具有任意径向轮廓的球坍缩中非线性横向与径向速度响应的近普适性
Near universality of nonlinear transverse and radial velocity responses in spherical collapse with arbitrary radial profiles
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中文总结 AI 辅助
本研究针对球坍缩中径向非均匀轮廓的速度响应问题，证明含宇宙学常数的无压物质壳层非线性响应可由顶帽函数确定，提出等龄构造方法和三项拟合公式，实现了高精度的膨胀响应计算，分离了密度轮廓与膨胀响应的关联。
中文摘要 AI 辅助
密度与膨胀之间的非线性关系通常是针对具有单一局域哈勃率的均匀球顶帽（top-hat）模型建立的。而平滑的球轮廓沿径向方向（$H_{\\\/\\/}$）和垂直径向方向（$H_{\perp}$）的膨胀方式不同——这是径向非均匀性的直接标志。我们证明，对于含宇宙学常数的增长模式无压物质，完整的壳层响应仍由一个顶帽函数确定。给定局域密度对比度$\delta(t,r)$和包围质量对比度$\Delta(t,r)$，横向响应及其导数可确定$H_{\perp}$和$H_{\\/\\/}$。线性和二阶极限是代数映射，其唯一动力学输入是通常的增长率$f$。一种精确的等龄构造无需积分演化方程即可给出非线性响应。我们验证了该方法能以数值精度重建完整的$\Lambda$LTB轮廓。我们还提供了一种考虑导数、不依赖宇宙学模型的三项符号拟合公式，仅需$f$、$\delta$和$\Delta$。在一组具有代表性的物质-曲率-红移组合中，对于位于补偿过渡区的壳层，横向和径向响应的最大相对误差分别为$0.3\\%$和$0.7\\%$。这种简洁公式将密度轮廓的生成与其膨胀响应分离开来，并明确了径向梯度的效应。
英文摘要
The nonlinear relation between density and expansion is usually formulated for a homogeneous spherical top hat, which has a single local Hubble rate. A smooth spherical profile instead expands differently along ($H_{\parallel}$) and across ($H_{\perp}$) the radial direction-a direct signature of radial inhomogeneity. We show that, for growing-mode pressureless matter with a cosmological constant, the complete shellwise response is nevertheless fixed by one top-hat function. Given the local density contrast $δ(t,r)$ and the enclosed contrast $Δ(t,r)$, a transverse response and its derivative determine $H_{\perp}$ and $H_{\parallel}$. The linear and second-order limits are algebraic maps whose only dynamical input is the usual growth rate $f$. An exact equal-age construction supplies the nonlinear response without integrating an evolution equation. We check that it reconstructs full $Λ$LTB profiles to numerical precision. We also provide a derivative-aware, cosmology-independent three-term symbolic fit that requires only $f$, $δ$, and $Δ$. Across a representative set of matter-curvature-redshift combinations and for a shell located in the compensated transition, the maximum relative errors are $0.3\%$ and $0.7\%$ in the transverse and radial responses, respectively. This compact formulation separates the production of a density profile from its expansion response and makes the effect of radial gradients explicit.
