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Reconstructing $f(R)$ gravity from viscous cosmology constrained by observations

In this paper, we reconstruct the $f(R)$ theory using the viscous $ω$CDM scenario (V-$ω$CDM) constrained by several differently astrophysical observations, including Type Ia supernovae (SNe Ia), observational Hubble parameter data (OHD), Wilkinson Microwave Anisotropy Probe 9-year data (WMAP-9), Planck observations and the single data point from the newest event GW150914 (GW). We find that the joint constraints from SNe Ia+OHD+Planck+GW data-sets give out a tighter restriction for the parameters of the V-$ω$CDM scenario than other constraints. Subsequently, we find that there exists a substantially high degeneracy among these reconstructed $f(R)$ theories in the past, and that they start exhibiting differently evolutional behaviors around the present epoch as well as much different in the distant future. Furthermore, we make a comparison between the reconstructed $f(R)$ theory with the V-$ω$CDM scenario and that with the $ω$CDM scenario constrained by using the SNe Ia+OHD+Planck+GW data-sets in the $f(R)-R$ plane, and find that the two scenarios share an extremely high degeneracy, which implies that the effect of bulk viscosity plays a negligible role in the V-$ω$CDM model during the whole evolutionary processes of the universe. In addition, one can also find that the reconstructed $f(R)$ theories using the two models just deviate from the standard Einstein gravity a little.