Black-Hole mimickers in GR and $f(R)$ gravity

TL;DR

The paper investigates horizonless ultracompact objects as black-hole mimickers, focusing on solitonic boson stars (SBS) with a sextic potential and incompressible perfect-fluid ultracompact objects (IPFUCO) in GR, then compares these to SBS and UCOs within $f(R)=R+aR^2$ gravity. It maps the SBS families in the two-parameter plane $(\tilde{\phi}_0,\sigma)$, showing that a second stable branch with light rings emerges at ultracompact configurations for small $|\sigma|$, and discusses stability debates surrounding LR-bearing SBS. IPFUCOs are shown to reach the Buchdahl limit $\mathcal{C}=4/9$ in GR and develop light rings beyond a critical central pressure, illustrating how a simple fluid model can mimic BHM features. In $R^2$ gravity, IPFUCOs do not exceed the GR Buchdahl bound; the effective density contributions tend to reduce compactness, indicating that higher $a$ does not automatically yield more compact UCOs for these configurations. Together, these results clarify the viability and limitations of SBS/IPFUCO as BH mimickers and highlight stability questions and future directions for SBS in modified gravity.

Abstract

Black hole mimickers (BHMs) are horizonless globally regular ultracompact relativistic self-gravitating objects (UCOs) of mass $M$ and radius $R$ with compactness $C = M/R$ higher than that of a neutron star and that produce an effective potential for null geodesics (photons) that possesses a local maximum, which is usually accompanied by an inner local minimum. The presence of a local maximum allows for unstable circular orbits to exist similar to light rings present in actual BH solutions, while it has been conjectured that the presence of a local minimum is symptomatic of potential instabilities. One such candidate for a BHM is a solitonic boson star (SBS) which is a boson star endowed with a sextic potential. In this paper we investigate further solutions of static and spherically symmetric SBSs in general relativity with a larger set of parameter values, and argue that such solutions are very similar to UCOs composed of an incompressible perfect fluid (IPF) with a sufficiently large pressure (the mimicker of a BHM). These IPFUCOs reach the Buchdahl limit $C= 4/9$ for arbitrarily large pressures. We investigate the extent to which the IPFUCOs constructed within a quadratic model in $f(R)$ gravity can overcome this limit or not, and thus pave the way for possibly building SBSs (or other kind of UCO) within this (or other alternative theory of gravity). We further elaborate about the stability properties of SBSs which have been the subject of some controversy recently.

Figures (16)

• Figure 1: Radial structure and global properties for a representative sequence of SBS with $\sigma = 0.05$ and different $\tilde{\phi}_0$ (indicated by colors; rows are labeled with numbers and columns with letters). Panel (1a): scalar field solution for different $\tilde{\phi}_0$; Panel (1b): effective potential $V_{\text{eff}}/L^2=e^{\nu(r)}/r^2$. Panel (1c): compactness $\mathcal{C}$ as a function of the central field amplitude $\tilde{\phi}_0$. Panels (2a) and (2b): metric potentials $e^{u(r)}$ and $e^{\nu(r)}$, respectively. Panel (2c): total mass $M(\Lambda\mu)$ as a function of $\tilde{\phi}_0$. Panels: (3a) and (3b): (rescaled) energy density $\tilde{\rho}(r)$ and radial pressure $\tilde{P}_\text{rad}(r)$. Panel (3c): total mass $M(\Lambda\mu)$ as a function of the field frequency $\omega/\mu$. The bottom row displays additional global quantities along the sequence: Panel (4a): Noether charge $Q(\Lambda\mu)^2$ versus $\tilde{\phi}_0$. Panel (4b): frequency $\omega/\mu$ as a function of $\tilde{\phi}_0$. Panel (4c): total mass $M(\Lambda\mu)$ as a function of the effective radius $R_{99}(\Lambda\mu)$. Open black circles mark the turning points of the mass curves $M(\tilde{\phi}_0)$--Panel (2c)--, which are replicated across the corresponding panels.
• Figure 2: (Left) Domain of existence of SBSs for different values of the solitonic parameter $\sigma$. (Right) Corresponding mass–radius relations $M$–$R_{99}$, together with the diagonal line marking the black–hole limit; the shaded region above this line is excluded by $\mathcal{C}>1/2$. The colour bar encodes the value of $\sigma$. Dotted lines indicate configurations with LRs.
• Figure 3: Compactness diagnostics for SBS sequences parameterized by the parameter $\sigma$. Left panel: compactness $\mathcal{C}$ as a function of the (rescaled) central scalar amplitude $\tilde{\phi}_0$ for each value of $\sigma$ (darker colors indicate smaller $\sigma$). The hexagon green markers indicate the configuration of maximum mass $M_{\max}$ along each sequence. An inset highlights the region around $\tilde{\phi}_0\simeq 1$ where several sequences backbend indicating that $\tilde{\phi}_0$ no longer uniquely determine the solutions. Right panel: Relevant compactness solutions as a function of $\sigma$: inverted triangles denote the maximum compactness $\mathcal{C}_{\max}(\sigma)$ attained along each sequence, while hexagons denote the compactness evaluated at the maximum-mass configuration, $\mathcal{C}(M_{\max})$. Horizontal lines indicate compactness reference values (Buchdahl bound $\mathcal{C}_{\rm BD}=4/9$, causal Buchdahl-bound $\mathcal{C}_{\rm CBD}= 0.354$, black-hole limit $\mathcal{C}_{\rm BH}= 1/2$, the $\Lambda$-threshold $\mathcal{C}_{\rm \Lambda}= 0.158$amaro-seoane_constraining_2010, and the mini-boson-star reference $\mathcal{C}_{\rm mini}= 0.08$kaup_klein-gordon_1968).
• Figure 4: Compactness--radius--frequency relations for SBS. (Left) Compactness $\mathcal{C}$ as a function of the effective radius $R_{99}(\Lambda\mu)$. (Middle) $\mathcal{C}$ versus the dimensionless scalar field frequency $\omega$. (Right) $R_{99}(\Lambda\mu)$ versus $\omega$. Each curve corresponds to an equilibrium sequence with fixed coupling $\sigma$, with color encoding the value of $\sigma$ as indicated by the vertical color bar. Dotted lines indicate configurations with LRs.
• Figure 5: Mass, radius and frequency of SBS equilibrium sequences as a function of the central scalar amplitude $\tilde{\phi}_0$. (Left) Mass (rescaled by $\Lambda\mu$). (Middle) Radius $R_{99}(\Lambda\mu)$ enclosing $99\%$ of the total mass. (Right) eigenfrequency $\omega/\mu$. Each curve corresponds to a fixed value of the coupling parameter $\sigma$ (indicated by the color bar). Mass and radius are plotted on logarithmic scales, whereas $\omega/\mu$ is shown on a linear scale. Note that the three panels share a sharp change near $\tilde{\phi}_0=1$. Dotted lines indicate configurations with LRs.