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

TL;DR

ECHO21 delivers a fast analytic tool to model the global 21-cm signal from the dark ages through cosmic dawn to reionization, enabling rapid parameter inference from upcoming data. It solves the coupled thermal and ionization histories in a two-zone IGM, incorporating Ly$\alpha$ heating and Ly$\alpha$ coupling explicitly, and ties emissivities to three distinct SFRD models, while allowing flexible cosmological parameters. The paper demonstrates fiducial results, assesses the impact of SFRD modelling, and shows scalable exploration of large parameter spaces with MPI parallelization, highlighting the importance of Ly$\alpha$ heating for unbiased inferences. As an open-source, extensible package, ECHO21 provides a practical bridge between 21-cm observations and broader cosmological and galaxy-formation probes, with potential synergy for pipeline testing and cross-disciplinary analyses.

Abstract

We introduce a Python package called ECHO21 for modelling the global 21-cm signal from the dark ages through cosmic dawn to the end of reionization. Leveraging its analytical framework, ECHO21 generates a single model in $\mathcal{O}(1)\,$s, allowing a large number of signals to be generated efficiently by distributing models across multiple cores. Thus, it is ideal for performing astrophysical or cosmological inference from a given 21-cm dataset. We offer six astrophysical parameters that control the Lyman-$α$ (Ly$α$) emissivity, X-ray emissivity, emissivity of ionizing photons, and star formation rate. Beyond its efficiency some of the attractive and novel features in ECHO21 relative to previously published codes are inclusion of Ly$α$ heating, ability to vary the standard cosmological parameters as easily as the astrophysical parameters, and different models of star formation rate density (physically-motivated, a semi-empirical, and an empirically-motivated). With a number of 21-cm experiments soon to provide cosmic dawn 21-cm data, ECHO21 is a flexible and extensible new open-source package for making quick but sufficiently realistic astrophysical inferences. We make our code publicly available.

Figures (10)

• Figure 1: Comoving cosmic star formation rate density in the redshift range $0\leqslant z\leqslant59$. For all physically-motivated models (equation \ref{['eq:phy_sfrd']}) we adopt 10 % SFE and $e4K$ minimum virial temperature. The solid-blue, dashed-red, and dotted-green curves correspond to Press-Schechter ('press74'), Sheth-Tormen ('sheth99'), and Tinker ('tinker08') HMF, respectively. The brown long-dashed curve is our semi-empirical model of SFRD (equation \ref{['eq:semi_sfrd']}) for a press74 HMF. The dash-dotted-black curve is the empirically-motivated model of SFRD (equation \ref{['eq:emp_sfrd']}).
• Figure 2: Bulk IGM electron fraction ($x_{\mathrm{e}}$, dotted pink), volume-filling factor ($Q$, dashed brown), and two-zone-model-averaged ionization fraction ($\bar{x}_{\mathrm{i}}$, solid cyan) computed using ECHO21. In this fiducial model, 50% reionization is complete by $7.42$ and 100% by $z=5.3$. We also show constraints on the state of ionization from some of the recent literature, namely McGreer_2014, Greig_2016Greig_2019, Davies_2018, Ning_2022, Nakane_2024, and Umeda_2024.
• Figure 3: Electron-scattering optical depth as a function of maximum integration redshift $z$ for our fiducial model generated using ECHO21. For all three cases we work with a physically-motivated SFRD model. The optical depth for press74 HMF out to the beginning of cosmic dawn comes out to be $0.0592$ which is within 1$\sigma$ limits of Planck 2018 results.
• Figure 4: Left: gas kinetic temperature (solid blue), CMB temperature (dotted green), and spin temperature (dashed red). Right: the global 21-cm signal. For fiducial set of parameters (see Table \ref{['tab:par']}) the strongest signal is $-142.4mK$, observed at $z = 19.21$. The redshift range is $5\leqslant z\leqslant1500$, covering dark ages until the end of EoR.
• Figure 5: Collisional (solid orange) and Ly$\upalpha$ (dashed purple) coupling for our fiducial set of parameters. Fig. \ref{['fig:t21']} shows the corresponding spin temperature and 21-cm signal.