MNRAS 540, 483–497 (2025) h ps://doi.o g/10.1093/mn as/s a 699
Ad ance Access publica ion 2025 Ap il 30
Semi-analy ical modelling o Pop. III s a o ma ion and me allici y
e olu ion – II. Impac on 21 cm powe spec um
Emanuele M. Ven u a ,
1 , 2 ‹Yuxiang Qin ,
2 , 3 S eedha Balu
1 , 2 , 4 and J. S ua B. Wyi he
2 , 3
1
School o Physics, Uni e si y o Melbou ne, Pa k ille, VIC 3010, Aus alia
2
ARC Cen e o Excellence o All Sky As ophysics in 3 Dimensions (ASTRO 3D), Canbe a, ACT 2611, Aus alia
3
Resea c h Sc hool o As onomy and As ophysics, Aus alian Na ional Uni e si y, Canbe a, ACT 2611, Aus alia
4
Facul ad de F
´
ısicas, Mul idisciplina y Uni o Ene gy Science, Uni e sidad de Se illa, 41012 Se ille, Spain
Accep ed 2025 Ap il 27. Recei ed 2025 Ap il 11; in o iginal o m 2025 Feb ua y 12
A B S T R A C T
Simula ing Popula ion (Pop.) III s a o ma ion in mini-haloes in a la ge cosmological simula ion is an ex emely challenging
ask bu i is c ucial o es ima e i s impac on he 21 cm powe spec um. In his wo k, we de elop a amewo k wi hin he
semi-analy ical code MERAXES o es ima e he adia i e backg ounds om Pop. III s a s needed o he compu a ion o he 21 cm
signal. We compu ed he 21 cm global signal and powe spec um o di e en Pop. III models a ying s a o ma ion e iciency,
ini ial mass unc ion and speci ic X- ay luminosi y pe uni o s a o ma ion (L
X
/SFR). In all he models conside ed, we ind
Pop. III s a s ha e li le o no impac on he eioniza ion his o y bu signi ican ly a ec he he mal s a e o he in e galac ic
medium (IGM) due o he s ong injec ion o X- ay pho ons om hei emnan s ha hea he neu al IGM a z ≥15. This is
e lec ed no only on he 21 cm sk y-a e aged global signal du ing he Cosmic Dawn bu also on he 21 cm powe spec um a z ≤
10 whe e models wi h s ong Pop. III X- ay emission ha e la ge powe han models wi h no o mild Pop. III X- ay emission.
We es ima e obse a ional unce ain ies on he powe spec um using 21CMSENSE and ind ha models whe e Pop. III s a s ha e
a s onge X- ay emission han Pop. II a e dis inguishable om models wi h no o mild Pop. III X- ay emission wi h 1000 h
obse a ions o he upcoming SKA1-low.
Key wo ds: s a s: Popula ion III – galaxies: high- edshi – cosmology: da k ages, eioniza ion, i s s a s.
1 INTRODUCTION
When and whe e did Popula ion III (Pop. III) s a s o m? Wha ole
did hey play in he Cosmic Dawn and Epoch o Reioniza ion (EoR)?
And wha is he bes way o de ec hem? These ques ions emain
open as no de ini i e obse a ion o a mini-halo o Pop. III s a
has been epo ed. To gain insigh , small size and high- esolu ion
hyd odynamical simula ions ha e been pe o med (e.g. G ei e al.
2011 ; Hi ano e al. 2018 ; Chon, Omukai & Schneide 2021 ; Chon
e al. 2022 ; Toyouchi e al. 2023 ; Sadana i e al. 2024 ) ha sugges
ha me al- ee (o poo ) mini-haloes a ou he o ma ion o Pop.
III s a s wi h a mo e op-hea y ini ial mass unc ion (IMF) and
wi h lowe s a o ma ion e iciencies han obse ed oday. I is also
hough ha Pop. III s a o ma ion migh occu down o he end o
he EoR a z 6 in p is ine me al ee pocke s o gas (e.g. Vendi i
e al. 2023 ).
The a ie y and complexi y o he p ocesses in ol ed in Pop.
III s a o ma ion and he esolu ion equi ed o keep ack o he
e olu ion o he gas pa icles, limi s he size o hese hyd odynamical
simula ions o ∼100 kpc. In o de o mi iga e his p oblem, semi-
analy ical models ha accoun o Pop. III s a o ma ion ha e been
de eloped (e.g. Visbal, B yan & Haiman 2020 ; Hegde & Fu lane o
E-mail:
e[email p o ec ed].edu.au
2023 ; Liu e al. 2024 ). These models allow a s a is ical s udy o Pop.
III s a o ma ion in mini-haloes ou o scales o ∼10 Mpc. While
hese olumes s a o in es iga e he chemical en ichmen o he
in e galac ic medium (IGM) and he Pop. III/II ansi ion, hey a e
s ill oo small o s udy he EoR as olumes o a leas ∼200 Mpc a e
equi ed (Ilie e al. 2014 ; Kau , Gille & Mesinge 2020 ; Balu e al.
2023a ).
Obse a ions and models a e con e ging on a scena io whe e he
Uni e se was comple ely ionized by z ∼5 . 3 (e.g. Fan, Ca illi &
Kea ing 2006 ; Ouchi e al. 2010 ; McG ee , Mesinge & D’Odo ico
2015 ; Qin e al. 2021a , 2024 ; Bosman e al. 2022 ) wi h eioniza ion
likely d i en by low-mass haloes (e.g. Kuhlen & Fauche -Gigu
`
e e
2012 ; Qin e al. 2021b ; Mu ch e al. 2024 ; Saxena e al. 2024 ).
Ho we e , he impac o Pop. III s a s and mini-haloes on he EoR
is unclea . Pop. III s a s a e likely o be he dominan con ibu ion
o he o al s a o ma ion a e densi y (SFRD) a z > 15 −20 and,
i hei IMF is mo e op-hea y han he p esen day one, Pop. III
could signi ican ly con ibu e o he hea ing and he ioniza ion o
he IGM which de e mines he e olu ion and shape o he 21 cm
signal (e.g. Qin e al. 2020 ; Gessey-Jones e al. 2022 ; Sa o io e al.
2023 ).
The 21 cm signal ep esen s ou mos p omising ool o pu
cons ain s on he he mal s a e o he IGM du ing he Cosmic Dawn
and EoR. E en hough no con i med de ec ion has been epo ed
so a , he i s uppe limi s on he 21 cm powe spec um ob ained
© 2025 The Au ho (s).
Published by Ox o d Uni e si y P ess on behal o Royal As onomical Socie y. This is an Open Access a icle dis ibu ed unde he e ms o he C ea i e
Commons A ibu ion License ( h ps:// c ea i ecommons.o g/ licenses/ by/ 4.0/ ), which pe mi s un es ic ed euse, dis ibu ion, and ep oduc ion in any medium,
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484 E. M. Ven u a e al.
MNRAS 540, 483–497 (2025)
Table 1. Main ee pa ame e s o galaxy o ma ion.
Pa ame e Desc ip ion Fiducial alue
αSF , II Pop. II S a o ma ion e iciency 0.1
αSF , III Pop. III S a o ma ion e iciency see Table 4
η0 Mass loading no maliza ion 7.0
0 Supe no a ene gy coupling no maliza ion 1.5
Z
c i C i ical me allici y o Pop III s a o ma ion 10
−4
Z
no m C i ical su ace densi y o cold gas o s a o ma ion 0.37 M
pc
−2
Pop. III IMF Shape o Pop. III IMF Sal [1, 500] M
E
PISN Ene gy om pai ins abili y SN 10
52
e g
E
CCSN Ene gy om co e collapse SN 10
51
e g
Table 2. Main ee pa ame e s o eioniza ion.
Pa ame e Desc ip ion Fiducial alue
0
esc , III
Pop. III escape ac ion no maliza ion 0.14
0
esc , II
as abo e o Pop. II 0.14
αesc , III Pop. III escape ac ion edshi scaling 0.2
αesc , II as abo e o Pop. II 0.2
L
X < 2keV , III
/ SFR Speci ic Pop. III X- ay luminosi y pe uni s a o ma ion see Table 4
L
X < 2keV , II
/ SFR as abo e o Pop. II 3.16 ×10
40
e g s
−1
M
−1
y
wi h HERA phase I s ongly dis a ou cold eioniza ion scena ios
(HERA Collabo a ion 2023 ). In he las ew yea s, he impac o Pop.
III s a s on he 21 cm signal has been s udied using bo h analy ical
and semi-analy ical models (e.g. Cohen e al. 2017 ; Cha e jee e al.
2020 ; Mebane, Mi ocha & Fu lane o 2020 ). Ho we e hese models
ei he did no compu e eioniza ion (e.g. Magg e al. 2022 ; Hegde &
Fu lane o 2023 ; C uz e al. 2024a ), ocusing only on he abso p ion
ough o he 21 cm signal occu ing a z ∼13 −20, o used a e y
simple analy ical app oach o compu e eioniza ion (e.g. Ven u a
e al. 2023 ). On he o he hand, Cohen e al. ( 2017 ), Qin e al.
( 2021a ), and Mu
˜
noz e al. ( 2022 ) used a simple analy ical model
o modelling Pop. III s a o ma ion bu compu ed he eioniza ion
sel -consis en ly.
In his wo k, we o e come hese challenges using a ealis ic
Pop. III s a o ma ion and mini-halo model (Ven u a e al. 2024 )
de eloped wi hin he semi-analy ical model MERAXES designed o
sel -consis en ly couple galaxy o ma ion and eioniza ion. While in
Ven u a e al. ( 2024 ) we an his model on a small ( L = 10 h
−1
cMpc) and high- esolu ion box, he e we ex end i o a signi ican ly
la ge olume simula ion ( L = 210 h
−1
cMpc) enabling he s udy o
cosmic eioniza ion. Since a such la ge olumes we canno di ec ly
esol e mini-haloes, we implemen ed scaling ela ions be ween he
SFRD and he da k ma e densi y ield calib a ed on he esul s om
he small and high- esolu ion box discussed in Ven u a e al. ( 2024 ).
Wi h his new model we a e able o accu a ely ollow he e olu ion
o he adia i e backg ounds ele an o he EoR and 21 cm signal
(X- ays, Lyman- α, ionizing UV, Lyman–We ne ) and o disen angle
he con ibu ion o Pop. III s a o ma ion o he 21 cm global signal
and powe spec um. Pop. III s a s a e expec ed o ha e a s onge
impac a z ≥15 whe e hey domina e s a o ma ion and ioniza ion.
Di e en ly om p e ious wo ks who explo ed he di e ences in he
21 cm signal a Cosmic Dawn due o a ious Pop. III models, he e
we ocus ou a en ion on he esidual signa u e o Pop. III on he
21 cm powe spec um a z ≤10 whe e he sensi i i y o he Squa e
Kilome e A ay (SKA) is expec ed o be signi ican ly be e and a
de ec ion is mo e plausible. To achie e his, i is c ucial o model
bo h Pop. III s a o ma ion and eioniza ion in a sel -consis en
amewo k. This s udy allows us o assess unde which condi ions
an ea ly hea ing o he IGM om Pop. III s a s lea es a de ec able
imp in on he 21cm powe spec um a z ≤10.
This pape is s uc u ed as ollows: In Sec ion 2 , we gi e a b ie
o e iew o Pop. III s a o ma ion in MERAXES . In Sec ion 3 ,
we p esen he scaling ela ion be ween he SFRD in mini-haloes
and he densi y ield calib a ed om he small and high- esolu ion
box which is implemen ed in he la ge (210 h
−1
cMpc)
3 box. In
Sec ion 4 , we discuss he impac o di e en Pop. III s a o ma ion
models on he 21 cm powe spec um and in Sec ion 5 we make
o ecas s on he obse abili y o hese powe spec a wi h SKA.
Finally, we summa ize ou main esul s and conclusions in Sec ion 6 .
Ou simula ions use he bes - i ing pa ame e s om he Planck
Collabo a ion XIII ( 2016 ): h = 0.6751, m
= 0.3121, b
= 0.0490,
= 0.6879, σ8
= 0.8150, and n
s
= 0.9653.
2 POP. III GALAXIES IN MERAXES
MERAXES
1
is a semi-analy ical model designed o s udy he in e play
be ween galaxy o ma ion and eioniza ion (Mu ch e al. 2016 ; Qin
e al. 2017 ; Qiu e al. 2019 ; Ven u a e al. 2024 ). MERAXES includes
a numbe o ee pa ame e s ha a e calib a ed agains obse a ions
(see Tables 1 and 2 ). Values in Table 1 a e calib a ed agains obse ed
luminosi y unc ions and s ella mass unc ions a z ∼5 −8 while
hose in Table 2 a e calib a ed agains cons ain s on he neu al
hyd ogen ac ion, ionizing emissi i y and he Thomson sca e ing
op ical dep h τe om Planck Collabo a ion VI ( 2020 ). The mos
ecen e sion o MERAXES (Ven u a e al. 2024 , V24) includes Pop.
III s a o ma ion and mini-halo physics. As shown in V24, he Pop.
III pa ame e s wi h he la ges impac a e he s a o ma ion e iciency
αSF , III and he shape o he IMF. The la e has a la ge impac
on galaxy e olu ion as i de e mines he s eng h o he eedback
and he emission p ope ies o he Pop. III s ella popula ion. In
he ollowing sec ions, we quickly summa ize he main ea u es o
MERAXES ele an o his wo k.
1
h ps:// gi hub.com/ me axes-de s/ me axes
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21 cm PS wi h Pop. III s a o ma ion 485
MNRAS 540, 483–497 (2025)
2.1 Galaxy o ma ion
MERAXES pos -p ocesses he ou pu o an N -body da k ma e
only simula ion, eading he spa ial and physical in o ma ion o
da k ma e haloes and compu ing he ba yonic physics o galaxy
o ma ion. In pa icula p ocesses included a e: (i) gas in all on o
da k ma e haloes, (ii) adia i e cooling o he in alling gas, (iii) s a
o ma ion, and (i ) supe no a and ac i e galac ic nuclei eedback.
In V24, he cooling p esc ip ions we e upda ed o accoun o
H
2 cooling ( he main cooling channel in mini-haloes) and a mo e
de ailed me al en ichmen model o keep ack o he me allici y o
each gas ese oi in a halo (which is c ucial o dis inguish be ween
Pop. III and Pop. II s a o ma ion episodes). We also accoun o he
e ec s o bo h ba yon-da k ma e s eaming eloci y and H
2
pho o-
dissocia ion by he Lyman–We ne backg ound which inc eases he
minimum mass o a mini-halo capable o hos ing s a s (e.g. Schaue
e al. 2021 ). Ou model accoun s o spa ial a ia ions only o he
LW backg ound, while o he ela i e eloci y we assume a mean
alue h oughou he en i e box. Fo his wo k, we upda ed MERAXES
by adding he e ec o H
2 sel -shielding which coun e ac s he H
2
pho o-dissocia ion, inc easing he Pop. III SFRD by up o one o de
o magni ude a z ∼10 (see e.g. Fea he s e al. 2024 ). We discuss
he de ails o he implemen a ion and impac o H
2
sel -shielding in
MERAXES in he Appendix A .
We e e he eade o Mu ch e al. ( 2016 ) o a mo e de ailed
explana ion o he main a chi ec u e o MERAXES , Qiu e al. ( 2019 )
o he supe no a model and V24 o he mini-halo model.
The main ee pa ame e s ha egula e he galaxy o ma ion in
MERAXES a e summa ized in Table 1 . The Pop. II ela ed ones a e
aken om Balu e al. ( 2023a ) whe e MERAXES was un on a cosmo-
logical olume o L = 210 h
−1
cMpc esol ing all a omic cooling
haloes and calib a ed in o de o ma ch he obse ed ul a iole (UV)
luminosi y unc ions a z ∼4 −7 ( he ag eemen holds up o z ∼13
as shown in Qin, Balu & Wyi he 2023 ) and he s ella mass unc ions
a z ∼5 −8. Gi en he lack o obse a ions o Pop. III s a s, he Pop.
III pa ame e s a e la gely uncons ained. The iducial alues adop ed
in his wo k a e aken om V24 and hei alues a e sugges ed om
hyd odynamical simula ions (e.g. Chon e al. 2021 ).
2.2 Reioniza ion and adia i e backg ounds
Toge he wi h galaxy o ma ion, MERAXES sel -consis en ly com-
pu es he eioniza ion and he mal e olu ion o he IGM using a
modi ied e sion o he seminume ical code 21 cm FAST (Mesinge ,
Fu lane o & Cen 2011 ). In his wo k, we compu e he backg ounds
ele an o he compu a ion o he 21 cm signal: he UV ionizing, X-
ays, L yman- αand L yman–We ne (LW). The i s is c ucial o s udy
he e olu ion o he eioniza ion, while he X- ay and he Lyman- α
backg ounds de e mine he he mal s a e o he IGM. In pa icula
he X- ay backg ound is likely o be he dominan con ibu ion o
he hea ing o he IGM once he i s galaxies o m and he lyman- α
backg ound is esponsible o he coupling be ween he kine ic and
he spin empe a u e o he neu al hyd ogen. The LW backg ound
does no di ec ly a ec he IGM empe a u e, bu de e mines whe he
o no mini-haloes ha e enough molecula hyd ogen o cool he
gas and o m Pop. III s a s. He ea e , we b ie ly summa ize he
key quan i ies ha de e mine he e olu ion o hese backg ounds.
Fo a mo e de ailed explana ion on he implemen a ions o hese
backg ounds, we e e he eade o Balu e al. ( 2023a ) o he UV,
Lyman- αand X- ay and o V24 o he LW.
The ionizing backg ound is mos ly dependen on he SFRD, he
a e age numbe o ionizing pho ons pe s ella ba yon N
γand he
escape ac ion o he UV pho ons
esc
. The second quan i y is mos ly
de e mined by he IMF: o Pop. II s a s we adop a K oupa IMF
which leads o N
γ∼6000. Since he Pop. III IMF is a ee pa ame e
in ou model, N
γ, III
is compu ed om he IMF adop ed using he Pop.
III s ella spec a om (Rai e , Schae e & Fosbu y 2010 ; N
γ, III
∼
20 000 −70 000).
esc is uned o ep oduce he EoR his o ies in
ag eemen wi h obse a ions. As pe Balu e al. ( 2023a ), we adop a
edshi -dependen escape ac ion de ined as ollowed:
esc
=
0
esc
1 + z
6 αesc
. (1)
X- ay emission is mos ly associa ed wi h high mass X- ay bina ies
(HMXB) and i s con ibu ion is p opo ional o he SFRD. In his
wo k we use he widely adop ed app oxima ion o he como ing
X- ay speci ic emissi i y (e g s
−1
Mpc
−3
) x ∝ L
X
/ SFR ×SFRD.
Finally, we need o accoun o he ac ha only pho ons wi h an
ene gy below 2 keV (so X- ays) a e able o hea he IGM. As a
esul he main ee pa ame e ha egula es X- ay emissi i y is he
so X- ay luminosi y pe uni s a o ma ion L
X < 2keV
/SFR. Fo Pop.
II s a s, his quan i y is es ima ed om heo e ical s udies o emission
spec a o HMXBs in low-me allici y en i onmen s (e.g. F agos e al.
2013 ; Das e al. 2017 ; Madau & F agos 2017 ; Qin e al. 2020 ; Kau
e al. 2022 ). Fo Pop. III s a s he e a e no obse a ional cons ain s
as his quan i y depends on he unknown Pop. III IMF. Recen ly,
Sa o io e al. ( 2023 ) es ima ed he L
X
/SFR o Pop. III s a s and
ound ha o mo e op-hea y IMFs his quan i y can be up o wo
o de s o magni ude highe han he Pop. II alue.
We highligh ha in his wo k we sepa a ely compu e he
backg ounds om Pop. III and Pop. II s a s due o he di e en
spec a, p ope ies, and s a o ma ion a e densi y o he wo dis inc
popula ions.
2.3 21 cm physics
Using he adia i e backg ounds compu ed om he galaxy popula-
ion in MERAXES , we can es ima e he 21 cm signal. We encou age
he eade o see Fu lane o, Peng Oh & B iggs ( 2006 ), Mo ales &
Wyi he ( 2010 ), P i cha d & Loeb ( 2012 ), and Liu & Shaw ( 2020 )
o e ie ws on he opic. He ea e , we only summa ize he key
equa ions used in his wo k (see also Balu e al. 2023a ).
We s a wi h he 21 cm b igh ness empe a u e ield ( δT
b
) which
measu es he de ia ion o he spin empe a u e o he neu al
hyd ogen ( T
S
) om he cosmic backg ound T
γ(i.e. he CMB). This
is gi en by (Fu lane o e al. 2006 ):
δT
b
=
T
S
−T
γ
1 + z
(1 −e
−τν0
)
≃ 27 x
HI
(1 + δnl
)
H
d
/ d + H
1 −T
γ
T
S
×1 + z
10
0 . 15
m
h
2
b
h
2
0 . 023
mK,
(2)
whe e τν0
is he op ical dep h a he 21 cm ansi ion equency
ν0
, x
H is he neu al hyd ogen ac ion, 1 + δnl is he densi y
con as in he da k ma e ield, H ( z) is he Hubble pa ame e a
he edshi z, and d
/ d is he adial de i a i e o he line-o -sigh
componen o he peculia eloci y. Once he cosmological model
(Planck Collabo a ion XIII 2016 ) and he eloci y and densi y ield
( om he N -body simula ion) a e ixed, δT
b is de e mined by he
ioniza ion and he spin empe a u e ields. The la e quan i ies he
popula ion a io o he wo H
I
hype ine ene gy le els and is sensi i e
o he he mal s a e (i.e. he kine ic empe a u e T
K
) o he gas as
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486 E. M. Ven u a e al.
MNRAS 540, 483–497 (2025)
Table 3. Simula ion pa ame e s.
Label Box side (cMpc) Mass esolu ion (M
) Pixel side
L10 10 h
−1 4.71 ×10
5 0.2 h
−1
L210 210 h
−1 3.16 ×10
7 0.2 h
−1
ollows:
T
−1
S
=
T
−1
γ+ x
αT
−1
α+ x
c
T
−1
K
x
α+ x
c
+ 1
, (3)
whe e T
αis he colou empe a u e which we ake equal o T
K
while x
αand x
c
a e he Lyman- αand collisional coupling coe icien s,
espec i ely. These coe icien s quan i y he s eng h o he p ocesses
( esonan sca e ing o Lyman- αpho ons, Wou huysen 1952 , and
collisions wi h ee elec ons) ha d i e he spin empe a u e owa ds
he kine ic empe a u e (when x
α+ x
c >> 1, T
S ∼T
K o he wise
T
S ∼T
γ). T
K is sensi i e o he adiaba ic cooling and o all he
p ocesses able o hea up (o cool) he IGM wi h he mos dominan
coming om he X- ay emission. Hence, in his wo k we will
conside only he X- ay hea ing neglec ing he o he sou ce o hea ing
such as p imo dial magne ic ields (Minoda, Tashi o & Takahashi
2019 ; Be a, Da a & Samui 2020 ; C uz e al. 2024b ), Lyman- α
(Cia di, Sal a e a & Di Ma eo 2010 ; Mi al & Kulka ni 2021 ;
Reis, Fialko & Ba kana 2021 ), shocks (Fu lane o & Loeb 2004 ;
Gnedin & Sha e 2004 ; Ma e al. 2021 ), cosmic ays (Be a, Samui &
Da a 2023 ), ea ly acc e ing black holes (Mebane e al. 2020 ; Ven u a
e al. 2023 ) and decaying o annihila ing da k ma e (Liu & Sla ye
2018 ; Sun e al. 2023 ; Facchine i e al. 2024 ; Hou & Mack 2025 ).
Ul ima ely, he e olu ion o T
S
du ing he Cosmic Dawn and he EoR
is mos ly de e mined by he Lyman- α( o he coupling be ween T
S
and T
K
) and X- ay lux.
Using equa ion ( 2 ) we can es ima e bo h he all-sky a e aged
global signal and i s luc ua ions (i.e. he powe spec um). In
his wo k, we will o en use he educed powe spec um
2
21
( k )
= k
3
/2 π2
P
21
(k) unless o he wise s a ed.
3 PUTTING POP. III GALAXIES AND
MINI-HALOES IN A LARGE-SCALE
SIMULATION
In his sec ion, we p esen a no el app oach ha enables us o
e icien ly es ima e he SFRD om mini-haloes in a la ge box ( ha
does no di ec ly esol e hese objec s) using esul s om a small,
high- esolu ion box. In Table 3 , we summa ize he key pa ame e s
o bo h he small (L10) and la ge (L210) box.
3.1 Calib a ing scaling ela ions om he L10 box
Ou s a ing poin is he small ( L = 10 h
−1
cMpc) high- esolu ion
(halo mass esolu ion o M ∼4 . 7 ×10
5 M
) simula ion used in
V24. When building a scaling ela ion be ween he s a o ma ion
a e (SFR) and o he physical quan i y, he i s ob ious choice is
he da k ma e densi y ield δ. Fo ins ance Mu
˜
noz ( 2023 ) showed,
as a i s -o de app oxima ion, SFR scales as e
δR whe e δR is he
densi y ield smoo hed o e a ce ain adius R and his ela ionship
wo ks qui e well o δ∼0 and la ge R ( ≥3 Mpc). To link ou SFR
in mini-haloes wi h he densi y ield, we i s compu e densi y, δ( x ,
z), and SFR g ids o bo h Pop. III, SFR
MC , III
( x , z), and Pop. II,
SFR
MC , II
( x , z), in he L10 box using he same g id esolu ion used
in Balu e al. ( 2023a ) o compu e eioniza ion ( L
pixel
∼0 . 3 cMpc)
and accoun ing o he SFR wi hin mini-haloes. We highligh ha
ou L
pixel
is qui e small compa ed o he smoo hing adius R adop ed
by Mu
˜
noz ( 2023 ), hence we expec a signi ican sca e in he abo e
ela ion. We also spli he con ibu ion be ween Pop. III and Pop. II
s a s (a chemically en iched mini-halo will o m Pop. II s a s). In he
le panel o Fig. 1 we show ou Pop. III SFR dis ibu ion as a unc ion
o he o e densi y δand wi h he g e y line we highligh he SFR
∝ e
δR
ela ion as in Mu
˜
noz ( 2023 ). Despi e he signi ican sca e o
he easons ou lined abo e ( σ∼0 . 65), he analy ical app oxima ion
ag ees wi h ou esul s. The esul s shown he ea e a e ob ained
om ou iducial simula ion in V24.
We s a by in es iga ing he dis ibu ion unc ion o log
10
(SFR)
a a ixed o e densi y δand edshi log
10
(SFR( δ, z)) inding ha i
ollows a Gaussian dis ibu ion (o logno mal in he linea space).
We show esul s o log
10
(SFR
III
) and selec ed alues o δin he
smalle panels in Fig. 1 . Hence, we can w i e:
( log
10
(SFR
MC
| δ, z)) = A ( δ, z) e
( log
10
(SFR) −log
10
( SFR ( δ,z)))
2
2 σ( δ,z)
2 , (4)
whe e he no maliza ion A , he mean log
10
( SFR ) and he s anda d
de ia ion σall depend on he o e densi y and edshi . The no mal-
iza ion is de ined as he a io be ween he numbe o s a o ming
pixels and he o al numbe o pixels. We ound he bes - i ing
pa ame e s o each δ(g ouped in bins o wid h = 0.1) and snapsho
o he simula ion. We es ed whe he he log
10
(SFR) dis ibu ion
unc ion is indeed Gaussian by conduc ing a K-S es . P - alues
a e calcula ed o each cell wi h SFR > 0 and aking a p ede ined
signi icance le el o 0.05 below which he null hypo hesis will be
ejec ed. Resul s a e shown in Fig. 2 o bo h SFR
III
and SFR
II
. P -
alues al w ays exceed he signi icance le el o bo h Pop. III (le
panel) and Pop. II ( igh panel) SFR sugges ing ha he Gaussian
dis ibu ion ep oduces ( log
10
(SFR
MC
| δ, z)) bo h in he Pop. III
and Pop. II cases. As expec ed, we see ha he e a e a mo e Pop. III
s a o ming pixels han Pop. II ones as mini-haloes a e mo e likely
o o m Pop. III s a s.
The nex s ep is o s udy how he mean, s anda d de ia ion, and
no maliza ion e ol e wi h δand z. In Fig. 3 , we show he edshi
e olu ion o hese pa ame e s o δ= 0.5 (black), 1.0 (g ey), 1.5
(pu ple), 2.0 ( ed), 2.5 (g een), and 3.0 (blue). SFR
MC , III
exhibi s an
almos cons an end in edshi and a co ela ion wi h δ(highe
δ esul s in highe SFR
MC , III
). This demons a es ha SFR
MC , III is
mos ly de e mined by he numbe o Pop. III s a o ming haloes in a
pixel, which is highe o mo e o e dense e gions. Since Pop. III s a
o ma ion episodes in mini-haloes a e o en he i s episode o s a
o ma ion expe ienced by a galaxy, i is no impac ed by supe no a
eedback
2
so he Pop. III SFR is almos cons an a all edshi . This
also explains why σis cons an o all z and δ( σMC , III
∼0 . 65). The
pa ame e ha is mo e sensi i e o bo h δand z is he no maliza ion.
F o e y o e dense e gions ( δ≥2) i is almos one, meaning ha
almos all he o e dense pix els hos Pop. III s a o ma ion mini-
haloes. Fo lowe δ he e is also an e olu ion in z as, wi h cosmic
ime, lowe densi y egions will hos a la ge numbe o mini-haloes
abo e he minimum mass o s a o ma ion. We epea ed he same
analysis o Pop. II s a o ming pixels (see Fig. 4 ). In his case,
he e olu ion is mo e sca e ed as Pop. II s a o ma ion episodes
2
E en hough supe no a eedback can be neglec ed, his is no ue o he
Lyman–We ne backg ound ha hal s s a o ma ion in mini-haloes. Fo his
eason we add a u he condi ion ha i a pixel is i adia ed by a LW
lux J
LW
≥J
c i abo e a c i ical h eshold de ined as M
c i , MC
= M
a o ( he
a omic cooling h eshold), (log
10
(SFR
MC
)) = 0. M
a o
is he i ial mass
co esponden o a halo wi h a i ial empe a u e T
i
= 10
4
K. While M
c i , MC
is de ined in equa ion ( A2 ).
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21 cm PS wi h Pop. III s a o ma ion 487
MNRAS 540, 483–497 (2025)
Figu e 1. Le panel shows he densi y dis ibu ion o Pop. III s a o ma ion a e in mini-haloes (M
y
−1
in loga i hm scale) e sus he da k ma e o e densi y
δ o each pixel a z = 15. The hick g ey line shows he analy ical i SFR ∝ e
δsimila o he one adop ed by Mu
˜
noz ( 2023 ) oge he wi h he 1 σde ia ion ( hin
lines). Fo di e en alues o δ(highligh ed wi h he black ec angles) we show he dis ibu ion o Pop. III SFR in mini-haloes (in loga i hm scale) oge he wi h
he bes Gaussian i .
Figu e 2. P - alue dis ibu ion o K-S es s conduc ed on he Pop. III (le )
and Pop. II ( igh ) s a o ming pixels. Ve ical line highligh s he signi ican
le el o 0.05.
a e no he i s s a o ming episodes wi hin a galaxy and so will
be a ec ed by bo h mechanical and chemical eedback om he
p e ious his o y o he galaxy. This is also demons a ed by he la ge
s anda d de ia ion ( σMC , II
∼0 . 8). The a e age alue o SFR
MC , II
is
∼1 o de o magni ude highe han o Pop. III. This e lec s he
highe Pop. II s a o ma ion e iciency . Finally , A
II
is al w ays smalle
han A
III
showing ha i is less likely o a mini-halo o o m Pop. II
s a s.
We es his pa ame iza ion on he small box by es ima ing he
mini-halo con ibu ion o he Pop. III and Pop. II SFRD om he
ma e densi y ield. To do his, we ead he densi y g id a each z
and o each pixel assign a alue o Pop. III SFR d awn andomly
om ( log
10
(SFR
MC , III
)) using δo he pixel. We epea he same
p ocedu e o SFR
MC , II adding he cons ain ha in o de o ha e
SFR
MC , II
> 0 ha pixel needs o al eady ha e expe ienced a Pop. III
s a o ma ion episode. This la e condi ion ensu es a mo e ealis ic
en ichmen model (a pixel canno ha e Pop. II s a o ma ion i i has
no p e iously hos ed Pop. III s a o ma ion). We an wen y di e en
ealiza ions and o each ealiza ion es ima ed he SFRD
MC , III and
SFRD
MC , II and compa ed wi h esul s o he simula ions. Resul s
a e shown in Fig. 5 . In he uppe panels, we show he Pop. III (le )
and Pop. II ( igh ) SFRD om MERAXES ou pu (black line) and om
each ealiza ion using he me hod ou lined abo e (c yan shaded lines).
In he lowe panels, we show he a io be ween he a e age o he 20
ealiza ions and he ue SFRD om MERAXES . All he ealiza ions
a e in easonable ag eemen wi h he da a and he a io is al w ays
lowe han ∼10 pe cen o bo h popula ions. This demons a es he
alidi y o he me hod ou lined abo e.
The main ad an age o his me hod is ha i allows es ima ion
o he SFRD om mini-haloes in a simula ion whe e hese a e no
di ec ly esol ed. P a ame izing SFRD wi h a Gaussian dis ibu ion
enables us o accoun o s ochas ic s a o ma ion (di e en pixels
wi h same δcan ha e di e en SFR), wi hou losing he co ela ion
wi h he ma e densi y ield ( SFR , σand A all depend on δ). This
me hod can be applied as long as he densi y ield om bo h he
low- and he high- esolu ion simula ion sha e he same p ope ies
(mean, s anda d de ia ion) and i can be calib a ed o any choice o
pa ame e s.
3.2 Applying scaling ela ions o he L210 box
We can now apply he me hodology ou lined in he p e ious sec ion o
he L210 box ha can only esol e he a omic cooling haloes by
eading he densi y ield and applying ( log
10
(SFR
MC
)) calib a ed
on di e en models. In Fig. 6 , he mini-halo con ibu ion o he
Pop. III SFRD shows a good ag eemen be ween he wo di e en
simula ions.
We highligh ha ou scaling ela ions do no explici ly depend
on he Lyman–We ne backg ound (excep o he egions ha a e
s ongly i adia ed by LW lux o which (log
10
(SFR
MC
)) = 0).
This implici ly assumes ha bo h he L10 and L210 box ha e
simila LW backg ounds. We e i y his assump ion by compu ing
he a e age LW backg ound and LW maps in bo h simula ions (see
Fig. 7 ). In he bo om igh panel, we show he edshi e olu ion
o he LW backg ound (in uni s o 10
−21
e g s
−1
cm
−2
Hz
−1
s
−1
) in
he L10 (black line) and in he L210 box ( ed line). The wo lines
sha e simila ends showing ha bo h simula ions ha e a simila
a e age LW backg ound. The le and op igh panels show he 2D
p ojec ions o he LW ield in he la ge and small box. These maps
illus a e ha he LW backg ound is oughly uni o m (as expec ed
gi en ha he mean ee pa h o LW pho ons is ∼100 Mpc). These
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488 E. M. Ven u a e al.
MNRAS 540, 483–497 (2025)
Figu e 3. Redshi e olu ion o log
10
( SFR ) ( op panel), σ(mid), and A
(bo om) o Pop. III o δ= 0.5 (black), 1.0 (g ey), 1.5 (pu ple), 2.0 ( ed),
2.5 (g een), and 3.0 (blue).
Figu e 4. Same as Fig. 3 o Pop. II.
plo s demons a e ha he LW ields in he small and la ge box a e
indeed compa able showing ha he es ima ion o he s a o ma ion
in mini-haloes wi h he scaling ela ions accoun s o he adia i e
eedback.
3
Di e en ly om wha has been done by Hazle e al. ( 2024 ) who
calib a ed a semi-analy ical model o he Reinassance simula ion in
o de o accoun o Pop. III s a o ma ion by adding he p e ious
s a o ma ion his o y o each a omic cooling galaxy esol ed in
he simula ion, he me hodology desc ibed in his sec ion allows us
3
In his discussion, we neglec ed he UV pho o-ionizing eedback. This is
jus i ied by he ac ha his e ec impac s mo e he low-mass a omic cooling
haloes a z 10 a he han mini-haloes.
Figu e 5. ( op) Pop. III (le ) and Pop. II ( igh ) SFRD e sus z om
MERAXES
(black) and es ima ed om he densi y ield (cyan). (bo om) a io
be ween he a e age o he 20 ealiza ions o he SFRD es ima ed om he
densi y ield and he SFRD om MERAXES .
Figu e 6. SFRD
MC , III
e sus z om he L10 (solid) and L210 (do ed) box
o wo di e en Pop. III s a o ma ion models (see mo e de ails in ex and
Table 4 ).
o es ima e only he o al SFRD occu ing in mini-haloes wi hin
a ce ain pixel o he simula ion. In his wo k, we ocus on h ee
models o Pop. III s a o ma ion by a ying h ee main pa ame e s:
he s a o ma ion e iciency, he IMF, and he speci ic Pop. III X- ay
luminosi y pe uni s a o ma ion while all he o he ee pa ame e s
(e.g. he escape ac ion) a e ixed a he iducial alue (see Tables 1
and 2 ). We chose o ocus only on hese h ee pa ame e s since hese
ha e he s onges impac on bo h he e olu ion o he Pop. III SFRD
and he amoun o UV and X- ay pho ons emi ed. The alues chosen
o he speci ic Pop. III X- ay luminosi y pe uni s a o ma ion a e
simila o hose ound in Sa o io e al. ( 2023 ) o he di e en
IMFs explo ed in hei wo k. He ea e , we analyse h ee di e en
Pop. III models designed o ha e he minimum, in e media e, and
maximum impac om Pop. III s a o ma ion in mini-haloes, each
model is summa ized in Table 4 . The IMFs conside ed in his model
a e a Salpe e be ween 1 and 500 sola masses and a logno mal IMF
cen ed a 60 M
(see V24 o mo e de ails). We highligh ha in
ou ex eme Pop. III model we enhance s a o ma ion e iciency,
L
X
/SFR and he op-hea iness o he IMF a he same ime. Each o
hese pa ame e s has a di e en impac on he e olu ion o he 21 cm
signal (see Appendix B o a mo e de ailed discussion o how each
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21 cm PS wi h Pop. III s a o ma ion 489
MNRAS 540, 483–497 (2025)
Figu e 7. Le panel shows he 2D p ojec ions o he LW backg ound (uni s o 10
−21
e g s
−1
cm
−2
Hz
−1
s
−1
in he L210 box a z = 15. Top igh panel shows
he same map bu in he L10 simula ion. Bo om igh panel shows he edshi e olu ion o he a e age LW backg ound (same uni s as abo e) in bo h he L10
(black line) and L210 ( ed line) simula ion.
Table 4. Pop. III model pa ame e s.
Label IMF ype
a αSF , III L
X < 2keV , II
/ SFR
Weak Pop. III Salpe e 0.008 3 ×10
40
Mode a e Pop. III Salpe e 0.008 3 ×10
41
Ex eme Pop. III logE 0.08 3 ×10
42
High SFE Salpe e 0.08 3 ×10
40
LogE logE 0.008 3 ×10
40
a
See Table 2 in V24 o he de ails.
Pop. III pa ame e changes he e olu ion o he 21 cm global signal
and powe spec um).
4 IMPACT OF POP. III STAR FORMATION ON
21 CM PHYSICS
We can now es ima e how di e en Pop. III s a o ma ion models in
MERAXES a ec he 21 cm signal. We s a ed by e i ying ha , a e
in oducing he addi ional Pop. III con ibu ion o he iducial Pop. II
only model (Balu e al. 2023a ) we s ill ob ain eioniza ion his o ies
consis en wi h he obse a ional cons ain s on he Thomson sca e -
ing op ical dep h τe
(Fig. 8 ) and
¯
x
HI
(Fig. 9 ). The eioniza ion his o ies
om he models wi h Pop. III s a s a e only sligh ly modi ied and
his negligible con ibu ion comes om he seconda y ioniza ions
om X- ays. This is expec ed gi en ha a z ≤15 he Pop. III SFRD
is a leas one o de o magni ude lowe han Pop. II and hei main
con ibu ion is expec ed om he X- ay emission a he han he UV.
We compu e he sk y-a e aged 21 cm global signal (see Fig. 10 )
wi hou (black line) and wi h (g e y, c yan, and ed line o weak,
mode a e and ex eme Pop. III espec i ely) Pop. III s a o ma ion.
Figu e 8. In eg a ed Thomson sca e ing op ical dep h τe compu ed o
model wi h weak (g ey), mode a e (cyan), ex eme ( ed) Pop. III, and
Balu e al. ( 2023a ) (black). The g een cu e and shaded egion show he
measu emen o τe
om he Planck 2018 collabo a ion (Planck Collabo a ion
VI 2020 ).
As expec ed, in oducing a Pop. III popula ion wi h he same X-
ay p ope ies as Pop. II ones (i.e. weak Pop. III), simply shi s
he abso p ion h ough o ea lie epochs in i ue o he s onge
coupling a highe - z (see also Hegde & Fu lane o 2023 ; Ven u a
e al. 2023 ). Ho we e , i Pop. III s a s ha e a s onge X- ay emission
(i.e. mode a e and ex eme models) as sugges ed by Sa o io e al.
( 2023 ), he abso p ion signal is quickly supp essed u ning in o an
emission signal as ea ly as z ∼18 o he ex eme Pop. III model and
a z ∼13 o he mode a e Pop. III one. We no e ha a simila esul
has been ound by a con empo aneous wo k by Gessey-Jones e al.
( 2025 ) who ound an analogous a ia ion in he iming ( z ∼3) and
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490 E. M. Ven u a e al.
MNRAS 540, 483–497 (2025)
Figu e 9. Cons ain s on he eioniza ion his o y (neu al hyd ogen ac ion
esus z ) o model wi h weak (g ey), mode a e (cyan), ex eme ( ed) Pop. III,
and Balu e al. ( 2023a ) (black). The obse a ional da a a e om analyses o
da k pixels (McG ee e al. 2015 ; Jin e al. 2023 ), damping-wing abso p ion
in quasa spec a (Ba
˜
nados e al. 2018 ; Da ies e al. 2018 ; Wang e al. 2020 ;
G eig e al. 2022 ; Spina e al. 2024 ) and equi alen wid h measu emen s
(Mesinge e al. 2014 ; Hoag e al. 2019 ; Mason e al. 2019 ; Jung e al. 2020 ;
Whi le e al. 2020 ).
Figu e 10. E ec o Pop. III s a o ma ion on he 21 cm global signal ( δT
b
e sus z). Pop. III models wi h small X- ay hea ing cause a s onge abso p ion
a ea lie imes, while ha ing a s onge Pop. III X- ay hea ing causes he
signal o be seen in emission ea lie . Colou coding as in he p e ious igu es.
B own and yellow dashed lines a e aken om Gessey-Jones e al. ( 2025 ) o
a Salpe e ( Sal ) and la ( In -0 ) IMF.
dep h ( δT
b
∼50 mK) o he abso p ion ough when conside ing
a s onge X- ay con ibu ion om Pop. III s a s (in hei model
he L
X
/SFR is sel -consis en ly modelled om he IMF so ha
he di e ence be ween hei In -0 and Sal model is o 2 o de s o
magni udes.)
As shown in Fig. 11 ea lie coupling and hea ing om Pop. III
impac s he 21 cm powe spec um bo h a he la ge and small scales.
The models conside ed he e p oduce 21 cm signals ha a e di e en
no only du ing he coupling and hea ing epoch ( z ∼10 −20), bu
also a lowe edshi when he eioniza ion is in p og ess. The impac
is s onge a smalle scales whe e models wi h a s onge hea ing
exhibi a la ge powe spec um a z ∼7 −10. This is o c ucial
impo ance as cu en and upcoming acili ies a e imp o ing hei
obse a ions a z ≤10 (see mo e discussion in Sec ion 5 ).
We compa e ou esul s wi h he ones in Mu
˜
noz e al. ( 2022 ) who
s udied ho w di e en Pop. III pa ame e s impac he 21 cm signal
using 21cm FAST . In he le panel o Fig. 11 , he yellow and b own
dashed lines a e ob ained by changing he speci ic Pop. III X- ay
luminosi y by a ac o o 9. The global ends a e simila wi h ou
models wi h s onge Pop. III X- ay emission (dashed yellow line and
ed/cyan lines) showing an ea lie and weake i s peak compa ed
o low X- ay models (dashed b own and g ey lines). In hei model,
di e en Pop. III X- ay p ope ies s ongly impac he posi ion and
ampli ude also o he second peak. In ou models ins ead, he posi ion
o he second peak is only sligh ly an icipa ed in he s ong X- ay
models. This di e ence is likely due o he ac ha ou second peak
occu s a much lowe edshi ( z ∼10) when mos o he emission
comes om Pop. II s a s. Finally, di e en ly om Mu
˜
noz e al.
( 2022 ) a z ∼10 simula ions wi h a s onge Pop. III X- ay hea ing
ha e a signi ican ly la ge powe spec um compa ed o weak X- ay
models (a simila esul has been ound also by Gessey-Jones e al.
2025 ).
This analysis shows ha Pop. III s a o ma ion no only impac s
he 21cm global signal du ing he cosmic dawn as p e iously assessed
(e.g. Qin e al. 2020 ; Gessey-Jones e al. 2022 ; Hegde & Fu lane o
2023 ; Ven u a e al. 2023 ; C uz e al. 2024a ) bu also ha an ea ly
( z ≥15) hea ing o he IGM p o ided by his popula ion lea es a
s ong signa u e on he powe spec um du ing he EoR ( z ≤10).
The impac on he powe spec um is s onge o models ha ha e
a la ge Pop. III X- ay emissi i y (i.e. mode a e and ex eme Pop. III
models) while models wi h a milde X- ay emission (i.e. weak Pop.
III) ha e a s onge e ec on he global signal. Ul ima ely, his ells
us ha he powe spec um du ing he EoR can be used o disen angle
di e en hea ing models and po en ially cons ain he p ope ies o
Pop. III s a s. We highligh ha he e we conside ed only he X- ay
emission om s ella emnan s. While he e migh be o he sou ces
ha signi ican ly hea he IGM a z ≥15 (see discussion a end o
Sec ion 2.3 ), he o he e ec s a e likely o be ei he subdominan ,
s ill ela ed o Pop. III s a s (e.g. cosmic ays) o dominan only a
he da k-ages (e.g. da k ma e annihila ion).
Finally, we no e ha in his wo k we did no include he e ec o he
X- ay eedback on Pop. III s a o ma ion. As no ed by Rico i ( 2016 )
and Pa k, Rico i & Sugimu a ( 2021 ), X- ays ha e bo h a posi i e
and ne ga i e e ec on Pop. III s a o ma ion as he y hea he gas and
inc ease he elec on ac ion. The hea ing makes gas acc e ion mo e
di icul (hence delaying s a o ma ion) and ee elec ons p omo e
he o ma ion o H
2
making he molecula cooling mo e e icien . In
p esence o a s ong X- ay backg ound his la e e ec is dominan
a 10 z 20 (see e.g. ig. 9 in Hegde & Fu lane o 2023 ). Hence,
including he X- ay eedback would likely inc ease he Pop. III SFRD
making he impac o Pop. III s a s e en s onge han wha p edic ed
he e especially o he mode a e and ex eme models.
5 OBSERVABILITY WITH SKA
In he p e ious sec ion, we showed ha an ea ly hea ing o he
IGM signi ican ly a ec s he 21cm powe spec um du ing he EoR.
Now we wan o in es iga e how ou models compa e wi h he
cu en ly a ailable uppe limi s and whe he hei di e ences will
be obse able wi h SKA. In his sec ion, we conside he powe
spec um a z ≤10 as he cu en and upcoming in e e ome e s a e
signi ican ly less sensi i e a highe edshi . To be e app ecia e his,
in Fig. 12 we show he powe spec um noise (mK
2
) wi h SKA as
a unc ion o edshi a k ∼0 . 2 and 0.9 cMpc
−1
assuming a 1000 h
(solid lines) and 180 h (dashed lines) obse a ions wi h SKA. A
highe edshi he noise s eadily inc eases and al eady a z ≥10 he
noise is o he o de o 10s o 100s mK
2
wi h 1000 h obse a ion.
Fi s , we compa e ou p edic ions wi h cu en uppe limi s om
a numbe o acili ies including he Mu chison Wide ield A ay
(MWA), LOw-F equency AR ay (LOFAR), Gian Me ewa e Radio
Telescope (GMRT), P ecision A ay o P obing he Epoch o
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21 cm PS wi h Pop. III s a o ma ion 491
MNRAS 540, 483–497 (2025)
Figu e 11. E ec o Pop. III s a o ma ion on he 21 cm powe spec um (
21
e sus z ) a la ge (k ∼0 . 1 Mpc
−1
) and small (k ∼0 . 9 Mpc
−1
) scales. Colou
coding as in he p e ious igu es. B o wn and yello w dashed lines a e aken om Mu
˜
noz e al. ( 2022 ) o k = 0 . 23 Mpc
−1
assuming a weak and s ong X- ay
luminosi y pe uni o Pop. III SFR (bo om igh panel o ig. 17).
Figu e 12. 21 cm powe spec um sensi i i y as a unc ion o edshi k ∼0 . 2
(black line) and 0.9 cMpc
−1
(o ange line) assuming a 1000 (solid lines) and
180 h (dashed lines) obse a ion wi h SKA.
Reioniza ion (PAPER) and Hyd ogen Epoch o Reioniza ion A ay
(HERA) (we ocus a z = 7 −10 whe e mos o he measu emen s
ha e been aken). In Fig. 13 , we show he 21 cm powe spec um
a z = 10, 9, 8, and 7 o all he ou models oge he wi h he
a ailable uppe limi s (see label). All ou models a e below hese
uppe limi s. Ho we e , he mode a e and e x eme Pop. III models
a e close o he cu en HERA cons ain s a z = 8 sugges ing
ha soon hese models can be ei he de ec ed o uled ou . In ou
model he main e ec o di e en Pop. III models is o change he
iming and ampli ude o he peaks in he 21 cm powe spec um
a he han in oducing a speci ic ea u e. We no e ha we did no
accoun o spa ial a ia ions in he eloci y acous ic oscilla ions
(see Sec ion 2.1 ) which would in oduce wiggles in he
2
21
a scales
k ∼0 . 1 Mpc
−1
(C uz e al. 2024a ). While his e ec can be impo an
when ocusing on he 21 cm powe spec um du ing he coupling and
hea ing epochs, hese luc ua ions a e quickly washed ou a z 13
(see Fig. 13 and sec ion 6C in C uz e al. 2024a ).
We nex conside he upcoming SKA. In o de o es ima e he
obse abili y o he ou models analysed so a , we pe o m a
simila analysis as in Balu, G eig & Wyi he ( 2023b ) ha we b ie ly
summa ize he ea e . The sensi i i y o a adio in e e ome e is
mos ly egula ed by he he mal noise (
N
) and he cosmic a iance
wi h he o me domina ing he noise a small scales and he la e
a la ge scales. The he mal noise is ela ed o he bandwid hs o
he ins umen , beam ac o (see Pa sons e al. 2014 ), he in eg a ion
ime o he mode k and he empe a u e o he sys em (gi en as he
sum o he sky and ecei e empe a u e) (Mo ales 2005 ; McQuinn
e al. 2006 ; Pa sons e al. 2012 ). We can hence w i e he o al noise
by summing hese wo componen s:
1
σ[
2
21
( k)]
2
=
i
1
2
N
+
2
21
2
. (5)
By doing so, we a e e ec i ely assuming ha he e o s a e Gaussian
dis ibu ed, which is easonable o he ele an scales in his
wo k (Qin e al. 2021a ; P elogo i
´
c & Mesinge 2023 ). Finally,
a 21 cm de ec ion is hea ily limi ed by he abili y o emo ing
he o eg ounds. We used he py hon package 21CMSENSE
4 (Pobe
e al. 2013 , 2014 ) which, gi en he speci ics o an in e e ome e ,
a mock 21 cm powe spec um and an obse a ional campaign,
compu es he in e e ome e sensi i i y o he 21 cm powe spec um
unde di e en assump ions o o eg ound emo als. We used he
assump ion ‘mode a e’ o eg ound emo als and we ocused on he
i s phase o SKA (i.e. SKA1-low), in pa icula we included he
s a ions in he ‘Cen al A ea’ o he SKA1-low,
5 esul ing in 296
s a ions o diame e 35 m dis ibu ed ac oss a ci cula a ea wi h
1.7 km diame e . We assumed wo di e en obse a ional campaigns:
4
h ps:// gi hub.com/ asg-a ilia es/ 21cmSense
5
See he o icial SKA1 Sys em Baseline Design documen in h ps://www.
skao.in /en o u he de ails.
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