Constructing Regularized Cosmic Propagators

TL;DR

The paper tackles the challenge of predicting nonlinear cosmic structure by unifying perturbation theory at low wave numbers with high-$k$ resummation through the Γ-expansion, which uses multi-point propagators as fundamental building blocks. It introduces a regularized, scale-dependent interpolation scheme that preserves the correct low-$k$ perturbative expansion while reproducing the exponential damping from the displacement field in the high-$k$ limit, and extends this approach to multi-point propagators and non-Gaussian initial conditions. Explicitly, it provides a concrete regularization formula for two-point and higher-order propagators, demonstrates equivalence to existing RPT results at one-loop, and shows how to incorporate higher-loop information in a controlled manner; PNG effects are handled by adjusted counter-terms and higher cumulants. The scheme is validated against numerical simulations and applied to compute the matter bispectrum at one-loop, offering robust, model-ready building blocks for predicting power spectra and bispectra across cosmologies. This framework enhances the accuracy and generality of analytic predictions for large-scale structure and can adapt to non-standard cosmological scenarios.

Abstract

We present a new scheme for the general computation of cosmic propagators that allow to interpolate between standard perturbative results at low-k and their expected large-k resummed behavior. This scheme is applicable to any multi-point propagator and allows the matching of perturbative low-k calculations to any number of loops to their large-k behavior, and can potentially be applied in case of non-standard cosmological scenarios such as those with non-Gaussian initial conditions. The validity of our proposal is checked against previous prescriptions and measurements in numerical simulations showing a remarkably good agreement. Such a generic prescription for multi-point propagators provides the necessary building blocks for the computation of polyspectra in the context of the so-called Gamma-expansion introduced by Bernardeau et al. (2008). As a concrete application we present a consistent calculation of the matter bispectrum at one-loop order.

Figures (11)

• Figure 1: Diagrammatic representation of the series expansion of $\Psi_{a}({\bf k})$ up to fourth order in the initial conditions $\Phi$. Time increases along each segment according to the arrow and each segment bears a factor $g_{cd}({\eta_{f}}-{\eta_{i}})$ if ${\eta_{i}}$ is the initial time and ${\eta_{f}}$ is the final time. At each initial point and each vertex point there is a sum over the component indices; a sum over the incoming wave modes is also implicit and, finally, the time coordinate of the vertex points is integrated from ${\eta_{i}}=0$ to the final time ${\eta_{f}}$ according to the time ordering of each diagram. For instance, at fourth order there are two different possible topologies.
• Figure 2: Representation of the one-loop correction to the two-point propagator. The value of this diagram is obtained by the contraction of two incoming lines of $\Psi^{(3)}$ multiplied by the initial power spectrum value. The expression of $G_{ab}^{{\rm 1-loop}}$ is then obtained after integration over the internal indices $c,\dots, j$, the momentum $q$ and the times $\eta'_{1}$ and $\eta'_{2}$.
• Figure 3: Representation of the first two terms of the multi-point propagator $\Gamma^{(n)}$ in a perturbative expansion. $\Gamma^{(n)}$ represents the average value of the emerging nonlinear mode ${\bf k}$ given $n$ initial modes in the linear regime. Here we show the first two contributions: tree-level and one-loop. Note that each object represents a collection of (topologically) different diagrams : each black dot represents a set of trees that connect respectively $n+1$ lines for the first term, $n+2$ for the second.
• Figure 4: Representation of the resummation rule given by Eq. (\ref{['BkExpansion']}). For Gaussian initial conditions, the bispectrum can be seen as a sum of product of $\Gamma^{(p)}$ functions.
• Figure 5: In all the diagrams contributing to $G_{ab}(k)$, as the one depicted here, there is a line connecting directly the initial time to the final time. This is the principal line, drawn here with a straight thick solid line. The dominant loops contributing to the resummed propagator are those drawn by dashed lines, while the sub-dominant loops are those in dotted lines.