Multi-Point Propagators in Cosmological Gravitational Instability

TL;DR

The paper generalizes the two-point propagator result to multi-point propagators within cosmological perturbation theory, deriving the one-loop corrections to the three-point propagator and showing that the dominant high-$k$ contributions can be resummed into a Gaussian damping factor multiplying the tree-level propagators. It demonstrates that any n-point correlator can be reconstructed from multi-point propagators and the initial power spectrum, establishing a direct link between small-scale nonlinear corrections and large-scale higher-order statistics. The authors validate the large-$k$ predictions with N-body simulations for the three-point propagator and apply the formalism to reconstruct the power spectrum and bispectrum at one-loop, finding good agreement and reduced triangle-shape dependence. These results provide a rigorous foundation for improved perturbative modeling of nonlinear structure formation, with potential impacts on precision cosmology and interpretation of large-scale structure surveys.

Abstract

We introduce the concept of multi-point propagators between linear cosmic fields and their nonlinear counterparts in the context of cosmological perturbation theory. Such functions express how a non-linearly evolved Fourier mode depends on the full ensemble of modes in the initial density field. We identify and resum the dominant diagrams in the large-$k$ limit, showing explicitly that multi-point propagators decay into the nonlinear regime at the same rate as the two-point propagator. These analytic results generalize the large-$k$ limit behavior of the two-point propagator to arbitrary order. We measure the three-point propagator as a function of triangle shape in numerical simulations and confirm the results of our high-$k$ resummation. We show that any $n-$point spectrum can be reconstructed from multi-point propagators, which leads to a physical connection between nonlinear corrections to the power spectrum at small scales and higher-order correlations at large scales. As a first application of these results, we calculate the reduced bispectrum at one-loop in renormalized perturbation theory and show that we can predict the decrease in its dependence on triangle shape at redshift zero, when standard perturbation theory is least successful.

Figures (6)

• 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 $g_{cd}(s_{f}-s_{i})$ if $s_{i}$ is the initial time and $s_{f}$ 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 $s=0$ to the final time $s$ according to the time ordering of each diagram. For instance, at fourth order there are two different possible topologies.
• Figure 2: 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.
• Figure 3: The one-loop contribution to $G_{ab}(k,s_2,s_1)$. The $\otimes$ represents a primordial power spectrum $P_0(q)$ with the corresponding "loop" momentum ${\bf q}$ integrated over with weight $(2\pi)^{-3}\int {\rm d}^3 {\bf q}$ . See 2006PhRvD..73f3520C for an explicit calculation of this diagram.
• Figure 4: The large-$k$ limit of the two-point density propagator $\Gamma^{(1)}$. Symbols correspond to measurements in numerical simulations at redshifts $z=1,0.5$ and $z=0$ (top to bottom), see text for details. The solid lines correspond to the large-$k$ limit expression given in Eq. (\ref{['GabHighk']}). The linear relation obtained by plotting $\log G$ vs. $k^2$ makes evident that the suppression of $G$ is indeed Gaussian in the high-$k$ limit. Moreover, the slope is very well predicted by Eqs. (\ref{['GabHighk']},\ref{['sigmavdef']}).
• Figure 5: Example of diagrams contributing to $G_{ab}(k)$ (top) and $\Gamma^{(3)}_{abcd}({\bf k},{\bf k}_{1},{\bf k}_{2},{\bf k}_{3})$ (bottom). The dominant contribution after resumming all possible configurations is expected to come from those diagrams where all loops are directly connected to the principal line (top) or principal tree (bottom). The principal line and tree are drawn with a thick solid line. The dominant loops are those drawn by dashed lines, while the sub-dominant loops are those in dotted lines.