A Uniform Search for Nearby Planetary Companions to Hot Jupiters in TESS Data Reveals Hot Jupiters are Still Lonely

We present the results of a uniform search for additional planets around all stars with confirmed hot Jupiters observed by the Transiting Exoplanet Survey Satellite (TESS) in its Cycle 1 survey of the southern ecliptic hemisphere. Our search comprises 184 total planetary systems with confirmed hot Jupiters with $R_{p}$ > 8$R_\oplus$ and orbital period < 10 days. The Transit Least Squares (TLS) algorithm was utilized to search for periodic signals that may have been missed by other planet search pipelines. While we recovered 169 of these confirmed hot Jupiters, our search yielded no new statistically-validated planetary candidates in the parameter space searched (P < 14 days). A lack of planet candidates nearby hot Jupiters in the TESS data supports results from previous transit searches of each individual system, now down to the photometric precision of TESS. This is consistent with expectations from a high eccentricity migration formation scenario, but additional formation indicators are needed for definitive confirmation. We injected transit signals into the light curves of the hot Jupiter sample to probe the pipeline's sensitivity to the target parameter space, finding a dependence proportional to $R_{p}^{2.32}P^{-0.88}$ for planets within 0.3$\leq$$R_{p}$$\leq$4 $R_\oplus$ and 1$\leq$$P$$\leq$14 days. A statistical analysis accounting for this sensitivity provides a median and $90\%$ confidence interval of $7.3\substack{+15.2 \\ -7.3}\%$ for the rate of hot Jupiters with nearby companions in this target parameter space. This study demonstrates how TESS uniquely enables comprehensive searches for nearby planetary companions to nearly all the known hot Jupiters.