Breaking cosmic degeneracies: Disentangling neutrinos and modified
  gravity with kinematic information
  Searches for modified gravity in the large-scale structure try to detect the
enhanced amplitude of density fluctuations caused by the fifth force present in
many of these theories. Neutrinos, on the other hand, suppress structure growth
below their free-streaming length. Both effects take place on comparable
scales, and uncertainty in the neutrino mass leads to a degeneracy with
modified gravity parameters for probes that are measuring the amplitude of the
matter power spectrum. We explore the possibility to break the degeneracy
between modified gravity and neutrino effects in the growth of structures by
considering kinematic information related to either the growth rate on large
scales or the virial velocities inside of collapsed structures. In order to
study the degeneracy up to fully non-linear scales, we employ a suite of
$N$-body simulations including both $f(R)$ modified gravity and massive
neutrinos. Our results indicate that velocity information provides an excellent
tool to distinguish massive neutrinos from modified gravity. Models with
different values of neutrino masses and modified gravity parameters possessing
a comparable matter power spectrum at a given time have different growth rates.
This leaves imprints in the velocity divergence, which is therefore better
suited than the amplitude of density fluctuations to tell the models apart. In
such models with a power spectrum comparable to $\Lambda$CDM today, the growth
rate is strictly enhanced. We also find the velocity dispersion of virialised
clusters to be well suited to constrain deviations from general relativity
without being affected by the uncertainty in the sum of neutrino masses.
