# Beamforming approaches toward detecting the 21-cm global signal from Cosmic
Dawn with radio array telescopes

D. C. Price  International Centre for Radio Astronomy Research, Curtin
University, Bentley, WA 6102, Australia  [

(dd Mmm YYYY; dd Mmm YYYY; dd Mmm YYYY)

###### Abstract

The formation of the first stars and galaxies during ‘Cosmic Dawn’ is thought
to have imparted a faint signal onto the 21-cm spin temperature from atomic
Hydrogen gas in the early Universe. Observationally, an absorption feature
should be measurable as a frequency-dependence in the sky-averaged (i.e.
global) temperature at meter wavelengths. This signal should be separable from
the smooth—but orders of magnitude brighter—foregrounds by jointly fitting a
log-polynomial and absorption trough to radiometer spectra. A majority of
approaches to measure the global 21-cm signal use radiometer systems on
dipole-like antennas. Here, we argue that beamforming-based methods may allow
radio arrays to measure the global 21-cm signal. We simulate an end-to-end
drift-scan observation of the radio sky at 50–100 MHz using a zenith-phased
array, and find that the complex sidelobe structure introduces a significant
frequency-dependent systematic. However, the $\lambda/D$ evolution of the beam
width with frequency does not confound detection. We conclude that a
beamformed array with a median sidelobe level $\sim-50$ dB may offer an
alternative method to measure the global 21-cm signal. This level is
achievable by arrays with $O(10^{5})$ antennas.

###### doi:

10.1017/pasa.2020.32

###### keywords:

astronomical instrumentation: radio telescopes; astronomical techniques: time
domain astronomy; radio frequency interference

D. C<EMAIL_ADDRESS>22 September 2020

##  1 INTRODUCTION

Cosmic Dawn is a cosmological epoch spanning from the formation of the first
stars ($z\sim 30$) until reionization started in the intergalactic medium
($z\sim 10$). The physics and cosmology of this epoch is imprinted upon the
spin temperature of neutral Hydrogen gas (HI), making the detection of HI
21-cm emission from the Cosmic Dawn one of the best probes of the early
Universe (Furlanetto et al., 2006; Morales & Wyithe, 2010; Pritchard & Loeb,
2012; Cohen et al., 2017).

Analogous to the cosmic microwave background (CMB), the 21-cm emission from
Cosmic Dawn is expected to be measurable as a global, sky-averaged signal: an
absorption trough in the global sky spectrum at observing frequencies of
45–130 MHz, corresponding to redshifts pf $30\gtrsim z\gtrsim 10$ (Shaver et
al., 1999; Furlanetto et al., 2006; Morales & Wyithe, 2010; Pritchard & Loeb,
2012). The depth, width, and central frequency of the absorption trough depend
on astrophysical processes and cosmology, but it is expected to have a
magnitude of $O(100)$ mK and $O(10)$ MHz width (Cohen et al., 2017). This
$\sim$100 mK signal must be separated from astrophysical foregrounds, which
are dominated by synchrotron emission that is many orders of magnitude
brighter. Fortunately, synchotron emission is spectrally smooth, and can be
accounted for using a low-order polynomial in log-frequency space. The 21-cm
signal can therefore be recovered by jointly fitting a low-order polynomial
and Gaussian absorption feature to measured data.

Using this approach, the Experiment to Detect the Global EoR Signal (EDGES)
reported the detection of a $\sim$18.7 MHz wide, $\sim$530 mK absorption
feature at 78.1 MHz (Bowman et al., 2018). Remarkably, this signal is 2–3
times brighter than that expected from optimistic models (Cohen et al., 2017),
which would point toward gas temperatures during Cosmic Dawn being far cooler
than predicted. However, there is ongoing debate as to whether or not the
EDGES detection is bona fide, or the result of unmodelled systematics. Hills
et al. (2018) and Singh & Subrahmanyan (2019) suggest that the feature may be
explained by an unaccounted for sinusoidal ripple within the EDGES receiver;
Bradley et al. (2019) suggests that the finite ground plane under the antenna
could give rise to a resonant spectral structure. Sims & Pober (2020) show via
a Bayesian reanalysis that models including previously unmodelled systematics
are decisively preferred, but do not rule out models which include a 21-cm
signal with a revised $\sim$200 mK amplitude (in closer agreement with
predictions). Most concerning is that results from the Shaped Antenna
Measurement of the Background Radio Spectrum 3 (SARAS 3) experiment reject the
EDGES best-fit profile to 95.3% (Singh et al., 2022).

Acronym | Experiment
---|---
EDGES-hi |  Experiment to Detect the Global EoR Signal1
EDGES-lo |  Experiment to Detect the Global EoR Signal, low-band2
BIGHORNS |  Broadband Instrument for Global HydrOgen ReioNisation Signal3
LEDA |  Large-aperture Experiment to detect the Dark Ages4
SARAS |  Shaped Antenna measurement of the background RAdio Spectrum5
SCI-HI |  Sonda Cosmológica de las Islas para la Detección de Hidrógeno Neutro6
PRIZM |  Probing Radio Intensity at high-$z$ from Marion7
REACH |  Radio Experiment for the Analysis of Cosmic Hydrogen8
DARE |  Dark Ages Radio Explorer9
1Rogers & Bowman (2008); Bowman & Rogers (2010); Rogers & Bowman (2012)
2Bowman et al. (2018) 3Sokolowski et al. (2015) 4Greenhill & Bernardi (2012);
Price et al. (2018)
5Patra et al. (2013); Singh et al. (2018, 2022) 6Voytek et al. (2014) 7Philip
et al. (2019)
8Cumner et al. (2022) 9Burns et al. (2021)
Table 1: Radiometer experiments to detect the global 21-cm signal.

This tension, and difficulty in ruling out low-level systematic effects,
highlights the need for alternative experimental approaches. A majority of
global 21-cm experiments, including EDGES, use a simple dipole-style antenna
connected to an exquisitely calibrated receiver (see Table 1). One alternative
approach is to use radio interferometers. Under normal assumptions and
conditions, interferometers are insensitive to the global signal, as they do
not sample the origin of the $uv$-plane. However, the interferometric response
of a pair of closely-spaced antennas to the global signal is not zero
(Vedantham et al., 2015; Presley et al., 2015; Singh et al., 2015), and
efforts to exploit this to detect the 21-cm signal are underway (McKinley et
al., 2020; Thekkeppattu et al., 2022). For these experiments, effects such as
mutual coupling and cross-talk must be understood and accounted for. An
alternative method, in which the occultation of the Moon in front of a patch
of sky is used to make a differential measurement of the 21-cm signal
(Vedantham et al., 2015; McKinley et al., 2018). This approach suffers from
comparably low sensitivity and is susceptible to reflected radio interference
from Earth, although improved techniques and instruments may solve these.

Another observing strategy for radio arrays is to use beamforming techniques.
In Dilullo et al. (2020) and Dilullo et al. (2021), the Long Wavelength Array
at Sevilleta (LWA-SV) was used in beamforming mode to simultaneously observe a
calibrator source (Virgo A) and a cold patch of sky. The main advantages of
this multi-beam technique is are a) a calibrator can be observed at the same
time as a science target field to counteract time-dependent systematics, b) a
cold patch of sky can be chosen, decreasing overall radiometer noise, and c)
the foreground model within the beam is much simpler, by virtue of the smaller
angular size. While this approach shows promise, systematics such as the
ionosphere, and uncertainty in the beam response, are challenges that must be
overcome.

Indeed, the complex beam response of a radio array is one of the primary
arguments put forth against their use. Any variations of an antenna’s beam
power pattern as a function of frequency will lead to chromatic mixing of
spatial structure into spectral structure (Vedantham et al., 2014; Bernardi et
al., 2015; Mozdzen et al., 2017). If the sky’s spatial structure is not taken
into account, or if the beam pattern is not known to enough accuracy, an
unwanted spectral structure will be imprinted upon calibrated spectrum. Given
an array of diameter $D$, the width of its primary beam is proportional to
observing wavelength by $\lambda/D$. Consequently, there is considerable
evolution of the beam across 45–130 MHz. Because of this, Dilullo et al.
(2021) employ a novel achromatic beamforming technique to counteract the
evolution of the beam with increased frequency.

The frequency-dependence of an antenna beam is often referred to as beam
chromaticity. The mixing of spatial structures into the observed spectrum
requires a more complex treatment of foregrounds. Bernardi et al. (2015)
showed that beam chromaticity for a realistic dipole introduces spectral
structure, but that it could be accounted for by using a higher-order
polynomial fit. This concern drove EDGES to change their antenna to a simpler
design that could be more accurately modelled in electromagnetic simulation
packages (Mozdzen et al., 2016; Monsalve et al., 2017a). For the same reason,
SARAS introduced new antennas at each iteration (Patra et al., 2013; Singh et
al., 2018, 2022). High-gain antennas were considered for the upcoming REACH
experiment, but simulations showed that their complex beam patterns resulted
in larger residuals than low-gain antenna designs (Cumner et al., 2022).

As presented by Hibbard et al. (2020), beam chromaticity can be accounted for
using eigenvectors formed from beam-weighted foregrounds. In agreement with
previous findings (Vedantham et al., 2014; Bernardi et al., 2015; Monsalve et
al., 2017b), the eigenvectors are shown to be more dependent upon the beam
model than the foreground model. The methodology offers a practical approach
to generating accurate beam-weighted foreground models that would be required
for beamforming approaches.

In this paper, we use simulations to investigate how sidelobes and $\lambda/D$
frequency dependence of a beamformed beam affect global 21-cm signal
experiments. For a $\sim$degree-sized beam, we find that frequency dependence
of the primary beam does not confound detection; rather, the sidelobe
structure introduces complex spectral features. If beam sidelobes can be
reduced to $\sim$50 dB of the primary beam, a detection of the global 21-cm
signal may be possible.

##  2 METHODS

Figure 1: An aerial view of the Engineering Development Array v2 (EDA2).

We investigate the plausibility of beamformer-based methods, by simulating
observations that incorporated realistic beam patterns and sky foregrounds.
Here, we simulated 24-hour drift-scan observations across 50–100 MHz. While
beamformed arrays allow for alternative pointing strategies, a drift-scan with
the array pointed toward zenith is the most similar to current radiometer
approaches. Further, by fixing to zenith, projection effects and effects due
inter-antenna beam pattern variations are minimized.

Our aim is to model the antenna temperature of a beamformed array. This is
given by the average of the sky brightness $T_{\rm{sky}}(\theta,\phi,\nu)$ as
seen from the antenna’s location, weighted by the antenna’s gain pattern
$B(\theta,\phi,\nu)$:

$T_{\rm{ant}}(\nu)=\frac{\int
d\Omega\,B(\theta,\phi,\nu)T_{\rm{sky}}(\theta,\phi,\nu)}{\int
d\Omega\,B(\theta,\phi,\nu)}.$ (1)

Separating the $T_{\rm{sky}}$ into a ‘foreground’ component, $T_{\rm{fg}}$,
and a cosmological term $T_{\rm{cosmo}}$ consisting of the sky-averaged 21-cm
emission and the CMB background, we have

$T_{\rm{ant}}(\nu)=\frac{\int
d\Omega\,B(\theta,\phi,\nu)T_{\rm{fg}}(\theta,\phi,\nu)}{\int
d\Omega\,B(\theta,\phi,\nu)}+T_{\rm{cosmo}}(\nu).$ (2)

We chose to use the antenna layout of Engineering Development Array (EDA,
Wayth et al., 2017, 2021) as a primary example (Figs. 1 and 2). The EDA is a
precursor instrument for the Square Kilometre Array SKA-low aperture array,
which will operate across 50–350 MHz. We chose the EDA2 as our example as is
under active development, has a well-understood beam response, and is located
close to EDGES within the Murchison Radio Observatory.

Figure 2: EDA2 antenna layout, as used in OSKAR simulations.

###  2.1 Station beam

We used the OSKAR111https://github.com/OxfordSKA/OSKAR aperture array
simulator (Dulwich et al., 2009) to simulate the beam pattern of EDA2. OSKAR
was designed to produce simulated data from aperture array telescopes that
employ beamforming, interferometry, or both methods. OSKAR employs the Radio
Interferometer Measurement Equation (RIME, Hamaker et al., 1996; Smirnov,
2011) to produce simulated visibility data. In the RIME framework,
beamforming—the weighted sum of the response from each antenna with complex
beamforming weight—is equivalent to computing a ‘station beam’ Jones matrix.

We used the OSKAR task oskar_sim_beam_pattern to generate zenith-phased
station beams across 50–100 MHz in 1 MHz steps, and saved these in FITS format
(Wells et al., 1981). Simulated station beams at 50 MHz and 100 MHz are shown
in Fig. 3. Evident in Fig. 3 is both the scaling of beam width with wavelength
$\lambda/D$, and the complex sidelobe response. Cuts through the beam centres
are shown in Fig. 4, with Gaussian fits to the primary beams overlaid. The
first sidelobe peaks at $\sim 15$ dB below the primary beam response.

Figure 3: Simulated station beam patterns for EDA2 (all-sky orthographic
projection), at 50 MHz (left) and 100 MHz (right). Figure 4: Cuts of antenna
gain patterns for EDA2, at 50 and 100 MHz. Gaussian fits to the primary beam
are shown as dashed grey lines.

###  2.2 Astrophysical Foregrounds

Figure 5: Diffuse sky model at 50 MHz, generated by PyGDSM using the de
Oliveira-Costa et al. (2008) GSM. Colorscale is $log_{2}(T)$. The sky area not
visible to EDA2 has been masked.

In order to simulate astrophysical foregrounds, we used the PyGDSM package
(Price, 2016), which provides an interface to three global diffuse sky models
(GDSMs):

  * •

GSM08: The Global Sky Model (de Oliveira-Costa et al., 2008).

  * •

GSM17: The ‘improved’ Global Sky Model (Zheng et al., 2017).

  * •

LFSM: The Low Frequency Sky Model (Dowell et al., 2017).

All three GDSMs can be used to estimate the diffuse foregrounds across the
50–100 MHz range.

During our investigations we noted that spectra generated with GSM17 showed a
discontinuity around 45 MHz, which we believe arises due to a scale offset in
the underlying Guzmán et al. (2011) map at 45 MHz. Indeed, Monsalve et al.
(2021) reports a $-160\pm 78$ K zero-level correction is required for the
Guzmán et al. (2011) map, based on EDGES measurements of the sky spectra. A
similar correction was applied to the LFSM to remove an inflection at 45 MHz
(J. Dowell, personal comms). Regardless, the models show broad agreement to
within $\sim 10\%$. For comparison with Bernardi et al. (2015), we used the
GSM08, and generated sky brightness maps (Fig. 5) over 50–100 MHz in 1 MHz in
HEALpix format (Górski et al., 2005) with NSIDE=256.

###  2.3 Drift scan simulation

The antenna temperature for a given pointing was simulated by combining
HEALpix maps generated with PyGDSM for $T_{\rm{fg}}(\theta,\phi,\nu)$, and
beam models from OSKAR for $B(\theta,\phi,\nu)$. We then investigated the
effect of the sidelobes by comparison against a simple Gaussian beam fitted to
the primary beam (Fig. 4).

The zenith-pointed OSKAR beam models are output in equatorial coordinates. To
simulate a drift scan, we converted the beam patterns celestial coordinates
corresponding to zenith at 48 equally-spaced times across a sidereal day, then
regridded into HEALpix arrays with NSIDE=256. $T_{\rm{cosmo}}$, the 21-cm
absorption trough, is modelled as a Gaussian with 5 MHz and 200 mK amplitude,
centered at 78 MHz. These values are based on EDGES, using the lower revised
amplitude from Sims & Pober (2020). We then numerically computed
$T_{\rm{ant}}(\nu)$ (Eq. 2).

Fig. 6 shows the resulting simulated drift-scan observations for the EDA2. In
comparison to a dipole, the directional beam of EDA2 results in lower
$T_{\rm{ant}}$ for most LSTs due to the suppression of the hotter Galactic
plane foregrounds.

##  3 RESULTS

Figure 6: Simulated observations with the EDA2, using a simulated beam model
(blue) and Gaussian beam model (orange). Figure 7: Contribution of sidelobes
to antenna temperature, as given by the difference between simulated and
Gaussian beam models shown in Fig 6, at LST=4 hr (top panel, black curve),
with $N$=5 log-polynomial fit (dashed red). The residual after subtraction of
the fit is shown in the bottom panel.

Our drift-scan simulation of EDA2 (Fig. 6) shows that antenna temperature
changes markedly with sidereal time, from a minimum of 2703 K at 50 MHz (4 hr
LST) to a maximum of 39,650 K as the Galactic plane crosses the primary beam
(LSTs 15–20 hr). In contrast, EDGES measures a much more gradual change in
$T_{\rm{ant}}$ with LST (see Tab.2, Monsalve et al., 2021), reaching a maximum
16,375$\pm$308 K at 45 MHz at LST 17.84 hr, and minimum 5607 $\pm$ 107 K at
LST 2.56 hr. The lower antenna temperature of the beamformer approach would
mean that a target radiometer noise level can be reached within a shorter
integration time.

Nevertheless, the sidelobe response of EDA2 introduces a chromatic systematic
that depends upon both frequency and time. In the right panel of Fig. 6, the
antenna temperature $T_{\rm{ant}}(\nu)$ at an LST of 4.0 hr is shown, for both
EDA2 and an ideal Gaussian beam with the same width. The contribution of the
antenna temperature due to the sidelobes is given by the difference between
the two spectra (Fig. 7). The sidelobes introduce complex frequency structure,
which remain after a $N$=5 log-polynomial fit. The systematic term shown in
Fig. 7 has a 1.6 K RMS: larger than the reported 500 mK EDGES 21-cm signal.

We further investigated the effect of the sidelobes, by modifying the EDA2
beam pattern to suppress the sidelobes by 10 dB and 20 dB (i.e. factors of 10
and 100) and re-running the simulation. The sidelobes were first isolated by
fitting and subtracting the primary beam from the beam pattern. We then
multipled the sidelobes by an attenuation factor, then added the result back
to the primary beam. The residuals for a $N=5$ log-polynomial fit are reduced
to 208 mK RMS and 20 mK RMS, respectively. The markedly lower residuals
suggest that sidelobes are a far larger source of systematic noise than the
$\lambda/D$ evolution of beam width.

To see if a the 21-cm absorption feature can be extracted from the simulated
antenna temperature, we used HIBAYES (Zwart et al., 2016; Bernardi et al.,
2016) to jointly fit the 21-cm spectrum and polynomial foreground. HIBAYES
uses a Bayesian framework to explore the signal-plus-foreground posterior
probability distribution and evaluate the Bayesian evidence for a given model.
Within HIBAYES, we model foregrounds as a 5-term log-polynomial with
coefficients $p_{i}$, and model the 21-cm absorption term as a Gaussian with
amplitude $A_{HI}$, central frequency $\nu_{HI}$ and width $\sigma_{HI}$. We
assigned uncertainties based on the radiometer equation,
$\sigma=T_{\rm{sys}}/\sqrt{\Delta\nu\tau}$, where $\Delta\nu$=1 MHz is channel
bandwidth, and $\tau$=1 hr, centered around LST=4 hr. The system temperature,
defined as $T_{\rm{sys}}=T_{\rm{ant}}+T_{\rm{rx}}$, was computed using a
receiver temperature $T_{\rm{rx}}$=500 K.

The 21-cm feature is not successfully recovered by HIBAYES for the EDA2 beam
pattern, or the EDA2 pattern with 10 dB sidelobe suppression. The 21-cm
feature is mostly recovered, however, when 20 dB sidelobe suppression is used;
HIBAYES reports $A_{\rm{HI}}=-169_{-38}^{+75}$ K,
$\nu_{\rm{HI}}=76.8_{-1.9}^{+0.8}$ MHz, and
$\sigma_{\rm{HI}}=4.6_{-1.1}^{+2.2}$ (95% CI). The global log evidence for the
fit is 30.8, which indicates that the fit is ‘very strong’ (Kass & Raftery,
1995). The posterior probability distribution for the EDA2 simulated data is
shown in Fig. 8. The recovered values for $\nu_{\rm{HI}}$ are offset from the
true values (i.e. the input to simulation), placing them outside the 95%
confidence region. This is likely due to the 1 MHz channel resolution in our
simulation, overfitting due to inflection points caused by the higher-order
coefficients of the log-polynomial model, and spectral structure introduced by
the sidelobes. In Fig. 7, an local maxima in the residuals is visible close to
78 MHz; despite the downweighted sidelobe contribution, spectral structure
such as this will affect the fit.

Figure 8: Posterior probability distribution, for a N=5 order polynomial and a
200 mK Gaussian absorption feature at 78 MHz with 5 MHz width, fitted to the
simulated data using the EDA2 beam pattern with sidelobes suppressed by 20 dB.
True values are indicated in red; the dark and light shaded regions indicate
the 68- and 95-percent confidence regions.

##  4 DISCUSSION

In this article, we investigated whether a beamformed array could be used to
measure the 21-cm absorption feature from Cosmic Dawn. Previous studies have
suggested that beam chromaticity is a major systematic, and as such a majority
of radiometer experiments have favored simple dipole-like antennas. Our
simulations suggest that the $\lambda/D$ evolution of an array beam does not
preclude detection of the Cosmic Dawn. However, sidelobe response is indeed an
issue.

Using the EDA2 as a case study, our simulations suggest that its sidelobes
would need to be attenuated by an additional 20 dB in order to detect the
21-cm feature. The mean sidelobe level of the EDA2 is -31 dB, so a mean
sidelobe level of $\sim-50$ dB is therefore needed. Simulations and
theoretical predictions show that the average sidelobe level of a quasi-random
array decrease as $1/\sqrt{N}$, when $N$ is the number of antenna elements. In
the SKA-low context, Sinclair (2015) showed that stations with $O(10^{3})$,
$O(10^{4})$ and $O(10^{5})$ elements would have median sidelobe levels of
$\sim-32,-42$ and $-52$ dB, respectively; the $-50$ dB average sidelobe
requirement could thus be achieved by an $O(10^{5})$ element array. We note
that early designs for the SKA included 11,200 antennas (van Ardenne et al.,
2012), which was originally driven by sidelobe requirements for imaging; this
was ultimately abandoned in favor of smaller 256-antenna stations so that
shorter inter-station baselines were possible.

A complementary approach to increasing station size for sidelobe suppression
is to apply apodization (also known as ‘windowing’) functions when
beamforming. These generally have the effect of decreasing sidelobe levels,
but also decrease directivity and widen the main beam. Methods such as that of
Taylor (1960) allow for a trade-off between main beam width and sidelobe
levels. However, the sidelobe level that a given array can achieve depends
upon the number of antennas in the array and how accurate the phasing
solutions are—so larger arrays may still be required to reach the $-50$ dB
sidelobe requirement. Efforts with the LWA-SV are already using apodization,
but to mitigate the $\lambda/D$ evolution of main beam width (Dilullo et al.,
2020, 2021). This has the effect of increasing sidelobe levels; our results
suggest that it may be preferable to focus on sidelobe reduction instead, and
allow the beam width to change. However, Dilullo et al. (2020) note that while
their approach increases sidelobes, it produces a ‘smoother’ beam by
decreasing the depth of nulls in the beam pattern, and does result in improved
residuals. While comparison between sidelobe smoothness and null depth is
beyond the scope of this work, determining optimal apodization functions for
low residuals will be an important area of research for beamformed global
signal experiments.

Here, we simulated a zenith-phased array, in part to avoid effects due to
variations in beam patterns at different pointing angles. Our OSKAR
simulations do not account for mutual coupling between antenna elements, which
causes the beam pattern for each antenna in a closely-packed aperture array to
differ. Mutual coupling effects for SKA-low stations have been investigated
using Electromagnetic simulation packages (Bolli et al., 2022; Sutinjo et al.,
2022); while considerably more computationally expensive, such simulations
offer a more comprehensive method to determine antenna beam patterns.
Alternatively, Kiefner et al. (2021) detail a holographic method to measure an
array’s antenna beam patterns. These approaches could be used in future work.

We also—and purposefully—did not account for the aperture array’s beam
pattern. Better results (i.e. lower residuals) are likely achievable if
a-priori information about beam pattern and foregrounds are taken into
account. For example, the optimal eigenvector methodology of Hibbard et al.
(2020), which accounts for beam-weighted foregrounds, could be applied to
beamformer-based experiments. The limitations of log-polynomials for
foreground modelling have been previously noted, and maximally smooth
functions with no inflection points have been proposed as an alternative to
polynomial fitting (Sathyanarayana Rao et al., 2015; Bevins et al., 2021).

An advantage of a beamformed array over a single antenna is that the beam can
be steered, and bright astrophysical sources can be used to experimentally
determine the beam pattern (as is done in Dilullo et al., 2021). However, far-
out sidelobes are hard to measure, so it may be challenging to reach the
required accuracy. Another advantage is that the array itself can be used to
measure the sky foreground—particularly if interferometric synthesis imaging
is used. We argue that the ability to make in-situ measurements of both the
beam and the sky are a promising capability that should be explored further
for 21-cm studies.

A huge amount of care is taken to characterize and stabilize radiometric 21-cm
experiments, as gain fluctuations must be controlled for the radiometer
equation to hold over $\sim 100$ hr timescales. A beamformed system can
instead use multiple pointings toward astrophysical sources with known
frequency spectra for bandpass calibration and to monitor gain variations.
Indeed, this approach is used in Dilullo et al. (2021) at LWA-SV, reaching an
RMS of 2.47–5.26 K using several fitting techniques. While they do not use a
drift scan approach, and use a different telescope, this RMS is similar in
magnitude to our simulated sidelobe noise.

In summary, we argue that beamforming approaches to 21-cm cosmology are a
promising avenue which should be revisited. We encourage further studies and
experiments using existing radio arrays with beamforming capability across
45–130 MHz, and design studies for next-generation aperture arrays with
$O(10^{4}-10^{5})$ elements.

## Software

This work made use of the following software packages: Numpy (Harris et al.,
2020), Astropy (Astropy Collaboration et al., 2013, 2018), Matplotlib (Hunter,
2007), Healpy (Zonca et al., 2019), OSKAR (Mort et al., 2017), PyGDSM (Price,
2016), and HIBAYES (Zwart et al., 2016; Bernardi et al., 2016).

D. Price thanks R. Wayth for comments on the manuscript, and C. Dillulo for
comments and contributions to PyGDSM.

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