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Beyond Linear Bias Expansions for AbacusSummit Halos at z = 8

Using AbacusSummit simulations at redshift $z=8$ , this study demonstrates that the large-scale clustering of high-redshift halos is well-described by linear and quadratic bias terms within a Gaussian framework, revealing a significant detection of the $\delta^2$ term while finding tidal bias to be negligible, with these coefficients evolving consistently to $z=5$ under linear theory.

Original authors: Kyle K. Boone, Daniel J. Eisenstein

Published 2026-03-20

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Kyle K. Boone, Daniel J. Eisenstein

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, cosmic ocean. In this ocean, there are invisible currents of "dark matter" that swirl and clump together. Sometimes, these clumps get so dense that they collapse under their own gravity to form "halos"—the invisible scaffolding where galaxies are born.

This paper is like a detective story about how these galaxy nurseries (halos) are arranged in the very early universe, specifically when the universe was just a toddler (about 800 million years old, or redshift $z=8$ ).

Here is the breakdown of what the scientists found, using simple analogies:

1. The Problem: The "Super-Clumpy" Universe

In our local neighborhood today, galaxies are spread out somewhat evenly. But in the early universe, the only places galaxies could form were the absolute densest, most crowded spots in the cosmic ocean.

Think of it like a party.

Today's party: Guests are scattered around the room. If you want to find a guest, you just look around. This is "linear" behavior; it's predictable.
The early universe party: Guests only show up if the room is packed to the brim. They are so rare and so crowded that finding one is a huge deal. Because they are so clumped together, their arrangement isn't a simple, smooth pattern. It's "non-Gaussian," which is a fancy way of saying the pattern is weird, lumpy, and unpredictable using simple math.

2. The Old Map vs. The New Map

Scientists have a standard "map" (called a Bias Expansion) to predict how these halos are arranged.

The Old Map: It assumes the universe is mostly smooth and only has small ripples. It tries to predict the clumps using a simple formula: "If the density goes up a little, the number of halos goes up a little."
The Reality: In the early universe, the "ripples" are huge mountains. The old map breaks because it doesn't account for the fact that these halos are formed in extreme conditions.

The authors asked: Do we need to add more complex terms to our map to get it right? Specifically, do we need to worry about the "shape" of the clumps (tidal forces) or just how "dense" they are?

3. The Experiment: Simulating the Cosmos

The researchers didn't have a time machine, so they used a supercomputer (AbacusSummit) to run a simulation of the universe. They created a virtual box 2 billion light-years wide and filled it with billions of particles to see how halos formed at $z=8$ .

They looked at two main things to measure the "clumpiness":

The Power Spectrum: Like measuring the average size of the clumps.
The Bispectrum: Like measuring how the clumps are arranged in triangles. (If three galaxies form a triangle, how does that shape tell us about the underlying physics?)

4. The Big Discovery: Density Matters More Than Shape

The scientists tested their "map" against the simulation data. Here is what they found:

The "Shape" Theory (Tidal Bias) Failed: They thought maybe the shape of the surrounding space (like being squeezed between two mountains) was the key factor. They looked for this "tidal bias."
- The Result: It was basically zero. It's like trying to find a specific flavor of ice cream in a bowl, but the flavor doesn't exist. The "shape" of the environment didn't matter much for these early halos.
The "Density" Theory Won: Instead, they found that the intensity of the density was everything.
- They found that terms like $\delta^2$ (density squared) and $\delta^3$ (density cubed) were huge.
- The Analogy: Imagine you are looking for a needle in a haystack. In the early universe, the haystack isn't just a haystack; it's a mountain of hay. The more hay you have (density), the more likely you are to find the needle. The shape of the haystack doesn't matter; only the amount of hay does.

5. The "Non-Gaussian" Surprise

The paper confirms that to understand these early galaxies, you can't just use a straight line (linear bias). You need a curve that gets steeper and steeper.

They found that the "non-linearity" of the matter itself (how the dark matter clumps together) was actually more important than the "shape" of the clumps.
Even though they used a massive simulation box (2 billion light-years), they still couldn't detect the "tidal" effect. It was too weak compared to the sheer power of the density.

6. Checking the Clock: Time Travel to $z=5$

To make sure their findings weren't a fluke, they ran a test at a slightly later time ( $z=5$ ).

They found that as the universe got older, the "clumpiness" decreased, and the bias coefficients changed in a predictable way.
Interestingly, at this slightly later time, the "tidal" (shape) effect started to show up a tiny bit, but the "density" effect was still the boss.

The Bottom Line

If you want to understand how galaxies formed in the very early universe, stop worrying about the shape of the cosmic web and start worrying about how crowded it is.

The old models that tried to account for complex shapes were overcomplicating things. The universe at that time was so extreme that only the sheer density of the matter mattered. The scientists concluded that for future surveys (like those from the James Webb Space Telescope), we can use a simpler, more powerful model that focuses on density squared and cubed, ignoring the complex "tidal" shapes that turn out to be irrelevant for these ancient, rare objects.

In short: In the early universe, it wasn't about where you were in the room; it was about how packed the room was. And that packedness was the only thing that mattered.

1. Problem Statement

High-redshift halos (e.g., those hosting early galaxies at $z \sim 8-10$ ) are rare objects forming in the most extreme overdense regions of the universe. Consequently, they exhibit high bias ( $b \sim 9-10$ ), meaning their clustering amplitude is significantly larger than the underlying matter field.

The Challenge: Standard perturbation theory assumes the matter overdensity $\delta \ll 1$ . However, for highly biased tracers, the ratios of bias coefficients (e.g., $b_2/b_1$ , $b_3/b_1$ ) can be large enough that higher-order terms in the bias expansion (powers of $\delta$ ) become comparable to or larger than lower-order terms.
The Question: Which terms in the standard bias expansion are necessary to accurately model the clustering of these high-redshift halos? Specifically, are non-linear density terms ( $\delta^2, \delta^3$ ) more important than tidal shape terms ( $K_{ij}$ ), and is the underlying matter field effectively Gaussian on these scales?

2. Methodology

The authors utilized the AbacusSummit suite of N-body simulations to analyze halo clustering.

Simulation Setup:
- Redshifts: Primary analysis at $z=8$ ; comparative test at $z=5$ .
- Volume: 25 independent simulation boxes, each with a side length of $2\,h^{-1}\text{Gpc}$ .
- Resolution: $6912^3$ particles with mass $2.11 \times 10^9\,h^{-1}M_\odot$ .
- Halo Selection: Halos identified using the CompaSO algorithm. The primary sample used a mass threshold of $M > 1 \times 10^{11}\,h^{-1}M_\odot$ (comoving number density $\approx 8.2 \times 10^{-5}\,h^3\text{Mpc}^{-3}$ ).
Observables:
- Measured the Halo Power Spectrum ( $P_h(k)$ ) and Bispectrum ( $B_h(k_1, k_2, k_3)$ ).
- Applied Triangular Shaped Cloud (TSC) mass assignment to a $2048^3$ mesh (downsampled to $256^3$ for efficiency) to minimize aliasing.
- Corrected for shot noise and discrete sampling effects.
Bias Expansion Model:
- Constructed a bias expansion including Local-in-Matter-Density (LIMD) terms and Tidal terms.
- Renormalization: Applied renormalized bias operators (following Assassi et al.) to handle ultraviolet divergences and ensure $\langle \delta_h \rangle = 0$ .
- Terms Included:
  - Linear: $b_1 \delta$
  - Quadratic: $b_2 \delta^2$ and $b_{K^2} K^2$ (tidal)
  - Cubic: $b_3 \delta^3$
  - Non-linear Matter: $b_1 \delta^{(2)}$ (second-order matter evolution)
- Fitting Strategy: Performed a $\chi^2$ minimization using the Optax package. They fitted the bias coefficients ( $b_1, b_2, b_3, b_{K^2}$ ) and nuisance noise parameters ( $b_0, b'_0, b''_0$ ) simultaneously to the power spectrum and bispectrum data.
- Covariance: Used a diagonal approximation for the bispectrum covariance matrix derived from the 25 realizations.

3. Key Contributions

Re-evaluation of Perturbative Ordering: The paper challenges the traditional ordering of bias expansions where tidal terms are often considered significant. It demonstrates that for high-redshift, highly biased halos, the expansion should be ordered by the magnitude of the bias coefficients relative to the perturbation parameter, rather than just the order of $\delta$ .
Renormalized Operator Implementation: The authors rigorously applied renormalization to the bias operators ( $\delta^2, \delta^3, K^2$ ) within the context of the Effective Field Theory of Large Scale Structure (EFTofLSS), ensuring robust predictions insensitive to small-scale cutoffs.
Joint Power Spectrum and Bispectrum Analysis: By combining constraints from both the power spectrum and bispectrum, the authors broke degeneracies between bias parameters that exist when using either observable alone.

4. Key Results

Dominance of Density Terms over Tidal Terms:
- The tidal bias ( $b_{K^2}$ ) was not detected at $z=8$ , with a deviation of only $\sim 0.1\sigma$ from zero.
- In contrast, the quadratic density term ( $b_2 \delta^2$ ) was detected at a $17\sigma$ level.
- The cubic density term ( $b_3 \delta^3$ ) was weakly detected at the $1.3\sigma$ level.
Non-linear Matter Evolution:
- The non-linear matter term ( $\delta^{(2)}$ ) was detected at the $1.3\sigma$ level, suggesting that matter non-linearity is more significant than tidal effects for these tracers.
Bias Coefficients at $z=8$ :
- $b_1 \approx 10.0 \pm 0.1$
- $b_2 \approx 40 \pm 2$
- $b_3 \approx 40 \pm 30$
- $b_{K^2} \approx -1 \pm 7$ (consistent with 0)
- Note: The hierarchy $b_3 \approx b_2 > b_1$ confirms that higher-order terms are crucial.
Evolution to $z=5$ :
- At $z=5$ (using a mass threshold to match the variance of the $z=8$ sample), the bias coefficients were smaller ( $b_1 \approx 7.6$ ).
- Tidal bias became detectable at the $1\sigma$ level at $z=5$ , whereas it was undetectable at $z=8$ .
- The $z=5$ coefficients, when linearly evolved to $z=8$ (scaling by the growth factor), matched the $z=8$ measurements within $\sim 10\%$ , confirming that the same physical halos are being tracked and that the bias evolution is driven by the growth of the matter field.
Model Fit: The proposed bias expansion (including $b_1, b_2, b_3$ ) successfully fit the data with no significant systematic residuals in the power spectrum or bispectrum.

5. Significance and Implications

Refinement of High-Redshift Bias Models: The study provides strong evidence that for high-redshift tracers ( $z \gtrsim 5$ ), standard bias expansions focusing on tidal terms are insufficient. Instead, expansions must prioritize higher-order local density terms ( $\delta^2, \delta^3$ ).
Physical Interpretation: The results support the physical intuition that high-redshift halos form in the most extreme peaks of the density field. In these regions, the magnitude of the overdensity is the dominant factor, while the shape (tidal field) of the environment is less relevant.
Future Surveys: For upcoming high-redshift galaxy surveys (e.g., with JWST or future spectroscopic missions), the authors suggest that efficient and accurate modeling can be achieved using bias expansions consisting primarily of linear and quadratic terms ( $b_1, b_2$ ) and assuming a Gaussian underlying matter field, provided the expansion is truncated appropriately. Including $b_3$ may be necessary for precision, but tidal terms can likely be neglected to simplify analysis.
Methodological Validation: The work validates the use of AbacusSummit and the BSKIT package for high-precision bias measurements, demonstrating that large-volume simulations ( $2\,h^{-1}\text{Gpc}$ ) are sufficient to constrain these parameters even at very high redshifts.