  We present numerical measurements of the power spectrum response function of
the gravitational growth of cosmic structures, defined as the functional
derivative of the nonlinear spectrum with respect to the linear counterpart,
based on $1,400$ cosmological simulations. We develop a simple analytical model
based on a regularization of the standard perturbative calculation. Using the
model prediction, we show that this function gives a natural way to interpolate
the nonlinear power spectrum over cosmological parameter space from single or
multi-step interpolations. We demonstrate that once an accurate numerical
spectrum template is available for one (or a small number of) cosmological
model(s), it doubles the range in $k$ for which percent level accuracy can be
obtained even for large change in the cosmological parameters. The python
package RESPRESSO we developed to make those predictions is publicly available.
