Abstract The correlation between the inner small planets and cold giants encodes the formation and evolution of planetary systems. It remains unclear if the correlation differs on the two sides of the radius valley. In this work, we compute the conditional frequency of cold Jupiters in systems with only inner sub-Neptunes P ( CJ ∣ SN ) and those with only inner super-Earths P (CJ∣SE). We find that, around the transiting sample around metal-rich stars, P ( CJ ∣ SN , Fe / H > 0 ) and P (CJ∣SE, Fe/H > 0) are 42 . 6 − 9.9 + 10.6 % and 14 . 5 − 6.9 + 12.7 % . Comparing with the field giant frequency ( 14 . 3 − 1.8 + 2.0 % ), we show that inner sub-Neptunes and cold Jupiters exhibit a significant positive correlation for metal-rich systems with a confidence level of 99.95%, whereas this correlation is absent for systems with super-Earths. We also consider a homogeneous Kepler–Keck subsample and derive similar results, with P ( CJ ∣ SN , Fe / H > 0 ) of 45 . 8 − 16.3 + 18.6 % and P (CJ∣SE, Fe/H > 0) of 13 . 3 − 6.8 + 17.0 % . A radial velocity sample shows consistent results, with metal-rich systems hosting massive inner planets exhibiting a strong positive correlation (confidence level of 99.11%) with outer cold Jupiters ( P ( CJ ∣ M p >
Chen et al. (Fri,) studied this question.