New methods to constrain the radio transient rate: results from a survey of four fields with LOFAR

We report on the results of a search for radio transients between 115 and 190\,MHz with the LOw-Frequency ARray (LOFAR). Four fields have been monitored with cadences between 15 minutes and several months. A total of 151 images were obtained, giving a total survey area of 2275 deg$^2$. We analysed our data using standard LOFAR tools and searched for radio transients using the LOFAR Transients Pipeline (TraP). No credible radio transient candidate has been detected; however, we are able to set upper limits on the surface density of radio transient sources at low radio frequencies. We also show that low-frequency radio surveys are more sensitive to steep-spectrum coherent transient sources than GHz radio surveys. We used two new statistical methods to determine the upper limits on the transient surface density. One is free of assumptions on the flux distribution of the sources, while the other assumes a power-law distribution in flux and sets more stringent constraints on the transient surface density. Both of these methods provide better constraints than the approach used in previous works. The best value for the upper limit we can set for the transient surface density, using the method assuming a power-law flux distribution, is 1.3$\cdot$10$^{-3}$ deg$^{-2}$ for transients brighter than 0.3 Jy with a time-scale of 15 min, at a frequency of 150 MHz. We also calculated for the first time upper limits for the transient surface density for transients of different time-scales. We find that the results can differ by orders of magnitude from previously reported, simplified estimates.