11institutetext:  ETH Zurich, Institute for Particle Physics & Astrophysics,
Wolfgang-Pauli-Str. 27, 8093 Zurich, Switzerland
e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>22institutetext: National
Center of Competence in Research PlanetS (www.nccr-planets.ch)
33institutetext: NASA Postdoctoral Program Fellow, NASA Goddard Space Flight
Center, Greenbelt, MD, USA 44institutetext: ETH Zurich, Department of Earth
Sciences, Sonneggstrasse 5, 8092 Zurich, Switzerland 55institutetext: NASA
Goddard Space Flight Center, Greenbelt, MD, USA 66institutetext: American
University, 4400 Massachusetts Ave NW, Washington, DC 20016 77institutetext:
Max-Planck-Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany
88institutetext: Department of Astronomy, The University of Michigan West Hall
323, 1085 S. University Avenue Ann Arbor, MI 48109 99institutetext: Lunar &
Planetary Laboratory, University of Arizona, Tucson, AZ, USA
1010institutetext: Department of Physics and Astronomy, York University, 4700
Keele St., Toronto, ON M3J 1P3, Canada

# Large Interferometer For Exoplanets (LIFE):

XIII. The Value of Combining Thermal Emission and Reflected Light for the
Characterization of Earth Twins

E. Alei 112233 S. P. Quanz 112244 B. S. Konrad 1122 E. O. Garvin 11 V. Kofman
5566 A. Mandell 55 D. Angerhausen 1122 P. Mollière 77 M. R. Meyer 88 T.
Robinson 99 S. Rugheimer and the LIFE Collaboration Webpage: www.life-space-
mission.com1010

(Received ; accepted )

###### Abstract

Context. Following the recommendations to NASA (in the Astro2020 Decadal
survey) and ESA (through the Voyage2050 process), the search for life on
exoplanets will be a priority in the next decades. Two concepts for direct
imaging space missions are being developed for this purpose: the Habitable
Worlds Observatory (HWO), and the Large Interferometer for Exoplanets (LIFE).
These two concepts operate in different spectral regimes: HWO is focused on
reflected light spectra in the ultraviolet/visible/near-infrared (UV/VIS/NIR),
while LIFE will operate in the mid-infrared (MIR) to capture the thermal
emission of temperate exoplanets.

Aims. In this study we aim to assess the potential of HWO and LIFE in
characterizing a cloud-free Earth twin orbiting a Sun-like star at 10 parsec
distance both as separate missions and in synergy with each other. We aim to
quantify the increase in information that can be gathered by joint atmospheric
retrievals on a habitable planet.

Methods. We perform Bayesian retrievals on simulated data obtained by a HWO-
like and a LIFE-like mission separately, then jointly. We consider the
baseline spectral resolutions currently assumed for these concepts and use two
increasingly complex noise simulations, obtained using state-of-the-art noise
simulators.

Results. A HWO-like concept would allow to strongly constrain H2O, O2, and O3,
in the atmosphere of a cloud-free Earth twin, while the atmospheric
temperature profile is not well constrained (with an average uncertainty
$\approx$ 100 K). LIFE-like observations would strongly constrain CO2, H2O, O3
and provide stronger constraints on the thermal atmospheric structure and
surface temperature (down to $\approx$ 10 K uncertainty). For all the
investigated scenarios, both missions would provide an upper limit on CH4. A
joint retrieval on HWO and LIFE data would accurately define the atmospheric
thermal profile and planetary parameters. It would decisively constrain CO2,
H2O, O2, and O3 and find weak constraints on CO and CH4. The significance of
the detection is in all cases greater or equal than the single-instrument
retrievals.

Conclusions. Both missions provide specific information that is relevant for
the characterization of a terrestrial habitable exoplanet, but the scientific
yield can be maximized by considering synergistic studies of UV/VIS/NIR+MIR
observations. The use of HWO and LIFE together will provide stronger
constraints on biosignatures and life indicators, with the potential of being
transformative for the search for life in the universe.

###### Key Words.:

Methods: statistical – Planets and satellites: terrestrial planets – Planets
and satellites: atmospheres

##  1 Introduction

The characterization of temperate terrestrial exoplanets and the search for
signatures of biology on these worlds are two primary goals that the
astronomical community will strive to achieve in the next decades. In general,
current and near-future observatories will not be able to directly detect the
signal of the atmospheres of habitable terrestrial planets: their flux is too
small compared to their host stars, and their small angular separation poses a
challenge for current high-contrast imaging instrumentation. Only some
specific targets will be available for near-term studies (e.g., planets
orbiting nearby stars, or systems orbiting M dwarfs) through transit
spectroscopy (using, e.g., JWST, see Greene et al. (2016); Dyrek et al.
(2024)) or direct imaging (using e.g., ELT/METIS, see Quanz et al. (2015);
Bowens et al. (2021)). Nevertheless, none of the currently planned missions
will provide direct measurements of the atmospheric composition of a
statistically significant sample of terrestrial, temperate exoplanets around
Sun-like stars. We will require more sensitive observatories and instruments
to directly image a sample of rocky temperate exoplanets and determine what
are considered the prime targets for the search for life beyond the Solar
System.

The US Astro2020 Decadal survey (National Academies of Sciences, Engineering,
and Medicine, 2021) recommended the pursuit of a technical and scientific
study for the Habitable Worlds Observatory (HWO), an ultraviolet/visible/near-
infrared (UV/VIS/NIR) “high-contrast direct imaging mission with a target off-
axis inscribed diameter of approximately 6 meters”, which shares design and
technology heritage with the pre-Decadal mission concept studies HabEx
(Habitable Exoplanet Observatory, Gaudi et al., 2020) and LUVOIR (Large
UV/Optical/IR Surveyor, Peterson et al., 2017). At the same time, the ESA
Voyage 2050 Senior Committee report (Voyage 2050 Senior Committee, 2021)
recommended the study of temperate exoplanets in the mid-infrared (MIR) as a
potential strategy for the upcoming decade. A European-led effort to develop
the MIR space-based nulling interferometer LIFE (Large Interferometer For
Exoplanets, Quanz et al., 2022a, b) is currently being pursued for this
purpose. In virtue of their different starlight suppression technologies,
these two facilities will likely cover partly different regions of the target
parameter space. LIFE will offer more flexibility to detect and characterize
planets orbiting F, G, K, and M stars, thanks to the large effective aperture
size enabled by ultra-stable nulling interferometry. HWO will be focusing on
F, G, and K stars since the inner working angle of a coronagraph impacts the
detection of habitable zones closer to the star (e.g., Earth-like planets
orbiting late K and M dwarfs). Still, there is an overlap in planets that will
be studied by both missions – even in the realm of already-known exoplanets,
as shown in Carrión-González et al. (2023).

Both concepts are still at the early stages of their development, and we
therefore rely on notional designs and approximations to better define the
scope and the current technical challenges related to both missions. The
community has been implementing strategies to simulate observations and data
analyses of specific planet archetypes to gather information on the minimum
and preferred mission requirements.

In this context, Bayesian atmospheric retrieval frameworks have been and
remain a valuable methodology (see, e.g., Madhusudhan, 2018). Such frameworks
apply Bayes’ theorem to infer probability distributions of planetary
properties from simulated empirical data and provide a statistically robust
method to select the best theoretical model that best explains the data. This
makes them not only the gold standard for analyzing and interpreting
observations but also essential analysis tools when it comes to designing
future missions. Simulated observations that take into account different
architectures of the instruments can be fed to a Bayesian retrieval framework,
to know what the retrieval process would infer from each hypothetical
observation. This allows us to evaluate different architectures in terms of
wavelength range, spectral resolution $R$, and signal-to-noise ratio $S/N$, as
a function of specific mission goals.

Various studies of this kind have been carried out for generic/idealized
versions of HabEx/LUVOIR-type instruments (e.g., Feng et al., 2018; Damiano &
Hu, 2022; Robinson & Salvador, 2023; Latouf et al., 2023a, b), focusing on
specific planet archetypes and simplifying assumptions about bandpass
limitations and noise simulation. For HWO, a similar approach is being used to
define a more accurate observing strategy for various planetary scenarios
(e.g., Young et al., 2023).

In a previous study assessing the capabilities of the LIFE mission concept by
Konrad et al. (2022) (LIFE Paper III), we performed atmospheric retrievals on
a cloud-free Earth twin at a 10 pc distance to understand the minimum
requirements to correctly characterize biosignatures of a living planet. We
expanded our work in Alei et al. (2022) (LIFE Paper V), where we analyzed the
potential of a LIFE-like mission to characterize the Earth at various stages
of its bio-geological evolution. Further studies have also been performed on a
Venus twin (Konrad et al., 2023, i.e., LIFE Paper IX) and on real Earth
satellite data (Mettler et al., 2024). Exploratory studies on the
detectability of phosphine and capstone signatures have also been carried out
within the LIFE series (Angerhausen et al., 2023, 2024, i.e., LIFE Papers VIII
and XII).

When it comes to characterization, the study of exoplanetary atmospheres at
various wavelengths provides us with separate unique windows into the physics
and chemistry of the target. From the chemical point of view, specific
molecules can be more spectrally active in one wavelength range and only be
constrained through observations at those characteristic wavelengths (e.g., O2
in the UV/VIS/NIR and CO2 in the MIR). When it comes to physical quantities
and processes, reflected light observations could provide us with knowledge of
the dynamics, cloud composition, and albedo of the planet, as well as having
direct access to surface biosignatures and ocean glint (see, e.g., Williams &
Gaidos, 2008; Sagan et al., 1993). On the other hand, thermal emission
observations would provide information on the planetary dimensions, the
thermal structure of the atmosphere, and its composition by directly measuring
the planetary radiation, without having to rely on the knowledge of the
stellar spectrum.

Observations in each wavelength range come with their drawbacks and
degeneracies. In reflected light, radius and albedo are extremely coupled,
especially in the case where clouds are present (see, e.g., Feng et al.,
2018). Yet, while complicated, observations in this wavelength range could
still infer some information on hazes and cloud production in planetary
atmospheres (see, e.g., Robinson & Salvador, 2023), which is essential for an
accurate energy balance calculation and for ultimately understanding the
nature of a given atmosphere. On the other hand, observations in the MIR would
be less sensitive to patchy clouds (see, e.g., LIFE Paper V), allow for higher
confidence in the characterization of the thermal and chemical composition of
the atmosphere, and provide us with a more precise measurement of the radius
(e.g., from the search phase of LIFE, see Dannert et al. (2022) i.e., LIFE
Paper II) which would help to disentangle the radius-albedo degeneracy.

In this work, we provide a first qualitative assessment of the scientific
impact of a synergistic approach between the two potential future missions. We
further provide an overview of the unique strengths of missions exploring the
UV/VIS/NIR and MIR wavelength range to characterize a cloud-free Earth-like
planet.

We aim to answer the following research questions: 1) What is the scientific
potential of these two separate missions that explore two different wavelength
regions? 2) To what extent would a joint atmospheric retrieval of UV/VIS/NIR
and MIR data improve the characterization of a cloud-free Earth twin? Would
the detection of the most relevant biosignatures and habitability-related
species improve by considering data from both wavelength ranges? 3) How would
the assumed noise model affect the results when analyzing data from different
instruments separately and jointly?

These are urgent questions to answer to provide realistic mission
requirements, inform the design phase, and plan precursor studies accordingly.
Such a study is critical at this point in time, as the community working on
future-generation missions is being formed and organized. Understanding
synergies between US-led and Europe-led missions is relevant to motivating and
uniting the community in supporting both.

In the remainder of the manuscript, we approximate the HWO concept with a
6-meter inscribed LUVOIR-B-like instrument, to be able to leverage the
knowledge acquired and the tools developed for the LUVOIR Final Report (The
LUVOIR Team, 2019), which are yet to be modified in any significant way for
HWO. For the LIFE concept, we consider the currently-assumed baseline
architecture (Quanz et al., 2022b; Dannert et al., 2022, i.e., Papers I and II
of the LIFE Series).

We describe how we produced the simulated observations and the updates we made
on the atmospheric retrieval framework in Section 2. We describe in Section 3
our results for the sets of retrievals, considering both single-instrument and
joint retrievals in each set. We compare the single-instrument results with
previous studies and we discuss the impact of multi-instrument observations,
as well as our limitations in Section 4. We draw our conclusions in Section 5.

##  2 Methods

In this section, we describe the calculation of the input spectrum and the
simulated observed spectra, updates to the Bayesian retrieval framework
compared to previous studies (see LIFE Papers III and V), and the details of
the sets of retrievals we consider in this study.

###  2.1 Input spectrum

For all retrievals performed in this study, we assumed a simplified cloud-free
Earth-like atmosphere. We assumed a pressure-independent atmospheric
composition (i.e., the composition was the same for all atmospheric layers)
and the pressure-temperature (P$-$T) profile was assumed to be a fourth-order
polynomial (5 parameters) that represented the best fit to the U.S. Standard
Atmosphere 1976 (United States Committee on Extension to the Standard
Atmosphere, 1976). The use of such simplified atmospheric profiles allowed us
to reduce the uncertainty linked to the variability of the atmospheric
profiles and abundances. The input values for each parameter used in the
retrieval are shown in Table 4.

Assuming this simplified atmosphere, we produced a theoretical spectrum using
petitRADTRANS (Mollière et al., 2019). To calculate the spectrum, we used
HITRAN 2020 opacity tables, calculated assuming air broadening and a line
cutoff of 25 cm-1 (see Table 5) and UV opacities for CO2, O2, H2O, O3, CH4,
N2O, and CO from the MPI-MAINZ UV/VIS Spectral Atlas (Keller-Rudek et al.,
2013). We also considered all possible collision-induced absorption (CIA)
opacities and Rayleigh scatterers (see Table 6).

The star-planet system is assumed to be at a 10 pc distance. This distance has
been used in previous studies (see LIFE Papers III and V). The planet was
assumed to be illuminated on only one hemisphere by a Sun-like star
($T_{\star}=5778$ K, $R_{\star}=\leavevmode\nobreak\ 1\leavevmode\nobreak\
\mathrm{R_{\odot}}$) as if the system was face-on, or an edge-on planetary
orbit seen at quadrature. To do this, we employ the scattering of direct light
treatment implemented in petitRADTRANS, which is explained in more detail in
Appendix A.

###  2.2 Simulating observed spectra

In this work, we simulate three sets of observations assuming 1) a high-
resolution low-noise case (used for validation), 2) a baseline resolution and
simplified noise scenario, and 3) a baseline resolution with a higher-fidelity
noise calculation. For each set, we produce two simulated observed spectra
(one for the UV/VIS/NIR and one for the MIR wavelength ranges respectively),
assuming different spectral resolutions and noise instances.

For the validation set (see Appendix B), we assume a spectral resolution of
$R=1000$ and we only consider photon noise, whose $S/N$ is 50 at a reference
wavelength point ($0.55$ µm for HWO and $11.2$ µm for LIFE). Such a high-
resolution low-noise scenario was chosen to both validate the retrieval
framework after the updates that were performed, as well as to simulate the
proof of concept of what could be the performance of such idealized missions.
The choice of the reference wavelength points was based on the continuum
emission of the spectrum and it has been previously adopted by other studies,
such as Feng et al. (2018) for LUVOIR and Konrad et al. (2022) and Alei et al.
(2022) for LIFE (LIFE Papers III and V).

The other two sets of retrievals assume more realistic spectral resolutions
and noise values, considering two different noise instances. The first set
considers a more simplified noise scenario, while the second set assumes
higher-fidelity noise simulations obtained with the available noise simulators
for the two missions. The simulated observations and the corresponding
wavelength-dependent $S/N$ for both missions are shown in Figures 1 and 2.

Figure 1: Simulated observation of an Earth twin by the HWO concept assuming a
simplified noise and a higher-fidelity noise instances. _Top panel_ : The
simulated observed flux at 10 pc distance between 0.2 and $2$ µm for the two
noise scenarios: simplified noise at constant error bars (dark gray shaded
area with solid edges) and higher-fidelity simulated noise obtained with PSG
(light gray shaded area with dotted edges). _Bottom panel_ : Signal-to-noise
ratio ($S/N$) as a function of wavelength for the two noise instances:
simplified noise (solid line) and PSG-simulated noise (dotted line).  Figure
2: Simulated observation of an Earth twin by the LIFE concept assuming a
simplified noise and a higher-fidelity noise instances. _Top panel_ : The
simulated observed flux at 10 pc distance between 4 and $18.5$ µm for the two
noise scenarios: simplified noise at constant error bars (dark gray shaded
area with solid edges) and higher-fidelity simulated noise obtained with
LIFEsim (light gray shaded area with dotted edges). _Bottom panel_ : Signal-
to-noise ratio ($S/N$) as a function of wavelength for the two noise
instances: simplified noise (solid line) and LIFEsim-simulated noise (dotted
line).

Concerning the simplified noise set, we assumed the current baseline
resolutions for the instruments ($R=7$ between $0.2-0.51$ µm, $R=140$ between
$0.51-1$ µm, $R=70$ between $1-2.0$ µm for HWO considering the values from the
LUVOIR-B concept; $R=50$ between $4-18.5$ µm for LIFE). We considered constant
error bars (i.e., the same flux value for uncertainty) at each flux point. The
value of the flux uncertainty was calculated to correspond to a value of
$S/N=10$ at the two reference wavelength points. Then, that constant value was
assumed throughout the whole wavelength range for the two instruments. This
implementation has been used in the literature for UV/VIS/NIR studies (see,
e.g., Feng et al., 2018; Damiano & Hu, 2022; Robinson & Salvador, 2023; Young
et al., 2023).

In the higher-fidelity simulated noise set, we assumed more realistic
uncertainty values using state-of-the-art noise simulators for the two
concepts. The theoretical input spectrum was fed to both the LIFE simulator
LIFEsim (as described in Dannert et al., 2022, i.e., LIFE Paper II) and the
HWO noise simulator included in the Planetary Spectrum
Generator111https://psg.gsfc.nasa.gov/index.php (Villanueva et al., 2022). The
settings used in LIFEsim to produce a simulated LIFE observation are shown in
Table 7. The settings used in the PSG LUVOIR-B simulator are collected in
Table 8.

We set up the simulators to be as similar as possible, though these are at a
different state of their development. The LIFE simulator LIFEsim considers
astrophysical noise, while the PSG HWO simulator includes also instrumental
noise. However, given the outcome of the Decadal Survey, the HWO simulator
will probably undergo significant changes to be adapted to the HWO concept.
Still, these two simulators represent the current knowledge of the
architecture and the expected noise from the two missions, which will be
further refined in the upcoming years.

The S/N generated by LIFEsim is scaled to a reference value of 10 at $11.2$
µm, similar to the previous papers in the LIFE series (LIFE Papers III and V).
For HWO, we calculated the S/N for each of the wavelength portions of the
spectrum separately, so that it would have a value of $S/N=10$ at three
reference wavelength points: $0.35$ µm for the UV range ($R=7$ between
$0.2-0.51$ µm); $0.55$ µm for the VIS range ($R=140$ between $0.51-1$ µm);
$1.2$ µm for the NIR range ($R=70$ between $1-2.0$ µm). Also in this case, the
choice of the wavelength reference points for the UV and NIR was based on the
peak of the continuum emission in the bands.

Compared to the simplified noise scenario, the HWO $S/N$ changes up to a
factor of about two at wavelength bands that are featured by specific spectral
signatures (see bottom panel of Figure 1): the noise around the UV O3 line
between $0.25-0.4$ µm and in the NIR range between $1-1.5$ µm is lower
compared to the previous set of runs that considered a simplified noise
scenario. For LIFE, the noise almost doubles at wavelengths between $15-18.5$
µm (see bottom panel of Figure 2) compared to the simplified noise scenario.

For all retrievals, we consider the same physical parameters of the star-
planet system (Table 9). These values are consistent with previous studies in
the LIFE series except for the exozodi level. While previous works in the
series assumed a value of 3 times the zodiacal dust level (see, e.g., LIFE
Papers III and V), the PSG LUVOIR-B simulator assumed a default value of 4.5
times the local zodiacal dust level. We therefore assumed the higher value of
the two to compare results.

In both instruments, the throughput acting on a point source is dependent on
the source position relative to its host star and the wavelength of the
observation. Therefore, the shape of the observed spectra will also depend on
the planet’s position. In this work, for the sake of correcting for this
effect, we assume perfect knowledge of the position of the planet. Hence, the
observed flux and its related noise are normalized by the wavelength-dependent
throughput of the system. We do not randomize each flux data point by the
noise. Rather, we assume that the noise is the uncertainty on the theoretical
data flux. We will briefly discuss the implications of this approximation in
Section 4.4.

###  2.3 Updates on the Bayesian retrieval framework and setup

The Bayesian atmospheric retrieval framework used in previous studies (e.g.,
LIFE Papers III and V) has been updated to allow the correct simultaneous
retrieval of both MIR and UV/VIS/NIR spectra. Generally, any Bayesian
atmospheric retrieval framework is composed of two main modules. First, a
parameter estimation module (pyMultiNest in this case, Buchner et al., 2014)
to iteratively sample the prior space to retrieve the best subset of
parameters that explains the data (the posteriors). Second, a forward model
(petitRADTRANS in this case Mollière et al., 2019) is used to calculate the
spectrum corresponding to a set of parameters. The majority of the updates
were performed to allow the use of features that were implemented in
petitRADTRANS for LIFE Paper V (see Appendix A of Alei et al., 2022) but that
were not considered in the previous studies.

Firstly, we adapted the existing retrieval framework to handle multiple input
spectra simultaneously to account for different resolutions, signal-to-noise
ratios, and wavelength ranges that are specific to the different instruments.
We achieve this by calculating theoretical spectra at a resolution $R=200$,
which is higher than the maximum foreseen resolution, and then binning these
down to the input wavelength points at each retrieval iteration. For this
step, we used the spectral rebinning tool SpectRes (Carnall, 2017).

Secondly, we consider the atmospheric and surface scattering of both direct
light and thermal atmospheric radiation as additional processes compared to
previous studies. We assume to know the stellar radius and temperature,
incident angle, and the distance of the stellar-planet system fixed in the
retrievals. We now introduce the surface reflectivity $r_{s}$ as a parameter
to be retrieved in the various runs. In the case of a thin, cloud-free
atmosphere, the surface reflectance makes up for the majority of the albedo of
the planet. Therefore, having $r_{s}$ as a free parameter allows us to
qualitatively analyze any degeneracy between albedo and radius at UV-VIS
wavelengths (see, e.g., Gaudi et al., 2020; Feng et al., 2018). We refer to
Appendix A for more details on the scattering treatment and the definition of
the surface reflectance.

In the retrieval, we consider the same set of opacities that were used to
calculate the input spectrum (see Table 5). The retrieval framework also takes
into account collision-induced absorption and Rayleigh scattering (see Table
6). This choice was made to minimize the biases that the use of different
opacity line lists might cause. We refer to Alei et al. (2022) for more
details on such systematic errors.

The parameters considered in all retrievals and the assumed priors are
described in Table 4. Here, the priors are displayed as follows:
$\mathcal{U}(x,y)$ represents a uniform (boxcar) prior with equal probability
between a lower threshold $x$ and upper threshold $y$;
$\mathcal{G}(\mu,\sigma)$ denotes a prior shaped like a Gaussian distribution
with mean $\mu$ and standard deviation $\sigma$). The priors are generally
consistent with our previous studies (see LIFE Papers III and V), with
differences in the assumed radius and mass priors. The radius prior is uniform
between 0.5 and 2 $R_{\oplus}$, and the mass prior is a Gaussian of mean 1 and
sigma 0.5 $M_{\oplus}$. These priors are wider than what we assumed in
previous papers of the LIFE series. While LIFE will be able to gather narrower
priors on the radius from the search campaign by measuring the emitted flux of
the planet (see LIFE Paper II for details), the same assumptions could not be
made in the case of reflected light measurements obtained with HWO. To allow
both concepts to be comparable fairly, we selected larger priors that are
realistic estimates of what could be a product of prior measurements that
precede a prior detailed characterization campaign.

As in LIFE Papers III and V, in the retrieval we assume the presence of a
filling gas whose abundance is $1-\Sigma(X_{i})$ where $X_{i}$ is the
abundance of the chemical species included as free parameters (see Table 4).
The filling gas has a molecular weight of 28, assuming an N2-dominated mixture
which is realistic for terrestrial planets. The filling gas only contributes
to the mean molecular weight of the atmosphere and it is not spectroscopically
active.

The retrieval also includes N2 as a free parameter to allow the inclusion of
N2 Rayleigh scattering cross sections and N2-related collision-induced
absorption (see Table 6). In principle, we could expect any gas (or mixture)
of similar abundance and cross-section to be able to model the same Rayleigh
feature in the reflection spectrum. The use of N2 in our retrievals is to be
interpreted as a representation of a non-interacting bulk Rayleigh scatterer
of an average molecular weight of 28. To ensure this distinction is clear, we
label this parameter in our plots as X2.

###  2.4 Methods for the significance analysis

Ultimately, when performing retrievals on either simulated or real data, we
are interested in the significance of the detection of the main free
parameters. In this work, we use established methodologies to quantify the
significance of the detection of each atmospheric species with the single-
instrument runs and the joint retrieval run.

####  2.4.1 Bayes factor

One well-known metric to quantify the robustness of a detection is the Bayes
factor (see, e.g., Parviainen, 2018, and references therein). The Bayes factor
analysis allows us to compare two models to determine which one best
reproduces the data with the least amount of parameters. We can derive the
Bayes factor by calculating the ratio of the Bayesian evidences
$\mathcal{Z}(\mathbf{D}|\mathcal{M}_{i})$ of two sets of models, given the
data $\mathbf{D}$:

$\mathcal{K}=\frac{\mathcal{Z}(\mathbf{D}|\mathcal{M}_{1})}{\mathcal{Z}(\mathbf{D}|\mathcal{M}_{2})}$
(1)

By choosing as $\mathcal{M}_{1}$ ($\mathcal{M}_{2}$) a model that includes
(excludes) a specific species in the atmosphere and comparing the Bayesian
evidences, we can determine if the presence of that atmospheric species as a
free parameter in the model is justified by the data. The Jeffreys’ scale
(Jeffreys, 1998) allows us to then determine which model is preferred and with
which confidence (see Table 1).

Table 1: Jeffrey’s scale (Jeffreys, 1998).

$\log_{10}\left(\mathcal{K}\right)$ | Evidence Strength
---|---
$<0$ | Support for $\mathcal{M}_{2}$
$(0,0.5)$ | Very weak support for $\mathcal{M}_{1}$
$[0.5,1)$ | Substantial support for $\mathcal{M}_{1}$
$[1,2)$ | Strong support for $\mathcal{M}_{1}$
$[2,\infty)$ | Decisive support for $\mathcal{M}_{1}$
222Scale for the interpretation of the Bayes factor. Adapted from LIFE Papers
III and V.

####  2.4.2 “Sigma” interpretation of the Bayes factor

Following Benneke & Seager (2013) we are also able to calibrate between
Bayesian and frequentist detections by relating the value of $\mathcal{K}$ to
a level of significance. This conversion is possible provided the priors are
either unimodal symmetric distributions or uniform, as it is in this study. In
addition, the priors have to be the same for both models involved in the test.
Then, the tests in the Bayesian and frequentist regimes become equivalent,
since the priors cancel out (see Trotta, 2017). However, the tests do not have
the same meaning and interpretation due to the different frameworks. With the
Bayesian interpretation, the posterior probability $P(\mathcal{M}_{1}|D)$
gives probabilistic support that $\mathcal{M}_{1}$ is more likely to be
correct in comparison to $\mathcal{M}_{2}$, provided that the two models
exhaust the space of possible models. On the other hand, the frequentist
P-value gives the probability of observing a test statistic value that is
larger or equal to the one we observe, if the null hypothesis
$\mathcal{H}_{0}$ (here associated with $\mathcal{M}_{2}$) were to be correct.
The P-value allows to reject $\mathcal{H}_{0}$ for a chosen type I error
value. To ease up interpretation, the P-values are translated to a “sigma”
value as frequently used in astronomy (see, e.g., Benneke & Seager, 2013).
This is to be interpreted as a complementary metric to the Bayes factor to
determine the strength of the detection. In this case, only positive values
are allowed, since no conclusive statement on the strength of the detection
can be given when $\log_{10}(\mathcal{K})<0$ (i.e., when $\mathcal{M}_{2}$,
the model that excludes that species, is preferred). In Table 2, we show some
relevant frequentist significance values and the corresponding Bayes factors
and evidence strengths. We used these reference values to convert the
$\mathcal{K}$ values into “sigma”.

Table 2: Translation table between some relevant frequentist significance values and their Bayesian interpretation. Adapted from Benneke & Seager (2013) and Jeffreys (1998). Bayesian | Frequentist
---|---
$\mathcal{K}$ | $\log_{10}(\mathcal{K})$ | $P(\mathcal{M}_{1}|D)$ | Ev. Strength | P-value | “sigma”
2.5 | 0.40 | 0.714 | Weak | $0.05$ | $2.0\sigma$
2.9 | 0.46 | 0.744 | Weak | $0.04$ | $2.1\sigma$
8.0 | 0.90 | 0.889 | Substantial | $0.01$ | $2.6\sigma$
12.0 | 1.08 | 0.923 | Strong | $0.006$ | $2.7\sigma$
21.0 | 1.32 | 0.955 | Strong | $0.003$ | $3.0\sigma$
53.0 | 1.72 | 0.981 | Strong | $0.001$ | $3.3\sigma$
150.0 | 2.18 | 0.993 | Decisive | $0.0003$ | $3.6\sigma$
43000.0 | 3.63 | 0.99998 | Decisive | $6\cdot 10^{-7}$ | $5.0\sigma$

####  2.4.3 Mean-squared error

Since we are dealing with a simulated data scenario, we can exploit the
knowledge of the true value to determine the precision and accuracy of the
various retrievals. We use the mean-squared error, a metric that takes into
account the variance of the posterior distribution and the distance of the
mean of the posterior distribution from the true value. The mean-squared error
of an “estimator” parameter $\hat{\theta}$ compared to the true parameter
$\theta$ is defined as

$\mathrm{MSE}(\hat{\theta})=Bias(\theta,\hat{\theta})^{2}+Var(\hat{\theta})$
(2)

Where $Bias$ is the bias of the estimator, defined as the difference between
the estimated value and the true value, and $Var$ is the variance of the
estimator. In our scenario, this translates into:

$\mathrm{MSE}=(\mu-\tau)^{2}+\sigma^{2}$ (3)

Where $\mu$ is the mean of the posterior distribution, $\tau$ is the true
value, and $\sigma$ is the standard deviation of the posterior distribution.

Since the priors assumed for various parameters span different ranges, we
divided the posterior distribution of each parameter by the corresponding
prior, so that the normalized posterior would become a value in the [0, 1]
interval instead of the [x,y] interval. For the mass, whose prior was
Gaussian, we assumed the edges of the prior to be 0 and the 5$\sigma$ value
from the mean of the Gaussian (3.5 $M_{\oplus}$). This allows us to represent
all of the parameters with a single metric.

While this metric will not be available when dealing with real observations
(since there will not be any true value to compare against), it can be useful
at this stage of the development of the two future-generation concepts. By
looking for the smallest $\mathrm{MSE}$, we can identify the scenario that
provides the most precise and accurate estimates of relevant parameters. To
better appreciate such small variations, we used the square root of the mean
squared error for our analyses. A similar metric to evaluate the performance
of different retrieval runs has been used in other recent works such as, for
example, Hayoz et al. (2023).

Table 3: Details on the models run for each retrieval set and their color. Color |  Range [$\mathrm{\mu}m$] |  Description
---|---|---
|  [0.2, 2.0] |  Retrievals on HWO simulated spectra (UV/VIS/NIR)
|  [4.0, 18.5] |  Retrievals on LIFE simulated spectra (MIR)
|  [0.2, 2.0] $\cup$ [4.0, 18.5] |  Joint retrievals on HWO and LIFE (UV/VIS/NIR+MIR)

##  3 Results

We performed retrievals on the sets described in Section 2. Each set in turn
contains three retrievals performed on either UV/VIS/NIR data (HWO-like), MIR
data (LIFE-like), or joint retrievals on UV/VIS/NIR+MIR data (HWO+LIFE-like).
These have been associated with the same colors throughout the manuscript (see
Table 3 for details). We describe our results for the idealized high-
resolution low-noise scenario (used for validation) in Appendix B. Here, we
focus on the simplified noise scenario in Section 3.1, and the high-fidelity
noise scenario in Section 3.2.

###  3.1 Simplified noise scenario

Simplified Noise

Figure 3: Retrieved spectra for the second retrieval set _Left panel_ : pure
HWO retrieval (magenta); _Central panel_ : pure LIFE retrieval (blue); _Right
panel_ : HWO+LIFE retrieval (yellow). In all panels, the 2-$\sigma$ and the
1-$\sigma$ intervals are shown in increasingly darker hues, as well as the
input spectra (black lines) with error bars (gray-shaded areas) for
comparison.

Simplified Noise

Figure 4: Retrieved pressure-temperature profiles for the second retrieval set
(simplified noise): _Left panel_ : pure HWO retrieval (magenta); _Central
panel_ : pure LIFE retrieval (blue); _Right panel_ : HWO+LIFE retrieval
(yellow). In all panels, the 2-$\sigma$ and the 1-$\sigma$ intervals are shown
in increasingly darker hues, as well as the input profiles (black lines) for
comparison. Inside each panel, the inset plot shows the 2D posterior space of
the ground pressure and temperature. In all panels and the inset plots, the
surface pressure and temperature point in the P$-$T space is shown as a red
square marker.

Simplified Noise

Figure 5: Posterior density distributions from the second set of retrievals
(simplified noise). The black lines indicate the expected values for every
parameter. HWO posteriors are shown in magenta with diagonal hatching; LIFE
posteriors are shown in cyan with crossed hatching; HWO+LIFE posteriors are
shown as fully colored gold histograms.

Simplified Noise

Figure 6: Bayes factor and confidence levels values for each retrieved
spectroscopically active species in the second set of retrievals (simplified
noise). The heatmap is color-coded according to the Bayes factor values
$\log_{10}(\mathcal{K})$ and the confidence levels (in “sigma” values) are
labeled within each cell whenever $\log_{10}(\mathcal{K})$ is positive. The
Bayes factor values were obtained by performing a retrieval including and
excluding the molecule of interest and then using Eq. 1. These were then
converted into $\sigma$ through Table 2. The Bayes factor values can be found
in Table 10.

Simplified Noise

Figure 7: Square root of the mean squared error (see Equation 3) for relevant
parameters in the second set of retrievals (simplified noise).

In this set of retrievals, we assume the current baseline resolutions for LIFE
and HWO. These are: $R=50$ from $4-18.5$ µm for LIFE; $R=7$ ($0.2-0.515$ µm),
$R=140$ ($0.515-1$ µm) and $R=70$ ($1.01-2$ µm) for HWO. For this set of
retrievals, we keep a simplified noise instance (for more details see Section
2.2). In Figures 3, 4, and 5 we show a comparison of the retrieved spectra,
the P$-$T profiles, and the posteriors for the three models in this set. The
Bayes factors and confidence levels in $\sigma$ values are shown in Figure 6
and Table 10. The values of the square root of the $\mathrm{MSE}$ are shown as
bar plots in Figure 7. The corner plot and a table containing the retrieved
estimates with their 1-$\sigma$ uncertainty for each parameter can be found in
Figure 18.

As shown in Figure 3, the retrieved spectrum is always within the noise
uncertainty (gray shaded area) for all models. For the single-instrument
retrievals, the retrieved spectra show larger uncertainties at short
wavelengths where the noise is higher, as well as in some prominent lines.
Such uncertainties on the retrieved spectrum are smaller in the HWO+LIFE
retrieval, as we would expect from a retrieval performed on a larger amount of
data which exploits a larger information content compared to the retrievals on
a single wavelength range.

Regarding the characterization of the thermal structure of the atmosphere, the
retrieval on LIFE data infers a more precise and accurate characterization of
the atmosphere, though with increasing uncertainties towards lower pressures.
This is related to the physics of the problem: as the majority of the
radiation in the thermal emission spectrum comes from the deeper layers of the
atmosphere and the surface, it is much easier to constrain the deeper layers
of the atmosphere rather than the upper layers. The retrieved value of the
ground pressure is around $\log_{10}(P_{0})=-0.13$ which corresponds to about
0.74 bar, with an uncertainty of $\approx$0.3 dex. The true surface pressure
value is within the 1-$\sigma$ uncertainty. The surface temperature is
estimated as $T_{0}=284\pm 10$ K, very close to the true value of 286 K.

The HWO run does not obtain the same accuracy on the thermal profile, with
much wider uncertainties across all pressures. The ground pressure is
overestimated as $\log_{10}(P_{0})=0.33\pm 0.25$, which corresponds to
$\approx$ 2.1 bar, a factor two larger than the expected value. The surface
temperature is also much more loosely constrained at $T_{0}=335\pm 55$ K,
about 50 K hotter than the true value.

For the joint HWO+LIFE retrieval, the surface pressure is also slightly
overestimated (retrieved $\log_{10}(P_{0})=0.27\pm 0.2$, which corresponds to
around 1.8 bar), though less than the HWO-only run.

The ground temperature retrieval obtained with a joint retrieval is more
precise and accurate than any of the single-instrument retrievals, with a
retrieved ground temperature of $T_{0}=286\pm 6$ K. The P$-$T profile is also
retrieved correctly, with similar uncertainties as the LIFE retrieved profile
(see Figure 4) at lower pressures, but even smaller uncertainties in the
deeper layers of the atmosphere.

When it comes to the characterization of the mass and radius, there is no
substantial difference between the various runs (see Figures 5 and 18). The
radius is accurately retrieved by all models with an uncertainty of 0.1
$R_{\oplus}$ or less. Specifically, the retrieval of LIFE and HWO are
comparable with each other, while the HWO+LIFE one gets a more accurate
constraint on the radius up to 0.05 $R_{\oplus}$ uncertainty. The mass is
consistently retrieved in all three retrievals, but its estimate is not
significantly more precise than the input prior.

Regarding the reflectance, the HWO and HWO+LIFE retrievals correctly retrieve
the value with comparable uncertainty, while LIFE alone does not manage to
constrain this parameter. This shows that only the reflected-light portion of
the spectrum contains enough information to correctly retrieve the reflectance
of the planet, which is to be expected and is confirmed by the idealized high-
resolution low-noise scenario (see Appendix B).

By observing the posteriors in Figure 5 we note that, generally, the
posteriors of the joint retrievals are significantly smaller than the
posteriors of the single models, especially when the LIFE and HWO retrievals
are not consistently retrieving a parameter. The HWO+LIFE posteriors place
themselves at the intersection of the posteriors of the single-instrument
runs; this translates into smaller uncertainties on the estimates of most
parameters and a greater decoupling of all correlated parameters. In other
words, compared to single-instrument retrievals, a joint retrieval reduces the
subset of possible parameters that can reproduce the data, particularly so
when two or more parameters are correlated (e.g., the pressure and the main
absorbing species of the atmosphere). This means that the joint retrieval
correctly leverages the total amount of information available in the two
wavelength ranges, as one would expect from the retrieval of complementary
data.

Figures 5, 6, and 7 quantify what we observed so far. The retrieval of the
atmospheric species with LIFE is consistent with the previous papers in the
series: LIFE can decisively constrain the abundance of H2O and CO2 (with more
than 5$\sigma$ confidence level), as well as strongly constrain O3 ($3\sigma$
confidence). However, it is not sensitive to the bulk scatterer X2, O2, and
CO, and can only retrieve upper limits on N2O and CH4. On the other hand, the
HWO run poses strong constraints on O2 and H2O ($5\sigma$), while O3 is weakly
constrained (at a $<2\sigma$ confidence level). The other molecules (CH4, CO,
CO2, N2O) are unconstrained or constrained with very broad upper limits.
Through HWO retrievals, it is also possible to estimate the abundance of the
bulk absorber X2 (see Section 4.1). The joint retrieval can retrieve with
higher precision the species that were already retrieved in both the single-
instrument retrievals. For all the molecules we retrieved, the HWO+LIFE
retrieval obtains equal or higher confidence levels (Figure 6) compared to the
single-instrument cases. Notably, it is possible to have a decisive detection
of O3 compared to a weak and strong detection of the HWO and LIFE retrievals
respectively. This result is however dependent on the noise simulation, as
will be seen in the third set of retrievals and further discussed in Section
4.3.

The mean-squared-error metric is also valuable to quantify the improvement of
the joint retrieval compared to the single-instrument scenarios (Figure 7):
for all the parameters considered in the retrieval, the estimate provided by
the multi-instrument retrieval is comparable or significantly better in
precision and accuracy compared to the single-instrument estimates (i.e., the
value of $\sqrt{\mathrm{MSE}}$ of the HWO+LIFE retrieval for each parameter is
the same or lower than the one from the single-instrument retrievals).

###  3.2 Higher-fidelity PSG/LIFEsim simulated noise scenario

PSG/LIFEsim Noise

Figure 8: Retrieved spectra for the third retrieval set (PSG/LIFEsim noise):
_Left panel_ : pure HWO retrieval (magenta); _Central panel_ : pure LIFE
retrieval (blue); _Right panel_ : HWO+LIFE retrieval (yellow). In all panels,
the 2-$\sigma$ and the 1-$\sigma$ intervals are shown in increasingly darker
hues, as well as the input spectra (black lines) with error bars (gray-shaded
areas) for comparison.

PSG/LIFEsim Noise

Figure 9: Retrieved pressure-temperature profiles for the third retrieval set
(PSG/LIFEsim noise): _Left panel_ : pure HWO retrieval (magenta); _Central
panel_ : pure LIFE retrieval (blue); _Right panel_ : HWO+LIFE retrieval
(yellow). In all panels, the 2-$\sigma$ and the 1-$\sigma$ intervals are shown
in increasingly darker hues, as well as the input profiles (black lines) for
comparison. Inside each panel, the inset plot shows the 2D posterior space of
the ground pressure and temperature. In all panels and the inset plots, the
surface pressure and temperature point in the P$-$T space is shown as a red
square marker.

PSG/LIFEsim Noise

Figure 10: Posterior density distributions from the third set of retrievals
(PSG/LIFEsim noise). The black lines indicate the expected values for every
parameter. HWO posteriors are shown in magenta with diagonal hatching; LIFE
posteriors are shown in cyan with crossed hatching; HWO+LIFE posteriors are
shown as fully colored gold histograms.

PSG/LIFEsim Noise

Figure 11: Bayes factor and confidence levels values for each retrieved
spectroscopically active species in the second set of retrievals (PSG/LIFEsim
noise). The heatmap is color-coded according to the Bayes factor values
$\log_{10}(\mathcal{K})$ and the confidence levels (in “sigma” values) are
labeled within each cell whenever $\log_{10}(\mathcal{K})$ is positive. The
Bayes factor values were obtained by performing a retrieval including and
excluding the molecule of interest and then using Eq. 1. These were then
converted into $\sigma$ through Table 2. The Bayes factor values can be found
in Table 11.

PSG/LIFEsim Noise

Figure 12: Square root of the mean squared error (see Equation 3) for relevant
parameters in the third set of retrievals (PSG/LIFEsim noise).

In this set of retrievals, we used the same “baseline” resolutions as the
previous set but assumed a more complex noise instance, which was generated by
the currently available noise simulators for the LIFE and the HWO concepts (as
described in Section 2.2). The results of this set of retrievals are shown in
Figures 8, 9, and 10. The Bayes factor and confidence level values of the
atmospheric species are shown in Figure 11. The numeric values of the Bayes
factor are reported in Table 11. The square root of the mean squared error for
the retrieved parameters is shown in Figure 12. The corner plot and the
retrieved estimates with their 1-$\sigma$ uncertainty for each parameter can
be found in Figure 19.

The P$-$T profile is accurately retrieved with LIFE. On the other hand, the
HWO run overestimates the ground pressure and temperature and constrains the
atmospheric profile more loosely. The joint retrieval overestimates the ground
pressure, but still retrieves a more accurate and precise estimate of the
ground temperature compared to the single-instrument runs ($T_{0}=287\pm 6$ K
for the HWO+LIFE run compared to $T_{0}=336\pm 60$ K for the HWO run and
$T_{0}=285\pm 11$ K for the LIFE run, compared to the true value of 286 K).

Regarding the atmospheric characterization (see Figures 10, 11, and 12) we
observe some differences compared to the previous set of models. The retrieval
on HWO data estimates with strong or decisive confidence ($\geq 3\sigma$
confidence level) the abundances of O2, H2O, and O3. The confidence on the O3
detection is a definite improvement compared to the previous set, though at
the expense of a lack of confidence in the retrieval of an upper limit of CO.

The performance of the LIFE retrieval does not significantly change in this
scenario. Also for this set of retrievals, the joint retrieval of UV/VIS/NIR +
MIR data produces results at least as confident as the single-instrument runs,
estimating all free parameters with higher accuracy and precision compared to
HWO- and LIFE-only retrievals (see Figure 12). However, the joint retrieval is
not able to confidently estimate CH4, CO, or N2O.

##  4 Discussion

In this section we discuss the results, focusing on the single-instrument
observations (Section 4.1), the joint observations (Section 4.2), and the
impact of noise (Section 4.3). We discuss the limitations of our work in
Section 4.4. To benchmark our results with the literature whenever possible,
we compare the single-instrument runs we performed in this work with other
HWO- and LIFE-centered studies in Appendix C.

###  4.1 Strengths of single-instrument observations

In all the retrieval sets performed in this study, we can directly compare the
potential of the two mission concepts in characterizing a Modern Earth-twin
exoplanet. Most of the differences lie in the intrinsic planetary spectral
features that are observable in different wavelength bands. Molecular oxygen
has sharp features in the visible reflected-light spectrum, while ozone is
prominent both in the UV (¡$0.3$ µm) and in the MIR (around $9.6$ µm). The CO2
band at $15$ µm is also a readily detectable feature, while H2O absorbs across
most wavelengths. Because of these differences, we can retrieve with higher
confidence specific molecules in the two wavelength ranges (see Figures 5, 10,
and 15, as well as Tables 10, 11). In all baseline scenarios (simplified noise
and high-fidelity noise), from the HWO retrievals we performed we can estimate
O2, H2O, and O3; on the other hand, LIFE retrievals allow us to estimate CO2,
H2O and O3. In both wavelength ranges, we also have upper limits on CH4 and
N2O (see Figures 5 and 10), though with very weak or no confidence for the
baseline values. This is in agreement with what was found in the literature
and our previous studies (see Appendix C). Importantly, even in the idealized
scenario (photon noise only, $R=1000$ and $S/N=50$, see Appendix B), the
single-instrument runs would still not be able to constrain CO and N2O for the
HWO case, or O2 for the LIFE case (see Figure 17).

Observations in the UV/VIS/NIR are especially sensitive to the scattering
processes (both Rayleigh and surface scattering), while MIR measurements would
directly retrieve an estimate of the thermal structure of the atmosphere.
These are results that can be observed clearly in all the runs (see, e.g.,
Figures 5, 10, and 15). Specifically, in the HWO retrievals the Rayleigh slope
is well reproduced by the forward model (see, e.g., Figures 3, 8, and 13). The
Rayleigh slope in the UV/VIS/NIR spectrum is mainly impacted by the most
abundant molecules, which could be spectrally active (e.g., O2) or inactive
(e.g., the bulk scatterer X2, which would correspond to N2 in an Earth-twin
scenario). While the abundance of spectrally active species can be constrained
through absorption lines in the spectrum, X2 can only be constrained through
the modeling of the Rayleigh slope. With HWO-like observations, we would be
able to infer the presence of a bulk scatterer X2, though further studies
would be needed to properly assess the nature of the absorber itself (see Hall
et al., 2023).

A correct retrieval of X2 and its contribution to scattering at short
wavelengths in turn breaks the degeneracy between radius and reflectance: at
short wavelengths, the planetary spectrum would become opaque because of
Rayleigh scattering, so most of the incident flux never reaches the surface
(see, e.g., Gaudi et al., 2020) – this allows these parameters to be retrieved
with greater accuracy. On the other hand, retrievals of data at UV/VIS/NIR
wavelengths cannot constrain the pressure-temperature profile to more than
tens of degrees of uncertainty, even in the best-case scenario (see Figure
14). This result informs us that there is simply not enough information in the
reflected-light spectrum to accurately pinpoint the thermal profile of the
atmosphere. At the baseline resolutions, HWO-like retrievals have the
potential of also overestimating the ground pressure (see Figures 4 and 9).
This biased result is coupled with a lower estimate of the abundance of some
molecules in the HWO-only runs (e.g., O2, H2O, O3), as a result of the well-
known pressure-abundance degeneracy, observed in many previous studies (see
Appendix C).

From the MIR point of view, strong absorption lines of CO2, H2O and O3 allow
for precisely fitting the P$-$T profile in the denser layer of the atmosphere
(see, e.g., Figures 4, 9, and 14). LIFE retrievals fail to constrain the
surface reflectance $r_{s}$. This is expected since the stellar and thermal
radiation that is scattered by the surface at MIR wavelengths is so little to
be below the noise level. The only exception is the high-resolution case: the
impact of the scattered light on the spectrum is greater than the noise level
for surface reflectances higher than 0.3-0.4, which allows the retrieval
framework to provide an upper limit on this parameter for MIR-only data.
Still, other estimates of the albedo of the planet are accessible through
LIFE-like retrievals. For example, from an accurate estimate of the radius,
which LIFE would gather from the search campaign, it would be possible to get
an estimate of the equilibrium temperature of the planet and, as a
consequence, the Bond albedo of the planet (see, e.g., Konrad et al., 2023,
i.e., LIFE Paper IX, for details).

When it comes to the interpretation of these results, single-instrument
retrievals would allow us to gather plenty of information to characterize the
atmosphere of a planet and its habitability. Retrievals performed considering
only UV/VIS/NIR data would in principle allow us to determine that the planet
has water vapor in the atmosphere, though the high uncertainty on the surface
temperature and pressure would still make it unclear whether there could be
liquid water on the surface. It would also be possible to detect oxygen, which
is a key biosignature for Earth-like life, but the lack of accurate
characterization of additional molecules such as CH4 and CO to constrain the
redox state of the atmosphere would not necessarily rule out the abiotic false
positive case, where oxygen is generated by non-biological processes such as
photolysis (see, e.g., Domagal-Goldman et al., 2014; Meadows et al., 2018).

On the other hand, retrievals on MIR data (LIFE retrievals, in blue) could
provide very strong constraints on CO2, H2O, and O3, which could in principle
also point to the presence of a potentially inhabited planet. However, the
relationship between O2 and its photo-chemical byproduct O3 is not linear (see
Segura et al., 2003; Kozakis et al., 2022). Also in this case, a low-quality
estimate of CH4 might hinder the vetting of O2 in a potential abiotic
scenario. Nevertheless, assuming higher baseline resolution and signal-to-
noise levels for LIFE (see, e.g., LIFE Papers III and IX) would mitigate this
problem by allowing a stronger constraint on the CH4 abundance. Earth-like
abundances of N2O would also be not possible to detect at this resolution and
noise level. A more accurate and precise characterization of the thermal
profile of the atmosphere, possible through MIR retrievals, would provide more
stringent results on the potential for liquid water on the surface, though no
direct measurement of the surface reflectivity (and potential water and
vegetation features) would be available.

These statements only apply in the case of modern Earth analogs, which are the
focus of this study. Other scenarios might behave differently, depending on
the atmospheric composition. An example is the Archean Earth scenario, when
CO2 and CH4 were likely much more abundant in the atmosphere. The simultaneous
detection of CO2 and CH4 is believed to be a signature of life on early Earth
(Krissansen-Totton et al., 2018). The higher abundance of these two molecules
in the early Earth would make the detection of these within reach of both HWO
(see, e.g., Young et al., 2024a, b; Damiano & Hu, 2022) and LIFE (see, e.g.,
Alei et al., 2022, i.e., LIFE Paper V). Another example would be an atmosphere
with enhanced biogenic N2O. Angerhausen et al. (2024, i.e., LIFE Paper XII)
found that this molecule could be detectable with LIFE-like observations if
present at plausible biological production fluxes.

###  4.2 Complementarity of multi-instrument observations

From what we have seen in this study, multi-instrument observations are
clearly helpful for reducing biases and overall enhancing the quality of the
results. For each set, the posterior distributions of the joint retrieval run
are mostly at the intersection of the single-instrument results (see the
corner plots in Figures 17, 18, 19). The posteriors also show a reduced bias
and a smaller variance around the true value (see the mean-squared-error bar
plots in Figures 7, 12, and 16).

Having data spanning a larger wavelength range translates into an increase in
the number of spectral features to compare the forward model against when
performing a Bayesian retrieval. It further becomes easier to disentangle
overlapping features, by including more bands where each specific molecule can
be more unambiguously detected. This limits the range of parameter space that
can accurately reproduce the data, thus shrinking uncertainties (see Figures
5, 12, and 15). From the point of view of the atmospheric structure, different
wavelength bandpasses sample different pressures throughout the atmosphere,
allowing us to have a more precise thermal structure when including data from
HWO and from LIFE, compared to the single-instrument counterparts (see Figures
4, 9, and 14).

At the baseline resolution values (simplified noise and high-fidelity noise
scenarios) a joint HWO+LIFE retrieval on a cloud-free Earth-like planet would
yield very strong constraints on O2, O3, CO2, and H2O, as well as weak
constraints on CH4 and CO. In our runs, the confidence level for the detection
of each of the considered species in the joint retrieval scenario is at least
equal to the larger one between the two single-instrument retrievals and it
increases considerably in some cases (e.g., O3, see Figures 7 and 12). This
shows that, whenever enough information is stored in one portion of the
wavelength spectrum, this is also similarly detected with a joint retrieval.
If, however, both portions of the spectrum contain information on a specific
parameter, then the joint retrieval performance is magnified.

The joint retrievals also slightly overestimate the ground pressure. This is
probably due to the VIS+NIR spectrum which, as we have discussed in the
previous subsection, is not sensitive to the atmospheric thermal profile but
that has a marginally higher resolution than the MIR thermal emission
spectrum. Therefore, the forward model might favor the strong O2 lines in the
reflected spectrum, which tend to skew the result towards high pressures and
low abundances, as it happens in the HWO-only retrievals. The overestimation
of the ground pressure would in principle slightly increase the range of
temperatures at which water can be liquid. However, the precise and accurate
estimate of the surface temperature (retrieved with an uncertainty of $\approx
5$ K when considering joint retrievals) would be more convincing about the
habitability of a promising target, compared to the single-instrument runs.

A joint retrieval would therefore provide the highest quality of information
available for a target. While an increase in the quality of the results
compared to the single-instrument retrievals may seem quite obvious since the
joint retrieval can leverage all the available information throughout the
wavelength range, this represents a strong validation of our base assumption
that the sum of both missions would be greater than each separately. This set
of runs also serves as a “proof of concept” for a joint characterization of a
terrestrial planet since it highlights how transformational the multi-
wavelength observation of a promising planet could be, especially when trying
to characterize the habitability of a potential candidate host for life.

Furthermore, a detailed characterization of both the reflected and the thermal
portions of the planetary spectrum would help break the radius-albedo
ambiguity by enabling an energy-budget analysis of the planet. By constraining
both the albedo and the abundance of various greenhouse gases, it would be
possible to estimate the expected greenhouse effect given the observed surface
temperature and atmospheric profile, which would be a prerequisite for an
accurate assessment of the habitability of the planet. Estimating the surface
temperature with high precision would also refine the search for life by
determining the likelihood of the presence of surface water; this would be a
key piece of information needed to identify a promising target for further in-
depth study, and it would likely not be prior available information.

Finally, the population-level results will be much more robust with both
reflected light and thermal emission. Observations in both wavelength ranges
will help us characterize the diversity of the planetary atmospheres that both
missions will observe, paving the way for an unbiased, complete picture of the
demographics of the sample.

###  4.3 Impact of noise

At this stage, accurate noise models that are specific to the two concepts are
not yet fully developed. The development of these tools is strongly coupled
with the architecture trades and the technological assessments that are
currently in progress and will be executed over the coming years. However, it
is assumed that we will be dealing with very small signals. In this “low-
quality” data regime, different noise instances might be the reason for the
detection or non-detection of specific molecules.

In this study, we used various noise instances to explore the impact that
noise has in the retrievals. As explained in Section 2.1, the noise level was
the only difference between the “simplified noise” and “PSG/LIFEsim noise”
retrievals. By comparing these sets of retrievals (see Section 3) we find that
a more complex noise modeling causes some differences in the results. The most
noticeable one is the detection of O3, which can be decisively detected thanks
to the lower noise level in the UV band in the high-fidelity noise set
compared to the very weak confidence in the detection in the simplified noise
set. This highlights the great impact that accurate noise modeling could have
in defining the requirements of future-generation missions and in prioritizing
some architectures compared to others. As we progress in the definition of the
various concepts, the noise models will need to be updated and these results
will likely change. It is therefore important at this stage of the concepts’
maturation to benchmark results and to simulate observations through
atmospheric retrievals, since these will provide information that will be fed
back into the definition of the preferred architectures.

###  4.4 Known limitations of this study

The results discussed in this work are valid within the assumptions that were
made. To simplify the problem and to provide a first-order overview, we used a
simplified, cloud-free scenario that is not purely motivated by physics and is
unlikely to be found in reality. In this scenario, it is possible to
disentangle the correlation between the radius and the albedo through a high-
resolution characterization of the Rayleigh cross section in the UV/VIS/NIR.
However, this would most likely not be the case when clouds and hazes are
present (see, e.g., Feng et al., 2018; Robinson & Salvador, 2023; Damiano &
Hu, 2022). A MIR observation should be less prone to such issues (see LIFE
Paper IX).

Furthermore, we used a simplified retrieval framework to allow for reasonable
computing time (see Section 2.3 for details). Some further biases in the
detection of the most abundant species in the atmosphere (the bulk scatterer
X2 and O2) could have been caused by pyMultiNest since this nested sampling
algorithm has been found to undersample the edges of the prior space (see
Appendix D of Himes, 2022), which is the case for these two molecules. This
bias could have especially negatively influenced the retrievals in the
UV/VIS/NIR range, for which it was possible to constrain such molecules.
Retrievals on the MIR portion of the spectrum would not have suffered from
this bias, since not enough information about these molecules is contained in
this range.

We assumed that we would perfectly know the geometry of the planet-star
system, set at quadrature, and that the planet signal is fixed: rotation and
atmospheric dynamics that happen on daily or seasonal timescales can be
ignored. In reality, the phase of the system is likely to be unknown in this
kind of observation and will need to be accurately modeled in retrievals.

Finally, we do not randomize the individual spectral points according to the
noise, but rather we consider the noise as an uncertainty to the theoretically
simulated flux points. This has been done in previous studies (see Appendix
C), but it could lead to an overestimation of the results. In the Appendix of
LIFE Paper III we compared the results of retrievals on non-randomized and
randomized flux points, showing that results on non-randomized spectra could
be interpreted as average estimates of the randomized-spectra results.

These issues will be overcome and improved in future studies. Some other
limitations are however much more rooted in the development of these concepts
themselves and the technical challenges that stem from that.

First of all, the noise treatment should be improved and updated as long as
the iterations for various architecture trades proceed. Given the impact that
noise could have on the confidence of the detection of important life-related
molecules (as shown by comparing the two baseline sets of retrievals), it is
mandatory to keep refining the instrumental parameters that have an impact on
the noise. This will also help define the observing time required for each
observation to achieve the desired $S/N$. Once we get a better understanding
of the processes that cause the spectral shape of the noise to change with
exposure time, it will be possible to simulate accurate noise instances at the
required observing time, instead of scaling a template noise to the desired
$S/N$ as it was done in this study. Retrieval benchmark studies will then need
to be repeated to identify the architecture that yields the best possible
results.

The Habitable Worlds Observatory will very likely include a coronagraph. This
will require the presence of specific, multiple spectral bandpasses in the
three wavelength ranges. We would not necessarily expect to have access to a
complete spectrum of the planet in the UV/VIS/NIR range, like it was assumed
here. Although some initial studies on the observation strategy (Young et al.,
2023) and on how these bandpasses should be defined for HWO (Latouf et al.,
2023a, b) were performed, these are still to be confirmed. Since acquisitions
in the same wavelength range are limited to one bandpass at a time (Young et
al., 2023) and since we would deal with long exposure times (of the order of
hours, or even days), it is likely that the geometry of the planet-star system
would change between one acquisition and another. This would complicate the
modeling and the retrieval of realistic results.

We used a relatively large prior on the radius to allow for comparable
retrieval results for both wavelength ranges of interest. However, a LIFE-like
mission should be able to provide more stringent constraints on the radius
from the search campaign (see LIFE Paper II) which could be used as prior for
retrievals in both wavelength ranges and therefore potentially allow a more
accurate retrieval of the planetary radius. Similarly, we could imagine that
an improvement in the prior estimate of the mass (e.g., leveraging prior
observations and/or the search campaign of a MIR LIFE-like mission) would
improve the results.

At this stage, it is entirely plausible that the two missions will not fly at
the same time, but rather one before the other. The added complexity of multi-
epoch observations might bring its own set of challenges, as the geometry of
the observation has likely changed and phase modeling will need to be taken
into account. In the context of finding more robust clues for the presence of
life on an exoplanet, joint retrievals on UV/VIS/NIR+MIR data might not be the
best strategy, as one would have to merge data at different ephemerids. It
might be more efficient to use the posteriors of a retrieval performed on data
from one mission as priors for retrievals performed in the other spectral
range. However, as we have seen in this study, biases are present in single-
instrument retrievals. Therefore, it would be more realistic to select only
the results we are more confident in to feed into future retrievals. For
example, one should be more confident in the radius estimate given by
retrievals in the MIR, rather than the UV/VIS/NIR. Similarly, the detection of
O2 through UV/VIS/NIR could be fed into retrievals on MIR data to improve the
determination of the molecular weight of the atmosphere. Given the strong
complementarity of these results, such strategies might play a role in
maximizing the yield from these missions. Further studies are required to
assess which parameters are more likely to be constrained by which concept
(and wavelength range) over a variety of case scenarios that go beyond a
cloud-free Earth twin.

##  5 Conclusions

In this study, we performed atmospheric Bayesian retrievals on simulated data
from a HWO-like and a LIFE-like mission to assess the science potential of the
two concepts separately and in synergy with each other for the
characterization of a simplified Earth twin. To do this, we produced simulated
observations of a cloud-free terrestrial planet both in reflected light and
thermal emission. Furthermore, the retrieval routine that we developed for
previous LIFE-related studies (e.g., Konrad et al., 2022; Alei et al., 2022;
Konrad et al., 2023; Mettler et al., 2024) has been updated to consider both
UV/VIS/NIR and MIR spectra.

We considered two scenarios: a simplified noise model at constant error bars
considering the baseline resolutions for the various wavelength ranges of
interest, and a more complicated noise model using the current noise
simulators available for LIFE and “6-m LUVOIR-B” as template for HWO.

We conclude that:

  1. 1.

Retrievals considering purely data from one of the two instruments would not
provide a full characterization of the atmosphere and thermal structure of the
planet. A retrieval on UV/VIS/NIR data alone strongly constrains O2 and H2O,
and retrieves an upper limit on CH4. A retrieval on MIR data strongly
constrains CO2, H2O, and O3, and also an upper limit on CH4. Neither of the
two concepts can sufficiently constrain Earth-like abundances of CO and N2O,
both relevant to rule out abiotic processes (for the former) and assess
metabolic activity (for the latter) (see, e.g., Schwieterman et al., 2018).
LIFE retrievals would constrain the thermal profile of the atmosphere, while
HWO retrievals would allow to constrain the surface reflectance (in the
absence of clouds).

  2. 2.

Independent of the noise and the spectral resolution, a joint UV/VIS/NIR+MIR
retrieval can improve the confidence level of the detection of the main
potential biosignatures and bioindicators. Retrievals on UV/VIS/NIR+MIR
constrain all the species that the single-instrument runs estimate, but
increasing the confidence of the estimates itself: O2, CO2, H2O, and O3 would
be detected with decisive confidence, while CO and CH4 could be weakly
detected. Joint retrievals also improve the estimate of the surface
temperature and pressure, as well as the radius.

  3. 3.

The different noise assumptions can critically impact the results: the most
realistic noise scenario (PSG/LIFEsim noise) would allow to strongly retrieve
O3 on UV/VIS/NIR only data, while it would be only possible to get a low-
significance constraint on that species in the simplified noise case. This
result suggests that a higher-fidelity noise model could allow more accurate
results that could drive the definition of the requirements of these missions.
Further, the spectral resolution and $S/N$ reference values are still up for
discussion and in this work we only explored a small section of the $R$-$S/N$
parameter space. The refinement of science requirements will be an iterative
process that will continue in time as technological progress is made and as
noise simulators take more and more realistic instrumental processes into
account.

This has been the first study on the possible synergies between the Habitable
Worlds Observatory and the Large Interferometer for Exoplanets in
characterizing habitable exoplanets, in support of the development of both by
the community. Both wavelength ranges provide us with their specific set of
unique information and come with specific drawbacks. Yet, the scientific yield
of synergistic observations in the UV/VIS/NIR+MIR range has the potential of
being greater than the sum of its parts. Having access to multiple spectral
windows into the atmosphere of a potentially habitable planet could be
transformative for the search for life in the universe.

###### Acknowledgements.

E. Alei’s work has been partly carried out within the framework of the NCCR
PlanetS supported by the Swiss National Science Foundation under grants
51NF40_182901 and 51NF40_205606. E. Alei’s research was partly supported by an
appointment to the NASA Postdoctoral Program at the NASA Goddard Space
Research, administered by Oak Ridge Associated Universities under contract
with NASA. S. P. Quanz’s work has been carried out within the framework of the
NCCR PlanetS supported by the Swiss National Science Foundation under grants
51NF40_182901 and 51NF40_205606. E. O. Garvin gratefully acknowledges the
financial support from the Swiss National Science Foundation (SNSF) under
project grant number 200020_200399. V. Kofman is supported by the GSFC Sellers
Exoplanet Environments Collaboration (SEEC) and the Exoplanets Spectroscopy
Technologies (ExoSpec). P. Mollière acknowledges support from the European
Research Council under the European Union’s Horizon 2020 research and
innovation program under grant agreement No. 832428. E. Alei acknowledges T.
Birbacher, F. Dannert, S. Gaudi, S. Mahadevan, B. Mennesson, A. Roberge, G.
Villanueva, and A. Young for their expertise and useful discussions. We thank
an anonymous reviewer for the helpful comments.

CRediT author statement. E. Alei: Conceptualization, Methodology, Software,
Validation, Formal Analysis, Investigation, Data Curation, Writing - Original
Draft, Visualization; S. P. Quanz: Conceptualization, Resources, Writing -
Review and Editing, Supervision, Project Administration, Funding Acquisition;
B. S. Konrad: Methodology, Software, Writing – Review and Editing; E. Garvin:
Formal Analysis, Writing – Review and Editing; V. Kofman: Methodology, Writing
– Review and Editing; A. Mandell: Methodology, Writing – Review and Editing;
D. Angerhausen: Writing – Review and Editing; P. Mollière: Writing – Review
and Editing; M. Meyer: Writing – Review and Editing; T. Robinson: Writing –
Review and Editing; S. Rugheimer: Writing – Review and Editing.

_Software._ This research made use of: Astropy333http://www.astropy.org, a
community-developed core Python package for Astronomy (Astropy Collaboration
et al., 2013, 2018); Matplotlib444https://matplotlib.org (Hunter, 2007);
pandas (pandas development team, 2020); petitRADTRANS (Mollière et al., 2019);
PSG (Villanueva et al., 2022).

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##  Appendix A Scattering of terrestrial exoplanets in reflected light

petitRADTRANS (Mollière et al. 2019) performs radiative transfer calculations
including scattering through the Feautrier method (Feautrier 1964). We used
the Feautrier method in LIFE Paper V only considering the thermal planetary
radiation and we refer to Appendix A of LIFE Paper V for details on the
implementation of that process, as well as Mollière (2017). Here, we report
the main equations and we discuss the additional terms that are now taken into
account to calculate the scattering by direct stellar light.

The radiative transfer equation (see Eq. 4) depends linearly on the intensity,
so planetary and stellar radiation fields can be treated in an additive way.

$\mu\frac{dI}{d\tau}=-I+S.$ (4)

In this equation, $\mu=\cos{\theta}$ where $\theta$ is the angle between a
light ray and the surface normal, $\tau$ is the optical depth, $I$ is the
intensity, and $S$ is the source function.

In the Feautrier algorithm, the intensity vectors parallel and anti-parallel
to the direction of a light ray ($I_{+}$ and $I_{-}$ respectively) are
defined. The radiative transfer equation is then solved for the sum and the
difference of these two rays (see LIFE Paper V). Two boundary conditions need
to be defined at the top of the atmosphere (TOA; corresponding to $P=0$) and
at the surface ($P=P_{s}$). In the case of planetary radiation scattering only
(used in LIFE Paper V), the conditions are:

$I_{+}(P=0,\mu)=0\quad\forall\mu$ (5)

$I_{-}(P=P_{s},\mu)=e_{s}\ B(T_{s})+r_{s}J^{scat}(P_{s}).$ (6)

Where $e_{s}$ is the surface emissivity; $r_{s}$ is the surface reflectance;
$B(T_{s})$ is the blackbody emission of the surface; and $J^{scat}$ the
average scattered intensity of the radiation that comes from the top layers.
The interpretation of these conditions is that: 1) there is no planetary
radiation coming from the top of the atmosphere and 2) the light going upwards
from the surface is composed by the thermal emission of the surface itself (as
the blackbody radiation of the surface scaled by the surface emissivity) and
by a portion of the incoming planetary radiation that is reflected by the
surface (as the average scattered intensity of the radiation that comes from
the top layers scaled by the surface reflectance). $J^{scat}$ is calculated as
follows:

$J^{scat}(P_{s})=\int_{0}^{1}I_{+}(P_{s})d\mu$ (7)

In the case of scattering of stellar radiation, it is possible to split the
incoming stellar intensity into two components: the incoming stellar radiation
$I_{*}^{dir}$ along the TOA incidence angle that survives down to a given
layer of interest and which gets attenuated through absorption and scattering,
and the stellar light that is scattered (potentially multiple times) in the
atmosphere $I_{*}^{scat}$.

$I_{*}=I_{*}^{dir}+I_{*}^{scat}$ (8)

The radiative transfer equation for these components becomes ($\epsilon$ being
the photon destruction probability):

$\frac{dI_{*}^{dir}}{d\tau}+\frac{dI_{*}^{scat}}{d\tau}=-I_{*}^{dir}-I_{*}^{scat}+(1-\epsilon)(J_{*}^{dir}+J_{*}^{scat})$
(9)

Which can be split into two equations:

$\frac{dI_{*}^{dir}}{d\tau}=-I_{*}^{dir}$ (10)

$\frac{dI_{*}^{scat}}{d\tau}=-I_{*}^{scat}+(1-\epsilon)(J_{*}^{dir}+J_{*}^{scat})$
(11)

Equation 10 is the regular attenuation of the direct light while crossing the
atmosphere. Its solution is:

$I_{*}^{dir}=I_{*}^{dir}(0,\mu)\ e^{-\tau/\mu}$ (12)

Where $I_{*}^{dir}(0,\mu)$ is the irradiation at the TOA.

For Eq. 11, the calculation of $J_{*}^{dir}$ is dependent on the geometry of
the problem.

In the case of isotropic stellar irradiation, $I_{*}^{dir}(0,\mu)$ is
integrated over all possible angles ($-90^{\circ}$ to $+90^{\circ}$ angle
between the ray and the direction normal to the surface, which corresponds to
the light that comes from the top). This translates as:

$I_{*}^{dir}(0)=\begin{cases}F_{*}^{dir}(0)/\pi,&\mbox{if }\mu<0\\\
0,&\mbox{otherwise}\end{cases}$ (13)

Where $F_{*}^{dir}(0)$ is the stellar flux scaled at the distance of the
planet. The average direct intensity over all angles is then:

$J_{*}^{dir}=\frac{1}{2}\int_{-1}^{1}I_{*}^{dir}d\mu=\int_{0}^{1}I_{*}^{dir}d\mu$
(14)

For non-isotropic irradiation (i.e., the stellar irradiation is incident under
an angle $\theta_{*}$ of cosine $\mu_{*}$), the value of $F_{*}^{dir}(0)$ is:

$F_{*}^{dir}(0)=\int_{\Delta\Omega_{*}}I_{*}^{dir}\mu d\Omega\approx
I_{*}^{dir}(0)\mu_{*}\Delta\Omega_{*}$ (15)

Where $\Delta\Omega_{*}$ is the solid angle subtended by the stellar disk from
the atmospheric location. In this case, $J_{*}^{dir}$ is:

$J_{*}^{dir}=\frac{I_{*}^{dir}\Delta\Omega_{*}}{4\pi}=\frac{F_{*}^{dir}(0)\
e^{-\tau/\mu_{*}}\Delta\Omega_{*}}{4\pi\mu_{*}}$ (16)

The stellar light that can be scattered at the surface is then calculated
analogously to Eq. 7 for the stellar radiation field and added to the
planetary radiation field. $J^{scat}$ will now automatically include both
contributions from the atmospheric thermal radiation, and the stellar one,
scattered at least once inside the atmosphere.

To meet the boundary condition, the mean direct radiation must be included in
the term that takes into account the portion of radiation reflected by the
surface. Therefore, Equation 6 becomes

$I_{-}(P_{surf})=e_{s}\
B(T_{surf})+r_{s}\left[J^{scat}(P_{surf})+J_{*}^{dir}(P_{surf})\right]$ (17)

In our study, we simulate the quadrature phase of the planet by assuming a
non-isotropic scenario with a specific angle $\theta_{*}=77.756$ degrees. In
principle, every region of the planet would have a specific angle of incidence
of the stellar light, so we should take into account the reflected spectrum of
each pixel encompassed by the field of view. In the case of a Lambertian
surface, which is what we assume in this study, this calculation can be
simplified by averaging the angles across the field of view. This produces an
“average” incident direct light angle of 77.756 degrees, which allows us to
run a single radiative transfer simulation with reasonable results. For more
information on this treatment, used widely in the Planet Spectrum Generator
suite, we refer to the PSG Handbook (Villanueva et al. 2022).

The surface reflectance $r_{s}$ assumed in the scattering calculation is a
free parameter in our runs. It represents the specific reflectance of the
surface material. In principle, petitRADTRANS can assume a wavelength-
dependent reflectance of a custom surface. The user can take advantage of
known reflectivity databases (e.g., the United States Geological Survey
Spectral Library, Kokaly et al. 2017) to fill this variable. In this study, we
assume a wavelength-independent value of $r_{s}$ of 0.1 (an average value for
habitable water-rich terrestrial planets) both when producing the input
spectrum and when retrieving the $r_{s}$ estimate.

The value of $r_{s}$ is therefore different from what is commonly assumed to
be the albedo of the planet as a whole, as no contribution from the atmosphere
is considered aside from the extinction of the incoming ray that is reflected
by the surface. However, in the case of thin enough atmospheres with no clouds
such as the one considered in this study, the majority of the scattered light
that would be detected originates from the surface. For this reason, in this
specific case, $r_{s}$ can be considered a proxy for planetary albedo. The
Bond albedo of the planet a posteriori can be retrieved from the planetary
radius estimate and the stellar luminosity, as it was done in Konrad et al.
(2023); Mettler et al. (2024).

##  Appendix B Idealized high-resolution low-noise scenario

Idealized high-resolution low-noise scenario

Figure 13: Best-fit spectra retrieved in the first retrieval set (photon noise
only at $S/N=50$, $R=1000$) for the three scenarios: _Left panel_ : pure HWO
retrieval (magenta); _Central panel_ : pure LIFE retrieval (blue); _Right
panel_ : HWO+LIFE retrieval (yellow). In all panels, the best-fit spectra are
compared with the input spectra (black lines) with error bars (gray-shaded
areas).

Idealized high-resolution low-noise scenario

Figure 14: Retrieved pressure-temperature profiles for the first retrieval set
(photon noise only at $S/N=50$, $R=1000$): _Left panel_ : pure HWO retrieval
(magenta); _Central panel_ : pure LIFE retrieval (blue); _Right panel_ :
HWO+LIFE retrieval (yellow). In all panels, the 2-$\sigma$ and the 1-$\sigma$
intervals are shown in increasingly darker hues, as well as the input profiles
(black lines) for comparison. Inside each panel, the inset plot shows the 2D
posterior space of the ground pressure and temperature. In all panels and the
inset plots, the surface pressure and temperature point in the P$-$T space is
shown as a red square marker.

Idealized high-resolution low-noise scenario

Figure 15: Posterior density distributions from the first set of retrievals
(photon noise only at $S/N=50$, $R=1000$). The black lines indicate the
expected values for every parameter. HWO posteriors are shown in magenta with
diagonal hatching; LIFE posteriors are shown in cyan with crossed hatching;
HWO+LIFE posteriors are shown as fully colored gold histograms.

Idealized high-resolution low-noise scenario

Figure 16: Square root of the mean squared error (see Equation 3) for relevant
parameters in the first set of retrievals (photon noise only at $S/N=50$,
$R=1000$).

Idealized high-resolution low-noise scenario

Figure 17: Corner plot for the posterior distributions from the first set of
retrievals (photon noise only at $S/N=50$, $R=1000$). The black lines indicate
the expected values for every parameter. The median and 1-$\sigma$
uncertainties for relevant retrieved and derived parameters are shown in the
table in the top right corner. The scenarios are color-coded according to
Table 3.

In this set of runs, we simulated the spectrum of a cloud-free Earth at
$R=1000$ with uncertainties based on a $S/N=50$ at $0.55$ µm and $11.2$ µm,
only taking photon noise into account. This serves as a validation of the
updated retrieval routine and represents an idealistic best-case scenario. In
Figures 13, 14, and 15, we show the best fit of the spectrum compared to the
input data, the retrieved P$-$T profiles, and the posteriors of the main
physical and chemical parameters for the first retrieval set. The
corresponding corner plot is shown as Figure 17. The square root of the mean
squared error for various parameters is shown in Figure 16.

As can be noticed in Figure 13, we are able to accurately and precisely
reproduce the input spectra in all the scenarios. The 1-, 2-, and 3-$\sigma$
envelopes of the retrieved spectrum lie well within the photon noise
uncertainty. For better clarity of this set of plots, we do not show the
$\sigma$ envelopes for the best-retrieved spectra in this case, but only the
best fit.

Retrieving an accurate P$-$T profile is satisfactory with LIFE: the profile is
correctly determined down to pressures of approximately $10^{-3}$ bar. This is
expected, since for a cloud-free thin-atmosphere scenario the majority of the
contribution to the thermal emission spectrum is coming from the lower, denser
layers of the atmosphere. In the LIFE retrieval, the surface pressure and
temperature are also retrieved with high precision and accuracy (with a
1-$\sigma$ uncertainty of 0.06 dex for the pressure, and only 0.5 K for the
temperature). The HWO retrieval is less accurate, as can be seen from the
larger envelopes in the P$-$T profile. The retrieval of surface temperature
and pressure is relatively accurate (the 1-$\sigma$ interval uncertainty of
$T_{0}$ is around 15 K in this case). The joint retrieval can estimate the
P$-$T profile with better accuracy and precision than the single-instrument
retrievals. Figure 15 shows that, in the case of the idealized HWO+LIFE
retrieval, the uncertainties on the surface pressure and temperature are
reduced (up to a 1-$\sigma$ uncertainty of 0.03 dex for $P_{0}$ and 0.3 K for
$T_{0}$). This result stems from the high quality of the spectrum, which
allows the retrieval framework to correctly model pressure and thermal
broadening of each line – and especially the deeper layers of the atmosphere
that contribute the most to the spectrum.

The retrieval of the radius is very accurate and precise in all three
retrievals: the estimate of the radius and corresponding 1-$\sigma$
uncertainty is $1\pm 0.01\ R_{\oplus}$ or lower for all scenarios. Retrievals
that include MIR data (LIFE and HWO+LIFE retrievals) can leverage the thermal
emission of the planet, which provides a strong constraint on the radius of
the planet (see, e.g., LIFE Paper III). On the other hand, the result of the
HWO retrieval is dependent on the assumptions made in this study. Since we
neglect clouds and assume a perfect knowledge of the geometry of the system,
the radius/albedo degeneracy in the UV/VIS/NIR can be easily disentangled.
This will likely not be the case in reality, as discussed in Section 4.4.

Concerning the planetary mass, even the most idealized scenario of a MIR LIFE-
like retrieval only slightly further constrains the mass of the planet
compared to the prior (retrieving a mass of $M_{pl}=1.08\pm 0.30\
M_{\oplus}$). This happens because of the well-known degeneracy between
gravity and chemical abundances, which contribute to the scale height of the
atmosphere (see LIFE Papers III and V for details). On the other hand, the
UV/VIS/NIR retrieval allows for a more precise estimate of the mass with an
uncertainty of about 10%, but a less accurate result (estimating
$M_{pl}=0.88\pm 0.10\ M_{\oplus}$). Considering both wavelength ranges
together would instead improve the mass estimate to $M_{pl}=0.94\pm 0.07\
M_{\oplus}$, with the true value within the 1-$\sigma$ confidence interval.

The surface reflectance is accurately and precisely determined by the HWO
retrieval. On the other hand, the LIFE retrieval can only rule out surface
reflectances higher than 0.3-0.4. Further, there are slight correlations
between $r_{s}$ and the major atmospheric absorbers in the MIR. The combined
HWO+LIFE retrieval retrieved this parameter correctly. The surface reflectance
is also impacted, like the radius, by the assumptions we make in these studies
regarding geometry and clouds (see Section 4.4).

At such high spectral resolution and signal-to-noise ratio, the retrievals are
generally precise and accurate at retrieving the abundances of the main
atmospheric species. There are, however, a few exceptions. Even in the most
idealized HWO retrieval, it is not possible to have a detection of CO and N2O
(only upper limits are retrieved). These two molecules do not have significant
lines in this wavelength range (the opacity being 3-4 orders of magnitude
fainter than the main absorbers in this range, such as O2 and H2O). In the
idealized LIFE retrieval, O2 cannot be sufficiently constrained, because of
the lack of significant lines in the MIR wavelength range. CO has a strong
detection peak around the true value, but lower abundances cannot be entirely
ruled out. This is due to its main spectral features being at short MIR
wavelengths ($\leq$$4.5$ µm), where the noise is large due to the lower
thermal emission of the planet. These spectral lines also overlap with
features of more abundant species (mainly H2O, CO2, and O3) which makes the
less abundant species harder to detect.

When retrieving over the entire wavelength range, not only are all the species
correctly retrieved, but there is a considerable increase in the precision of
the retrieved estimate (a factor 2-10 on the 1-$\sigma$ uncertainty between a
retrieval performed on only one portion of the spectrum, compared to the joint
retrieval, see Figure 16).

##  Appendix C Comparison with previous works

While the current study is, to our knowledge, the first retrieval study that
considers the joint performance of a direct imaging UV/VIS/NIR mission (HWO-
like) together with a MIR mission (LIFE-like), many retrieval studies in the
last few years have tackled the potential of future-generation instruments in
characterizing Earth-like planets. We deem it useful to compare the single-
instrument retrievals we performed against previous studies in the field.

###  C.1 HWO-like simulations

From the UV/VIS/NIR point of view, none of the previous studies can be
directly compared with ours because of strong differences in the setup.
However, we will discuss similarities between our results and the ones in Feng
et al. (2018), Damiano & Hu (2022), Robinson & Salvador (2023), and Young et
al. (2023).

The work performed in Feng et al. (2018) concerned a study on the preferred
resolution and $S/N$ to allow the characterization of an Earth twin. The
authors considered an albedo model coupled with a direct-imaging noise
simulator to provide the observed planet-to-star flux ratio. Their retrieval
framework included cloud parameters, at the expense of a simplified atmosphere
parameterization (isothermal P$-$T profile with only H2O, O2, O3 considered in
the retrieval). The parameter estimation module used Markov Chain Monte Carlo
instead of Nested Sampling. Furthermore, in Feng et al. (2018) only a
wavelength range between $0.4-1.0$ µm was considered, at resolutions 70 and
140. Like in our third set of retrievals, the $S/N$ was wavelength-dependent
and set to a reference value of 5, 10, 15, 20 at $0.55$ µm. To compare with
our results, we will focus on the $S/N=10$ and $R=140$ results, which match
the quality of the VIS band in this work.

In Feng et al. (2018) the correlation between gravity, pressure, and
atmospheric abundances is also present. However, the authors notice a strong
radius-albedo degeneracy, which is not noticeable in our case given the
absence of clouds. Gravity and mass remained unconstrained compared to the
priors in Feng et al. (2018), which is consistent with our results. In both
studies H2O, O3, and O2 are detected successfully at this resolution and
$S/N$, though wider uncertainties are present in Feng et al. (2018) compared
to our study. This can be explained by the larger information content that we
exploit in our retrieval, since we consider both UV and NIR wavelengths, while
Feng et al. (2018) is focused on the VIS range.

Damiano & Hu (2022) performed retrievals using a reflected light forward model
and considering H2O, CO2, O2, O3, and CH4 as free parameters, assuming N2 as a
filler gas. In their work, the atmosphere is assumed to be isothermal and
clouds were also taken into account. The authors considered a variety of
scenarios: an Earth-like atmosphere, an early Earth, a CO2-dominated
atmosphere with O2 and clouds, and a dry CO2-dominated atmosphere. For every
scenario, they focused their attention on the information content of the NIR
band to correctly characterize the atmosphere. They explore various
resolutions and $S/N$ values. The authors retrieve their error bars by fixing
the $S/N$ at the reference wavelength bin corresponding to the maximum value
of the spectrum, which may vary depending on the scenarios. To compare our
results with this work, we focus on the modern Earth scenario with $R=140/40$
for VIS and NIR and $S/N=10$, which is the closest setup to the one we
simulate. These retrievals correctly identify the main components of the
atmosphere (N2, O2, H2O, O3) and do not constrain CO2 and CH4, which is
consistent with our results. According to the authors, to have an upper limit
on the methane abundance, it would be necessary to increase the $S/N$ to 20.
Despite the differences in the parameterization of the P$-$T profile, also in
Damiano & Hu (2022) the ground pressure is found to be overestimated.

A more recent study by Robinson & Salvador (2023) explored the detectability
of an Earth twin through HabEx/LUVOIR varying the wavelength range and $S/N$
in each band (considering three cases: optical only at $S/N=35$, UV/VIS/NIR at
$S/N=20$, and VIS/NIR at $S/N=10/45$). The authors used a single-scene albedo
model for the reflected light spectrum and a constant error bar $S/N$ whose
reference points were at $0.4$ µm for the UV, $0.55$ µm for the VIS, and $1$
µm in the NIR. Their results on the VIS retrieval at $S/N=35$ were consistent
with higher $S/N$cases in Feng et al. (2018). As for our comparison, we
discuss our results (specifically, the HWO retrieval in the simplified noise
scenario) with the Robinson & Salvador (2023) retrieval on the whole
UV/VIS/NIR range, which is the most similar setup despite a factor 2
difference in the $S/N$. The authors successfully retrieved O2, H2O, O3, and
only retrieve upper limits for CO2 and CH4, in line with our results. By
comparing our results with Figure 10 in Robinson & Salvador (2023) we can
observe similar variances in the posteriors of the constrained species, except
for the O3 posterior. In our model, we cannot rule out abundances lower than
$10^{-7}$, while the species is constrained in the earlier study. This is
probably a result to be expected when increasing the $S/N$ to 20, though such
retrieval was not performed in this work.

Finally, we compare our results with a recent publication by Young et al.
(2023). Here, the authors used atmospheric retrievals to identify the best
observing strategy for the Habitable Worlds Observatory coronagraph. They
performed retrievals on various combinations of wavelength bandpass regions,
defining a decision tree that would allow us to differentiate between a modern
Earth-like planet and an Archean Earth-like planet. They used the same
retrieval framework as Robinson & Salvador (2023) with a nominal $S/N=10$ and
the current baseline resolutions for the UV/VIS/NIR range ($R=7/140/70$). When
performing retrievals for all the considered bandpasses (three in total,
centered at $0.75$ µm, $0.85$ µm, and $1.65$ µm with a 20% bandpass width
each), Young et al. (2023) retrieve H2O, O2, and get upper limits on CO2 and
CH4. This is consistent with what we find in the HWO retrievals. On the other
hand, we find better constraints on the albedo and radius of the planet
compared to Young et al. (2023). This is to be expected considering that our
retrievals can leverage the full UV/VIS/NIR spectrum and neglect clouds, which
have been known to impact these parameters. Young et al. (2023) also
considered different priors on these parameters compared to our studies, which
can explain the differences in the results. The mass is unconstrained
(compared to the chosen prior) in both studies, and the ground temperature and
pressure show similar uncertainties.

###  C.2 LIFE-like simulations

Regarding the MIR portion of the spectrum, we compare our results with Konrad
et al. (2022) i.e., LIFE Paper III. The forward model and the retrieval
routine used in LIFE Paper III is overall consistent with the one used in this
study, apart from the updates mentioned in Alei et al. (2022) (LIFE Paper V)
and Section 2.3 of this manuscript. In Konrad et al. (2022), a grid of
spectral resolutions, $S/N$ values, and wavelength ranges were explored. The
$S/N$ was calculated through LIFEsim, in a very similar way as described in
Section 2.1. To compare with the present work, we focus on the retrieval
performed in LIFE Paper III considering $R=50$, $S/N=10$, and $4-18.5$ µm. We
find very similar results between this model and the LIFE retrieval in the
third set of models (PSG/LIFEsim noise). N2, O2, CO, N2O are unconstrained in
both models. On the other hand, CO2, O3 and H2O are constrained though the
mean of the posterior in our set of models is slightly shifted towards higher
pressures by about 0.5 dex, a result of the slightly underestimated ground
pressure that we observe in our run. In both cases, an upper limit on methane
can be retrieved, though it is impossible to rule out abundances below
$10^{-6}$. This happens because the methane abundance is right at the
sensitivity limit of the instrument at this resolution and $S/N$ (see Konrad
et al. 2022, for more details).

The setup of the framework validation is also consistent with the first grid
of results shown in this manuscript (i.e., $R=1000$, $S/N=50$ only considering
photon noise). The precise and accurate P$-$T profile retrieval in our model
is consistent with the previous work, as are the estimates of the major
components of the atmosphere. As expected in the MIR, the validation run in
Konrad et al. (2022) also does not detect O2 nor significantly reduces the
estimate of the mass of the planet.

##  Appendix D Ancillary Plots and Tables

Table 4: Summary of the parameters used in the retrievals, their expected
values, and their prior distributions.

Parameter | Description | Priors | Input values
---|---|---|---
$a_{4}$ |  P$-$T Parameter (Degree 4) | $\mathcal{U}(0,10)$ | 1.674
$a_{3}$ |  P$-$T Parameter (Degree 3) | $\mathcal{U}(0,100)$ | 23.120
$a_{2}$ |  P$-$T Parameter (Degree 2) | $\mathcal{U}(0,500)$ | 99.703
$a_{1}$ |  P$-$T Parameter (Degree 1) | $\mathcal{U}(0,500)$ | 146.626
$a_{0}$ |  P$-$T Parameter (Degree 0) | $\mathcal{U}(0,1000)$ | 285.218
$\log_{10}\left(P_{0}\left[\mathrm{bar}\right]\right)$ | Surface Pressure | $\mathcal{U}(-4,3)$ | 0.006
$R_{\text{pl}}\,\left[R_{\oplus}\right]$ | Planet Radius (bulk value) | $\mathcal{U}(0.5,2)$ | 1.000
$M_{\text{pl}}\,\left[M_{\oplus}\right])$ | Planet Mass (bulk value) | $\mathcal{G}(1.0,0.5)$ | 1.000
$\log_{10}(\mathrm{N_{2}})$ |  N2 Mass Fraction | $\mathcal{U}(-15,0)$ | -0.103
$\log_{10}(\mathrm{O_{2}})$ |  O2 Mass Fraction | $\mathcal{U}(-15,0)$ | -0.679
$\log_{10}(\mathrm{H_{2}O})$ |  H2O Mass Fraction | $\mathcal{U}(-15,0)$ | -3.000
$\log_{10}(\mathrm{CO_{2}})$ |  CO2 Mass Fraction | $\mathcal{U}(-15,0)$ | -3.387
$\log_{10}(\mathrm{CH_{4}})$ |  CH4 Mass Fraction | $\mathcal{U}(-15,0)$ | -5.770
$\log_{10}(\mathrm{O_{3}})$ |  O3 Mass Fraction | $\mathcal{U}(-15,0)$ | -6.523
$\log_{10}(\mathrm{CO})$ |  CO Mass Fraction | $\mathcal{U}(-15,0)$ | -6.903
$\log_{10}(\mathrm{N_{2}O})$ |  N2O Mass Fraction | $\mathcal{U}(-15,0)$ | -6.495
$r_{s}$ | Surface Reflectance | $\mathcal{U}(0,1)$ | 0.1
555$\mathcal{U}(x,y)$ denotes a boxcar prior with a lower threshold $x$ and
upper threshold $y$; $\mathcal{G}(\mu,\sigma)$ represents a Gaussian prior
with mean $\mu$ and standard deviation $\sigma$).

In this section, we collect some ancillary plots and tables. In Table 4 we
list all the parameters, the priors, and the true values assumed in the
retrievals. In Tables 5, 6, 7, and 8 we show details concerning the molecular
opacities, the CIA and the Rayleigh opacities used in the forward model, and
the simulation settings used in LIFEsim and the PSG simulators. We report in
Tables 10 and 11 the Bayes factor for the two baseline sets of retrievals. In
Figures 18, and 19 we show the corner plots and the retrieved estimates of
relevant parameters for the two baseline retrieval runs.

Table 5: References for the molecular opacities used in the retrievals.

Species | Line List | Broadening | Line Cutoff
---|---|---|---
O2 | HITRAN 2020 _(1)_ | $\gamma_{\mathrm{air}}$ | 25 $\mathrm{cm^{-1}}$
CO2 | HITRAN 2020 _(1)_ | $\gamma_{\mathrm{air}}$ | 25 $\mathrm{cm^{-1}}$
O3 | HITRAN 2020 _(1)_ | $\gamma_{\mathrm{air}}$ | 25 $\mathrm{cm^{-1}}$
CH4 | HITRAN 2020 _(1)_ | $\gamma_{\mathrm{air}}$ | 25 $\mathrm{cm^{-1}}$
CO | HITRAN 2020 _(1)_ | $\gamma_{\mathrm{air}}$ | 25 $\mathrm{cm^{-1}}$
H2O | HITRAN 2020 _(1)_ | $\gamma_{\mathrm{air}}$ | 25 $\mathrm{cm^{-1}}$
N2O | HITRAN 2020 _(1)_ | $\gamma_{\mathrm{air}}$ | 25 $\mathrm{cm^{-1}}$

_(1)_ Gordon et al. (2022)

Table 6: References for the CIA and Rayleigh opacities used in the retrievals.

CIA | References | Rayleigh | References
---|---|---|---
N2-N2 | HITRAN _(1)_ |  N2 | _(3,4)_
O2-O2 | HITRAN _(1)_ |  O2 | _(3,4)_
O2-N2 | HITRAN _(1)_ |  CO2 | _(5)_
CO2-CO2 | HITRAN _(1)_ |  H2O | _(6)_
CH4-CH4 | HITRAN _(1)_ |  CH4 | _(5)_
H2O-H2O | MT_CKD _(2)_ |  CO | _(5)_
H2O-N2 | MT_CKD _(2)_ |  |

(1) Karman et al. (2019); (2) Kofman & Villanueva (2021); (3) Thalman et al.
(2014); (4) Thalman et al. (2017); (5) Sneep & Ubachs (2005); (6) Harvey et
al. (1998).

Table 7: Simulation settings used in LIFEsim.

Parameter | LIFEsim
---|---
Interferometric Baseline (m) | 14.5
Aperture Diameter (m) | 2 m
Wavelength (µm) | 4.0–18.5
Spectral Resolution | 50
Reference $S/N$ aafootnotemark: $a$ | 10
Reference Wavelength (µm) aafootnotemark: $a$ | 11.2
Detector quantum efficiency | 0.7
Total instrument throughput | 0.05
666See LIFE Paper I and LIFE Paper II for details. aafootnotemark: $a$The
$S/N$ is initially calculated for a set exposure time (1 hr), then scaled to
the reference $S/N$ value at the reference wavelength.

Table 8: Simulation settings used in the PSG “6-m LUVOIR-B” simulator for each
wavelength bandpass.

Parameter | HWO
---|---
Aperture Diameter (m) | 6
Number of exposures | 1
Number of pixels | 10
Reference $S/N$ aafootnotemark: $a$ | 10
Emissivity of the optics | 0.1
Temperature of the optics (K) | 270
Contrast | $10^{-10}$
Inner Working Angle | $\approx 4\lambda/D$
| UV | VIS | NIR
Wavelength (µm) | 0.2–0.5 | 0.5–1.0 | 1.0–2.0
Spectral Resolution | 7 | 140 | 70
Reference Wavelength (µm) aafootnotemark: $a$ | 0.35 | 0.55 | 1.20
Average instrument efficiency | 0.05 | 0.15 | 0.2
Read noise ($e^{-}/px$) | 0 | 0 | 2.5
Dark Current ($e^{-}/s/px$) | $3\cdot 10^{-5}$ | $3\cdot 10^{-5}$ | $2\cdot 10^{-3}$
Beam [FWHM] (arcsec) | 0.011 | 0.024 | 0.047
777aafootnotemark: $a$The $S/N$ is initially calculated for a set exposure
time (1 hr), then scaled to the reference $S/N$ value at the reference
wavelength.

Table 9: Physical parameters used in the noise simulation. Parameter | Value
---|---
Stellar class | G-type
Stellar temperature (K) | 5778
Planet-star separation (AU) | 1
Planet radius (${R_{\oplus}}$) | 1
Distance to the system (pc) | 10
Exozodi level | 4.5 $\times$ local zodiacal dust
Table 10: Bayes factor for each retrieved spectrally active species in the second set of retrievals (simplified noise). Values were obtained by performing a retrieval including and excluding the molecule of interest and then calculating the Bayes factor (eq. 1).  Parameter | HWO | LIFE | HWO+LIFE
---|---|---|---
O2 | 4.6 | -0.1 | 5.0
H2O | 38.3 | 8.4 | 62.7
CO2 | -0.3 | 16.8 | 31.1
CH4 | -0.4 | -0.2 | 0.0
O3 | 0.0 | 1.5 | 2.9
CO | 0.0 | -0.2 | 0.1
N2O | -0.3 | -0.3 | -0.1
Table 11: Bayes factor for each retrieved spectrally active species in the second set of retrievals (PSG/LIFEsim noise). Values were obtained by performing a retrieval including and excluding the molecule of interest and then calculating the Bayes factor (eq. 1).  Parameter | HWO | LIFE | HWO+LIFE
---|---|---|---
O2 | 3.1 | 0.0 | 3.5
H2O | 48.0 | 3.5 | 84.4
CO2 | -0.1 | 8.9 | 15.4
CH4 | -0.3 | -0.1 | 0.0
O3 | 5.7 | 1.8 | 9.9
CO | -0.2 | -0.1 | -0.1
N2O | -0.2 | -0.1 | -0.2

Simplified Noise

Figure 18: Corner plot for the posterior distributions from the second set of
retrievals (simplified noise). The black lines indicate the expected values
for every parameter. The median and 1-$\sigma$ uncertainties for relevant
retrieved and derived parameters are shown in the table in the top right
corner. The scenarios are color-coded according to Table 3.

PSG/LIFEsim Noise

Figure 19: Corner plot for the posterior distributions from the third set of
retrievals (PSG/LIFEsim noise). The black lines indicate the expected values
for every parameter. The median and 1-$\sigma$ uncertainties for relevant
retrieved and derived parameters are shown in the table in the top right
corner. The scenarios are color-coded according to Table 3.