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Advanced Three Level Approximation for Numerical Treatment Of Mon. Not. R. Astron. Soc. 000, 1–10 (2011) Printed October 18, 2018 (MN LATEX style file v2.2) Advanced Three Level Approximation for Numerical Treatment of Cosmological Recombination Kholupenko E.E.1,2, Ivanchik A.V.1,2, Balashev S.A.1,2, and Varshalovich D.A.1,2 1Ioffe Physical-Technical Institute, St.-Petersburg 194021, Russia 2 St.-Petersburg State Polytechnical University, Russia October 18, 2018 ABSTRACT New public numerical code for fast calculations of the cosmological recombi- nation of primordial hydrogen-helium plasma is presented. The code is based on the three-level approximation (TLA) model of recombination and allows us to take into account some fine physical effects of cosmological recombi- nation simultaneously with using fudge factors. The code can be found at http://www.ioffe.ru/astro/QC/CMBR/atlant/atlant.html Key words: Keywords: cosmological recombination, CMBR, anisotropy, spectral distortion, hydrogen, helium 1 INTRODUCTION tigation of cosmological recombination and development of applied numerical codes for modelling of this process Cosmological recombination is one of the key pro- (e.g. Zeldovich et al. (1968), Peebles (1968), Matsuda et al. cesses in the early Universe. It determines the epoch of (1969), Jones & Wyse (1985), Grachev & Dubrovich (1991), decoupling of radiation from matter and thereby deter- Seager et al. (1999)). Increasing accuracy of experiments in mines the epochs at which baryonic matter can start to fall the last decade leads to increase of efforts of theorists in into gravitational potential wells created by clustering cold the study of details of cosmological recombination (see e.g. dark matter (CDM). Afterwards these CDM+baryonic mat- Dubrovich & Grachev (2005), Chluba et al. (2007, 2010) ter clouds develop into non-relativistic gravity bound sys- Chluba & Sunyaev (2009a, 2010b), Rubi˜no-Mart´ın et al. tems like galaxies (Peebles (1965, 1968); Doroshkevich et al. (2008), Switzer & Hirata (2008b), Hirata & Switzer (2008), (1967); Ma & Bertschinger (1995)). The sizes of these proto- Ali-Ha¨ımoud et al. (2010)) and perfection of numerical objects also depend on cosmological recombination via the codes. In the light of soon releases of experimental data from kinetics of divergence of radiation and matter temperatures Planck mission this task becomes more and more important arXiv:1104.3050v1 [astro-ph.CO] 15 Apr 2011 which determines the critical Jeans length. Cosmological re- and urgent (for overview of efforts of investigators to find combination affects primordial chemistry (Dalgarno & Lepp exact recombination scenario and to estimate remain uncer- (1987)) and correspondingly rate of radiative cooling of col- tainties see e.g. Sunyaev & Chluba (2009); Shaw & Chluba lapsing clouds (via emission in resonant lines of molecules (2011)). Today for successful treatment of Planck data the which depends on the abundances of primordial molecules). minimal required accuracy of numerical codes evaluating From observational point of view the cosmological re- cosmological recombination is about 0.1% (on free electron combination is also very important process because its ki- fraction) for the epoch of hydrogen recombination and 1% netics determines the position of last scattering surface for the epoch of helium recombination. Desirable accuracy (Sunyaev & Zeldovich (1970), Hu et al. (1995)). This in is about 0.01% and 0.1% correspondingly. turn affects the cosmic microwave background radiation The next possible step of the investigations of (CMBR) anisotropy which is one of the main sources of in- the early Universe in the epochs z = 800 − 104 formation about evolution of the early Universe, its com- is the experimental study of CMBR spectral distor- position and other properties. A great number of exper- tions originated from cosmological recombination of hy- iments on CMBR anisotropy have been carried out in drogen and helium. Such experiments would be pow- the last thirty years (Relikt-1 1983, COBE 1989, QMAP- erful sources of information about history of the Uni- Toco 1996, BOOMERanG 1997-2003, MAXIMA 1998-1999, verse in these epochs. Thanks to the numerous theoretical WMAP 2001-present day, and many others). Treatment works in this field (Dubrovich (1975), Lyubarsky & Sunyaev of the results of these experiments demands clear under- (1983); Fahr & Loch (1991); Rybicki & dell’Antonio (1993); standing of cosmological recombination physics. That is Boschan & Biltzinger (1998); Dubrovich & Grachev (2004); why many efforts have been made for theoretical inves- Rubi˜no-Mart´ın et al. (2008) and references therein) one can c 2011 RAS 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 de- fined 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).
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