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. 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 particle and covariant Galileon[3] as special cases.

Action

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

S [ g μ ν , ϕ ] = ∫ d 4 x − g [ ∑ i = 2 5 1 8 π G N L i [ g μ ν , ϕ ] + L m [ g μ ν , ψ M ] ] {\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]} {\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

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

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

L 4 = G 4 ( ϕ , X ) R + G 4 , X ( ϕ , X ) [ ( ◻ ϕ ) 2 − ϕ ; μ ν ϕ ; μ ν ] {\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}}_{4}=G_{4}(\phi ,\,X)R+G_{4,X}(\phi ,\,X)\left[\left(\Box \phi \right)^{2}-\phi _{;\mu \nu }\phi ^{;\mu \nu }\right]}

L 5 = G 5 ( ϕ , X ) G μ ν ϕ ; μ ν − 1 6 G 5 , X ( ϕ , X ) [ ( ◻ ϕ ) 3 + 2 ϕ ; μ ν ϕ ; ν α ϕ ; α μ − 3 ϕ ; μ ν ϕ ; μ ν ◻ ϕ ] {\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]} {\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 G N {\displaystyle G_{N}} {\displaystyle G_{N}} is Newton's constant, L m {\displaystyle {\mathcal {L}}_{m}} {\displaystyle {\mathcal {L}}_{m}} represents the matter Lagrangian, G 2 {\displaystyle G_{2}} {\displaystyle G_{2}} to G 5 {\displaystyle G_{5}} {\displaystyle G_{5}} are generic functions of ϕ {\displaystyle \phi } {\displaystyle \phi } and X {\displaystyle X} {\displaystyle X} , R , G μ ν {\displaystyle R,G_{\mu \nu }} {\displaystyle R,G_{\mu \nu }} are the Ricci scalar and Einstein tensor, g μ ν {\displaystyle g_{\mu \nu }} {\displaystyle g_{\mu \nu }} is the Jordan frame metric, semicolon indicates covariant derivatives, commas indicate partial derivatives, ◻ ϕ ≡ g μ ν ϕ ; μ ν {\displaystyle \Box \phi \equiv g^{\mu \nu }\phi _{;\mu \nu }} {\displaystyle \Box \phi \equiv g^{\mu \nu }\phi _{;\mu \nu }} , X ≡ − 1 / 2 g μ ν ϕ ; μ ϕ ; ν {\displaystyle X\equiv -1/2g^{\mu \nu }\phi _{;\mu }\phi _{;\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, L 1 {\displaystyle {\mathcal {L}}_{1}} {\displaystyle {\mathcal {L}}_{1}} from the coupling of the scalar field to the top field and L 2 {\displaystyle {\mathcal {L}}_{2}} {\displaystyle {\mathcal {L}}_{2}} via coupling to jets down to low coupling values with proton collisions at the ATLAS experiment.[5] L 4 {\displaystyle {\mathcal {L}}_{4}} {\displaystyle {\mathcal {L}}_{4}} and L 5 {\displaystyle {\mathcal {L}}_{5}} {\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]

See also

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. 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 s = 13 {\displaystyle {\sqrt {s}}=13} {\displaystyle {\sqrt {s}}=13} TeV p p {\displaystyle pp} {\displaystyle pp} 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.