  Massive neutrinos make up a fraction of the dark matter, but due to their
large thermal velocities, cluster significantly less than cold dark matter
(CDM) on small scales. An accurate theoretical modelling of their effect during
the non-linear regime of structure formation is required in order to properly
analyse current and upcoming high-precision large-scale structure data, and
constrain the neutrino mass. Taking advantage of the fact that massive
neutrinos remain linearly clustered up to late times, this paper treats the
linear growth of neutrino overdensities in a non-linear CDM background. The
evolution of the CDM component is obtained via N-body computations. The smooth
neutrino component is evaluated from that background by solving the Boltzmann
equation linearised with respect to the neutrino overdensity. CDM and neutrinos
are simultaneously evolved in time, consistently accounting for their mutual
gravitational influence. This method avoids the issue of shot-noise inherent to
particle-based neutrino simulations, and, in contrast with standard
Fourier-space methods, properly accounts for the non-linear potential wells in
which the neutrinos evolve. Inside the most massive late-time clusters, where
the escape velocity is larger than the neutrino thermal velocity, neutrinos can
clump non-linearly, causing the method to formally break down. It is shown that
this does not affect the total matter power spectrum, which can be very
accurately computed on all relevant scales up to the present time.

Cosmology and Nongalactic Astrophysics