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Early-type main-sequence pulsating stars such as \ds{} variables are one ofthe least explored class of targets in the search for exoplanets. Pulsationtiming (PT) is an alternative technique to the most effective search methods.It exploits the light-travel-time effect (LTTE) to infer the presence ofadditional massive bodies around a pulsating star by measuring a periodic phasemodulation of its signal. PT has been extremely successful in discovering andcharacterizing stellar binaries when it was applied to high-precision lightcurves over long temporal baselines, such as those delivered by the Keplermission. In favorable conditions, the sensitivity of PT can reach theplanetary-mass regime; one such candidate has already been claimed. The adventof TESS, with its nearly full-sky coverage and the availability of full-frameimages, opens a great opportunity to expand this field of research. In thiswork, we present a pilot study aimed to understand the potential of PT asapplied to TESS data, which are considerably different from Kepler data interms of photometric noise, sampling cadence, and temporal baseline. We focusedon the most favorable class of \ds{}, that is, those showing strong pulsationsand very simple frequency spectra. After the development of a customizedpipeline, we were able to detect candidate companions for two targets in the(sub-)stellar mass regime: Chang~134 ($43\pm 5$~$M_\mathrm{jup}$, $P\simeq82$~d) and V393 Car ($\gtrsim 100$~$M_\mathrm{jup}$, $P\gtrsim 700$~d). Ourresults also highlight the limiting factors of this technique and theimportance of an accurate absolute time calibration for future missions such asPLATO.

TESS search for substellar companions through pulsation timing of $\delta$ Scuti stars. I. Discovery of companions around Chang 134 and V393 Car