Astronomy Reports, Vol. 47, No. 9, 2003, pp. 709–716. Translated from Astronomicheski˘ı Zhurnal, Vol. 80, No. 9, 2003, pp. 771–779. Original Russian Text Copyright c 2003 by Burgin. Hydrogen Subordinate Line Emission at the Epoch of Cosmological Recombination M. S. Burgin Astro Space Center, Lebedev Physical Institute, Russian Academy of Sciences, Profsoyuznaya ul. 84/32, Moscow, 117997 Russia Received December 10, 2002; in final form, March 14, 2003 Abstract—The balance equations for the quasi-stationary recombination of hydrogen plasma in a black- body radiation field are solved. The deviations of the excited level populations from equilibrium are computed and the rates of uncompensated line transitions determined. The expressions obtained are stable for computations of arbitrarily small deviations from equilibrium. The average number of photons emitted in hydrogen lines per irreversible recombination is computed for plasma parameters corresponding to the epoch of cosmological recombination. c 2003 MAIK “Nauka/Interperiodica”. 1. INTRODUCTION of this decay is responsible for the appreciable differ- When the cosmological expansion of the Universe ence between the actual degree of ionization and the lowered the temperature sufficiently, the initially ion- value correspondingto the Saha –Boltzmann equi- ized hydrogen recombined. According to [1, 2], an librium. A detailed analysis of processes affectingthe appreciable fraction of the photons emitted at that recombination rate and computations of the temporal time in subordinate lines survive to the present, lead- behavior of the degree of ionization for various sce- ingto the appearance of spectral lines in the cosmic narios for the cosmological expansion are presented background (relict) radiation. Measurements of the in [5, 6]. wavelengths, intensities, and profiles of these lines The second task involves computingthe rate of can provide information about the plasma tempera- emission of suprathermal photons in excess of the ture, density, and redshift correspondingto the epoch Planck background radiation, that is, the rate of of recombination, as well as the duration of this epoch. uncompensated transitions. To first approximation, In a general formulation, computing the intensities the rate of uncompensated transitions in subordinate and profiles of recombination lines requires the solu- lines is proportional to the rate of uncompensated tion of coupled non-stationary differential equations transitions 2 ⇒ 1.Hereandbelow,nu ⇒ nl denotes describingthe radiative transfer and the evolution of the transition between states with the principal quan- the distributions of atoms over their energy levels. tum numbers nu and nl. The technique of computing However, in reality, the problem is drastically simpli- the proportionality coefficient by modelinga random fied by the smallness of two dimensionless parame- walk of electrons between energy levels was proposed ters: the ratio of the numbers of baryons and photons in [7], while [8, 9] present an approximate analytical and the ratio of the relaxation time for the distri- solution for the correspondingequation. bution of hydrogen atoms over levels with principal quantum numbers n ≥ 2 to the characteristic time Finally, the third task of solvingfor the non- for variations of the degree of ionization. These small stationary radiative transfer in the recombination parameters enable us to separate the computation of lines must be carried out in order to compute the the emission coefficients into three subtasks that can distortions in the contemporary relict-radiation spec- be considered independent in a first approximation. trum. Since these distortions are extremely weak and their reverse effect on the recombination and First, the recombination kinetics can be computed population of excited levels is negligible, computing for a given baryon density, radiation temperature, and the radiation intensity reduces to integrating the cosmological expansion rate. According to [3, 4], the functions determined previously. two-photon decay of the 2S state is the main process responsible for the growth of the ground-state popu- The present work is devoted to solvingthe second lation of the hydrogen atoms with time, since the Uni- task. A numerical solution of the balance equation verse is virtually completely opaque in the resonance for the recombination of plasma exposed to the relict lines and Lyman continuum, and the small probability radiation yields the populations of excited hydrogen 1063-7729/03/4709-0709$24.00 c 2003 MAIK “Nauka/Interperiodica” 710 BURGIN levels and the rates of uncompensated transitions in boundary beingdetermined by the parameters M and subordinate lines. M ,where2 <M≤ M .Forn(i) ≤ M, we deter- mine the densities Ni from the stationarity conditions, (0) ≤ 2. FUNDAMENTAL ASSUMPTIONS and assume Ni = Ni for M<n(i) M . In the approximation used, the stationarity condi- We adopt below the physical scenario proposed tions take the form in [7, 8]. An ensemble of hydrogen atoms located in discrete levels with n ≥ 2 is considered. The ensemble K − interacts with a blackbody radiation field and hydro- RjiNj LiNi + Bi =0, (1) gen plasma, with the radiation and plasma tempera- j=1 tures specified to be equal. The populations in levels where K = M − 1 is the order of the stationarity with the same n but different orbital quantum num- equations solved below. The term RjiNj describes bers l are proportional to the correspondingstatistical transitions within the ensemble, L N processes that weights. i i extract atoms from the ensemble, and Bi processes Radiative (both spontaneous and stimulated) that add new atoms to the ensemble. transitions and transitions induced by inelastic elec- The coefficients of this equation are determined by ≥ tron collisions occur between levels n 2. Radiative the relations recombinations to levels n ≥ 2 add new atoms to r e the ensemble, while photoionization by the relict Rji = Rji + Rji (i = j), (2) r radiation removes atoms from the ensemble. Rji = An(j)n(i) (1 + nph(Eij)) (j>i), (3) In addition, emptyingand replenishment of the r (0) r (0) ensemble occur due to transitions to and from the RjiNj = RijNi (i = j), (4) n =1level. Photons emitted in resonance lines are Re = N q (j<i), (5) absorbed almost immediately, with the transition ji e n(j)n(i) e (0) e (0) rates of emission and absorption beingequal for each RjiNj = RijNi (i = j), (6) L , L ,... line, so that these processes completely β γ i−1 K compensate each other. Rii = − Rji − Rji, There is not complete compensation for L ,and α j=1 j=i+1 the net effect of these transitions is modeled by the ⇒ (0) (1) outflow 2 1, with the rate beingproportional to the Li = Li + σLi , (7) population in the second level. In the approximation M used, this outflow of atoms from the ensemble is re- L(0) = R (T )+ R , sponsible for the nonequilibrium distribution of atoms i n(i)c ik over the n ≥ 2 levels and, ultimately, for the formation k=M+1 of lines. 1 for i =1 L(1) = The relaxation of the atomic distribution within the i 0 for i>1 n ≥ 2 ensemble is determined by allowed transitions, −8 with characteristic times ti ≈ 10 s. The character- and istic time for the evolution of external processes coin- (0) (0) cides with the duration of the epoch of recombination Bi = Li Ni . (8) 4 5 te;thatis,10 –10 yrs. The small ratio ti/te enables Here, Aul is the Einstein coefficient for the sponta- us to consider the distribution of the ensemble atoms neous transition u ⇒ l, nph(E) is the number filling over their levels to be stationary. factor for relict photons with energy E, Eij is the energy of the transition n(i) ⇒ n(j), qlu is the coeffi- l ⇒ u 3. INPUT EQUATIONS cient for collisional excitation of the transition , ∞ Let Ne be the electron density, T the tempera- Rnc = c an(Eph)ρph(Eph)dEph (9) ture of the recombiningplasma, and Ni the number density of atoms excited to the level with principal E(n) (0) quantum number n(i)=i +1.WedenoteNi to be is the rate of relict photoionization from the level n, c is the correspondingnumber density for the case when the speed of light, a is the cross section for ionization ≥ n the plasma, radiation, and atoms with n 2 are in full from level n, Eph is the photon energy, ρph(Eph) is thermodynamic equilibrium. the spectral density of relict photons with temperature In accordance with [5, 7], we will divide the atomic T , Rcn is the coefficient for recombination to the hydrogen levels considered into two groups, with their level n (takinginto account transitions stimulated by ASTRONOMY REPORTS Vol. 47 No. 9 2003 HYDROGEN SUBORDINATE LINE EMISSION 711 the relict radiation), and E(n) is the photoionization where m =0, 1. With this notation, Eqs. (1) take the threshold for the level with principal quantum number form n. Equations (4), (6), and (8) result from the principle (Q − σL(1))N + B =0, (10) of detailed equilibrium. (0) where Q = R† − L0 and R† denotes the transposi- The quantities Li describe those processes that remove atoms from the ensemble and are precisely tion of the matrix R. It can easily be shown that compensated by the addition of new atoms to the Q − σL(1) is a nondegenerate matrix for any σ ≥ 0, ensemble in a complete equilibrium, while Eq. (7) so that Eq. (10) has the unique solution formally expresses the fact that the uncompensated − N(σ)=−(Q − σL(1)) 1B. (11) outflow from the level i =1(i.e., n =2)attherate (1) After N(σ) is found, the rate of uncompensated ra- σL1 N1 = σN1 is the only origin of deviations from equilibrium. diative transitions Zij takes the form ⇒ r − r If the two-photon transition 2 1 were the sole Zij = NiRij NjRji. (12) process resultingin transitions between the ground state and excited levels, and the populations of sub- Formally, (11) and (12) enable us to compute the levels with various angular momenta were propor- rate of uncompensated radiative transitions for ar- tional to the correspondingstatistical weights,the bitrary plasma parameters, and thereby obtain a full value of σ could be easily computed in the explicit solution of the problem in the framework of the ap- form σ = g2SA2S⇒1S/(g2S + g2P).