Impulsive radio bursts that are detectable across cosmological distances constitute extremely powerful probes of the ionized Inter-Galactic Medium (IGM), intergalactic magnetic fields, and the properties of space-time itself. Their dispersion measures (DMs) will enable us to detect the missing baryons in the low-redshift Universe and make the first measurements of the mean galaxy halo profile, a key parameter in models of galaxy formation and feedback. Impulsive bursts can be used as cosmic rulers at redshifts exceeding 2, and constrain the dark energy equation-of-state parameter, $w(z)$ at redshifts beyond those readily accessible by Type Ia SNe. Both of these goals are realisable with a sample of $sim 10^4$ fast radio bursts (FRBs) whose positions are localized to within one arcsecond, sufficient to obtain host galaxy redshifts via optical follow-up. It is also hypothesised that gravitational wave events may emit coherent emission at frequencies probed by SKA1-LOW, and the localization of such events at cosmological distances would enable their use as cosmological standard sirens. To perform this science, such bursts must be localized to their specific host galaxies so that their redshifts may be obtained and compared against their dispersion measures, rotation measures, and scattering properties. The SKA can achieve this with a design that has a wide field-of-view, a substantial fraction of its collecting area in a compact configuration (80% within a 3,km radius), and a capacity to attach high-time-resolution instrumentation to its signal path.
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