AI 中文总结
该研究在曲率引力族中构建正则模糊暗物质致密天体，推导场方程并分析其视界结构，对比不同暗物质密度分布的结果，揭示两种暗物质模型的物理差异。
AI 中文摘要
我们在单参数曲率引力族 $f(R)=R+\eta R^n$ 中，构建完全由自引力暗物质流体支撑的正则、允许视界的致密天体。采用Einasto密度分布，我们推导了静态球对称度规的各向异性流体场方程，并在类德西特型物态方程下得到了精确的广义相对论极限，其中中心奇点被正则德西特核取代，解要么是无视界的液滴，要么是具有1或2个Killing视界的黑洞，具体取决于单一的约化质量参数。我们计算了所得的霍金温度和测地线有效势，并利用 $\eta=0$ 场方程的线性性，对一般n的曲率修正进行了闭式微扰求解，通过在固定β下直接比较n=2（Starobinsky引力）和n=3（立方引力），表明其符号和径向形状确实依赖于模型。我们针对非局域物态方程重复了该构建，并通过Kretschmann标量证实所得液滴是曲率正则的。最后，将Einasto分布替换为核心型Burkert分布保留了定性的正则机制，但由于Burkert晕缺乏有限总质量，产生的视界结构的内外半径相差近三个数量级——这是两种相当现实的暗物质模型之间的真正物理区别。
英文摘要
We construct regular, horizon-admitting compact objects supported entirely by a self-gravitating dark-matter fluid, in the one-parameter curvature-gravity family $f(R)=R+βR^n$. Working with the Einasto density profile, we derive the anisotropic fluid field equations for a static, spherically symmetric metric and obtain the exact General Relativity limit under a de~Sitter-type equation of state, in which the central singularity is replaced by a regular de~Sitter core and the solution is either a horizonless droplet or a black hole with one or two Killing horizons, depending on a single rescaled-mass parameter. We compute the resulting Hawking temperature and geodesic effective potential, and -- exploiting the linearity of the $β=0$ field equation -- solve the curvature correction perturbatively in closed form for general $n$, showing that its sign and radial shape are genuinely model-dependent by direct comparison at fixed $β$ between $n=2$ (Starobinsky) and $n=3$ (cubic) gravity. We repeat the construction for a non-local equation of state and confirm the resulting droplets are curvature-regular via the Kretschmann scalar. Finally, replacing the Einasto profile with the cored Burkert profile preserves the qualitative regularity mechanism but, because the Burkert halo lacks a finite total mass, produces a horizon structure with inner and outer radii separated by nearly three orders of magnitude -- a genuine physical distinction between two comparably realistic dark-matter models.
Comments14 pages
