Researchers have uncovered a near-universal relationship in the transverse and radial velocity responses during the spherical collapse of pressureless matter. Traditionally, the nonlinear relation between density and expansion is formulated for a homogeneous spherical top-hat model, which assumes a single local Hubble rate. However, a smooth spherical profile with a density distribution expands differently along ($H_{\parallel}$) and across ($H_{\perp}$) the radial direction—a direct signature of radial inhomogeneity. This new work demonstrates that, for growing-mode pressureless matter with a cosmological constant, the complete shellwise response is nevertheless fixed by one top-hat function.

The study reveals that, given the local density contrast $\delta(t,r)$ and the enclosed contrast $\Delta(t,r)$, the transverse response and its derivative determine both $H_{\perp}$ and $H_{\parallel}$. The linear and second-order limits behave as algebraic maps whose only dynamical input is the usual growth rate $f$. An exact equal-age construction method provides the nonlinear response without the need to integrate an evolution equation. The authors verified that this method reconstructs full $\Lambda$LTB (Lemaître-Tolman-Bondi with cosmological constant) profiles to numerical precision.

Additionally, a derivative-aware, cosmology-independent three-term symbolic fit has been developed, requiring only $f$, $\delta$, and $\Delta$. This fit shows maximum relative errors of 0.3% and 0.7% in the transverse and radial responses, respectively, across a representative set of matter-curvature-redshift combinations and for a shell located in the compensated transition. This compact formulation separates the production of a density profile from its expansion response, making the effect of radial gradients explicit. The finding is crucial for understanding large-scale structure formation in the universe and for refining cosmological models.