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Upper Limit on the Central Density of Dark Matter in the Eddington inspired Born-Infield (EiBI) Gravity

We investigate the stability of circular material orbits in the analytic galactic metric recently derived by Harko \textit{et al.} (2014). It turnsout that stability depends more strongly on the dark matter central density $%ρ_{0}$ than on other parameters of the solution. This property then yields an upper limit on $ρ_{0}$ for each individual galaxy, which we call here $ρ_{0}^{\text{upper}}$, such that stable circular orbits are possible \textit{only} when the constraint $ρ_{0}\leq ρ_{0}^{\text{upper}}$ is satisfied. This is our new result. To approximately quantify the upper limit, we consider as a familiar example our Milky Way galaxy that has a projected dark matter radius $R_{\text{DM}}\sim 180$ kpc and find that $ρ_{0}^{\text{upper}}\sim 2.37\times 10^{11}$ $M_{\odot }$kpc$^{-3}$. This limit turns out to be about four orders of magnitude larger than the latest data on central density $ρ_{0}$ arising from the fit to the Navarro-Frenk-White (NFW) and Burkert density profiles. Such consistency indicates that the EiBI solution could qualify as yet another viable alternative model for dark matter.