Near universality of nonlinear transverse and radial velocity responses in spherical collapse with arbitrary radial profiles
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 $\delta(t,r)$ and the enclosed contrast $\Delta(t,r)$, 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 $\Lambda$LTB profiles to numerical precision. We also provide a derivative-aware, cosmology-independent three-term symbolic fit that requires only $f$, $\delta$, and $\Delta$. 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.