Limits on Fast Radio Bursts from Four Years of the V-FASTR Experiment

The V-FASTR experiment on the Very Long Baseline Array was designed to detect dispersed pulses of milliseconds duration, such as fast radio bursts (FRBs). We use all V-FASTR data through February 2015 to report V-FASTR's upper limits on the rates of FRBs, and compare these with re-derived rates from Parkes FRB detection experiments. V-FASTR's operation at lambda=20 cm allows direct comparison with the 20 cm Parkes rate, and we derive a power-law limit of \gamma<-0.4 (95% confidence limit) on the index of FRB source counts, N(>S)\propto S^\gamma. Using the previously measured FRB rate and the unprecedented amount of survey time spent searching for FRBs at a large range of wavelengths (0.3 cm > \lambda > 90 cm), we also place frequency-dependent limits on the spectral distribution of FRBs. The most constraining frequencies place two-point spectral index limits of \alpha_{20cm}^{4cm} < 5.8 and \alpha_{90cm}^{20cm} > -7.6, where fluence F \propto f^\alpha if we assume true the burst rate reported by Champion et al. (2016) of R(F~0.6 Jy ms) = 7 x 10^3 sky^{-1} day^{-1} (for bursts of ~3 ms duration). This upper limit on \alpha suggests that if FRBs are extragalactic but non-cosmological, that on average they are not experiencing excessive free-free absorption due to a medium with high optical depth (assuming temperature ~8,000 K), which excessively invert their low-frequency spectrum. This in turn implies that the dispersion of FRBs arises in either or both of the intergalactic medium or the host galaxy, rather than from the source itself.