Although spectral retrieval studies have been used extensively to explore the detectability of atmospheric compositions for the direct imaging of exoplanets for multiple telescopes both current and future (e.g., R. E. Lupu et al. 2016; M. Nayak et al. 2017; A. J. R. W. Smith et al. 2020; M. Damiano & R. Hu 2022; A. V. Young et al. 2024), these retrievals are extremely computationally expensive. Most Bayesian retrievals use real-time radiative transfer calculations, combined with exploring large parameter spaces, mission capabilities, and constant model improvement that increase computation time to a rate that is not easily usable. Different methods to accelerate retrievals have been explored through efficient radiative transfer schemes or machine learning (e.g., P. Márquez-Neila et al. 2018; T. Zingales & I. P. Waldmann 2018; A. D. Cobb et al. 2019; C. Fisher et al. 2020; M. D. Himes et al. 2022; T. D. Robinson & A. Salvador 2023).

In order to facilitate the efficient production of new optimized spectral grids for our grid-based retrievals, we developed a Python package called Gridder, which is a generalized grid-building scheme based on the methodology of S23 (M. D. Himes et al. 2024, in preparation). Gridder produces arbitrary spectral grids using the Planetary Spectrum Generator (PSG), a publicly available radiative transfer model for creating planetary spectra (G. L. Villanueva et al. 2018, 2022). PSG can be used to calculate spectra over an ultrabroad wavelength range (50 nm–100 mm) and includes planetary atmospheres, surfaces, and bulk properties such as aerosols, atomic, continuum, and molecular scattering/radiative processes implemented layer by layer. The Gridder parameter structure is highly customizable, allowing any input parameter to PSG to be used as a parameter in the optimized grid, including parameterized thermal profiles (e.g., isothermal or adiabatic; M. R. Line 2013) and atmospheric chemistry (e.g., constant with altitude or thermochemical equilibrium). In addition to the error metric of S23, Gridder offers other common metrics, such as the maximum absolute percent error and mean squared error, for greater control over the resulting grid’s accuracy. Gridder features various optimizations (e.g., parallelization and checkpoints) to ensure efficient production of optimized spectral grids in a consistent, reproducible manner. Using Gridder, we have built a new set of spectral grids, each with six parameters to decrease the grid-building computational time, but over a far wider wavelength range from the UV to NIR (0.2–2.0 μm), and we have added the additional molecular atmospheric constituents CH₄, CO₂, CO, SO₂, and N₂O.

Our newly built grid set, hereafter the KEN grids (no acronym; they are just ken), allow for a direct comparison between grids since they cover the same wavelength range and the three base parameters of P₀, A_s, and g with R_p fixed to 1 R_⊕. However, they reduce the computational time by taking advantage of the different spectral regions covered by the absorption features of different molecules. Any molecules that have overlapping spectral features (for the range of atmospheric pressures examined) are housed within the same grid, while any that are orthogonal to each other do not need to be simultaneously retrieved within one grid, such as CH₄ and O₃. The background gas in each grid is set to N₂ = 1 − P₁ – P₂ – P₃ where P₁ – P₃ are the molecules per grid. In this way, we can also investigate the effects of false positive or negative molecular detection due to overlapping features. We developed four different spectral grids at a resolving power of 500 (R = 500), which can then be binned to lower resolving powers, with the ability to vary the cloud fraction C_f at any value between 0% cloudy to 100% cloudy. We describe the molecular parameters in each named grid in the KEN grid set, along with a modern-Earth-like spectrum per grid and the highlighted wavelength regimes for use per grid, in Figure 2.

The KEN grids will allow us to investigate varying atmospheric compositions beyond O₂-dominated atmospheres, such as CH₄ or CO₂ dominated, to develop observational procedures for multiple planetary archetypes through wavelength.

We follow an identical methodological approach to that of BARBIE1 and BARBIE2 for the CH₄ case study analysis. Herein we present a brief summary of the main steps in our analysis. For more detailed information, please see BARBIE1 and BARBIE2.

• 1.  
  We set a modern-Earth twin as our fiducial data spectrum following Y. K. Feng et al. (2018), with isotropic volume mixing ratios (VMRs) of H₂O = 3 × 10⁻³, O₃ = 7 × 10⁻⁷, O₂  =  0.21, constant temperature profile at 250 K, A_s of 0.3, P₀ of 1 bar, and a planetary radius fixed at R_p = 1 R_⊕. We bin our grid from the native resolving power of 500–140 and 70 for our simulations, and split the spectrum in 25 evenly spaced bandpasses across 20%, 30%, and 40% widths. R = 140 is generally accepted in the optical regime, while R = 70 is generally accepted in the NIR (The LUVOIR Team 2023). These ranges mimic the simultaneous bandpasses that may be achieved with high-performance coronagraphs in the future, or dual coronagraphs that can be used in combination (G. J. Ruane et al. 2015; E. H. Por 2020; R. Juanola-Parramon et al. 2022).
• 2.  
  We run a series of Bayesian nested sampling retrievals using PSGnest¹⁰ housed in PSG. PSGnest is a Bayesian spectral retrieval methodology adapted from the original Fortran version of the Multinest retrieval algorithm (F. Feroz et al. 2009) and designed for application to exoplanetary observations (i.e., incorporating grid multidimensional interpolation, grid retrievals, the fitting methods, as well as the Multinest retrievals).
• 3.  
  We receive the highest-likelihood values, the average values from the posterior distributions, uncertainties, and the log evidence (logZ; G. L. Villanueva et al. 2022) as outputs. We then calculate the log-Bayes factor (lnB; B. Benneke & S. Seager 2013); this compares retrievals with and without a given molecule to estimate the likelihood of the molecule’s presence in the exoplanet’s atmosphere. For our purposes, lnB < 2.5 is unconstrained (no detection), 2.5 ≤ lnB < 5.0 is a weak detection, and lnB ≥ 5.0 is a strong detection (see Table 2 of B. Benneke & S. Seager 2013). We also calculate the median and upper and lower limit values of the 68% credible region (J. Harrington et al. 2022). However lnB is a better estimation of detection rather than the 68% credible region since it is directly investigating the molecular presence versus absence and calculating which iteration yields the best fit, while the 68% credible region, by definition, includes the true value 68% of the time. lnB is not calculated for nongaseous components, since those factors cannot be absent.

This process is then repeated for varying abundances. For this work, we vary CH₄ according to Table 1, following J. Kasting (2005) and L. Kaltenegger et al. (2007), while also adding values to bridge the large gap between abundance values (i.e., between Hadean and Archean values). The level of CH₄ present in any specific habitable exoplanet atmosphere is uncertain, since CH₄ production is significantly driven by biological activity; the same is true for any other biologically produced species. Since we are not running fully physically consistent chemistry models, we therefore use modern-Earth values as the default for other species as well as to isolate the effect of a single molecule, as in prior Bayesian Analysis for Remote Biosignature Identification on exoEarths (BARBIE) works. We also investigate the effects of H₂O on CH₄ detectability by testing a range of combined H₂O and CH₄ values, shown in Table 2. All retrievals were performed between 0.8 and 1.5 μm with 20%, 30%, and 40% bandpass widths, and R = 140 and R = 70. We carefully select a narrower wavelength range in order to investigate the detectability of CH₄ in the optical or at the critical 0.9 μm H₂O feature; this is currently poised as the first step in observations following the decision tree technique in A. V. Young et al. (2024).

In S23 Figure 8, they investigate the interpolation error in the developed grid to ensure that the grid is not introducing a large error past any known interpolation error. We have replicated this in Figure 3 using the same technique as in S23 using our M-KEN grid, as that has the same parameters as in the S23 grid. To summarize, we ran 6D retrievals for each value of each parameter in the test grid. The test grid in use consists of the midpoint values of the M-KEN grid as the interpolation error is typically maximized near the midpoints, and thus to test the error we want to investigate the area of highest error. For each parameter set, a “true” spectrum was created (i.e., pulling the data spectrum directly from the midpoints grid) at an SNR of 20. The retrievals were then repeated, but the data spectrum was created by interpolating from the main M-KEN grid (i.e., the way that all retrievals will work when using KEN grids with PSGnest). We use the offset between the values retrieved for the “true” and interpolated spectra to assess the impact of interpolation error on these retrievals.

Although the M-KEN grid covers a wider wavelength range than the S23 grid, Figure 3 covers the wavelength range of 0.4–1 μm in order to directly replicate the range in S23. The error from retrieving on interpolated data is shown in pink dots, while the error from retrieving on “true” data is shown in purple stars. We also calculate the 68% credible region and portray those as error bars. The left column shows the full extent of the offset for every point, while the right column zooms in to the regions shown in the black dotted lines. We select these areas based on where there is a high concentration of overlapping points. Any differences between the results are due to an interpolation error.

We find that the retrieval error is generally very similar for both the “true” and interpolated data retrievals, with the derived retrieval error for both always within 1σ of the input value. The outlier errors, as in S23, are at small P₀, and at large O₃, although even those error ranges are no more than 15%. The average error hovers at below 5%, while the error between the interpolated and true data is never larger than 2% (with the singular exception of 10% at low O₃). In fact, our error spread is much smaller than S23, validating that the new optimization procedures of Gridder yield higher fidelity results with a much faster overall runtime. This confirms that the Gridder selection of grid points has been successful, with the interpolation error not serving as a significant inhibitor for performing retrievals in this region.

To further isolate the impact of the retrieval error induced by interpolation within the grid apart from the Bayesian inference routine, we performed 4800 retrievals using the UltraNest package (J. Buchner 2016, 2019, 2021) on PSG-simulated spectra for values near the midpoints across the B-KEN grid’s parameter space. We use the grid’s native R = 500 spectral resolution and the full 0.2–2 μm wavelength range and assume an SNR of 15 for each wavelength channel. We find that the mean absolute error (MAE) between the maximum likelihood and the known true value is typically ∼0.02 for A_s and ∼0.2 for all other parameters when there is sufficient information content to constrain the parameter. This finding is generally consistent with Figure 3. When H₂O, CH₄, and CO₂ can simultaneously be constrained, the MAE for the extrema of the 3σ region of all parameters is nearly equal to the MAE of the maximum likelihood, indicating that the 1D marginalized posteriors are offset from the known truth. This bias is generally negligible except in the case of g, where the mean absolute percentage error can be >60% at low values of and ∼20% for Earth-like gravities. These results represent worst-case scenarios; under typical grid usage conditions (R = 140, reduced wavelength coverage, at least one parameter not at a grid midpoints), these errors will be reduced.

We present another analysis of grid efficacy in Figure 4, looking specifically at the deep CH₄ feature centered at 1.1 μm. This presents modern-Earth values of all parameters except CH₄, which is at an Archean value—increasing the CH₄ value is necessary to investigate grid efficacy, as we expect it to be well constrained. This is using the B-KEN grid, which is also the grid used for all the science cases explored in this work. The B-KEN grid is the only KEN grid that contains CH₄, thus it is our only option to investigate CH₄ detectability. We find that there are no sharp cutoffs in our 1D histograms except for g, which can especially be seen in the 2D marginalized posterior for g and P₀. However, g around the Earth value is the primary use case for these grids, which is well retrieved. CO₂ and g also have significant posterior density at 1 × 10⁻¹⁰ VMR and 1 m s⁻², respectively, indicating that the 68% credible region is underestimated. However, any molecule at such low abundances does effect spectral change, and CO₂ is unconstrained in this wavelength region where there is no CO₂. We also see known and expected degeneracies with A_s and C_f, with every parameter, especially the molecular parameters, heavily impacted, but every parameter is also retrieved within the 68% credible region.

For further analysis on grid, and Gridder, validation, please see M. D. Himes et al. (2024, in preparation) for a more thorough description.

We begin by presenting the detectability of CH₄ as a function of SNR for the fiducial modern-Earth case, as examined by S23, BARBIE1, and BARBIE2, to maintain consistency (all CH₄ data and calculated lnB factors across abundance, SNR, and wavelength will be available to the community on Zenodo at doi:10.5281/zenodo.13760695). All other parameters were left to modern-Earth values, including a modern-Earth value of H₂O. However, a modern level of CH₄ (1.65 × 10⁻⁶) is completely undetectable at at all SNRs and bandpass widths tested at a resolving power of 140 and 70. Many other works have previously found the same result, due to the low concentration and thus weak absorption features (E. W. Schwieterman et al. 2018; M. Damiano & R. Hu 2022; S. Gilbert-Janizek et al. 2024; A. V. Young et al. 2024). Since a modern-Earth CH₄ was not possible to detect, we quickly proceeded to the varying abundances listed in Table 1 to explore the minimum value at which CH₄ could be detected.

For our abundance case study, we vary the abundance of CH₄ above modern-Earth values up to a cap of the Archean-Earth value (7.07 × 10⁻³). We also vary the SNR for each abundance level to fully investigate the trade-off between higher SNR and molecular detectability. The CH₄ abundances investigated are presented in Table 1, along with intermediate values to fill in the large gap between the Hadean and Archean epochs.

In Figures 5(a), (b), and (c), we present the strong detectability of the three highest CH₄ abundances in our study, along with a modern H₂O abundance for context, in purple triangles and pink squares, respectively. SNR is on the y-axis, with bandpass centers on the x-axis. We see that, as expected, as the abundance of CH₄ increases, the required SNR for detection decreases, with SNRs of 9, 5, and 3 for CH₄ abundances of 4.15 × 10⁻⁴, 1.65 × 10⁻³, and 7.07 × 10⁻³, respectively. Notably, at 4.15 × 10⁻⁴, the lowest SNR (9) is accessible at ∼1.05 μm, whereas at 1.65 × 10⁻³ and 7.07 × 10⁻³ abundances, the lowest SNRs are accessible at ∼0.9 μm. In order to achieve a detection of CH₄ at 4.15 × 10⁻⁴ at 0.9 μm, an SNR of 13 is necessary.

Looking at the full range of abundance values in our study, we present a heat map showing detection strength as a function of SNR (y-axis) and CH₄ abundance (x-axis) in Figure 6(a). The color bars represent the range of 0–5 lnB, between unconstrained and strong detection. We see here that CH₄ is not strongly detectable until 4.15 × 10⁻⁴ (Phanerozoic), at which point an SNR of 9 is required. Moving to higher abundances significantly drops the required SNR for strong detection, with 1.65 × 10⁻³ (early Phanerozoic) requiring an SNR of 5 and 7.07 × 10⁻³ (Archean) detectable at all SNRs. At low abundances, such as a modern-Earth value of 1.65 × 10⁻⁶, CH₄ is not detectable at any SNR. Interestingly, at 1 × 10⁻⁴ a weak detection is possible at an SNR of 20, with the next value of 4.15 × 10⁻⁴ only requiring an SNR of 9 for strong detection as mentioned, and an SNR of 6 for weak detection. This results in a very drastic change in CH₄ detection over a relatively small change in abundance of CH₄.

However, returning to Figure 5, we see a significant change in detectability for H₂O across CH₄ abundance. A higher SNR is required to detect H₂O as CH₄ increases. Also as the CH₄ abundance increases, the range of strong H₂O detection as a function of wavelength decreases, with a sharp cutoff at ∼1.1 μm. For wavelengths longer than 1.1 μm, it requires a high SNR (≥15) to detect H₂O although there is a very large H₂O feature at 1.35 μm. This indicates that the H₂O features are being masked by strong CH₄ features, thus H₂O detectability is impacted.

This is confirmed in Figure 7, which presents a spectrum of modern H₂O and three increasing values of CH₄ in three panels (Phanerozoic and early and late Archean from top to bottom) as a function of wavelength and geometric albedo (x-axis and y-axis, respectively). As CH₄ increases in abundance, the absorption feature aligns more closely with that of H₂O, particularly between ∼1.1 μm and 1.2 μm. This results in an inflection point, wherein H₂O is more difficult to detect due to the absorption feature being masked by the overpowering CH₄. This led us to investigate the opposite effect—does H₂O mask the detection of CH₄ at lower CH₄ abundance?

We reran all values of CH₄ in Table 1, however we set the abundance of H₂O to zero in our fiducial spectrum. We find that, indeed, our detection of CH₄ changes dramatically. In Figure 6(b), we again present a detectability heat map across all abundances of CH₄ as in Figure 6(a), although with no H₂O present. We see that CH₄ is strongly detectable down to 7 × 10⁻⁵ VMR at an SNR of 14, and 4.15 × 10⁻⁴ VMR was detectable at an SNR of 6. Thus, with the absence of H₂O, our detectable CH₄ abundance shifts down over an order of magnitude. We can see the effect of H₂O on CH₄ detectability, and this can suggest that the abundance of CH₄ cannot accurately be determined unless you also separately constrain the H₂O abundance. However, we want to further understand the interactions between varying the abundances of H₂O and CH₄.

We ran a suite of retrievals with all combined H₂O and CH₄ values shown in Table 2, at 20%, 30%, and 40% bandpass widths—note that these values were selected purely to understand the inflection point of molecular abundance and detectability, with the highest H₂O abundance at modern levels (3 × 10⁻³ VMR) and the highest CH₄ abundance at Archean levels (7.07 × 10⁻³ VMR). Our lowest tested value of CH₄ is (1 × 10⁻⁴ VMR), the first abundance at which any detection is possible.

In Figure 8, we present the results of our degeneracy study as six heat maps at 1.1 μm and the longest possible bandpass (top and bottom rows, respectively), wherein the color corresponds to the SNR, with the required SNR for strong detection labeled on the plot. The x-axis is the H₂O abundance, the y-axis is the CH₄ abundance, with a 20%, 30%, and 40% bandpass width, respectively, in columns. All results are with R = 70, the current fiducial resolving power in the NIR. We can see as a trend in Figure 8 that the bandpass width has a very large impact on detectability for CH₄ as a function of abundance: at high CH₄ abundances, especially at 1.1 μm where there is a large CH₄ absorption feature, the Archean levels of CH₄ remain essentially unchanged through bandpass width. This reaffirms our earlier results that high CH₄ did not vary through bandpass width and that H₂O abundance is a large factor in detectability. Looking first to Figures 8(a)–(c), at high CH₄ values, 3 × 10⁻³ VMR and above, the H₂O abundance minimally influences the CH₄ detectability. We begin to see an influence in detectability at 1 × 10⁻³ VMR, where at the lowest H₂O abundance (3 × 10⁻⁵ VMR) the required SNR for CH₄ detection is ∼4, through the central H₂O abundances, the required SNR is ∼6 for CH₄ detection, and at the highest H₂O abundance (1 × 10⁻² VMR), the required SNR for CH₄ detection is up to 11 at a 20% bandpass, and an 8 for a 40% bandpass. The CH₄ abundance is unchanging across this range of H₂O abundances; thus it is clear that the influence must be H₂O. This influence continues to grow stronger as the CH₄ abundance decreases, with 3 × 10⁻⁴ VMR CH₄ abundance requiring an SNR of 8 at the lowest H₂O abundances for a 20% bandpass and 6 for 30% and 40%, and an SNR higher than 20 (i.e., undetectable in our study) at the highest H₂O abundances. The degeneracy is most obvious at 1 × 10⁻⁴ VMR CH₄ abundance, with each increasing abundance of H₂O increasing the required SNR for strong CH₄ detection—and resulting in no detections.

In Figures 8(d)–(f), we adjust the wavelength that is being investigated. Since the bandpass centers change as the bandpass width increases, the final bandpass is investigated in Figures 8(d)–(f) with different bandpass centers. The emphasis in this last bandpass is in the 1.3 μm feature that appears for both CH₄ and H₂O. We find a similar trend to the above, with higher SNR requirements at each bandpass. With the 20% bandpass width (Figure 8(d)) specifically, the required SNR is higher than in Figures 8(a)–(c), especially at high H₂O. In fact, at the highest value of H₂O (1 × 10⁻² VMR), CH₄ is not detectable at 3 × 10⁻³ VMR and 1 × 10⁻³ VMR although both of those abundances were still detectable at shorter wavelengths. Thus longer wavelengths (e.g., 1.35 μm) are not well suited for CH₄ detection, especially with narrower bandpasses.

Next, we investigate H₂O detectability. We present in Figure 9 another set of heat maps to investigate how the detection of H₂O changes as a function of CH₄ abundance. All plot aspects remain the same as Figure 8, with the exception that we are investigating H₂O detectability. We again see that there is a strong correlation with the bandpass width, with detectability increasing as the bandpass width increases, especially at shorter wavelengths. At shorter wavelengths, such as 1.1 μm, H₂O is detectable at all but the lowest abundances, with changing SNR. At a modern-Earth level of H₂O, detectability at a 20% bandpass width requires at most an SNR of 12 and more than halves to an SNR ≤ 5 for a bandpass width of 40%. However, this does not hold for long wavelengths. We find that at longer wavelengths, the high CH₄ abundances completely mask the H₂O features leading to an inability to detect H₂O at high CH₄ or low H₂O. However, we did find that at low H₂O and sufficiently low CH₄, the presence of both molecules together increases the detectability of H₂O, likely due to the degeneracy between the two molecules being broken in this wavelength regime.