TIaRA TESS 1: Estimating exoplanet yields from Year 1 and Year 3 SPOC lightcurves

We present a study of the detection efficiency for the TESS mission, focusing on the yield of longer-period transiting exoplanets ($P > 25$ days). We created the Transit Investigation and Recoverability Application (TIaRA) pipeline to use real TESS data with injected transits to create sensitivity maps which we combine with occurrence rates derived from Kepler. This allows us to predict longer-period exoplanet yields, which will help design follow-up photometric and spectroscopic programs, such as the NGTS Monotransit Program. For the TESS Year 1 and Year 3 SPOC FFI lightucurves, we find $2271^{+241}_{-138}$ exoplanets should be detectable around AFGKM dwarf host stars. We find $215^{+37}_{-23}$ exoplanets should be detected from single-transit events or "monotransits". An additional $113^{+22}_{-13}$ detections should result from "biennial duotransit" events with one transit in Year 1 and a second in Year 3. We also find that K dwarf stars yield the most detections by TESS per star observed. When comparing our results to the TOI catalogue we find our predictions agree within $1\sigma$ of the number of discovered systems with periods between 0.78 and 6.25\,days and agree to $2\sigma$ for periods between 6.25 and 25\,days. Beyond periods of 25 days we predict $403^{+64}_{-38}$ detections, which is 3 times as many detections as there are in the TOI catalogue with $>3\sigma$ confidence. This indicates a significant number of long-period planets yet to be discovered from \tess\ data as monotransits or biennial duotransits.