Abstract
We reveal the existence of a certain hidden symmetry in general ghost-free scalar-tensor theories which can only be seen when generalizing the geometry of the spacetime from Riemannian. For this purpose, we study scalar-tensor theories in the metric-affine (Palatini) formalism of gravity, which we call scalar-metric-affine theories for short, where the metric and the connection are independent. We show that the projective symmetry, a local symmetry under a shift of the connection, can provide a ghost-free structure of scalar-metric-affine theories. The ghostly sector of the second-order derivative of the scalar is absorbed into the projective gauge mode when the unitary gauge can be imposed. Incidentally, the connection does not have the kinetic term in these theories and then it is just an auxiliary field. We can thus (at least in principle) integrate the connection out and obtain a form of scalar-tensor theories in the Riemannian geometry. The projective symmetry then hides in the ghost-free scalar-tensor theories. As an explicit example, we show the relationship between the quadratic order scalar-metric-affine theory and the quadratic U-degenerate theory. The explicit correspondence between the metric-affine (Palatini) formalism and the metric one could be also useful for analyzing phenomenology such as inflation.
Highlights
In the last decade, a great deal of attention has been paid to scalar-tensor theories including second-order derivatives of a scalar field in the Lagrangian for building models of inflation and dark energy
We reveal the existence of a certain hidden symmetry in general ghost-free scalar-tensor theories which can only be seen when generalizing the geometry of the spacetime from Riemannian
We discussed the possibility that the Ostrogradsky ghost-free property of scalar-tensor theories is guaranteed by symmetry and show that the projective symmetry could be an important ingredient for ghost-free theories
Summary
A great deal of attention has been paid to scalar-tensor theories including second-order derivatives of a scalar field in the Lagrangian for building models of inflation and dark energy. After the discovery of the Gleyzes-Langlois-Piazza-Vernizzi (GLPV) theory [8,9], it is recognized that the assumption of keeping the second-order equation of motion is too strong for general Ostrogradsky ghost-free theories. This led to the idea of degeneracy, in which a constraint is imposed onto the Lagrangian and eliminates the Ostrogradsky ghost. More general scalar-metric-affine theories with nonminimal coupling to the curvature are discussed in Sec. IV, and some ghost-free couplings are found. General Hamiltonian analysis of scalar-metric-affine theories is discussed in the Appendix
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