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The Most Probable Outer Density Profile from Excursion Set Theory

This paper re-derives the most probable outer density profile from excursion set theory by relaxing key simplifying assumptions and finding that while the new analytic model agrees with numerical simulations of the theory, it still diverges from N-body simulation results due to differences in window functions, suggesting the original analytic profile should only be used as an effective description with parameters fitted to cosmological simulations.

Original authors: Ericka Florio, Vasiliki Pavlidou

Published 2026-08-14

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CC BY 4.0

Original authors: Ericka Florio, Vasiliki Pavlidou

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 matter, stretching out in every direction. Most of this ocean is smooth and calm, but occasionally, gravity pulls the water together to form massive islands called galaxy clusters. These clusters are the biggest things in the cosmos that are held together by their own gravity. Now, picture a specific boundary around these islands: a point where the universe's expansion is trying to push things away, but the cluster's gravity is pulling them back. This tug-of-war creates a "turnaround radius," a magical edge where the flow of the universe stops and the cluster's own gravity takes over. Scientists are obsessed with measuring this edge because it acts like a cosmic ruler, helping them figure out the secret ingredients of the universe, like how much invisible "dark energy" is pushing everything apart.

To understand this edge, researchers use a clever mathematical trick called "excursion set theory." Think of this theory as a way to predict the shape of a storm cloud by watching how raindrops fall in a random walk. It treats the density of matter in the universe as a wandering path that steps up and down randomly. By following this path, scientists can predict where the "turnaround" happens and what the density of matter looks like just outside the cluster. For a while, a simplified version of this math seemed to match perfectly with giant computer simulations of the universe. But the big question remained: Was that match just a lucky coincidence caused by taking shortcuts in the math, or was it the real truth?

In this paper, astronomers Ericka Florio and Vasiliki Pavlidou decided to take those shortcuts away to see what happens. They went back to the original, messy math of the "double distribution"—a complex map showing how likely it is to find a certain amount of matter at a certain distance from a cluster. They carefully removed five specific simplifying assumptions that previous researchers had made to make the equations easier to solve. They expected that by fixing the math, their new, more accurate prediction would match the computer simulations even better.

However, the result was a bit of a plot twist. When they relaxed the assumptions and let the math run its full, complicated course, their new prediction actually drifted further away from the computer simulations. The "perfect" match they had with the simulations only existed when they used the simplified, shortcut version of the math. The authors found that their corrected, more rigorous math agreed perfectly with their own internal calculations, but it failed to match the simulated universe.

So, what went wrong? The authors suggest the culprit isn't the math itself, but the "window" they used to look at the data. In the simplified theory, the math assumes that each step of the random walk is independent of the last, like flipping a coin where the previous flip doesn't matter. This is called a "Markovian" process. But in the real computer simulations, the way matter is grouped together creates a "top-hat" window that links steps together, making the walk "correlated" and dependent on history. The authors argue that the simplified math accidentally worked because it ignored these correlations, effectively canceling out errors in a way that made it look right.

The paper concludes that the simplified formula, known as the "universal-scaling profile," shouldn't be treated as a fundamental law of physics derived from first principles. Instead, the authors propose we should treat it as a useful tool with a few adjustable knobs. If we take that formula and tweak its parameters to fit the results of computer simulations, it works great as a description of the universe's outer edges. But if we try to use the pure, un-simplified math from excursion set theory to predict these edges directly, it seems to hit a wall, likely because the theory's assumption of independent steps doesn't hold up in the complex, correlated reality of the universe. The authors suggest that future work needs to build a new version of the theory that accounts for these "memory" effects in the random walk to truly solve the puzzle.

Technical Summary: The Most Probable Outer Density Profile from Excursion Set Theory

Problem Statement
Precision cosmology requires breaking degeneracies between cosmological parameters, such as matter density ( $\Omega_m$ ) and dark energy ( $\Omega_\Lambda$ ). The turnaround radius ( $R_{ta}$ ) of galaxy clusters offers a potential probe for this, defined as the radius where cosmic expansion is balanced by gravitational infall. Recent work by Korkidis & Pavlidou (2024, 2025) established that $R_{ta}$ coincides with the point where outer density profiles in N-body simulations deviate from the "most probable outer density profile" derived analytically via Excursion Set Theory (EST).

However, the analytic profile proposed by Korkidis & Pavlidou (2024), known as the Universal-Scaling (US) profile, relies on five specific simplifying assumptions to achieve a tractable, mass-independent form. It remains unclear whether the agreement between this analytic profile and N-body simulations is a result of the underlying physics of EST or a cancellation of errors introduced by these assumptions. This paper investigates whether relaxing these assumptions improves the correspondence between the analytic EST prediction and simulated density profiles.

Methodology
The authors re-derive the mode of the "double distribution" (DD)—the joint probability distribution of collapsed mass and environmental overdensity—by systematically relaxing the five assumptions used in the original US profile derivation:

Order of Operations: Instead of differentiating the DD with respect to linearized overdensity ( $\tilde{\delta}_l$ ) and then converting to non-linear density ( $\hat{\rho}$ ), the authors first transform the DD to a distribution in terms of $\hat{\rho}$ and then find the mode.
Structure-in-Structure Term: The term correcting for halos embedded in larger structures is retained rather than dropped.
Spherical Collapse Approximation: The authors replace the approximate conversion between linear and non-linear overdensity with the full spherical collapse model solution.
Power-Law Approximation: The mass variance $S(m)$ is calculated using the Bardeen transfer function rather than a fitted power-law approximation.
Sub-leading Mass Dependence: The assumption that a specific term in the polynomial equation for the mode is negligible is removed; the full equation is solved directly.

The authors implemented a Python pipeline to numerically resolve the DD across a range of masses and clustering parameters ( $\beta$ ). They compared the numerically realized modes against the re-derived semi-analytic modes and against the original US profile and N-body simulation data from Korkidis & Pavlidou (2024).

Key Contributions and Results

Analytic-Numeric Consistency: The authors demonstrate excellent agreement between their re-derived semi-analytic mode (Eq. 6) and the numerically resolved mode of the transformed double distribution. This validates the mathematical correctness of their transformation and the code implementation.
Divergence from Simulations: Contrary to the hypothesis that relaxing assumptions would improve accuracy, the authors find that the corrected analytic profile diverges more strongly from N-body simulation profiles than the original US profile.
Assumption Sensitivity:
- Including the "structure-in-structure" term and using the Bardeen transfer function (Assumptions 2 and 4) alters the shape of the distribution but does not account for the primary discrepancy with simulations.
- The approximate spherical collapse conversion (Assumption 3) introduces only a few percent error in the relevant density range, confirming previous estimates.
- The most significant finding is that the profile derived from the "correct" (relaxed) EST formalism fails to match simulations, whereas the profile derived from the "approximate" (simplified) formalism (the US profile) provides a good fit.
The Window Function Hypothesis: The authors attribute this mismatch to the fundamental difference in window functions. EST relies on a sharp $k$ -space window function, which yields a Markovian random walk (independent steps). In contrast, extracting profiles from simulations requires a top-hat window in configuration space, which induces correlations between Fourier modes (non-Markovian). The authors argue that the US profile's success is likely an artifact of the simplifying assumptions compensating for the inherent limitations of the Markovian EST framework when applied to non-Markovian simulation data.

Significance and Conclusions
The paper concludes that the analytic US profile proposed by Korkidis & Pavlidou (2024) should not be interpreted as a rigorous first-principles prediction of the outer density profile from Excursion Set Theory. Instead, the authors posit that it serves effectively as a "few-parameter effective description" of the most probable outer density profile.

The significance of this work lies in identifying that the agreement between EST-based analytic profiles and N-body simulations is not a validation of the EST framework's ability to describe outer density profiles on large scales, but rather a result of specific approximations that inadvertently mask the theory's inability to handle the non-Markovian nature of structure formation induced by configuration-space window functions. The authors suggest that future work should focus on deriving outer density profiles from non-Markovian stochastic frameworks (e.g., Maggiore & Riotto 2010; Lapi & Danese 2020) to resolve this discrepancy fundamentally. Until then, the US profile remains a useful phenomenological tool, provided its parameters are fitted to simulation results.

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