Multipoint propagators for non-Gaussian initial conditions

Francis Bernardeau, Martín Crocce, and Emiliano Sefusatti

Phys. Rev. D 82, 083507 – Published 7 October 2010

Abstract

We show here how renormalized perturbation theory calculations applied to the quasilinear growth of the large-scale structure can be carried on in presence of primordial non-Gaussian (PNG) initial conditions. It is explicitly demonstrated that the series reordering scheme proposed in Bernardeau, Crocce, and Scoccimarro [Phys. Rev. D 78, 103521 (2008)] is preserved for non-Gaussian initial conditions. This scheme applies to the power spectrum and higher-order spectra and is based on a reorganization of the contributing terms into the sum of products of multipoint propagators. In case of PNG, new contributing terms appear, the importance of which is discussed in the context of current PNG models. The properties of the building blocks of such resummation schemes, the multipoint propagators, are then investigated. It is first remarked that their expressions are left unchanged at one-loop order irrespective of statistical properties of the initial field. We furthermore show that the high-momentum limit of each of these propagators can be explicitly computed even for arbitrary initial conditions. They are found to be damped by an exponential cutoff whose expression is directly related to the moment generating function of the one-dimensional displacement field. This extends what had been established for multipoint propagators for Gaussian initial conditions. Numerical forms of the cutoff are shown for the so-called local model of PNG.

Received 15 July 2010

DOI:https://doi.org/10.1103/PhysRevD.82.083507

Authors & Affiliations

Francis Bernardeau¹, Martín Crocce², and Emiliano Sefusatti¹

¹Institut de Physique Théorique, CEA/DSM/IPhT, Unité de Recherche Associée au CNRS, CEA/Saclay 91191 Gif-sur-Yvette, Cédex, France
²Institut de Ciències de l’Espai, IEEC-CSIC, Facultat de Ciències, Torre C5 par-2, Barcelona 08193, Spain

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Vol. 82, Iss. 8 — 15 October 2010

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Figure 1
Example of partitions of $n_{1} + n_{2}$ points (here $n_{1} = 13$ and $n_{2} = 10$ ). On each ensemble the elements have been numbered. For this choice of partitions there are 5 elements out of 13 that are in subsets that span both column and 4 out of 10; therefore $p_{1} = 5$ , $q_{1} = 8$ and $p_{2} = 4$ , $q_{2} = 6$ . The left-hand side shows a generic partition of that kind. The right panel shows a partition which is identical to the left one up to a renumbering of the indices. These two contributing diagrams take the same values. When the order of the indices in each subset is preserved, the number of such contributing terms (e.g. the number of diagrams that lead to the same right hand side figure) is $(\binom{p_{1} + q_{1}}{p_{1}}) (\binom{p_{2} + q_{2}}{p_{2}})$ . The summation over all possible partitions can then be restricted to those of the type of the left-hand side with a weight given by the aforementioned symmetry factor. Such terms are defined by three new partitions that are elements, respectively, of $P_{q_{1}}$ , $P_{q_{2}}$ , and $P_{p_{1}, p_{2}}^{X}$ (see text for definitions).Reuse & Permissions
Figure 2
$Γ$ expansion for the power spectrum up to one-loop. The shaded regions indicate the initial (or “primordial”) statistics corresponding to each diagram, that is: one initial power spectra (2-points) in the left most diagram, one initial bispectrum in the center-left one, two power spectra in the center-right, and one initial trispectrum in the right most figure.Reuse & Permissions
Figure 3
$Γ$ expansion for the bispectrum up to one-loop. The first two contributions are proportional to $P_{0}^{2}$ (tree level), the remaining ones to $P_{0}^{3}$ , assuming that the bispectrum $B_{0}$ and the trispectrum $T_{0}$ scale, respectively, as $P_{0}^{2}$ and $P_{0}^{3}$ . The meaning of the shaded regions is described in the caption to Fig. 2 (see text for details).Reuse & Permissions
Figure 4
The non-Gaussian transition function $f (k)$ in the large- $k$ limit in ratio to its expression for Gaussian statistics (i.e. setting $f_{NL} = g_{NL} = 0$ ). For currently accepted values of non-Gaussian parameters (i.e. close to those shown) the correction to the damping is remarkably subdominant, even more so at higher redshift (e.g. bottom panel).Reuse & Permissions
Figure 5
The damping function $\log f (k)$ as a function of $k$ for Gaussian and non-Gaussian initial conditions, with $f_{NL} = 5 \times 10^{3}$ and $g_{NL} = 10^{9}$ , respectively.Reuse & Permissions
Figure 6
The $Γ$ expansion for equilateral (left) and squeezed (right) bispectrum configurations in the local model of PNG, up to one-loop. The “gravitational” tree-level bispectrum is shaped by $Γ^{(2)}$ while the PNG one arises from the primordial bispectrum and is exponentially suppressed to high- $k$ by $Γ^{(1)}$ . At one-loop there are terms $\sim P_{0}^{3}$ (grav.), $\sim P_{0} \times B_{0}$ (PNG, $f_{NL}$ ), and $\sim T_{0}$ (PNG, $f_{NL}^{2}$ and $g_{NL}$ ). Inset labels regarding various analytical contributions are shared for both panels.Reuse & Permissions

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