Cosmic Degeneracies II: Structure formation in joint simulations of Warm
  Dark Matter and $f(R)$ gravity
  We present for the first time the outcomes of a cosmological N-body
simulation that simultaneously implements a Warm Dark Matter (WDM) particle
candidate and a modified gravitational interaction in the form of $f(R)$
gravity, and compare its results with the individual effects of these two
independent extensions of the standard $\Lambda $CDM scenario, and with the
reference cosmology itself. We consider a rather extreme value of the WDM
particle mass ($m_{\rm WDM}=0.4$ keV) and a single realisation of $f(R)$
gravity with $|\bar{f}_{R0}|=10^{-5}$, and we investigate the impact of these
models and of their combination on a wide range of cosmological observables
with the aim to identify possible observational degeneracies. In particular, we
focus on the large-scale matter distribution, as well as on the statistical and
structural properties of collapsed halos and cosmic voids. Differently from the
case of combining $f(R)$ gravity with massive neutrinos -- previously
investigated in Baldi et al. (2014) -- we find that most of the considered
observables do not show any significant degeneracy due to the fact that WDM and
$f(R)$ gravity are characterised by individual observational footprints with a
very different functional dependence on cosmic scales and halo masses. In
particular, this is the case for the nonlinear matter power spectrum in real
space, for the halo and sub-halo mass functions, for the halo density profiles
and for the concentration-mass relation. However, other observables -- like
e.g. the halo bias -- do show some level of degeneracy between the two models,
while a very strong degeneracy is observed for the nonlinear matter power
spectrum in redshift space, for the density profiles of small cosmic voids --
with radius below $\approx 5$ Mpc$/h$ -- and for the voids abundance as a
function of the void core density.
