Anisotropic Dark Matter Bosonic Stars in regularized 4D Einstein$-$Gauss$-$Bonnet gravity

TL;DR

The paper addresses how anisotropy and regularized 4D Einstein–Gauss–Bonnet gravity affect equilibrium and stability of bosonic dark-matter stars. It derives the modified TOV equations for a self-interacting complex scalar field with a polytropic EoS and an anisotropy profile, then solves the system numerically over ranges $α$ and $β$. Key findings show that larger $α$ increases the maximum mass and compactness, while more negative anisotropy $β$ reduces these values, with all configurations remaining statically stable and causal and satisfying energy conditions. These results offer observationally testable predictions for dark-matter compact objects in modified gravity and set the stage for extensions to rotation or alternative gravity theories.

Abstract

In this work, we have constructed anisotropic bosonic dark-matter star (DMS) solutions in the context of a regularized four-dimensional Einstein$-$Gauss$-$Bonnet (4D EGB) gravity theory. Using dimensional regularization, we solve modified Tolman$-$Oppenheimer$-$Volkoff equations for a self-interacting complex scalar field in the dilute polytropic regime, $p_r = K ρ^2$, with anisotropy parameterized as $σ= β\, p_r \left( 1 - e^{-2λ} \right)$. We perform a comprehensive numerical analysis across the \((α,β)\) parameter domain, where \(α\in [0,8]~\mathrm{km}^2\) and \(β\in [-2,0]\), to examine mass$-$radius relations and evaluate multiple stability indicators including static equilibrium \(dM/dp_c\), sound-speed causality, the radial adiabatic index \(Γ_r\), and energy conditions. Positive Gauss$-$Bonnet coupling enhances both the maximum mass and compactness (e.g., \(M_{\rm max} \approx 1.62\, M_\odot\) at \(α=0\) rising to \(\approx 2.09\, M_\odot\) at \(α= 8~\mathrm{km}^2\)), while negative anisotropy reduces them (e.g., from \(\approx 2.21\, M_\odot\) at \(β=0\) to \(\approx 1.73\, M_\odot\) at \(β= -2\)). The resulting configurations remain statically stable up to the mass peak and satisfy physical criteria. This work extends previous isotropic boson-star analyses by systematically incorporating anisotropy within a regularized 4D EGB framework. These findings provide observationally relevant predictions for compact dark-matter objects under modified gravity.

Figures (9)

• Figure 1: From top to bottom, we present the energy density $\rho$, radial pressure $P_r$, and transverse pressure $P_\perp$ as functions of the radial coordinate $r$. The range of values for $\alpha \in [0,8]\ \mathrm{km}^2$, while the other parameters are fixed as $B = 205~\mathrm{MeV/fm^3}$, $\beta = -0.5$, and $z = 0.05$. A black dashed line represents the anisotropic solution of Einstein's gravity.
• Figure 2: The effect of the coupling constant on the mass-radius relation and the maximum compactness of dark matter compact stars is investigated. These results are obtained using the parameter values employed in Fig. \ref{['fig1']}. The general relativity (anisotropic solution of Einstein's gravity) result is also indicated by a black dashed line.
• Figure 3: From top to bottom, we present the energy density $\rho$, radial pressure $p_r$, and transverse pressure $p_t$ as functions of the radial coordinate $r$. The range of values for $\beta \in [-2,0]$, while the other parameters are fixed as $B = 205~\mathrm{MeV/fm^3}$, $\alpha=8km^2$, and $z = 0.05$. A black dashed line represents the isotropic solution of 4D EGB gravity (weak-field limit).
• Figure 4: The effect of the anisotropy parameter $\beta$ on the mass-radius and maximum compactness relations of dark matter compact stars is examined. The results are obtained using the parameter values presented in Table \ref{['t2']}. The isotropic case (4D EGB gravity) is illustrated by a black dashed line.
• Figure 5: The $M-p_{c}$ curves for a family of anisotropic dark matter stars with variations in $\alpha$ and $\beta$ are presented.