Constraints on the Thompson optical depth to the CMB from the Lyman-$α$ forest

Abstract

We present the first constraints on the electron optical depth to reionization, $τ_{\mathrm{e}}$, from the Lyman-$α$ forest alone for physically motivated reionization models that match the reionization's end-point, $z_{\rm{end}}$, required by the same astrophysical probe, and for symmetric reionization models with fixed duration, $Δz$, commonly adopted in CMB reionization analyses. Compared to traditional estimates from the latter, the Lyman-$α$ forest traces the ionization state of the IGM through its coupling with the thermal state. We find an explicit mapping between the two solving the chemistry and temperature evolution equations for hydrogen and helium. Our results yield $τ_{\mathrm{e}}$=$0.042^{+0.047}_{-0.02}$ (95% C.L) and $τ_{\mathrm{e}}$=$0.042^{+0.024}_{-0.015}$ for reionization models with $z_{\rm{end}}$ and $Δz$-fixed, respectively, disfavoring a high $τ_{\mathrm{e}}$=0.09 by 2.57$σ$ and 4.31$σ$. With mock Lyman-$α$ forest data that mimics the precision of future larger quasar sample datasets, we would potentially obtain tighter $τ_{\mathrm{e}}$ constraints and exclude such a high $τ_{\mathrm{e}}$ with a higher significance, paving the way for novel constraints on the epoch of reionization from a large-scale structure probe independent of the CMB.

Figures (5)

• Figure 1: H I photo-ionization rate $\Gamma_{\rm HI}$ (left), electron scattering optical depth $\tau_{\mathrm{e}}$ (middle) and H II ionized fraction $x_{\rm{HII}}$ (right) for reionization models that keep $z_{\rm{end}}$=${z}_{\rm{end}}^{\rm{ P19\,late}}$ fixed (varying $z_{\rm{mid}}$) and $\Delta z$$\approx$ 6.65 fixed (varying $z_{\rm{start}}$) in the top and bottom rows, respectively. We highlight in the middle column $\tau_{\mathrm{e}}$ inferred by previous works (Planck18belsunce21sailer25). We further show puchwein19's late and hm12's reionization models in black and orange, respectively. The gray dashed line on the right column shows the Lyman-$\alpha$ forest end-point of reionization requirement from the former: $x_{\rm{HII}}$$\approx$ 1.
• Figure 2: Posterior in the $u_{0}-T_{0}$ plane at $z$=5.0 for the default analysis in gg25 (Gaussian $T_{0}$ priors) shown as black and red solid contours. The posteriors resulting from the chains using mock data with smaller relative error bars of 5% are shown as dashed contours also in black and red. The colormap shows isocontours for $\tau_{\mathrm{e}}$. We highlight the $\tau_{\mathrm{e}}$ = [0.06, 0.09] contours in white. The grid points show the simulated reionization models used to infer the extrapolation scheme for $\tau_{\mathrm{e}}$: fixing $z_{\rm{end}}$ in the left and $\Delta z$ in the right.
• Figure 3: 1D posterior distribution for $\tau_{\mathrm{e}}$ obtained with $z_{\rm{end}}$-fixed in black and $\Delta z$-fixed in red reionization models. Solid and dashed lines correspond to the analysis that uses boera19 and GHOSTLy-like data, respectively. Vertical lines indicate the median of each distribution.
• Figure S1: 2D posterior in ${u_0}-{T_0}$ plane at $z$=5.0 for four analyses with $z_{\rm{end}}$-fixed reionization models, using different priors on $\tau_{\mathrm{e}}$ with the data from boera19. The blue contour is equivalent to that shown in solid black and red lines in Figure \ref{['isocontours']}. Red, yellow and green contours, respectively, are obtained for chains with the following $\tau_{\mathrm{e}}$ priors: $\tau_{\mathrm{e}} \geq 0.034$, $\tau_{\mathrm{e}}=0.054 \pm 0.007$ (Planck13), $\tau_{\mathrm{e}}=0.09 \pm 0.0007$ (sailer25).
• Figure S2: Evolution of the ionized electron fraction for $z_{\rm{end}}$ and $\Delta z$-fixed models on the left and right, respectively. We further show puchwein19's late and hm12's reionization models in black and orange, respectively.