{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/spherical-collapse-formation-hysteresis-and","title":"Spherical collapse, formation hysteresis and the deeply non-linear cosmological power spectrum","arxiv_id":"1606.05345","date":"2016-06-16","proceeding":null,"authors":["Alexander Mead"],"abstract":"I examine differences in non-linear structure formation between cosmological models that share a $z=0$ linear power spectrum in both shape and amplitude, but that differ via their growth history. $N$-body simulations of these models display an approximately identical large-scale-structure skeleton, but reveal deeply non-linear differences in the demographics and properties of haloes. I investigate to what extent the spherical-collapse model can help in understanding these differences, in both real and redshift space. I discuss how this is difficult to do if one attempts to identify haloes directly, because in that case one is subject to the vagaries of halo finding algorithms. However, I demonstrate that the halo model of structure formation provides an accurate non-linear response in the power spectrum, but only if results from spherical collapse that include formation hysteresis are properly incorporated. I comment on how this fact can be used to provide per cent level accurate matter power spectrum predictions for dark energy models for $k\\leq5\\,h\\mathrm{Mpc}^{-1}$ by using the halo model as a correction to accurate $\\Lambda$CDM simulations. In the appendix I provide some fitting functions for the linear-collapse threshold ($\\delta_\\mathrm{c}$) and virialized overdensity ($\\Delta_\\mathrm{v}$) that are valid for a wide range of dark energy models. I also make my spherical-collapse code available at https://github.com/alexander-mead/collapse","url_abs":"http://arxiv.org/abs/1606.05345v2","url_pdf":"http://arxiv.org/pdf/1606.05345v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"spherical-collapse-formation-hysteresis-and","repo_url":"https://github.com/alexander-mead/collapse","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}