Horndeski's theory

Horndeski's theoryis the most general theory of gravity in four dimensions whose Lagrangian is constructed out of themetric tensorand ascalar fieldand leads tosecond order equationsof motion.The theory was first proposed byGregory Horndeskiin 1974and has found numerous applications, particularly in the construction ofcosmological modelsofInflationanddark energy.Horndeski's theory contains many theories of gravity, includingGeneral relativity,Brans-Dicke theory,Quintessence,Dilaton,Chameleonand covariant Galileonas special cases.

Action

Horndeski's theory can be written in terms of anactionas

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

with theLagrangian densities

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

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

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

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

Here$G_{N}$isNewton's constant,${\mathcal {L}}_{m}$represents the matter Lagrangian,$G_{2}$to$G_{5}$are generic functions of$\phi$and$X$,$R,G_{\mu \nu }$are theRicci scalarandEinstein tensor,$g_{\mu \nu }$is theJordan framemetric, semicolon indicatescovariant derivatives,commas indicatepartial derivatives,$\Box \phi \equiv g^{\mu \nu }\phi _{;\mu \nu }$,$X\equiv -1/2g^{\mu \nu }\phi _{;\mu }\phi _{;\nu }$and repeated indices are summed over followingEinstein's convention.

Constraints on parameters

Many of the free parameters of the theory have been constrained,${\mathcal {L}}_{1}$from the coupling of the scalar field to the top field and${\mathcal {L}}_{2}$via coupling to jets down to low coupling values with proton collisions at theATLAS experiment.${\mathcal {L}}_{4}$and${\mathcal {L}}_{5}$,are strongly constrained by the direct measurement of the speed of gravitational waves followingGW170817.

See also

• Classical theories of gravitation
• General relativity
• Brans–Dicke theory
• Dual graviton
• Massive gravity
• Lovelock theory of gravity
• Alternatives to general relativity