# A dynamics-based density profile for dark haloes -- II. Fitting function

**Authors:** Benedikt Diemer  
**arXiv:** [2205.03420](https://arxiv.org/abs/2205.03420)  
**Date:** 2022-05-06  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/2205.03420v2/

## Key points

- Existing models (NFW or Einasto profiles) fail in the transition region of dark matter haloes dominated by newly accreting matter
- Authors propose a new fitting function that combines orbiting and infalling components of halo density profiles
- Orbiting term modeled as truncated Einasto profile with five parameters: normalization, scale and truncation radii, alpha and beta (control steepness)
- Infalling profile represented by power law transitioning to constant at center of halo
- Fitting function accurately describes averaged total density profiles in simulations with 5% accuracy
- Three-parameter model outperforms Einasto profile when fitting individual halos (alpha = 0.18, beta = 3)
- Detailed mathematical descriptions provided for previous fitting functions and new models
- Derivatives with respect to parameters included
- Appendix discusses alternative model variants and additional mathematical properties

## Summary

This paper presents a new fitting function for the density profiles of dark matter haloes. Existing models such as the NFW or Einasto profiles commonly describe these profiles but fail in the transition region where halos are dominated by newly accreting matter. The authors propose a simple and accurate fitting function that combines orbiting and infalling components of halo density profiles. The orbiting term is modeled as a truncated Einasto profile with five parameters including normalization, scale and truncation radii, and two parameters (alpha and beta) that control the steepness of the profiles. The infalling profile is represented by a power law in overdensity that transitions to a constant at the center of the halo. The authors demonstrate that their fitting function can accurately describe averaged total density profiles in simulations with an accuracy of about 5%. When alpha and beta are set to 0.18 and 3 respectively, this three-parameter model outperforms the Einasto profile on average when fitting individual halos. Detailed mathematical descriptions of previous fitting functions as well as their new models for both orbiting and infalling components are provided along with derivatives with respect to their parameters. Alternative model variants are discussed in an appendix along with additional mathematical properties. This study offers a refined understanding of dark matter halo density profiles through its novel fitting function which captures both orbiting and infalling components accurately across different scenarios.