Physics: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]
[math]\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] }[/math]
with the Lagrangian densities
[math]\displaystyle{ \mathcal{L}_{2} = G_{2}(\phi,\, X) }[/math]
[math]\displaystyle{ \mathcal{L}_{3} = G_{3}(\phi,\,X)\Box\phi }[/math]
[math]\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] }[/math]
[math]\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] }[/math]
Here [math]\displaystyle{ G_N }[/math] is Newton's constant, [math]\displaystyle{ \mathcal{L}_m }[/math] represents the matter Lagrangian, [math]\displaystyle{ G_2 }[/math] to [math]\displaystyle{ G_5 }[/math] are generic functions of [math]\displaystyle{ \phi }[/math] and [math]\displaystyle{ X }[/math] , [math]\displaystyle{ R,G_{\mu\nu} }[/math] are the Ricci scalar and Einstein tensor, [math]\displaystyle{ g_{\mu\nu} }[/math] is the Jordan frame metric, semicolon indicates covariant derivatives, commas indicate partial derivatives, [math]\displaystyle{ \Box\phi \equiv g^{\mu\nu}\phi_{;\mu\nu} }[/math] ,[math]\displaystyle{ X\equiv -1/2g^{\mu\nu}\phi_{;\mu}\phi_{;\nu} }[/math] and repeated indices are summed over following Einstein's convention.
Constraints on parameters
Many of the free parameters of the theory have been constrained, [math]\displaystyle{ \mathcal{L}_{1} }[/math] from the coupling of the scalar field to the top field and [math]\displaystyle{ \mathcal{L}_{2} }[/math] via coupling to jets down to low coupling values with proton collisions at the ATLAS experiment.[5] [math]\displaystyle{ \mathcal{L}_{4} }[/math] and [math]\displaystyle{ \mathcal{L}_{5} }[/math], are strongly constrained by the direct measurement of the speed of gravitational waves following GW170817.[6][7][8][9][10][11]
See also
- Classical theories of gravitation
- General relativity
- Brans–Dicke theory
- Dual graviton
- Massive gravity
- Alternatives to general relativity
References
- ↑ Horndeski, Gregory Walter (1974-09-01). "Second-order scalar-tensor field equations in a four-dimensional space" (in en). International Journal of Theoretical Physics 10 (6): 363–384. doi:10.1007/BF01807638. ISSN 0020-7748. Bibcode: 1974IJTP...10..363H.
- ↑ Clifton, Timothy; Ferreira, Pedro G.; Padilla, Antonio; Skordis, Constantinos (March 2012). "Modified Gravity and Cosmology". Physics Reports 513 (1–3): 1–189. doi:10.1016/j.physrep.2012.01.001. Bibcode: 2012PhR...513....1C.
- ↑ Deffayet, C.; Esposito-Farese, G.; Vikman, A. (2009-04-03). "Covariant Galileon". Physical Review D 79 (8): 084003. doi:10.1103/PhysRevD.79.084003. ISSN 1550-7998. Bibcode: 2009PhRvD..79h4003D.
- ↑ 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. doi:10.1143/PTP.126.511. ISSN 0033-068X. Bibcode: 2011PThPh.126..511K.
- ↑ ATLAS Collaboration (2019-03-04). "Constraints on mediator-based dark matter and scalar dark energy models using [math]\displaystyle{ \sqrt{s}=13 }[/math] TeV [math]\displaystyle{ pp }[/math] collision data collected by the ATLAS detector". Jhep 05: 142. doi:10.1007/JHEP05(2019)142.
- ↑ Lombriser, Lucas; Taylor, Andy (2016-03-16). "Breaking a Dark Degeneracy with Gravitational Waves". Journal of Cosmology and Astroparticle Physics 2016 (3): 031. doi:10.1088/1475-7516/2016/03/031. ISSN 1475-7516. Bibcode: 2016JCAP...03..031L.
- ↑ 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. doi:10.1103/PhysRevD.95.084029. ISSN 2470-0010. Bibcode: 2017PhRvD..95h4029B.
- ↑ Creminelli, Paolo; Vernizzi, Filippo (2017-10-16). "Dark Energy after GW170817". Physical Review Letters 119 (25): 251302. doi:10.1103/PhysRevLett.119.251302. PMID 29303308. Bibcode: 2017PhRvL.119y1302C.
- ↑ 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. doi:10.1103/PhysRevLett.119.251303. PMID 29303345. Bibcode: 2017PhRvL.119y1303S.
- ↑ Ezquiaga, Jose María; Zumalacárregui, Miguel (2017-12-18). "Dark Energy After GW170817: Dead Ends and the Road Ahead". Physical Review Letters 119 (25): 251304. doi:10.1103/PhysRevLett.119.251304. PMID 29303304. Bibcode: 2017PhRvL.119y1304E.
- ↑ Grossman, Lisa (2017-10-24). "What detecting gravitational waves means for the expansion of the universe" (in en). Science News. https://www.sciencenews.org/article/what-detecting-gravitational-waves-means-expansion-universe.
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