On the Soft Limit of the Large Scale Structure Power Spectrum: UV
  Dependence
  We derive a non-perturbative equation for the large scale structure power
spectrum of long-wavelength modes. Thereby, we use an operator product
expansion together with relations between the three-point function and power
spectrum in the soft limit. The resulting equation encodes the coupling to
ultraviolet (UV) modes in two time-dependent coefficients, which may be
obtained from response functions to (anisotropic) parameters, such as spatial
curvature, in a modified cosmology. We argue that both depend weakly on
fluctuations deep in the UV. As a byproduct, this implies that the renormalized
leading order coefficient(s) in the effective field theory (EFT) of large scale
structures receive most of their contribution from modes close to the
non-linear scale. Consequently, the UV dependence found in explicit
computations within standard perturbation theory stems mostly from
counter-term(s). We confront a simplified version of our non-perturbative
equation against existent numerical simulations, and find good agreement within
the expected uncertainties. Our approach can in principle be used to precisely
infer the relevance of the leading order EFT coefficient(s) using small volume
simulations in an `anisotropic separate universe' framework. Our results
suggest that the importance of these coefficient(s) is a $\sim 10 \%$ effect,
and plausibly smaller.
