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Galaxy formation in modified gravity -- II. galaxy halo connection and assembly bias

Using state-of-the-art hydrodynamical simulations for Stage-IV surveys, this study demonstrates that while modified gravity and assembly bias complicate the galaxy-halo connection, incorporating environmental density into the Halo Occupation Distribution model effectively reduces assembly bias effects to 2–3%, providing a robust framework for testing non-standard cosmologies with large-scale structure data.

Original authors: Michael Collier, Sownak Bose, Baojiu Li

Published 2026-06-12

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CC BY 4.0

Original authors: Michael Collier, Sownak Bose, Baojiu Li

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: Mapping the Universe's Invisible Skeleton

Imagine the universe as a giant, invisible city. The "buildings" of this city are galaxies (like our Milky Way), but they are built on top of an invisible skeleton made of Dark Matter. We can't see the skeleton directly, but we can see the buildings.

Cosmologists want to understand how this city was built and what laws of physics (gravity) govern its construction. To do this, they use massive surveys (like DESI and Euclid) to map millions of galaxies. However, there is a tricky problem: How do we know which building sits on which piece of the invisible skeleton?

This paper tackles that problem, specifically asking: Does the way gravity works change how galaxies form on their dark matter skeletons?

The "Gravity" Question: Is Einstein Right?

For a long time, we've assumed gravity works exactly as Albert Einstein described (General Relativity). But some scientists think gravity might act differently on huge scales, perhaps explaining why the universe is expanding faster than expected. This idea is called Modified Gravity (MG).

Think of it like this:

Standard Gravity (Einstein): Gravity is a constant rulebook.
Modified Gravity: The rulebook has a "secret mode." In empty, quiet neighborhoods (low density), gravity might get a little stronger. But in crowded cities (high density), a "shield" kicks in to hide this extra strength so we don't notice it in our solar system. This shield is called the Chameleon Mechanism.

The Experiment: A Cosmic Simulation

The authors didn't just look at the sky; they built a massive video game simulation of the universe.

They created a virtual universe with different gravity rules (Standard vs. Modified).
They let the simulation run for billions of years, watching dark matter clump together and form "haloes" (the invisible skeletons).
They then "painted" galaxies onto these haloes based on real physics (how gas cools, stars form, etc.).

They focused on two types of galaxies, like two different neighborhoods in the city:

LRGs (Luminous Red Galaxies): The "old, wealthy" neighborhoods. Massive, red, and quiet.
ELGs (Emission Line Galaxies): The "young, energetic" neighborhoods. Smaller, blue, and full of active star formation.

The Problem: The "Assembly Bias" Glitch

To analyze the data, scientists use a simple rule called the HOD (Halo Occupation Distribution).

The Simple Rule: "If a dark matter halo is heavy enough, it gets a galaxy. If it's too light, it doesn't."
The Glitch (Assembly Bias): In reality, it's not just about weight. It's about history and neighborhood. Two haloes with the exact same weight might have different galaxies because one formed earlier or lives in a crowded area.

The paper found that if you ignore this "history" and only look at weight, your map of the universe is wrong.

In Standard Gravity: Ignoring this history makes you underestimate how clumpy the galaxies are by about 10–20%.
In Modified Gravity: It gets even messier. Because of the "Chameleon Shield," galaxies in quiet neighborhoods behave differently than those in crowded ones. The simple "weight-only" rule fails even more spectacularly here.

The Discovery: A "Time-Travel" Sequence

The authors noticed something fascinating about how these galaxies behave in Modified Gravity. It's not random; it follows a specific evolutionary sequence, like a movie playing at different speeds for different characters:

The Unmasking: In Modified Gravity, the "shield" drops first in empty, quiet regions. Galaxies there start growing faster and forming stars earlier.
The Catch-Up: Eventually, the shield drops in crowded regions too. The galaxies there start growing faster, catching up to the ones that started early.
The Saturation: After a while, the effect hits a limit. No matter how strong the gravity gets, the universe can't form infinite galaxies. The models start to look similar again.

This means that at any given moment, a "strong" Modified Gravity model might be in a different stage of this sequence than a "weak" one, making them look very different from each other and from Standard Gravity.

The Solution: Adding a "Neighborhood" Variable

The paper asks: Can we fix our simple "weight-only" rule to make it accurate again?

They tried adding a second ingredient to the rule: Environment Density (how crowded the neighborhood is).

The Fix: Instead of just asking, "Is the halo heavy?", they asked, "Is the halo heavy AND is it in a crowded neighborhood?"

The Result:

When they added this "neighborhood" factor, the error dropped dramatically.
For the "young" galaxies (ELGs) and "old" galaxies (LRGs), the prediction error shrank from 10–20% down to just 2–3%.
Interestingly, they tried adding another factor (how "compact" the halo is), but that didn't help much. The "neighborhood" (environment) was the key.

The Bottom Line

If we want to use future telescopes to test if gravity is different from Einstein's theory, we cannot use the old, simple rules that only look at galaxy mass. We must account for the environment where the galaxy lives.

Without this fix: We might think we've discovered new physics (Modified Gravity) when we've just made a mistake in our mapping rules.
With this fix: We can accurately map the universe, allowing us to truly test if gravity behaves differently in the deep cosmos.

In short: To understand the universe's architecture, you can't just weigh the bricks; you have to know where they were built.

Technical Summary: Galaxy formation in modified gravity – II. galaxy halo connection and assembly bias

Problem Statement
Modern cosmological surveys (e.g., DESI, Euclid) aim to constrain cosmological parameters and test theories of gravity by mapping the large-scale structure (LSS) of the Universe. However, the interpretation of these data is limited by uncertainties in the "galaxy-halo connection"—the mapping between observed galaxies and the underlying dark matter haloes. Standard analyses often rely on the Halo Occupation Distribution (HOD) model, which assumes galaxy occupation depends solely on halo mass. This assumption neglects "assembly bias" (AB), where galaxy formation also depends on secondary halo properties such as formation history, environment, and concentration. While AB is known to affect clustering in the standard $\Lambda$ CDM model, its behavior in Modified Gravity (MG) theories, particularly $f(R)$ gravity, remains poorly understood. The complex interplay between MG screening mechanisms (e.g., chameleon screening) and galaxy formation physics could introduce significant biases in cosmological parameter estimation if not properly accounted for.

Methodology
The authors utilize state-of-the-art hydrodynamical simulations performed with the MG-AREPO code, which incorporates the full baryonic physics of the IllustrisTNG model but adapted for $f(R)$ gravity. The simulation suite includes a range of Hu-Sawicki $f(R)$ models with varying strengths of the fifth force, denoted by the background scalar field magnitude today, $|f_{R0}| = 10^{-x}$ (where $x \in \{4.0, 4.5, 5.0, 5.5, 6.0\}$ ), alongside a standard $\Lambda$ CDM control ( $f_{R0}=0$ ).

From these simulations, the authors generate mock catalogues for two galaxy populations relevant to Stage-IV surveys:

Luminous Red Galaxies (LRGs): Selected by stellar mass ranking.
Emission Line Galaxies (ELGs): Selected based on specific star formation rate (sSFR) and a stellar mass cut ( $M_\star > 2.5 \times 10^{10} M_\odot$ ) to mitigate resolution effects.

The study proceeds in three stages:

HOD Fitting: The authors fit standard HOD models (Zheng et al. 2005 for LRGs; Alam et al. 2020 for ELGs) to the simulation data to determine how HOD parameters evolve with redshift and gravity strength.
Secondary Dependency Analysis: The galaxy catalogues are split into quartiles based on secondary halo properties: environmental density (defined by overdensity in a spherical annulus) and halo concentration. HOD parameters are fitted independently for each quartile to quantify the dependency of galaxy occupation on these properties.
Assembly Bias Quantification: The authors generate mock galaxy catalogues using three HOD prescriptions: (i) the basic mass-only HOD, (ii) an environment-dependent HOD, and (iii) a concentration-dependent HOD. They compare the large-scale correlation function ( $\xi_{LS}$ ) of these mocks against the "ground truth" measured directly from the hydrodynamical simulations to quantify the residual assembly bias.

Key Contributions and Results

Evolutionary Sequence of MG Effects: The study identifies a distinct evolutionary sequence in how MG affects the galaxy-halo connection. As the fifth force "unscreeens" haloes over time, models progress through regimes of initial unscreening, catch-up, and quenching. This sequence explains the non-monotonic behavior of HOD parameters (e.g., $M_c$ , $p_{max}$ , $M_1$ ) across different gravity models and redshifts. For instance, in weaker MG models, the transition from screened to unscreened states occurs later, leading to different HOD behaviors compared to stronger models at the same redshift.
Saturation Effect: The authors confirm a "saturation" effect where the impact of MG on galaxy formation and clustering reaches a limit. Stronger MG models (e.g., F4.0, F4.5) eventually "catch up" with each other in terms of HOD parameters, suggesting that for a fixed galaxy selection criteria, the universe cannot indefinitely produce more galaxies; MG primarily accelerates the formation timeline rather than creating an infinite excess of galaxies.
Secondary Dependencies:
- Environment: There is a strong dependency of HOD parameters on environmental density in both $\Lambda$ CDM and MG models. In MG, this is exacerbated by the chameleon screening mechanism, which causes unscreening to occur first in low-density environments.
- Concentration: While HOD parameters show significant dependence on halo concentration, this dependency does not translate to a significant reduction in assembly bias when used to generate mocks.
Assembly Bias Magnitude:
- Neglecting assembly bias (using a basic mass-only HOD) underestimates galaxy clustering strength by 10–20% in $\Lambda$ CDM at $z \lesssim 1$ .
- In MG models, the AB effect is non-monotonic and can be significantly different from $\Lambda$ CDM. For example, in the F5.5 model, the AB effect is stronger than in GR, while in F4.0 and F4.5, it can be weaker.
- Crucially, the direction and magnitude of the AB effect in MG can differ from $\Lambda$ CDM, potentially leading to misidentification of the gravity model if standard analysis pipelines are used.
Mitigation Strategy: Extending the basic HOD model to include environmental density as a secondary variable reduces the assembly bias effect to 2–3% for $z \lesssim 0.5$ across all models (including $\Lambda$ CDM and various $f(R)$ scenarios). In contrast, extending the model with halo concentration does not significantly reduce the AB effect.

Significance and Claims
The paper claims that the basic HOD model is insufficient for precision cosmology tests involving Modified Gravity, particularly for Emission Line Galaxies (ELGs). The authors argue that the complex interplay between MG screening and galaxy formation creates assembly bias signatures that differ qualitatively and quantitatively from standard gravity.

The primary significance lies in demonstrating that incorporating environmental density into the HOD model provides a well-motivated and effective starting point for minimizing assembly bias systematics. This is particularly relevant for upcoming surveys like DESI and Euclid, which probe scales where assembly bias is strong. The authors caution that failing to account for these effects could bias cosmological parameter inference or lead to incorrect conclusions regarding the nature of gravity. They conclude that while the environment-dependent HOD reduces AB to a residual level ( $\lesssim 3\%$ at low $z$ ), further work with larger, higher-resolution simulations is needed to fully eliminate these systematics and test more sophisticated HOD extensions.