Primordial helium recombination. II. Two-photon processes

Interpretation of precision measurements of the cosmic microwave background (CMB) will require a detailed understanding of the recombination era, which determines such quantities as the acoustic oscillation scale and the Silk damping scale. This paper is the second in a series devoted to the subject of helium recombination, with a focus on two-photon processes in He i. The standard treatment of these processes includes only the spontaneous two-photon decay from the $2^{1}S$ level. We extend this treatment by including five additional effects, some of which have been suggested in recent papers but whose impact on He i recombination has not been fully quantified. These are: (i) stimulated two-photon decays; (ii) two-photon absorption of redshifted He i line radiation; (iii) two-photon decays from highly excited levels in He i ($n^{1}S$ and $n^{1}D$, with $n\ge 3$); (iv) Raman scattering; and (v) the finite width of the $2^1{P}^o$ resonance. We find that effect (iii) is highly suppressed when one takes into account destructive interference between different intermediate states contributing to the two-photon decay amplitude. Overall, these effects are found to be insignificant: they modify the recombination history at the level of several parts in ${10}^4$.

Figure 1

A comparison of the effect of stimulated two-photon emission and nonthermal two-photon absorption relative to the reference model. The net effect is a delay in recombination, $\Delta x_e>0$. The two peaks correspond to the effect on He ii recombination ($z\sim 5500$) and He i recombination ($z\sim 2500$). Note that in both cases the effect on the recombination history is small, i.e. a few parts in ${10}^5$.

Figure 2

The function $f(\Omega )$ appearing in Eq. (28), which describes the radial matrix elements for large $n$. Note that $f(\Omega )$ is largest for small $\Omega$, implying that $⟨n^{1}L||\mathbf{d}||n^{\prime 1}{P}^o⟩$ is largest for small $n^{\prime }-n$. The points mark integer values of $\Omega$.

Figure 3

The dipole matrix elements $⟨1^{1}S||\mathbf{d}||n^{\prime 1}{P}^o⟩$ and $⟨n^{1}S||\mathbf{d}||n^{\prime 1}{P}^o⟩$ for $n=25$, computed using the formula of Ref. 50 (for $1^{1}S$) and the Coulomb approximation (for ${25}^{1}S$). Note that the matrix elements with $n^{\prime }$ close to, but not equal to, $n$ are comparable in magnitude to $⟨n^{1}S||\mathbf{d}||n^1{P}^o⟩$ but have negative instead of positive sign. Therefore these contributions can interfere destructively with the $n^1{P}^o$ intermediate level in the matrix element ${\mathcal{M}}_{2\gamma }$.

Figure 4

The 2-photon spectrum from the $2^{1}S$ level of He i. Note that there are no resonances in the spectrum since there are no energetically allowed $1+1$ electric dipole decays from $2^{1}S$. The points are the more detailed calculations by Drake [38].

Figure 5

The 2-photon spectrum from the $3^{1}S$ and $3^{1}D$ levels of He i. Note the resonance at 21.2 eV corresponding to the $2^1{P}^o$ intermediate level. There are also resonances at much lower energies corresponding to the optical transitions in He i, $3^{1}S-2^1{P}^o$ and $3^{1}D-2^1{P}^o$. Also note the nulls in the two-photon rate from $3^{1}S$. The series of points are the results considering only the $n^{\prime }=n$ term in the matrix element, Eq. (13), for $3^{1}S$ (upper series) and $3^{1}D$ (lower series). Note that keeping only this term is a poor approximation except at the very ends of the spectrum.

Figure 6

The 2-photon spectrum from the $4^{1}S$ and $4^{1}D$ levels of He i. There are two pairs of resonances corresponding to the $1+1$ decays via the 2, $3^1{P}^o$ intermediate levels. The series of points are the results considering only the $n^{\prime }=n$ term in the matrix element, Eq. (13), for $4^{1}S$ (upper series) and $4^{1}D$ (lower series).

Figure 7

The contribution of the nonresonant part of the two-photon rate in He i for $n>2$ produces a maximum change in the free electron fraction of several $\times {10}^{-4}$.

Figure 8

An example of an incoherent scattering process in He i considered in Sec. 5a.

Figure 9

Representation of complete redistribution as a two-photon process, with one photon from the thermal distribution. In the left frame, a photon is absorbed on the red side of the line, then assisted to $3^{1}D$ by a blackbody photon. In the right frame, a photon is absorbed on the blue side of the line and assisted by a lower energy blackbody photon. The virtual levels have energies offset from $E(2^1{P}^o)$; the fractional difference between the forward and backward scattering rates is of order the frequency difference times $h/k_{\mathrm{B}}{T}_{\mathrm{r}}$. Because there are more low-energy thermal photons, scattering to the blue side of the line is slightly preferred.

Figure 10

The phase space density in the $2^1{P}^o-1^{1}S$ line using the parameters $z=2175$, $x_{\mathrm{HeI}}=0.02825$, ${\mathcal{N}}_{\mathrm{L}}=6.307\times {10}^{-16}$, $\mathcal{N}({\nu }_{+})=5.277\times {10}^{-19}$, ${\dot{x}}_{\mathrm{HeI}}=0.1596H$, ${\dot{\mathcal{N}}}_{\mathrm{L}}=-2.444\times {10}^{-15}H$, and $\dot{\mathcal{N}}({\nu }_{+})=-4.073\times {10}^{-17}H$. These parameters occurred during the first feedback iteration of the recombination history. The solid line shows the steady-state solution ${\mathcal{N}}_0$, while the long-dashed line is the first-order correction ${\mathcal{N}}_0+{\mathcal{N}}_1$. The short-dashed line is the analytic approximation to the steady-state solution from Eq. (b17); note that it is plotted only for $\Delta \nu <0$.

Figure 11

The change in the recombination history from the modified treatment of the $2^1{P}^o-1^{1}S$ resonance. Note that the effect on the electron abundance is very small: a few parts in ${10}^4$. This figure should only be interpreted as an estimate of the magnitude of the correction (see text).