Post-Collapse Perturbation Theory

• PCPT is an analytical framework that extends standard perturbation methods beyond shell crossing to capture the nonlinear dynamics of multi-stream gravitational collapse.
• It employs low-order polynomial expansions and adaptive smoothing to model S-shaped phase-space spirals, power-law density profiles, and nearly stationary halos.
• Recent extensions generalize PCPT to cosmological contexts and three-dimensional dynamics, linking microscopic collapse physics with macroscopic observables.

Post-Collapse Perturbation Theory (PCPT) is a family of analytical and semi-analytical frameworks that extend standard perturbation theory beyond the regime of shell crossing in collisionless gravitational systems. PCPT enables a controlled, fine-grained description of the nonlinear dynamics of cold matter after regions of the phase-space sheet have folded, entering the multi-stream regime. Originally developed in one dimension for Vlasov–Poisson systems, PCPT constructs perturbative corrections around the singularities and caustics that develop after collapse, accurately describing features such as S-shaped phase-space spirals, power-law density profiles, and the emergence of nearly stationary halos. The approach has subsequently been generalized to cosmological contexts and, recently, to fully three-dimensional dynamics.

1. Conceptual Foundation and Motivation

PCPT addresses a critical shortcoming of Standard Perturbation Theory (SPT) and Lagrangian Perturbation Theory (LPT): the breakdown of their validity at shell crossing, where the mapping from Lagrangian to Eulerian coordinates becomes multi-valued and the formal density diverges due to caustic formation. SPT and LPT describe only the single-stream (pre-collapse) evolution; however, gravitational dynamics inherently produce shell crossings and multi-stream flows—key for the emergence of halos, filaments, and other cosmic structures.

In the canonical PCPT framework, the post-collapse dynamics of the central region—now a tightly wound, S-shaped spiral in phase space—are modeled analytically using low-order polynomial expansions in the Lagrangian coordinate, supplemented by iterative corrections arising from self-gravity and the influence of the newly formed halo (Colombi, 2014). This allows one to propagate the solution through successive crossing events, each time updating the parameters that encode the coarse-grained shape and velocity profile of the system. PCPT thereby forms an essential link between microscopic, fine-grained phase-space dynamics and macroscopic observables such as the density profile, mass loss into the halo, and the emergence of universal behaviors.

2. PCPT in One-Dimensional Self-Gravitating Systems

The foundational formulation of PCPT was derived for 1D cold, self-gravitating Vlasov–Poisson systems (Colombi, 2014). Before shell crossing, the system's evolution is exactly described by the Zel’dovich approximation:

$$x(q, t) = q - (q - q^3)t^2, \quad v(q, t) = -2(q - q^3)t$$

At the first crossing time, the central region's phase-space curve is well-approximated by a cubic polynomial (the "S–shape" approximation):

$$x(q, t_c) = a_n\,q^3, \qquad v(q, t_c) = -b_n q + c_n q^3$$

where $a_n$, $b_n$, $c_n$ encode the spiral's shape at the nth crossing. To advance solutions through the post-collapse regime, PCPT posits that the S-shape is embedded in a nearly stationary, harmonic halo with background density $\rho_b$. The unperturbed motion then satisfies

$$\frac{\partial^2 x}{\partial t^2} = -\omega^2 x, \quad \omega = \sqrt{2 \rho_b}$$

$$x_b(q, h) = a q^3 \cos(\omega h) + \frac{-b q + c q^3}{\omega} \sin(\omega h)$$

$$v_b(q, h) = (-b q + c q^3) \cos(\omega h) - a \omega q^3 \sin(\omega h)$$

The self-gravity of the S-shape induces corrections $g(q, h)$ that are treated perturbatively:

$$x(q, h) = x_b(q, h) + g(q, h), \qquad v(q, h) = v_b(q, h) + \frac{\partial g}{\partial h}$$

Crucially, the force on each fluid element, including during multi-valued flow, is given as a function of the enclosed mass, with the correct treatment requiring attention to how multiple Lagrangian points map to the same Eulerian location.

Iterative updating of $(a_n, b_n, c_n)$ from one crossing to the next, including corrections for mass transfer into the halo and the evolving background potential, leads to semi-analytical recurrence relations. This approach quantitatively predicts features such as a power-law density profile $M_\mathrm{int}(x) \propto x^{1/2}$ and the approach to universal behavior observed in simulations. At late times, nontrivial evolution of the recurrence (e.g., sign changes in $a_n$) can indicate flattening of the central core, though such results are speculative beyond a few crossing times.

3. Extensions: Cosmological Context and Adaptive Smoothing

PCPT has been extended to describe large-scale structure formation in 1D cosmologies, emphasizing the necessity to accurately recover multi-stream regions after shell crossing (Taruya et al., 2017). Here the displacement field is locally expanded around Lagrangian densities that first collapse, using a third-order Taylor expansion in $\Delta q$ about the critical point. Corrections to the Zel’dovich solution are then computed by integrating the gravitational force induced by the multistream structure.

To generalize further and maintain accuracy in cosmological settings with merging halos and overlapping multi-stream regions, adaptive smoothing is employed. This procedure selects the smoothing scale of the input density field based on dynamical timescales—specifically, the "next-crossing time"—ensuring that the multi-stream approximation remains accurate and localized. This adaptive coarse-graining is essential for reproducing the correct power spectrum and phase-space structure deep into the nonlinear and stable clustering regime, as confirmed by comparisons with high-resolution N-body simulations.

4. Transfer Function Expansions, EFT, and Systematic Corrections

The necessity of going beyond SPT/LPT and consistently capturing post-collapse physics has motivated the incorporation of effective transfer function corrections and the application of effective field theory (EFT) language (Vlah et al., 2015). The nonlinear matter power spectrum is recast via a multiplicative transfer function $T(k)$ that absorbs the post-collapse ignorance:

$$P_\mathrm{dm}(k) = P_\mathrm{PT}(k)\left[1 + \alpha_1 k^2 + \alpha_2 k^4 + \cdots\right]$$

where $\alpha_i$ are free parameters mapped in EFT to the effects of shell crossing and multi-streaming. Explicitly, in 1D, these expansions are equivalent at leading order to the EFT counterterms and extend the convergence of perturbation expansions with minimal "dressing" of the baseline PT result.

Accurate modeling of nonlinear oscillatory structure, such as baryon acoustic oscillations (BAO), is achieved only by including such effective corrections, as both LPT and SPT are blind to the small-scale halo formation that damps these features. The transfer function is observed to be nearly universal (low-order expansions suffice across a range of cosmological models), but limitations remain as stochastic contributions and parameter "running" become relevant at higher $k$ or in fully three-dimensional settings.

5. Divergence, Nonperturbative Contributions, and the Limitation of Ordinary Perturbation Theory

Fundamental constraints on the reach of pre-collapse PT frameworks arise from their failure to recover correct statistics for real-space observables after shell crossing (Pajer et al., 2017). While the SPT expansion for the power spectrum $P(k)$ converges in Fourier space, the real-space two-point correlation function and count-in-cell cumulants become dominated by nonperturbative, exponentially suppressed contributions from rare, high-density events (collapsed or multi-stream regions). These effects cannot be recovered to any finite order in perturbative expansions and motivate the development of PCPT as the only analytic means to correctly interpolate between the smooth linear regime and the highly nonlinear, collapsed structures.

Such nonperturbative errors scale as $\exp[-(r/\sigma(r))^2]$ and are intrinsic to the physical process of collapse, independent of ultraviolet or infrared regularization. Any statistical averaging over the nonlinear field, especially in real space, integrates over these essential singularities, resulting in divergence of the perturbative series and requiring resummation or direct analytic modeling via PCPT.

6. Three-Dimensional Generalization and Pancake Caustics

Recent advances have enabled a systematic extension of PCPT to genuinely three-dimensional gravitational collapse (Saga et al., 23 Sep 2025). The formalism employs high-order LPT to evolve the background displacement field up to the first shell-crossing event. Near the collapse, the phase-space sheet folds pancaking along one axis, and a second-order Taylor expansion in the transverse (and cubic in the collapse) direction is performed:

$$x(\tau, q) = -B_x(\tau) q_x + [C_y(\tau) q_y^2 + C_z(\tau) q_z^2] q_x + C_x(\tau) q_x^3$$

$$y(\tau, q) = B_y(\tau) q_y, \quad z(\tau, q) = B_z(\tau) q_z$$

The caustic surface is then determined by setting $\partial x/\partial q_x = 0$, yielding an ellipsoidal boundary for the multi-streaming region in Lagrangian space. Within this thin pancake, the three-dimensional Poisson equation reduces to a one-dimensional form along $x$, allowing the use of established 1D post-collapse results for the gravitational backreaction.

This perturbative correction is then iteratively computed, updating both the background and the emergent halo's effect, ultimately producing explicit corrections to displacement and velocity fields in 3D after shell crossing. The predictions have been validated against high-resolution Vlasov-Poisson simulations, accurately reproducing thin pancakes, caustic evolution, and early nonlinear post-collapse dynamics. This marks an important step in semianalytical modeling of the transition from cosmic sheets to higher-order structures.

7. Cosmological Implications and Limitations

PCPT not only furnishes a fine-grained, physically grounded description of violent relaxation in cold collisionless systems but also bridges the gap to statistical cosmological observables. By providing explicit mappings between Lagrangian space singularities and observed (Eulerian) density profiles, PCPT yields the analytical origin of universal power-law scalings, the formation of pseudo-stationary halos, and, in principle, the development of cored or cusped density centers.

However, PCPT is intrinsically local in time and space—it is most accurate in the immediate aftermath of collapse and where phase-space folding is well captured by low-order expansions; at late times, when mixing, mergers, and virialization are important, the approach becomes less predictive. Adaptive smoothing and transfer function/EFT corrections remain essential to coherently join PCPT patches across varying dynamical time and mass scales, particularly in hierarchical cosmological models.

In sum, PCPT constitutes a critical analytical framework connecting the microphysics of post-shell-crossing evolution to macroscopic structure formation—a necessary advance beyond standard approximations, with broad ramifications for cosmological theory and interpretation of precision large-scale structure data.