arXiv:1810.06273v3 [astro-ph.CO] 24 Dec 2018 MNRAS n EA( HERA and 2015 ( PAPER ( (

ua Majumdar Suman aehMondal Rajesh mean the of history evolution fraction Hydrogen the determine neutral to method A rnil,mpteH the map principle, eim(G)i Dwt h ieo ih LS axis ⋆ (LoS) observer’s sight an of However, line redshift). the (or with frequency the 3D being in (IGM) medium 7Dcme 2018 December 27 feoti newyt eetteER2-msig- 21-cm interferometric EoR radio ( the upcoming GMRT detect and e.g. to ongoing amount experiments underway considerable using is A nal (EoR). effort reionization of of epoch the bevtoso h esitd2-msga rmneu- from signal 21-cm (H redshifted hydrogen the tral of Observations INTRODUCTION 1 c 2 3 4 5 1 omne l 2013 al. 2013 et Bowman al. et Haarlem van E-mail: eateto hsc etefrTertclSuis Ind Studies, Theoretical for Centre & Physics of Department eateto hsc,Peiec nvriy 61Colleg 86/1 University, Presidency Physics, of Department eteo srnm,Ida nttt fTcnlg Indore Technology of Institute Indian Astronomy, of Centre eateto hsc,Bakt aoaoy meilColl Imperial Laboratory, Blackett Physics, of Department srnm ete eateto hsc n srnm,Uni Astronomy, and Physics of Department Centre, Astronomy 08TeAuthors The 2018 sn h esitdH redshifted the Using ,SA( SKA ), 000 [email protected] asn ta.2014 al. et Parsons , eore l 2017 al. et DeBoer 1 elm ta.2013 al. et Mellema – 5 21)Pern 7Dcme 08Cmie sn NA L MNRAS using Compiled 2018 December 27 Preprint (2018) i r h otpoiigpoeof probe promising most the are ) ; igye l 2013 al. et Tingay 1 i ⋆ ; itiuini h intergalactic the in distribution ont Bharadwaj Somnath , 4 aaat ta.2013 al. et Yatawatta , ; aiae l 2013 al. et Paciga 5 i l ta.2015 al. et Ali bns .Shaw K. Abinash , ). 1c inloecn in can, one signal 21-cm h ut-rqec nua oe pcrm(MAPS) spectrum power angular multi-frequency The eododrsaitc of statistics second-order utain ln l h he pta ietosecddin encoded directions spatial homogene three statistically the the all also along and evolution fluctuations cosmological the to due hntema yrgnnurlfraction neutral hydrogen o Epoch mean the the during pronounced when particularly is effect The observer. ipemdlweetesseai frequency systematic the where model simple a h ih-oe(C ffc mrnstecsooia vlto fthe of evolution cosmological the imprints signal effect (LC) light-cone The ABSTRACT eosrt hti spsil orcvrterinzto itr a history reionization value the the recover have to we provided possible bandwidth is observational it that demonstrate e words: Key input. external eemn h eoiainhsoy osdrn Csmlto fthe of simulation elements diagonal LC the a use Considering we history. signal, reionization the determine ag-cl tutr fUies bevtos–mtos st methods: – observations – Universe of structure large-scale – nieydet h vlto of evolution the to due entirely ; opase l 2015 al. et Koopmans T ; b ilne l 2014 al. et Dillon ( n ˆ ν , ; ) aose al. et Jacobs omlg:ter akae,rinzto,fis tr iuer diffuse – stars first reionization, ages, dark – theory cosmology: ,LOFAR ), ln h rqec xswihi h ieo ih LS ieto fan of direction (LoS) sight of line the is which axis frequency the along ,MWA ), g,Lno W A,UK 2AZ, SW7 London ege, tet okt 003 India 700073, Kolkata Street, e ), irl noe435,India 453552, Indore Simrol, , ) a nttt fTcnlg hrgu,Kaapr710,I 721302, Kharagpur Kharagpur, Technology of Institute ian 2 2 T la .Iliev T. Ilian , est fSse,Biho N9H UK BN19QH, Brighton Sussex, of versity b n na .Sarkar K. Anjan and ( n ˆ n t aiu ttsis hshsbe ae noac- into taken been signal has 21-cm EoR This the statistics. effect by on count various (LC) impact ‘light-cone’ its significant the and to a rise has gives which This (LoS). sight nicuigteRDi o iuain by simulations EoR in invested RSD effort substantial the been including has in there Although, LoS. ( ( correl two-point function. the in tion anisotropies LC the modelling while light- backward the to H restricted the is and universe cone, the of view h rbe fhwt rpryicuetepcla veloci- peculiar the include properly to how of problem the nte2-msga ( signal 21-cm the in hf pc itrin(S)det h euirvelocities peculiar the to H due (RSD) distortion space shift interferometers. radio observable of primary generation the first is the which of spectrum power 3D 21-cm ν , 2012 2014 x ¯ i H iia oteL ffc S nrdcsanisotropies introduces RSD effect LC the to Similar . ) i nte motn ieo ih LS ffc stered- the is effect (LoS) sight of line important Another osdrn ohtesseai aito along variation systematic the both considering ( ); EoR the on effect this of impact the examined have ) ν ) audre al. et Majumdar hspoie e ehdt observationally to method new a provides This . ν 1 akn Loeb & Barkana = x ¯ ν at tal. et Datta H 2 i ( of 1 i ν aa .Datta K. Kanan , ) ( 1c inleovsaogteln of line the along evolves signal 21-cm ν C al ail steuies evolves. universe the as rapidly falls 1 ℓ ν , ( ν hrda l 2004 Ali & Bharadwaj 2 1 ( ) ν , x ¯ 2013 ( H eedneof dependence ( C 2012 2 2006 i ℓ 2 ) ( tasnl rqec san as frequency single a at , ν ovldt u oe.We model. our validate to 1 , n 2016 ˆ and ) ν , 2014 and 2 atistical. ) ); eoiain(EoR) Reionization f unie h entire the quantifies and ) aaae al. et Zawada esne al. et Jensen ν eew propose we Here . u n isotropic and ous esitd21-cm redshifted A rs h entire the cross T C aPat tal. et Plante La E ℓ tl l v3.0 file style X ( 3 ν ln the along ) , o 21-cm EoR 1 ν , a tal. et Mao ndia adiation 2 ) ( ( 2013 arises 2014 a- of ), ν ) L2 R. Mondal et al. ties of H i in LC simulation was addressed by Mondal et al. ulations are represented by particles whose H i masses were (2018). calculated from the neutral Hydrogen fraction xH i interpo- The statistical homogeneity (or ergodicity) along the lated from its eight adjacent grid points. The positions, pe- LoS gets destroyed by the LC effect. One of the main prob- culiar velocities and H i masses of these particles are then lems regarding the interpretation of the EoR 21-cm sig- saved for each such coeval cube. nal through its 3D power spectrum P(k) lies in the sig- To construct the LC map, we slice the coeval maps at nal’s non-ergodic nature. The 3D power spectrum P(k) 25 different radial distance ri, and construct the LC map assumes that the signal is ergodic and periodic, thus it for the region between ri to ri+1 with the H i particles from provides a biased estimate of the statistics of EoR signal corresponding slices of the coeval snapshot. Finally, we map (Trott 2016). In contrast, the multi-frequency angular power the H i particles within the LC box from r = rnˆ to observ- spectrum (hereafter MAPS) Cℓ(ν1, ν2) (Datta et al. 2007) ing frequency ν and direction nˆ which are the appropriate does not have any such intrinsic assumption in its defini- variables for the observations of redshifted 21-cm brightness tion. Mondal et al. (2018) have demonstrated that the en- temperature fluctuations δTb(nˆ, ν) in 3D. Note that for this tire second-order statistics of the non-ergodic LC EoR signal mapping, the cosmological expansion and the radial com- can be expressed by the MAPS. ponent of the H i peculiar velocity nˆ · v together determine In this Letter, we demonstrate, as a proof of concept, the observed frequency ν for the 21-cm signal originating how one can use the intrinsic non-ergodicity of the light-cone from the point nˆr. Our LC box is centered at the co-moving EoR 21-cm signal to uncover the underlying reionization his- distance rc = 9151.53 Mpc (νc = 157.78 MHz) which corre- tory. The reionization history is one of the most sought-after spond to the redshift ≈ 8. The mass-averaged H i fraction outcomes of any experiment aiming to observe the EoR. We x¯H i at the centre of the LC simulation is ≈ 0.51, and it propose and validate a formalism whereby the measured changes from x¯H i ≈ 0.65 (at farthest end) to x¯H i ≈ 0.35 MAPS can be used to extract the reionization history in (at nearest end), following the reionization history Fig. 2 of a model independent manner. In this Letter, we have used Mondal et al. (2018). the Planck+WP best fit values of cosmological parameters (Planck Collaboration et al. 2014). 3 MODELLING THE MULTI-FREQUENCY ANGULAR POWER SPECTRUM 2 SIMULATING THE LIGHT-CONE 21-CM The issue under consideration here is ‘How to quantify the SIGNAL FROM THE EoR statistics of the non-ergodic EoR 21-cm signal δTb(nˆ, ν) in In this section, we briefly summarise the simulation tech- 3D?’. We know that the LC effect makes the cosmological n nique used for generating the light-cone EoR 21-cm sig- 21-cm signal (δTb(ˆ, ν)) evolve significantly along the LoS nal. The reader is referred to the Section 2 of Mondal et al. direction ν. The 3D power spectrum P(k) is not accurate (2018) for a detailed description of the simulations. Here, we when the statistical properties of the signal evolve along a have considered a region that spans the comoving distance specific direction. Additionally, the Fourier transform im- range rn = 9001.45 Mpc (nearest) to rf = 9301.61 Mpc (far- poses periodicity on the signal, an assumption that cannot thest), which correspond to the frequencies νn = 166.91 Mhz be justified along the LoS when the LC effect has been taken and νf = 149.04 Mhz, respectively. We have simulated snap- into account. As a consequence, the 3D power spectrum fails shots of the H i distribution (coeval cubes) at 25 different to quantify the entire information in the signal and gives a co-moving distances ri in the aforesaid r range (see Fig. 2 biased estimate of the statistics (Trott 2016; Mondal et al. of Mondal et al. 2018) which were chosen so that the mean 2018). In contrast to this the MAPS Cℓ(ν1, ν2) quantifies the neutral Hydrogen fraction x¯H i varies approximately by an entire second order statistics of the EoR 21-cm signal even equal amount in each interval. in the presence of the LC effect (Mondal et al. 2018). We have used semi-numerical simulations to generate The redshifted 21-cm brightness temperature fluctua- the coeval ionization cubes with comoving volume V = tions are decomposed into spherical harmonics as [300.16 Mpc]3. These simulations involve three main steps. m δTb(nˆ, ν) = aℓm(ν)Y (nˆ), (1) N ℓ First step involves a particle-mesh (PM) -body code to ℓ,m simulate the dark matter distribution. The N-body run Õ has 42883 grids with 0.07 Mpc grid spacing using 21443 and these are used to define the MAPS (Datta et al. 2007) 8 dark matter particles (particle mass 1.09 × 10 M⊙). In the using next step, a Friends-of-Friends (FoF) algorithm is used to ∗ Cℓ(ν1, ν2) = aℓm(ν1) a (ν2) . (2) identify collapsed halos in the dark matter distribution. A ℓm fixed linking length of 0.2 times the mean inter-particle This takes into account the assumption that the EoR 21- distance is used for the FoF and we have set the crite- cm signal is statistically homogeneous and isotropic with rion that a halo should have at least 10 dark matter parti- respect to different directions in the sky, however it does not cles. In the third and last step, an ionization field following assume the signal to be statistically homogeneous along the an excursion set formalism (Furlanetto et al. 2004) is pro- LoS direction ν. Considering the particular situation where duced. For this we have adopted the ionization parameters the signal is ergodic (statistically homogeneous) along the 9 {Nion, Mhalo,min, Rmfp} = {23.21, 1.09 × 10 M⊙, 20 Mpc}, LoS, we have Cℓ(ν1, ν2) = Cℓ(∆ν) i.e. it depends only on the identical to Mondal et al. (2017). This final step closely fol- frequency separation ∆ν =| ν1 − ν2 |. low the assumption of homogeneous recombination adopted Under the assumption that the H i spin temperature by Choudhury et al. (2009). The H i distribution in our sim- is much larger than the CMB temperature i.e. Ts ≫ Tγ,

MNRAS 000, 1–5 (2018) Reionization history L3

the redshifted 21-cm brightness temperature fluctuations in the regime where the flat sky approximation holds true, (eq. 4 and A5 of Bharadwaj & Ali 2005) can be expressed we have converted the comoving displacement with respect as (Mondal et al. 2018) to the centre of the box x = (x⊥, xk) to angle and frequency ′ respectively using θ = x⊥/r and ν − νc = xk/r where we ρH i H0νe ∂r Tb(nˆ, ν) = T¯0 , (3) assume that the centre of the simulation box is located at ρ¯H c ∂ν   a redshift zc = 1420 MHz/νc with corresponding comoving

where distance r and with r′ = dr/dν evaluated at ν . The resulting c 2 δe(θ, ν) is statistically isotropic in θ and ergodic in ν. The Ωbh 0.7 T¯0 = 4.0mK , (4) ergodicity along the LoS is broken by the function f (ν) which 0.02 h     we have assumed to be of the form ρ i/ρ¯ is the ratio of the neutral hydrogen density to the H H ν − νc r f (ν) = 1 − a . (8) mean hydrogen density, and refers to the comoving dis- B tance from which the redshifted H i emission, observed at   = frequency ν, is originated. The factors ρH i/ρ¯H and ∂r/∂ν Here f (ν) is a linear function which has value f (νc) 1 at both evolve along the LoS direction (ν or z) due to a vari- the centre of the frequency bandwidth B, and it has values ety of factors including the evolution of various quantities f = 1 − a/2 and f = 1 + a/2 at the nearest and furthest edges pertaining to the background cosmological model and the of the band. In principle, one can choose different forms of growth of density perturbation in ρH i. However during the f (ν). The aim here is to mimic a situation where we are EoR the evolution of the mean mass weighted neutral hydro- analysing observations of a part of the reionization history gen fraction x¯H i = ρ¯H i/ρ¯H by far dominates over the other where the evolution of the neutral fraction is approximately factors that cause Tb(nˆ, ν) to evolve along the LoS direction. linear (see Fig. 2 of Mondal et al. 2018). Different values of Based on this we propose a model a correspond to different values of the slope or equivalently E different values of the reionization rate. The different pan- C (ν , ν ) = x¯ i(ν ) x¯ i(ν ) C (∆ν), (5) ℓ 1 2 H 1 H 2 ℓ els of Fig. 1 show δTb(θ, ν) for a single realization of δe(x) E ∆ considering different values of a. where Cℓ ( ν) is ergodic along the LoS, and the factor We have used the simulated δTb(θ, ν) to estimate x¯H i(ν1) x¯H i(ν2) which accounts for the evolution of the mean hydrogen neutral fraction breaks the ergodicity along the Cℓ(ν1, ν2) in the flat sky approximation (Mondal et al. 2018). = LoS. We expect the above relation to hold at small scales Here, we focus on the diagonal elements ν1 ν2 where the where the HI density traces the underlying DM density. MAPS signal peaks. In principle, one can use the full in- However, at scales larger than the typical bubble size the formation contained in MAPS matrix Cℓ (ν1, ν2) to analyse evolution is expected to be dominated by the evolution of the results. However for simplicity we have only considered ¯ bubble sizes and the above relation may not stay valid in the diagonal terms. We have used the ratio A Cℓ(ν)/Cℓ to determine f (ν) from our simulations. Here C (ν) ≡ C (ν, ν), that regime. This will provide a handle to measure the evo- ℓ p ℓ i ¯ = −1 B/2 lution of the H neutral fraction x¯H i(z) as reionization pro- Cℓ B −B/2 Cℓ (ν) dν and A is a normalisation constant ceeds. Unfortunately this will only allow us to determine the whose value∫ has to be externally specified. Here we use the ratio x¯H i(z2)/x¯H i(z1) at two different epochs, and it will not prior information that f (νc) = 1 to decide the value of A. allow us to uniquely determine x¯H i(z1) or x¯H i(z2). For the Fig. 2 shows the ratio A Cℓ (ν)/C¯ℓ evaluated at different ℓ purpose of this Letter we consider values which all have been shown as function of ν − ν . We p c have used 10 equally spaced logarithmic ℓ bins. We find that x¯ i(z2) C (ν2, ν2) H = ℓ , (6) the ratio is independent of ℓ i.e. they all overlap. We also see x¯H i(z1) s Cℓ (ν1, ν1) that the ratio is able to correctly recover the functional form which does not uniquely determine the H i reionization his- f (ν) from the simulations shown in Fig. 1. This validates our tory. However the reionization history is uniquely specified if method of analysis. we combine these measurements with a single measurement We next apply the same method to our LC simulations to test if our model (eq. 5) actually holds for the simulated of x¯H i at any particular epoch say z1 using an independent method (e.g. Majumdar et al. 2012). EoR 21-cm signal. The light-cone EoR 21-cm signal is un- doubtedly non-ergodic along the LoS. An earlier work (Fig. 9 of Mondal et al. 2018) demonstrates that Cℓ(ν)/C¯ℓ −1 shows C ( )/C¯ − 4 VALIDATING OUR MODEL a systematic variation with ν, the value of ℓ ν ℓ 1 is found to increases as we move from the nearest to the fur- As a first step towards validating our model we consider a thest end of the simulation box along LoS. This result cor- situation where the brightness temperature fluctuations are, relates well with the fact that x¯H i increases along the LoS by construction, of the form direction. This systematic variation, however, is only seen at the large ℓ bins (small angular scales) where we have a δT (nˆ, ν) = f (ν)× δ (nˆ, ν), (7) b e large number of Fourier modes in each bin. This systematic where f (ν) is a known function and δe(nˆ, ν) is a random variation is not seen in the small ℓ bins, partly because of field which is isotropic in nˆ and ergodic in ν. Using this we the fewer number of Fourier modes in each bin (leading to investigate whether our method of analysis can determine large sample variance) and partly due to the fact that the f (ν) from the estimated Cℓ(ν1, ν2). Here we have simulated evolution of the H i signal at these scales is dominated by the 10000 statistically independent realizations of homogeneous evolution of ionised bubbles not the x¯H i. To avoid this un- and isotropic Gaussian random fields δe(x) corresponding to certainty, we restrict the ℓ range to ℓ > 2571 for the present a realisation of the CDM power spectrum PDM(k). Working analysis.

MNRAS 000, 1–5 (2018) L4 R. Mondal et al.

8.67 5.98 2.99 0.00 -2.99 -5.98 -2.99 -5.98 -2.99 -5.98 -8.74 ν ν ν ν ν ν

Figure 1. This shows a section through one realization of the different Gaussian brightness temperature fluctuations fields for a = 0 (ergodic), 0.5 and 1.0 (see eq. 8). In the panels, the value of a increases from the left to the right.

1.6 0.7 Ergodic 1.4 a = 0.5 0.65 = a 1.0 0.6 ℓ

¯ 1.2 ℓ C ¯ C

)/ 0.55 )/ ν ( ν ℓ 1 (

ℓ 0.5 C C PSfrag replacementsp p

A 0.8 0.45 A 0.4 0.6 Fit 0.35 a = 0.1 Reionization history a = 0.2 0.4 0.3 8 6 4 2 0 -2 -4 -6 -8 0.04 ν − νc MHz 0.02 PSfrag replacements 0 -0.02 Residual Figure 2. This shows A Cℓ (ν)/C¯ℓ for different values of a as -0.04 shown in the figure. The points show results from our simulations p 8 6 4 2 0 -2 -4 -6 -8 and the solid lines show the function f (ν) (eq. 8). ν − νc MHz

We divide the ℓ range corresponding to our LC simula- tion into equally spaced logarithmic bins, and we compute Figure 3. The upper panel shows A Cℓ (ν)/C¯ℓ estimated from ¯ our LC EoR simulation. The red points show the results at differ- Cℓ(ν)/Cℓ for all of these bins for which ℓ > 2571. We see p ℓ ℓ > A C(ν)/C¯ that the values of C (ν)/C¯ display a large scatter (Fig. 3) ent bins with 2571. The blue points show which p ℓ ℓ have been estimated by combining all the modes with ℓ > 2571 i.e. ¯ p for a fixed ν pwe find a range of values of Cℓ(ν)/Cℓ into a single bin. The light blue solid line shows the 2nd order across the different ℓ bins. We attempt to mitigate the ef- ¯ p polynomial fit to the values of A C(ν)/C and the lower panel fect of these variations by estimating C(ν) which is obtained shows the residual after subtracting out the fit. The black solid p by combining the signal in all the modes with ℓ > 2571 into line shows the values ofx ¯H i(ν) corresponding to the reionization a single bin. Note that the EoR 21-cm signal is highly non- history of our LC simulation. The horizontal green line shows the Gaussian (Bharadwaj & Pandey 2005; Mondal et al. 2015; valuex ¯H i(νc) = 0.51. Majumdar et al. 2018) which makes it quite non-trivial to predict errors for the estimated C(ν) (Mondal et al. 2016), and we have not attempted this here. Apart from the cosmic parently random fluctuation with varying frequency (Fig. 3). variance there will be instrumental noise which will further In order to model this systematic variation we have fitted a χ2 nd ¯ = + ν−νc + ν−νc 2 worsen our predictions (i.e. the goodness of fit of our 2 order polynomial C(ν)/C a0 a1 B a2 B model). to the values estimated from the LC simulation. We have p

We find that in addition to a systematic increase with used a least-squares fit to obtain the best fit a0, a1 and a2. decreasing frequency, the values of C(ν)/C¯ exhibit an ap- We find that the best fit curve captures the systematic varia- p MNRAS 000, 1–5 (2018) Reionization history L5 tion of C(ν)/C¯ quite well and the residuals after subtracting ACKNOWLEDGEMENTS out the fit appear to be consistent with random fluctuations p This work was supported by the Science and Technol- around zero (lower panel of Fig. 3). If our model (eq. 5) holds ogy Facilities Council [grant numbers ST/F002858/1 and i( ) = C( )/C¯ we then have x¯H ν A ν . As mentioned earlier, it ST/I000976/1] and the Southeast Physics Network (SEP- is necessary to introduce one additional input to determine p Net). the value of A. Here we use the information that we have x¯H i ≈ 0.51 at νc = 157.78 MHz. We use this in conjunction with the polynomial fit to determine the value of A. 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MNRAS 000, 1–5 (2018)