Memory of Initial Conditions in Gravitational Clustering

TL;DR

The paper reframes gravitational clustering by treating the nonlinear propagator as a measure of how strongly density and velocity fields remember their initial conditions. It develops a renormalized perturbation theory (RPT) framework to sum nonlinear contributions, yielding a nearly Gaussian decay of the propagator with a characteristic memory scale that defines the breakdown of linear theory. The authors derive analytic expressions for the one-loop propagator, perform a physically motivated resummation, and show that the RPT predictions match N-body simulations remarkably well across nonlinear scales, while highlighting strong finite-volume effects due to missing large-scale modes. The work provides a robust, parameter-free method to compute the nonlinear power spectrum and clarifies the role of cosmology through the linear growth factor. It also establishes practical cautions for simulations, especially at high redshift where box size and mode coupling significantly influence memory decay.

Abstract

We study the nonlinear propagator, a key ingredient in renormalized perturbation theory (RPT) that allows a well-controlled extension of perturbation theory into the nonlinear regime. We show that it can be thought as measuring the memory of density and velocity fields to their initial conditions. This provides a clean definition of the validity of linear theory, which is shown to be much more restricted than usually recognized in the literature. We calculate the nonlinear propagator in RPT and compare to measurements in numerical simulations, showing remarkable agreement well into the nonlinear regime. We also show that N-body simulations require a rather large volume to recover the correct propagator, due to the missing large-scale modes. Our results for the nonlinear propagator provide an essential element to compute the nonlinear power spectrum in RPT.

Figures (7)

• Figure 1: Cross-correlation coefficients between initial and final conditions $r_a$ as a function of wavenumber $k$ for density ($a=1$, solid lines) and velocity divergence ($a=2$, dashed lines) fields at redshift $z=3$ (left panel) and $z=0$ (right panel). Initial conditions are at redshift $z=35$.
• Figure 2: Diagrams for the non linear propagator $G(k,\eta)$ up to two loops.
• Figure 3: Tree diagrams for $\Psi_a({\hbox{\bf k}},\eta)$, up to 4 vertices, that lead to the non linear propagator's resummable subset.
• Figure 4: The density (left panel) and velocity divergence (right panel) propagators at redshift $z=0$ from initial conditions at $z=5$ (corresponding to $D_+=4.68$). The symbols represent measurement in numerical simulations, from implementation of the functional derivative (crosses) and from the relation to the cross-correlation coefficient (circles). The solid lines show the predictions of one-loop PT, Eq. (\ref{['proponeloop1']}).
• Figure 5: Predictions for the propagator of density (left panel) and velocity divergence (right panel) from Eq. (\ref{['model3one']}) against measurements in N-body simulations. The three cases correspond (from left to right in each panel) to $z=0,2,5$, with the initial conditions at $z_{\rm initial}=35$. The predictions shown in solid lines have no free parameters.