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.

Download

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.

Resources

ascl:1808.010

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.

Contact

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. Initial conditions for the Galileon dark energy C. Germani 1609.06598
  6. The Observational Future of Cosmological Scalar-Tensor Theories
    D. Alonso, E. Bellini, P. G. Ferreira, M. Zumalacarregui PRD95 (2017) 063502
  7. Early Cosmology Constrained L. Verde, E. Bellini, C. Pigozzo, A. Heavens, R. Jimenez 1611.00376
  8. Hiding neutrino mass in modified gravity cosmologies N. Bellomo, E. Bellini, B. Hu, R. Jimenez, C. Pena-Garay 1612.02598
  9. Galileon Gravity in Light of ISW, CMB, BAO and H0 data
    J. Renk, M. Zumalacarregui, F. Montanari, A. Barreira JCAP 1710 (2017) 020
  10. A comparison of Einstein-Boltzmann solvers for testing General Relativity
    E. Bellini et al. PRD97 (2018) 023520
  11. The impact of relativistic effects on cosmological parameter estimation
    C. Lorenz, D. Alonso, P. Ferreira PRD97 (2018) 023537
  12. Dark Energy after GW170817: Dead Ends and the Road Ahead
    J. M. Ezquiaga, M. Zumalacarregui Phys.Rev.Lett. 119 251304 (see Physics Viewpoint )
  13. 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
  14. Dark energy from α-attractors: phenomenology and observational constraints C. García-García, E. Linder, P. Ruíz-Lapuente, M. Zumalacárregui JCAP 1808 (2018) 022
  15. Investigating scalar-tensor-gravity with statistics of the cosmic large-scale structure R. Reischke, A. Spurio Mancini, B. Malte Schäfer, P. Merkel 1804.02441
  16. Testing Horndeski gravity as dark matter with hi_class A. Casalino, M. Rinaldi 1807.01995
  17. Dark Energy in light of Multi-Messenger Gravitational-Wave astronomy JM Ezquiaga, M. Zumalacárregui 1807.09241
  18. Gravity's Islands: Parametrizing Horndeski Stability M. Denissenya, E. Linder 1808.00013
  19. Inflation and Early Dark Energy with a Stage II Hydrogen Intensity Mapping experiment Cosmic Visions 21 cm Collaboration 1810.09572
  20. No Slip CMB M. Brush, E. Linder, M. Zumalacarregui 1810.12337
  21. Radiative stability and observational constraints on dark energy and modified gravity J. Noller, A. Nicola 1811.03082
  22. Cosmological parameter constraints for Horndeski scalar-tensor gravity J. Noller, A. Nicola 1811.12928
  23. KiDS+GAMA: Constraints on Horndeski gravity from combined large-scale structure probes A Spurio Mancini et al. 1901.03686
  24. The phenomenology of beyond Horndeski gravity D. Traykova, E. Bellini, P. Ferreira 1902.10687
  25. Positivity in the sky S. Melville J. Noller 1904.05874
  26. Designing Horndeski and the effective fluid approach R. Arjona, W. Cardona, S. Nesseris 1904.06294
  27. Modified Gravity Away from a ΛCDM Background G. Brando et al. 1904.12903
  28. Alpha-attractor dark energy in view of next-generation cosmological surveys C. Garcia-Garcia et al. 1905.03753
  29. The Shape Dependence of Vainshtein Screening in the Cosmic Matter Bispectrum C. Burrage, J. Dombrowski, D. Saadeh 1905.06260
  30. Dark sector evolution in Horndeski models F. Pace et al. 1905.06795
  31. Testing modified gravity at cosmological distances with LISA standard sirens LISA Cosmology Working Group 1906.01593
  32. hi_class: Background Evolution, Initial Conditions and Approximation Schemes
    E. Bellini, I. Sawicki, M. Zumalacarregui
    1909.01828
  33. Theoretical priors in scalar-tensor cosmologies: Thawing quintessence C. Garcia-Garcia, E. Bellini, P. Ferreira, D. Traykova, M. Zumalacarregui 1911.02868
  34. Cosmological constraints on dark energy in light of gravitational wave bounds J. Noller 2001.05469
  35. Gravity in the Era of Equality: Towards solutions to the Hubble problem without fine-tuned initial conditions M. Zumalacarregui 2003.06396
  36. The H0 tension: ΔG_N vs. ΔN_eff G. Ballesteros, A. Notari, F. Rompineve 2004.05049
  37. A larger value for H0 by an evolving gravitational constant Matteo Braglia et al. 2004.11161
  38. Improvements in cosmological constraints from breaking growth degeneracy L. Perenon, S. Ilic, R. Maartens, A. de la Cruz-Dombriz 2005.00418
  39. Information entropy in cosmological inference problems A. Pinho, R. Reischke, M. Teich, B. Malte Schäfer 2005.02035
  40. 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
  41. Relativistic Corrections to the Growth of Structure in Modified Gravity G. Brando, K. Koyama, D. Wands 2006.11019
  42. Constraining Scalar-Tensor Modified Gravity with Gravitational Waves and Large Scale Structure Surveys T. Baker, I. Harrison 2007.13791
  43. Scalar-tensor cosmologies without screening J. Noller, L. Santoni, E. Trincherini, L. Trombetta 2008.08649
  44. Early modified gravity in light of the $H_0$ tension and LSS data M. Braglia, M. Ballardini, F. Finelli and K. Koyama 2011.12934
  45. Can Conformally Invariant Modified Gravity Solve The Hubble Tension? T. Abadi and E. Kovetz 2011.13853
  46. Testing modified (Horndeski) gravity by combining intrinsic galaxy alignments with cosmic shear R. Reischke, V. Bosca, T. Tugendhat, B. Malte Schäfer 2103.01657
  47. Positivity Bounds on Dark Energy: When Matter Matters C. de Rham, S. Melville and J. Noller 2103.06855
  48. 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
  49. 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
  50. On tachyonic stability priors for dark energy R. Gsponer, J. Noller 2107.01044
  51. Positivity bounds from multiple vacua and their cosmological consequences S. Melville, J. Noller 2202.01222
  52. Enabling matter power spectrum emulation in beyond-ΛCDM cosmologies with COLA G. Brando, B. Fiorini, K. Koyama, H. Winther 2203.11120
  53. Neutrino mass and kinetic gravity braiding degeneracies G. Garcia-Arroyo, J. L. Cervantes-Cota, U. Nucamendi 2205.05755
  54. A Forecast for Large Scale Structure Constraints on Horndeski Gravity with Line Intensity Mapping B. Scott, K. Karkare, S. Bird, 2209.13029
  55. Testing gravity with gravitational wave friction and gravitational slip I. Matos, E. Bellini, M. Calvão, M. Kunz, 2210.12174
  56. The Effective Fluid Approach for Modified Gravity and Its Applications S. Nesseris, 2212.12768
  57. Testing gravity on cosmological scales: theoretical predictions with the COLA method B. Fiorini (PhD thesis), 2303.07121
  58. Revisiting Vainshtein Screening for fast N-body simulations G. Brando, K. Koyama, H. Winther, 2303.09549
  59. Probing Dark Energy and Modifications of Gravity with Ground-Based Millimeter-Wavelength Line Intensity Mapping A. Moradinezhad Dizgah, E. Bellini, G. Keating, 2304.08471
  60. Euclid: Constraining linearly scale-independent modifications of gravity with the spectroscopic and photometric primary probes Euclid Collaboration: N. Frusciante, F. Pace, V. F. Cardone et al. 2306.12368
  61. Machine learning unveils the linear matter power spectrum of modified gravity J. B. Orjuela-Quintana, S. Nesseris, D. Sapone, 2307.03643
  62. Probing Early Modification of Gravity with Planck, ACT and SPT G. F. Abellán, M. Braglia, M. Ballardini, F. Finelli, V. Poulin, 2308.12345
  63. Constraining dark energy with the integrated Sachs-Wolfe effect E. Seraille, J. Noller, B. Sherwin, 2401.06221
  64. A simple prediction of the non-linear matter power spectrum in Brans-Dicke gravity from linear theory H. Sletmoen, H. Winther, 2403.03786
  65. Modified gravity interpretation of the evolving dark energy in light of DESI data A. Chudaykin, M. Kunz, 2407.02558
  66. mochi_class: Modelling Optimisation to Compute Horndeski In class M. Cataneo, E. Bellini, 2407.11968
  67. Interpretable and physics-informed emulator for the linear matter power spectrum from machine learning J. B. Orjuela-Quintana, D. Sapone, S. Nesseris 2407.16640
  68. Scant evidence for thawing quintessence W. Wolf, C. García-García, D. Bartlett, P. Ferreira 2408.17318
  69. Constraints on dark energy and modified gravity from the BOSS Full-Shape and DESI BAO data P. Taule, M. Marinucci, G. Biselli, M. Pietroni, F. Vernizzi 2409.08971
  70. Fast radio bursts as a probe of gravity on cosmological scales D. Neumann, R. Reischke, S. Hagstotz, H. Hildebrandt 2409.11163
  71. Testing gravity with the full-shape galaxy power spectrum: First constraints on scale-dependent modified gravity A. Aviles 2409.15640
  72. Matching current observational constraints with nonminimally coupled dark energy W. Wolf, P. Ferreira, C. Garcia-Garcia 2409.17019
  73. Modified gravity constraints from the full shape modeling of clustering measurements from DESI 2024 DESI Collaboration: M. Ishak, J. Pan, R. Calderon et al. 2411.12026
  74. Exploring cosmological imprints of phantom crossing with dynamical dark energy in Horndeski gravity Y. Tiwari, U. Upadhyay, R. Jain 2412.00931
  75. Robustness of Dark Energy Phenomenology Across Different Parameterizations W. Wolf, C. Garcia-Garcia, P. Ferreira 2502.04929
  76. Preference for evolving dark energy in light of the galaxy bispectrum Z. Lu, T. Simon, P. Zhang 2503.04602
  77. Modified gravity constraints with Planck ISW-lensing bispectrum A. Chudaykin, M. Kunz, J. Carron 2503.09893
  78. The Cosmological Evidence for Non-Minimal Coupling W. Wolf, C. Garcia-Garcia, T. Anton, P. Ferreira 2504.07679
  79. Nonlinear dynamics in Horndeski gravity: a renormalized approach to effective gravitational coupling L. Amendola, C. Bernal, R. Gannouji 2505.03999
  80. Euclid preparation. Constraining parameterised models of modifications of gravity with the spectroscopic and photometric primary probes Euclid Collaboration: I. S. Albuquerque, N. Frusciante, Z. Sakr et al. 2506.03008
  81. Monodromic Dark Energy and DESI S. Goldstein, M. Celoria, F. Schmidt 2507.16970
  82. Dark energy constraints in light of theoretical priors N. Shah, K. Koyama, J. Noller 2507.19450
  83. Cosmological constraints on Galileon dark energy with broken shift symmetry W. Wolf, P. Ferreira, C. Garcia-Garcia 2509.17586
  84. From Dark Radiation to Dark Energy: Unified Cosmological Evolution in K-essence Models E. Moreno, J. De-Santiago 2509.25088
  85. KGB-evolution: a relativistic N-body code for kinetic gravity braiding models A. Nouri-Zonoz, F. Hassani, E. Bellini, M. Kunz 2511.04676
  86. A Dynamical Scalar Field Model for Dark Energy: Addressing the Hubble Tension and Cosmic Evolution A. Kottur, J. Mahajan, R. Dabhade 2511.10317
  87. New multiprobe analysis of modified gravity and evolving dark energy Z. Lu, T. Simon 2511.10616
  88. Clustering in dynamical dark energy: observational constraints from DESI, CMB, and supernovae Y. Yang, Q. Wang, X. Ren, E. Saridakis, Y. Cai 2511.22478
  89. Euclid preparation. Review of forecast constraints on dark energy and modified gravity Euclid Collaboration: N. Frusciante, M. Martinelli et al. 2512.09748
  90. KiDS-Legacy: Constraints on Horndeski gravity from weak lensing combined with galaxy clustering and CMB B. Stölzner, R. Reischke, M. Grasso, M. Cataneo, B. Joachimi et al. 2512.11039
  91. Non-parametric exploration of minimally coupled gravity with phantom crossing M. Cataneo, K. Koyama 2512.13691
  92. Abundance of cosmic voids in EFT of dark energy T. Takadera, S. Hirano, T. Kobayashi 2512.20171
  93. On the Difficulties with Late-Time Solutions for the Hubble Tension P. Bansal, D. Huterer 2602.06293
  94. H-EFTCAMB: A Cobaya-Integrated, Python-Wrapped Extension of EFTCAMB for Covariant Horndeski Gravity G. Ye et al 2603.01662
  95. PySCo-EFT and ECOSMOG-EFT: a tandem of N-body simulation codes for the Effective Field Theory of Dark Energy H. Ganjoo, Y. Rasera, E. Bellini, M.-A. Breton et al. 2604.15434
  96. cloelib: A Flexible Python Library for Computing Cosmological Observables in the Euclid Era M. Bonici, G. Canas-Herrera, P. Carrilho et al. 2605.23839
  97. Dark energy perturbations and the robustness of cosmological neutrino-mass constraints Y.-H. Yang, E. N. Saridakis, Y.-F. Cai, H.-R. Yu 2606.28074
  98. Constraining dark energy with complementary probes of large-scale structure N. Shah, K. Koyama, J. Noller, L. Yi 2606.31977
  99. The Status of Single Scalar Field Dark Energy C. García-García, P. G. Ferreira, W. J. Wolf 2607.07777