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

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, $p_r = K \rho^2$, with anisotropy parameterized as $\sigma = \beta\, p_r \left( 1 - e^{-2\lambda} \right)$. We perform a comprehensive numerical analysis across the \((\alpha,\beta)\) parameter domain, where \(\alpha \in [0,8]~\mathrm{km}^2\) and \(\beta \in [-2,0]\), to examine mass$-$radius relations and evaluate multiple stability indicators including static equilibrium \(dM/dp_c\), sound-speed causality, the radial adiabatic index \(\Gamma_r\), 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 \(\alpha=0\) rising to \(\approx 2.09\, M_\odot\) at \(\alpha = 8~\mathrm{km}^2\)), while negative anisotropy reduces them (e.g., from \(\approx 2.21\, M_\odot\) at \(\beta=0\) to \(\approx 1.73\, M_\odot\) at \(\beta = -2\)). 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.