Cosmological hydrogen recombination: The effect of extremely high-$n$ states

Calculations of cosmological hydrogen recombination are vital for the extraction of cosmological parameters from cosmic microwave background (CMB) observations, and for imposing constraints to inflation and reionization. The Planck mission and future experiments will make high precision measurements of CMB anisotropies at angular scales as small as $\ell \sim 2500$, necessitating a calculation of recombination with fractional accuracy of $\approx {10}^{-3}$. Recent work on recombination includes two-photon transitions from high excitation states and many radiative transfer effects. Modern recombination calculations separately follow angular momentum sublevels of the hydrogen atom to accurately treat nonequilibrium effects at late times ($z<900$). The inclusion of extremely high-$n$ ($n\gtrsim 100$) states of hydrogen is then computationally challenging, preventing until now a determination of the maximum $n$ needed to predict CMB anisotropy spectra with sufficient accuracy for Planck. Here, results from a new multilevel-atom code (RecSparse) are presented. For the first time, “forbidden” quadrupole transitions of hydrogen are included, but shown to be negligible. RecSparse is designed to quickly calculate recombination histories including extremely high-$n$ states in hydrogen. Histories for a sequence of values as high as $n_{\max }=250$ are computed, keeping track of all angular momentum sublevels and energy shells of the hydrogen atom separately. Use of an insufficiently high $n_{\max }$ value (e.g., $n_{\max }=64$) leads to errors (e.g., $1.8\sigma$ for Planck) in the predicted CMB power spectrum. Extrapolating errors, the resulting CMB anisotropy spectra are converged to $\sim 0.5\sigma$ at Fisher-matrix level for $n_{\max }=128$, in the purely radiative case.

Figure 1

Schematic of the sparse rate matrix $\mathbf{T}$ with components given by Eq. (15) and submatrix building blocks given by Eq. (40). Boldface zeros denote block matrices of all zeros, and enforce the dipole selection rule that the initial state $l^{\prime }$ angular momentum obeys $l^{\prime }=l\pm 1$, where $l$ is the final state angular momentum. The submatrix ${\mathbf{M}}_{l{l}^{\prime }}$ has dimension $(n_{\max }-n_{\min }+1)\times (n_{\max }-n_{\min }^{\prime }+1)$, where $n_{\min }=2$ if $l=0$, and $n_{\min }=l+1$ if $l>0$. Note that submatrices ${\mathbf{M}}_{l,l}$ on the block diagonal of the larger rate matrix $T$ are themselves diagonal, as seen from Eq. (40) and the fact that in the purely radiative case, ${\Gamma }_{n,n^{\prime }}^{l,l^{\prime }}=0$ if $n\ne n^{\prime }$ and $l=l^{\prime }$.

Figure 2

Early time deviations from statistical equilibrium between different $l$ at fixed $n$ and $n_{\max }$, as computed by RecSparse.

Figure 3

Deviations from statistical equilibrium between different $l$ at fixed $n$ and $n_{\max }$, shown as computed by RecSparse at a variety of times through the recombination process. The left panel shows results for states with $n=25$, while the right panel shows results for states with $n=140$.

Figure 4

The origin of the $l=2$ dip in Figs. 2, 3 is illustrated. Deviations from statistical equilibrium between different $l$ at fixed $n$ and $n_{\max }$ are shown at a variety of times through the recombination process. The left panel shows standard results with RecSparse. The right panel shows the results obtained if $l=2$ Balmer rates are artificially set to zero in the code. This figure highlights the relatively rapid $l=2$ Balmer transitions as the origin of the $l=2$ dip.

Figure 5

Actual population of the $n$th shell compared to its population in Boltzmann equilibrium with $n=2$, as computed by RecSparse at a variety of times through the recombination process.

Figure 6

Actual population of the $n$th shell compared to the Saha equilibrium population, as computed by RecSparse at a variety of times through the recombination process.

Figure 7

Actual population of energy shells compared to Saha equilibrium values, shown for several $n$ values as an explicit function of cosmological redshift $z$.

Figure 8

The left panel shows relative errors between successively more accurate recombination histories with the indicated values of $n_{\max }$. Higher values of $n_{\max }$ make recombination more efficient and yield lower freeze-out values of $x_e(z)$. As $n_{\max }$ increases, relative errors shrink, indicating that recombination is convergent with $n_{\max }$. The right panel contains the absolute recombination histories $x_e(z)$ and a legend. The relative error $\Delta x_e^i$ is defined in Eq. (60).

Figure 9

Relative errors between successively more accurate recombination histories. Values are shown here for 3 different values of redshift $z$. Errors shrink with $n_{\max }$, indicating relative convergence. Note, however, that this figure gives no scale for the absolute error.

Figure 10

Relative errors between temperature anisotropy spectra $C_{\ell }^{\mathrm{TT}}$ computed using CMBFast, modified to include successively more accurate RecSparse recombination histories. Pairs of $n_{\max }$ values used for the comparison are indicated in the legend. $C_{\ell }^{\mathrm{TT}}$ increases with $n_{\max }$, as discussed in Sec. 5b. The correction shrinks with increasing $n_{\max }$. The long dashed line indicates the cosmic variance target for $\Delta C_{\ell }/C_{\ell }$, as discussed in the text.

Figure 11

Relative errors between E-mode polarization anisotropy spectra $C_{\ell }^{\mathrm{EE}}$ computed using CMBFast, modified to include successively more accurate RecSparse recombination histories. Pairs of $n_{\max }$ values used for the comparison are indicated in the legend of Fig. 10. $C_{\ell }^{\mathrm{EE}}$ increases with $n_{\max }$, as discussed in Sec. 5b. The correction shrinks with increasing $n_{\max }$. The long dashed line indicates the cosmic variance target for $\Delta C_{\ell }/C_{\ell }$, as discussed in the text.

Figure 12

Fractional difference between recombination histories with/without E2 quadrupole transitions included for different values of $n_{\max }$. The net effect is always to speed up recombination.