Source-linked AI summary
HMFcalc: An Online Tool for Calculating Dark Matter Halo Mass Functions
Steven Murray, Chris Power, Aaron Robotham
TL;DR
Computing the halo mass function is not straightforward, so the paper presents HMFcalc and its hmf engine for flexible calculation and analysis. The implementation supports efficient variation of cosmological parameters and fitting functions, while worked examples demonstrate practical effects such as finite simulation box size.
Problem
Calculating the halo mass function from cosmological inputs and fitting functions is not straightforward.
Method
The paper presents the HMFcalc web application and hmf engine, combining theoretical calculations, CAMB transfer functions, configurable window functions, and an updating method for efficient recomputation.
Results
50% average processing-time reduction is achieved when each available parameter is changed once, while changing only redshift reduces computations by more than 94%.
Takeaways & Limitations
HMFcalc provides a standard for checking code, an interface for comparing generated HMFs with observations or simulations, and a tool for exploring fitting functions and cosmological parameters.
Takeaways & Limitations
Because most HMF fits are tied to specific cosmologies and universality remains under study, results should be compared with N-body simulations using the appropriate cosmology.
Abstract
from arXiv · showhide
The dark matter halo mass function (HMF) is a characteristic property of cosmological structure formation models, quantifying the number density of dark matter haloes per unit mass in the Universe. A key goal of current and planned large galaxy surveys is to measure the HMF and to use it to test theories of dark matter and dark energy. We present a new web application for calculating the HMF -- the frontend HMFcalc and the engine hmf. HMFcalc has been designed to be flexible, efficient and easy to use, providing observational and theoretical astronomers alike with the means to explore standard functional forms of the HMF or to tailor their own. We outline the theoretical background needed to compute the HMF, we show how it has been implemented in hmf, and finally we provide worked examples that illustrate HMFcalc's versatility as an analysis tool.
1. Introduction
The HMF quantifies dark-matter halo abundance as a function of mass and is sensitive to cosmological parameters, making it relevant to testing structure-formation models. The paper introduces HMFcalc and hmf as flexible tools for computing and exploring HMFs.
- Motivation: The HMF measures the number density of dark-matter haloes per unit mass and varies with cosmological parameters.It is used to connect theoretical structure formation with observable halo abundances.
- Motivation: Cosmological N-body simulations are the established tool for studying the HMF, but their information is commonly distilled into compact functional forms.The resulting representation is usually the comoving halo number density per unit logarithm of halo mass.
- Tool objectives: HMFcalc provides a standard for checking code and an interface for generating HMFs against observational or simulation data.These uses support both validation and comparison workflows.
- Tool objectives: HMFcalc offers a visually intuitive way to explore how cosmology affects the HMF.The web application is intended to make parameter-dependent exploration accessible to observers and theorists.
- Paper scope: The paper develops the theoretical background, describes hmf and HMFcalc implementation, and presents worked examples of the tool’s usefulness and versatility.Its structure covers fitting functions, CDM and WDM comparisons, algorithms, and example applications.
2. The Halo Mass Function (HMF)
The HMF is computed from the mass variance, power spectrum, window function, and a chosen fitting function, with numerical bounds ensuring converged integrals. The section compares fitting functions and explains how CDM and WDM assumptions alter predicted halo abundances.
- The HMF gives the number density of dark matter haloes per unit mass and comoving volume as a function of halo mass.
- The mass variance σ is computed by integrating the linear power spectrum against the Fourier-space top-hat window function W(kR).The window is the transform of a real-space sphere with sharp edges.
- Numerical integration requires kR bounds near 0.1 and 3 to capture 95% of the window-function integral, and hmf warns when these bounds are violated.The limits are designed to ensure convergence when integrating over k.
- The linear power spectrum uses P(k) = A k^n T^2(k), with normalization from σ8 and transfer functions computed using CAMB.The transfer function captures scale-dependent modifications sensitive to dark-matter and baryon parameters.
- 2.1. Fitting Functions: The fitting function f(σ) determines the HMF fit, with HMFcalc compiling literature forms and comparing them against the Sheth-Tormen fit.Press-Schechter underestimates high-mass halo abundance and overestimates low-mass abundance relative to simulations.
- 2.3. Warm Dark Matter Models: WDM suppresses small-scale power relative to CDM, reducing the number density of lower-mass haloes as the WDM particle mass decreases.HMFcalc models WDM using an additional transfer function and allows the particle mass mX to vary while fixing other parameters to fiducial values.
3. Implementation
HMFcalc combines a cross-platform Django web interface with the Python hmf backend, whose dependency-aware calculations support efficient, extensible HMF generation.
- HMFcalc is designed as a visually intuitive, easy-to-learn, flexible, and extensible web application.
- 3.1. HMFcalc’s engine, hmf: The Python hmf backend provides modules for transfer functions, cosmography, HMF calculations, power spectra, and mass variance.
- 3.1. HMFcalc’s engine, hmf: Perturbations computes quantities lazily through properties, including power, mass variance, growth, fitting functions, and differential and cumulative mass functions.
- 3.1. HMFcalc’s engine, hmf: Users can extend the available fitting functions by adding a method and registering it under a unique string identifier.
- 3.1. HMFcalc’s engine, hmf: Dependency tracking enables efficient updates when parameters change, reducing processing time by 50% on average and by more than 94% when only redshift changes.The method was used to generate 20,000 HMF realisations for each of 11 fitting functions.
- 3.1. HMFcalc’s engine, hmf: The calculation pipeline imports or generates transfer functions, interpolates and re-grids them, normalises the power spectrum, applies WDM and growth corrections, and computes the HMF.
- 3.3. HMFcalc’s web application: The Django frontend passes parsed form inputs to the hmf backend, producing one HMF for each parameter combination and storing results in a keyed Python dictionary.
4. Example Applications
HMFcalc is used to study simulation box requirements and finite-box effects on halo abundances, including the box size needed to contain massive haloes and the expected number of haloes.
- HMFcalc generates HMFs across fitting functions, cosmological parameters, and cosmic epochs, with particular utility for cosmological simulation studies.
- 4.1. Box Size for One Halo of Mass M: The box-size example asks how large a periodic simulation box must be to contain at least one halo above mass M.
- 4.1. Box Size for One Halo of Mass M: The hmf backend derives the one-halo box-size relation through the how_big property, which is exposed as an optional HMFcalc plot.
- 4.1. Box Size for One Halo of Mass M: The web application plots the required box size L against halo mass on a logarithmic y-axis, using the mass grid for the x-axis.
- 4.1. Box Size for One Halo of Mass M: The WMAP7 z=0 example computes the one-halo box-size result by default, while the expected halo count for a box of size L is not yet implemented.
- 4.2. The Impact of Finite Box Size: Finite box size omits perturbations longer than the box, making the measured mass variance smaller and affecting the HMF at higher masses.
- 4.2. The Impact of Finite Box Size: A 50 h^-1Mpc box enhances the number density of 10^14M☉h^-1 haloes by approximately 50%, the effect weakens at 100 h^-1Mpc, and is negligible at 250 h^-1Mpc.
5. Future Development
Future development will broaden HMFcalc’s physical model coverage, connect halo populations to observed galaxies, and provide more interactive control over generated data.
- Extending Range of Dark Matter and Dark Energy Models: Future versions will update Warm Dark Matter modelling and add alternative dark energy models.The current version includes fiducial Cold Dark Matter and a straightforward Warm Dark Matter extension based on Bode et al. (2001).
- HOD-Based Galaxy Populations: HOD extensions will calibrate galaxy–halo relations against observational constraints to reproduce observed galaxy populations.Planned inputs include cosmological parameters, dark matter and dark energy models, and observables such as two-point correlation functions.
- Interactive Data Generation: HMFcalc is being developed as a dynamic tool that updates HMF predictions in real time as users vary cosmological parameters.A similar interactive framework is planned for exploring how HOD-derived galaxy observables change with cosmology and dark matter or dark energy models.
6. Summary
HMFcalc is presented as a flexible, efficient, cross-platform web application for calculating and exploring the dark matter halo mass function. The authors plan broader model coverage, more flexible data interaction, and infrastructure for bespoke synthetic galaxy surveys.
- Summary: HMFcalc and its Python engine hmf calculate the dark matter halo mass function as a community-oriented, extensible resource.The application is intended for both observational and theoretical users.
- Summary: Users can explore how cosmological parameters, redshift, and fitting functions affect predicted HMFs.The tool can also help simulators understand biases introduced by finite simulation volume.
- Summary: Future work will broaden dark matter and dark energy models and support fast approximate generation of bespoke synthetic galaxy surveys.The authors also plan to give users greater flexibility in interacting with their data.