Physics and papers

Physics and papers

The papers behind hi_class: what to cite, where the formalism is defined, how the code was validated, and the reference papers for each shipped model. This is a link hub — for the physics itself, follow the links.

The code papers

Cite these if you use hi_class in a publication or preprint — at least the original CLASS paper and the two hi_class papers:

• hi_class: Horndeski in the Cosmic Linear Anisotropy Solving System — Zumalacárregui, Bellini, Sawicki, Lesgourgues, Ferreira · arXiv:1605.06102 · JCAP 1708 (2017) 019. The main code paper: the EFT/α-function implementation, stability tests, and code structure.
• hi_class: Background Evolution, Initial Conditions and Approximation Schemes — Bellini, Sawicki, Zumalacárregui · arXiv:1909.01828 · JCAP 2002 (2020) 008. Covariant Lagrangian models, background shooting, perturbation initial conditions, and the quasi-static approximation schemes.
• CLASS II: Approximation schemes — Blas, Lesgourgues, Tram · arXiv:1104.2933 · JCAP 1107 (2011) 034. The underlying Boltzmann code.

The formalism

• Maximal freedom at minimum cost: linear large-scale structure in general modifications of gravity — Bellini & Sawicki · arXiv:1404.3713 · JCAP 1407 (2014) 050.

hi_class is built on this reformulation of the effective field theory of dark energy: linear perturbations in any Horndeski model are fully specified by the expansion history H(a), the effective Planck mass M*², and four functions of time — α_K (kineticity), α_B (braiding), α_M (Planck-mass run rate) and α_T (tensor speed excess). See Models: Parametrized alphas for the definitions and the parametrizations the code implements.

Code validation

• A comparison of Einstein–Boltzmann solvers for testing general relativity — Bellini et al. · arXiv:1709.09135. Cross-validates hi_class against EFTCAMB, COOP and other codes on parametrized and covariant models (sub-percent agreement on observables).

Per-model references

References harvested from the headers of the shipped example files in gravity_models/. The models themselves are documented in Models: Covariant theories; parametrized α-functions and their background choices are in Models: Parametrized alphas and Models: Expansion history.

Model (example file)	References
Brans–Dicke (brans_dicke.ini)	Avilez & Skordis · 1303.4330; Bellini et al. · 1709.09135
Covariant Galileon, cubic/quartic/quintic (galileon_3.ini … galileon_5nu.ini)	Renk et al. · 1707.02263; Barreira et al. · 1406.0485 — the *nu variants use the massive-neutrino best fits of 1707.02263
nKGB — kinetic gravity braiding (nkgb.ini)	Kimura et al. · 1110.3598; Deffayet et al. · 1008.0048
Monomial quintessence (quintessence_monomial.ini)	Marsh et al. · 1406.2301
Tracker quintessence (quintessence_tracker.ini)	Amalgamation of standard tracker potentials; see e.g. astro-ph/9910214
α-attractor dark energy (alpha_attractor_canonical.ini)	Linder · 1505.00815; García-García et al. · 1803.00661 (nomenclature and best fits) and 1905.03753
Conformally coupled cubic Galileon (cccg_exp.ini, cccg_exp_canonical.ini)	Zumalacárregui · 2003.06396 · PRD 102 (2020) 023523

Learning materials

• hiclass-code.net — Resources — lecture notes, introductory slides (Corfu, IFT Madrid), exercises from the IFT and Aachen Cosmo Tools schools, the CLASS course video, and the CosmoSIS wrapper. Kept up to date there; not duplicated here.
• CLASS wiki and class-code.net — documentation and courses for the underlying CLASS code; everything there about standard CLASS input, output and internals applies to hi_class too.

Everything built with hi_class

A curated list of ~99 publications that used hi_class — constraints, forecasts, model implementations and code extensions — is maintained at hiclass-code.net/#pubs.