Can the degeneracies in the gravity sector be broken?
  It has been long known that by appropriately modifying gravity one can always
reproduce the expansion history of any dark energy model, e.g. the
$\Lambda$CDM. This degeneracy cannot be broken with geometric probes like the
Type Ia supernovae, the cosmic microwave background or the baryon acoustic
oscillations since they are based on the measurement of distances and scales
and therefore require the expansion history $H(z)$, so one may only hope to
break this degeneracy by using dynamic probes like the growth rate data that
track the matter density perturbations. We demonstrate that breaking this
degeneracy is not currently possible by explicitly constructing $f(R)$, $f(G)$
and $f(T)$ theories that mimic exactly the $\Lambda$CDM model at the background
level and confronting them against the latest observational cosmological data.
We also determine the necessary improvement in the growth rate data in order to
discriminate these theories from $\Lambda$CDM. We found that at least a
threefold improvement is necessary, something which will be possible with a
survey like Euclid.
