Responses in Large-Scale Structure

We introduce a rigorous definition of general power-spectrum responses as resummed vertices with two hard and $n$ soft momenta in cosmological perturbation theory. These responses measure the impact of long-wavelength perturbations on the local small-scale power spectrum. The kinematic structure of the responses (i.e., their angular dependence) can be decomposed unambiguously through a "bias" expansion of the local power spectrum, with a fixed number of physical response coefficients, which are only a function of the hard wavenumber $k$. Further, the responses up to $n$-th order completely describe the $(n+2)$-point function in the squeezed limit, i.e. with two hard and $n$ soft modes, which one can use to derive the response coefficients. This generalizes previous results, which relate the angle-averaged squeezed limit to isotropic response coefficients. We derive the complete expression of first- and second-order responses at leading order in perturbation theory, and present extrapolations to nonlinear scales based on simulation measurements of the isotropic response coefficients. As an application, we use these results to predict the non-Gaussian part of the angle-averaged matter power spectrum covariance ${\rm Cov}^{\rm NG}_{\ell = 0}(k_1,k_2)$, in the limit where one of the modes, say $k_2$, is much smaller than the other. Without any free parameters, our model results are in very good agreement with simulations for $k_2 \lesssim 0.06\ h/{\rm Mpc}$, and for any $k_1 \gtrsim 2 k_2$. The well-defined kinematic structure of the power spectrum response also permits a quick evaluation of the angular dependence of the covariance matrix. While we focus on the matter density field, the formalism presented here can be generalized to generic tracers such as galaxies.