On modelling the Fast Radio Burst (FRB) population and event rate predictions

Assuming that Fast Radio Bursts (FRBs) are of extragalactic origin, we have developed a formalism to predict the FRB detection rate and the redshift distribution of the detected events for a telescope with given parameters. We have adopted FRB 110220, for which the emitted pulse energy is estimated to be $E_0 = 5.4 \times 10^{33}J$, as the reference event. The formalism requires us to assume models for (1) pulse broadening due to scattering in the ionized inter-galactic medium - we consider two different models for this, (2) the frequency spectrum of the emitted pulse - we consider a power law model $E_{\nu} \propto \nu^{-\alpha}$ with $-5 \leq \alpha \leq 5$, and (3) the comoving number density of the FRB occurrence rate $n(E,w_i,z)$ - we ignore the z dependence and assume a fixed intrinsic pulse width $w_i = 1$ms for all the FRBs. The distribution of the emitted pulse energy $E$ is modelled through (a) a delta-function where all the FRBs have the same energy $E = E_0$ , and (b) a Schechter luminosity function where the energies have a spread around $E_0$. The models are all normalized using the 4 FRBs detected by Thornton et al. (2013). Our model predictions for the Parkes telescope are all consistent with the inferred redshift distribution of the fourteen FRBs detected there to date. We also find that scattering places an upper limit on the redshift of the FRBs detectable by a given telescope; for the Parkes telescope this is $z \sim 2$. Considering the upcoming Ooty Wide Field Array, we predict a FRB detection rate of $\sim 0.01$ to $\sim 10^3$ per day.