Classical density–velocity relations in the nonlinear regime are formulated for a homogeneous spherical 'top hat', which has a single expansion rate. But real profiles are smooth: different shells expand at different rates along and across the radius. This anisotropy is a direct signature of radial inhomogeneity. Nearby work has already constructed exact Lemaitre–Tolman–Bondi solutions and quasilocal identities, but a compact response expressed in terms of observable density contrasts was missing. This work fills that gap, relying on the idea of equal shell age.
The Lemaitre–Tolman–Bondi geometry is used for dust matter with Λ. Two contrasts are introduced: the local δ and the internal Δ (averaged over the volume inside the shell). From mass conservation and the condition of equal age of shell and background, the function A(Δ) = ΔΘ⊥(Δ) is obtained, where Θ⊥ is the nonlinear correction to the transverse response. The exact relation between the radial gradient of Δ and the difference δ−Δ (the q-scalar identity) allows expressing the radial and local rates and the anisotropy in terms of A and its derivative A′. For quick estimates, a three-term approximation M26 is proposed, calibrated on a grid of cosmologies including the derivative.
The main result: the transverse response depends only on Δ, while the radial response additionally includes a term with δ−Δ and A′. In the linear limit, the response is purely algebraic and depends only on the growth rate f. In second order, there is also no need to solve the age equation. The exact equal-age construction reproduces full ΛLTB profiles to machine precision. The M26 approximation gives maximum relative errors of 0.3% for the transverse and 0.7% for the radial response on a representative set of cosmologies and a shell in the compensated transition zone. On full profiles, absolute errors are of order 10⁻³.
This work separates the evolution of the density profile from the expansion response: knowing δ and Δ, one can immediately obtain all velocities without re-solving the dynamics. This simplifies the analysis of spherical models and allows using arbitrary profiles from simulations or phenomenological models. Unlike neighboring works that focus on the homogeneous top hat or exact solutions, here a compact response is given for an arbitrary smooth profile with explicit dependence on the gradient.
Generalization to non-spherical configurations is possible, where strain and tidal tensors will be required. Also promising is the inclusion of baryonic physics and accounting for shell crossing. For applications to data, observables on the light cone and realistic selection functions are needed.
The result is applicable in cosmological simulations and analytical models of structure formation where spherical profiles are important. It can be used to interpret measurements of peculiar velocities and to constrain cosmological parameters.
Test the M26 approximation for other dark energy equations of state (w ≠ −1). Develop a similar formalism for triaxial collapse.
In cosmology, the exact relation between density and velocity in the nonlinear regime for arbitrary geometries remains an open question. This work provides an exact solution for spherical symmetry, but the problem of virialization and shell crossing remains unsolved.
This section is addressed to the author of the paper and their colleagues. Neighbouring works were found by meaning-based search across our archive, not by word overlap. We do not review: we show the field around the work.
What the work does
The paper derives compact expressions for the transverse and radial expansion response of an arbitrary smooth spherical profile of collisionless matter with Λ, using the Lemaitre–Tolman–Bondi geometry and the condition of equal shell ages. Local and internal density contrasts are introduced, through which the expansion anisotropy is expressed, and a three-term approximation M26 is proposed that does not require numerical solution of the equations.
Where its strength is
The strength of the work lies in separating the evolution of the density profile from the expansion response: knowing only δ and Δ, one can immediately obtain all velocities without re-solving the dynamics. This opens the way to analyzing arbitrary profiles from simulations or phenomenological models.
Where it could go next
Worth trying
Why this may matter
If the proposed method can be extended to more realistic asymmetric structures or linked to observable effects, it could simplify the interpretation of peculiar velocity data and provide a new tool for testing dark matter models.
Nearby works in our archive
The archive is sparse around this work: the closest review is only 0.54, against a usual 0.64. It looks like few have drilled nearby.
For a shell in a void wall, the local contrast can be positive while the internal contrast is negative: the shell is denser than average, but inside it there is still a mass deficit. This difference drives the expansion anisotropy.