Primordial helium recombination. I. Feedback, line transfer, and continuum opacity

Precision measurements of the cosmic microwave background temperature anisotropy on scales $\ell >500$ will be available in the near future. Successful interpretation of these data is dependent on a detailed understanding of the damping tail and cosmological recombination of both hydrogen and helium. This paper and two companion papers are devoted to a precise calculation of helium recombination. We discuss several aspects of the standard recombination picture, and then include feedback, radiative transfer in He i lines with partial redistribution, and continuum opacity from H i photoionization. In agreement with past calculations, we find that He ii recombination proceeds in Saha equilibrium, whereas He i recombination is delayed relative to Saha due to the low rates connecting excited states of He i to the ground state. However, we find that at $z<2200$ the continuum absorption by the rapidly increasing H i population becomes effective at destroying photons in the He i $2^1{P}^o-1^{1}S$ line, causing He i recombination to finish around $z\approx 1800$, much earlier than previously estimated.

Figure 1

Formation of neutral helium: a Grotrian diagram (up to $n=3$) for He i. Singlet ($S=0$ parahelium) and triplet ($S=1$ orthohelium) levels and higher-order transitions give a rich system of low-lying transitions. Marked are two-photon transition (light dashed lines from $2^1{S}_0$ and $3^1{S}_0$), (allowed) electric dipole transitions (solid lines from $2^1{P}_1^o$ and $3^1{P}_1^o$, like the Lyman series in H i), the intercombination lines (heavy dashed lines from $2^3{P}_1^o$ and $3^3{P}_1^o$), and quadrupole transitions (from $3^1{D}_2$). The dipole transitions are treated in Sec. 5d using a Monte Carlo method with partial redistribution; forbidden one-photon lines are treated in Sec. 4 using an analytic method for complete redistribution. (The energy levels are not drawn to scale.)

Figure 2

The fractional difference in the matter temperature relative to the radiation temperature, $1-T_{\mathrm{m}}/T_{\mathrm{r}}$. Note the sign: matter is cooler than radiation at late times because of its different adiabatic index ($5/3$ versus $4/3$). During He i recombination for $1600<z<3000$, the fractional difference is $<{10}^{-6}$.

Figure 3

The ionization/relaxation timescales of H i during the period of He i recombination, assuming each He i recombination generates a photon that photoionizes a hydrogen atom. Here we consider two He i recombination histories: one in equilibrium and the history derived here (Fig. 12). The “H i relaxation rate” is the inverse-time scale $t_{\mathrm{Saha}}^{-1}$ (Eq. (29)) for H i to return to Saha equilibrium if its abundance is perturbed. In either He i history, the ionizing radiation from He i is not sufficient to push H i evolution out of Saha equilibrium.

Figure 4

The continuum optical depth $d{\tau }_c/d\nu ={\eta }_{\mathrm{c}}$ times the Doppler width of He i $2^1{P}^o-1^{1}S$ as a function of redshift.

Figure 5

A comparison of the effect of feedback of a spectral distortion produced by higher-lying states on lower-lying states in the same species after several iterations, relative to the reference model, for He ii, He i, and H i, where $\Delta x$ is the abundance with feedback minus without feedback. Here, continuous opacity from hydrogen photoionization between He i is included (discussed in Fig. 6). In all cases, feedback has the effect of retarding the formation of the neutral species. A larger number of iterations ($\sim 5$) are needed for He i recombination than for H i or He ii. Here, the uppermost line is the first iteration, moving down with further iterations and better convergence.

Figure 6

The effect of hydrogen continuum absorption on the feedback between transitions to the ground state in He i. Feedback slows He i recombination, but becomes increasingly less significant as more hydrogen recombines, increasing the bound-free opacity and absorbing the distortion.

Figure 7

Inverse of the differential optical depth ${\eta }_{\mathrm{c}}$ from hydrogen photoionization as a function of redshift, compared to the optically thick line-width due to incoherent processes in the $2^1{P}^o-1^{1}S$ line. Continuum processes start to become important over scales inside the (incoherent) optically thick part within the line around $z=2100$. Also plotted is the Doppler width of the line, emphasizing that the line is optically thick out into the wings. Continuum processes do not act on scales smaller than the Doppler core until $z<1800$.

Figure 8

The radiation profile near the intercombination line $1^1{S}_0\to 2^3{P}_1^o$ resonance with Voigt parameter $a={\Gamma }_{\mathrm{line}}/(4\pi \Delta {\nu }_{\mathrm{D}})={10}^{-5}$ for the Sobolev optical depth ${\tau }_{\mathrm{S}}=2.8$ and ${\mathcal{N}}_{\mathrm{C}}=0$, for several sample continuum optical depths ${\eta }_{\mathrm{c}}\Delta {\nu }_{\mathrm{D}}$, showing the effect of continuous opacity. Because of the low optical depth and small natural linewidth, very little radiation extends more than three doppler widths above the line, and the effect of the continuum is significant in relaxing the radiation phase space density near line center.

Figure 9

The transition $2^1{P}^o-1^{1}S$ interpreted as a two-photon process with $2^1{P}_1^o$ as an intermediate resonant state. In an incoherent scattering through He i $1^1{S}_0\leftrightarrow 2^1{P}_1^o$, the incoming photon excites $1^1{S}_0\to 2^1{P}_1^o$. The atom absorbs a second photon and explores several intermediate states before decaying through $2^1{P}^o-1^{1}S$, emitting a $2^1{P}^o-1^{1}S$ photon with complete redistribution. In coherent scattering, only $1^1{S}_0$ and $2^1{P}_1^o$ are involved, and the outgoing photon is emitted with the same energy as the incoming photon in the rest frame of the atom.

Figure 10

Modification to the He i recombination history due to continuum opacity in the $n^{1}D-1^{1}S$, $n^3{P}^o-1^{1}S$, and $n^1{P}^o-1^{1}S$ lines (and where coherent scattering through $2^1{P}^o-1^{1}S$ is neglected) relative to a model where only continuum opacity in the $2^1{P}^o-1^{1}S$ line is accounted for. The total effect on $x_e$ is at the level of ${10}^{-4}$. Without feedback, $n^1{P}^o-1^{1}S$ slightly speeds recombination relative to just $2^1{P}^o-1^{1}S$—the allowed rates from $n>2$ to the ground state are accelerated. Coherent scattering through $2^1{P}^o-1^{1}S$ is also seen to be negligible for the recombination history.

Figure 11

The modified escape probability from ${}^4\mathrm{He}$ $2^1{P}^o-1^{1}S$ during He i recombination, comparing the results of the standard Sobolev approximation and the modification due to continuous opacity with and without coherent scattering. Once coherent scattering is introduced, photon diffusion effects increase the escape probability by increasing the span of frequencies traversed by the scattering photon before it escapes. These probabilities are log-interpolated over the grid of $x_{\mathrm{HeI}}$ and $z$ used in the level code. We find that the effect of doubling the grid resolution and with it the smoothness of the interpolated probability is negligible.

Figure 12

Modifications to the He i recombination history from the inclusion of feedback and continuum opacity in He i lines, compared to a Saha He i recombination history and the standard He i recombination history. Continuous opacity starts to become important at $z\sim 2100$ (as suggested by Fig. 7), and pushes the He i evolution to a Saha by $z\sim 1800$. The beginning of H i recombination is visible here starting at $z\sim 1700$, and for later times $x_e$ drops precipitously. The modifications to He i recombination suggested here are twofold: at early times during He i recombination, feedback slows recombination, and at later times, continuous opacity accelerates recombination relative to standard models.

Figure 13

Recombination directly to He i $1^{1}S$ can occur because He i recombination photons can ionize neutral hydrogen, before they ionize He i. This would tend to accelerate recombination. Here we show that this effect leads to a maximum fractional deficit of electrons of around ${10}^{-4}$.