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A Universal Relation Between Primordial Density-Potential Cross-correlation Coefficient and Spin Factor Distribution

This paper establishes a universal analytic relationship between the primordial density-potential cross-correlation coefficient and the statistical properties of the primordial spin factor, enabling the reconstruction of early universe physics from observable galaxy size distributions without cosmological degeneracies.

Original authors: Jun-Sung Moon (ASIAA), Jounghun Lee (Seoul Nat'l U.), Juhan Kim (KIAS)

Published 2026-07-16

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CC BY 4.0

Original authors: Jun-Sung Moon (ASIAA), Jounghun Lee (Seoul Nat'l U.), Juhan Kim (KIAS)

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 of dark matter, a mysterious substance that holds everything together but refuses to shine. Long ago, before stars and galaxies existed, this ocean was mostly smooth, but it had tiny ripples—slight bumps and dips in density. As gravity pulled these ripples together, they spun up, much like a figure skater pulling in their arms to spin faster. This spinning motion is called "angular momentum," and it's the reason galaxies aren't just round blobs of stars but often beautiful, swirling spirals.

Scientists have known for a while that the way a galaxy spins is linked to how it was born. But there's a tricky puzzle: the initial "bumps" in the dark matter ocean (density) and the "pull" of gravity (potential) aren't perfectly lined up. Think of it like two dancers trying to hold hands; if they are slightly out of step, the dance gets messy, and the spin changes. This misalignment is measured by something called the "primordial spin factor." Recently, researchers found that if you know how much a galaxy spins, you can guess its size and shape. But to understand the very beginning of the universe, we need to know why those initial dancers were out of step. That's where a new study comes in, trying to write a universal rulebook for this cosmic dance.

The Cosmic Dance of Misalignment

In this new study, a team of astronomers led by Jun-Sung Moon, Jounghun Lee, and Juhan Kim decided to crack the code of how the universe's initial spin was determined. They focused on a specific relationship: how the "misalignment" between the density bumps and the gravitational pull affects the spin of dark matter clouds. They call this misalignment the primordial spin factor (let's call it $\tau$ ).

Imagine you have two transparent sheets of paper. On one, you draw a map of where the dark matter is thickest (the density). On the other, you draw the map of the gravitational pull (the potential). If you stack them perfectly, the lines match up, and the spin is zero. But in reality, the lines are slightly crooked. The more crooked they are, the more the dark matter spins. The researchers wanted to find a simple math formula that could predict exactly how crooked these lines are based on a single number: the cross-correlation coefficient (let's call it $q$ ). Think of $q$ as a "match score" between the two maps. If $q$ is high, the maps match well; if $q$ is low, they are very different.

The Heuristic Guess and the Super-Computer Test

The authors didn't just guess; they built a "heuristic" model. In science speak, this means they used a clever, educated guess based on how things should work, rather than solving a massive, impossible equation from scratch. They proposed that the average spin and the "spread" of spins (variance) follow a specific pattern as the match score ( $q$ ) changes. Their formula suggests that as the match score gets closer to perfect (1), the spin drops rapidly, like a car slamming on the brakes. But if the match score is lower, the spin stays higher.

To test if this guess was right, they didn't look at real galaxies first. Instead, they went to the Multiverse simulations. These are massive computer experiments where they created virtual universes inside a box 1,024 $h^{-1}$ Mpc wide, filled with 2048 $^3$ particles of dark matter. They ran these simulations for four different types of universes:

Two versions of our standard universe (called $\Lambda$ CDM) with slightly different amounts of dark energy.
Two versions where the "dark energy" behaves differently (called wCDM), with different speeds of expansion.

They watched these virtual universes form, measured the spin factors ( $\tau$ ) at the very beginning (redshift $z=99$ ), and checked if their formula matched the computer data.

The Universal Rule

The results were surprisingly consistent. The computer simulations showed that the spin factors followed a specific statistical shape called a Gamma distribution (a curve that looks like a skewed hill). More importantly, the authors' formula agreed quite well and showed excellent agreements with the numerical results for all four types of universes they tested.

Whether the universe had a different amount of dark energy or if that energy acted differently, the relationship between the match score ( $q$ ) and the spin ( $\tau$ ) proved to be robustly constant. The three numbers in their formula (the peak spin, the threshold where it drops, and the rate of the drop) remained stable against the specific changes in dark energy density and equation of state. This suggests that the rule is universal. It doesn't matter if you are in a universe with one type of dark energy or another; the way the initial density and potential fields misalign to create spin follows the same mathematical dance steps.

Why This Matters

So, what's the big deal? The authors point out that we can actually see the sizes of galaxies today. Previous research showed that the size of a galaxy is linked to its spin. If we can measure the sizes of galaxies, we can work backward to figure out the spin, and now, thanks to this new formula, we can work backward even further to find the cross-correlation coefficient ( $q$ ).

This is a powerful tool because $q$ tells us about the very early universe, before galaxies even formed. It's like finding a fossil that tells us about the weather on the day the Earth was born. The authors suggest that by measuring galaxy sizes, we could determine $q$ and use it to test theories about the early universe—like whether the universe had weird fluctuations or if dark energy behaved strangely—without getting confused by other factors that usually make these measurements tricky.

In short, the paper doesn't just describe a relationship; it provides a universal key. It says that no matter how the universe expands or what kind of dark energy it has, the link between the initial cosmic "match score" and the resulting spin is fixed. This allows astronomers to potentially use the sizes of galaxies we see today as a time machine to peek at the conditions of the universe when it was just a baby.

Technical Summary: A Universal Relation Between Primordial Density-Potential Cross-correlation Coefficient and Spin Factor Distribution

Problem Statement
Recent numerical studies have established that the physical properties of visible galaxies (e.g., size, morphology, star formation rates) are closely linked to the angular momenta of their host dark matter (DM) halos. According to Linear Tidal Torque Theory (LTTT), halo angular momentum is conserved and proportional to the primordial spin factor, $\tau$ . This factor quantifies the degree of misalignment between the principal axes of the initial density Hessian tensor and the initial potential Hessian tensor (tidal tensor). While numerical experiments indicate that $\tau$ follows a Gamma distribution characterized by its mean and variance, a robust analytical model linking these statistical properties to the initial conditions of the universe has been lacking. Specifically, there is a need for an analytic expression that describes how the mean and variance of $\tau$ depend on the initial density-potential cross-correlation coefficient, $q$ , independent of smoothing scales and cosmological parameters.

Methodology
The authors develop a heuristic analytical model and validate it using high-resolution N-body simulations from the "Multiverse" project.

Heuristic Modeling:
- The study assumes the initial density and potential fields are Gaussian.
- It posits that the dependence of the mean ( $\bar{\tau}$ ) and variance ( $S_\tau$ ) of the spin factor on initial conditions can be expressed solely through the cross-correlation coefficient $q$ , defined as the ratio of the density-potential cross-correlation to the geometric mean of their auto-correlations.
- The authors propose a functional form where $\bar{\tau}$ and $S_\tau$ decrease exponentially as $q$ approaches 1 (perfect correlation/anti-correlation) and follow a power-law decrease for lower $q$ . The proposed forms are:
  $\bar{\tau} = \tau_0 \left[ 1 - \left(\frac{q}{q_{c}}\right)^a \exp\left(\frac{q}{q_{c}} - 1\right) \right]$
  $S_\tau = S_{\tau,0} \left[ 1 - \left(\frac{q}{q_{s,c}}\right)^b \exp\left(\frac{q}{q_{s,c}} - 1\right) \right]$
- This approach avoids the computationally expensive 12-dimensional integration required for a full analytic evaluation of the conditional multivariate-Gaussian distribution.
Numerical Validation:
- Simulations were performed using $2048^3$ particles in a periodic box for four distinct cosmological scenarios: two $\Lambda$ CDM models (with $\Omega_\Lambda = 0.74$ and $0.64$) and two $w$ CDM models (with dark energy equation of state $w = -0.5$ and $-1.5$).
- Initial density ( $\delta$ ) and potential ( $\Phi$ ) Hessian tensors were constructed at redshift $z=99$ on various smoothing scales ( $R_{f1}$ and $R_{f2}$ ).
- The primordial spin factor $\tau$ was computed at each grid point, and its distribution was fitted to a Gamma distribution to extract the mean and variance.
- The cross-correlation coefficient $q$ was calculated for the same smoothing scales using the linear density power spectrum.

Key Contributions and Results

Universal Analytic Relation: The study successfully derives and validates an analytic expression relating the mean and variance of the primordial spin factor $\tau$ to the initial density-potential cross-correlation coefficient $q$ .
Robustness Across Scales and Cosmologies: Numerical tests demonstrate that the derived functional forms (Eqs. 10 and 11) accurately describe the behavior of $\bar{\tau}(q)$ and $S_\tau(q)$ across different smoothing scales. Crucially, the best-fit parameters ( $\tau_0, q_c, a$ , etc.) remain constant across different cosmological models, including variations in the dark energy density ( $\Omega_\Lambda$ ) and equation of state ( $w$ ).
Gamma Distribution Confirmation: The numerical data confirms that the distribution of $\tau$ is well-approximated by a Gamma distribution for all tested cosmologies and smoothing scales.
Independence from $\sigma_8$ : Since $q$ is independent of the amplitude of initial density fluctuations ( $\sigma_8$ ) and the analytic model depends only on $q$ , the resulting distribution of $\tau$ is also independent of $\sigma_8$ .

Significance and Claims
The paper claims that this universal analytic expression serves as a new, complementary diagnostic tool for early universe physics. By leveraging the previously established link between observable galaxy sizes and the primordial spin factor distribution, this model allows for the reconstruction of the initial density-potential cross-correlation coefficient $q$ directly from galaxy observations.

The authors highlight several potential implications:

Breaking Degeneracies: Because the diagnostic relies on $q$ (which is independent of $\sigma_8$ ) and the model is universal across dark energy parameters, it may suffer less from cosmological degeneracies compared to other large-scale structure probes.
Probing Early Universe Physics: Determining $q$ on galactic scales could provide constraints on early universe phenomena such as running spectral indices, scale-dependent primordial non-Gaussianity, primordial black hole formation, and early dark energy, offering information beyond what is available from power spectra alone.
Scale Applicability: Unlike previous proxies for density-potential correlations that were limited to scales larger than galaxy clusters due to non-linear effects, this heuristic model enables the estimation of these correlations on galactic scales.

The authors conclude by noting their intention to apply this model to real observational data to test for deviations from standard physics, though they present the current work as a theoretical and numerical validation of the heuristic framework.

Technical Summary: A Universal Relation Between Primordial Density-Potential Cross-correlation Coefficient and Spin Factor Distribution

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