Degenerate Higher-Order Scalar-Tensor theories

DHOST theories were introduced in 2015 by David Langlois and Karim Noui.[3][4] They are a generalisation of Beyond Horndeski (or GLPV) theories, which are themselves a generalisation of Horndeski theories. The equations of motion of Horndeski theories contain only two derivatives of the metric and the scalar field, and it was believed that only equations of motion of this form would not contain an extra scalar degree of freedom (which would lead to unwanted ghosts).[5] However, it was first shown that a class of theories now named Beyond Horndeski also avoided the extra degree of freedom. Originally theories which were quadratic in the second derivative of the scalar field were studied, but DHOST theories up to cubic order have now been studied.[5]

All DHOST theories depend on a scalar field ${\displaystyle \phi }$. The general action of DHOST theories is given by[5]

${\displaystyle {\begin{aligned}S[g,\phi ]=\int d^{4}x{\sqrt {-g}}\left[f_{0}(X,\phi )+f_{1}(X,\phi )\square \phi +f_{2}(X,\phi )R+C_{(2)}^{\mu \nu \rho \sigma }\phi _{\mu \nu }\phi _{\rho \sigma }+\right.f_{3}(X,\phi )G_{\mu \nu }\phi ^{\mu \nu }+C_{(3)}^{\mu \nu \rho \sigma \alpha \beta }\phi _{\mu \nu }\phi _{\rho \sigma }\phi _{\alpha \beta }],\end{aligned}}}$

where ${\displaystyle X}$ is the kinetic energy of the scalar field, ${\displaystyle \phi _{\mu \nu }=\nabla _{\mu }\nabla _{\nu }\phi }$, and the quadratic terms in ${\displaystyle \phi _{\mu \nu }}$ are given by

${\displaystyle C_{(2)}^{\mu \nu \rho \sigma }\phi _{\mu \nu }\phi _{\rho \sigma }=\sum _{A=1}^{5}a_{A}(X,\phi )L_{A}^{(2)},}$

where

${\displaystyle {\begin{array}{l}{L_{1}^{(2)}=\phi _{\mu \nu }\phi ^{\mu \nu },\quad L_{2}^{(2)}=(\square \phi )^{2},\quad L_{3}^{(2)}=(\square \phi )\phi ^{\mu }\phi _{\mu \nu }\phi ^{\nu }},\quad {L_{4}^{(2)}=\phi ^{\mu }\phi _{\mu \rho }\phi ^{\rho \nu }\phi _{\nu },\quad L_{5}^{(2)}=\left(\phi ^{\mu }\phi _{\mu \nu }\phi ^{\nu }\right)^{2}},\end{array}}}$

and the cubic terms are given by

${\displaystyle C_{(3)}^{\mu \nu \rho \sigma \alpha \beta }\phi _{\mu \nu }\phi _{\rho \sigma }\phi _{\alpha \beta }=\sum _{A=1}^{10}b_{A}(X,\phi )L_{A}^{(3)},}$

where

${\displaystyle {\begin{array}{l}{L_{1}^{(3)}=(\square \phi )^{3},\quad L_{2}^{(3)}=(\square \phi )\phi _{\mu \nu }\phi ^{\mu \nu },\quad L_{3}^{(3)}=\phi _{\mu \nu }\phi ^{\nu \rho }\phi _{\rho }^{\mu },\quad L_{4}^{(3)}=(\square \phi )^{2}\phi _{\mu }\phi ^{\mu \nu }\phi _{\nu }},\\{L_{5}^{(3)}=\square \phi \phi _{\mu }\phi ^{\mu \nu }\phi _{\nu \rho }\phi ^{\rho },\quad L_{6}^{(3)}=\phi _{\mu \nu }\phi ^{\mu \nu }\phi _{\rho }\phi ^{\rho \sigma }\phi _{\sigma },\quad L_{7}^{(3)}=\phi _{\mu }\phi ^{\mu \nu }\phi _{\nu \rho }\phi ^{\rho \sigma }\phi _{\sigma }},\\{L_{8}^{(3)}=\phi _{\mu }\phi ^{\mu \nu }\phi _{\nu \rho }\phi ^{\rho }\phi _{\sigma }\phi ^{\sigma \lambda }\phi _{\lambda },\quad L_{9}^{(3)}=\square \phi \left(\phi _{\mu }\phi ^{\mu \nu }\phi _{\nu }\right)^{2},\quad L_{10}^{(3)}=\left(\phi _{\mu }\phi ^{\mu \nu }\phi _{\nu }\right)^{3}}.\end{array}}}$

The ${\displaystyle a_{A}}$ and ${\displaystyle b_{A}}$ are arbitrary functions of ${\displaystyle \phi }$ and ${\displaystyle X}$.