Leptogenesis, $\mu -\tau$ Symmetry and ${\theta }_{13}$

We show that in theories where neutrino masses arise from type I seesaw formula with three right handed neutrinos and where large atmospheric mixing angle owes its origin to an approximate leptonic $\mu -\tau$ interchange symmetry, the primordial lepton asymmetry of the Universe, ${ϵ}_l$ can be expressed in a simple form in terms of low energy neutrino oscillation parameters as $\epsilon_l = (a \Delta m^2_\odot+ b \Delta m^2_A \theta^2_{13})$,where$a$and$b$ are parameters characterizing high scale physics and are each of order $\le {10}^{-2}$ eV${}^{-2}$. We also find that for the case of two right handed neutrinos, ${ϵ}_l\propto {\theta }_{13}^2$ as a result of which, the observed value of baryon to photon ratio implies a lower limit on ${\theta }_{13}$. For specific choices of the CP phase $\delta$ we find ${\theta }_{13}$ is predicted to be between $0.10-0.15$.