Simulating cosmologies beyond $\Lambda$CDM with PINOCCHIO
  We present a method that extends the capabilities of the PINpointing
Orbit-Crossing Collapsed HIerarchical Objects (PINOCCHIO) code, allowing it to
generate accurate dark matter halo mock catalogues in cosmological models where
the linear growth factor and the growth rate depend on scale. Such cosmologies
comprise, among others, models with massive neutrinos and some classes of
modified gravity theories. We validate the code by comparing the halo
properties from PINOCCHIO against N-body simulations, focusing on cosmologies
with massive neutrinos: $\nu\Lambda$CDM. We analyse the halo mass function,
halo two-point correlation function, halo power spectrum and the moments of the
halo density field, showing that PINOCCHIO reproduces the results from
simulations with the same level of precision as the original code
($\sim5-10\%$). We demonstrate that the abundance of halos in cosmologies with
massless and massive neutrinos from PINOCCHIO matches very well the outcome of
simulations, and point out that PINOCCHIO can reproduce the
$\Omega_\nu-\sigma_8$ degeneracy that affects the halo mass function. We show
that the clustering properties of the halos from PINOCCHIO matches accurately
those from simulations both in real and redshift-space, in the latter case up
to $k=0.3~h~{\rm Mpc}^{-1}$. We finally point out that the first moments of the
halo density field from simulations are precisely reproduced by PINOCCHIO. We
emphasize that the computational time required by PINOCCHIO to generate mock
halo catalogues is orders of magnitude lower than the one needed for N-body
simulations. This makes this tool ideal for applications like covariance matrix
studies within the standard $\Lambda$CDM model but also in cosmologies with
massive neutrinos or some modified gravity theories.
