The TESS-Keck Survey XXIV: Outer Giants may be More Prevalent in the Presence of Inner Small Planets

We present the results of the Distant Giants Survey, a three-year radial velocity (RV) campaign to search for wide-separation giant planets orbiting Sun-like stars known to host an inner transiting planet. We defined a distant giant to have $a$ = 1--10 AU and $M_{p} \sin i = 70-4000$ \mearth~ = 0.2-12.5 \mj, and required transiting planets to have $a<1$ AU and $R_{p} = 1-4$ \rearth. We assembled our sample of 47 stars using a single selection function, and observed each star at monthly intervals to obtain $\approx$30 RV observations per target. The final catalog includes a total of twelve distant companions: four giant planets detected during our survey, two previously known giant planets, and six objects of uncertain disposition identified through RV/astrometric accelerations. Statistically, half of the uncertain objects are planets and the remainder are stars/brown dwarfs. We calculated target-by-target completeness maps to account for missed planets. We found evidence for a moderate enhancement of distant giants (DG) in the presence of close-in small planets (CS), P(DG|CS) = $30^{+14}_{-12}\%$, over the field rate of P(DG) = $16^{+2}_{-2}\%$. No enhancement is disfavored ($p \sim$ 8%). In contrast to a previous study, we found no evidence that stellar metallicity enhances P(DG|CS). We found evidence that distant giant companions are preferentially found in systems with multiple transiting planets and have lower eccentricities than randomly selected giant planets. This points toward dynamically cool formation pathways for the giants that do not disturb the inner systems.