The Most Probable Outer Density Profile from Excursion Set Theory

The paper The Most Probable Outer Density Profile from Excursion Set Theory by Ericka Florio and Vasiliki Pavlidou investigates the validity of an analytic profile for the outer density distribution around galaxy clusters, derived from excursion set theory. The context is that measurements of the turnaround radius around galaxy clusters can break degeneracies between matter and dark energy densities, and prior work by Korkidis & Pavlidou (2024) showed that this radius coincides with the first deviation between N-body simulation profiles and an analytic profile from excursion set theory. However, that analytic profile relied on several simplifying assumptions.

The paper's aim is to evaluate the effect of these assumptions on the shape of the analytic profile and its correspondence with simulated outer density profiles. The authors relax the key simplifying assumptions and re-derive the mode of the outer density profile from excursion set theory. They then numerically resolve the double distribution (DD) across a range of masses and clustering parameters, and compare the numerical mode estimate to the re-derived analytic profile.

The five assumptions relaxed are: (1) the order of operations (differentiating before transforming from linear to non-linear overdensity); (2) dropping the structure-in-structure term; (3) using an approximation to spherical collapse; (4) approximating the mass variance as a power law; and (5) dropping a sub-leading mass-dependent term.

The main results are: We find excellent agreement between our analytic profile and the numerically-realized DD. However, our analytic profiles diverge from N-body profiles, and this divergence grows as we relax successive assumptions. Specifically, the paper states: Counterintuitively, we find that it does not: the corrected prediction agrees with the profile derived numerically directly from the double distribution, but departs more strongly from the simulated profiles.

The authors conclude: "We relate this mismatch to the differing window functions used in the analytic and simulation-based approaches, which respectively yield Markovian and correlated density trajectories. We conclude that the analytic profile proposed by Korkidis & Pavlidou should only be used as a few-parameter effective description of the most-probable outer density profile, with parameters fitted to results of cosmological simulations."

They argue that the divergence arises from a deeper assumption within excursion set theory itself: the use of a sharp k-space window function, which makes the random walk Markovian, whereas simulations use a top-hat window function in configuration space, which induces correlations. The paper notes: "Excursion set theory has been questioned by other authors on both theoretical grounds... One particular aspect of excursion set theory that has been highlighted as the source of such issues is the choice of window function." They suggest that a re-derivation from a non-Markovian framework, such as those by Maggiore & Riotto (2010) or Lapi & Danese (2020), might explain the good fit of the universal scaling profile to simulations.

Improvements to AI Systems Based on This Paper:

1. Uncertainty-Aware Profile Fitting for Cosmological Simulations

• Improvement: Integrate the finding that analytic profiles (e.g., from excursion set theory) diverge from N-body simulations due to window-function-induced correlations. Train an AI model to automatically detect when an analytic approximation is being over-extrapolated, and to flag cases where Markovian assumptions break down.

• Capability: The AI can now provide calibrated confidence intervals for density profile fits, explicitly warning when a fitted analytic form (like the Korkidis & Pavlidou profile) is only valid as an effective description, not a physical prediction.

2. Hybrid Simulation-Analytic Emulator

• Improvement: Use the paper’s conclusion to build a hybrid emulator that combines the fast, analytic profile (with fitted parameters) with a correction term learned from N-body simulations. The AI learns the residual between the analytic and simulated profiles as a function of mass, clustering parameter, and window function choice.

• Capability: The AI can generate accurate outer density profiles for galaxy clusters at a fraction of the computational cost of full simulations, while explicitly quantifying where the analytic baseline fails.

3. Window-Function-Aware Generative Models

• Improvement: Modify generative models (e.g., diffusion or normalizing flows) for cosmological density fields to incorporate the choice of window function (sharp-k vs. top-hat) as a conditioning variable. This directly addresses the paper’s root cause of divergence—Markovian vs. correlated trajectories.

• Capability: The AI can generate realistic density profiles that match either analytic or simulation-based statistics, and can interpolate between them, enabling users to test how different smoothing kernels affect structure formation predictions.

4. Automated Assumption-Auditing for Theoretical Models

• Improvement: Build an AI tool that, given a theoretical derivation (like the five assumptions relaxed in this paper), automatically identifies which assumptions are most likely to cause divergence from numerical or observational data, by cross-referencing with known simulation results.

• Capability: The AI can suggest which simplifying assumptions to relax first in future analytic models, prioritizing those that have the largest impact on predictive accuracy (as shown here: the order of operations and the structure-in-structure term).

5. Robust Parameter Inference for Turnaround Radius Measurements

• Improvement: Use the paper’s finding that the analytic profile is only an effective description to develop an AI-based inference pipeline that marginalizes over the systematic uncertainty in the profile shape when estimating the turnaround radius. This prevents biased estimates of matter and dark energy densities.

• Capability: The AI can provide unbiased cosmological parameter constraints from cluster outskirts, with error bars that correctly account for the known analytic-simulation mismatch.

6. Non-Markovian Excursion Set Solver

• Improvement: Implement a neural solver that learns the first-crossing distribution for non-Markovian random walks (e.g., using correlated trajectories from top-hat windows), as suggested by the paper’s reference to Maggiore & Riotto or Lapi & Danese. This would replace the sharp-k assumption.

• Capability: The AI can produce theoretically consistent analytic profiles that match simulations, eliminating the need for ad-hoc fitting parameters and enabling first-principles predictions of outer density profiles.

Measurements of the turnaround radius around galaxy clusters can be used to break the degeneracy between measurements of the present day energy density of matter and dark energy. Korkidis & Pavlidou showed that the turnaround radius coincides with the first point of deviation between outer density profiles in N-body simulations and the analytic profile predicted by excursion set theory. However, their analytic profile relied on a number of simplifying assumptions, which may each introduce systematic error. We evaluate the effect of these assumptions on the shape of the analytic profile and its correspondence with simulated outer density profiles. We relax the key simplifying assumptions and re-derive the mode of the outer density profile from excursion set theory. We then numerically resolve the double distribution (DD) across a range of masses and clustering parameters, and compare the numerical mode estimate to the re-derived analytic profile. We find excellent agreement between our analytic profile and the numerically-realized DD. However, our analytic profiles diverge from N-body profiles, and this divergence grows as we relax successive assumptions. We relate this mismatch to the differing window functions used in the analytic and simulation-based approaches, which respectively yield Markovian and correlated density trajectories. We conclude that the analytic profile proposed by Korkidis & Pavlidou should only be used as a few-parameter effective description of the most-probable outer density profile, with parameters fitted to results of cosmological simulations.