Semi-analytical modelling of Pop. III star formation and metallicity evolution - II. Impact on 21 cm power spectrum

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, p o ided he o iginal wo k is p ope ly ci ed. Downloaded om h ps://academic.oup.com/mn as/a icle/540/1/483/8123416 by gues on 08 July 2025 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 Downloaded om h ps://academic.oup.com/mn as/a icle/540/1/483/8123416 by gues on 08 July 2025 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 Downloaded om h ps://academic.oup.com/mn as/a icle/540/1/483/8123416 by gues on 08 July 2025 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 ). Downloaded om h ps://academic.oup.com/mn as/a icle/540/1/483/8123416 by gues on 08 July 2025 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 Downloaded om h ps://academic.oup.com/mn as/a icle/540/1/483/8123416 by gues on 08 July 2025 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 Downloaded om h ps://academic.oup.com/mn as/a icle/540/1/483/8123416 by gues on 08 July 2025 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 Downloaded om h ps://academic.oup.com/mn as/a icle/540/1/483/8123416 by gues on 08 July 2025 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 Downloaded om h ps://academic.oup.com/mn as/a icle/540/1/483/8123416 by gues on 08 July 2025 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. Downloaded om h ps://academic.oup.com/mn as/a icle/540/1/483/8123416 by gues on 08 July 2025