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. The theory was first proposed by Gregory Horndeski in 1974 and has found numerous applications, particularly in the construction of cosmological models of Inflation and dark energy. Horndeski's theory contains many theories of gravity, including General relativity, Brans-Dicke theory, Quintessence, Dilaton, Chameleon and covariant Galileon as special cases.

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

${\displaystyle }$

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

with the Lagrangian densities

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

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

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

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

Here $G_{N}$ is Newton'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 the Ricci and Einstein tensors, $g_{\mu \nu }$ is the Jordan frame metric, semicolon indicates covariant derivatives, commas indicate partial derivatives, $\Box \phi \equiv g^{\mu \nu }\phi _{;\mu \nu }$ ,$X\equiv g^{\mu \nu }\phi _{;\mu }\phi _{;\nu }$ and repeated indices are summed over following Einstein's convention.

The free parameters of the theory, especially the contributions from ${\mathcal {L}}_{4}$ and ${\mathcal {L}}_{5}$, are strongly constrained by the direct measurement of the speed of gravitational waves following GW170817.

See also

• Classical theories of gravitation
• General relativity
• Brans–Dicke theory