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        "description": "The halo mapping assumes a homogeneous isotropic Gaussian initial field and a specified collapse threshold. Sharp-k filtering supplies independent increments and the Brownian first-crossing calculation; other filters produce correlated, non-Markov histories and require approximations or numerics. Filter-radius/mass calibration and finite simulation comparisons have limits. This is not an exact physical collapse theory, complete merger-history model or independently verified prediction for every mass range.",
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      "statement": "# Excursion sets: follow the first threshold crossing to avoid nested double counting\n\nBond, Cole, Efstathiou and Kaiser replace single-scale threshold counting with first crossings across smoothing scales, derive the Press-Schechter factor of two for sharp-k Gaussian walks, and extend the framework to conditional mass histories.\n\nA small dense region can sit inside a larger region that also collapses. Counting each scale separately risks assigning the same matter more than once. This paper follows the smoothed initial density at each position as the smoothing scale changes, and assigns it using the first crossing of a collapse threshold. With a sharp cutoff in Fourier space and Gaussian fluctuations, the changes are independent: a Brownian first-passage calculation recovers the earlier Press-Schechter mass formula, including its factor of two. Other filters give correlated histories, so the same analytic answer does not automatically apply; the authors use bounds and numerical trajectories instead. A two-threshold construction also describes conditional mass distributions at two epochs. Comparisons with existing simulations support some average mass statistics over their tested range, while particle-by-particle identification, especially for small groups, performs poorly. The statistical construction is a reusable cosmological model, with filter, mass-calibration and collapse assumptions explicitly retained.\n\n## Scientific scope\n\nThe halo mapping assumes a homogeneous isotropic Gaussian initial field and a specified collapse threshold. Sharp-k filtering supplies independent increments and the Brownian first-crossing calculation; other filters produce correlated, non-Markov histories and require approximations or numerics. Filter-radius/mass calibration and finite simulation comparisons have limits. This is not an exact physical collapse theory, complete merger-history model or independently verified prediction for every mass range.\n\n## People and systems\n\nJ. R. Bond; S. Cole; G. Efstathiou; N. Kaiser\n\n\n\n## Original sources\n\n- [Bond/Cole/Efstathiou/Kaiser1991: Excursionsetmassfunctions, complete440-460](https://adsabs.harvard.edu/pdf/1991ApJ...379..440B)\n- [Maggiore/Riotto2009v3: nonMarkovexcursion/boundary formulation](https://arxiv.org/abs/0903.1249v3)\n",
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