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Cosmological Constraints on Non-flat Exponential $f(R)$ Gravity

We explore the viable $f(R)$ gravity models in FLRW backgrounds with a free spatial curvature parameter $Ω_{K}$. In our numerical calculation, we concentrate on the exponential $f(R)$ model of $f(R) = R - λR_{ch}(1-\exp{(-R/R_{ch}}))$, where $R_{ch}$ is the characteristic curvature scale, which is independent of $Ω_K$, and $λ$ corresponds to the model parameter, while $R_{ch}λ=2Λ$ with $Λ$ the cosmological constant. In particular, we study the evolutions of the dark energy density and equation of state for exponential $f(R)$ gravity in open, flat and closed universe, and compare with those for $Λ$CDM. From the current observational data, we find that $λ^{-1}=0.42927^{+0.39921}_{-0.32927}$ at 68$\%$ C.L and $Ω_K=-0.00050^{+0.00420}_{-0.00414}$ at 95$\%$ C.L. in the exponential $f(R)$ model. By using Akaike information criterion (AIC), Bayesian information criterion (BIC) and Deviance Information Criterion (DIC), we conclude that there is no strong preference between the exponential $f(R)$ gravity and $Λ$CDM models in the non-flat universe.