Title: Responses in Large-Scale Structure,
Abstract: 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.