Consistent Extension of Quasidilaton Massive Gravity

Abstract

This paper is devoted to the Hamiltonian analysis of extension of the quasidilaton massive gravity as was proposed recently in [arXiv:1306.5502]. We show that, for given formulation of the theory, the additional primary constraint that is responsible for the elimination of the Boulware-Deser ghost is missing. We compare this situation with the quasidilaton massive gravity. Finally, we propose the ghost-free extension of quasidilaton massive gravity.

1. Introduction and Summary

Recently, new version of the full nonlinear massive gravity was found by de Rham, Gabadadze and Toley (dRGT) [1, 2] that provides the positive answer to the question of whether graviton can have a nonzero mass. In fact, among many remarkable properties, there is the crucial one which is the absence of the Boulware-Deser ghost [3, 4].

The consistent massive gravity could also provide a possible explanation of the observed acceleration of the cosmic expansion which is one of the greatest mysteries in modern cosmology. It is tempting to speculate that the finite graviton mass could be a source of the accelerated expansion of the universe. For that reason, it is of great interest to formulate theoretically consistent cosmological scenario in massive gravity that is also in agreement with the observations. Unfortunately, it was recently shown that all homogeneous and isotropic cosmological solutions in dRGT theory are unstable [5]; see also [6–8].

In order to resolve this problem, we have two possible options: either to break homogeneity [9] or isotropy [10, 11] or to extend the theory as in [12, 13]. Recently in [14], de Felice and Mukohyama proposed new extension of the quasidilaton massive gravity that could provide stable and self-accelerating homogeneous and isotropic cosmological solution. They further argued that given extension belongs to the class of models studied in [15] that are free from the Boulware-Deser ghosts. However, the explicit Hamiltonian analysis of given theory was not performed in [14].

The goal of this paper is to reconsider the problem of the Boulware-Deser ghost in the model [14]. We present an evidence that, for the action that was introduced in [14], the Boulware-Deser ghost cannot be eliminated. More precisely, performing the Hamiltonian analysis of this model with the time dependent quasidilaton, we find that the primary constraint which is responsible for the elimination of the ghost in Stückelberg formulation of nonlinear massive gravity [16–19] is missing (for related work, see [20]). This result implies that generally Boulware-Deser ghost is present. On the other hand we show that this additional constraint emerges in two cases when 𝜔≠0 and when 𝛼_𝜎=0. We also propose model of consistent extension of the quasidilaton nonlinear massive gravity that can be considered as the generalization of the coupling between massive gravity and the galileon [21]. Then, we argue that given theory is ghost-free, following [19].

This paper is organized as follows. In next Section 2, we review the basic facts about extension of the quasidilaton nonlinear massive gravity as was proposed in [14]. Then, we proceed to the Hamiltonian analysis of given theory and argue that there is no scalar primary constraint that could eliminate the Boulware-Deser ghost. We also discuss the analysis performed recently in [22] and we argue that the constraints introduced there cannot be considered as the gauge fixing constraint. In Section 3, we perform the Hamiltonian analysis of quasidilaton nonlinear massive gravity when we find that there is an additional primary constraint. This result shows that the quasidilaton massive theory as was proposed in [12] is ghost-free at least in their minimal version. Finally in Section 4, we propose an extension of the quasidilaton massive gravity that is ghost-free and that can be considered as the generalization of proposal [14]. It would be extremely interesting to analyze cosmological consequences of this theory.

2. Extension of Quasidilaton Massive Gravity

In this section, we review basic facts about extension of quasidilaton massive gravity as was proposed in [14]. For simplicity, we restrict ourselves to the minimal form of the massive gravity keeping in mind that its generalization is straightforward.

Explicitly, let us consider the following action: 𝑆=𝑆_m.g.+𝑆_𝜎,𝑆_m.g.=𝑀²_𝑝𝑑⁴𝑥−𝑔⁽⁴⁾𝑅+2𝑚²3−Ω(Φ)𝑔⁻¹𝑓, (1) where 𝑓_𝜇𝜈 was introduced in [14], 𝑓_𝜇𝜈=𝑓_𝜇𝜈−𝛼_𝜎𝑀²_𝑝𝑚²𝑒^{−2𝜎/𝑀𝑝}𝜕_𝜇𝜎𝜕_𝜈𝜎,𝑓_𝜇𝜈=𝜕_𝜇𝜙^𝑎𝜕_𝜈𝜙^𝑏𝜂_𝑎𝑏, (2) where 𝜙^𝑎,𝑎,𝑏=0,1,2,3 are Stückelberg fields and where 𝜂_𝑎𝑏=diag(−1,1,1,1). Further, 𝑆_𝜎 is defined as 𝑆_𝜎𝜔=−2𝑑⁴𝑥−𝑔𝑔^𝜇𝜈𝜕_𝜇𝜎𝜕_𝜈𝜎. (3) Note that Ω(Φ) is a function of 𝜎 which is necessary for the invariance of the theory under global symmetry 𝜎⟶𝜎+𝜎₀,𝜙^𝑎⟶𝑒^{−𝜎0/𝑀𝑝}𝜙^𝑎 (4) so that, under (4), 𝑓_𝜇𝜈 and 𝑓_𝜇𝜈 transform as 𝑓_𝜇𝜈⟶𝑒^{−2𝜎0/𝑀𝑝}𝑓_𝜇𝜈,𝑓_𝜇𝜈⟶𝑒^{−2𝜎0/𝑀𝑝}𝑓_𝜇𝜈. (5) The massive term contains square root of the expression 𝑔^𝜇𝜈𝑓_𝜈𝜌 that, under (4), transforms as 𝑔⁻¹𝑓⟶𝑒^{−𝜎0/𝑀𝑝}𝑔⁻¹𝑓 (6) which implies that Ω(𝜎) has to have the form Ω(𝜎)=𝑒^{𝜎/𝑀𝑝}. (7) Now, we are ready to proceed to the Hamiltonian analysis of given theory. Due to the presence of the square root in the action, we perform the redefinition of the shift functions [23, 24] 𝑁^𝑖=𝑀𝑛^𝑖+𝑓^𝑖𝑘𝑓_0𝑘+𝑁𝐷^𝑖_𝑗𝑛^𝑗, (8) where 𝑀²=−𝑓₀₀+𝑓_0𝑘𝑓^𝑘𝑙𝑓_𝑙0,𝑓_𝑖𝑗𝑓^𝑗𝑘=𝛿^𝑗_𝑖, (9) and where 𝐷^𝑖_𝑗 obeys the equation √𝑥𝐷^𝑖_𝑗=𝑔^𝑖𝑘−𝐷^𝑖_𝑚𝑛^𝑚𝐷^𝑘_𝑛𝑛^𝑛𝑓_𝑘𝑗,𝑥=1−𝑛^𝑖𝑓_𝑖𝑗𝑛^𝑗 (10) and also the following important identity: 𝑓_𝑖𝑘𝐷^𝑘_𝑗=𝑓_𝑗𝑘𝐷^𝑘_𝑖. (11) We also use 3 + 1 decomposition of the four-dimensional metric 𝑔_𝜇𝜈 [25, 26]: 𝑔₀₀=−𝑁²+𝑁_𝑖𝑔^𝑖𝑗𝑁_𝑗,𝑔_0𝑖=𝑁_𝑖,𝑔_𝑖𝑗=𝑔_𝑖𝑗,𝑔⁰⁰1=−𝑁²,𝑔^0𝑖=𝑁^𝑖𝑁²,𝑔^𝑖𝑗=𝑔^𝑖𝑗−𝑁^𝑖𝑁^𝑗𝑁². (12) Note that, in 3 + 1 formalism, the kinetic term for 𝜎 has the form −𝜔𝑀²_𝑝𝑔^𝜇𝜈𝜕_𝜇𝜎𝜕_𝜈𝜔𝜎=𝑀²_𝑝∇_𝑛𝜎²−𝜔𝑀²_𝑝𝜕_𝑖𝜎𝑔^𝑖𝑗𝜕_𝑗𝜎, (13) where, after redefinition (8), ∇_𝑛𝜎 has the form ∇_𝑛1𝜎=𝑁𝜕_𝑡𝜎−𝑀𝑛^𝑖+𝑓^𝑖𝑘𝑓_0𝑘+𝑁𝐷^𝑖_𝑗𝑛^𝑗𝜕_𝑖𝜎. (14) With the help of these expressions, we rewrite action (1) into the form 𝑆=𝑀²_𝑝𝑑³𝐱𝑑𝑡𝑁√𝑔𝐾_𝑖𝑗𝒢^{𝑖𝑗𝑘𝑙}𝐾_𝑘𝑙+𝑁√𝑔𝑅−√𝑔𝑀𝑈−−2𝑚²𝑁Ω(Φ)√𝑔√𝑥𝐷^𝑖_𝑖−3𝑁√𝑔++𝑁√𝑔𝜔2𝑀²_𝑝∇_𝑛𝜎²−𝑁√𝑔𝜔2𝑀²_𝑝𝜕_𝑖𝜎𝑔^𝑖𝑗𝜕_𝑗𝜎, (15) where 𝑈=2𝑚²Ω(Φ)√𝑥, (16) and where we used the 3 + 1 decomposition of the four-dimensional scalar curvature ⁽⁴⁾𝑅=𝐾_𝑖𝑗𝒢^{𝑖𝑗𝑘𝑙}𝐾𝑘𝑙+𝑅, (17) where 𝑅 is three-dimensional scalar curvature. We also introduced de Witt metric 𝒢^{𝑖𝑗𝑘𝑙}=12𝑔^𝑖𝑘𝑔^𝑗𝑙+𝑔^𝑖𝑙𝑔^𝑗𝑘−𝑔^𝑖𝑗𝑔^𝑘𝑙 (18) with inverse 𝒢_{𝑖𝑗𝑘𝑙}=12𝑔_𝑖𝑘𝑔_𝑗𝑙+𝑔_𝑖𝑙𝑔_𝑗𝑘−12𝑔_𝑖𝑗𝑔_𝑘𝑙,𝒢_{𝑖𝑗𝑘𝑙}𝒢^{𝑘𝑙𝑚𝑛}=12𝛿^𝑚_𝑖𝛿^𝑛_𝑗+𝛿^𝑛_𝑖𝛿^𝑚_𝑗. (19) Note that, in (17), we ignored the terms containing total derivatives. Finally, note that 𝐾_𝑖𝑗 is defined as 𝐾_𝑖𝑗=1𝜕2𝑁^𝑡𝑔_𝑖𝑗−∇_𝑖𝑁_𝑗𝑛,𝑔−∇_𝑗𝑁_𝑖𝑛,𝑔, (20) where 𝑁_𝑖 depends on 𝑛^𝑖 and 𝑔 through relation (8).

Now, we could proceed to the Hamiltonian formulation of given theory. However, the structure of the derivative ∇_𝑛𝜎 (14) suggests very complicated relations between momenta and velocities. For that reason, we consider simpler case when we presume that 𝜎 depends on time only. Note that this is the reasonable approximation that does not spoil the physical content of the theory. From (15), we find the momenta conjugate to 𝑁, 𝑛^𝑖, and 𝑔_𝑖𝑗: 𝜋_𝑁≈0,𝜋_𝑖≈0,𝜋^𝑖𝑗=𝑀²_𝑝√𝑔𝒢^{𝑖𝑗𝑘𝑙}𝐾_𝑘𝑙. (21) While, in case of 𝜙^𝑎 and 𝜎, we find 𝑝_𝑎=ℳ_𝑎𝑏𝜕_𝑡𝜙^𝑏𝑀𝑛^𝑖ℛ_𝑖+𝑀²_𝑝√𝑔𝑈−𝑓^𝑖𝑗𝜕_𝑗𝜙_𝑎ℛ_𝑖,𝑝_𝜎1=−𝑀𝛼_𝜎𝑀²_𝑝𝑚²𝑒^{−2𝜎/𝑀2𝑝}𝜕_𝑡𝜎𝑛^𝑖ℛ_𝑖+𝑀²_𝑝√𝑔𝑈+𝜔√𝑔𝜕_𝑡𝜎, (22) where 𝑀²=𝑀²₀+𝛼_𝜎𝑀²_𝑝𝑚²𝑒^{−2𝜎/𝑀2𝑝}𝜕_𝑡𝜎²,ℛ_𝑖=−2𝑔_𝑖𝑘∇_𝑗𝜋^𝑘𝑗,𝑀²₀=−𝜕_𝑡𝜙^𝑎ℳ_𝑎𝑏𝜕_𝑡𝜙^𝑏,ℳ_𝑎𝑏=𝜂_𝑎𝑏−𝜕_𝑖𝜙_𝑎𝑓^𝑖𝑗𝜕_𝑗𝜙_𝑏. (23) These relations imply 𝑀²₀1=−Π_𝑎ℳ^𝑎𝑏Π_𝑏+𝑛^𝑖ℛ_𝑖+𝑀²_𝑝√𝑔𝑈²Π_𝑎ℳ^𝑎𝑏Π_𝑏𝛼_𝜎𝑀²_𝑝𝑚²×𝑒^{−2𝜎/𝑀2𝑝}𝜕_𝑡𝜎², (24) where Π_𝑎=𝑝_𝑎+𝑓^𝑖𝑗𝜕_𝑗𝜙_𝑎ℛ_𝑖. Then, it is easy to find relation between momenta and velocities: 𝑝_𝜎+𝛼_𝜎𝑀²_𝑝𝑚²𝑒^{−𝜎/𝑀2𝑝}Π_𝑎ℳ^𝑎𝑏Π_𝑏+𝑛^𝑖ℛ_𝑖+𝑀²_𝑝√𝑔𝑈²=1𝑁𝜔√𝑔𝜕_𝑡𝜎×Π_𝑎Π_𝑎ℳ^𝑎𝑏Π_𝑏+𝑛^𝑖ℛ_𝑖+𝑀²_𝑝√𝑔𝑈²=𝑀_𝑝𝑚√𝛼_𝜎𝑒^{𝜎/𝑀2𝑝}ℳ_𝑎𝑏𝜕_𝑡𝜙^𝑏𝜕_𝑡𝜎. (25) It is crucial that these relations do not imply an existence of the scalar primary constraint which is in sharp contrast with the case of the dRGT massive gravity [16] or dRGT massive gravity coupled to the galileon [19]. On the other hand, using the property of the matrix ℳ_𝑎𝑏, we find three constraints: 𝜕_𝑖𝜙^𝑎Π_𝑎≡Σ_𝑖=𝜕_𝑖𝜙^𝑎𝑝_𝑎+ℛ_𝑖≈0. (26) With suitable extension of these constraints by terms proportional to the primary constraints 𝜋_𝑖≈0, we find that they are the first class constraints whose smeared forms are the generator of spatial diffeomorphism.

Now, using (25), we determine corresponding Hamiltonian 𝐻=𝑑³𝐱𝑁ℋ₀, (27) where ℋ₀=12𝜔√𝑔𝑝_𝜎+𝛼_𝜎𝑀²_𝑝𝑚²𝑒^{−𝜎/𝑀2𝑝}×Π_𝑎ℳ^𝑎𝑏Π_𝑏+𝑛^𝑖ℛ_𝑖+𝑀²_𝑝√𝑔𝑈²²+1√𝑔𝑀²_𝑝𝜋^𝑖𝑗𝒢_{𝑖𝑗𝑘𝑙}𝜋^𝑘𝑙−√𝑔⁽³⁾𝑅−2𝑚²Ω√𝑔√𝑥𝐷^𝑖_𝑖−2√𝑔+𝐷^𝑖_𝑗𝑛^𝑗ℛ_𝑖. (28) The requirement of the preservation of the constraint 𝜋_𝑁≈0 implies that ℋ₀ is a constraint as well. The succeeding analysis is straightforward. We have six first class constraints Σ_𝑖≈0, ℋ₀≈0, 𝜋_𝑁≈0 and six second class constraints 𝜋_𝑖≈0, 𝒞_𝑖≈0 where 𝒞_𝑖≈0 are the secondary constraints that arise from the requirement of the preservation of the constraints 𝜋_𝑖≈0. These constraints can be solved for 𝜋_𝑖 and 𝑛^𝑖. Further, the first class constraint 𝜋_𝑁≈0 can be gauge fixed that leads to the elimination of 𝜋_𝑁 and 𝑁 as dynamical variables. Finally, the constraints Σ_𝑖, ℋ₀ can be again gauge fixed which leads to the elimination of the Stückelberg fields and conjugate momenta. As a result, we are left with 12 degrees of freedom coming from the gravity sector that can be identified with 10 degrees of freedom corresponding to the massive graviton and two degrees of freedom corresponding to the scalar at least at the linearized approximation. Note that this scalar mode cannot be eliminated due to the absence of the scalar constraint. As a result the Boulware-Deser ghost is generally present. Finally, there are two phase space degrees of freedom 𝜎 and 𝑝_𝜎.

2.1. Remark about the Paper [22]

After the first version of this paper was published, Mukohyama argued in his paper [22] that the extension of the quasidilaton theory is ghost-free. His arguments is based on the existence of the additional constraints that he imposed on the scalar fields: 𝜙⁰=−𝑒^{−𝜎/𝑀𝑝},𝜙^𝑖=𝛿^𝑖_𝜇𝑥^𝜇,𝑖=1,2,3. (29) He called this constraint as the gauge fixing constraint and then he was able to derive the Hamiltonian in the form 𝐻=∫𝑑³𝐱𝑁𝒞₀, where the specific form of 𝒞₀ is given in [22]. According to this result, he claimed that there exists an additional constraint in the gauge fixed theory so that this constraint is responsible for the elimination of the Boulware-Deser ghost.

In this section, we reconsider (29) from different point of view and argue that this is not a gauge fixing constraint at all. We do not impose the fixing of spatial diffeomorphism and consider the following relation between 𝜙₀ and 𝜎 (note that we do not use the word “gauge fixing condition”): 𝜎=−𝑀_𝑝ln𝜙⁰ (30) that is equivalent to (29). Inserting this relation to the definition of 𝑓_𝜇𝜈, we obtain 𝑓_𝜇𝜈=𝜂_𝑎𝑏𝜕_𝜇𝜙^𝑎𝜕_𝜈𝜙^𝑏−𝛼_𝜎𝑚²_𝑔𝜕_𝜇𝜙⁰𝜕_𝜈𝜙⁰. (31) Then, inserting (31) into the dRGT massive gravity, we obtain 𝑆=𝑀²_𝑝𝑑⁴𝑥−𝑔⁽⁴⁾𝑅+2𝑚²13−𝜙⁰²𝑔⁻¹𝑓−𝜔2𝜙⁰²𝑔^𝜇𝜈𝜕_𝜇𝜙⁰𝜕_𝜈𝜙⁰. (32) Now we see that given action is manifestly diffeomorphism invariant. In other words, while we reduce the number of degrees of freedom by imposing (30), the number of gauge symmetries is the same. However we mean that the fact that we have less degrees of freedom than the original theory while the number of gauge symmetries is the same implies that these two theories have different physical content and should not be considered as equivalent. In other words, condition (29) is not a gauge fixing condition.

From given analysis, it is clear that the constraint 𝒞₀ that was identified in [22] corresponds to the Hamiltonian constraint in the theory with fixed spatial diffeomorphism. Clearly, this constraint should have vanishing Poisson bracket {𝒞₀(𝐱),𝒞₀(𝐲)} which implies that it is the first class constraint. Clearly, there is no way how to generate the additional constraint by imposing the requirement of the preservation of the constraint 𝒞₀ during the time evolution of the system since the Hamiltonian is equal to 𝐻=∫𝑑³𝐱𝑁𝒞₀. However, since 𝒞₀ is the first class constraint, it can be gauge fixed and, hence, we reduce the number of physical degrees of freedom by two. But we should again stress that the theory represented by action (33) is not equivalent to action (1) that represents the extension of the quasidilaton dRGT theory. In summary, we mean that the arguments that were presented in [22] do not prove that the extension of quasidilaton massive gravity is ghost-free.

Let us compare this situation with the imposing static gauge for the Hamiltonian (27). This gauge fixing is represented by four gauge fixed functions: 𝒢₀=𝜙⁰−𝑡≈0,𝒢_𝑖=𝜙^𝑖−𝑥^𝑖≈0. (33) These constraints together with ℋ₀, Σ_𝑖 form the second class constraints that can be explicitly solved for 𝑝_𝑎. We solve ℋ₀ for 𝑝₀ since we can identify the gauge fixed Hamiltonian with 𝑝₀=−ℋ_g.f.. The resulting Hamiltonian describes the dynamic of the physical degrees of freedom 𝑔_𝑖𝑗, 𝜋^𝑖𝑗, and 𝜎, 𝑝_𝜎. The detailed counting of the physical degrees of freedom was presented in the end of previous section and we will not repeat it here.

3. The Case 𝛼_𝜎=0

In previous section, we saw that the extension of the quasidilaton theory that was suggested in [14] suffers from the presence of Boulware-Deser ghost due to the absence of two additional scalar constraints in the Hamiltonian formulation of given theory. It is instructive to see whether these constraints emerge in the case of quasidilaton massive gravity which corresponds to the situation when 𝛼_𝜎=0. Explicitly, we consider the action [11, 12] 𝑆=𝑀²_𝑝𝑑³𝐱𝑑𝑡𝑁√𝑔𝐾_𝑖𝑗𝒢^{𝑖𝑗𝑘𝑙}𝐾_𝑘𝑙+𝑁√𝑔𝑅−√𝑔𝑀₀𝑈−2𝑚²𝑁√𝑔√𝑥Ω𝐷^𝑖_𝑖−3𝑁√𝑔+𝑁√𝑔𝜔𝑀²_𝑝∇_𝑛𝜎²−𝑁√𝑔𝜔𝑀²_𝑝𝜕_𝑖𝜎𝑔^𝑖𝑗𝜕_𝑗𝜎. (34) Let us now perform the Hamiltonian analysis of action (34). First of all, we find the canonical momenta 𝑝_𝑎=−−ℳ_𝑎𝑏1𝑀₀𝜕_𝑡𝜙^𝑏𝑛^𝑖+𝑓^𝑖𝑗𝜕_𝑗𝜙_𝑎ℛ_𝑖+1𝑀₀𝑀²_𝑝√𝑔ℳ_𝑎𝑏𝜕_𝑡𝜙^𝑏𝑈+1𝑀₀ℳ_𝑎𝑏𝜕_𝑡𝜙^𝑏𝑛^𝑖𝜕_𝑖𝜎𝑝_𝜎−𝜕_𝑖𝜎𝑓^𝑖𝑘𝜕_𝑘𝜙_𝑎𝑝_𝜎,𝑝_𝜎=𝜔√𝑔∇_𝑛𝜎, (35) so that it is easy to find the scalar primary constraint Σ_𝑝≡𝑝_𝑎+ℛ_𝑖+𝜕_𝑖𝜎𝑝_𝜎𝑓^𝑖𝑘𝜕_𝑘𝜙_𝑎𝜂^𝑎𝑏×𝑝_𝑏+ℛ_𝑖+𝜕_𝑖𝜎𝑝_𝜎𝑓^𝑖𝑘𝜕_𝑘𝜙_𝑏+𝑛^𝑖ℛ_𝑖+𝑝_𝜎𝜕_𝑖𝜎+𝑀²_𝑝√𝑔𝑈²≈0, (36) and also, using the fact that 𝜕_𝑖𝜙^𝑎ℳ_𝑎𝑏=0, we find additional three constraints: Σ_𝑖=𝑝_𝑎𝜕_𝑖𝜙^𝑎+ℛ_𝑖+𝜕_𝑖𝜎𝑝_𝜎. (37) Now, using these constraints, we can simplify (36) so that it has the form Σ_𝑝=𝑝_𝑎ℳ^𝑎𝑏𝑝_𝑏+𝑛^𝑖ℛ_𝑖+𝑝_𝜎𝜕_𝑖𝜎+𝑀²_𝑝√𝑔𝑈²≈0. (38) This has exactly the same form as the scalar constraint that emerges in case of dRGT massive gravity written in the Stückelberg formalism. The minimal form of this gravity was analyzed in [16] and this analysis can be easily applied to our case. From (15), we find the Hamiltonian with all primary constraints included 𝐻_𝐸=𝑑³𝐱𝑁𝒞₀+𝑣_𝑁𝜋_𝑁+𝑣^𝑖𝜋_𝑖+Ω_𝑝Σ_𝑝+Ω^𝑖Σ_𝑖, (39) where we introduced the constraint Σ_𝑖=Σ_𝑖+𝜕_𝑖𝑛^𝑗𝜋_𝑗+𝜕_𝑗𝑛^𝑗𝜋_𝑖 (40) and where 𝒞₀=1√𝑔𝑀²_𝑝𝜋^𝑖𝑗𝒢_{𝑖𝑗𝑘𝑙}𝜋^𝑘𝑙−𝑀²_𝑝√𝑔𝑅+2𝑚²𝑀²_𝑝√𝑔√𝑥Ω𝐷^𝑖_𝑖−6𝑚²𝑀²_𝑝√𝑔+𝐷^𝑖_𝑗𝑛^𝑗ℛ_𝑖+𝑝_𝜎𝜕_𝑖𝜎+1√𝑝𝑔𝜔^2𝜎+𝜔√𝑔𝑔^𝑖𝑗𝜕_𝑖𝜎𝜕_𝑗𝜎. (41) As the next step, we have to perform the analysis of the stability of the primary constraints 𝜋_𝑖≈0, 𝜋_𝑁≈0 and Σ_𝑝≈0. In case of the constraint 𝜋_𝑖≈0, we find 𝜕_𝑡𝜋_𝑖=𝜋_𝑖,𝐻=−ℛ_𝑗+𝑝_𝜎𝜕_𝑗𝜎−2𝑚²𝑀²_𝑝Ω√𝑔√𝑥𝑛^𝑘𝑓_𝑘𝑗×𝑁𝛿𝐷^𝑗_𝑚𝑛^𝑚𝛿𝑛^𝑖+𝛿^𝑗_𝑖𝑛^𝑖ℛ_𝑖+𝑝_𝜎𝜕_𝑖𝜎+𝑀²_𝑝√𝑔𝑈=0 (42) so that we impose the following secondary constraint 𝒞_𝑖=ℛ_𝑖+𝑝_𝜎𝜕_𝑖𝜎−2𝑚²𝑀²_𝑝Ω√𝑔√𝑥𝑓_𝑖𝑗𝑛^𝑗. (43) Now, with the help of this constraint and the constraint Σ_𝑖, we can simplify Σ_𝑝 in a similar way as in [16]: Σ_𝑝=4𝑚⁴𝑀⁴_𝑝𝑔Ω²+𝑝_𝐴𝜂^𝐴𝐵𝑝_𝐵. (44) Then, it is easy to show that {Σ_𝑝(𝐱),Σ_𝑝(𝐲)}=0 and the requirement of the preservation of given constraint leads to the emergence of an additional constraint. These constraints are responsible for the elimination of Boulware-Deser ghost; see again [16] for more details. In other words, the presence of the kinetic term for the quasidilaton that minimally couples to gravity does not spoil the property that the given theory is ghost-free.

4. Ghost-Free Extension of Quasidilaton Massive Gravity

We argued, in Section 2, that the extension of quasidilaton theory as was formulated in [14] is plagued by the presence of the Boulware-Deser ghost. On the other hand, given theory has many nice properties so that it is desirable to propose its ghost-free version. In this section, we propose such a formulation when we replace the kinetic term for 𝜎 by following tadpole galileon term: 𝑆_𝜎=−𝑇𝑑⁴𝑥ΨΦ^𝐴−det𝑓_𝜇𝜈=−𝑇𝑑⁴𝑥ΨΦ^𝐴𝑀𝑓, (45) where Φ^𝐴=(𝜙^𝑎,𝜎), 𝑓=det𝑓_𝑖𝑗 and where the function Ψ(Φ^𝐴) was chosen in such a way that action (45) is invariant under (4). We claim that the quasidilaton theory formulated as the dRGT massive theory with 𝑓_𝜇𝜈 and with the kinetic term for the galileon given by (45) is ghost-free.

To see this explicitly, it is useful to introduce the following notation. Let s write 𝑓_𝜇𝜈 as 𝑓_𝜇𝜈=𝜕_𝜇Φ^𝐴𝒢_𝐴𝐵𝜕_𝜈Φ^𝐵, (46) where we introduced the metric 𝒢_𝐴𝐵: 𝒢_𝐴𝐵=𝜂_𝐴𝐵0𝛼0−^𝜎𝑀²_𝑝𝑚²𝑒^{−2𝜎/𝑀𝑝}. (47) We see that our proposal has the form of the galileon coupled to dRGT massive gravity [21] whose Hamiltonian analysis was performed in [19]. On the other hand, the action defined by (45) is more complicated since the metric 𝜂_𝐴𝐵 is replaced with the more general metric 𝒢_𝐴𝐵 that depends on Φ^𝐴 and there are also additional scalar fields Ω(𝜙^𝐴), Ψ(Φ^𝐴). However, we can expect that this fact will not modify the constraint structure of given theory.

To see this explicitly, let us briefly review the Hamiltonian analysis of the nonlinear massive gravity with the term (45) keeping in mind that more detailed analysis can be found in [19]. As usual, the momenta conjugate to 𝑁, 𝑛^𝑖, and 𝑔_𝑖𝑗 are 𝜋_𝑁≈0,𝜋_𝑖≈0,𝜋^𝑖𝑗=𝑀²_𝑝√𝑔𝒢^{𝑖𝑗𝑘𝑙}𝐾_𝑘𝑙 (48) while the momentum conjugate to Φ^𝐴 has the form 𝑝_𝐴=−𝛿𝑀𝛿𝜕_𝑡Φ^𝐴𝑛^𝑖+𝒢_𝐴𝐵𝑓^𝑖𝑗𝜕_𝑗Φ^𝐵ℛ_𝑖−𝑀²_𝑝√𝑔𝛿𝑀𝜕_𝑡Φ^𝐴𝑈^′, (49) where 𝑈^′=2𝑚²Ω(Φ)√𝑇𝑥+𝑀²_𝑝Ψ(Φ)𝑓 (50) and where 𝑀² has the form 𝑀²=−𝜕_𝑡Φ^𝐴ℳ_𝐴𝐵𝜕_𝑡Φ^𝐵,ℳ_𝐴𝐵=𝒢_𝐴𝐵−𝒢_𝐴𝐶𝜕_𝑖Φ^𝐶𝑓^𝑖𝑗𝜕_𝑗Φ^𝐷𝒢_𝐷𝐵. (51) Note that the matrix ℳ_𝐴𝐵 obeys following relations: ℳ_𝐴𝐵𝒢^𝐵𝐶ℳ_𝐶𝐷=ℳ_𝐴𝐷,𝜕_𝑖Φ^𝐴ℳ_𝐴𝐵=0. (52) Then, it is easy to determine the following primary constraints: Σ_𝑝=𝑛^𝑖ℛ_𝑖+𝑀²_𝑝√𝑔𝑈^′2+𝑝_𝐴+ℛ_𝑖𝑓^𝑖𝑗𝒢_𝐴𝐶𝜕_𝑗Φ^𝐶𝒢^𝐴𝐵×𝑝_𝐵+ℛ_𝑖𝑓^𝑖𝑗𝒢_𝐵𝐷𝜕_𝑗Φ^𝐷𝜕≈0,_𝑖Φ^𝐴𝑝_𝐴+ℛ_𝑖=Σ_𝑖≈0. (53) Now, we are ready to write the extended Hamiltonian which includes all the primary constraints: 𝐻_𝐸=𝑑³𝐱𝑁𝒞₀+𝑣_𝑁𝜋_𝑁+𝑣^𝑖𝜋_𝑖+Ω_𝑝Σ_𝑝+Ω^𝑖Σ_𝑖, (54) where 𝒞₀=1√𝑔𝑀²_𝑝𝜋^𝑖𝑗𝒢_{𝑖𝑗𝑘𝑙}𝜋^𝑘𝑙−𝑀²_𝑝√𝑔𝑅+2𝑚²𝑀²_𝑝√𝑔Ω(Φ)√𝑥𝐷^𝑖_𝑖−6𝑚²𝑀²_𝑝√𝑔+𝐷^𝑖_𝑗𝑛^𝑗ℛ_𝑖 (55) and where we introduced the constraints Σ_𝑖 defined as Σ_𝑖=Σ_𝑖+𝜕_𝑖𝑛^𝑖𝜋_𝑖+𝜕_𝑗𝑛^𝑗𝜋_𝑖. (56)

To proceed further, we have to check the stability of all constraints. The procedure is the same as in [19] so that we find that Σ_𝑖 are the first class constraints while the requirement of the preservation of the constraints 𝜋_𝑖≈0 implies following secondary constraints [23, 24]: 𝒞_𝑖≡ℛ_𝑖−2𝑚²𝑀²_𝑝Ω(Φ)√𝑔√𝑥𝑓_𝑖𝑗𝑛^𝑗≈0. (57) Further, the requirement of the preservation of the constraint 𝜋_𝑁≈0 implies an existence of the secondary constraint 𝒞₀≈0. Using the constraints 𝒞_𝑖 and Σ_𝑖, we replace the constraint Σ_𝑝 by new independent constraint Σ_𝑝: Σ_𝑝=4𝑚⁴𝑀⁴_𝑝Ω²𝑔+𝑝_𝐴𝒢^𝐴𝐵𝑝_𝐵+2𝑇Ψ𝑓×𝑝_𝐴𝜕_𝑖𝜙^𝐴𝑓^𝑖𝑗𝜕_𝑗𝜙^𝐵𝑝_𝐵+4𝑚⁴𝑀⁴_𝑝Ω²𝑔+𝑇²Ψ²𝑓=0. (58) Then, the total Hamiltonian with all constraints included has the form 𝐻_𝑇=𝑑³𝐱𝑁𝒞₀+𝑣_𝑁𝜋_𝑁+𝑣^𝑖𝜋_𝑖+Ω_𝑝Σ_𝑝+Ω^𝑖Σ_𝑖+Γ^𝑖𝒞_𝑖. (59) Now, we are ready to analyze the stability of all constraints that appear in (59). Again, the analysis is the same as in [19] with slight complication that now there are additional terms Ψ(Φ), Ω(Φ) together with 𝒢_𝐴𝐵(Φ^𝐴) in the definition of the action. However, these terms are local functions of Φ^𝐴 so that they do not affect the result that {Σ_𝑝(𝐱),Σ_𝑝(𝐲)}≈0. As a result, the requirement of the preservation of the constraint Σ_𝑝≈0 implies new constraint Σ_𝑝^𝐼𝐼≈0. These two constraints are the second class constraints that can be used for the elimination of the Boulware-Deser ghost mode and its conjugate momenta.

In this section, we proposed an extension of the quasidilaton massive gravity that is ghost-free. This proposal can be generalized in different ways, either consider the most general potential of the dRGT massive gravity or more complicated kinetic term for 𝜎. It would be also very interesting to analyze the cosmological consequences of the model with action (45) following [14].

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was supported by the Grant agency of the Czech republic under the Grant P201/12/G028.