Simulating the Detection of the Global 21 cm Signal with MIST for Different Models of the Soil and Beam Directivity

The Mapper of the IGM Spin Temperature (MIST) is a new ground-based, single-antenna, radio experiment attempting to detect the global 21 cm signal from the Dark Ages and Cosmic Dawn. A significant challenge in this measurement is the frequency dependence, or chromaticity, of the antenna beam directivity. MIST observes with the antenna above the soil and without a metal ground plane, and the beam directivity is sensitive to the electrical characteristics of the soil. In this paper, we use simulated observations with MIST to study how the detection of the global 21 cm signal from Cosmic Dawn is affected by the soil and the MIST beam directivity. We simulate observations using electromagnetic models of the directivity computed for single- and two-layer models of the soil. We test the recovery of the Cosmic Dawn signal with and without beam chromaticity correction applied to the simulated data. We find that our single-layer soil models enable a straightforward recovery of the signal even without chromaticity correction. Two-layer models increase the beam chromaticity and make the recovery more challenging. However, for the model in which the bottom soil layer has a lower electrical conductivity than the top layer, the signal can be recovered even without chromaticity correction. For the other two-layer models, chromaticity correction is necessary for the recovery of the signal, and the accuracy requirements for the soil parameters vary between models. These results will be used as a guideline to select observation sites that are favorable for the detection of the Cosmic Dawn signal.


INTRODUCTION
The sky-averaged, or global, redshifted 21 cm signal from neutral hydrogen in the intergalactic medium (IGM) is expected to reveal how the Universe evolved before and during the formation of the first stars (Furlanetto et al. 2006; Corresponding author: Raul A. Monsalve raul.monsalve@berkeley.eduPritchard & Loeb 2008).Models for this signal primarily consist of two absorption features in the radio spectrum: one at ν ≲ 50 MHz, produced during the Dark Ages (Mondal & Barkana 2023), and the second one in the range 50 MHz ≲ ν ≲ 150 MHz, due to the appearance of the first stars at Cosmic Dawn (Tozzi et al. 2000;Furlanetto 2006).
The Mapper of the IGM Spin Temperature (MIST) is a new experiment designed to measure the global 21 cm signal.The MIST instrument is a ground-based, single-antenna, total-power radiometer measuring in the range 25-105 MHz, which encompasses the Dark Ages and Cosmic Dawn.The antenna used by MIST is a horizontal blade dipole, which operates without a metal ground plane in order to avoids systematics from ground plane resonances (Bradley et al. 2019) and edge effects (Mahesh et al. 2021;Rogers et al. 2022;Spinelli et al. 2022).MIST runs on 12 V batteries, has a power consumption of only 17 W, and is very compact, with all the electronics and batteries housed in a single receiver box located under the antenna.The low power consumption and compactness enable MIST to be easily transported to multiple observation sites.The instrument design and initial performance are described in Monsalve et al. (2023).
One of the greatest instrumental challenges in detecting the 21 cm signal is the "beam chromaticity", that is, the change in the antenna beam directivity as a function of frequency.The beam chromaticity changes the spatial weighting of the sky brightness temperature distribution across frequency.This change in spatial weighting produces structure in the spectrum that could mask or mimic the global 21 cm signal (e.g., Vedantham et al. 2014;Bernardi et al. 2015;Mozdzen et al. 2016;Sims et al. 2023).Furthermore, the beam chromaticity of ground-based instruments is sensitive to the properties of the soil, which increases the complications and uncertainties in the modeling of the observations (e.g., Mahesh et al. 2021;Singh et al. 2022;Spinelli et al. 2022).For MIST, which operates without a ground plane, understanding the influence of the soil on the beam chromaticity and the detectability of the 21 cm signal is critical.
This paper studies the extraction of the Cosmic Dawn absorption feature from simulated observations with MIST.Specifically, we address the following questions: (1) How does the MIST beam chromaticity bias the 21 cm parameter estimates?(2) How does this bias depend on the electrical properties of the soil?(3) To what extent could this bias be reduced by correcting the data for beam chromaticity?And (4) how accurately do we need to know the soil parameters when computing the chromaticity correction?To address these questions, we use the nine models of the MIST beam directivity introduced in Monsalve et al. (2023).These directivity models were produced through electromagnetic simulations that incorporated different models for the soil.

Simulated Observations
We simulate the sky-averaged antenna temperature spectrum that MIST would measure from the McGill Arctic Research Station (MARS) in the Canadian High Arctic (79.37980 • N, 90.99885 • W).MARS is one of the sites from where MIST has already conducted observations (Monsalve et al. 2023).We simulate the observations in the restricted frequency range 45-105 MHz to focus on the absorption fea-Table 1. Soil models used in the FEKO simulations of MIST.These models were introduced in Monsalve et al. (2023).Five of these models ("nominal" and 1L xx) are single-layer models and four (2L xx) are two-layer models.The layers are characterized in terms of their electrical conductivity (σ) and relative permittivity (ϵ r ).In the two-layer models, the thickness of the top layer is L = 1 m.

Model
where T 21 is the Cosmic Dawn 21 cm absorption feature, T f g is the contribution from the astrophysical foreground, ν is frequency, and t is time.Noise is added to the simulated observations as described in Section 2.5.We simulate the time-independent 21 cm signal as a Gaussian, that is, where a 21 , ν 21 , and w 21 are the Gaussian amplitude, center frequency, and full width at half maximum (FWHM), respectively.We have chosen a phenomenological model for the 21 cm signal because here we focus on the detectability of the signal instead of on its physical interpretation, similarly to previous works (e.g., Bernardi et al. 2016;Monsalve et al. 2017;Bowman et al. 2018;Spinelli et al. 2019Spinelli et al. , 2022;;Anstey et al. 2023).Existing physical models differ in their astrophysical parameters, values, and assumptions, as well as in their computational implementation, but the shape of most of them can be well approximated by a Gaussian (e.g., Furlanetto 2006;Pritchard & Loeb 2008;Mesinger et al. 2011;Mirocha 2014;Cohen et al. 2017;Mirocha et al. 2021;Muñoz 2023).A Gaussian analytical model can be quickly evaluated in a likelihood computation and has parameters that are easy to interpret.In our simulations, the input Gaussian parameter values are a 21 = −200 mK, ν 21 = 80 MHz, and w 21 = 20 MHz.These values are consistent with standard physical models (Furlanetto et al. 2006), and our choices for the center frequency and width are close to the values reported by the EDGES experiment (Bowman et al. 2018).
The time-dependent foreground contribution to Equation 1 is computed as Here, D is a model for the antenna beam directivity, which we discuss in Section 2.2; θ and ϕ are the zenith and azimuth angles, respectively, of both the antenna and the sky; and T GS M is a model for the spatially dependent foreground brightness temperature distribution, for which we use the Global Sky Model (GSM; de Oliveira-Costa et al. 2008;Price 2016).We compute T f g across 24 hr of local sidereal time (LST) with a cadence of 6 minutes.In the computations, the excitation axis of the dipole antenna is aligned north-south and the horizontal blades of the antenna are perfectly level.
The simulated observations assume perfect receiver calibration and correction of losses, in particular radiation loss, balun loss, and ground loss.Effects from the ionosphere (Vedantham et al. 2014), polarized diffuse foreground emission (Spinelli et al. 2019), and mountains in the horizon (Bassett et al. 2021;Pattison et al. 2023) are assumed to be perfectly calibrated out.

Beam Directivity
We simulate sky observations with the nine beam directivity models introduced in Monsalve et al. (2023).The directivity models were obtained from electromagnetic simulations with FEKO 1 that incorporated different models for the soil.The characteristics of the soil models are shown in Table 1, reproduced from Monsalve et al. (2023).In the FEKO simulations, the soil model extends to infinity in the horizontal direction and in depth.Five of the soil models ("nominal" and 1L_xx) are single-layer models.These models intend to mimic the optimistic scenario of a soil that, from the point of view of the antenna, is effectively homogeneous and well characterized in terms of only two parameters: its electrical conductivity (σ 1 ) and relative permittivity (ϵ r1 ).The other four soil models (2L_xx) are two-layer models, in which the top layer has a thickness L = 1 m and the bottom layer extends to infinite depth.Except for one parameter of the bottom layer (either σ 2 or ϵ r2 ), the parameters of the two-layer models take the same values as in the nominal model.The two-layer models are used to study the more realistic scenario in which the characteristics of the soil change below 1 https://www.altair.com/feko a certain depth.An example of this scenario is the soil at MARS during the summer, which consists of an unfrozen top layer and a permanently frozen, or "permafrost," bottom layer (e.g., Pollard et al. 2009;Wilhelm et al. 2011).
For the nominal model we use σ 1 = 0.01 Sm −1 and ϵ r1 = 6.In the other models, the conductivity of the single-or bottomlayer is changed to 0.1 and 0.001 Sm −1 , and the relative permittivity is changed to 10 and 2. These values are motivated by values reported for geological materials that we could encounter at our observation sites.For instance, snow, freshwater ice, and permafrost have σ and ϵ r in the ranges 0-0.01 Sm −1 and 2-6, respectively.For sand, silt, and clay, σ and ϵ r have a strong sensitivity to moisture and span the wide ranges 0-1 Sm −1 and 2-40 (Reynolds 2011).Our parameter values are consistent with the values reported by Sutinjo et al. (2015) for the soil at the Inyarrimanha Ilgari Bundara, the CSIRO Murchison Radio-astronomy Observatory, with different moisture levels.Our values are also consistent with the soil measurements done at the Owens Valley Radio Observatory for dry and wet conditions, presented in Spinelli et al. (2022).
As shown in Monsalve et al. (2023), for single-layer models the directivity has a smooth frequency evolution.Twolayer soil models, in contrast, produce ripples in the directivity as a function of frequency, which are expected to complicate the extraction of the 21 cm signal.The period of the ripples depends on the depth of the interface between the two layers.The amplitude of the ripples depends on the difference in parameter values between the two layers as well as on the sign of the change.Among our two-layer models, 2L_cproduces the smallest ripples, followed by 2L_p+, 2L_p-, and 2L_c+.
The nine soil models considered in this paper are used to provide initial intuition about the impact of the soil properties on the detection of the Cosmic Dawn signal with MIST.We leave to future work an in-depth study of soil effects with a wide range of models.Additional models to consider would include (1) two-layer models with different top-layer thicknesses; (2) models with more than two layers (which were discussed in Spinelli et al. (2022) in the context of the LEDA experiment); and (3) models with frequency dependence in the parameters.

Chromaticity Correction
The simulated observations computed with Equation 3 are affected by the chromaticity of the beam.In this paper, we study the extraction of the Cosmic Dawn signal from the observations affected by this chromaticity, as well as after applying a correction to remove this effect.The chromaticity correction used is the one introduced in Monsalve et al. (2017): Here, T GS M is the foreground brightness temperature distribution from the GSM, D c is a model for the beam directivity used for chromaticity correction, and ν r is the reference frequency for chromaticity correction.We use ν r = 75 MHz, which is the center of our frequency range.Applying the chromaticity correction involves dividing by C the data produced by Equation 3. In Equation 4, the numerator (denominator) corresponds to the foreground antenna temperature measured by a frequency-dependent (-independent) beam directivity.Therefore, C is the factor by which the frequencydependent directivity modifies the sky-averaged foreground spectrum that would be measured if the directivity were frequency independent and equal to D c (θ, ϕ, ν r ).Note that C is computed assuming that there is no 21 cm signal.We account for the effect of the chromaticity correction on the 21 cm signal at the signal extraction stage (Section 2.6).
We study the extraction of the signal with three types of chromaticity correction: (1) no correction (NC), which is equivalent to making C = 1; (2) imperfect correction, where D c in Equation 4 has been computed with errors in the soil model relative to D in Equation 3; and (3) perfect correction (PC), where D c = D.In this paper we only address complications related to beam chromaticity.Therefore, C is always computed with a perfect model for the foreground, that is, the GSM, also used to simulate the observations with Equation 3.

Errors in Soil Parameters
We implement the second type of chromaticity correction introduced above as follows.For each of the soil models used to produce the simulated observations, we compute several D c that are imperfect due to errors in the soil model assumed for chromaticity correction.In the assumed soil models, the number of layers is the same as in the input soil models, that is, they are both single layer or both two layer.The only error in the assumed soil models is in the value of one of the soil parameters.For the single-layer models, we assign incorrect values to either σ 1 or ϵ r1 .For the two-layer models, we assign incorrect values to σ 1 , σ 2 , ϵ r1 , ϵ r2 , or L, one at a time.The incorrect values correspond to values higher than those in Table 1 used to compute D. We explore three levels of errors: 10%, 5%, and 1%.The same general approach was used by Spinelli et al. (2022) to study the sensitivity of the LEDA experiment to soil parameter errors.
Running a large number of FEKO simulations in which several parameters are varied simultaneously to robustly sample the soil parameter space is very computationally intensive.We leave this task to future work.We also leave to future work the exploration of errors in which the assumed number of soil layers is wrong.

LST-averaging and Noise
For each soil model, chromaticity correction, and error type tested, we study the Cosmic Dawn signal extraction after averaging the simulated spectra over 24 hr of LST.Leveraging the time dependence of the beam-convolved foreground is expected to improve the constraints on the timeindependent Cosmic Dawn signal (e.g., Liu et al. 2013;Tauscher et al. 2020;Anstey et al. 2023).However, here we choose to work with the LST-averaged spectrum because (1) this choice simplifies our analysis, which represents an initial effort to quantify the effects from the soil and beam chromaticity; (2) averaging in LST reduces the measurement noise for a given observation time and is often a useful step when working with a limited amount of real data, motivating this approach in simulation; and (3) the variations with LST of the beam-convolved foreground observed from MARS are not as large as from lower latitudes (up to ≈ 20% in our simulations of the GSM), and, therefore, their leverage is expected to have a lower impact.
For each case being tested, the final simulated spectrum, d(ν), is produced by applying chromaticity correction to the data (with C = 1 for the NC cases), averaging the data in LST, and adding random noise, that is, where ⟨. ..⟩ represents average over LST and n(ν) is the noise.
For practicality, we add n(ν) in a single step after LST averaging the noiseless spectra.This is equivalent to LST averaging noise previously added to the 6 minute cadence spectra.The noise has a Gaussian distribution, no correlation between frequency channels, and a standard deviation of 3 mK at 75 MHz, which evolves with frequency proportionally to the noiseless term.According to the radiometer equation, for a reference system temperature of 1500 K, a channel width of 1 MHz, and 100% on-sky duty cycle, this noise level would be obtained with 2.9 days of observations.

Spectral Fitting
We fit the following model to the simulated spectra: where T 21 is the model for the Cosmic Dawn signal.In this paper, we are interested in the recovery of the 21 cm param-eters assuming that the 21 cm model itself is perfect.Therefore, T 21 in Equation 6is the Gaussian model of Equation 2, also used for the input 21 cm signal.The 21 cm fit parameters are a 21 , ν 21 , and w 21 , which are encapsulated by µ 21 .Dividing T 21 by ⟨C(ν, t)⟩ in Equation 6accounts for the effect of the beam chromaticity correction on the 21 cm signal (Sims et al. 2023).The variable m f g is a model for the beam-convolved, chromaticity-corrected, and LST-averaged foreground contribution, for which in this paper we use the "LinLog" expression (Hills et al. 2018;Bradley et al. 2019;Mahesh et al. 2021;Sims et al. 2023) Here, µ f g encapsulates the a i linear fit parameters, N f g is the number of terms in the expansion, and ν f g = 75 MHz is a normalization frequency used to improve the numerical stability of the fit.We use the LinLog expression because it can efficiently model the MIST observations for different levels of chromaticity correction.We leave for future work the exploration of other analytical models, including different linear and nonlinear power law-based expansions (Voytek et al. 2014;Bernardi et al. 2016;Monsalve et al. 2017;Bowman et al. 2018;Hills et al. 2018;Singh et al. 2022).
In the simulated observations, the level of spectral structure from the beam-convolved, LST-averaged foreground is expected to vary across soil models and chromaticity correction cases.To optimize the fitting of this structure, we sweep N f g between 4 and 8, and keep for our analysis the results for the N f g that minimizes the Bayesian information criterion (BIC): where N µ = N f g +3 is the number of foreground plus 21 cm fit parameters and N ν = 61 is the number of frequency channels.
To fit the model parameters -in particular µ 21 -in a computationally efficient way, we use the technique developed in Monsalve et al. (2018).In this technique, only the three nonlinear 21 cm parameters are sampled, while the linear foreground parameters are fitted using the matrix operations of the linear least squares method.Despite only sampling the three-dimensional µ 21 space, the resulting probability density functions (PDFs) for µ 21 do account for covariances with the foreground parameters.We sample the µ 21 space using the pocoMC code, which implements the preconditioned Monte Carlo method for accelerated Bayesian inference (Karamanis et al. 2022(Karamanis et al. , 2023)).In the preconditioned Monte Carlo fit, we use the following uninformative uniform priors for the 21 cm parameters: [−1, 1] K for a 21 , [45,105] MHz for ν 21 , and [1, 60] MHz for w 21 .Our prior for a 21 is wide and would enable the detection of the input sig-nal as well as of a large absorption feature produced by exotic physics (e.g., Bowman et al. 2018;Feng & Holder 2018;Muñoz & Loeb 2018).More importantly for this paper, our prior for a 21 enables the detection of absorption and emission features corresponding to artifacts produced by soil effects combined with the limitations of our analysis approach.Although forcing the fitted 21 cm model to be found in absorption could be warranted in some types of analyses, in this paper we choose to transparently expose the results affected by systematic effects.Our priors for ν 21 and w 21 are chosen to match our frequency range.

RESULTS AND DISCUSSION
The results for all the soil models, chromaticity corrections, and error types are presented in Figures 1-3.
Figure 1 shows the 21 cm parameter estimates in terms of best-fit values and 1-sigma2 error bars.The best-fit values correspond to the median or 50th percentile of the posterior PDFs.The "+" and "−" sigmas correspond to the 84th-50th and 16th-50th percentiles of the PDFs, respectively.
Because of the low noise in our simulated observations, in most cases the 21 cm estimates are limited by the accuracy of the best-fit value rather than by statistical precision.Therefore, as another performance metric, in Figure 2 we report the results in terms of the error of the best-fit value.Specifically, we report the absolute percentage error, ∆, of each estimate.As an example, for a 21 this quantity is computed as where â21 is the best-fit value and a 21,input = −200 mK is the input value.
Figure 3 shows the N f g used to fit the foreground contribution to the spectrum, as well as the BICs of the fits.The reduced χ 2 corresponding to the BICs of the fits, computed as N ν − N µ −1 χ 2 , are in the range 0.81-1.17.These values indicate that the LinLog model is reasonable across all cases and that there is no significant over-or underfitting.
The results for the five single-layer soil models are similar and shown in the first column of the figures.The second through fifth columns of the figures show the results for each of the four two-layer soil models.

Perfect Chromaticity Correction
We start the discussion of our results with the PC cases.In the PC cases, the impact from beam chromaticity has been perfectly removed and the corrected spectra represent obser- Estimates for the parameters of our Gaussian 21 cm absorption model, a 21 , ν 21 , and w 21 , obtained from fits to simulated observations with the MIST instrument over nine soil models.The first column shows the results for the five single-layer soil models.These results are similar and depicted with the same color for the same chromaticity correction case and error type.The second through fifth columns show the results for each of the two-layer models.On each panel, the x-axis indicates the different chromaticity correction cases.The results without chromaticity correction (NC) are depicted in black.The results with perfect chromaticity correction (PC) are shown in green.Other colors represent cases where the chromaticity correction has been applied but the value assumed for one of the soil parameters during the computation of the beam directivity has an error (10%, 5%, or 1%).The legend for the two-layer soil models is contained in panel (g).The reduced χ 2 of the fits across all cases is in the range 0.81-1.17,indicating that the LinLog model is reasonable and that there is no significant over-or underfitting.
vations conducted with beams that are effectively achromatic.Therefore, the PC results serve as the reference.
The 21 cm estimates in the PC cases are obtained with errors below 1% for â21 , 0.2% for ν21 , and 0.3% for ŵ21 .These estimates are less than 2 sigma away from the input values.For the nine soil models, the achromatic effective beams are different.As Figure 7 of Monsalve et al. (2023) shows, the FWHM of the original beams at ν r = 75 MHz (the frequency from which the achromatic effective beams are produced) spans the ranges [57 • , 63 • ] and [103 • , 121 • ] in the E and H plane, respectively.The differences in the beams produce differences in the beam-convolved foreground contribution to the spectra.However, these differences do not lead to significant differences in the 21 cm estimates, which remain consistent across soil models.Further, to model the beam-convolved foreground contribution, all the PC spectra need N f g = 5.In the absence of beam chromaticity, N f g = 5 quantifies the complexity of the spectra due to the foreground alone, in our case from the GSM.N f g = 5 is consistent with results from EDGES, in which five terms are needed to describe the measured spectra in the ranges 50-100 MHz and 90-190 MHz after applying chromaticity correction (Monsalve et al. 2017;Bowman et al. 2018).The BICs of all the PC fits are ≈ 81.24.For N µ = N f g + 3 = 8, this BIC corresponds to a χ 2 of 48.35 and a reduced χ 2 of 0.91.

Single-layer Models
The figures show that there is high consistency in the 21 cm estimates across our single-layer models.This consistency reflects that the estimates are not very sensitive to the absolute value of the beam directivity, which for our single-layer models can vary by more than 100% at some angles and frequencies (Monsalve et al. 2023).Further, the estimates closely match the input values, even without chromaticity correction.In the NC cases, the errors are < 20% for â21 , 0.1% for ν21 , and 4% for ŵ21 , and the estimates are less than 2 sigma away from the input values.The NC spectra require N f g = 6.Although this value represents a higher spectral complexity than from the foreground alone, it is low compared to the number of terms used by other experiments when modeling real data without chromaticity correction.For instance, LEDA, measuring above soil and a 3 m × 3 m ground plane, needed eight terms to model their spectrum over 50-85 MHz (Bernardi et al. 2016).SARAS 3, observing on a lake, needed seven terms over 55-85 MHz (Singh et al. 2022).The need for only six terms without chromaticity correction over 45-105 MHz reflects the low beam chromaticity of MIST when observing over uniform soils.
Applying a chromaticity correction significantly improves the accuracy and precision of the 21 cm estimates, even when the assumed soil parameters have errors.Specifically, when the chromaticity correction is affected by errors in the soil parameters of 10% or less, the errors in the 21 cm estimates are comparable to the PC cases: < 1% for â21 , 0.2% for ν21 , and 0.6% for ŵ21 .Further, for these erroneous chromaticity corrections, N f g is 5 and the BIC is almost the same as for the PC cases.These values for N f g and the BIC indicate that the correction errors do not significantly increase the complexity of the spectra above the intrinsic complexity of the foreground.

Two-layer Models
As expected from the more complex directivity they produce, two-layer soil models make the extraction of the 21 cm signal more difficult and, in general, impose requirements on the accuracy of the soil parameter values.

Changes in Bottom-layer Conductivity
Model 2L_c+ is the most challenging among our two-layer soil models because it produces the largest ripples in the simulated directivity (Monsalve et al. 2023).Without chromaticity correction, the complexity in the spectrum produced by this soil model requires increasing N f g to 8, as shown in panel (c) of Figure 3.The errors in the estimates are ≈ 300% for â21 , 10% for ν21 , and 30% for ŵ21 .The large and uneven error bars seen in Figure 1, in particular for â21 and ν21 , occur because in the NC case the posterior PDFs are wide and multimodal.When applying chromaticity correction with 10% and 5% errors in the soil parameters, the accuracy of the 21 cm estimates does not improve compared to the NC case.In some of these cases, fitting the spectrum also requires an N f g as high as 8, indicating that the imperfection of the correction leaves complex structure behind.The error bars of the estimates that require N f g = 8 are also large due to wide and multimodal PDFs.When the soil parameter errors in the chromaticity correction are reduced to 1%, the errors in ν21 are reduced to 1%.However, the errors in â21 and ŵ21 remain large: ≈ 60% and 20%, respectively.Therefore, conducting observations with MIST over the type of soil represented by model 2L_c+ would require a soil parameter accuracy better than 1%.Reaching this accuracy is anticipated to be extremely challenging and, hence, this type of soil should be avoided.
Model 2L_c-enables the recovery of the 21 cm parameters with the highest accuracy among our two-layer models.This occurs because the simulated directivity for 2L_c-has the smallest ripples compared with the other two-layer models (Monsalve et al. 2023).For this model, when chromaticity correction is not applied, the errors in the estimates are ≈ 24% for â21 , 0.3% for ν21 , and 9% for ŵ21 , and the fitting of the spectrum requires N f g = 6.Applying chromaticity correction with increasing accuracy consistently reduces the errors in the 21 cm estimates.In particular, with 5% (1%) errors in the soil parameters, the errors in â21 and ŵ21 are reduced to ≈ 8% (5%) and 2.3% (1.2%), respectively.Fitting the corrected spectra in most cases requires an N f g of 5 or 6.In one case, however, a value of 4 is preferred.This case occurs when the correction is computed with a 10% error in ϵ r2 .Requiring an N f g < 5, as in this case, indicates that the spectral structure introduced by the imperfect chromaticity correction cancels out some of the structure in the intrinsic foreground spectrum.

Changes in Bottom-layer Permittivity
When the simulated observations are conducted over the 2L_p+ two-layer soil model, applying chromaticity correction is necessary for extracting the 21 cm feature.Fitting the spectrum in the NC case requires N f g = 5.This value would initially suggest that the beam chromaticity does not significantly increase the complexity of the spectrum beyond that due to the foreground.However, in this case the Gaussian is found in emission instead of absorption and at an incorrect center frequency.The best-fit amplitude is 193 mK (error of 196.5%) and the best-fit center is 67.5 MHz (error of 15.6%).When chromaticity correction is applied, the Gaussian is found in absorption and the estimates cluster about the input parameter values.In particular, with 10% errors in the soil parameters, the errors in the 21 cm estimates are 30% for â21 , 2.4% for ν21 , and 7% for ŵ21 .Among our two-layer soil models, 2L_p+ follows 2L_c-in terms of fidelity in the extraction of the Cosmic Dawn feature as long as chromaticity correction is applied.This result is consistent with the fact that 2L_p+ produces the second smallest ripples in the simulated directivity after 2L_c- (Monsalve et al. 2023).
For model 2L_p-, the extraction of the Cosmic Dawn feature is again challenging and requires chromaticity correction with a soil parameter accuracy of 1%.Without correction, the Gaussian is found in absorption but with large errors: 230% for â21 , 6% for ν21 , and 20% for ŵ21 .When correction is applied, a noticeable decrease in the errors of ν21 and ŵ21 , to 1.6% and 11% respectively, is achieved if the errors in the soil parameters are reduced to 5%.However, sufficiently decreasing the error in â21 , specifically to 14%, requires reducing the soil parameter errors to 1%.This is the secondstrictest accuracy requirement among our models, consistent with 2L_p-producing the second largest ripples in the simulated directivity following 2L_c+ (Monsalve et al. 2023).Because of this tight accuracy requirement, this type of soil should also be avoided.

SUMMARY AND FUTURE WORK
In this paper, we use simulated observations to gain intuition about the effects of the MIST beam chromaticity on the detection of the global 21 cm signal from the Cosmic Dawn.We attempt the detection of this signal for the nine models of the MIST beam directivity introduced in Monsalve et al. (2023).These directivity models come from electromagnetic simulations that incorporate different models for the soil.Five of the soil models are single-layer, uniform models.The other four are two-layer models in which either the electrical conductivity or relative permittivity changes below 1 m from the surface.We study one phenomenological model for the Cosmic Dawn absorption feature.This model corresponds to a Gaussian with an amplitude of −200 mK, a center at 80 MHz, and a full width at half maximum of 20 MHz, consistent with standard physical models.
Our five single-layer soil models yield accurate and precise estimates for the 21 cm parameters even without making a correction for beam chromaticity before fitting the spectrum.Nonetheless, if chromaticity correction is applied, the estimates improve noticeably, becoming highly consistent with the input values even if the correction is imperfect due to errors in the parameters assumed for the soil.These results are very encouraging and strongly motivate us to optimize for soil uniformity when choosing an observation site.
Two-layer soil models produce ripples in the beam directivity as a function of frequency, which make the extraction of the 21 cm signal more challenging.The best results among our soil models are obtained for model 2L_c-, when the bottom soil layer has a lower conductivity than the top layer, which is the case that produces the smallest ripples.In this case, the 21 cm parameters can be determined (with an accuracy that for the absorption amplitude is 24% or better) even without chromaticity correction.The second best results are obtained for model 2L_p+, in which the bottom layer has a higher permittivity than the top layer and which produces larger ripples.In this case, however, chromaticity correction is required for the 21 cm estimates to approach the input values.In models 2L_p-and 2L_c+ the bottom layer has a lower permittivity and higher conductivity than the top layer, respectively, and the ripples in the directivity are even larger.In these two cases it is possible to recover the 21 cm parameters with acceptable accuracy (in particular, an amplitude error within a few tens of a percent) only if chromaticity correction is applied with a soil parameter accuracy equal to and better than 1%, respectively.Meeting these requirements is expected to be extremely challenging; therefore, these two types of soils must be avoided.
Natural extensions to this analysis include the exploration of a wide range of models for the 21 cm signal as well as for the soil.In the future, we will incorporate soil models with more than two layers and different layer thicknesses.We will also explore errors in the chromaticity correction due to assuming an incorrect number of layers for the soil.Sampling the soil parameter space in a statistically robust way can be computationally intensive, especially for soil models with two or more layers (i.e., five or more parameters), when each sample requires a new electromagnetic simulation of the instrument.To overcome this computational bottleneck, we are developing an emulator that will quickly generate realizations of the beam directivity after being trained on a sufficiently large set of precomputed electromagnetic sim-ulations.This emulator will enable the efficient sampling of the soil parameter space and help reveal the covariances between soil and 21 cm parameters.
For each soil model, chromaticity correction, and error type tested, we studied the extraction of the Cosmic Dawn signal using a single spectrum.This spectrum corresponds to a 24 hr average of observations simulated with a cadence of 6 minutes from the latitude of MARS (79.38 • N).To fit the foreground contribution to this spectrum we use the LinLog analytical model.In the cases with perfect chromaticity correction, this model requires five expansion terms, consistent with theoretical predictions and experimental results.The reduced χ 2 of the PC fits is 0.91 and across all cases is in the range 0.81-1.17,which indicates that the LinLog model is a reasonable choice for this analysis.In the future we will explore other analytical models for the foreground contribution.Leveraging the time dependence of the beam-convolved foreground, in addition to improving the constraints on the timeindependent 21 cm signal (e.g., Liu et al. 2013;Tauscher et al. 2020;Anstey et al. 2023), could relax the accuracy requirements on the soil parameters.We also leave the exploration of this possibility for future work.
We acknowledge support from ANID Chile Fondo 2018 QUIMAL/180003, Fondo 2020 ALMA/ASTRO20-0075, Fondo 2021 QUIMAL/ASTRO21-0053.We acknowledge support from Universidad Católica de la Santísima Concepción Fondo UCSC BIP-106.We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC), RGPIN-2019-04506, RGPNS 534549-19.This research was undertaken, in part, thanks to funding from the Canada 150 Research Chairs Program.This research was enabled in part by support provided by SciNet and the Digital Research Alliance of Canada.We acknowledge the support of the Canadian Space Agency (CSA) [21FAMCGB15].We also acknowledge the Polar Continental Shelf Program for providing funding and logistical support for our research program, and we extend our sincere gratitude to the Resolute staff for their generous assistance and bottomless cookie jars.
Figure1.Estimates for the parameters of our Gaussian 21 cm absorption model, a 21 , ν 21 , and w 21 , obtained from fits to simulated observations with the MIST instrument over nine soil models.The first column shows the results for the five single-layer soil models.These results are similar and depicted with the same color for the same chromaticity correction case and error type.The second through fifth columns show the results for each of the two-layer models.On each panel, the x-axis indicates the different chromaticity correction cases.The results without chromaticity correction (NC) are depicted in black.The results with perfect chromaticity correction (PC) are shown in green.Other colors represent cases where the chromaticity correction has been applied but the value assumed for one of the soil parameters during the computation of the beam directivity has an error (10%, 5%, or 1%).The legend for the two-layer soil models is contained in panel (g).

Figure 2 .
Figure 2. Absolute percentage error of the estimates shown in Figure 1 computed using Equation 9.

Figure 3 .
Figure 3. (Top) Number of terms in the LinLog foreground model used to produce the results of Figure 1.These numbers were determined by minimizing the BIC.(Bottom)BICs corresponding to the N f g in the top row.The reduced χ 2 of the fits across all cases is in the range 0.81-1.17,indicating that the LinLog model is reasonable and that there is no significant over-or underfitting.