# Early Science with the Large Millimeter Telescope: Constraining the Gas
Fraction of a Compact Quiescent Galaxy at $z=$1.883

Joyce N Caliendo Physics Department, University of Connecticut, Storrs, CT
06269  Katherine E Whitaker Department of Astronomy, University of
Massachusetts, Amherst, MA 01003  Cosmic Dawn Center, Copenhagen, Denmark
Mohammad Akhshik Physics Department, University of Connecticut, Storrs, CT
06269  Grant Wilson Department of Astronomy, University of Massachusetts,
Amherst, MA 01003  Christina C. Williams Steward Observatory, University of
Arizona, Tucson, AZ 85721  NSF Fellow  Justin S. Spilker NHFP Hubble Fellow
Department of Astronomy, University of Texas at Austin, 2515 Speedway, Stop
C1400, Austin, TX 78712]  Guillaume Mahler Department of Astronomy, University
of Michigan, 1085 S. University Ave, Ann Arbor, MI 48109, USA  Alexandra Pope
Department of Astronomy, University of Massachusetts, Amherst, MA 01003  Keren
Sharon Department of Astronomy, University of Michigan, 1085 S. University
Ave, Ann Arbor, MI 48109, USA  Emmaly Aguilar Corporación Mexicana de
Investigación en Materiales S.A. de C.V., Ciencia y Tecnología No 790 Col.
Saltillo 400, C.P. 25290 Saltillo, Coahuila. México  Rachel Bezanson
Department of Physics and Astronomy and PITT PACC, University of Pittsburgh,
Pittsburgh, PA, 15260, USA  Miguel Chavez Dagostino  Instituto Nacional de
Astrofísica Óptica y Electrónica, Luis Enrique Erro 1, CP 72840, Tonantzintla,
Puebla, México  Arturo I. Gómez-Ruiz Instituto Nacional de Astrofísica Óptica
y Electrónica, Luis Enrique Erro 1, CP 72840, Tonantzintla, Puebla, México
Consejo Nacional de Ciencia y Tecnología, Av. Insurgentes Sur 1582, Col.
Crédito Constructor, Alcaldía Benito Juárez, C.P. 03940, Ciudad de México
Alfredo Montaña Instituto Nacional de Astrofísica Óptica y Electrónica, Luis
Enrique Erro 1, CP 72840, Tonantzintla, Puebla, México  Consejo Nacional de
Ciencia y Tecnología, Av. Insurgentes Sur 1582, Col. Crédito Constructor,
Alcaldía Benito Juárez, C.P. 03940, Ciudad de México  Sune Toft Cosmic Dawn
Center, Copenhagen, Denmark  Niels Bohr Institute, University of Copenhagen,
Jagtvej 128, DK-2200 Copenhagen, Denmark  Miguel Velazquez de la Rosa
Instituto Nacional de Astrofísica Óptica y Electrónica, Luis Enrique Erro 1,
CP 72840, Tonantzintla, Puebla, México  Milagros Zeballos Universidad de las
Américas Puebla, Ex Hacienda Sta. Catarina Mártir S/N. San Andrés Cholula,
Puebla, México

###### Abstract

We present constraints on the dust continuum flux and inferred gas content of
a gravitationally lensed massive quiescent galaxy at $z$=1.883$\pm$0.001 using
AzTEC 1.1mm imaging with the Large Millimeter Telescope. MRG-S0851 appears to
be a prototypical massive compact quiescent galaxy, but has evidence that it
experienced a centrally concentrated rejuvenation event in the last 100 Myr
(see Akhshik et al. 2020a). This galaxy is undetected in the AzTEC image but
we calculate an upper limit on the millimeter flux and use this to estimate
the H2 mass limit via an empirically calibrated relation that assumes a
constant molecular gas-to-dust ratio of 150. We constrain the 3$\sigma$ upper
limit of the H2 fraction from the dust continuum in MRG-S0851 to be
${M_{\mathrm{H_{2}}}/M_{\mathrm{\star}}}$ $\leq$ 6.8%. MRG-S0851 has a low gas
fraction limit with a moderately low sSFR owing to the recent rejuvenation
episode, which together results in a relatively short depletion time of $<$0.6
Gyr if no further H2 gas is accreted. Empirical and analytical models both
predict that we should have detected molecular gas in MRG-S0851, especially
given the rejuvenation episode; this suggests that cold gas and/or dust is
rapidly depleted in at least some early quiescent galaxies.

##  1 Introduction

One of the biggest unanswered questions in galaxy evolution is what physical
mechanism(s) cause galaxies to cease forming stars. Many studies have been
performed with the goal of better understanding what drives and halts star
formation (e.g., Man & Belli 2018). It has been established that there are two
main types of galaxy populations spanning from the local to the distant
Universe: less massive star-forming galaxies with blue colors owing to their
young stars, and quiescent galaxies that are generally more massive with red
colors due to older stellar populations (Strateva et al. 2001; Whitaker et al.
2011). The most massive galaxies in the local Universe have predominantly old
stellar populations, having formed most of their stars many Gyr ago (McDermid
et al. 2015). By looking closer to their epoch of formation at $z>1$, massive
recently quenched galaxies are a useful target population to search for
observational signatures of quenching.

The role of H2 molecular gas depletion in quenching is not well understood
(Janowiecki et al. 2020). Because quenching for most massive galaxies seems to
occur on a rapid timescale, the number of galaxies experiencing this process
at any time will be small (Schawinski et al. 2014; Wild et al. 2016). However,
identifying these rare targets allows one to study the H2 gas properties
closest to the transition epoch, enabling investigations of poorly explored
processes. There is a long list of physical mechanisms that could contribute
to both halting and/or fueling star-formation in galaxies. In particular,
molecular gas inflow and outflow may be a major factor in explaining their
star formation histories. Gas heating and/or removal can impact quenching
through processes like strangulation (e.g., Peng et al. 2015), galactic
merging events (e.g., Hopkins et al. 2006), halo quenching (e.g., Dekel &
Birnboim 2006), and morphological quenching (e.g., Martig et al. 2009).

Most star-forming galaxies follow the log(SFR)-log(M${}_{\star})$ main
sequence (Speagle et al. 2014), and as galaxies fall below this relation we
observe a drop in the molecular gas fraction and star formation efficiency
(Tacconi et al. 2020). Quantifying the gas reservoir and measuring its
physical properties are important first steps towards developing an
understanding of the role of cold gas and star-formation efficiency in driving
quenching (e.g., Saintonge et al. 2017). H2 is difficult to detect directly,
therefore two proxies are commonly used. One methodology includes using CO as
a proxy; CO studies measure flux emitted by rotational transitions in the
tracer molecules and relies on assumptions about the CO excitation and the CO
to H2 ratio. The second uses the dust continuum flux measurements assuming a
dust temperature and H2 to dust ratio. Here, we use dust to quantify molecular
gas reservoirs and include comparisons to studies that utilize both
methodologies. Quantifying the H2 molecular gas reservoir is an important
first step toward understanding what happens during the quenching process in
massive high-$z$ galaxies.

In this letter, we place an upper limit on the H2 reservoir of MRG-S0851, a
strongly  gravitationally lensed massive galaxy at $z=1.88$ that is overall
quiescent but experienced a burst of new star formation within the last 100
Myr. Our study contributes constraints on the H2 gas content, as traced by
1.1mm dust continuum, associated with a rare example of a massive compact
rejuvenating galaxy at $z\sim 2$; currently, very few such measurements exist
in the literature at z >1.5. We assume a standard $\Lambda$CDM cosmology with
$\Omega_{M}$ = 0.3, $\Omega_{\Lambda}$ = 0.7 and H0 = 70 km s-1 Mpc-1.

Figure 1: MRG-S0851 imaged by LMT/AzTEC. The composite color images are from
HST/WFC3 HF160W (red), IF814W (green), and UF390W (blue). There are five
images within the $8\farcs 5$ beamsize (white circle), but only E1, E2, and E3
were used in calculating the magnification factor, as E4 and E5 are negligible
demagnified images. The red line is the critical curve, which marks regions in
the image plane with theoretically infinite magnification.

##  2 Data

Galaxy MRG-S0851 is a red galaxy at $z$=1.88 quintuply imaged via strong
gravitational lensing by the cluster SDSS J0851$+$3331 identified in the Sloan
Giant Arc Survey (Figure 1; see also Sharon et al. 2020). As a target in the
REsolving QUIEscent Magnified (REQUIEM) galaxy survey, Akhshik et al. (2020b)
present the analysis of the stellar population gradients of our main target
with surprising evidence supporting a rejuvenation phase in the last 100 Myr
(Akhshik et al. 2020a). MRG-S0851 is otherwise a prototypical $z\sim 2$
quiescent galaxy, with an intrinsic stellar mass of
$\log{M_{*}}/M{{}_{\odot}}$ = 11.02 $\pm$ 0.04, a low SFR of 6$\pm$1M⊙/yr,
sSFR log(SFR/M⋆) = -10.32${}_{-0.05}^{+0.07}$ yr-1, a Sérsic index of
$n=2.35\pm 0.05$, and a compact circularized effective radius of
1.7${}^{+0.3}_{-0.1}$ kpc. The SFR estimate of MRG-S0851 from Akhshik et al.
(2020b) was made using joint spectro-photometric fit assuming a non-parametric
star formation history (SFH). These estimates are generally $\sim$0.1-1 dex
lower than UV+IR estimates (Leja et al. 2019, see also Section 4). Fitting the
grism spectrum along with photometry decreases the SFR from $\sim$14 M⊙/yr of
photometry-only to $\sim$6 M⊙/yr from the spectro-photometric fit.

MRG-S0851 was observed for 5.5 hours with the 1.1 mm continuum camera AzTEC on
the Large Millimeter Telescope (Early Science Phase with 32m of active surface
configuration) between February 12th and 22nd in 2016. The LMT is a 50 m
single dish millimeter-wave radio-telescope located on the summit of Sierra
Negra at an altitude of 4600 m above the sea level in the Mexican state of
Puebla. AzTEC is 144 pixel bolometric camera with high mapping speed and high
sensitivity which proves to be essential to observing dust obscured galaxies
(Wilson et al. 2008). The 1.1mm bandpass probes the Rayleigh-Jeans tail of the
dust distribution in galaxies up to $z<$3.4, where the dust continuum has been
shown to be a proxy for the molecular gas mass in star-forming galaxies
(Scoville et al. 2016).

The 5.5 hour integration was taken in good to excellent weather conditions
($\tau_{225GHz}=0.04-0.07$). Observations of the cluster SDSS J0851+3331,
centered on our target MRG-0851, were taken with the AzTEC small-map observing
mode, covering an area of 7.5 arcmin2. The instrument has a point spread
function (PSF) with a full width at half-maximum (FWHM) beam size of $8\farcs
5$. The fives images of MRG-S0851 span roughly $7\farcs 0$ on the sky, as
shown in Figure 1, such that the images are unresolved within the $8\farcs 5$
LMT/AzTEC beam (white circle). Two (demagnified) images (E4 and E5) are
negligibly faint in the optical, with the remaining three brighter images
still resulting in a null detection in AzTEC. Full details about the lens
model can be found in Sharon et al. (2020) and Appendix A of Akhshik et al.
(2020b). The calibration and analysis of the AzTEC data follow the procedure
described in Wilson et al. (2008). Our map covers 7.5 arcmin2, with a mean RMS
of 0.24 mJy. We find excellent agreement between the 1$\sigma$ limits at the
locations of E1-3, with an average value consistent at the $\pm$0.001 mJy
level. There are no sources detected above an SNR > 4 within a 105" radius and
we determine the 3 $\sigma$ upper limit on the 1.1mm flux from the RMS and the
standard deviation of the weight map.

##  3 Methodology

Figure 2: This figure plots log(SFR) as a function of stellar mass. Galaxies
are color coded by their sSFR. The black lines represent the SFR-M∗ main
sequence from Speagle et al. (2014) for $z\sim$0 and from Whitaker et al.
(2014) at $z>$0.5. The dotted lines are 0.3 dex above and below the main
sequence. *See Tacconi et al. (2018) for full list of previous PHIBBS survey
data included.

As noted in Section 2, our target MRG-S0851 went undetected in the deep 5.5
hour AzTEC/LMT integration. To calculate noise we found the RMS of the signal
map and as a confirmation also compared to the standard deviation from the
weight map. We constrain the 3$\sigma$ upper limit of the dust continuum to be
0.72 mJy. At the redshift of MRG-S0851, the 1.1 mm imaging traces rest-frame
382$\mu$m. Imaging at rest-frame wavelengths >250 $\mu$m to sample the
Rayleigh Jeans tail where the dust continuum is a good proxy for the mass of
the interstellar medium. At these wavelengths, the dust emission is optically
thin so that the observed flux density is proportional to the mean temperature
of dust, the mass of dust, and the dust opacity coefficient (Scoville et al.
2016). We assume a luminosity-weighted dust temperature of 25 K, which is the
average temperature of galaxies at $z$ <2.3 with little variation (see, e.g.,
Scoville et al. 2016). In this methodology, the observed millimeter flux
density is calibrated to estimate the molecular gas mass component of the ISM
via observed galaxies in Scoville et al. (2014). Adopting our 3$\sigma$ upper
limit, we use this dust continuum method to calculate a gas mass of
$\log{M_{H_{2}}}$ = 10.69M⊙, before correction for magnification for
MRG-S0851, following Equation 16 which is corrected in the erratum to Scoville
et al. (2016).

As the LMT beam size at the time of our observations was $8\farcs 5$, all five
images of our main target remain unresolved and therefore contribute to the
null detection. Of these five images, we use three to calculate the
magnification correction, as the other two are negligible demagnified images
(see Figure 1). To determine the upper limit for a single image of MRG-S0581,
we calculate the ratio of the total area in the HST image plane relative to
the source plane to determine a magnification correction. Specifically, we
define three regions in the image plane corresponding to E1, E2, and E3, and
we map these regions back to the source plane using the lens model. The ratio
of the total projected area of these regions in the image plane relative to
the source plane is calculated to be $\mu$=9.6, and is used as an estimate of
the total magnification due to strong gravitational lensing. As a
confirmation, we also estimate the magnification using the GALFIT model (Peng
et al. 2002) of images E1-3 (Appendix A; Akhshik et al. 2020b), calculating a
light-weighted average magnification of $\mu=9.46$; both estimates of
magnification are therefore consistent. Flux contamination from the nearby
cluster galaxies at rest-frame 800$\mu$m (observed 1.1mm at $z$=0.368) is
negligible.

We calculate the upper limit on the intrinsic gas mass of MRG-S0851 by
dividing the uncorrected measurement by the total magnification, finding
<5.04$\times$109 M⊙. This results in a 3$\sigma$ upper limit of the gas
fraction for galaxy MRG-S0851 of $M_{\mathrm{H_{2}}}/M_{\mathrm{\star}}$<6.8%.
Given the SFR corrected for systematic offsets (see Section 4), we therefore
calculate a gas depletion timescale for MRG-S0851 to be
M${}_{\mathrm{H_{2}}}$/SFR < 0.6 Gyr.

##  4 Comparison to Literature

In order to compare to the LMT data presented herein, we conduct a literature
search to collate a sample of galaxies with measured H2 reservoirs and
SFRs111Compilation available at https://www.astrowhit.com/data-products. Below
we summarize our composite reference sample, including the redshift and sample
selection. The surveys use different SFR indicators owing to heterogeneous
data sets. For all tracers, we homogenize measurements to a Chabrier IMF
(Chabrier 2003) and an assumption of $\alpha_{\mathrm{CO}}$ = 4.4 where
applicable. To trace H2 gas, the surveys have either used CO emission lines or
dust continuum, and target both individual galaxies and stacked samples. When
using CO emission lines, all measurements are in lower transition levels
(CO(2-1), CO(3-2) CO(1-0)). The adopted SFR indicators include: UV+IR SFRs
(based on either 24$\mu$m empirical calibrations or low resolution far-
infrared photometry), extinction corrected H$\alpha$ emission lines (or in one
case [OII]), and SED fitting at ultraviolet to optical wavelengths, and in
some cases extending to the MIR. Several of the surveys follow the Wuyts et
al. (2011) ‘SFR ladder’, prioritizing first FIR photometry for high mass/star-
forming galaxies, then empirically calibrated deeper 24$\mu$m as a proxy for
SFRs for less massive galaxies, and finally parameterized SED modeling of the
ultraviolet/optical light at the lowest sSFRs.

Most surveys of molecular gas in quiescent galaxies have been limited to the
relatively nearby Universe. Most notably, the xCOLD GASS survey did an
extensive survey of galaxies at a redshift of $z\sim$0, including 532 galaxies
selected to span a dynamic range in stellar mass and SFR (Saintonge et al.
2017). xCOLD GASS measurements of CO(1-0) are especially useful, as the sample
serves as an unbiased reference to predict the behavior of the H2 gas
properties at higher redshift.

At intermediate redshifts (0.5 $<z<$ 1.0) between 0.5 and 1.0, we include data
from the following studies: PHIBSS2 survey by Tacconi et al. (2018), Spilker
et al. (2018), Decarli et al. (2016), and Suess et al. (2017). Whereas the
PHIBSS2 survey targets typical star-forming galaxies, Spilker et al. (2018)
and Suess et al. (2017) both target intermediate redshift quiescent galaxies.

For redshifts ranging from 1.0 <$z$ <1.5, we include data from nine different
studies. Sargent et al. (2015); Hayashi et al. (2018); Decarli et al. (2016)
and Daddi et al. (2010) are all studies using CO(2-1) to trace molecular
hydrogen. Sargent et al. (2015) presented a measurement for one quiescent
galaxy. Hayashi et al. (2018) present 18 gas-rich galaxies at $z$ = 1.46,
including one that is otherwise quiescent. Daddi et al. (2010) include three
star-forming galaxies. We include two starbursting galaxies from Silverman et
al. (2015). We include 37 star-forming galaxies from the PHIBSS2 survey
(Tacconi et al. 2018). We also include galaxies from Saintonge et al. (2013)
and Scoville et al. (2016), who utilize data from the IRAM Plateau de Bure
Interferometer and Herschel PACS and SPIRE to study the dust and gas content.
Finally, we include galaxies from the ASPECS survey (Aravena et al. 2019) and
GOODS-ALMA (Franco et al. 2020).

For redshifts between 1.5 to 2.5, we include data from 14 different studies.
From Saintonge et al. (2013); Daddi et al. (2010); Silverman et al. (2015);
Decarli et al. (2016); Scoville et al. (2016); Franco et al. (2020); Tacconi
et al. (2018) and Aravena et al. (2019), we include galaxies spanning all
classifications. Data collected from Spilker et al. (2016) includes compact
star-forming galaxies with low gas fractions. Rudnick et al. (2017) includes
cluster galaxies at $z$=1.62 with substantial gas reservoirs. Gobat et al.
(2018) present a stacking analysis of 977 quiescent galaxies at $z\approx 1.8$
at far-infrared to sub-mm wavelengths by deblending _Herschel_ and SCUBA2
photometry to estimate the molecular gas content. From Zavala et al. (2019) we
include protocluster galaxies at $z$=2.1, some of which are gas poor and in a
pre-quenching phase. Lastly, Bezanson et al. (2019) place a strong upper-limit
on the H2 gas mass of a quiescent galaxy at $z$ = 1.522.

Non-parametric SFHs are known to yield systematically lower sSFRs (Leja et al.
2019); to ensure an apples-to-apples literature comparison, we thus hereafter
correct the stellar mass of MRG-S0851 by -0.15 dex and the SFR by +0.15 dex,
yielding a +0.3 dex shift in sSFR following Leja et al. (2019). Even with this
more conservative systematic correction, our conclusions remain unchanged.

##  5 Results

In Figure 2, we show log(SFR) as a function of log(M⋆) for four different
redshift bins in the range $0.0<z<2.5$. Given the corrected specific SFR,
-9.9${}_{-0.05}^{+0.07}$ yr-1, MRG-S0851 is 1.0 dex below the average
log(SFR)-log(M⋆) relation at $z=$1.88, and thereby classified as a quiescent
galaxy. As relatively few observations exist for galaxies below the main
sequence at $z>1$, our new measurement contributes to poorly explored
parameter space.

Figure 3: This figure shows the gas fraction
$M_{\mathrm{H_{2}}}/M_{\mathrm{\star}}$ as a function of redshift, color coded
by the average log(sSFR) with a minimum of one galaxy per bin. Our target
MRG-S0851 has the strongest limits placed to date for a quiescent galaxy at
$z$>1.6. The thick black line is the gas fraction redshift main sequence
adopted from Carilli & Walter (2013). The thin black line represents the
combination of the measured star formation history of MRG-S0851 with empirical
molecular gas scaling relations from Tacconi et al. (2018). Comparisons to
analytical equilibrium “bathtub models” by Davé et al. (2012) (dotted/dashed
lines) demonstrate that only the most massive halos reach similarly low gas
fractions by $z\sim 2$.  Figure 4: Depletion timescale
(M${}_{\mathrm{H_{2}}}$/SFR) as a function of redshift (left) and distance
from the average log(SFR)-log(M⋆) relation ($\Delta$log(SFR); right), color
coded by sSFR. Evolutionary tracks for MRG-S0851 (thin black line) are
inferred from the combination of the measured star formation history and
empirical molecular gas scaling relations (Tacconi et al. 2018) The depletion
timescale for MRG-S0851 is an upper limit of 0.6 Gyr. The gas depletion
timescale of this galaxy is relatively short, similar to star-forming galaxies
at the same epoch (left), and a factor of 3 shorter than predicted by
empirical relations.

Figure 3 shows the redshift evolution of the molecular gas fraction, defined
as $f_{H_{2}}$ = ${M_{\mathrm{H_{2}}}/M_{\mathrm{\star}}}$. While the average
$f_{H_{2}}$ in galaxies increases with redshift, MRG-S0851 has a relatively
low gas fraction of $<$6.8% compared to co-eval galaxies of similar stellar
mass. For context, we compare to the predictions of analytical “bathtub"
models that balance gas inflow, outflow, and star formation in galaxies (Davé
et al. 2012, see Section 6 for more discussion). We additionally use the
detailed star formation history for MRG-S0851 measured in Akhshik et al.
(2020a) to explore its preceding evolution in gas fraction. The thin black
line shows the gas fraction inferred using the empirical scaling relation of
Tacconi et al. (2018) as a function of the projected past stellar mass and SFR
of MRG-S0851, including systematic corrections as described in Section 4. The
late-time increase in gas fraction reflects the rejuvenation in the SFH,
presenting a tension with our strong upper limit on $f_{H2}$ and suggesting
relatively rapid gas consumption.

The molecular gas depletion timescale, $M_{\mathrm{H2}}$/SFR, as a function of
the redshift (left) and difference in SFR relative to the average
log(SFR)-log(M⋆) relation ($\Delta$log(SFR); right) are shown in Figure 4. Our
value is 1.0 dex below the SFR-M∗ Main Sequence. While many quiescent galaxies
have relatively low gas masses, they have similarly low SFRs which drive the
longer depletion times. MRG-S0851 has a low gas fraction limit with an
intermediate sSFR owing to the recent rejuvenation episode, which together
results in a relatively short depletion time upper limit of 0.6 Gyr. Our
measured short depletion time is in line with recent observations that suggest
that scaling relations overestimate $f_{H2}$ and depletion time in the poorly-
explored regime of low sSFR and high redshift (Spilker et al. 2018; Williams
et al. 2020). Akhshik et al. (2020a) estimate that less than 1% of the stellar
mass (0.5$\pm$0.1%) formed in the last 100 Myr as a result of this central
rejuvenation. Given the depletion timescale of $<$0.6 Gyr, if MRG-S0851 no
longer accretes fresh H2 gas, by $z\sim 1.6$ the new stars will have evolved
and faded such that this target likely returns to resembling a compact, old
quiescent galaxy. To agree with the general trend in the right panel of Figure
4, where galaxies that lie below the average log(SFR)-log(M⋆) relation have
significantly longer depletion times, the depletion timescale for MRG-S0851
would need to increase significantly with time.

##  6 Discussion

Studying the molecular gas reservoirs in quiescent galaxies can provide
insight into the formation and evolution of massive galaxies. While very few
constraints exist for quiescent galaxies at $z>0$, there are observations of
quiescent systems having surprisingly large H2 reservoirs (e.g. Suess et al.
2017; Gobat et al. 2018), however, this may not be the norm. This study
provides the strongest limits to date of a quiescent galaxy at $z>1.6$ using
1.1mm imaging with AzTEC. At $z=1.883$, we constrain the gas fraction of
MRG-S0851 to $<$6.8%. Results from similar galaxies range from $f_{gas}$ of
$<$2-10% (Williams et al. 2020; Bezanson et al. 2019; Sargent et al. 2015;
Zavala et al. 2019), to 16% (Gobat et al. 2018) to 40% (Rudnick et al. 2017;
Hayashi et al. 2018), indicating a diversity in molecular gas contents for
high redshift quiescent galaxies.

While the large scatter in observed molecular gas fractions may reflect the
intrinsic variation within the quiescent population, it is also possible that
the adopted methodologies are contributing to the apparent scatter. There are
two differing methodologies for quantifying molecular gas and each have their
respective uncertainties. In the case of dust tracing gas mass, Gobat et al.
(2018) perform the first statistical study of the H2 gas properties of
quiescent galaxies at a similar redshift to our target. However, this study
used far-IR to sub-mm data from the _Herschel Space Telescope_ and SCUBA2
whose large PSF complicates the analysis, with deblending issues lending
considerable uncertainty to the analysis. While the dust continuum methodology
can efficiently push to deeper sensitivity limits, uncertainties remain on the
applicability of the calibrations presented in Scoville et al. (2016) for low
sSFR galaxies. The calibrations in Scoville et al. (2016) are based on data
for galaxies in the local Universe, with studies showing that they hold on
average for star-forming galaxies at $z\sim 2$ (Kaasinen et al. 2019).
However, Li et al. (2019) show that the dust-to-gas ratios at low sSFRs can
vary dramatically (see also Privon et al. 2018). Perhaps the reason we find
f${}_{H_{2}}$ is low for MRG-S0851 is rather that the assigned dust-to-gas
ratio is too high. While adopting the CO methodology would better constrain
this ratio, uncertainties in the $\alpha_{\mathrm{CO}}$ conversion factor
remain a source of much debate (Hodge & da Cunha 2020; Tacconi et al. 2020).

MRG-S0851 has a rapid depletion time with an upper limit of 0.6 Gyr, about 10
times faster than similar quiescent galaxies at $z$=0. This implies that gas
stabilization from collapse, a mechanism which has been invoked by Suess et
al. (2017) to explain high gas fractions, does not play a key role, at least
for MRG-S0851. As discussed more extensively in Williams et al. (2020), the
most massive halos (those that grow to $>10^{14}$ by $z=0$) could instead
reach equivalently low gas fractions at $z\sim 2$ by passing a critical mass
threshold truncating gas accretion at early times. Abundance matching suggests
MRG-S0851 is hosted in a dark matter halo of $\sim 10^{12.6}$ M⊙ at $z\sim 2$
(Behroozi et al. 2010), which would naturally increase by a factor of 10 by
$z\sim 0$ (i.e., midway between the two analytical models in Figure 3). Some
tension of our low $f_{H2}$ measurement is therefore mitigated if MRG-S0851
resides in a more massive halo (e.g., dashed line in Figure 3), but only if
the recently accreted gas is rapidly depleted.

It is interesting to note that the low gas fractions found by Spilker et al.
(2016) at $z\sim 2-2.5$ are for similarly compact galaxies (e.g., effective
surface densities of 7.8x$10^{9}$-3.18x$10^{10}M_{\odot}$kpc-2), but these
targets are still actively forming new stars (see Figures 3 and 4). The
depletion times in Spilker et al. (2016) are analogous to MRG-S0851 in the
sense that they are both outliers among their star formation class. Perhaps
their high stellar density increases the star-formation efficiency, as
Bezanson et al. (2019) suggest.

The flat age gradient measured by Akhshik et al. (2020a) and the compactness
of MRG-S0851 are consistent with an early-formation pathway. In the early-
formation scenario, a galaxy forms the bulk of its stellar mass early in the
Universe, consumes its gas and quenches (Williams et al. 2014; Wellons et al.
2015). MRG-S0851 started forming stars between $z\sim$9 to $z\sim$3 and
quenched by $z$=2. MRG-S0851’s rejuvenation phase occurred between $z$=2 and
$z$=1.88 (epoch of observation). Tentative evidence suggests that MRG-S0851
therefore accreted pristine gas at $z$=2, fueling a short-lived reignition.

This letter presents only the first measurement of an individual quiescent
galaxy at $z\sim 2$ and we caution that this may not be representative of the
quiescent population. Continuing to study the molecular gas content of
quiescent galaxies using a range of methodologies at high redshift is
important both in understanding the basic gas property demographics of this
population, as well as to test limitations in methodologies. A larger sample
size is also needed to determine how rapidly massive galaxies deplete and/or
replenish their molecular gas and the mechanisms by which star formation halts
at early times. The upcoming Legacy surveys with the TolTEC instrument on the
50m LMT will cover large extragalactic fields allowing for dust mass estimates
for thousands of quiescent galaxies out to $z\sim 4$ with a spatial resolution
of 5$\farcs$5\. Specifically, the Ultra-Deep Survey will cover 0.8 sq. deg in
the CANDELS fields to a detection limit of 0.1mJy (4$\sigma$) at 1.1mm. We
estimate that there are $\sim$2,000 quiescent galaxies above a stellar mass of
$\log{M_{*}}/M{{}_{\odot}}$ = 10 at $0<z<4$ in CANDELS, and we can constrain
their gas fractions individually as low as 2% for more massive galaxies or
equivalently low through stacking analyses. Such a survey provides the
required sensitivity and sample statistics to facilitate a revolution in our
understanding of the role of gas fueling in truncating SF in massive galaxies.

We would like to thank the anonymous referee for sharing valuable comments
that helped improve the letter significantly. K.W. wishes to acknowledge
contributions made by Warren Sharpp to help shape this project. We gratefully
acknowledge support by NASA under awards No80NSSC19K1418, HST-GO-14622, and
HST-GO-15663. This work is based on observations made with the NASA/ESA Hubble
Space Telescope, obtained at the Space Telescope Science Institute, which is
operated by the Association of Universities for Research in Astronomy, Inc.,
under NASA contract NAS 5-26555. Financial support for M.A. and K.W. is
gratefully acknowledged. K.W. wishes to acknowledge funding from the Alfred P.
Sloan Foundation. The Cosmic Dawn Center is funded by the Danish National
Research Foundation under grant No. 140. C.C.W. acknowledges support from the
National Science Foundation Astronomy and Astrophysics Fellowship grant
AST-1701546 and NIRCam Development Contract NAS50210 from NASA Goddard Space
Flight Center to the University of Arizona. AM thanks the support from Consejo
Nacional de Ciencia y Technología (CONACYT) project A1-S-45680.

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# Feasibility of Detecting Interstellar Panspermia in Astrophysical
Environments

Claudio Grimaldi  Laboratory of Statistical Biophysics, Ecole Polytechnique
Fédérale de Lausanne - EPFL, 1015 Lausanne, Switzerland  Centro Studi e
Ricerche Enrico Fermi, 00184 Roma, Italy  Manasvi Lingam  Department of
Aerospace, Physics and Space Sciences, Florida Institute of Technology,
Melbourne, FL 32901, USA  Amedeo Balbi  Dipartimento di Fisica, Universita‘ di
Roma “Tor Vergata”, 00133 Roma, Italy

###### Abstract

The proposition that life can spread from one planetary system to another
(interstellar panspermia) has a long history, but this hypothesis is difficult
to test through observations. We develop a mathematical model that takes
parameters such as the microbial survival lifetime, the stellar velocity
dispersion, and the dispersion of ejecta into account in order to assess the
prospects for detecting interstellar panspermia. We show that the correlations
between pairs of life-bearing planetary systems (embodied in the pair-
distribution function from statistics) may serve as an effective diagnostic of
interstellar panspermia, provided that the velocity dispersion of ejecta is
greater than the stellar dispersion. We provide heuristic estimates of the
model parameters for various astrophysical environments, and conclude that
open clusters and globular clusters appear to represent the best targets for
assessing the viability of interstellar panspermia.

##  1 Introduction

The modern conception of ‘lithopanspermia’, i.e. the idea that life can travel
across planetary systems carried by meteoroids and other minor bodies, dates
back to at least the 19th century (Kamminga, 1982), but it has regained
serious consideration in scientific discussions (Melosh, 1988; Wesson, 2010;
Wickramasinghe, 2010). This revival was motivated by a number of developments.
First, data concerning the resistance of radiation-tolerant organisms in deep
space, as well as experimental tests of hypervelocity impacts, show that it is
plausible that some organisms can survive the accidental transfer from an
inhabited planet to another location (Horneck et al., 2010; Onofri et al.,
2012; Merino et al., 2019). Second, there is now strong evidence that rock
fragments have in fact been exchanged between nearby planets in the Solar
System (Nyquist et al., 2001) and it is conceivable that similar mechanisms
can be even more efficient in more densely packed planetary systems, such as
the TRAPPIST-1 system (Lingam & Loeb, 2017; Krijt et al., 2017). Finally, the
direct observation of at least two objects of interstellar origin transiting
the Solar Systems has confirmed that the exchange of material is possible even
between different planetary systems (Meech et al., 2017; Guzik et al., 2020).

The actual feasibility of lithopanspermia on interstellar distances has been
addressed in the past, and shown to be achievable, at least in principle
(Zubrin, 2001; Wallis & Wickramasinghe, 2004; Napier, 2004; Ginsburg et al.,
2018; Siraj & Loeb, 2020), especially in crowded environments, such as in
star-forming clusters (Adams & Spergel, 2005; Valtonen et al., 2009; Belbruno
et al., 2012) or in the Galactic bulge (Chen et al., 2018; Balbi et al.,
2020); furthermore, its probability can be significantly enhanced by
interactions with binary systems (Lingam & Loeb, 2018). Although the issue is
still debated and, admittedly, rather speculative, there are strong
theoretical motivations to explore how a viable lithopanspermia mechanism
could impact the distribution of life in the Galaxy, particularly if compared
to the case where life originates independently in separate locations.

There are at least two major implications that warrant a careful analysis of
the problem (Balbi, 2021). First, lithopanspermia may change the estimated
extent of the Galactic Habitable Zone (GHZ), since it can act as a
compensating factor with respect to potentially catastrophic events that can
eradicate life from planetary surfaces (Balbi et al., 2020). Second, it can
alter the statistical impact and interpretation of future biosignature
detections via surveys of nearby planetary systems as shown by Balbi &
Grimaldi (2020).

Therefore, we argue it is timely to develop models that can shed some light on
the consequences of interstellar lithopanspermia, assuming it is a practical
mechanism. In this Letter, we present a theoretical treatment of the expected
statistical distribution of inhabited planets in a lithopanspermia scenario,
with a particular focus on its correlation properties. Our work generalizes
and quantifies the prior analyses by Lin & Loeb (2015) and Lingam (2016a, b),
which dealt with this topic.

##  2 Formalism

To model the spreading of life over interstellar distances, we suppose that
impacts of meteoroids and other minor celestial bodies on rocky planets
collectively engender a steady ejection rate of matter escaping the
gravitational pull of the host star. This assumption has been employed in
several publications on interstellar lithopanspermia. Melosh (2003) predicted,
for example, that $\sim 15$ rocks of size $>10$ cm originating from impacts on
inner planets exit the Solar System each year on average. On the other hand,
when it comes to micron-sized fragments, they are thought to be expelled at an
average rate of $\gtrsim 10^{14}$ per year (Napier, 2004). While the ejection
rate is likely to have been boosted during an epoch of intense bombardment as
a result of the higher impact rates (Adams & Spergel, 2005; Belbruno et al.,
2012), our model does not necessitate knowledge of the exact value of this
parameter, as long as it is not infinitesimally small.

A fraction of the rocks ejected from planets harboring a biosphere may
encapsulate simple forms of life which, provided that they are appropriately
shielded from UV and ionizing radiation as well as desiccation and other
extremes, can possibly survive in space over timescales of millions of years
(Mileikowsky et al., 2000; Lingam & Loeb, 2021). On a related note, as per
some (rather controversial) studies, Earth-based microbes might be capable of
survival for intervals as high as $\sim 100$ Myr under suitable conditions
(Cano & Borucki, 1995; Vreeland et al., 2000; Morono et al., 2020).

A world harboring a biosphere could thus be envisioned as being surrounded by
a spherical region of radius $R$ sparsely populated by life-bearing ejecta
(Wallis & Wickramasinghe, 2004). Under favorable circumstances, these objects
could potentially seed life on an initially sterile planet located within such
spheres of influence, each of which is termed a _Lebenssphäre_ (life-sphere).
We roughly estimate the average radius of a Lebenssphäre by $R=\langle
v_{o}\rangle\tau_{o}$, where $\tau_{o}$ is the mean lifetime of the microbial
populations encapsulated in the ejecta and $\langle v_{o}\rangle$ is the mean
velocity of the objects in the local frame of the Lebenssphäre. By adopting a
Maxwellian velocity distribution with dispersion $\sigma_{o}$, the average
radius can be expressed as follows:

$R=\sqrt{8/\pi}\sigma_{o}\tau_{o}\simeq\textrm{5.3
ly}\left(\frac{\sigma_{o}}{\textrm{1 km
s${}^{-1}$}}\right)\left(\frac{\tau_{o}}{\textrm{1 Myr}}\right),$ (1)

yielding, for example, $R\simeq 50$-$100$ ly in the event $\sigma_{o}=10$-$20$
km s-1 and $\tau_{o}$ as large as $1$ Myr. A sphere of influence of this size
located in the solar neighborhood would contain $\sim 10^{3}$-$10^{4}$ stars
and a comparable number of rocky planets, while it would accommodate a number
of stars up to four orders of magnitude greater if located in the Galactic
bulge. We further note that a seeding sphere of $\sim 50$ ly in radius is much
smaller than most typical length scales in the Galaxy. This allows us to
consider volume samples of the Galaxy of linear size larger than $R$ but, at
the same time, sufficiently small that the distribution of stars in the sample
may be treated as homogeneous and isotropic.

If a planet to which life has been successfully transferred through
lithopanspermia can sustain an enduring biosphere, it will after some time
develop a new Lebenssphäre surrounding it, which will potentially seed life on
other planets in turn. We make the ostensibly reasonable hypothesis that this
mechanism engenders Lebenssphären with a constant birthrate per planet,
denoted by $\tau^{-1}$, whereas Lebenssphären formed by spontanous and
independent abiogenesis events grow at a constant rate $\gamma$ per unit
volume. Furthermore, we assume that the proliferation of life-harboring
planets is counterbalanced by random sterilizing events such as nearby
supernovas or other cataclysms (Gowanlock & Morrison, 2018), so that within a
sample volume there exists a constant number density $\rho$ of Lebenssphären
formed around planets on which life arose spontaneously or by lithopanspermia.
As shown in Appendix A, such steady-state regimes are reached as long as the
inequality $L/\tau<1$ holds true, where $L$ denotes the average persistence
lifetime of a Lebenssphäre, consequently yielding $\rho=\gamma L/(1-\tau/L)$.

Figure 1: Schematic illustration of Lebenssphären (circles) centered around
life-bearing planets (points). Case a: Independent abiogenesis results in a
uniform spatial distribution of Lebenssphären. Their centers are spatially
uncorrelated and the pair distribution function is $g(r)=1$. Case b: When
panspermia sets in and the stellar velocities are minimal, the Lebenssphären
are aggregated in clusters formed by life-bearing planets at pair distances
smaller than $R$. In this scenario, $g(r)$ peaks at $r\lesssim R$.

In the following statistical analysis, it is mathematically convenient to
first evaluate the static limit where all the stars in the sample have fixed
positions that do not change over time; this is not a bad assumption as long
as the stellar velocity dispersion is sufficiently low. The Lebenssphären,
which we assume all possess the same mean radius $R$ for simplicity, will
therefore have their center at given immutable positions. The average number
of the centers of Lebenssphären within a distance $R$ from a randomly chosen
seeding world is:

$N(R)=4\pi\rho\int_{0}^{R}\\!dr\,r^{2}g(r)$ (2)

where $g(r)$ represents the pair-distribution function defined in such a way
that $\rho^{2}g(r)$ leads to the probability distribution function associated
with two Lebenssphäre centers being situated at a distance $r$ from each
other. If we suppose that life arose independently on different planetary
systems (i.e., independent abiogenesis), the Lebenssphäre centers would be
spatially uncorrelated, as schematically shown in Figure 1a. In this case, it
follows that $g(r)=1$ and therefore

$N(R)=\frac{4}{3}\pi\rho R^{3}\equiv\eta.$ (3)

We now suppose that a panspermia mechanism along the lines described
previously is able to spread life. In the static limit, a planet fertilized
through panspermia (which in turn develops its own Lebenssphäre) must be
located at a distance $r\leq R$ from the center of a Lebenssphäre, as
illustrated in Figure 1b. The coordination number $N(R)$ must therefore be
larger than its value in the absence of panspermia, Eq. (3), because there now
exists, within $R$, a higher probability of finding a planet to which life has
been transferred:

$N(R)=4\pi\rho\int_{0}^{R}\\!dr\,r^{2}g(r)=\chi\eta,$ (4)

where $\chi\geq 1$ accounts for the enhanced population of life-bearing
planets located within $R$; we hereafter christen $\chi$ the panspermia
amplification factor (PAF). Equation (4) simply states that, compared to the
case of spontaneous and independent abiogenesis, the pair-distribution
function is accordingly enhanced in the neighborhood of a seeding planet. We
specify $g(r)=\chi$ for $r\leq R$, which amounts to assuming that the
probability of seeding a planet is uniform for $r\leq R$. Although a more
realistic ansatz is feasible, we wish to primarily explicate the qualitative
features of this model.

Figure 2: Pair-distribution function $g(r)$ calculated in the static limit for
$\eta=0.4$ and different values of the panspermia intensity parameter $\chi$.
Inset: log-linear plot of the pair-connectedness function demonstrating that
$P(r)$ decays exponentially with $r$ when the system is below the percolation
threshold $\eta\chi=4$. At the critical point, $P(r)\propto 1/r$.

At greater distances, we expect the pair-distribution function to vary on a
length scale of order $\xi$, eventually reaching the uniform limit $g(r)=1$
for $r\gg\xi$, where $\xi$ measures the typical size of clusters formed by
life-bearing planets at pair distances smaller than $R$. The spatial
distribution of such “connected” planets is given by the so-called pair-
connectedness function $P(r)$, defined such that $\rho^{2}P(r)$ gives the
probability distribution function associated with finding two Lebenssphäre
centers separated by a distance $r$ and belonging to the same cluster
(Torquato, 2002). From this definition it follows that $P(r)=g(r)$ for $r\leq
R$, while $P(r)$ decays exponentially over distances larger than $\xi$, when
no giant cluster of connected planets emerges; in the language of percolation
theory, this regime amounts to being below the percolation threshold (Stauffer
& Aharony, 1994).

The above considerations suggest the following ansatz:

$g(r)=1+\frac{\chi-1}{\chi}P(r),$ (5)

from which we recover the equality $g(r)=\chi\geq 1$ for $r\leq R$, which we
had posited earlier. To find $g(r)$ for $r>R$ we calculate $P(r)$ by following
standard techniques developed in continuum percolation theory and briefly
described in Appendix B. Figure 2 depicts $g(r)$ calculated for $\eta=0.4$ and
for values of the PAF ranging from $\chi=1$ (no panspermia) up to $\chi=10$,
at which point the system reaches the percolation threshold $\eta_{c}\chi=4$
(Appendix B.2). Aside from the enhanced $g(r)$ for $r\leq R$, we see from
Figure 2 that the higher is the PAF (i.e., larger $\chi$), the greater is the
distance above which $g(r)\rightarrow 1$ is applicable.

This can be understood by noticing that the correlation length scales as
$\xi\propto\eta\chi/(1-\eta\chi)$; see Eq. (B13) of Appendix B.3, which
increases monotonically with $\chi$ and eventually diverges at the percolation
threshold of $\eta\chi=4$. At this point, $P(r)$, and therefore $g(r)-1$ due
to Eq. (5), decays as a power-law (inset of Figure 2). It is worth pointing
out that our panspermia model predicts a critical density of percolation that
is inversely proportional to $\chi$ – specifically, $\eta_{c}=4/\chi$ as shown
in Appendix B.2 – meaning that a sufficiently high PAF is capable of inducing
an entire sample-spanning transfer of life even if life-bearing planets are,
overall, very rare.

The ansatz (5) is strictly applicable only to the subcritical regime wherein
$\eta<\eta_{c}$. Once we exceed the percolation threshold, characterized by
the formation of a giant cluster (Stauffer & Aharony, 1994), we expect that
the pair distribution function generated by the panspermia process would
gradually become less peaked as $\eta$ increases beyond the percolation
threshold, eventually tending toward $g(r)=1$ for $\eta\gg 1/\chi$. In this
limit, indeed, the Lebenssphären would essentially cover the entire sample
volume in a uniform fashion.

Figure 3: Pair-distribution function $g(r)$ calculated for $\eta=0.4$,
$\chi=10$ in the case of non-zero stellar velocity dispersion $\sigma_{s}$.
The parameter $\sigma_{r}=\sqrt{2}\sigma_{s}\tau$ measures the dispersion of
the relative separation of Lebenssphäre centers; note that $\sigma_{r}=0$
corresponds to the static limit. As $\sigma_{r}$ increases, the discontinuity
of $g(r)$ at $r=R$ is smeared out and the peak at $r\lesssim R$ is weakened.
For $\sigma_{r}\gtrsim R$ the pair distribution function becomes flat as in
the uncorrelated case. Inset: Correlation length $\xi$ plotted as a function
of $\sigma_{r}/R$.

We will now extend the static panspermia mechanism discussed above to the more
realistic case in which stars hosting planetary systems have nonzero relative
velocities. Intuitively, even if the static panspermia model may evince
pronounced spatial correlations, we would expect them to be nevertheless
weakened by the accompanying stellar motion. To explore this effect, we note
that for Lebenssphären that are generated via panspermia at a constant rate
$\tau^{-1}$, the position of a newly formed Lebenssphäre relative to that of
the seeding planet will be translated by $\sim\bm{v}\tau$, where $\bm{v}$ is
the relative velocity of the two planetary systems. We capture this effect by
rewriting the ansatz (5) as:

$g(r)=1+\frac{\chi-1}{\chi}\int\\!d\bm{v}\,f(\bm{v})P(|\bm{r}+\bm{v}\tau|),$
(6)

where $f(\bm{v})$ is the distribution function of the relative velocities.
Assuming that in the local frame of the sample volume the stars move in random
directions with velocities obeying a Maxwell distribution with dispersion
$\sigma_{s}$, Eq. (6) can be recast in the following form:

$g(r)=1+\frac{\chi-1}{\chi}\\!\int\\!d\bm{r}^{\prime}\,\frac{1}{(2\pi\sigma_{r}^{2})^{3/2}}\exp\\!\left(\\!-\dfrac{|\bm{r}-\bm{r}^{\prime}|^{2}}{2\sigma_{r}^{2}}\right)P(r^{\prime}),$
(7)

where $\sigma_{r}=\sqrt{2}\sigma_{s}\tau$ measures the dispersion of the
relative distance between two centers of Lebenssphären. Clearly, for
$\sigma_{r}\rightarrow 0$ we recover the ansatz (5) while for $\sigma_{r}\gg
R$ the pair-distribution of the Lebenssphäre centers reaches the uniform limit
as $g(r)\rightarrow 1+\mathcal{O}(R^{3}/\sigma_{r}^{3})$.

We show the crossover from the correlated to the uncorrelated regime in Figure
3, where we report the numerical calculations of $g(r)$ obtained by self-
consistently solving Eq. (7) and the integral equation of $P(r)$ (Appendix
B.1). The enhancement of $g(r)$ for $r\leq R$ is rapidly weakened as
$\sigma_{r}$ increases and, simultaneously, the correlation length declines
towards the uncorrelated limit (inset of Figure 3). In fact, even for
$\sigma_{r}/R\approx 1$ the pair-distribution function is essentially
indistinguishable from that of a fully uncorrelated system; the latter refers
to the case with $\chi=1$ in Figure 2.

For $\sigma_{r}/R\gtrsim 1$, therefore, the Lebenssphäre centers are uniformly
distributed, thereby hindering our ability to infer any information about the
extent and intensity of panspermia events if one were to simply measure the
distribution of life-bearing planets in a given sample volume through an
appropriate survey.

##  3 Discussion

Table 1: Heuristic estimates of model parameters for various astrophysical
environments

System | $\sigma_{o}$ | $\sigma_{s}$ | $\tau_{o}$ | $\tau$ | $R$ | $\sigma_{r}$ | $\sigma_{r}/R$ | $\rho_{\star}$ | $\zeta_{c}$
---|---|---|---|---|---|---|---|---|---
Solar neighborhood | $10$ | $20$ | $1$ | $1$ | $53$ | $94$ | $1.8$ | $2.9\times 10^{-3}$ | $2.2\times 10^{-3}$
Globular clusters | $10$ | $5$ | $1$ | $1$ | $53$ | $23.5$ | $0.4$ | $2.9$ | $2.2\times 10^{-6}$
Open clusters | $10$ | $1$ | $1$ | $1$ | $53$ | $4.7$ | $9\times 10^{-2}$ | $0.3$ | $2.2\times 10^{-5}$
Galactic bulge | $10$ | $100$ | $1$ | $1$ | $53$ | $470$ | $8.9$ | $0.3$ | $2.2\times 10^{-5}$
Galactic halo | $10$ | $150$ | $1$ | $1$ | $53$ | $705$ | $13.3$ | $1.4\times 10^{-5}$ | $4.6\times 10^{-1}$

Additional notes: $\sigma_{o}$ and $\sigma_{s}$ are expressed in units of km
s-1, $R$ and $\sigma_{r}$ in units of light years (ly), $\tau$ and $\tau_{0}$
in units of Myr, and $\rho_{\star}$ in units of stars per cubic ly. The
parameters for the above systems were chosen based on the following
references: (i) solar neighborhood (Holmberg et al., 2009), (ii) globular
clusters (Baumgardt & Hilker, 2018), (iii) open clusters (Valtonen et al.,
2009; Foster et al., 2015), (iv) Galactic bulge (Zhu et al., 2017; Balbi et
al., 2020), and (v) Galactic halo (Helmi, 2008).

It is now instructive to carry out some fiducial estimates for select
astrophysical systems, and consequently assess the efficacy of panspermia. Of
the various parameters at play, one of the most ambiguous among them is
$\tau$, which is the time elapsed from a seeding event to the formation of a
Lebenssphäre of radius $R$, owing to which it cannot be smaller than
$\tau_{o}$. If we suppose that life could germinate quickly after the seeding
event and form a biosphere, given habitable conditions, it is conceivable that
$\tau$ is of the same order as $\tau_{o}$, owing to which we will employ the
condition $\tau\gtrsim\tau_{o}$.

One other crucial variable is $\rho$ because it regulates $\eta$, as seen from
Eq. (3). Given that $\rho$ is the number density of Lebenssphären, this is not
an easy quantity to gauge since it requires us to know the fraction of
planetary systems with habitable worlds on which life blossomed into a
biosphere ($\zeta_{\star}$); in essence, therefore, it corresponds to two
weakly constrained factors in the Drake equation (Drake, 1965). As per our
postulate, we have $\rho=\zeta_{\star}\rho_{\star}$, where $\rho_{\star}$ is
the stellar density. In the static limit, we have shown that the percolation
threshold is $\eta_{c}=4/\chi$ in Appendix B.2, which implies that
$\eta_{c}<4$ because $\chi>1$ by construction. It is therefore feasible to
determine an upper bound on $\zeta_{\star}$ (denoted by $\zeta_{c}$) in order
for $\eta$ to stay below the percolation threshold and in the subcritical
regime from Eq. (3) as follows:

$\zeta_{c}=\frac{3}{\pi\rho_{\star}R^{3}}$ (8)

If $\rho_{\star}$ and/or $R$ are sufficiently small, then $\zeta_{c}>1$ is
mathematically possible, but not physically realizable because we have defined
$\zeta$ such that it is smaller than unity. Hence, environments with
$\zeta_{c}<1$ are liable to always exist in the subcritical regime with
$\eta<\eta_{c}$. Estimating the required value of $\zeta_{c}$ is helpful
because it functions as a heuristic signpost for determining whether we are
below or above the percolation threshold. In the latter scenario, as explained
in Sec. 2, the panspermia correlations manifested in the pair-distribution
function could become indistinguishable if $\eta$ is high.

At this juncture, we emphasize that some of our ensuing results are not
sensitive to $\eta$. For instance, the crucial ratio of $\sigma_{r}/R$ is
expressible as

$\frac{\sigma_{r}}{R}=\frac{\sqrt{\pi}}{2}\left(\frac{\sigma_{s}}{\sigma_{o}}\right)\left(\frac{\tau}{\tau_{o}}\right),$
(9)

which implies that a sufficient condition for $\sigma_{r}/R\gtrsim 1$ to hold
true is $\sigma_{s}\gtrsim 1.13\,\sigma_{o}$; this relation follows after
utilizing $\tau\gtrsim\tau_{o}$ from the preceding paragraph. Thus, broadly
speaking, if the stellar velocity dispersion is higher than that of the
ejecta, the panspermia correlations are effectively washed out. If the
opposite is true, then the correlated regime might be manifested, enabling us
to discern panspermia through observations, but only provided that $\tau$ is
roughly comparable to $\tau_{o}$.

In what follows, we will suppose that the characteristic dispersion of ejecta
from planetary systems is $\sigma_{o}\sim 10$ km s-1, which is close to the
mean value of $6.2\pm 2.7$ km s-1 for ejection speeds estimated in Adams &
Spergel (2005). However, it should be noted that a small fraction of ejecta
may have speeds of order $0.1$ km s-1 (Belbruno et al., 2012). The estimate
for $\tau_{o}$ is not well understood, since it depends on the size of the
object, among other factors, but survival timescales of $\gtrsim 1$ Myr are
possible for microbial populations in ejecta with sizes of $\gtrsim 1$ m
(Mileikowsky et al., 2000; Valtonen et al., 2009; Lingam & Loeb, 2021) and
intervals of $\sim 10$-$100$ Myr are not impossible (Cano & Borucki, 1995;
Morono et al., 2020); we will therefore specify $\tau_{0}\sim 1$ Myr
hereafter. Although we adopt this fiducial value, our results regarding the
viability of detecting panspermia are sensitive to the ratio $\tau/\tau_{o}$,
as seen Eq. (9), and are consequently not directly dependent on the magnitude
of $\tau_{o}$.

We have presented heuristic estimates for the salient parameters of our model
in Table 1. As $\tau$ is subject to significant uncertainty, we have
considered the limiting case of $\tau\sim\tau_{o}$ herein and adopted the
values delineated in the prior discussion. There are two broad inferences that
can be drawn from this table:

  1. 1.

We find that $\zeta_{c}\ll 1$ is valid for all astrophysical environments
except the Galactic halo. Thus, insofar as the static limit is concerned, the
subcritical regime (with $\eta$ below the percolation threshold) is feasible
when the density of Lebenssphären is very low. To put it differently, if only
a very small fraction of planetary systems develop Lebenssphären in these
astrophysical settings, the subcritical regime is valid.

  2. 2.

It was argued earlier that panspermia correlations are discernible only when
$\sigma_{r}/R\lesssim 1$. We notice from Table 1 that this criterion is
comfortably satisfied only in the case of open stellar clusters, although
globular clusters and the solar neighborhood are not far removed from this
desired limit.

##  4 Conclusions

We presented the results of a mathematical model describing the dissemination
of life over interstellar distances through lithopanspermia. The model depends
on a number of parameters that are known to varying degrees of accuracy, but
all of them can be empirically constrained in principle.

We have focused on the predicted correlation properties of life-bearing
planetary systems. Our calculations show that an active panspermia process
could lead to a distinct amplification of the population of life-bearing
planets within a certain characteristic distance compared to the case of
independent abiogenesis. This correlation distance is sensitive to the details
of the lithopanspermia mechanism and is therefore capable of serving as an
observational diagnostic to constrain various scenarios. However, we also
demonstrated that the correlations can become attenuated, or even nullified
altogether, depending on the astrophysical environment under investigation. As
this attenuation is related to dynamical parameters, in particular to the
velocity dispersion of stellar systems, it may be predicted to an extent in
specific settings.

Hence, based on our formalism, we found that stellar clusters are more
promising insofar as detecting the instantiation of panspermia is concerned,
although it might still be discernible in our solar neighborhood. On the other
hand, crowded environments endowed with high stellar dispersions, such as the
galactic bulge, could be so effective at spreading life through
lithopanspermia that they are essentially indistinguishable from the case with
minimal panspermia and a high abiogenesis rate; in other words, the
correlations would be washed out.

Given our findings, at least two further directions are worthy of pursuit in
future investigations. First, our work can be expanded and refined by carrying
out detailed numerical simulations and proceeding beyond some of the
simplifying assumptions adopted, most notably the criterion of being below the
percolation limit. Such numerical simulations are expected to yield further
insights concerning the feasibility of distinguishing between independent
abiogenesis and panspermia in a given astrophysical system, and what number of
inhabited worlds need to be detected toward this end. From an observational
standpoint, it has been suggested that confirming panspermia at $5\sigma$
confidence requires the sampling of $>25$ life-bearing worlds in the optimal
scenario (Lin & Loeb, 2015).

It is necessary, in the same vein, to investigate the relative weights of
proliferation versus sterilization as these process govern the prevalence of
life in planetary systems, as shown in Appendix A. Second, as a separate line
of inquiry, experimental studies of the viability of panspermia would pave the
way for estimating some of the parameters involved (e.g., survival time of
extremophile populations in space), consequently evaluating the prospects of
detecting life on other planets and gauging the feasibility of interstellar
panspermia.

The authors thank the reviewer for the constructive feedback, which was
helpful for improving the paper. A.B. was partially funded by the Italian
Space Agency through the Life in Space project (ASI N. 2019-3-U.0) and by
grant number FQXi-MGA-1801 and FQXi-MGB-1924 from the Foundational Questions
Institute and Fetzer Franklin Fund, a donor advised fund of Silicon Valley
Community Foundation.

##  Appendix A Rate equation for the density of Lebenssphären

We make the assumption that within a given volume sample there is a number
density $\rho_{0}$ of planets on which life arose spontaneously and
independently. If we denote by $\gamma$ the rate of formation per unit volume
of Lebenssphären created by spontaneous abiogenesis and $L$ embodies their
typical lifetime, the rate equation for $\rho_{0}$ reads

$\frac{d\rho_{0}}{dt}=\gamma-\frac{\rho_{0}}{L}.$ (A1)

In the presence of panspermia, however, there will be an additional number
density $\rho_{p}$ contribution arising because life has been transferred and
new Lebenssphären have been accordingly formed. Given that life can be
transferred from planets on which life arose spontaneously or via prior
panspermia events, the corresponding rate equation is

$\frac{d\rho_{p}}{dt}=\frac{\rho_{0}+\rho_{p}}{\tau}-\frac{\rho_{p}}{L}$ (A2)

where $\tau^{-1}$ is the formation rate of Lebenssphären (per planet) due to
panspermia and in the last term we have presumed that sterilizing events
affect $\rho_{p}$ to effectively the same degree as $\rho_{0}$. The rate
equation for $\rho=\rho_{0}+\rho_{p}$ is therefore given by

$\frac{d\rho}{dt}=\gamma+\frac{\rho}{\tau}-\frac{\rho}{L}.$ (A3)

The above differential equation admits two types of solutions depending on
whether the formation rate of Lebenssphären is greater or smaller than the
rate of sterilizing events. The solution for $L/\tau>1$ gives rise to a number
density of life-harboring planets which increases exponentially with time; the
special limit of $L/\tau=1$ would result in the number density growing
linearly with time. In these cases, the rate of panspermia events is such that
all habitable planets within the sample volume eventually develop a biosphere,
thence leading to a spatially uniform distribution of Lebenssphären. On the
contrary, when it comes to $L/\tau<1$, the solution of Eq. (A3) for $t\gg
L/(1-L/\tau)$ reaches the steady state regime in which $\rho$ is finite and
expressible as

$\rho\rightarrow\frac{\gamma L}{1-L/\tau}.$ (A4)

It is in such a regime of dynamical equilibrium that the spatial distributions
of Lebenssphären is anticipated to show non-trivial correlations.

##  Appendix B Connectedness model of panspermia

###  B.1 Pair-connectedness function

In our model of panspermia, we define two planets as being “connected” if
their relative distance is smaller than $R$, namely, the radius of their
Lebenssphären. In this fashion, the pair-distribution function can be
decomposed into a “connected” part and a “disconnected” part:
$g(r)=P(r)+B(r)$. Here, $P(r)$ is the pair-connectedness function associated
with finding two Lebenssphären, with centers separated by a distance $r$,
within the same cluster. $B(r)$ is the pair-blocking function associated with
finding two Lebenssphären at distance $r$ _not_ belonging to the same cluster
(Torquato, 2002). From these definitions it follows that $P(r\leq R)=g(r)$ and
$B(r\leq R)=0$. It is worth pointing out that the ansatz (5) adopted for the
static limit is actually an approximation for $B(r)$; it can be obtained from
from $g(r)=P(r)+B(r)$ by replacing $B(r)$ with $1-P(r)/\chi$, which satisfies
the condition $B(r\leq R)=0$.

Since the pair-connectedness function fully defines our model $g(r)$, Eq. (5),
we exploit the standard integral equation method of continuum percolation
theory to find $P(r)$ via the solution of the Ornstein-Zernike relation (Chiew
& Glandt, 1983; DeSimone et al., 1986):

$P(r)=D(r)+\rho\int\\!d\bm{r}D(|\bm{r}-\bm{r}^{\prime}|)P(r^{\prime})$ (B1)

where $\rho$ is the number density of the Lebenssphären and $D(r)$ is the
direct-connectedness function which is applicable to Lebenssphären that are
directly connected (i.e., the relative distance between their centers is
$r\leq R$). Equation (B1) is solvable by imposing a suitable closure relation.
Here, we use $D(r)=0$ for $r>R$, which is basically the well-known Percus-
Yevick closure relation utilized in the theory of liquids (Hansen & McDonald,
2006).

We implement the so-called Baxter factorization to solve Eq. (B1) under the
condition

$\begin{array}[]{ll}P(r)=g(r),&r\leq R,\\\ D(r)=0,&r>R.\end{array}$ (B2)

This amounts to decoupling $P(r)$ from $D(r)$ by introducing a new function
$q(r)$, having the property $q(r)=0$ for $r<0$ and $r>R$, which is related to
the pair-connectedness and direct-connectedness functions by:

$\left[\hat{q}(k)\hat{q}(-k)\right]^{-1}=1+\rho\hat{P}(k)=\left[1-\rho\hat{D}(k)\right]^{-1}$
(B3)

where $\hat{P}(k)$ and $\hat{D}(k)$ are the Fourier transforms of $P(r)$ and
$D(r)$, respectively, and

$\hat{q}(k)=1-2\pi\rho\int_{o}^{R}\\!dr\,q(r)e^{ikr}.$ (B4)

The factorization enables us to express $P(r)$ in terms of $q(r)$ [see Hansen
& McDonald (2006) for a thorough derivation]:

$rP(r)=-q^{\prime}(r)+2\pi\rho\int_{0}^{R}\\!dt\,(r-t)P(|r-t|)q(t),$ (B5)

where $q^{\prime}(r)=dq(r)/dr$. The function $q(r)$ can be found by solving
Eq. (B5) for $0<r<R$, since in this range $P(r)=g(r)$:

$rg(r)=-q^{\prime}(r)+2\pi\rho\int_{0}^{R}\\!dt\,(r-t)g(|r-t|)q(t),\,\,\,\,0<r<R.$
(B6)

In the static limit of our panspermia model, the pair-distribution function
for $0<r<R$ is simply $g(r)=\chi$, so that the solution Eq. (B6) is of the
form $q^{\prime}(r)=\alpha r+\beta$. Given the boundary condition $q(R)=0$, we
find therefore:

$q(r)=\frac{1}{2}\alpha(r^{2}-R^{2})+\beta(r-R).$ (B7)

Substitution of this result into Eq. (B6) allows us to find the coefficients
$\alpha$ and $\beta$:

$\alpha=-\chi\frac{1-\eta\chi/4}{(1+\eta\chi/8)^{2}},\,\,\,\,\beta=-\frac{3R}{16}\frac{\eta\chi^{2}}{(1+\eta\chi/8)^{2}},$
(B8)

where $\eta=\frac{4}{3}\pi\rho R^{3}$. At this point, $P(r)$ can be calculated
by numerically solving Eq. (B5) in the range $r>R$.

When we consider the effects of the stellar velocities, we follow essentially
the same steps of the static case with, however, the important difference that
now $g(r)$ and $P(r)$ must be calculated self-consistently. In practice, we
first integrate numerically Eq. (7) using the pair-connectedness function
$P(r)$ calculated for the static limit as described above. The resulting
$g(r)$ is inserted into Eq. (B6) and the integro-differential equation for
$q(r)$ is solved numerically. Next, $P(r)$ is calculated by solving Eq. (B5)
across the entire range of $r$ and inserted back into Eq. (7). We repeat this
procedure until convergence is reached.

###  B.2 Percolation threshold

It should be noted that the Ornstein-Zernike relation (B1) applies below the
percolation threshold where no giant cluster of connected centers of
Lebenssphären exists. The percolation threshold is the point at which the mean
cluster size

$S=1+\rho\int\\!d\bm{r}P(r)=1+\rho\hat{P}(0)$ (B9)

diverges. Using Eqs. (B3) and (B4), $S$ can be expressed in terms of the
function $q(r)$:

$S=\left[1-2\pi\rho\int_{0}^{R}\\!dr\,q(r)\right]^{-2}.$ (B10)

In the static limit, $S$ can be calculated analytically using Eqs. (B7) and
(B8), yielding $S=(1+\eta\chi/8)^{4}/(1-\eta\chi/4)^{2}$, which diverges at
the critical point $\eta\chi=4$.

###  B.3 Correlation length

We calculate the correlation length $\xi$ by solving Eq. (B5) for $r\gg R$.
Letting $f(r)=rP(r)$ we can recast $\eqref{OZ2}$ as a differential equation
for $f(r)$:

$f(r)=2\pi\rho\int_{0}^{R}\\!dt\,f(r-t)q(t)\simeq 2\pi\rho
f(r)\int_{0}^{R}\\!dt\,q(t)-2\pi\rho
f^{\prime}(r)\int_{0}^{R}\\!dt\,tq(t),\,\,\,r\gg R,$ (B11)

where we have used $f(r-t)\simeq f(r)-f^{\prime}(r)t$ for $t\leq R\ll r$. It
is straightforward to solve Eq. (B11) to find $P(r)\propto\exp(-r/\xi)/r$,
where the correlation length is given by:

$\xi=\dfrac{2\pi\rho\int_{0}^{R}\\!dt\,tq(t)}{1-2\pi\rho\int_{0}^{R}\\!dt\,q(t)}.$
(B12)

In the static limit, $\xi$ can be calculated analytically by inserting Eqs.
(B7) and (B8) into the above expression:

$\xi/R=\frac{3}{16}\frac{\eta\chi}{1-\eta\chi/4}.$ (B13)

Note that the percolation length also diverges at the percolation threshold of
$\eta\chi=4$ derived in Appendix B.2, which is a classic feature of
percolation theory (Stauffer & Aharony, 1994).

When the stellar motion is taken into consideration, the resulting correlation
length is readily obtained by inserting into Eq. (B12) the function $q(r)$,
which is calculated numerically as described in Appendix B.1.

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# KS-Net: Multi-band joint speech restoration and enhancement network for 2024
ICASSP SSI Challenge

###### Abstract

This paper presents the speech restoration and enhancement system created by
the 1024K team for the ICASSP 2024 Speech Signal Improvement (SSI) Challenge.
Our system consists of a generative adversarial network (GAN) in complex-
domain for speech restoration and a fine-grained multi-band fusion module for
speech enhancement. In the blind test set of SSI, the proposed system achieves
an overall mean opinion score (MOS) of 3.49 based on ITU-T P.804 and a Word
Accuracy Rate (WAcc) of 0.78 for the real-time track, as well as an overall
P.804 MOS of 3.43 and a WAcc of 0.78 for the non-real-time track, ranking
$1^{st}$ in both tracks.

Index Terms—  speech signal improvement, speech restoration GAN, multi-band
fusion, speech enhancement

##  1 Introduction

During audio communication, speech signals recorded by microphones may be
severely degraded by various distortions, including environmental noise,
reverberation, coloration, discontinuity and loudness. Although existing
methods have shown effectiveness in the noise suppression, restoring high-
quality speech with the presence of multiple simultaneous distortions still
remains challenging. The ICASSP SSI Challenge aims to restore the high-qulity
speech signal from the captured signal distorted by the aforementioned complex
acoustic conditions in a real-time communication system [1]. In this paper, we
propose a multi-band joint restoration-and-enhancement framework called KS-Net
to address the aforementioned distortions. Firstly, the framework adopts a
complex-domain based generative adversarial network (GAN) to restore the
coarse-grained high-quality speech, aiming to remove various types of
distortions. Subsequently, the remaining transient noise and artifacts after
the restoration GAN are eliminated by a multi-band fusion network for speech
enhancement. The proposed KS-Net approach achieves the $1^{st}$ places in both
real-time and non-real-time tracks in the 2024 SSI challenge.

Fig. 1: The flowchart of the proposed KS-Net.

##  2 System Architecture

As illustrated in Fig 1, the training procedure of our system consists of two
stages. The first stage involves the primary restoration GAN, which aims to
address various distortions, such as band-width extension (BWE), speech
distortion restoration, loudness adjustment, packet loss concealment (PLC),
noise suppression (NS) and dereverberation (Dereverb). In the second stage,
the multi-band fusion SE (MF-Net) focuses specifically on removing residual
transient noise and artifacts generated by the restoration GAN.

###  2.1 Complex-domain based restoration GAN

Inspired by GAN-related methods presented in ICASSP 2023 SSI-Challenge[2, 3],
we proposed a restoration GAN to restore the coarse-grained high-quality
speech in the complex domain. To facilitate information exchange between low
frequency bands and high frequency bands, we divide the full-band complex
spectrum into three subbands and concatenate them along the channel dimension
as inputs to the generator. The restoration GAN’s generator follows a
convolutional encoder-decoder architecture. It includes five groups of
squeezed temporal convolution modules (S-TCMs) and two residual time-frequency
LSTM (TF-LSTM) for sequence modeling in the bottleneck. The encoder consists
of five convolutional layers, with a dilated dense-block inserted after the
first convolutional layer, which contains five dilated convolutional layers.
On the other hand, the decoder comprises five corresponding transposed
convolutional layers and a corresponding dilated dense-block. To improve
temporal modeling, each group of S-TCMs is composed of four S-TCN blocks with
exponentially increasing dilation rates. After the last S-TCMs, two TF-LSTM
are stacked. The time-LSTM focuses on capturing long-term dependencies along
the time axis, while the bi-directional frequency-LSTM targets the long-term
dependencies along the frequency axis.

Regarding the discriminators, multi-resolution STFT discriminators are used to
capture various spectral patterns in speech signals[4]. The training loss is a
combination of a reconstruction loss, an adversarial loss, and a feature match
loss. The reconstruction loss includes both a multi-resolution STFT loss and a
waveform loss.

###  2.2 Multi-band fusion SE

The restoration GAN’s output lacks certain spectral details in the low-
frequency parts and still has some leftover transient noise. To address this,
we introduce the MF-Net, a multi-band fusion enhancement module. This module
aims to restore the spectral details and eliminate the remaining noise.

Based on the insights from our prior study [5], we have developed MF-Net - a
two-stage solution that simultaneously restores the overall full-band spectral
pattern and the intricate sub-band spectral details. In the first stage, we
apply coarse denoising to the compressed ERB-scaled spectrum. This step
effectively reduces noise and enhances the general features. In the second
stage, due to the varying spectral characteristics among different frequency
bands, we employ two distinct sub-networks to refine the low-frequency and
high-frequency bands separately.

##  3 Experiments

###  3.1 Data Augmentation

The training set for this project is constructed using multiple sources: the
DNS Challenge dataset[6], the VCTK corpus[7] and our own private datasets.
Specifically, we create a large dataset of 800 hours of paired data. In this
dataset, we add different distortions, such as various types of noise,
reverberation, clipping, packet loss, low-pass filtering, traditional noise
reduction algorithms, and low bit-rate codecs, to the original clean full-band
speech. We add these distortions with different probabilities to simulate
real-life scenarios. During the training stage for MF-Net, transient noises
are further added to the outputs of the restoration GAN with signal-to-noise
ratios (SNRs) ranging from -5dB to 10dB.

###  3.2 Experimental Setup

In our experiments, we utilized a Hanning window with a 20 ms window length
and a 10 ms frame shift. It is worth noting that the difference between the
submitted system for SSI real-time track (KS-Net-1) and non-real-time track
(KS-Net-2) is that the generator and MF-Net in KS-Net-2 is designed with a
large amount of network parameters and computational complexity. KS-Net-1 has
a total of 15.64 million(M) trainable parameters and its real time factor
(RTF) is 0.42 on an Intel Core i5 Quadcore CPU clocked at 2.4 GHz, while KS-
Net-2 has a total parameter number of 19.15 M and an RTF of 0.62.

###  3.3 Results and analysis

Table 1 presents the results of the P.804 subjective test conducted on the
official blind set. It is evident that both KS-Net-1 and KS-Net-2 exhibit
considerable improvements across all subjective metrics when compared to the
noisy signals. This suggests that our proposed system effectively enhances
speech quality by alleviating distortions introduced by various factors such
as noise, reverberation, coloration, discontinuity, and loudness. Moving
forward, our future research endeavors will focus on further exploring the
underlying factors contributing to the superior performance of real-time KS-
Net-1 in contrast to non-real-time KS-Net-2.

Table 1: ITU-T P.804 MOS and WAcc results on the SSI 2024 blind test set.

System | MOS | WAcc
---|---|---
COL | DISC | LOUD | NOISE | REVERB | SIG | ORVL | (%)
Noisy | 3.34 | 3.70 | 3.78 | 3.21 | 3.40 | 3.05 | 2.58 | 82.68
KS-Net-1 | 4.08 | 3.89 | 4.34 | 4.30 | 4.27 | 3.83 | 3.49 | 78.23
KS-Net-2 | 4.01 | 3.89 | 4.34 | 4.28 | 4.24 | 3.74 | 3.43 | 77.30

## References

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