  Context. The overdensity inside a cosmological sub-volume and the tidal
fields from its surroundings affect the matter distribution of the region. The
resulting difference between the local and global power spectra is
characterized by the response function.
  Aims. Our aim is to provide a new, simple, and accurate formula for the power
spectrum overdensity response at highly nonlinear scales based on the results
of cosmological simulations and paying special attention to the lognormal
nature of the density field.
  Methods. We measured the dark matter power spectrum amplitude as a function
of the overdensity ($\delta_W$) in $N$-body simulation subsamples. We show that
the response follows a power-law form in terms of $(1+\delta_W)$, and we
provide a new fit in terms of the variance, $\sigma(L)$, of a sub-volume of
size $L$.
  Results. Our fit has a similar accuracy and a comparable complexity to
second-order standard perturbation theory on large scales, but it is also valid
for nonlinear (smaller) scales, where perturbation theory needs higher-order
terms for a comparable precision. Furthermore, we show that the lognormal
nature of the overdensity distribution causes a previously unidentified bias:
the power spectrum amplitude for a subsample with an average density is
typically underestimated by about $-2\sigma^2$. Although this bias falls to the
sub-percent level above characteristic scales of $200Mpch^{-1}$, taking it into
account improves the accuracy of estimating power spectra from zoom-in
simulations and smaller high-resolution surveys embedded in larger
low-resolution volumes.

Cosmology and Nongalactic Astrophysics