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Absorption effects in the expanding Universe: spectral transmittance functions of intergalactic medium for distant sources

This paper analyzes the spectral transmittance of the intergalactic medium for high-redshift sources ( $5 \le z \le 15$ ) by modeling absorption from neutral hydrogen and helium across various reionization histories, ultimately providing a compact analytic prescription for effective transmission and demonstrating its impact on the observed spectra of distant starless halos.

Original authors: Anguohao Yang, Bohdan Novosyadlyj, Pavlo Kopach, Bohdan Melekh, Gennadii Milinevsky

Published 2026-04-08

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CC BY 4.0

Original authors: Anguohao Yang, Bohdan Novosyadlyj, Pavlo Kopach, Bohdan Melekh, Gennadii Milinevsky

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the Universe as a giant, expanding room filled with invisible fog. In the very beginning, this fog was thick with neutral hydrogen and helium gas. As the first stars and galaxies began to light up, their light had to travel through this fog to reach us today.

This paper is essentially a guidebook for understanding how that fog dims and distorts the light from the earliest objects in the Universe.

Here is a breakdown of the paper's main ideas using simple analogies:

1. The "Fog" of the Early Universe

Think of the early Universe (between 5 and 15 billion years ago) as a dense, cosmic mist. This mist is made of two main ingredients: Hydrogen and Helium.

The Problem: When light from a distant star or galaxy tries to pass through this mist, the gas atoms act like tiny sponges. They soak up specific colors (wavelengths) of light.
The Result: By the time that light reaches Earth, huge chunks of its spectrum are missing. It's like looking at a bright white light through a pair of sunglasses that only block specific colors, leaving you with a dark, patchy view.

2. Two Different Histories: The "Early" vs. "Late" Party

The scientists in this paper are trying to figure out exactly when this fog cleared up (a process called reionization). They looked at the latest data from space telescopes (like the James Webb Space Telescope) and the Planck satellite to create two possible stories:

The "Late Reionization" Scenario: The fog lingered longer. The Universe stayed dark and misty until about 6.5 billion years after the Big Bang.
The "Early Reionization" Scenario: The fog cleared up faster. The Universe became transparent about 7.7 billion years after the Big Bang.

The paper creates mathematical formulas to describe these two scenarios, acting like a "weather forecast" for the early Universe's atmosphere.

3. The Three Layers of Absorption

The authors didn't just look at Hydrogen; they realized the fog is a three-layer cake of absorption:

Hydrogen (HI): This is the main layer. It creates a big "blackout" on the red end of the spectrum (the long wavelengths). It's like a heavy curtain blocking the back of the room.
Neutral Helium (HeI): This is a secondary layer that absorbs light in the middle of the spectrum.
Singly Ionized Helium (HeII): This is the "sharp edge" layer. It eats up the shortest, most energetic wavelengths (ultraviolet light).

The Analogy: Imagine a radio station playing music.

Hydrogen cuts off the bass notes (low frequencies).
Helium cuts off the high-pitched squeals (high frequencies).
The result is that the "music" (the light spectrum) from distant sources gets muffled in the middle, creating a wide, deep "valley" of silence where no light gets through.

4. The "Transmission Map"

The biggest achievement of this paper is creating a universal "transmittance function."

Think of this as a filter map. If you know how bright a star should be, this map tells you exactly how dim it will look after traveling through the cosmic fog.
The authors created a simple, easy-to-use formula (a "recipe") that astronomers can plug into their computers. Instead of running complex, slow simulations every time, they can just use this formula to predict how the fog will change the light from any distant object.

5. Testing the Theory: The "Cloud 9" Experiment

To show how their map works, the authors imagined a specific type of object: a "Cloud 9" halo.

What is it? Imagine a giant, hot ball of gas floating in space that is too hot to form stars. It's a "starless" cloud.
The Experiment: They calculated what these clouds would look like at different distances (redshifts).
The Finding: When they applied their "fog filter," the beautiful, smooth glow of the cloud turned into a jagged, dark spectrum with deep holes.
The Catch: They found that even with the best telescopes we have today (like JWST), these specific "starless clouds" are too faint to see directly. However, the shape of their light (if we could see it) would tell us exactly when the fog cleared up.

Why Does This Matter?

This paper is like giving astronomers a new pair of glasses.

Before, we knew the fog existed, but we weren't sure exactly how thick it was or when it disappeared.
Now, with these new formulas, astronomers can look at the light from the very first galaxies and say, "Ah, the fog cleared up at this specific time," or "The helium cleared up before the hydrogen."

In summary: The universe was once a foggy room. This paper provides the math to understand how that fog filtered the light from the first stars, helping us figure out exactly when the lights finally turned on in the cosmic dark ages.

1. Problem Statement

The paper addresses the challenge of modeling the absorption of radiation from high-redshift sources ( $5 \le z \le 15$ ) by the diffuse intergalactic medium (IGM) during the Cosmic Dawn and Epoch of Reionization.

Context: The IGM, dominated by neutral hydrogen (HI) and helium (HeI, HeII), absorbs radiation via Lyman-series lines and continua. This creates broad absorption troughs in the spectra of distant sources, suppressing their detectability.
Limitations of Existing Models: Current treatments often rely on:
1. Effective optical depth prescriptions based on the statistical distribution of discrete absorbers (e.g., Lyman-alpha forests, Lyman-limit systems) like Madau (1995) or Inoue et al. (2014).
2. Numerical radiative-transfer simulations.
Gap: There is a need for a closed-form analytic description of IGM transmission that incorporates observationally constrained reionization histories (specifically for both Hydrogen and Helium) and treats the IGM as a smoothly distributed medium. This is crucial for distinguishing between different reionization scenarios (early vs. late) and understanding the specific impact of Helium absorption, which is often less emphasized than Hydrogen.

2. Methodology

The authors developed a comprehensive analytical framework to compute wavelength-dependent optical depths and spectral transmittance functions.

A. Reionization Histories (Ionization Fractions)

The authors constructed two analytic approximations for the neutral hydrogen fraction, $x_{HI}(z)$ , based on observational data:

KBBI (Late Reionization): Fits data from Kageura et al. (2025), Bosman et al. (2022), Becker et al. (2013), and Inoue et al. (2014). Predicts $x_{HI} \approx 0.5$ at $z \approx 6.5$ .
GBBI (Early Reionization): Fits data from Glazer et al. (2018), Bosman et al. (2022), and lower-redshift Lyman-alpha forest data. Predicts $x_{HI} \approx 0.5$ at $z \approx 7.7$ .

For Helium, they developed a phenomenological model:

HeI: Assumed to track HI at high redshift ( $x_{HeI} \approx x_{HI}$ ) and transition to a residual fraction ( $x_{HeI} = \eta x_{HI}$ ) at low redshift.
HeII: Modeled via a smooth transition function representing the second ionization of helium (HeII $\to$ HeIII), driven by quasars, completing around $z \approx 2.7$ .
Validation: The models were validated by computing the Thomson scattering optical depth ( $\tau_{Th}$ ) and comparing it with Planck 2018 CMB constraints.

B. Optical Depth Calculations

The authors derived analytical expressions for optical depths ( $\tau$ ) in the Lyman-series lines and continua for HI, HeI, and HeII:

HI: Generalized the Gunn-Peterson formula for the first 40 Lyman-series lines and the Lyman continuum.
HeI: Calculated optical depths for the first 10 Lyman-series lines and the continuum using specific cross-section approximations.
HeII: Treated as a hydrogen-like ion, scaling HI data by atomic mass and charge factors.
Assumptions: The IGM is homogeneous; atoms reside predominantly in ground states; thermal and turbulent broadening are accounted for via Voigt profiles.

C. Spectral Transmittance Functions

The spectral transmittance function, $S(\lambda; z_s) = \exp[-\tau(\lambda; z_s)]$ , was derived. This function is independent of the intrinsic source spectrum and represents the fraction of flux transmitted through the IGM.

D. Illustrative Application

To demonstrate the effects, the authors modeled the spectral flux from "Cloud9" type objects: starless, virialized gas halos ( $M \sim 10^9 M_\odot$ ) emitting via free-free and free-bound processes at $T \approx 2.5 \times 10^5$ K.

3. Key Contributions

Analytic Prescription: Provided a compact, closed-form analytic prescription for effective IGM spectral transmittance over $0 \le z \le 15$ , calibrated to observational constraints.
Dual Reionization Scenarios: Explicitly compared "early" and "late" reionization histories, quantifying their impact on spectral features, particularly in the range $5.5 \lesssim z \lesssim 10$ .
Helium Absorption Analysis: Systematically quantified the role of HeI and HeII absorption, showing that while HI dominates the long-wavelength edge (Ly $\alpha$ ), HeII Lyman continuum absorption sets the sharp short-wavelength cutoff for high-redshift sources.
Physical Distinction: Clarified the physical difference between "smooth IGM" models (used here) and "discrete absorber" models (Madau/Inoue). The authors show that in a smooth medium, the $\nu^{-3}$ dependence of the photoionization cross-section can cause transmission to increase at shorter wavelengths, whereas discrete absorber models predict increasing opacity due to the cumulative path length of optically thick systems.

4. Key Results

Absorption Troughs: The combined absorption by HI, HeI, and HeII creates deep, broad absorption troughs.
- The long-wavelength edge is defined by HI Ly $\alpha$ absorption.
- The short-wavelength edge is defined by HeII Lyman continuum absorption.
Sensitivity to Reionization History: For sources at $z \approx 5.5 - 7$ , the residual transmission within the broad trough is sensitive to the specific reionization history (early vs. late). Differences are subtle (percent-level) but potentially distinguishable with high-sensitivity future instruments.
Thomson Scattering: Thomson scattering by free electrons uniformly suppresses the spectrum by $\sim 2.8\%$ at $z=5$ and $\sim 5\%$ at $z=15$ , a small but non-negligible effect.
Comparison with Discrete Models:
- At $z \le 5$ and wavelengths $\lambda > 1216$ Å, the results agree well with Inoue et al. (2014).
- At shorter wavelengths (Lyman continuum), the smooth IGM model predicts higher transmission (lower optical depth) compared to discrete absorber models because the cross-section drops rapidly with frequency, outweighing the increased path length.
Flux Estimates: The predicted fluxes from starless halos at $z > 5$ are well below the sensitivity limits of current facilities like JWST (by orders of magnitude). However, the study confirms that intergalactic absorption fundamentally reshapes the intrinsic emission, creating characteristic spectral signatures.

5. Significance

Tool for Early Universe Cosmology: The derived transmittance functions provide a physically motivated, computationally efficient tool for modeling the Cosmic Dawn and Reionization epochs without relying on complex numerical simulations for every source.
Helium's Role: The work highlights that neglecting Helium absorption (specifically HeII) leads to an incomplete understanding of the UV spectrum of high-redshift sources.
Future Probes: The paper suggests that future high-sensitivity observations could use the subtle variations in residual transmission within the absorption troughs to indirectly probe the timing and topology of hydrogen and helium reionization.
Theoretical Clarification: It resolves conceptual differences between smooth-medium and discrete-absorber approaches, offering a clearer physical picture of how the diffuse IGM interacts with radiation in the early Universe.

In summary, this paper provides a rigorous analytical framework for understanding how the expanding, ionizing IGM filters the light of the first cosmic sources, offering new insights into the specific contributions of Helium and the differences between various reionization scenarios.