Horndeski in the Cosmic Linear
Anisotropy Solving System

Einstein-Boltzmann solver for dark energy and modified gravity

The Science

hi_class implements Horndeski's theory in the modern Cosmic Linear Anisotropy Solving System. It can be used to compute any linear cosmological observable in seconds, including FRW distances, CMB, matter power and number count spectra. hi_class can be readily interfaced with Monte Python to test Gravity and Dark Energy models.

Horndeski is the most general scalar-tensor theory described by second-order equations of motion, and contains many well known models, including (but by no means limited to) covariant Galileons, Brans-Dicke, f(R), chameleons, k-essence and quintesssence. hi_class relies on a reformulation of the Effective Field Theory for Dark Energy developed by E. Bellini and I. Sawicki (see JCAP 1407 (2014) 050).

The publicly available version (hi_class v2.0) is presented and described in:

hi_class has been used to obtain results in a number of publications, including

See the full list below (if your article is not listed, please contact us).

The Code

hi_class computes the cosmological predictions of alternative theories of gravity. The code solves the linear equations starting deep in the radiation era, and can compute any cosmological observable, including (but not limited to) cosmological distances, the matter power spectrum, Cosmic Microwave Background temperature and polarization, as well as their correlation with the matter distribution. The publicly available version incorporates parameterized models based on the Effective Field Theory of Dark Energy.

hi_class has been tested for a range of models against several codes in a dedicated paper A comparison of Einstein-Boltzmann solvers for testing General Relativity . The codes validated include EFTCAMB , COOP and the Galileon code developed by Barreira et al. (based on CAMB). The results agree within 0.1% for CMB and matter power spectra, as good as for base CLASS/CAMB using default precision parameters (at low multipoles the agreement is within 0.5%, well within cosmic variance). The achieved precision is sufficient for tests of gravity with next-generation surveys.


hi_class is freely available to the scientific community. If you use it in a publication/preprint please cite at least the original CLASS paper and

The code can be cloned from the GitHub repository or downloaded as a compressed file. To get started and find detailed information on the available models and code functionality please read the hi_class.ini file.



The Team

hi_class is being developed by

We are very grateful to Thomas Tram for his invaluable advice and the many users who have offered suggestions, found bugs and contributed to improve the code.


If you are interested in using a beta version or for other inquiries about hi_class please contact emilio - bellini -- physics.ox.ac.uk or miguelzuma -- berkeley.edu

Publications using hi_class

hi_class has been used to obtain results in the following publications:

  1. Nonlinear evolution of the BAO scale in alternative theories of gravity
    E. Bellini, M. Zumalacarregui PRD92 (2015) 063522
  2. Constraints on deviations from LCDM within Horndeski gravity
    E. Bellini, A. Cuesta, R. Jimenez, L. Verde JCAP 1602 (2016) 053
  3. Gravity at the horizon: on relativistic effects, CMB-LSS correlations and ultra-large scales in Horndeski's theory J. Renk, M. Zumalacarregui, F. Montanari JCAP 1607 (2016) 040
  4. hi_class: Horndeski in the Cosmic Linear Anisotropy Solving System
    M. Zumalacarregui, E. Bellini, I. Sawicki, J. Lesgourgues, P. Ferreira
    JCAP 1708 (2017) 019
  5. The Observational Future of Cosmological Scalar-Tensor Theories
    D. Alonso, E. Bellini, P. G. Ferreira, M. Zumalacarregui PRD95 (2017) 063502
  6. Galileon Gravity in Light of ISW, CMB, BAO and H0 data
    J. Renk, M. Zumalacarregui, F. Montanari, A. Barreira JCAP 1710 (2017) 020
  7. A comparison of Einstein-Boltzmann solvers for testing General Relativity
    E. Bellini et al. PRD97 (2018) 023520
  8. The impact of relativistic effects on cosmological parameter estimation
    C. Lorenz, D. Alonso, P. Ferreira PRD97 (2018) 023537
  9. Dark Energy after GW170817: Dead Ends and the Road Ahead
    J. M. Ezquiaga, M. Zumalacarregui Phys.Rev.Lett. 119 251304 (see Physics Viewpoint )
  10. Testing (modified) gravity with 3D and tomographic cosmic shear A. Spurio Mancini, R. Reischke, V. Pettorino, B.M. Schaefer, M. Zumalacarregui MNRAS 480 (2018) 3725
  11. Dark energy from α-attractors: phenomenology and observational constraints C. García-García, E. Linder, P. Ruíz-Lapuente, M. Zumalacárregui JCAP 1808 (2018) 022
  12. Investigating scalar-tensor-gravity with statistics of the cosmic large-scale structure R. Reischke, A. Spurio Mancini, B. Malte Schäfer, P. Merkel 1804.02441
  13. Testing Horndeski gravity as dark matter with hi_class A. Casalino, M. Rinaldi 1807.01995
  14. Dark Energy in light of Multi-Messenger Gravitational-Wave astronomy JM Ezquiaga, M. Zumalacárregui 1807.09241
  15. Gravity's Islands: Parametrizing Horndeski Stability M. Denissenya, E. Linder 1808.00013
  16. No Slip CMB M. Brush, E. Linder, M. Zumalacarregui 1810.12337
  17. Radiative stability and observational constraints on dark energy and modified gravity J. Noller, A. Nicola 1811.03082
  18. Cosmological parameter constraints for Horndeski scalar-tensor gravity J. Noller, A. Nicola 1811.12928
  19. KiDS+GAMA: Constraints on Horndeski gravity from combined large-scale structure probes A Spurio Mancini et al. 1901.03686
  20. The phenomenology of beyond Horndeski gravity D. Traykova, E. Bellini, P. Ferreira 1902.10687
  21. Positivity in the sky S. Melville J. Noller 1904.05874
  22. Designing Horndeski and the effective fluid approach R. Arjona, W. Cardona, S. Nesseris 1904.06294
  23. Modified Gravity Away from a ΛCDM Background G. Brando et al. 1904.12903
  24. Alpha-attractor dark energy in view of next-generation cosmological surveys C. Garcia-Garcia et al. 1905.03753
  25. Dark sector evolution in Horndeski models F. Pace et al. 1905.06795
  26. The Shape Dependence of Vainshtein Screening in the Cosmic Matter Bispectrum C. Burrage, J. Dombrowski, D. Saadeh 1905.06260
  27. Testing modified gravity at cosmological distances with LISA standard sirens LISA Cosmology Working Group 1906.01593
  28. hi_class: Background Evolution, Initial Conditions and Approximation Schemes
    E. Bellini, I. Sawicki, M. Zumalacarregui
  29. Theoretical priors in scalar-tensor cosmologies: Thawing quintessence C. Garcia-Garcia, E. Bellini, P. Ferreira, D. Traykova, M. Zumalacarregui 1911.02868
  30. Cosmological constraints on dark energy in light of gravitational wave bounds J. Noller 2001.05469
  31. Gravity in the Era of Equality: Towards solutions to the Hubble problem without fine-tuned initial conditions M. Zumalacarregui 2003.06396
  32. The H0 tension: ΔG_N vs. ΔN_eff G. Ballesteros, A. Notari, F. Rompineve 2004.05049
  33. A larger value for H0 by an evolving gravitational constant Matteo Braglia et al. 2004.11161
  34. Improvements in cosmological constraints from breaking growth degeneracy L. Perenon, S. Ilic, R. Maartens, A. de la Cruz-Dombriz 2005.00418
  35. Information entropy in cosmological inference problems A. Pinho, R. Reischke, M. Teich, B. Malte Schäfer 2005.02035
  36. Cross-bispectra Constraints on Modified Gravity Theories from Nancy Grace Roman Space Telescope and Rubin Observatory Legacy Survey of Space and Time C. Heinrich, O. Doré 2006.03138
  37. Relativistic Corrections to the Growth of Structure in Modified Gravity G. Brando, K. Koyama, D. Wands 2006.11019
  38. Constraining Scalar-Tensor Modified Gravity with Gravitational Waves and Large Scale Structure Surveys T. Baker, I. Harrison 2007.13791
  39. Scalar-tensor cosmologies without screening J. Noller, L. Santoni, E. Trincherini, L. Trombetta 2008.08649
  40. Early modified gravity in light of the $H_0$ tension and LSS data M. Braglia, M. Ballardini, F. Finelli and K. Koyama 2011.12934
  41. Can Conformally Invariant Modified Gravity Solve The Hubble Tension? T. Abadi and E. Kovetz 2011.13853
  42. Positivity Bounds on Dark Energy: When Matter Matters C. de Rham, S. Melville and J. Noller 2103.06855
  43. Testing modified (Horndeski) gravity by combining intrinsic galaxy alignments with cosmic shear R. Reischke, V. Bosca, T. Tugendhat, B. Malte Schäfer 2103.01657
  44. Theoretical priors in scalar-tensor cosmologies: Shift-symmetric Horndeski models D. Traykova, E. Bellini, P. G. Ferreira, C. García-García, J. Noller, M. Zumalacárregui 2103.11195
  45. Fully relativistic predictions in Horndeski gravity from standard Newtonian N-body simulations G. Brando, K. Koyama, D. Wands, M. Zumalacárregui, I. Sawicki, E. Bellini 2105.04491
  46. On tachyonic stability priors for dark energy R. Gsponer, J. Noller 2107.01044
  47. Positivity bounds from multiple vacua and their cosmological consequences S. Melville, J. Noller 2202.01222
  48. Enabling matter power spectrum emulation in beyond-ΛCDM cosmologies with COLA G. Brando, B. Fiorini, K. Koyama, H. Winther 2203.11120
  49. Neutrino mass and kinetic gravity braiding degeneracies G. Garcia-Arroyo, J. L. Cervantes-Cota, U. Nucamendi 2205.05755
  50. A Forecast for Large Scale Structure Constraints on Horndeski Gravity with Line Intensity Mapping B. Scott, K. Karkare, S. Bird, 2209.13029
  51. Testing gravity with gravitational wave friction and gravitational slip I. Matos, E. Bellini, M. Calvão, M. Kunz, 2210.12174
  52. Revisiting Vainshtein Screening for fast N-body simulations G. Brando, K. Koyama, H. Winther, 2303.09549
  53. Probing Dark Energy and Modifications of Gravity with Ground-Based Millimeter-Wavelength Line Intensity Mapping A. Moradinezhad Dizgah, E. Bellini, G. Keating, 2304.08471
  54. Machine learning unveils the linear matter power spectrum of modified gravity J. B. Orjuela-Quintana, S. Nesseris, D. Sapone, 2307.03643
  55. Probing Early Modification of Gravity with Planck, ACT and SPT G. F. Abellán, M. Braglia, M. Ballardini, F. Finelli, V. Poulin, 2308.12345
  56. Constraining dark energy with the integrated Sachs-Wolfe effect E. Seraille, J. Noller, B. Sherwin, 2401.06221
  57. A simple prediction of the non-linear matter power spectrum in Brans-Dicke gravity from linear theory H. Sletmoen, Hans A. Winther, 2403.03786
  58. Modified gravity interpretation of the evolving dark energy in light of DESI data A. Chudaykin, M. Kunz, 2407.02558