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$\begin{figure} \par\includegraphics[width=7.85cm,clip]{eps/7921f001.eps}\end{figure}$ Figure 1: Non-resonant two-photon emission spectra, Eq. (8), for several transitions. All curves are normalized to unity at y=1/2. The values of $\phi ^{\rm nr}(1/2)$ can be found in Appendix B.
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In the text

$\begin{figure} \par\includegraphics[width=7.85cm,clip]{eps/7921f002.eps}\hspace*{2mm} \includegraphics[width=7.85cm,clip]{eps/7921f003.eps}\end{figure}$ Figure 2: Two-photon emission spectra for the $3{\rm s}\rightarrow 1{\rm s}$ and $3{\rm d}\rightarrow 1{\rm s}$ transitions. The non-resonant, cascade, and combined spectra are shown as labeled. Also we give the analytic approximation based on the equations discussed in Sect. 2.1.3, together with the fitting formula for the 2s and 3d non-resonant matrix element (see Appendix B). The resonances correspond to the Balmer- $\alpha$ and Lyman- $\alpha$ transitions.
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In the text

$\begin{figure} \par\includegraphics[width=7.9cm,clip]{eps/7921f004.eps}\hspace*{... ...s}\hspace*{1mm} \includegraphics[width=7.9cm,clip]{eps/7921f007.eps}\end{figure}$ Figure 3: Two-photon emission spectra for the $5{\rm s}\rightarrow 1{\rm s}$ and $5{\rm d}\rightarrow 1{\rm s}$ transitions. The non-resonant, cascade, and combined spectra are shown as labeled. Also we give the analytic approximation based on the equations discussed in Sect. 2.1.3, together with the fitting formula for the 5s and 5d non-resonant matrix element (see Appendix B). The resonances correspond to the Brackett- $\alpha$ , Paschen- $\beta$ , and Balmer- $\gamma$ transitions at y<1/2, and Lyman- $\alpha$ , Lyman- $\beta$ , and Lyman- $\gamma$ at y>1/2.
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In the text

$\begin{figure} \par\includegraphics[width=7.85cm,clip]{eps/7921f008.eps}\end{figure}$ Figure 4: Non-resonant contribution to the total two-photon decay rate in vacuum for the ns and nd-states of the hydrogen atom. The results were computed using the first 200 terms above n_i.
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In the text

$\begin{figure} \par\includegraphics[width=7.6cm,clip]{eps/7921f009.eps}\hspace*{1mm} \includegraphics[width=7.6cm,clip]{eps/7921f010.eps}\end{figure}$ Figure 5: Comparison of the two-photon emission profiles for the 4s and 4d states. We show, $\phi ^{2\gamma }_{4\rm s/d\rightarrow 1s}$ , following from the full $2\gamma$ treatment according to Eq. (7), and, $\phi ^{1+1\gamma }_{4\rm s/d \rightarrow 1s}$ , using the 1+1-single photon description, as given by Eq. (19). The first 200 terms above n=4 were included in the sum for $\phi ^{2\gamma }_{4\rm s/d\rightarrow 1s}$ .
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In the text

$\begin{figure} \par\includegraphics[width=7.6cm,clip]{eps/7921f011.eps}\hspace*{1mm} \includegraphics[width=7.6cm,clip]{eps/7921f012.eps}\end{figure}$ Figure 6: Normalized 1+1-two-photon profile $\phi ^{n\rm s/d, 1+1\gamma }_{2\rm p\rightarrow 1s}$ , see Eq. (17), close to the Lyman- $\alpha$ frequency in comparison with the full two-photon-profiles, $\phi ^{2\gamma }_{n\rm s/d\rightarrow 1s}$ according to Eq. (7), for several initial s and d-states. We divided the two-photon spectra by their partial one-photon transition rate to the 2p-state ( $\Gamma _{n\rm s/d \rightarrow 2p}$ ) and transformed to the variable $y_\alpha =\nu /\nu _{\alpha }$ . For the given curves we included the first 500 terms above n_i. The dashed curve corresponds to the usual Lorentzian profile of the Lyman- $\alpha$ transition.
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In the text

$\begin{figure*} \par\mbox{\includegraphics[width=7.6cm,clip]{eps/7921f013.eps}\hspace*{1mm} \includegraphics[width=7.6cm,clip]{eps/7921f014.eps} } \end{figure*}$ Figure 7: Relative difference, $[\nu_\alpha~\phi^{2\gamma}_{n\rm s/d \rightarrow 1{\rm s}}(y_\alpha)/\nu_{n\rm ... ...rrow 1s}(y_\alpha)]/\phi^{n\rm s/d, 1+1\gamma}_{\rm 2p\rightarrow 1s}(y_\alpha)$ , of the curves given in Fig. 6 with respect to normalized 1+1-two-photon profile. Very close to $y_\alpha \sim 1$ , the curves should be considered as extrapolated estimates.
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In the text

$\begin{figure} \par\includegraphics[width=7.7cm,clip]{eps/7921f015.eps}\hspace*{1mm} \includegraphics[width=7.7cm,clip]{eps/7921f016.eps}\end{figure}$ Figure 8: Rate of photon production at frequencies below $x_\Gamma$ according to Eq. (22) for several initial s and d-states. To convert to the variable $[\nu -\nu _{\alpha }]/\nu _{\alpha }$ , one should multiply $x_\Gamma$ by $\sim$ $2.54 \times 10^{-7}$ .
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In the text

$\begin{figure} \par\includegraphics[width=7.7cm,clip]{eps/7921f017.eps}\hspace*{1mm} \includegraphics[width=7.7cm,clip]{eps/7921f018.eps}\end{figure}$ Figure 9: Effective change in the rate of photon production (real profile minus Lorentzian) at frequencies below $x_\Gamma$ , according to Eq. (24). To convert to the variable $[\nu -\nu _{\alpha }]/\nu _{\alpha }$ , one should multiply $x_\Gamma$ by $\sim$ $2.54 \times 10^{-7}$ .
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In the text

$\begin{figure} \par\includegraphics[width=7.6cm,clip]{eps/7921f019.eps}\end{figure}$ Figure 10: Combined effective two-photon photon production rate, $\Delta A_{n\rm s+\it n\rm d \rightarrow 1s}(\nu _{\rm c})$ , computed according to Eq. (26). To convert to the variable $[\nu -\nu _{\alpha }]/\nu _{\alpha }$ , one should multiply $x_\Gamma$ by $\sim$ $2.54 \times 10^{-7}$ .
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In the text

$\begin{figure} \par\includegraphics[width=7.6cm,clip]{eps/7921f020.eps}\end{figure}$ Figure 11: Relative change in the free electron fraction. Here we only included the additional two-photon process for the 3s and 3d-states. The computations were performed for a 30-shell hydrogen atom. The effective two-photon rates for three different values of $\nu _{\rm c}$ according to Table 2 were used. For comparison, we show the results that were obtained using the 3s and 3d two-photon decay rates of Dubrovich & Grachev (2005) and Wong & Scott (2007).
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In the text

$\begin{figure} \par\includegraphics[width=7.6cm,clip]{eps/7921f021.eps}\end{figure}$ Figure 12: Relative change in the free electron fraction for different values of $n_{2\gamma }$ . The computations were performed including 30 shells.
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In the text

$\begin{figure} \par\includegraphics[width=7.6cm,clip]{eps/7921f022.eps}\end{figure}$ Figure 13: Relative change in the free electron fraction when taking the additional two-photon emission for up to 10 shells into account. The computations were performed including 30 shells, for three different values of $\nu _{\rm c}$ . For comparison, the result that was obtained within the approach of Wong & Scott (2007) is shown, but for $n_{2\gamma }=40$ and using a 50-shell model for the hydrogen atom.
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In the text

$\begin{figure} \par\includegraphics[width=7.85cm,clip]{eps/7921f001.eps}\end{figure}$	Figure 1: Non-resonant two-photon emission spectra, Eq. (8), for several transitions. All curves are normalized to unity at y=1/2. The values of $\phi ^{\rm nr}(1/2)$ can be found in Appendix B.
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$\begin{figure} \par\includegraphics[width=7.85cm,clip]{eps/7921f008.eps}\end{figure}$	Figure 4: Non-resonant contribution to the total two-photon decay rate in vacuum for the ns and nd-states of the hydrogen atom. The results were computed using the first 200 terms above n_i.
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$\begin{figure} \par\mbox{\includegraphics[width=7.6cm,clip]{eps/7921f013.eps}\hspace{1mm} \includegraphics[width=7.6cm,clip]{eps/7921f014.eps} } \end{figure*}$	Figure 7: Relative difference, $[\nu_\alpha~\phi^{2\gamma}_{n\rm s/d \rightarrow 1{\rm s}}(y_\alpha)/\nu_{n\rm ... ...rrow 1s}(y_\alpha)]/\phi^{n\rm s/d, 1+1\gamma}_{\rm 2p\rightarrow 1s}(y_\alpha)$ , of the curves given in Fig. 6 with respect to normalized 1+1-two-photon profile. Very close to $y_\alpha \sim 1$ , the curves should be considered as extrapolated estimates.
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$\begin{figure} \par\includegraphics[width=7.7cm,clip]{eps/7921f015.eps}\hspace*{1mm} \includegraphics[width=7.7cm,clip]{eps/7921f016.eps}\end{figure}$	Figure 8: Rate of photon production at frequencies below $x_\Gamma$ according to Eq. (22) for several initial s and d-states. To convert to the variable $[\nu -\nu _{\alpha }]/\nu _{\alpha }$ , one should multiply $x_\Gamma$ by $\sim$ $2.54 \times 10^{-7}$ .
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$\begin{figure} \par\includegraphics[width=7.7cm,clip]{eps/7921f017.eps}\hspace*{1mm} \includegraphics[width=7.7cm,clip]{eps/7921f018.eps}\end{figure}$	Figure 9: Effective change in the rate of photon production (real profile minus Lorentzian) at frequencies below $x_\Gamma$ , according to Eq. (24). To convert to the variable $[\nu -\nu _{\alpha }]/\nu _{\alpha }$ , one should multiply $x_\Gamma$ by $\sim$ $2.54 \times 10^{-7}$ .
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$\begin{figure} \par\includegraphics[width=7.6cm,clip]{eps/7921f019.eps}\end{figure}$	Figure 10: Combined effective two-photon photon production rate, $\Delta A_{n\rm s+\it n\rm d \rightarrow 1s}(\nu _{\rm c})$ , computed according to Eq. (26). To convert to the variable $[\nu -\nu _{\alpha }]/\nu _{\alpha }$ , one should multiply $x_\Gamma$ by $\sim$ $2.54 \times 10^{-7}$ .
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$\begin{figure} \par\includegraphics[width=7.6cm,clip]{eps/7921f021.eps}\end{figure}$	Figure 12: Relative change in the free electron fraction for different values of $n_{2\gamma }$ . The computations were performed including 30 shells.
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