The Baryon Budget of Galaxies across the First Billion Years -- Theoretical Predictions for Gas Phases, Depletion Times, and Stellar Return Fractions

This paper utilizes ColdSIM hydrodynamical simulations to provide a comprehensive census of baryon phases, depletion times, and stellar return fractions in early galaxies, revealing that the warm gas phase dominates post-reionization and that stellar return fractions are significantly lower than previously assumed.

The Gist

The Big Picture: The Universe's First Billion Years as a Construction Site

Imagine the early Universe (the first billion years after the Big Bang) as a massive, chaotic construction site. The "builders" are galaxies, and the "materials" they use are baryons—basically, normal matter like gas and stars.

For a long time, astronomers thought they knew how this construction site worked: Gas falls in, cools down, and turns into stars. But this new paper, using super-computer simulations, suggests the reality is much more dynamic and messy. It's not just a straight line from gas to stars; it's a complex cycle of heating, cooling, and recycling.

The authors, Umberto Maio and Céline Péroux, used a digital time machine (called ColdSIM) to watch how this construction site evolved from the very beginning (redshift $z > 5$) up to the time when the first galaxies were fully formed.

The Three States of Matter: The "Weather" of the Galaxy

To understand the paper, you have to understand the three "weather conditions" of the gas in these early galaxies:

1. Cold Gas (The Raw Lumber): This is the fuel. It's dense, cool, and ready to be turned into stars. Think of this as the raw timber waiting to be cut into planks.
2. Warm Gas (The Construction Dust): When stars are born, they blast energy out, heating up the surrounding gas. This is the "warm" phase. It's like the dust and heat generated by a busy construction crew.
3. Hot Gas (The Exhaust Fumes): This is gas that has been blasted so hard by supernovae (exploding stars) that it's superheated and flying away. It's the exhaust fumes of the construction site.

The Big Discovery:
In the very beginning (before the first billion years were over), the construction site was mostly Cold Gas. It was a quiet place waiting to get to work. But as stars started forming and exploding, they heated everything up. By the time the Universe was about 1 billion years old, the Warm Gas had taken over. The "dust" became more common than the "raw lumber."

The "Recycling" Problem: How Much Stuff Comes Back?

One of the most important findings in the paper is about recycling.

When stars die, they don't just disappear. They explode or shed their outer layers, throwing a lot of their mass back into the gas cloud to be used again. Astronomers call this the Stellar Return Fraction.

• The Old Belief: Scientists used to think that for every 100 tons of stars born, about 30–40 tons would be recycled back into gas eventually.
• The New Reality: The paper says, "Not so fast!" Because the Universe was so young, the stars hadn't lived long enough to die and recycle their mass yet. The actual recycling rate was only about 15–20%.

The Analogy: Imagine a factory that just opened. The old plan assumed the factory had been running for 50 years, so they expected a huge pile of scrap metal (recycled mass) to be available. But in reality, the factory has only been open for 5 years. There is very little scrap metal yet. This means we have been overestimating how much "new" gas is available to be turned into stars in the early Universe.

The "Depletion Time": How Fast Do They Burn the Fuel?

Another key concept is Depletion Time. This asks: "If a galaxy has a certain amount of cold gas, how long will it take to turn all of it into stars?"

• The Old Belief: It takes a long time (billions of years) to burn through the fuel.
• The New Reality: In the early Universe, galaxies were like race cars, not sedans. They burned their fuel incredibly fast. The paper finds that in the earliest epochs, galaxies could burn through their gas in as little as 10 to 100 million years.

The Analogy: Think of a campfire.

• Low Redshift (Later Universe): A slow-burning log fire. It lasts for hours.
• High Redshift (Early Universe): A pile of dry kindling and gasoline. It flares up instantly and burns out in minutes.

This explains why we see so many stars forming so quickly in the early Universe, but also why those galaxies might run out of fuel and stop forming stars sooner than we thought.

The "Main Sequence": The Growth Chart

The paper also looked at the relationship between how much gas a galaxy has and how many stars it has. They found a "Main Sequence"—a predictable pattern where bigger galaxies have more gas and make more stars.

However, they found that metallicity (how "dirty" the gas is with heavy elements like gold or iron) plays a huge role.

• Clean Gas (Low Metallicity): Harder to cool down, harder to form stars.
• Dirty Gas (High Metallicity): Cools down easily, forms stars very fast.

The simulations showed that as galaxies got "dirtier" (more metals), they became much more efficient at turning gas into stars, shortening their fuel supply even further.

Why Does This Matter?

This paper is a correction to the "instruction manual" for the early Universe.

1. It fixes the math: By realizing that recycling is slower and fuel burning is faster, we can better calculate how many stars should exist in the early Universe.
2. It matches new telescopes: The James Webb Space Telescope (JWST) and ALMA are currently looking at these ancient galaxies. This paper provides the theoretical "ruler" to measure what they are seeing.
3. It explains the "Missing" Mass: It helps explain where all the gas is. It's not all in stars; a huge chunk is in that "warm" phase, heated up by the stars themselves, which is hard to see with current telescopes.

The Bottom Line

The early Universe wasn't a slow, steady builder. It was a frantic, high-energy construction site.

• Gas started cold and quiet.
• Stars formed in a frenzy, burning fuel at a breakneck pace.
• Feedback from those stars heated the gas up, turning the "cold fuel" into "warm dust."
• Recycling was slower than we thought because the stars were too young to die yet.

This study gives us a clearer, more accurate picture of how the first galaxies grew up, helping us understand the cosmic dawn that led to the Universe we see today.

Technical Summary

1. Problem Statement and Motivation

The paper addresses the critical gap in understanding the baryon cycle during the epoch of reionization ($z \gtrsim 5$), specifically the first billion years of the Universe. While recent observations from JWST and ALMA have constrained the existence of early galaxies, the quantitative distribution of baryons across different phases (cold, warm, hot, stellar) remains uncertain.

Key challenges identified include:

• Uncertain Gas Phases: The relative contributions of cold neutral gas (HI, H$_2$), warm ionized gas, and hot X-ray emitting gas to the total cosmic mass budget at high redshift are debated.
• Inaccurate Stellar Return Fractions ($R$): Standard values for the fraction of stellar mass returned to the interstellar medium (ISM) are typically derived from low-redshift ($z < 1$) assumptions using instantaneous recycling approximations and specific Initial Mass Functions (IMFs). These may be invalid for high-$z$ populations where stellar ages are young.
• Depletion Timescales: The timescales for converting gas into stars ($t_{\text{depl}}$) and the resulting Star Formation Efficiency (SFE) at high redshift are poorly constrained, with observational data often conflicting or lacking.
• Lack of High-$z$ Theory: There is a scarcity of theoretical studies that explicitly model non-equilibrium chemistry and the interplay between gas phases and stellar feedback in the early Universe.

2. Methodology

The authors utilized the ColdSIM project, a suite of advanced hydrodynamical simulations designed for primordial structure formation.

• Simulation Setup:

  • Cosmology: $\Lambda$CDM model with parameters consistent with Planck 2020 ($\Omega_{0,\Lambda}=0.726$, $\Omega_{0,m}=0.274$, $\Omega_{0,b}=0.0458$, $h=0.702$, $\sigma_8=0.816$).
  • Volumes: Three simulation boxes were used: a Reference (Ref) run ($L=10$ Mpc/$h$, $2\times 512^3$ particles), a High-Resolution (HR) run ($L=10$ Mpc/$h$, $2\times 1000^3$), and a Large-Box (LB) run ($L=50$ Mpc/$h$, $2\times 1000^3$).
  • Physics: The code includes smoothed-particle hydrodynamics (SPH) with non-equilibrium atomic and molecular chemistry (tracking H, He, electrons, H$_2$, HD, etc.).
  • Feedback & Evolution: Explicit modeling of Population III and Population II-I stars with a Salpeter IMF. It includes metal-dependent yields from Type II/Ia SNe and AGB stars, thermal feedback, and galactic winds (350 km/s).
  • Phase Definitions: Gas is categorized by temperature:
    • Cold: $T < 10^4$ K
    • Warm: $10^4 \le T \le 10^7$ K
    • Hot: $T > 10^7$ K

• Analysis: The authors computed mass density parameters ($\Omega$) for bound structures (galaxies + CGM) and the Intergalactic Medium (IGM). They derived scaling relations between gas masses (HI, H$_2$) and galaxy properties (Stellar Mass $M_{\star}$, Star Formation Rate SFR, Metallicity $Z$). Crucially, they calculated the stellar return fraction ($R$) as a function of stellar age ($t_{\text{age}}$) rather than assuming instantaneous recycling.

3. Key Contributions

• Explicit Phase Tracking: Unlike many simulations that rely on sub-grid models for molecular gas, this work explicitly computes HI and H$_2$ masses based on non-equilibrium chemistry, allowing for a direct census of cold vs. warm gas.
• Time-Dependent Stellar Return Fraction: The paper provides a self-consistent calculation of $R$ as a function of stellar age at high redshift, challenging the standard assumption of $R \approx 0.3-0.4$.
• High-$z$ Depletion Times: It establishes robust theoretical constraints on HI and H$_2$ depletion timescales ($t_{\text{depl}}$) and Star Formation Efficiency (SFE) across $z=5-20$.
• Scaling Relations: The study provides fitting functions for gas-to-star fractions, mass functions, and depletion times, offering tools to interpret future JWST and ALMA data.

4. Key Results

A. Baryon Budget and Phase Evolution

• Cold vs. Warm Dominance: Prior to reionization ($z \gtrsim 6$), the cold phase dominates the cosmic mass budget, closely tracing the total baryon density ($\Omega_{\text{cold}} \approx \Omega_{0,b}$).
• Transition at Reionization: As reionization completes ($z \lesssim 6$), the cold phase drops by two orders of magnitude ($\Omega_{\text{cold}} \sim 10^{-3}$) due to UV heating. The warm phase becomes the dominant mass reservoir at later epochs, driven by star formation feedback.
• Hot Gas: Hot gas ($T > 10^7$ K) remains subdominant in the early Universe, primarily tracing rare, massive halos.

B. Stellar Return Fraction ($R$)

• Lower than Expected: The study finds that at high redshifts ($z \gtrsim 5$), the stellar return fraction is $R \approx 0.15 - 0.20$.
• Age Dependence: This is roughly half the values typically adopted ($R \approx 0.3-0.4$) for low-$z$ galaxies. The lower value is due to the young ages of stellar populations at high-$z$; they have not yet evolved through the full stellar lifecycle to eject significant mass via AGB winds or low-mass SN.
• Implication: Standard SFR estimates based on high-$R$ assumptions may be overestimating the recycled gas contribution in the early Universe.

C. Gas Depletion Times and Efficiency

• Short Depletion Times: H$_2$ depletion times are extremely short, ranging from 0.01 to 0.1 Gyr at $z \sim 15$, increasing to $\sim 1$ Gyr at $z \sim 5$.
• Metallicity Dependence: Depletion times show a weak dependence on metallicity but a strong dependence on SFR and stellar mass.
• Star Formation Efficiency (SFE): The median SFE remains low and constant at $\sim 1-4\%$ across redshifts, with no evidence for a significant increase in efficiency at early times compared to the present day.

D. Scaling Relations

• Mass Functions: Cold and warm gas mass functions evolve from narrow distributions at high-$z$ to broader, Schechter-like forms at lower-$z$.
• HI and H$_2$ Tracers: HI mass is a reliable tracer of the total cold gas mass. H$_2$ mass correlates tightly with SFR ($M_{\text{H}_2} \propto \text{SFR}^{0.5}$).
• Gas-to-Star Fractions: The gas-to-stellar mass ratios ($\mu_{\text{HI}}$ and $\mu_{\text{H}_2}$) decrease monotonically with cosmic time and stellar mass. At high-$z$, galaxies are gas-rich, but the warm component can constitute $>50\%$ of the baryonic mass in bound structures, meaning HI/H$_2$ alone do not account for the total gas budget.

5. Significance and Conclusions

This work provides a comprehensive theoretical framework for the baryon cycle in the first billion years of the Universe. Its primary significance lies in:

1. Correcting High-$z$ Assumptions: It demonstrates that standard low-redshift parameters (specifically $R$) are invalid for the early Universe, necessitating a revision of Star Formation Rate (SFR) density calculations.
2. Validating Observations: The results are consistent with recent ALMA, VLA, and IRAM surveys (e.g., ALFALFA, xCOLDGASS, ASPECS) regarding gas depletion times and mass functions, bridging the gap between low-$z$ observations and high-$z$ theory.
3. The Role of Feedback: It highlights that the transition from cold to warm gas is driven not by mass transfer, but by the heating energy injected by feedback (SNe, winds) into small amounts of gas.
4. Predictive Power: The provided fitting functions for phase mass relations, return fractions, and depletion times serve as essential tools for interpreting upcoming deep-field observations from JWST and next-generation radio telescopes (SKA).

In conclusion, the paper establishes that the early baryon cycle is characterized by a cold-dominated phase transitioning to a warm-dominated phase, governed by short depletion times and a lower-than-expected stellar return fraction, fundamentally altering our quantitative understanding of early galaxy formation.