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 $\rho_{0}\leq\rho_{0}^{\text{upper}}\propto$ ($R_{\text{WR}}$)$^{-2}$. 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 $\left\langle \rho\right\rangle ^{\text{lower}}$. 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.