The Lagrangian-space Effective Field Theory of Large Scale Structures

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 an expansion 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.