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.

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.

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]}

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