Advanced Three Level Approximation for Numerical Treatment Of

arXiv:1104.3050v1 [astro-ph.CO] 15 Apr 2011 o.Nt .Ato.Soc. Astron. R. Not. Mon. coe 8 2018 18, October 2

hlpnoE.E. Kholupenko Recombination Cosmological Numerical of for Treatment Approximation Level Three Advanced h ayeot aebe aefrtertclinves- theoretical for made is been That have under- physics. efforts clear recombination demands many experiments why cosmological Treatment these of others). of standing many results the and in of day, QMAP- out 1989, 2001-present 1998-1999, COBE carried WMAP MAXIMA 1983, 1997-2003, been exper- BOOMERanG (Relikt-1 1996, have years of Toco thirty number anisotropy last great CMBR com- the A its on properties. Universe, iments early other in- the of and sources of position main radiation evolution in the of about background This one formation microwave is (1995)). which surface cosmic anisotropy al. scattering (CMBR) the et Hu last affects of (1970), turn ki- Zeldovich position its & because the (Sunyaev process determines important very netics also is combination molecules). molecules primordial of of lines abundances the resonant on in depends co emission which of (via cooling clouds radiative of lapsing rate correspondingly Lepp and & (1987)) (Dalgarno r chemistry primordial Cosmological affects length. combination Jeans temperature critical matter the and the determines radiation which via of recombination divergence cosmological of on proto- kinetics these depend of sizes also The objects (1995)). a Bertschinger et sys & Doroshkevich Ma 1968); bound (1967); (1965, gravity (Peebles galaxies non-relativistic like into tems develop clouds mat- CDM+baryonic c ter these fall clustering Afterwards to by deter- (CDM). matter start created of dark thereby can wells matter epoch potential and baryonic gravitational the which into matter at determines epochs from the It mines radiation Universe. of early decoupling the in cesses INTRODUCTION 1 c 1 oePyia-ehia nttt,S.Ptrbr 19402 St.-Petersburg Institute, Physical-Technical Ioffe t-eesugSaePltcnclUiest,Russia University, Polytechnical State St.-Petersburg 01RAS 2011 rmosrainlpito iwtecsooia re- cosmological the view of point observational From omlgclrcmiaini n ftekypro- key the of one is recombination Cosmological 000 1 , 2 –0(01 rne coe 8 08(NL (MN 2018 18, October Printed (2011) 1–10 , vnhkA.V. Ivanchik , ainsmlaeul ihuigfdefcos h oecnb fou r be cosmological can of code effects The physical an factors. words: fine recombination fudge Key r some of using code cosmological account model with The the into http://www.ioffe.ru/astro/QC/CMBR/atlant/atlant.html (TLA) simultaneously presented. of take approximation is nation calculations to plasma three-level fast us hydrogen-helium the for primordial on code of numerical nation public New ABSTRACT itrin yrgn helium hydrogen, distortion, ewrs omlgclrcmiain MR nstoy spect anisotropy, CMBR, recombination, cosmological Keywords: 1 ,Russia 1, , 2 aahvS.A. Balashev , old e- l- l. s - uin -atı ta.(08 n eeecsteen n c one (2004); therein) Grachev references & and Dubrovich Rubiño-Mart´ın (2008) (1993); (1998); dell’Antonio al. Biltzinger Uni- & et & Rybicki Boschan (1991); the Sunyaev Loch theoretical & & of numerous pow- Lyubarsky Fahr (1975), the (1983); (Dubrovich history to be field this Thanks about in would works epochs. distor- information these experiments in of spectral hy-verse Such sources of CMBR helium. recombination erful of and cosmological study from drogen originated experimental tions the is h al nvrei h epochs the in Universe early the correspondingly. accuracy 0.1% Desirable and recombination. 1% 0.01% helium and about of is recombination hydrogen epoch of the electron for epoch free the evaluating (on for 0.1% codes fraction) about numerical is the of recombination data cosmological accuracy Planck Chluba of required & treatment minimal Shaw successful (2009); for Chluba Today & find (2011)). Sunyaev to uncer- e.g. remain investigators see estimate of tainties to and efforts scenario of recombination overview exact important more (for and urgent more numerical fro and becomes data experimental task of of this releases mission soon Planck perfection of light the and In (2008), codes. Switzer (2010)) & Rubiño-Mart´ın Hirata 2010) al. al. (2008b), Ali-Ha¨ımoud et et Hirata 2010b), (2007, & Switzer (2009a, al. (2008), in et Chluba Sunyaev theorists e.g & of (see (2005), Chluba recombination efforts Grachev cosmological of & of Dubrovich increase details of to study i leads the experiments decade of accuracy last Increasing the (1991), (1999)). Dubrovich process al. al & et this et Grachev Seager (1985), Matsuda of Wyse (1968), & Peebles modelling Jones (1968), (1969), for al. development et codes Zeldovich and (e.g. numerical recombination applied cosmological of of tigation h etpsil tpo h netgtosof investigations the of step possible next The A 1 T , E 2 tl l v2.2) file style X n asaoihD.A. Varshalovich and , z 800 = allows d sbased is ecombi- ecombi- dat nd − 1 , 2 10 ral an m n 4 . . 2 Kholupenko, Ivanchik, Balashev, and Varshalovich understand parameters of these spectral distortions clearly The total concentrations of hydrogen and helium de- enough. Of course the cosmological recombination is an in- pend on redshift by the following formulas: tegral part of modelling of these distortions. Note that some N = N (1 + z)3 , N = N (1 + z)3 (4) simple methods for this problem (Bernshtein et al. (1977); H H0 He He0 Burgin (2003)) demand the knowledge of derivatives of ion- where NH0 and NHe0 are the values of concentrations at ization fractions of hydrogen and helium, so calculated ion- present epoch. The total concentration of the primordial ization fractions should be smooth as possible (i.e. without hydrogen atoms and ions at present epoch is given by the sharp numerical features). In spite of the fact that detection following relation: of CMBR spectral distortions by cosmological recombina- ρc NH0 = ΩbXp (5) tion is impossible today, the rapid progress of measurement mH equipment for CMBR anisotropy allows us to hope that such where m is the hydrogen atom mass, X is the primordial experiments will be possible in the near future. H p hydrogen mass fraction (i.e. hydrogen mass fraction after Thus the main aim of this work is to present nu- the primordial nucleosynthesis). merical code covering investigations of cosmological recom- The total concentration of the primordial helium atoms bination widely as possible within simple model which and ions at present epoch is given by the following relation: we used. Our code called atlant (advanced three level ρc approximation for numerical treatment of cosmological re- NHe0 = ΩbYp (6) combination) may be useful for regular calculations of free mHe electron fraction for treatment of CMBR anisotropy data, where mHe is the helium atom mass, Yp = (1 − Xp) is the further investigations of cosmological recombination, theo- primordial helium mass fraction. Fractions of other elements retical predictions of new observational cosmological effects are considered negligible. (e.g. Dubrovich et al. (2009); Grachev & Dubrovich (2010)), The temperature T of equilibrium radiation background and comparison with other numerical codes (e.g. recfast depends on redshift according to: by Seager et al. (1999) and Wong et al. (2008), RICO by T = T (1 + z) (7) Fendt et al. (2009), RecSparse by Grin & Hirata (2010), 0 HyRec by Ali-Ha¨ımoud & Hirata (2010a), CosmoRec by Chluba & Thomas (2010)). 3 HYDROGEN RECOMBINATION 3.1 Main equations 2 COSMOLOGICAL MODEL The time-dependent behaviour of hydrogen ionization frac- The standard cosmological model is used. Hubble constant tion in the isotropic homogeneous expanding Universe is de- H as a function of redshift z is given by the following equa- scribed by the following kinetic equation (Zeldovich et al. tion: (1968); Peebles (1968)): 2 3 4 H(z)= H0 ΩΛ + ΩK (1 + z) + Ωm(1 + z) + Ωrel(1 + z) (1) hνα x˙ HII = −CHI [αHII (Tm) NexHII − βHI (T )exp − xHI ] (8) p kBT where H0 is the value of Hubble constant in present epoch,

ΩΛ is the vacuum-like energy density, ΩK is the energy den- where xHII = NHII / (NH + NHe), CHI is the factor by sity related to the curvature of the Universe, Ωm is the en- which the ordinary recombination rate is inhibited by the ergy density of non-relativistic matter, which includes con- presence of HI Lyα resonance-line radiation, αHII is the to- tributions from CDM and baryonic matter Ωm = ΩCDM + tal HII→HI recombination coefficient to the excited states Ωb. Ωrel is the energy density of relativistic matter, which of HI, Ne = xeNH is the free electron concentration (xe includes contributions from photons (mainly CMBR) and is the free electron fraction in common notation, see e.g. neutrinos Ωrel = Ωγ + Ων . recfast), βHI is the total HI→HII ionization coefficient The photon energy density is related to the temperature from the excited states of HI, να is the Lyα transition fre- of CMBR: quency, Tm is the kinetic temperature of the electron gas, 4 aRT0 xHI = NHI / (NH + NHe) is the neutral hydrogen fraction. Ωγ = 2 (2) ρcc Note that ionization fractions of ionic components are defined relative to the total number of hydrogen and helium where a is the radiation constant, T is the CMBR tem- R 0 atoms and ions (N + N ) while free electron fraction x perature at present epoch, ρ = 3H2/ (8πG) is the critical H He e c 0 is normalized to the concentration of hydrogen atoms and density of the Universe, c is the speed of light, G is the ions, N , as it is accepted commonly (so it is necessary for gravitational constant. H possible use of atlant results by other numerical codes). At the present epoch the relativistic neutrino energy The specific form of inhibition coefficient depends on density is related with the photon energy density by the fine effects which are taken into account. In the current ver- following formula: sion (1.0) of the code only radiative feedbacks for resonant 7 4 4/3 transitions are taken into account from the whole list of Ων = Nν Ωγ (3) 8 11 known (considered until now) fine effects. Other fine effects are planned to be included in future works. So, the inhibition where N is the effective number of neutrino types. ν coefficient is given by the formula: In this paper (and current version of code) we consider r Ωtot = ΩΛ + ΩK + Ωm + Ωrel = 1, so ΩK is fixed by relation Aeff + A2s1s CHI = r (9) ΩK = 1 − (ΩΛ + Ωm + Ωrel). βHI + Aeff + A2s1s

c 2011 RAS, MNRAS 000, 1–10 0,4

ATLANT0,3 for Cosmological Recombination 3

r , % where Aeff is the total eﬀective coeﬃcient of np↔1s tran- e

/x 0,2

−1 e sitions, A2s1s = 8.22458 s is the coeﬃcient of 2s→1s two- x photon spontaneous transition (e.g. Goldman (1989)). ∆ 0,1 The recombination coeﬃcient αHII is given by the following approximation:

b 0 aT4 αHII (T )= FH (10) 1+ cT d 4 -3 10 where FH is the hydrogen fudge factor by Seager et al. 4 −13 3 −1 (1999), T4 = T [K]/10 , and a = 4.309 · 10 cm s , b = −0.6166, c = 0.6703, d = 0.5300 are the parameters

e -4

ﬁtted by Pequignot et al. (1991). /x 10 e x

The ionization coeﬃcient βHI can be found by using ∆ principle of detailed balance:

-5 hνc2 10 βHI (T )= αHII (T ) ge (T )exp − (11) kBT

3/2 3 where ge (T ) = (2πmekB T ) /h is the partition function -6 10 of free electrons, νc2 is theHI c→2 transition frequency (here 200 400 600 800 1000 1200 1400 1600 symbol “c” denotes continuum state). z Note that recombination and ionization coefficients included in the kinetic equation (8) should be calculated Figure 1. Top panel: the relative difference ∆xe/xe between at different temperatures (see e.g. Ali-Ha¨ımoud & Hirata free electron fractions calculated by atlant and recfast for the (2010a)). Since the kinetics of recombination process is period of hydrogen recombination. Bottom panel: (i) solid curve determined by the free electron distribution function, the corresponds to the relative difference ∆xe/xe between free electron fractions calculated by atlant and recfast [i.e. the same recombination coefficient depends on kinetic temperature as in the top panel but in logarithmic scale], (ii) dashed curve of free electrons Tm. The kinetics of ionization is deter- corresponds to the relative difference ∆xe/xe between free elec- mined by the photon distribution function, therefore ion- tron fractions calculated by atlant and recfast with corrected ization coefficient depends on temperature of photons T . ionization rate [i.e. ionization coefficient βHI (T ) and exponen- Thus the detailed balance does not take place in considered tial term exp (−hνα/ [kBT ]) in Eq. (8) of recfast are calculated case, and relation (11) has mathematical sense only and al- at the temperature of radiation T ], (iii) dotted curve corresponds lows us to avoid direct calculation of ionization coefficient, to the relative difference ∆xe/xe between free electron fractions atlant βHI (T ), from integral of collisions for photons and hydro- calculated by with modified ionization rate [i.e. ionization gen atoms. Also exponential term in ionization part of (8), coefficient βHI (Tm) and exponential term exp (−hνα/ [kBTm]) atlant exp(−hν / [k T ]), should be calculated at the temperature in Eq. (8) of are calculated at the temperature of matter α B T ] and recfast. Dashed and dotted curves are partially over- of radiation. The distinction of temperatures used for cal- m lapped. culation of ionization coefficients in the present code and in recfast (Seager et al. (1999); Wong et al. (2008), there the 3 temperature of matter is used for this aim) leads to a little r 8πHνα Aeff = 3 δA (12) but important difference in the free electron fraction (see NHI c Fig. 1). where δ is given by the following expression The equation (8) is solved numerically together with A other main kinetic equations (19), (35) by using second- nmax 3 νn En − E2 order method of integration of ODE system. The relative δA = Cn 3 exp − (13) να kB T deviation between results obtained by atlant and recfast nX>2 for the period of hydrogen cosmological recombination is where νn is the frequency of n → 1 transition (ν2 = να), presented in Fig. 1. En = hνn, and coefficients Cn are: ′ Cn = 1 − ΓH (zn)/ΓH (z) (14) 3.2 Radiative Feedbacks where in turn ΓH is the relative overheating of Lyα radiation (occupation number ηα) in comparison with its equilibrium Due to a great number of fine effects the radiative feed- 0 backs for resonant transitions have been chosen for includ- value (occupation number ηα): ing in the first published version of recombination code. It 0 ΓH = ηα/ηα − 1 (15) is because physics of this effect is clear (it is difficult to state this is true about many other fine effects) and in- and dependently obtained results of calculation of this effect ′ z = ((1+ z)νn+1/νn − 1) (16) (Chluba & Sunyaev (2010a); Kholupenko et al. (2010)) con- n firm each other. Inclusion of feedbacks into the code is based The set of formulas (12-16) allows us to calculate ioniza- on formulas suggested by Kholupenko et al. (2010). Accord- tion history taking into account the feedbacks for hydrogen ing this the total effective coefficient of np↔1s transitions n → 1 resonant transitions with principal quantum num- is: bers n 6 nmax. For taking into account the feedback effect

c 2011 RAS, MNRAS 000, 1–10 4 Kholupenko, Ivanchik, Balashev, and Varshalovich

0,25 calc sulting file), xe is the solution of ODE’s system describing ionization fractions and free electron fraction. Note that function δX affects final result xe but not 0,2 ODE’s system. This allows user to control changes of free electron fraction strictly by including fudge factors. Analyzing previous works devoted to the fine effects 0,15 of cosmological recombination (e.g. Grachev & Dubrovich (2008); Chluba & Sunyaev (2009b)) one may note that typ- ical form of corrections to the ionization fraction can be de- , % e

/x 0,1 scribed by a bell-shaped function (e.g. Lorentzian or Gaus- e x

∆ sian proﬁle or others). In this work the Lorentzian function has been chosen to describe uncertain deviations (until now) 0,05 of free electron fraction from well known ODE’s solution:

Ap δX = 2 (18) 1+[(z − zp) /∆zp] 0 where Ap is the relative amplitude of perturbation of ionization fraction, zp is the redshift of perturbation maximum, ∆z is the half-width of perturbation function at half- -0,05 p 600 800 1000 1200 1400 1600 altitude. z Result of use of fudge function is shown in the Fig. 2 where we have plotted ∆x /x = δ as function of redshift Figure 2. Dashed line corresponds to the relative difference e e X z. Here we show how fudge function can mimic real fine between atlant with fudge function δX and base result [i.e. corrections by means of example of feedback correction for ∆xe/xe = δX ]. Parameters of fudge function are the following: −3 Ap = 2.166 · 10 , zp = 1019, ∆zp = 180. For comparison: solid nmax = 10. line corresponds to the relative difference between feedback result for nmax = 10 [i.e. feedbacks 11 ⇒ ... ⇒ 2 are taken into account] and the base result [i.e. without feedbacks]. 4 HEII→HEI HELIUM RECOMBINATION The time-dependent behaviour of HeII fraction in the we use simple perturbation theory: at the first stage (unper- isotropic homogeneous expanding Universe is described by turbed equations) atlant calculates the relative overheating the following kinetic equation (Kholupenko et al. (2007); ΓH and stores this, at the second stage atlant solves equa- Wong et al. (2008)): tions perturbed by the feedback effect. Relative deviation of ga Eag free electron fraction between perturbed (nmax = 10) and x˙ HeII = −Cpar[αparNexHeII − βpar exp − xHeI ] gg kB T unperturbed calculations is presented in Fig. 2. ga′ Ea′g −Cor[αorNexHeII − βor exp − xHeI ] (19) gg kB T

3.3 Fudge factors where xHeII = NHeII / (NH + NHe) is the fraction of HeII ions relative to the total number of hydrogen and helium Developed recombination code allows users to use fudge fac- atoms and ions, Cpar is the factor by which the ordi- tors as well as add physical effects. This opportunity has nary recombination rate is inhibited by the presence of been included because of the following reasons: 1) list of fine 1 1 HeI 2 p → 1 s resonance-line radiation, αpar is the total effects leading to 0.01 - 0.1 % corrections in the ionization HeII→HeI recombination coefficient to the excited para- fraction may be incomplete, and it is difficult to say how long states of HeI, subscript a denotes 21s state of HeI atom, this list will be in final form and when this will be achieved; Ne is the free electron concentration, ga = 1 is the statis- 2) considerations of some fine effects give contradictory re- 1 tical weight of 2 s state of HeI, gg = 1 is the statistical sults due to not completely clear physical picture (e.g. two- 1 weight of 1 s state of HeI, βpar is the total HeI→HeII ion- photon transitions from high excited states [Wong & Scott ization coefficient from the excited para-states of HeI, Eag is ( 2008), Hirata (2008), and Labzowsky et al. (2009)], recoil 1 1 the 2 s → 1 s transition energy, Cor is the factor by which [Grachev & Dubrovich (2008) and Hirata & Forbes (2009)] the ordinary recombination rate is inhibited by the presence and others). 3 1 of HeI 2 p → 1 s resonance-line radiation, αor is the total First fudge factor is the common (see recfast) hydro- HeII→HeI recombination coefficient to the excited ortho- ′ gen fudge factor FH introduced in the expression (10) for states of HeI, subscript a denotes state 23s of HeI atom hydrogen recombination coefficient αHII . 3 ga′ = 3 is the statistical weight of 2 s state of HeI, βor is Second fudge factor is the perturbation function δX the total HeI→HeII ionization coefficient from the excited modifying free electron fraction: 3 1 ortho-states of HeI, Ea′g is the 2 s → 1 s transition energy, res calc xHeI = NHeI / (NH + NHe) is the neutral helium fraction. xe = xe (1 + δX ) (17) The para- and ortho- recombination coefficients res where xe is the free electron fraction (normalized by the are given by widely used approximation formulas (e.g. total concentration of hydrogen atoms and ions) being the Verner & Ferland (1996)) parameters of which are based on final result of code running (i.e. it is value shown in the re- data by Hummer & Storey (1998)):

c 2011 RAS, MNRAS 000, 1–10 ATLANT for Cosmological Recombination 5

−1/2 −(1−pS ) −(1+pS ) T T T Table 1. H αS (T )= qS 1+ 1+ (20) Parameters of the approximation of PD T2 T2 T1 where subscript S takes values ‘par’ or ‘or’, and qS , pS, T1 = Range of γ p q 105.114 K, T = 3 K (Seager et al. (1999)) are the parameters 2 0 6 γ 6 5 · 102 0.66 0.9 of approximation. For the recombination via para-states we 5 · 102 < γ 6 5 · 104 0.515 0.94 −10.744 3 −1 have qpar = 10 cm s , ppar = 0.711 (Seager et al. 5 · 104 < γ 6 5 · 105 0.416 0.96 (1999)). For the recombination via ortho-states we have 5 · 105 < γ 0.36 0.97 −10.306 3 −1 qor = 10 cm s , por = 0.761 (Wong et al. (2008)). The para- and ortho- recombination and ionization coefficients are related by the following formula H The value Pfg is given by the following: gc ES βS = αS ge(T )exp − , (21) H H H H gS kB T Pfg = PD + PG + PR (28) ′′ H where subscript “c denotes continuum state of HeI atom, where in turn PD approximately is: gc = 4 is the statistical weight of continuum state of H q −1 + − 1 PD = (1+ pγ ) (29) (He +e ), Epar is the c → 2 s transition energy, Eor is the c → 23s transition energy. where parameters p, q are given in the Tab. 1. H The inhibition factor Cpar is given by the following ex- The value PG is given by the following approximate pression: formula: H 1/4 −5/8 −3/4 (gb/ga) Abg Pbg exp(−Eba/kB T )+ Aag PG = (8λa) π γ · Cpar = (22) −1/2 βpar +(gb/ga) AbgPbg exp(−Eba/kB T )+ Aag · [1 +exp(−1.07 ln (s + 1.5) − 0.45)] (s + 1.28) (30) −1 1 1 where Abg is the Einstein coefficient [s ] of 2 p ↔ 1 s spon- where a = Γf /4π∆νD,f is the Voigt parameter (here Γf taneous transitions, Pbg is the probability of the uncompen- is the natural line width), λ is the single-scattering albedo 1 1 1 1 sated 2 p → 1 s transitions, Eba is the 2 p → 2 s transition in f → g line, and parameter s is given by the following 1 1 energy, Aag is the coefficient of two-photon 2 s → 1 s spon- relation (Grachev (1988)): taneous decay. − − − s = 2 3/2π 1/4(1 − λ)λ 1/2a1/2γ1/2 − 1/4 (31) The inhibition factor Cor is given by the following expression The single-scattering albedo λ can be calculated by:

′ ′ ′ ′ ′ ′ (gb /ga ) Ab gPb g exp(−Eb a /kB T ) Afg Cor = (23) λ ≃ (32) ′ ′ ′ ′ ′ ′ βor +(gb /ga ) Ab gPb g exp(−Eb a /kB T ) Afg + Rf ′ 3 → where subscript b denotes state 2 p of HeI, gb′ = 9 is the where Rf = n Rf n is the total coefficient of radia- 3 statistical weight of 2 p state of HeI, where Abg is the Ein- tive transitionsP (including induced transitions, while colli- stein coefficient [s−1] of 23p ↔ 11s spontaneous transitions, sion transitions are considered negligible) from f state to 3 1 Pb′g is the probability of the uncompensated 2 p → 1 s excited states of HeI atom. 3 3 H transitions, Eb′a′ is the 2 p → 2 s transition energy. The value PR is approximately: The probabilities Pbg and P ′ take into account the b g H a π escape of HeI resonant photons from the line profiles due to PR = 2(1−λ) − arctan (f(a,λ,γ)) (33) r(1 − λ)γπ3/2 2 the cosmological expansion and destruction of these photons by neutral hydrogen. They can be found by the following where f(a,λ,γ) is defined by the expression formula: −1/2 1/4 −1/4 f(a,λ,γ) ≃ 0.69 · (1 − λ) λ (aγ) ln γ (34) r H p Pfg = Pfg + Pfg (24) The equation (19) is solved numerically together with other r main kinetic equations (8) and (35) to determine free elec- where Pfg is approximately given by the following: tron fraction xe. The relative deviation of free electron frac- r −1 −2 −1 Pfg = 1+ γ τHe,f (1 − exp(−τHe,f )) (25) tion between the results calculated by using atlant and recfast for the period of HeII→HeI recombination is presented where γ is the ratio of the helium and hydrogen absorption in Fig. 3. coefficients at the central frequency of the f → g line (here symbol f = b or b′ depending on what transition is considered), τ is the Sobolev optical depth. The value γ is He,f 5 HEIII→HEII HELIUM RECOMBINATION given by the following relation:

2 In the diﬀerence with recfast we use non-equilibrium (i.e. (g /gg) A NHeI c γ = f fg (26) kinetic) approach for consideration of HeIII→HeII helium σ (ν ) 8π3/2ν2 ∆ν N H fg fg D,f HI recombination. The time-dependent behaviour HeIII frac- where σH is the ionization cross-section of hydrogen ground tion in the isotropic homogeneous expanding Universe is de- 2 state, parameter ∆νD,f = νfg 2kB T/ (mHec ) is the scribed by the following kinetic equation: Doppler line width. p x˙ HeIII = −CHeII [αHeIII NexHeIII − The optical depth τHe,f is: hνHeII,21 3 3 βHeII exp − xHeII ] (35) τHe,f = gf AfgNHeI c / gg8πHνfg (27) kB T c 2011 RAS, MNRAS 000, 1–10 6 Kholupenko, Ivanchik, Balashev, and Varshalovich

0,25

0,2 -0,05

0,15 -0,1 , % , % e e /x /x e e x x ∆ ∆ 0,1 -0,15

-0,2 0,05

-0,25 0 1600 1800 2000 2200 2400 2600 2800 3000 4800 5200 5600 6000 6400 6800 z z

Figure 3. The relative diﬀerence ∆xe/xe between free elec- Figure 4. The relative diﬀerence ∆xe/xe between free electron fractions calculated by atlant and recfast for the period tron fractions calculated by atlant and recfast for the period of HeII→HeI recombination. of HeIII→HeII recombination.

where xHeIII = NHeIII / (NH + NHe), CHeII is the factor sitions due to escape of HeII 2p↔1s resonant photons from by which the ordinary recombination rate is inhibited by the the line profile because of cosmological redshift is given by presence of HeII 2p → 1s resonance-line radiation, αHeIII the following formula: is the total HeIII→HeII recombination coefficient to the ex- 3 r 8πHνHeII,21 cited states of HeII, βHeII is the total HeII→HeIII ioniza- AHeII,2p1s = 3 (39) NHeII c tion coefficient from the excited states of HeII, νHeII,21 is the HeII 2p → 1s transition frequency. The value AHeII,2s1s is found from charge scaling for the 6 The inhibition coefficient is given by the expression: hydrogenic ions AZ,2s1s = Z A2s1s (see e.g. Shapiro & Breit r (1959), Zon & Rapoport (1968), Nussbaumer & Schmutz AHeII,2p1s + AHeII,2s1s (1984)). In the considered case this gives us AHeII,2s1s = CHeII = r (36) βHeII + A + AHeII,2s1s 6 HeII,2p1s 2 A2s1s. r The equation (35) is solved numerically together with where AHeII,2p1s is the effective coefficient of HeII 2p↔1s other main kinetic equations (8) and (19) to determine free transitions, AHeII,2s1s is the coefficient of HeII 2s→1s two- photon spontaneous transition. electron fraction xe. The relative deviation between results Since HeII is the hydrogenic ion the recombination coef- by atlant and recfast for the period of HeIII→HeII recombination is presented in Fig. 4. ficient αHeIII can be found by using simple scaling relation (see e.g. Verner & Ferland (1996)):

2 αZ (T )= Zα1 T/Z (37) 6 EVOLUTION OF MATTER TEMPERATURE where Z is the nuclear charge for the hydrogenic ions. In The behaviour of the matter temperature in the isotropic ho- considered case Eq. (37) yields α (T ) = 2α (T/4), HeIII HII mogeneous expanding Universe is described by the following where α is taken from (10) without hydrogen fudge factor HII equation (e.g. Peebles (1968); Scott & Moss (2009)): FH . 4 The ionization coeﬃcient βHeII can be found by using 8σT aRT xe T˙m = (T − Tm) − 2HTm (40) principle of detailed balance: 3mec 1+ xe + NHe0/NH0

hνHeII,c2 where σT is the Thomson scattering cross section. βHeII (T )= αHeIII (T ) ge (T )exp − (38) kB T Defining relative deviation, δT , of the matter temperature from the radiation temperature via T = T (1 − δ ) ν is the HeII c→2 transition frequency. m T HeII,c2 and substituting this into (40) one can obtain: Note that in Eq (35) the recombination and ionization coefficients are calculating at the same temperature T . This δ˙T = − (RT + H) δT + H (41) is because the matter temperature Tm is very close to the − where R is the rate of energy transfer between matter and radiation one T (relative deviation is less than 10 5) dur- T radiation via Compton scattering: ing HeIII→HeII recombination and calculation at different 4 temperatures has no sense at required level of accuracy. 8σT aRT xe r RT = (42) The effective coefficient AHeII,2p1s of HeII 2p↔1s tran- 3mec 1+ xe + NHe0/NH0

c 2011 RAS, MNRAS 000, 1–10 ATLANT for Cosmological Recombination 7

-1 10 -2 10

-2 10

-3 -3 10 10

-4 e

10 /x T e δ x

∆ -4 10 -5 10

-6 10 -5 10

-7 10

-8 -6 10 10 200 400 600 800 1000 1200 200 400 600 800 1000 1200 z z

Figure 5. The corrections, δT , to the matter temperature, Tm, Figure 6. The relative difference ∆xe/xe of the free electron relative to the radiation temperature, T , as functions of redshift z: fractions as a function of redshift z: dotted curve corresponds to dashed curve corresponds to the correction, δT,0, dashed-dotted the difference, ∆xe/xe, between results for δT = 0 (i.e. Tm = T ) curve corresponds to |δT,1| (δT,1 < 0), and solid curve corre- by atlant and recfast corrected (for the details of corrections, see sponds to δT,0 + δT,1. section 3.1 of the present paper), dashed curve corresponds to the difference, ∆xe/xe, between results for δT = δT,0 by atlant and corrected recfast, and solid curve corresponds to the difference, To solve equation (41) we applied a perturbation ap- ∆xe/xe, between results for δT = δT,0 + δT,1 by atlant and proach. In the early stages of the Universe history (z & 200) corrected recfast. the rate of energy transfer between matter and radiation via Compton scattering is much larger than the rate of the tem- δ = 0.5. Equation (46) corresponds to the equation of perature change (the latter is about Hubble expansion rate T,0 state of non-relativistic matter. H), i.e. δ˙ / (R δ ) ≪ 1. Thus we can use expansion of the T T T Results of calculations of δ and δ are presented in solution over this smallness: T,0 T,1 Fig. 5. The influence of taking into account different approx- ∞ imations for δ [depending on the number of kept members δ = δ (43) T T T,i of expansion (43)] on the free electron fraction is shown in Xi=0 Fig. 6. where zeroth-order approximation is determined as qua- sistacionary solution of Eq. (41) (see also Hirata (2008); Ali-Ha¨ımoud & Hirata (2010b)): 7 VARIATION OF THE FUNDAMENTAL −1 δT,0 =(RT /H + 1) (44) CONSTANTS and the next members of expansion (43) are related by the Today the opportunity to vary of the fundamental constants following equation: becomes essential at the analysis of CMBR anisotropy (e.g. − Scóccola et al. (2008)) and we decided to include this in our δ = − (R + H) 1 δ˙ (45) T,i+1 T T,i code. Thus the current version of the code allows user to see In the present version of the code we keep only two first how recombination occurs at different values of the funda- corrections δT,0 and δT,1 in Eq. (43). mental constants (this means that changes of fundamental For the period z . 200 the quasistationary condition constants lead to corresponding changes of derived physi- is violated, so expansion (43) loses convergence and cannot cal values, e.g. ionization energies of atoms, Thomson cross represent the solution of Eq. (41). In this redshift range we section and others). use the following dependence of matter temperature on red- Since some of used physical values are given only nu- shift: merically (e.g. level energies and transition probabilities for 2 HeI atom, see www.nist.gov) we use simple scalings to take dec 1+ z Tm = Tm (46) into account the influence of variation of fundamental con- 1+ zdec stants on these values. These scalings are the following: dec 1) For the level energies: where Tm is the matter temperature from (43) at zdec (in- dex “dec” means “decoupling”) which is considered as the 4 −2 e me ~ moment of decoupling of the matter temperature from radia- E = Est (47) est me,st ~st tion the one (at value zdec we should make join of solutions). In present version of code we determine zdec from condition 2) For the one-photon transition coefficients:

c 2011 RAS, MNRAS 000, 1–10 8 Kholupenko, Ivanchik, Balashev, and Varshalovich

non-equilibrium (kinetic) process in the whole range of red- 1 shifts and sets an initial conditions corresponding to the fully ionized plasma at the moment given by user (in our case zbegin = 8000). The sharp peak of ∆xe/xe in the pe- − riod z = 1550 − 1650 (maximal value is about 5 · 10 4) 0,8 is due to transition from recombination according to Saha formula to non-equilibrium recombination which occurs in recfast at free electron fraction xe = 0.985 (that corre- 0,6 sponds to z ≃ 1555 in considered case). There is a simple e

x method of modelling of CMBR spectral distortion due to cosmological recombination (e.g. Bernshtein et al. (1977); Dubrovich & Stolyarov (1995, 1997); Burgin (2003)). This 0,4 method is based on the formalism of matrix of efficiency of radiative transitions (ERT-matrix) and it does not demand direct calculation of atomic level populations. On the 0,2 other hand this method demands use of the following values dxHII /dz, dxHeII /dz, and dxHeIII /dz to determine the rates of resonance photon emission and correspondingly the 0 shape of cosmological recombination lines. Thus for using 1000 2000 3000 ERT-matrix method the sharp numerical features may be z significant defects of cosmological recombination modelling. The second difference of results by atlant and recfast ap- Figure 7. Free electron fractions xe as functions of redshift z pears in the period z = 300 − 1000. It arises due to different at different values of the fine-structure constant αfine: dashed 0 structure of ionization items of hydrogen kinetic equation curve corresponds to αfine/αfine = 0.95, solid curve corre- 0 (8) in atlant and recfast: in recfast the ionization coeffi- sponds to αfine/αfine = 1, dashed-dotted curve corresponds to 0 cient βHI and Boltzmann exponential term are calculated at αfine/αfine = 1.05 the temperature of matter while in atlant at the radiation temperature. Most part of this difference is due to distinc- 10 −6 −3 tion of ionization coefficients including in the denominator of e m ~ c Aγ = Aγ e (48) inhibition coefficient C . The maximum of ∆x /x in men- st e m ~ c HI e e st e,st st st tioned range of redshifts is about 0.27% at z ≃ 770. This is 3) For the two-photon transition coefficients: little but maybe important difference in the context of future analysis of Planck data. The third difference appears at 16 ~ −9 −6 γγ γγ e me c low redshifts (z . 300). It is due to the different approaches A = Ast ~ (49) est me,st st cst to evaluating matter temperature in recfast (where ODE atlant 4) For the recombination coefficients: for temperature is solved) and in (where perturbation theory for temperature estimate is used). In the bottom 6 −3/2 −1 −3 e me ~ c panel of Fig. 1 we present three curves. Solid curve is the α = αst (50) est me,st ~st cst same as in the top panel but in a logarithmic scale. Dashed curve corresponds to the relative difference ∆xe/xe between In the expressions (47)-(50) subscript “st” denotes current free electron fractions calculated by atlant and recfast with values of physical quantities while symbols without subscript corrected ionization rate [i.e. ionization coefficient βHI (T ) denote values which user suggests to be valid during the and exponential term exp (−hνα/ [kB T ]) in Eq. (8) of the recombination epoch. recfast model are calculated at the temperature of radi- The influence of variations of the fine structure constant ation, T ]. Dotted curve corresponds to the relative differ- αfine (via varying elementary charge) on the free electron ence ∆xe/xe between free electron fractions calculated by fraction is shown in Fig. 7. atlant with modified ionization rate [i.e. ionization coefficient βHI (Tm) and exponential term exp (−hνα/ [kB Tm]) in Eq. (8) of the atlant model are calculated at the temperature of matter Tm] and recfast. These curves show that 8 RESULTS in the frame of identical physical models the accordance of The main numerical results of this paper are the relative results by atlant and recfast is wholly satisfactory. Resid- −5 differences of free electron fractions ∆xe/xe calculated by ual difference which does not exceed 5 · 10 for the period atlant and recfast (figs 1, 3, 4). In the top panel of Fig. 1 z = 400−1500 can be explained by a little different values of the relative difference ∆xe/xe between result of atlant and physical constants and different integration methods which current version of recfast (version 1.5, Wong et al. (2008)) have been used in atlant and recfast. for the hydrogen recombination epoch. The first significant In Fig. 2 we demonstrate how feedbacks for resonant difference appears in the period z = 1550 − 1650. This is transitions of hydrogen affect free electron fraction (solid because at early epochs z & 1550 recfast considers re- curve) and show an example of using fudge function, δX combination as quasi-equilibrium process (according to Saha (dashed curve). The change due to feedbacks has the maxi- formula) and sets initial condition for the kinetic equation mum about 0.2166% at redshift z ≃ 1019 in full accordance correspondingly, while atlant considers recombination as a with Chluba & Sunyaev (2010a). This calculation has been

c 2011 RAS, MNRAS 000, 1–10 ATLANT for Cosmological Recombination 9 done for nmax = 10. Parameters of the fudge function have curve corresponds to the correction δT,0, dashed-dotted −3 been choosen as following: Ap = 2.166·10 , zp = 1019, and curve corresponds to |δT,1|, and solid curve corresponds to ∆zp = 180. This is to show how use of the fudge function (δT,0 + δT,1). One can see that difference between the ra- allows us to imitate the influence of real physical effects on diation and matter temperature is less than 10−4 down to − the free electron fraction. z ≃ 940 and less than 10 3 down to z ≃ 790. Taking into In Fig. 3 the relative difference of the results by atlant account that Tm is included in kinetic equation (8) only and recfast for the epoch of HeII→HeI recombination is via recombination coefficient αHII (Tm) one can expect the presented. The breaks in the range z = 2700 − 2900 and similar changes in hydrogen ionization fraction at transition break at z ≃ 2300 are the numerical features connected from approximation Tm = T to more exact estimate of Tm. with switch in recfast code. The maximum of difference From accuracy point of view the correction δT,1 becomes is about -0.23% at z ≃ 1950. The difference between re- important beginning from z ≃ 480 where/when it achieves −3 sults by atlant and recfast for the epoch of HeII→HeI values about 10 . At the low z (z . 300) the correction δT,1 recombination arises because different approximations for is larger than 20% of δT,0 and should be taken into account estimate of effective eacape probability are used: in at- for correct determination of residual ionization fraction as lant the approach based on analytical consideration of res- possible. In Fig. 6 we show how including of δT,i’s affects the onance radiation transfer in the presence of continuum ab- free electron fraction. We have compared our results with the sorption is used (Kholupenko et al. (2008)) while recfast results by recfast corrected in the following way: ionization uses the simple approximation with an additional fudge items of Eq. (8), i.e. ionization coefficient βHI and exponen- factor bHe (Wong et al. (2008)). It leads to the numerical tial term exp (−hνα/ [kB Tm]) are calculated at temperature differences because used approximations are not identical. of radiation T . The dotted curve corresponds to the relative Also there is physical reason for divergence of these ap- difference ∆xe/xe of the result by atlant for approximation proximations: Wong et al. (2008) have determined bHe from Tm = T (i.e. δT = 0) and the mentioned result by recfast, the best agreement with result by Switzer & Hirata (2008a) the dashed one corresponds to ∆xe/xe at δT = δT,0, and who took into account not only hydrogen continuum opac- the solid one does ∆xe/xe at δT = (δT,0 + δT,1). From Fig. ity but also feedbacks for resonant transitions of HeI and 6 one can see that rough approximation Tm = T is quite found this effect is about 0.46% at z ≃ 2045. Later works valid (at required level of accuracy) down to z ≃ 670 where −3 (e.g. Chluba & Sunyaev (2010a)) show that this feedback ef- ∆xe/xe achieves values about 10 . Thus using this rough fect is about 0.17% at z ≃ 2300, i.e. negligible for the most approximation biases the results not so dramatically as it of modern problems connected with cosmological recombi- seems when this approximation is used in recfast. More ad- −3 nation (e.g. CMBR anisotropy analysis). Thus we do not vanced approximations are valid (∆xe/xe 6 10 ) down to take helium feedbacks into account in the present version z ≃ 310 (δT = δT,0) and z ≃ 250 (δT =(δT,0 + δT,1)). of the code. One more important point of calculations of Fig. 7 illustrates one of the additional opportunities of effective escape probability for HeI is the estimate of num- atlant: here the changes of free electron fraction at varia- ber of neutral hydrogen at the high redshifts z & 1600 tion of the fine-structure constant are presented. Presented (i.e. when hydrogen ionization fraction close to unity [e.g. results are obtained for the following values of the fine- 0 0 0 greater than 0.985]): recfast uses Saha formula for this aim, structure constant αfine: 0.95αfine, αfine, and 1.05αfine 0 while in atlant the kinetic equation is solved in the whole (where αfine is the current value of the fine-structure con- range of redshifts to determine the fractions of all considered stant). Obtained results are very similar to the results by plasma components. This explains the discrepancy between Scóccola et al. (2008). results of the present paper and Kholupenko et al. (2008) where approximation by Wong et al. (2008) has also been Acknowledgements Authors are grateful to participants investigated by using earlier version of our code (i.e. Wong- of Workshops “Physics of Cosmological Recombination” Moss-Scott approximation has been calculated, but number (Max Planck Institute for Astrophysics, Garching 2008 and of neutral hydrogen has been calculated from kinetic equa- University Paris-Sud XI, Orsay 2009) for useful discussions tion). on cosmological recombination. In Fig. 4 the relative difference of the results by atlant This work has been partially supported by Ministry and recfast for the epoch of HeIII→HeII recombination is of Education and Science of Russian Federation (contract presented. The break at z = 5000 is the numerical feature # 11.G34.31.0001 with SPbSPU and leading scientist G.G. connected with switches in recfast code again. In the range Pavlov), RFBR grant 11-02-01018a, and grant “Leading Sci- of redshifts z = 5000 − 7000 the difference between the re- entific Schools of Russia” NSh-3769.2010.2. Balashev S. A. sults by atlant and recfast arises because HeIII→HeII re- also thanks the Dynasty Foundation. combination is treated by atlant in non-equilibrium way while recfast treats this according to Saha formula. The maximum of difference is about 0.225% at z ≃ 5790. Such References difference is too small to be a reason of any observational effects at the current level of experimental accuracy, but Ali-Ha¨ımoud Y., Grin D., Hirata C. M., 2010, Phys. Rev. non-equilibrium treatment of HeIII→HeII allows us to avoid D, 82, 123502 artificial numerical features (that is important for the mod- Ali-Ha¨ımoud Y., Hirata C. 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