Analytic estimate for the number of spectral distortion photons

4.3 The Lyman alpha line

4.3.2 Interaction with the Deuterium Lyman-α line

4.3.2.3 Analytic estimate for the number of spectral distortion photons

We now compute the number of spectral distortion photons. We will treat the deuterium line in the Sobolev approximation, i.e., assuming that its profile is a delta-function. This approximation is well justified since the deuterium line is dominated by its Doppler core: the damping wings are only marginally optically thick, and the differential optical depth in the deuterium wings is always much smaller than that of hydrogen anyway (see for example Fig. 7 of Ref. [42]). In this case complete redistribution is a good approximation since both partial and complete redistribution have a similar

6This section is not part of the published paper [40], where the interested reader will find a more elaborate treatment that accounts for frequency diffusion. Here I have opted for a simpler and less complete, but hopefully more enlightening calculation.

characteristic frequency width, the Doppler width of the line (see Appendix 4.B). For the hydrogen line, we only consider true absorption and emission and neglect frequency diffusion due to resonant scattering. We will discuss the validity of this approximation at the end of the section. Finally, since νD−νH∼10⁻⁴νH, we can work in the steady-state approximation in the vicinity of the Lyαlines.

In the blue damping wing of the hydrogen Lyαline, and forν 6=νD, the steady-state radiative transfer equation for the distortion ∆fν ≡fν−e^−hν/(kTr) can be written

∂∆fν

∂ν = W

(ν−νH)² ∆fν−∆f_eq^H

, (4.67)

where ∆f_eq^H ≡x_H(2p)/(3x_H(1s))−e^{−hνH/(kTr)}. To arrive at Eq. (4.67), we have used Eq. (4.18) and (4.31) in the damping wing approximation, used Eq. (4.22) to approximate f_(em)^H ≈ f_eq^H, and the definition ofW, Eq. (4.35). We have also assumed equal absorption and emission profiles, which is a good approximation sinceh(νD−νH)/(kTr)1.

The optical depth in the H Lyαdamping wing blueward of the deuterium line is

τ≡ W

νD−νH

. (4.68)

We find that 0.02∼< τ ∼<3 for 700< z <1600, with a maximum ofτ≈3.2 atz≈1300. The photon occupation number incoming on the blue side of the deuterium line is therefore

∆f₊^D= ∆f_eq^H 1−e^−τ

, (4.69)

where we assumed that the photon occupation number at ν → ∞ is a blackbody (i.e., we neglect feedback from Lyβ).

In the Sobolev approximation, ∆f₋^D= ∆f_eq^D ≡xD(2p)/(3xD(1s))−e^{−hνD/(kTr)}since the deuterium line is optically thick. The radiation field between the two lines is therefore

∆f_ν^(H+D)= ∆f_eq^H + ∆f_eq^D−∆f_eq^H exp

W

νD−νH − W ν−νH

, νH< ν < νD, (4.70)

whereas in the absence of the deuterium line, it would be

∆f_ν^(H)= ∆f_eq^H

1−exp

− W ν−νH

. (4.71)

The number of distortion photons [Eq. (4.65)] is therefore

U =8πν_Lyα² c³NH

(νD−νH)

∆f_eq^D−∆f_eq^H

+ ∆f_eq^He^−τ

I(τ), (4.72)

where we have defined

I(τ)≡τe^τ Z ∞

τ

e^−udu

u². (4.73)

We find that the functionI(τ) is fit to better than 2% accuracy over the range of interest 0.02<

τ <3.3 by the simple formula

I(τ)≈ 1

1 + 1.5τ^0.8. (4.74)

Using Eq. (4.69) and the definition of RLyα (the rate of Lyα escape per atom in the 2p state), Eq. (2.20), we can rewrite Eq. (4.72) in the form

U =H⁻¹ νD−νH

νH I(τ)RLyα3xH(1s)

∆f_eq^D−∆f₊^D

. (4.75)

We now need to estimate the population of the excited state in deuterium in order to obtain the difference ∆f_eq^D−∆f₊^D. The population of then= 2 shell of deuterium is controlled by net recombi- nations to the excited states and Lyman-αdecays (as mentioned earlier, decays to the ground state through the Lyman-α channel are dominant over two-photon decays from the 2sstate because of the relatively low optical depth of the D Lyαline). We can obtain the contribution of the former with a Peebles-like estimate [31]:

˙ xD(2p)

rec= 3

4αBNHxexD⁺−βBxD(2p)= 3

4αBNHxDx²_e−βBxD(2p), (4.76) whereαB(Tm) is the case-B recombination coefficient, and we have usedxD⁺ ≈xDxe in the second equality. The net rate of Lyαdecays in the deuterium line is, in the Sobolev approximation:

˙ xD(2p)

Lyα=8πHν_Lyα³ c³NH

∆f₊^D−∆f_eq^D

= 3RLyαxH(1s) ∆f₊^D−∆f_eq^D

, (4.77)

where RLyα is the rate of escape of Lyα photons per atom in the 2p state and was defined in Eq. (2.20). We can now solve for the the population of the excited state, in the steady-state approximation:

˙ xD(2p)

rec+ ˙xD(2p)

Lyα≈0, (4.78)

from which we obtain [recalling that ∆f_eq^D ≡xD(2p)/(3xD(1s))−e^{−hνD/(kTr)}]

∆f_eq^D−∆f₊^D=

1

4αBNHx²_e−βBxH(1s)e^{−hνH/(kTr)}−βBxH(1s)∆f₊^D xH(1s)

βB+^R_x^Lyα

D

. (4.79)

We can simplify Eq. (4.79) with the following considerations. First, we can simplify the denominator since RLyα ∼>10⁻³βB xDβB (i.e., the rate of Lyα escape is much larger than the rate of pho- toionizations from the excited state in deuterium). Secondly, using Eq. (4.69) we can simplify the

numerator. Thirdly, assuming Boltzmann equilibrium between excited states of hydrogen (which is a good approximation whenever radiative transfer effects are important), we arrive at

∆f_eq^D−∆f₊^D= 1 4

xD

RLyαxH(1s)

hαBNHx²_e−βBxH(n=2)+ e^−τβB

xH(n=2)−4xH(1s)e^{−hνH/(kTr)}i . (4.80) Finally, using again a simple Peebles’ estimate for hydrogen (see Section 2.2.1), we arrive at

∆f_eq^D−∆f₊^D=xD3RLyα+ Λ2s,1s

4RLyα

1 + e^−τ 4βB

3RLyα+ Λ2s,1s

∆f_eq^H, (4.81) or, equivalently,

∆f_eq^D−∆f₊^D=−1 4

xD

RLyαxH(1s)

1 + e^−τ 4βB

3RLyα+ Λ2s,1s

˙

xe, (4.82)

and our final estimate for the number of distortion photons is U = 3

4H⁻¹xD

νD−νH

νH I(τ)

1 + e^−τ 4βB

3RLyα+ Λ2s,1s

|x˙e|. (4.83) We show the number of distortion photons per deuterium atom, U/xD, as a function of redshift, in Fig. 4.5. We see that this number is already relatively small at all times, and sincexD∼10⁻⁵, we conclude that the effect of deuterium on the recombination history is completely negligible (for comparison, two-photon decays and Raman scattering events, which represent a∼1% correction to the recombination history, lead to a distortionU2γ ∼0.01, see Fig. 9 of Ref. [49]).

To summarize the essential points, the distortion is very small because

(i) the separation between the lines is very small (νD−νH)/νH1 and there is just not much space (in the frequency domain) that can be filled with distortion photons, see Eq. (4.75) or Fig. 4.4, (ii) the optical depth in the deuterium Lyαline is low enough (although still1) that Lyαescape dominates over photoionizations from the excited states and therefore the excited state population equilibrates such that ∆f_eq^D−∆f₊^D∼xD∆f_eq^H, see Eq. (4.81), and

(iii) even if (i) and (ii) are already sufficient to largely suppress any distortion, the fact that the damping wing of hydrogen is marginally optically thick (τ∼>1) adds an additional suppression (see Fig. 4.5), throughI(τ)<1 and the exponential factor e^−τ in Eq. (4.81), because the the radiation field is already nearly in equilibrium with the H Lyαline at the D Lyαfrequency.

Finally, we have not considered the effect of frequency diffusion here. We treat this case in Ref. [40], where we find that frequency diffusion suppresses the distortion even more, essentially by enhancing effect (iii). The reason is that the deuterium line lies with the diffusion-dominated region, (νD−νH)³ S³.

!

H

!

D

! f f f

DDH+eqeq

f

^!

Figure 4.4: Schematic representation of the deuterium problem: the distortion induced by the presence of the deuterium line is shown as the hatched area.