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Pressure-regulated feedback-modulated star formation as a subgrid model for galaxy formation simulations

This paper introduces and validates a physically grounded, pressure-regulated feedback-modulated (PRFM) subgrid model for star formation in galaxy simulations, demonstrating that its volumetric and integrated implementations yield robust, resolution-independent results that outperform existing empirical prescriptions like IllustrisTNG.

Original authors: Sarah M. R. Jeffreson, Eve C. Ostriker, Chang-Goo Kim, Jan Burger

Published 2026-06-10

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CC BY 4.0

Original authors: Sarah M. R. Jeffreson, Eve C. Ostriker, Chang-Goo Kim, Jan Burger

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 trying to predict how a city grows. You can't simulate every single brick, every person, and every car. Instead, you use a simplified rule: "If there are enough people and resources, a new building goes up."

For decades, astronomers have used similar simplified rules to simulate how galaxies form stars. They assumed that if a cloud of gas is dense enough, stars will form at a steady, predictable rate. But this paper argues that those old rules are too simple. They miss the real "traffic jams" and "construction crews" happening inside galaxies.

The authors, led by Sarah Jeffreson, have created a new, smarter set of rules called PRFM (Pressure-Regulated, Feedback-Modulated). Here is how it works, explained through everyday analogies:

1. The Old Way vs. The New Way

The Old Way (The "Static Rule"):
Imagine a construction site where the foreman says, "If we have 100 bricks, we build 1 house." It doesn't matter if the wind is howling, if the ground is shaking, or if the workers are tired. The rule is fixed. In astronomy, this meant assuming star formation depends only on how dense the gas is.

The New Way (The "Dynamic Balance"):
The PRFM model realizes that building a house (making a star) is a tug-of-war.

Gravity is trying to crush the gas cloud together to make a star.
Feedback (explosions from new stars, radiation, and winds) is trying to blow the cloud apart.

The paper argues that the rate of star formation isn't fixed; it's a balance. If the gas gets too heavy (high pressure), the "construction crew" (feedback) works harder to blow the cloud apart, but the gravity is so strong that the stars still form faster than before. It's like a pressure cooker: the more you heat it (gravity/pressure), the more violently it releases steam (feedback), but the cooking happens faster.

2. The Two Versions of the New Rule

The authors realized that computer simulations come in different "resolutions" (like different camera zoom levels). You can't use the same rule for a blurry, wide-angle photo as you can for a high-definition close-up. So, they built two versions of their new model:

PRFM-vol (The "High-Definition" Version):
- When to use it: When the computer is powerful enough to see the "thickness" of the galaxy's gas disk in 3D detail.
- How it works: It looks at the gas in every tiny 3D cell, measures the pressure right there, and calculates how fast stars should form based on the local tug-of-war.
- Analogy: It's like a construction manager walking around the site, checking the wind and weight on every single beam before ordering more bricks.
PRFM-int (The "Smart Estimate" Version):
- When to use it: When the computer isn't powerful enough to see the 3D thickness of the disk (it looks like a flat sheet).
- How it works: Since it can't see the height, it uses math to guess the height and pressure based on the total weight of the gas and stars above it. It calculates the "average" pressure of the whole column.
- Analogy: It's like a construction manager who can't see the beams, so they look at the total weight of the building and the ground beneath it to estimate how fast construction can happen. They don't need to see every brick to know the rules.

3. What They Found

The team tested these new rules on a simulated galaxy that looks like our own Milky Way, running the simulation at three different levels of detail (high, medium, and low resolution).

The "Pressure Cooker" Effect: They found that in high-pressure areas (like the centers of galaxies or in the early universe), the new model predicts stars form much faster than the old models did. The old models were too slow because they didn't account for the fact that high pressure actually boosts efficiency.
The Resolution Problem:
- The old "High-Definition" model (PRFM-vol) worked great when the computer was powerful. But when they lowered the resolution (made the simulation blurrier), it started to fail. It couldn't "see" the gas properly, so it got confused and calculated the wrong star formation rates.
- The "Smart Estimate" model (PRFM-int) was robust. Even when the simulation was blurry and low-resolution, it kept giving the right answer. It didn't matter if the computer couldn't see the 3D thickness; the math of the "total weight" still worked.

4. Why This Matters

Think of the old models as a map that only works if you are standing right next to the building. If you zoom out, the map becomes useless.

The new PRFM-int model is like a GPS that works whether you are standing on the street or looking at the city from a satellite. It allows astronomers to run massive simulations of the entire universe (which require low resolution because the universe is so big) without losing the physics of how stars actually form.

In short: The authors have built a new, physics-based rulebook for galaxy simulations. It replaces "guessing" with a balance of forces, and they created a special version of this rulebook that works perfectly even when the computer simulation isn't detailed enough to see the fine print. This paves the way for more accurate simulations of how galaxies have evolved from the beginning of the universe to today.

Technical Summary: Pressure-Regulated Feedback-Modulated Star Formation as a Subgrid Model

Problem Statement
Large-volume cosmological simulations of galaxy formation (e.g., IllustrisTNG, EAGLE, SIMBA) successfully reproduce statistical properties of galaxy populations but cannot resolve the physical scales of giant molecular clouds (GMCs) or the multiphase interstellar medium (ISM). Consequently, these simulations rely on empirically calibrated subgrid prescriptions for star formation rates (SFR) and the effective equation of state (EoS). Traditional models typically assume the specific SFR scales with gas density as $\dot{m}_*/m_g \propto \rho^{1/2}$ (inverse free-fall time), with a normalization set by local universe observations.

The authors identify two primary limitations in this approach:

Lack of Physical Connection: The empirical scaling lacks a direct link to the physical processes regulating star formation, specifically the balance between gravity and stellar feedback.
Resolution and Environmental Sensitivity: These prescriptions often fail in regimes where gas density and pressure are significantly higher than in the local universe (e.g., high-redshift galaxies). Furthermore, models relying on local volumetric density are sensitive to numerical resolution; if the gas disk scale height is not resolved, the measured volume density becomes inaccurate, leading to inconsistent star formation predictions across different resolutions.

Methodology
The paper introduces a new subgrid framework based on the Pressure-Regulated, Feedback-Modulated (PRFM) theory of star formation (Ostriker & Kim 2022). In this framework, the SFR adjusts to maintain thermal and vertical dynamical equilibrium in the ISM, where the total midplane pressure ( $P_{tot}$ ) balances the weight of the ISM ( $W$ ) and is proportional to the SFR per unit area ( $\Sigma_{SFR}$ ) via a feedback yield ( $\Upsilon_{tot}$ ).

The authors implement two complementary versions of this model in the moving-mesh code Arepo, calibrated using high-resolution magnetohydrodynamic simulations from the TIGRESS suite:

PRFM-vol (Volumetric): Designed for simulations where the vertical thickness of the gas disk is numerically resolved (typically mass resolution $\sim 10^5 M_\odot$ ).
- It calculates the effective pressure ( $P_{eff}$ ) in each cell using a TIGRESS-calibrated EoS ( $P_{tot} \propto n_H^{1.8}$ ).
- The dynamical time ( $t_{dyn}$ ) is derived from local volume densities of gas, stars, and dark matter.
- The depletion time ( $t_{dep}$ ) is set based on the local pressure and a renormalization factor ( $R_f \approx 2$ ) to account for the vertical integration of the volumetric law.
PRFM-int (Integrated): Designed for coarser resolutions where the disk scale height is unresolved (typically mass resolution $\gtrsim 10^6 M_\odot$ ).
- Instead of relying on local volume density, it reconstructs the midplane density and pressure from vertical equilibrium considerations using surface densities ( $\Sigma_g$ , $\Sigma_*$ ) and the ISM weight ( $W$ ).
- It solves for the equilibrium gas scale height ( $H_g$ ) and derives $t_{dep}$ directly from these integrated quantities, avoiding the use of resolution-sensitive volume densities in the SFR prescription.

Key Contributions

Physically Grounded Calibration: The model derives the SFR and EoS from the PRFM equilibrium condition ( $W = P_{tot} = \Upsilon_{tot}\Sigma_{SFR}$ ), calibrated against TIGRESS simulations rather than empirical fits to local galaxies.
Resolution Robustness: The development of the PRFM-int approach specifically addresses the failure of volumetric models at low resolution by utilizing surface densities and equilibrium constraints, which are less sensitive to the inability to resolve the vertical disk structure.
Variable Efficiency: Unlike traditional models that assume a constant efficiency per free-fall time, the PRFM model predicts that star formation efficiency per dynamical time ( $\epsilon_{dyn}$ ) increases with pressure ( $\epsilon_{dyn} \propto P_{tot}^{0.43}$ ).

Results
The authors tested both implementations in isolated, Milky-Way-like disk simulations across mass resolutions of $10^5$ , $10^6$ , and $10^7 M_\odot$ , comparing them against the standard IllustrisTNG prescription.

High Resolution Agreement: At $10^5 M_\odot$ , both PRFM-vol and PRFM-int agree closely with each other and with TIGRESS results for SFRs and depletion times. Both successfully recover the TIGRESS-calibrated Ostriker-Kim relation ( $\Sigma_{SFR} \propto P_{tot}^{1.2}$ ).
Resolution Sensitivity of Volumetric Models: The PRFM-vol and TNG models show significant sensitivity to resolution. As resolution coarsens to $10^6$ and $10^7 M_\odot$ , the inability to resolve the disk scale height leads to inaccurate volume density measurements. This causes the PRFM-vol model to overestimate depletion times and fail to match the high-resolution SFR history.
Robustness of PRFM-int: The PRFM-int implementation remains robust across all tested resolutions. It successfully tracks the global star formation history and surface density distributions of the high-resolution ( $10^5 M_\odot$ ) PRFM-vol simulation, even at $10^7 M_\odot$ .
Comparison with TNG: Compared to the TNG model, the PRFM models predict shorter gas depletion times, particularly in high-pressure and high-density regions. The PRFM models exhibit a much wider dynamic range in depletion times and a steeper dependence on pressure, whereas the TNG model maintains a constant efficiency per free-fall time.

Significance
The paper claims that the PRFM-derived subgrid prescriptions provide a physically grounded and numerically stable framework for star formation across the dynamic range of galaxy formation simulations.

For High-Resolution Simulations: The PRFM-vol model offers a more realistic treatment of the ISM physics than traditional empirical laws, linking SFR directly to the dynamic balance of the galaxy.
For Large-Scale Cosmological Simulations: The PRFM-int model solves the critical problem of resolution dependence. It allows large-box cosmological simulations (which typically cannot resolve disk scale heights) to utilize physically motivated star formation laws that remain consistent with high-resolution physics.
High-Redshift Implications: The authors note that the pressure-dependent efficiency of the PRFM model suggests that star formation in high-redshift galaxies (where pressures are higher) may be significantly more efficient than predicted by current TNG-like models, potentially impacting the interpretation of luminous galaxies observed by JWST.

The work is presented as a step toward the "Learning the Universe Collaboration" goal of developing predictive, rather than empirically tuned, subgrid models that can be updated as high-resolution ISM simulations (like TIGRESS) expand to cover broader parameter spaces (e.g., varying metallicity).

1. The Old Way vs. The New Way

2. The Two Versions of the New Rule

3. What They Found

4. Why This Matters

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