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The Lagrangian-space Effective Field Theory of Large Scale Structures

Rafael A. Porto, Leonardo Senatore, Matias Zaldarriaga

TL;DR

The paper introduces LEFT, a Lagrangian-space EFT for large-scale structure that treats long-wavelength dynamics as an EFT of extended objects described by center-of-mass motion and multipole moments. Finite-size effects are encoded via a controlled multipole expansion and renormalized through UV counter-terms, enabling a finite, systematic $k/k_{\rm NL}$ perturbative expansion with resummation of large IR displacements. A detailed one-loop calculation in a power-law EdS universe demonstrates that divergences are absorbed by LEFT's counter-terms, with a minimal set of renormalized parameters (e.g., $l_{s,\rm comb}$, $l_{\delta_m,\rm comb}$) governing displacement and density correlations. The work also presents an effective action formulation and discusses resummation techniques, showing LEFT’s potential to improve BAO modeling and extend to broader cosmological applications, including biased tracers and redshift-space distortions.

Abstract

We introduce a Lagrangian-space Effective Field Theory (LEFT) formalism for the study of cosmological large scale structures. Unlike the previous Eulerian-space construction, it is naturally formulated as an effective field theory of extended objects in Lagrangian space. In LEFT the resulting finite size effects are described using a multipole expansion parameterized by a set of time dependent coefficients and organized in powers of the ratio of the wavenumber of interest $k$ over the non-linear scale $k_{\rm NL}$. The multipoles encode the effects of the short distance modes on the long-wavelength universe and absorb UV divergences when present. There are no IR divergences in LEFT. Some of the parameters that control the perturbative approach are not assumed to be small and can be automatically resummed. We present an illustrative one-loop calculation for a power law universe. We describe the dynamics both at the level of the equations of motion and through an action formalism.

The Lagrangian-space Effective Field Theory of Large Scale Structures

TL;DR

The paper introduces LEFT, a Lagrangian-space EFT for large-scale structure that treats long-wavelength dynamics as an EFT of extended objects described by center-of-mass motion and multipole moments. Finite-size effects are encoded via a controlled multipole expansion and renormalized through UV counter-terms, enabling a finite, systematic perturbative expansion with resummation of large IR displacements. A detailed one-loop calculation in a power-law EdS universe demonstrates that divergences are absorbed by LEFT's counter-terms, with a minimal set of renormalized parameters (e.g., , ) governing displacement and density correlations. The work also presents an effective action formulation and discusses resummation techniques, showing LEFT’s potential to improve BAO modeling and extend to broader cosmological applications, including biased tracers and redshift-space distortions.

Abstract

We introduce a Lagrangian-space Effective Field Theory (LEFT) formalism for the study of cosmological large scale structures. Unlike the previous Eulerian-space construction, it is naturally formulated as an effective field theory of extended objects in Lagrangian space. In LEFT the resulting finite size effects are described using a multipole expansion parameterized by a set of time dependent coefficients and organized in powers of the ratio of the wavenumber of interest over the non-linear scale . The multipoles encode the effects of the short distance modes on the long-wavelength universe and absorb UV divergences when present. There are no IR divergences in LEFT. Some of the parameters that control the perturbative approach are not assumed to be small and can be automatically resummed. We present an illustrative one-loop calculation for a power law universe. We describe the dynamics both at the level of the equations of motion and through an action formalism.

Paper Structure

This paper contains 23 sections, 177 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Effective description of long-wavelength modes in Lagrangian space
  3. Lagrangian space dynamics
  4. Effective Field Theory in Lagrangian space
  5. Expectation values, Response & Noise
  6. Renormalization, Matching & Power Counting
  7. Illustrative one-loop calculation
  8. Basics
  9. Quadrupole moment & composite operators
  10. Displacement
  11. Combined parameters
  12. Resummed correlation function
  13. Expansion parameters of perturbation theory
  14. Higher order terms in the displacement
  15. Effective action approach
  16. ...and 8 more sections

Figures (3)

  • Figure 1: Parameters measuring the amplitude of non-linear correction on a mode of wavenumber $k$. They quantify the motions created by modes longer ($\epsilon_{s<}$) and shorter ($\epsilon_{s>}$) than $k$ and the tides from larger scales ($\epsilon_{\delta <}$).
  • Figure 2: Left panel: finite sized regions of size $R_0\sim 1/k_{\rm NL}$ in Lagrangian space. Right panel: Eulerian space. The vector $\vec{z}_L(\vec{q},\eta)$ gives the center of mass position of each Lagrangian region. Notice that upon evolution the regions will eventually overlap. See sec. \ref{['sec:reno']} and appendix \ref{['uvmatch']} for more details.
  • Figure 3: Left panel: a region of size $V$ in Lagrangian space containing several cells of size $R_0 \simeq k_{\rm NL}^{-1}$, each evolving with its own quadrupole moment $Q^{ij}(\vec{q},\eta)$, as shown in Fig. \ref{['figL1']}. Right panel: The same region in Eulerian space. The physical value of the quadrupole moment of the entire region, $Q^{ij}_V$, must be independent of the sizes of the Lagrangian cells, i.e. it must be cutoff independent. Determining the value of the quadrupole and other multipoles in LEFT involves free parameters that may be obtained from data or comparison with N-body simulations.