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Informative Priors on Primordial Non-Gaussianity Bias $b_{\phi}$ From Galaxy Formation

This paper presents a framework using the CAMELS-SAM simulation suite to construct observationally conditioned priors on the galaxy bias parameter $b_{\phi}$ , demonstrating that conditioning on the stellar mass function or stellar-to-halo mass relationship can reduce galaxy formation uncertainties by up to 97%, thereby significantly improving constraints on primordial non-Gaussianity ( $f^{\rm{loc}}_{\rm{NL}}$ ) for next-generation spectroscopic surveys like DESI.

Original authors: Anne Moore, Lucia A. Perez, Elisabeth Krause

Published 2026-04-24

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CC BY 4.0

Original authors: Anne Moore, Lucia A. Perez, Elisabeth Krause

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

The Big Picture: Trying to Hear a Whisper in a Storm

Imagine the universe is a giant, quiet room. Scientists are trying to hear a very faint whisper from the very beginning of time (the Big Bang). This whisper is called Primordial Non-Gaussianity (or $f_{NL}$ ). It tells us exactly how the universe started and which "recipe" of physics created it.

However, there is a problem. The room is full of a loud, chaotic storm: the formation of galaxies. As galaxies form, they create their own noise that sounds exactly like the whisper we are trying to hear. In scientific terms, there is a "degeneracy"—a perfect confusion—between the signal we want ( $f_{NL}$ ) and a parameter called $b_\phi$ (which describes how galaxies react to the early universe).

If you don't know what the storm sounds like, you can't isolate the whisper.

The Old Way: Guessing the Storm

For a long time, scientists tried to solve this by assuming a "Universal Rule." They thought, "If we know how many galaxies there are ( $b_1$ ), we can just use a simple formula to guess the storm's noise ( $b_\phi$ )."

The Analogy: Imagine you are trying to guess how loud a specific car engine is just by looking at the car's color.

The Old Rule: "All red cars have engines that are 50 decibels loud."
The Problem: In reality, a red sports car has a loud engine, but a red minivan has a quiet one. If you assume they are all the same, your guess will be wrong. In cosmology, this "Universal Rule" breaks down because different types of galaxies (red vs. blue, big vs. small) react to the early universe in different ways. This leads to systematic errors—like trying to tune a radio but getting static instead of music.

The New Solution: The "Galactic Detective" Kit

This paper introduces a new, smarter way to figure out the noise ( $b_\phi$ ) without making bad guesses. Instead of assuming a universal rule, they build a physically motivated prior.

The Analogy: Instead of guessing the engine noise based on color, you ask the car owner: "What kind of engine do you have? How much fuel does it burn? How fast does it accelerate?" You then use those specific details to predict the noise level.

Here is how they did it, step-by-step:

1. The Simulation Lab (The "Car Factory")

The authors used a massive computer simulation suite called CAMELS-SAM. Think of this as a giant virtual car factory.

They didn't just build one car; they built 1,000 different versions of the universe.
In each version, they tweaked the "factory settings" (parameters like how much gas stars blow out, or how black holes eat gas).
This created a huge library of different possible universes, each with different galaxy properties.

2. The "Separate Universe" Trick (The Sound Test)

To measure the specific noise parameter ( $b_\phi$ ), they used a technique called Separate Universe Simulations.

The Analogy: Imagine you have a perfect model of a car. To test how loud the engine is, you put the car in a slightly different room where the air pressure is just a tiny bit higher. You see how the engine reacts to that change.
In the simulation, they slightly tweaked the "background cosmology" (the air pressure of the universe) and watched how the number of galaxies changed. This tells them exactly how sensitive galaxies are to the early universe's whispers.

3. The "Translator" (The Emulator)

They have 1,000 simulations, but they can't run a test for every single possible combination of settings (that would take forever).

The Analogy: They built a super-smart AI translator (a Gaussian Process Emulator).
They fed the AI data from the 1,000 simulations. Now, if you ask the AI, "What happens if I change the gas setting by 0.1%?", the AI can instantly predict the result without running a new simulation.

4. Matching Reality (The "Fingerprint")

Now, they took real data from the sky (from surveys like DESI) regarding things we can actually see:

SMF: How many galaxies of different sizes exist?
SHMR: How heavy are the galaxies compared to their invisible dark matter halos?
Metallicity: How "dirty" (heavy element-rich) are the stars?

They used their AI translator to find which of the 1,000 simulated universes looked most like our real universe.

The Results: From a Blur to a Laser Beam

When they combined these tools, the results were dramatic:

Before: Their uncertainty on the noise parameter ( $b_\phi$ ) was huge (like trying to guess a number between 0 and 10).
After: By conditioning their guess on real observations (like the Stellar Mass Function), they reduced the uncertainty by 88% to 97%.
The Result: They went from a blurry guess to a laser-sharp measurement.

Why this matters:
Even though their computer model (the SC-SAM) had some flaws when trying to match the biggest galaxies (the "high mass" problem), the final result for the noise parameter ( $b_\phi$ ) remained consistent. This proves that their method is robust.

The Takeaway

This paper is like upgrading from a crystal ball (guessing based on a simple rule) to a forensic lab (using detailed evidence to reconstruct the past).

By acknowledging that galaxy formation is complex and uncertain, and by using massive simulations to "marginalize" (average out) those uncertainties, they have created a much more reliable way to listen for the whispers of the Big Bang. This paves the way for future telescopes (like DESI and SPHEREx) to finally tell us which theory of the universe's birth is correct.

1. Problem Statement

The primary goal of modern cosmology is to constrain primordial non-Gaussianity (PNG), specifically the local type parameter $f_{\text{loc}}^{\text{NL}}$ , to distinguish between different inflationary models.

The Degeneracy: In Large Scale Structure (LSS) analyses, the imprint of local PNG on the galaxy power spectrum introduces a scale-dependent bias term proportional to $b_\phi f_{\text{loc}}^{\text{NL}}$ . At leading order, $f_{\text{loc}}^{\text{NL}}$ and the bias parameter $b_\phi$ are exactly degenerate.
Current Limitation: Standard analyses break this degeneracy using the "universality relation" ( $b_\phi = 2\delta_c(b_1 - 1)$ ), which assumes a universal halo mass function. However, this relation is known to fail for physically motivated galaxy selections (e.g., based on stellar mass or star formation rate), introducing systematic errors in inferred $f_{\text{loc}}^{\text{NL}}$ .
The Gap: Without independent, physically motivated constraints on $b_\phi$ , next-generation surveys (like DESI, Euclid, SPHEREx) cannot achieve the precision required to rule out single-field inflation models. Current approaches often use uninformative priors, leading to projection effects and biased results.

2. Methodology

The authors propose a framework to construct observation-conditioned priors on $b_\phi$ by marginalizing over uncertainties in galaxy formation physics. The core equation for the prior is:
$p(b_\phi|\{Obs\}) = \int d\theta_{gf} \, p(b_\phi|\theta_{gf}) \, p(\theta_{gf}|\{Obs\})$
Where $\theta_{gf}$ represents galaxy formation parameters and $\{Obs\}$ are observed galaxy properties.

The methodology involves three main components:

A. Simulation Suites

CAMELS-SAM (Latin Hypercube): A suite of 1,005 N-body simulations ( $100 h^{-1}\text{cMpc}$ $100 h^{- 1} cMpc$ volume) using the Santa Cruz Semi-Analytic Model (SC-SAM).
- Parameters Varied: Cosmological ( $\Omega_M, \sigma_8$ ) and three galaxy formation feedback parameters: $A_{\text{AGN}}$ (AGN feedback), $A_{\text{SN1}}$ (stellar feedback normalization), and $A_{\text{SN2}}$ (stellar feedback slope).
- Purpose: To generate a dense training set for predicting galaxy formation observables (Stellar Mass Function, Stellar-to-Halo Mass Relation, etc.) across the parameter space.
Separate Universe (SU) Simulations: A suite of 50 simulations ( $205 h^{-1}\text{Mpc}$ $205 h^{- 1} Mpc$ volume) using the same SC-SAM but with fixed cosmology and varying $\sigma_8$ $σ_{8}$ to simulate long-wavelength perturbations.
- Purpose: To directly measure $b_\phi$ using the Separate Universe formalism, which quantifies the response of galaxy number counts to changes in the primordial power spectrum amplitude.

B. Emulation and Interpolation

To bridge the gap between the dense CAMELS-SAM sampling and the sparse SU sampling:

Gaussian Process Emulators (GPE): Trained on the CAMELS-SAM suite to predict galaxy formation observables (e.g., Stellar Mass Function - SMF) at arbitrary points in the parameter space $\theta_{gf}$ . This allows for Markov Chain Monte Carlo (MCMC) sampling of the posterior $p(\theta_{gf}|\{Obs\})$ .
Radial Basis Function (RBF) Interpolator: Trained on the 50 SU simulations to map specific galaxy formation parameters $\theta_{gf}$ to the corresponding $b_\phi$ value.

C. Analysis Workflow

MCMC Sampling: The GPEs are used to sample the posterior distribution of galaxy formation parameters conditioned on specific observables (SMF, Stellar-to-Halo Mass Relation - SHMR, or Stellar Metallicity-Stellar Mass relation).
Mapping to $b_\phi$ : Each sampled set of parameters is passed through the RBF interpolator to generate a corresponding $b_\phi$ value.
Resulting Prior: The distribution of these mapped values constitutes the observation-conditioned prior $p(b_\phi|\{Obs\})$ .

3. Key Contributions

Framework for Informative Priors: The first framework to explicitly construct priors on $b_\phi$ by marginalizing over galaxy formation uncertainties conditioned on non-clustering observables (SMF, SHMR).
Breaking the Degeneracy: Provides a physically motivated alternative to the universality relation, which is invalid for many galaxy samples.
Handling Model Discrepancies: Demonstrates that even when the underlying galaxy formation model (SC-SAM) has known discrepancies with observations (specifically at high halo masses), the resulting $b_\phi$ distributions remain robust and consistent across different conditioning observables.

4. Results

The study was validated using simulated data and then applied to real observational data (DESI-like Emission Line Galaxy sample).

Uncertainty Reduction:
- The uninformative prior (unweighted SU distribution) has a standard deviation $\sigma_{b_\phi} \approx 0.69$ .
- Conditioning on the Stellar Mass Function (SMF) reduces $\sigma_{b_\phi}$ to 0.08 (an 88% reduction).
- Conditioning on the Stellar-to-Halo Mass Relation (SHMR) reduces $\sigma_{b_\phi}$ to 0.02 (a 97% reduction).
- Conditioning on the Stellar Metallicity-Stellar Mass relation yields a 33% reduction.
Consistency: Despite significant shifts in the inferred galaxy formation parameters (e.g., $A_{\text{AGN}}$ shifts to the prior boundary when conditioning on SHMR due to SC-SAM model deficiencies at high masses), the resulting $b_\phi$ distributions from all three observables are mutually consistent.
Information Gain: The Kullback-Leibler (KL) divergence indicates that the MCMC-based conditioning extracts significantly more information (KL $\approx$ 1.6–3.2) compared to simple simulation reweighting (KL $\approx$ 0.05).
Bias: All analyses prefer lower values of $b_\phi$ compared to the unweighted prior, suggesting the universality relation may overestimate $b_\phi$ for these samples.

5. Significance

Enabling Next-Generation Constraints: By reducing the uncertainty in $b_\phi$ by nearly an order of magnitude, this method removes a major systematic bottleneck. This is crucial for future spectroscopic surveys (DESI, Euclid, SPHEREx) to reach the sensitivity ( $\sigma(f_{\text{loc}}^{\text{NL}}) \sim 1$ ) required to rule out single-field inflation models.
Robustness to Systematics: The approach demonstrates that one can obtain precise constraints on cosmological parameters even if the specific astrophysical model used has known tensions with data, provided the uncertainties are properly marginalized.
Future Directions: The authors suggest that future work should explore more flexible inference methods (e.g., neural density estimation) and expanded simulation suites to further refine these priors and account for inter-model discrepancies.

In summary, this paper provides a critical statistical tool to decouple astrophysical uncertainties from cosmological constraints, ensuring that the search for primordial non-Gaussianity is not compromised by imperfect knowledge of galaxy formation physics.