ECHO21: a tool for modelling global 21-cm signal from dark ages to reionization

Technical Summary of "ECHO21: A tool for modelling global 21-cm signal from dark ages to reionization"

Problem and Motivation
The dark ages and cosmic dawn are poorly understood periods of the cosmic timeline due to a lack of direct observational probes. The 21-cm hyperfine transition line of neutral hydrogen offers a potential probe to unravel the physics of the first stars, the intergalactic medium (IGM), and cosmological parameters. While numerical and semi-numerical codes like 21cmFAST and BEORN provide accurate, physically rich predictions, they are computationally slow, hindering efficient astrophysical or cosmological inference from large datasets. Conversely, existing analytical codes like ARES and Zeus21 offer speed but lack certain physical components, specifically the heating effects of Lyman-α (Lyα) photons. The authors identify that the omission of Lyα heating can lead to strong biases in signal modeling. To address the need for a fast, sufficiently realistic, and flexible tool capable of handling large parameter spaces for inference, the authors introduce ECHO21 (Exploring the Cosmos with Hydrogen Observation).

Methodology
ECHO21 is a Python package utilizing an analytical framework to model the global 21-cm signal from redshift $z \approx 1500$ down to the present day. The code operates on a two-zone model of the IGM, distinguishing between the bulk IGM and fully ionized H II regions.

• Physical Framework: The code solves coupled differential equations for the gas kinetic temperature ($T_k$), the bulk IGM electron fraction ($x_e$), and the volume-filling factor of H II regions ($Q$).

  • Thermal History: The evolution of $T_k$ accounts for adiabatic cooling, Compton heating, X-ray heating, and, crucially, Lyα heating. The authors note that while previous analytical codes ignored Lyα heating, ECHO21 includes it, calculating the heating rate based on the Lyα specific intensity ($J_{Ly}$) derived from the star formation rate density (SFRD).
  • Ionization History: The code tracks the recombination of the bulk IGM and the ionization driven by X-rays and UV photons. It models the volume-filling factor of H II regions by comparing the emission rate of ionizing photons against radiative recombinations.
  • Star Formation Rate Density (SFRD): A novel feature of ECHO21 is the inclusion of three distinct SFRD models:
    1. Physically-motivated: Based on the collapse of baryons into dark matter halos (using Press-Schechter, Sheth-Tormen, or Tinker halo mass functions) with a constant star formation efficiency.
    2. Semi-empirical: A variant of the physical model incorporating a characteristic star formation timescale and a sharp cutoff at a minimum virial temperature.
    3. Empirically-motivated: A model constrained by UV luminosity function data (Madau-Dickinson form at low $z$, exponential drop at high $z$).

• Parameters: The model is controlled by six free astrophysical parameters: Lyα normalization ($f_{Ly}$), Lyman series spectral index ($s$), X-ray normalization ($f_X$), X-ray spectral index ($w$), escape fraction of ionizing photons ($f_{esc}$), and minimum virial temperature ($min(T_{vir})$). Additionally, the code allows for the variation of standard cosmological parameters ($H_0, \Omega_m, \Omega_b, \sigma_8, n_s, T_{\gamma0}, Y_p$).

• Performance: The analytical approach allows ECHO21 to generate a single global 21-cm signal instance in approximately $O(1)$ second on a single CPU. The code is designed for MPI parallelization, enabling the generation of thousands of models in minutes.

Key Contributions

1. Inclusion of Lyα Heating: ECHO21 is the first analytical code to explicitly include Lyα heating in the thermal evolution of the IGM. The authors demonstrate that neglecting this term can introduce significant biases, particularly in scenarios with low X-ray emissivity and strong Lyα emission.
2. SFRD Model Diversity: The code uniquely offers three different approaches to modeling the SFRD (physical, semi-empirical, and empirical), allowing users to assess how the choice of star formation prescription impacts the resulting 21-cm signal.
3. Flexibility in Cosmology: Unlike many specialized codes, ECHO21 treats cosmological parameters with the same ease as astrophysical parameters, facilitating the study of cosmological uncertainties (e.g., the Hubble tension) on the 21-cm signal.
4. Speed and Scalability: The code is optimized for rapid inference, capable of generating 15,625 models across a six-dimensional astrophysical parameter space in ~17 minutes using 50 CPUs.

Results

• Fiducial Model: Using a physically-motivated SFRD and Press-Schechter halo mass function, the fiducial model predicts a reionization history complete by $z \approx 5.3$ with a 50% ionization redshift of $z_{re} = 7.42$. The electron-scattering optical depth is calculated as $\tau_e \approx 0.0592$, consistent with Planck 2018 constraints.
• Signal Characteristics: The fiducial global 21-cm signal exhibits a first absorption peak of $-41.1$ mK at $z=87.5$ (dark ages) and a second, deeper absorption peak of $-142.4$ mK at $z=19.2$ (cosmic dawn), followed by an emission peak of $22.2$ mK at $z=11.2$.
• Impact of Lyα Heating: Comparisons with models excluding Lyα heating show that the omission leads to a stronger (deeper) 21-cm absorption signal. The authors emphasize that Lyα heating is a crucial component for accurate modeling, as it can counteract cooling effects and alter the signal strength non-monotonically depending on X-ray parameters.
• Impact of SFRD Models: The choice of SFRD model significantly alters the signal. Empirically-motivated models, which predict a later onset of star formation compared to physically-motivated models, result in signals that are delayed by $\sim 8.8$ in redshift and shallower by $\sim 39.4$ mK. Similarly, the choice of Halo Mass Function (HMF) affects the timing and depth of the signal features.
• Parameter Sensitivity: The signal is most sensitive to $f_{Ly}$, $f_X$, $w$, and $min(T_{vir})$. Variations in cosmological parameters within current observational uncertainties result in relatively small variations in the signal amplitude (e.g., a range of $\sim 18$ mK in the peak absorption depth).

Significance and Claims
The authors position ECHO21 as a flexible and extensible open-source tool designed for making "quick but sufficiently realistic astrophysical inferences" from upcoming 21-cm datasets (e.g., from REACH, SARAS, or future space-based experiments). They claim the code is particularly well-suited for:

• Performing robust astrophysical and cosmological inference via large-scale parameter exploration.
• Testing data analysis pipelines (specifically for the REACH collaboration) with physically-motivated signals rather than purely empirical fits.
• Building synergies between 21-cm observations and other cosmological probes, such as BBN predictions, CMB anisotropies, and the Hubble tension.

The paper modestly acknowledges that while ECHO21 is less accurate than semi-numerical models due to its analytical approximations (e.g., neglecting multiple Lyα scatterings), its speed and modularity make it an ideal tool for initial inferences and for incorporating new physics in future versions. The code is publicly available via PyPI and GitHub.