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: ETH Centre for Origin and Prevalence of Life, Wolfgang-Pauli-
Str. 27, 8093 Zurich, Switzerland 44institutetext: Max-Planck-Institut für
Astronomie, Königstuhl 17, 69117 Heidelberg, Germany 55institutetext:
Department of Astronomy and Astrophysics, University of California, Santa
Cruz, CA, USA 95064 66institutetext: KU Leuven, Institute of Astronomy,
Celestijnenlaan 200D, 3001 Leuven, Belgium 77institutetext: Royal Observatory
of Belgium, Ringlaan 3, 1180 Brussels, Belgium 88institutetext: Institute of
Astronomy, University of Cambridge, CB3 0HA, UK 99institutetext: University of
Bern, Center for Space and Habitability, Gesellschaftsstrasse 6, 3012 Bern,
Switzerland 1010institutetext: Department of Physics and Astronomy, York
University, 4700 Keele Street, North York, Ontario 3MJ 1P3, Canada
1111institutetext: Department of Earth Sciences, University of Cambridge, CB2
3EQ, UK 1212institutetext: School of Engineering and Applied Sciences, Harvard
University, Cambridge, MA 02138, USA

# Large Interferometer For Exoplanets (LIFE):

IX. Assessing the impact of clouds on atmospheric retrievals at mid-infrared
wavelengths with a Venus-twin exoplanet

B.S. Konrad 1122 E. Alei 1122 S.P. Quanz 112233 P. Mollière 44 D. Angerhausen
1122 J.J. Fortney 55 K. Hakim 6677 S. Jordan 88 D. Kitzmann 2299 S. Rugheimer
1010 O. Shorttle 881111 R. Wordsworth 1212 the LIFE Collaboration Webpage:
www.life-space-mission.com

(Received -; accepted -)

###### Abstract

Context. Terrestrial exoplanets in the habitable zone are likely a common
occurrence. The long-term goal is to characterize the atmospheres of dozens of
such objects. The Large Interferometer For Exoplanets (LIFE) initiative aims
to develop a space-based mid-infrared (MIR) nulling interferometer to measure
the thermal emission spectra of such exoplanets.

Aims. We investigate how well LIFE could characterize a cloudy Venus-twin
exoplanet. This allows us to: (1) test our atmospheric retrieval routine on a
realistic non-Earth-like MIR emission spectrum of a known planet, (2)
investigate how clouds impact retrievals, and (3) further refine the LIFE
requirements derived in previous Earth-centered studies.

Methods. We ran Bayesian atmospheric retrievals for simulated LIFE
observations of a Venus-twin exoplanet orbiting a Sun-like star located $10$
pc from the observer. The LIFEsim noise model accounted for all major
astrophysical noise sources. We ran retrievals using different models (cloudy
and cloud-free) and analyzed the performance as a function of the quality of
the LIFE observation. This allowed us to determine how well the atmosphere and
clouds are characterizable depending on the quality of the spectrum.

Results. At the current minimal resolution ($R=50$) and signal-to-noise
($S/N=10$ at $11.2$ µm) requirements for LIFE, all tested models suggest a
CO2-rich atmosphere ($\geq 30\%$ in mass fraction). Further, we successfully
constrain the atmospheric pressure-temperature (P$-$T) structure above the
cloud deck (P$-$T uncertainty $\leq\pm 15$ K). However, we struggle to infer
the main cloud properties. Further, the retrieved planetary radius
($R_{\text{pl}}$), equilibrium temperature ($T_{\mathrm{eq}}$), and Bond
albedo ($A_{\mathrm{B}}$) depend on the model. Generally, a cloud-free model
performs best at the current minimal quality and accurately estimates
$R_{\text{pl}}$, $T_{\mathrm{eq}}$, and $A_{\mathrm{B}}$. If we consider
higher quality spectra (especially $S/N=20$), we can infer the presence of
clouds and pose first constraints on their structure.

Conclusions. Our study shows that the minimal $R$ and $S/N$ requirements for
LIFE suffice to characterize the structure and composition of a Venus-like
atmosphere above the cloud deck if an adequate model is chosen. Crucially, the
cloud-free model is preferred by the retreival for low spectral qualities. We
thus find no direct evidence for clouds at the minimal $R$ and $S/N$
requirements and cannot infer the thickness of the atmosphere. Clouds are only
constrainable in MIR retrievals of spectra with $S/N{}\geq 20$. The model
dependence of our retrieval results emphasizes the importance of developing a
community-wide best-practice for atmospheric retrieval studies.

###### Key Words.:

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

##  1 Introduction

One major goal for the future of exoplanet science is to constrain the
atmospheric structure and composition of a statistically significant number of
terrestrial exoplanets. Special attention will be given to planets within or
close to the habitable zone (HZ; Kasting et al., 1993; Kopparapu et al., 2013)
of the host star. Such exoplanets are expected to be common within our galaxy
(Petigura et al., 2013; Foreman-Mackey et al., 2014; Dressing & Charbonneau,
2015; Bryson et al., 2020), and have been detected within 20 pc of the Sun
(e.g., Anglada-Escudé et al., 2016; Gillon et al., 2016, 2017; Gilbert et al.,
2020). A powerful approach to characterize exoplanets is to analyze their
spectra, which contain important information about relevant properties such as
the atmospheric pressure-temperature (P$-$T) structure, the chemical
composition, and the possible existence of clouds and their properties. If and
how well an exoplanet property can be constrained depends on the wavelength
regime covered by the spectrum and the accuracy with which the spectrum is
measured.

For terrestrial exoplanets orbiting their host star close to or within the HZ,
detections are challenging, but possible, with current and approved future
ground- and space-based observatories. However, these instruments will not be
capable of obtaining detailed spectroscopic measurements for several dozens of
such terrestrial exoplanets. Partially motivated by this goal, there is great
interest in the community to develop a new generation of observatories. HabEx
(Gaudi et al., 2020) and LUVOIR (The LUVOIR Team, 2019), two flagship mission
concepts that aim to directly detect and characterize HZ terrestrial
exoplanets in reflected light (at ultraviolet, optical, and near-infrared or
UV/O/NIR wavelengths), were evaluated in the Astro 2020 Decadal Survey in the
United States (National Academies of Sciences, Engineering, and Medicine,
2021). As a result, the space-based, UV/O/NIR flagship Habitable Worlds
Observatory (HWO) was recommended. Additionally, the Voyage 2050 plan of the
European Space Agency (ESA; Voyage 2050 Senior Committee, 2021) recommended
considering a large-scale, mid-infrared (MIR), space-based mission to
characterize HZ terrestrial exoplanets via their thermal emission. The Large
Interferometer For Exoplanets (LIFE) initiative aims to achieve this goal
using a space-based MIR nulling interferometer (Kammerer & Quanz, 2018; Quanz
et al., 2021, 2022).

A first step in the LIFE design phase is to derive the requirements necessary
to adequately characterize the atmospheres of nearby HZ terrestrial
exoplanets. This includes constraining the wavelength coverage, spectral
resolution, and instrument sensitivity. Previous studies in the LIFE series
(Konrad et al., 2022; Alei et al., 2022a, hereafter Paper III and Paper V)
derive first estimates for the required spectral quality. These studies use
atmospheric retrievals (for recent reviews on retrievals, see e.g.,
Madhusudhan, 2018; Deming et al., 2018; Barstow & Heng, 2020) to derive
quantitative estimates for important atmospheric and planetary parameters from
a simulated or observed exoplanet spectrum. Both studies focus on the
characterizing Earth-like planets (Paper III – modern Earth, Paper V – Earth
at various stages of its evolution; both assume the LIFEsim observation noise
simulator from Dannert et al., 2022, hereafter Paper II). However, a future
observatory should not only be able to characterize Earth-like exoplanets, but
also discern Earth-like from non Earth-like HZ exoplanets. In addition to
being Earth-centric, our previous studies do not systematically investigate
the effect of clouds on exoplanet characterization. Yet, since clouds
influence an exoplanet’s spectrum (e.g., Kitzmann et al., 2011; Rugheimer et
al., 2013; Vasquez et al., 2013; Komacek et al., 2020; Feinstein et al.,
2022), a more detailed study of the impact of clouds is required to derive
robust requirements for LIFE.

Earth-centered retrieval studies and retrieval studies on theoretical spectra
of habitable worlds are often used to investigate the characterization
performance for different quality spectra (e.g., von Paris et al., 2013;
Brandt & Spiegel, 2014; Feng et al., 2018; Quanz et al., 2021; Léger et al.,
2019; Carrión-González et al., 2020; Robinson & Salvador, 2022). Venus – to
our knowledge – has not yet been considered in a comparable retrieval study.
However, terrestrial exoplanets with a Venus-like insolation could maintain
habitable conditions if a surface ocean is present (e.g., Yang et al., 2014;
Way et al., 2016)111Yet, the existence of large bodies of surface water on
early Venus and Venus-like exoplanets is uncertain and heavily debated (e.g.,
Kasting & Harman, 2021; Turbet et al., 2021). If Venus ever had liquid surface
water, it was lost in a runaway greenhouse process (Kasting, 1988).. Further,
exoplanets at the inner edge of the HZ (i.e., potentially Venus-like planets)
are ideal targets for LIFE (Quanz et al., 2022). Finally, in contrast to
theoretical planet models, Venus and its atmosphere are a known outcome of
planet formation and evolution and thus provide a realistic ground-truth for a
retrieval study.

Despite Venus being Earth-like in size and mass, its atmospheric state and
surface conditions are vastly different. In addition to a CO2 dominated
atmosphere, the mean surface pressure reaches $93$ bar, significantly
exceeding that on Earth. Further, at atmospheric pressures of 0.05 bar to 1
bar, a layer of opaque H2SO4 clouds covers the planet. This opaque cloud layer
can lead to an ambiguity in characterization between thick and cloudy or
tenuous and cloud-free atmospheres (Barstow et al., 2016; Lustig-Yaeger et
al., 2019). Further, the detectability of H2SO4 clouds is of great interest,
since it could provide constraints on the amount of liquid surface water
(Loftus et al., 2019). Finally, the high atmospheric CO2 content leads to
strong atmospheric greenhouse heating, which raises the mean surface
temperature to a hostile $730$ K (for a recent review on Venus’ atmosphere,
see, e.g., Taylor et al., 2018).

In this study, we reevaluate the LIFE requirements for wavelength range,
spectral resolution ($R$), and signal-to-noise ratio ($S/N$) from Paper III.
We focus on the key science application of distinguishing a cloudy Venus-like
planet from an Earth-like planet. To this purpose, we studied a Venus-twin
exoplanet using a modified version of our retrieval framework. The opaque
cloud layer in Venus’ atmosphere required us to model clouds in our
retrievals. This also allowed us to investigate how clouds affect exoplanet
characterization with LIFE. Hence, this study improves the robustness of our
instrument requirements and provides new insights into difficulties in
exoplanet characterization via atmospheric retrievals.

In Sect. 2, we introduce the model used to simulate the Venus-twin MIR
emission spectrum, our retrieval framework, and the LIFEsim observation noise
model. The retrieval results for different quality spectra are presented in
Sect. 3. In Sect. 4, we discuss implications of our results for the LIFE
requirements. Important takeaway points are summarized in Sect. 5.

##  2 Methods

In Sect. 2.1, we introduce the atmosphere model used to simulate Venus’ MIR
thermal emission spectrum and compare our spectrum to the literature. Next, we
introduce our Bayesian retrieval routine and discuss updates with respect to
previous versions (Sect. 2.2). In Sect. 2.3, we introduce the noise model used
to generate the input spectra for the retrievals, the different atmospheric
models fitted in the retrieval, and the assumed model parameter priors.

###  2.1 Cloudy Venus-twin model

Figure 1: Schematic structure of Venus’ atmospheric P$-$T profile and position
of the opaque H2SO4 cloud layer. Table 1: Parameters used in the different
(retrieval) models, assumed true values, and prior distributions.

Parameter | Description | Truth | Prior | Model Configuration
---|---|---|---|---
$\mathcal{M}^{\ce{H2SO4}}_{\mathrm{Op.}}$ |  $\mathcal{M}^{\ce{H2SO4}}_{\mathrm{Tr.}}$ |  $\mathcal{M}^{\ce{H2O}}_{\mathrm{Op.}}$ |  $\mathcal{M}^{\mathrm{CF}}$
$a_{3}$ |  P$-$T parameter (degree 3) |  $2.30$11$1$Obtained by fitting Eq. (1) to the Venus P$-$T profile in Figure 1 of Mueller-Wodarg et al. (2008); | $\mathcal{U}(0,10)$ |  $\boldsymbol{\checkmark}$ |  $\boldsymbol{\checkmark}$ |  $\boldsymbol{\checkmark}$ |  $\boldsymbol{\checkmark}$
$a_{2}$ |  P$-$T parameter (degree 2) |  $29.83$11$1$Obtained by fitting Eq. (1) to the Venus P$-$T profile in Figure 1 of Mueller-Wodarg et al. (2008); | $\mathcal{U}(0,500)$ |  $\boldsymbol{\checkmark}$ |  $\boldsymbol{\checkmark}$ |  $\boldsymbol{\checkmark}$ |  $\boldsymbol{\checkmark}$
$a_{1}$ |  P$-$T parameter (degree 1) |  $125.64$11$1$Obtained by fitting Eq. (1) to the Venus P$-$T profile in Figure 1 of Mueller-Wodarg et al. (2008); | $\mathcal{U}(0,1000)$ |  $\boldsymbol{\checkmark}$ |  $\boldsymbol{\checkmark}$ |  $\boldsymbol{\checkmark}$ |  $\boldsymbol{\checkmark}$
$a_{0}$ |  P$-$T parameter (degree 0) |  $344.94$11$1$Obtained by fitting Eq. (1) to the Venus P$-$T profile in Figure 1 of Mueller-Wodarg et al. (2008); | $\mathcal{U}(0,1000)$ |  $\boldsymbol{\checkmark}$ |  $\boldsymbol{\checkmark}$ |  $\boldsymbol{\checkmark}$ |  $\boldsymbol{\checkmark}$
$\log_{10}(P_{0}{})$ | Surface pressure $\left[\mathrm{bar}\right]$ |  $1.97$22$2$Near surface abundances from NASA’s planet factsheet: https://nssdc.gsfc.nasa.gov/planetary/factsheet/venusfact.html; | $\mathcal{U}(-4,3)$ |  $\boldsymbol{\times}$ |  $\boldsymbol{\checkmark}$ |  $\boldsymbol{\times}$ |  $\boldsymbol{\checkmark}$
$R_{\text{pl}}{}$ | Planet radius $\left[R_{\oplus}\right]$ |  $0.95$22$2$Near surface abundances from NASA’s planet factsheet: https://nssdc.gsfc.nasa.gov/planetary/factsheet/venusfact.html; | $\mathcal{G}(0.95,0.20)$ |  $\boldsymbol{\checkmark}$ |  $\boldsymbol{\checkmark}$ |  $\boldsymbol{\checkmark}$ |  $\boldsymbol{\checkmark}$
$\log_{10}(M_{\text{pl}}{})$ | Planet mass $\left[M_{\oplus}\right]$ |  $-0.09$22$2$Near surface abundances from NASA’s planet factsheet: https://nssdc.gsfc.nasa.gov/planetary/factsheet/venusfact.html; | $\mathcal{G}(-0.09,0.40)$ |  $\boldsymbol{\checkmark}$ |  $\boldsymbol{\checkmark}$ |  $\boldsymbol{\checkmark}$ |  $\boldsymbol{\checkmark}$
$\log_{10}(\mathrm{CO_{2}})$ |  CO2 mass fraction |  $-0.015$22$2$Near surface abundances from NASA’s planet factsheet: https://nssdc.gsfc.nasa.gov/planetary/factsheet/venusfact.html; | $\log_{10}\left(1-10^{\mathcal{U}(-7,0)}\right)$ |  $\boldsymbol{\checkmark}$ |  $\boldsymbol{\checkmark}$ |  $\boldsymbol{\checkmark}$ |  $\boldsymbol{\checkmark}$
$\log_{10}(\mathrm{H_{2}O})$ |  H2O mass fraction |  $-4.699$22$2$Near surface abundances from NASA’s planet factsheet: https://nssdc.gsfc.nasa.gov/planetary/factsheet/venusfact.html; | $\mathcal{U}(-7,0)$ |  $\boldsymbol{\checkmark}$ |  $\boldsymbol{\checkmark}$ |  $\boldsymbol{\checkmark}$ |  $\boldsymbol{\checkmark}$
$\log_{10}(\mathrm{CO})$ |  CO mass fraction |  $-4.770$22$2$Near surface abundances from NASA’s planet factsheet: https://nssdc.gsfc.nasa.gov/planetary/factsheet/venusfact.html; | $\mathcal{U}(-7,0)$ |  $\boldsymbol{\checkmark}$ |  $\boldsymbol{\checkmark}$ |  $\boldsymbol{\checkmark}$ |  $\boldsymbol{\checkmark}$
$\log_{10}(\mathrm{H_{2}SO_{4}^{cloud}})$ |  H2SO4 cloud mass fraction |  $-5.097$33$3$Mean of values from Oschlisniok et al. (2012) and Krasnopolsky (2015); | $\mathcal{U}(-7,-1)$ |  $\boldsymbol{\checkmark}$ |  $\boldsymbol{\checkmark}$ |  $\boldsymbol{\times}$ |  $\boldsymbol{\times}$
$\log_{10}(\mathrm{H_{2}O^{cloud}})$ |  H2O cloud mass fraction | $-$ | $\mathcal{U}(-7,-1)$ |  $\boldsymbol{\times}$ |  $\boldsymbol{\times}$ |  $\boldsymbol{\checkmark}$ |  $\boldsymbol{\times}$
$\log_{10}(P\mathrm{{}^{cloud}_{top}}{})$ | Cloud top pressure $\left[\mathrm{bar}\right]$ |  $-1.30$44$4$Mean values for the mode 1 cloud particles in Titov et al. (2018). | $\mathcal{U}(-6,3)$ |  $\boldsymbol{\checkmark}$ |  $\boldsymbol{\checkmark}$ |  $\boldsymbol{\checkmark}$ |  $\boldsymbol{\times}$
$\log_{10}(P\mathrm{{}^{cloud}_{span}}{})$ | Cloud thickness $\left[\mathrm{bar}\right]$ |  $0.161$44$4$Mean values for the mode 1 cloud particles in Titov et al. (2018). | $\mathcal{U}(-6,3)$ |  $\boldsymbol{\checkmark}$ |  $\boldsymbol{\checkmark}$ |  $\boldsymbol{\checkmark}$ |  $\boldsymbol{\times}$
$\log_{10}(\bar{R}\mathrm{{}^{cloud}}{})$ | Mean cloud particle radius $\left[\mathrm{cm}\right]$ |  $-4.30$44$4$Mean values for the mode 1 cloud particles in Titov et al. (2018). | $\mathcal{U}(-8,-3)$ |  $\boldsymbol{\checkmark}$ |  $\boldsymbol{\checkmark}$ |  $\boldsymbol{\checkmark}$ |  $\boldsymbol{\times}$
$\sigma\mathrm{{}^{cloud}}{}$ | Log-normal particle size spread |  $1.95$44$4$Mean values for the mode 1 cloud particles in Titov et al. (2018). | $\mathcal{U}(1,5)$ |  $\boldsymbol{\checkmark}$ |  $\boldsymbol{\checkmark}$ |  $\boldsymbol{\checkmark}$ |  $\boldsymbol{\times}$

222In the third column we provide the values used to generate the Venus-twin
spectrum introduced in Sect. 2.1. These values provide the ground truth for
all retrievals. The fourth column lists the prior distributions used in the
retrievals. With $\mathcal{U}(x,y)$, we denote a boxcar prior with lower
threshold $x$ and upper threshold $y$; $\mathcal{G}(\mu,\sigma)$ indicates a
Gaussian prior with mean $\mu$ and standard deviation $\sigma$. The last four
columns summarize the model parameters used by the different forward models in
the retrievals ($\boldsymbol{\checkmark}$ $=$ used, $\boldsymbol{\times}$ $=$
unused; see Sect. 2.3.2) and the model color-coding used throughout the paper.
$\mathcal{M}^{\ce{H2SO4}}_{\mathrm{Op.}}$ – opaque H2SO4 clouds (84% H2SO4,
16% H2O by weight); $\mathcal{M}^{\ce{H2SO4}}_{\mathrm{Tr.}}$ – transparent
H2SO4 clouds (84% H2SO4, 16% H2O by weight);
$\mathcal{M}^{\ce{H2O}}_{\mathrm{Op.}}$ – opaque H2O clouds (100% H2O);
$\mathcal{M}^{\mathrm{CF}}$ – cloud-free.

As in Papers III and V, we used the 1D radiative transfer code petitRADTRANS
(Mollière et al., 2019, 2020; Alei et al., 2022a) to model the MIR thermal
emission spectrum of our Venus-twin exoplanet. petitRADTRANS passes a
featureless black-body spectrum at the surface temperature through discrete
atmospheric layers and models the interaction of each layer with the
radiation. Further, it accounts for the scattering of photons by the
atmosphere and the surface. This yields the MIR emission spectrum at the top
of the atmosphere. Each layer is characterized by its temperature, pressure,
and the opacity sources that are present. We provide a list of all model
parameters along with the assumed true values in Table 1.

We parametrized Venus’ P$-$T structure of using a polynomial. In Papers III
and V, this approach allowed us to minimize the number of model parameters and
thus reduce the computational complexity of our retrieval. An extensive
discussion, justifying this choice of P$-$T parametrization, is provided in
the appendix of Paper III. Since, in contrast to Earth, Venus’ P$-$T profile
does not exhibit a temperature inversion, a third order polynomial (four
parameters) is sufficient for the present study:

$T(P)=\sum_{i=0}^{3}a_{i}P^{i}.$ (1)

Here, $P$ is the pressure and $T$ the corresponding temperature of each
atmospheric layer. The $a_{i}$ terms are the parameters of the polynomial
P$-$T model. In Fig. 1, we show our ground-truth Venus P$-$T profile, which we
obtained by fitting the polynomial model to the Venus P$-$T profile from
Mueller-Wodarg et al. (2008). The corresponding $a_{i}$ values are given in
Table 1.

To simulate the MIR emission of a Venus-twin, we accounted for different
opacity sources. First, we modeled the MIR absorption and emission features of
CO2, H2O, and CO (see Table 1 for the assumed mass fractions and Table 2 for
the used line lists, broadening coefficients, and line cutoffs). For all three
gases, we assumed constant vertical abundance profiles. Second, we modeled
spectral features from collision-induced absorption (CIA) by CO2, and Rayleigh
scattering by all three molecules (see Table 3 for the used opacities). Third,
we considered the opaque H2SO4 clouds (see Fig. 1), which is essential to
accurately model the MIR thermal emission of Venus. We accounted for the H2SO4
clouds by adding a cloud slab to the atmosphere, which spanned multiple
atmospheric layers. The cloud slab was characterized using five parameters:
the pressure at the cloud-top $P\mathrm{{}^{cloud}_{top}}$, the thickness of
the cloud layer $P\mathrm{{}^{cloud}_{span}}$ in bar, the mass fraction of the
cloud forming substance, the mean cloud-particle radius
$\bar{R}\mathrm{{}^{cloud}}$, and the standard deviation
$\sigma\mathrm{{}^{cloud}}$ of the log-normal cloud-particle size
distribution. The parameter $P\mathrm{{}^{cloud}_{top}}$ defined the uppermost
atmospheric layer that contained clouds, while the difference
$P\mathrm{{}^{cloud}_{top}}$$-$$P\mathrm{{}^{cloud}_{span}}$ the corresponding
lowermost layer. Throughout the cloud slab defined by these two parameters, we
assumed a constant mass fraction of the cloud forming H2SO4$-$H2O solution
(84% H2SO4, 16% H2O by weight). All other atmospheric layers were modeled to
be cloud-free (mass fraction of the cloud forming substance set to zero). Our
cloud model further assumed homogeneous, spherical cloud particles of variable
size. We assumed both $\bar{R}\mathrm{{}^{cloud}}$ and
$\sigma\mathrm{{}^{cloud}}$ to be constant throughout the cloud deck. We
calculated the pressure- and temperature-dependent opacities for the different
cloud particle sizes from the wavelength-dependent index of refraction (see
Table 3 for sources) using Mie scattering theory. For this calculation, we
relied on the software presented in Min et al. (2005), which uses the codes
from Toon & Ackerman (1981). Thereafter, we used the standard petitRADTRANS
cloud modeling pathway to include clouds in the MIR Venus-twin spectrum.

Table 2: Molecular line opacities used to calculate MIR spectra.

Molecule | Line List | Pressure broadening | Line Cutoff
---|---|---|---
CO2 | HITEMP | $\gamma_{\mathrm{air}}$ | BU69
H2O | HITEMP | $\gamma_{\mathrm{air}}$ | HA02
CO | HITEMP | $\gamma_{\mathrm{air}}$ | HA02

(HITEMP) Rothman et al. (2010); (BU69) Burch et al. (1969); (HA02) Hartmann et
al. (2002).

Table 3: Continuum opacities used to calculate MIR spectra.

Opacity type | Material | Reference
---|---|---
CIA |  CO2-CO2 | KA19
Cloud |  H2SO4 (liquid) | PW75
Cloud |  H2O (liquid) | SE81
Rayleigh |  CO2 | SU05
Rayleigh |  H2O | HA98
Rayleigh |  CO | SU05
333The cloud opacities were calculated from the indices of refraction of the
cloud species via Mie scattering theory.

(KA19) Karman et al. (2019); (PW75) Palmer & Williams (1975); (SE81)
Segelstein (1981); (SU05) Sneep & Ubachs (2005); (HA98) Harvey et al. (1998);

In Fig. 2 we compare our MIR Venus-twin spectrum, to the simulated Venus
spectra from NASA’s Planetary Spectrum Generator
(PSG444https://psg.gsfc.nasa.gov; Villanueva et al., 2018) and Arney & Kane
(2018). In contrast to our Venus-twin atmosphere, both models included
additional atmospheric isotopes and trace gases and assumed altitude-dependent
abundance profiles for all gases. Additionally, both models assumed a more
complex atmospheric cloud structure (PSG: altitude-dependent volcanic clouds;
Arney & Kane (2018): altitude-dependent multilayer H2SO4 solution clouds). We
observe that the general shape of all three spectra is comparable and the
differences are smaller or of similar magnitude as the assumed noise level.
Furthermore, our MIR Venus-twin spectrum is not missing any significant
spectral absorption features, despite not taking into account various
atmospheric species and isotopes. There are two possible explanations for this
finding: either these species have no significant spectral lines in the MIR
(e.g., O2 and N2) or their atmospheric abundance above the opaque cloud layer
is too low to cause a noticeable signature in Venus’ MIR spectrum (e.g., SO2
and O3). This finding justifies our approach of excluding these additional
molecules from our Venus-twin model. Between $6$ µm and $12$ µm, we observe
minor differences between the spectra. Since the spectrum in this wavelength
range is predominantly determined by the clouds, the observed variance is
likely rooted in the differences between the three cloud models. Additionally,
we observe differences in the CO2 absorption feature between $13$ µm and $17$
µm. While the PSG and the Arney & Kane (2018) models yield similar results,
our model deviates. This deviance is most likely evoked by differences in the
assumed P$-$T profiles, but might also be partially due to differences between
the line lists, pressure broadening coefficients, or line cutoffs. However,
since we are interested in assessing the impact of clouds on exoplanet
characterization and whether the first requirements from Papers III and V are
sufficient, the deviances are negligible for this study.

Figure 2: Comparison of our Venus-twin MIR spectrum (opaque H2SO4 clouds) to
other models. We show the flux received by an observer located 10 pc from the
planet. The solid line is the MIR thermal emission spectrum of our model
(calculated assuming the true values listed in Table 1). The gray-shaded
region indicates the $S/N=10$ LIFEsim uncertainty considered for our
retrievals (see Sect. 2.3.1). The black-dashed line is the standard Venus
emission spectrum from the NASA-GSFC Planetary Spectrum Generator (PSG)
(https://psg.gsfc.nasa.gov; Villanueva et al., 2018).The brown-dashed-dotted
line is the MIR Venus spectrum from Arney & Kane (2018).

###  2.2 Bayesian retrieval framework

The study we present here relied on our Bayesian retrieval routine, which we
first introduced in Paper III. Our routine relied on two subroutines. First,
it applied the 1D radiative transfer code petitRADTRANS (Mollière et al.,
2019, 2020; Alei et al., 2022a) to calculate the theoretical MIR spectrum
corresponding to a given combination of values of the forward model parameters
listed in Table 1. Second, the routine used pyMultiNest (Buchner et al.,
2014), which efficiently samples the parameter space spanned by the prior
probability distributions (or ”priors”) of the forward model parameters to
determine parameter combinations that fit the simulated Venus-twin observation
well. This yielded posterior probability distributions (or ”posteriors”) for
the model parameters and estimates for the Bayesian evidence $\mathcal{Z}$ of
the model. The posteriors tell us how likely different combinations of model
parameter values are. The evidence $\mathcal{Z}$ measures how well the model
fits the input spectrum and can be used for model selection (see Sect. 2.3.2).
The pyMultiNest package is based on the MultiNest (Feroz et al., 2009)
implementation of the Nested Sampling algorithm (Skilling, 2006). In all
retrievals we performed throughout this study, we ran pyMultiNest using 700
live points and a sampling efficiency of 0.3 (suggested for evidence
evaluation by the MultiNest
documentation555https://github.com/farhanferoz/MultiNest). An in depth
description of our atmospheric retrieval routine can be found in Paper III of
the LIFE series. Here, we focus on the updates and improvements implemented
since Paper III.

In Paper III, we did not consider the impact surface, atmospheric, and cloud
scattering processes have on the MIR thermal emission. Including scattering in
the forward model, would have significantly increased the required computation
time per spectrum, making the study presented in Paper III unfeasible.
However, in both Papers III and V, the surface and atmospheric scattering
effects were shown to be negligible for the MIR thermal emission. Further,
only cloud-free forward models were considered, which justified neglecting
cloud scattering. However, for the cloudy Venus-twin retrievals we performed
in this study, cloud scattering effects were no longer negligible. The
limiting factor for the retrievals presented in Paper III was that
petitRADTRANS calculated spectra at a predetermined resolution of $R=1000$.
Our retrieval routine then binned down the spectra to the resolution of the
input spectrum. More recent improvements to petitRADTRANS enabled us to
compute spectra directly at the resolution of the input spectrum (see Paper V
for further information). With this update, the computation time required for
one cloudy spectrum including scattering at $R=50$ dropped from $\approx$ 20.0
seconds to just $\approx$ 1.5 seconds. This reduction in the computation time
per spectrum enabled us to run retrievals accounting for scattering effects,
making the present study feasible. We validate the updated retrieval routine
in Appendix A.

###  2.3 Retrieval setup

In Sect. 2.3.1, we discuss the Venus-twin input spectrum and the noise model
used in the retrievals. Thereafter, we introduce four atmospheric models we
used as forward models during the retrievals (Sect. 2.3.2). Lastly, the prior
distributions assumed in our retrievals are motivated in Sect. 2.3.3.

####  2.3.1 Input spectra and noise terms

We generated the Venus-twin input spectra for our retrieval study using
petitRADTRANS and the cloudy Venus-twin model introduced in Sect. 2.1. We
defined the resolution $R$ of a spectrum as $\lambda/\Delta\lambda$, where
$\lambda$ was the wavelength at the center of a wavelength bin and
$\Delta\lambda$ was the bin width. Further, we used noise models to estimate
the wavelength dependent $S/N$. We defined the $S/N$ of the input spectrum as
the $S/N$ at the $11.2$ µm reference bin, since this bin did not coincide with
strong absorption features from the considered atmospheric species.

For all LIFE retrievals in Sect. 3, we used the LIFEsim noise model introduced
in Paper II. LIFEsim provides estimates for the wavelength-dependent $S/N$
expected for observations with LIFE by accounting for noise contributions from
the photon noise of the planet’s emission, stellar leakage, and local- as well
as exozodiacal dust emission. We hence implicitly assumed that a large future
space mission like LIFE will not be dominated by instrumental noise terms
(Dannert et al., in prep.). Possible consequences of this assumption are
mentioned in Sect. 4.2. For our study, we assumed a Venus-twin exoplanet
orbiting a G2V Star on an $0.72$ AU orbit at a distance of $10$ pc from the
observer. We further set the exozodiacal dust emission to be three times the
level of the local zodiacal light. This value corresponds to the median
exozodi level found for Sun-like stars in the HOSTS survey (Ertel et al.,
2020).

In all retrievals, we interpreted the noise as uncertainty to the simulated
spectral points and assumed that the noise does not impact the predicted flux
values. As discussed in Feng et al. (2018) and the Appendix of Paper III,
randomizing the individual spectral data points according to the $S/N$ would
simulate more accurate observational instances. However, a retrieval study
based on a single noise instance will result in biased estimates for the
retrieval’s characterization performance due to the random placement of the
few spectral points. An ideal retrieval study should thus consider multiple
($\geq 10$) different noise realizations of each input spectrum and evaluate
the instrument performance by considering the average retrieved parameter
posterior. However, the vast number of different retrievals (12 retrievals for
each of the four models introduced in Sect. 2.3.2 resulting in a total of 48
retrievals) we executed for this study and the average computation time per
retrieval ($\sim$ 1 day on 20 CPUs) made such a study computationally
unfeasible ($\geq 5$ months of total cluster time). In addition, in the
Appendix of Paper III, we motivated that by retrieving the unrandomized input
spectra we obtain reliable estimates for the average expected retrieval
performance.

####  2.3.2 Atmospheric forward models in the retrievals

To test our retrieval framework’s sensitivity for Venus’ clouds, we analyzed
how our routine performed for different atmospheric forward models. This
approach enabled us to test if the LIFE design requirements from Papers III
and V are sufficient to infer the presence of clouds and to accurately
characterize the clouds in the atmosphere of a Venus twin. Additionally, this
approach provided us with important new insights into the biases that arise
when assuming an incorrect atmospheric model in a retrieval study. We
considered four atmospheric forward models (see Table 1 for the parameter
configuration of each model):

  1. 1.

Opaque H2SO4 clouds – (14 parameters;
$\mathcal{M}^{\ce{H2SO4}}_{\mathrm{Op.}}$): As is true for Venus, we assumed
that opaque H2SO4 clouds blocked the contributions from the lower atmospheric
layers and surface to the outgoing MIR emission spectrum. By fixing the
surface pressure $P_{0}$ to an arbitrary $10^{4}$ bar, we forced the retrieval
to add an opaque cloud layer to the atmosphere.

  2. 2.

Transparent H2SO4 clouds – (15 parameters;
$\mathcal{M}^{\ce{H2SO4}}_{\mathrm{Tr.}}$): In contrast to the opaque model,
we assumed that contributions from the lower atmosphere are not fully blocked.
Therefore, we tried to retrieve for the surface pressure $P_{0}$.

  3. 3.

Opaque H2O clouds – (14 parameters; $\mathcal{M}^{\ce{H2O}}_{\mathrm{Op.}}$):
Similar to the opaque H2SO4 model, we assumed an opaque cloud layer to be
present and fixed the surface pressure $P_{0}$ to $10^{4}$ bar. However, we
assumed pure H2O clouds to determine if we could identify the correct cloud
species in retrievals.

  4. 4.

Cloud-free – (ten parameters; $\mathcal{M}^{\mathrm{CF}}$): We assumed no
clouds to be present in the atmosphere. With this model, we investigated
whether the presence of clouds in an atmosphere can be inferred at the
considered input qualities.

We used Bayesian model selection to determine which model performed best as a
function of the quality of the input spectrum. If we run two retrievals that
assume different atmospheric models $\mathcal{A}$ and $\mathcal{B}$, both
retrieval results will be characterized by their evidences
$\ln\left(\mathcal{Z_{A}}\right)$ and $\ln\left(\mathcal{Z_{B}}\right)$. We
can use the evidences to identify the better fitting model via the Bayes
factor $K$:

$\log_{10}(K)=\frac{\ln\left(\mathcal{Z}_{\mathcal{A}}\right)-\ln\left(\mathcal{Z}_{\mathcal{B}}\right)}{\ln\left(10\right)}.$
(2)

The Jeffreys scale (Jeffreys, 1998, see Table 4) provides a possible
interpretation for the value of the Bayes factor $K$.

Table 4: Jeffreys scale (Jeffreys, 1998).

$\log_{10}(K)$ | Probability | Strength of Evidence
---|---|---
$<0$ | $<0.5$ | Support for $\mathcal{B}$
$0-0.5$ | $0.5-0.75$ | Very weak support for $\mathcal{A}$
$0.5-1$ | $0.75-0.91$ | Substantial support for $\mathcal{A}$
$1-2$ | $0.91-0.99$ | Strong support for $\mathcal{A}$
$>2$ | $>0.99$ | Decisive support for $\mathcal{A}$
666Scale for interpreting the values of the Bayes’ factor
$\log_{10}(K)=\left(\ln\left(\mathcal{Z}_{\mathcal{A}}\right)-\ln\left(\mathcal{Z}_{\mathcal{B}}\right)\right)/\ln\left(10\right)$.
The scale is symmetrical, i.e., negative values of $\log_{10}(K)$ correspond
to very weak, substantial, strong, or decisive support for model
$\mathcal{B}$.

####  2.3.3 Prior distributions

In Table 1, we provide a summary of the assumed prior distributions, which
define the range of parameter space sampled by pyMultiNest. For the P$-$T
parameters $a_{i}$ and the surface pressure $P_{0}$, we chose broad uniform
priors such that the corresponding P$-$T profiles covered a wide range of
atmospheric structures. For the abundances of the atmospheric species and the
cloud parameters, we assumed broad and uniform priors that spanned large
regions of parameter space. In contrast to Paper III, we assumed narrower
abundance priors for H2O and CO, since the lowest abundance detectable by our
retrieval routine for the quality of input spectra considered is approximately
$10^{-7}$ in mass fraction (cf. Paper III). For CO2, we used a prior that
covers the full abundance range, but samples high abundances more densely.
This prior allows us to better estimate the CO2 abundance and better identify
a potential upper limit.

As in Papers III and V, we chose a Gaussian prior for $R_{\text{pl}}$. For the
mean, we assumed Venus’ true radius, for the standard deviation of 20% of the
true value. This choice for $R_{\text{pl}}$ was motivated by findings
presented in Paper II, which demonstrated that the detection of a planet
during LIFE’s search phase would yield such constraints for $R_{\text{pl}}$
(for a terrestrial planet around the HZ, we expect a radius estimate
$R_{\mathrm{est}}$ for the true radius $R_{\mathrm{true}}$ with
$R_{\mathrm{est}}/R_{\mathrm{true}}=0.97\pm 0.18$). We then used the
$R_{\text{pl}}$ prior to derive a Gaussian prior for the planet mass
$\log_{10}(M_{\text{pl}}{})$ using the statistical mass-radius relation
Forecaster777https://github.com/chenjj2/forecaster (Chen & Kipping, 2016).

##  3 Retrieval results

Here, we present the retrieval results for the Venus-twin mock observations
with LIFE for the different assumed forward models (see Sect. 2.3.2 and Table
1). In Sect. 3.1, we discuss the results obtained for a Venus-twin spectrum at
the minimal LIFE requirements determined in Paper III ($4-18.5$ µm, $R=50$,
and $S/N=10$). Thereafter, in Sect. 3.2, we discuss if and how the retrieval’s
characterization performance is improved when considering higher quality
spectra. To this purpose, we ran retrievals for various spectra of different
wavelength coverage ($4-18.5$ µm, $3-20$ µm), $R$ ($50$, $100$), and $S/N$
($10$, $15$, $20$).

###  3.1 Results for the current minimal LIFE requirements

Figure 3: Residuals of the spectra corresponding to the posteriors retrieved
for the $4-18.5$ µm, $R=50$, LIFEsim $S/N=10$ Venus-twin spectrum (opaque
H2SO4 clouds). Color-shaded areas represent different residual quantiles. The
gray area marks the 1$\sigma$ LIFEsim noise level. Each panel contains the
results for a different forward model (see Sect. 2.3.2). From top to bottom:
Opaque H2SO4 clouds, transparent H2SO4 clouds, opaque H2O clouds, and cloud-
free. Figure 4: Residuals of the P$-$T profiles retrieved for the $4-18.5$ µm,
$R=50$, LIFEsim $S/N=10$ Venus-twin spectrum (opaque H2SO4 clouds) relative to
the retrieved median P$-$T profile. Color-shaded areas indicate percentiles of
the P$-$T residuals. If the model includes clouds, the gray shaded regions
indicate percentiles of the retrieved cloud-top pressure. The solid black
line, the orange circular marker, and the red square marker represent the true
profile relative to the retrieved median (for the transparent H2SO4 cloud and
cloud-free models, we cannot plot the true surface and the P$-$T profile at
pressures higher than retrieved surface pressure). In the top right, we plot
the 2D $P_{0}$-$T_{0}$ posterior (if retrieved; otherwise
$P\mathrm{{}^{cloud}_{top}}$-$T\mathrm{{}^{cloud}_{top}}$). Each panel
summarizes the result for one of the four different forward models (see Sect.
2.3.2). From top-left to bottom-right: Opaque H2SO4 clouds, transparent H2SO4
clouds, opaque H2O clouds, and cloud-free. Figure 5: Posteriors retrieved for
the $4-18.5$ µm, $R=50$, LIFEsim $S/N=10$ Venus-twin spectrum (opaque H2SO4
clouds) for different forward models (see Sect. 2.3.2). Here, $L(\cdot)$
abbreviates $\log_{10}(\cdot)$. We include the planetary equilibrium
temperature $T_{\mathrm{eq}}$ and the Bond albedo $A_{\mathrm{B}}$, which we
derived from the posteriors (see Appendix B). Dashed black lines indicate the
true values (see Table 1). For $\mathrm{Species^{cloud}}$, the true value is
the H2SO4 mass fraction in the Venus-twin model. Solid green area $-$ opaque
H2SO4 clouds; thin-blue outlined area $-$ transparent H2SO4 clouds; orange-
hatched area $-$ opaque H2O clouds; thick-yellow outlined area $-$ cloud-free.

We present the retrieval results for the different forward models at the
minimal LIFE design requirements ($4-18.5$ µm, $R=50$, $S/N=10$) in Figs. 3
(spectrum residuals), 4 (P$-$T profile residuals), and 5 (posteriors). The
full corner plots, the absolute retrieved P$-$T profiles and spectra, the
wavelength- and pressure-dependent contribution to the emission spectrum, and
tables with the retrieved values can be found in Appendix C.

####  3.1.1 Fit to the Venus-twin spectrum

The spectrum residuals in Fig. 3 show that the fit of all four forward models
to the Venus-twin spectrum lies well within the LIFEsim noise level. This
indicates that all models can reproduce the Venus-twin input with sufficient
accuracy. Above $8$ µm, the retrieved quantile envelopes of all forward models
are similar, roughly centered on the truth, and smaller than the LIFEsim
noise. Below $8$ µm, the quantile envelopes become larger and deviate more
from the truth as the LIFEsim noise level increases. We discuss the origin of
these deviations in Sect. 3.1.3. The spread of the quantiles is largest for
the true model (opaque H2SO4 clouds). For the other models, the spread is
smaller, but the residual deviates from the input. This indicates that the
three models cannot reproduce the input accurately. However, due to the large
LIFEsim errors in this wavelength range, these deviations will not affect the
retrieval performance significantly.

####  3.1.2 Retrieved P–T structure

The P$-$T profile residuals in Fig. 4 show that the fit of all four models is
best above Venus’ cloud layer (roughly between $10^{-1}$ bar and $10^{-4}$
bar). The retrieved means for the cloud-free and H2SO4 cloud forward models
lie within $\pm 15$ K of the truth. For the opaque H2O cloud model, the
deviations from the truth are larger (within $\pm 25$ K). The uncertainty on
the retrieved P$-$T structure in this pressure range is roughly $\pm 15$ K for
all four forward models. When considering higher or lower pressures, the
deviations from the truth increase and the uncertainties grow. This behavior
is due to a lack of significant spectral features from the high and low
pressure atmospheric layers in the Venus-twin MIR spectrum (see emission
contribution plots in Appendix C). The constraints on the percentile envelopes
for these layers stem from extrapolation of the P$-$T model (nonphysical
polynomial model). Thus, we cannot trust the P$-$T predictions for these
atmospheric layers and cannot estimate Venus’ surface pressure $P_{0}$ and
temperature $T_{0}$ accurately.

For the cloud-free and the transparent H2SO4 cloud model, the retrieved
$P_{0}$ roughly corresponds to the position of the cloud-top in Venus’
atmosphere and the retrieved $T_{0}$ slightly overestimates the cloud-top
temperature (by roughly 10 K for both models). Additionally, for the
transparent H2SO4 cloud model, the large spread in the retrieved cloud-top
pressure and temperature indicates that these two parameters are no longer
well constrained. In contrast, $P\mathrm{{}^{cloud}_{top}}$ is accurately
retrieved and $T\mathrm{{}^{cloud}_{top}}$ is slightly overestimated (by
roughly 25 K for H2O and 10 K for H2SO4 clouds) in the retrievals that assume
opaque cloud forward models. Thus, we find accurate estimates for the position
of the cloud-top in all retrievals.

####  3.1.3 Retrieved parameter posteriors

Lastly, we consider the retrieved posteriors displayed in Fig. 5. The figure
further includes distributions for the planetary equilibrium temperature
$T_{\mathrm{eq}}$ and Bond albedo $A_{\mathrm{B}}$, which we derived from the
posteriors using the method outlined in Appendix B.

First, we consider the results for the planetary surface pressure $P_{0}$. We
see, that if $P_{0}$ is a model parameter (transparent H2SO4 clouds or cloud-
free models, see Table 1), the posterior is strongly constrained. However, the
retrieved value does not correspond to Venus’ true surface pressure, but
coincides with the cloud-top pressure
($\log_{10}(P\mathrm{{}^{cloud}_{top}}{}[\mathrm{bar}])=-1.3$). This is in
agreement with the findings for the P$-$T profiles we outlined in the previous
section. The forward models that assumed an opaque cloud layer did not
retrieve for $P_{0}$ and thus yielded no estimates.

The $M_{\text{pl}}$ posterior is roughly Gaussian in log space ($\mu=-0.18$,
$\sigma=0.33$) for all forward models and is not strongly constrained over the
assumed Gaussian prior ($\mu=-0.09$, $\sigma=0.4$). This failure to further
constrain $M_{\text{pl}}$ was also observed in Papers III and V and is due to
the well known degeneracy between the planet mass (surface gravity) and the
abundances of the atmospheric trace gases (see also, e.g., Mollière et al.,
2015; Feng et al., 2018; Madhusudhan, 2018; Quanz et al., 2021).

For $R_{\text{pl}}$, the retrieved posterior strongly depends on the forward
model. For the transparent H2SO4 and the cloud-free model, the $R_{\text{pl}}$
posterior is roughly centered on the truth, approximately Gaussian
($\mu=0.97\,\mathrm{R}_{\oplus}$, $\sigma=0.05\,\mathrm{R}_{\oplus}$), and
significantly constrained over the Gaussian prior
($\mu=0.95\,\mathrm{R}_{\oplus}$, $\sigma=0.20\,\mathrm{R}_{\oplus}$). In
contrast, the $R_{\text{pl}}$ posteriors for the forward models assuming
opaque clouds are broader, non-Gaussian, and not centered on the truth. When
assuming opaque H2O clouds, the posterior is shifted relative to the true
value and roughly Gaussian (slightly asymmetric, with a tail to larger radii).
The retrieved median strongly underestimates the planet radius by
$0.2\,\mathrm{R}_{\oplus}$. For the opaque H2SO4 cloud forward model, the
resulting $R_{\text{pl}}$ posterior is significantly broader. We observe two
separate peaks, one of which is centered on the truth. The other is shifted to
the left and underestimates $R_{\text{pl}}$ by approximately
$0.18\,\mathrm{R}_{\oplus}$.

Since the $T_{\mathrm{eq}}$ and $A_{\mathrm{B}}$ distributions are derived
from the $R_{\text{pl}}$ posterior (see Appendix B), they inherit the forward
model dependence of $R_{\text{pl}}$. Thus, retrievals using the transparent
H2SO4 or the cloud-free forward model result in accurate, Gaussian-shaped
estimates for $T_{\mathrm{eq}}$ ($\mu=233$ K, $\sigma=5$ K) and
$A_{\mathrm{B}}$ ($\mu=0.74$, $\sigma=0.04$). For the forward models that
assume an opaque cloud layer, the underestimation of $R_{\text{pl}}$ results
in an overestimation of $T_{\mathrm{eq}}$. A higher $T_{\mathrm{eq}}$ can only
occur if the planet retains more of the incident stellar radiation, which
manifests itself in a lower Bond albedo $A_{\mathrm{B}}$. As a result, we
overestimate $T_{\mathrm{eq}}$ by roughly 20 K and underestimate
$A_{\mathrm{B}}$ by approximately $0.1$ for the opaque H2O cloud model. For
the opaque H2SO4 clouds, the $T_{\mathrm{eq}}$ posterior is similar to the
$R_{\text{pl}}$ posterior. It is non-Gaussian in shape, exhibits a peak at the
true $T_{\mathrm{eq}}$ value, and extends significantly toward higher
$T_{\mathrm{eq}}$ values. For the $A_{\mathrm{B}}$ distribution, the peak
coincides with the truth, but the distribution shows a significant tail toward
lower $A_{\mathrm{B}}$ values.

We retrieve high atmospheric CO2 abundances ($\geq 30\%$ in mass fraction) for
all forward models. Thus, we can easily differentiate between a Venus-like,
CO2 dominated atmosphere and an Earth-like atmosphere with lower CO2
abundances. In contrast, H2O and CO are not detected at the considered input
quality, since the signatures in Venus’ spectrum lie below $8$ µm and are
therefore not significant compared to the high LIFEsim noise level. For H2O,
the drop in the posterior at high abundances rules out abundances $\gtrsim
10^{-3}$. This limit on H2O and the unconstrained CO abundance cause the drop
in the spectrum residual below $8$ µm observed in Sect. 3.1.1. Both posteriors
extend to abundances significantly above the truth. Thus, on average, the
spectra corresponding to the retrieved parameters have stronger H2O and CO
absorption features than the true Venus-twin spectrum, which leads to the
observed drop in the residual below $8$ µm.

Last, we consider the cloud parameter posteriors. The cloud-top pressure
($P\mathrm{{}^{cloud}_{top}}$) posterior for both opaque cloud forward models
provides a good approximation of the truth and is well described by a Gaussian
($\mu=-1.5$, $\sigma=0.2$). Furthermore, we manage to retrieve a value for the
minimal possible cloud thickness ($P\mathrm{{}^{cloud}_{span}}$;
$\log_{10}(P\mathrm{{}^{cloud}_{span}}{}[\mathrm{bar}])\gtrsim-1.5\pm 0.8$).
Interestingly, even with the opaque H2O forward model, which assumes a wrong
cloud composition, we obtain accurate estimates for the position of the cloud
deck in the atmosphere. In contrast, for the transparent H2SO4 cloud model, we
do not manage to significantly constrain either the cloud-top position or the
cloud thickness. The posteriors are flat and unconstrained with respect to the
assumed priors. Similarly, also the cloud particle mass fraction
($\mathrm{Species^{cloud}}$) is unconstrained for the transparent H2SO4
forward model.

The $\mathrm{Species^{cloud}}$ posterior for the opaque H2SO4 forward model is
strongly bimodal. The lower of the two peaks is centered on the true H2SO4
mass fraction, while the other overestimates the abundance by roughly $3$ dex.
A more in-depth analysis of the posterior distributions (see corner plots in
Appendix C) reveals a strong correlation between the retrieved cloud species
abundance and $R_{\text{pl}}$. Interestingly, an overestimated cloud particle
abundance is linked to an underestimated $R_{\text{pl}}$ and thus also
correlated with the $T_{\mathrm{eq}}$ and $A_{\mathrm{B}}$ posteriors. For the
opaque H2O forward model, the retrieved median abundance lies roughly $2.5$
dex above the true H2SO4 cloud particle abundance.

Finally, the parameters describing the cloud particle size,
$\bar{R}\mathrm{{}^{cloud}}$ and $\sigma\mathrm{{}^{cloud}}$, are not well
constrained for any of the forward models. In an in-depth analysis of the
posteriors for the two opaque cloud models (see corner plots in Appendix C),
we find a degeneracy between these two parameters. This indicates that a
smaller $\bar{R}\mathrm{{}^{cloud}}$ can be compensated with a larger
$\sigma\mathrm{{}^{cloud}}$ for the considered spectral quality. For the
transparent H2SO4 model, we observe no degeneracy between the two parameters.
The lack of constraints on all cloud parameters for the transparent H2SO4
model is caused by the addition of the surface pressure $P_{0}$ to the
retrieval. The retrieval sets the surface at the cloud-top, which alleviates
the need to model an opaque cloud layer.

###  3.2 Retrieval results for higher quality spectra

Figure 6: Model parameter posteriors for increased quality $4-18.5$ µm Venus-
twin spectra (opaque H2SO4 clouds). Here, $L(\cdot)$ abbreviates
$\log_{10}(\cdot)$. Dashed black lines indicate the true values (see Table 1).
For $\mathrm{Species^{cloud}}$, the true value is the H2SO4 mass fraction in
the Venus-twin model. Solid green area $-$ opaque H2SO4 clouds; thin-blue
outlined area $-$ transparent H2SO4 clouds; orange-hatched area $-$ opaque H2O
clouds; thick-yellow outlined area $-$ cloud-free. Columns (left to right)
show the results for the planet radius $R_{\text{pl}}$, cloud species
abundance $\mathrm{Species^{cloud}}$, equilibrium temperature
$T_{\mathrm{eq}}$, Bond albedo $A_{\mathrm{B}}$ ($T_{\mathrm{eq}}$ and
$A_{\mathrm{B}}$ were calculated following Appendix B; for the posteriors of
all model parameters see Appendix D). Rows (top to bottom) represent different
input qualities: $R=50$, $S/N=10$; $R=100$, $S/N=10$; $R=50$, $S/N=20$;
$R=100$, $S/N=20$.

We now investigate how the retrieval results change if we consider higher
quality Venus-twin spectra. For most model parameters, both increases in $R$
and $S/N$ do not significantly change the retrieval results. Generally, most
parameters are better constrained as we move to higher quality spectra, but
the general shape of the posterior distributions remains unchanged. Further,
increasing the wavelength coverage from $4-18.5$ µm to $3-20$ µm does not
significantly impact the results either. In Fig. 6, we focus on the parameter
posteriors that significantly change when increasing the quality of the
LIFEsim input spectrum ($R_{\text{pl}}$, abundance of cloud species,
$T_{\mathrm{eq}}$, and $A_{\mathrm{B}}$). We plot the results for the $4-18.5$
µm input spectra and exclude the intermediate $S/N=15$ retrievals to increase
readability. We provide the retrieved posteriors for all remaining model
parameters and the retrieval results for the $S/N=15$ and $3-20$ µm input
spectra in Appendix D.

For the opaque H2SO4 cloud model, the correlated bimodal nature of the
$R_{\text{pl}}$ and the H2SO4 abundance posteriors (see Sect. 3.1) diminishes
strongly as we consider higher quality spectra. While neither of the two modes
in these posteriors is preferred for the $R=50$, $S/N=10$ input spectrum (both
peaks are equally high), the modes centered on the true values are clearly
preferred for the higher quality spectra. For the $S/N=20$ input spectra, the
bimodalities disappear completely and both $R_{\text{pl}}$ and the H2SO4
abundance are accurately determined.

For retrievals with the opaque H2O cloud model, $R_{\text{pl}}$ is
underestimated by roughly $0.2\,R_{\oplus}$ independent of the quality of the
spectrum. Similarly, the retrieved median cloud particle abundance lies
roughly 3 dex above the true H2SO4 abundance, irrespective of the $R$ and
$S/N$ of the spectrum. However, for the $S/N=20$ inputs, bimodalities in the
$R_{\text{pl}}$ and the H2O cloud species posteriors emerge. These
bimodalities are likely also present at lower $S/N$, but are not observable
due to the larger uncertainties on the individual posterior modes.

For the transparent H2SO4 and the cloud-free model, the constraint on
$R_{\text{pl}}$ is increased for higher $R$ and $S/N$ spectra. However, while
the posterior is centered on the true value for the $R=50$, $S/N=10$
retrieval, $R_{\text{pl}}$ is overestimated in retrievals of higher quality
spectra. This bias is stronger for the cloud-free model ($\approx
0.05\,R_{\oplus}$) and is likely due to differences between the model assumed
to generate the Venus-twin input spectrum and the forward model. It is not
observable at $R=50$, $S/N=10$, due to the larger uncertainties on the
posterior. Finally, for the transparent H2SO4 cloud forward model, the
retrieved cloud species abundance converges toward abundances below the true
H2SO4 abundances as we consider higher quality LIFEsim spectra.

Lastly, we consider the $T_{\mathrm{eq}}$ and $A_{\mathrm{B}}$ distributions.
When assuming the opaque H2SO4 model, the tail toward high $T_{\mathrm{eq}}$
observed at $R=50$, $S/N=10$ diminishes analogously to the $R_{\text{pl}}$
bimodality as we consider higher quality inputs. Consequentially, the tail
toward low $A_{\mathrm{B}}$ decreases. Especially for $S/N=20$ spectra, the
$A_{\mathrm{B}}$ and $T_{\mathrm{eq}}$ distributions provide accurate Gaussian
estimates that are centered on the truth. In contrast, systematic offsets from
the truth emerge for $T_{\mathrm{eq}}$ for the transparent H2SO4 and the
cloud-free forward models as we consider higher quality spectra. These offsets
are linked to the offsets in the $R_{\text{pl}}$ posteriors discussed above.
Albeit less prominent, the systematic shifts also appear in the
$A_{\mathrm{B}}$ distributions. They are less noticeable since the offset in
$T_{\mathrm{eq}}$ is small compared to the uncertainties on the other
parameters used to calculate $A_{\mathrm{B}}$ (see Appendix B). Finally, for
the opaque H2O cloud model, the $T_{\mathrm{eq}}$ and $A_{\mathrm{B}}$
estimates are not improved significantly as we move to higher $R$ and $S/N$.

In summary, we observe significant changes in the posteriors of some
parameters when considering higher quality spectra. While the transparent
H2SO4 and the cloud-free forward models perform well at $R=50$, $S/N=20$,
biases emerge for higher resolutions. This indicates that for high quality
spectra, these models are likely not sufficient. In contrast, results for the
opaque H2SO4 forward model are further refined with the input quality
increase. While the estimates for many parameters are weak and biased at
$R=50$, $S/N=10$, they improve significantly as we consider higher quality
spectra (especially for $S/N=20$). Finally, the results for the wrong opaque
H2O cloud model do not improve significantly when considering higher quality
spectra.

##  4 Discussion

After summarizing the main results from our retrieval analysis in Sect. 3, we
discuss how well one can characterize a Venus-twin exoplanet from simulated
LIFE MIR observations of different quality. In Sect. 4.1, we compare the
performance of the different forward models to see whether we can find
evidence for clouds by analyzing Venus’ MIR thermal emission spectrum. We
further discuss potential alternative pathways for cloud inference.
Thereafter, in Sect. 4.2, we address the limitations of our approach and
motivate potential future studies.

###  4.1 Forward model selection and interpretation

In Sect. 3, we find that the retrieved P$-$T profile shape and the posterior
distributions of the atmospheric gases (CO2, H2O, and CO) exhibit only minor
variations with the forward model and input quality. In contrast, the
posteriors for $R_{\text{pl}}$ and the cloud parameters, as well as the
inferred distributions for $T_{\mathrm{eq}}$ and $A_{\mathrm{B}}$ depend
significantly on the forward model and the input quality. This shows that
incorrect model assumptions or an inadequate level of model complexity can
result in incorrect exoplanetary characterization. These dependencies of the
posteriors are problematic, because the true atmospheric structure and
composition is unknown for an observed exoplanet. Consequentially, we will not
be able to verify if the parameter values we retrieve assuming a forward model
characterize the observed exoplanet correctly. Therefore, we require a method
of determining an adequate forward model for a given exoplanet spectrum (Sect.
4.1.1). Furthermore, once an adequate forward model is determined, we need to
understand how to link the obtained retrieval results to the conditions
present on the exoplanet (Sect. 4.1.2).

####  4.1.1 Forward model selection via the Bayes factor

Table 5: Model comparison via the Bayes factor $K$.

Compared models | $\log_{10}(K)$ | Preferred model
---|---|---
$\mathcal{M}^{\ce{H2SO4}}_{\mathrm{Op.}}$ versus $\mathcal{M}^{\ce{H2SO4}}_{\mathrm{Tr.}}$ | $-0.4\pm 0.1$ | $\mathcal{M}^{\ce{H2SO4}}_{\mathrm{Tr.}}$
$\mathcal{M}^{\ce{H2SO4}}_{\mathrm{Op.}}$ versus $\mathcal{M}^{\ce{H2O}}_{\mathrm{Op.}}$ | $0.1\pm 0.1$ | Either
$\mathcal{M}^{\ce{H2SO4}}_{\mathrm{Op.}}$ versus $\mathcal{M}^{\mathrm{CF}}$ | $-1.0\pm 0.1$ | $\mathcal{M}^{\mathrm{CF}}$
$\mathcal{M}^{\ce{H2SO4}}_{\mathrm{Tr.}}$ versus $\mathcal{M}^{\ce{H2O}}_{\mathrm{Op.}}$ | $0.5\pm 0.1$ | $\mathcal{M}^{\ce{H2SO4}}_{\mathrm{Tr.}}$
$\mathcal{M}^{\ce{H2SO4}}_{\mathrm{Tr.}}$ versus $\mathcal{M}^{\mathrm{CF}}$ | $-0.6\pm 0.1$ | $\mathcal{M}^{\mathrm{CF}}$
$\mathcal{M}^{\ce{H2O}}_{\mathrm{Op.}}$ versus $\mathcal{M}^{\mathrm{CF}}$ | $-1.1\pm 0.1$ | $\mathcal{M}^{\mathrm{CF}}$
888Performance comparison of forward models for retrievals of the $4-18.5$ µm,
$R=50$, $S/N=10$ Venus-twin (opaque H2SO4 clouds) LIFEsim spectrum. We
calculate $\log_{10}(K)$ for pairs of models with Eq. 2 and interpret its
value via the Jeffreys scale (see Table 4, Jeffreys, 1998).

Figure 7: Comparison of the retrieval performance of the true opaque H2SO4
cloud model with the other models via the Bayes factor $K$. We use the
Jeffreys scale as metric for comparison (see Table 4, Jeffreys, 1998) and
consider different wavelength coverages, $R$, and $S/N$ of the simulated
Venus-twin LIFE observations. Green squares (positive $\log_{10}(K)$ values)
indicate that the correct model (opaque H2SO4 clouds) is preferred, red
squares (negative $\log_{10}(K)$ values) mark preference for the incorrect
model. The intensity of coloring represents the preference strength.

As outlined in Sect. 2.3.1, the Bayes’ factor $K$ allows us to compare the
retrieval performance of different forward models for a given exoplanet
spectrum. Importantly, the Bayes’ factor $K$ does not tell us if a model is
correct or not. It provides a metric that measures which model out of a set of
models is best suited to explain an observed exoplanet spectrum. Here, we
investigate if clouds can be detected and characterized in retrievals for
different quality input spectra. We do this by comparing the retrieval
performance of the correct opaque H2SO4 cloud forward model to the other
tested forward models using the Bayes’ factor $K$.

First, we compare the performance of different forward models at the minimal
LIFE requirements from Paper III ($4-18.5$ µm, $R=50$, and $S/N=10$). We list
the Bayes factor $K$ for all combinations of forward models in Table 5. No
forward model we tested is decisively ruled out or preferred
($\left|K\right|\leq 2$ for all comparisons). We only find slight performance
differences between the forward models. Overall, the cloud-free model performs
best (always preferred; $\left|K\right|\geq 0.6$), while the opaque cloud
models perform worst (never preferred, despite the opaque H2SO4 cloud model
being the true model). Importantly, the cloud-free model also uses fewer
parameters than the cloudy scenarios (ten versus 14–15 parameters). The
finding that the cloud-free forward model yields the best retrieval
performance indicates that the additional parameters required to model clouds
are not justifiable.

For the Venus twin, this suggests that MIR retrievals at the current minimal
LIFE requirements are not sufficient to find evidence for atmospheric clouds.
Crucially, this does not rule out clouds. It merely indicates, that the
considered spectrum is satisfactorily described by the cloud-free model. This
agrees well with the findings for cloud-free retrievals on cloudy input
spectra presented in Paper V.

Second, we investigate if our findings change for higher quality LIFEsim
spectra. In Fig. 7, we compare the retrieval performance of the opaque H2SO4
cloud model (model used to produce the input spectra) to the other models via
the Bayes factor $K$. Green fields mark positive $\log_{10}(K)$ values and
thus preference for the true opaque H2SO4 cloud model. Red fields (negative
$\log_{10}(K)$) indicate a preference for the incorrect model.

For spectra with $S/N{}\leq 20$, we observe that both the cloud-free and the
transparent H2SO4 models perform better than (or comparable to) the opaque
H2SO4 cloud model. Further, the cloud-free model outperforms the transparent
H2SO4 model, as we see from the lower $\log_{10}(K)$ values. The two models
assuming opaque clouds perform equally well. As before, these findings suggest
that no evidence for clouds in Venus’ atmosphere can be found via retrieval
studies of spectra with $S/N{}<20$.

For $S/N=20$ spectra, a preference for the opaque H2SO4 forward model emerges.
This preference is generally stronger for spectra with larger wavelength
coverage and higher $R$, since they contain more information. This suggests
that an $S/N$ of at least 20 is required to infer the presence of clouds in a
retrieval study on the MIR thermal emission spectrum of a Venus-twin
exoplanet.

Crucially, the $S/N$-dependent model preferences agree with our findings for
the posteriors (see Sect. 3). For $S/N{}<20$ spectra, the preferred cloud-free
model yields good estimates for Venus’ atmospheric structure and composition
above the cloud-top. The planetary parameters $R_{\text{pl}}$,
$T_{\mathrm{eq}}$, and $A_{\mathrm{B}}$ are correctly retrieved. Similarly,
the transparent H2SO4 cloud model (second-best performance) approximates the
aforementioned parameters well. The retrieved $P_{0}$ corresponds to the
cloud-top, which alleviates the need to model an opaque cloud layer. Thus, the
cloud parameters are unconstrained. In contrast, the opaque cloud retrievals
(lowest preference) yield weak and biased estimates for $R_{\text{pl}}$,
$T_{\mathrm{eq}}$, $A_{\mathrm{B}}$, and the cloud parameters. This suggests,
that for spectra with $S/N{}<20$, the spectral information content is not
sufficient to constrain these additional parameters. Hence, the cloud-free
model with fewer parameters is preferred.

For $S/N=20$ spectra, we notice significant changes in both the posteriors and
the model preference. For the transparent H2SO4 and the cloud-free model, the
estimates for $R_{\text{pl}}$, $T_{\mathrm{eq}}$, and $A_{\mathrm{B}}$ are
offset from the true value and thus yield biased estimates. For the opaque H2O
cloud model, we see no significant improvements in the posteriors over the
$S/N{}<20$ retrievals. For the opaque H2SO4 cloud model, which performs best
on the $S/N=20$ spectra, the biases we find for $S/N{}<20$ spectra are no
longer present. The posteriors for cloud and planet parameters are unbiased
and provide good estimates. This suggests that at $S/N=20$, the information
content of the input spectrum is sufficient to justify the additional cloud
parameters. These observations for the posteriors agree well with the shift in
forward model preference from the cloud-free ($S/N{}<20$) to the opaque H2SO4
cloud ($S/N=20$) forward model.

In conclusion, we find that for low quality LIFEsim Venus-twin spectra
($S/N{}<20$) cloud presence is not inferrable via MIR retrievals. For these
inputs, the cloud-free model yields accurate estimates for fundamental
planetary and atmospheric parameters. The accuracy of the constraints on these
parameters are in accordance with the findings presented in Papers III and V.
For the $S/N=20$ spectra (especially if $R=100$), we manage to find weak
evidence for clouds in the atmosphere of the Venus twin and to constrain the
cloud properties. We emphasize that our findings are based on the assumption
of a Venus twin. However, the conclusion that clouds are hard to infer and
constrain via retrievals of low quality MIR thermal emission spectra is likely
generalizable to arbitrary terrestrial exoplanets. Further testing of this
important result is foreseen for the future.

####  4.1.2 Interpretation of model selection results

Inferring cloud presence for terrestrial exoplanets via MIR thermal emission
retrievals is challenging. At the minimal LIFE specifications, we find the
cloud-free model to perform best, and thus no direct evidence for a cloud
deck. There are two simple interpretations of the retrieval’s preference for
the cloud-free forward model. In the first interpretation, the retrieved
surface pressure $P_{0}$ is incorrect. The true $P_{0}$ is larger and not
retrieved correctly, since the atmospheric high pressure layers are optically
thick and thus leave no signatures in the spectrum. For our study, this would
suggest that the emission spectrum contains no information about the
exoplanet’s lower atmosphere ($\gtrsim 0.05$ bar) and surface conditions. In
the second interpretation, the retrieved $P_{0}$ corresponds to the truth and
the surface contributes to the emission spectrum. In this case, the exoplanet
would possess a thin atmosphere ($\approx 0.05$ bar). In both cases, the
exoplanet is characterized by a high bond albedo ($A_{\mathrm{B}}{}=0.74\pm
0.4$).

By considering the $A_{\mathrm{B}}$ of Solar System objects, we attempt to
link the retrieved $A_{\mathrm{B}}$ to planet properties. In Table 6, we list
the $A_{\mathrm{B}}$ of selected objects along with the main source of the MIR
continuum emission. If an object has a low $A_{\mathrm{B}}$ ($\lesssim 0.3$),
the continuum emission typically originates from the planet’s surface (rocky
objects with no or an optically thin atmosphere). For high $A_{\mathrm{B}}$
objects ($\gtrsim 0.5$), the MIR continuum emission stems from either clouds
(e.g., Venus) or predominantly ice and frost covered surfaces (e.g., Europa).
Thus, our retrieval results suggest an exoplanet with either a cloudy
atmosphere (first scenario) or an icy surface (second scenario). In the second
scenario, the retrieved $P_{0}$ ($\approx 0.05$ bar) and the corresponding
temperature ($T_{0}{}=235\pm 4$ K) would allow for a water ice surface999This
possibility of strongly underestimating the surface temperature of a cloudy
exoplanet has already been discussed for habitable Earth-like exoplanets
(e.g., Kitzmann et al., 2011).. Such ”snowball” states have occurred on Earth
(Kirschvink, 1992; Hoffman et al., 1998, 2017) and are also conceivable toward
the inner edge of the HZ (Wordsworth, 2021; Graham, 2021). In a snowball
state, the majority of the incident stellar radiation is reflected due to the
high albedo of the planet’s surface, which leads to low surface temperatures.
Even for high incident radiation from the host star and large concentrations
of atmospheric greenhouse gases, the planet surface can remain in a stable
frozen state (Budyko, 1969; Sellers, 1969).

Yet, the icy scenario appears improbable given the exoplanet-star separation
($0.72$ AU) and the high retrieved levels of the strong greenhouse gas CO2
($\geq 30$% in mass fraction). In addition, the low retrieved surface pressure
($\approx 0.05$ bar) seems unlikely for an evolved Venus-sized exoplanet
(e.g., Ortenzi et al., 2020). Lastly, the long-term stability of such a planet
is uncertain and depends on various factors such as the rate of volcanic
outgassing (e.g., Pierrehumbert, 2010). However, the icy scenario cannot be
ruled out solely via low quality MIR observations ($S/N{}<20$). Increasing the
spectrum’s $S/N$ to at least 20 allows us to infer cloud presence, yet still
not robustly. Furthermore, an increase in $S/N$ would require significantly
more observation time (for $S/N=20$ roughly four times longer than for
$S/N=10$). Thus, alternative cloud inference pathways are desirable to resolve
this ambiguity in interpretation.

Table 6: Bond albedos $A_{\mathrm{B}}$ of selected Solar System objects.

Object | $A_{\mathrm{B}}$ | MIR Continuum Emission | Reference
---|---|---|---
Mercury | 0.08 | Rocky surface | 1
Venus | 0.76 | Clouds | 2
Earth | 0.30 | Surface and clouds | 3
Moon | 0.14 | Rocky surface | 4
Mars | 0.24 | Rocky surface | 5
Jupiter | 0.53 | Clouds | 6
Europa | 0.55 | Icy surface | 7
Saturn | 0.34 | Clouds | 8
Tethys | 0.67 | Icy surface | 9
Enceladus | 0.81 | Icy surface | 9
101010The third column specifies the main source of the MIR continuum emission
(for clear atmospheres the continuum originates from the surface, for opaque
cloudy atmospheres it stems from the cloud deck).

(1) Mallama (2017); (2) Haus et al. (2016); (3) Pallé et al. (2003); (4)
Matthews (2008); (5) Pleskot & Kieffer (1977); (6) Li et al. (2018); (7)
Spencer et al. (1999); (8) Hanel et al. (1983); (9) Howett et al. (2010)

Figure 8: Flux difference between a cloudy Venus-twin exoplanet (opaque H2SO4
clouds) and an icy world with a thin CO2-dominated atmosphere (see Appendix E
for a comparison of the absolute fluxes). In brown, we plot the flux residual
of the icy world with respect to the Venus twin as a function of wavelength.
The gray shaded region indicates the 1$\sigma$ LIFEsim noise level at
$S/N=10$. The thin, hatched, and black area represents the expected
wavelength-dependent noise for the LUVOIR A mission concept (The LUVOIR Team,
2019); the thick, solid black line represents the expected noise for the HabEx
+ Starshade mission concept (Gaudi et al., 2020).

A potential remedy to the aforementioned ambiguity, is to leverage 1D or 3D
(photo-)chemistry and climate models (e.g.,
Atmos111111https://github.com/VirtualPlanetaryLaboratory/atmos,
ROCKE-3D121212https://www.giss.nasa.gov/projects/astrobio/, or
PlaSim131313https://www.mi.uni-hamburg.de/en/arbeitsgruppen/theoretische-
meteorologie/modelle/plasim.html). Such a simulative approach can help us
identify and rule out nonphysical planetary states. For example, for the icy
exoplanet scenario motivated above, there are two crucial questions that could
be studied via climate simulations. First, one has to investigate if an icy
surface together with the retrieved atmospheric structure and composition
describes a physically possible and stable state. Since our retrieval
framework does not model the atmospheric physics (e.g., convection,
photochemistry), not all points in the posterior distribution result in stable
atmospheres. In the case of the icy exoplanet, the atmosphere described by the
retrieved posteriors might enter a rapid runaway greenhouse phase, which would
lead to melting and subsequent evaporation of the surface H2O. Under these
circumstances, the icy scenario would be highly unlikely, and thus the cloudy
scenario preferred. Studies similar to Boukrouche et al. (2021), Chaverot et
al. (2022), or Graham et al. (2022) could help us better understand the
stability of the icy scenario. If we find that the icy state to be realistic,
the second question to tackle is if it can be reached, given the exoplanets
proximity to the host star and the high atmospheric CO2 abundance. To answer
this question, studies similar to Wordsworth (2021) or Graham (2021), that
investigate a wide range of different atmospheres, could provide answers. Such
studies are an example of potential future synergies between atmospheric
retrievals and the theoretical modeling of atmospheres. However, recent
intercomparison efforts show that both retrievals (e.g., Barstow et al., 2020)
and atmospheric modeling (e.g., Sergeev et al., 2022) depend on the model
choice. Community efforts, such as the CUISINES Working
Group141414https://nexss.info/cuisines/, that benchmark, compare, and validate
different models, are vital to the studies proposed above.

An alternative approach is to not only consider a terrestrial exoplanet’s MIR
emission, but also the stellar light it reflects in the UV/O/NIR. While we do
not expect the scenarios to differ noticeably in MIR emission, the different
reflective properties of clouds and ice will significantly impact the UV/O/NIR
reflected light spectrum. To test this approach, we simulated spectra between
$0.3$ µm and $21$ µm using petitRADTRANS. For the first scenario, we used our
cloudy Venus-twin exoplanet (opaque H2SO4 clouds; see Sect. 2.1 and Table 1).
For the second scenario, we modeled an icy surface at a pressure of 0.05 bar
($P_{0}$ from cloud-free retrievals) and a cloud-free atmosphere with the same
composition and P$-$T profile as for the first scenario. We used data from the
ECOSTRESS Spectral Library151515https://speclib.jpl.nasa.gov (Baldridge et
al., 2009; Meerdink et al., 2019) to model the wavelength-dependent
reflectivity of the icy surface. An 80% frost- and 20% ice-covered surface
yielded a total reflectance of 0.75 in the UV/O/NIR ($\approx$ retrieved
$A_{\mathrm{B}}$). In Fig. 8, we plot the residual of the ice-planet spectrum
relative to the cloudy Venus-twin spectrum and the LIFEsim noise ($S/N=10$).
We provide an analogous plot comparing the absolute fluxes in Appendix E. In
the UV/O/NIR, we show the noise expected for observations of the same planets
with LUVOIR A (The LUVOIR Team, 2019) or HabEx + Starshade (Gaudi et al.,
2020) ($S/N=10$ at $0.6$ µm, calculated with the NASA-GSFC Planetary Spectrum
Generator161616https://psg.gsfc.nasa.gov; Villanueva et al., 2018). While the
residual lies below the LIFEsim noise level in the MIR, it is significantly
larger than both the expected LUVOIR and HabEx noise in the UV/O/NIR below
$0.7$ µm. This indicates that the reflected light spectrum is more suitable to
differentiate between the icy and cloudy scenario than the MIR thermal
emission spectrum. It further exemplifies the complementarity of UV/O/NIR
reflected light and MIR thermal emission observations and highlights the
importance of following both strategies. Retrieval studies on combined
UV/O/NIR and MIR spectra are foreseen for the future.

###  4.2 Limitations and future work

The study we present here provides us with first estimates for how well a
Venus-like exoplanet could be characterized by LIFE. Further, we obtain
insights into how atmospheric clouds can complicate retrieval studies and the
interpretation of their output. As we are making several assumptions in our
approach, our findings cannot readily be generalized to arbitrary science
cases. Here, we discuss these limitations in detail.

First, we restricted ourselves to the study of a Venus twin. While the
performance for individual model parameters will not generalize well, the more
general findings provide insights into biases inherent to retrievals (e.g.,
the dependence of the parameter posteriors on the forward model). However, as
suggested in Robinson & Salvador (2022), Solar System planets provide an
excellent benchmark for retrievals, because these atmospheres are known to be
physical and estimates for the ground-truth values of the model parameters are
available. Performing retrievals on different Solar System planets will help
us generalize our predictions for the characterization performance.

Second, there are limitations inherent to our theoretical Venus-twin input
spectrum. We assumed a fully mixed atmosphere (vertically constant
abundances). While the same assumption was made in Paper III, the input
spectra in Paper V were based on variable abundance profiles. Nevertheless,
the retrieval results from Papers III and V are comparable, indicating that
this simplification does not heavily impact retrievals at the spectral
qualities considered. We also treated the clouds in a simplified manner. More
realistic cloud models, which consider P$-$T-dependent cloud particle sizes
and abundances or spatial variations in the cloud deck, could affect the MIR
emission spectrum measurably. Further, we neglected both temporal and spatial
variances in the atmospheric structure and composition. Real (exo)planet
emission spectra can vary with time and depend on the viewing geometry (e.g.,
Mettler et al., 2020, 2022). Retrieval studies on more realistic input spectra
will provide more reliable estimates for LIFE’s performance and are foreseen
in the future.

Third, the limitations above are also valid for the forward models used in the
retrievals. However, only limited increases in the forward model complexity
are possible, as they lead to a substantial rise in the retrievals’
computational complexity. For example, we only retrieved for molecules present
in the Venus-twin atmosphere. Including additional molecules could lead to
false positive detections of gases and a mischaracterization of the
atmosphere. However, a first robustness study for false positive detections in
the appendix of Paper III justifies our approach. Another simplification is
that we use a 1D forward model. While this is not problematic here, since the
Venus-twin input is also calculated with a 1D model, it will be wrong for
retrievals of real spectra and spectra from 3D models. However, in a recent
study, Robinson & Salvador (2022) compared the performance of their 1D
retrieval suite (rfast) to results from a computationally expensive 3D
retrieval (Feng et al., 2018). They concluded that for the $R$ and $S/N$ we
considered here, 1D retrievals suffice to obtain a first order understanding
of how the spectral quality affects the exoplanet characterization
performance.

Fourth, we used petitRADTRANS to generate the Venus-twin input spectrum and as
radiative transfer model in the retrieval. As discussed in Paper V and Barstow
et al. (2020), systematic differences between the radiative transfer model
used to generate the input spectrum and in the retrievals (e.g., differences
in the used line lists, Alei et al., 2022b) can lead to biases in the
posteriors. For retrievals on real exoplanet spectra, similar problems are
unavoidable, since the radiative transfer model will never capture the full
atmospheric physics and chemistry of the observed exoplanet. Thus, our results
might be overly optimistic and it is indispensable to investigate the nature
and magnitude of the resulting biases in future studies.

Last, important limitations are rooted in the LIFEsim noise model. Currently,
LIFEsim models the dominant astrophysical noise terms but neglects systematic
instrumental effects (Paper II). Ideally, instrumental noise contributions
will not dominate LIFE’s noise budget. Nevertheless, they will contribute to
the observational noise by altering the relative distribution of noise across
the wavelength range (Dannert et al., in prep.), which might affect the
retrieval results. More accurate estimates will be possible once LIFE’s
optical, thermal, and detector designs have matured and are accounted for by
LIFEsim. Also, we interpreted the LIFEsim noise as uncertainty on the Venus-
twin spectrum. Crucially, we did not randomize the values of individual
spectral points. This decision might lead to overly optimistic results (see
Sect. 2.3.1). We expect the low $R$ and $S/N$ cases to be more strongly
affected by randomization. Further, the cloud inference and characterization
capabilities could also be overly optimistic. For a detailed discussion on
potential impacts of this simplification on the characterization performance,
we refer to the appendix of Paper III.

##  5 Conclusions and outlook

In this study, we ran retrievals for a cloudy Venus twin orbiting a G2V star
at a distance of 10 pc. The goal was to investigate how the minimal $R$ and
$S/N$ requirements for LIFE defined in Paper III and verified in Paper V are
affected by clouds.

We approximated Venus’ MIR thermal spectrum using the 1D radiative transfer
model petitRADTRANS (Mollière et al., 2019, 2020; Alei et al., 2022a) assuming
a third order polynomial P$-$T structure, vertically constant CO2, H2O, and CO
abundances, and a uniform, Mie scattering H2SO4–H2O cloud slab. The LIFEsim
tool (Paper II) simulated LIFE observations of the Venus twin. Using an
updated version of the retrieval suite from Papers III and V, we ran
retrievals for variable quality spectra (from the $4-18.5$ µm, $R=50$,
$S/N=10$ minimal LIFE requirements to $3-20$ µm, $R=100$, $S/N=20$ spectra)
assuming different (cloud-free and cloudy) forward models.

At the minimal LIFE requirements, we correctly retrieve the P$-$T structure
above the cloud top (for pressures $\leq 0.03\pm 0.02$ bar, $T$ lies within
$\leq\pm 25$ K of the truth) and find high CO2 levels ($\geq 30\%$ in mass
fraction). These findings allow us to discern Venus- from Earth-like
exoplanets. Further, Venus’ surface conditions are not constrainable via its
MIR thermal emission, since the opaque atmospheric clouds block contributions
from the lower atmospheric layers. The results for the planet radius
$R_{\text{pl}}$, equilibrium temperature $T_{\mathrm{eq}}$, Bond albedo
$A_{\mathrm{B}}$, and the cloud parameters depend strongly on the forward
model choice. Overall, the cloud-free model yields the best estimates
($R_{\text{pl}}{}=0.97\pm 0.05\leavevmode\nobreak\ \mathrm{R}_{\oplus}$,
$T_{\mathrm{eq}}{}=233\pm 5$ K, $A_{\mathrm{B}}{}=0.74\pm 0.04$) and is
favored by the Bayes factor analysis. This suggests that cloud presence cannot
be inferred at the minimal LIFE requirements. For high quality spectra
($S/N{}\geq 20$), the parameter constraints increase and the model preference
shifts toward the correct opaque H2SO4 cloud model. While this suggests that
retrieval based cloud inference is possible with LIFE, other approaches, such
as followup UV/O/NIR observations with a HWO-like telescope, offer an
alternative and synergistic approach.

Crucially, we find that our retrieval results for important planetary
parameters ($R_{\text{pl}}$, $T_{\mathrm{eq}}$, $A_{\mathrm{B}}$) strongly
depend on the chosen forward model. An incorrect forward model or an
inadequate level of forward model complexity (e.g., too complex given the
quality of the input spectrum) can heavily bias retrieval results. This is a
major concern since, for observations of real exoplanets, the atmospheric
state will be unknown. Furthermore, the work we presented here suggests that
model selection via the Bayes factor will likely be hard and thus the risk of
over- or misinterpretation of the available data is high. While sufficient
quality MIR spectra of Earth- or Venus-like exoplanets will not be available
in the near future, the James Webb Space Telescope will measure transmission
spectra as well as thermal emission of many exoplanets in the upcoming years.
Atmospheric retrievals will be used to analyze these spectra (Cowan et al.,
2015; Greene et al., 2016; Krissansen-Totton et al., 2018; Nixon &
Madhusudhan, 2022). Therefore, working toward a community-wide common approach
for retrieval studies of exoplanet spectra is of great importance, as it would
mitigate the risk of false characterization significantly and augment the
comparability of different studies. Applied to empirical data from powerful
future space missions, such as LIFE, the in-depth characterization of
different types of terrestrial exoplanets seems within reach.

###### Acknowledgements.

This 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. S.P.Q. and E.A. acknowledge the financial support from the
SNSF. S.R. acknowledges support from the Natural Sciences and Engineering
Research Council of Canada (NSERC) Discovery Grant, [2022-04588]. _Author
contributions._ B.S.K. carried out the analyses, created the figures, and
wrote the manuscript. S.P.Q. initiated this project. S.P.Q. and E.A. provided
regular guidance. All authors discussed the results and commented on the
manuscript.

## References

  * Alei et al. (2022a) Alei, E., Konrad, B. S., Angerhausen, D., et al. 2022a, A&A, 665, A106
  * Alei et al. (2022b) Alei, E., Konrad, B. S., Mollière, P., et al. 2022b, in SPIE Conference Series, Vol. 12180, Space Telescopes and Instrumentation 2022: Optical, Infrared, and Millimeter Wave, 121803L
  * Anglada-Escudé et al. (2016) Anglada-Escudé, G., Amado, P. J., Barnes, J., et al. 2016, Nature, 536, 437
  * Arney & Kane (2018) Arney, G. & Kane, S. 2018, Venus as an Analog for Hot Earths
  * Baldridge et al. (2009) Baldridge, A., Hook, S., Grove, C., & Rivera, G. 2009, Remote Sensing of Environment, 113, 711
  * Barstow et al. (2016) Barstow, J. K., Aigrain, S., Irwin, P. G. J., Kendrew, S., & Fletcher, L. N. 2016, MNRAS, 458, 2657
  * Barstow et al. (2020) Barstow, J. K., Changeat, Q., Garland, R., et al. 2020, MNRAS, 493, 4884
  * Barstow & Heng (2020) Barstow, J. K. & Heng, K. 2020, Space Science Reviews, 216, 82
  * Boukrouche et al. (2021) Boukrouche, R., Lichtenberg, T., & Pierrehumbert, R. T. 2021, ApJ, 919, 130
  * Brandt & Spiegel (2014) Brandt, T. D. & Spiegel, D. S. 2014, Proceedings of the National Academy of Sciences, 111, 13278
  * Bryson et al. (2020) Bryson, S., Kunimoto, M., Kopparapu, R. K., et al. 2020, AJ, 161, 36
  * Buchner et al. (2014) Buchner, J., Georgakakis, A., Nandra, K., et al. 2014, A&A, 564, A125
  * Budyko (1969) Budyko, M. I. 1969, Tellus, 21, 611
  * Burch et al. (1969) Burch, D. E., Gryvnak, D. A., Patty, R. R., & Bartky, C. E. 1969, J. Opt. Soc. Am., 59, 267
  * Carrión-González et al. (2020) Carrión-González, Ó., García Muñoz, A., Cabrera, J., et al. 2020, A&A, 640, A136
  * Chaverot et al. (2022) Chaverot, G., Turbet, M., Bolmont, E., & Leconte, J. 2022, A&A, 658, A40
  * Chen & Kipping (2016) Chen, J. & Kipping, D. 2016, ApJ, 834, 17
  * Cowan et al. (2015) Cowan, N. B., Greene, T., Angerhausen, D., et al. 2015, PASP, 127, 311
  * Dannert et al. (2022) Dannert, F. A., Ottiger, M., Quanz, S. P., et al. 2022, A&A, 664, A22
  * Deming et al. (2018) Deming, D., Louie, D., & Sheets, H. 2018, PASP, 131, 013001
  * Dressing & Charbonneau (2015) Dressing, C. D. & Charbonneau, D. 2015, ApJ, 807, 45
  * Ertel et al. (2020) Ertel, S., Defrère, D., Hinz, P., et al. 2020, AJ, 159, 177
  * Feinstein et al. (2022) Feinstein, A. D., Radica, M., Welbanks, L., et al. 2022, arXiv e-prints, arXiv:2211.10493
  * Feng et al. (2018) Feng, Y. K., Robinson, T. D., Fortney, J. J., et al. 2018, AJ, 155, 200
  * Feroz et al. (2009) Feroz, F., Hobson, M. P., & Bridges, M. 2009, MNRAS, 398, 1601–1614
  * Foreman-Mackey et al. (2014) Foreman-Mackey, D., Hogg, D. W., & Morton, T. D. 2014, ApJ, 795, 64
  * Gaudi et al. (2020) Gaudi, B. S., Seager, S., Mennesson, B., et al. 2020, arXiv e-prints, arXiv:2001.06683
  * Gilbert et al. (2020) Gilbert, E. A., Barclay, T., Schlieder, J. E., et al. 2020, AJ, 160, 116
  * Gillon et al. (2016) Gillon, M., Jehin, E., Lederer, S. M., et al. 2016, Nature, 533, 221
  * Gillon et al. (2017) Gillon, M., Triaud, A. H. M. J., Demory, B.-O., et al. 2017, Nature, 542, 456
  * Graham (2021) Graham, R. J. 2021, Astrobiology, 21, 1406
  * Graham et al. (2022) Graham, R. J., Lichtenberg, T., & Pierrehumbert, R. T. 2022, Journal of Geophysical Research: Planets, 127, e2022JE007456
  * Greene et al. (2016) Greene, T. P., Line, M. R., Montero, C., et al. 2016, ApJ, 817, 17
  * Hanel et al. (1983) Hanel, R., Conrath, B., Kunde, V., Pearl, J., & Pirraglia, J. 1983, Icarus, 53, 262
  * Hartmann et al. (2002) Hartmann, J. M., Boulet, C., Brodbeck, C., et al. 2002, J. Quant. Spec. Radiat. Transf., 72, 117
  * Harvey et al. (1998) Harvey, A. H., Gallagher, J. S., & Sengers, J. M. H. L. 1998, Journal of Physical and Chemical Reference Data, 27, 761
  * Haus et al. (2016) Haus, R., Kappel, D., Tellmann, S., et al. 2016, Icarus, 272, 178
  * Hoffman et al. (2017) Hoffman, P. F., Abbot, D. S., Ashkenazy, Y., et al. 2017, Science Advances, 3, e1600983
  * Hoffman et al. (1998) Hoffman, P. F., Kaufman, A. J., Halverson, G. P., & Schrag, D. P. 1998, Science, 281, 1342
  * Howett et al. (2010) Howett, C., Spencer, J., Pearl, J., & Segura, M. 2010, Icarus, 206, 573
  * Jeffreys (1998) Jeffreys, H. 1998, The Theory of Probability, Oxford Classic Texts in the Physical Sciences (OUP Oxford), 432–441
  * Kammerer & Quanz (2018) Kammerer, J. & Quanz, S. P. 2018, A&A, 609, A4
  * Karman et al. (2019) Karman, T., Gordon, I. E., van der Avoird, A., et al. 2019, Icarus, 328, 160
  * Kasting (1988) Kasting, J. F. 1988, Icarus, 74, 472
  * Kasting & Harman (2021) Kasting, J. F. & Harman, C. E. 2021, Nature, 598, 259
  * Kasting et al. (1993) Kasting, J. F., Whitmire, D. P., & Reynolds, R. T. 1993, Icarus, 101, 108
  * Kirschvink (1992) Kirschvink, J. L. 1992, in The Proterozoic Biosphere: A Multidisciplinary Study, ed. J. W. Schopf, C. Klein, & D. Des Maris (Cambridge University Press), 51–52
  * Kitzmann et al. (2011) Kitzmann, D., Patzer, A. B. C., von Paris, P., Godolt, M., & Rauer, H. 2011, A&A, 531, A62
  * Komacek et al. (2020) Komacek, T. D., Fauchez, T. J., Wolf, E. T., & Abbot, D. S. 2020, ApJ, 888, L20
  * Konrad et al. (2022) Konrad, B. S., Alei, E., Quanz, S. P., et al. 2022, A&A, 664, A23
  * Kopparapu et al. (2013) Kopparapu, R. K., Ramirez, R., Kasting, J. F., et al. 2013, ApJ, 765, 131
  * Krasnopolsky (2015) Krasnopolsky, V. A. 2015, Icarus, 252, 327
  * Krissansen-Totton et al. (2018) Krissansen-Totton, J., Garland, R., Irwin, P., & Catling, D. C. 2018, AJ, 156, 114
  * Léger et al. (2019) Léger, A., Defrère, D., Muñoz, A. G., et al. 2019, Astrobiology, 19, 797, pMID: 30985192
  * Li et al. (2018) Li, L., Jiang, X., West, R. A., et al. 2018, Nature Communications, 9, 3709
  * Loftus et al. (2019) Loftus, K., Wordsworth, R. D., & Morley, C. V. 2019, ApJ, 887, 231
  * Lustig-Yaeger et al. (2019) Lustig-Yaeger, J., Meadows, V. S., & Lincowski, A. P. 2019, ApJ, 887, L11
  * Madhusudhan (2018) Madhusudhan, N. 2018, Handbook of Exoplanets, 2153–2182
  * Mallama (2017) Mallama, A. 2017, The Spherical Bolometric Albedo of Planet Mercury
  * Matthews (2008) Matthews, G. 2008, Appl. Opt., 47, 4981
  * Meerdink et al. (2019) Meerdink, S. K., Hook, S. J., Roberts, D. A., & Abbott, E. A. 2019, Remote Sensing of Environment, 230, 111196
  * Mettler et al. (2020) Mettler, J.-N., Quanz, S. P., & Helled, R. 2020, AJ, 160, 246
  * Mettler et al. (2022) Mettler, J.-N., Quanz, S. P., Helled, R., Olson, S. L., & Schwieterman, E. W. 2022, Earth as an Exoplanet: II. Earth’s Time-Variable Thermal Emission and its Atmospheric Seasonality of Bio-Indicators
  * Min et al. (2005) Min, M., Hovenier, J. W., & de Koter, A. 2005, A&A, 432, 909–920
  * Mollière et al. (2020) Mollière, P., Stolker, T., Lacour, S., et al. 2020, A&A, 640, A131
  * Mollière et al. (2019) Mollière, P., Wardenier, J. P., van Boekel, R., et al. 2019, A&A, 627, A67
  * Mollière et al. (2015) Mollière, P., Boekel, R. v., Dullemond, C., Henning, T., & Mordasini, C. 2015, ApJ, 813, 47
  * Mueller-Wodarg et al. (2008) Mueller-Wodarg, I. C. F., Strobel, D. F., Moses, J. I., et al. 2008, Neutral Atmospheres (New York, NY: Springer New York), 191–234
  * National Academies of Sciences, Engineering, and Medicine (2021) National Academies of Sciences, Engineering, and Medicine. 2021, Pathways to Discovery in Astronomy and Astrophysics for the 2020s (Washington, DC: The National Academies Press)
  * Nixon & Madhusudhan (2022) Nixon, M. C. & Madhusudhan, N. 2022, ApJ, 935, 73
  * Ortenzi et al. (2020) Ortenzi, G., Noack, L., Sohl, F., et al. 2020, Scientific Reports, 10, 10907
  * Oschlisniok et al. (2012) Oschlisniok, J., Häusler, B., Pätzold, M., et al. 2012, Icarus, 221, 940
  * Pallé et al. (2003) Pallé, E., Goode, P. R., Yurchyshyn, V., et al. 2003, Journal of Geophysical Research: Atmospheres, 108
  * Palmer & Williams (1975) Palmer, K. F. & Williams, D. 1975, Appl. Opt., 14, 208
  * Petigura et al. (2013) Petigura, E. A., Howard, A. W., & Marcy, G. W. 2013, Proceedings of the National Academy of Science, 110, 19273
  * Pierrehumbert (2010) Pierrehumbert, R. T. 2010, Principles of Planetary Climate (Cambridge University Press)
  * Pleskot & Kieffer (1977) Pleskot, L. K. & Kieffer, H. H. 1977, Icarus, 30, 341
  * Quanz et al. (2021) Quanz, S. P., Absil, O., Angerhausen, D., et al. 2021, Experimental Astronomy
  * Quanz et al. (2022) Quanz, S. P., Ottiger, M., Fontanet, E., et al. 2022, A&A, 664, A21
  * Robinson & Salvador (2022) Robinson, T. D. & Salvador, A. 2022, Exploring and Validating Exoplanet Atmospheric Retrievals with Solar System Analog Observations
  * Rothman et al. (2010) Rothman, L., Gordon, I., Barber, R., et al. 2010, J. Quant. Spec. Radiat. Transf., 111, 2139
  * Rugheimer et al. (2013) Rugheimer, S., Kaltenegger, L., Zsom, A., Segura, A., & Sasselov, D. 2013, Astrobiology, 13, 251
  * Segelstein (1981) Segelstein, D. J. 1981, The complex refractive index of water
  * Sellers (1969) Sellers, W. D. 1969, Journal of Applied Meteorology and Climatology, 8, 392
  * Sergeev et al. (2022) Sergeev, D. E., Lewis, N. T., Lambert, F. H., et al. 2022, Bistability of the atmospheric circulation on TRAPPIST-1e
  * Skilling (2006) Skilling, J. 2006, Bayesian Anal., 1, 833
  * Sneep & Ubachs (2005) Sneep, M. & Ubachs, W. 2005, J. Quant. Spec. Radiat. Transf., 92, 293
  * Spencer et al. (1999) Spencer, J. R., Tamppari, L. K., Martin, T. Z., & Travis, L. D. 1999, Science, 284, 1514 – 1516
  * Taylor et al. (2018) Taylor, F. W., Svedhem, H., & Head, J. W. 2018, Space Science Reviews, 214, 35
  * The LUVOIR Team (2019) The LUVOIR Team. 2019, The LUVOIR Mission Concept Study Final Report
  * Titov et al. (2018) Titov, D. V., Ignatiev, N. I., McGouldrick, K., Wilquet, V., & Wilson, C. F. 2018, Space Sci. Rev., 214, 126
  * Toon & Ackerman (1981) Toon, O. B. & Ackerman, T. P. 1981, Appl. Opt., 20, 3657
  * Turbet et al. (2021) Turbet, M., Bolmont, E., Chaverot, G., et al. 2021, Nature, 598, 276
  * Vasquez et al. (2013) Vasquez, M., Schreier, F., Gimeno García, S., et al. 2013, A&A, 557, A46
  * Villanueva et al. (2018) Villanueva, G., Smith, M., Protopapa, S., Faggi, S., & Mandell, A. 2018, J. Quant. Spec. Radiat. Transf., 217, 86–104
  * von Paris et al. (2013) von Paris, P., Hedelt, P., Selsis, F., Schreier, F., & Trautmann, T. 2013, A&A, 551, A120
  * Voyage 2050 Senior Committee (2021) Voyage 2050 Senior Committee. 2021, Voyage 2050 – Final Recommendations from the Voyage 2050 Senior Committee
  * Way et al. (2016) Way, M. J., Del Genio, A. D., Kiang, N. Y., et al. 2016, Geochim. Res. Lett., 43, 8376
  * Wordsworth (2021) Wordsworth, R. 2021, ApJ, 912, L14
  * Yang et al. (2014) Yang, J., Boué, G., Fabrycky, D. C., & Abbot, D. S. 2014, ApJ, 787, L2

##  Appendix A Cloud retrieval validation

Figure 9: Results from the validation retrieval on the $3-20$ µm, $R=1000$,
and photon noise $S/N=50$ Venus-twin spectrum (opaque H2SO4 clouds) using the
opaque H2SO4 cloud forward model. _Top left_ : Corner plot of the posterior
distribution of the forward model parameters. Here, $L(\cdot)$ stands for
$\log_{10}(\cdot)$. We derived the equilibrium temperature $T_{\mathrm{eq}}$
and the Bond albedo $A_{\mathrm{B}}$ from the other posteriors (see Appendix
B). The dotted black lines indicate the true values. In the 1D posteriors, we
show the 16th, 50th, and 84th percentiles as black dashed lines. _Top right_ :
P$-$T profile residuals relative to the retrieved median P$-$T profile. Color-
shaded regions indicate percentiles of the retrieved P$-$T profiles. The gray
shaded regions indicate percentiles of the retrieved cloud-top pressure. The
solid black line, the orange circular marker, and the red square marker
represent the true Venus-twin P$-$T profile relative to the median retrieved
P$-$T profile. In the inset figure, we plot the 2D $P_{0}$-$T_{0}$ posterior.
_Bottom_ : Residuals of the spectra corresponding to the retrieved posteriors
relative to the Venus-twin input. Color-shaded areas represent different
quantiles of the residuals. The gray area marks the 1$\sigma$ photon noise
level.

Here, we validate the updated retrieval routine and test if the newly added
cloud model is constrainable. For this retrieval test, we considered a high
resolution ($R=1000$), low noise (only photon noise, $S/N=50$), and $3-20$ µm
cloudy Venus-twin MIR spectrum. For this validation retrieval, we used the
opaque H2SO4 forward model and the priors specified in Table 1. We summarize
the results in Fig. 9.

The top left panel of Fig. 9, shows the corner plot of the posterior
distribution, excluding the P$-$T parameters $a_{i}$ for clarity (the full
corner plot is in Appendix C). Additionally, we derived the equilibrium
temperature $T_{\mathrm{eq}}$ and the Bond albedo $A_{\mathrm{B}}$ from the
posteriors following the method outlined in Appendix B. We see that most
posteriors are well constrained and centered on the true values. We especially
emphasize the strong constraints on both $R_{\text{pl}}$ and $A_{\mathrm{B}}$.
The ability to constrain $R_{\text{pl}}$ and $A_{\mathrm{B}}$ simultaneously
is unique to studies of the MIR thermal emission spectrum. Comparable
constraints are not inferrable from reflected light spectra, due to a strong
degeneracy between $R_{\text{pl}}$ an $A_{\mathrm{B}}$ (Feng et al. 2018;
Carrión-González et al. 2020).

Further, we find that the atmosphere is CO2 dominated (abundance $\geq$85%)
and manage to detect both H2O and CO. Most cloud parameters are well
retrieved, which indicates that there are spectral features from the clouds.
Only $P\mathrm{{}^{cloud}_{span}}$ is not well constrained. The posterior is
best described by a step function, where any
$P\mathrm{{}^{cloud}_{span}}{}\geq 0.5$ bar is possible. This lower limit on
$P\mathrm{{}^{cloud}_{span}}$ corresponds to the thickness at which the cloud
layer becomes opaque. Thicker cloud layers are not excluded since the spectrum
contains no signatures from the lower atmospheric layers. For thinner layers,
there are contributions from the lower atmospheric layers, which result in a
bad fit to the input spectrum.

The model parameters are generally well retrieved. However, there are
interesting features in some posteriors. First, the rectangular shape of the
$P\mathrm{{}^{cloud}_{top}}$ posterior is linked to the fixed atmospheric
pressure grid of the opaque H2SO4 cloud forward model. All
$P\mathrm{{}^{cloud}_{top}}$ values in the rectangular peak fall within the
same atmospheric layer and thus result in the same cloud-top. Their posterior
probability is therefore equal. Thus, at the high $R$ and $S/N$ considered
here, we are sensitive to the discrete nature of our atmosphere model. Cloud-
tops in lower or higher layers are very unlikely and are linked to the
secondary peaks seen in both the 2D and 1D posteriors of other parameters
(e.g., $M_{\text{pl}}$).

Second, the strength of the $M_{\text{pl}}$ constraint opposes the results
from Papers III and V, where $M_{\text{pl}}$ was less strongly constrained
relative to the prior due to a known degeneracy between the mass (surface
gravity) and the abundances of the atmospheric gases ($M_{\text{pl}}$ and
abundances simultaneously over- or undersetimated; see also, e.g., Mollière et
al. 2015; Feng et al. 2018; Madhusudhan 2018; Quanz et al. 2021). This
degeneracy is visible for CO2 and H2O, but not for CO (due to overall weaker
abundance constraints). However, the CO2 posterior converges toward the upper
edge of the prior range and is more strongly constrained (tail to high
abundances is cut off). This breaks the degeneracy and thus also
$M_{\text{pl}}$ is strongly constrained. The secondary peaks in the
$M_{\text{pl}}$ posterior correspond to the secondary peaks in the
$P\mathrm{{}^{cloud}_{top}}$ posterior (cloud top in different layers).

The top right panel of Fig. 9 shows the constraints on the atmospheric P$-$T
structure relative to the retrieved median (we plot the relative retrieved
P$-$T profile, since deviances from the truth are more visible). The absolute
P$-$T plot is given in Appendix C. We visualize percentiles of the retrieved
profiles as green-shaded regions and compare them to the true profile (red
square, orange circle, solid black line). Further, we plot the percentiles of
the retrieved cloud-top as gray areas. The inset plot shows the 2D histogram
of the retrieved $P\mathrm{{}^{cloud}_{top}}$ and $T\mathrm{{}^{cloud}_{top}}$
(cloud-top temperature; calculated from the retrieved P$-$T profiles).

Both the P$-$T profile above the cloud-top and the cloud-top are accurately
retrieved. The retrieved P$-$T structure above the cloud-top is centered on
the true profile, and roughly $90\%$ of the profiles lie within $\leq\pm 2$ K
of the truth. Similarly, roughly $90\%$ of the retrieved
$P\mathrm{{}^{cloud}_{top}}$ and $T\mathrm{{}^{cloud}_{top}}$ values are
distributed symmetrically around the true cloud-top and lie within $\leq\pm
0.01$ bar and $\leq\pm 3$ K of the truth. The $5-95\%$ P$-$T envelope shows an
asymmetry toward positive relative temperatures. In the
$P\mathrm{{}^{cloud}_{top}}$-$T\mathrm{{}^{cloud}_{top}}$ posterior, the
$5-95\%$ envelope exhibits a second isolated region (same
$T\mathrm{{}^{cloud}_{top}}$, lower $P\mathrm{{}^{cloud}_{top}}$). These two
outliers correspond to the cases discussed above where the cloud-top falls
within a different atmospheric layer.

The uncertainty on the retrieved P$-$T profile grows with decreasing pressure.
This increase is due to a lack of spectral features from these optically thin,
low-pressure layers in Venus’ MIR spectrum at the $R$ and $S/N$ considered
here. Similarly, the uncertainty on the retrieved P$-$T structure increases
below the cloud-top, because the optically thick cloud deck blocks spectral
contributions from these atmospheric regions. This lack of spectral features
can be seen in the emission contribution function provided in Appendix C.
Constraints on the high- and low-pressure regions are obtained by
extrapolating the P$-$T model.

In the bottom panel of Fig. 9, we show the residual of the emission spectra
corresponding to the retrieved parameter posteriors relative to our Venus-twin
input spectrum. We indicate percentiles of the residuals as green-shaded areas
and the $1\sigma$ photon noise level as a gray area. Overall, the retrieval
output is well centered on the input spectrum and the deviations from the
truth mostly lie below $1\%$. The observed deviations grow larger for
wavelengths where the flux is low and the photon noise is large (e.g., below
$6$ µm, in the CO2 band between $14-16$ µm). However, in general, the assumed
forward model accurately reproduces the input spectrum, despite the points
mentioned above. This demonstrates that our cloudy Venus-twin model is suited
for our retrieval study.

##  Appendix B Equilibrium temperature and Bond albedo

The planetary equilibrium temperature $T_{\mathrm{eq}}$ and the Bond albedo
$A_{\mathrm{B}}$ are two parameters, which were of particular interest to us.
These parameters provide important information about the energy budget of the
planet. However, both parameters were not directly determined by our retrieval
framework. Therefore, we required a method to derive their values from the
retrieved posterior distributions of the model parameters.

In a first step, we generated a set of MIR spectra that was representative of
the retrieved parameter posteriors. Next, we determined the planet’s
$T_{\mathrm{eq}}$ by fitting the black body emission,
$F_{BB}\left(\lambda\right)$, expected for a spherical planet with radius
$R_{\text{pl}}$ to the continuum (between $8$ µm and $11$ µm) of these MIR
spectra:

$F_{BB}\left(\lambda\right)=\frac{2\pi
hc^{2}}{\lambda^{5}}{e^{1-\frac{hc}{\lambda
k_{B}T_{\mathrm{eq}}{}}}}\cdot\frac{R_{\text{pl}}{}^{2}}{d^{2}}.$ (3)

Here, $d$ is the distance to the observer (10 pc in our case), $\lambda$ the
wavelength, $h$ the Planck constant, $c$ the speed of light, and $k_{B}$ the
Boltzmann constant. This provided us with a probability distribution for the
planetary equilibrium temperature $T_{\mathrm{eq}}$. From the
$T_{\mathrm{eq}}$ distribution, we then calculated the probability
distribution for $A_{\mathrm{B}}$ using the following relation:

$A_{\mathrm{B}}=1-16\,\pi\sigma\frac{a^{2}_{P}T_{\mathrm{eq}}^{4}}{L_{*}}.$
(4)

Here, $\sigma$ is the Stefan–Boltzmann constant, $a_{P}$ is the semi-major
axis of the planetary orbit around the host star, and $L_{*}$ is the
luminosity of the host star. For the $A_{\mathrm{B}}$ calculation, we assumed
to know both $a_{P}$ (Venus-like: $a_{P}=0.72$ AU) and $L_{*}$ (Sun-like:
$L_{*}=1\,L_{\odot}$) with an uncertainty of $\pm$5%. For each sample from the
$T_{\mathrm{eq}}$ distribution, we drew a random $a_{P}$ and $L_{*}$ value
from two uncorrelated normal distributions and used these random values to
calculate $A_{\mathrm{B}}$. This yielded a probability distribution for the
planetary Bond albedo $A_{\mathrm{B}}$.

This approach for determining $T_{\mathrm{eq}}$ and $A_{\mathrm{B}}$ is
advantageous, because correlations between the posteriors of the retrieved
planetary parameters are taken into account when deriving the
$T_{\mathrm{eq}}$ and $A_{\mathrm{B}}$ distributions. Furthermore, the
resulting correlations between the distributions of $T_{\mathrm{eq}}$,
$A_{\mathrm{B}}$, and the retrieved parameter posteriors are correct and can
be analyzed.

##  Appendix C Additional plots for retrieval results

Additional plots for the results from the validation retrieval (see Sect. A)
and the retrievals of the $4-18.5$ µm, $R=50$, and $S/N=10$ LIFEsim spectrum
(see Sect. 3.1). We plot the retrieval results from the validation retrieval
in Fig. 10. For the $4-18.5$ µm, $R=50$, and $S/N=10$ LIFEsim spectrum, we
plot the results for the opaque H2SO4 forward model in Fig. 11, for the
transparent H2SO4 forward model in Fig. 12, for the opaque H2O forward model
in Fig. 13, and for the cloud-free forward model in Fig. 14.

Figure 10: Results from the validation retrieval on the $3-20$ µm, $R=1000$,
and photon noise $S/N=50$ Venus-twin spectrum (opaque H2SO4 clouds) using the
opaque H2SO4 cloud forward model (see Sect. A). _Lower left half_ : Corner
plot of the posterior distribution of the forward model parameters. Here,
$L(\cdot)$ stands for $\log_{10}(\cdot)$. We derived the equilibrium
temperature $T_{\mathrm{eq}}$ and the Bond albedo $A_{\mathrm{B}}$ from the
other posteriors (see Appendix B). The dotted black lines indicate the true
values. In the 1D posteriors, we show the 16th, 50th, and 84th percentiles as
black dashed lines. _Top center_ : P$-$T profiles corresponding to the
retrieved P$-$T parameters. Color-shaded regions indicate percentiles of the
retrieved P$-$T profiles. The gray shaded regions indicate percentiles of the
retrieved cloud-top pressure. The solid black line, the orange circular
marker, and the red square marker represent the true Venus-twin P$-$T profile.
In the inset figure, we plot the 2D $P_{0}$-$T_{0}$ posterior. _Top right_ : A
table summarizing the true values of the forward model parameters and the
16th, 50th, and 84th percentiles of the parameter posteriors. _Center right_ :
Comparison of the retrievals best fit to the Venus-twin input spectrum. The
thin colored line represents the best fit, the thick black line the Venus-twin
input spectrum. The uncertainties on the best fit are too small to be visible.
Below the spectrum plot, we show the mean wavelength-dependent contribution of
the atmospheric layers to the emission spectrum corresponding to the retrieved
parameter posteriors. Darker colors indicate higher contributions. The dashed
black line indicates the position of the cloud-deck assumed to simulate the
input spectrum (opaque H2SO4 clouds).

Figure 11: Results from the retrieval on the $4-18.5$ µm, $R=50$, and LIFEsim
noise $S/N=10$ Venus-twin spectrum (opaque H2SO4 clouds) using the opaque
H2SO4 cloud forward model (see Sect. 3.1). _Lower left half_ : Corner plot of
the posterior distribution of the forward model parameters. Here, $L(\cdot)$
stands for $\log_{10}(\cdot)$. We derived the equilibrium temperature
$T_{\mathrm{eq}}$ and the Bond albedo $A_{\mathrm{B}}$ from the other
posteriors (see Appendix B). The dotted black lines indicate the true values.
In the 1D posteriors, we show the 16th, 50th, and 84th percentiles as black
dashed lines. _Top center_ : P$-$T profiles corresponding to the retrieved
P$-$T parameters. Color-shaded regions indicate percentiles of the retrieved
P$-$T profiles. The gray shaded regions indicate percentiles of the retrieved
cloud-top pressure. The solid black line, the orange circular marker, and the
red square marker represent the true Venus-twin P$-$T profile. In the inset
figure, we plot the 2D $P_{0}$-$T_{0}$ posterior (if retrieved; otherwise
$P\mathrm{{}^{cloud}_{top}}$-$T\mathrm{{}^{cloud}_{top}}$). _Top right_ : A
table summarizing the true values of the forward model parameters and the
16th, 50th, and 84th percentiles of the parameter posteriors. _Center right_ :
Spectra corresponding to the retrieved posteriors in comparison to the Venus-
twin input. Color-shaded areas represent different quantiles of the retrieved
spectra. The solid black line represents the Venus-twin input spectrum. The
gray area marks the 1$\sigma$ LIFEsim noise level. Below the spectrum plot, we
show the mean wavelength-dependent contribution of the atmospheric layers to
the emission spectrum corresponding to the retrieved parameter posteriors.
Darker colors indicate higher contributions. The dashed black line indicates
the position of the cloud-deck assumed to simulate the input spectrum (opaque
H2SO4 clouds).

Figure 12: Same as Fig. 11, but for the retrieval on the $4-18.5$ µm, $R=50$,
and LIFEsim noise $S/N=10$ Venus-twin spectrum (opaque H2SO4 clouds) using the
transparent H2SO4 cloud forward model (see Sect. 3.1).

Figure 13: Same as Fig. 11, but for the retrieval on the $4-18.5$ µm, $R=50$,
and LIFEsim noise $S/N=10$ Venus-twin spectrum (opaque H2SO4 clouds) using the
opaque H2O cloud forward model (see Sect. 3.1).

Figure 14: Same as Fig. 11, but for the retrieval on the $4-18.5$ µm, $R=50$,
and LIFEsim noise $S/N=10$ Venus-twin spectrum (opaque H2SO4 clouds) using the
cloud-free forward model (see Sect. 3.1).

##  Appendix D Supplementary results from retrievals on higher quality input
spectra

Here we plot the posteriors for all parameters found in the retrieval analysis
for higher quality input spectra discussed in Sect. 3.2. We include the
$T_{\mathrm{eq}}$ and $A_{\mathrm{B}}$ distributions, which we calculated from
the posteriors following the method introduced in Appendix B. Further, we add
the results for the intermediate $S/N=15$ noise levels. In Figs. 15 and 16, we
provide the retrieval results for the $3-20$ µm input spectra. In Figs. 17 and
18, we display the results obtained for the $4-18.5$ µm input spectra.

Figure 15: Model parameter posteriors for increased quality $3-20$ µm Venus-
twin spectra (opaque H2SO4 clouds). Here, $L(\cdot)$ abbreviates
$\log_{10}(\cdot)$. Dashed black lines indicate the true values (see Table 1).
Solid green area $-$ opaque H2SO4 clouds; thin-blue outlined area $-$
transparent H2SO4 clouds; orange-hatched area $-$ opaque H2O clouds; thick-
yellow outlined area $-$ cloud-free. Rows (top to bottom) represent different
input qualities: $R=50$, $S/N=10$; $R=100$, $S/N=10$; $R=50$, $S/N=15$;
$R=100$, $S/N=15$; $R=50$, $S/N=20$; $R=100$, $S/N=20$. Continuation in Fig.
16.

Figure 16: Continuation of Fig. 15. For $\mathrm{Species^{cloud}}$, the true
value is the H2SO4 mass fraction in the Venus-twin model.

Figure 17: Model parameter posteriors for increased quality $4-18.5$ µm Venus-
twin spectra (opaque H2SO4 clouds). Here, $L(\cdot)$ abbreviates
$\log_{10}(\cdot)$. Dashed black lines indicate the true values (see Table 1).
Solid green area $-$ opaque H2SO4 clouds; thin-blue outlined area $-$
transparent H2SO4 clouds; orange-hatched area $-$ opaque H2O clouds; thick-
yellow outlined area $-$ cloud-free. Rows (top to bottom) represent different
input qualities: $R=50$, $S/N=10$; $R=100$, $S/N=10$; $R=50$, $S/N=15$;
$R=100$, $S/N=15$; $R=50$, $S/N=20$; $R=100$, $S/N=20$. Continuation in Fig.
18.

Figure 18: Continuation of Fig. 17. For $\mathrm{Species^{cloud}}$, the true
value is the H2SO4 mass fraction in the Venus-twin model.

##  Appendix E Cloudy versus icy scenario - Comparison of absolute fluxes

In Fig. 19, we compare the absolute fluxes of the cloudy and icy exoplanet
scenarios discussed in Sect. 4.1.2. In the UV/O/NIR, we show the noise
expected for observations of the cloudy Venus-twin exoplanet with the HabEx +
Starshade (Gaudi et al. 2020) or LUVOIR A (The LUVOIR Team 2019) instruments
($S/N=10$ at $0.6$ µm, calculated with the NASA-GSFC Planetary Spectrum
Generator; Villanueva et al. 2018). In the MIR, we show the wavelength-
dependent expected LIFEsim noise ($S/N=10$ at $11.2$ µm).

The increase in the HabEx noise at roughly $1$ µm is caused by the HabEx +
Starshade instrument design. Above $1$ µm, the throughput of the instrument is
strongly decreased. Thus, the $R$ of the spectrum above $1$ µm is decreased to
increase the $S/N$ of the measured spectrum. However, the Venus-twin spectrum
plotted in Fig. 19 is constant in $R$ over the full wavelength range
considered. Further, the integration time is also constant over the full
spectrum. Thus, the HabEx + Starshade noise above $1$ µm is significantly
larger than at shorter wavelengths.

Figure 19: Flux of a cloudy Venus-twin exoplanet (opaque H2SO4 clouds; dashed
green line) compared to an icy world with a thin CO2-dominated atmosphere
(solid brown line). In the top panel, we plot the UV/O/NIR wavelength range.
The hatched area represents the expected wavelength-dependent $S/N=10$ noise
for the LUVOIR A mission concept (The LUVOIR Team 2019). The thick, solid
black line represents the expected $S/N=10$ noise for the HabEx + Starshade
mission concept (Gaudi et al. 2020). In the bottom panel, we plot the MIR
wavelength range. The gray shaded region indicates the 1$\sigma$ LIFEsim noise
level at $S/N=10$.