Hamiltonian analysis of Mimetic gravity with higher derivatives

Two kinds of mimetic gravity model with higher derivatives of the mimetic field are analyzed in the Hamiltonian formalism. We first perform the Hamiltonian analysis for the mimetic gravity with a general higher derivative function and show the degrees of freedom (DOFs) is 3 which is consistent with the previous result of the Hamiltonian analysis at the perturbation level. We then perform the Hamiltonian analysis for the extended mimetic gravity with higher derivatives directly couples to the Ricci scalar in both Einstein frame and Jordan frame, and we show that different from our previous research at the cosmological perturbation level where only 3 propagating DOFs show up, this generalized mimetic model in general has 4 DOFs. To solve this discrepancy, we find out that the DOFs is reduced to 3 in the unitary gauge while the extra mode is eliminated by appropriate boundary conditions (homogeneous scalar field profile). What makes the system so special is that the Dirac matrix becomes singular in the special unitary gauge, generating extra secondary constraints and reducing the number of DOFs, so we give a similar but simpler example to illustrate how gauge choice affects the number of secondary constraints and the DOFs when the rank of the Dirac matrix is gauge dependent.


Introduction
Standard cosmology based on dark energy and dark matter is very successful so far. Despite its observational success, the origins of dark matter and dark energy are still puzzles in modern cosmology and particle physics, and a number of scenarios including modifying gravity have developed.
Recently a novel interesting model dubbed mimetic dark matter has been proposed [1] as a modification of general relativity, where the physics metric is related to a scalar field and an auxiliary metric via g µν = (g αβ φ α φ β )g µν , (1.1) where φ α ≡ ∇ α φ denotes the covariant derivative of the scalar field with respect to spacetime. This transformation separates the conformal mode of gravity in the covariant manner. The resulting gravitational equations by varying the usual Einstein-Hilbert action plus matter sector, which are constructed from the physical metric, contains the usual Einstein equation plus the extra contribution of the mimetic field which can mimics the cold dark matter. One can see the kinetic term of scalar field is subject to the constraint Actually the number of degrees of freedom remains unchanged under a general invertible disformal transformation [2], one may wonder how does the new DOF arise in mimetic scenario. It has been shown that mimetic scenario can be viewed as a singular limit of general disformal transformation and therefore a new DOF φ arises in this setup [3][4][5].
Even being offered a potential, there is no nontrival dynamics for scalar perturbation, i.e. the propagation velocity is zero c s = 0. This may rise to caustic singularities. Besides, the notion of quantum fluctuations is lost as there is no propagating degree of freedom for the scalar perturbation. Hence, when applied to the early universe, such model fails to produce the primordial perturbations which seeds the formation of large scale structure. To remedy these issues, higher derivative terms (2φ) 2 are introduced in [7] to promote the scalar degree of freedom to be dynamical with a non-zero sound speed. Although the equation for the scalar perturbation has the wave-like form by choosing appropriate coefficient, the analysis in the action formalism shows that the mimetic scenario with higher derivatives always suffer from ghost instability or gradient instability [48]. Actually, the mimetic model with higher derivative terms can be produced as a certain limit of the projective version of the Horava-Lifshitz gravity and such instability has already been pointed out [49]. It has been shown that simply generalizing the quadratic higher derivative terms to arbitrary function f (2φ) [50] or introducing the non-minimal coupling of mimetic field to the Ricci scalar f (φ)R [51] can not cure this pathology. To find a way out of the ghost and gradient instabilities, in [51] we show that it is possible to circumvent both the ghost and gradient instabilities by introducing the direct couplings of the higher derivatives of the mimetic field to the curvature. Similar couplings are also proposed in [52,53]. The extended action in our previous work [51] has the form From the reduced quadratic action of the perturbations, one scalar and two tensor modes are obtained, and we showed it is indeed possible to avoid all the instabilities. It seems that we have achieved the goal to construct a healthy model without any instabilities. However, since the action (1.4) contains the direct coupling between the higher derivative terms of mimetic field and the curvature, one might be concerned whether the model has 3 DOFs exactly. Besides, the modified dispersion relation [54] (involving k 4 term) of scalar perturbation may imply the existence of extra DOF which do not show up at the perturbation level with cosmological background. The aim of this paper is to identify the number of DOFs for an extended mimetic model (3.1) which is slightly different from (1.4). As we shall see, generally such kind of theories has 4 DOFs, of which 3 are propagating and one is non-propagating and will be eliminated in the unitary gauge.
The paper is organized as follows. In the next section, we perform the full Hamiltonian analysis for the mimetic model with a general higher derivative function and show the DOFs is 3 which is consistent with the previous result of the Hamiltonian analysis at perturbation level in [50]. In section 3, the full Hamiltonian analysis for the extended mimetic gravity with higher derivatives directly couples to the Ricci scalar is performed in both Einstein frame and Jordan frame, and we find 4 DOFs in general. To clarify the confusion why only 3 DOFs show up at the cosmological perturbation level, we also perform the Hamiltonian analysis in the unitary gauge where only 3 DOFs appear. Finally, we give a simple example where the rank of the Dirac matrix is gauge dependent in section 4 followed by conclusion and discussions in section 5. A special case of mimetic gravity with higher derivative terms is discussed in the Appendix.

Mimetic gravity with higher derivative terms
We start from the following action of mimetic theory where R is the Ricci scalar, λ is the Lagrange multiplier enforcing the mimetic constraint (1.2), g(2φ) is the general higher derivative function and we have considered the non-minimal coupling of the mimetic field to the curvature. This model can be viewed as a generalization of the model in [50], and is slightly different from the model considered in [51] which includes terms φ µν φ µν . Recently, the detecion of the gravitational wave event GW170817 [55] has provided strict constraints on the sound speed of gravitational waves c t , which has to be equal to the light speed c=1, up to very high accuracy |c 2 t /c 2 − 1| 5 × 10 −16 . As one can see from the quadratic action of perturbation in [51], the inclusion of terms φ µν φ µν will change the sound speed of gravitational waves and leads to the deviation from the light speed, thus the φ µν φ µν terms will not be considered in this paper. The main goal of this section is to identify the number of DOFs of the theory (2.1). Introducing a new variable ϕ = 2φ, one can rewrite the action as where the Lagrange multiplier Λ in the last term fixes ϕ. To get rid of the appearance of higher derivatives of the mimetic field in the action, we simplify the action and drop the boundary term One can switch the action of Jordan frame to the Einstein frame by weyl scaling g µν = Ω 2ḡ µν where Ω 2 = f (φ) −1 . The final action in the Einstein frame is To identify the number of DOFs in this model, We shall perform the full Hamiltonian analysis. Although the Hamiltonian analysis of this model in the case of f (φ) = 1 has been studied at the perturbation level [50], there may be extra DOF not showing up at the perturbation level with cosmological FRW background and thereby the analysis of the general non-perturbation theory is necessary.

Hamiltonian analysis: Einstein frame
We use bar to distinguish the variables in the Einstein frame from the ones in the Jordan frame. However, all the bars over the variables have been omitted in this subsection for briefness. Under ADM decomposition the action becomes where R denotes the 3-dimensional Ricci scalar and K ij = (ḣ ij − N i|j − N j|i )/2N is the extrinsic curvature. In the following, we perform a Hamiltonian analysis of the theory (2.1). There are 14 coordinate variables Q a = {N, N i , h ij , φ, λ, ϕ, Λ} and for each coordinate variable Q a , define the conjugate momentum as π a = ∂L ∂Qa . As the coordinates N, N i , ϕ and λ have no time derivative in the action, this leads to six primary constraints Other conjugate momentums are Following the standard route , we obtain the total Hamiltonian where and Imposing the conservation of the primary constraints, enables us to determine six corresponding secondary constraints [56,57]. Using Eq. (2.8) together with the primary constraints in (2.6), we find where the weak equality sign "≈" denotes an identity up to terms that vanish on the constraint surface. By employing the constraint equation Φ 6 , one can express ϕ in term of Λ. The conservation of constraint Φ 6 determines the Lagrange multiplier v ϕ and so the chain of constraints for primary constraint Φ 5 determinates here. Writing the constraints in smeared form we have To recognise that H i is indeed the diffeomorphism constraint, we can verify the following Poisson bracket where A is a scalar quantity such as φ, h ij φ i Λ j and so on, A i is a covariant vector quantity such as φ i , and Π can be the conjugate momentum quantities such as π φ or scalar densities with wight 1 like √ hA. We assume that A, A i , π a in the above equations only depend on φ, θ, Λ, λ, h ij and their conjugate momentums (without N, N i dependence). Therefore the Poisson bracket of any constraints Φ (without N, N i dependence) with D[N i ] vanish after imposing the constraint equation, i.e. D[N i ] or H i is first class. This property greatly simplify the subsequent process of calculating the secondary constraints.
The following functional derivatives of H[N ] will be useful to derive the time evolution of variables including constraints (2.14) The time evolution of constraint Φ 2 is given bẏ where the new constraint (2.16) can be derived. The next consistency condition generates another new constraint (2.17) By requiring the conservation of the constraint Φ 4 , the Lagrange multiplier v λ is determined in terms of other variables and so the chain of constraints for the primary constraint Φ 1 determinates here. Note that H ≈ 0 and H i ≈ 0 are expected to correspond to the Hamiltonian and momentum constraints respectively. With some manipulation the following Poisson brackets are found to be the usual ones (2.18) We emphasize here that H is not first-class, but one can construct a new Hamiltonian con-straintH [58] as a linear combination of H, π λ and π ϕ such that (up to boundary term) where the Lagrange multipliers v ϕ and v λ are solved in terms of other variables by requiring all the above consistency conditions. One can easily see that v ϕ and v λ are linearly dependent on lapse function N or its derivative, therefore N is not involved inH, thus the new Hamiltonian constraintH is first-class. Besides, it is natural to expect 8 first-class constraint due to the diffeomorphism invariance of the starting theory. The time evolution ofH and H i do not yield any new constraints and the chain of constraints for primary constraints π N and π i determinate here.
To sum up, The above considerations show that there are 14 constraints: These constraints reduce the dimension of phace space and thus the physical DOFs of the model (2.1) are which is consistent with the Hamiltonian analysis in [50] and [46]. Besides, there exists a very special case in the general theory (2.1). This special case can be found by requiring that in (2.17) Φ 4 doesn't contain λ in the unitary gauge, i.e.
which gives f (φ) = 1 and g(2φ) = 1 3 (2φ) 2 by taking account of the constraint equation Φ 6 . As the independence of Φ 4 on λ in the unitary gauge will lead to more secondary constraints than in the general gauge, thus less DOFs show up in the unitary gauge than in this special case. More discussion about this special case f (φ) = 1 and g(2φ) = 1 3 (2φ) 2 can be found in the Appendix.

Mimetic gravity with higher derivative terms couples to the curvature
In this section, we shall consider the following extended action of mimetic theory with higher derivative terms directly couples to the curvature which is slightly different from the model (1.4) considered in [51]. The aim of this section is to identify the number of DOFs of the theory (3.1). Similar to the previous section, one can introduce another Lagrange multiplier Λ which impose the constraint equation ϕ = 2φ and rewrite the action as To avoid the higher derivativesin the action, we simplify the action and drop the boundary term To simplify the calculation we define χ = f (φ, ϕ) and the inverse function ϕ = F (φ, χ), then we have where Ω 2 = χ −1 = exp ( 2 √ 6 θ). The final action in the Einstein frame is We will first perform the full Hamiltonian analysis in the Einstein frame which is simpler and then do the similar analysis in the Jordan frame. We shall see the results in both frames are consistent with each other.

Hamiltonian analysis: Einstein frame
To make the notation concise, we will drop all the bars of variables in the action (3.5) After ADM decomposition, the action becomes The coordinate N , N i and λ have no time derivatives in the action, which means we have five primary constraints Other non-vanishing conjugate momentums are defined as After some calculations we obtain the total Hamiltonian where and The time evolution of primary constraints generate the corresponding secondary constrains, which are the Hamiltonian constraint the diffeomorphism constraint and mimetic constraint Imposing the consistency condition of mimetic constraint yields and other useful functional derivatives of Hamiltonian are Plugging the above formulae in the integral we have where the new constraint With the constraint equations H, Φ 2 , Φ 3 , one can eliminate the dependence on π φ , π Λ , π θ in the later calculation. It will be useful to compute the following Poisson bracket The functional derivatives of Φ 3 needed are given by where we have used the constraint equation Φ 2 ≈ 0. The time evolution of Φ 3 leads to another new constraint where the new constraint is here the explicit expression of J function is tediously long and not important for our purpose.
The key point is that direct calculation shows Φ 4 does not depend on N . Because of the dependence of Φ 4 on λ, the time evolution of Φ 4 involves Lagrange multiplier v λ , thus the chain of constraints for primary constraint Φ 1 terminates here. Similar to the previous section, the time evolution of H, H i are automatically satisfied and yield nothing. The above considerations show that five primary constraints {Φ 1 , π N , π i } yield seven secondary constraints {Φ 2 , Φ 3 , Φ 4 , H, H i }, therefore we have the standard 8 first-class constraints and 4 second-class constraints in all. According to the usual counting degrees of freedom for constraint systems, the number of degrees of freedom in our theory (3.1) is 14 − 8 − 1 2 × 4 = 4. However, one has while Φ 1 commutes with all the other constraints. We can see that the Dirac matrix of the 4 secondary constraints {Φ 1 , Φ 2 , Φ 3 , Φ 4 } become singular and the rank will be reduced by two if the mimetic scalar field is homogeneous ∇φ = 0 (which is related to the coordinate choice). We have to redone the analysis because the conservation of Φ 4 will yield further constraints rather than fix v λ . This indeed implies that the number of DOFs becomes 3 for homogeneous case, as will be shown in the subsection below.

Hamiltonian analysis : Einstein frame in the unitary gauge
If we consider our model S 2 in the special unitary gauge φ = t from the beginning, i.e. the effective field theory (EFT) S which is just the former Hamiltonian plus one additional term imposing the unitary gauge condition. The primary constraints now are given by Here we use tilde to distinguish the constraints in the unitary gauge. The time evolution of those constraints generate the following new constraints where these expressions have been simplified by employing the constraints equation. Requiring which determines the Lagrange multiplier v N and so the chain of constraints forΦ 7 terminate here.
The time evolution of mimetic constraintΦ 2 gives us a new constraint Through a direct calculation We find out that {Φ 2 ,Φ 3 } ≈ 0 just as expected. Then the time evolution ofΦ 3 generates a new constraint which is independent of λ. With the constraint equations H,Φ 2 ,Φ 3 , one can eliminate the dependence on π φ , π Λ , π θ in the later calculation. Through a direct calculation one can find out that the Poisson bracket ofΦ 2 andΦ 4 weakly vanish , thus the new generated constraint by imposing the conservation ofΦ 4 also has no dependence on λ. The exact expression ofΦ 4 andΦ 5 is complicated, but fortunately for our purpose we only care which variables they depend on. As λ is not involved iñ Φ 5 , the time evolution ofΦ 5 yield another constraintΦ 6 . The time evolution ofΦ 6 involves the Lagrangian multiplier v λ because of the dependence ofΦ 6 on λ , therefore the chain of constraints forΦ 1 = π λ ≈ 0 determinates. Further more, as we have set the unitary gauge which breaks the first-class property of energy constraint, the time evolution of H gives u = 0 while the time evolution of H i are still automatically satisfied. Therefore the chain of constraints for π N and π i terminate here.
Above considerations show that six primary constraints yield nine secondary constraints, therefore the system admits 15 constraints which are 6 first − class : π i , H i , According to the usual counting of DOFs for constraint systems, the number of independent degrees of freedom in our theory (3.48) is 14 − 6 − 1 2 × 10 = 3 . We emphasize here that the number of DOFs is indeed different between the general case and the homogeneous field configurations [46,58,59]. Normally it is supposed that gauge choice should not affect the physics and the number of DOFs. What is special in our theory is that the associated Dirac matrix happens to be singular for the unitary gauge, resulting in further constraints (Φ 5 ,Φ 6 ) besides the usual unitary gauge fixing conditions (Φ 7 ,Φ 8 ).
As one can always chooce the gauge invariant quantities to fully describe the perturbations of the system, the linear perturbation theory should be the same between the general case and the homogeneous field configurations. This leads to the conclusion that we can only see 3 degrees of freedom (1 scalar and 2 tensor modes) in the perturbation theory of our model S 2 , and the other one scalar degree of freedom don't appear in the cosmological background. This is consistent with our previous paper [51] which works in the Lagrangian formalism and only consider the second order action.

Hamiltonian analysis: Jordan frame
Start with the action (3.4) in the Jordan frame and one can rewrite it in the ADM formalism The aim of this subsection is to obtain the number of DOFs of the model (3.1) in the Jordan frame and compare it with the result in the Einstein frame. As the action does not include time derivatives of N , N i , and λ, we have the primary constraints where we use Ψ A to denote the constraints in Jordan frame. Other conjugate momentums are The total Hamiltonian is then given by where and With the primary constraints (3.36), the corresponding secondary constraints are found to be the Hamiltonian constraint, diffeomorphism constraint and mimetic constraint Again, one can write the constraints in smeared form as before. We will frequently use the property in the subsequent calculations that the Poisson bracket of any constraint The following functional derivatives will be useful for the subsequent calculations The time evolution of mimetic constraint is With the constraint equations H, Ψ 2 , Ψ 3 , one can eliminate the dependence on π φ , π Λ , π θ in the later calculation. The time evolution of Ψ 3 leads to another constraint Using the result of the following Poisson bracket the new constraint is obtained to be here the explicit expression of J 2 is tedious and not important for us. The key point is that through direct calculation we find out all the terms involving N cancel exactly, i.e. Ψ 4 does not depend on N . Requiring this constraint to be time independent, determines the Lagrangian multiplier v λ in terms of phase space variables and the chain of constraints for primary constraint Ψ 1 = π λ ≈ 0 determinates. Besides, the time evolution of H i are automatically satisfied and yield no extra constraint. Therefore the system admits 12 constraints Ψ A = {Ψ 1 , Ψ 2 , Ψ 3 , Ψ 4 , π N , H, π i , H i }, of which 8 are first class and 4 are second class. Thus, the number of independent physical degrees of freedom in the model (3.1) is 14 − 8 − 1 2 × 4 = 4 which is consistent with the analysis in the Einstein frame. Similar to the Einstein frame discussed above, one can see that if the mimetic field is homogeneous ∇φ = 0, the Dirac matrix will become singular and the rank will be reduced. Now Let's work out the Hamiltonian analysis in the special unitary gauge to see how the degrees of freedom change.

Hamiltonian analysis: Jordan frame in the unitary gauge
Consider the action S J in the special unitary gauge φ = t, and then one obtain the new total Hamiltonian H (u) (3.48) which is the former Hamiltonian plus one additional term imposing the unitary gauge condition. The primary constraints now are given by The time evolution of those constraints generate the following new constraints where these expressions have been simplified by using the constraints equation. RequiringΨ 8 to be time independent gives which determines the Lagrange multiplier v N and so the chain of constraints for unitary gaugẽ Ψ 7 terminate here. The time evolution of mimetic constraintΨ 2 gives us a new constraint One can easily see {Ψ 2 ,Ψ 3 } ≈ 0 just as expected. The time evolution ofΨ 3 generates a new constraintΨ One can verify that the Poisson bracket ofΨ 2 andΨ 4 vanish just as the case in the Einstein frame. The time evolution of this new constraintΨ 4 also gives another constraint which has no dependence on λ. Although the exact expression ofΨ 4 andΨ 5 is complicated, the key point isΨ 4 and Ψ 5 do not include λ and the time evolution ofΨ 5 yield another constraintΨ 6 involving λ. ThereforeΨ 6 ≈ 0 involves the Lagrangian multiplier v λ and so the chain of constraints forΨ 1 = π λ ≈ 0 determinates here. Further more, as we have set the unitary gauge which satisfy {φ − t, H[N ]} = 0, the time evolution of H determines u = 0 and so the chain of constraints for π N ≈ 0 determinates here. The first class property of spatial diffeomorphism is unspoiled in the unitary gauge, and the time evolution of H i is automatically preserved, so the chain of constraints for π i determinates here.
The Hamiltonian analysis in the Jordan frame is consistent with the analysis in the Einstein frame, thus we have shown that our conclusion is independent of the frame: the number of DOFs according to Dirac in the general theory (3.1) is 4, and it is reduced to 3 in the unitary gauge. We deduce the extra DOF is automatically eliminated by the homogeneous field configurations.

Gauge dependence of the rank of the Dirac matrix: a simple example
To better understand the results above, consider a simple constrained system with countable degrees of freedom in which the rank of the Dirac matrix is gauge dependent. The ideal is to construct a Hamiltonian system with one first-class f 1 (gauge system) and two second-class constraints f 2 and f 3 , and the rank of the associated Dirac matrix is related to the gauge choice, i.e. where the sign ≈ denotes an inequality up to terms that vanish on the constraint surface.
To realize the ideal, we can first assume the total Hamiltonian of the system is given by where f 1 ,f 2 are two primary constraints and u, v are the corresponding Lagrange multiplier. As f 1 is supposed to be first-class, we require where the Lagrange multiplier w enforces the gauge fixing (4.5). We have 3 primary constraints The time evolutions of f 1 determine w = 0, the time evolutions of D 23 involves Lagrange multiplier u, while the time evolutions of f 2 yield the secondary constraint f 3 ≈ 0. The consistency relation of f 3 generate a new constraint after using the gauge condition D 23 ≈ 0. Thus in the gauge fixing (4.5), we have at least 5 constraints {f 1 , f 2 , f 3 , f 4 , D 12 } while only 4 constraints exist in the general gauge. Such a simple example is a good demonstration that there exists some special systems where some DOF may not like the usual DOF and will be eliminated by appropriate gauge fixing. We shall further analyze this issue in the future work.

Conclusion and discussions
Recently, there is an increasing investigation in exploring the instability issue [48] of mimetic model with higher derivative terms. In the previous work [51] we pointed out that it is possible to overcome this pathology by introducing the direct coupling of the higher derivatives of the mimetic field to the Ricci scalar of the spacetime. Although it seems that our setup have one scalar mode and two tensor modes by analyzing the quadratic actions of perturbation, the modified dispersion relation of scalar perturbation may imply the existence of extra DOF which do not show up at the cosmological perturbation level. In this paper we first confirmed that the mimetic gravity with a general higher derivative function of the mimetic field (2.1) has 3 DOFs which is consistent with the previous result of the Hamiltonian analysis in [50] and [46]. Then we perform a detailed Hamiltonian constraint analysis for the extended mimetic model (3.1) (which is slightly different from the model considered in [51]) in both Einstein frame and Jordan frame. The conclusion is consistent with each other in both frames: generally such kind of theories has 4 DOFs while only 3 propagating DOFs show up at the cosmological perturbation level [51]. To clarify the discrepancy, we reanalyze the model after fixing the unitary gauge. Interestingly, the DOFs is reduced to 3. Therefore, we conclude that the number of propagating DOFs in the model (3.1) is 4 in general, and the extra DOF is automatically eliminated by the homogeneous field configurations.. This gives us a hint that there exist some kind of special theories in which the DOFs of the space-time covariant version may not always be equivalent to the DOFs of its effective spatially covariant version, and some DOF may not show up on the FRW cosmological background. Actually this situation has already been studied in [60][61][62], and it was argued that this apparently dangerous mode is non-propagating and can be eliminated by choosing appropriate boundary conditions. For our case, the unitary gauge leads to the elimination of this extra mode. This also can explain the reason why the XG3 theory [63] is larger than the DHOST theory [59,64] : some gravitational theories, which belong to the XG3 theory, may have extra DOFs when recovering the spacetime diffeomorphism.
Another comment is that even the number of DOFs according to Dirac in the spacetime covariant version and the spatially covariant version are not equal in some cases, the perturbative theory in FRW universe is always the same. Furthermore, we should point out the appearance of higher power of w and k than two in the dispersion relation (such as in the case of the XG3 theory and Horava gravity [65]) may suggests the existence of extra non-propagating DOF. The relation between the modification of dispersion relation and the existence for extra DOF deserves detailed investigations in the future.

A a special case
Here we consider the case of f (φ) = 1 and g(2φ) = α(2φ) 2 , the action S 1 reduces to In the special case α = 1 3 , one has the Poisson bracket {Ψ 2 , Ψ 3 } ∝ (∇φ) 2 , and Ψ 4 = λ(∇φ) 2 + J 0 . As Ψ 4 involves λ, the time evolution will fix the Lagrangian multiplier v λ and the chain of constraints for π λ determinates here. The number of DOFs will be 3 according previous analysis. But if we set the unitary gauge from the beginning, the situation will change. the evolution of Ψ 4 will generate two secondary constraints Ψ 5 and Ψ 6 . This will reduce the DOFs to be 2 which means there are only two tensor modes at the corresponding perturbation theory.
The quadratic action for scalar perturbation in the model (A.1) is [48] S (2) where the coefficient of time derivative term happens to be vanishing in the special case α = 1 3 . The EOM gives ζ = 0. Therefore indeed only two tensor perturbations contribute to the DOFs. However, it is strange that the background equation in this special case becomes [48] 0 = V (t), (A.3) which will be not self-consistent unless the model have no potential term. But if the potential is vanishing, the background equation (A.3) will be automatically satisfied and gives us nothing, i.e. we don't have the evolution equation for the scale factor at all !