Lunar Orbit Measurement of the Cosmic Dawn's 21 cm Global Spectrum

The mock observational data are generated with a physically motivated sky model. The input sky map can be obtained at a given frequency by extrapolation from observed data. A few programs are available for this purpose, e.g., the Global Sky Model (GSM; de Oliveira-Costa et al. 2008; Zheng et al. 2017), the Cosmology in the Radio Band (CORA; Shaw et al. 2014), and the Self-consistent whole-Sky foreground Model (SSM; Huang et al. 2019). However, interstellar medium (ISM) absorption becomes quite significant at low frequencies, which may affect the foreground 21 cm signal separation. In this work, we make use of the high-resolution Ultra-Long wavelength Sky model with Absorption (ULSA; Cong et al. 2021), which incorporated the free–free absorption effect of the ISM. ⁶

The sky maps at 30 MHz, 60 MHz, and 120 MHz are shown in Figure 1. The galactic free–free absorption lowers the temperature of the sky at lower frequencies toward H ii regions around massive stars or supernova remnants—an especially obvious one is the Gum Nebula (Woermann et al. 2001) at the lower-left corner (around l ∼ 150^°) at 30 MHz. In comparison to the conventional absorption-free model, the inclusion of the free–free absorption effect results in decrements of 2.5% at 30 MHz and 0.5% at 120 MHz in sky brightness temperature.

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The ULSA model considers both cases of a constant spectral index and a direction-dependent spectral index for the emission. In the case of a direction-dependent spectral index, the spectral index map is obtained by combining the Haslam 408 MHz map (Haslam et al. 1974, 1981, 1982; Remazeilles et al. 2015), the LWA maps (Dowell et al. 2017), and the Guzman 45 MHz map (Guzmán et al. 2011; through Equation (29) in Cong et al. 2021), after smoothing all maps with a Gaussian beam with FWHM = 5°. The whole-sky-averaged spectrum is shown in Figure 2. The solid lines are the absorption-included sky brightness, and the dashed lines are the absorption-free sky brightness. Here we show the result for both a constant and a direction-dependent spectral index of emission. We can see that there are significant differences at the lower frequencies due to absorption, which may affect the fitting of the spectrum for a given foreground model. It is therefore important to take into account the ISM absorption if one is to fit the spectrum at lower frequencies. Also, considering the importance of incorporating the spatial variation in the spectral index when accounting for the chromatic beam effect (Anstey et al. 2021), in this work, we shall use the model with absorption and direction-dependent spectral index to simulate the mock data.

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Then, in the data analysis, the foreground contribution to the global spectrum could be modeled as polynomials of frequency or its logarithm. In Table 2, we list some foreground models used in the literature along with short names (Pritchard & Loeb 2008, 2010; Harker et al. 2012; Bowman et al. 2018; Hills et al. 2018).

Table 2. Foreground Fitting Models, ν_c = 75 MHz

Model	Fitting Function
(a) five-term Poly	${T}_{{\rm{F}}}(\nu )=\displaystyle \sum _{n\,=\,0}^{4}{a}_{n}{\left(\tfrac{\nu }{{\nu }_{{\rm{c}}}}\right)}^{n-2.5}$
(b) seven-term Poly	${T}_{{\rm{F}}}(\nu )=\displaystyle \sum _{n\,=\,0}^{6}{a}_{n}{\left(\tfrac{\nu }{{\nu }_{{\rm{c}}}}\right)}^{n-2.5}$
(c) three-term LogPoly	${T}_{F}(\nu )={\left(\tfrac{\nu }{{\nu }_{c}}\right)}^{-2.5}\displaystyle \sum _{n\,=\,0}^{2}{a}_{n}{\left(\mathrm{log}\tfrac{\nu }{{\nu }_{c}}\right)}^{n}$
(d) five-term LogPoly	${T}_{F}(\nu )={\left(\tfrac{\nu }{{\nu }_{c}}\right)}^{-2.5}\displaystyle \sum _{n\,=\,0}^{4}{a}_{n}{\left(\mathrm{log}\tfrac{\nu }{{\nu }_{c}}\right)}^{n}$
(e) three-term LogLogPoly	$\mathrm{log}{T}_{{\rm{F}}}(\nu )=\displaystyle \sum _{n\,=\,0}^{2}{a}_{n}{\left(\mathrm{log}\tfrac{\nu }{{\nu }_{{\rm{c}}}}\right)}^{n}$
(f) five-term LogLogPoly	$\mathrm{log}{T}_{{\rm{F}}}(\nu )=\displaystyle \sum _{n\,=\,0}^{4}{a}_{n}{\left(\mathrm{log}\tfrac{\nu }{{\nu }_{{\rm{c}}}}\right)}^{n}$

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Figure 3 shows the residuals of fitting the global spectrum using the models listed in Table 2. Here we show both a fit in the frequency range of 30–120 MHz (solid lines) and a fit in the frequency range of 60–120 MHz (dash lines) for each model. The rms of residuals of models (a)–(f) are 78.37 mK (0.06 mK), 7.58 mK (0.01 mK), 172.74 mK (33.69 mK), 8.14 mK (0.17 mK), 117.60 mK (1.21 mK), and 16.08 mK (0.01 mK) for the solid lines (dashed–dotted lines), respectively.

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As shown by the figure and the residues, the primary source of foreground, the galactic synchrotron radiation, has a nearly power-law form for its spectrum, which can be easily fitted with a few terms in a polynomial expansion. However, the ISM free–free absorption at low frequency breaks the power law. So when the fitting is expanded to the 30–120 MHz frequency range, the five-term Poly and three-term LogLogPoly models, which worked well in the 60–120 MHz, range become inadequate. The five-term LogPoly, five-term LogLogPoly, and seven-term Poly models can fit well in all cases. We take the five-term LogPoly model as our fiducial model for the foreground below.

We consider here two models of the 21 cm global signal: one is the form adopted by the EDGES experiment in Bowman et al. (2018), another is a Gaussian fit for the 21 cm through with adjustable amplitude and width (Hills et al. 2018).

EDGES 21 cm model: This is motivated by the EDGES experiment observation. We assume the signal is

$$\begin{eqnarray}&&{T}_{21}(\nu )=-A\left(\displaystyle \frac{1-{{\rm{e}}}^{-\tau {{\rm{e}}}^{B}}}{1-{{\rm{e}}}^{-\tau }}\right),\end{eqnarray} \tag{ 3 }$$

where

$$\begin{eqnarray}&&B=\displaystyle \frac{4{\left(\nu -{\nu }_{0}\right)}^{2}}{{w}^{2}}\mathrm{log}\left[-\displaystyle \frac{1}{\tau }\mathrm{log}\left(\displaystyle \frac{1+{{\rm{e}}}^{-\tau }}{2}\right)\right].\end{eqnarray} \tag{ 4 }$$

Gaussian model: as demonstrated in Cohen et al. (2018), different astrophysical parameters can produce a variety of global 21 cm spectra with absorption trough depths in the range of −200 mK < T_b < −25 mK. Xu et al. (2021) also took into account the inevitable nonlinear density fluctuations in the intergalactic medium, shock heating, and Compton heating and found that these effects can reduce the maximum absorption signal by 15% at redshift 17. Here we use a simpler three-parameter Gaussian form given in Equation (5) as our global signal model:

$$\begin{eqnarray}&&{T}_{21}(\nu )=-{{Ae}}^{\tfrac{-{\left(v-{\nu }_{0}\right)}^{2}}{2{w}^{2}}}.\end{eqnarray} \tag{ 5 }$$

Then we consider four cases for the global 21 cm signal profile:

Case 1: No detectable 21 cm global signal.

Case 2: The EDGES 21 cm signal given by Equation (3), with A = 0.5 K, ν₀ = 78.0 MHz, w = 19.0 MHz, τ = 7.

Case 3: A Gaussian signal given by Equation (5), with A = 0.20 K, ν₀ = 75.0 MHz, w = 6.0 MHz.

Case 4: A Gaussian signal given by Equation (5), with A = 0.155 K, ν₀ = 75.0 MHz, w = 10.0 MHz.

Figure 4 illustrates the 21 cm global signal for these four scenarios.

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The thermal noise on the observed signal is

$$\begin{eqnarray}&&{\sigma }_{n}=\displaystyle \frac{{T}_{\mathrm{sys}}}{\sqrt{N{\rm{\Delta }}\nu {t}_{\mathrm{int}}}},\end{eqnarray} \tag{ 6 }$$

where N is the number of independent measurements, Δν is the channel bandwidth, which we set to 0.4 MHz, and t_int is the integration time. The system temperature is given by

$$\begin{eqnarray}&&{T}_{\mathrm{sys}}={T}_{\mathrm{sky}}+{T}_{\mathrm{rcv}},\end{eqnarray} \tag{ 7 }$$

where the average sky temperature is given by

$$\begin{eqnarray}&&{T}_{\mathrm{sky}}\sim 2300{\left(\displaystyle \frac{\nu }{75\,\mathrm{MHz}}\right)}^{-2.5}.\end{eqnarray} \tag{ 8 }$$

Note that while it is important to account for the ISM absorption when fitting the low-frequency spectrum, which would affect the extracted 21 cm signal several orders of magnitude lower than the foregrounds, the absorption effect makes little difference when estimating the thermal noise. Therefore, a simple power-law model is assumed here. T_rcv gives the equivalent antenna temperature induced by the noise in the receiver. As the antenna and receiver impedances are mismatched, the actual power that goes into the receiver is (1 − ∣Γ∣²)P_ant, where Γ is the reflection coefficient between the antenna and receiver, and P_ant is the power of the antenna output. So,

$$\begin{eqnarray}&&{T}_{\mathrm{rcv}}=\displaystyle \frac{{T}_{\mathrm{rcv}}^{0}}{1-| {\rm{\Gamma }}{| }^{2}},\end{eqnarray} \tag{ 9 }$$

where ${T}_{\mathrm{rcv}}^{0}$ is the noise temperature of the receiver. For our current receiver design, we have measured ${T}_{\mathrm{rcv}}^{0}\approx 200\,{\rm{K}}$ in the given band. Based on antenna simulation, we find that

$$\begin{eqnarray}&&\displaystyle \frac{1}{1-| {\rm{\Gamma }}{| }^{2}}\approx 16.809\exp \left(-\displaystyle \frac{{\left(\tfrac{\nu }{{\nu }_{{\rm{c}}}}-0.4\right)}^{1.277}}{0.0808}\right)+1.034.\end{eqnarray} \tag{ 10 }$$

We plot T_rcv in Figure 5. It is about 200 K near 80 MHz.

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With such a system temperature, for a single antenna (N = 1), after about 24 hr (10 orbits) of integration time, the rms noise level is ≤15 mK at 75 MHz, assuming a channel bandwidth of 0.4 MHz. Even if we assume that only the data obtained during the "doubly good time," when both Earth and Sun are shielded, can be used, which is about 10% of the total orbit time, it takes only 10 days for the thermal noise to be suppressed down to this level according to Equation (6).

A detailed study of the global spectrum in the 1–30 MHz band will be presented elsewhere, but here we note in passing that at 25 MHz, the sky temperature is about 3.6 × 10⁴ K, according to Equation (8). After a full three years of integration, the thermal noise of a single antenna could be suppressed to a level of about 10 mK for 0.4 MHz channel bandwidth. There are five to eight satellites in the DSL mission, and each is equipped with three pairs of antennas of different polarizations. So the DSL mission has the potential to measure the 21 cm global spectrum from the dark ages and cosmic dawn. However, we note that this estimate accounts only for the thermal noise; a number of other factors or systematic errors, such as the self-generated RFI, stability, dynamic range, and spectral response of the system will probably limit the precision of the measurement. It is necessary to design the system such that these systematic errors will be smaller than the expected 21 cm signal.

The global spectrum in the 30–120 MHz band will be measured by an antenna on a single satellite in the DSL mission. The primary requirement of the design is to have low chromaticity, i.e., its beam pattern should be as frequency independent as possible, to avoid convolving the spatial structure of the sky into the frequency domain. This can be achieved by using an antenna that is electrically small, i.e., with a physical size smaller than the half-wavelength, so that the resonance frequency is above the observational band. The small size of the antenna is also consistent with the requirement on the payload of the mission.

In the present design, we consider a disk-cone antenna model, which is composed of a radiating cone and a reflective disk, as shown in Figure 6. The size parameters of the cone and disk are labeled on the figure. The design of this antenna will be presented elsewhere; here, we take a particular design as our fiducial model. The beam of this antenna is simulated using the software FEKO. ⁷ The cross section of the axial-symmetric beam profiles, cut at an arbitrary ϕ with varying θ, is shown in the top panel of Figure 7; the actual beam in the 3D space can be generated by rotating the cross-section figure around the central axis (θ = 0°). The beam pattern of these electrically small disk-cone antennas is quite similar to that of a short dipole; there is a single lobe of the beam (it peaks at θ = 90° and 270° on the cross-section plot). To see the effect of frequency variation of the beam more clearly, we also show the logarithmic frequency gradient of the beam $\partial B(\nu ,\theta )/\partial \mathrm{log}\nu =\nu \partial B(\nu ,\theta )/\partial \nu$ in the bottom panel of Figure 7. The gradient for this disk-cone antenna is fairly small. In the next section, we shall mainly use this antenna beam model, but to explore how the chromatic beam affects the global spectrum measurement result, we shall also consider beam models that are more frequency dependent.

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As there are large-scale variations in the foreground, and the Moon blocks different parts of the sky for the satellite orbiting to the different positions, we first check how much the variation is for different locations. For this purpose, we select four points, marked as A, B, C, and D, which are located 90° apart on the lunar orbit. These are essentially the cross-over points of the orbit with the ecliptic plane and the highest and lowest points away from the elliptic plane along the orbit. The observed global spectrum at these points are the averaged sky temperature as weighted by S( n )B( n ) at these locations. In Figure 8, we show the product of the antenna beam and the Moon blockage function, S( n )B( n ), at the four locations. As expected from the symmetry, the pair A and C have exactly opposite patterns, and so does the pair B and D.

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We then calculate the foreground spectra as would be observed at these points with our fiducial disk-cone antenna model described above. In Figure 9, we plot the global spectra in the 30–120 MHz range. There are obvious differences in the amplitude for the four locations, but the shapes of the spectra are still quite similar to each other, and the spectra from locations A and C almost coincide.

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Just as shown by the spectra at these four points, there is a large variation in the total intensity. In Figure 10, we plot the simulated intensity variation at 60 MHz, as the satellite takes an observation over 24 hr. In this plot, the red solid line marks the part of data obtained when Earth is shielded, while the black dashed line are for the rest of the time. The figure shows that the signal received at a fixed frequency has a regular periodic variation, with a period of 8246 s corresponding to one cycle of the orbit, and an amplitude of ∼10% level.

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Actually the global spectrum is measured over a period of time as the satellite orbits the Moon. We can compute the sky average for a segment of the orbit. Despite large variations in the overall amplitude of the antenna temperature at different parts of the orbit, as shown in Figure 10, the variation in the shape of the spectrum is not large.

We fit each of the integrated spectrum for the AB, BC, CD, and DA segments of the orbit separately with the five-term LogPoly foreground model and plot the residuals in Figure 11. Here the input sky model includes only the foreground, while the mock observation is performed with simulated noise, Moon blockage, and the fiducial model of antenna response. For clarity, the results are vertically shifted by different amounts in the figure. We find that the beam-weighted signals can be reconstructed with a high accuracy. The four segments yield similar residuals, and these residuals (0.012 K, 0.009 K, 0.011 K, 0.008 K, respectively) are all noticeably smaller in magnitude than the expected global 21 cm signal in typical cosmic dawn models, showing that despite the variation of the foreground with sky direction, this would not seriously affect the 21 cm signal extraction.

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In Figure 12, we plot the averaged spectrum at 30–120 MHz after a total of 30 orbits, but take observations only when Earth is shielded, which is about one-third of the effective time. In the same figure we also plot the fitting residual of the foreground model for both the noise-free case and with noise given at a level corresponding to t_obs = 10 orbits. The data are assumed to be taken with our fiducial antenna model.

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We find that the LogPoly model is able to account for the global spectral structure by expanding around the expected power-law spectral index of −2.5. Here a₀ is an overall foreground scale factor, $\tfrac{{a}_{n}}{{a}_{0}}\ll 1\ (n\geqslant 1)$ is the correction to the typical spectral index of the foreground by capturing anisotropies of the spectral index and other higher-order spectral corrections, such as the ISM absorption. The five-term LogPoly model can provide a good fit in this foreground-only sky model, with the foreground model parameters taking reasonable values as shown in Figure 13. There is also no serious degeneracy in these parameters. Note that with a microsatellite on the lunar orbit, our fitting implicitly assumes that the parameter dependencies encode only the foreground properties and does not account for ionospheric effects.

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Next we consider the 21 cm signal extraction from the mock data simulated with the foreground, the 21 cm signal, and the noise. We plot in Figure 14 the five-term LogPoly model0fitting results for the EDGES 21 cm model (left panel) and the two Gaussian 21 cm models (middle and right panels). We see that in all cases, the 21 cm model signals can be well recovered by fitting the mock observation data.

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We also list the fitted model parameters for the EDGES 21 cm model in Table 3 and for the two Gaussian 21 cm models in Table 4 (the four rows correspond to the satellite completing 1, 2, 5, and 10 orbits, respectively). The thermal noise level drops with observation time as ${t}_{\mathrm{obs}}^{-1/2}$. For the frequency resolution of 0.4 MHz, the residual rms induced by the thermal noise drops to ∼0.05 K after 10 orbits, allowing the extraction of the 21 cm signal with good precision. The uncertainties of the parameters are estimated by Markov Chain Monte Carlo simulations. If the experiment is properly designed so that the error is dominated by thermal noise on this timescale, one should be able to discriminate between the EDGES model and the Gaussian models, and also between the two Gaussian models with different parameters introduced earlier. We also checked the results generated at different starting positions of precession for a 10 effective-orbit observation; the recovered 21 cm signals generally agree with each other within 0.01 K.

Table 3. The Best-fit Parameters of the 21 cm Global Signal and rms Values for the EDGES Model

21 cm model	tobs [orbit]1	rms [K]	A	τ	ν0	ω
Input	⋯	⋯	0.50	7.00	78.00	19.00
Recovered	1	0.133	${0.501}_{-0.013}^{0.013}$	${7.2}_{-1.1}^{1.5}$	${77.96}_{-0.14}^{0.13}$	${19.00}_{-0.31}^{0.31}$
	2	0.077	${0.5017}_{-0.0093}^{0.0092}$	${6.84}_{-0.82}^{0.94}$	${77.89}_{-0.10}^{0.10}$	${19.00}_{-0.23}^{0.22}$
	5	0.061	${0.5001}_{-0.0067}^{0.0061}$	${6.92}_{-0.54}^{0.67}$	${77.960}_{-0.065}^{0.065}$	${19.02}_{-0.15}^{0.14}$
	10	0.041	${0.5004}_{-0.0044}^{0.0043}$	${7.02}_{-0.40}^{0.46}$	${77.979}_{-0.045}^{0.046}$	${19.02}_{-0.10}^{0.10}$

Note. Here, as a simple model for the circular orbit, the satellite will orbit the Moon with a period of 8248.7 s (∼2.3 hr).

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Table 4. The Best-fit Parameters of the 21 cm Global Signal and the rms Values for the Two Gaussian Models

21 cm model	tobs [orbit]	rms [K]	A	ν0	ω
Input	⋯	⋯	0.20	75.00	6.00
Recovered	1	1.149	${0.201}_{-0.015}^{0.015}$	${74.85}_{-0.52}^{0.49}$	${6.11}_{-0.55}^{0.61}$
	2	0.095	${0.200}_{-0.010}^{0.010}$	${75.04}_{-0.36}^{0.34}$	${5.93}_{-0.39}^{0.38}$
	5	0.074	${0.2000}_{-0.0067}^{0.0067}$	${74.90}_{-0.23}^{0.22}$	${5.90}_{-0.24}^{0.26}$
	10	0.041	${0.2020}_{-0.0049}^{0.0048}$	${74.91}_{-0.17}^{0.16}$	${6.04}_{-0.18}^{0.20}$
Input	⋯	⋯	0.155	75.00	10.00
Recovered	1	0.117	${0.170}_{-0.026}^{0.032}$	${75.3}_{-1.4}^{1.2}$	${10.2}_{-1.5}^{1.6}$
	2	0.092	${0.160}_{-0.020}^{0.017}$	${75.27}_{0.85}^{-0.94}$	${9.6}_{-1.0}^{1.1}$
	5	0.060	${0.162}_{-0.012}^{0.014}$	${75.11}_{-0.67}^{0.62}$	${10.22}_{-0.70}^{0.76}$
	10	0.047	${0.1582}_{-0.0082}^{0.0089}$	${75.07}_{-0.44}^{0.42}$	${10.06}_{-0.48}^{0.50}$

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To explore how the chromatic beam affects the global spectrum measurement, we now consider some nonideal cases where the beam is more frequency-dependent. To be concrete, we still consider disk-cone antennas but with different disk diameters (D₃ in Figure 6). However, we find that as long as the size of the antenna is smaller than the half-wavelength, the beam is pretty similar to that of the short dipole and the frequency dependence is weak. This is good news for the design of the antenna for the space experiment. However, for this illustration, we decide to employ two "villain" models, which are intentionally designed to have a larger frequency variation of the beam. These are realized by keeping the cone part unchanged, while increasing the disk to a size comparable with the half-wavelength. In the first case, we increase the disk to 200 cm from the 55 cm in the fiducial model while keeping the cone unchanged. In the second case, the disk size is increased to 300 cm. One can obviously make antenna less frequency dependent, but we use them as illustrative examples because they are derived from electromagnetic simulations of an antenna configuration.

Figure 15 shows the beam profile of these two villain models in the top row and the corresponding logarithmic frequency gradient of the beam in the bottom row. Comparing with Figure 7, in these two cases, the beams vary with frequency much more significantly, and the logarithmic frequency gradient is also much larger. When convoluted with the spatial variations of the foreground, this may generate fake features in the global spectrum. Furthermore, in the model with a 300 cm disk (right panels of Figure 15), an additional sidelobe appears. The presence of a sidelobe could induce even larger features in the global spectrum from the sky variation, as we will find out below.

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To examine how this affects the measurement of the global spectrum, we simulate the observation of a total of 10 orbits with the foreground-only model without noise. Here for simplification we have used the data from the full orbit but ignored any possible RFI.

In Figure 16, we plot the residuals after fitting and subtracting a five-term LogPoly model for our fiducial model, as well as for the two villain models. The residuals for the two villain models are about two orders of magnitude higher than our fiducial model, so we multiply a factor of 100 to the fiducial model residue to show them on the same plot. The rms of the residuals of our fiducial model without adding the noise is about 10 mK.

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As was shown in Figure 7, ν∂B/∂ν ∼ 10⁻² for our fiducial antenna beam model, and as shown in Figure 10 the amplitude of the global spectrum varies at 10% level. We may then expect the chromatic beam may introduce a deviation at the 10⁻³ level over the observation band, or ∼1 K for the 10³ ∼ 10⁴ K sky temperature of this band. However, the variation in the spectral shape or spectral index induced by the varying position/orientation is much smaller. As a result, this deviation varies smoothly over the frequencies, and so when we fit with the five-term LogPoly model, the bulk of it is removed. In the end, we obtain a residual with an rms of only 10 mK. This is still a very substantial and important source of error for the global spectrum measurement.

The two villain models are both exaggerated bad cases, but these illustrate how the beam may vary with frequency over the observational band. Even for better designed cases, the frequency gradient of the beam may have a pattern similar to them, though with smaller gradient values. In particular, we note that the additional sidelobe may greatly increase the error. Thus, as shown in Figure 16, the disk-cone antenna with a 300 cm disk, which has an additional sidelobe in its beam, has much larger rms residuals than the one with a 200 cm disk.

It is likely that the real antenna has a beam more like that of our fiducial model than that of the two villain models shown above. Nevertheless, there may be frequency-dependent deviations due to imperfect manufacturing or additional components necessary for a satellite, which will modify the antenna beam pattern. Such deviations could adversely affect the performance of the global spectrum measurement. It is critical to minimize such effects for the experiment to succeed.