) profile equation']}) for four different choices of $\mu^2$. The finite boundary conditions (\ref{['boundary conditions']}) are connected only in special cases when the solution curves intersect at a singular point. This happens for special values of $\beta\!=\!\beta_c$, shown in black. Curves for $\beta<\beta_{c}$ are in blue, and curves for $\beta>\beta_{c}$ are in red.
• Figure 3: Dependence of $f_{c}$ and $\beta_{c}^{2}$ on the mass parameter $\mu^{2}$ in the range $-30 \leq \mu^{2} \leq 1$.
• Figure 4: Dimensionless modulus field profile $\sigma_{0}(f)$(left) and its $f$-derivative (right) for the $P(X)$ model, plotted for different values of the mass parameter $\mu^2$, including tachyonic values. Bullet points on the curves represent the location of the critical point, which coincides with the acoustic horizon Frolov:2004vm.
• Figure 5: Bounds on the applicability of the perturbative approach. Left: Solid curve depicts the relative correction to the dimensionless flux (\ref{['beta_1']}) in units of $\xi^2$, as a function of $\mu^2$. For values $\xi^2$ greater than this relative correction (shaded region) the perturbative expansion in (\ref{['pert expansion']}) is justified, while for the values of $\xi^2$ lower than this relative correction the perturbative expansion is not justified (white region). Right: Relative corrections to the modulus field profile to leading order in $1/\xi^{2}$, in units of $1/\xi^2$. The values of the curves at $f\!=\!0$ correspond to $\beta_1/\beta_0$.