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 $rho _{0}^{text{upper}}$ 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.