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Testing halo models for constraining astrophysical feedback with multi-probe modeling: I. 3D Power spectra and mass fractions

This paper evaluates existing analytical halo models against the Magneticum hydrodynamical simulation to assess their ability to jointly model 3D power spectra and mass fractions of matter and baryons for upcoming Stage-IV surveys, finding that while they achieve sub-percent to few-percent accuracy in reproducing power spectra, their reliability in recovering underlying halo properties varies significantly and requires further refinement.

Original authors: Pranjal R. S., Shivam Pandey, Dhayaa Anbajagane, Elisabeth Krause, Klaus Dolag

Published 2026-03-31

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CC BY 4.0

Original authors: Pranjal R. S., Shivam Pandey, Dhayaa Anbajagane, Elisabeth Krause, Klaus Dolag

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, invisible ocean. Most of this ocean is made of "dark matter," which we can't see but know is there because it holds galaxies together with its gravity. Floating in this dark ocean are islands of normal stuff—stars, gas, and dust—which we call baryons.

For a long time, scientists have been trying to map this ocean. But there's a problem: the normal stuff isn't just sitting still. It's being kicked around by cosmic "storms" like exploding stars and supermassive black holes. These storms (called astrophysical feedback) heat up the gas, push it out of galaxies, and change the shape of the universe's structure.

If we want to use the universe to measure the fundamental laws of physics (like dark energy), we need to understand these storms perfectly. If we get the physics wrong, our measurements of the universe will be wrong, too.

The Problem: Guessing the Recipe

Scientists have built "recipe books" (mathematical models) to predict how this cosmic gas behaves. The paper you asked about tests three of the most popular recipe books:

Mead20+
Schneider19+
Arico24+

The authors wanted to know: Do these recipes actually work? And more importantly, if we use them to guess what the gas looks like inside a galaxy cluster, do we get the right answer?

To test them, they didn't just look at the real sky (which is messy and hard to measure). Instead, they used a supercomputer simulation called Magneticum. Think of this as a "video game universe" where the laws of physics are known perfectly. They ran the simulation, then tried to use the three recipe books to reverse-engineer what the simulation did.

The Test: Listening to the Symphony

In the real world, we can't see the gas directly. Instead, we listen to the "music" it makes. The paper looks at three different "instruments" (probes):

Weak Lensing (The Gravity Lens): How much the dark matter bends light. This tells us about the total mass.
Sunyaev-Zel'dovich Effect (The Heat Map): How the hot gas scatters light from the Big Bang. This tells us about the gas pressure and temperature.
Fast Radio Bursts (The Density Counter): How much the gas slows down radio signals. This tells us about the gas density.

The researchers asked the models: "Can you predict the sound of all three instruments at the same time?"

The Results: Who Passed the Test?

Here is the breakdown of how the three models performed, using some everyday analogies:

1. The "Good Enough" Fit (The Power Spectra)

When the models were asked to just match the general "volume" of the music (the power spectra), all three models did surprisingly well. They could predict the overall shape of the data with an error of less than 1-2%.

Analogy: Imagine three different chefs trying to recreate a complex soup. If you just ask them to match the saltiness of the soup, all three chefs can do it perfectly.

2. The "Secret Ingredient" Problem (The Profiles)

But then, the scientists asked a harder question: "Okay, you matched the saltiness. Now, tell me exactly how much carrot is in the soup, and how hot it is in the middle versus the edges."
This is where things got messy.

The Schneider Model: This chef was the best. It could guess the temperature and density of the gas inside the galaxy clusters with high accuracy (within 10%). It was like a chef who actually tasted the soup and knew exactly how the ingredients were layered.
The Mead and Arico Models: These chefs could match the saltiness, but when asked about the specific ingredients, they got confused. They predicted the gas was too cold or too spread out.
Why? The paper suggests these models have "degeneracies." This is like a chef who can make the soup taste right by adding either a lot of salt and no pepper, or no salt and a lot of pepper. The math works, but the physical reality is wrong. The models are "flexible" enough to fake the data, but not physically accurate enough to tell the truth about the gas.

The Big Takeaway

The paper concludes that while these mathematical models are getting better at matching the numbers we see in the sky, they are still struggling to tell the true story of what the gas is actually doing.

The Good News: We have tools that can match the data very closely.
The Bad News: If we use these tools to learn about the physics of the universe, we might be fooled. The models might be "cheating" by adjusting their knobs in ways that don't make physical sense.

The Future

The authors say we need to stop just trying to fit the numbers and start building models based on first principles (the actual laws of physics). We need to understand why the black holes push the gas out, rather than just guessing how much gas is pushed out to make the math work.

In a nutshell: The paper is a "stress test" for our cosmic recipe books. They passed the easy test (matching the data), but failed the hard test (understanding the physics). To unlock the secrets of the universe, we need better recipes that don't just look right on paper, but actually taste like the real universe.

1. Problem Statement

Upcoming Stage-IV cosmological surveys (e.g., Euclid, LSST, DESI) will measure the distribution of matter with unprecedented precision, requiring theoretical models accurate at the percent level. A primary source of uncertainty in these models is astrophysical feedback (from supernovae and Active Galactic Nuclei), which alters the distribution and thermodynamics of baryons on scales relevant to cosmological probes.

Current analytical models (Halo Models) struggle to simultaneously and self-consistently describe multiple observables that trace different aspects of baryonic physics:

Weak Lensing: Traces total matter distribution.
Thermal Sunyaev-Zel'dovich (tSZ): Probes electron pressure.
Kinematic Sunyaev-Zel'dovich (kSZ) & Fast Radio Bursts (FRBs): Probe electron density.

The central question is whether existing analytical Halo Models are sufficiently flexible to jointly model the 3D power spectra of matter, electron pressure, and electron density while accurately recovering the underlying physical properties of halos (e.g., gas fractions, thermodynamic profiles) without being tuned specifically to the simulation data.

2. Methodology

The authors benchmark three distinct analytical Halo Models against high-resolution hydrodynamical simulations using a multi-probe approach.

Simulations: The study utilizes the Magneticum hydrodynamical simulation suite (specifically the Box2b/hr run, $640 \, \text{Mpc}/h$ box size). This simulation was not used to calibrate any of the tested models, serving as an independent testbed.
Models Tested:
1. Mead20+: An updated version of Mead et al. (2020). It modifies halo concentration to account for feedback and uses a polytropic equation of state.
2. Schneider19+: An extended version of the "baryonification" method (Schneider et al. 2019). It models gas as a flexible density profile with adiabatic relaxation of dark matter and solves for hydrostatic equilibrium.
3. Arico24+: An extension of Aricò & Angulo (2024) to include thermodynamic quantities. It divides gas into bound, reaccreted, and ejected components.
Observables: The models are fitted to the 3D power spectrum response ( $R(k)$ $R (k)$ ), defined as the ratio of the power spectrum with baryons to a dark-matter-only counterpart. The specific tracers analyzed are:
- Matter-Matter ( $R_{mm}$ )
- Matter-Pressure ( $R_{mp}$ )
- Electron Density ( $R_{ee}$ )
- Mass Fractions (Bound Gas $f_{BG}$ and Stars $f_{star}$ )
Analysis Strategy:
- The authors perform joint fits using different combinations of the above tracers.
- They test two parameter sets: a minimal 6-parameter set and an extended set (11–15 parameters) to assess model flexibility.
- They use a Bayesian nested sampler (Nautilus) with Gaussian likelihoods.
- Validation: After fitting the power spectra, the models are used to predict physical quantities (gas fractions, pressure, temperature, and density profiles) for group- and cluster-sized halos, which are then compared directly to the Magneticum simulation measurements.

3. Key Contributions

Unified Framework: This is one of the first studies to jointly test Halo Models against the full suite of 3D power spectra (matter, pressure, and density) required for next-generation multi-probe cosmology.
Model Flexibility Assessment: The paper rigorously quantifies the "prediction space" of different models, showing how well one model's parameter variations can be represented by another (via basis vector projection).
Physical Recovery vs. Statistical Fit: The study distinguishes between a model's ability to fit summary statistics (power spectra) and its ability to recover the true physical state of the universe (thermodynamic profiles).
Identification of Systematics: The authors identify specific physical assumptions (e.g., constant ejected gas temperature, mass-independent non-thermal pressure) that limit the physical realism of certain models, even when they fit the power spectra well.

4. Key Results

A. Power Spectrum Fitting Accuracy

Six-Parameter Set: All three models can reproduce the matter ( $R_{mm}$ ) and matter-pressure ( $R_{mp}$ ) power spectra with sub-percent to few-percent accuracy. However, the Arico24+ model struggled slightly with $R_{mm}$ (percent-level error) in the minimal set.
Extended Parameter Set: With $\sim15$ free parameters, all models achieved percent-level accuracy across all three power spectra ( $R_{mm}, R_{mp}, R_{ee}$ ).
Degeneracy Breaking: Combining different power spectra (e.g., adding $R_{ee}$ or mass fractions) significantly breaks parameter degeneracies, particularly separating constraints on stellar components from gas components.

B. Recovery of Physical Properties

Despite achieving excellent fits to the power spectra, the recovery of underlying physical profiles varied significantly:

Schneider19+: Performed best overall. When jointly fitting power spectra and mass fractions, it recovered thermodynamic profiles (pressure, temperature, density) with <10% RMS error for cluster-sized halos and reasonable accuracy for group-sized halos.
Mead20+ and Arico24+: Showed poorer performance, with RMS errors ranging from 25% to 80% for inferred profiles.
- Mead20+: Underpredicted bound gas in massive halos and failed to capture the correct density profiles, likely due to its assumption that gas and dark matter share the same modified concentration profile (which contradicts simulation results showing gas concentration increases with mass).
- Arico24+: Severely underestimated gas content in group-sized halos when fitting $R_{ee}$ , suggesting its prescription for ejected gas temperature ( $T_w$ ) or mass dependence was insufficient.

C. The "Overfitting" Paradox

The authors observed that increasing the number of free parameters (from 6 to ~15) improved the fit to the power spectra but degraded the accuracy of the inferred physical profiles (especially for group-sized halos). This suggests that without strong physical priors or additional constraints, the models can "compensate" for physical inaccuracies by shifting parameters to fit the aggregate power spectrum, leading to unphysical profile predictions.

5. Significance and Outlook

Necessity of Multi-Probe Data: The study demonstrates that fitting only the matter power spectrum is insufficient to constrain baryonic feedback. Joint modeling of tSZ (pressure), kSZ/FRB (density), and weak lensing (matter) is essential to break degeneracies and constrain physical parameters.
Limitations of Current Models: While current Halo Models are statistically flexible enough to fit current and near-future data, they often lack the physical grounding to reliably predict the internal thermodynamic structure of halos across all mass scales.
Future Directions:
- Models require more physically motivated prescriptions (e.g., mass-dependent non-thermal pressure, realistic ejected gas temperatures).
- Future work must move from 3D power spectra to observable space (mock lightcones) to account for projection effects and survey systematics.
- There is a need for "first-principles" models that co-evolve galaxies and the circumgalactic medium rather than relying solely on empirical scaling relations.

In conclusion, while analytical Halo Models are powerful tools for cosmological inference, this paper highlights that their current formulations may not be robust enough to simultaneously extract precise cosmological parameters and detailed astrophysical feedback physics without further refinement and the inclusion of diverse multi-wavelength data.