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How I stop worrying about non-universality and $b_\phi$ : Constraining local $f_{\rm NL}$ with $b_\phi$ priors from HOD posteriors

This paper demonstrates that deriving priors for the galaxy response parameter $b_\phi$ from halo occupation distribution (HOD) fits to small-scale clustering data enables unbiased constraints on local primordial non-Gaussianity ( $f_{\rm NL}$ ), effectively overcoming the dominant uncertainty caused by $b_\phi$ even in the presence of assembly bias.

Original authors: Jiaxi Yu, Nhat-Minh Nguyen

Published 2026-07-03

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Jiaxi Yu, Nhat-Minh Nguyen

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: Listening to the Universe's Echo

Imagine the universe as a giant, quiet room. When the universe began (the Big Bang), it wasn't perfectly smooth; it had tiny ripples, like dust motes dancing in a sunbeam. Scientists call these ripples "primordial density fluctuations."

Most of the time, these ripples behave very predictably, like a standard drumbeat. But sometimes, the drumbeat gets a little "off-key." This is called Non-Gaussianity. Specifically, this paper focuses on a type called "local" non-Gaussianity. If we can measure how "off-key" this drumbeat is, we learn secrets about the very first split-second of the universe (inflation).

The problem? The "off-key" sound is very faint. To hear it, we need to listen to the "echoes" left behind by galaxies.

The Problem: The Muffled Microphone

The paper argues that we have a major obstacle in hearing this echo clearly.

When we look at galaxies, we try to measure a specific signal (let's call it the Echo, or $f_{NL}$ ). However, the strength of the echo we hear depends on two things:

How loud the original sound was (the actual physics of the early universe, $f_{NL}$ ).
How sensitive the microphone is (how galaxies react to the ripples, a factor called $b_\phi$ ).

The paper explains that for a long time, scientists assumed all microphones were identical. They thought, "If we know how loud the galaxy is ( $b_1$ ), we automatically know how sensitive it is to the echo ( $b_\phi$ )." They called this the "Universality Relation."

The Analogy: Imagine you are trying to measure the volume of a concert by asking people in the crowd how loud it is. You assume everyone has the same hearing sensitivity. But in reality, some people are wearing earplugs, some have hearing aids, and some are just naturally more sensitive. If you don't know who has earplugs, you can't tell if the concert is actually loud or if the crowd just has bad hearing.

The paper says: "We can't assume the microphones are identical." Real galaxies are complex; they form in messy environments. Assuming they all react the same way introduces a huge error.

The Solution: Calibrating the Microphone

The authors propose a clever new way to fix this. Instead of guessing how sensitive the galaxies are, or relying on complex computer simulations of how galaxies form (which might be wrong), they use the galaxies' own behavior to calibrate the microphone.

The Method in Simple Steps:

Look at the Neighborhood: The team looks at how galaxies cluster together on small scales (like neighbors chatting in a backyard). This clustering tells them exactly how the galaxies are distributed around "halos" (invisible clouds of dark matter that hold galaxies). They use a model called HOD (Halo Occupation Distribution) to map this out.
Create Two Universes: They take this map and create two sets of fake universes (mocks) in a computer:
- Universe A: Normal strength.
- Universe B: Slightly weaker strength (simulating a different "sensitivity").
Measure the Difference: They count the galaxies in both universes. The difference in the count tells them exactly how sensitive the galaxies are to the "echo" ( $b_\phi$ ).
Build a "Cheat Sheet": They turn this measurement into a prior (a "cheat sheet" or a rule of thumb). Now, when they look at real data, they don't have to guess the sensitivity; they use their cheat sheet.

The Metaphor: Instead of guessing how sensitive your microphone is, you play a test tone at two different volumes. You see how much the reading changes. Now you know exactly how your microphone reacts. You can then use that knowledge to measure the real concert accurately.

The Results: Clearer Hearing

The authors tested this method using fake data that included the "off-key" signal (simulated with specific values like $f_{NL} = 30$ or $-30$).

Without the Cheat Sheet: The measurements were all over the place. The "microphone" was so uncertain that they couldn't tell if the signal was there or not.
With the Cheat Sheet: The measurements snapped into focus. They recovered the correct signal almost perfectly.
The Gain: The precision improved by 9% to 55%, depending on the type of galaxy. The higher the redshift (the farther away the galaxy), the better the improvement.

They also checked if "Assembly Bias" (a fancy term for the idea that galaxies in crowded neighborhoods might act differently than those in lonely spots) would ruin their cheat sheet. It didn't. The method remained robust even when the galaxies behaved slightly differently based on their environment.

Why This Matters

This paper doesn't just say "we found a better number." It offers a third way to solve a problem that has two other difficult options:

Simulation Route: Rely on computer models of galaxy formation (which might be wrong).
Observation Route: Try to calculate sensitivity from how galaxy numbers change over time (which is very hard to model perfectly).
This Paper's Route: Use the small-scale clustering of the specific galaxies you are studying to build your own calibration.

The Bottom Line:
By using the galaxies' own neighborhood patterns to calibrate their sensitivity, the authors have created a more reliable way to listen to the faint echoes of the Big Bang. This allows future telescopes (like DESI and others mentioned in the paper) to measure the physics of the universe's birth with much greater confidence, without having to blindly trust assumptions about how galaxies behave.

Technical Summary: Constraining Local $f_{\text{NL}}$ with $b_\phi$ Priors from HOD Posteriors

Problem Statement
Local primordial non-Gaussianity (local PNG), characterized by the amplitude parameter $f_{\text{NL}}$ , induces a scale-dependent bias in galaxy clustering proportional to the product $b_\phi f_{\text{NL}}$ . Here, $b_\phi$ represents the response of galaxy formation to long-wavelength primordial potential perturbations. A primary obstacle to obtaining robust constraints on $f_{\text{NL}}$ from galaxy surveys is the uncertainty in $b_\phi$ . Without precise knowledge of $b_\phi$ , clustering data can only constrain the degenerate product $b_\phi f_{\text{NL}}$ , preventing the reliable inference of the central value or uncertainty of $f_{\text{NL}}$ itself.

Common approaches to resolve this include assuming a "universality relation" (linking $b_\phi$ to the linear bias $b_1$ via halo mass functions) or calibrating $b_\phi$ against cosmological simulations. However, the universality relation is a poor approximation for observed, color- and flux-selected galaxies, while simulation-based calibrations often rely on generic galaxy-formation observables rather than the specific clustering statistics of the target sample. Alternatively, inferring $b_\phi$ directly from the redshift evolution of tracer abundance ($dn/dz$) requires complex modeling of survey selection effects.

Methodology
The authors propose a third route that combines cosmological simulations with the small-scale clustering statistics of the target galaxy samples via the Halo Occupation Distribution (HOD) model. The core methodology involves the following steps:

HOD Posterior Sampling: The authors utilize HOD constraints derived from the DESI Early Data Release (EDR) One-Percent Survey (SV3) for Luminous Red Galaxies (LRGs) across three redshift bins (LRG1: $0.4<z<0.6$ , LRG2: $0.6<z<0.8$ , LRG3: $0.95<z<1.1$ ). They sample the HOD parameter posteriors, inflating the uncertainties to $3\sigma$ (or $1\sigma$ for LRG3) to account for potential variations across the full DESI dataset.
Separate Universe Mock Generation: Using the sampled HOD parameters, they generate pairs of mock galaxy catalogs from the AbacusSummit simulation suite. One pair corresponds to the fiducial cosmology ( $c000$ , $\sigma_8 = 0.807952$ ) and the other to a low- $\sigma_8$ cosmology ( $c004$ , $\sigma_8 = 0.749999$ ), with all other parameters fixed.
$b_\phi$ Measurement: For each HOD realization, $b_\phi$ is measured using the "separate universe" technique. This treats the difference in $\sigma_8$ as a proxy for the response to a long-wavelength mode, calculating $b_\phi \approx 2 \frac{\ln N - \ln N_{\text{fid}}}{\ln \sigma_8 - \ln \sigma_{8,\text{fid}}}$ .
Prior Construction: The distribution of measured $b_\phi$ values across the 100 sampled HOD realizations for each redshift bin is fitted to a Gaussian distribution, creating a data-driven prior $b_\phi \sim \mathcal{N}(\mu, \sigma^2)$ .
Validation and Inference: These priors are applied to an inference framework using AbacusPNG simulations (which include explicit local PNG signals of $f_{\text{NL}} = 0, \pm 30$ $f_{NL} = 0, \pm 30$ ). The authors perform Bayesian inference on the redshift-space power spectrum monopole and quadrupole, comparing three modes:
- $b_{\text{fnl}}$ mode (No prior): Constrains the product $B_i = b_{\phi,i} f_{\text{NL}}$ with a flat prior.
- $b_{\text{fnl}}$ mode (With prior): Marginalizes over a common $f_{\text{NL}}$ while enforcing the Gaussian $b_\phi$ priors.
- $b$ - $p$ mode (Universality): Assumes the universality relation $b_\phi = 2\delta_c(b_1 - 1)$ .
Assembly Bias Test: The robustness of the method is tested against halo assembly bias (where galaxy occupation depends on halo environment at fixed mass) using a decorated HOD model.

Key Results

Recovery of $f_{\text{NL}}$ : When applied to AbacusPNG mocks with known $f_{\text{NL}}$ , the method successfully recovers the true input values. The inclusion of the HOD-derived $b_\phi$ priors tightens the $68\%$ confidence intervals on $f_{\text{NL}}$ by 9% to 55% compared to the no-prior case, with the largest improvements occurring at higher redshifts (LRG3) where linear bias is higher.
Comparison with Universality: The constraints obtained using the HOD priors are slightly broader than those obtained by assuming the strict universality relation ( $p=1$ ). However, the authors note that the universality relation is theoretically expected to fail for color-selected galaxies, making the HOD prior a more flexible and physically motivated alternative that avoids unverified assumptions.
Joint Constraints: Combining data from all three LRG redshift bins (LRG1–LRG3) improves the $f_{\text{NL}}$ constraint by approximately 65% relative to using LRG3 alone, driven by the increased effective volume and reduced cosmic variance.
Robustness to Assembly Bias: The inferred $f_{\text{NL}}$ remains stable even when the data vector includes assembly bias effects. The analysis shows that assembly bias primarily shifts the Gaussian linear bias parameter $b_1$ (by $2\text{--}3\sigma$ ) rather than the scale-dependent PNG term ( $b_\phi f_{\text{NL}}$ ), indicating that the method is robust against this specific systematic.

Significance and Claims
The paper claims to offer a "third route" to constraining local PNG that bridges the gap between simulation calibration and direct observational inference. By mapping small-scale clustering (via HOD) to large-scale bias ( $b_\phi$ ), the method generates sample-specific priors without relying on the universality relation or complex $dn/dz$ modeling.

The authors emphasize that this approach provides a flexible alternative to the universality assumption. While the universality relation yields tighter constraints, those constraints rely on a relation that observed galaxies are not expected to obey. The HOD prior trades a small amount of precision for robustness against model assumptions.

The work is presented as a pilot study with specific limitations:

It is currently restricted to LRGs; extending to Emission Line Galaxies (ELGs) or quasars is non-trivial due to less understood HODs and systematics.
The constraints are derived from cubic mocks with hypothetical effective volumes, not actual DESI data (though DESI DR1/DR2 fits are identified as the natural next step).
The prior construction treats HOD parameters as independent Gaussians, neglecting cross-covariances, which the authors acknowledge as a source of approximation.

Ultimately, the paper argues that as small-scale clustering measurements and galaxy formation understanding improve, these $b_\phi$ priors will sharpen, enhancing the prospects for Stage-V surveys (such as MUST, Spec-S5, and WST) to probe inflation physics via local PNG.

The Big Picture: Listening to the Universe's Echo

The Problem: The Muffled Microphone

The Solution: Calibrating the Microphone

The Results: Clearer Hearing

Why This Matters

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