An emulator for the ionizing photon mean free path in ultra-high resolution simulations: the implications of mean free path measurements for the reionization history

TL;DR

The study presents a high-fidelity neural emulator for the ionizing photon mean free path $\lambda_{\rm mfp}$, trained on ultra-high-resolution radiative-transfer simulations that resolve gas clumping down to $\sim$kpc scales. By predicting $\lambda_{\rm mfp}$ as a function of $z$, $z_{\rm re}$, $\Gamma_{-12}$, $\delta/\sigma$, and photon energy, and by integrating over extended, patchy reionization histories, the authors constrain the reionization timeline and estimate the global ionizing emissivity without assuming simple opacity scaling. Their results favor late or extended reionization with non-negligible neutral fractions around $z\sim6$, and reveal a decline in the ionizing emissivity by a factor of $2$–$3$ from $z\sim6$ to $4.8$, aligning with but refining previous analyses. The emulator offers a fast, physically grounded alternative to large-volume simulations and can act as a subgrid opacity model in future reionization studies, aiding interpretation of JWST and future survey data.

Abstract

Measurements of the mean free path of ionizing photons from high-redshift quasar spectra at $z \sim 5$-$6$ constrain the reionization history, but interpreting them requires modeling the kiloparsec-scale clumping that large-volume reionization simulations cannot resolve. We present a deep learning emulator for the mean free path (MFP) trained on high-resolution cosmological radiative transfer simulations of ionization fronts sweeping through small 2 comoving~Mpc/h volumes. Using a residual multi-layer perceptron neural network, we predict the MFP at a given redshift as a function of the reionization redshift, photoionization rate, wavelength, and box-scale density, achieving a median relative error of 1.6\% across nearly four orders of magnitude in MFP. Integrating its predictions over box-scale overdensity and an extended reionization history allows the emulator to predict the global MFP. We apply the emulator to extended reionization histories constrained by observed photoionization rates, finding that models prefer late reionization with substantial neutral fractions persisting at $z \lesssim 6$. Fitting a parametric ionization history yields a midpoint of reionization of $z_{\rm re} = 6.8\pm 1.2$ for reionization durations consistent with Planck and kinetic Sunyaev-Zeldovich constraints, and the universe being $10\%$ neutral still at $z_{\rm re} < 5.8 ~(6.3)$ at 1~(2)$σ$. Global ionizing emissivity inferences using measurements of the photoionization rate and MFP plus our emulator, which avoids common power-law assumptions, suggest a factor of $2-3$ decline between $z = 6$ and $4.8$, in agreement with previous studies. Our method provides an efficient (and more converged) alternative to large-volume radiative-hydrodynamic simulations of reionization for interpreting MFP measurements, and can also serve as a subgrid prescription for the ionizing opacity within such simulations.

Figures (8)

• Figure 1: Gas density distribution from our simulation suite showing the diversity of IGM environments at $z=5.5$. Each panel shows a 2D slice through a $2\;h^{-1}$ cMpc simulation volume with different combinations of redshift ($z$), reionization redshift ($z_{\rm re}$), photoionization rate ($\Gamma_{-12}$), and box-scale overdensity ($\delta/\sigma$). The colorbar highlights gas densities that range from voids (blue) to dense filaments and minihalos (red/orange).
• Figure 2: Neural network emulator performance and sensitivity to reionization parameters. Left: Predicted versus true MFP on the held-out test set, showing excellent agreement with a median relative error of 1.6%. The red dashed line indicates perfect agreement. Middle: MFP evolution as a function of redshift for fixed $\Gamma_{-12} = 0.3$ and varying reionization redshift $z_{\rm re} = \{5, 6, 6.5, 7, 9, 12, 15\}$. Points show direct simulation measurements, while dashed lines show emulator predictions. The black star shows a validation simulation at $z_{\rm re} = 6.5$ not included in the training suite, which the emulator predicts with 1.7% error at 13.6 eV. Right: MFP evolution for fixed $z_{\rm re} = 7$ and varying photoionization rate $\Gamma_{-12} = \{0.03, 0.1, 0.3, 1.0, 3.0\}$. The emulator accurately captures both the amplitude and redshift evolution of MFP across the full parameter space, demonstrating its sensitivity to the physical parameters.
• Figure 3: Grid search results constraining $\Gamma_{-12}$ and $z_{\rm re}$ from MFP observations at $z > 5.0$, assuming instantaneous reionization. Left: Best-fit model (black line) compared to observations from Worseck2014 (purple diamonds), Becker2021 (brown squares), and 2023ApJ...955..115Z (pink triangles). Several other parameter combinations are shown as dashed lines. The sharp drop in the best-fit model at $z \approx 5.9$ is unphysical for a global average and indicates that dynamical relaxation alone cannot explain the rapid MFP evolution. Right:$\chi^2$ landscape showing the best-fit at $\Gamma_{-12} = 0.36 \times 10^{-12}$ s$^{-1}$ and $z_{\rm re} = 5.82$ (red star). Dashed red and dotted blue contours show 68% and 95% confidence regions.
• Figure 4: Mean free path predictions for three reionization histories compared to observations at $z > 4.5$. Each panel shows a different reionization scenario: early-early (left), early-late (center), and late-late (right) shown on the first panel. Colored points show observations from multiple studies, and solid lines show emulator predictions using measured photoionization rates from the same studies. Shaded bands show uncertainties propagated from $\Gamma_{-12}$ measurement errors. All three models provide reasonable agreement with observations when using measured $\Gamma_{-12}$ values.
• Figure 5: Constraints on the ionization history from a tanh model fit to MFP observations. Top Left: Best-fit model (red line) compared to observed MFP values (blue points with error bars). Sample parameter combinations are shown as thin colored lines, whose parameter values are the corresponding stars in the other panels wit the red star marking the best fit. Top Right:$\chi^2$ landscape in the $(z_{\rm end}, \Delta z)$ plane. Dashed magenta and dotted black lines show 68% and 95% confidence regions from the MFP fit. The blue band shows Planck+kSZ (using SPT and ACT data) CMB constraints PlanckXLVII2016 ($z_{\rm re} = 7.8 \pm 0.9$, $\Delta z < 2.8$ at 68% confidence) with $z_{\rm re} = 7.8$ represented by the navy blue line. The grey shaded region shows parameter space excluded by dark gap observations Davies_2025, which constrain $z_{\rm end} > 5$ at $2 \, \sigma$. Colored stars mark sample parameter combinations shown in the other panels. Bottom Left: Best-fit ionization history (red line) with sample parameter combinations shown as thin colored lines matching the stars in the right panels. The vertical navy blue line and blue shaded region indicate $z_{\rm re} = 6.84$ (midpoint) and the $\pm\Delta z$ range. Red circles mark the ionization fractions at the observed redshifts. Bottom Right: Same $\chi^2$ landscape shown in the $(z_{\rm re}, \Delta z)$ plane for comparison with Planck+kSZ constraints. The grey shaded region again shows the dark gap exclusion. Our MFP-based constraints are consistent with Planck+kSZ at the 1$\sigma$ level.