© The Authors 2025

1 Introduction

Whether the presence of short-period small planets (SPs) with P ≲ 100 d (corresponding to a semimajor axis of a ≲ 0.4 AU) and 1 < M_p < 20 M_⊕ around solar-type stars correlates with that of cold Jupiters (CJs) – gaseous giant planets with a ≃ 1–10 AU and mass 0.5 ≤ M_p ≤ 20 M_Jup (or 0.3 ≤ M_p ≤ 13 M_Jup, depending on the definition) – is an intriguing question in exoplanetary science because it could potentially reveal crucial features of planet formation and evolution. However, there are strong discrepancies about this possible correlation in the literature from both theoretical and observational perspectives.

Theoretically, an anticorrelation between SPs and CJs is expected if CJs either act as a dynamical barrier to the inward migration of icy SPs from beyond the water snowline at ~ 1–3 AU (Izidoro et al. 2015), and/or if they considerably reduce the inward flux of pebbles to the inner regions of the protoplanetary disk, which is required to form dry SPs within the water snowline (Lambrechts et al. 2019). From the Generation 3 Bern simulations of planet formation and evolution, Schlecker et al. (2021) instead find no or a weak SP-CJ correlation and predict a strong architecture-composition link, with more icy (rocky) SPs in the absence (presence) of one or more CJs, due to the same dynamical barrier effect of the CJ (Izidoro et al. 2015). Furthermore, Schlecker et al. (2021) predict a scarcity of SPs at high stellar metallicity due to the disruption of inner planetary systems by more dynamically active CJs (see their Fig. 19). A recent study by Bitsch & Izidoro (2023) shows that a correlation between the presence of SPs and that of CJs is possible, depending on the gas contraction rate for the formation of CJs: for slow gas contraction rates, the cores that form in the proximity of the water snowline are too small to effectively accrete large envelopes and can migrate inward, becoming SPs, while cores at greater distances can grow into cold gas giants.

Observationally, occurrence rates of CJs in SP systems (f_CJ|SP¹) as high as ~40% reported by both Zhu & Wu (2018) and Bryan et al. (2019) disagree with a more recent estimate of $f_{\text{CJ|SP}}={8.7}_{-2.7}^{+7.3}\%$ for M_p = 0.5–20 M_Jup (or $f_{\text{CJ|SP}}={9.3}_{-2.9}^{+7.7}\%$ for M_p = 0.3–13 M_Jup) by Bonomo et al. (2023) (hereafter B23). To reconcile these estimates, both Zhu (2024) and Bryan & Lee (2024) (hereafter BL24) explore the SP-CJ relation as a function of stellar metallicity and report a correlation at super-solar metallicity ([Fe/H] > 0). Specifically, BL24 analyzed a large sample of 184 SP systems detected by either transit or radial velocity (RV) surveys and compared their derived f_CJ|SP with the frequency of CJs (f_CJ²) from the California Legacy Survey (CLS, Rosenthal et al. 2021). For this purpose, they considered the confirmed CJs and homogeneously computed the sensitivity to CJs for both their sample and the CLS survey. They find no correlation at [Fe/H] ≤ 0 and an SP-CJ correlation at [Fe/H] > 0 with a 2.7σ significance level ($f_{\text{CJ|SP}}={28.0}_{-4.6}^{+4.9}\%$ vs $f_{\text{CJ}}={14.3}_{-1.8}^{+2.0}\%$). After submission of this paper, Bryan & Lee (2025) (hereafter BL25) report a positive correlation with a significance level greater than 2σ in metal-rich stars, with mass M_★ > 0.8 M_⊙. BL25 interpret this as evidence for the critical role of stellar metallicity and/or mass (regarded as proxies for the disk solid and gas budget, respectively) in nucleating both inner low-mass and outer large-mass planets.

The studies by Zhu (2024), BL24, and BL25 were partly motivated by the observation that several stars in the Kepler and K2 sample from B23 are metal-poor, which might bias the B23 estimate of f_CJ|SP toward lower values for the well-known relation between giant planets and stellar metallicity (e.g., Fischer & Valenti 2005; Johnson et al. 2010; Sousa et al. 2011; Mortier et al. 2013). However, the average metallicity of the B23 sample $\left(\overline{\left[\text{Fe/H}\right]}=-0.066\pm 0.014\right)$ and that of the Anglo Australian Telescope (AAT) RV survey $\left(\overline{\left[\text{Fe/H}\right]}=-0.045\pm 0.018\right)$ (Wittenmyer et al. 2020), which B23 used as a comparison sample, are consistent within 1σ and are both lower than the peak metallicity of approximately –0.1 dex in a volume-limited sample of the solar neighborhood (e.g., Sousa et al. 2011).

Based on the work of BL24, in the present study we recompute: (i) f_CJ|SP with a larger and more carefully selected sample of SP systems and (ii) f_CJ, using three different blind RV campaigns, namely the AAT, CLS, and HARPS (Mayor et al. 2011; Trifonov et al. 2020) surveys, and merging the stars of these three surveys into a single large sample. We then search for SP-CJ relations as a function of stellar metallicity, mass, or planet properties and draw our conclusions about their possible existence.

2 Samples and radial velocity data

2.1 Small planet transit and radial velocity sample

The sample of SP systems to compute f_CJ|SP was built through queries to the NASA Exoplanet Archive³ on 2 December 2024 following the same criteria as BL24. The systems must have (1) at least one confirmed SP with 1 < M_p < 20 M_⊕⁴ for transiting planets and 1 < M_p sin i < 20 M_⊕ for planets discovered by RV surveys; (2) a solar-type (i.e., FGK dwarf or sub-giant) host star, thereby excluding M dwarfs with M_★ ≤ 0.6 M_⊙; and (3) at least one publicly available RV dataset with more than 20 measurements and baseline ΔT ≥ 1 yr. However, unlike BL24, we initially discarded “mixed systems” containing both SP and short-period (P < 100 d) planets with M_p ≥ 20 M_⊕, such as the WASP-47 system, which also hosts a hot Jupiter (WASP-47b) between the two SPs WASP-47e and d (e.g., Vanderburg et al. 2017; Nascimbeni et al. 2023). It is unclear whether the theoretical predictions mentioned in Sect. 1 apply to such systems in which giant planets, especially the hot Jupiters, may have undergone a substantial migration process (e.g., Dawson & Johnson 2018). Nonetheless, we also computed the occurrence rates by taking the mixed systems into account and noted no significant differences (see Sect. 3). We removed a few outliers from the BL24 sample, such as TOI-1853b, which has M_p = 73 ± 3 > 20 M_⊕ (Naponiello et al. 2023), and Kepler-22b, whose orbital period is P = 290 > 100 d (Borucki et al. 2012).

Our SP sample comprises 217 systems, of which 134 were detected by transit surveys and 83 by RVs. It is therefore considerably larger than that of BL24 (184 mixed systems) and contains 148 systems in common (see Table 1 in Additional Tables). Including the mixed systems leads to 242 systems, 151 and 91 of which were found with transits and RVs, respectively (Tables 1 and 2 in Additional Tables). As in BL24, for each system we selected the available RV dataset with the best sensitivity for the detection of CJs (see Sect. 3).

2.2 Radial velocity comparison samples

As comparison samples to estimate f_CJ, we considered the three largest RV surveys with high-precision RVs known to us, namely the AAT, CLS, and HARPS surveys, after applying the same criteria (2) and (3) as above for consistency. This yielded 196, 402, and 782 solar-type stars for the AAT, CLS, and HARPS campaigns, respectively. The number of solar-type stars from the CLS survey (402) is lower than that considered by BL24 (562), likely because BL24 applied the third selection criterion only to their SP sample and not to the CLS sample. Applying the third criterion (Sect. 2.1) to the CLS survey reduced the number of solar-type stars by 160, while excluding only one system out of 52 with CJs, showing that a baseline ΔT ≥ 1 yr and more than 20 RV measurements for the best dataset of a given target are key to discovering CJs. The reduced number of CLS stars resulted in a slightly higher estimate of f_CJ (see Table F.1), which is more consistent with the f_CJ determinations from the other two surveys, AAT and HARPS (see Table F.2).

The mass, metallicity, and uncertainty on the metallicity of the selected solar-type stars from the three surveys were all taken from the Hypatia catalog⁵ (Hinkel et al. 2014) and are listed in Table 3 of Additional Tables.

We also merged the three aforementioned RV surveys to create a large comparison sample with 1167 stars, 213 of which are common to two or all three surveys. In this way, we could compute a more precise and accurate f_CJ (see Tables D.1 and F.2).

2.3 Comparison of the small planet and radial velocity merged samples

The distributions of the stellar metallicity and mass for both the SP and RV merged samples are compared in Fig. A.1.

The Kolmogorov-Smirnov (K-S) test between the two empirical cumulative distribution functions (eCDF) of metallicity yields a relatively low p-value of 0.021, as the SP sample contains a lower fraction of metal-poor stars at [Fe/H] ≲ –0.2 than the comparison sample (see Fig. A.1, left panel).

The same K-S test for the stellar mass cumulative distributions returns a p-value of 3E-09, indicating that the two distributions are highly incompatible. This is due to an excess of lower-mass stars in the SP sample (see Fig. A.1, right panel), given that SPs are more easily detected around smaller stars with both the transit and RV methods.

In Fig. A.2, we show the stellar metallicity distributions and eCDFs of both the SP and RV merged samples for the three intervals of stellar mass considered in Sect. 3, namely 0.6–0.8, 0.8–1.0, and 1.0–1.2 M_⊙. While the eCDFs of metallicity are compatible in the 0.6–0.8 Μ_⊙ and 0.8–1.0 Μ_⊙ intervals, with p-values of 99% and 27%, respectively, they differ considerably at super-solar mass (p-value < 1%) because of the considerably larger number of metal-poor stars in the RV merged sample (right panels of Fig. A.2).

3 Statistical analyses

We determined f_CJ|SP from the SP sample (Sect. 2.1) and f_CJ from both the individual RV surveys and the comparison sample (Sect. 2.2), first at the average metallicity of the different samples, and then by considering three intervals in stellarmetallicity, i.e., sub-solar ([Fe/H] < –0.1), solar (–0.1 ≤ [Fe/H] ≤ 0.1), and super-solar ([Fe/H] > 0.1) metallicity. Similarly to BL24, we also split the samples into two broad intervals, namely [Fe/H] ≤ 0 and [Fe/H] > 0. However, given the typical uncertainties of ~0.04–0.07 dex on [Fe/H], a star with 0.0 < [Fe/H] < 0.1 (–0.1 < [Fe/H] < 0.0) should be better classified as solar in metallicity rather than as metal-rich (poor).

Given the different distributions of stellar mass in the SP and comparison samples (Sect. 2.3), we further considered three different intervals in stellar mass, namely 0.6 ≤ M_★ < 0.8 M_⊙, 0.8 ≤ M_★ < 1.0 M_⊙, and 1.0 ≤ M_★ < 1.2 Μ_⊙⁶. This is of interest because the frequency of giant planets also depends on stellar mass, though to a lesser extent than on metallicity (e.g., Johnson et al. 2010; Fulton et al. 2021).

For each sample and its relative subsamples in metallicity, we used binomial statistics: $$b\left(d|N_{★,\text{eff},f\text{\hspace{0.17em}CJ|SP}}\right)=\frac{N_{★,}\text{eff\hspace{0.17em}!}}{d!\left(N_{★,\text{eff}}-d\right)!}{f}_{\text{CJ|SP}}^d{\left(1-f_{\text{CJ|SP}}\right)}^{N_{★,\text{\hspace{0.17em}eff}}-d},$$(1)

where d is the number of systems with at least one confirmed CJ, and N_★,eff is the “effective” number of stars, given by the product of the number of stars in the survey, N_★, and the average survey sensitivity (or completeness) to CJs, C, i.e., N_★,eff = N_★ ∙ C.

The completeness of each individual star, C_i, was computed in the same way as in B23 and BL24 (see their Sect. 3), that is, through injection and recovery experiments of CJ RV signals in a logarithmic grid of 30×30 ΔΜ_p-Δa cells covering the ranges 0.3–20 M_Jup and 1–10 AU, respectively. Specifically, for each cell, we simulated 200 RV signals of CJs at the epochs of the RV dataset by randomly varying: (i) M_p and a within the cell bounds, (ii) T_c within the range of P corresponding to a and the stellar mass M_★ (Tables 1–3 in Additional Tables) from Kepler’s third law; (iii) cos i from 0 to 1, where i is the orbital inclination; and iv) the argument of periastron ω from 0 to 2π, while drawing the orbital eccentricity e from a β distribution (Kipping 2013). We then shifted every RV point at time t following a Gaussian distribution with mean and standard deviation equal to the RV value and its uncertainty, respectively. We estimated C_i as the recovery rate of the simulated signals (Keplerian orbits or linear and quadratic trends if ΔΤ was considerably shorter than the simulated CJ orbital period) across the whole grid using the ΔΒIC criterion (see B23 and BL24). The average completeness C of each sample (or subsample) was then obtained by the mean of the completenesses of the stars in that sample: C = (1/N_★) ${\displaystyle {\sum }_{i=1}^{N_{★}}}$ C_i. The average completenesses of all the samples described in Sect. 2 are displayed in Fig. B.1. We note a very good agreement when comparing our average completenesses of the SP RV and transit samples (upper panels in Fig. B.1) with those shown in Fig. A.1 of BL24 (for a = 1–10 AU).

We considered two ranges of CJ mass⁷, namely M_p = 0.5–20 M_Jup and M_p = 0.3–13 M_Jup (for comparison with B23), but we mainly rely on the former because the latter suffers from incompleteness issues between 0.3 and 0.5 M_Jup (Fig. B.1). For each CJ mass range and the aforementioned bins of stellar metal-licity and/or mass, we provide the number of solar-type stars (N_★), the number of stars with confirmed CJs (d), the average completenesses (C), and CJ occurrence rates from (i) the SP and the large comparison samples in Tables D.1 and D.2, (ii) the SP transit, RV, and transit+RV samples as well as the SP and mixed transit, RV, and transit+RV samples in Table E.1, and (iii) the AAT, CLS, and HARPS surveys in Table F.2. Tables 4–6 in Additional Tables list the known CJs and their properties in the SP, mixed, and comparison samples. Figure C.1 shows the architectures of systems in the SP sample: on the left, the 27 systems with CJs of mass 0.3–13 M_Jup, and on the right, the 23 systems with CJs of mass 0.5–20 M_Jup.

4 Results

4.1 Frequency of CJs at average stellar metallicity and mass

At the average stellar metallicity $\left(\overline{\left[\text{Fe/H}\right]}=-0.011\pm 0.005\right)$ and mass $\left(\overline{M_{★}}=0.916\pm 0.012{\text{M}}_{\odot }\right)$ of the SP (transit+RV) stars, we determine $\left(\overline{M_{\star }}=0.916\pm 0.012{\text{M}}_{\odot }\right)$ for the CJ mass range M_p = 0.5–20 M_Jup. Including the 25 mixed systems yields $f_{\text{CJ|SP+Mixed}}={12.5}_{-1.8}^{+2.5}\%$. Returning to the original sample of BL24, which has $\overline{\left[\text{Fe/H}\right]}=-0.001\pm 0.005$ and $\overline{M_{★}}=0.923\pm 0.014M_{\odot }$ and includes the mixed systems, and accounting for its average com-pleteness⁸, we find a fully consistent result: ${15.2}_{-2.3}^{+3.1}\%$. Both occurrence rates are lower than previous claims of f_CJ|SP ~ 40% (Sect. 1). They agree within 1σ with the estimate of $f_{\text{CJ|SP}}={8.7}_{-2.7}^{+7.3}\%$ by B23, whose smaller sample is on average slightly more metal-poor by ~0.06 dex.

For the comparison sample, at its average metallicity $\left(\overline{\left[\text{Fe/H}\right]}=-0.072\pm 0.009\right)$ and mass $\left(\overline{M_{★}}=0.924\pm 0.004M_{\odot }\right)$, we find $f_{\text{CJ}}={9.8}_{-0.8}^{+0.9}\%$, which is fully compatible with f_CJ|SP⁹.

From our occurrence rates f_CJ|SP and f_CJ, and the frequency of SPs $f_{\text{SP}}={28.1}_{-5.1}^{+6.6}\%$ from Rosenthal et al. (2022) (see also Zhu et al. 2018), we can derive the conditional probability that a solar-type star with a CJ also hosts a SP (Zhu & Wu 2018), namely f_SP|CJ = f_SP × f_CJ|SP / f_CJ = 32 ± 11%. This agrees within 1σ with the f_SP|CJ computed from a sample of 115 solar-type stars known to host a CJ (Ruggieri et al., in prep.), and is substantially lower than the estimate of f_SP|CJ ~ 100% in Bryan et al. (2019).

4.2 Frequency of CJs as a function of stellar metallicity

We report f_CJ|SP and f_CJ for the three subintervals of stellar metallicity (Sect. 3) in Table D.1 and display them in Fig. 1 (see also Fig. C.3 for M_p = 0.3–13 M_Jup). For the same metallicity bins, we also provide the estimates of f_CJ from the AAT, CLS, and HARPS surveys separately in Table F.2; these are fully consistent with each other (see Fig. C.2).

By comparing f_CJ|SP and f_CJ in Table D.1, we find that (i) at sub-solar and solar metallicities, they are fully compatible within 1σ, and (ii) at super-solar metallicity, the value of f_CJ|SP is higher than f_CJ by approximately 3/2 (~30% vs ~20%; Fig. 1), but the significance of this difference is low, at 1.7σ (compared to 2.7σ in BL24 for [Fe/H] > 0).

Fig. 1 Occurrence rates of CJs with Mp = 0.5–20 MJup in the SP sample (red solid line) and the merged RV comparison sample (black dashed line) at sub-solar ([Fe/H] < –0.1), solar (–0.1 ≤ [Fe/H] ≤ 0.1), and super-solar ([Fe/H] > 0.1) metallicity. See Table D.1.

4.3 Frequency of CJs as a function of stellar metallicity and mass

We report f_CJ|SP and f_CJ for the same bins of stellar metallicity as above and for the three ranges of stellar mass discussed in Sect. 3, in Table D.2. We show these values in Fig. 2 (see also Fig. C.4 for M_p = 0.3–13 M_Jup).

We find an excess of CJs in SP systems at 2.5σ significance level only for super-solar mass and metallicity, namely M_★ ≥ 1.0 M_⊙ and [Fe/H] > 0 (see the right panel in Fig. 2, and Table D.2), in agreement with the recent findings of BL25. This is the origin of the higher f_CJ|SP compared to f_CJ at super-solar metallicity and integrated stellar masses (Sect. 4.2 and Fig. 1).

4.4 SP-CJ relations as a function of SP and CJ multiplicity

We explored possible SP-CJ trends as a function of SP multiplicity. For this analysis, we adopted the 0.3–13 M_Jup mass range (instead of 0.5–20 M_Jup), given the higher number of systems with both CJs and SPs (27 compared to 23). If the presence of CJs hinders the formation or migration of SPs, one would expect a lower degree of SP multiplicity in the presence of CJs. For example, low-mass “jumpers” overtaking the CJ towards inner orbits are more likely to form single SP systems (see Fig. 3 in Izidoro et al. 2015). By comparing the frequency of multiple systems with a number of SPs n_p ≥ 2 in the SP subsamples with and without CJs, we find $f_{n_{p\ge 2,\text{w/CJ}}}={37}_{-8}^{+10}\%$ (10 out of 27 systems) and $f_{n_{p\ge 2,\text{w/o\hspace{0.17em}CJ}}}={43}_{-3}^{+4}\%$ (81 out of 190 systems), respectively. These rates are statistically consistent within uncertainties.

We also searched for possible (anti)correlations as a function of CJ multiplicity to investigate whether two or more CJs may hinder SP formation more than a single CJ system (Sect. 1). To this end, we compared the frequency of single SPs in systems with one CJ to that in systems with two (or more) CJs. The latter should be higher if two or more CJs have a more deleterious effect than a single CJ on the formation of multiple SP systems. We find no evidence for this: the rates of single SP systems with one CJ and two CJs are compatible, namely ${63}_{-12}^{+9}\%$ (12 out of 19 systems) and ${62}_{-18}^{+13}\%$ (5 out of 8 systems), respectively (see Fig. C.1).

4.5 SP-CJ relation as a function of SP composition

We investigated the possible presence of an architecture-composition link by determining the frequency of rocky SPs¹⁰ in CJ systems. This analysis tests whether the formation of rocky planets is favored in the presence of CJs if CJs block the migration of icy sub-Neptunes from beyond the water snowline (Schlecker et al. 2021). We derive $f_{\text{rocky,w/CJ}}={33}_{-9}^{+13}\%$ from five transiting systems with rocky planets (HD 219134, Kepler-139, Kepler-407, K2-312, and K2-364) out of fifteen in total (including three RV systems that were later found transiting). This indicates that systems with both rocky planets and CJs do not represent the majority of the SP-CJ transiting systems.

4.6 SP multiplicity as a function of CJ eccentricity

When ordering the SP systems with CJs as a function of the CJ eccentricity (Fig. 3), only single or, more rarely, two SPs are found with eccentric (e ≳ 0.4) CJs, the most extreme case being K2-312/HD 80653 (B23). In contrast, higher multiplicity (up to n_p = 4 for HD 219134) SP systems are found in the company of CJs in quasi-circular orbits. This is consistent with the expectation of stronger dynamical perturbations at inner orbital separations by eccentric CJs (e.g., Huang et al. 2017).

5 Discussion and conclusions

5.1 Cold Jupiters and small planets: friends, foes, or indifferent?

In this work, we first investigated the possible SP-CJ relation as a function of stellar metallicity and mass. This did not reveal any correlation or anticorrelation (i.e., SPs and CJs are neither friends nor foes) in eight of the nine mass and metallicity bins, that is, at sub-solar mass for any metallicity and at supersolar mass for sub-solar and solar metallicity. We instead find an enhancement of CJs in SP systems (i.e., SPs and CJs could be friends) at super-solar mass and metallicity, though at less than 3σ, in general agreement with BL25.

We do not find any SP-CJ trends with planet multiplicity and composition, which further challenges the anticorrelation models predicting a dearth of CJs in SP systems (Sect. 1). Nonetheless, it should be noted that (i) the limited subsamples of SP systems with CJs (27 systems) inevitably result in large uncertainties in CJ occurrence rates, making it difficult to detect possible weak correlations or anticorrelations with planet properties; and (ii) the SP multiplicity considered in this work is based on the detected SPs and may be different from the true multiplicity, especially if non-transiting SPs with relatively long periods (P ~ 25–100 d) and/or low mass were not detected by RV follow-up.

Our investigation of the architecture-composition relation in Sect. 4.5 must be considered with caution because it only concerned rocky planets. In principle, it may also apply to rocky planets with tenuous H/He atmospheres, if some available gas was accreted by the super-Earths at the time of their formation, and was not subsequently stripped away by post-formation processes such as atmospheric photoevaporation (e.g., Lopez & Fortney 2014) or core-powered mass loss (e.g., Ginzburg et al. 2018). However, the well-known degeneracy in SP composition from the measurement of the bulk density alone does not allow us to distinguish whether or which non-rocky planets in the ten transiting systems with CJs of our SP sample are actually dry (rocky worlds shrouded by thin H/He atmospheres, which formed within the water snowline) or icy (water worlds with possible H/He envelopes, which formed beyond the water snowline and then migrated inwards). This makes the investigation of the architecture-composition link challenging.

We show that higher multiplicity SP systems are found in systems with CJs in low-eccentricity orbits, being immune to dynamical instabilities generated by highly eccentric CJs. In this sense, SPs are not indifferent to CJs, even in the majority of cases where f_CJ|SP ~ f_CJ.

Fig. 2 Same as Fig. 1 for three different bins of stellar mass: 0.6–0.8 M⊙ (left), 0.8–1.0 M⊙ (center), and 1.0–1.2 M⊙ (right). See Table D.2.

Fig. 3 Architectures of 27 systems with both SPs and CJs in the mass range 0.3 ≤ Mp ≤ 13 MJup, ordered according to increasing orbital eccentricity of the CJs (from top to bottom; for multiple CJs, the highest eccentricity is considered). The planet circles have sizes proportional to their mass (see legend in the top right corner), and colors indicating orbital eccentricity (more eccentric planets are shown in yellow). Host stars are color coded as a function of stellar metallicity (metallicity increases from blue to red). All systems with three or more inner SPs are accompanied by CJs with low eccentricities.

5.2 Caveats of this work, BL24, and BL25

The occurrence rates derived in this work, BL24, and BL25 may be slightly underestimated due to two primary factors. However, these factors should affect both the SP and comparison samples approximately equally, making the search for correlations, in principle, reliable. Firstly, the completeness values computed in our work and in BL24 are optimistic because they only account for the formal RV uncertainties and not for uncorrected jitter (as addressed comprehensively by B23), which in some cases may be considerably higher than the former¹¹. Secondly, a small number of linear or quadratic RV trends, which might later result in CJ signals, were not taken into account in the computation of the occurrence rates, which is based only on confirmed CJs. Nevertheless, among the targets in the SP sample with relatively short temporal baselines, that is 1 ≤ ΔT ≤ 4 yr, we identified only three targets with RV long-term trends still compatible with CJs (namely, TOI-1279, TOI-1742, and TOI-1778). Moreover, such trends are very often induced by brown dwarfs or low-mass stellar companions, or they correlate with activity indicators (see B23).

Conversely, occurrence rates may decrease if the minimum masses (M_p sin i) of some of the considered CJs turn out to be true masses (M_p) above the 20 (or 13) M_Jup upper limit, corresponding to brown dwarfs or low-mass stellar objects, due to low orbital inclinations (e.g., Xiao et al. 2023).

5.3 Perspectives

As more and more SP systems are characterized with high-precision RV follow-up and monitored over extended periods, more precise and accurate determinations of f_CJ|SP can be derived, allowing further investigation of the SP-CJ relation, even in the lower CJ mass range 0.3–13 M_Jup. We plan to update our results over time by incorporating the increasing number of well-characterized SP systems.

Crucial progress in this regard will come from the Gaia (e.g., Sozzetti 2024) and PLATO (Rauer et al. 2014, 2025) space missions. The Gaia mission, particularly with Data Release 4 anticipated in 2026, is expected to (i) provide crucial information on the presence of CJs (e.g., Holl et al. 2023), especially around relatively bright stars; (ii) provide CJ true masses (note that about 30% of the CJs in the SP sample already have measured M_p owing to the combination of RVs and Hipparcos-Gaia proper motion anomalies; see, e.g., Kervella et al. 2022; Errico et al. 2022; and Table 4 in Additional Tables); and (iii) help us clarify the origin of certain RV long-term trends (e.g., Bardalez Gagliuffi et al. 2021). PLATO is expected to detect ≳8000 SP systems orbiting bright solar-type stars with V ≤ 11¹², which will be well suited to high-precision RV follow-up. Moreover, PLATO will reveal new SPs in known TESS systems, enabling more in-depth studies of f_CJ|SP as a function of SP multiplicity.

Data availability

Tables listing the stars and their properties, as well as the cold Jupiters and their parameters, in the small planet, mixed, and comparison samples are available in Additional Tables.

Acknowledgements

We are grateful to the anonymous referee for her/his valuable comments, which allowed us to considerably improve this work. We are thankful to Profs. A. Morbidelli and B. Bitsch for useful discussions about theoretical predictions on SP-CJ relations. We also thank Dr. K. Biazzo for her suggestion to use the Hypatia catalog. We acknowledge financial contribution from the INAF Large Grant 2023 “EXODEMO” and from the European Union – Next Generation EU RRF M4C2 1.1 PRIN MUR 2022 project 20229R43BH. D.G. gratefully acknowledges the financial support from the grant for internationalization (GAND_GFI_23_01) granted by the University of Turin (Italy). This research has made use of the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program. The research shown here acknowledges use of the Hypatia Catalog Database, an online compilation of stellar abundance data, which was supported by NASA’s Nexus for Exoplanet System Science (NExSS) research coordination network and the Vanderbilt Initiative in Data-Intensive Astrophysics (VIDA).

Appendix A Comparison of the stellar properties of the small planet and merged radial velocity samples

Fig. A.1 Distributions of stellar metallicity (left panel) and stellar mass (right panel) in the SP sample (red solid line) and merged (AAT+CLS+HARPS) RV comparison sample (black dashed line). Note that the relative stellar counts of the SP and RV merged samples cannot be directly compared as they were normalized from different distributions.

Fig. A.2 Upper panels: same as left panel of Fig. A.1 for the three different bins of stellar mass 0.6–0.8 M⊙ (left), 0.8 – 1.0 M⊙ (center), and 1.0–1.2 M⊙ (right). Note that the relative stellar counts of the SP and RV merged samples cannot be directly compared as they were normalized from different distributions. Lower panels: empirical cumulative distribution functions for the same bins of stellar mass as above. The p-values of the Kolmogorov-Smirnov test are also reported.

Appendix B Completenesses of the considered samples

Fig. B.1 Average completenesses of all the samples considered in this work. As indicated in the color bars, the detection rate/sensitivity increases from dark blue (0%) to green (~50–80%) and yellow (100%).

Appendix C Cold Jupiters and their occurrence rates in the small planet and radial velocity comparison samples

Fig. C.1 The architectures of the systems with both SPs and CJs considered in this work and ordered according to increasing semimajor axis of the innermost planet (from bottom to top). The CJs in the left (27 systems) and right (23 systems) panels have masses 0.3 ≤ Mp ≤ 13 MJup and 0.5 ≤ Mp ≤ 20 MJup, respectively, with a large overlap for CJs with 0.5 ≤ Mp ≤ 13 MJup. The planet circles have sizes proportional to their mass (see legend in the top right corner), and colors indicating orbital eccentricity (more eccentric planets are shown in yellow). Host stars are color coded as a function of stellar metallicity (metallicity increases from blue to red).

Fig. C.2 Similarly to Fig. 1, occurrence rates of CJs with Mp = 0.5–20 MJup in the AAT (orange), CLS (violet), and HARPS (green) RV surveys, at sub-solar ([Fe/H] < –0.1), solar (–0.1 ≤ [Fe/H] ≤ 0.1), and super-solar ([Fe/H] > 0.1) metallicity. Almost identical occurrence rates are obtained for Mp = 0.3–13 MJup. See Table F.2.

Fig. C.3 Same as Fig. 1 for Mp = 0.3–13 MJup. See also Table D.1.

Fig. C.4 Same as Fig. 2 for Mp = 0.3–13 MJup. See also Table D.2.

Appendix D Occurrence rates of cold Jupiters in the small planet and in the large radial velocity comparison samples as a function of stellar metallicity and/or mass

Appendix E Occurrence rates of cold Jupiters in the small planet transit, radial velocity, and transit+radial velocity samples, without and with mixed systems, as a function of stellar metallicity

Appendix F Occurrence rates of cold Jupiters in the AAT, CLS, and HARPS radial velocity surveys

References

All Tables

All Figures

	Fig. 1 Occurrence rates of CJs with Mp = 0.5–20 MJup in the SP sample (red solid line) and the merged RV comparison sample (black dashed line) at sub-solar ([Fe/H] < –0.1), solar (–0.1 ≤ [Fe/H] ≤ 0.1), and super-solar ([Fe/H] > 0.1) metallicity. See Table D.1.
In the text

	Fig. 2 Same as Fig. 1 for three different bins of stellar mass: 0.6–0.8 M⊙ (left), 0.8–1.0 M⊙ (center), and 1.0–1.2 M⊙ (right). See Table D.2.
In the text

Fig. 3 Architectures of 27 systems with both SPs and CJs in the mass range 0.3 ≤ Mp ≤ 13 MJup, ordered according to increasing orbital eccentricity of the CJs (from top to bottom; for multiple CJs, the highest eccentricity is considered). The planet circles have sizes proportional to their mass (see legend in the top right corner), and colors indicating orbital eccentricity (more eccentric planets are shown in yellow). Host stars are color coded as a function of stellar metallicity (metallicity increases from blue to red). All systems with three or more inner SPs are accompanied by CJs with low eccentricities.

In the text

	Fig. A.1 Distributions of stellar metallicity (left panel) and stellar mass (right panel) in the SP sample (red solid line) and merged (AAT+CLS+HARPS) RV comparison sample (black dashed line). Note that the relative stellar counts of the SP and RV merged samples cannot be directly compared as they were normalized from different distributions.
In the text

Fig. A.2 Upper panels: same as left panel of Fig. A.1 for the three different bins of stellar mass 0.6–0.8 M⊙ (left), 0.8 – 1.0 M⊙ (center), and 1.0–1.2 M⊙ (right). Note that the relative stellar counts of the SP and RV merged samples cannot be directly compared as they were normalized from different distributions. Lower panels: empirical cumulative distribution functions for the same bins of stellar mass as above. The p-values of the Kolmogorov-Smirnov test are also reported.

In the text

	Fig. B.1 Average completenesses of all the samples considered in this work. As indicated in the color bars, the detection rate/sensitivity increases from dark blue (0%) to green (~50–80%) and yellow (100%).
In the text

Fig. C.1 The architectures of the systems with both SPs and CJs considered in this work and ordered according to increasing semimajor axis of the innermost planet (from bottom to top). The CJs in the left (27 systems) and right (23 systems) panels have masses 0.3 ≤ Mp ≤ 13 MJup and 0.5 ≤ Mp ≤ 20 MJup, respectively, with a large overlap for CJs with 0.5 ≤ Mp ≤ 13 MJup. The planet circles have sizes proportional to their mass (see legend in the top right corner), and colors indicating orbital eccentricity (more eccentric planets are shown in yellow). Host stars are color coded as a function of stellar metallicity (metallicity increases from blue to red).

In the text

	Fig. C.2 Similarly to Fig. 1, occurrence rates of CJs with Mp = 0.5–20 MJup in the AAT (orange), CLS (violet), and HARPS (green) RV surveys, at sub-solar ([Fe/H] < –0.1), solar (–0.1 ≤ [Fe/H] ≤ 0.1), and super-solar ([Fe/H] > 0.1) metallicity. Almost identical occurrence rates are obtained for Mp = 0.3–13 MJup. See Table F.2.
In the text

	Fig. C.3 Same as Fig. 1 for Mp = 0.3–13 MJup. See also Table D.1.
In the text

	Fig. C.4 Same as Fig. 2 for Mp = 0.3–13 MJup. See also Table D.2.
In the text