  The clustering properties of sterile neutrinos are studied within an
extension of the minimal standard model, where these are produced via the decay
of a gauge singlet scalar. The distribution function after decoupling is
strongly out of equilibrium. (DM) abundance and phase space density constraints
from (dSphs) constrain the mass in the $\mathrm{keV}$ range consistent with a
gauge singlet with mass and vacuum expectation value $\sim 100,\mathrm{GeV}$
decoupling at this temperature. The (DM) transfer function and power spectrum
are obtained from the solution of the non-relativistic Boltzmann-Vlasov
equation in the matter dominated era. The small momentum enhancement of the
distribution function leads to long range memory of gravitational clustering
and a \emph{substantial enhancement of the power spectrum at small scales
compared to a thermal relic or sterile neutrino produced via non-resonant
mixing with active neutrinos}. The scale of suppression of the power spectrum
for such sterile neutrino with $m\sim \mathrm{keV}$ is $\lambda \sim 488
,\mathrm{kpc}$. At large scales $T(k)\sim 1-C, k^2/k^2_{fs}(t_{eq}) +...$ with
$C \sim \mathrm{O}(1)$. At small scales $65 \mathrm{kpc} \lesssim \lambda
\lesssim 500 \mathrm{kpc}$ corrections to the fluid description and memory of
gravitational clustering become important, and we find $T(k) \simeq 1.902
e^{-k/k_{fs}(t_{eq})}$, where $k_{fs}(t_{eq}) \sim 0.013/\mathrm{kpc}$ is the
free streaming wavevector at matter-radiation equality. The enhancement of
power at small scales may provide a possible relief to the tension between the
constraints from X-ray and Lyman-$\alpha$ forest data.

Astrophysics;High Energy Physics - Phenomenology