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 particle and covariant Galileon as special cases.
Action Horndeski's theory can be written in terms of an action as
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]}
with the Lagrangian densities
L 2 = G 2 ( ϕ , 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 }
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]}
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