# Detection of HCN and diverse redox chemistry in the plume of Enceladus

Jonah S. Peter<EMAIL_ADDRESS>Jet Propulsion Laboratory, California
Institute of Technology, Pasadena, California 91109, USA  Biophysics Program,
Harvard University, Boston, Massachusetts 02115, USA  Tom A. Nordheim  Jet
Propulsion Laboratory, California Institute of Technology, Pasadena,
California 91109, USA  Kevin P. Hand  Jet Propulsion Laboratory, California
Institute of Technology, Pasadena, California 91109, USA

###### Abstract

The Cassini spacecraft observed that Saturn’s moon Enceladus possesses a
series of jets erupting from its South Polar Terrain. Previous studies of in
situ data collected by Cassini’s Ion and Neutral Mass Spectrometer (INMS) have
identified H2O, CO2, CH4, NH3, and H2 within the plume of ejected material.
Identification of minor species in the plume remains an ongoing challenge,
owing to the large number of possible combinations that can be used to fit the
INMS data. Here, we present the detection of several new compounds of strong
importance to the habitability of Enceladus, including HCN, C2H2, C3H6, and
C2H6. Our analyses of the low velocity INMS data, coupled with our detailed
statistical framework, enable discrimination between previously ambiguous
species in the plume by alleviating the effects of high dimensional model
fitting. Together with plausible mineralogical catalysts and redox gradients
derived from surface radiolysis, these compounds could potentially support
extant microbial communities or drive complex organic synthesis leading to the
origin of life.

Shortly after its arrival in the Saturn system, the Cassini spacecraft
detected intense plume activity at the mid-sized moon, Enceladus [1, 2].
Cassini in situ and remote sensing observations have confirmed that the plume
consists primarily of H2O gas [3, 4, 5, 6] as well as H2O-ice grains that feed
Saturn’s E-ring [7, 8, 9, 10]. CO2 has also been detected in both the gaseous
phase within the plume itself [4, 5, 6] and as a condensate in plume deposits
on Enceladus’ surface [11]. In situ measurements of the plume’s neutral gas
component made by Cassini’s Ion and Neutral Mass Spectrometer (INMS) further
indicate the presence of CH4, NH3, and H2 within Enceladus’ subsurface ocean
[4]. Although early publications identified several additional species within
the plume [5], more recent work suggests that many of these compounds resulted
from incidental high velocity impact fragmentation of larger molecules within
the instrument antechamber [12, 4]. Analyses of plume material sampled during
the lower velocity flybys of Enceladus (for which this fragmentation was less
significant) do imply the existence of additional plume species, however no
study to date has been able to verify the identity of any other intrinsic
compounds.

Fig. 1: New compounds identified in the Enceladus plume indicate a potentially
habitable environment. (a) Jets emanating from ice fissures in Enceladus’
South Polar Terrain feed a plume of ejected material containing organic
molecules with varying oxidation states. Electron bombardment of the surface
might help facilitate the production of oxidants and prebiotic feedstock
molecules observed in the plume. These compounds could potentially support
biologically-mediated redox metabolisms or polymerize to form nucleic and
amino acid precursors leading to the origin of life. (b) The average oxidation
state of carbon for organic compounds confirmed or suspected in the plume.
Plume-derived H2 and O2 could act as strong reducing and oxidizing agents,
respectively, and may be responsible for the diverse redox chemistry seen at
Enceladus.

Difficulty in resolving minor plume constituents stems from the large number
of plausible compounds relative to the low mass resolution of INMS. When
training statistical models in this high dimensional regime, simple regression
techniques tend to form overly complex models that produce specious results
[13, 14, 15, 16]. Models of INMS spectra suffer from an additional complexity
in that the signals produced by individual molecules are not necessarily
linearly independent. As such, there may be multiple different combinations of
species that appear to fit the data equally well. The resulting large
correlations between model components generally reduce model performance and
can mask the importance of any particular component by limiting statistical
power [17, 13]. Studies that leave these issues unaddressed are likely to
encounter model ambiguities that preclude reliable statistical inference about
the plume’s composition.

In this work, we seek to resolve the apparent compositional ambiguity of
Enceladus’ plume. By characterizing the information content of the average low
velocity spectrum obtained during the E14, E17, and E18 flybys, we determine
constraints on the number of species that can reliably be extracted from the
INMS dataset under spacecraft conditions that were least modified by
fragmentation. We then use relative entropy minimization to assess the
likelihood of tens of billions of potential models and show that multi-model
inference allows for the identification of several new compounds not
previously confirmed at Enceladus. Our results indicate the presence of a
rich, chemically diverse environment that could support complex organic
synthesis and possibly even the origin of life (Fig. 1).

##  I Constraints on the complexity of plume models

Deconvolving the overlapping signals of each species in the plume requires
comparing features in the INMS flight data to a library of known mass spectra.
Consequently, species in the plume cannot be identified unless explicitly
included within the models used for comparison. Previous studies have
constructed model INMS spectra using a variety of methods, including singular
value decomposition [18] and the application of custom-defined fit statistics
[19, 20, 21, 22] (see Supplementary Information). In all cases, these fitting
procedures seek to minimize the training error associated with the residual
counts between the INMS spectrum and the reconstructed model fit. Crucially,
however, the training error is not an appropriate metric for evaluating model
performance. In fact, it is a well-known concept in statistical modeling that
the training error will continue to decrease with the inclusion of additional
model components, regardless of whether those components are genuinely related
to the observed data. Instead, it is necessary to approximate how each model
would perform on a set of independent observations. This is the preferred
method of model validation when additional data sets are unavailable for
explicit model testing [13]. Analysis techniques that utilize only the
training error risk developing overly complex models and claiming false
species detections.

Fig. 2: Model performance as a function of model complexity. (a) Maximum
relative likelihoods across all models with $d$ species. Blue circles indicate
best-fit (highest $\lambda$) models identified via an exhaustive search for
$d<16$. Orange circles represent models constructed using a benchmark forward
modeling procedure extending to $d=50$ (see Methods). There is good agreement
between the trends observed with forward modeling and the exhaustive search.
Models with $\lambda>1/e^{3}$, $1/e^{2}$, and $1/e$ (dashed lines) exhibit
increasingly strong predictive power. All models with $d<10$ species underfit
the data, while those with $d>15$ overfit. (b) The maximum magnitude of
pairwise correlations
$r_{ij}=\mathrm{cov}(\beta_{i}\beta_{j})/\sqrt{\mathrm{var}(\beta_{i})\mathrm{var}(\beta_{j})}$
between regression coefficients in the best-fitting models at each value of
$d$. Blue and orange circles represent the same models as in panel (a).
Correlations loosely follow a logistic curve demonstrating model redundancy
for $d>15$. The inflection point occurs within the optimal performance
interval $d\in[10,15]$ characterized by $\lambda>1/e^{3}$ (green shaded area).

Here, we evaluate model performance using the small sample bias-corrected
Akaike Information Criterion (AICc) [23, 24, 25, 26, 27] which estimates the
relative entropy between a given model and the unknown, true distribution that
produced the observed data. We construct each candidate model as a linear
combination of end-member spectra representing the different cracking patterns
of individual molecules. A model $M$ is given by

$M:\quad\hat{y}_{i}=\sum^{d}_{k=1}\beta_{k}x_{k,i},$ (1)

where $\hat{y}_{i}$ is the total modeled counts in mass channel $i$, $x_{k,i}$
is the cracking pattern of species $k$ at mass channel $i$, $\beta_{k}\geq 0$
is the regression coefficient for species $k$, and $d$ is the total number of
species in the model. The observed INMS counts at each mass channel, $y_{i}$,
are treated as independent data points with unequal Gaussian uncertainties,
$\sigma_{i}$ (see Methods). The AICc for each model is then given by

$\text{AICc}=2d-2\ln[\mathcal{L}(M_{0}|y)]+\frac{2d(d+1)}{n-d-1},$ (2)

where $n$ is the total number of mass channels and $\mathcal{L}(M|y)$ is the
model’s likelihood function,

$\mathcal{L}(M|y)=\prod^{n}_{j=1}\left(\frac{1}{\sigma_{j}\sqrt{2\pi}}\right)\exp\left\\{-\frac{1}{2}\sum^{n}_{i=1}\left(\frac{y_{i}-\hat{y}_{i}}{\sigma_{i}}\right)^{2}\right\\},$
(3)

evaluated at the maximum likelihood estimate, $M_{0}$, obtained by optimizing
the set of $\\{\beta_{k}\\}$. The last term in Eq. (2) is a correction factor
that penalizes overly complex models when the sample size is small
($n/d\lesssim 40$) [27]. The model with the minimium AICc
(AICc${}_{\text{min}}$) asymptotically approximates the model with the lowest
Kullback-Leibler information loss relative to the observed data [28, 29, 27].
Whereas standard maximum likelihood estimation seeks to minimize the training
error quantified by the variance-weighted sum of squared residuals,
$\sum^{n}_{i=1}((y_{i}-\hat{y}_{i})/\sigma_{i})^{2}$, minimization of the AICc
accounts for the bias introduced by estimating the regression coefficients via
the training data. The relative likelihood, or evidence ratio, of each model
follows as [30]

$\lambda=\exp\left\\{-(\text{AICc}-\text{AICc}_{\text{min}})/2\right\\}$ (4)

and can be used to compare models of differing complexity. Generally, models
with $\lambda<1/e^{3}\approx 0.05$ exhibit little to no predictive power while
those above this threshold form a $\sim$95–99% model confidence interval
suitable for multi-model inference [27, 31, 32, 33].

To capture the full range of possible plume constituents, we composed a large
spectral library containing the most recently published list of plausible
INMS-detectable plume species [34] as well as several additional compounds
found in organic synthesis and laboratory experiments simulating icy
satellites (Extended Data Table 1). Using this library, we performed an
exhaustive search up to $d=15$ of tens of billions of potential models for the
plume’s composition. Fig. 2a demonstrates that optimal model performance is
achieved with only 10–15 species. The maximum relative likelihood across all
models peaks sharply at $d=13$ and drops precipitously for more complex
models. A benchmark forward modeling procedure extending to $d=50$ confirms
this trend for large $d$ (see Methods). Models with greater than 15 species
drastically overfit the INMS data and incorrectly extract false signal from
statistical noise. Large correlations
$r_{ij}=\mathrm{cov}(\beta_{i}\beta_{j})/\sqrt{\mathrm{var}(\beta_{i})\mathrm{var}(\beta_{j})}$
between regression coefficients in these models indicate the erroneous
inclusion of redundant species with similar mass spectra (Fig. 2b). This leads
to overfitting and poor model performance, despite a monotonic decrease in the
training error with increasing complexity. By contrast, significantly simpler
models exhibit low correlations between model parameters, indicating the
presence of additional features within the INMS spectrum that have not yet
been fit by a candidate species (i.e., underfitting). Optimal performance
occurs near the inflection point after all major data features have been
explained but before redundant species are incorporated.

##  II New species detected in the plume

Table 1: Volume mixing ratios for the Enceladus plume. Probabilities and
mixing ratios are shown for all compounds with probability $>0.1$. Evidence
for the remaining library species is poor (Extended Data Table 2). Values are
calculated based on the exhaustive set of 26332 high likelihood
($\lambda>1/e^{3}$) models using a multi-model averaging procedure that
incorporates uncertainties due to model ambiguities (see Methods). Mixing
ratios are given as mean $\pm$ SE and are scaled to incorporate the 0.9% H2
number abundance reported in ref. [4]. Upper limits (mean + 3 SE) are reported
for species with mean values below their corresponding SE. Species listed
under “Confirmed” are detected with $>2$ SE precision and represent the most
conservative model of the plume. Species listed under “Strong” are very likely
detected but with mixing ratios that are less certain. Species listed in
brackets are included in the alcohol mixing ratio. Species listed in
parentheses are included in the 43 u fragment mixing ratio.

Evidence | Species | Probabilitye | Mixing Ratio (%)e | Previous Limit (%)f
---|---|---|---|---
Confirmed | H2O | 1 | $95.9\pm 0.3$ | 96–99
| CO2 | 1 | $0.45\pm 0.04$ | 0.3–0.8
| COa | 1 | $0.72\pm 0.07$ | 0.01–0.2, $<0.05$gh
| CH4 | 1 | $0.11\pm 0.02$ | 0.1–0.3
| NH3 | 1 | $1.8\pm 0.1$ | 0.4–1.3
| HCN | 1 | $0.11\pm 0.02$ | 0.01–0.2
| C2H2 | 0.92 | $0.023\pm 0.005$ | 0.01–0.2
| C3H6 | 0.87 | $0.004\pm 0.002$ | $<0.01$
| H2b | $-$ | $-$ | 0.4–1.4, 0.9g
Strong | C2H6 | 0.70 | $0.013\pm 0.010$ | 0.01–0.2
| Alcoholsc | 0.82 | $<0.02$ | $-$
| [CH3OH] | 0.44 | $<0.005$ | $<0.01$
| [C2H7NO] | 0.23 | $<0.002$ | $<0.01$
| [C2H6O2] | 0.20 | $<0.01$ | $<0.01$
| [C3H8O] | 0.11 | $<0.005$ | $<0.01$
| 43 u fragmentsd | 0.72 | $<0.01$ | $-$
| (C2H6N2) | 0.24 | $<0.002$ | $<0.01$
| (C3H6O) | 0.17 | $<0.001$ | $<0.01$
| O2 | 0.64 | $<0.008$ |  $<0.01$, $<0.004$gh
|  40Ar | 0.58 | $<0.004$ | $<0.01$
Moderate | H2S | 0.37 | $<0.003$ | $<0.01$
| PH3 | 0.24 | $<0.002$ | $<0.01$
| C3H8 | 0.23 | $<0.005$ | $<0.01$
| C3H5Cl | 0.23 | $<0.001$ | $<0.01$
Poor | CH3CN | 0.16 | $<0.003$ | $<0.01$
| C3H4 | 0.16 | $<0.002$ | $<0.01$
| CH5N | 0.14 | $<0.003$ | $<0.01$

  * a

The vast majority of CO is likely produced through incidental impact
fragmentation.

  * b

Determination of the H2 mixing ratio requires analysis of additional INMS data
not examined here (see Methods).

  * c

C4H10O might also contribute to this mixing ratio (see Supplementary
Information).

  * d

Other mass 43 fragments produced by C4H10, C4H6O2, C4H9N, C5H9N, C8H18, or
C5H12 might also contribute to this mixing ratio.

  * e

Denotes values calculated in this work.

  * f

Denotes values presented in ref. [34] unless otherwise specified.

  * g

Denotes values presented in ref. [4].

  * h

Denotes number density relative to H2O.

The most recently published list of neutral gas species confirmed in the plume
consists only of H2O, CO2, CH4, NH3, and H2 [34, 4]. According to that work,
no other compounds could be definitively detected by INMS during the low
velocity flybys due to model ambiguities at low mixing ratios. Here, we
account for those ambiguities via a multi-model averaging procedure that
treats the minimum Kullback-Leibler divergence as a statistical random
variable and weights each model by its probability of minimizing this
information loss (see Methods). Whereas previous studies have relied on ad-hoc
assessments of model ambiguity, the procedure here is rooted in the
fundamental concepts of information theory and explicitly incorporates
uncertainties resulting from the modeling process into the standard error (SE)
of each mixing ratio.

In addition to H2O, CO2, CH4, and NH3, we find significant evidence—beyond 2
SE precision—for HCN, C2H2, C3H6, and CO in the plume (Table 1). Our results
are agnostic to the presence of H2 which requires analysis of additional INMS
data not examined here (see Methods). We also demonstrate strong evidence for
C2H6 at 1 SE precision, as well as the very probable detection of an alcohol
(most likely CH3OH), molecular O2, 40Ar, and an unidentified 43 u fragment
(possibly from C2H6N2 or C3H6O). Aside from CO, which is known to be produced
as a fragmentation product within the INMS antechamber [5, 4, 12, 35], these
species are most likely intrinsic to the plume itself. In general, our mixing
ratios are in excellent agreement with the most recently published limits [34,
4]. The reconstructed model fit is plotted alongside the INMS spectrum in Fig.
3.

Fig. 3: Average low velocity INMS spectrum and reconstructed model fit. The
black silhouette shows the 12-47 u range of the average INMS spectrum obtained
during the E14, E17, and E18 flybys. Data was adapted from ref. [12]. Counts
at all other mass channels are at or below the minimum count uncertainty.
Error bars show 1$\sigma$ Gaussian uncertainty in the observed counts. Red
circles indicate the model fit based on the mixing ratios in Table 1.

Interestingly, the NH3 abundance derived here is somewhat higher than that
reported by Waite et al. [4] and leaves a large negative residual at 16 u
(Extended Data Fig. 1). This larger mixing ratio stems from the residual
signal at masses 14–16 left by fitting H2O, CO2, and CH4 (Fig. 4a). However,
we note that masses 17 and 18 are predominantly dictated by H2O, which is used
to standardize the count rates across the E14, E17, and E18 flybys (see
Methods). As such, the small count uncertainties at these mass channels are
likely an underestimate of the true signal variability at Enceladus and may
artificially enhance the NH3 mixing ratio (see Supplementary Information).
Chemisorption of NH3 molecules collected during previous flybys onto the INMS
instrument antechamber could also contribute to elevated count rates at these
mass channels [35]. Nevertheless, some amount of NH3 is likely required to
explain the presence of nitrogen-bearing ions observed in Saturn’s inner
magnetosphere, consistent with a source from Enceladus [36].

A major finding of this work is that nitrogen is also definitively present at
Enceladus in the form of HCN. Previous studies have been unable to resolve the
HCN abundance due to confounding signals from fragmentation products at mass
28. In their analysis of the E2 flyby, Waite et al. [6] quote an upper limit
of $<0.5$%, whereas upper limits of $<0.58$% and $<0.74$% are given by Waite
et al. [5], based on the high velocity E3 and E5 flybys. For the faster flybys
in particular, the authors note that the elevated count rate at 28 u
introduces a model ambiguity at masses 27 and 28 (N2+HCN versus C2H4) that
precludes the identification of HCN. By contrast, our procedure allows for the
unambiguous identification of CO as the major 28 u fragment (CO is present in
all high likelihood models—see Extended Data Table 2). This conclusion is
strongly supported by the fragmentation signatures observed by INMS at higher
velocities, and by INMS ice grain spectra [12]. Properly accounting for the CO
signal (and the minor contribution from C2H6) at mass 28 eliminates
alternative model fits and reveals that HCN is required at 27 u (see Fig. 4b
and Extended Data Fig. 2). The mixing ratio reported here ($0.11\pm 0.02\%$)
is consistent with that observed in cometary comae [37, 38, 39].

We also report the first strong evidence for native alcohols and additional
evidence for O2 in the plume. Alcohols contribute significantly to the
observed signal in the 30–32 u range, while O2 maintains a strong contribution
at mass 32 (Fig. 4c). The likelihood that at least one alcohol is present is
$>0.82$, though similarities between their cracking patterns introduce
uncertainty in their respective mixing ratios (see Supplementary Information).
In their analysis of the low velocity flybys, Waite et al. [4] note that mass
32 exhibits an elevated count rate, likely in part due to surface processing
between H2O and the Ti instrument antechamber [35]. Accounting for this
instrument effect, they estimated an upper limit on the intrinsic plume O2 $/$
H2O number density of $<0.004\%$. In accounting for additional species, our
statistical analysis confirms that the excess at mass channel 32 is best
explained by O2, though the overlapping signal from dissociation products of
alcohol and CO2 parent molecules precludes the quantification of a precise
mixing ratio. That the O2 detected by INMS is in fact native to Enceladus
would be consistent with the suite of partially oxidized organics reported
here (Fig. 1), and is expected if radiolysis or photolysis of H2O-ice
contribute to the observed plume chemistry [40]. Nevertheless, further
characterization of potential instrument artifacts is warranted to confirm the
source of the O2 signal beyond the circumstantial evidence provided here.

Fig. 4: Contributions of individual species to the model fit. Black
silhouettes show different mass ranges for the average low velocity INMS
spectrum plotted in Fig. 3. Error bars show $1\sigma$ Gaussian uncertainty in
the observed counts. (a) Blue circles show the total contribution of
H2O+CO2+CH4. Green circles show the contribution of NH3. (b) Model
contributions from HCN, C2H2, C2H6, and CO (cyan, orange, purple, and magenta
circles, respectively). (c) Navy circles show the contribution from O2 added
to the total contribution from alcohols. The blue circle shows the dissociated
O2 contribution from cracked CO2. Pink circles show the tentatively proposed
contribution of H2S+PH3. Low signal-to-noise ratios at these mass channels
preclude the firm identification of H2S or PH3. (d) Brown and indigo circles
show contributions from C3H6 and 40Ar, respectively. The teal circle shows the
total contribution from 43 u fragments.

Our analysis also indicates moderate (Table 1) evidence for either H2S or PH3
in the plume, with probabilities of 0.37 and 0.24, respectively. This
conclusion stems primarily from mass 34, where the omission of these species
leaves a small residual signal that cannot be explained by other library
spectra (Fig. 4c). However, this signal alone is not enough to unambiguously
conclude their presence within the INMS data. Both species improve the model
fit when added to the minimal model (“Confirmed” species in Table 1), but not
when additional higher likelihood species are included (“Strong” species in
Table 1). Our results are therefore only suggestive of a 34 u compound in the
plume. Given the profound astrobiological implications of finding sulfur or
phosphorous compounds at Enceladus, mass spectra for alternative 34 u species
(such as H2O2) should be characterized experimentally and investigated in
future studies.

##  III Discussion

In aggregate, the results presented here indicate that Enceladus is host to a
multiphasic and compositionally diverse chemical environment that is
consistent with a habitable subsurface ocean. The new species identified in
this work also suggest that this ocean may contain the necessary building
blocks required to synthesize compounds important to the origin of life.

The detections of partially oxidized hydrocarbons along with the very likely
detections of an alcohol and molecular O2 are particularly interesting as
these species potentially imply a diverse redox environment within the ocean
(Fig. 1). Past measurements of CH4 and H2 in the plume [6, 5, 4] support the
hypothesis that Enceladus may be hydrothermally active and could be a source
of biologically useful reductants. In particular, methanogenesis via the
reduction of CO2 has been proposed as a potential pathway that could support
extant microbial communities near the sea floor [41, 42]. However, without
additional oxidants, reductants in the ocean would be of little biochemical
utility, as no electron transfer mechanism beyond methanogenesis would be
available to yield a negative change in Gibbs free energy. The presence of O2
and partially oxidized carbon compounds may solve this problem, as they
implicate a multitude of highly exergonic redox pathways that could help power
life in Enceladus’ subsurface ocean. Such compounds, for example, could serve
as direct substrates for biological growth [43, 44, 45, 46], or be
intermediaries of other metabolic reactions involving additional organics and
oxidants (see e.g., refs. [47, 48, 49]).

Our results also provide the first conclusive evidence for HCN, C2H2, and C3H6
in the plume, which, in addition to their significance for habitability, are
particularly intriguing for their relevance to prebiotic chemistry. HCN
polymerization is implicated in a number of potential pathways for the
formation of nucleobases and amino acids [50, 51, 52, 53]. Although these
reactions might be limited within a dilute subsurface ocean, concentrated
conditions favorable for polymerization could be achieved within the ice shell
via eutectic freezing [54, 51, 55]. Indeed, Levy et al. [56] demonstrated the
production of alanine, glycine, aspartic acid, and adenine from HCN and NH3
under conditions similar to those of icy satellites. Repeated freezing and
thawing caused by the cycling of material between the ice shell and the warmer
interior fissures of the plume vents may provide conditions favorable for
organic synthesis.

In accordance with this scenario, there does exist evidence that the plume
contains at least some material that has not been diluted by the ocean. Liquid
water facilitates the rapid hydrolysis of HCN into CH3NO, which subsequently
decays into CH2O2 and NH3 [54]. The simultaneous presence of ppt amounts of
HCN and absence of CH3NO and CH2O2 suggests that the plume may be sourced, in
part, by solid-phase material residing in the ice shell [5]. HCN at Enceladus
could be primordial in nature or produced via the radiolysis of nitrogen-
bearing surface ices by magnetospheric electrons. Laboratory experiments of
hydrocarbon-rich ices simulating the temperature and surface radiolysis
conditions of Enceladus support the evidence for cyano group-containing
species. Hand [57] and Hand et al. [58] found that the warming of H2O+NH3+C3H6
ice films from 70 K to 300 K after irradiation by 10 keV electrons produces
HCN and nitrile species, including CH3CN. Irradiation of H2O+CO2 ice in the
presence of hydrocarbons can also produce alcohols, including CH3OH [57].
Thus, the detection of HCN and an alcohol (as well as O2 and possibly CH3CN)
in the plume could potentially be explained by the aerosolization of
radiolytically processed material on or near the surface. Additional evidence
from organic-rich ice grains detected in the plume further suggests the
presence of solid-phase organic films that exist above the water table and are
transported to the surface via the plume vents [12]. The macromolecular nature
of these compounds, some in excess of 200 u, might be evidence of ongoing
synthetic chemistry. Accumulation and isolation of buoyant organic compounds
at the cold ice-water interface between the ocean and ice shell may also
promote longer lifetimes and inhibit hydrolysis [59].

Of course, the availability of organic compounds at Enceladus to support or
facilitate the origin of life likely depends strongly on the geochemistry of
the subsurface ocean. Although the mineral composition of the ocean floor is
unknown, the simultaneous detections of SiO2 particles by CDA [60] and gaseous
H2 by INMS [4] indicate the presence of a complex hydrothermal environment
[61]. Similarities between the inferred temperature and pH of Enceladus’ ocean
and those of the Lost City hydrothermal vents point towards serpentinization
reactions as a possible source for the observed H2 abundance [62, 4, 63]. Such
an explanation would also be consistent with the tentative evidence for H2S
presented here. The presence of C2H2 and C2H6 in the plume further implicates
ongoing catalytic reactions driven by metal-bearing minerals within the ocean
[64, 65, 66]. If ferrous iron is present at Enceladus (as it is at the Lost
City sites) along with H2S, sufficient reducing power may be available for a
metabolic pathway to abiogenesis [67, 51, 68]. Laboratory evidence suggests
that confirmation of C3H6 in the plume might allow for the formation of
vesicle-type structures at Enceladus, which could then shelter the burgeoning
proto-metabolism [57].

An alternative scenario for complex organic synthesis could be realized
through the photochemical processing of ocean material ejected by the plume
onto the surface. HCN might be sequestered as ferrocyanide by sodium or
potassium salts, both of which have been found in plume ice grains [9, 8].
Ferrocyanide is an important feedstock molecule for the cyanosulfidic
prebiotic chemistry paradigm, which relies on reductive homologation of HCN to
form the sugars needed for ribonucleotide assembly [69, 53, 70]. If there
exists sufficient UV photon processing of organic material on Enceladus’
surface, the selective production of RNA and amino acid precursors from plume-
derived HCN and C2H2 (and possibly C3H6O, H2S, and PH3), together with
plausible mineralogical catalysts might occur [53, 71]. This surface-based
production could then feed back into the plume or ocean via downward transport
through the ice shell. Whether this type of chemistry is efficient under
Enceladus-like conditions could be explored in future experimental studies,
while more detailed examination of Enceladus’ oceanic material will require
future robotic missions.

##  IV Methods

###  IV.1 Instrument overview

The Cassini Ion and Neutral Mass Spectrometer (INMS) was a quadrupole mass
spectrometer designed primarily for neutral gas analysis [72]. When INMS was
operating in its Closed Source Neutral (CSN) mode, neutral molecules were
accumulated in the instrument antechamber before being directed towards a 70
eV electron source for subsequent ionization. Detections of ionized molecules
at the instrument target were then recorded sequentially at individual mass
channels corresponding to mass-to-charge ratios (m/z), where z is typically
assumed to be 1. The INMS mass channels ranged from 1 to 99 u at a resolution
of 1 u, excluding 9, 10, and 11 u.

When neutral molecules enter the ionization region of a mass spectrometer such
as INMS, emitted electrons from the electron source both ionize incoming
parent molecules and produce dissociated, ionized fragments. For a given
electron energy (e.g., 70 eV), each incoming parent molecule fragments
according to a specific cracking pattern that describes the relative
proportions of the dissociated products. As such, a mass spectrum obtained
from a single molecular species will exhibit counts at mass channels
pertaining to the molecular weight of the ionized parent molecule, as well as
those of each of the ionized dissociation products. INMS spectra therefore
consist of a combination of overlapping spectral features resulting from each
parent molecule’s cracking pattern.

INMS was not able to detect parent molecules or fragments with masses above
the maximum mass of 99 u. However, larger molecules may have impacted the
instrument walls and fragmented into smaller molecules that were within the
detectable mass range [5]. The rate of these impact fragmentation events
depends heavily on the kinetic energy of the spacecraft [34]. As such, spectra
generated from faster flybys ($>$ 10–15 km/s) exhibit a suite of spectral
features (associated with impact fragments of higher mass molecules) that are
not observed at lower speeds [12, 34, 4]. As a result, measurements obtained
during the lower velocity flybys of Enceladus provide the best opportunity to
study the intrinsic plume composition in the absence of fragmentation effects.

###  IV.2 Data set selection

During the E14, E17, and E18 flybys of Enceladus, Cassini made a series of
three low velocity ($<7.5$ km/s) passages through the plume along nearly
identical trajectories. These flybys, referred to as the “slow” flybys [4],
produced the most consistent INMS data with which to analyze the plume.
Previous studies [4, 5, 6, 12, 19] have developed a robust data processing
procedure for converting the raw time series data collected by INMS into
instrument-corrected mass spectra. The post-processed slow flyby INMS spectra
have been used in other compositional analyses [4, 12] and provide the best
opportunity to study the plume’s intrinsic neutral gas composition in the
absence of instrument artifacts. Here, we use the averaged CSN spectrum
obtained across the three slow flybys, as done in refs. [4, 12]. This averaged
slow flyby spectrum is identical to that presented in ref. [12], with minor
differences described below.

In order to quantify variability in the INMS data between flybys, we assumed a
Gaussian random error on the counts at each mass channel. The $2\sigma$ width
for each count distribution was estimated based on the range of observed
counts at each mass channel across the three flybys (as depicted in ref.
[34]), similar to that described by Waite et al. in their own analysis of the
slow flyby spectrum [4]. However, mass 18 is the anchor mass used to
standardize the E14, E17, and E18 flybys and therefore, by definition, does
not exhibit any inter-flyby variability. Other H2O-dominated mass channels at
17, 19, and 20 u also demonstrate reduced variability as a result of this
standardization. To account for this effect, we instead used a standard
Poisson uncertainty equal to the square root of the average number of counts
on all mass channels where the inter-flyby variability is smaller than the
uncertainty due to counting statistics (note that in the limit of large
counts, the Poisson distribution approaches the Gaussian distribution).
Moreover, mass channels beyond 46 u exhibit significantly lower signal-to-
noise ratios than lower mass channels. To avoid erroneously fitting to these
data points, we implemented a larger fractional uncertainty on peaks where
both the inter-flyby variability and the counting statistics error is less
than the typical count value measured at these mass channels ($\sim$2 counts).
This larger relative uncertainty was set equal to the magnitude of this noisy
signal and serves as the minimum uncertainty across all mass channels.

When INMS was operating in the CSN mode, the vast majority of counts at 1 and
2 u arose due to interactions between gas phase H2O molecules and the walls of
the instrument antechamber [4]. Proper assessment of these mass channels
requires a careful analysis of data from the instrument’s Open Source Neutral
Beam (OSNB) mode. This problem has been explored in detail by Waite et al. [4]
who attribute $\sim$98% of the total signal at 1 and 2 u to H2 (at a mixing
ratio of $\sim$0.9%) and H2O. The cracking pattern for H2 does not contribute
to any other mass channels and is therefore independent from the deconvolution
of the remainder of the spectrum. Furthermore, the mixing ratio of H2O is
predominantly dictated by high leverage points at masses 18 and 17 and not
significantly affected by these lower mass channels. For these reasons, we
chose to adopt the H2 mixing ratio of ref. [4] and exclude masses 1 and 2 from
our analysis (Extended Data Fig. 1).

###  IV.3 Spectral library

As discussed in the main text, species that are present within the INMS
spectrum cannot be identified unless explicitly included within a reference
library used for comparison. However, large spectral libraries suffer from
high dimensionality; the inclusion of too many library species leads to models
that are unnecessarily complex and prone to overfitting. This phenomenon is
similar to how any univariate relationship can be fit with zero residual error
using a sufficiently high order polynomial. Although overfitting can be
ameliorated by using regularizing heuristics such as the AICc, chemically
implausible species may still be incorporated into the final model if they are
included within the initial library. As an additional hindrance, the inclusion
of many (possibly collinear) model parameters reduces statistical power and
raises the computational intensity of the analysis. It is therefore desirable
to constrain the library of candidate species as much as possible based on
prior knowledge and the results of previous studies.

Our library consists of a carefully chosen set of 50 compounds, including the
most recent list of INMS-detectable candidate plume species presented in ref.
[34]. All library species and their reasons for inclusion are presented in
Extended Data Table 1. Because all counts observed above 46 u during the slow
flybys are below the estimated minimum uncertainty, no species with base peaks
above this threshold were included in this analysis. The spectra in our
library are 70 eV electron ionization mass spectra collected from the INMS
Refurbished Engineering Unit (REU) and the National Institute of Standards and
Technology (NIST) online database. We adopted the method used in refs. [19,
73] to prioritize the REU spectra when available. REU spectra were obtained
from the most recent INMS calibration file provided on the Planetary Data
System. All spectra were analyzed at 1 u resolution and matched to the mass
detection range of INMS. All spectra were base peak normalized to ensure a
uniform method of comparison.

###  IV.4 Benchmarking model complexity

In deducing limits on INMS model complexity, we constructed all possible
combinations of plume species up to $d=15$ from a candidate library of 50
plausible compounds (Extended Data Table 1). To reduce the computational
intensity associated with this analysis, we assumed H2O, CO2, and CH4 to be
present in each model. H2O and CO2 have been verified at Enceladus by
independent Cassini instruments [74, 75], and CH4 is among the most consistent
INMS observations, having been detected on every flyby that sampled the plume
[6, 5, 4]. This resulted in a total of
$\sum_{d=3}^{15}\frac{47!}{(d-3)!(50-d)!}\approx 7.7\times 10^{10}$ models for
direct comparison. In order to efficiently sample the parameter space of more
complex models, we implemented a forward modeling stepwise selection procedure
[13]. Starting from the three-component model of H2O$+$CO2$+$CH4, the next
most complex model was constructed by including an additional species that,
when added to the model, resulted in the greatest decrease in the variance-
weighted sum of squared residuals. This selection process was
repeated—sequentially incorporating new species based on their influence on
the sum of squared residuals—to produce a set of nested models, each
containing one more species than the last. Notably, the sum of squared
residuals (i.e., the training error) was only used to compare models of equal
complexity and therefore selects the same species as would comparisons based
on the AICc. As shown in Fig. 2, the results of this forward modeling
algorithm resemble those of the exhaustive model search for $d<16$.
Extrapolation of these results to more complex models confirms a monotonic
decrease in relative model likelihood for $d>13$.

###  IV.5 Multi-model inference

Initial consideration of the exhaustive set of possible models reduces user
bias and ensures that all plausible compounds are accounted for. Such a
complete analysis also allows for a robust determination of optimal model
complexity (Fig. 2). However, sound statistical inferences cannot—and should
not—be made using the exhaustive model list. The inclusion of extraneous, low
likelihood models into the final inferential set precludes reliable parameter
estimation due to the well-known bias-variance tradeoff phenomenon (for a
complete discussion see, e.g., ref. [27]). Therefore, in order to extract a
reduced set of high likelihood models suitable for multi-model inference, we
eliminated all models with $\lambda<1/e^{3}$, as they are exponentially less
likely to explain the INMS data. Monte Carlo studies have demonstrated that
this threshold corresponds to a $\sim$95–99% model confidence interval,
leading to its broad acceptance within the literature [27, 31, 32, 33]. An
extensive post-hoc analysis further suggests that our results are stable to
alternative model selection criteria (see Supplementary Information).

In order to determine mixing ratios that properly account for the ambiguity
induced by model degeneracy, we computed model-averaged regression
coefficients based on the relative likelihood of each model,

$\bar{\beta}_{k}=\sum^{R}_{j=1}w_{j}\beta_{k,j}.$ (5)

Here, $\beta_{k,j}$ is the regression coefficient for species $k$ in model
$j$, $R$ is the total number of most probable models (those with
$\lambda>1/e^{3}$), and $w_{j}=\lambda_{j}/\sum_{m}^{R}\lambda_{m}$ is the
Akaike weight [27] of each model. Because the observed INMS spectrum consists
of sample data taken from the unknown population distribution of plume
contents, the minimum AICc model is a statistical random variable that
estimates the actual (also unknown) model that minimizes the Kullback-Leibler
divergence with the true distribution. Therefore, the Akaike weight may be
interpreted as the probability that a given model minimizes this information
loss between the model and the unknown, true distribution from which the
observed INMS data were sampled. The probability that each species is present
in the true minimum AICc model follows accordingly as,

$\mathcal{P}_{k}=\sum^{R}_{j=1}w_{j}\Theta(\beta_{k,j})$ (6)

where $\Theta(\beta_{k,j})$ is the Heaviside step-function that equals 1 when
$\beta_{k,j}>0$ and is zero otherwise. We then computed uncertainties in the
model parameters using the unconditional standard error estimator [27],

$\mathrm{SE}(\bar{\beta}_{k})=\sum^{R}_{j=1}w_{j}\sqrt{\mathrm{var}(\beta_{k,j})+(\beta_{k,j}-\bar{\beta}_{k})^{2}}$
(7)

where $\mathrm{var}(\beta_{k,j})$ is the variance of the regression
coefficient conditional on model $M_{j}$. Here, $\mathrm{var}(\beta_{k,j})$
characterizes the intra-model uncertainty associated with maximum likelihood
estimation, whereas the term $(\beta_{k,j}-\bar{\beta}_{k})^{2}$ quantifies
inter-model variability due to the presence of additional high likelihood
models. Importantly, the results presented in this work incorporate the
relative probabilities of each model and account for ambiguities in the model
selection process.

###  IV.6 Conversion to mixing ratios

The formula for converting INMS counts to ambient densities is described in
ref. [76] and given by,

$n_{k}=\left(\frac{T_{0}}{T_{a}}\right)^{1/2}\frac{1}{D_{k}}\frac{X_{k}}{s_{k}}$
(8)

where $X_{k}$ is the count rate of species $k$ (measured at the base peak),
$s_{k}$ is the INMS sensitivity, $D_{k}$ is the ram enhancement factor,
$T_{a}$ is the ambient temperature, and $T_{0}$ is room temperature (293 K).
When available, INMS sensitivities were obtained from refs. [73, 19, 18] and
otherwise estimated based on the electron impact ionization cross section
procedure of Fitch and Sauter (see Eq. 2 of ref. [77]) adapted to INMS data in
refs. [73, 19]. For one species (PH3), cross section data was taken from ref.
[78]. As done in ref. [73], a 30% uncertainty was implemented on all
sensitivities estimated from NIST spectra.

The count rate is related to the model-averaged regression coefficient of
Equation (5) by $X_{k}=\bar{\beta}_{k}c_{0}$ where $c_{0}$ is the standardized
(species-independent) base peak count rate of each library spectrum. The
mixing ratio for each species $m_{k}$ (scaled to include the H2 mixing ratio
$m_{\text{H}_{2}}=0.9\%$ given in ref. [4]) is then,

$\displaystyle m_{k}$
$\displaystyle=(100-m_{\text{H}_{2}})\frac{n_{k}}{\sum_{l}n_{l}}$
$\displaystyle=(100-m_{\text{H}_{2}})\frac{\bar{\beta}_{k}}{D_{k}s_{k}}\sum_{l}\frac{D_{l}s_{l}}{\bar{\beta}_{l}}.$
(9)

At suprathermal spacecraft speed and small ram angle (conditions valid during
the E14, E17, and E18 flybys), the ram enhancement factor is approximately [4,
76],

$D_{k}\sim 0.7u\sqrt{\frac{2\pi\mu_{k}}{k_{B}T_{a}}}$ (10)

for spacecraft speed $u$ and molecular mass $\mu_{k}$. The final expression
for the mixing ratio of each species is therefore,

$m_{k}=(100-m_{\text{H}_{2}})\frac{\bar{\beta}_{k}}{\sqrt{\mu_{k}}s_{k}}\sum_{l}\frac{\sqrt{\mu_{l}}s_{l}}{\bar{\beta}_{l}}.$
(11)

Future updates to INMS sensitivity coefficients or ram enhancement factors may
change the relative mixing ratios presented here but will not affect which
species have been identified—or the evidence for their detection—as these
depend only on the set of $\\{\bar{\beta}_{k}\\}$.

##  V Data availability

All INMS REU spectra are publicly available from the Planetary Data System at
https://doi.org/10.17189/1519605. All NIST spectra are publicly available from
the NIST Chemistry WebBook at https://doi.org/10.18434/T4D303.

##  VI Code availability

The code for this work was implemented using open source Python libraries
available at https://pypi.org/ and is available upon request.

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##  VII Additional Information

Correspondence and requests for materials should be addressed to J.S.P.

##  VIII Acknowledgements

The authors thank J. Hunter Waite and Brian A. Magee for their help on
interpreting INMS instrument effects. J.S.P. thanks Masao Sako and Kiri L.
Wagstaff for useful discussions and statistical insight, as well as Dimitar D.
Sasselov for helpful discussions on prebiotic chemistry. All authors
acknowledge the support of the Cassini Data Analysis Program (NNN13D466T) and
the Jet Propulsion Laboratory, California Institute of Technology, under a
contract with NASA. T.A.N. was also supported by an appointment to the NASA
Postdoctoral Fellowship Program at the Jet Propulsion Laboratory administered
by Oak Ridge Associated Universities and Universities Space Research
Association through a contract with NASA. K.P.H. also acknowledges support
from the NASA Astrobiology Program (80NSSC19K1427) and the Europa Lander Pre-
Project, managed by the Jet Propulsion Laboratory, California Institute of
Technology, under a contract with NASA.

##  IX Author contributions

J.S.P, T.A.N., and K.P.H contributed to the conceptual development of the
study, the interpretation of results, and to the final version of the
manuscript. J.S.P. developed the methodology, performed the mathematical
analysis, and wrote the first version of the manuscript.

##  X Competing interests

The authors declare no competing interests.

Extended Data Table 1: Complete list of species included in the analysis. Those taken from ref. [34] comprise the most recently published list of INMS-detectable species in the plume. REU: INMS Refurbished Engineering Unit; NIST: National Institute of Standards and Technology online database. Species | Name | Mass (u) | Source | $s_{k}$ |  $s_{k}$ Ref. | Reason/Ref.
---|---|---|---|---|---|---
CH4 | Methane | 16 | REU | 6.01$\times 10^{4}$ |  [73] | [34, 4]
NH3 | Ammonia | 17 | REU | 4.77$\times 10^{4}$ |  [73] | [34, 4]
H2O | Water | 18 | REU | 4.34$\times 10^{4}$ |  [73] | [34, 4]
C2H2 | Acetylene | 26 | REU | 8.81$\times 10^{4}$ |  [73] | [34, 6]
HCN | Hydrogen Cyanide | 27 | REU | 5.20$\times 10^{4}$ |  [73] | [34, 5]
C2H4 | Ethylene | 28 | REU | 6.21$\times 10^{4}$ |  [73] | [34, 4]
CO | Carbon Monoxide | 28 | REU | 6.60$\times 10^{4}$ |  [73] | [34, 4]
N2 | Nitrogen | 28 | REU | 6.29$\times 10^{4}$ |  [73] | [34, 4]
CH2O | Formaldehyde | 30 | NIST | 3.21$\times 10^{4}$ |  [73] | [34, 5]
NO | Nitric Oxide | 30 | NIST | 5.28$\times 10^{4}$ | $-$ | [34]
C2H6 | Ethane | 30 | REU | 6.94$\times 10^{4}$ |  [73] | [34, 5]
CH5N | Methylamine | 31 | NIST | 4.07$\times 10^{4}$ |  [73] | [34]
CH3OH | Methanol | 32 | NIST | 4.17$\times 10^{4}$ |  [73] | [34, 4]
H2S | Hydrogen Sulfide | 34 | NIST | 5.38$\times 10^{4}$ |  [73] | [34, 5]
PH3 | Phospine | 34 | NIST | 5.98$\times 10^{4}$ | $-$ | [34]
O2 | Oxygen | 36 | NIST | 5.03$\times 10^{4}$ |  [73] | [34, 4]
36Ar | Argon 36 | 36 | REU | 7.87$\times 10^{4}$ |  [18] | [34, 4]
C3H4 | Propyne | 40 | REU | 4.19$\times 10^{4}$ |  [73] | [34, 5]
40Ar | Argon 40 | 40 | REU | 7.29$\times 10^{4}$ |  [19] | [34, 5]
CH3CN | Acetonitrile | 41 | REU | 5.28$\times 10^{4}$ |  [73] | [34, 57]
C2H2O | Ketene | 42 | NIST | 4.37$\times 10^{4}$ |  [73] | [34]
C3H6 | Propene | 42 | NIST | 4.06$\times 10^{4}$ |  [73] | [34, 5]
CH2N2 | Cyanamide | 42 | NIST | 7.55$\times 10^{4}$ | $-$ | Organic synthesis [71, 69]
C2H4O | Acetaldehyde | 44 | NIST | 4.42$\times 10^{4}$ |  [73] | [34, 5]
C3H8 | Propane | 44 | REU | 5.11$\times 10^{4}$ |  [73] | [34, 6]
CO2 | Carbon Dioxide | 44 | REU | 7.09$\times 10^{4}$ |  [73] | [34, 4]
C2H7N | Ethylamine | 45 | NIST | 1.71$\times 10^{4}$ |  [73] | [34]
CH3NO | Formamide | 45 | NIST | 6.22$\times 10^{4}$ |  [73] | HCN decomposition [54]
C2H6O | Ethanol | 46 | NIST | 1.62$\times 10^{4}$ |  [73] | [34, 5]
CH2O2 | Formic Acid | 46 | NIST | 2.94$\times 10^{4}$ |  [73] | HCN decomposition [54]
C4H8 | 1-Butene | 56 | NIST | 3.73$\times 10^{4}$ |  [73] | [34, 5]
C2H6N2 | Azomethane | 58 | NIST | 8.46$\times 10^{4}$ | $-$ | [34]
C3H6O | Acetone | 58 | NIST | 6.70$\times 10^{4}$ |  [73] | [34, 5]
C4H10 | Isobutane | 58 | NIST | 3.24$\times 10^{5}$ |  [73] | [34, 5]
C2H4O2 | Acetic Acid | 60 | NIST | 4.51$\times 10^{4}$ |  [73] | [34, 5]
C3H8O | 1-Propanol | 60 | NIST | 8.80$\times 10^{5}$ |  [73] | [34, 5]
C2H7NO | Monoethanolamine | 61 | NIST | 1.67$\times 10^{3}$ | $-$ | [34]
C2H6O2 | 1,2-Ethanediol | 62 | NIST | 5.77$\times 10^{5}$ |  [73] | [34, 5]
C5H10 | Cyclopentane | 70 | NIST | 4.39$\times 10^{4}$ |  [73] | [34]
C4H9N | Pyrrolidine | 71 | NIST | 1.18$\times 10^{3}$ | $-$ | [34]
C5H12 | Pentane | 72 | NIST | 1.53$\times 10^{4}$ |  [73] | [34, 5]
C4H10O | 1-Butanol | 74 | NIST | 6.14$\times 10^{5}$ | $-$ | [34]
C2H5NO2 | Glycine | 75 | NIST | 6.14$\times 10^{5}$ |  [73] | [34, 5]
C3H5Cl | Allyl Chloride | 76 | NIST | 9.63$\times 10^{4}$ | $-$ | [34]
C5H9N | Butyl Isocyanide | 83 | NIST | 8.34$\times 10^{4}$ | $-$ | Hydrocarbon irradiation [57]
C4H6O2 | 2,3-Butanedione | 86 | NIST | 3.91$\times 10^{4}$ |  [73] | [34, 5]
C3H7NO2 | Alanine | 89 | NIST | 1.73$\times 10^{3}$ | $-$ | [34]
C8H18 | Octane | 114 | NIST | 1.65$\times 10^{3}$ | $-$ | [34]
C6H12N4 | Methenamine | 140 | NIST | 3.86$\times 10^{3}$ | $-$ | [34]
C6H14N2O2 | Lysine | 146 | NIST | 1.34$\times 10^{3}$ | $-$ | Amino acid [79]
Extended Data Table 2: Modeling results for the complete spectral library.
Mixing ratios are mean $\pm$ SE and scaled to include the 0.9% H2 abundance of
ref. [4] unless otherwise specified. For compounds with upper limits (mean + 3
SE) less than the minimum INMS count uncertainty (2 counts), upper limits were
instead estimated relative to H2O based on a 2 count signal after correcting
for ram enhancement and instrument sensitivities. The quantities $N_{d}$
indicate the number of high likelihood ($\lambda>1/e^{3}$) models containing
each species for models of dimension $d$. Values in parentheses denote the
total number of high likelihood models for each $d$.

Species | Probability | Mixing Ratio (%) |  $N_{10}$ (4) |  $N_{11}$ (146) |  $N_{12}$ (1348) |  $N_{13}$ (5956) |  $N_{14}$ (11134) |  $N_{15}$ (7744)
---|---|---|---|---|---|---|---|---
CH4 | 1 | $0.11\pm 0.02$ | 4 | 146 | 1348 | 5956 | 11134 | 7744
NH3 | 1 | $1.8\pm 0.1$ | 4 | 146 | 1348 | 5956 | 11134 | 7744
H2O | 1 | $95.9\pm 0.3$ | 4 | 146 | 1348 | 5956 | 11134 | 7744
C2H2 | 0.92 | $0.023\pm 0.005$ | 3 | 110 | 1103 | 5098 | 10048 | 7646
HCN | 1 | $0.11\pm 0.02$ | 4 | 146 | 1348 | 5956 | 11134 | 7744
C2H4 | 0.10 | $<0.05$ | 1 | 36 | 260 | 965 | 1326 | 323
CO | 1 | $0.72\pm 0.07$ | 4 | 146 | 1348 | 5956 | 11134 | 7744
N2 | 0.02 | $<0.005$ | 0 | 0 | 15 | 114 | 255 | 255
CH2O | 0.07 | $<0.004$ | 0 | 2 | 73 | 446 | 921 | 666
NO | 0.05 | $<0.002$ | 0 | 7 | 99 | 349 | 695 | 340
C2H6 | 0.70 | $0.013\pm 0.010$ | 4 | 102 | 838 | 3850 | 7484 | 5757
CH5N | 0.14 | $<0.003$ | 0 | 6 | 102 | 647 | 1673 | 1329
CH3OH | 0.44 | $<0.005$ | 2 | 66 | 691 | 2844 | 4634 | 3285
H2S | 0.37 | $<0.003$ | 0 | 16 | 226 | 1396 | 3701 | 4362
PH3 | 0.24 | $<0.002$ | 0 | 12 | 147 | 963 | 2825 | 2493
O2 | 0.64 | $<0.008$ | 1 | 42 | 511 | 3206 | 7445 | 5575
36Ar | 0.03 |  $<4\times 10^{-4}$a | 0 | 1 | 21 | 155 | 370 | 396
C3H4 | 0.16 | $<0.002$ | 0 | 3 | 127 | 919 | 2416 | 1000
40Ar | 0.58 | $<0.004$ | 0 | 12 | 257 | 2021 | 6293 | 6443
CH3CN | 0.16 | $<0.003$ | 0 | 5 | 133 | 931 | 2226 | 1234
C2H2O | 0.02 |  $<7\times 10^{-4}$a | 0 | 0 | 15 | 116 | 260 | 226
C3H6 | 0.87 | $0.004\pm 0.002$ | 4 | 143 | 1258 | 5293 | 9278 | 6461
CH2N2 | 0.06 | $5\times 10^{-4}$ | 0 | 0 | 23 | 250 | 780 | 801
C2H4O | 0.08 | $<0.002$ | 0 | 11 | 119 | 574 | 1056 | 442
C3H8 | 0.23 | $<0.005$ | 0 | 21 | 295 | 1332 | 2684 | 1669
CO2 | 1 | $0.45\pm 0.04$ | 4 | 146 | 1348 | 5956 | 11134 | 7744
C2H7N | 0.02 |  $<0.002$a | 0 | 0 | 15 | 144 | 255 | 255
CH3NO | 0.02 |  $<5\times 10^{-4}$a | 0 | 0 | 15 | 114 | 255 | 255
C2H6O | 0.02 |  $<0.002$a | 0 | 0 | 18 | 127 | 268 | 227
CH2O2 | 0.02 |  $<0.001$a | 0 | 0 | 15 | 144 | 255 | 255
C4H8 | 0.03 |  $<7\times 10^{-4}$a | 0 | 1 | 15 | 130 | 330 | 313
C2H6N2 | 0.24 | $<0.002$ | 0 | 15 | 167 | 978 | 2295 | 2587
C3H6O | 0.17 | $<0.001$ | 0 | 14 | 141 | 748 | 1739 | 1707
C4H10 | 0.08 | $<0.01$ | 0 | 7 | 99 | 520 | 933 | 689
C2H4O2 | 0.02 |  $<6\times 10^{-4}$a | 0 | 1 | 16 | 116 | 261 | 225
C3H8O | 0.11 | $<0.005$ | 0 | 5 | 74 | 481 | 1347 | 1132
C2H7NO | 0.23 | $<0.002$ | 0 | 25 | 299 | 1432 | 2744 | 1799
C2H6O2 | 0.20 | $<0.01$ | 1 | 19 | 156 | 944 | 2203 | 1899
C5H10 | 0.03 |  $<6\times 10^{-4}$a | 0 | 0 | 17 | 155 | 377 | 323
C4H9N | 0.06 |  $<2\times 10^{-4}$a | 0 | 7 | 83 | 415 | 732 | 565
C5H12 | 0.07 |  $<0.002$a | 0 | 4 | 66 | 410 | 897 | 553
C4H10O | 0.03 |  $<4\times 10^{-4}$a | 0 | 3 | 36 | 203 | 403 | 305
C2H5NO2 | 0.08 | $<0.02$ | 0 | 14 | 148 | 603 | 911 | 553
C3H5Cl | 0.23 | $<0.001$ | 0 | 2 | 103 | 845 | 2749 | 2789
C5H9N | 0.05 |  $<3\times 10^{-4}$a | 0 | 4 | 58 | 313 | 666 | 418
C4H6O2 | 0.08 |  $<6\times 10^{-4}$a | 0 | 7 | 100 | 545 | 902 | 607
C3H7NO2 | 0.05 | $<5\times 10^{-4}$ | 0 | 2 | 41 | 267 | 668 | 387
C8H18 | 0.05 |  $<1\times 10^{-4}$a | 0 | 4 | 58 | 315 | 591 | 374
C6H12N4 | 0.05 |  $<5\times 10^{-5}$a | 0 | 0 | 20 | 218 | 660 | 670
C6H14N2O2 | 0.02 |  $<1\times 10^{-4}$a | 0 | 1 | 15 | 116 | 261 | 226

  * a

Denotes number density relative to H2O.

Extended Data Fig. 1: Full mass range INMS spectrum and analysis of residuals.
(a) The black silhouette shows the full mass range of the INMS spectrum used
in this work. Shaded gray bars indicate count values at 1 and 2 u that were
omitted from this work (see Methods). The spectrum was adapted from ref. [12].
Error bars show 1$\sigma$ Gaussian uncertainty in the observed count rates.
The minimum count uncertainty was estimated as $\sim$2 counts from the count
rates of noisy mass channels at masses $>46$ u. Red circles show the
reconstructed model fit based on the mixing ratios of Extended Data Table 2.
(b) Scatterplot of the standardized residuals produced by fitting the model.
Only mass channels with counts above the minimum uncertainty are shown. There
is no discernable pattern amongst the residuals or evidence of
heteroscedasticity. (c) Histogram of the standardized residuals (red bars)
compared to a reference Gaussian distribution with zero mean (black curve).
The residuals show good agreement with the Gaussian distribution, indicating a
robust model fit. The black arrow indicates a potential outlier at mass 16,
which likely results from the standardization of the slow flyby count rates at
neighboring mass channels. Extended Data Fig. 2: Comparison of model
performance without HCN. Blue circles show the maximum relative likelihoods
across all models for each value of $d$ species (as in Fig. 2a). Red diamonds
indicate the highest likelihood models without HCN. All models without HCN
exhibit exceptionally poor performance, with a maximum relative likelihood
peaking at $\lambda=6.1\times 10^{-4}$ for $d=12$.


# Gauge-invariant formulation for the gravitational wave equations

Junpei Harada  Health Sciences University of Hokkaido, 1757 Kanazawa, Tobetsu-
cho, Ishikari-gun, Hokkaido 061-0293, Japan<EMAIL_ADDRESS>
###### Abstract

A gauge-invariant formulation for the gravitational wave equations is
presented. Using this approach, weak, plane wave solutions in a vacuum are
derived in various theories. These include general relativity with two modes
of polarization with helicity $\pm 2$, Yang’s theory with three modes of
polarization with helicity $\pm 2$ and 0, and so-called “general metric
theories” with six modes of polarization with helicity $\pm 2$, $\pm 1$, and
two 0’s. To identify the polarizations of gravitational waves, it is
explicitly demonstrated how the gauge-invariant approach reproduces the
earlier results.

  * May 2024

##  1 Introduction

To identify the polarizations of gravitational waves, the conventional
approach uses the wave equations expressed in terms of the metric perturbation
$h_{\mu\nu}$, along with some convenient gauge-fixing condition. This is the
standard approach adopted in textbooks (e.g., [1, 2, 3]). However,
gravitational waves are observables, suggesting it should be possible to
express the gravitational wave equation in a gauge-invariant manner.
Nevertheless, the author realized that this approach has been less frequently
mentioned in the literature or textbooks.

Motivated by this observation, in this work, a gauge-invariant formulation for
the gravitational wave equations is presented. Due to its gauge invariance,
this formulation avoids the need for gauge fixing when dealing with the wave
equations. Using this formulation, we derive weak, plane wave solutions in a
vacuum for various theories: general relativity, Yang’s gravitational theory
[4], and “general metric theories” [5, 6] (explained later). Our results align
with earlier results obtained through conventional methods. Thus, the results
presented in this work are not new, while the formulation is original. The
gauge-invariant approach and the conventional one are complementary, and both
are useful.

In Sec. 2, we discuss general relativity as an elementary example to formulate
gauge-invariant wave equations. This is an elementary topic, yet the author
could not find it in the standard textbooks. In Sec. 3, Yang’s theory [4] is
chosen to provide the simplest example for describing a polarization mode
distinct from those in general relativity. In Sec. 4, more general theories
are considered. It is commonly stated in the literature that there are six
possible modes of polarization (two tensors, two vectors, and two scalars) in
“general metric theories” for gravity [5, 6]. We clarify the exact meaning of
“general metric theories” in this context, and then derive six modes of
polarization solely from the gauge-invariant wave equations. Section 5
provides a conclusion.

##  2 Gravitational wave equation in general relativity

###  2.1 Gauge-invariant wave equation

In electromagnetism, the wave equations are given by

$\displaystyle\opensquare A_{\mu}$ $\displaystyle=$ $\displaystyle 0,$ (1a)
$\displaystyle\partial_{\mu}A^{\mu}$ $\displaystyle=$ $\displaystyle 0$ (1b)

where $\opensquare=\partial^{\mu}\partial_{\mu}$ and $A_{\mu}$ is a vector
potential. Here, (1a) represents Maxwell’s equations in a vacuum, and (1b) is
the gauge fixing condition proposed by Ludvig Lorenz. The equation (1a) holds
only for $A_{\mu}$ in the Lorenz gauge.

The gauge-invariant equations for describing the electromagnetic waves can be
derived from the identities:

$\displaystyle\partial_{\rho}F_{\mu\nu}+\partial_{\mu}F_{\nu\rho}+\partial_{\nu}F_{\rho\mu}=0$
(1b)

where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$. Applying
$\partial^{\rho}\equiv\eta^{\rho\mu}\partial_{\mu}$ to (1b) and using
Maxwell’s equations ($\partial_{\mu}F^{\mu\nu}=0$), we obtain

$\displaystyle\opensquare F_{\mu\nu}=0.$ (1c)

Unlike (1a), this wave equation is gauge invariant because $F_{\mu\nu}$ is an
Abelian field strength and thus gauge invariant.

In the case of gravity, when gravity is weak, we can suppose that the metric
is close to the flat metric $\eta_{\mu\nu}$:

$\displaystyle g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ (1d)

where $|h_{\mu\nu}|\ll 1$. To first order in $h$, the Ricci tensor is

$\displaystyle
R_{\mu\nu}=\frac{1}{2}\left(\partial_{\mu}\partial_{\rho}h^{\rho}{}_{\nu}+\partial_{\nu}\partial_{\rho}h^{\rho}{}_{\mu}-\partial_{\mu}\partial_{\nu}h^{\lambda}{}_{\lambda}-\opensquare
h_{\mu\nu}\right).\quad$ (1e)

The Einstein equation $R_{\mu\nu}=0$ reduces to

$\displaystyle\opensquare h_{\mu\nu}$ $\displaystyle=$ $\displaystyle 0,$
(1fa) $\displaystyle\partial_{\mu}h^{\mu}{}_{\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\partial_{\nu}h^{\lambda}{}_{\lambda}$ (1fb)

where (1fb) is known as the harmonic coordinate condition or the Hilbert gauge
condition (sometimes referred to as the Lorenz gauge condition, but
historically, “Hilbert” may be more appropriate). The wave equation (1fa)
holds only for $h_{\mu\nu}$ satisfying (1fb).

Equations (1fa) and (1fb) correspond to (1a) and (1b), respectively. Our aim
now is to find a gravitational analogue of $\opensquare F_{\mu\nu}=0$.

Similar to electromagnetism, the gauge-invariant equations can be derived from
the Bianchi identities:

$\displaystyle\partial_{\lambda}R_{\mu\nu\rho\sigma}+\partial_{\rho}R_{\mu\nu\sigma\lambda}+\partial_{\sigma}R_{\mu\nu\lambda\rho}=0$
(1fg)

where terms of ${\cal O}(h^{2})$ are ignored. Throughout this paper, the
covariant derivative $\nabla_{\mu}$ will be replaced with the partial
derivative $\partial_{\mu}$, as the differences between the two derivatives
are of higher order in $h$.

Applying $\partial^{\lambda}$ to equation (1fg), we obtain

$\displaystyle\opensquare
R_{\mu\nu\rho\sigma}+\partial_{\rho}\partial^{\lambda}R_{\mu\nu\sigma\lambda}+\partial_{\sigma}\partial^{\lambda}R_{\mu\nu\lambda\rho}=0.$
(1fh)

Using the identities
$\partial^{\lambda}R_{\mu\nu\lambda\rho}=\partial_{\mu}R_{\nu\rho}-\partial_{\nu}R_{\mu\rho}+{\cal
O}(h^{2})$, equation (1fh) reads:

$\displaystyle\opensquare
R_{\mu\nu\rho\sigma}-\partial_{\mu}\partial_{\rho}R_{\nu\sigma}+\partial_{\nu}\partial_{\rho}R_{\mu\sigma}+\partial_{\mu}\partial_{\sigma}R_{\nu\rho}-\partial_{\nu}\partial_{\sigma}R_{\mu\rho}=0.$
(1fi)

When $R_{\mu\nu}=0$ holds, equation (1fi) reduces to

$\displaystyle\opensquare C_{\mu\nu\rho\sigma}=0$ (1fj)

where $C_{\mu\nu\rho\sigma}$ is the Weyl tensor. This is the wave equation in
general relativity, expressed in terms of the curvature tensor rather than the
metric perturbation $h_{\mu\nu}$.

To first order in $h$, it is straightforward to confirm that the Riemann
tensor (and consequently the Weyl tensor) is invariant under the gauge
transformations:

$\displaystyle x^{\mu}\rightarrow x^{\prime\mu}$ $\displaystyle=$
$\displaystyle x^{\mu}+\varepsilon^{\mu}(x),$ (1fka) $\displaystyle
h_{\mu\nu}\rightarrow h_{\mu\nu}^{\prime}$ $\displaystyle=$ $\displaystyle
h_{\mu\nu}-\partial_{\mu}\varepsilon_{\nu}(x)-\partial_{\nu}\varepsilon_{\mu}(x)$
(1fkb)

where $|h_{\mu\nu}^{\prime}|\ll 1$. Therefore, equation (1fj) is gauge
invariant. In this sense, the wave equation (1fj) is a gravitational analogue
of $\opensquare F_{\mu\nu}=0$.

The Weyl tensor $C_{\mu\nu\rho\sigma}$ vanishes identically in two or three
dimensions. Therefore, according to the wave equation (1fj), gravitational
waves do not propagate in two or three dimensions. However, in four or more
dimensions, gravitational waves do propagate. We will now examine the plane
wave solutions in four dimensions.

###  2.2 Polarizations of gravitational wave

The general solution of equation (1fj) is a linear superposition of solutions
of the form

$\displaystyle
C_{\mu\nu\rho\sigma}=e_{\mu\nu\rho\sigma}\exp(ik_{\lambda}x^{\lambda})+{\rm
c.c.}$ (1fkl)

where “c.c.” denotes the complex conjugate, and $e_{\mu\nu\rho\sigma}$ is the
polarization tensor. This satisfies equation (1fj) if

$\displaystyle k_{\lambda}k^{\lambda}=0.$ (1fkm)

The Weyl tensor $C_{\mu\nu\rho\sigma}$ consists of ten independent components
in four dimensions, and so does the polarization tensor
$e_{\mu\nu\rho\sigma}$. Without loss of generality, we can choose, for
example, the ten independent components as:

$\displaystyle
e_{0101},e_{0102},e_{0123},e_{0131},e_{0231},e_{1223},e_{1231},e_{1212},e_{1201},e_{1202}.$
(1fkn)

To first order in $h$, these components are invariant under the
transformations (1fka) and (1fkb). When $R_{\mu\nu}=0$, the number of
independent components of $e_{\mu\nu\rho\sigma}$ should reduce from ten to
two, corresponding to two physical degrees of freedom in general relativity.
This will be demonstrated below.

When $R_{\mu\nu}=0$, the Bianchi identities (1fg) read

$\displaystyle\partial_{\lambda}C_{\mu\nu\rho\sigma}+\partial_{\rho}C_{\mu\nu\sigma\lambda}+\partial_{\sigma}C_{\mu\nu\lambda\rho}=0.$
(1fko)

Contracting the indices of this equation yields

$\displaystyle\partial^{\mu}C_{\mu\nu\rho\sigma}=0.$ (1fkp)

The solution (1fkl) satisfies (1fko) if

$\displaystyle
k_{\lambda}e_{\mu\nu\rho\sigma}+k_{\rho}e_{\mu\nu\sigma\lambda}+k_{\sigma}e_{\mu\nu\lambda\rho}=0$
(1fkq)

and satisfies (1fkp) if

$\displaystyle k^{\mu}e_{\mu\nu\rho\sigma}=0.$ (1fkr)

Thus, the polarization tensor $e_{\mu\nu\rho\sigma}$ is orthogonal to the wave
vector $k^{\mu}$, indicating that the wave is transverse.

For instance, consider a wave traveling in the $+z$ direction, with the wave
vector

$\displaystyle k^{\mu}=(k,0,0,k),\quad k>0.$ (1fks)

In this case, the conditions (1fkq) allow us to express the
$e_{\mu\nu\rho\sigma}$ listed in (1fkn) in terms of $e_{0101}$ and $e_{0102}$:

$\displaystyle e_{0131}=-e_{0101},$ $\displaystyle
e_{0123}=-e_{0231}=e_{0102},$ $\displaystyle
e_{1223}=e_{1231}=e_{1212}=e_{1201}=e_{1202}=0.$ (1fkt)

Hence, only $e_{0101}$ and $e_{0102}$ are independent.

To find the helicity of gravitational waves, we consider a rotation about the
$z$-axis:

$\displaystyle
e_{\mu\nu\rho\sigma}^{\prime}=\Lambda_{\mu}{}^{\alpha}\Lambda_{\nu}{}^{\beta}\Lambda_{\rho}{}^{\gamma}\Lambda_{\sigma}{}^{\delta}e_{\alpha\beta\gamma\delta},$
(1fku)

where

$\displaystyle\Lambda_{1}{}^{1}=\Lambda_{2}{}^{2}=\cos\theta,\quad\Lambda_{1}{}^{2}=-\Lambda_{2}{}^{1}=\sin\theta,$
$\displaystyle\Lambda_{0}{}^{0}=\Lambda_{3}{}^{3}=1,\quad{\rm other}\
\Lambda_{\mu}{}^{\nu}=0.$ (1fkv)

From equations (1fku) and (1fkv), we find that

$\displaystyle e_{0101}^{\prime}\mp ie_{0102}^{\prime}=\exp(\pm
2i\theta)(e_{0101}\mp ie_{0102}),$ (1fkw)

representing that the waves have helicity $\pm 2$.

In summary, we have shown that the wave equation in general relativity is
expressed by (1fj), and the plane wave solution (1fkl), along with (1fkt),
represents two transverse modes of polarization with helicity $\pm 2$. To
first order in $h$, the polarization tensor $e_{\mu\nu\rho\sigma}$ is
invariant under the gauge transformation. Thus, this formulation is free from
gauge fixing, as specified in (1fb).

##  3 Scalar wave in Yang’s theory

In general relativity, the Ricci tensor vanishes in a vacuum, and
gravitational waves are described by the Weyl tensor $C_{\mu\nu\rho\sigma}$,
as shown in (1fj). However, in certain gravitational theories, the Ricci
tensor may not necessarily vanish even in a vacuum. In such theories,
additional modes of polarization can arise. In this section, we take Yang’s
gravitational theory as an example of such cases.

Yang’s field equations [4] in a vacuum are given by

$\displaystyle\nabla_{\mu}R_{\nu\rho}=\nabla_{\nu}R_{\mu\rho}.$ (1fkx)

From the Bianchi identities, this is equivalent to

$\displaystyle\nabla_{\lambda}R^{\lambda}{}_{\rho\mu\nu}=0.$ (1fky)

Furthermore, this can be expressed as

$\displaystyle
2\nabla_{\lambda}C^{\lambda}{}_{\rho\mu\nu}+\frac{1}{6}(g_{\rho\mu}\partial_{\nu}-g_{\rho\nu}\partial_{\mu})R=0.$
(1fkz)

We can readily observe that all vacuum solutions of the Einstein equations
($R_{\mu\nu}=0$ or $R_{\mu\nu}=\Lambda g_{\mu\nu}$) satisfy (1fkx), and all
vacuum solutions of Nordström’s theory ($C_{\mu\nu\rho\sigma}=0,R=0$) satisfy
(1fkz).

When Yang’s equation holds, Eq. (1fh) simplifies to

$\displaystyle\opensquare R_{\mu\nu\rho\sigma}=0$ (1fkaa)

where $R_{\mu\nu\rho\sigma}$ satisfies (1fky). This is the wave equation in
Yang’s theory. To first order in $h$, this is gauge invariant.

###  3.1 Riemann tensor

We consider weak, plane wave solutions to Eq. (1fkaa). The general solution of
(1fkaa) is as follows:

$\displaystyle
R_{\mu\nu\rho\sigma}=e_{\mu\nu\rho\sigma}\exp(ik_{\lambda}x^{\lambda})+{\rm
c.c}.$ (1fkab)

This satisfies (1fkaa) if $k_{\lambda}k^{\lambda}=0$, and satisfies (1fky) if

$\displaystyle k^{\mu}e_{\mu\nu\rho\sigma}=0.$ (1fkac)

Thus, the wave is null and transverse. The solution (1fkab) also satisfies the
Bianchi identities (1fg), and then we have

$\displaystyle
k_{\lambda}e_{\mu\nu\rho\sigma}+k_{\rho}e_{\mu\nu\sigma\lambda}+k_{\sigma}e_{\mu\nu\lambda\rho}=0.$
(1fkad)

The Riemann tensor $R_{\mu\nu\rho\sigma}$ consists of twenty independent
components in four dimensions, as does the polarization tensor
$e_{\mu\nu\rho\sigma}$. However, when Yang’s equation holds, the number of
independent components of $e_{\mu\nu\rho\sigma}$ reduce from twenty to three,
corresponding to three physical degrees of freedom. Hence, another
polarization, distinct from the two tensor modes, appears as follows.

Similarly to Sec. 2, consider a wave propagating in the $+z$ direction, with
the wave vector $k^{\mu}=(k,0,0,k)$. In this case, Eqs. (1fkac) and (1fkad)
enable us to express twenty independent $e_{\mu\nu\rho\sigma}$ in terms of
$e_{0101}$, $e_{0102}$, and $e_{0202}$:

$\displaystyle e_{3131}=-e_{0131}=e_{0101},$ $\displaystyle
e_{0123}=-e_{0231}=-e_{2331}=e_{0102},$ $\displaystyle
e_{0223}=e_{2323}=e_{0202},\quad{\rm other}\ e_{\mu\nu\rho\sigma}=0.$ (1fkae)

Hence, $e_{0101}$, $e_{0102}$, and $e_{0202}$ are independent, corresponding
to three modes of polarization.

To find the helicity of the waves, we consider a rotation about the $z$-axis.
From (1fku) and (1fkv), we find that

$\displaystyle\frac{e_{0101}^{\prime}-e_{0202}^{\prime}}{2}\mp
ie_{0102}^{\prime}=\exp(\pm 2i\theta)\left(\frac{e_{0101}-e_{0202}}{2}\mp
ie_{0102}\right)$ (1fkaf)

representing two tensor modes with helicity $\pm 2$, and

$\displaystyle e_{0101}^{\prime}+e_{0202}^{\prime}=e_{0101}+e_{0202}$ (1fkag)

representing a scalar mode with helicity 0. Thus, in Yang’s theory, there is a
scalar mode in addition to two tensor modes. These three modes are null and
transverse.

###  3.2 Ricci tensor

Another convenient approach to derive the scalar mode is available as follows.

When $R_{\mu\nu}=0$, equation (1fkaa) for the Riemann tensor simplifies to
equation (1fj) for the Weyl tensor. In this case, the solutions are identical
to those of general relativity. Therefore, to identify additional polarization
modes beyond those in general relativity, it is sufficient to consider the
case when $R_{\mu\nu}\not=0$.

Contracting the indices in equation (1fkaa) yields

$\displaystyle\opensquare R_{\mu\nu}=0$ (1fkah)

where $R_{\mu\nu}$ satisfies (1fkx). The plane wave solution of (1fkah) is a
linear superposition of solutions of the form

$\displaystyle R_{\mu\nu}=e_{\mu\nu}\exp(ik_{\lambda}x^{\lambda})+{\rm c.c}.$
(1fkai)

This satisfies (1fkah) if $k_{\lambda}k^{\lambda}=0$ and satisfies (1fkx) if

$\displaystyle k_{\mu}e_{\nu\rho}=k_{\nu}e_{\mu\rho}.$ (1fkaj)

The Ricci tensor consists of ten independent components. This number should
reduce from ten to one, corresponding a scalar mode as follows. Consider a
wave propagating in the $+z$ direction with the wave vector
$k^{\mu}=(k,0,0,k)$. The condition (1fkaj) gives:

$\displaystyle e_{33}=-e_{03}=e_{00},$ $\displaystyle
e_{01}=e_{02}=e_{11}=e_{22}=e_{23}=e_{31}=e_{12}=0.$ (1fkak)

Thus, only $e_{00}$ is independent, and $e_{\mu\nu}$ is traceless.

Under a rotation about the $z$-axis, $e_{\mu\nu}$ transforms as

$\displaystyle
e_{\mu\nu}^{\prime}=\Lambda_{\mu}{}^{\alpha}\Lambda_{\nu}{}^{\beta}e_{\alpha\beta}$
(1fkal)

where $\Lambda_{\mu}{}^{\nu}$ is given by (1fkv), and so we find that

$\displaystyle e_{00}^{\prime}=e_{00}.$ (1fkam)

Hence, $e_{00}$ describes a scalar mode with helicity 0.

In summary, we have shown that plane wave solutions in Yang’s theory have
three modes of polarization: two tensor modes with helicity $\pm 2$, and a
scalar mode with helicity 0. This scalar mode is described by the traceless
Ricci tensor. These three modes are null and transverse.

##  4 Six polarizations in general metric theories

In the literature, it is commonly stated that there are six modes of
polarization (two tensors, two vectors, and two scalars) in “general metric
theories” of gravitation. This conclusion was initially reached in Ref. [5, 6]
using the Newman–Penrose formalism [7]. In this section, we begin by
clarifying the meaning of “general metric theories” in this context. Then, we
derive six modes of polarization solely from the gauge-invariant wave
equations.

###  4.1 Riemann tensor

We find from (1fh) that if the following relation

$\displaystyle\partial_{\rho}\partial^{\lambda}R_{\mu\nu\sigma\lambda}+\partial_{\sigma}\partial^{\lambda}R_{\mu\nu\lambda\rho}=0$
(1fkan)

is satisfied, then equation (1fh) reduces to

$\displaystyle\opensquare R_{\mu\nu\rho\sigma}=0.$ (1fkao)

Gravitational theories that necessarily satisfy Eq. (1fkan), such as general
relativity or Yang’s theory, are referred to as “general metric theories” in
this context.

The plane wave solution of (1fkao) takes the form

$\displaystyle
R_{\mu\nu\rho\sigma}=e_{\mu\nu\rho\sigma}\exp(ik_{\lambda}x^{\lambda})+{\rm
c.c.}$ (1fkap)

This satisfies (1fkao) if

$\displaystyle k_{\lambda}k^{\lambda}=0$ (1fkaq)

and satisfies (1fkan) if

$\displaystyle
k_{\rho}k^{\lambda}e_{\mu\nu\sigma\lambda}+k_{\sigma}k^{\lambda}e_{\mu\nu\lambda\rho}=0.$
(1fkar)

From the Bianchi identities (1fg), we also have

$\displaystyle
k_{\lambda}e_{\mu\nu\rho\sigma}+k_{\rho}e_{\mu\nu\sigma\lambda}+k_{\sigma}e_{\mu\nu\lambda\rho}=0.$
(1fkas)

These conditions (1fkaq), (1fkar), and (1fkas) are not independent: When
(1fkao) holds, equation (1fkan) is automatically satisfied via the identities
(1fh). Therefore, for a null wave vector $k^{\mu}$ satisfying (1fkaq),
considering (1fkas) alone is sufficient. It reduces the number of independent
components of $e_{\mu\nu\rho\sigma}$ from twenty to six as follows.

Consider a wave propagating in the $+z$ direction with the wave vector
$k^{\mu}=(k,0,0,k)$. In this case, equation (1fkas) allows us to express
twenty independent components of $e_{\mu\nu\rho\sigma}$ in terms of six
$e_{\mu\nu\rho\sigma}$:

$\displaystyle e_{3131}=-e_{0131}=e_{0101},$ $\displaystyle
e_{2323}=e_{0223}=e_{0202},$ $\displaystyle e_{0331}=-e_{0301},$
$\displaystyle e_{0323}=e_{0203},$ $\displaystyle
e_{0123}=-e_{0231}=-e_{2331}=e_{0102},$ $\displaystyle
e_{0112}=e_{0212}=e_{0312}=e_{2312}=e_{3112}=e_{1212}=0.\quad$ (1fkat)

Six independent components, $e_{0101}$, $e_{0102}$, $e_{0202}$, $e_{0203}$,
$e_{0301}$, and $e_{0303}$, correspond to six modes of polarization.

Under a rotation about the $z$-axis, the transformations of three transverse
modes, $e_{0101}$, $e_{0202}$, and $e_{0102}$, are given in equations (1fkaf)
and (1fkag) in Sec. 3. They represent two tensor modes with helicity $\pm 2$,
and a scalar mode with helicity 0.

The other three non-transverse components, $e_{0203}$, $e_{0301}$, and
$e_{0303}$, transform as

$\displaystyle e_{0203}^{\prime}\pm ie_{0301}^{\prime}$ $\displaystyle=$
$\displaystyle\exp(\pm i\theta)(e_{0203}\pm ie_{0301})$ (1fkau)

representing two vector modes with helicity $\pm 1$, and

$\displaystyle e_{0303}^{\prime}$ $\displaystyle=$ $\displaystyle e_{0303}$
(1fkav)

representing a scalar mode with helicity 0. Thus, there are six modes of
polarization (three of them are transverse, ant the others are non-
transverse).

###  4.2 Ricci tensor

As in Sec. 3.2, another convenient approach is to consider the wave equation
associated with the Ricci tensor. This method is useful for clarifying
additional modes beyond two tensor modes in general relativity.

Contracting the indices in (1fkao) yields

$\displaystyle\opensquare R_{\mu\nu}=0$ (1fkaw)

where $R_{\mu\nu}$ satisfies

$\displaystyle\partial_{\mu}\partial_{\rho}R_{\nu\sigma}-\partial_{\nu}\partial_{\rho}R_{\mu\sigma}-\partial_{\mu}\partial_{\sigma}R_{\nu\rho}+\partial_{\nu}\partial_{\sigma}R_{\mu\rho}=0.\quad$
(1fkax)

To first order in $h$, this relation is identical to (1fkan).

The weak, plane wave solution of (1fkaw) takes the form

$\displaystyle R_{\mu\nu}=e_{\mu\nu}\exp(ik_{\lambda}x^{\lambda})+{\rm c.c.}$
(1fkay)

This satisfies (1fkaw) if $k_{\lambda}k^{\lambda}=0$, and satisfies (1fkax) if

$\displaystyle
k_{\mu}k_{\rho}e_{\nu\sigma}-k_{\nu}k_{\rho}e_{\mu\sigma}-k_{\mu}k_{\sigma}e_{\nu\rho}+k_{\nu}k_{\sigma}e_{\mu\rho}=0.\quad$
(1fkaz)

Two vector modes and two scalar modes should be described by (1fkay). Thus,
the number of independent $e_{\mu\nu}$ should reduce from ten to four as
follows.

Consider a wave propagating in the $+z$ direction with the wave vector
$k^{\mu}=(k,0,0,k)$. Equation (1fkaz) gives six relations:

$\displaystyle e_{01}=-e_{31},$ (1fkba) $\displaystyle e_{02}=-e_{23},$
$\displaystyle e_{03}=-\frac{1}{2}(e_{00}+e_{33}),\quad$ $\displaystyle
e_{11}=e_{22}=e_{12}=0.$

Hence, $10-6=4$ components of $e_{\mu\nu}$ are independent, corresponding to
four additional modes of polarization.

Under a rotation about $z$-axis, we find that

$\displaystyle e_{23}^{\prime}\pm ie_{31}^{\prime}=\exp(\pm i\theta)(e_{23}\pm
ie_{31}),$ (1fkbba) $\displaystyle e_{00}^{\prime}=e_{00},\quad
e_{33}^{\prime}=e_{33}$ (1fkbbb)

representing two vector modes with helicity $\pm 1$, and two scalar modes with
helicity 0.

##  5 Conclusion

The gauge-invariant formulation for the gravitational wave equations has been
presented. Using this approach, weak, plane wave solutions have been derived
to identify polarization modes in various theories. These theories include
general relativity with two tensor modes of polarization including helicity
$\pm 2$, Yang’s theory with three polarization modes including helicity $\pm
2$ and 0, and so-called “general metric theories” with six polarization modes,
including helicity $\pm 2$, $\pm 1$, and two $0$’s. Our results agree with the
earlier results. Thus, the results presented in this study are not new, while
the formulation is an original work. The gauge-invariant formulation presented
here complements the conventional one using metric perturbation $h_{\mu\nu}$,
and both approaches are useful.

## Acknowledgments

This work was supported by JSPS KAKENHI Grant No. JP22K03599.

## References

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