Minimal theory of quasidilaton massive gravity

We introduce a quasidilaton scalar field to the minimal theory of massive gravity with the Minkowski fiducial metric, in such a way that the quasidilaton global symmetry is maintained and that the theory admits a stable self-accelerating de Sitter solution. We start with a precursor theory that contains three propagating gravitational degrees of freedom without a quasidilaton scalar and introduce Stückelberg fields to covariantize its action. This makes it possible for us to formulate the quasidilaton global symmetry that mixes the Stückelberg fields and the quasidilaton scalar field. By the Hamiltonian analysis we confirm that the precursor theory with the quasidilaton scalar contains 4 degrees of freedom, three from the precursor massive gravity and one from the quasidilaton scalar. We further remove one propagating degree of freedom to construct the minimal quasidilaton theory with three propagating degrees of freedom, corresponding to two polarizations of gravitational waves from the minimal theory of massive gravity and one scalar from the quasidilaton field, by carefully introducing two additional constraints to the system in the Hamiltonian language. Switching to the Lagrangian language, we find self-accelerating de Sitter solutions in the minimal quasidilaton theory and analyze their stability. It is found that the self-accelerating de Sitter solution is stable in a wide range of parameters.

Figure 1

Four examples of allowed regions in the parameter space of the theory, for the values $\alpha =-7$, $\alpha =-2$, $\alpha =0$, and $\alpha =2$. The shaded areas denote the regions in the $(\omega ,r)$ plane for which both ${\nu }_{dS}^2>0$ and ${\mu }_{dS}^2>0$, under the condition $c_{dS}^2=1$ and $c_4=0$. Lifting one or both of these two conditions would only enlarge the allowed region. Full lines correspond to either one of the masses becoming zero and are thus included in the allowed region. Dashed lines are not permitted. In particular, the parameter space has the following restrictions: $0<\omega <6$ due to a no-ghost condition and positivity of the effective gravitational constant, $r>0$ from the positivity of the lapse function and the scale factor. The dotted line at $r=1$ corresponds to the Minkowski limit.