Horndeski's theory

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Horndeski's theory is the most general theory of gravity in four dimensions whose Lagrangian is constructed out of the metric tensor and a scalar field and leads to second order equations of motion.[clarification needed] The theory was first proposed by Gregory Horndeski in 1974[1] and has found numerous applications, particularly in the construction of cosmological models of Inflation and dark energy.[2] Horndeski's theory contains many theories of gravity, including General relativity, Brans-Dicke theory, Quintessence, Dilaton, Chameleon and covariant Galileon[3] as special cases.


Horndeski's theory can be written in terms of an action as[4]

with the Lagrangian densities

Here is Newton's constant, represents the matter Lagrangian, to are generic functions of and , are the Ricci scalar and Einstein tensor, is the Jordan frame metric, semicolon indicates covariant derivatives, commas indicate partial derivatives, , and repeated indices are summed over following Einstein's convention.

Constraints on parameters[edit]

Many of the free parameters of the theory have been constrained, from the coupling of the scalar field to the top field and via coupling to jets down to low coupling values with proton collisions at the ATLAS experiment.[5] and , are strongly constrained by the direct measurement of the speed of gravitational waves following GW170817.[6][7][8][9][10][11]

See also[edit]


  1. ^ Horndeski, Gregory Walter (1974-09-01). "Second-order scalar-tensor field equations in a four-dimensional space". International Journal of Theoretical Physics. 10 (6): 363–384. Bibcode:1974IJTP...10..363H. doi:10.1007/BF01807638. ISSN 0020-7748. S2CID 122346086.
  2. ^ Clifton, Timothy; Ferreira, Pedro G.; Padilla, Antonio; Skordis, Constantinos (March 2012). "Modified Gravity and Cosmology". Physics Reports. 513 (1–3): 1–189. arXiv:1106.2476. Bibcode:2012PhR...513....1C. doi:10.1016/j.physrep.2012.01.001. S2CID 119258154.
  3. ^ Deffayet, C.; Esposito-Farese, G.; Vikman, A. (2009-04-03). "Covariant Galileon". Physical Review D. 79 (8): 084003. arXiv:0901.1314. Bibcode:2009PhRvD..79h4003D. doi:10.1103/PhysRevD.79.084003. ISSN 1550-7998. S2CID 118855364.
  4. ^ Kobayashi, Tsutomu; Yamaguchi, Masahide; Yokoyama, Jun'ichi (2011-09-01). "Generalized G-inflation: Inflation with the most general second-order field equations". Progress of Theoretical Physics. 126 (3): 511–529. arXiv:1105.5723. Bibcode:2011PThPh.126..511K. doi:10.1143/PTP.126.511. ISSN 0033-068X. S2CID 118587117.
  5. ^ ATLAS Collaboration (2019-03-04). "Constraints on mediator-based dark matter and scalar dark energy models using TeV collision data collected by the ATLAS detector". Jhep. 05: 142. arXiv:1903.01400. doi:10.1007/JHEP05(2019)142. S2CID 119182921.
  6. ^ Lombriser, Lucas; Taylor, Andy (2016-03-16). "Breaking a Dark Degeneracy with Gravitational Waves". Journal of Cosmology and Astroparticle Physics. 2016 (3): 031. arXiv:1509.08458. Bibcode:2016JCAP...03..031L. doi:10.1088/1475-7516/2016/03/031. ISSN 1475-7516. S2CID 73517974.
  7. ^ Bettoni, Dario; Ezquiaga, Jose María; Hinterbichler, Kurt; Zumalacárregui, Miguel (2017-04-14). "Speed of Gravitational Waves and the Fate of Scalar-Tensor Gravity". Physical Review D. 95 (8): 084029. arXiv:1608.01982. Bibcode:2017PhRvD..95h4029B. doi:10.1103/PhysRevD.95.084029. ISSN 2470-0010. S2CID 119186001.
  8. ^ Creminelli, Paolo; Vernizzi, Filippo (2017-10-16). "Dark Energy after GW170817". Physical Review Letters. 119 (25): 251302. arXiv:1710.05877. Bibcode:2017PhRvL.119y1302C. doi:10.1103/PhysRevLett.119.251302. PMID 29303308. S2CID 206304918.
  9. ^ Sakstein, Jeremy; Jain, Bhuvnesh (2017-10-16). "Implications of the Neutron Star Merger GW170817 for Cosmological Scalar-Tensor Theories". Physical Review Letters. 119 (25): 251303. arXiv:1710.05893. Bibcode:2017PhRvL.119y1303S. doi:10.1103/PhysRevLett.119.251303. PMID 29303345. S2CID 39068360.
  10. ^ Ezquiaga, Jose María; Zumalacárregui, Miguel (2017-12-18). "Dark Energy After GW170817: Dead Ends and the Road Ahead". Physical Review Letters. 119 (25): 251304. arXiv:1710.05901. Bibcode:2017PhRvL.119y1304E. doi:10.1103/PhysRevLett.119.251304. PMID 29303304. S2CID 38618360.
  11. ^ Grossman, Lisa (2017-10-24). "What detecting gravitational waves means for the expansion of the universe". Science News. Retrieved 2017-11-08.