Cosmic Degeneracies I: Joint N-body Simulations of Modified Gravity and Massive Neutrinos

We present the first suite of cosmological N-body simulations that simultaneously include the effects of two different and theoretically independent extensions of the standard $\Lambda$CDM cosmological scenario - namely an $f(R)$ theory of Modified Gravity (MG) and a cosmological background of massive neutrinos - with the aim to investigate their possible observational degeneracies. We focus on three basic statistics of the large-scale matter distribution, more specifically the nonlinear matter power spectrum, the halo mass function, and the halo bias, for which we determine the deviation with respect to the fiducial $\Lambda$CDM cosmology in the context of both separate and combined simulations of $f(R)$ MG and massive neutrinos scenarios. Our results show that while these two extended models separately determine very prominent and potentially detectable features in all the three statistics, when we allow them to be simultaneously at work these features are strongly suppressed, resulting in much weaker deviations from the standard model's predictions. In particular, when an $f(R)$ gravity model with $f_{R0}=-1\times 10^{-4}$ is combined with a total neutrino mass of $\Sigma _{i}m_{\nu _{i}}=0.4$ eV, the resulting matter power spectrum, halo mass function, and bias at z=0 are found to be consistent with the standard model's predictions at the 10%, 20%, and 5% accuracy levels, respectively. Therefore, our results imply an intrinsic theoretical limit to the effective discriminating power of present and future observational data sets with respect to these widely considered extensions of the standard cosmological scenario in the absence of independent measurements of the neutrino masses from laboratory experiments, even though the high-redshift evolution might still allow to partially break the degeneracy [Abridged].