The heating rate of the Intergalactic Medium by Lyman-$α$ photon scattering

Abstract: The strength of the 21-cm Cosmic Dawn radio absorption signature against the Cosmic Microwave Background or against a bright background radio source depends on the temperature of the neutral hydrogen in the intergalactic medium. While x-rays from forming galaxies will likely dominate the heating rate, recent models suggest scenarios in which heating by the scattering of Lyman-$α$ photons sourced by the galaxies contributes non-negligibly as well, and may even dominate. The efficiency of Lyman-$α$ photon heating for a point source at high redshift using a numerical solution to the exact radiative transfer equation is provided here. It is found to be a factor of several smaller than the rate computed in the commonly used diffusion approximation to the radiative transfer equation.

• The paper presents a full radiative transfer solution which finds that the Lyα recoil heating rate is approximately 3.8 times lower than estimates from the diffusion approximation.
• It carefully treats the resonant scattering and photon redistribution, highlighting the limitations of simplified diffusion models in conserving photon number and energy.
• The findings imply that more accurate Lyα heating models are crucial for reliable 21-cm signal predictions and interpretation of Cosmic Dawn observations.

The Heating Rate of the Intergalactic Medium by Lyman-$\alpha$ Photon Scattering

Introduction

This paper addresses the contribution of Lyman-$\alpha$ (Ly$\alpha$) photon recoil heating to the thermal evolution of the Intergalactic Medium (IGM) during Cosmic Dawn, specifically its impact on the 21-cm absorption signature in cosmological radio observations. The thermal state of neutral hydrogen in the IGM plays a critical role in determining the amplitude of the 21-cm differential brightness temperature signal against the Cosmic Microwave Background (CMB). While x-ray heating driven by early galaxies and high-mass X-ray binaries is conventionally regarded as the dominant mechanism, recent modeling has highlighted scenarios where Ly$\alpha$ scattering can provide a non-negligible, or even dominant, heating contribution.

Radiative Transfer Framework and Methodology

The evolution of the radiation field and its energy deposition into the IGM by Ly$\alpha$ scattering is governed by the radiative transfer (RT) equation, which requires careful treatment of resonant scattering and atomic recoil effects. Standard approaches frequently invoke the diffusion approximation as a computational expedient, replacing the integro-differential RT equation with a second-order partial differential equation for the radiation field. However, this procedure suffers from limitations in the vicinity of the line core: specifically, it cannot simultaneously enforce both photon number and total energy conservation over all frequencies. Alternative strategies like the Fokker-Planck approximation offer photon and energy conservation, but are prone to non-physical artifacts within the line core.

The author numerically solves the full angle-averaged, frequency-resolved RT equation for a point continuum source in an expanding universe at $z=10$, taking into account precise re-distribution of photon frequencies due to scattering, recoil, and gas temperature effects. The approach yields self-consistent solutions for the local Ly$\alpha$ radiation field, the emergent light temperature $T_L$, and consequently, the local rate of energy exchange between Ly$\alpha$ photons and the IGM.

Main Results

A key result of this work is the demonstration that the Ly$\alpha$ photon recoil heating rate, when computed via the exact RT equation, is significantly lower than standard estimates from the diffusion approximation. For an IGM temperature $\alpha$0 K at $\alpha$1, the calculated heating efficiency factor $\alpha$2 for a uniform source distribution is $\alpha$3, with $\alpha$4 K nearly constant with distance from the source. In contrast, the widely used diffusion approximation (parameterized following prior work) yields a value near $\alpha$5—a factor of approximately 3.8 higher than the rigorous result.

The analysis clarifies that, at large distances from a source, $\alpha$6 asymptotes to a nearly spatially constant value, and the discrepancy between approximations is strictly within the resonance core. The results highlight the limitations of current semi-analytic treatments, particularly in Cosmic Dawn modeling where absolute temperature offsets at the sub-Kelvin level are consequential for 21-cm signal predictions.

Implications for Cosmic Dawn Modeling

The findings have direct implications for IGM heating prescriptions in large-scale cosmological simulations and semi-numerical post-processing schemes. Since the Ly$\alpha$7 recoil heating is systematically overestimated by the conventional diffusion approximation, models employing such estimates may incorrectly predict earlier or more efficient heating of the IGM, artificially suppressing the depth of potential 21-cm absorption features. This discrepancy affects predictions for the timing and morphology of the 21-cm global signal, as well as the interpretation of constraints from ongoing and future radio observations.

Practically, the results motivate the adoption of more accurate numerical treatments or recalibrations of the Ly$\alpha$8 heating module in 21-cm simulation codes. Theoretical implications also include the importance of precision RT in accurately modeling the microphysics of resonance scattering during the Cosmic Dawn and the necessity of accounting for breakdowns in simple diffusion closures, especially in regimes characterized by strong resonant coupling and low kinetic temperatures.

Future Prospects

Further investigation is warranted into alternative approximations that may provide computationally efficient yet accurate recoil heating rates suitable for inclusion in large-volume simulations. Cross-validation with Monte Carlo RT methods, as well as with new analytic closures maintaining energy and photon conservation, may enable the development of improved subgrid Ly$\alpha$9 heating models. These advances will facilitate more robust interpretation of ongoing and planned 21-cm cosmology experiments probing the physics of the first luminous sources.

Conclusion

This study rigorously quantifies the Ly$\alpha$0 photon recoil heating rate in the IGM using the full radiative transfer equation, demonstrating that diffusion-based estimates routinely used in the literature overpredict the heating efficiency by factors of several. The work underscores the necessity for precision modeling of resonant radiative transfer effects when simulating the pre-reionization IGM, and indicates that current approaches may require substantial revision. These insights are essential for refining theoretical models of the Cosmic Dawn and for the interpretation of upcoming redshifted 21-cm observations.