Effective conductance method for the primordial recombination spectrum
Yacine Ali-Ha¨ımoud∗ Institute for Advanced Study, Einstein Drive, Princeton, New Jersey 08540 (Dated: June 7, 2021) As atoms formed for the first time during primordial recombination, they emitted bound-bound and free-bound radiation leading to spectral distortions to the cosmic microwave background. These distortions might become observable in the future with high-sensitivity spectrometers, and pro- vide a new window into physical conditions in the early universe. The standard multilevel atom method habitually used to compute the recombination spectrum is computationally expensive, im- peding a detailed quantitative exploration of the information contained in spectral distortions thus far. In this work it is shown that the emissivity in optically thin allowed transitions can be fac- tored into a computationally expensive but cosmology-independent part and a computationally cheap, cosmology-dependent part. The slow part of the computation consists in pre-computing temperature-dependent effective “conductances”, linearly relating line or continuum intensity to departures from Saha equilibrium of the lowest-order excited states (2s and 2p), that can be seen as “voltages”. The computation of these departures from equilibrium as a function of redshift is itself very fast, thanks to the effective multilevel atom method introduced in an earlier work. With this factorization, the recurring cost of a single computation of the recombination spectrum is only a fraction of a second on a standard laptop, more than four orders of magnitude shorter than standard computations. The spectrum from helium recombination can be efficiently computed in an identical way, and a fast code computing the full primordial recombination spectrum with this method will be made publicly available soon.
I. INTRODUCTION photons. The seminal papers on primordial recombina- tion by Peebles and Zeldovich et al. [9, 10] already eval- uated the distortion due to Ly-α transitions and 2s 1s A significant part of our knowledge about the universe − at early times and on large distance scales is derived two-photon decays. Even though this represents a large from the observation and analysis of its spatial inhomo- distortion to the Wien tail of the CMB blackbody spec- geneities. In particular, observations of the temperature trum, it lies many orders of magnitude below the cosmic and polarization anisotropies of the cosmic microwave infrared background (CIB, [11]), which renders its detec- background (CMB) have allowed cosmologists to deter- tion very challenging, if not hopeless. mine the geometry, contents, and initial conditions of the In addition, exactly one free-bound photon per hydro- universe to an exquisite level of precision (see for example gen atom and two per helium atom were emitted, as well Ref. [1]). as a few bound-bound photons per atom from transi- 1 Additional, and perhaps complementary information, tions between highly excited states [12]. It was first is hidden in the tiny but unavoidable spectral distortions pointed out by Dubrovich [13] that these primordial re- to the nearly perfectly thermal radiation background combination lines could be observed today as broadened [2, 3]. On the one hand, broad spectral distortions can be features in the centimeter to decimeter wavelength range. generated by energy injection in the early universe, tak- A few authors have since then tackled the computation ing the form of chemical potential (µ type) or Compton of the recombination spectrum, with various degrees of (y type) distortions (and more generally− continuously approximation or numerical convergence [14–17], and it interpolating− between these two analytic cases [4]), de- is only quite recently that highly-accurate computations pending on the redshift of energy injection. Physical were carried out by Rubi˜no-Mart´ın, Chluba and Sunyaev mechanisms causing such energy injections include the [12, 18–20]. dissipation of small-scale acoustic waves, which occurs in Thus far the study of the primordial recombination arXiv:1211.4031v1 [astro-ph.CO] 16 Nov 2012 the standard cosmological picture (see e.g. [5]), and pos- spectrum has remained the niche domain of a few afi- sibly the decay or annihilation of dark matter into stan- cionados. First and foremost, the observational prospects dard model particles that then deposit their energy into may seem meager, as the predicted signal in the cm- the primeval plasma [6]. A vast literature exists on the wavelengths lies a billion times below the undistorted subject, and we refer the interested reader to the recent CMB spectrum, far below the current best upper bounds works [4, 7, 8] and references therein. on spectral distortions from FIRAS [21]. In addition, On the other hand, the primordial recombinations of galactic and extragalactic foregrounds would need to be helium and hydrogen lead to a few distortion photons per atom, in the form of free-bound and bound-bound
1 Throughout this paper we refer to bound-bound transitions be- tween “highly excited” states as those for which the lower state ∗ [email protected] itself is excited, n0 → n ≥ 2. 2 understood and subtracted very precisely. Finally, the Currently the fastest code using this method takes about machinery required to compute a recombination spec- one hour for nmax = 100 and one full day for nmax = 250 trum with the standard multilevel method is, although on a standard laptop, with the computation time scaling conceptually not very difficult, somewhat cumbersome to as t n3.7 [28]. ∝ max implement and too computationally expensive for a sys- The method that we introduce here allows us to fac- tematic analysis of its information content. torize the problem into a computationally expensive part The reward in finding such a needle-in-a-haystack sig- (for which large linear systems need to be solved) that nal is, however, potentially significant. Ref. [22] showed is cosmology-independent and can be pre-computed once that the recombination spectrum is a sensitive thermome- and for all, and a computationally cheap part that does ter and baryometer. It could also provide a clean mea- depend on cosmology. This method builds on and ex- surement of the primordial helium abundance, before the tends the effective multilevel atom (EMLA) method in- formation of the first stars [3, 20]. Finally, pre-existing troduced in an earlier work [29] (hereafter AH10; see also spectral distortions could lead to a significant increase Refs. [30, 31]); it is, however, not a trivial extension, of the recombination radiation even if the initial distor- since the EMLA method is designed for computing the tions are small in absolute value [15, 23]. The recombina- free electron fraction xe(z) and essentially collapses all tion spectrum could therefore be a probe of non-standard the transitions between excited states into effective tran- physics such as dark-matter annihilations [24]. sition rates into and out of 2s and 2p. In this process, Let us point out, in addition, that technological ad- information not directly necessary to the evolution of the vances should make it possible to reach sensitivities three free electron fraction is lost, whereas our present goal is orders of magnitude below that of FIRAS, corresponding to go beyond xe(z) and compute the full recombination 8 to distortions at the level of 10− (see the proposal for ∼ spectrum. the PIXIE instrument [25]). Spectral distortions from re- We shall lay down our method in detail in the remain- combination are only one order of magnitude weaker than 8 der of this paper, but the main idea can be intuitively this sensitivity (and even get to the 10− level around 10 understood if one pictures the system of radiatively con- GHz [12]), and it is not unlikely that they will be within nected excited levels as a circuit, where the currents are reach of the next generation instruments. The feasibility the line intensities, the voltages are the departures of the of foreground subtraction at the required level has yet excited states populations from Saha equilibrium, and to be demonstrated, but one may hope that the spatial the conductances are the transition rates (this insight- isotropy of the signal and its very specific spectral fea- ful analogy is due to Chris Hirata [32]). The linearity of tures should allow us to disentangle it from foreground Kirchhoff’s laws (the steady-state rate equations for the emission. excited state) ensures that the “current” in any transi- Before any of the truly challenging issues of instrumen- tion is proportional to the outer “voltages”, i.e. the de- tal sensitivity and foreground subtraction are addressed, partures from Saha equilibrium of the 2s and 2p states. it seems that the first task is to undertake a detailed The proportionality coefficients, which we shall call effec- quantitative study of the information content of the re- tive conductances, moreover only depend on the temper- combination spectrum. In order to do so, a fast and ature of the ambient blackbody radiation that is nearly accurate computational method is required, so that the undistorted at the relevant frequencies (this can in princi- large space of cosmological parameters can be efficiently 2 ple be generalized to include simple parameterizations of explored . Introducing such a fast method is the purpose the ambient spectrum, as well as collisional transitions). of the present work. Once the effective conductances are pre-computed as a The main task in obtaining the spectrum resides in function of transition energy and temperature, the re- computing the populations of the excited states, or, more combination spectrum can be computed very efficiently precisely, their small departures from equilibrium with for any cosmology, by using the EMLA method to eval- one another, since the line and continuum emissions are uate the recombination history and the outer “voltages” proportional to the latter. High-precision spectra require as a function of redshift. This method is very similar in accounting for excited states up to principal quantum spirit to the widely used line-of-sight integration method number nmax of a few hundred, resolving the angular mo- for CMB anisotropies [33], which factorizes the compu- mentum substates. With the standard multilevel atom 2 2 tation of the CMB power-spectrum into a geometric, method, one has to invert a large nmax nmax matrix 2 × 2 cosmology-independent part and a cosmology-dependent at each timestep in order to compute the populations of but multipole-independent source term. We illustrate our the excited states (although this matrix is sparse due to method graphically in Fig. 1. selection rules, so only (n3 ) elements are nonzero). O max This paper is organized as follows. In Sec. II, we write down the general equations that need to be solved for the computation of the recombination spectrum. The effec- 2 It was brought to my attention by J. Chluba that Fendt [26] con- tive conductance method is described in Sec. III. We dis- ducted a preliminary study of cosmological parameter estimation cuss our numerical implementation and results in Sec. IV from spectral distortions, using the fast interpolation algorithm and give our conclusions and future research directions Pico [27]. in Sec. V. 3
+ e + p B. Bound-bound emission from transitions between excited states
n0 The emissivity due to bound-bound transitions be- I tween excited states is given by V n0n V x2s x n 2p hν R(Tr) j (ν) = n bb H 4π × X X [xn l Rn l nl xnlRnl n l ] δ(ν νn n),(1) 0 0 0 0→ − → 0 0 − 0 2s 2p 2 n where ν is the frequency of the n0l0 nl transitions n0n → and Rnl n l represents the radiative transition rate from → 0 0 nl to n0l0. The recombination process adds at most a few 1s photons per atom, and the transitions between excited states are mostly below the peak of the blackbody spec- FIG. 1. Schematic representation of the main idea of this trum (except for Balmer transitions, but their energy is paper, highlighting the circuit analogy of Hirata [32]. All only a few times above the blackbody peak, where there transitions between excited states are mediated by blackbody is still a very large number of thermal photons per hydro- photons and their rates (and corresponding resistances R(Tr) gen atom). As a consequence, the radiation field medi- shown as solid resistor symbols) only depend on the blackbody ating the transitions is, to 10 8 accuracy, a blackbody temperature. The linearity of the steady-state rate equations − at temperature T . One can∼ check the validity of this as- (Kirchhoff’s laws in the circuit analogy) ensures that the cur- r rent I in any transition between excited states is linear in sumption a posteriori once the distortions are computed n0n (see for example Fig. 2 of Ref. [18]). Note, however, that the departures from Saha equilibrium ∆x2s and ∆x2p, which play the role of externally imposed voltages. The proportion- small y-distortions to the blackbody spectrum (at the 6 ality coefficient is a function of temperature only. The values level of y 10− ) may significantly enhance the hydro- ∼ of ∆x2s and ∆x2p are determined from the balance between gen and helium line emission [15, 23]. We defer the study the net downward current from the highly excited states (ob- of the effect of such pre-existing distortions on the recom- tained through the effective rates defined in AH10) and the bination spectrum to future work, and shall here assume non-thermal transition rates to the ground state (represented that transitions between excited states are mediated by by dashed resistor symbols). We have not represented all al- thermal photons only. lowed transitions in order to keep the graph readable. With this assumption, the radiative transition rates satisfy the detailed balance relations, qnl Rn l nl = Rnl n l , (2) 0 0→ → 0 0 qn0l0 II. GENERAL EQUATIONS where we have defined En/kTr q (2l + 1)e− , (3) nl ≡ A. Notation 2 where En 13.6 eV/n is the (negative) energy of the bound states≡ − with principal quantum number n. We see that if the excited states were in Boltzmann We denote by nH the total number density of hydro- equilibrium, so that xn0l0 /xnl = qn0l0 /qnl, the net emis- gen in all its forms (ionized and neutral), xe the ratio of sivity would vanish. The emissivity therefore scales lin- the free-electron abundance to the total hydrogen abun- early with the small departures from equilibrium. To dance, xp the fraction of ionized hydrogen, and xnl (or make this apparent, let us define the departures from in some cases Xnl) the fractional abundance of neutral Saha equilibrium at temperature Tr with the free elec- hydrogen in the excited state of principal quantum num- tron and protons, ber n and angular momentum number l. The matter qnl and radiation temperatures are denoted by Tm and Tr, ∆xnl xnl nHxexp, (4) ≡ − qe respectively. We denote emissivities by jν (with units of energy per unit time per frequency interval per unit where we have defined volume per unit solid angle) and specific intensities by Iν (with units of energy per unit time per frequency interval 2πm kT 3/2 q e r , (5) per unit area per unit solid angle). All our derivations e ≡ h2 are for hydrogen atoms but the generalization to helium is straightforward. We only consider radiative transitions where me is the reduced mass of the electron-proton sys- here and neglect the effect of collisions. tem. We can now rewrite the bound-bound emissivity in 4 the following form: free-bound emissivity would vanish if the excited states were in Saha equilibrium with the ionized plasma and if hν j (ν) = n the matter and radiation temperature were identical. We bb H 4π × can rewrite the free-bound emissivity in terms of small X X qnl departures from equilibrium as follows: ∆xn0l0 ∆xnl Rnl n0l0 δ(ν νn0n),(6) qn l − → − 2 n The emissivity due to free-bound transitions to excited D. Radiative transfer equation states is given by Strictly speaking, the computation of the specific in- hν X X dαnl dβnl j (ν) = n n x x x , (7) tensity requires the knowledge of not only the emissivity fb H 4π H e p dν − nl dν but also the absorption coefficient [36]. However, the n 2 l III. THE EFFECTIVE CONDUCTANCE ∂αK X 0 = nHxexp∆T + ∆xiRi K METHOD − ∂T Tm=Tr → m i X + ∆XLRL K ∆XK ΓK . (19) A. Populations of the excited states → − L=K 6 In order to obtain the bound-bound and free-bound In the standard multilevel atom method, the large sys- emissivities, we see that we need to evaluate the popula- tem (19) is solved at every timestep, which makes the tions of the excited states, or, more precisely, their small computation very slow. departures from Saha equilibrium with the plasma. Following AH10, we define the matrix M of elements The populations of excited states can be obtained to MKL ΓK δKL (1 δKL)RK L. (20) very high accuracy in the steady-state approximation, be- ≡ − − → cause of the large ratio of internal transition rates to the X overall recombination rate. This assumption was checked We also define the vector ∆ of elements ∆XK and the S explicitly in Ref. [28] and found to be extremely accurate, source vector ∆ of elements for the computation of both the recombination history ∂αK X and the recombination spectrum. ∆SK nHxexp∆T + ∆xiRi K . (21) ≡ − ∂Tm Tm=Tr → Following AH10, we separate the excited states in “in- i terior” states, only connected radiatively to other excited The system (19) can be rewritten in compact matrix form states and the continuum, and “interface” states, essen- as tially 2s and 2p (and potentially any additional “weak interface” states, such as 3s, 3p, 3d...), which are radia- M T(∆X) = ∆S, (22) tively connected to the ground state. We denote the pop- ulations of the former by a capital XK , where K stands where M T is the transpose of M. We showed in AH10 for both quantum numbers of the state, and those of the that the matrix M is nonsingular, and this system has latter by xi, where i = 2s, 2p (and 3s, 3p, 3d... if needed). the formal solution The steady-state rate equation for the population of T 1 1 T the interior state K is ∆X = (M )− (∆S) = (M − ) (∆S), (23) ˙ X 0 XK = nHxexpαK + xiRi K i.e., explicitly, ≈ → i X 1 X ∆XK = (M − )LK ∆SL + XLRL K XK ΓK , (16) → − L L=K 6 X 1 ∂αL = nHxexp∆T (M − )LK where αK (Tm,Tr) is the total recombination coefficient to − ∂Tm Tm=Tr the state K (accounting for stimulated recombinations) L X X 1 and + ∆xi (M − )LK Ri L. (24) → X X i L ΓK βK + RK L + RK i (17) ≡ → → We showed in AH10 that detailed balance relations be- L=K i 6 tween radiative transition rates ensure that is the total rate of transitions out of the state K, where 1 qK 1 (M − )LK = (M − )KL. (25) βK is the total photoionization rate from K. Note that qL here again we have assumed that the Sobolev escape probability is unity in all transitions (or that the Sobolev Using the detailed balance relation for Ri L, we may → optical depth is zero). If this were not the case the net rewrite the last term of Eq. (24) as transition rates would depend non-linearly on the state X 1 qK X 1 populations, which would significantly complicate mat- (M − )LK Ri L = (M − )KLRL i → qi → ters. L L Provided the transitions between excited states are me- qK i PK , (26) diated by blackbody photons, the transition rates satisfy ≡ qi the detailed balance relation Eq. (2). The recombination where P i is the probability that an excited atom initially coefficients and photoionization rates are related through K in the interior state K eventually reaches the interface qK state i (after possibly many transitions in the interior), αK (Tm = Tr) = βK (Tr). (18) qe the complementary events being to eventually reach one 6 of the other interface states or to be photoionized. The where we have defined the coefficients probabilities P i were defined in AH10, where they were K X qnl h i e (T ) P˜e P˜e R , (32) an intermediate step to compute the effective recombina- n0n r n0l0 nl nl n0l0 G ≡ qe − → tion coefficients to and effective transition rates between l,l0 the interface states. In the last line of Eq. (26), we have X qnl i (T ) P i P i R . (33) n0n r nl n0l0 nl n0l0 used the formal solution of the defining equation for the G ≡ qi − → i l,l0 probabilities PK . Let us now deal with the first term of Eq. (24). We Note the minus sign in Eq. (31): we have chosen this con- define the dimensionless coefficients vention because the excited states are in general under- ∂ log α populated with respect to Saha equilibrium, and the co- γ K . (27) K efficients n n defined in Eq. (33) are positive (in general ≡ − ∂ log Tm Tm=Tr 0 the probabilityG of reaching interface states decreases as Using again detailed balance relations, we obtain n increases). If one thinks of the departures from Saha equilibrium X ∂αL 1 qK X M 1 M 1 ( − )LK = ( − )KLβLγL ∆xi as voltages and of the net decay rate in the n0 n ∂Tm Tm=Tr −Tr qe → L L transitions as a current, then the coefficients i can be Gn0n 1 qK ˜e thought of as effective conductances linearly relating the PK , (28) ≡ −Tr qe two. A similar analogy can be made for the first term: there, the voltage is the fractional temperature difference where the last equality defines the dimensionless coeffi- e ∆T/T and the conductance would be nHxexp . Note ˜e n0n cient PK . If all the coefficients γL were equal to unity, e 3 1 G i e e that the have units of cm s− whereas the have then we would have P˜ = P , the probability that an G1 G K K units of s− . excited atom initially in the interior state K eventually Similarly, the net free-bound decay rate per unit fre- gets photoionized before reaching an interface state. In quency can be rewritten as general, however, γL = 1 (but is in general positive and 6 ˜e of order unity), so the numbers PK do not have a clear X X qnl ∆T dβnl nHxexpγn(ν) ∆xnl physical significance but are numerically of the same or- qe T − dν e n 2 l qnl ∆T X qnl ∆x = P˜e n x x + P i ∆x . (30) nl q nl H e p T q nl i e i i C. Bound-bound and free-bound emissivities B. Effective conductances The expressions for the emissivities can now be rewrit- ten in terms of the effective conductances: We can now rewrite the net decay rate in the n0 n hν → jbb(ν) = nH transitions (with n < n0), per hydrogen atom, in the form 4π × " # X ∆T X X qnl e n x x i ∆x δ(ν ν )(37), ∆xn l ∆xnl Rnl n l n0n H e p n0n i n0n 0 0 → 0 0 G T − G − qn0l0 − n We can rewrite the total emissivity formally as of the problem, as they require solving large (of order n2 /2 n2 /2) matrix equations. Due to selection hν max × max jν jbb(ν) + jfb(ν) = nH rules for radiative transitions, these large matrices are ≡ 4π × 3 " # very sparse, and only have of order nmax nonvanishing d e ∆T X d i elements. We can therefore use a sparse matrix tech- G (ν, T ) n x x G (ν, T )∆x , (39) dν r H e p T − dν r i nique identical to the one introduced in Ref. [38]. i We show some of the computed effective conductances where in Fig. 2, for a temperature Tr = 3800 K (corresponding e,i e,i roughly to the peak of the emission). d X e,i d fb G δ(ν νn n) + G . (40) dν ≡ Gn0n − 0 dν n where νb are the bin centers, and The computation of the effective conductances requires e,i solving large (formally infinite) linear systems. One must e,i X e,i d fb 1b(νn n) + ∆νb G (νb), (42) impose some truncation criterion to make the system fi- Gb ≡ Gn0n 0 dν n 4 Other truncation schemes could be imagined, such as assuming 5 This can be understood from Eq. (13): a fractional error that the excited states above some threshold are in Saha equi- ∆ log ν in the rest-frame frequency translates into the same frac- librium with the continuum; in practice, as long as a clear con- tional error in the redshift of emission, hence a fractional error −1 vergence is exhibited with increasing nmax, we need not worry ∆ log ν/(Hτ) on the emissivity, where τ ∼ H /few is the char- about the detailed truncation prescription. acteristic time of evolution of the populations. 3 8 2 3 4 10 Balmer T = 3800 K r Paschen 1 nmax = 100 ) 2 1 2 nmax = 200 10 (s transitions n ↵ 0 p 2 transitions n G 3 4 0 + 100 n 0 0 s 20 40 60 80 100 2 1 n G 1 4 10-2 0 0 20 40 60 80 100 10-4 0.00001 0.00010 0.00100 0.01000 0.10000 1.00000 10.00000 h⌫n0n (eV) Tr = 3800 K Balmer 10000 nmax = 100 Paschen ) nmax = 200 1 (s 1000 p 2 fb ⌫ G d d ⌫ 3 4 + 100 s 2 fb ⌫ G d d ⌫ 1 4 10 1 0.001 0.010 0.100 1.000 10.000 h⌫nh0n⌫ (eV) FIG. 2. Upper panel: Weighted average of effective conductances 1 G2s + 3 G2p at T = 3800 K, as a function of transition 4 n0n 4 n n energy hν , for n = 100 (open circles) and n = 200 (dots). The nearly vertical0 families of points correspond to a given n0n max max series n0 → n with fixed n, such as the Balmer series n0 → 2, the Paschen series n0 → 3, etc..., with n increasing from right to left. The families of diagonal lines correspond to a given order (n + ∆n) → n with fixed ∆n, such as the α−transitions (n + 1) → n, the β−transitions (n + 2) → n etc..., with lower conductances for higher order transitions. Lower panel: Weighted 2s 2p 1 dGfb 3 dGfb average of effective free-bound conductances per log-frequency interval 4 ν dν + 4 ν dν at T = 3800 K, as a function of photon energy hν, for nmax = 100 (dotted line) and nmax = 200 (solid line). The edges correspond to photoionization thresholds from various shells. We only show the effective conductances for these two values of nmax in order not to cluster the figure; proper convergence with nmax is discussed in Section IV. 9 from Fig. 2, effective conductances indeed decrease with the effective recombination coefficients (T ,T ), pho- Ai m r increasing transition order (see also Table 1 of Ref. [18]). toionization rates i(Tr) and transition rates i j(Tr) With this discretization, the emissivity for bound- for i, j = 2s, 2p. AllB the complication of theR recombi-→ bound and free-bound transitions between excited states nation computation then resides in the slow transitions becomes to the ground state, where a proper time-dependent ra- diative transfer calculation is required for percent-level 1 high-n hν X n e ∆T jν used = b nHxexp precision. Here we are not worried about such subtleties, nH 4π × G T b and use the following simple prescriptions for decay rates X i o to the ground state. For the two-photon transitions from ∆xi δ(ν νb).(43) − Gb − 2s, we neglect stimulated decays and absorptions of non- i=2s,2p thermal photons, so the net decay rate is h i E21/kTr C. Lyman-α and 2s − 1s emission x˙ 1s = x˙ 2s = Λ2s,1s x2s x1se− , (48) 2γ − 2γ − 1 The net emissivity in the Lyman-α line is given by where Λ 8.22 s− is the spontaneous two-photon 2s,1s ≈ decay rate. For the net decay rate in the Lyman-α line, jν Lyα hν hνLyα/kTr we use the Sobolev approximation, = x 3x e− n 4π 2p − 1s H h i E21/kTr PescA2p,1sδ(ν νLyα), (44) x˙ 1s = x˙ 2p = RLyα x2p 3x1se− , × − Lyα − Lyα − where Pesc is the escape probability for the optically thick (49) where R A P is the net decay rate in the Lyman-α line. In the Sobolev approximation (see for Lyα ≡ 2p,1s esc example Ref. [41] for a detailed derivation), it is given by Ly-α line, accounting for the small escape probability of resonant photon, given in Eq. (45). 8πHν3 The steady-state rate equations for 2s and 2p then read P = . (45) esc 3c3n x A H 1s 2p,1s E21/kTr 0 x˙ = n x x + x e− Λ ≈ 2s H e pA2s 1s 2s,1s The 2s 1s two-photon emissivity is given by + x2p 2p 2s Γ2sx2s, (50) − R → − E21/kTr jν 0 x˙ 2s = nHxexp 2p + 3x1se− RLyα 2γ hν dΛ2s,1s ≈ A = [x2s(1 + fν )(1 + fν ) x1sfν fν ] , 0 0 + x2s 2s 2p Γ2px2p, (51) nH 4π dν − R → − (46) where the effective inverse lifetimes of the 2s and 2p where ν0 = νLyα ν and fν , fν are the values of the pho- − 0 states are given by ton occupation number at ν, ν0. This expression properly accounts for stimulated decays and absorption of non- Γ2s 2s + 2s 2p + Λ2s,1s, (52) thermal photons (emitted in the Lyα line for example). ≡ B R → Γ2p 2p + 2p 2s + RLyα. (53) In order to be consistent with our simple “standard” ≡ B R → treatment of two-photon decays (see next section), we This simple 2 by 2 system (which is exact in the limit shall neglect these two effects here and use the approxi- that 2s and 2p are the only interface states) can be solved mate expression analytically. Using detailed balance relations satisfied by the effective rates, we find that the departures from Saha jν h i 2γ hν dΛ2s,1s hνLyα/kTr x2s x1se− . (47) equilibrium are given by nH ≈ 4π dν − 2p 2s R → s2s + s2p Γ We approximated the differential two-photon decay rate ∆x = 2p (54) 2s 2p 2s dΛ2s,1s/dν with the fitting formula of Ref. [42]. R → Γ2s 2s 2p Γ Note that neither Eq. (44) nor Eq. (47) are accurate − R → 2p 2s 2p s + s R → at the percent level. If needed, it would be relatively 2p 2s Γ2s ∆x2p = , (55) straightforward to include the appropriate corrections. 2s 2p Γ2p 2p 2s R → − R → Γ2s where we have defined D. Fast part of the computation E21/kTr s n x x ∆ + ∆x e− Λ , (56) 2s ≡ H e p A2s 1s 2s,1s E21/kTr In this section we briefly recall how the recombination s2p nHxexp ∆ 2p + 3 ∆x1s e− RLyα, (57) history can be very efficiently computed with the EMLA ≡ A method [29] and give explicit equations for the “voltages” where ∆ i i(Tm,Tr) i(Tr,Tr) and ∆x1s is the de- that source the emissivities. parture ofA the≡ Aground state−A population from Saha equilib- The EMLA formulation of the problem collapses all rium with the plasma, defined as in Eq. (4). Similar ex- the fast and thermally-mediated interior transitions into pressions can easily be obtained for the departures from 10 Boltzmann equilibrium with the ground state, needed for procedure was used (with additional complications due to the Lyman-α and 2s 1s emissivities. large optical depths and frequency diffusion) in Refs. [35, The rate of change− of the free-electron fraction can be 37]. This method forces the timestep ∆ log a to be no obtained, for example, from the departures from Saha greater than the smallest bin separation ∆ log ν. equilibrium computed above: In our fiducial computation, we have grouped effective conductances in bins of width ∆ log ν = 10 2 (and we re- X − x˙ = [n x x ∆ ∆x ] . (58) call that the central frequency assigned to each bin is cho- e − H e p Ai − iBi i=2s,2p sen to coincide with the frequency of the lowest-order line 3 it contains), and used a time-step ∆ log a = 5 10− . We Finally, the matter temperature is obtained from the have checked that reducing the bin width and× timestep Compton-heating equation, by a factor of 10 leads to maximum changes of at most 4 a percent over the whole range of frequencies considered, 8xeσTarTr T˙m = 2HTm + (Tr Tm), (59) with a root-mean-square difference of the order of 0.15%. − 3(1 + xe + fHe)mec − We also checked that the spectrum is converged with re- spect to the redshift range over which it is computed – where σ is the Thomson cross-section, a is the radiation T r this stems from the fact that the emissivities are rela- constant, m is the electron mass and f is the He:H e He tively sharply peaked around z 1300, as we show in abundance ratio. At high redshifts where T is locked to m the next section. ∼ Tr, one can use the quasi-steady-state solution [37] ∆T 3H(1 + x + f )m c e He e . (60) 4 F. Results T ≈ 8xeσTarTr As an aside, we point out that it is straightforward to include the effect of dark matter annihilations in this All quantities shown in this section are evaluated for a system of equations (see for example Refs. [24, 43]). One flat universe with fiducial cosmological parameters con- sistent with the latest WMAP results [1]: Tcmb = 2.728 simply has to add an additional photoionization term to 2 2 2 the free-electron fraction evolution equation, a heating K, Ωbh = 0.022, Ωmh = 0.13, ΩΛh = 0.34, YHe = term to the matter-temperature evolution equation, and 0.24, Nν,eff = 3.04. properly account for the additional excitations when solv- In Fig. 3, we show the number of photons emitted ing for the populations of the excited states. per hydrogen atom per logarithmic redshift interval or per logarithmic interval of observed frequency, for the first few transitions of the Balmer series, and for the α- E. Numerical solution of the radiative transfer transitions of the first few series. We see that in general equation the number of emitted photons decreases rapidly with the order of the transition within a series, and decreases as We first solve for the ionization history and matter well (but less rapidly) for higher series. Note that some temperature as described in Section IV D, using the code transitions may show absorption, as the Hβ transition HyRec in EMLA mode, which we have adapted to also [18]. We find that a total of 0.63, 0.019, 0.036, 0.32 and extract the populations of the excited states (more pre- 0.13 photons are emitted per hydrogen atom in the Hα, cisely, their departures form equilibrium). We store these Hβ,Hγ,Pα and Brα transitions, respectively. quantities on a fine redshift grid ranging from z = 2500 Figure 4 shows the convergence of the high-n bound- to z = 400 for future interpolation. bound and free-bound spectra with nmax. We find that We follow the spectral distortion from z = 2500 to the fractional difference in the total spectrum between 5 nmax = 250 and nmax = 500 is less than a percent for z = 400 on a constant energy range 10− eV < E < 10.2 eV, where the upper limit corresponds to the Lyman-α ν & 0.5 GHz. Nothing formally limits us from going frequency. We assume a purely thermal spectrum blue- beyond nmax = 500 with our method, since the tabu- ward of Ly-α. Below z = 400, we set the emissivity to lation of effective conductances needs to be done only zero and freely redshift the distortion down to z = 0. once. However, consistently computing the spectrum at We discretize the 2s 1s emissivity on the same grid low frequencies would also require accounting for free- as the high-n transitions− emissivity. Our total discretized free absorption and collisional transitions [19, 28], which emissivity therefore has the form we do not include here. We therefore limit ourselves to nmax = 500 for now, keeping in mind that the spectrum X obtained is not accurate below a few tenths of GHz. jν = Jbδ(ν νb). (61) used − Figure 5 shows the total recombination spectrum to- b day, as well as its subcomponents: bound-bound, free- At each time step, we first redshift the pre-existing bound, two-photon emission from 2s 1s decays, and spectral distortion from one bin to the next lower one, Lyman-α emission. Note that in the− present work we using Eq. (12). We then update it by adding the emission have not accounted for any of the radiative transfer ef- from the “lines” at frequencies νb, as in Eq. (14). This fects recently investigated with the purpose of obtaining 11 2.5 2.5 H (3 2) H (3 2) H (4 2) H (4 2) H (5 2) H (5 2) 2.0 P (4 3) 2.0 P (4 3) Br (5 4) Br (5 4) 1.5 1.5 ) per H atom obs 1.0 1.0 /dlog( /dlog(1+z) per H atom photon photon dN dN 0.5 0.5 0.0 0.0 600 800 1000 1200 1400 1600 1800 100 1000 z obs (GHz) FIG. 3. Number of photons per hydrogen atom emitted per logarithmic redshift interval (or equivalently per logarithmic interval of today’s observation frequency), for several bound-bound transitions. In the left panel, this quantity is plotted as a function of emission redshift. In the right panel, it is plotted as a function of observed frequency today. The two are related by νobs = νem/(1 + zem). high-accuracy recombination histories (see for example The recurring time for computing a full recombina- Refs. [37, 41, 44–49] and many more references therein). tion spectrum, independent of nmax used for the effective It is to be expected that these corrections will lead to a conductances, is approximately 0.1 second on a standard few percent corrections to the recombination spectrum; workstation. This is four to six orders of magnitude faster nevertheless, we are far from even detecting the recombi- than the standard multilevel atom method8, depending nation spectrum, and such refinements are not yet needed on the value used for nmax [28]. for this purpose. The effective conductance method is, moreover, oblivious to all the complications that may occur between interface states and the ground state, and V. CONCLUSIONS AND FUTURE could be very easily adapted to include these effects6. DIRECTIONS If needed for some reason, a high-accuracy Lyman-lines and 2s 1s spectrum can be extracted from the modern In this paper we have introduced a new and highly ef- recombination− codes HyRec [35] and CosmoRec7 [50], ficient method to compute the primordial recombination that do account for these radiative transfer effects. spectrum. Our method relies on the factorization of the Let us point out that even though we have performed problem in a computationally expensive but cosmology- all computations with the correct matter temperature, independent part and a very fast cosmology-dependent we found that setting Tm = Tr (and doing so consis- part – the computational efficiency of the latter part is tently, including when computing the ionization history itself due to a similar method of factorization, the ef- and departures from Saha equilibrium), leads to an error fective multilevel atom method, introduced in an earlier of at most 0.4% for ν 0.1 GHz. It would therefore work [29]. The recurring cost of a spectrum computation be sufficient, at the percent≥ level of accuracy, to assume with this method is a fraction of a second, at least four Tm = Tr for the spectrum computation (this assumption orders of magnitude faster than computations carried out is not valid if one wishes to compute the low-redshift tail with the standard multilevel atom method. of the recombination history, when the two temperatures The computations carried out here relied on several may differ significantly). simplifying assumptions. First, we have restricted our- Finally, we have compared our results with those of selves to the “standard” (i.e. simplified) transitions from Ref. [12] and found a very good agreement. More thor- 2s and 2p to the ground state, and have neglected a whole ough comparisons will be made once we implement emis- suite of radiative transfer effects that were shown to be sion from helium as well. 8 One can speed up the computation of the spectrum by a factor of ∼ 10 if precomputing the recombination history with the EMLA 6 A subtlety might arise when dealing with two-photon decays method [29] and then computing the spectrum with the standard from higher levels, as one must avoid double-counting of the low- MLA method but with a larger timestep (J. Chluba, private frequency photons. communication). This would still be a few orders of magnitude 7 http://www.cita.utoronto.ca/∼jchluba slower than the method we present here. 12 10-26 nmax = 500 Balmer nmax = 250 Paschen nmax = 125 n = 60 max Bracket ) -27 -1 10 sr -1 Hz -1 s -2 Bound-bound spectrum (J m -28 I 10 Free-bound spectrum 10-29 0.1 1.0 10.0 100.0 1000.0 (GHz) FIG. 4. Bound-bound (not including 2s → 1s and Ly-α photons) and free-bound spectra for various values of the cutoff principal quantum number nmax. 10-26 bound-bound free-bound 2s-1s Ly- total ) -27 -1 10 sr -1 Hz -1 s -2 (J m -28 I 10 10-29 0.1 1.0 10.0 100.0 1000.0 (GHz) FIG. 5. Total spectral distortion created by hydrogen emission (neglecting the influence of helium), as well as individual 0 contributions from various processes, using nmax = 500. The “bound-bound” curve only accounts for n → n ≥ 2 transitions. We display ∆Iν in the same units as Refs. [12, 18] for an easier comparison by eye. 13 important for a highly accurate recombination history is a non-negligible contribution to the spectrum, and can and are efficiently computed by modern recombination be used to probe the primordial helium abundance be- codes [35, 50]. To our knowledge, no computation of the fore the first stars formed [20]. Computing the emission primordial recombination spectrum thus far has also in- from helium recombination can be achieved with the ex- cluded all these effects consistently. Including them with act same method, the only additional work would be to our formalism should not raise any major issues, but they compute effective conductances for HeI (those for HeII are expected to induce corrections to the spectrum at the can be easily obtained from the ones computed for hy- level of a few percent at most (though in principle one drogen by simple rescalings). Such additions are essential should check this statement precisely). The second as- and we shall include them in the near future. pect we neglected is collisional transitions. They were An additional aspect that we have not treated here is shown to modify the spectrum mostly in the low fre- the processing of pre-existing spectral distortions by the quency regime ν . 0.1 GHz, where the recombination recombining atoms. These can lead to large enhance- distortions are very smooth and therefore probably dif- ments of the line emission, even if the initial smooth ficult to observe [28]. Finally, we neglected broadening spectral distortion is tiny [23]. Formally, one can gener- of the lines by electron scattering, as well as free-free ab- alize our method to account for non-thermally mediated sorption. The former effect should increase the width of transitions, as long as the underlying spectrum can be the broad spectral features by less than a percent of their described by a small set of parameters. We defer such central frequency, and the latter strongly affects the spec- a study to future work. Once the latter two aspects are trum at frequencies below 0.1 GHz [19]. Since all these implemented, we shall release a public code to efficiently ∼ effects are small in the frequency regime ν & 0.1 GHz, we compute the recombination spectrum. do not consider them a priority before the observability Besides having a certain aesthetic appeal, we believe of the recombination spectrum is solidly established. that the factorization method presented here will prove In this work we have only dealt with the line and con- very useful in order to quantitatively study the infor- tinuum emission from recombining hydrogen atoms, and mation content of the recombination spectrum. Most of have neglected the effect of helium on the recombina- our knowledge about the early universe currently comes tion spectrum (but we did account for helium when com- from the study of spatial variations of the thermal photon puting the ionization history of the plasma). The pres- background, through CMB anisotropy observations. An ence of about 0.08 helium nuclei per hydrogen nucleus exciting avenue is to probe the spectral variations of the will modify our results in the two following ways. First, CMB, which might inform us about the universe before photons emitted in the HeI 21P 11S line (the equiva- the last-scattering redshift. A fast and accurate code to lent of the hydrogen Ly-α line) with− an energy of 21.2 compute the primordial spectrum will be of great help to eV can photoionize some of the few neutral hydrogen study precisely if, what and how new information can be atoms already present at the epoch of HeII HeI recom- gained from the primordial spectrum. bination [51–54]. This effect is already included→ in our code when computing the ionization fraction of helium [35], but we have not accounted here for the small depar- ACKNOWLEDGMENTS ture from Saha equilibrium that it induces for hydrogen, and the resulting line emission. This process was studied I thank Chris Hirata for pointing out the analogy of in detail in Ref. [54], who showed that it leads to pre- the system of fast high-n transitions with a circuit. I recombination emission features in the hydrogen spec- am grateful to Jens Chluba for providing recombination trum. Inclusion of this effect with our method presents spectra for comparison with the results presented here no additional complication: one only needs to properly and to Matias Zaldarriaga for discussions on this work. compute the small departures from equilibrium in hy- I am indebted to Simeon Bird, Jens Chluba and Daniel drogen, given the helium ionizing photons. The second Grin for reading the draft of this manuscript and provid- and more direct consequence of having helium nuclei is ing many useful comments. 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