# Horndeski's theory

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.

## Action

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

${\displaystyle S[g_{\mu \nu },\phi ]=\int \mathrm {d} ^{4}x\,{\sqrt {-g}}\left[\sum _{i=2}^{5}{\frac {1}{8\pi G_{\text{N}}}}{\mathcal {L}}_{i}[g_{\mu \nu },\phi ]\,+{\mathcal {L}}_{\text{m}}[g_{\mu \nu },\psi _{M}]\right]}$

with the Lagrangian densities

${\displaystyle {\mathcal {L}}_{2}=G_{2}(\phi ,\,X)}$

${\displaystyle {\mathcal {L}}_{3}=G_{3}(\phi ,\,X)\Box \phi }$

${\displaystyle {\mathcal {L}}_{4}=G_{4}(\phi ,\,X)R+G_{4,X}(\phi ,\,X)\left[\left(\Box \phi \right)^{2}-\phi _{;\mu \nu }\phi ^{;\mu \nu }\right]}$

${\displaystyle {\mathcal {L}}_{5}=G_{5}(\phi ,\,X)G_{\mu \nu }\phi ^{;\mu \nu }-{\frac {1}{6}}G_{5,X}(\phi ,\,X)\left[\left(\Box \phi \right)^{3}+2{\phi _{;\mu }}^{\nu }{\phi _{;\nu }}^{\alpha }{\phi _{;\alpha }}^{\mu }-3\phi _{;\mu \nu }\phi ^{;\mu \nu }\Box \phi \right]}$

Here ${\displaystyle G_{N}}$  is Newton's constant, ${\displaystyle {\mathcal {L}}_{m}}$  represents the matter Lagrangian, ${\displaystyle G_{2}}$  to ${\displaystyle G_{5}}$  are generic functions of ${\displaystyle \phi }$  and ${\displaystyle X}$  , ${\displaystyle R,G_{\mu \nu }}$  are the Ricci scalar and Einstein tensor, ${\displaystyle g_{\mu \nu }}$  is the Jordan frame metric, semicolon indicates covariant derivatives, commas indicate partial derivatives, ${\displaystyle \Box \phi \equiv g^{\mu \nu }\phi _{;\mu \nu }}$  ,${\displaystyle X\equiv 1/2g^{\mu \nu }\phi _{;\mu }\phi _{;\nu }}$  and repeated indices are summed over following Einstein's convention.

## Constraints on parameters

Many of the free parameters of the theory have been constrained, ${\displaystyle {\mathcal {L}}_{1}}$  from the coupling of the scalar field to the top field and ${\displaystyle {\mathcal {L}}_{2}}$  via coupling to jets down to low coupling values with proton collisions at the ATLAS_experiment[5]. ${\displaystyle {\mathcal {L}}_{4}}$  and ${\displaystyle {\mathcal {L}}_{5}}$ , are strongly constrained by the direct measurement of the speed of gravitational waves following GW170817.[6][7][8][9][10][11]

## References

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.
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.
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.
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.
5. ^ ATLAS Collaboration (2019-03-04). "Constraints on mediator-based dark matter and scalar dark energy models using ${\displaystyle {\sqrt {s}}=13}$  TeV ${\displaystyle pp}$  collision data collected by the ATLAS detector". arXiv:1903.01400. Cite journal requires |journal= (help)
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.
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.
8. ^ Creminelli, Paolo; Vernizzi, Filippo (2017-10-16). "Dark Energy after GW170817". Physical Review Letters. 119 (25): 251302. arXiv:1710.05877. doi:10.1103/PhysRevLett.119.251302. PMID 29303308.
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. doi:10.1103/PhysRevLett.119.251303. PMID 29303345.
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.
11. ^ Grossman, Lisa (2017-10-24). "What detecting gravitational waves means for the expansion of the universe". Science News. Retrieved 2017-11-08.