# DEEP FUNCTIONAL MULTIPLE INDEX MODELS WITH AN APPLICATION TO SER

Saumard Matthieu  VISION-AD Team, Labisen
ISEN Yncréa Ouest
Brest, France
<EMAIL_ADDRESS>El Haj Abir  VISION-AD Team, Labisen
ISEN Yncréa Ouest
Caen, France
<EMAIL_ADDRESS>Napoleon Thibault  VISION-AD Team, Labisen
ISEN Yncréa Ouest
Brest, France
<EMAIL_ADDRESS>
###### Abstract

Speech Emotion Recognition (SER) plays a crucial role in advancing human-
computer interaction and speech processing capabilities. We introduce a novel
deep-learning architecture designed specifically for the functional data model
known as the multiple-index functional model. Our key innovation lies in
integrating adaptive basis layers and an automated data transformation search
within the deep learning framework. Simulations for this new model show good
performances. This allows us to extract features tailored for chunk-level SER,
based on Mel Frequency Cepstral Coefficients (MFCCs). We demonstrate the
effectiveness of our approach on the benchmark IEMOCAP database, achieving
good performance compared to existing methods.

###### Index Terms:

speech emotion recognition, human-computer interaction, functional data
analysis, deep learning

##  I Introduction

Emotion recognition is an essential aspect of human-robot interaction, as it
allows for a more natural and effective means of communication. Humans
communicate their emotions through various modalities, including facial
expressions, body language, and speech. However, among these modalities,
speech plays a crucial role in emotion recognition, as it is one of the most
reliable and informative ways to convey emotional information. Through speech,
individuals can convey not only the content of their message but also the
underlying emotional state, including tone, pitch, and intonation. Moreover,
speech can also provide insights into the speaker’s personality, cognitive
state, and overall well-being. As such, developing accurate and efficient
methods for speech emotion recognition is critical for creating more effective
and responsive human-robot interfaces.

Support Vector Machines have been shown to be effective for speech emotion
recognition. They are able to learn complex decision boundaries and can handle
high-dimensional data, such as speech. Other types of classifiers that have
been used for speech emotion recognition include decision trees, k-nearest
neighbors, and neural networks. Speech emotion recognition has made rapid
progress in recent years with the use of deep learning and convolutional
neural networks (CNN) [1], transformer, attention and self-supervised learning
[2, 3, 4, 5] methods.

Since the pioneer monographs [6] and [7], functional data analysis (FDA) has
become a vibrant field of research in the statistical community due to it vast
applications in various sciences, including astronomy, chemo-metrics, health
and finance [8, 9, 10]. The core objects of FDA are curves or functions of a
separable Hilbert space like $L^{2}[0,1]$. However, statistical study of
random variables of more general functions space like Banach space [11] or
curves on manifold [12] falls within the spectrum of FDA. The fundamental
frequency curve of an utterance has already been considered as a functional
object [13, 14]. Recently, [15] analysed dialect sound variations across Great
Britain using a spatial modeling approach that employs MFCC. In this article,
we extend this point of view to speech emotion recognition. Each coefficient
of MFCC can be interpreted as a functional data variable. With a collection of
coefficients, it is therefore natural to establish a correspondence between a
speech recording and a multivariate functional data object. The number of
covariates of the multivariate functional data object is the number of
coefficients of the MFCC used. A transformation of the multivariate functional
object is employed to serve as the final multivariate object, which is
considered in a functional multiple-index model.

In our work, we propose a novel approach for speech emotion recognition that
involves treating Mel Frequency Cepstral Coefficients (MFCCs) as a functional
data object. By doing so, we can represent each coefficient as a function of
time, and thus extract information from the speech signal. However, to compare
functional data objects between samples with different duration, we need to
preprocess the MFCC. This is achieved by splitting the MFCC in chunks, which
allows us to represent each sample as a multivariate functional object. An
abundant literature on chunk-level SER exists, see [16],[17] and [18] for
example. This multivariate object can then be transformed into another
multivariate object using a suitable transformation, enabling us to use the
functional multiple-index model for multivariate functional covariate to
classify the emotions of the speaker.

This novel approach is particularly advantageous as it allows us to consider
each MFCC as a functional variable, which captures the dynamic nature of
speech and its relationship to emotions. By using a multivariate functional
object, we can compare it across samples with different duration. Moreover,
the functional multiple-index model allows us to consider the interdependence
between the different coefficients of the MFCC, providing a more accurate and
comprehensive representation of the speech signal. Overall, our approach shows
interesting perspectives for improving speech emotion recognition.

The paper is organized as follows. We highlight the previous work in the
following section. Next, we describe the method in details with our
contributions. We propose to evaluate our method on IEMOCAP database described
in the following fourth section along with the results and comparison with
other methods. We conclude and discuss the results in the final section.

##  II Related work

###  II-A On functional data models

The functional single index models have been studied both from a theoretical
and practical point of view in [19, 20, 21]. Some authors [22] and [23] use
functional additive models that can be more stable than functional multiple-
index models. The authors of [24] use deep neural network strategy to learn
the parameter of the functional linear model by using a new functional neuron
that can learn the functional representation with functional inputs. The
article of [25] proposes a novel neural network that learns the best basis
functions for supervised learning tasks with functional inputs. They called it
AdaFNN. The AdaFNN network parameterizes each basis node with a micro neural
network that outputs a score of the input function $X(t)$, which is the inner
product between the basis function $\theta(t)$ and the input function:

$c=\langle\theta,X\rangle=\int\theta(t)X(t)\text{d}t.$

They introduce neural networks that employ a new Basis Layer whose hidden
units are each basis functions themselves implemented as a micro neural
network. On the link between deep learning and FDA, there exist other approach
based on a functional layer developed by [26] investigated in details by [27]
and [24]. We present in the next section our model and a simulation comparison
with [24]. The advantage of the approach in [25] is that we can use all the
deep learning tools without rewriting the back-propagation algorithm which is
necessary in [24].

Let us introduce the functional single index model in the regression context.
Let $(Y_{i},X_{i})_{i=1,\ldots,n}$ be the set of data where $X_{i}$ represent
the $i$-th functional data variable in $L^{2}[0,1]$, the space of square
integrable function on $[0,1]$, and $Y_{i}\in\mathbb{R}$.

$Y_{i}=g\left(\langle X_{i},\theta\rangle\right)+\varepsilon_{i},$ (1)

where $\varepsilon_{i}$ are the error terms, $\langle.,.\rangle$ is the inner
product in $L^{2}[0,1]$. $g$ and $\theta$ are unknown function to be
estimated. Generally, $g$ is estimated from a nonparametric framework with a
kernel and $\theta$ is estimated from a basis decomposition or a functional
principal component analysis. We propose to extend this definition to a
functional multiple-index model for multivariate functional data covariate,
see section 3.3.

###  II-B On speech emotion recognition

There is a rich literature on speech emotion recognition using various deep
learning architectures. We can cite [28, 29, 30, 31] which use self-attention,
Bayesian neural networks and positional encoding. The article of [32] proposes
to make silence representations of speech. The article of [33] makes progress
by creating self-supervised learning for SER tasks.

##  III Method

Figure 1: Proposed method. SA: Self-Attention, FF: Feed-Forward network, DFN:
Deep Functional Network, FCL: Fully Connected Layers.

###  III-A MFCC

Let us recall the method to calculate the MFCC. With at hand a raw audio
signal data representing by a time series $s(t)$ for $t=1,\ldots,T.$ We can
consider that $s$ is defined for $t\in\mathbb{Z}$ by adding $0$ to non-value.
Let $w_{M}(t)$ for $t\in\mathbb{Z}$ be a window function of width $M$, we can
define the spectrogram of the audio signal $s(t)$ by

$\text{Spec}(t,\omega)=|\sum_{u=1}^{T}s(t-u)w_{M}(u)\exp(-i\omega u)|,$ (2)

for $t=1,\ldots,T,\omega\in[0,2\pi]$.

Hence, we can define the Mel spectrogram, which is a filtered version of the
spectrogram to represent the human ear auditory system:

$\text{MelSpec}(t,f)=\sum_{k=0}^{N-1}\text{Spec}\left(t,\frac{2k\pi}{N}\right)b_{f,k}$
(3)

for $f=0,\ldots,F$, with $b_{f,k}$ representing the set of the Mel-scale
filter bank. Recall that the Mel scale is $m=2595\log_{10}(1+\frac{f}{700})$.
The MFCC are then:

$\text{MFCC}(t,m)=\frac{1}{F}\sum_{f=0}^{F}\log\left(\text{MelSpec(t,f)}\right)\exp(i(2\pi\frac{m-1}{F+1})f),$
(4)

for $m=1,\ldots,n_{MFCC}$.

Note that there exist variations of the definition of MFCC in the literature.
For example, we can use a Discrete Cosinus Transform (DCT) to return to the
time scale.

###  III-B Functional multiple index models

For extracting new features based on MFCC, we employ a deep functional
multiple index model. Let $(Y_{i},X_{i})_{i=1,\ldots,n}$ be the set of data
where $X_{i}$ represent the $i$-th multivariate functional object and $Y_{i}$
is its associated label. Let us introduce the model.

$\displaystyle Z_{i}=$ $\displaystyle T(X_{i})$ (5) $\displaystyle Y_{i}=$
$\displaystyle g\left(\langle Z_{i}^{1},\theta_{1}\rangle,\ldots,\langle
Z_{i}^{p},\theta_{K}\rangle\right)+\varepsilon_{i},$ (6)

where $T:(L^{2}[0,1])^{p}\rightarrow(L^{2}[0,1])^{p}$ is a transformation of
the multivariate functional data, $\theta_{j}$ are the indexes (functions of
$L^{2}[0,1]$), $g:\mathbb{R}^{p\times K}\rightarrow\\{0,\ldots,C-1\\}$ is the
link function, $C$ is the number of classes, $\langle.,.\rangle$ is the inner
product of $L^{2}[0,1]$ functions and the power $j$ of the variable $Z_{i}$ is
the $j$-th component function of $Z_{i}$ as $Z_{i}$ is a multivariate
functional data.

The unknown functions are the transformation $T$, the link function $g$ and
the indexes $\theta_{j}$, $j=1,\ldots,p\times K.$ The transformation $T$ is
inferred by a transformer encoder architecture. Next, we apply an AdaFNN
network to each functional variable, the output represents new features
extracted from the MFCCs. Thus, the $\theta_{j}$ are estimated by a deep
functional network (DFN) based on a concatenation of the AdaFNN module. Then,
we apply a fully connected layer to estimate the link function $g$.

###  III-C Deep neural network of the model

With a speech recording, we calculate the MFCC associated and cut it with an
overlapping method. Then, we can feed the Deep Neural Network with this chunk
of MFFC seen as a multivariate functional object. First, we make a
transformation of the MFCC chunk. There is undoubtedly a link between the size
of MFCC and the information about the emotion of the speech. But we do not
know how is this link. So we apply a transformation $T$ on the resized MFCC to
accurately predict the emotion contained in the speech. The transformation
module is made of a stack of $N$ transformer encoders with self-attention and
Feed-Forward network. The second module is the neural network of the paper
[25] which is generalized to adapt to the multivariate context. In few words,
the paper of [25] proposes to integrate the data by an adaptive function by a
numerical scheme . So, the outputs of this second module are simply the
$L^{2}$ product between each component of the transformed MFCC and an adaptive
function. The third module of our network is a classical fully connected
layer. The proposed method is represented globally in Figure 1.

The three modules of our new network architecture are:

  1. 1.

The transformation module made of $N$ transformer encoders.

  2. 2.

The Deep Functional Network that outputs the scores of multivariate functional
variable.

  3. 3.

A Fully Connected Layers to classify the emotions.

###  III-D Simulations

In order to show the ability of our proposed method, we study three different
scenarios. For completing the study on other potential models using adaptive
layer, we refer to the simulation section of [25]. We simulate four functional
covariates $X^{(j)}\,j=1,2,3,4$ coming from four different processes:
exponential variogram ($j=1$), Brownian ($j=2$), Fractional Brownian ($j=3$)
and Gaussian process with Matérn covariance function. These curves are
evaluated at $30$ equally-spaced time points from $[0;1]$. The unknown
parameter functions are $\beta_{1}(t)=5\sin(2\pi t)$, $\beta_{2}(t)=5\sin(3\pi
t)$, $\beta_{3}(t)=3\cos(2\pi t)$ and $\beta_{4}(t)=3\cos(3\pi t)$. A Gaussian
error of variance $0.04$ has been added to the regression term. Let us
introduce the three different scenarios:

$(S1)\,Y_{i}=g\left(\sum_{j=1}^{4}\langle\beta_{1},X_{i}^{(j)}\rangle,\sum_{j=1}^{4}\langle\beta_{2},X_{i}^{(j)}\rangle\right)+\varepsilon_{i}$

with

$\displaystyle g(a,b)=a^{2}+b^{2}.$

$(S2)\,Y_{i}=g\left(\sum_{j=1}^{4}\langle\beta_{1},X_{i}^{(j)}\rangle,\cdots,\sum_{j=1}^{4}\langle\beta_{4},X_{i}^{(j)}\rangle\right)+\varepsilon_{i}$

with

$\displaystyle g(a,b)=a^{2}+b^{2}+c^{2}+d^{2}.$

$\begin{split}(S3)\,Y_{i}=&(\langle\beta_{1},\tilde{X_{i}}^{(1)}\rangle+\langle\beta_{2},\tilde{X_{i}}^{(2)}\rangle+\\\
\langle\beta_{3},X_{i}^{(3)}\rangle\times&\langle\beta_{4},X_{i}^{(4)}\rangle)^{2}+\\\
(\langle\beta_{1},\tilde{X_{i}}^{(1)}\rangle\times&\langle\beta_{2},\tilde{X_{i}}^{(2)}+\langle\beta_{3},X_{i}^{(3)}\rangle+\langle\beta_{4},X_{i}^{(4)}\rangle)^{2}+\varepsilon_{i}\end{split}$

TABLE I: Performance on simulations. We report RMSE. Scenario | RMSE
---|---
$S(1)$ | $0.085$
$S(2)$ | $0.074$
$S(3)$ | $0.031$

To make an evaluation of the transformation part, we change $X^{(1)}$ by
$\tilde{X}^{(1)}=(X^{(1)})^{2}$ and take $\tilde{X}^{(2)}=|X^{(2)}|$ in the
last scenario. We do not have the code of the paper [24], so we could not
compare to their method. In table I, the simulations reflect a good behaviour
of our approach, even with complex behaviour and transformations in variables.

##  IV Application to SER

###  IV-A Database

The dataset IEMOCAP [34] contains approximately 12 hours of speech from 10
speakers. The literature selects a total of 5531 utterances that are labeled
with one of five categories: Happy, Angry, Neutral, Sad, Exited. This set is
reduced to four categories by merging Exited and Happy into a single category.

For evaluating our method, we use the protocol designed in [35]. Namely, we
perform a $10$-fold Speaker Independent cross-validation with one speaker as
test, eight as training and one as validation set.

###  IV-B Implementation and hyperparameters

The code is written in python and use the pytorch library including its
modules. We use the Adam optimizer with a learning rate of $3\times 10^{-4}$
and a focal loss with the $L^{2}$ penalty on the basis parameter. We set the
batch size to $32$ with $15$ epoch and return the best model on the validation
set. We do not tune the number of basis $K$, we use the best result of [25],
namely $4$ basis functions. We set the number of MFCC $p$ to $40$. And, we
make chunks of the MFCC with a duration of $64$. So, finally, the size of the
final MFCC is $(64,40)$. We use $N=2$ transformer encoder layers. We use in
the DFN of the basis layers an hidden FF of three connected layers of $128$
each. The subsequent FCL network is two dense layer with an $\tanh$ activation
function, dropout of $0.2$, follow up by a projection to the $4$ classes.

###  IV-C Results

We calculate two metrics namely WA for weighted accuracy (overall accuracy)
and UA for unweigthed accuracy (average of the recall). We compare with [36,
2] which are the best results on IEMOCAP with four emotions and speech only,
see Table II. It is worth mentioning that [36] do not only use MFCC but also
Zero-crossing rate (ZCR), root mean square (RMS), Mel vector, chroma. And [2]
use MFCC and first and second order frame-to-frame difference.

TABLE II: Performance of different approaches on the IEMOCAP testing set. We report WA and UA. Method | WA | UA
---|---|---
GRU [36] | $64.95$ | .
AUDIO-CNN [2] | $66.6$ | $68.4$
DFN (ours) | 57.27 | 56.43

The results cannot be compared directly because we make our analysis on chunk-
level. We think that the performance of our approach with the protocol of [35]
can be improve by considering all the chunks of the audio record considered.

##  V Discussion

In this method, we choose to make chunks with an overlap of $25\%$ of the
duration of the chunk. Recently, [16] and [18] choose dynamically the
percentage of overlapping with a significant improvement in accuracy. We can
enhance our method by dynamically choosing the chunk overlapping by using the
method in [16] and [18]. Moreover, the features can be passed to a recurrent
neural network (RNN) (like LSTM, Bi-LSTM, GRU) to make a global decision on
the whole audio input. Adding a RNN on top of the model may enhance the
results, and that can be done in a two-stage or end-to-end learning.

##  VI Conclusion

In conclusion, the article presents a promising advancement in SER using a
novel model. The results, validated through simulations and tested on the
IEMOCAP dataset at the chunk-level, demonstrate satisfactory performance. The
model leverages new features extracted from MFCC and relies on functional
data, showcasing an innovative approach to emotion detection in speech. This
development not only contributes to the ongoing evolution of SER methodologies
but also highlights the potential for further exploration of functional data
in enhancing emotion recognition systems.

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# Spectroscopic follow-up of Gaia exoplanet candidates:
Impostor binary stars invade the Gaia DR3 astrometric exoplanet candidates.

Marcus L. Marcussen Stellar Astrophysics Centre, Department of Physics and
Astronomy, Aarhus University, Ny Munkegade 120, 8000 Aarhus C, Denmark
E-mail<EMAIL_ADDRESS>Simon H. Albrecht Stellar Astrophysics
Centre, Department of Physics and Astronomy, Aarhus University, Ny Munkegade
120, 8000 Aarhus C, Denmark

###### Abstract

In this paper we report on the follow-up of five potential exoplanets detected
with Gaia astrometry and provide an overview of what is currently known about
the nature of the entire Gaia astrometric exoplanet candidate sample, 72
systems in total. We discuss the primary false-positive scenario for
astrometric planet detections: binary systems with alike components that
produce small photocenter motions, mimicking exoplanets. These false positives
can be identified as double-lined SB2 binaries through analysis of high
resolution spectra. Doing so we find that three systems, Gaia DR3
1916454200349735680, Gaia DR3 2052469973468984192, and Gaia DR3
5122670101678217728 are indeed near equal mass double star systems rather than
exoplanetary systems. The spectra of the other two analyzed systems, HD 40503
and HIP 66074, are consistent with the exoplanet scenario in that no second
set of lines can be found in the time series of publicly available high
resolution spectra. However, their Gaia astrometric solutions imply radial-
velocity semi-amplitudes $\sim$ 3 (HD 40503) and $\sim$ 15 (HIP 66074) larger
than what was observed with ground based spectrographs. The Gaia astrometry
orbital solutions and ground-based radial-velocity measurements exhibit
inconsistencies in six out of a total of 12 exoplanet candidate systems where
such data are available, primarily due to substantial differences between
observed ground-based radial-velocity semi-amplitudes and those implied by the
Gaia orbits. We investigated various hypotheses as to why this might be the
case, and though we found no clear perpetrator, we note that a mismatch in
orbital inclination offers the most straightforward explanation.

##  1 Introduction

The detection of exoplanets through the astrometric reflex motion of their
host stars has been predicted to revolutionize the field of exoplanetary
science by increasing the total number of known exoplanets by a factor
${\sim}$ 4 after the nominal 5 yr mission of Gaia (Gaia Collaboration et al.,
2016) and a factor ${\sim}$ 14 for the extended 10 yr mission (Perryman et
al., 2014). A benefit of the astrometric planet detection method over the
Radial-Velocity (RV) method is that it provides the full orbital solution and
the mass of the planet, though obtaining the precision needed for detecting
planetary-mass companions is comparatively harder with this method. The
astrometric detection sensitivity increases with orbital separation, opposite
to the RV and transit methods which are most suitable for shorter period
systems. This makes the methods highly complementary.

Figure 1: Astrometric exoplanet candidate systems in an HR diagram. All Gaia
DR3 systems where the astrometric two-body motion of the source is small
enough to indicate a companion with a mass $<20M_{\rm Jup}$, assuming the
companion does not contribute any flux, are shown here as circles. The
released Gaia astrometric exoplanet candidates are shown using closed circles
while open circles indicate systems where a lower bound on the flux of the
companion is non-zero as per the binary_masses table. Large circles indicate
systems where the nature of the companion is confirmed, either here or in the
literature, see also table 1. The systems are colored according to the current
knowledge of the nature of the companion, where brown indicates a planetary
companion, orange indicates a stellar companion and cyan indicates that it is
unknown. The two white dots indicate two white dwarf hosts where the companion
nature is unknown. The comparison stars shown in the background are the 1 000
000 nearest to us Gaia DR3 stars whose estimated parallax is more than 10
times its uncertainty. The density of these stars is represented as a purple-
to-yellow heatmap, where purple (yellow) indicates a low (high) density. Since
most of the planet candidates are too faint to have Gaia DR3 reddening and
extinction parameters, candidates and comparison stars alike are shown using
their uncorrected colors and magnitudes. The black dashed line represents the
best fit to the Main Sequence. The six systems discussed in detail in section
4 are indicated by their name.

On June 13 2022 the Gaia Data Release 3 (DR3) was published (Gaia
Collaboration et al., 2022a). This is the first data release where non-single
source solutions are included. A single Keplarian astrometric orbit model is
used. The 169 277 orbital solutions released were compiled from 34 months of
Gaia observation (Halbwachs et al., 2022), illustrating the potential of
astrometric planet searches. However, the majority of these solutions stem
from double star systems, which produce much larger astrometric signals than
planetary systems. A small number of the released orbital solutions (1162) are
from the more resource-intensive ”exoplanet” pipeline (Holl et al., 2022a)
aimed at low signal-to-noise ratio targets with substellar and particularly
exoplanetary mass companions. The inputs for this exoplanet pipeline were
either targets where the two body orbit model of the ”binary” pipeline (Gaia
Collaboration et al., 2022b) did not improve enough the single star fit (the
orbitalAlternative solutions) or were pre-selected targets
(orbitalTargetedSearch). Many of the pre-selected targets were chosen because
they were known to harbor planets or brown dwarfs beforehand. In total 1843
astrometric brown dwarf and 72 exoplanet candidates were published in DR3
(Gaia Collaboration et al., 2022b).

Figure 2: Exoplanet candidate parameters. Top panel: Companion masses of the
same sample as shown in Fig. 1 are calculated under the assumption that their
flux contribution is negligible compared to their host and assuming Gaussian
uncertainties in photocenter semi-major axis a0, orbital period P and primary
mass $M_{1}$. Middle panel: The G magnitudes of the sample. Bottom panel: The
size of the photocenter semi-major axis relative to its uncertainty, which can
be taken as a proxy for the robustness of the Keplarian orbit detection.

An overview of all 102 systems with astrometric signals small enough to
naively imply a companion mass $<20M_{\rm Jup}$ are shown in Fig. 1 in a
color-magnitude diagram. In order to get a sense of the prevalence of binary
systems to exoplanetary systems, the figure includes 30 systems that are not
among the 72 released exoplanet candidates
111https://www.cosmos.esa.int/web/gaia/exoplanets, due to their high
likelihood of being binaries. For the masses of the primary stars we used the
$m_{1}$ masses from the binary_masses table where available. For a few systems
where these were not available we used the single-source mass_flame from the
astrophysical_parameters table. The systems are colored according to the
current status of their nature, also available in table 1. A subset of the
systems in this table have already been studied for consistency between the
astrometric and radial velocity data by Holl et al. (2022a), Gaia
Collaboration et al. (2022b) and Winn (2022). We discuss their results in the
sections of the specific systems. In Fig. 2 we show for these same systems the
companion mass estimates, apparent magnitudes, and the relative uncertainty of
their astrometric orbits. We note that only 17 of the systems have companion
mass estimates below the often-used planet/brown dwarf limit of $13M_{\rm
Jup}$.

With only one previous discovery (Curiel et al., 2022), astrometry is not yet
a well-developed technique for the detection of exoplanets. The method comes
with its own set of unique challenges. In particular, the photocenter (the
apparent position of the objects) motion of a binary system can be arbitrarily
small, and thereby mimic a low-mass companion, as the flux ratio of the two
stars approaches their mass ratio. In order for Gaia to achieve its full
planet detection potential we need to understand better the nature of these
challenges, their prevalence as well as provide paths to overcome them. In
this work we discuss possible steps to confirm or reject the planet status of
an astrometric candidate system using high resolution spectra. Specifically we
are employing both publicly available spectra (of the three potentially new
astrometric exoplanet discoveries discussed in Holl et al. 2022a and Gaia
Collaboration et al. 2022b) as well as new FIES spectra (of another three
exoplanet candidates) that we obtained from the Nordic Optical
Telescope.222These spectra have been obtained through the Fast-Track channel
available at the observatory. http://www.not.iac.es/observing/proposals/

Our paper is structured as follows: We first summarize important aspects of
astrometric binary signals, section 2, and discuss how to differentiate
between genuine planet detections and false positives, section 3. In section 4
we analyze six exoplanet candidate systems, and discuss our findings in
section 5.

##  2 Astrometric signal

A star and its companion orbit their common barycenter. If the companion is an
exoplanet, i.e. dark and low mass, the observed astrometric reflex motion of
the source is small and coincides with the motion of the star. If the
companion is another star though, i.e. it has a higher mass and contributes
flux, the actual motion of the primary star is then larger, but the
observable, the photocenter of the system, is located between the two
components. Its astrometric amplitude is smaller than the actual motion of the
primary star and can be arbitrary small given the system parameters. In this
section we outline how the observed semi-major axis relates to the flux and
mass ratios of the components in systems where the primary component has a
single companion (subsection 2.1) and in systems where an inner pair of stars
orbit an outer companion (subsection 2.2).

The total astrometric motion of a star orbited by a single companion is the
linear combination of the 5-parameter single source astrometric model and a
Keplarian 7-parameter model. The single-source model consists of the two
position parameters RA & Dec, the proper motions, $\mu_{\rm RA}$ & $\mu_{\rm
Dec}$, and the parallax, $\varpi$. The Keplarian model consists of the orbital
period $P$, time of perisastron passage $T_{0}$, eccentricity $e$, semi-major
axis of the star’s orbit $a_{1}$, the orbital inclination $i$, the argument of
periastron $\omega$, and the right ascension of the ascending node $\Omega$.
The latter four parameters are the Campbell elements that may also be
expressed by the Thiele Innes coefficients A, B, F, G. For more information on
the Thiele Innes coefficients and how they are derived from the Gaia along-
scan measurements, see Gaia Collaboration et al. (2022b).

###  2.1 Binary systems

For systems with a single companion, the semi-major axis of the relative orbit
$a$ is the sum of the semi-major axes $a_{1}$ and $a_{2}$ around the
barycenter of the star and companion respectively, see Fig. 3. The ratio of
the semi-major axes is equal to the inverse ratio of the masses,
$M_{2}/M_{1}=a_{1}/a_{2}\equiv q$, giving

Figure 3: Photocenter. Not-to-scale illustration of three different geometries
of face-on, circular orbits that the same astrometric signal may originate
from. Left panel: An exoplanet orbiting a star. The semi-major axis of the
exoplanet orbit, $a_{2}$, is much larger than the orbit of the star, $a_{1}$,
due to the large difference in mass. The flux contribution from the planet is
negligible and thus the photocenter closely follows the star. Middle panel:
Twin stars (i.e. almost equal flux and mass stars) orbit their barycenter
(black dot). The primary component, the yellow star, is slightly more massive
than the secondary component, the red star, leading to a smaller semi-major
axis $a_{1}$ than $a_{2}$. The ratio of the masses is slightly higher than the
ratio of the received fluxes, $M_{2}/M_{1}>F_{2}/F_{1}$, resulting in a small
photocenter semi-major axis $a_{0}$ that traces out (a diminished version of)
the orbit of the primary component. Right panel: Two stars with significantly
different masses and fluxes in a hierarchical triple system, where the center-
of-mass of the red and yellow star orbit around a larger and brighter third
component that is very far away and shown in white. The flux of the third
component diminishes the size of the photocenter semi-major axis but does not
contribute to the astrometric signal due to its very long period. The
photocenter semi-major axis would be larger and more closely follow the yellow
star if not for the third component.

$a_{1}=a\frac{M_{2}}{M_{2}+M_{1}}=a\frac{q}{q+1}\,.$ (1)

However, in completely unresolved systems where the secondary component has a
non-negligible flux, we observe the motion of system’s photocenter, i.e. the
flux-weighted mean position of the two components. This gives the following
definition for the semi-major axis of the photocenter $a_{0}$:

$a_{0}=\left|\frac{F_{1}a_{1}-F_{2}a_{2}}{F_{1}+F_{2}}\right|.$ (2)

Here $F_{i}$ indicate the flux of the i’th component (in the band-pass in
which the astrometric signal is measured). Trivially, the astrometric
observable, $a_{0}$, is equal to $a_{1}$ when the companion is dark
($F_{2}=0$). This scenario is illustrated in the left panel of Fig. 3. A ratio
of the fluxes of $\epsilon\equiv F_{2}/F_{1}$ equal to one leads to
$a_{0}=a_{1}-a_{2}$. If $q$ is also $1$, then the photocenter remains in the
barycenter of the system. In the case that $F_{2}$ is non-zero the semi-major
axis of the photocenter is given by:

$\displaystyle a_{0}$
$\displaystyle=\left|\frac{F_{1}a_{1}-F_{2}a_{2}}{F_{1}+F_{2}}\right|=\left|\frac{a_{1}-\epsilon
a_{2}}{1+\epsilon}\right|$ $\displaystyle=\left|\frac{a-a_{2}-\epsilon
a_{2}}{1+\epsilon}\right|$
$\displaystyle=\left|\frac{a}{1+\epsilon}\left(1-\frac{1}{1+q}\left(1+\epsilon\right)\right)\right|$
$\displaystyle=\left|a\left(\frac{1}{1+\epsilon}-\frac{1}{1+q}\right)\right|.$
(3)

And making use of Kepler’s third law we find:

$\displaystyle a_{0}^{3}$
$\displaystyle=\left|GM_{1}\left(\frac{P}{2\pi}\right)^{2}\left(1+q\right)\left(\frac{1}{1+q}-\frac{1}{1+\epsilon}\right)^{3}\right|\,.$
(4)

The gravitational constant is indicated by $G$. A tiny photocenter movement
may thus originate from two common but astrometrically indistinguishable
scenarios: I) A single star orbited by a dark, low-mass companion or II) A
double-star system with $\epsilon\approx q$, illustrated in the left and
middle panels of Fig. 3 respectively. Therefore in order to confirm an
astrometric exoplanet candidate detected by the small astrometric reflex
motion of its host star, additional measurements using different methods are
needed.

We note that if $q$ exceeds $\epsilon$, the location of the photocenter is
between the barycenter and the secondary component rather than the primary.
This will result in a $180^{\circ}$ phase-shift in $\omega$ of the photocenter
orbit. With RV measurements of the primary component, one would in principle
be able to distinguish between $q>\epsilon$ and $q<\epsilon$ by the opposing
sign of the RV signal to that inferred by the astrometric photocenter motion.
However, because the astrometric motion is the sky-projection of the three-
dimensional orbit one cannot a priori distinguish between the configurations
($\Omega$, $\omega$) and ($\Omega+180^{\circ}$, $\omega+180^{\circ}$). Even
though we can resolve this ambiguity of the photocenter orbital configuration
by combining RV data with an astrometric solution, we thus still face
uncertainty about whether the orbit corresponds to the orbit of the primary or
secondary component of the system. In other words, if the RV signal’s sign
contradicts our expectations from the photocenter orbit, it remains unclear
whether this discrepancy arises from an incorrect assumption about $\omega$ or
from a larger value of $q$ compared to $\epsilon$. In order to truly lift this
degeneracy, second order effects like the light-travel time (light has a
shorter travel distance when the component is on ”our side” of the sky-plane)
and local perspective effects (the parallax is larger when the component is on
”our side” of the sky-plane) would need to be taken into account (Halbwachs,
2009). These effects are not modeled in Gaia DR3. If we do not know whether
the photocenter orbit traces out the orbit of the secondary or primary, i.e.
if $q$ is larger or smaller than $\epsilon$, and if we have no prior on the
flux ratio of the components, i.e. $\epsilon$ is anywhere between 0 and 1,
knowledge of $a_{0}$ alone is not enough to determine $M_{2}$.

###  2.2 Triple systems

Another configuration that could conceivably produce a small photocenter semi-
major axis, $a_{0}$, is a hierarchical triple star system where a large
amplitude astrometric signal from a double star system is muted by the
presence of a brighter third component further away. If the distance to the
third component is sufficiently large (but not so large as to be resolvable),
its long term contribution to the astrometry would be undetectable in the Gaia
data but the amplitude of $a_{0}$ would be diminished by a factor

$(F_{1}+F_{2})/F_{3}.$ (5)

Where $F_{3}$ denotes the flux of the third star in the relevant band pass.
This is illustrated in the right hand panel of Fig. 3. The third component
might also be a non associated source i.e., a chance alignment.

##  3 Differentiating between stellar and planetary companions

In this section we describe ways to determine the nature of a companion given
the detection of an astrometric Keplarian signal with a small photocenter.

###  3.1 Color-magnitude relation

The luminosity of a star on the Main Sequence can be well constrained by its
effective temperature. A system of two identical, unresolved stars has twice
the luminosity of a similar single star but the same effective temperature, in
turn elevating its position relative to the single star Main-Sequence in an
HR-diagram. This fact allows for an estimate of the flux ratio $F_{1}/F_{2}$
using isochrones (Gaia Collaboration et al., 2022b) giving a first indication
of whether an astrometric two-body orbit is caused by a stellar or substellar
companion. In Fig. 1 we can observe how all the systems whose flux ratio have
been estimated to be non-zero this way (indicated by open circles), are
located clearly above the single star Main Sequence, i.e. their luminosity is
higher than expected, given its observed color, if the system were a single
star. For DR3 systems with both astrometric and spectroscopic solutions, first
$M_{2}$ is derived from the RV semi-amplitude, K, via equation 3 in Gaia
Collaboration et al. (2022b), and then the flux ratio is inferred from how
much smaller the astrometric signal is (our equation 2.1). The masses shown in
Fig. 2 are all calculated using eq. 4 from the astrometric semi-major axis
alone, i.e. using $F_{2}=0$ for all systems. We do this in order to get a
better sense of the ratio of stellar companions to substellar companions among
all orbital solutions with small $a_{0}$.

###  3.2 Spectra

High resolution spectroscopic follow-up observations can be used to reveal the
impostor binaries among the astrometric exoplanet candidate systems. Spectra
from star-planet systems are fundamentally different from star-star systems:
only one set of absorption lines are present and the radial velocity of the
host star is much smaller due to the lower mass of the companion. This means
the Doppler-shift of the spectrum over time is much smaller. Depending on
whether an impostor binary system consists of two truly identical twins (i.e.
have very similar effective temperatures, luminosities, rotation speeds and
masses) or are significantly different stars whose $q$ happen to be close to
$\epsilon$ (e.g. a red giant and a Main Sequence star), the two sets of
oppositely shifting absorption lines produced by the binary system may present
themselves differently:

#### Uneven twin impostors

If one of the component is of a very different stellar class and/or has a very
different rotation rate, the absorption lines from a single component may
dominate the spectrum of the system. Such a system would be classified as a
single lined binary (SB1) system. Impostors of this type allow us to measure
the orbital RV semi-amplitude of the component most visible in the spectra,
which will be significantly larger than if it were an exoplanetary system. If
an astrometric exoplanet candidate is in fact an SB1 binary, a few
observations at different orbital phases are enough to identify it as such. If
the orbital period and time of periastron passage are well determined from the
astrometry such that the follow-up observations are ensured to be undertaken
at advantageous points of the orbit, just two observations may be enough.

#### Identical twin impostors

If, on the other hand, the two stars are very alike in every way, their
absorption lines will be similar in shape and size. Additionally, the RV shift
of each stars absorption lines will be equal in amplitude but with opposite
signs. These impostors may be identified as double lined (SB2) binaries rather
than exoplanets with a single observation, if the RV separation of the two
stars is of the order of (or larger than) the combined absorption line-width
of the stars and the spectral resolution is sufficient. In that case, their
identification as false positive candidates is trivial.

However, depending on the spectral resolution of the instrument, the orbital
phase of the observations, the orbital inclination, the period, the masses of
the components and the absorption line-widths, each stars absorption lines may
not always be clearly separated. When that is the case, the RV shift of the
two sets of absorption lines may be identified as a change of the line width,
i.e., the FWHM (Full Width at Half Maximum), of the combined line. Observing a
small RV semi-amplitude of what appears to be a line of a single star does
therefore not rule out the possibility that the companion is an identical
stellar twin. The midpoint of this combined line from two unresolved sets of
absorption lines will RV shift slightly, given by small flux and/or rotation
speed differences of the two stars. We simulated the RV shift of a combined
line from two almost identical twin binaries to test whether its small RV
semi-amplitude could mimic that of a star orbited by a planetary companion.

Fig. 4 illustrates this. Here we assumed a simple $\epsilon=q^{4}$ mass-
luminosity relation, though as long as the mass ratio is always closer to $1$
than the flux ratio, the steepness of the relation did not qualitatively
change our results. We then varied the $\epsilon/q$ ratios of the simulated
systems from $0$ to $1$ and calculated the mass of the companion as inferred
by the astrometric method and the RV method, see the left panel of Fig. 4. The
width of the simulated absorption lines of the primary and secondary
components were kept equal. In the right panel we show that as long as the RV
semi-amplitude is $\lesssim$ $v\sin i$ of the components, the determined
companion mass as determined by the two methods is very similar. However, the
ratio of the RV determined mass to the astrometric mass,
$M_{2,RV}/M_{2,astrometry}$ becomes increasingly sensitive to small changes in
rotation speed differences between the components and in the mass ratio of the
components as $\epsilon/q$ approaches $1$. In conclusion, if the spectral
resolution is low, both the astrometric and RV data may falsely – but
consistently between the data sets – indicate the presence of an orbiting
exoplanet for some systems. In those cases, changes in the line width will be
a better indicator for identifying the stellar nature of the companion.

Figure 4: Astrometry and RV simulations. Left: The mass of a companion as
determined by astrometry (dashed line) and RVs (solid lines) relative to its
actual mass as a function of the flux ratio divided by the mass ratio,
$\epsilon/q=(F_{2}/F_{1})/(M_{2}/M_{1})$. The line colors indicate the ratio
of the RV semi-amplitude to the absorption line widths, $v\sin i$. Right: The
ratio of the RV determined mass to the astrometry determined mass.

The Gaia spectral line broadening parameter vbroad, which is the median of
epoch vbroad measurements from the Gaia Radial Velocity Spectrometer, may be
useful for identifying twin binary systems without the need for ground-based
follow-up. The median vbroad of the 412 Gaia non-single-source solutions with
flux ratios above 0.25 is 15.9 km s-1, whereas it is 10.9 km s-1 for the 19
799 systems with flux ratios below 0.25. Unfortunately, vbroad is only
available for stars brighter than G${}_{\rm mag}12$ and most current
astrometric exoplanet candidates are found around fainter stars (see Fig. 2).

Figure 5: HD 68638. Top left: The geometry of the Gaia two-body orbital
solution for the photocenter is shown with north pointing up and east pointing
left. The Gaia parameters are printed as well. The barycenter is shown as a
black dot and the star is shown at periastron as an orange dot with the
direction of motion indicated by the black arrow. The line through the
barycenter represents the orbital angular momentum vector. The positive
direction of the angular momentum is marked with an arrow that is occluded by
the orbital disk in this particular view. The light gray part of the orbit
with a solid orange outline indicates the near side of the sky plane and the
dark gray part with a dashed outline indicates the far side of the sky plane.
The blue circles are the location on the orbit of the RV measurements. Bottom
left: CCFs from all 27 publicly available ELODIE spectra are shown and colored
according to their FWHM. Top right: The solid black line indicates the
expected radial velocity curve of the primary component based on the Gaia-
derived orbital and stellar parameters assuming two equal-mass companions. The
dashed line represents an orbit with a higher eccentricity of $0.6$. Middle
right: The FWHM of the CCFs are shown as blue dots as a function of orbital
phase. The solid black line is the FWHM as a function of orbital phase
inferred from the Gaia orbit. The dashed black line represents the FWHM of a
more eccentric orbit that is a better fit to the observed FWHM. Bottom right:
CCFs plotted with a horizontal offset so their midpoints are located under
their respective FWHM data points of the middle right panel.

##  4 Example systems

In this section we analyze spectra of six astrometric exoplanet candidate
systems to clarify the nature of the companions that cause a Keplarian motion
of their host star. The first three systems, HD 68638, HD 40503 and HIP 66074,
also highlighted by Holl et al. (2022a), have publicly available archival high
resolution spectra that we reanalyzed. The latter three systems, Gaia DR3
2052469973468984192, Gaia DR3 1916454200349735680 and Gaia DR3
5122670101678217728, were investigated using high resolution spectra that we
obtained with the FIES spectrograph installed at the Nordic Optical Telescope
(NOT). The RVs inferred from these spectra enable us to supply the non-
degenerate orbital solutions for the photocenters of the former three systems.
Below we discuss our findings for each system separately.

###  4.1 HD 68638

HD 68638 (Gaia DR3 1035000055055287680) is a G magnitude $7.3$ system whose
orbital solution was highlighted in Holl et al. (2022a) and Gaia Collaboration
et al. (2022b) because of its small photocenter semi-major axis. Despite its
slightly elevated position in the HR diagram (Fig. 1), its Gaia DR3 lower
bound flux ratio is $0$. Although this system has previously been
characterized as a binary system (Busà et al., 2007), in order to test our
methodology and to better understand the characteristics of false-positives,
we reanalyzed the 27 publicly available ELODIE spectra of this system. In the
bottom left panel of Fig. 5 we present the CCFs (Cross-Correlation Functions)
obtained from these spectra along with the Gaia astrometric two-body orbital
solution in the top left panel. Where the relative RVs are largest the CCF
displays two peaks with similar widths and heights, confirming the twin SB2
status of this system. The same conclusion was reached by Holl et al. (2022a)
and Gaia Collaboration et al. (2022b). In the top right panel of the same
figure we show the RV curve implied by the Gaia astrometric solution. To
illustrate how the FWHM correlates with orbital phase, we show in the middle
right panel the expected FWHM of the average absorption line as implied by the
Gaia orbit. The measured FWHM of the ELODIE spectra are superimposed on this.
By increasing the eccentricity of the Gaia orbit (shown as a dashed line) from
0.32 to 0.6 we get a better fitting orbit solution with the observed FWHM. For
the sake of consistency we calculated the FWHM of the total CCF even when the
CCF splitting is apparent.

###  4.2 HD 40503

Figure 6: HD 40503. Shown here is the geometry and orbital parameters of HD
40503 as well as RV measurements in red and blue from the HARPS and CORALIE
spectrographs respectively. The black dashed line represents the best fit to
the RV data alone.

HD 40503 (Gaia DR3 2884087104955208064, HIP 28193) was discussed in Holl et
al. (2022a) and Gaia Collaboration et al. (2022b) as a potential new exoplanet
discovery, because the $826$d orbital period derived by Gaia is consistent
with the periodicity in publicly available Coralie and HARPS spectra. In Fig.
6 we show CCFs from these spectra as well as the FWHM as a function of orbital
phase, none of which indicate binarity. Because we see no indications that the
companion is a binary, we compare the Gaia RV curve to the HARPS and CORALIE
data points (top and right panels of Fig. 6). Although we also find the Gaia
period of $826\pm 50$ d to be consistent within $2\sigma$ of our $758$ d best
fit to the RV data, the RV semi-amplitudes are highly inconsistent: The Gaia
photocenter semi-major axis of $a_{0}=0.010$ au ($0.25$ mas) implies a
companion mass of $5.18\pm 0.59\,M_{\rm Jup}$, assuming $F_{2}=0$, whereas the
observed RV amplitude would imply a companion mass of $1.55\pm 0.18\,M_{\rm
Jup}$. The observed discrepancies are possibly due to stellar activity
affecting the RVs as noted by Holl et al. (2022a) and Winn (2022). However, it
is interesting that the grounds-based RVs of HIP 66074 (see also below) as
well as HR 810 (Winn, 2022) also have much smaller RV semi-amplitudes than
their Gaia orbit counterparts.

###  4.3 HIP 66074

Figure 7: HIP 66074. Shown here is the Gaia geometry and orbital parameters of
HIP 66074 as well as RV measurements in blue derived from HIRES spectra. The
Gaia astrometry and RV data are in excellent agreement except for a large
factor 15 disagreement in the RV amplitudes between the data sets. In the
bottom left we show CCFs that we obtained from the HIRES spectra. In contrast
to Fig. 5, the CCFs from the different observations are so similar that they
look like one, ruling a stellar companion.

The system HIP 66074 (Gaia DR3 1712614124767394816) was discussed by Holl et
al. (2022a) and Winn (2022) because of its small photocenter semi-major axis
of $0.0075$ au (0.21 mas), which is consistent with a $6.9\pm 1.1M_{\rm Jup}$
mass companion. In Fig. 7 we show our CCFs derived from the public HIRES
spectra that show no sign of double lines or FWHM changing with orbital phase,
ruling out a twin binary system.

The orbital period, eccentricity, time of periastron passage, and argument of
periastron from the Gaia orbital solution and the best RV fit to the ground-
based data are all within just $1\sigma$ of each other (Winn, 2022). However,
there is an enormous factor 15 semi-amplitude discrepancy between the ground-
based RV data and Gaia astrometric data as highlighted by Holl et al. (2022a)
and Winn (2022). Interestingly, although they caution its interpretation, Winn
(2022) do find an excellent fit between the two data sets with the only caveat
that the companion needs a substantial flux ratio of $\epsilon=1\pm 0.12\%$.
This flux ratio is well above the mass ratio of $q=0.063\%$ which would imply
that the photocenter orbit is a scaled-down version of the companion’s orbit
and not the host star’s orbit. It also implies a seemingly unrealistic
luminosity of the planet: assuming the companion has a radius of $1R_{\rm
Jup}$, radiating 1% of the flux seen from HIP 66074 as a blackbody puts its
effective temperature at around $3\,400$ K! To find more plausible
explanations to the large semi-amplitude discrepancy, we considered several
hypotheses. Below we go through these and discuss the challenges each has:

#### There is no companion

The independent detections of an orbit with similar orbital parameters by Gaia
astrometry and HIRES RVs makes it very unlikely that the Keplarian motion of
the source is a simple case of a false-positive detection.

#### Stellar activity affecting the RVs

Perhaps, as may be the case for HD 40503, stellar activity affects and mutes
the RVs? If that were the case, we would not expect the RV semi-amplitude to
be the only affected parameter. We might also expect deformations in the CCFs,
which we do not see (Fig. 7).

#### Additional planets

Would affect the reflex motion of the host star and thus the RVs and could
erase or mute a two-body orbital signal. However, because the astrometric and
RV data would be affected equally by additional planets, it does not describe
the large observed RV semi-amplitude difference. Additionally, if the motion
of the host star is the linear combination of several Keplarian orbits, the
simple two-body model would not fit the data as well as it does.

#### The Gaia semi-major axis is overestimated

If the actual size of $a_{0}$ was a factor 15 smaller than the reported value
of $0.21\pm 0.03$ mas then the two independent data sets would agree. The
issue with this explanation is that an astrometric two-body orbit with an
angular diameter of $0.014$ mas would then have been detected, which is well
below the precision of Gaia.

#### Triple system

Another potential explanation is that we are observing the orbit of a binary
system which itself is part of a hierarchical triple system (Fig. 3, right
panel). In such a system the astrometric and RV signature of the binary orbit
could be muted by different amounts, preserving all observed characteristics
of the orbit other than their apparent amplitudes. The mechanism is as
follows: The hypothetical spectrum from such a system would be a combination
of lines from all three components, but only the spectra from the two binaries
would shift in velocity space. If the components are not separately identified
in the spectrum, the obtained RV signal would be muted relative to a signal
from just the binary system. The astrometric signal of the close binary would
also be muted by the polluting light from the third component, but not
necessarily by the same amount as the RV signal. As shown in section 2, the
astrometric signal is muted by a factor $(F_{1}+F_{2})/F_{3}$ by a bright,
third component on a long orbit.

The boundary conditions of this scenario are i) the third component must be on
a distant enough orbit that the corrections to the two-body orbit of the
double-stars is undetectable and ii) it is close enough to the binary system
that it cannot be spatially resolved by Gaia. To get an estimate of the
possible range of orbital periods where both of these criteria are fulfilled,
we conservatively assume a total mass of the triple system equal to the
binary_masses $m_{1}$ mass. This puts an upper limit on the orbital period of
the third component’s orbit of around 16.5 years. To calculate this, we used
its distance to us of 35 pc and Gaia’s 0.23 arcsec two-body resolving
limit333https://www.cosmos.esa.int/web/gaia/science-performance. On the near-
end the orbital period would need to be several times the 34 month baseline of
DR3 to not significantly impact the two-body orbital solution. However, any
unresolved companion with an intermediate orbital period ($\sim 4-75$y, Gaia
Collaboration et al. (2022b); Kervella et al. (2022)) can also be probed by
the difference in proper motion it will have caused between Gaia and HIPPARCOS
(HIP, 1997) single source proper motions. The proper motion anomaly between
these catalogues for HIP 66074 is only $\chi^{2}=0.072$ (Brandt, 2021), where,
for reference, a three sigma difference corresponds to $\chi^{2}=11.8$,
leaving little room to hide for a third component. Even so, to test the
scenario we extended our simulation of RV and astrometric measurements of
binary systems, described in section 3, to encompass a third component. We
also allowed the color, rotation speed and flux ratio of the components to
vary significantly. Lastly, we took into account the slightly redder band pass
of Gaia compared to HIRES. Depending on the configuration of stars, the band
pass differences can amplify the relative brightness and thus the astrometric
signal of two red binary stars relative to a hotter and bluer third component
as much as a factor $\sim$ 1.9. The effects on RVs by the inclusion of a third
component in the spectrum depends not only on the flux ratios of the
components but also the widths of the absorption lines. A faster (slower)
rotation of the third star compared to the two close components diminishes
(increases) the impact of the third star on the combined spectrum, and thereby
on the extracted RVs from that spectrum. The exact influence is further
modified by the ratio of the rotational to orbital velocities at the epoch of
observations and the RV extraction method, e.g. template matching, CCF, or the
iodine method. Our simulations showed RV signals diminished up to $\sim$5-6
times (1.9 from the band pass and $\sim$2.5-3 from differences in flux and
rotation) relative to the astrometric one but none of the configurations we
tested were able to reproduce all the observables to a satisfying degree.

Additionally, our simulations showed that the RVs were muted in an
increasingly non-linear fashion as the influence of the third component
increased. The resulting deformed shape of the RV curve meant worse fitting
RVs at best and an unrecognizable RV curve at worst. We note that the HIRES
observations employed the iodine method (e.g. Butler et al., 1996) and we
tested here the effect of the third star on the CCFs and its RVs as measured
by a Gaussian. We do not expect that the effect of the third star on the HIRES
RVs to be exactly the same.

#### Nearly face-on orbit

An ”easy” way to get a large difference in semi-amplitude between the
astrometric and RV orbits is if the orbit of HIP 66074 is seen nearly face-on
rather than edge-on. The orbital inclination of $i=90.3\pm 4.67^{\circ}$, as
reported in Gaia DR3, does make this explanation seem unlikely. However, the
one-dimensionality of the Gaia along-scan abscissa measurements for targets
dimmer than G = 13 can give rise to degenerate solutions of the two-body
models that are not necessarily close in parameter-space (Holl et al., 2022b).
It may also be worth noting that the majority of exoplanet candidates lie
above this exact magnitude threshold. The issue will be more prominent if the
number of data points is not much larger than the number of free parameters of
the model. As mentioned in section 1, the Gaia nss_two_body_orbit solutions
have 12 free parameters. In the case of HIP 66074, 38 measurements spread out
over 28 distinct visibility periods444A visibility period is defined as ”a
group of observations separated from other groups by a gap of at least 4 days”
as paraphrased from the Gaia online documentation,
https://gea.esac.esa.int/archive/documentation/GDR3/ . were used to calculate
the astrometric solution. 28 visibility periods is in fact higher than the
exoplanet candidate average, giving no obvious reason to distrust the quoted
orbital inclination. Without access to the epoch data, a more in-depth
analysis is not currently possible.

###  4.4 Gaia DR3 1916454200349735680

Figure 8: NOT targets. Each row corresponds to one of the three NOT fast track
targets. These systems are, from top to bottom: Gaia DR3 052469973468984192,
Gaia DR3 916454200349735680, and Gaia DR3 5122670101678217728. Left column:
The most likely orbits from the Gaia astrometric two-body solution are shown
alongside the orbital parameters and their uncertainties. Middle column: RV
curve of the primary component as inferred from the Gaia orbital solution. The
left y-axis scaling assumes the companion to have a mass equal to the primary.
The right y-axis scaling assumes the companion to be dark. The two NOT
observations are shown as solid and dotted lines. The gray shaded areas
indicate the 1$\sigma$ uncertainties in orbital phase of these observations.
The rectangles are darker where the uncertainties overlap. The large
uncertainties in orbital phase timing are caused by the long baseline between
the Gaia and NOT observations. Right column: CCFs obtained from the FIES
spectra, using line styles corresponding to those from the middle column.

Gaia DR3 1916454200349735680 (G5680) is the first of three systems we observed
with the FIES spectrograph at the NOT. We obtained two spectra of the system
three weeks apart. G5680 is characterized in Gaia as a K-dwarf and has a G
magnitude of $10.9$. Its location in the HR-diagram is visibly elevated, see
Fig. 1. This elevation was the reason for including it in our small sample as
it serves as a test of whether the location in an HR diagram can be used to
gauge the likelihood of a binary false positive. The suspicion that this
system was a binary was confirmed right away by inspecting the CCFs (Fig. 8,
top panel). The CCF of both spectra display a clear double peak structure
indicating it is an SB2 binary. The similar heights and widths of the CCFs
show that the two stars are very similar and that this system is a
prototypical twin binary impostor. The difference in the separation between
the CCF peaks implies a decrease in the radial component of the primary
component’s orbital velocity from $\sim 7.7$ km s-1 to $\sim 5.8$ km s-1 in
the 21 days between our observations of the system. Such a decrease in RV fits
with the first observation being at an orbital phase of around 0.92 and the
second at 1, corresponding to 18.3 days. Here we assumed equal mass of the
components equal to the $M_{1}$ binary_masses value of $0.63M_{\sun}$ and
using Gaia’s orbital inclination of $127^{\circ}$. The period and inclination
of the nss_two_body_orbit orbital solution are thus consistent with our
observations.

###  4.5 Gaia DR3 2052469973468984192

Gaia DR3 2052469973468984192 (G4192), our second FIES target, is classified as
an M-dwarf with a G magnitude of $10.7$. It is located right on the single
star Main-Sequence in the HR-diagram (Fig. 1), so the expectation would be
that the small astrometric motion of the photocenter, $a_{0}=0.02$ au ($1.14$
mas) would originate from an orbiting Brown Dwarf with a mass of $22.9\pm
1.6M_{\rm Jup}$, as per equation 2.1. However, despite the first observation
showing only a single CCF peak, this system is also a twin binary system,
evident by the two-peaked CCF of the second observation obtained a few weeks
later, see Fig. 8 middle panel. The separation between the CCF peaks increases
from 0 km s-1 to 4.6 km s-1 between the two observations. Using the quoted
inclination of $141^{\circ}$ and $M_{1}=M_{2}=0.47M_{\sun}$, where the mass
once again taken to be the binary_masses$M_{1}$ mass, this RV separation would
be expected to take $33.2$ days. The actual gap between the observations was
21 days. While not wildly different, we do not attempt to investigate whether
this discrepancy is caused by an underestimate of $M_{1}$ or an overestimate
of either the inclination or the orbital period. We do note however that
$1/\sin 141^{\circ}\approx 33.2/21$, or in other words, if the configuration
of the orbit is edge-on it would exactly account for the discrepancy.

###  4.6 DR3 5122670101678217728

The third and last of our FIES Fast Track targets is the system Gaia DR3
5122670101678217728 (G7728). It is classified as a G-dwarf by Gaia and has a G
magnitude of $8.9$. The observed photocenter semi-major axis is $a_{0}=0.029$
au ($0.28\pm 0.03$ mas), which amounts to a companion mass of $15.6\pm
1.7M_{\rm Jup}$, assuming the companion is dark and using the mass estimate of
the primary provided by Gaia. However, just like the two other FIES systems,
the binary nature of the system is revealed by the double peak feature of the
CCF obtained at a larger RV separation, see the bottom panel of Fig. 8. The
two peaks are extremely similar in height and width indicating that this
system hosts two twin stars and not a star and a planet. The separation
between the CCF peaks corresponds to an RV increase of the primary of 6.4 km
s-1 over the 31 day period between our observations of this target. Using the
same equal-mass assumption as the other systems and using $M_{1}=1.04M_{\sun}$
and $i=110^{\circ}$ corresponds to a $32.7$ day gap starting from an orbital
phase of 0.4. Our observations are thus consistent with the orbital solution
of this system.

##  5 Conclusion

With 9 of the 72 astrometric exoplanet candidates being previously confirmed
exoplanets, Gaia’s ability to detect photocenter movements small enough to
find exoplanets is validated by DR3 and shows the current and future impact of
the Gaia spacecraft on exoplanetary science. In this work we investigated the
nature of six of these candidate systems using information contained in
archival and newly obtained high resolution spectra. We found four systems, HD
68638, Gaia DR3 1916454200349735680, Gaia DR3 2052469973468984192, and Gaia
DR3 5122670101678217728 to be false positives. The double-peaked CCFs of these
systems show that they all consist of two nearly equal flux binaries. In the
remaining two systems, HD 40503 and HIP 66074, the candidates are possibly
exoplanets, although inconsistencies between the RV and astrometric solutions
exist. The astrometric solutions imply in both cases an RV signal
significantly larger than observed. Yet, all other orbital parameters are
fully consistent between RV and astrometry. It is not clear what causes this,
but a more face-on geometry of the orbits than implied by the astrometric
solution could solve the disagreements.

We found 30 systems in the Gaia two-body orbital solutions with photocenter
semi-major axes small enough to indicate a star-planet system that were not
among the released exoplanet candidates. These systems are likely binary stars
as per Gaia’s flux ratio estimates, see Fig. 1. These 30 systems along with
the four false-positive exoplanets discussed in this paper highlight the
difficulties of detecting exoplanets using astrometry. In this work we
emphasize the need for spectra of high spectral resolution to supplement the
astrometric data in order to reveal impostors by their double-lined nature or
by the correlation of the FWHM with orbital phase. The high resolution is
needed because the total wavelength shift of two sets of unresolved absorption
lines from a binary system is dimmed in almost perfect proportion to the
astrometric dimming, imitating an exoplanet signal in both data sets. The
silver lining of the twin binaries issue is the fact that the similarity of
the components makes it easier to identify the splitting in two of the
absorption lines because the absorption lines have similar depths.

Table 1: Apparent low-mass companions.

Gaia DR3 id | Name | solution | Status | RV data | $M_{1}$ | $M_{2}$ | Flux ratio | GMag | Distance | References |
---|---|---|---|---|---|---|---|---|---|---|---
|  | _type |  |  | ($M_{\sun}$) | ($M_{J}$) | $F_{2}/F_{1}$ |  | (pc) |  |
1035000055055287680 | HD 68638 | OTSV | SB2 | Consistent | ${}^{1}0.95^{+1.00}_{-0.82}$ | 31${}^{+48}_{-14}$ | $\sim 1$ | 7.3 | 32.5 | 1, 2, 3 |
2884087104955208064 | HD 40503 | OTSV | Unknown | Inconsistent | ${}^{1}0.79^{+0.85}_{-0.57}$ | 5${}^{+8}_{-3}$ | $[0-1]$ | 9.0 | 39.2 | 2, 3 |
1712614124767394816 | HIP 66074 | OTSV | Unknown | Inconsistent | ${}^{1}0.71^{+0.76}_{-0.51}$ | 7${}^{+11}_{-3}$ | $[0-1]$ | 9.7 | 35.4 | 2, 3, 4, 5 |
1916454200349735680 |  | Orbital | SB2 | Consistent | ${}^{1}0.63^{+0.68}_{-0.48}$ | 17${}^{+26}_{-8}$ | $\sim 0.88$ | 10.9 | 37.1 | This work |
2052469973468984192 |  | Orbital | SB2 | Consistent | ${}^{1}0.47^{+0.52}_{-0.29}$ | 18${}^{+27}_{-9}$ | $\sim 0.85$ | 10.7 | 17.4 | This work |
5122670101678217728 | HD 12357 | Orbital | SB2 | Consistent | ${}^{1}1.04^{+1.10}_{-0.97}$ | 21${}^{+33}_{-10}$ | $\sim 1$ | 8.9 | 102.8 | This work |
4062446910648807168 | HR 164604 | OTSV | Exoplanet | No | ${}^{2}0.75^{+0.79}_{-0.71}$ | 14${}^{+24}_{-5}$ | $\sim 0$ | 9.3 | 40.0 | 12 |
1594127865540229888 | HD 132406 | OTSV | Exoplanet | Inconsistent | ${}^{2}0.97^{+1.01}_{-0.93}$ | 6${}^{+11}_{-3}$ | $\sim 0$ | 8.3 | 70.5 | 4, 8 |
2367734656180397952 | BD-17 63 | OTSV | Exoplanet | Consistent | ${}^{1}0.74^{+0.79}_{-0.56}$ | 4${}^{+7}_{-2}$ | $\sim 0$ | 9.2 | 34.5 | 4, 6 |
4745373133284418816 | HR 810 | OTS | Exoplanet | Inconsistent | ${}^{1}1.09^{+1.15}_{-0.93}$ | 6${}^{+10}_{-3}$ | $\sim 0$ | 5.3 | 17.4 | 4, 5, 11, 13, 14 |
637329067477530368 | HD 81040 | OTSV | Exoplanet | Consistent | ${}^{1}0.98^{+1.03}_{-0.68}$ | 8${}^{+12}_{-4}$ | $\sim 0$ | 7.6 | 34.4 | 4, 7 |
4901802507993393664 | HD 3221 | OTS | SB2 | No | ${}^{3}1$ | 14 | $\sim 1$ | 9.1 | 44.4 | 17 |
5855730584310531200 | HD 111232 | OTSV | Exoplanet | Inconsistent | ${}^{1}0.93^{+0.99}_{-0.68}$ | 8${}^{+13}_{-4}$ | $\sim 0$ | 7.4 | 28.9 | 4, 9, 10, 11 |
2603090003484152064 | GJ 876 | OTSV | Exoplanet | No | ${}^{1}0.38^{+0.43}_{-0.21}$ | 4${}^{+5}_{-2}$ | $\sim 0$ | 8.9 | 4.7 | 16 |
4976894960284258048 | HD 142 | OTS | Exoplanet | No | ${}^{1}1.19^{+1.25}_{-1.06}$ | 7${}^{+11}_{-3}$ | $\sim 0$ | 5.6 | 26.2 | 15 |
6421118739093252224 | HD 175167 | OTSV | Exoplanet | Inconsistent | ${}^{2}1.09^{+1.13}_{-1.05}$ | 10${}^{+16}_{-5}$ | $\sim 0$ | 7.8 | 71.2 | 4, 12 |
4764340705296117120 |  | Orbital | Unknown | No | ${}^{1}0.34^{+0.39}_{-0.18}$ | 13${}^{+19}_{-6}$ | $[0-1]$ | 14.8 | 61.8 |  |
1984587671751337600 |  | Orbital | Unknown | No | ${}^{1}0.36^{+0.41}_{-0.24}$ | 19${}^{+29}_{-9}$ | $[0.34-1]$ | 14.2 | 54.0 |  |
4392798821280870912 |  | Orbital | Unknown | No | ${}^{1}0.42^{+0.47}_{-0.26}$ | 15${}^{+23}_{-7}$ | $[0.05-1]$ | 13.4 | 42.6 |  |
5220375041387610880 |  | Orbital | Unknown | No | ${}^{1}0.74^{+0.79}_{-0.53}$ | 20${}^{+31}_{-9}$ | $[0-1]$ | 12.8 | 183.6 |  |
6418925831870553472 |  | Orbital | Unknown | No | ${}^{1}0.37^{+0.42}_{-0.20}$ | 15${}^{+23}_{-8}$ | $[0-1]$ | 15.0 | 84.0 |  |
2571855077162098944 |  | Orbital | Unknown | No | ${}^{1}0.54^{+0.59}_{-0.37}$ | 15${}^{+23}_{-7}$ | $[0-1]$ | 14.5 | 114.6 |  |
5036787935627755520 |  | Orbital | Unknown | No | ${}^{1}0.22^{+0.26}_{-0.10}$ | 12${}^{+17}_{-6}$ | $[0.21-1]$ | 14.9 | 33.5 |  |
932447162423519232 |  | ASSB1 | Unknown | No | ${}^{1}0.63^{+0.68}_{-0.48}$ | 17${}^{+27}_{-8}$ | $[0.22-1]$ | 11.6 | 52.5 |  |
5654515588409756160 |  | Orbital | Unknown | No | ${}^{1}0.26^{+0.31}_{-0.12}$ | 23${}^{+34}_{-11}$ | $[0-1]$ | 16.4 | 88.4 |  |
5271515801094390912 |  | Orbital | Unknown | No | ${}^{1}0.56^{+0.61}_{-0.39}$ | 21${}^{+32}_{-10}$ | $[0-1]$ | 12.2 | 55.6 |  |
5085864568417061120 |  | Orbital | Unknown | No | ${}^{1}0.46^{+0.51}_{-0.28}$ | 16${}^{+24}_{-8}$ | $[0-1]$ | 13.7 | 55.8 |  |
423297927866697088 |  | Orbital | Unknown | No | ${}^{1}0.41^{+0.46}_{-0.24}$ | 21${}^{+32}_{-10}$ | $[0-1]$ | 14.8 | 79.2 |  |
2845310284780420864 |  | Orbital | Unknown | No | ${}^{1}0.37^{+0.42}_{-0.20}$ | 10${}^{+15}_{-5}$ | $[0-1]$ | 13.8 | 41.5 |  |
4188996885011268608 |  | Orbital | Unknown | No | ${}^{1}0.23^{+0.28}_{-0.09}$ | 8${}^{+12}_{-4}$ | $[0-1]$ | 13.5 | 17.8 |  |
4842246017566495232 |  | Orbital | Unknown | No | ${}^{1}0.36^{+0.41}_{-0.19}$ | 8${}^{+12}_{-4}$ | $[0-1]$ | 13.0 | 31.9 |  |
1336053176328998144 |  | Orbital | Unknown | No | ${}^{1}0.44^{+0.49}_{-0.33}$ | 19${}^{+29}_{-9}$ | $[0.48-1]$ | 14.3 | 83.2 |  |
6081071334868194176 |  | Orbital | Unknown | No | ${}^{1}0.61^{+0.66}_{-0.43}$ | 20${}^{+31}_{-10}$ | $[0-1]$ | 12.9 | 99.0 |  |
405316961377489792 |  | Orbital | Unknown | No | ${}^{1}0.27^{+0.32}_{-0.13}$ | 22${}^{+33}_{-10}$ | $[0-1]$ | 15.7 | 77.4 |  |
6671454584430500864 |  | Orbital | Unknown | No | ${}^{1}0.57^{+0.62}_{-0.43}$ | 14${}^{+24}_{-6}$ | $[0.17-1]$ | 13.9 | 112.4 |  |
3621891774065137408 |  | Orbital | Unknown | No | ${}^{1}0.36^{+0.41}_{-0.22}$ | 19${}^{+29}_{-9}$ | $[0.19-1]$ | 15.5 | 92.3 |  |
4159075462792179456 |  | Orbital | Unknown | No | ${}^{1}0.35^{+0.40}_{-0.20}$ | 8${}^{+13}_{-4}$ | $[0.07-1]$ | 14.9 | 64.4 |  |
3665298981300771200 |  | Orbital | Unknown | No | ${}^{1}0.69^{+0.74}_{-0.54}$ | 20${}^{+31}_{-9}$ | $[0.29-1]$ | 10.7 | 48.7 |  |
4810127839810808576 |  | Orbital | Unknown | No | ${}^{1}0.55^{+0.60}_{-0.38}$ | 16${}^{+24}_{-7}$ | $[0.07-1]$ | 13.0 | 61.4 |  |
2998643469106143104 |  | Orbital | Unknown | No | ${}^{1}0.37^{+0.41}_{-0.20}$ | 20${}^{+30}_{-10}$ | $[0-1]$ | 14.0 | 48.3 |  |
557717892980808960 |  | Orbital | Unknown | No | ${}^{1}0.55^{+0.60}_{-0.37}$ | 8${}^{+13}_{-4}$ | $[0-1]$ | 12.0 | 40.0 |  |
6381440834777420928 |  | Orbital | Unknown | No | ${}^{1}0.28^{+0.33}_{-0.13}$ | 21${}^{+32}_{-10}$ | $[0-1]$ | 15.1 | 60.6 |  |
4596564611107874944 |  | Orbital | Unknown | No | ${}^{1}0.32^{+0.37}_{-0.17}$ | 19${}^{+29}_{-9}$ | $[0.10-1]$ | 14.8 | 54.9 |  |
2271703211828512896 |  | Orbital | Unknown | No | ${}^{1}0.16^{+0.21}_{-0.07}$ | 13${}^{+20}_{-6}$ | $[0.30-1]$ | 16.4 | 42.3 |  |
5612039087715504640 |  | Orbital | Unknown | No | ${}^{1}0.28^{+0.33}_{-0.13}$ | 14${}^{+21}_{-7}$ | $[0-1]$ | 13.9 | 32.4 |  |
834357565445682944 |  | Orbital | Unknown | No | ${}^{1}0.43^{+0.48}_{-0.25}$ | 17${}^{+25}_{-8}$ | $[0-1]$ | 13.7 | 53.6 |  |
5446516751833167744 |  | Orbital | Unknown | No | ${}^{1}0.27^{+0.32}_{-0.12}$ | 17${}^{+25}_{-8}$ | $[0.01-1]$ | 16.0 | 70.4 |  |
846058258950880384 |  | Orbital | Unknown | No | ${}^{1}0.60^{+0.65}_{-0.49}$ | 15${}^{+24}_{-7}$ | $[0.44-1]$ | 13.9 | 143.4 |  |
2117364283603705984 |  | Orbital | Unknown | No | ${}^{1}0.41^{+0.46}_{-0.29}$ | 15${}^{+23}_{-7}$ | $[0.37-1]$ | 13.5 | 48.9 |  |
2074815898041643520 |  | Orbital | Unknown | No | ${}^{1}0.36^{+0.41}_{-0.20}$ | 21${}^{+32}_{-10}$ | $[0-1]$ | 13.2 | 41.3 |  |
2047188847334279424 |  | OTS | Unknown | No | ${}^{1}0.96^{+1.01}_{-0.82}$ | 14${}^{+22}_{-7}$ | $[0-1]$ | 7.3 | 32.6 |  |
6685861691447769600 |  | Orbital | Unknown | No | ${}^{1}0.50^{+0.55}_{-0.32}$ | 20${}^{+31}_{-10}$ | $[0-1]$ | 14.3 | 99.6 |  |
6079316686107743488 |  | Orbital | Unknown | No | ${}^{1}0.38^{+0.43}_{-0.21}$ | 20${}^{+31}_{-10}$ | $[0-1]$ | 14.7 | 66.4 |  |
430892357759527424 |  | Orbital | Unknown | No | ${}^{1}0.19^{+0.24}_{-0.08}$ | 7${}^{+11}_{-3}$ | $[0.15-1]$ | 15.6 | 38.1 |  |
1298992006611690112 |  | Orbital | Unknown | No | ${}^{1}0.52^{+0.57}_{-0.36}$ | 14${}^{+21}_{-7}$ | $[0.11-1]$ | 13.6 | 73.9 |  |
1878822452815621120 |  | OTS | Unknown | No | ${}^{1}0.82^{+0.88}_{-0.61}$ | 16${}^{+25}_{-8}$ | $[0-1]$ | 9.8 | 62.3 |  |
1879554280883275136 |  | Orbital | Unknown | No | ${}^{1}0.43^{+0.48}_{-0.26}$ | 18${}^{+27}_{-9}$ | $[0-1]$ | 14.3 | 79.3 |  |
6471102606408911360 |  | Orbital | Unknown | No | ${}^{1}0.65^{+0.81}_{-0.49}$ | 23${}^{+36}_{-11}$ | $[0-1]$ | 14.3 | 62.1 |  |
5671384265738137984 |  | Orbital | Unknown | No | ${}^{1}0.15^{+0.20}_{-0.07}$ | 16${}^{+24}_{-8}$ | $[0.27-1]$ | 15.7 | 28.3 |  |
2104920835634141696 |  | Orbital | Unknown | No | ${}^{1}0.21^{+0.26}_{-0.08}$ | 13${}^{+19}_{-6}$ | $[0-1]$ | 15.4 | 46.2 |  |
373892712892466048 |  | Orbital | Unknown | No | ${}^{1}0.51^{+0.56}_{-0.33}$ | 17${}^{+25}_{-8}$ | $[0-1]$ | 13.9 | 75.3 |  |
5399010462168339456 |  | Orbital | Unknown | No | ${}^{1}0.47^{+0.52}_{-0.39}$ | 14${}^{+22}_{-6}$ | $[0.65-1]$ | 13.8 | 79.6 |  |
5618776310850226432 |  | Orbital | Unknown | No | ${}^{1}0.47^{+0.52}_{-0.29}$ | 21${}^{+32}_{-10}$ | $[0-1]$ | 14.7 | 94.1 |  |
6781298098147816192 |  | Orbital | Unknown | No | ${}^{1}0.32^{+0.37}_{-0.16}$ | 6${}^{+10}_{-3}$ | $[0-1]$ | 14.4 | 48.1 |  |
2050702366781291776 |  | Orbital | Unknown | No | ${}^{1}0.26^{+0.31}_{-0.13}$ | 18${}^{+27}_{-9}$ | $[0.22-1]$ | 16.4 | 85.2 |  |
1052042828882790016 |  | Orbital | Unknown | No | ${}^{1}0.47^{+0.52}_{-0.30}$ | 21${}^{+32}_{-10}$ | $[0-1]$ | 14.0 | 78.1 |  |
726588585356221568 |  | Orbital | Unknown | No | ${}^{1}0.27^{+0.32}_{-0.15}$ | 20${}^{+30}_{-9}$ | $[0.28-1]$ | 15.6 | 67.8 |  |
5486916932205092352 |  | Orbital | Unknown | No | ${}^{1}0.33^{+0.38}_{-0.17}$ | 11${}^{+16}_{-6}$ | $[0-1]$ | 12.2 | 17.1 |  |
5052449001298518528 |  | Orbital | Unknown | No | ${}^{1}0.47^{+0.52}_{-0.29}$ | 20${}^{+31}_{-10}$ | $[0-1]$ | 13.6 | 59.2 |  |
246890014559489792 |  | Orbital | Unknown | No | ${}^{1}0.36^{+0.41}_{-0.19}$ | 6${}^{+9}_{-3}$ | $[0-1]$ | 14.4 | 52.5 |  |
5055723587443420928 |  | Orbital | Unknown | No | ${}^{1}0.45^{+0.50}_{-0.27}$ | 19${}^{+30}_{-9}$ | $[0-1]$ | 14.6 | 137.7 |  |
3676303512147120512 |  | Orbital | Unknown | No | ${}^{1}0.23^{+0.28}_{-0.10}$ | 18${}^{+27}_{-9}$ | $[0-1]$ | 14.8 | 38.7 |  |
1059462676944293376 |  | Orbital | Unknown | No | ${}^{1}0.48^{+0.53}_{-0.35}$ | 16${}^{+24}_{-8}$ | $[0.33-1]$ | 14.7 | 111.6 |  |
5490183684330661504 |  | Orbital | Unknown | No | ${}^{1}0.14^{+0.19}_{-0.05}$ | 23${}^{+36}_{-11}$ | $[0-1]$ | 17.7 | 72.4 |  |
2277249663873880576 |  | Orbital | Unknown | No | ${}^{1}0.50^{+0.55}_{-0.32}$ | 11${}^{+17}_{-5}$ | $[0-1]$ | 12.9 | 51.1 |  |
2259699048817216256 |  | Orbital | Unknown | No | ${}^{1}0.62^{+0.67}_{-0.49}$ | 16${}^{+24}_{-7}$ | $[0.31-1]$ | 13.1 | 105.9 |  |
2259968811419624448 |  | Orbital | Unknown | No | ${}^{1}0.40^{+0.45}_{-0.23}$ | 22${}^{+33}_{-10}$ | $[0-1]$ | 13.9 | 54.9 |  |
4812716639938468992 |  | Orbital | Unknown | No | ${}^{1}0.31^{+0.36}_{-0.15}$ | 22${}^{+33}_{-11}$ | $[0-1]$ | 14.7 | 48.2 |  |
6521749994635476992 |  | Orbital | Unknown | No | ${}^{1}0.48^{+0.53}_{-0.30}$ | 15${}^{+22}_{-7}$ | $[0-1]$ | 13.6 | 62.4 |  |
1457486023639239296 |  | Orbital | Unknown | No | ${}^{1}0.64^{+0.69}_{-0.45}$ | 13${}^{+21}_{-6}$ | $[0-1]$ | 11.9 | 73.8 |  |
1462767459023424512 |  | Orbital | Unknown | No | ${}^{1}0.37^{+0.41}_{-0.19}$ | 8${}^{+12}_{-4}$ | $[0-1]$ | 15.1 | 72.0 |  |
73648110622521600 |  | Orbital | Unknown | No | ${}^{1}0.25^{+0.30}_{-0.11}$ | 20${}^{+30}_{-10}$ | $[0-1]$ | 15.5 | 58.6 |  |
5796338299045711232 |  | Orbital | Unknown | No | ${}^{1}0.18^{+0.23}_{-0.08}$ | 11${}^{+17}_{-5}$ | $[0.37-1]$ | 14.8 | 25.8 |  |
3937630969071148032 |  | Orbital | Unknown | No | ${}^{1}0.43^{+0.48}_{-0.26}$ | 20${}^{+30}_{-10}$ | $[0-1]$ | 14.7 | 84.7 |  |
2446599193562312320 |  | Orbital | Unknown | No | ${}^{1}0.50^{+0.55}_{-0.33}$ | 24${}^{+37}_{-11}$ | $[0-1]$ | 14.7 | 129.1 |  |
1862136504889464192 |  | Orbital | Unknown | No | ${}^{1}0.37^{+0.42}_{-0.21}$ | 16${}^{+23}_{-8}$ | $[0-1]$ | 15.2 | 83.1 |  |
3925216795598987264 |  | Orbital | Unknown | No | ${}^{1}0.40^{+0.45}_{-0.25}$ | 19${}^{+29}_{-9}$ | $[0.16-1]$ | 14.8 | 75.0 |  |
2824801747222539648 |  | Orbital | Unknown | No | ${}^{1}0.44^{+0.49}_{-0.27}$ | 19${}^{+29}_{-9}$ | $[0-1]$ | 14.0 | 70.4 |  |
6694115931396057728 |  | Orbital | Unknown | No | ${}^{1}0.33^{+0.38}_{-0.17}$ | 9${}^{+14}_{-5}$ | $[0-1]$ | 13.4 | 38.1 |  |
4698424845771339520 |  | OTS | Unknown | No | ${}^{1}0.65^{+0.81}_{-0.49}$ | 9${}^{+14}_{-4}$ | $[0-1]$ | 13.7 | 9.7 |  |
4702845638429469056 |  | Orbital | Unknown | No | ${}^{1}0.43^{+0.47}_{-0.26}$ | 7${}^{+10}_{-3}$ | $[0-1]$ | 13.9 | 68.2 |  |
6677563745912843776 |  | Orbital | Unknown | No | ${}^{1}0.39^{+0.44}_{-0.21}$ | 23${}^{+35}_{-11}$ | $[0-1]$ | 15.5 | 94.7 |  |
6354671987249126784 |  | ASSB1 | Unknown | No | ${}^{1}0.69^{+0.75}_{-0.64}$ | 13${}^{+21}_{-6}$ | $[0.12-1]$ | 9.5 | 32.5 |  |
522135261462534528 |  | OTS | Unknown | No | ${}^{1}1.09^{+1.15}_{-0.79}$ | 7${}^{+11}_{-3}$ | $[0-1]$ | 6.4 | 27.0 |  |
5375875638010549376 |  | Orbital | Unknown | No | ${}^{1}0.40^{+0.45}_{-0.23}$ | 14${}^{+21}_{-6}$ | $[0-1]$ | 14.4 | 67.1 |  |
1610837178107032192 |  | OTS | Unknown | No | ${}^{1}1.13^{+1.19}_{-0.97}$ | 13${}^{+21}_{-6}$ | $[0-1]$ | 8.2 | 73.7 |  |
4963614887043956096 |  | OTS | Unknown | No | ${}^{3}1$ | 46 | $[0-1]$ | 15.0 | 40.2 |  |
3120450116011961984 |  | Orbital | Unknown | No | ${}^{1}0.64^{+0.69}_{-0.49}$ | 16${}^{+25}_{-8}$ | $[0.21-1]$ | 12.8 | 94.7 |  |
4983571882081864960 |  | Orbital | Unknown | No | ${}^{1}0.47^{+0.52}_{-0.29}$ | 12${}^{+19}_{-6}$ | $[0-1]$ | 13.3 | 55.5 |  |
5236626338671861760 |  | Orbital | Unknown | No | ${}^{3}1$ | 21 | $[0-1]$ | 13.9 | 51.3 |  |
5437488554482255872 |  | Orbital | Unknown | No | ${}^{1}0.50^{+0.55}_{-0.37}$ | 18${}^{+27}_{-8}$ | $[0.29-1]$ | 13.4 | 67.4 |  |
198464052134353536 |  | Orbital | Unknown | No | ${}^{1}0.20^{+0.25}_{-0.09}$ | 11${}^{+17}_{-5}$ | $[0.22-1]$ | 14.4 | 24.1 |  |

Note. — The 102 listed systems are the union of the following subsets: A) the
released Gaia exoplanet candidates and B) all systems whose Gaia
nss_two_body_orbit implies $M_{2}<20M_{J}$ assuming a dark companion,
regardless of the binary_masses flux ratio value. The same systems are shown
in Fig. 1 and Fig. 2. The source of information for the mass of the primary
component $M_{1}$ and its uncertainty, was in order of decreasing priority:
$M_{1}$ from binary_masses, ”mass_flame” from astrophysical_parameters and
$M_{1}=1M_{\sun}$, denoted by $1,2$ and $3$ respectively in the corresponding
column of the table. The top six systems of the table are those analyzed in
this paper. The system name is shown systems previously studied in the
literature. Whether RV data from ground-based spectroscopy is available for a
system, and if so, whether it is consistent with the nss_two_body_orbit
solution and companion mass or not is listed in the ”RV data” column. We
define the data sets to be consistent if all the orbital parameters as well as
the RV semi-amplitudes are within $3\sigma$ of one another. However, for our
three systems with only two spectra each, only the orbital period and RV semi-
amplitudes are tested for consistency. The distance column is 1$/$parallax,
where the parallax is from the gaia_source table. The solution_type column
refers to the nss_solution_type of the nss_two_body_orbit table. OTS, OTSV,
and ASSB1 refer to orbitalTargetedSearch, OrbitalTargetedSearchValidated and
AstroSpectroSB1 respectively. References: (1) Busà et al. (2007); (2) Holl et
al. (2022a); (3) Gaia Collaboration et al. (2022b); (4) Winn (2022); (5)
Butler et al. (2017); (6) Moutou et al. (2009); (7) Sozzetti et al. (2006);
(8) da Silva et al. (2007); (9) Minniti et al. (2009); (10) Mayor et al.
(2004); (11) Trifonov et al. (2020); (12) Arriagada et al. (2010); (13)
Kürster et al. (2000); (14) Naef et al. (2001); (15) Wittenmyer et al. (2012);
(16) Marcy et al. (2001); (17) Bonavita et al. (2022).

This work has made use of data from the European Space Agency (ESA) mission
Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing
and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. We thank Berry Holl for
valuable clarification about the Gaia exoplanet pipeline and the exoplanet
sample. We thank Emil Knudstrup for insightful discussions concerning RV
measurements and FIES spectra in particular. Funding for the Stellar
Astrophysics Centre is provided by The Danish National Research Foundation
(Grant agreement no.: DNRF106). We acknowledge the support from the Danish
Council for Independent Research through a grant, No.2032-00230B.

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