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Constraint on dark matter central density in the Eddington inspired Born-Infeld (EiBI) gravity with input from Weyl gravity

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 $ρ_{0}^{\text{upper}}$ on the central density $ρ_{0}$ of dark matter such that stable circular orbits are possible only when the constraint $ρ_{0}\leq ρ_{0}^{\text{upper}}$ is satisfied in each galactic sample. To quantify different $ρ_{0}^{\text{upper}}$ 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 $ρ_{0}$ obey the constraint $ρ_{0}\leqρ_{0}^{\text{upper}}\propto $ ($R_{\text{WR}}$)$^{-2}$. Using the mass-to-light ratios giving $α$, we shall also evaluate $ρ_{0}^{\text{lower}}$ $\propto $ $(α-1)M_{\text{lum}}R_{\text{WR}}^{-3}$ and the average dark matter density $\left\langle ρ\right\rangle ^{\text{lower}}$. Quantitatively, it turns out that the interval $ρ_{0}^{\text{lower}}$ $\leq ρ_{0}\leq $ $ρ_{0}^{\text{upper}}$ verifies reasonably well against many dark matter dominated low surface brightness (LSB) galaxies for which values of $ρ_{0}$ are independently known. The interval holds also in the case of Milky Way galaxy. Qualitatively, the existence of a stability induced upper limit $ρ_{0}^{\text{upper}}$ is a remarkable prediction of the EiBI theory.