# Probing the Global 21-cm Signal via the Integrated Sachs-Wolfe Effect on the
21-cm Background

Kyungjin Ahn<EMAIL_ADDRESS>Minji Oh  Department of Earth Sciences,
Chosun University, Gwangju 61452, Korea

###### Abstract

We propose a novel method to probe the global 21-cm background. This
background experiences the integrated Sachs-Wolfe effect (ISW) as the cosmic
microwave background does. The 21-cm ISW is modulated by the spectral shape of
the global 21-cm signal, and thus the measure of the 21-cm ISW will be a probe
of the evolution of the global signal. This strategy naturally mitigates the
impact of the Milky Way foreground, which is a common and most significant
challenge in conventional 21-cm background probes. With the phase-1 SKA
telescope, probing the global 21-cm background would be feasible with a few
1000 hours of observation, enabling consistency checks with existing measures
of the global 21-cm signal by EDGES and SARAS that are conflicting with each
other.

cosmology

##  I introduction

The anisotropy of the cosmic microwave background (CMB) is caused by various
physical processes in the early and the late universe. Relatively weak yet
detected with high significance is the late-time integrated Sachs-Wolfe effect
[1] (ISW henceforth) on the CMB caused by the temporal variation of the large-
scale gravitational potential in the relatively nearby universe [2, 3, 4, 5,
6, 7, 8, 9, 10]. In a $\Lambda$CDM universe, after the matter-dominated epoch,
the gravitational potential changes in such a way that photons travel out of
the potential well (bump) with less energy loss (gain) than the energy gain
(loss) experienced while traveling into, causing the net blueshift (redshift)
[1]. The measured ISW on CMB, originating from the last scattering surface at
redshift $z\simeq 1000$, thus shows correlation with lower-redshift
($z\lesssim 2$) density tracers, or galaxies: the first proposal to use such a
cross-correlation is by [11]. Aside from the possibility of the curvature
density [12] or modified gravity effects [13, 14] for ISW, the measured ISW by
galaxy-CMB correlation is consistent with the present-day dark energy content
in units of the critical density at $\rho_{\Lambda}/\rho_{{\rm crit,0}}\sim
0.7$.

Redshifted 21-cm radiation from HI gas at $100\gtrsim z\gtrsim 6$ is another
radiation background that originates from high redshift and will experience
the ISW effect. Its sky-averaged brightness temperature, or the global 21-cm
background temperature ($\delta T_{b}$($z$) henceforth), has temporal
variation determined by cosmology (baryon temperature and the collisional
excitation of the spin-orbit angular momentum $J$ of hydrogen) and
astrophysics (impact on $T$ from X-ray heating, the ultraviolet background due
to early galaxies and the corresponding radiational excitation of $J$). Such a
temporal evolution is marked by the observed frequency $\nu$ that is related
to the originating redshift $z$ through the relation $\nu=1.42\,{\rm
GHz}/(1+z)$.

The main channel for the 21-cm background to be correlated with the low-
redshift galaxies is through the ISW effect (another subdominant channel in
[5]). As will be shown in this paper, the temperature fluctuation of a
frequency-dependent photon background caused by ISW is proportional to the
amplitude of the global value and the spectral feedback:

$\Delta T(\nu)\propto T(\nu)-\frac{\partial T(\nu)}{\partial\ln\nu}.$ (1)

Therefore, one can in principle measure the cross-correlation between the
21-cm background in different frequency (redshift) bins and the galaxy
distribution at low redshifts, and use the frequency variation in the cross-
correlation to probe $\delta T_{b}(z)$.

The 21-cm ISW as a measurement of $\delta T_{b}(\nu)-\partial\delta
T_{b}/\partial\ln\nu$ can then be used to check consistency with already
existing observations of the global 21-cm signal by monoblock receivers: EDGES
(Experiment to Detect the Global EoR Signature; [15]) and SARAS3 (Shaped
Antenna measurement of the background RAdio Spectrum, version 3; [16]). Both
experiments measure the whole sky without interferometry, and a successful
removal of the strong foreground composed of the Milky Way radio signal, the
extragalactic background and CMB should be performed. These two experiments
exhibit mutually conflicting results: the EDGES signal is reported to have
$\delta T_{b}(\text{$\nu\sim 80$\,{\rm MHz}})\sim-500\,{\rm mK}$, while the
SARAS3 signal is consistent with a null result and rejects the EDGES signal at
$\sim 1.8\sigma$. The absorption trough by the EDGES is much stronger than
what is expected in the $\Lambda$CDM cosmology and thus has been interpreted
as a possible signature for non-standard cosmology [17]. The refutation by
SARAS3 seems justified as we have better control of its systematics than of
the EDGES instrument [16]. However, the ambiguity in the foreground removal
process is quite substantial in these “monopole-only” observations [18] and
therefore an independent measurement of $\delta T_{b}(\nu)$ would be essential
for confirmation. Our proposition here is one of such independent
measurements.

Detecting the 21-cm ISW effect for the purpose of obtaining redundancy to the
already-measured CMB ISW effect for cosmology has been proposed by [19]. We
turn around their proposition and use the 21-cm ISW as a probe of $\delta
T_{b}(z)$ using the dependence of the ISW on the global temperature (Eq. 1).
We define an observable which is based on the 21-cm ISW and designed to probe
$\delta T_{b}(\nu)$, different from what was used in [19]. This will be found
practically equivalent to giving up the redundancy in cosmological information
but instead gaining the information on the amplitude and the spectral shape of
the global 21-cm background. We will show that such a probe would become
practical with the upcoming Square Kilometre Array (SKA) aiming at the 21-cm
background anisotropy from the $z\simeq 27$ – $6$ universe, contrary to the
pessimistic forecast by [19] who based the noise estimation at $z=30$ which is
too high for the sky noise to be controllable and used an observable different
from what we will use in this paper.

The paper is organized as follows. In section II, we show how the 21-cm
background is affected by ISW, and develop an observable based on the 21-cm
ISW but aimed at probing $\delta T_{b}(z)$. In section III, we present
estimated detectability of the observable with Phase-1 SKA specification to
show its encouraging prospect. We also include the detectability when only
using superstructures as 21-cm ISW drivers. In section IV, we discuss the
observational strategy to realize the measurement and some caveat in our error
estimates. Appendices are dedicated for technical details that were used for
estimating observational uncertainties.

##  II theory and observable

###  II.1 Theory

We start from the energy equation of a photon on a line of sight of an
observer, first calculated in [1]. All calculations henceforth are accurate to
the linear order. A freely traveling photon reaching an observer at present
experiences the temperature shift $\Delta T\equiv T-\langle T\rangle$ given by

$\displaystyle\left.\frac{\Delta T(\hat{n})}{\langle
T\rangle}\right|_{\eta_{0}}=$ $\displaystyle\left.\frac{\Delta
T(\hat{n})}{\langle
T\rangle}\right|_{\eta_{i}}-\Psi(\eta_{0})+\Psi(\eta_{i})+\left.{\bf
v}\cdot\hat{n}\right|_{\eta_{i}}^{\eta_{0}}$
$\displaystyle+\int_{\eta_{i}}^{\eta_{0}}2\Psi^{\prime}(\hat{n},\eta)d\eta,$
(2)

where $T$ is the temperature, $\langle T\rangle$ is the sky-averaged $T$
(monopole), $\eta_{i}$ and $\eta_{0}$ are the conformal times of some early
onset (usually at CMB decoupling) and the present respectively, $\hat{n}$ is
the sky-direction unit vector, $\Psi$ is the gravitational potential, and
${}^{\prime}\equiv\partial/\partial\eta$. With $\eta$ and $\Psi$, the
Friedmann-Lemaître-Robertson-Walker metric perturbed in the linear order reads

$ds^{2}=a^{2}(\eta)\left\\{-(1+2\Psi)d\eta^{2}+(1-2\Psi)\delta_{ij}dx^{i}dx^{j}\right\\},$
(3)

where $a$ is the scale factor, $x^{i}$ is the comoving spatial coordinate,
$\delta_{ij}$ is the Kronecker delta, and only one perturbation term $\Psi$
(as both the curvature perturbation and the gravitational potential) is
sufficient when ignoring the anisotropic stress tensor (a valid approximation
for the linear ISW: e.g. [3]). While taking $\eta_{i}$ as the conformal time
for the last scattering surface is usual in the CMB literature, we take
$\eta_{i}\equiv\eta(z)$ to generalize for the background that originates from
many different redshifts, such as the 21-cm background.

We now generalize Eq. (2) but focus only on the late-time ISW with a
frequency-dependent brightness temperature111One could instead formulate the
21-cm ISW in terms of the flux density or the specific intensity, whose
amplitude at frequency $\nu$ (redshift $z$ for spectral-line backgrounds)
defines the brightness temperature $T(\nu)$ ($T(z)$) that needs not be a
thermal temperature, in general, as in the case of the 21-cm brightness
temperature. field (contrary to the CMB temperature which is independent of
frequency). Also considering the global optical depth $\tau$ due to Thomson
scattering by free electrons [7] for a collection of photons, we have

$\left.\frac{\Delta T(\nu,\hat{n})}{\langle
T(\nu)\rangle}\right|_{\eta_{0}}=\int_{\eta_{i}}^{\eta_{0}}e^{-\tau(\eta)}2\Psi^{\prime}(\hat{n},\eta)d\eta$
(4)

where $\tau(\eta)\equiv\int_{\eta}^{\eta_{0}}n_{{\rm e}}(\eta)\sigma_{{\rm
T}}a(\eta)d\eta$, with the electron number density $n_{{\rm e}}$, the scale
factor $a$ and the Thomson scattering cross section $\sigma_{{\rm T}}$. In
practice, in the $\Lambda$CDM framework, ISW becomes effective only after
$\Omega_{\Lambda}\sim\Omega_{m}$ (or at $z\lesssim 2$) and thus the exact
value of $\eta_{i}$ does not matter. The observed frequency, excluding the
kinematically induced dipole [20], is shifted from the cosmological one as
222Note the disappearance of $\exp[-\tau(\eta)]$ term compared to Eq. (4).

$\nu(\hat{n},z)-\bar{\nu}(z)=\int_{\eta_{i}}^{\eta_{0}}2\Psi^{\prime}d\eta$
(5)

where $\bar{\nu}(z)\equiv\nu_{21}/(1+z)=1420\,{\rm MHz}/(1+z)$, whose
$z$-dependence will henceforth be dropped in notation for simplicity, is the
redshifted 21-cm line frequency in an unperturbed universe. As shown in [19],
combining Eqs. (4) and (5), the observed temperature difference at $\bar{\nu}$
becomes

$\displaystyle\Delta T$ $\displaystyle(\bar{\nu},\hat{n})$
$\displaystyle=\langle
T(\bar{\nu})\rangle\int_{\eta_{i}}^{\eta_{0}}\left(e^{-\tau(\eta)}-\frac{\partial\ln\langle
T(\bar{\nu})\rangle}{\partial\ln\bar{\nu}}\right)2\Psi^{\prime}(\hat{n},\eta)d\eta$
$\displaystyle\simeq\left(\langle T(\bar{\nu})\rangle-\frac{\partial\langle
T(\bar{\nu})\rangle}{\partial\ln\bar{\nu}}\right)\int_{\eta_{i}}^{\eta_{0}}2\Psi^{\prime}(\hat{n},\eta)d\eta,$
(6)

where the spectral feedback term including ${\partial\langle
T(\bar{\nu})\rangle}/{\partial\ln\bar{\nu}}$ arises when we Taylor-expand
$\Delta T(\nu,\hat{n})$ into terms in $\bar{\nu}$, with the help of Eq. (5).
Physically, this feedback occurs because, when an ISW driver (e.g. a potential
well) shifts the observed frequency and temperature, the observer probes the
temperature fluctuation situated at the other frequency modulated by the same
ISW effect. In comparison, the CMB-only shift becomes (we denote the CMB by
“$\gamma$” throughout this paper)

$\displaystyle\Delta T_{\gamma,0}(\hat{n})$ $\displaystyle=\langle
T_{\gamma,0}\rangle\int_{\eta_{i}}^{\eta_{0}}e^{-\tau(\eta)}2\Psi^{\prime}(\hat{n},\eta)d\eta$
$\displaystyle\simeq\langle
T_{\gamma,0}\rangle\int_{\eta_{i}}^{\eta_{0}}2\Psi^{\prime}(\hat{n},\eta)d\eta.$
(7)

In Eqs. (6) and (7), we approximated $\tau(\eta)\simeq 0$ because only the
late, low-density universe obtains the optical depth in the case of ISW. In
the concurrent $\Lambda$CDM model, the ISW-relevant (when $\Psi^{\prime}\neq
0$) regime limits $\tau(\eta)\lesssim\tau(z\leq 2)\simeq 0.0087$, and thus
this approximation is justified if one were not to calculate the ISW effect at
sub-percent level accuracy.

The low-frequency global temperature is the one shifted from the global CMB
temperature $\langle T_{{\gamma,0}}\rangle$ ($=2.725\,{\rm K}$) such that
$\langle T(\bar{\nu})\rangle=\langle T_{\gamma,0}\rangle+\delta
T_{b}(\bar{\nu})+T_{{\rm FG}}(\bar{\nu})$, where $\delta T_{b}(\bar{\nu})$ is
the global 21-cm brightness temperature and $T_{{\rm FG}}(\bar{\nu})$ is the
foreground temperature by the Milky Way, extragalaxies, and possibly some
unknown foreground (background). Let us assume for the sake of discussion that
$T_{{\rm FG}}$ comes mostly from the Milky Way and thus does not produce any
ISW effect. This will be realized by correlating the temperature field with
the galaxy distribution. Then, one can readily see that the ISW effect on the
low-frequency temperature deviates from that of the CMB according to

$\Delta T(\bar{\nu},\hat{n})=\Delta T_{\gamma,0}(\hat{n})\left(1+\frac{\delta
T_{21}(\bar{\nu})}{\langle T_{\gamma,0}\rangle}\right),$ (8)

where

$\delta T_{21}(\bar{\nu})\equiv\delta T_{b}(\bar{\nu})-\frac{\partial\delta
T_{b}(\bar{\nu})}{\partial\ln\bar{\nu}}.$ (9)

Therefore, measuring the low-frequency ISW effect at varing frequencies and
comparing this to the CMB ISW will enable one to probe $\delta
T_{b}(\bar{\nu})$ or more directly $\delta T_{21}(\bar{\nu})$.

The galaxy-temperature cross-correlation has been the only practical measure
of the ISW for CMB, and it will be much more so for the 21-cm due to the
weakness of the signal. The cross-correlation angular power spectrum of
$\Delta T(\bar{\nu},\hat{n})$ in the radio frequency and galaxy overdensity
$\delta_{{\rm g}}$ at observing frequency $\bar{\nu}$ is given by

$\displaystyle C_{\ell}^{gT}(\bar{\nu})=$ $\displaystyle\left\\{\langle
T_{\gamma,0}\rangle+\delta T_{21}(\bar{\nu})\right\\}$ $\displaystyle 4\pi\int
d\ln k\,\Delta_{0}^{2}(k)W_{\ell}^{{\rm g}}(k)W_{\ell}^{{\rm ISW}}(k),$ (10)

which is a specific case of the general correlation angular power spectrum of
fields $X$ and $Y$ (see Appendix A for details),

$C_{\ell}^{XY}(\bar{\nu})=N^{XY}(\bar{\nu})4\pi\int d\ln
k\,\Delta_{0}^{2}(k)W_{\ell}^{X}(k)W_{\ell}^{Y}(k),$ (11)

with $X=g$, $Y=T$ and $N^{gT}(\bar{\nu})=\langle T_{\gamma,0}\rangle+\delta
T_{21}(\bar{\nu})$. In equations (10) and (11), $W_{\ell}^{X}(k)$ is the
window function for a field $X$ relating the wavenumber $k$ of the field to
the multipole $\ell$ projected on the sky.
$\Delta_{0}^{2}(k)\equiv{k^{3}}P_{0}(k)/{2\pi^{2}}$ is the variance of the
present-day matter power spectrum $P_{0}(k)$,

$\displaystyle W_{\ell}^{{\rm g}}(k)$ $\displaystyle=\int
dz\,b(z)\phi(z)D(z)j_{\ell}[kr(z)],$ $\displaystyle W_{\ell}^{{\rm ISW}}(k)$
$\displaystyle=\int
dz\frac{3\Omega_{m,0}H_{0}^{2}}{k^{2}}\frac{dg}{dz}j_{\ell}[kr(z)],$ (12)

$\phi(z)$ is the normalized galaxy selection function [7], $b(z)$ is the
linear galaxy bias, $D(z)$ is the linear growth factor of the density
fluctuation, $j_{\ell}$ is the spherical Bessel function of order $\ell$,
$r(z)=\eta_{0}-\eta(z)$ is the comoving distance, $\Omega_{m,0}$ is the matter
fraction at present, $H_{0}$ is the Hubble parameter at present, and
$g(z)=D(z)(1+z)$ is the linear growth factor of the gravitational potential
333For a pedagogical derivation of the ISW cross power, see Ref. [3, 21],
which yields Eq. (12).. If the foreground comes only from the nearby ($z\simeq
0$) universe, $C_{\ell}^{gT_{{\rm FG}}}=0$ and thus
$C_{\ell}^{gT}(\bar{\nu})\propto\left(\langle T_{\gamma,0}\rangle+\delta
T_{21}(\bar{\nu})\right)$ as we put in Eq. (12). However, some fraction of the
foreground can originate from higher redshift range, which is the subject we
discuss in Section IV.

###  II.2 Observable

We define an observable as a combination of $C_{\ell}^{gT}(\bar{\nu})$ and
$C_{\ell}^{gT_{\gamma}}$. In principle, both quantities will have the
identical angular pattern on the sky, because ISWs on the CMB and the 21-cm
background are caused by the same gravitational potentials that are probed by
galaxies. This enables an observable which converges to a single scalar
quantity for any multipole $\ell$, as will be shown in the following. The key
forecast will be presented in Fig. 1 and Section III.1.

A special, frequency-independent version of Eq. (10) with $\delta T_{21}=0$ is
the usual high-frequency CMB one, $C_{\ell}^{gT_{\gamma}}$. It is obvious that
we can extract the global 21-cm signal by defining

$s_{\ell}(\bar{\nu})\equiv\langle
T_{\gamma,0}\rangle\left(\frac{C_{\ell}^{gT}(\bar{\nu})}{C_{\ell}^{gT_{\gamma}}}-1\right)=\delta
T_{21}(\bar{\nu}).$ (13)

Note that $s_{\ell}(\bar{\nu})$ turns out to be independent of $\ell$. The
cumulative signal

$S(\bar{\nu})\equiv\frac{\sum_{\ell=\ell_{{\rm min}}}^{\ell_{{\rm
max}}}s_{\ell}(\bar{\nu})}{(\ell_{{\rm max}}-\ell_{{\rm min}}+1)}=\delta
T_{21}(\bar{\nu})$ (14)

will maximize the net sensitivity with many harmonics, where we take
$\ell_{{\rm min}}=2$ and $\ell_{{\rm max}}=100$. In practice, $\ell_{{\rm
max}}\sim 20$ is already large enough to saturate SNR of $S(\bar{\nu})$, as
will be seen in Fig. 2. The cumulative SNR for $S(\bar{\nu})$ is given
by444This is a rough estimate; a more accuarate error estimate should be taken
for the effect of the masking at least. In practice,
$C_{l}^{T_{\gamma}T_{\gamma}}$ measured by the Planck telescope is limited
only by cosmic variance at $\ell\lesssim 1100$, and thus we may neglect the
detector noise in Eq. (15). (deriviation given in Appendix B.1)

$\left(\frac{{\rm S}}{{\rm N}}\right)^{2}(\bar{\nu})=\sum_{\ell=\ell_{{\rm
min}}}^{\ell_{{\rm max}}}\left(\frac{\delta
T_{21}(\bar{\nu})}{\sigma_{s_{\ell}}(\bar{\nu})}\right)^{2}=\frac{\delta
T_{21}^{2}(\bar{\nu})}{\langle T_{\gamma,0}\rangle^{2}}\sum_{\ell=\ell_{{\rm
min}}}^{\ell_{{\rm
max}}}\left[\frac{\left(C_{\ell}^{gg}+\bar{n}_{g}^{-1}\right)\left\\{C_{\ell}^{\delta
T_{b}\delta T_{b}}(\bar{\nu})+\epsilon_{\ell,T}(\bar{\nu})+\left(\frac{\delta
T_{21}(\bar{\nu})}{\langle
T_{\gamma,0}\rangle}\right)^{2}C_{\ell}^{T_{\gamma}T_{\gamma}}\right\\}}{(C_{\ell}^{gT_{\gamma}})^{2}f_{{\rm
sky}}(2\ell+1)}\right]^{-1},$ (15)

where $f_{{\rm sky}}$ is the sky coverage, $\bar{n}_{g}$ is the surface number
density of ISW galaxies (total number of galaxies per steradian),
$\epsilon_{\ell,T}$ is the 21-cm systematic error due to the sky brightness
given as

$\epsilon_{\ell,T}=\frac{2\pi\lambda^{2}(\bar{\nu})\left\\{180\left({\bar{\nu}}/{180\,{\rm
MHz}}\right)^{-2.6}\,{\rm K}\right\\}^{2}}{f_{{\rm cov}}^{2}D^{2}\Delta
t\Delta\nu}$ (16)

with the observing wavelength $\lambda$, the maximum baseline $D$ of the
densely spaced core of interferometers, the observing time $\Delta t$, the
land coverage $f_{{\rm cov}}$, and the bandwidth $\Delta\nu$ [22]. We have
nomalization coefficients (Eq. 11) $N_{\ell}^{\delta T_{b}\delta T_{b}}=\delta
T_{b}^{2}(\bar{\nu})$, $N_{\ell}^{gT_{\gamma}}=T_{\gamma,0}$ and
$N_{\ell}^{T_{\gamma}T_{\gamma}}=T_{\gamma,0}^{2}$. Especially,
$C_{\ell}^{\delta T_{b}\delta T_{b}}$ is dominated by the primordial power
spectrum with negligible contribution from the auto-correlation of the 21-cm
ISW (Appendix B.1), and thus the net power $C_{\ell}^{\delta T_{b}\delta
T_{b}}$ does not get boosted by the spectral feedback term $\partial\delta
T_{b}(\bar{\nu})/\partial\ln\bar{\nu}$. This is an important fact that renders
the noise controllable during the epoch when the noise is dominated not by
$\epsilon_{\ell,T}$ but by $C_{\ell}^{\delta T_{b}\delta T_{b}}(\bar{\nu})$.
The noise on $s_{\ell}(\bar{\nu})$, or $\sigma_{s_{\ell}(\bar{\nu})}$, depends
on the spectral shape of $\delta T_{b}(\bar{\nu})$ and thus SNR depends on
cosmology and astrophysics. Note that one should also consider the noise
$\sigma_{\langle T_{\gamma,0}\rangle}$ in the global CMB temperature due to
our definition of the signal (Eq. 13); however, we find it negligible already
with the FIRAS observation giving $\sigma_{\langle
T_{\gamma,0}\rangle}=0.57\,$mK [23, 24] while the net noise
$S(\bar{\nu})/({\rm S/N})(\bar{\nu})$ hardly reaches such a small value, and
always residing at $>20\,$mK with all the observational configurations
considered in this paper.

##  III Result

###  III.1 full-sky observation

We test three distinctive 21-cm models and estimate (S/N)($\bar{\nu}$) given
in Eq. (15). We consider two different strategies in measuring
$C_{\ell}^{gT}$: (1) the usual full-sky ISW measurement and (2) the ISW
measurement only through superstructures [25, 26, 27, 8, 28, 29, 30, 31], to
be described in Section III.2. The test models are (1) “V” (vanilla), a
typical $\Lambda$CDM model with galaxies without much feedback on their
formation [32], (2) “SRII” (self-regulated type II), a model under the
$\Lambda$CDM framework but with strong self-regulation of galaxy formation
during the cosmic dawn [33, 34], and (3) “NS” (non-standard): a model beyond
$\Lambda$CDM that has a large dip of $\delta T_{b}\simeq-500\,$mK around
$\nu\simeq 80\,$MHz, having resemblance to the EDGES signal. All these models
have $\delta T_{b}>0$ at $z\lesssim 9$ with steep upturns at higher
frequencies, which is believed to be natural with moderate X-ray heating and
consistent with what is indicated by the upper limits on the 21-cm fluctuation
put by some of the SKA precursors [35, 36, 37]. Therefore, all these models
have epochs with steep slopes in $\delta T_{b}(\bar{\nu})$ such that
$\left|\partial\delta T_{b}/\partial\ln\bar{\nu}\right|\simeq[2-8]\left|\delta
T_{b}\right|$ in some range of frequencies and the net signal $\delta T_{21}$
can be greatly amplified from $\delta T_{b}$.

Figure 1: Forecasts for measuring the global 21-cm background $\delta
T_{21}(\nu)=\delta T_{b}(\nu)-\partial\delta T_{b}/\partial\ln\nu$, through
observing the 21-cm ISW effect by the SKA1-LOW telescope with 3000-, 10000-
and 30000-hour observing times and 5-MHz bandwidth on 70% of the sky. Mock
signals from three different test models (V: left, SRII: middle, NS: right)
are used. From top to bottom, subpanels show the spectra of $\delta T_{b}$
(top), spectra of the observable (thick, red), $\delta T_{21}$, with
1-$\sigma$ error bars when $\Delta t$=3000 (2nd from top), 10000 (middle) and
30000 (4th from top) hours, and finally SNRs (bottom). Error bars and SNR
curves follow the same line-color convention (3000 hr: black; 10000 hr: blue;
30000 hr: blown). While there is model dependency, moderate to high-
significance detection of $\delta T_{21}$, and thus $\delta T_{b}$, from the
cosmic dawn and the epoch of reionization is feasible, with the exception for
the shallow-slope (in $\delta T_{b}$) case of SRII at $z\gtrsim 13$
($\nu\lesssim 100\,$MHz). Observation during the dark ages ($z\gtrsim 30$)
remains impossible with our strategy in all cases, due to the ever increasing
sky brightness as frequency decreases.

We take the following observational parameters that are suitable with the
$\sim$1-km core region of the Square Kilometre Array: $D=1$km, $f_{{\rm
cov}}=0.25$, $\Delta t$={3000, 10000, 30000} hr, $f_{{\rm sky}}=0.7$,
$\Delta\nu=5\,$MHz, and galaxy surveys satisfying:
$\bar{n}_{g}\geq(b/2)^{-2}\times 10^{7}{\rm sr}^{-1}$ where $b\equiv\langle
b(z)\rangle$ is the average galaxy bias. The last condition and the large
$f_{\rm sky}$ requirement are already met with the Wide-Field Infrared Survey
Explorer (WISE) galaxy catalogues [38, 8], and will get more surpassed by the
upcoming full-sky galaxy survey Spectro-Photometer for the History of the
Universe, Epoch of Reionization, and Ices Explorer (SPHEREX) [39]. We find
that uncertainty due to the galaxy shot noise starts to take effect, mostly at
$\bar{\nu}\gtrsim 160$MHz, when $\bar{n}_{g}$ fails to satisfy the above
criterion.

The prospect of using the observable $S(\bar{\nu})$ as a probe of the global
21-cm signal is encouraging (Fig. 1). Due to the non-exixtence of the
correlation between the Milky Way (dominating the radio sky) and the 21-cm
background, our method is free from the ambiguity in the foreground removal
555The impact by extragalactic and unknown foregrounds still exist, which is
addressed in Section IV.. Except for the dark ages ($z\gtrsim 30$;
$\bar{\nu}/{\rm MHz}\lesssim 46$), we can achieve reasonably high SNR in
probing both the cosmic dawn ($15\lesssim z\lesssim 30$;
$89\gtrsim\bar{\nu}/{\rm MHz}\gtrsim 46$) and the epoch of reionization
($6\lesssim z\lesssim 15$; $202\gtrsim\bar{\nu}/{\rm MHz}\gtrsim 89$) only
with the exception of SRII having low SNR in the cosmic dawn. Especially, the
NS case has $\gtrsim(2-3)\sigma$ detectability per frequency band, and thus
the 21-cm ISW observation would either rule out or detect the non-standard
universe with relatively high significance (compared to the expected
theoretical maximum SNR for CMB ISW [11, 3], SNR$\sim 7-9$, this can be
considered high significance). The 21-cm ISW observation would thus place a
significant constraint on the global 21-cm background, and allow consistency
checks with the previous, mutually conflicting observations of $\delta
T_{b}(\bar{\nu})$ by the EDGES and the SARAS.

Figure 2: Cumulative
(S/N)${}^{2}(\leq\ell)\equiv\sum_{\ell^{\prime}=2}^{\ell}({\rm
S/N})^{2}(\ell^{\prime})$ for the 21-cm probe $S(\bar{\nu})$ at a
representative frequency $\bar{\nu}=90\,{\rm MHz}$ (red, solid) and for the
CMB-ISW cross-correlation $C_{\ell}^{gT_{\gamma}}$ (blue, dotted). Curves are
normalized to show relative importance of modes for the two cases, such that
the saturated $({\rm S/N})^{2}$ reaches 1 in both cases. Most of the
contribution comes from modes at $\ell\lesssim 20$ in the 21-cm probe, while
from modes at $\ell\lesssim 200$ in the CMB-ISW cross-correlation. Therefore,
it is necessary to accurately probe the largest scales of the sky for the
21-cm-ISW correlation, and also for the CMB-ISW correlation because the net
observable is the combination of the two cross-correlations.

###  III.2 superstructure observation

Can we use the superstructures (superclusters and supervoids) as the 21-cm ISW
driver and probe the global 21-cm background? We find that such a strategy is
impractical even with planned telescopes at the moment. The superstructure ISW
effect on CMB has been found to be about $A_{\rm ISW}\simeq 1.5-5$, where
$A_{\rm ISW}$ is the ratio of the observed $\Delta T$ in superstructures to
the $\Delta T$ expected in $\Lambda$CDM due to ISW. Recent analyses such as
[28, 30] with carefully chosen convolution filters, which are used to identify
and characterize superstructures, find that ISW in these regions is only
marginally amplified, or $A_{\rm ISW}\sim 1.5$, and thus consistent with
$\Lambda$CDM. Analyses beforehand using a rather special convolution filter
used to find $A_{\rm ISW}\sim 5$, too large to be due to the ISW in
$\Lambda$CDM [25, 26, 27, 8, 29, 31]. Whether or not the superstructure ISW is
consistent with the standard cosmology, we may use these as the 21-cm ISW
drivers.

Figure 3: Forecasts for measuring the global 21-cm background $\delta
T_{21}(\nu)$ via stacked 21-cm ISW effect on $\sim 200$ superclusters, with
$\Delta nu=5\,{\rm MHz}$ and observing time of 10000 hours by SKA. In all the
tested cases with SKA, SNRs are too low for such observations to be practical,
leaving the full-sky observation (Fig. 1) the only viable strategy.

The 21-cm ISW by superstructures will have

$\Delta T^{\rm ISW}(\bar{\nu})=A_{\rm ISW}\frac{\delta
T_{21}(\bar{\nu})}{T_{\gamma,0}}\Delta\tilde{T}^{\rm ISW}_{\gamma},$ (17)

with the theoretically expected $\Delta{\tilde{T}}^{\rm ISW}_{\gamma}\simeq
2\,\mu{\rm K}$ on superstructures in $\Lambda$CDM [25, 28] and comparing this
signal to the expected noise inside superstructures, we obtain SNR which is
well below 1 in the full frequency range (Fig. 3; calculation of SNR described
in Appendix B.2). This is so even when we use $A_{\rm ISW}=5$. Note that the
signal is the average of the stacked $\Delta T^{\rm ISW}(\bar{\nu})$, where we
assumed stacking of 200 superstructures that is comparable to the number of
superstructures used in [25, 26] to reduce the noise in measuring the CMB ISW.

##  IV Discussion

We briefly remark on the impact of the extragalactic foreground and some
possible unknown foreground. As stressed earlier, we have null contribution
from any radiation of the Milky Way origin, or $C_{\ell}^{gT^{\rm
MW}}(\bar{\nu})=0$. Even so, foregrounds of other origin can be problematic.
First, extragalactic foreground can be correlated with the late-time galaxy
distribution simply because radio emission from these galaxies exist.
Therefore,

$C_{\ell}^{gT}(\bar{\nu})=C_{\ell}^{gT^{\rm
exgal}}(\bar{\nu})+C_{\ell}^{gT^{\rm unknown}}(\bar{\nu})+C_{\ell}^{g\delta
T_{b}}(\bar{\nu})$ (18)

in general. Further, both these foregrounds can be decomposed into the
intrinsic and ISW-driven parts, because if a part of radiation comes from high
redshift, $z\gtrsim 2$, that part of the foreground can also experience the
ISW by galaxies at $z\lesssim 2$. The same logic will apply to some unknown
foreground. Therefore, in practice, removal of extragalactic and unknown
foregrounds from the field of $C_{\ell}^{gT}(\bar{\nu})$ will still be
necessary. We will further investigate this issue in a future study.

According to what is shown in Section III, we conclude that the full-sky 21-cm
ISW measurement by radio interferometers is the only practical way to probe
the global 21-cm background through ISW. Observing the kinematically induced
multipole moments of the 21-cm background anisotropy due to the motion of the
earth is another independent measure of the global 21-cm background as we
suggested elsewhere [40, 20]. There are pros and cons of these two
observational strategies, which are complementary to each other. The 21-cm ISW
observation does not suffer from the imperfect removal of the Milky Way
foreground, thanks to the null cross-correlation with the 21-cm background,
while requires interferometric telescopes still in the future with very high
sensitivity, such as SKA and HERA. The multipole (practically dipole and
possibly quadrupole at best) observation requires much smaller telescopes,
while the foreground removal process in each multipole is still required.

For observation of the 21-cm ISW, both HERA (with a strong limitation
described below though) and SKA1-LOW (phase-1 SKA in low frequency) meet the
thermal noise requirement. However, SKA1-LOW will be the only choice at the
moment. HERA is a telescope with $D=300\,{\rm m}$ and $f_{\rm cov}\simeq 0.75$
for the collecting area in the core region to reach $51000\,{\rm m}^{2}$. This
is roughly the same as what we assumed in this paper for SKA in terms of
$Df_{\rm cov}$, but the 21-cm ISW observation with HERA would become practical
only if the full-sky observation can be performed with the addition of the
beam-forming capability. This is because the largest-scale modes should be
probed and also the sky coverage is essential to achieve high SNR (Eq. 15).
Currently, HERA is planned to observe $1440$-deg2 sky which is far too small
for the ISW observation. The core region of SKA1-LOW is what we have assumed
as the SKA-core configuration in this paper, planned to be spread over the
circular area of $D\simeq 1\,{\rm km}$ with the total collecting area $\sim
215000\,{\rm m}^{2}$, or $f_{\rm cov}\simeq 0.27$. We even find that 3000-hr
integration would still give moderate SNR at $\nu\gtrsim 90\,$MHz. Therefore,
the predicted SNR of this paper will become possible with $\gtrsim$1-year
operation of SKA1-LOW, which practically amounts to about $\gtrsim
2000$–$4000$-hr observation time depending on the observing condition
throughout the total observing time $\Delta t$. With the expected completion
of SKA1-LOW construction in year 2029, probing the global 21-cm background
through the 21-cm ISW observation could be realized within about a decade from
now.

We note that the currently planned observational strategy for SKA1-LOW is
likely to be unsuitable for the full-sky 21-cm ISW observation we propose
here. The survey area on the sky for SKA1-LOW 21-cm background observation is
expected to be limited, to gain very high sensitivity for scales of common
interest in the EoR science: $k\sim 0.1$–$1$ Mpc-1 [41]. However, our proposal
is quite promising in terms of both sensitivity and mitigated foreground
effect (that of the Milky Way especially). Therefore, we suggest that our
proposal for the low-angular-resolution, full-sky ($\ell\lesssim 20$)
observation be given serious consideration as a new science potential of the
SKA.

###### Acknowledgements.

KA is supported by NRF-2021R1A2C1095136 and a research grant from Chosun
University (2018).

##  Appendix A angular power spectrum of auto- and cross-correlation

For theoretical estimates of the signals and noises, the angular power spectra
of the auto- and cross-correlation are required. The angular power spectrum of
the cross-correlation between fields $X$ and $Y$ is given by

$C_{\ell}^{XY}(\bar{\nu})=N^{XY}(\bar{\nu})4\pi\int d\ln
k\,\Delta_{0}^{2}(k)W_{\ell}^{X}(k)W_{\ell}^{Y}(k),$ (19)

where $N^{XY}$ is the coefficient that renders $C_{\ell}^{XY}$ to be
dimensional when a temperature field is involved, and the frequency dependence
in $C_{\ell}^{XY}$ and $N^{XY}$ appears when the 21-cm background is involved.
Window functions $W_{\ell}(k)$ are shown in Eq. (12) of the main text. This
dimensional definition of $C_{\ell}^{XY}$ is convenient as we use two
different temperature fields (CMB and 21-cm background), and the 21-cm
background experiences the spectral feedback by the ISW while the CMB does
not. Eq. (19) can be applied for the auto-correlation of the galaxy
overdensity, and of the ISW-affected temperature fields. $N^{XY}$’s are given
by

$N^{gg}=1;\,\,N^{gT_{\gamma}}=T_{\gamma,0};\,\,N^{g\delta
T_{b}}(\bar{\nu})=\delta T_{21}(\bar{\nu}).$ (20)

In both backgrounds, the temperature autocorrelations
$C_{\ell}^{T_{\gamma}T_{\gamma}}$ and $C_{\ell}^{\delta T_{b}\delta T_{b}}$
are dominated by the primordial fluctuations, not by the ISW-driven
fluctuations [7]. We obtain $C_{\ell}^{T_{\gamma}T_{\gamma}}$ using the
Boltzmann solver “Code for Anisotropies in the Microwave Background” (CAMB;
[42]). With a reasonable assumption of linearity, we can still use Eq. (19) to
obtain $C_{\ell}^{\delta T_{b}\delta T_{b}}(\bar{\nu})$ using the matter
density fluctuation:

$\displaystyle N_{\ell}^{\delta T_{b}\delta T_{b}}(\bar{\nu})$
$\displaystyle=\delta^{2}T_{b}(\bar{\nu}),$ $\displaystyle W_{\ell}^{\delta
T_{b}}(\bar{\nu})$ $\displaystyle=\int^{\Delta
z(\bar{\nu})}dz\,D(z)j_{\ell}[kr(z)]/\Delta z(\bar{\nu}),$ (21)

where $\Delta z(\bar{\nu})$ is the redshift range defined by the given
frequency bandwidth around $\bar{\nu}$, and in effect $W_{\ell}^{g}$ (Eq. 12)
is used with $\phi(z)=1/\Delta z$ and $b(z)=1$. Because at high redshift, very
small wavenumbers $k\sim(\ell+1/2)/r(z=30\text{--}6)\lesssim
0.002\text{--}0.003\,h\,{\rm Mpc}^{-1}$ become relevant in the range of
harmonics relevant to the 21-cm ISW, $\ell\lesssim 20$ (see Fig. 2), and thus
the nonlinear effects from the photon-intensity fluctuations and the
inhomogeneous H II structure do not count much in calculating
$C_{\ell}^{\delta T_{b}\delta T_{b}}(\bar{\nu})$ [43]. As stressed in the main
text, it is important to notice that $N_{\ell}^{\delta T_{b}\delta
T_{b}}(\bar{\nu})$ is equal not to $\delta^{2}T_{21}(\bar{\nu})$ but to
$\delta^{2}T_{b}(\bar{\nu})$, because otherwise the noise would have been
uncontrollably large.

We use $b(z)=2$ and a smooth selection function [7]

$\phi(z)=\frac{1}{\Gamma\left(\frac{\alpha+1}{\beta}\right)}\beta\frac{z^{\alpha}}{z_{0}^{\alpha+1}}\exp\left[-\left(\frac{z}{z_{0}}\right)^{\beta}\right]$
(22)

with $\alpha=3.457$, $\beta=1.1$, $z_{0}=0.3$, peaking at $z\simeq 0.8$. We
experimented with different values of {$\alpha,$ $\beta$, $z_{0}$} and find
that our error estimation is hardly affected by these quantities. The
dependence on $b(z)$ roughly appears as the limit we find,
$\bar{n}_{g}\geq(b/2)^{-2}10^{7}\,{\rm sr}^{-1}$, that saturates the SNR at
the maximum values we find for values of {$\alpha,$ $\beta$, $z_{0}$} above
(Fig. 1 in the main text). Note that the observable $S(\bar{\nu})$ is hardly
affected by $b(z)$, because its dependence cancels out when the division
${C_{\ell}^{gT}(\bar{\nu})}/{C_{\ell}^{gT_{\gamma}}}$ is made. This limit on
$\bar{n}_{g}$ might be relaxed if $\phi(z)$ were peaked at lower redshifts
than we have chosen here, due to the relative increase in $C_{\ell}^{gg}$. We
have calculated the integration in $W_{\ell}^{X}$’s by using the “FFTLog”
algorithm developed elsewhere [44], instead of using the Limber approximation.
Fig. 4 shows dimensionless auto- and cross-correlations $C_{\ell}^{XY}/N^{XY}$
we obtained.

Figure 4: Dimensionless auto- and cross-correlation power spectrum of the
galaxy (“g”), the CMB temperature (“$T_{\gamma}$”) and the 21-cm temperature
(“$\delta T_{b}$”).

##  Appendix B Noise in observables

###  B.1 full-sky 21-cm ISW

The observable $s_{\ell}(\bar{\nu})$ defined in Eq. (13) of the main text has
the noise spectrum $\sigma_{s_{\ell}}(\bar{\nu})$ to be shown in Eq. (30)
below. We show how we obtained the expression. First, one should start from
the error propagation (we drop the frequency dependence for readability):

$\displaystyle\sigma_{s_{\ell}}^{2}=$ $\displaystyle
T_{\gamma,0}^{2}\left(\frac{C_{\ell}^{gT}}{C_{\ell}^{gT_{\gamma}}}\right)^{2}$
$\displaystyle\left(\frac{\sigma_{C_{\ell}^{gT}}^{2}}{\left(C_{\ell}^{gT}\right)^{2}}+\frac{\sigma_{C_{\ell}^{gT_{\gamma}}}^{2}}{\left(C_{\ell}^{gT_{\gamma}}\right)^{2}}-2\frac{\sigma_{C_{\ell}^{gT}C_{\ell}^{gT_{\gamma}}}}{C_{\ell}^{gT}C_{\ell}^{gT_{\gamma}}}\right),$
(23)

where we ignored the variance in the global CMB temperature
$\sigma_{T_{\gamma,0}}^{2}=0.57^{2}\,{\rm mK}^{2}$, which is much smaller than
the net noise we find (see Fig. 5). Realizing mutually related gaussian fields
at given harmonics ($\ell$, $m$) by combinations of independent gaussian
fields are useful [45, 46]:

$\displaystyle a_{\ell m}^{g}=$ $\displaystyle\sqrt{C_{\ell}^{gg}}\psi_{1,\ell
m},$ $\displaystyle a_{\ell m}^{T_{\gamma}}=$
$\displaystyle\frac{C_{\ell}^{gT_{\gamma}}}{\sqrt{C_{\ell}^{gg}}}\psi_{1,\ell
m}+\sqrt{C_{\ell}^{T_{\gamma}T_{\gamma}}-\frac{(C_{\ell}^{gT_{\gamma}})^{2}}{C_{\ell}^{gg}}}\psi_{2,\ell
m},$ $\displaystyle a_{\ell m}^{T}=$
$\displaystyle\frac{C_{\ell}^{gT}}{\sqrt{C_{\ell}^{gg}}}\psi_{1,\ell
m}+\left(C_{\ell}^{T_{\gamma}T}-\frac{C_{\ell}^{gT_{\gamma}}C_{\ell}^{gT}}{C_{\ell}^{gg}}\right)/\sqrt{C_{\ell}^{T_{\gamma}T_{\gamma}}-\frac{(C_{\ell}^{gT_{\gamma}})^{2}}{C_{\ell}^{gg}}}\psi_{2,\ell
m}$
$\displaystyle+\sqrt{C_{\ell}^{TT}-\frac{C_{\ell}^{T_{\gamma}T_{\gamma}}\left(C_{\ell}^{gT}\right)^{2}+C_{\ell}^{gg}\left(C_{\ell}^{T_{\gamma}T}\right)^{2}-2C_{\ell}^{gT_{\gamma}}C_{\ell}^{gT}C_{\ell}^{T_{\gamma}T}}{C_{\ell}^{gg}C_{\ell}^{T_{\gamma}T_{\gamma}}-\left(C_{\ell}^{gT_{\gamma}}\right)^{2}}}\psi_{3,\ell
m},$ (24)

where $\psi_{1,\ell m}$, $\psi_{2,\ell m}$, and $\psi_{3,\ell m}$ are mutually
independent, unit-variance gaussian random fields for harmonics ($\ell$, $m$)
that satifsy

$\displaystyle\left\langle\psi_{i,\ell m}\psi_{j,\ell m}\right\rangle$
$\displaystyle=\delta_{ij},$ $\displaystyle\left\langle\psi_{i,\ell
m}^{3}\psi_{j,\ell m}\right\rangle$ $\displaystyle=3\delta_{ij},$
$\displaystyle\left\langle\psi_{i,\ell m}^{2}\psi_{j,\ell m}^{2}\right\rangle$
$\displaystyle=3\delta_{ij}+(1-\delta_{ij}),$ (25)

where $\left\langle\;\right\rangle$ denotes the ensemble average. We also use
the fact

$\displaystyle C_{\ell}^{gT}$
$\displaystyle=C_{\ell}^{gT_{\gamma}}+C_{\ell}^{g\delta
T_{b}}=\left(1+\frac{\delta
T_{21}}{T_{\gamma,0}}\right)C_{\ell}^{gT_{\gamma}},$ $\displaystyle
C_{\ell}^{TT}$ $\displaystyle=C_{\ell}^{T_{\gamma}T_{\gamma}}+C_{\ell}^{\delta
T_{b}\delta T_{b}},$ $\displaystyle C_{\ell}^{T_{\gamma}T}$
$\displaystyle=C_{\ell}^{T_{\gamma}T_{\gamma}},$ (26)

where we assumed $C_{\ell}^{T_{\gamma}\delta T_{b}}=0$ in the last two
equations, which is just an approximation because the evolving Dopper effect
during reionization can give a substantial cross-correlation between the 21-cm
background and CMB, especially so when reionization proceeds fast, yet still
subdominant to $C_{\ell}^{T_{\gamma}T_{\gamma}}$ [47]. Also another cross-
correlation comes through the common ISW effect:

$C_{\ell}^{T_{\gamma}\delta
T_{b}}=\frac{C_{\ell}^{gT_{\gamma}}C_{\ell}^{g\delta T_{b}}}{C_{\ell}^{gg}}$
(27)

which is again quite subdominant to $C_{\ell}^{T_{\gamma}T_{\gamma}}$.

Using the equality

$\displaystyle\sigma_{C_{\ell}^{XY}}^{2}$ $\displaystyle=\left\langle a_{\ell
m}^{X}a_{\ell m}^{Y*}a_{\ell m}^{X*}a_{\ell
m}^{Y}\right\rangle-\left(C_{\ell}^{XY}\right)^{2},$
$\displaystyle\sigma_{C_{\ell}^{XY}C_{\ell}^{XZ}}$ $\displaystyle=\left\langle
a_{\ell m}^{X}a_{\ell m}^{Y*}a_{\ell m}^{X*}a_{\ell m}^{Z}\right\rangle-
C_{\ell}^{XY}C_{\ell}^{XZ},$ (28)

together with Eqs. (24 – 28), we obtain

$\displaystyle\sigma_{C_{\ell}^{gT_{\gamma}}}^{2}$
$\displaystyle=\left(C_{\ell}^{gT_{\gamma}}\right)^{2}+C_{\ell}^{gg}C_{\ell}^{T_{\gamma}T_{\gamma}},$
$\displaystyle\sigma_{C_{\ell}^{gT}}^{2}$
$\displaystyle=\left(C_{\ell}^{gT}\right)^{2}+C_{\ell}^{gg}C_{\ell}^{TT}=\left(1+\frac{\delta
T_{21}}{T_{\gamma,0}}\right)^{2}\left(C_{\ell}^{gT_{\gamma}}\right)^{2}+C_{\ell}^{gg}\left(C_{\ell}^{T_{\gamma}T_{\gamma}}+C_{\ell}^{\delta
T_{b}\delta T_{b}}\right)$
$\displaystyle\sigma_{C_{\ell}^{gT_{\gamma}}C_{\ell}^{gT}}$
$\displaystyle=C_{\ell}^{gT_{\gamma}}C_{\ell}^{gT}+C_{\ell}^{gg}C_{\ell}^{T_{\gamma}T}=\left(1+\frac{\delta
T_{21}}{T_{\gamma,0}}\right)\left(C_{\ell}^{gT_{\gamma}}\right)^{2}+C_{\ell}^{gg}C_{\ell}^{T_{\gamma}T_{\gamma}}.$
(29)

Finally, using Eqs. (23) and (29) we obtain

$\sigma_{s_{\ell}}^{2}(\bar{\nu})=\left\langle
T_{\gamma,0}\right\rangle^{2}\frac{\left(C_{\ell}^{gg}+\bar{n}_{g}^{-1}\right)\left\\{C_{\ell}^{\delta
T_{b}\delta T_{b}}(\bar{\nu})+\epsilon_{\ell,T}(\bar{\nu})+\left(\frac{\delta
T_{21}(\bar{\nu})}{\left\langle
T_{\gamma,0}\right\rangle}\right)^{2}C_{\ell}^{T_{\gamma}T_{\gamma}}\right\\}}{\left(C_{\ell}^{gT_{\gamma}}\right)^{2}f_{{\rm
sky}}(2\ell+1)},$ (30)

where systematic errors in auto-correlations are now included and the number
of modes with imperfect sky coverage is accounted for as well. The net noise
is given by

$N^{-2}(\bar{\nu})=\sum_{\ell=\ell_{{\rm min}}}^{\ell_{{\rm
max}}}\sigma_{s_{\ell}}^{-2}(\bar{\nu}).$ (31)

Fig. 1 shows how the observing time affects the SNR. In the case of
$\Delta\nu=5\,$MHz, the gain from having $\Delta t=30000\,$hr instead of
$\Delta t=10000\,$hr is not much, because the sky brightness noise is still
too high at $z\sim 30$ and the lower-redshift ($z\lesssim 15-20$) noise is
intrinsic, coming mostly from the 21-cm auto-correlation $C_{\ell}^{\delta
T_{b}\delta T_{b}}(\bar{\nu})$. See the relative noise contribution in Fig. 5.
The even easier $\Delta t=3000\,$hr case also gives moderate sensitivity
during the early EoR phase, and thus should be considered seriously in the
early run of the SKA1-LOW.

Figure 5: Composition of the noise in the case of $\Delta\nu=1\,$MHz.
Different observing times are also investigated (top: 3000 hr; middle: 10000
hr; bottom: 30000 hr). The sky brightness ($\epsilon_{T}$: orange, dot-dashed)
dominates the net noise (red, solid) at $z\gtrsim 15-20$, while the noise in
the lower-redshift epoch is dominated by the uncontrollable, intrinsic
quantity $C_{\ell}^{\delta T_{b}\delta T_{b}}$ (blue, dashed).

###  B.2 superstructure 21-cm ISW

If one were to use superstructures, this is not to use the power spectrum as
the full-sky cross-correlation scheme does. Instead, this is close to the full
imaging. Therefore, the automatic removal of the Milky Way foreground does not
apply to this strategy. Let us imagine that we have the temperature field
where all the foreground components (CMB, Milky Way, extragalactic) are
removed.

The noise is estimated as follows. Superstructures are characterized by their
large angular span, $\sim 4^{\circ}$ in radius, and the temperature field is
convoluted with specific smoothing or matching filters $\psi^{{\rm
MF}}(\left|\hat{n}-\hat{n}_{0}\right|)=\psi^{{\rm MF}}(\theta)$ where
$\hat{n}_{0}$ is the location of the center of a superstructure and $\theta$
is the angle away from $\hat{n}_{0}$. For example, [25] used a smoothing
filter that averages the temperature in the region where $\theta<\theta_{{\rm
th}}$ and further subtracts the average temperature where $\theta_{{\rm
th}}<\theta<\sqrt{2}\theta_{{\rm th}}$, and claimed the oddly strong ($A_{{\rm
ISW}}\simeq 5$) ISW effect on superstructures. We follow [28] to estimate the
noise: we take three different matching filters shown in [28] that would each
give the filtered 21-cm temperature field

$\delta T_{b}^{\prime}(\hat{n}^{\prime},\bar{\nu})=\int d\hat{n}\,\delta
T_{b}(\hat{n},\bar{\nu})\psi^{{\rm
MF}}(\left|\hat{n}^{\prime}-\hat{n}\right|).$ (32)

The variance of this filetered field will be given by

$\sigma_{\delta T_{b}}^{2}(\bar{\nu})=\sum_{\ell}\left(C_{\ell}^{\delta
T_{b}\delta T_{b}}+\epsilon_{\ell,T}\right)\left(\psi_{\ell 0}^{{\rm
MF}}\right)^{2},$ (33)

which is synonymous to the mass variance of filtered density field,
$\sigma_{M}$. Here, $\psi_{\ell 0}^{{\rm MF}}$ is the expansion coefficient in
Legendre polynomials of $\psi^{{\rm MF}}(\theta)$:

$\psi^{{\rm MF}}(\theta)=\sum_{\ell}\psi_{\ell 0}^{{\rm MF}}Y_{\ell
0}(\cos\theta).$ (34)

We found that even with the difference in the three filters, the variance
calculated by Eq. (33) all lead to about the same values at all frequencies.
We inverted Eq. (34) using orthogonality of $Y_{\ell 0}$, and calculated the
integral to obtain $\psi_{\ell 0}^{{\rm MF}}$ in a brute-force way. We took
adaptively fine samples in the Legendre polynomials, which was somewhat
computationally costly. Because $\psi^{{\rm MF}}$ is a low-$\ell$-pass filter,
we obtained $\psi_{\ell 0}^{{\rm MF}}$ only up to $\ell=200$ and set the rest
to zero. We found that $\psi_{\ell 0}^{{\rm MF}}$ obtained this way reproduces
$\psi^{{\rm MF}}(\theta)$ well enough.

Because multiple superstructure ISW signals are stacked, if there are $N_{{\rm
ss}}$ superstructures stacked then the final noise is given by

$\sigma_{\delta T_{b}}^{2}(\bar{\nu})=\sum_{\ell}\left(C_{\ell}^{\delta
T_{b}\delta T_{b}}+\epsilon_{\ell,T}\right)\left(\psi_{\ell 0}^{{\rm
MF}}\right)^{2}/N_{{\rm ss}},$ (35)

and the signal is given by Eq. (17) of the main text. We used Eq. (35) to
calculate the noise with $N_{{\rm ss}}=200$, twice as large as what was used
in [25] or about 1/2 of that in [26], and obtained Fig. 3 of the main text.

## References

  * Sachs and Wolfe [1967] R. K. Sachs and A. M. Wolfe, Astrophys. J. 147, 73 (1967), _Perturbations of a cosmological model and angular variations of the microwave background_.
  * Fosalba et al. [2003] P. Fosalba, E. Gaztañaga, and F. J. Castander, Astrophys. J. Lett. 597, L89 (2003), _Detection of the integrated sachs-wolfe and sunyaev-zeldovich effects from the cosmic microwave background-galaxy correlation_.
  * Afshordi et al. [2004] N. Afshordi, Y.-S. Loh, and M. A. Strauss, Phys. Rev. D 69, 083524 (2004), _Cross-correlation of the cosmic microwave background with the 2mass galaxy survey: Signatures of dark energy, hot gas, and point sources_.
  * Cabré et al. [2006] A. Cabré, E. Gaztañaga, M. Manera, P. Fosalba, and F. Castander, Mon. Not. Roy. Astron. Soc. 372, L23 (2006), _Cross-correlation of wilkinson microwave anisotropy probe third-year data and the sloan digital sky survey dr4 galaxy survey: new evidence for dark energy_.
  * Giannantonio and Crittenden [2007] T. Giannantonio and R. Crittenden, Mon. Not. Roy. Astron. Soc. 381, 819 (2007), _The effect of reionization on the cosmic microwave background-density correlation_.
  * Ilić et al. [2011] S. Ilić, M. Douspis, M. Langer, A. Pénin, and G. Lagache, Mon. Not. Roy. Astron. Soc. 416, 2688 (2011), _Cross-correlation between the cosmic microwave and infrared backgrounds for integrated sachs-wolfe detection_.
  * Giannantonio et al. [2012] T. Giannantonio, R. Crittenden, R. Nichol, and A. J. Ross, Mon. Not. Roy. Astron. Soc. 426, 2581 (2012), _The significance of the integrated sachs-wolfe effect revisited_.
  * Planck Collaboration et al. [2016] Planck Collaboration, P. A. R. Ade, N. Aghanim, M. Arnaud, M. Ashdown, J. Aumont, C. Baccigalupi, A. J. Banday, R. B. Barreiro, N. Bartolo, et al., A&A 594, A21 (2016), _Planck 2015 results. xxi. the integrated sachs-wolfe effect_.
  * Dong et al. [2021] F. Dong, Y. Yu, J. Zhang, X. Yang, and P. Zhang, Mon. Not. Roy. Astron. Soc. 500, 3838 (2021), _Measuring the integrated sachs-wolfe effect from the low-density regions of the universe_.
  * Bahr-Kalus et al. [2022] B. Bahr-Kalus, D. Parkinson, J. Asorey, S. Camera, C. Hale, and F. Qin, Mon. Not. Roy. Astron. Soc. 517, 3785 (2022), _A measurement of the integrated sachs-wolfe effect with the rapid askap continuum survey_.
  * Crittenden and Turok [1996] R. G. Crittenden and N. Turok, Phys. Rev. Lett. 76, 575 (1996), _Looking for a cosmological constant with the rees-sciama effect_.
  * Kamionkowski [1996] M. Kamionkowski, Phys. Rev. D 54, 4169 (1996), _Matter-microwave correlations in an open universe_.
  * Song et al. [2007] Y.-S. Song, W. Hu, and I. Sawicki, Phys. Rev. D 75, 044004 (2007), _Large scale structure of f(r) gravity_.
  * Cai et al. [2014] Y.-C. Cai, B. Li, S. Cole, C. S. Frenk, and M. Neyrinck, Mon. Not. Roy. Astron. Soc. 439, 2978 (2014), _The integrated sachs-wolfe effect in f(r) gravity_.
  * Bowman et al. [2018] J. D. Bowman, A. E. E. Rogers, R. A. Monsalve, T. J. Mozdzen, and N. Mahesh, Nature (London) 555, 67 (2018), _An absorption profile centred at 78 megahertz in the sky-averaged spectrum_.
  * Singh et al. [2022] S. Singh, N. T. Jishnu, R. Subrahmanyan, N. Udaya Shankar, B. S. Girish, A. Raghunathan, R. Somashekar, K. S. Srivani, and M. Sathyanarayana Rao, Nature Astronomy 6, 607 (2022), _On the detection of a cosmic dawn signal in the radio background_.
  * Barkana [2018] R. Barkana, Nature (London) 555, 71 (2018), _Possible interaction between baryons and dark-matter particles revealed by the first stars_.
  * Hills et al. [2018] R. Hills, G. Kulkarni, P. D. Meerburg, and E. Puchwein, Nature (London) 564, E32 (2018), _Concerns about modelling of the edges data_.
  * Raccanelli et al. [2016] A. Raccanelli, E. Kovetz, L. Dai, and M. Kamionkowski, Phys. Rev. D 93, 083512 (2016), _Detecting the integrated sachs-wolfe effect with high-redshift 21-cm surveys_.
  * Hotinli and Ahn [2023] S. C. Hotinli and K. Ahn, arXiv e-prints arXiv:2305.01672 (2023), _Probing the global 21-cm background by velocity-induced dipole and quadrupole anisotropies_.
  * Afshordi [2004] N. Afshordi, Phys. Rev. D 70, 083536 (2004), _Integrated sachs-wolfe effect in cross-correlation: The observer’s manual_.
  * Zaldarriaga et al. [2004] M. Zaldarriaga, S. R. Furlanetto, and L. Hernquist, Astrophys. J. 608, 622 (2004), _21 Centimeter Fluctuations from Cosmic Gas at High Redshifts_.
  * Fixsen et al. [1996] D. J. Fixsen, E. S. Cheng, J. M. Gales, J. C. Mather, R. A. Shafer, and E. L. Wright, Astrophys. J. 473, 576 (1996), _The Cosmic Microwave Background spectrum from the full COBE FIRAS data set_.
  * Fixsen [2009] D. J. Fixsen, Astrophys. J. 707, 916 (2009), _The temperature of the cosmic microwave background_.
  * Granett et al. [2008] B. R. Granett, M. C. Neyrinck, and I. Szapudi, Astrophys. J. Lett. 683, L99 (2008), _An imprint of superstructures on the microwave background due to the integrated sachs-wolfe effect_.
  * Nadathur et al. [2012] S. Nadathur, S. Hotchkiss, and S. Sarkar, J. Cosmol. Astropart. Phys. 2012, 042 (2012), _The integrated sachs-wolfe imprint of cosmic superstructures: a problem for $\Lambda$cdm_.
  * Flender et al. [2013] S. Flender, S. Hotchkiss, and S. Nadathur, J. Cosmol. Astropart. Phys. 2013, 013 (2013), _The stacked isw signal of rare superstructures in $\Lambda$cdm_.
  * Nadathur and Crittenden [2016] S. Nadathur and R. Crittenden, Astrophys. J. Lett. 830, L19 (2016), _A detection of the integrated sachs-wolfe imprint of cosmic superstructures using a matched-filter approach_.
  * Kovács et al. [2019] A. Kovács, C. Sánchez, J. García-Bellido, J. Elvin-Poole, N. Hamaus, V. Miranda, S. Nadathur, T. Abbott, F. B. Abdalla, J. Annis, et al., Mon. Not. Roy. Astron. Soc. 484, 5267 (2019), _More out of less: an excess integrated sachs-wolfe signal from supervoids mapped out by the dark energy survey_.
  * Hang et al. [2021] Q. Hang, S. Alam, Y.-C. Cai, and J. A. Peacock, Mon. Not. Roy. Astron. Soc. 507, 510 (2021), _Stacked cmb lensing and isw signals around superstructures in the desi legacy survey_.
  * Kovács et al. [2022] A. Kovács, R. Beck, A. Smith, G. Rácz, I. Csabai, and I. Szapudi, Mon. Not. Roy. Astron. Soc. 513, 15 (2022), _Evidence for a high-z isw signal from supervoids in the distribution of eboss quasars_.
  * Furlanetto [2006] S. R. Furlanetto, Mon. Not. Roy. Astron. Soc. 371, 867 (2006), _The global 21-centimeter background from high redshifts_.
  * Ahn et al. [2012] K. Ahn, I. T. Iliev, P. R. Shapiro, G. Mellema, J. Koda, and Y. Mao, Astrophys. J. Lett. 756, L16 (2012), _Detecting the Rise and Fall of the First Stars by Their Impact on Cosmic Reionization_.
  * Ahn and Shapiro [2021] K. Ahn and P. R. Shapiro, Astrophys. J. 914, 44 (2021), _Cosmic Reionization May Still Have Started Early and Ended Late: Confronting Early Onset with Cosmic Microwave Background Anisotropy and 21 cm Global Signals_.
  * Trott et al. [2020] C. M. Trott, C. H. Jordan, S. Midgley, N. Barry, B. Greig, B. Pindor, J. H. Cook, G. Sleap, S. J. Tingay, D. Ung, et al., Mon. Not. Roy. Astron. Soc. 493, 4711 (2020), _Deep multiredshift limits on epoch of reionization 21 cm power spectra from four seasons of murchison widefield array observations_.
  * Ghara et al. [2020] R. Ghara, S. K. Giri, G. Mellema, B. Ciardi, S. Zaroubi, I. T. Iliev, L. V. E. Koopmans, E. Chapman, S. Gazagnes, B. K. Gehlot, et al., Mon. Not. Roy. Astron. Soc. 493, 4728 (2020), _Constraining the intergalactic medium at z¡ 9.1 using lofar epoch of reionization observations_.
  * HERA Collaboration et al. [2023] HERA Collaboration, Z. Abdurashidova, T. Adams, J. E. Aguirre, P. Alexander, Z. S. Ali, R. Baartman, Y. Balfour, R. Barkana, A. P. Beardsley, et al., Astrophys. J. 945, 124 (2023), _Improved constraints on the 21 cm eor power spectrum and the x-ray heating of the igm with hera phase i observations_.
  * Yan et al. [2013] L. Yan, E. Donoso, C.-W. Tsai, D. Stern, R. J. Assef, P. Eisenhardt, A. W. Blain, R. Cutri, T. Jarrett, S. A. Stanford, et al., Astron. J. 145, 55 (2013), _Characterizing the mid-infrared extragalactic sky with wise and sdss_.
  * Doré et al. [2014] O. Doré, J. Bock, M. Ashby, P. Capak, A. Cooray, R. de Putter, T. Eifler, N. Flagey, Y. Gong, S. Habib, et al., arXiv e-prints arXiv:1412.4872 (2014), _Cosmology with the spherex all-sky spectral survey_.
  * Deshpande [2018] A. A. Deshpande, Astrophys. J. Lett. 866, L7 (2018), _Dipole anisotropy as an essential qualifier for the monopole component of the cosmic-dawn spectral signature, and the potential of diurnal pattern for foreground estimation_.
  * Koopmans et al. [2015] L. Koopmans, J. Pritchard, G. Mellema, J. Aguirre, K. Ahn, R. Barkana, I. van Bemmel, G. Bernardi, A. Bonaldi, F. Briggs, et al., in _Advancing Astrophysics with the Square Kilometre Array (AASKA14)_ (2015), p. 1, eprint 1505.07568.
  * Lewis and Challinor [2011] A. Lewis and A. Challinor, _Camb: Code for anisotropies in the microwave background_ , Astrophysics Source Code Library, record ascl:1102.026 (2011), eprint 1102.026, URL https://ui.adsabs.harvard.edu/abs/2011ascl.soft02026L.
  * Barkana and Loeb [2005] R. Barkana and A. Loeb, Astrophys. J. Lett. 624, L65 (2005), _A Method for Separating the Physics from the Astrophysics of High-Redshift 21 Centimeter Fluctuations_.
  * Fang et al. [2020] X. Fang, E. Krause, T. Eifler, and N. MacCrann, J. Cosmol. Astropart. Phys. 2020, 010 (2020), _Beyond limber: efficient computation of angular power spectra for galaxy clustering and weak lensing_.
  * Cabré et al. [2007] A. Cabré, P. Fosalba, E. Gaztañaga, and M. Manera, Mon. Not. Roy. Astron. Soc. 381, 1347 (2007), _Error analysis in cross-correlation of sky maps: application to the integrated sachs-wolfe detection_.
  * Giannantonio et al. [2008] T. Giannantonio, R. Scranton, R. G. Crittenden, R. C. Nichol, S. P. Boughn, A. D. Myers, and G. T. Richards, Phys. Rev. D 77, 123520 (2008), _Combined analysis of the integrated sachs-wolfe effect and cosmological implications_.
  * Alvarez et al. [2006] M. A. Alvarez, E. Komatsu, O. Doré, and P. R. Shapiro, Astrophys. J. 647, 840 (2006), _The cosmic reionization history as revealed by the cosmic microwave background doppler-21 cm correlation_.