Time-Sliced Perturbation Theory for Large Scale Structure I: General Formalism

TL;DR

Time-Sliced Perturbation Theory (TSPT) reframes large-scale structure perturbation theory by evolving a time-dependent probability distribution via a Liouville equation, isolating time evolution from statistical averaging. This yields a diagrammatic expansion with infrared-safe building blocks, since all equal-time quantities are protected by the equivalence principle and do not develop spurious IR divergences. The formalism reproduces standard results for the velocity divergence and density power spectra in both Zel'dovich and exact dynamics limits, while providing a natural route to infrared resummation and effective-field-theory–like UV handling. The approach offers a flexible framework to incorporate primordial non-Gaussianity, stochastic noise, redshift-space distortions, and biased tracers, with potential for rigorous renormalization-group treatment of UV sensitivity in LSS.

Abstract

We present a new analytic approach to describe large scale structure formation in the mildly non-linear regime. The central object of the method is the time-dependent probability distribution function generating correlators of the cosmological observables at a given moment of time. Expanding the distribution function around the Gaussian weight we formulate a perturbative technique to calculate non-linear corrections to cosmological correlators, similar to the diagrammatic expansion in a three-dimensional Euclidean quantum field theory, with time playing the role of an external parameter. For the physically relevant case of cold dark matter in an Einstein--de Sitter universe, the time evolution of the distribution function can be found exactly and is encapsulated by a time-dependent coupling constant controlling the perturbative expansion. We show that all building blocks of the expansion are free from spurious infrared enhanced contributions that plague the standard cosmological perturbation theory. This paves the way towards the systematic resummation of infrared effects in large scale structure formation. We also argue that the approach proposed here provides a natural framework to account for the influence of short-scale dynamics on larger scales along the lines of effective field theory.

Figures (2)

• Figure 1: Example of TSPT Feynman diagrams.
• Figure 2: Vertices appearing in the expansion of $\delta$ treated as a composite operator.