SO(3) massive gravity

In this paper, we propose a massive gravity theory with 5 degrees of freedom. The mass term is constructed by 3 Stuckelberg scalar fields, which respects SO(3) symmetry in the fields' configuration. By the analysis on the linear cosmological perturbations, we found that such 5 d.o.f are free from ghost instability, gradiant instability, and tachyonic instability.

Introduction. The search for a consistent theory of finite range gravity is a longstanding and well motivated problem. Whether there exists such a consistent extension of general relativity (GR) by a mass term is a basic question of classical field theory. After Fierz and Pauli's pioneering work in 1939 [1], this question has been attracting a great deal of interest. However, its consistency has been a challenging problem for several decades.
In Fierz and Pauli's model, the GR is extended by a linear mass term. However, such simplest massive gravity model gives rise to a discontinuity in the observables [2] [3]. This problem can be alleviated by nonlinear terms [4]. However, since the lack of Hamiltonian constraint and momentum constraint, it ends up with six d.o.f in the gravity sector. The poincare symmetry in the 3+1 space time implies that a massive spin-2 particle should only contain 5 helicities modes. The rest sixth mode is the so-called Boulware-Deser (BD) ghost [5], spoiling the stability of the theory.
Only recently, a non-linear massive gravity theory (which is dubbed as dRGT gravity) has just been found [6] [7], where the BD ghost is removed by construction in the decoupling limit. It was shown that Hamiltonian constraint and the associated secondary constraint are restored in this theory. As a result, away from decoupling limit, this theory is also free from BD ghost [8]. However, the following up cosmological perturbations analysis revealed a new ghost instability among the rest five d.o.f [9][10] [11]. On the other hand, this theory may also suffer from the conceptional problem of acausality [12].
By the lesson of dRGT gravity, we learn that if we start from the framwork of breaking the 4 space-time diffeomorphism invariance, it is very hard to get a healthy 5 d.o.f massive spin-2 theory. Inspired by this point, in this paper, we propose a massive gravity thoery by only breaking the 3 spatial diffeomorphism invariance, and keep the time reparameterization invariant, This theory can also be considered as one of the subcategories of Lorentz violation massive gravity, which was briefly discussed in [13]. By taking eq.(1) as our start-  [18]). By adopting the Stückelberg trick, we introduce 3 scalars, which respect residual SO(3) symmetry in the fields' configuration, to recover the general covariance. This paper is organized as follows: Firstly we write down a general action based on the time reparameterization invariance and residual SO(3) symmetry in the scalar fields' configuration. Then we apply our theory to cosmology. The linear cosmological perturbation analysis reveils 5 healthy d.o.f on the perturbation sepctrum.
Setup Taking the time reparameterization invariance and residual SO(3) symmetry as our building principle, a general action with a masss term can be written as Please notice that G µν is the Einstein tensor and G µν f µν is the so called Horndeski term [19]. The equation of motion is still second derivative. On the other hand, as for the M 2 p m 2 2 (...) part, in principle we can add an infinitely polynomial series inside of round bracket. However, for the simplicity of calculations, we truncate the higher order term, just consider a constant term, a linear term and two quadratic terms in this paper. The cosmology in the presence of matter content will be discussed in our future work [21].
In the unitary gauge, where α is an un-normalized quantity, which can be absorbed into the redefinetion of coefficients c 1 , c 2 , d 2 .
Without introducing any ambiguity, we set α = 1 and f µν = (0, 1, 1, 1). Assume we start from a FRW background, with physical metric, The horizontal axes denotes f ≡ g µν fµν , the blue curve denotes the mass term with c1(3c2 + d2) < 0, the green curve denotes the mass term with c1(3c2 + d2) > 0, and the red line, which overlaps with the horizontal axes, denotes an Einstein static universe which exhibit the infinitely strong coupling. Please notice f is positive by definition.
and f = 3a −2 in this case. It is important to notice that for the parameter region c 1 (3c 2 + d 2 ) < 0, when f approaches to the bottom of the potential, one gets an Einstein static universe, provided a proper cosmological constant(see FIG.1). The helicity 0 mode and helicity 1 mode exhibit the infinitely strong coupling in such Einstein static universe, thus it is not our interest (the detail will be present in [21]). The condition c 1 (3c 2 + d 2 ) > 0 is required to avoid such problem. Cosmological solution We choose the FRW ansatz, and the space-time metric can be written as eq.(4). By taking the variation of the action with respect to the metric g µν , we get such two background Einstein equations, where . In addition to a bare cosmological constant, we can see our mass term contributes a curvature-like term, and a radiationlike term. Noting that since the the SO(3) symmetry in the fields' configuration, the constraint equations of 3 Stückelberg scalars φ a are trivially satisfied. Scalar perturbations We perturb the space-time metric and define the scalar perturbations by Here we choose the unitary gauge, where φ a = x a . After integrating out the non-dynamical degree, the quadratic action of scalar perturbation is where The full expression of M s is quite bulky, and we are not going to show it here.
To check if the scalar mode is ghosty, let's substitute eq.(5) into the above formula, and then take the super horizon approximation, we get The scalar mode is ghost free at super horizon scale as long as a 2 ) is positive. To see the situation in the small scale, let's take the sub-horizon approximation, we get At late time epoch where a → ∞, K s > 0 requires and for the early stage where a ≪ 1, K s > 0 requires In order to check if our theory is free from gradiant instability and tachyonic instability, we define a new canonical variable where κ is defined in the following eq.(18) and eq. (20). The canonical normalized action can be rewritten in terms of this canonical variable as Under the super horizon approximation, at leading order we have, There is no tachyonic instability outside of horizon if m 2 2 (4c 1 + 3r 1 c 0 ) > 0. During late time epoch, under the sub horizon approximation, at the leading order we have We can see that during the late time epoch, at leading order the sound speed of scalar mode at subhorizon scale is 1. Although we start from an action break the Lorentz invariance, Lorentz violation effect doesn't show up at the leading oder of our calculation. Vector perturbations By perfoming the similar approach to vector perturbation, we can also check that the vector mode is also healthy under the same ghost free conditon. Firstly, let's define the vector perturbations of the metric as, where the vector perturbations satisfy the transverse condition, After integrating out the non-dynamical degree, we get the quadratic action of the vector mode as follows, We are not going to show the full expression of M v since it is too bulky. At super horizon scale, we have Again, similar to the scalar case, r 2 > 0 ensures that the kinetic term is positive. At subhorizon scale, we have By requiring that the vector mode is ghost free at two oppsite limit a → ∞ and a ≪ 1, we get exactly the same ghost free condition as in eq.(14) (15). In order to check the gradiant instability and tachyonic instability, we write down the canonical normalized action for vector perturbations, where at leading order, and The result is quite similar to the scalar perturbations.
Tensor perturbations Tensor perturbations on the metric can be defined as where the transverse condition and traceless condition are satisfied, The quadratic action of the tensor perturbations reads where Different from GR, the dispersion relation of tensor mode is modified by an effective mass term M 2 GW . To see the