Are falling planets spinning up their host stars?

We investigate the effects of tidal interactions on the planetary orbits and stellar spin rates of the WASP-18 and WASP-19 planetary systems using a forward integration scheme. By fitting the resulting evolutionary tracks to the observed eccentricity, semi-major axis and stellar rotation rate, and to the stellar age derived from isochronal fitting, we are able to place constraints on the stellar and planetary reduced tidal quality factors, Q's and Q'p. We find that for WASP-18, log(Q's)=8.21+0.90-0.52 and log(Q'p)=7.77+1.54-1.25, implying a system age of 0.579+0.305-0.250 Gyr. For WASP-19 we obtain values of log(Q's)=6.47+2.19-0.95 and log(Q'p)=6.75+1.86-1.77, suggesting a system age of 1.60+2.84-0.79 Gyr and a remaining lifetime of 0.0067+1.1073-0.0061 Gyr. We investigate a range of evolutionary histories consistent with these results and the observed parameters for both systems, and find that the majority imply that the stars have been spun up through tidal interactions as the planets spiral towards their Roche limits. We examine a variety of evidence for WASP-19 A's age, both for the value above and for a younger age consistent with gyrochronology, and conclude that the older estimate is more likely to be correct. This suggests that WASP-19 b might be in the final stages of the spiral-in process, although we are unable to rule out the possibility that it has a substantial remaining lifetime.