Cosmological tests to distinguish between dark energy (DE) and modifications to gravity are a promising route to obtain clues on the origin of cosmic acceleration. We study here the robustness of these tests to the presence of DE density, velocity, and anisotropic stress perturbations. We find that the dispersion in the growth index parameter remains small enough to distinguish between extreme cases of DE models and some commonly used modified gravity models. The sign of the slope parameter for a redshift-dependent growth index was found to be inconsistent as an additional test in extreme cases of DE models with perturbations. Next, we studied the effect of DE perturbations on the modified growth (MG) parameters that enter the perturbed Einstein equations. We find that while the dark energy perturbations affect the MG parameters, the deviations remain smaller than those due to modified gravity models. Additionally, the deviations due to DE perturbations with a nonzero effective sound speed occur at scale ranges that are completely different than those due to some modified gravity models such as the f(R) models. In the case of modified gravity models with zero anisotropic stress at late times, the simultaneous determination of the effective dark energy equation of state and the MG parameters can provide the distinction between these models and DE. The growth index test was found to be the most robust to these perturbations. The scale dependence of the MG parameters in some cases of modified gravity constitutes a clear-cut discriminant regardless of any DE perturbations. In summary, we find that the currently proposed cosmological tests to distinguish between DE and modified gravity are robust to DE perturbations even for extreme cases. This is certainly the case even for DE models with equations of state of DE that fall well outside of current cosmological constraints.
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