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 $%rho_{0}$ than on other par
ameters of the solution. This property then yields an upper limit on $rho _{0}$ for each individual galaxy, which we call here $rho _{0}^{text{upper}}$, such that stable circular orbits are possible textit{only} when the constraint $rho _{0}leq rho _{0}^{text{upper}}$ 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 $rho _{0}^{text{upper}}sim 2.37times 10^{11}$ $M_{odot }$kpc$^{-3}$. This limit turns out to be about four orders of magnitude larger than the latest data on central density $rho _{0}$ 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.
Recently, Harko et al. (2014) derived an approximate metric of the galactic halo in the Eddington inspired Born-Infeld (EiBI) gravity. In this metric, we show that there is an upper limit $rho _{0}^{text{upper}}$ on the central density $rho _{0}$ of
dark matter such that stable circular orbits are possible only when the constraint $rho _{0}leq rho_{0}^{text{upper}}$ is satisfied in each galactic sample. To quantify different $rho _{0}^{text{upper}}$ for different samples, we follow the novel approach of Edery & Paranjape (1998), where we use as input the geometric halo radius $R_{text{WR}}$ from Weyl gravity and equate it with the dark matter radius $R_{text{DM}}$ from EiBI gravity for the same halo boundary. This input then shows that the known fitted values of $rho _{0}$ obey the constraint $rho_{0}leqrho_{0}^{text{upper}}propto $ ($R_{text{WR}}$)$^{-2}$. Using the mass-to-light ratios giving $alpha $, we shall also evaluate $rho _{0}^{text{lower}}$ $propto $ $(alpha -1)M_{text{lum}}R_{text{WR}}^{-3}$ and the average dark matter density $leftlangle rhorightrangle ^{text{lower}}$. Quantitatively, it turns out that the interval $rho _{0}^{text{lower}}$ $leq rho _{0}leq $ $rho _{0}^{text{upper}}$ verifies reasonably well against many dark matter dominated low surface brightness (LSB) galaxies for which values of $rho _{0}$ are independently known. The interval holds also in the case of Milky Way galaxy. Qualitatively, the existence of a stability induced upper limit $rho _{0}^{text{upper}}$ is a remarkable prediction of the EiBI theory.
