On the stability conditions for theories of modified gravity in presence
  of matter fields
  We present a thorough stability analysis of modified gravity theories in the
presence of matter fields. We use the Effective Field Theory framework for Dark
Energy and Modified Gravity to retain a general approach for the gravity sector
and a Sorkin-Schutz action for the matter one. Then, we work out the proper
viability conditions to guarantee in the scalar sector the absence of ghosts,
gradient and tachyonic instabilities. The absence of ghosts can be achieved by
demanding a positive kinetic matrix, while the lack of a gradient instability
is ensured by imposing a positive speed of propagation for all the scalar
modes. In case of tachyonic instability, the mass eigenvalues have been studied
and we work out the appropriate expressions. For the latter, an instability
occurs only when the negative mass eigenvalue is much larger, in absolute
value, than the Hubble parameter. We discuss the results for the minimally
coupled quintessence model showing for a particular set of parameters two
typical behaviours which in turn lead to a stable and an unstable
configuration. Moreover, we find that the speeds of propagation of the scalar
modes strongly depend on matter densities, for the beyond Horndeski theories.
Our findings can be directly employed when testing modified gravity theories as
they allow to identify the correct viability space.
