Lyman-$\alpha$ transfer in primordial hydrogen recombination

Cosmological constraints from the cosmic microwave background (CMB) anisotropies rely on accurate theoretical calculations of the cosmic recombination history. Recent work has emphasized the importance of radiative transfer calculations due to the high optical depth in the H i Lyman lines. Transfer in the $\mathrm{Ly}\alpha$ line is dominated by true emission and absorption, Hubble expansion, and resonant scattering. Resonant scattering causes photons to diffuse in frequency due to random kicks from the thermal velocities of hydrogen atoms, and also to drift toward lower frequencies due to energy loss via atomic recoil. Past analyses of $\mathrm{Ly}\alpha$ transfer during the recombination era have either considered a subset of these processes, ignored time dependence, or incorrectly assumed identical emission and absorption profiles. We present here a fully time-dependent radiative transfer calculation of the $\mathrm{Ly}\alpha$ line including all of these processes, and compare it to previous results that ignored the resonant scattering. We find a faster recombination due to recoil enhancement of the $\mathrm{Ly}\alpha$ escape rate, leading to a reduction in the free electron density of 0.45% at $z=900$. This results in an increase in the small-scale CMB power spectrum that is negligible for the current data but will be a $0.9\sigma$ correction for Planck. We discuss the reasons why we find a smaller correction than some other recent computations.

Figure 1

A schematic representation of the photon phase space density near $\mathrm{Ly}\alpha$. The dashed line shows the chemical equilibrium solution, Eq. (10). The true solution (solid line) comes to equilibrium near line center. As one moves to the red side of the line, the rate of absorption, emission, and scattering decrease, and at some point ($\nu \sim {\nu }_{\mathrm{trans}}$) the photons fall out of chemical equilibrium and redshift out of the line.

Figure 2

The correction to the recombination history due to $\mathrm{Ly}\alpha$ diffusion. The numerical computation of Sec. 3 is shown with a solid line, and Sec. 4 with a dashed line. Recombination is accelerated due to the additional redshifting of photons via atomic recoil.

Figure 3

The correction factors ${\xi }_1(z)$ and ${\xi }_2(z)$. The “standard” result for infinitesimally narrow $\mathrm{Ly}\alpha$ line with no damping wing or diffusion effects is ${\xi }_1(z)={\xi }_2(z)=1$. Note that over most of the recombination history these are greater than 1, implying a faster $2p\to 1s$ decay rate but also more photons redshifting out of the $\mathrm{Ly}\alpha$ line. The latter effect will lead to more two-photon absorption at low redshifts. [The glitch in ${\xi }_1(z)$ at $z\approx 1360$ is a startup transient from $\mathrm{Ly}\beta$ at $z=z_{\mathrm{init}}$ redshifting into $\mathrm{Ly}\alpha$; given that the overall effect of the scattering correction is $<0.6\%$ in $C_{\ell }$, this is far too small to affect CMB results.]

Figure 4

The radiation spectrum in the vicinity of the $\mathrm{Ly}\alpha$ line at $z=1006$. The solid line shows the results of the diffusion code, and the dashed line shows the old code with no diffusion [19]. The vertical dotted lines show the boundaries of the $\mathrm{Ly}\alpha$ diffusion region. The “chemical equilibrium” curve shows the approximation $f_{\nu }\approx (x_{2p}/3x_{1s})e^{-h(\nu -{\nu }_{\mathrm{Ly}\alpha })/T_{\mathrm{r}}}$ that should be valid in the immediate vicinity of line center. Note that ${\nu }_{\mathrm{Ly}\alpha }=0.75$ Rydbergs.

Figure 5

(a) The analytic correction factors $\chi (W,S)$ and (b) $\mathcal{I}(W,S)$ associated with transport in the $\mathrm{Ly}\alpha$ line. The thick bold line shows the factors without frequency diffusion, i.e. for $S=0$. The thin lines show the factors for $S={10}^{-7}$, ${10}^{-6}$, ${10}^{-5}$, ${10}^{-4}$, and ${10}^{-3}$ from bottom to top. Note that both emission/absorption in the far damping wings (parametrized by $W$) and frequency diffusion (parametrized by $S$) tend to increase the escape rate and the number of distortion photons in the $\mathrm{Ly}\alpha$ blue wing, as expected.

Figure 6

The dimensionless $\mathrm{Ly}\alpha$ transport parameters $W(z)$ and $S(z)$.

Figure 7

The change in CMB power spectra for the temperature (top panel), polarization (middle panel), and the correlation coefficient (bottom panel) due to inclusion of $\mathrm{Ly}\alpha$ scattering. The overall tilt and oscillating behavior are both the result of a faster recombination and hence higher redshift of the surface of last scattering: the tilt from the reduced Silk damping, and the oscillations due to the smaller acoustic horizon.