2023

These authors contributed equally to this work.

]School of Physics and Astronomy, Tel Aviv University, Tel Aviv, 69978, Israel

# Precision cosmology with the 21-cm signal from the dark ages

Rajesh Mondal<EMAIL_ADDRESS>Rennan Barkana<EMAIL_ADDRESS>[

###### Abstract

The 21-cm signal from the dark ages provides a potential new probe of
fundamental cosmology. While exotic physics could be discovered, here we
quantify the expected benefits within the standard cosmology. A measurement of
the global (sky-averaged) 21-cm signal to the precision of thermal noise from
a 1,000 hour integration would yield a 10% measurement of a combination of
cosmological parameters. A 10,000 hour integration would improve this to 3.2%,
and constrain the cosmic Helium fraction to 9.9%. Precision cosmology with
21-cm fluctuations requires a collecting area of 10 km2 (which corresponds to
400,000 stations), which with a 1,000 hour integration would exceed the same
global case by a factor of $\sim$ 2\. Enhancing the collecting area or
integration time $\times$10 would yield a 0.5% parameter combination, a Helium
measurement five times better than Planck, and a constraint on the neutrino
mass as good as Planck. Our analysis sets a baseline for upcoming lunar and
space-based dark ages experiments.

##  1 Main

Observation of the redshifted 21-cm signal due to the hyperfine transition of
neutral hydrogen (HI) is a promising method to study its three-dimensional
(3D) distribution in the Universe Sunyaev1972 ; Hogan ; Scott1990 . The era
from the epoch of recombination (redshift $z\sim 1100$), when matter decoupled
from the radiation and these photons free-streamed as the cosmic microwave
background (CMB), until the formation of the first substantial population of
luminous objects ($z\sim 30$) is referred to as the ‘Dark Ages’. After
decoupling, the temperature of the gas ($T_{\rm g}$) declined adiabatically as
$(1+z)^{2}$ whereas the CMB temperature ($T_{\gamma}$) fell as $(1+z)$. The
spin temperature $T_{\rm s}$ (an effective temperature that measures the ratio
of the population densities of the upper and lower states of the 21-cm
transition) was strongly coupled to $T_{\rm g}$ through collisional coupling
until $z\sim 70$ madau97 . After this time, the collisional process became
ineffective and $T_{\rm s}$ began to approach $T_{\gamma}$. Thus, during the
dark ages $T_{\rm s}$ remained significantly lower than $T_{\gamma}$ over a
wide redshift range of $300\lesssim z\lesssim 30$, during which the HI is
expected to produce absorption features in the CMB spectrum.

The dark ages are potentially a critically important window in the
evolutionary history of the Universe. Since cosmic evolution at this time was
not yet significantly affected by astrophysical processes, the dark ages offer
a clean probe of fundamental cosmology similar to the CMB. This is in contrast
to cosmological probes in the modern Universe, such as those based on the
galaxy distribution or on the statistics of Lyman-$\alpha$ absorption lines,
that suffer an irreducible systematic uncertainty due to the potential
influence of complex astrophysics (including star formation, radiative
feedback from stars and stellar remnants, and supernova feedback). The dark
ages can be probed using the redshifted HI 21-cm signal, either by measuring
the global (or mean) signal or by measuring the fluctuations at various length
scales, i.e., the power spectrum. Unlike the CMB which comes to us from a
single cosmic time, the 21-cm signal can be observed over a range of cosmic
times, as each frequency 1420 MHz$/(1+z)$ corresponds to a different look-back
time. The 21-cm data set is thus 3D; moreover, since small-scale fluctuations
that are washed out in the CMB are available in the 21-cm power spectrum, the
latter contains potentially far more cosmological information Loeb2004 .

Prior to recombination, the coupling of the baryons to the photons kept the
baryon density and temperature fluctuations negligible on sub-horizon scales.
The 21-cm power spectrum during the dark ages probes the era of baryonic
infall into the dark matter potential wells barkana2005  and is quite
sensitive to the values of the $\Lambda$CDM cosmological parameters
barkana2005 , particularly on the scale of the baryon acoustic oscillations
(BAO) that are a remnant of the pre-recombination baryon-photon fluid.
Importantly, the fluctuations on these scales are still quite linear during
the dark ages, and thus modeling them does not need to deal with complex non-
linearity as is the case for probes in the more recent Universe. The 21-cm
fluctuations probe fluctuations of the baryon density, peculiar velocity
bharadwaj04 ; barkana05 , and baryon temperature (as determined by the
fluctuating sound speed naoz05 ; barkana2005 ). A number of smaller
contributions must be included in a precise calculation Lewis2007 ; Ali-Ha2014
. All of this assumes the standard cosmology. Besides this, there are various
studies on non-standard possibilities during the dark ages, such as DM-baryon
interactions Tashiro14 ; Munoz2015 ; Barkana2018  or features in the
primordial power spectrum 2016JCAP . It has also been shown that the 21-cm
signal can potentially be a powerful probe of primordial non-Gaussianity, at
the levels expected from cosmic inflation (see, e.g., Loeb2004 ; Pillepich2007
; Joudaki2011 ; Floss2022 ; Balaji2022 ), but below we find that it would be
difficult observationally to reach the high wavenumbers where this is most
promising. While some types of exotic (non-standard) cosmology would be easier
to detect, we focus in this work on the safest case of standard cosmology.

Observing the 21-cm signal from the dark ages using radio telescopes on Earth
would be nearly impossible due to the ionosphere, which heavily distorts and
eventually blocks very low frequencies. This necessitates lunar or space-based
experiments, and these are being rapidly developed as part of the
international race to return to the moon, with efforts including NCLE
(https://www.ru.nl/astrophysics/radboud-radio-lab/projects/netherlands-china-
low-frequency-explorer-ncle) (Netherlands-China), DAPPER
(https://www.colorado.edu/project/dark-ages-polarimeter-pathfinder) (USA),
FARSIDE (https://www.colorado.edu/project/lunar-farside) (USA), DSL
(https://www.astron.nl/dsl2015) (China), PRATUSH
(https://wwws.rri.res.in/DISTORTION/pratush.html) (India), FarView
(https://www.colorado.edu/ness/projects/farview-lunar-far-side-radio-
observatory) (USA), SEAMS (India), LuSee Night (https://www.lusee-
night.org/night) (USA), ALO
(https://www.astron.nl/dailyimage/main.php?date=20220131) (Europe), and ROLSES
(https://www.colorado.edu/ness/projects/radiowave-observations-lunar-surface-
photoelectron-sheath-rolses) (USA). These missions will probe either the
global signal or the spatial fluctuations of the dark ages 21-cm signal. We
note that most of these experiments are at the early design stage, and also
measuring the dark ages power spectrum is substantially more futuristic than
the global signal.

Given the great potential for precision cosmology, as well as the rapid
observational developments, in this paper we study the use of the 21-cm signal
(both the global signal and power spectrum) during the dark ages to constrain
the cosmological parameters. In this prediction it is important to account for
observational limitations, and not just assume the theoretical limiting case
of a full-sky, cosmic variance limited experiment. Indeed, an inevitable
obstacle is thermal noise, which rises rapidly with redshift as well as
wavenumber. For the power spectrum, another observational barrier is the
angular resolution, since for an interferometer with a given set of antennae,
reaching a higher angular resolution reduces the array’s sensitivity. In
general in 21-cm cosmology, an interferometer requires a much greater
investment of resources than a simple antenna for measuring the global signal;
the potential reward of the former is also substantially greater due to the
richer information content available in measuring the power spectrum versus
redshift. We note that the just-mentioned observational obstacles worsen more
rapidly with redshift for the 21-cm power spectrum; the signal also declines
faster for the power spectrum, since the dark ages universe was more
homogeneous, density fluctuations were smaller, and there were no galaxies
around to amplify the 21-cm fluctuations. These factors make the global signal
relatively advantageous at least as an initial probe of the dark ages.

There are other practical difficulties faced by 21-cm experiments, including
removing terrestrial radio-frequency interference (RFI), accounting for the
effect of the ionosphere, and removing or avoiding foreground emission (coming
from synchrotron radiation from our own Milky Way as well as other galaxies).
As these obstacles are increasingly overcome, this will allow for deeper
integrations for which the noise remains dominated by the thermal noise.
Techniques for achieving this are a matter of research that is achieving
continuous improvement, as reflected in the best current constraints from
global experiments such as EDGES Bowman:2018  and SARAS SARAS , and
interferometers such as LOFAR LOFAR-EoR:2020 , MWA Trott:2020  and HERA HERA .
Thus, for the 21-cm signal from the dark ages, the thermal noise for various
integration times serves as a fiducial benchmark for future experiments. Note
that going to the moon could present substantial practical advantages beyond
just avoiding the Earth’s ionosphere: a potentially benign environment that is
extremely dry and stable, plus the blocking out of terrestrial RFI (on the
lunar far-side).

##  2 Calculating the 21-cm signal

As previously noted, the 21-cm differential brightness temperature relative to
the CMB, ${T}_{\rm b}$, from each redshift $z$ is observed at a wavelength of
$21\times(1+z)$ cm. Global experiments measure the cosmic mean 21-cm
brightness temperature; since these are relatively simple and are advantageous
at the highest redshifts, we consider them first in Sec. 3.1. In addition, the
21-cm brightness temperature fluctuates spatially, mainly due to the
fluctuations in the gas density and temperature. As noted above, the gas
retains some memory of the early BAO, on the scale traversed by sound waves in
the photon-baryon fluid (wavenumber $k\sim 0.1\,{\rm Mpc}^{-1}$; scales are
comoving unless indicated otherwise). The signature of these oscillations can
be detected in the 21-cm power spectrum from the dark ages (see Sec. 3.2).

We use the standard CAMB (http://camb.info) Lewis2007 ; CAMB  cosmological
perturbation code to precisely generate the 21-cm global signal from the dark
ages and, after accounting for the anisotropy due to redshift space
distortions BLlos , the 3D 21-cm power spectrum. While the line-of-sight
anisotropy is in principle measurable, foreground removal is expected to make
this difficult, so here we conservatively consider only the spherically-
averaged power spectrum. We also add the Alcock-Paczyński effect AP1979 ;
AliAP ; Nusser ; APeffect  and the light-cone effect barkana06 ; mondal18  to
our power spectrum calculations, and account for the effect of the field of
view and angular resolution of radio interferometers (see Supplementary
Information). We use the latest measurements (based mainly on the CMB)
Planck:2018  to set our fiducial values of the cosmological parameters. The
21-cm global signal and power spectra during the dark ages for this fiducial
model are shown, respectively, in Figs. 1 and 2, which are explained below in
further detail.

##  3 Results

###  3.1 The global 21-cm signal

As noted above, measuring the 21-cm global signal requires a single, well-
calibrated antenna. Fig. 1 shows the 21-cm global signal from the dark ages as
a function of $\nu$ (and $z$), for the fiducial cosmological model. The
expected signal dips to a maximum absorption of $-40.2$ mK at $z=86$
($\nu=16.3$ MHz). Also shown in Fig. 1 is the instrumental noise for $t_{\rm
int}=1$,000 hrs (a standard fiducial integration time, also equal to 11.4% of
a year) and 100,000 hrs, for a bin of width $\Delta(\ln\nu)=1$ around each
$\nu$. The noise increases sharply with redshift, yielding a maximum signal-
to-noise ratio (S/N) of 11.6 for $t_{\rm int}=1$,000 hrs and 116 for 100,000
hrs (both at $z=41$ or $\nu=34$ MHz); in the latter case, the S/N for the
global signal remains above unity up to $z=207$ ($\nu=6.8$ MHz). We note that
if other difficulties are overcome and integration time becomes the main
issue, then multiple copies of a global experiment effectively increase
$t_{\rm int}$ in proportion to the number of copies (as long as they are not
placed spatially too close together).

Figure 1: The 21-cm global signal from the dark ages (blue solid line) as a
function of $\nu$ (or $z$ as the top $x$-axis). We also show the expected
thermal noise for a global signal experiment observing for integration time
1,000 hrs (orange dashed line) or 100,000 hrs (green dotted line) for a bin
with $\Delta(\ln\nu)=1$.

By fitting the standard cosmological model to the global signal with its
expected errors, we find the constraints that can be obtained on the
cosmological parameters. While the amplitude of the global signal depends
significantly on some of the parameters, the shape is highly insensitive,
which means that the signal essentially measures a single quantity, the
overall amplitude, which depends on a combination of cosmological parameters.
Specifically, the global signal depends significantly only on the parameters
$\Omega_{\rm b}h^{2}$ and $\Omega_{\rm m}h^{2}$, where $\Omega_{\rm b}$ and
$\Omega_{\rm m}$ are the cosmic mean densities of baryons and (total) matter,
respectively, in units of the critical density, and $h$ is the Hubble constant
in units of 100 km s-1 Mpc-1. Given the strong degeneracy between the two
parameters, the constraint is on the combination (see Supplementary
Information)

$C_{\rm Global}\equiv\frac{\Omega_{\rm b}h^{2}}{(\Omega_{\rm
m}h^{2})^{0.248}}\ .$ (1)

The relative errors in $C_{\rm Global}$ for three different values of $t_{\rm
int}$ are shown (along with our other main results) in Table 1 and Fig. 3. We
account for the fact that the presence of the bright synchrotron foreground
means that a signal component of the same shape cannot be distinguished from
the foreground (see Supplementary Information). A measurement of the global
21-cm signal to the precision of thermal noise from a 1,000 hour integration
would yield a 10.1% measurement. This would be a remarkable achievement for
cosmological concordance, since it would be independent of other cosmological
probes and come from a previously unexplored cosmological era. Increasing the
integration time would improve this precision as the inverse square root, so
that sub-percent precision in $C_{\rm Global}$ (comparable to the typical
Planck precision on each cosmological parameter) would require a considerable
integration time exceeding 100,000 hrs.

Table 1: The relative errors in % and the limits on the total mass of
neutrinos (all are $1\sigma$). For the Helium fraction ($Y_{\rm P}$) and the
neutrino mass, we compare to constraints based on Planck CMB measurements
alone and also (Planck + BAO) those that include BAO measurements from galaxy
clustering Planck:2018 .

Global | Planck | Planck |  Integration time
---|---|---|---
signal |  $+$ BAO |  | 100,000 hrs | 10,000 hrs | 1,000 hrs
$C_{\rm Global}$ |  |  | 1.01 | 3.18 | 10.1
$Y_{\rm P}$ | 4.96 | 5.44 | 3.14 | 9.94 | 31.4
$\sum m_{\nu}\,[{\rm eV}]$ | $<0.0578$ | $<0.108$ | $<0.746$ |  |

Power | Planck | Planck |  Configuration
---|---|---|---
spectrum |  $+$ BAO |  | D | C | B | A | G
$C_{\rm PowSpec}$ |  |  | 0.0457 | 0.382 | 0.462 | 4.59 | 10.1
$Y_{\rm P}$ | 4.96 | 5.44 | 0.116 | 0.981 | 1.20 | 11.9 | 26.6
$\sum m_{\nu}\,[{\rm eV}]$ | $<0.0578$ | $<0.108$ | $<0.0100$ | $<0.0839$ | $<0.107$ | $<1.06$ |
|  |  |  |  |  |  |

In addition to the minimal, standard set of cosmological parameters, the
global signal can also provide additional constraints, of which we consider a
couple examples. For these additional parameters, we consider the favorable
approach in which we fix the standard set of parameters at their fiducial
values (e.g., based on Planck with its small errors) and explore the power of
the 21-cm signal to constrain these specific extended parameters. In
particular, since the 21-cm signal depends directly on hydrogen and not just
the total baryon density, the first additional parameter is the fraction of
the baryonic mass in helium, usually denoted $Y_{\rm P}$. This parameter is
currently constrained by Planck (even when combined with galaxy clustering) at
a level that is an order of magnitude worse than the precision on the standard
parameters. For $t_{\rm int}=1$,000 hrs, the global 21-cm signal would yield
an independent 31.4% constraint on $Y_{\rm P}$; 10,000 hrs would measure
$Y_{\rm P}$ to 9.94%, and the best-case scenario of 100,000 hrs would beat
Planck by a factor of 1.7 (Table 1 and Fig. 3). Similarly, we find constraints
on the neutrino mass, though for this purpose the global signal would not be
competitive with Planck, even for $t_{\rm int}=100$,000 hrs.

###  3.2 The 21-cm power spectrum

As noted above, compared to the global 21-cm signal from the dark ages, it
would take a substantially larger effort to measure the power spectrum.
However, looking towards the future, the power spectrum has a far greater
potential to become a ground-breaking cosmological probe, as it is a much
richer dataset. Fig. 2 shows the spherically-averaged power spectrum of 21-cm
brightness fluctuations as a function of wavenumber at various redshifts
during the dark ages. The signal rises initially as the adiabatic expansion
cools the gas faster than the CMB, creating an absorption signal that
strengthens with time. Eventually, though, the declining density reduces the
collisional coupling so that $x_{c}$ drops below unity and the 21-cm signal
weakens. For example, the maximum squared fluctuation $\Delta^{2}$ at
$k=0.1\,{\rm Mpc}^{-1}$ is 0.44 mK2 at $z=51$.

Figure 2: The spherically-averaged (total) power spectrum of 21-cm brightness
fluctuations as a function of wavenumber $k$ during the dark ages, at
redshifts $z=[150,125,75,50,40,30]$. The dotted lines show the power spectrum
at $z=75$ and 40 when accounting for the effect of angular resolution (for our
A or B configuration). We also show the 1$\sigma$ noise (thermal plus cosmic
variance) for our A (short dashed lines) and B (long dashed lines)
configurations, at $z=75$ and 40 (for bins with $\Delta(\ln\nu)=1$ and
$\Delta(\ln k)$=1).

For the observational setup, we assume a minimal case for which a 1,000 hour
integration would significantly exceed (by a factor of $\sim$ 2) the
constraint level given by the same global case; this would be required to
justify the much greater effort involved in building an interferometer. We
find that this would (approximately) require a collecting area of $A_{\rm
coll}=10\,{\rm km}^{2}$, which along with $t_{\rm int}=1$,000 hrs we adopt as
our minimal, A configuration. The collecting area of 10 km2 corresponds to
400,000 stations, each with an effective collecting area of 25 m2 (see
Supplementary Information). This is quite futuristic but we hope that our
theoretical work helps motivate new ideas to achieve this more quickly. The
four observational configurations that we use to illustrate measurements of
the 21-cm power spectrum are listed in Table 2 (for reference, we also include
a G configuration that yields constraints roughly equal to the 1,000 hour
global case). Fig. 2 shows the 1$\sigma$ noise expected for our A and B
configurations, when we include the (dominant) thermal noise as well as cosmic
variance (see Supplementary Information). The figure also shows the power
spectrum when accounting for the effect of angular resolution. The thermal
noise increases rapidly with redshift, and so the maximum S/N (without the
effect of angular resolution) occurs at the minimum redshift we consider
($z=30$), and is 13.3 for the A configuration and 133 for the B configuration,
both at $k=0.091$ Mpc-1.

Table 2: The observational configurations that we use to illustrate
measurements of the 21-cm power spectrum, in terms of the collecting area
$A_{\rm coll}$ and integration time $t_{\rm int}$ (see Supplementary
Information).

| Configuration
---|---
| D | C | B | A | G
$A_{\rm coll}$ [km2] | 100 | 100 | 10 | 10 | 5
$t_{\rm int}$ [hrs] | 10,000 | 1,000 | 10,000 | 1,000 | 1,000
|  |  |  |  |

Figure 3: We show graphically the main results that are listed in Table 1,
namely the relative errors in % and the limits on the total mass of neutrinos
(all are $1\sigma$). For the Helium fraction ($Y_{\rm P}$) and the neutrino
mass, we compare to constraints based on Planck (with or without BAO from
galaxy clustering). Note that we abbreviate “configuration” as “Conf.”.

Given measurements of the 21-cm power spectrum throughout the dark ages
($z>30$), we carry out a Fisher analysis with five cosmological parameters
(see Supplementary Information). The relative errors in the $\Lambda$CDM
cosmological parameters are rather large due to significant degeneracies, and
even configuration D approaches the accuracy level of Planck only in some of
the parameters (see Supplementary Information). As with the global signal, it
is more useful to consider parameter combinations that are well constrained.
In particular, we focus on the minimum variance combination (see Supplementary
Information), which for configuration A is

$C_{\rm PowSpec}\equiv\Omega_{\rm b}h^{2}\frac{(A_{\rm
s}e^{-2\tau})^{0.307}(0.9950)^{n_{\rm s}}}{(\Omega_{\rm
m}h^{2})^{0.464}H_{0}^{0.0753}}\,.$ (2)

Here the additional parameters Planck:2018  are the Hubble constant $H_{0}$
(in units of km s-1 Mpc-1), the primordial amplitude $A_{\rm s}$, the total
reionization optical depth to the CMB $\tau$, and the scalar spectral index
$n_{\rm s}$. We note that the form of $C_{\rm PowSpec}$ (eq. 2) changes
slightly for different scenarios (see Supplementary Information).

The relative errors in $C_{\rm PowSpec}$ for the various observational
configurations are shown in Table 1 and Fig. 3. Configuration A would yield a
4.59% measurement of the parameter combination $C_{\rm PowSpec}$, which would
be observationally independent of the global signal constraint and thus
provide a powerful cross-check. More importantly, there would be a great
potential for future improvement, as Configurations B and (the slightly
better) C would improve this by an order of magnitude (reaching the typical
Planck precision on each cosmological parameter), and D by a further order of
magnitude. Just as for the global signal, we consider constraints on
additional parameters while fixing the standard set of parameters. Here
configuration A would measure $Y_{\rm P}$ to 11.9%, B and C would do 5 times
better than the Planck constraint, and D would do almost 50 times better than
Planck. It is reasonable to again consider these constraints while fixing the
standard parameters based on Planck, since Planck constrains the standard
parameters better than any of our 21-cm configurations. Finally, the
constraint on the total neutrino mass would not be competitive with Planck for
configuration A, but would roughly match Planck for configurations B or C, and
beat it by an order of magnitude for configuration D. This constraint is
driven by the suppression of small-scale power due to neutrino free-streaming.
Here we emphasize a major advantage for probing the neutrino effect on
gravitational clustering during the dark ages: the corresponding scales were
still in the regime of linear fluctuations, and were not yet affected by the
complex astrophysics of galaxies.

##  4 Conclusions

Observations of the redshifted 21-cm signal from the dark ages have great
cosmological potential. While various exotic, non-standard scenarios could be
easily detected (or ruled out), here we considered the safe, conservative case
of standard cosmology, studying the potential for creating a powerful new
cosmological probe. We found constraints on the $\Lambda$CDM cosmological
parameters, independently considering the two major types of 21-cm
measurements, the global (or mean) signal as a function of frequency and the
spherically-averaged power spectrum as a function of both frequency and scale.
We used CAMB and added to it redshift space distortions, the Alcock-Paczyński
effect, the light-cone effect, and the effect of angular resolution. For the
error estimates, we considered different levels of thermal noise (plus cosmic
variance), meant to serve as a benchmark for experiments which face additional
practical challenges including foreground removal.

With global 21-cm signal measurements, we found that a combination of
cosmological parameters, $C_{\rm Global}$ (eq. 1), can be effectively
constrained. An integration time of 1,000 hrs would yield a relative error in
$C_{\rm Global}$ of 10.1%, with improvement to a best-case precision of 1.01%
for 100,000 hrs. In the case of the 21-cm power spectrum, it would take a
greater effort to achieve comparable constraints, but there are better
prospects for future advances. The parameter combination $C_{\rm PowSpec}$
(eq. 2) can be constrained to 4.59% in our configuration A (a 1,000 hr
integration with an array of collecting area 10 km2), but the precision can
improve to 0.0457% in our configuration D (a 10,000 hr integration with a
collecting area of 100 km2).

Fixing the standard set of cosmological parameters to their fiducial values,
we found constraints on separately varying two other important parameters.
Given the direct dependence of the 21-cm signal on hydrogen, the fraction of
the baryonic mass in helium $Y_{\rm P}$ would be constrained to 31.4% with a
1,000 hr integration of the global signal; 10,000 hrs would measure it to
9.94%, and the best-case scenario of 100,000 hrs would beat Planck by a factor
of 1.7. Using the power spectrum, configuration A would measure $Y_{\rm P}$ to
11.9%, B and C would do 5 times better than the Planck constraint, and D would
do almost 50 times better than Planck. Regarding limits on the total mass of
neutrinos, constraints that are competitive with Planck would be possible only
with the 21-cm power spectrum, for which configurations B or C would roughly
match Planck, and configuration D would beat it by an order of magnitude.

Our analysis highlights the potential of the 21-cm signal as a probe of
cosmology, and suggests a focus on the global signal as the first step, with
the 21-cm power spectrum being much more promising in the long run. Our
results set a baseline reference for many upcoming and future lunar and space-
based dark ages experiments.

##

Correspondence and requests for materials should be addressed to R. Mondal
(rajeshmondal18@gmail.com).

## Acknowledgments

The authors would like to thank Antony Lewis and Léon V.E. Koopmans for their
useful discussions. RM is supported by the Israel Academy of Sciences and
Humanities & Council for Higher Education Excellence Fellowship Program for
International Postdoctoral Researchers. The authors acknowledge the support of
the Israel Science Foundation (grant No. 2359/20).

## Author contributions

R.B. initiated the project. R.M. performed the calculations, made the figures,
and wrote the paper, in consultation with R.B..

## Data availability

The data are available upon request from the corresponding author.

## Code availability

CAMB is available at http://camb.info. emcee is available at
https://github.com/dfm/emcee. corner is available at
https://github.com/dfm/corner.py. The analyses are done in Python, extensively
using publicly available routines in NumPy (https://numpy.org) and Matplotlib
(https://matplotlib.org). All other codes used are available upon request from
the corresponding author.

## Competing interests

The authors declare no competing interests.

##  Appendix A Supplementary note

In this Supplementary Note, we first (Sec. A.1) briefly summarize our methods
and add some details and technical notes. We next describe in detail how our
predicted signal accounts for several effects: the Alcock-Paczyński effect
(Sec. A.2), the light-cone effect (Sec. A.3), and the effect of angular
resolution (Sec. A.4). We then note our method for constructing a useful
(minimum variance) linear combination of correlated parameters (Sec. A.5), and
present some additional results and discussion (Sec. A.6). Finally, we briefly
discuss the effect of foregrounds (Sec. A.7).

###  A.1 Summary of our methods

The main quantity for 21-cm observations, the excess brightness temperature
relative to the CMB from redshift $z$, is

$T_{\rm b}=(T_{\rm s}-T_{\gamma})\frac{1-e^{-\tau_{21}}}{1+z}\ ,$ (3)

where $\tau_{21}$ is the optical depth of the 21-cm transition. Assuming
$\tau_{21}\ll 1$, this can be expressed in the simpler form madau97 ;
Furlanetto2006

${T}_{\rm b}\simeq 54.0\,{\rm mK}\,\frac{\rho_{\rm HI}}{\bar{\rho}_{\rm
H}}\left(\frac{\Omega_{\rm b}h^{2}}{0.02242}\right)\left(\frac{\Omega_{\rm
m}h^{2}}{0.1424}\right)^{-\frac{1}{2}}\left(\frac{1+z}{40}\right)^{\frac{1}{2}}\frac{x_{\rm
c}}{1+x_{\rm c}}\left(1-\frac{T_{\gamma}}{T_{\rm g}}\right),$ (4)

where $\rho_{\rm HI}$ is the neutral hydrogen density and $\bar{\rho}_{\rm H}$
is the cosmic mean density of hydrogen, and also $x_{\rm c}$ is the
collisional coupling coefficient. During the dark ages, CMB scattering pulls
$T_{\rm s}\xrightarrow[]{}T_{\gamma}$, whereas atomic collisions pull $T_{\rm
s}\xrightarrow[]{}T_{\rm g}$.

As noted in the main text, we use the standard CAMB (http://camb.info)
Lewis2007 ; CAMB  cosmological perturbation code to generate the predicted
21-cm signal. Note that CAMB does not directly yield the 21-cm global signal
(i.e., the mean 21-cm brightness temperature) as a function of redshift, so we
extract it indirectly by running once with temperature (mK) units on and once
with temperature units off, and taking the ratio of the transfer functions in
the two cases. Also, CAMB outputs the 2D angular power spectrum, inspired by
CMB analyses but less relevant to 21-cm data that naturally constitute a 3D
dataset both theoretically and observationally. CAMB yields the transfer
function for the 21-cm monopole power spectrum, to which we add by hand the
redshift space distortions using the transfer function of baryon density,
based on Ref. BLlos . Having obtained the anisotropic 3D power spectrum, we
then average over angle in this linear-theory case (this is shown explicitly
within the derivation in Sec. A.2 below). After this, we add several effects
that are presented in detail in the next few sections.

In CAMB and throughout the paper, for the cosmological parameters we use
fiducial values (based mainly on the CMB) Planck:2018  of $H_{0}=67.66$ km s-1
Mpc-1, $\Omega_{\rm b}h^{2}=0.02242$, $\Omega_{\rm m}h^{2}=0.14240$, $A_{\rm
s}e^{-2\tau}=1.881\times 10^{-9}$, and $n_{\rm s}=0.9665$. We note that
$\Omega_{\rm m}=\Omega_{\rm c}+\Omega_{\rm b}+\Omega_{\nu}$, where
$\Omega_{\rm c}h^{2}=0.11933$ is the contribution of cold dark matter and the
fiducial neutrino contribution (based on the minimal total mass of 0.06 eV
allowed by neutrino oscillation experiments) is $\Omega_{\nu}h^{2}=6.451\times
10^{-4}$.

In this paper, our variables in the global signal case are $\log(\Omega_{\rm
b}h^{2})$ and $\log(\Omega_{\rm m}h^{2})$. For the 21-cm power spectrum we
have three additional variables: $\log(A_{\rm s}e^{-2\tau})$, $n_{\rm s}$, and
$\log(H_{0})$. We assume a flat Universe, where the rest of the energy density
($1-\Omega_{\rm m}$) is given by a cosmological constant. We note that in the
21-cm power spectrum from the dark ages (which, like cosmic recombination in
the case of the CMB, occurred long before any significant reionization), the
amplitude that is directly probed is $(A_{\rm s}e^{-2\tau})$ and not $A_{\rm
s}$. This is since the re-scattering of the 21-cm photons during reionization
damps the fluctuations as in the case of sub-horizon CMB fluctuations, i.e.,
the brightness temperature relative to the mean gets multiplied by a factor of
$e^{-\tau}$. Unlike the CMB, there is no separate information on $A_{\rm s}$
and $\tau$, since in the CMB there is information on the largest scales and on
polarization, but both of these are not expected in 21-cm measurements (at
least in the near future). We also note that there is no logarithm on $n_{\rm
s}$ since it is a power, i.e., it effectively is already defined as a
logarithmic variable. For the power spectrum $P(k)$, we express the results in
terms of the squared fluctuation $\Delta^{2}\equiv k^{3}P(k)/(2\pi^{2})$.

The thermal noise in a global signal measurement is Shaver1999

$\Delta T=\frac{T_{\rm sys}}{\sqrt{\Delta\nu\,t_{\rm int}}}\ ,$ (5)

where $\Delta\nu$ is the bandwidth, $t_{\rm int}$ is the integration time, and
we assume that the system temperature $T_{\rm sys}$ is approximately equal to
the sky brightness temperature $T_{\rm sky}=180\times(\nu/180\,{\rm
MHz})^{-2.6}$ K Furlanetto2006 .

Next we estimate the observational errors for the 21-cm power spectrum.
Although it is negligible in most practical cases, for completeness (and for
comparison with previous theoretically-motivated work) we include the error
due to cosmic variance (CV), which for the power spectrum measured in a bin
centered at a wavenumber $k$ and frequency $\nu$ we can express as (following
eq. 30 of Ref. mondal16 )

$\delta P_{\rm cv}(k,\nu)=\frac{2\pi P(k,\nu)}{\sqrt{V(\nu)k^{3}\Delta(\ln
k)}}\ ,$ (6)

where the survey volume for the frequency (redshift) bin $V(\nu)$, in the
limit of a thin bin, is given by $\Omega_{\rm FoV}r_{\nu}^{2}\Delta
r_{\Delta\nu}$, where $\Omega_{\rm FoV}$ is the field of view, $r_{\nu}$ the
comoving distance to the bin center, and $\Delta r_{\Delta\nu}$ is the
comoving length corresponding to the bandwidth $\Delta\nu$. Note that the
field of view (FoV) is $[21(1+z)\,{\rm cm}]^{2}/A_{\rm eff}$ for an antenna
with an effective collecting area of $A_{\rm eff}$. For example, $\Omega_{\rm
FoV}=8.89$ at $z\sim 70$ assuming $A_{\rm eff}=25\,{\rm m}^{2}$, compared to
the whole sky which is $4\pi=12.6$. Thus, given the small effective area and
large $z$, for a dark ages array the FoV is typically a significant fraction
of the sky; we set a cutoff of half the sky as the maximum solid angle
available to an interferometer.

The dominant error that we include in the power spectrum measurement is that
due to thermal noise, which can be expressed as mellema13

$\delta P_{\rm thermal}=\frac{2}{\pi}\left(\frac{k^{3}\,V}{\Delta(\ln
k)}\right)^{1/2}\,\frac{T^{2}_{\rm sys}}{\Delta\nu~{}t_{\rm
int}}\,\frac{1}{N^{2}}\,\frac{A_{\rm core}}{A_{\rm eff}}\,,$ (7)

where $N$ is the total number of stations and $A_{\rm core}$ is the core area
of the telescope array. A reasonable plan for an upcoming lunar array (Leon
Koopmans, personal communication) consists of $N=128^{2}=16,384$ stations with
$A_{\rm eff}=25\,{\rm m}^{2}$ and a core area equal to the collecting area,
i.e., $A_{\rm core}=A_{\rm coll}=N\times A_{\rm eff}$. This gives a total
collecting area of 0.4096 km2. We keep all of these relations fixed but modify
$N$, getting a total dependence of $\delta P_{\rm thermal}\propto 1/N$. This
number must be increased by a factor of 24.4 to $N=400$,000 to give our A
configuration in the main text (with a collecting area of $10\,{\rm km}^{2}$).
We note though that the smaller collecting area would suffice to put new
strict limits on various non-standard models, and it would approach the
performance of our A configuration if used with an integration time of a few
tens of thousands of hours.

We showed noise estimates that are independent of binning, so that we gave the
overall S/N based on a bin size of order the central value, i.e.,
$\Delta(\ln\nu)=1$ as well as $\Delta(\ln k)=1$. For the Fisher matrix
predictions, we used 8 frequency/redshift bins in the range $5.81\leq\nu\leq
45.81$ with a bin width of $\Delta\nu=5\,{\rm MHz}$, which corresponds to
central redshifts of [170, 106, 76.6, 59.9, 49.2, 41.6, 36.1, 31.8]; the upper
end of the frequency range was chosen at $z=30$, the typical redshift where
galaxies at cosmic dawn form in sufficient numbers to significantly affect the
21-cm signal subtle . For the power spectrum, we used 11 logarithmic $k$ bins
covering the range $0.00779\leq k<1.91\,{\rm Mpc}^{-1}$ with bin width
$\Delta(\ln k)=0.5$. We checked that the results are insensitive to increasing
these binning resolutions. See also sec. A.6 where these ranges are varied.

###  A.2 The Alcock-Paczyński effect

When using the 21-cm power spectrum for constraining cosmological parameters,
it is important to account for the fact that the 3D power spectrum depends on
distances, but these are usually not directly measurable in cosmology.
Instead, redshifts are measured along the line of sight, while angles are
measured on the sky. The conversions of these quantities to comoving distances
depend on the values of the cosmological parameters, which themselves are
being constrained by the data. This leads to the so-called Alcock-Paczyński
effect AP1979 , which is important also for the 21-cm signal AliAP ; Nusser ;
APeffect .

Following the analysis of this effect on the 21-cm power spectrum in Ref.
APeffect , the setup is that we have a true cosmology (which we take as that
given by the central, fiducial values of the cosmological parameters), and a
different assumed cosmology (for example, where one of the parameters is
varied from its fiducial value in order to find the resulting derivative of
the signal, for the Fisher matrix calculation). The conversion to distances at
redshift $z$ involves (on the sky) the angular diameter distance $D_{A}$ and
(for small distances along the line of sight) the Hubble constant $H$ at $z$.
The ratio $D_{A}$(true)/$D_{A}$(assumed) we designate $1+\alpha_{\perp}$, and
the ratio $[HD_{A}]$(true)/$[HD_{A}]$(assumed) we designate $1+\alpha$. Note
that these standard scalings, as written, are for physical distances, while we
are interested in comoving distances, but this does not matter here since the
difference is a redshift factor which in 21-cm cosmology is known precisely,
independently of the cosmological parameters. Now, instead of using the full
(complicated) equations in Ref. APeffect , we show here how to implement the
effect of the changing distance measures in two steps. Note that to linear
order the effects of $\alpha_{\perp}$ and of $\alpha$ are independent APeffect
.

The first step is to include the effect of $\alpha_{\perp}$ assuming
$\alpha=0$, which corresponds to assuming that the scalings from angle to
perpendicular distance and from frequency to line-of-sight distance are the
same, in terms of the true parameters relative to the assumed parameters. This
case is isotropic and is simple to do exactly (without a linear approximation)
APeffect . The formulas simplify further when applied to the dimensionless
combination $k^{3}P(k)$ (which is proportional to $\Delta^{2}$):

$k^{3}P(k)=k_{\rm{true}}^{3}P_{\rm{true}}(k_{\rm{true}})\ ,$ (8)

where $k_{\rm{true}}=k/(1+\alpha_{\perp})$.

Figure 4: The logarithmic dependence of the squared fluctuation on the two
parameters of relevance for the Alcock-Paczyński (AP) effect; i.e.,
$\frac{1}{\Delta^{2}}\frac{\partial\Delta^{2}}{\partial(\ln H_{0})}$ (left
panel) and
$\frac{1}{\Delta^{2}}\frac{\partial\Delta^{2}}{\partial(\ln[\Omega_{\rm
m}h^{2}])}$ (right panel), shown at $z=40$. We show the results in three
cases: without the AP effect, with the first (isotropic $\alpha_{\perp}$) step
only, and with the full AP effect.

The second effect, that of $\alpha\neq 0$, is anisotropic, but here we insert
it only for the case of interest, i.e., the simplified case of the effect on
the spherically-averaged power spectrum, to linear order in changes of the
parameters (i.e., to first order in $\alpha$). The result uses the angular
decomposition of the 21-cm power spectrum in linear theory, including the
effect of line-of-sight velocity gradients BLlos :

$P(k,\mu)=\mu^{4}P_{\mu^{4}}(k)+\mu^{2}P_{\mu^{2}}(k)+P_{\mu^{0}}(k)\ ,$ (9)

where $\mu=k_{z}/k$ is the cosine of the angle between the $\vec{k}$ vector
and the line of sight. Here and subsequently, the components (in the
decomposition in powers of $\mu$) of $P(k,\mu)$ refer to the result of eq. 8
(note that $\mu$ is left unchanged by the $\alpha_{\perp}$ rescaling). We note
that $P_{\mu^{0}}(k)$ is the (monopole) 21-cm power spectrum without velocity
effects, $P_{\mu^{4}}(k)$ is simply the power spectrum of the baryon density
(the dimensionless power spectrum times the global temperature squared, to get
mK2 units), and $P_{\mu^{2}}(k)$ is the cross term of the 21-cm fluctuation
and the baryon density fluctuation. The total spherically-averaged 21-cm power
spectrum is then:

$P(k)=\frac{1}{5}P_{\mu^{4}}(k)+\frac{1}{3}P_{\mu^{2}}(k)+P_{\mu^{0}}(k)\ .$
(10)

Now, from Ref. APeffect  we find that the second effect, that of $\alpha\neq
0$, on the spherically-averaged 21-cm power spectrum, is the addition to
$k^{3}P(k)$ of:

$\alpha\,\frac{\partial}{\partial\log
k}\left[\frac{1}{7}\,k^{3}P_{\mu^{4}}(k)+\frac{1}{5}\,k^{3}P_{\mu^{2}}(k)+\frac{1}{3}\,k^{3}P_{\mu^{0}}(k)\right]\
,$ (11)

where again the use of the dimensionless combination $k^{3}P(k)$ simplified
the result. Note that here the $\partial/\partial\log k$ refers to a
derivative at fixed cosmological parameters (as the change in the parameters
is captured through $\alpha_{\perp}$ and $\alpha$).

As noted in the main text, the Alcock-Paczyński effect is important in fitting
the dark ages 21-cm power spectrum, since it introduces a dependence on
$H_{0}$ that is separate from the dependence on the other cosmological
parameters; it also modifies the dependence of the 21-cm signal on
$\Omega_{\rm m}h^{2}$. Fig. 4 illustrates the logarithmic dependence of the
power spectrum on these two parameters that are relevant for the Alcock-
Paczyński effect. Both steps (in the above two-step procedure) have a
comparable contribution to the $H_{0}$ dependence (which would otherwise be
completely absent), while mainly the first (isotropic $\alpha_{\perp}$) step
significantly enhances the dependence on $\Omega_{\rm m}h^{2}$.

###  A.3 The light-cone effect

In measurements of the 21-cm power spectrum, since different positions along
the line of sight correspond to different redshifts (i.e., what is observed
are points along our past light cone), this results in anisotropy in the 21-cm
power spectrum barkana06 ; mondal18 . Here we are interested only in the
spherically-averaged power spectrum, as averaged over the redshift span of
each frequency bin, and the effect is then simply to average the signal over
this redshift range.

To understand how this averaging works, it is easier to consider the
correlation function, which is of course closely related to the power
spectrum. For the correlation function at some (comoving) distance $r$, what
we do is average over all pairs separated by $r$ in the observed volume. Let
us call the line-of-sight comoving distance $x$ in this subsection (to avoid
confusion with the redshift $z$). In the pair, let us call the two points #1
and #2; for each point #1, we average over points #2 in a spherical shell at a
distance $r$ from point #1. Actually, the shell is partially cut off at the
near and far edges of the radial bin, but this can be neglected as long as the
bin is large compared to the scales $1/k$ that we are interested in. Each
spherical shell is symmetric about point #1, so there is a cancellation as
long as we can treat the power spectrum as a linear function of $x$, over
distances of order $1/k$. We indeed assume this linear case, consistently with
our overall approach.

What remains is the averaging over points #1, so the result is simply an
average of the power spectrum over comoving volume:

$\frac{\int x^{2}\Omega(x)P(k,x)dx}{\int x^{2}\Omega(x)dx}\ .$ (12)

Here the solid angle $\Omega(z)$ at each $z(x)$ is the same as before (a
function of $z$ but no more than half the sky). Also, $P(k,x)$ denotes the
power spectrum at $k$ at the redshift corresponding to comoving distance $x$
(note that $x=(1+z)D_{A}(z)$ in terms of the angular diameter distance). For
each frequency bin, the integrals are over the range of $x$ corresponding to
the bin.

We find that the light-cone effect in our analysis is fairly small given our
bandwidth of $\Delta\nu=5$ MHz. E.g., when fitting the power spectrum with
configuration A, if we do not include the light-cone effect, the error in
$C_{\rm PowSpec}$ changes from 4.59% to 4.95%.

###  A.4 Angular resolution

When radio interferometry is done at increasingly high redshifts, it becomes
more difficult to achieve a given angular resolution. Thus, the resolution is
a significant limiting factor in measuring the 21-cm fluctuations from the
dark ages. We account for the effect of angular resolution analytically, as
follows. Based on simulations of future radio arrays as well as experience
with current arrays (Koopmans, personal communication), it is a good
approximation to assume a Gaussian point-spread function (PSF), with a full-
width at half max (FWHM) corresponding to $0.6\lambda/D$, where $\lambda$ is
the wavelength and $D$ is the maximum diameter of the array (which we find
assuming that the collecting area is a full circle).

Thus, if the comoving coordinates are $X$, $Y$, and $Z$ (with the latter being
the line-of-sight direction in this subsection), angular resolution smooths
the 21-cm map with a window function

$W=\frac{1}{2\pi R^{2}}\,e^{-(X^{2}+Y^{2})/(2R^{2})}\delta_{D}(Z)\ ,$ (13)

where $\delta_{D}$ is a Dirac delta function, the pre-factor ensures
normalization to a volume integral of unity, and $R$ is the comoving distance
corresponding to angle $\theta_{D}$ (i.e., $R=(1+z)D_{A}(z)\theta_{D}$ in
terms of the angular diameter distance $D_{A}$), where the above yields an
angle

$\theta_{D}=\frac{0.6\lambda/D}{2\sqrt{2\ln{2}}}=0.25\,\frac{\lambda}{D}=9.\mkern-4.0mu^{\prime}1\left(\frac{1+z}{50}\right)\left(\frac{D}{1\,\mathrm{km}}\right)^{-1}\
.$ (14)

Then the Fourier transform of $W$ is

$\tilde{W}=\int
d^{3}r\,We^{-i\vec{k}\cdot\vec{r}}=e^{-\frac{1}{2}k^{2}R^{2}(1-\mu^{2})}\ ,$
(15)

where $\mu=\cos{\theta}$ in terms of the angle $\theta$ between $\vec{k}$ and
the line of sight. The power spectrum is multiplied by the square of
$\tilde{W}$.

Finally, since here we are only considering the spherically-averaged power
spectrum, we average over angle, which multiplies the power spectrum by the
factor $F(kR$), where

$F(\alpha)=\frac{1}{2}\int_{-1}^{1}d\mu\,e^{\alpha^{2}(\mu^{2}-1)}\ .$ (16)

This integral is related to the error function, but note that the coefficient
of $\mu^{2}$ in the exponent is positive. This function is shown in Fig. 5.

Figure 5: The function $F(\alpha)$ that captures the effect of angular
resolution, is shown as a function of $\alpha$.

As an example, when fitting the power spectrum with configuration A, if we do
not include the effect of angular resolution, the error in $C_{\rm PowSpec}$
changes from 4.59% to 3.53%. We note, however, that since in our calculations,
we compare the theoretically-predicted power spectrum with the effect of
angular resolution to the standard expression for the thermal noise, we are
being somewhat overly conservative since in reality the limited angular
resolution will also smooth out the power spectrum of the thermal noise (an
effect that we do not include).

###  A.5 Method for constructing a minimum variance linear combination of
correlated parameters

Given that the fitting of the 21-cm signal to cosmological parameters results
in significant degeneracies among the parameters, we found it useful to
construct combinations of the parameters that have a minimum variance. This
best captures the constraining power of the data, especially since these
combinations are unique to the 21-cm signal and are substantially different
from the combinations that are best constrained by other cosmological
datasets.

Fitting the 21-cm global signal is a case of two parameters. In general, let
the parameters be $x$ and $y$, and assume we know
$\sigma_{x}\equiv\sqrt{\langle(\Delta x)^{2}\rangle}$ (where $\Delta x\equiv
x-\langle x\rangle$), $\sigma_{y}\equiv\sqrt{\langle(\Delta y)^{2}\rangle}$,
and the correlation coefficient $r=\langle\Delta x\Delta
y\rangle/(\sigma_{x}\sigma_{y})$. We treat $x$ as the primary variable, which
in practice should be chosen as the parameter that the signal is most
sensitive to; naturally, this parameter will have the largest coefficient in
the linear combination below. Then the linear combination of $x$ and $y$ with
minimum variance, normalized so that $x$ has a coefficient of unity, is:

$C=x-\alpha y\ ,$ (17)

where

$\alpha=r\frac{\sigma_{x}}{\sigma_{y}}\ ,$ (18)

and $C$ has a standard deviation of

$\sigma_{C}=\sigma_{x}\sqrt{1-r^{2}}\ .$ (19)

Fitting the 21-cm power spectrum is a case of $n=5$ parameters. In general,
let the parameters be $x_{i}$, with $i=1$ through $n$. Then we desire to find
the weights $w_{i}$ so that the parameter combination

$C=\sum_{i}x_{i}w_{i}\,,$ (20)

has minimum variance, where in the weight vector $w$, we fix $w_{1}=1$, thus
treating $x_{1}$ as the primary variable. Assume we know the covariance matrix
$S_{ij}=\langle\Delta x_{i}\Delta x_{j}\rangle$. Then to get the solution, we
remove the first row and column and obtain the reduced $(n-1)\times(n-1)$
matrix $U$, which is simply $S_{ij}$ for $i,j>1$. Also the covariances of the
other $x_{j}$ (for $j>1$) with $x_{1}$, i.e., $S_{j1}$ for $j>1$, we will call
the $(n-1)\times 1$ vector $v$. In addition, the $n-1$ weights, $w_{j}$ for
$j>1$, are a reduced weight vector $z$. Now we solve: $Uz=-v$, so that the
solution is:

$z=-U^{-1}v\ .$ (21)

We construct the full vector $w$ from this solution for $z$, and the resulting
minimum variance is

$\sigma_{C}^{2}=w^{T}Sw=\sum_{i,j}S_{ij}w_{i}w_{j}\ ,$ (22)

where $w^{T}$ is the transpose of $w$, and $i,j$ go from 1 to $n$.

We note that it is a general result that if only the parameter $x_{1}$ is fit
to the data with all the other parameters held fixed, then the resulting error
$\sigma_{1}$ in $x_{1}$ is in fact equal to the just-written expression for
$\sigma_{C}$.

###  A.6 Additional results and discussion

In this section we present a number of additional results and checks, along
with additional discussion. We begin with the global signal. We focused on the
parameter combination $C_{\rm Global}$, where the power in the denominator
indicates the power-law dependence of the global signal amplitude on
$\Omega_{\rm m}h^{2}$ relative to $\Omega_{\rm b}h^{2}$. It is naturally
expected to be near 1/4, given eq. (4) (with its terms directly suggesting a
power of 1/2) plus the fact that most of the signal-to-noise comes from the
relatively low redshifts where $x_{c}$ is significantly below 1, and this
coefficient is proportional to the collision rate per atom, and thus to
$\Omega_{\rm b}h^{2}$; this suggests a total dependence roughly proportional
to $(\Omega_{\rm b}h^{2})^{2}/(\Omega_{\rm m}h^{2})^{1/2}$, and $C_{\rm
Global}$ then goes as the square root of this (since we fix the dependence on
the primary parameter, $\Omega_{\rm b}h^{2}$, to the power of unity). In Table
3 we also show the global signal constraints on the two relevant cosmological
parameters. The errors are very large, in general and also compared to the
Planck measurements. There is a nearly complete degeneracy, in that the
correlation coefficient between $\Omega_{\rm b}h^{2}$ and $\Omega_{\rm
m}h^{2}$, for example for $t_{\rm int}=10$,000 hrs, is 0.994. We note that for
any parameter $x$, $\sigma[\ln(x)]$ equals the relative error in $x$ when
$\sigma\ll 1$; this relation is only approximately true when $\sigma$ is not
small, but for simplicity, we always quote $\sigma[\ln(x)]$ as the relative
error in $x$.

Table 3: For the global signal, the 1$\sigma$ relative errors (in %) on cosmological parameters, compared to Planck. Global | Planck | Planck |  Integration time
---|---|---|---
signal |  $+$ BAO |  | 100,000 hrs | 10,000 hrs | 1,000 hrs
$\Omega_{\rm b}h^{2}$ | 0.624 | 0.671 | 9.76 | 30.9 | 97.6
$\Omega_{\rm m}h^{2}$ | 0.611 | 0.769 | 39.2 | 124 | 392
|  |  |  |  |
Figure 6: Global 21-cm signal constraints based on MCMC fitting for $t_{\rm
int}=10$,000 hrs. Here we use two basic variables, $\ln(\Omega_{\rm b}h^{2})$
and the logarithm of $C_{\rm Global}\equiv\Omega_{\rm b}h^{2}/(\Omega_{\rm
m}h^{2})^{0.248}$. The panels show the posterior distributions (1D and 2D) of
the two parameters.

As the errors on the parameters are large while that on $C_{\rm Global}$ is
small, we run an MCMC chain in one case ($t_{\rm int}=10$,000 hrs) to verify
that the non-linear individual errors are not leading to a breakdown of the
Fisher matrix approach as applied to the important parameter, $C_{\rm
Global}$. The results are shown in Fig. 6. In the 2D posterior panel, we see
that there is almost no correlation between $\Omega_{\rm b}h^{2}$ and $C_{\rm
Global}$, which justifies the choice of $C_{\rm Global}$ as the second
parameter. We find a 1$\sigma$ constraint on $\ln{C_{\rm Global}}$ of
$-3.310^{+0.033}_{-0.035}$, equivalent to a relative error of 3.4% in $C_{\rm
Global}$, compared to the Fisher approach that gave a relative uncertainty of
$3.18\%$, an error that is close to the MCMC limits. Also the 2$\sigma$ MCMC
constraint on $\ln{C_{\rm Global}}$ is $-3.310^{+0.067}_{-0.070}$, which is
roughly double the 1$\sigma$ range but shows slight asymmetry. We conclude
that the Fisher approach is good enough for approximate answers in this first
analysis, but full MCMC is needed for higher precision when the errors in some
of the underlying parameters are large. Fig. 6 was generated using the python
packages emcee Foreman-Mackey2013  and corner corner .

Moving to the 21-cm power spectrum, in Fig. 2 in the main text we showed
slices through this 2D dataset at fixed redshifts. Fig. 7 shows slices in the
opposite direction, namely the variation with $\nu$ (or $z$), at fixed
wavenumber values $k=[0.01$, 0.04, 0.1, 0.4, 1.0, $4.0]\,{\rm Mpc}^{-1}$, for
the fiducial cosmological model. As expected, the power increases as we go
from large scales to small scales. The power (for all the curves that are
shown) peaks at $z=51$. These slices show the smooth evolution with redshift
at each $k$. We also show the 1$\sigma$ noise curves (thermal plus cosmic
variance) for the A and B configurations.

Figure 7: The spherically-averaged (total) power spectrum of 21-cm brightness
fluctuations as a function of $\nu$ (or $z$ as the top $x$-axis) at wavenumber
values $k=[0.01$, 0.04, 0.1, 0.4, 1.0, $4.0]\,{\rm Mpc}^{-1}$. We also show
the 1$\sigma$ noise (thermal plus cosmic variance) for our A (short dashed
lines) and B (long dashed lines) configurations, at $k=0.1$ Mpc-1 and 1.0
Mpc-1. The effect of the angular resolution is not shown here.

We now consider the results for the cosmic variance (CV) only case, which
corresponds to the limit of infinite collecting area or integration time. This
is a purely theoretical limit of some interest as a comparison case, given its
role in some previous work Floss2022 ; mondal17 . We assume in this limiting
case no thermal noise, perfect angular resolution, and a full sky (i.e.,
$\Omega_{\rm FoV}=4\pi$). The relative error in $C_{\rm PowSpec}$ for the CV-
only case would be $7.72\times 10^{-5}\,$%. The relative error in $Y_{\rm P}$
(fixing all other parameters) would be $2.18\times 10^{-4}\,$%, and the sum of
the neutrino masses would be constrained to $\sum m_{\nu}<2.40\times 10^{-5}$
eV. Fixing the other parameters would not be an appropriate assumption in this
case with such minuscule errors, but we include this here for comparison with
the other cases considered in the main text.

As noted in the main text, there are strong correlations among the
cosmological parameters, which is what led us to focus on the combination
$C_{\rm PowSpec}$. The values of the correlation coefficients are illustrated
in Table 4, for Configuration A and for the CV-only case. Some of the
coefficients approach unity in absolute value.

We summarize the coefficients for configuration A as [0.307, 0.9950, 0.464,
0.0753] for the power of $(A_{\rm s}e^{-2\tau})$, the base of $n_{\rm s}$, and
the powers in the denominator of $\Omega_{\rm m}h^{2}$ and $H_{0}$,
respectively. While the dependence of the 21-cm power spectrum on the
cosmological parameters is complex, we can try to roughly understand what
drives the various powers in the combination $C_{\rm PowSpec}$. As discussed
in the first paragraph of this section, the global signal is roughly
proportional to $(\Omega_{\rm b}h^{2})^{2}/(\Omega_{\rm m}h^{2})^{1/2}$. The
power spectrum goes as the global signal squared times the dimensionless
(i.e., relative) squared fluctuation level. This is proportional to the
primordial amplitude $A_{\rm s}$, reduced by post-reionization scattering (as
for all sub-horizon scales in the CMB) by the factor $e^{-2\tau}$. Then, the
growth of fluctuations (squared) from the early Universe down to the cosmic
dark ages is roughly proportional to the growth factor (squared) at the dark
ages relative to matter-radiation equality (which is when significant matter
fluctuation growth begins). Fixing as before the dependence on the primary
parameter, $\Omega_{\rm b}h^{2}$, to a power of unity, this would suggest a
power of 0.25 for $A_{\rm s}e^{-2\tau}$ and 0.75 in the denominator for
$\Omega_{\rm m}h^{2}$. The actual powers are changed by various additional
complications, including a strong scale dependence in the sensitivity to
$\Omega_{\rm m}h^{2}$ and a weak separate sensitivity to the Hubble constant
introduced by the Alcock-Paczyński effect (see Sec. A.2). In addition, the
weak dependence on $n_{\rm s}$ in $C_{\rm PowSpec}$ means that the effective
scale that is being constrained by the 21-cm power spectrum is close to the
pivot scale $k=0.05\,{\rm Mpc}^{-1}$ at which $A_{\rm s}$ is defined
Planck:2018 .

As we noted in the main text, the form of $C_{\rm PowSpec}$ changes for
different scenarios. The coefficients for configuration G are [0.304, 0.0488,
0.484, 0.0698], for configuration B: [0.307, 0.9986, 0.461, 0.0751], for
configuration C: [0.311, 1.118, 0.382, 0.0811], for configuration D: [0.315,
1.233, 0.300, 0.0760], and for the CV-only case: [0.335, 2.97, -0.292,
0.0223]. Thus, the coefficients for configurations A and B are nearly
identical (since both are strongly dominated by the thermal noise), but things
change with C and D (the angular resolution is now higher, and the CV plays
some role, particularly for D), and big changes happen for CV-only (as the
detailed shape of the power spectrum now plays a major role, and much smaller
scales come into play).

Table 4: The correlation coefficients in the fits of the 21-cm power spectrum. Note that the actual parameters used in the fitting are the logarithms of the parameters listed here (except for $n_{\rm s}$). | CV only | Configuration A
---|---|---
$\Omega_{\rm b}h^{2}$ | 0.538 |  |  |  | -0.661 |  |  |
$\Omega_{\rm m}h^{2}$ | -0.965 | -0.423 |  |  | -0.779 | 0.317 |  |
$A_{\rm s}e^{-2\tau}$ | 0.917 | 0.271 | -0.897 |  | 0.633 | -0.992 | -0.225 |
$n_{\rm s}$ | 0.814 | 0.260 | -0.911 | 0.674 | -0.0812 | 0.575 | -0.487 | -0.658
| $H_{0}$ | $\Omega_{\rm b}h^{2}$ | $\Omega_{\rm m}h^{2}$ | $A_{\rm s}e^{-2\tau}$ | $H_{0}$ | $\Omega_{\rm b}h^{2}$ | $\Omega_{\rm m}h^{2}$ | $A_{\rm s}e^{-2\tau}$
|  |  |  |  |  |  |  |

In fitting the 21-cm power spectra from the dark ages, in the main text we
focused on $C_{\rm PowSpec}$ as well as constraints on Helium and neutrinos.
The relative errors in the standard cosmological parameters are listed in
Table 5 and shown in Fig. 8. We do not show configuration A (for which the
errors are even significantly larger than for the 1,000 hr global signal
case). For configuration D some of the errors approach Planck levels, while
the ultimate CV-only case is in principle better than Planck by between 1 and
3 orders of magnitude.

Table 5: For the 21-cm power spectrum, the relative 1$\sigma$ errors in %,
compared to Planck. Note that we include the CV-only case (which has an extra
factor of $10^{-2}$ as indicated). We also list here the errors on
$\Omega_{\rm c}h^{2}$ since this is one of the standard input parameters in
CAMB.

| Planck | Planck | CV only |  Configurations
---|---|---|---|---
|  $+$ BAO |  | ($\times 10^{-2}$) | D | C | B
$H_{0}$ | $0.621$ | 0.802 | 8.10 | 4.76 | 40.4 | 42.4
$\Omega_{\rm b}h^{2}$ | 0.624 | 0.671 | 0.105 | 1.62 | 13.8 | 18.4
$\Omega_{\rm m}h^{2}$ | 0.611 | 0.769 | 1.21 | 0.968 | 7.85 | 8.60
$A_{\rm s}e^{-2\tau}$ | 0.532 | 0.584 | 0.859 | 5.82 | 49.3 | 62.9
$n_{\rm s}$ | $0.393$ | 0.435 | 0.237 | 0.687 | 5.58 | 7.32
$\Omega_{\rm c}h^{2}$ | 0.762 | 1.00 | 1.46 | 1.09 | 8.73 | 9.69
|  |  |  |  |  |

Figure 8: The relative 1$\sigma$ errors in % from fitting to the 21-cm power
spectrum from the dark ages. We show graphically the main results listed in
Table 5. Note that ‘Conf.’ stands for configuration.

Finally, we explore the dependence of our power spectrum results on varying
the assumed observational ranges, for configuration A. For $k$ this is of
interest since observational limitations (such as foreground removal) could
limit the available range. Table 6 shows that the results are insensitive as
long as we include the scales around the first few BAO, where the S/N is
maximized. We also explore various $\nu$ ranges, keeping $\Delta\nu=5$ MHz and
removing low redshifts. This is interesting since in rare models the 21-cm
signal can be affected by galaxies at redshifts almost up to 35 subtle , plus
it is of interest to understand to what degree the lower redshifts dominate
the fitting. As shown in Table 7, since the S/N is maximized at the lowest
redshift, the cutoff redshift indeed has a substantial effect on the results;
the minimum redshifts corresponding to the tabulated cases are 30, 33.8, 38.7,
and 45.1. The precise high-redshift cutoff is less important given the low S/N
at that end.

Table 6: The relative (1$\sigma$) errors in %, for various $k$ ranges, when fitting the power spectrum with configuration A. In all cases we maintain an integer number of bins with $\Delta\ln k=0.5$. |  $k$ range [${\rm Mpc}^{-1}$]
---|---
| Fiducial [0.00779 - 1.91] | [0.0234 - 2.10] | [0.0779 - 2.58] | [0.00779 - 5.18]
$C_{\rm PowSpec}$ | 4.59 | 4.65 | 7.76 | 4.59
|  |  |  |
Table 7: The relative (1$\sigma$) errors in %, for various $\nu$ ranges, when
fitting the power spectrum with configuration A. In all cases we maintain an
integer number of bins with $\Delta\nu=5$ MHz.

|  $\nu$ range [MHz]
---|---
| Fiducial $[5.81-45.81]$ | $[5.81-40.81]$ | $[5.81-35.81]$ | $[5.81-30.81]$
$C_{\rm PowSpec}$ | 4.59 | 6.57 | 12.2 | 32.7
|  |  |  |

###  A.7 Discussion of foregrounds

The brightness temperature of the foreground sky emission at $z=40$ is
expected to be around 13,070 K. While this is significantly higher than at
lower redshifts, the thermal noise is proportional to the sky brightness for
the global signal (eq. 5), and the square of the sky brightness for the power
spectrum (eq. 7); thus, the relative accuracy needed for foreground removal,
in order for the foreground residuals to fall below the thermal noise, is
independent of redshift (for a fixed integration time and frequency bin size).
For example, for the global signal with $t_{\rm int}$ = 1,000 hrs, the
foreground must be removed to an accuracy of a part in $10^{6}$ or better
(depending on the frequency bin size). This is challenging, but the cosmic
dawn experiments are making steady progress, and as explained in the
introduction of the main text, we expect the lunar environment to make this
task significantly easier than for the terrestrial environment.

We wish to account for foreground removal while fitting the global signal, at
least in the best-case scenario. Thus we add a free parameter $A$ to the model
that we fit to the data, in the shape of the synchrotron foreground, i.e.,
$A\,\nu^{-2.6}$. In practice, in current global signal experiments more
polynomial terms are usually added for a more realistic foreground modeling.
As we noted, there are reasons to hope that less of this will be required on
the moon, but even in the best-case scenario, a signal component of the same
shape as the foreground cannot be distinguished from it. To illustrate the
impact, we note that the error in $C_{\rm Global}$ for $t_{\rm int}=1$,000
hrs, which is 10.1%, would instead be 5.54% without this additional foreground
term.

In the case of the 21-cm power spectrum, in addition to foreground removal,
which can never be perfect, another method of dealing with foregrounds is to
avoid them. Foreground contamination is expected to be largely restricted to
within a wedge-shaped region in the 2D $(k_{\perp},k_{\parallel})$ Fourier
space, where these are the components of the wavevector perpendicular and
parallel to the line of sight, respectively. Thus, it may be easier to achieve
an extremely high accuracy of foreground removal outside the wedge. For a
rough estimate of the effect of foreground avoidance, we consider this
foreground wedge with different levels of contamination. We calculate the
wedge boundary using Datta2010 ; dillon14 ; pober14 ; jensen15

$k_{\parallel}=\left(\frac{r_{\nu}\sin{\theta_{\rm
L}}}{r_{\nu}^{\prime}\nu}\right)k_{\perp}\,,$ (23)

where $r_{\nu}$ is the comoving distance to the bin center,
$r_{\nu}^{\prime}=\frac{dr}{d\nu}$ at the bin center, and $\theta_{\rm L}$ is
the angle on the sky with respect to the zenith from which the foregrounds
contaminate the power of the 21-cm signal. At $z=40$, we assume two scenarios.
The first assumes $\theta_{\rm L}=2\times$ FWHM (optimistic). We estimate that
roughly $1/10$ of the $(k_{\perp},k_{\parallel})$ space is affected in this
case. Assuming a more pessimistic case of $\theta_{\rm L}=\pi/2$, we find that
roughly 1/2 of the S/N can be lost due to foreground contamination. Thus, the
effect of foreground avoidance can be significant but is most likely not a
game changer.

## References

  *   * (1) Sunyaev, R. A. & Zeldovich, Y. B.  Formation of Clusters of Galaxies; Protocluster Fragmentation and Intergalactic Gas Heating.  _A &A_ 20, 189 (1972) .
  * (2) Hogan, C. J. & Rees, M. J.  Spectral appearance of non-uniform gas at high z.  _MNRAS_ 188, 791–798 (1979).  10.1093/mnras/188.4.791 .
  * (3) Scott, D. & Rees, M. J.  The 21-cm line at high redshift: a diagnostic for the origin of large scale structure.  _MNRAS_ 247, 510 (1990) .
  * (4) Madau, P., Meiksin, A. & Rees, M. J.  21 Centimeter Tomography of the Intergalactic Medium at High Redshift.  _ApJ_ 475, 429 (1997).  arXiv:astro-ph/9608010 .
  * (5) Loeb, A. & Zaldarriaga, M.  Measuring the Small-Scale Power Spectrum of Cosmic Density Fluctuations through 21cm Tomography Prior to the Epoch of Structure Formation.  _Phys. Rev. Lett._ 92 (21), 211301 (2004).  10.1103/PhysRevLett.92.211301, arXiv:astro-ph/0312134 [astro-ph].
  * (6) Barkana, R. & Loeb, A.  Probing the epoch of early baryonic infall through 21-cm fluctuations.  _MNRAS_ 363 (1), L36–L40 (2005).  10.1111/j.1745-3933.2005.00079.x, arXiv:astro-ph/0502083 [astro-ph].
  * (7) Bharadwaj, S. & Ali, S. S.  The cosmic microwave background radiation fluctuations from HI perturbations prior to reionization.  _MNRAS_ 352, 142–146 (2004).  arXiv:astro-ph/0401206 .
  * (8) Barkana, R. & Loeb, A.  A Method for Separating the Physics from the Astrophysics of High-Redshift 21 Centimeter Fluctuations.  _ApJ_ 624, L65–L68 (2005).  arXiv:astro-ph/0409572 .
  * (9) Naoz, S. & Barkana, R.  Growth of linear perturbations before the era of the first galaxies.  _MNRAS_ 362 (3), 1047–1053 (2005).  10.1111/j.1365-2966.2005.09385.x, arXiv:astro-ph/0503196 [astro-ph].
  * (10) Lewis, A. & Challinor, A.  21cm angular-power spectrum from the dark ages.  _Phys. Rev. D_ 76 (8), 083005 (2007).  10.1103/PhysRevD.76.083005, arXiv:astro-ph/0702600 [astro-ph].
  * (11) Ali-Haïmoud, Y., Meerburg, P. D. & Yuan, S.  New light on 21 cm intensity fluctuations from the dark ages.  _Phys. Rev. D_ 89 (8), 083506 (2014).  10.1103/PhysRevD.89.083506, arXiv:1312.4948 [astro-ph.CO].
  * (12) Tashiro, H., Kadota, K. & Silk, J.  Effects of dark matter-baryon scattering on redshifted 21 cm signals.  _Phys. Rev. D_ 90 (8), 083522 (2014).  10.1103/PhysRevD.90.083522, arXiv:1408.2571 [astro-ph.CO].
  * (13) Muñoz, J. B., Kovetz, E. D. & Ali-Haïmoud, Y.  Heating of baryons due to scattering with dark matter during the dark ages.  _Phys. Rev. D_ 92 (8), 083528 (2015).  10.1103/PhysRevD.92.083528, arXiv:1509.00029 [astro-ph.CO].
  * (14) Barkana, R.  Possible interaction between baryons and dark-matter particles revealed by the first stars.  _Nature_ 555 (7694), 71–74 (2018).  10.1038/nature25791, arXiv:1803.06698 [astro-ph.CO].
  * (15) Chen, X., Meerburg, P. D. & Münchmeyer, M.  The future of primordial features with 21 cm tomography.  _J. Cosmology Astropart. Phys_ 2016 (9), 023 (2016).  10.1088/1475-7516/2016/09/023, arXiv:1605.09364 [astro-ph.CO].
  * (16) Pillepich, A., Porciani, C. & Matarrese, S.  The Bispectrum of Redshifted 21 Centimeter Fluctuations from the Dark Ages.  _ApJ_ 662 (1), 1–14 (2007).  10.1086/517963, arXiv:astro-ph/0611126 [astro-ph].
  * (17) Joudaki, S., Doré, O., Ferramacho, L., Kaplinghat, M. & Santos, M. G.  Primordial Non-Gaussianity from the 21 cm Power Spectrum during the Epoch of Reionization.  _Phys. Rev. Lett._ 107 (13), 131304 (2011).  10.1103/PhysRevLett.107.131304, arXiv:1105.1773 [astro-ph.CO].
  * (18) Flöss, T., de Wild, T., Meerburg, P. D. & Koopmans, L. V. E.  The Dark Ages’ 21-cm trispectrum.  _J. Cosmology Astropart. Phys_ 2022 (6), 020 (2022).  10.1088/1475-7516/2022/06/020, arXiv:2201.08843 [astro-ph.CO].
  * (19) Balaji, S., Ragavendra, H. V., Sethi, S. K., Silk, J. & Sriramkumar, L.  Observing Nulling of Primordial Correlations via the 21-cm Signal.  _Phys. Rev. Lett._ 129 (26), 261301 (2022).  10.1103/PhysRevLett.129.261301, arXiv:2206.06386 [astro-ph.CO].
  * (20) Bowman, J. D., Rogers, A. E. E., Monsalve, R. A., Mozdzen, T. J. & Mahesh, N.  An absorption profile centred at 78 megahertz in the sky-averaged spectrum.  _Nature_ 555 (7694), 67–70 (2018).  10.1038/nature25792, arXiv:1810.05912 [astro-ph.CO].
  * (21) Singh, S. _et al._ On the detection of a cosmic dawn signal in the radio background.  _Nature Astronomy_ 6, 607–617 (2022).  10.1038/s41550-022-01610-5, arXiv:2112.06778 [astro-ph.CO].
  * (22) Mertens, F. G. _et al._ Improved upper limits on the 21 cm signal power spectrum of neutral hydrogen at z $\approx$ 9.1 from LOFAR.  _MNRAS_ 493 (2), 1662–1685 (2020).  10.1093/mnras/staa327, arXiv:2002.07196 [astro-ph.CO].
  * (23) Trott, C. M. _et al._ Deep multi-redshift limits on Epoch of Reionisation 21 cm Power Spectra from Four Seasons of Murchison Widefield Array Observations.  _MNRAS_ (2020).  10.1093/mnras/staa414, arXiv:2002.02575 [astro-ph.CO].
  * (24) The HERA Collaboration _et al._ Improved Constraints on the 21 cm EoR Power Spectrum and the X-Ray Heating of the IGM with HERA Phase I Observations.  _arXiv e-prints_ arXiv:2210.04912 (2022).  arXiv:2210.04912 [astro-ph.CO].
  * (25) Lewis, A. & Bridle, S.  Cosmological parameters from CMB and other data: A Monte Carlo approach.  _Phys. Rev. D_ 66, 103511 (2002).  10.1103/PhysRevD.66.103511, arXiv:astro-ph/0205436 [astro-ph].
  * (26) Barkana, R. & Loeb, A.  A Method for Separating the Physics from the Astrophysics of High-Redshift 21 Centimeter Fluctuations.  _ApJ_ 624 (2), L65–L68 (2005).  10.1086/430599, arXiv:astro-ph/0409572 [astro-ph].
  * (27) Alcock, C. & Paczynski, B.  An evolution free test for non-zero cosmological constant.  _Nature_ 281, 358 (1979).  10.1038/281358a0 .
  * (28) Ali, S. S., Bharadwaj, S. & Pandey, B.  What will anisotropies in the clustering pattern in redshifted 21-cm maps tell us?  _MNRAS_ 363 (1), 251–258 (2005).  10.1111/j.1365-2966.2005.09444.x, arXiv:astro-ph/0503237 [astro-ph].
  * (29) Nusser, A.  The Alcock-Paczyński test in redshifted 21-cm maps.  _MNRAS_ 364 (2), 743–750 (2005).  10.1111/j.1365-2966.2005.09603.x, arXiv:astro-ph/0410420 [astro-ph].
  * (30) Barkana, R.  Separating out the Alcock-Paczyński effect on 21-cm fluctuations.  _MNRAS_ 372 (1), 259–264 (2006).  10.1111/j.1365-2966.2006.10882.x, arXiv:astro-ph/0508341 [astro-ph].
  * (31) Barkana, R. & Loeb, A.  Light-cone anisotropy in 21-cm fluctuations during the epoch of reionization.  _MNRAS_ 372, L43–L47 (2006).  10.1111/j.1745-3933.2006.00222.x, astro-ph/0512453 .
  * (32) Mondal, R., Bharadwaj, S. & Datta, K. K.  Towards simulating and quantifying the light-cone EoR 21-cm signal.  _MNRAS_ 474, 1390–1397 (2018).  10.1093/mnras/stx2888, arXiv:1706.09449 .
  * (33) Planck Collaboration _et al._ Planck 2018 results. VI. Cosmological parameters.  _A &A_ 641, A6 (2020).  10.1051/0004-6361/201833910, arXiv:1807.06209 [astro-ph.CO].
  * (34) Furlanetto, S. R., Oh, S. P. & Briggs, F. H.  Cosmology at low frequencies: The 21 cm transition and the high-redshift Universe.  _Phys. Rep._ 433 (4-6), 181–301 (2006).  10.1016/j.physrep.2006.08.002, arXiv:astro-ph/0608032 [astro-ph].
  * (35) Shaver, P. A., Windhorst, R. A., Madau, P. & de Bruyn, A. G.  Can the reionization epoch be detected as a global signature in the cosmic background?  _A &A_ 345, 380–390 (1999).  arXiv:astro-ph/9901320 [astro-ph].
  * (36) Mondal, R., Bharadwaj, S. & Majumdar, S.  Statistics of the epoch of reionization 21-cm signal \- I. Power spectrum error-covariance.  _MNRAS_ 456, 1936–1947 (2016).  10.1093/mnras/stv2772, arXiv:1508.00896 .
  * (37) Mellema, G. _et al._ Reionization and the Cosmic Dawn with the Square Kilometre Array.  _Experimental Astronomy_ 36, 235–318 (2013).  10.1007/s10686-013-9334-5, arXiv:1210.0197 [astro-ph.CO].
  * (38) Reis, I., Fialkov, A. & Barkana, R.  The subtlety of Ly $\alpha$ photons: changing the expected range of the 21-cm signal.  _MNRAS_ 506 (4), 5479–5493 (2021).  10.1093/mnras/stab2089, arXiv:2101.01777 [astro-ph.CO].
  * (39) Foreman-Mackey, D., Hogg, D. W., Lang, D. & Goodman, J.  emcee: The MCMC Hammer.  _PASP_ 125 (925), 306 (2013).  10.1086/670067, arXiv:1202.3665 [astro-ph.IM].
  * (40) Foreman-Mackey, D.  corner.py: Scatterplot matrices in python.  _The Journal of Open Source Software_ 1 (2), 24 (2016).  URL https://doi.org/10.21105/joss.00024.  10.21105/joss.00024 .
  * (41) Mondal, R., Bharadwaj, S. & Majumdar, S.  Statistics of the epoch of reionization (EoR) 21-cm signal - II. The evolution of the power-spectrum error-covariance.  _MNRAS_ 464, 2992–3004 (2017).  10.1093/mnras/stw2599, arXiv:1606.03874 .
  * (42) Datta, A., Bowman, J. D. & Carilli, C. L.  Bright Source Subtraction Requirements for Redshifted 21 cm Measurements.  _ApJ_ 724 (1), 526–538 (2010).  10.1088/0004-637X/724/1/526, arXiv:1005.4071 [astro-ph.CO].
  * (43) Dillon, J. S. _et al._ Overcoming real-world obstacles in 21 cm power spectrum estimation: A method demonstration and results from early Murchison Widefield Array data.  _Phys. Rev. D_ 89 (2), 023002 (2014).  10.1103/PhysRevD.89.023002, arXiv:1304.4229 [astro-ph.CO].
  * (44) Pober, J. C. _et al._ What Next-generation 21 cm Power Spectrum Measurements can Teach us About the Epoch of Reionization.  _ApJ_ 782, 66 (2014).  10.1088/0004-637X/782/2/66, arXiv:1310.7031 .
  * (45) Jensen, H. _et al._ The wedge bias in reionization 21-cm power spectrum measurements.  _Monthly Notices of the Royal Astronomical Society_ 456 (1), 66–70 (2015).  10.1093/Monthly Notices of the Royal Astronomical Society/stv2679 .