Radiative transfer and escape probabilities

Numerically solving the radiative transfer problem is computationally intensive, but tremendous simplification can be achieved with the Sobolev escape probability formalism, also known as the Sobolev approximation [279]. The Hubble flow can be used to define a lengthscale over which the bulk flow induces a velocity change equal to the thermal velocity: L=p

3kTM/matom/H(TR), where

H(TR) is the value of the Hubble expansion parameter when the radiation has temperatureTRand

matomis the mass of an atom [276]. The conditions of the Sobolev approximation are [276, 279, 292]:

(i)Lis much smaller than the typical length scales over which cosmological quantities vary, (ii)L/c is much smaller than the typical time scales over which cosmological quantities vary, (iii) complete frequency distribution— the rest-frame frequency of an outgoing scattered photonνdoes not depend on the incoming frequency ν0_{— and (iv) no other emission, absorption, or scattering processes}

occur in the vicinity of the line. Corrections to the Sobolev approximation result from diffusion around resonance lines [295, 296], atomic recoil [292, 297], Thomson scattering near resonances [298, 299], and overlap of the higher Ly series lines, leading to important corrections to cosmological recombination calculations. In this work, however, we work in the Sobolev approximation to focus on other physical effects.

In the Sobolev approximation, the escape probability for photons produced in the downward transition [n0_{, l}0_{]→[n, l] is [276]} Pl,l 0 n,n0 = 1₋e−τl,l 0 n,n0 τ_n,nl,l00 , (4.11)

where the Sobolev optical depth is given by τl,l 0 n,n0 = c3_η H 8πHν3 n,n0 Al,l 0 n,n0 gl0 glxn,l−xn0,l0 , (4.12)

with transition frequency

νn,n0 = En,n0 h = IH h 1 n2 − 1 n02 . (4.13)

Correct expressions forn0_{< n are obtained by reversing arguments. During cosmological recombi-}

nation, transitions between excited states are optically thin (Pl,l

0

P_n,nl,l00 = 1 in our calculations for non-Lyman lines. The solution for the radiation field near isolated

Lynlines has the approximate analytic solution [300]

N− n1=N eq n1+ Nn+1− N eq n1 e−τll 0 n1 Nneq1≡ xnp 3x1s . (4.14)

Transitions in the Lyman (Ly) series (n0 _{> n= 1, l}0 _{= 1,l = 0) are optically thick (τ}l,l0

n,n0 1)

[283], and so P₁0_,n,10 ' 1/τ

0,1

1,n0. Ly transitions cannot, however, be ignored in the recombination

calculation, as the rate at which atoms find their way to the ground state through the redshifting of resonance photons,P₁0_,n,10A

0,1

1,n0 is comparable to Λ2s→1sand other two-photon rates [283]. Strictly

speaking,τ₁0_,n,10 depends onxn0_,1, and so one should solve forxn0_,1 and then iteratively improve the

solution. The populations of the excited states, however, are very small and the maximum resulting correction to the optical depth is 2×10−12 _(forn0 _{= 2, z= 1600) [283]. We thus drop the second}

term in Eq. (4.12), simplifying our computation by working in the approximation where the Lyman- n(Lyn) line optical depth depends only on the ground-state population and not on the excited-state populations.

Another aspect of the Lyman-series lines is feedback: a photon that escapes from the Lyn (np_→1s) line will redshift into the Ly(n₋1) line and be reabsorbed. RecSparse_{has the ability}

to implement the resulting feedback, using the iterative technique of Ref. [300]. This slows down the code by a factor of a few, however, and so to efficiently focus on thenmax problem, we turned

feedback off. For the high Lyman lines, feedback is almost instantaneous: the Universe expands by a factor of ∆ lna ≈2n−3 _{during the time it takes to redshift from Lyn to Ly(n−1). In the}

instantaneous-feedback limit, the Lynlines do not lead to a net flux of H atoms to the ground state. To approximate this net effect we turned off Lyman transitions with n >3; this leads to a smaller error than would result from leaving these transitions on but disabling feedback. Previous tests using the code of Ref. [292] show resulting errors in the recombination history at the_≈1% level; in any case, this should only weakly be related to thenmax problem. All of the recombination histories

and plots in this chapter were produced by running RecSparse _{with both feedback and Lyman}

transitions fromn >3 disabled, unless noted otherwise. Using the toolkit provided byRecSparse_,

we are also exploring extensions to the Sobolev approximation. 4.2.2.1 Line overlap

It is a well known fact that at highn, the separation between adjacent Ly series lines shrink:

νn+1−νn = IH h " 1 n2 − 1 (n+ 1)2 # ' 2I_hnH3. (4.15)

The thermal motion of atoms leads to Doppler broadening of the line, and so it is clear that at any given temperature, there is a transition value n =no above which neighboring lines overlap.

In other words, the fates accessible to a photon produced in the overlapping regime of the Ly-n resonance extend beyond re-absorption by the Ly-nline or escape (until red-shifting brings it into the next Lyman line). This photon may also be immediately re-absorbed by other nearby Ly lines [e.g. Ly-(n+ 1), Ly-(n₋1), and so on]. The Doppler width of a Lyman line is given by

∆νn=νn

s

2kTM

mpc2, (4.16)

wheremp is the proton mass. We can obtain the line-overlap condition by demanding that ∆νn _∼> νn+1−νn, thus yielding n_∼> no≡44 _T M 3000 K −1/6 (4.17) as the requirement for overlap of adjacent Ly lines [291]. For yet highern, high-lying Ly lines will even overlap with the continuum. Similar arguments then lead to the condition

n_∼> nc≡206 _T M 3000 K −1/4 (4.18) for Ly line overlap with the continuum. Once this condition kicks in, the large reservoir of ionizing photons could feed the recombination network through an additional channel due to overlap between the continuum and the Lyman series, pumping atoms into excited states, but also providing a new channel to the ground state. Modern recombination codes routinely proben > nmax= 100 [7–9, 44].

Indeed, in the work described in this chapter and later work by others, substantial effort was devoted to determining the value of nmax required for adequate convergence in the recombination history.

Since Ly line overlap becomes important at highn, it is important to properly treat the effect of line overlap on recombination.

Line overlap explicitly breaks assumption iv) of the Sobolev approximation: (iv) no other emis- sion, absorption, or scattering processes occur in the vicinity of the line. Fortunately, the same techniques used to solve for the occupation number N(En,n0±) in the Sobolev approximation

may be readily generalized. The results may be written in a form that lends itself to straightforward numerical integration, as shown in recent work by Y. Ali-Ha¨ımoud [291]. Qualitatively speaking, for n_∼> no(though it turns out small corrections due to overlap already appear atn_∼>22), the solution for the occupation number [Eq. (4.14)] must be replaced with a sum over overlapping Lyman lines, and the portion of the rate matrix corresponding tonp_→n0_{ptransitions must be adjusted to include}

overlap terms. Using theRecSparse _{toolkit, we are working on the inclusion of these corrections}

[291]. Preliminary results indicate that the correction toxe(z)_∼<10−5_forn

the conclusions in this chapter unchanged.