Inevitable Ghost and the Degrees of Freedom in f(r, G) Gravity

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1 503 Progress of Theoretical Physics, Vol., No. 3, September 00 Inevitable Ghost and the Degrees of Freedom in fr, G Gravity Antonio De Felice and Takahiro Tanaka Department of Physics, Tokyo University of Science, Tokyo 6-860, Japan Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto , Japan Received July, 00; Revised August 6, 00 The study of linear perturbation theory for general functions of the Ricci and Gauss- Bonnet scalars is carried out over an empty anisotropic universe, i.e., the Kasner-type background, in order to show that an anisotropic background in general has ghost degrees of freedom, which are absent on Friedmann-Lemaître-Robertson-Walker FLRW backgrounds. The study of the scalar perturbation reveals that on this background the number of independent propagating degrees of freedom is four and reduces to three on FLRW backgrounds, as one mode becomes highly massive to decouple from the physical spectrum. When this mode remains physical, there is inevitably a ghost mode. Subject Index: 53. Introduction Modifications of gravity have been used in many contexts. Perhaps the most famous is the first model of inflation introduced by Starobinsky in 980. However, in the last few years, modifications of gravity have been attracting attention as an alternative to quintessence, 3 in order to explain the late-time acceleration at large cosmological scales. 6 The most general modifications of gravity whose Lagrangian is built using only the metric tensors so as not to introduce any extra vector or spin- degrees of freedom other than the graviton are such that the Lagrangian is given by S = M P d x gfr, G,. 6π where R and G R R μν R μν + R μναβ R μναβ are the Ricci R and Gauss-Bonnet G scalars, respectively. We have only two nonzero Lovelock scalars in four dimensions. 7 There are several recent studies on the models of this type. 8, 9 In Ref. 8, the linear cosmological perturbation theory of these general modifications on a Friedmann-Lemaître-Robertson-Walker FLRW background was studied, and the result is that only one propagating mode for the scalar-type perturbations is present for a general function f. Furthermore, such a mode has a scale-dependent speed of propagation, depending on the wave number k as well as the time dependence through the background quantities. Furthermore, the background can be chosen so that this scalar mode is not a ghost. Each term in the Lagrangian of this scalar mode has only two temporal differentiations, whereas the square of the Laplacian operator appears, leading to a dispersion relation such as ω = Bk /a,where B is a function of the background quantities. Even though the Lagrangian looks Downloaded from by guest on December 08

2 50 A. De Felice and T. Tanaka exotic, there is no ghost as long as the signature of the second temporal derivative term is normal. 0 However, a few questions naturally arise by considering this result carefully. For example, in Ref. 8, the studied background was special, i.e., spatially homogeneous and isotropic. If we consider other inhomogeneous/anisotropic backgrounds, the term proportional to k, i.e., the operator, may shift to the term proportional to ω, i.e., the operator t, which would imply the existence of ghost degrees of freedom. Therefore, it is interesting to study the behavior of this same theory on one of the simplest anisotropic backgrounds, the Kasner-type solution. There is another point that is tightly connected to the previous one. The action. can be rewritten as S = M P 6π d x g [ FR+ ξ G V F, ξ ],. where F and ξ are auxiliary fields. By integrating out these auxiliary fields, we can verify that the action. is equivalent to.. On the FLRW background among the two extra scalar fields introduced in Eq.., one linear combination of their perturbations does not have a kinetic term. Here, it is totally unclear if such a feature is common to all backgrounds or it only happens on the FLRW one. In fact, as we shall see later, both modes propagate independently on Kasner-type backgrounds. In this paper, we will also see that having one more propagating field implies that the ω k dispersion relation returns back to a normal one, i.e., ω k. But how can we explain this apparent reduction of the propagating degrees of freedom from one background to another? As we shall see later, the determinant of the kinetic operator for the perturbation modes reduces to zero as we take the limit of the FLRW backgrounds. At the same time, the mass for the mode whose kinetic term vanishes on the FLRW backgrounds in general increases to infinity. Therefore, in the FLRW limit, this extra mode decouples from the physical spectrum to be integrated out, and the number of effective propagating modes is reduced by one.. Background models.. Kasner-type backgrounds For the reasons mentioned above, we study a simplified Kasner-type anisotropic background, which can be written as ds = dt + f t dx + f tdy + dz.. Downloaded from by guest on December 08 We analyze the behavior of the perturbations at linear order, using + + decomposition according to the symmetry of the background metric. From the symmetry in y-z plane, in general, the perturbation can be decomposed into three even-parity modes and one odd-parity mode. One of the three even-parity modes has a vanishing kinetic term in the FLRW limit, where the mode decouples since its mass becomes infinitely large.

3 Inevitable Ghost and the Degrees of Freedom in fr, G Gravity 505 The equations of motion for the system. are where Σ μν is the tensor defined by V F = R, V ξ = G,. FG μν = Σ μν,.3 Σ μν = μ ν F g μν F +R μ ν ξ g μν R ξ R μ λ λ ν ξ R ν λ λ μ ξ +R μν ξ +g μν R αβ α β ξ +R μαβν α β ξ g μνv.. On the Kasner-type background, we have R= f f f f + f f f + f f f f f f f f f,.5 f f f f f f f f +f f f f f f f G= f f 3..6 We can use the tt-component of Einstein equations in order to write V t interms of the other background quantities as V t = F f f f +6 ξ f f + F f f +F f f f +F f f f f F = f f f f f f..7 Furthermore, one can use the xx- and yy-components of Einstein equations in order to write F and ξ in terms of the other background quantities as follows: [ f f f f F +f f f F f f F f f f ξf f ξ = f f f f f f f.. Almost FLRW backgrounds + f 3 f ξ f f f f F f f f f F + f f ξf f f ξ f f F f f 3 +f 3 f F +f f f f F [ F f f f +F f f f f F f f 3 +f ξ f f f f ], + f ξ f f f + f Ff 3 +8f ξ f f f f ξ f 3 f F f f 3 F f f f f f f ξ f f f f ξ f ] f. For the practical construction of cosmological models, we are particularly interested in the behavior of these theories on the backgrounds close to FLRW. Therefore, it is sometimes convenient to consider Kasner-type backgrounds written in the form of homogeneously perturbed FLRW as f = at + δf t,.8 f = at + δf t,.9 F = F 0 t+δft,.0 ξ = ξ 0 t+δξt.. Downloaded from by guest on December 08

4 506 A. De Felice and T. Tanaka At the linear order in this perturbation, combining the xx- and yy-components of the Einstein equations leads to a closed differential equation for the field δ δf δf,. which reads where δ +ν δ + M δ =0,.3 νt = H + F 0 +H ξ 0 M t = Ḣ H H F 0 +H ξ 0 d dt F 0 +H ξ 0,. d dt F 0 +H ξ 0,.5 with H := ȧ/a. The differential equation admits the general solution δ = c at t dt at 3 [F 0 t +H ξ 0 t ] + c at..6 The second solution represents just the constant shift of the scale of spatial coordinates. Namely, it is just a gauge mode. The first physical solution decays, compared with the a term, when the term a 3 F 0 +H ξ 0 increases in time faster than t.in this case, f gets closer to f, and the Kasner-type background reduces to FLRW. 3. Perturbation analysis 3.. Reduction of the action for the even-parity modes We begin with the even-parity modes, which here mean the scalar-type perturbations in the two-dimensional space spun by the y-z plane. In order to achieve this goal, we consider the perturbation given by F = F t+δft, x, 3. ξ = ξt+δξt, x, 3. ds = [ + αt, x] dt + x χt, x dt dx + y ζdtdy + z ζdtdz+[f + βt, x] dx + f tdy + dz. 3.3 We have fixed the gauge so that the perturbations of the spatial metric in the twodimensional subspace spun by y and z vanish. We will find it convenient to make a field redefinition, ψ = δf + f f δξ. 3. Expanding the action to the second order, and introducing a Fourier expansion for the fields with the wavenumber k, we eliminate the perturbations of lapse and shift α, χ, andζ by using the constraint equations obtained from the variations of the action with respect to these nondynamical variables. Then, one reaches the result S = dt d 3 kf f / [ A ij φi φj B ij φi φ j q C ij φ i φ j D ij φ i φ j ], 3.5 Downloaded from by guest on December 08

5 Inevitable Ghost and the Degrees of Freedom in fr, G Gravity 507 where φ i =β,ψ,δξ and we have introduced q k y + k z /. All the matrices A, B, C, andd depend on the momentum only through the ratio γ k x /q. Only the matrix B is antisymmetric, while the others are all symmetric. 3.. Degrees of freedom and the signature of kinetic term It is important to establish the sign of the eigenvalues of the kinetic matrix A. If they are not all positive, it means that there exist ghostlike degrees of freedom. ThefirstkeyissueisthatthematrixA is not degenerate on general backgrounds. Namely, det A = f Δ f f f J f, 3.6 f does not vanish. Here, we defined Δ = f F + f ξ, 3.7 J = Fγ f + Ff f + f Ff +6γ f ξ +Ff γ f +6 f ξ f + f Ff. 3.8 Thus, the number of independent degrees of freedom present in the even-parity perturbation is three. We notice that det A vanishes in the FLRW limit, f f. This implies a reduction of the number of degrees of freedom in this limit. In fact, in this limit, both A 3 and A 3 become of order Oδ whereasa 33, like the determinant, is of order Oδ, as shown by the following expressions: Hc A 3 = a 5 F +H ξγ + FH + F + ξh 0 H 3 F ξ HFFγ 8 H γ ξ H ξ 8 H Fγ ξ 6 HF F + F γ H F F 6 H 3 Fγ ξ 3 FH ξ, 3.9 A 3 = c H 8 Fγ H ξγ H +γ F 36 ξh 8 FH F FH + F + ξh γ +F +H ξ, 3.0 a 3 A 33 = 6c H a 6 F +H ξ 3 γ + FH + F + ξh F γ 8 H 3 Fγ ξ +HF Fγ 6 FH 3 γ ξ +H Fγ ξ 8 H γ ξ H γ F HF F 96 H ξ +H FFγ +6H Fγ ξ f FH ξ 8 F γ H H 3 F ξ 5 H F, 3. Downloaded from by guest on December 08 recall that c is the magnitude of δ = δf δf, whereas the determinant tends to 6H c det A = a 0 F +H ξγ + FH + F + ξh. 3. The fact that the sign of det A is opposite to the one of F +H ξ shows that on Kasner-type backgrounds close to FLRW models, at least one mode becomes a ghost

6 508 A. De Felice and T. Tanaka as F +H ξ > 0 is required for the tensor modes not to be a ghost in FLRW backgrounds. 8 The matrix A, in the FLRW limit c 0, becomes block diagonal, i.e., it is composed of a matrixanda matrix. The latter is zero valued. The submatrix is nondegenerate and finite. The signs of its eigenvalues are read from A = A A A = H F +H ξ FH + F + ξh, 3.3 3F +H ξ H γ + FH + F + ξh a. 3. Since both factors are positive assuming F +H ξ >0, the unique ghost mode in this limit is identified with the third field, which is δξ in the present calculation. We note that the important point to obtain the property that the kinetic term of the third field vanishes as δ is in how we choose the first and second fields β and ψ. As we will see below, the ghost mode δξ can be integrated out from the action near the FLRW backgrounds The Laplacian operator and the q behavior on FLRW Let us restrict our attention to better understand the FLRW limit for the Laplacian matrix C. In particular, we are interested in the elements regarding the coupling of δξ with itself and with the other two fields. We find and det C = 6 F + ξ+γ Ḣ H FH + F +H ξ a, F C 3 = + ξh Ḣ ξh +FH + F a, 3.6 6γ C 3 = ξh +FH + F a, 3.7 6c Ḣγ +H C 33 = ξh +FH + F F +H ξ a Therefore, also the q -self-coupling vanishes on FLRW, but the q -couplings with the other fields do not. It is of crucial importance whether the remaining term quadratic in δξ, the mass term D 33, vanishes or not. D 33 is given by Downloaded from by guest on December 08 D 33 = F + ξh ξ + 8HḢ ξ V F 6 F + ξh Ḧ ξ +HḢ 768H Ḣ F +H ξ ξh +FH + F, 3.9

7 Inevitable Ghost and the Degrees of Freedom in fr, G Gravity 509 which does not vanish in general. This implies that in the Lagrangian in the FLRW limit, the only term quadratic in δξ is its mass. It should be noted that in constructing cosmological models, we have to require that the model is sufficiently close to the Einstein gravity at least at early times, i.e., F andξ 0. Then, we have F H and H ξ. This implies that D 33 /H in general becomes very large at early times. This fact may help in integrating out the ghost, as it naturally gives a very large mass to δξ compared with its kinetic term. The part of the Lagrangian related to δξ is written as L a3 D 33 δξ a 3 δξ q C 3 β + B 3 β + q C 3 ψ + B 3 ψ + + Oδ. 3.0 Integrating out δξ, withtheaidoftherelationc 3 B 3 = C 3 B 3, we obtain the terms proportional to q, which were found in Ref Odd-parity modes Let us consider the odd-parity modes, i.e., the modes that are composed of divergence-less vectors in terms of two dimensions spun by y and z. Perturbations of all four-dimensional scalars are zero for odd-parity modes. The perturbation of odd-parity modes appears only in the metric perturbation as v i = h 0i, w i = h xi, h ij =0, 3. where i, j = y or z, andv i and w i are divergence-less vectors. We used one gauge degree of freedom belonging to the odd-parity modes to eliminate the perturbation in ij-components. As in the case of even-parity modes, we erase the shift perturbation v i by using the constraint equation obtained by taking the variation with respect to v i itself. Then, we obtain S = dtd 3 kf / f A o [ẇ i ẇ i c k ] o w i w i m o f w iw i, 3. where Δ Δ A o = f f Δ + γ Δ, 3.3 c o = + γ Δ + γ Δ Δ Δ f f F f f f f f f + Ff f f +Ff f f 3 f Ff 3 f m o = follows F f f f +f ξ f 3 8f ξ f f f +f f ξ f f, 3. f ξ f f f +f f ξ f f +f F f f 3 f ξ f f 8f F f f f ξ f +8f f f ξ f f Ff f f 3 γ ξ f f F f f γ ξ f V F ξξ The expression V = f F ξξ GG /f RR f GG frg, where f RR can be rewritten in terms of second derivatives of f in action, as f etc. R GG Downloaded from by guest on December 08

8 50 A. De Felice and T. Tanaka +F f γ f f f 3 8f f f ξ f f F f f f 3 γ f +F f 3 f 3 γ f +F f f 3 γ ξ F f f γ f f +8f Ff ξ f f 8f f 3 f γ ξ f f f γ ξ f Ff 3 +8f f f γ ξ f F +Ff 3 f f γ f F 6Ff f 3 f γ ξ f +f F f f ξ f f γ Ff 3 f f γ f ξ f Ff γ f Ff +Ff 3 f γ ξ f 3 Ff f f 3 γ F f f 3 f 3 γ ξ F 3F f f f γ f +8f f ξ f f 3 +F f f 3 f f F f f f 3 f + F f γ f 3 [ Ff + f ξff + ξγ f + f ξ + f Fγ f ] f f f f f, 3.5 and Δ = f F + f ξ. Here, co looks divergent in the FLRW limit, but in fact, it is not. In the FLRW limit, we find and c V A o = c o = F +H ξ a + γ, 3.6 F + ξ F +H ξ, 3.7 m o = 8ḢH ξ + ḢF +H3 ξ + H F + HF +H ξ F +H ξ, 3.8 reduces here to the speed of the tensor modes in the FLRW background.. Implication of the perturbation analysis around Kasner-type backgrounds.. Inevitable ghost Here, we show that the no-ghost condition A o > 0withA o given in Eq. 3.3 is incompatible with another no-ghost condition det A>0for the even-parity modes. From Eq. 3.6, we find that Δ < 0 is required for the absence of a ghost. Then, if Δ < 0, A o becomes negative for any γ, and hence, the odd-parity modes become ghost. Even if Δ > 0, A o becomes negative for a sufficiently large γ, and hence, at least a part of odd-parity modes becomes ghost. Therefore, we conclude that the presence of a ghost is inevitable for generic models of fr, G once we consider Kasner-type backgrounds. This is the main claim of this paper. However, as we have seen above, the perturbative action for such a mode does not completely vanish in general even when the kinetic term vanishes, e.g., the matrix D is not degenerate even in the FLRW limit. Therefore, this ghost mode does not become strongly coupled in the FLRW limit, but it becomes infinitely massive. Thus, such a mode is to be integrated out as long as backgrounds are chosen to be sufficiently close to the FLRW universe. In this sense, such models might be harmless in spite of the presence of a ghost. Downloaded from by guest on December 08

9 Inevitable Ghost and the Degrees of Freedom in fr, G Gravity 5.. Particular cases with only three degrees of freedom We will consider here the case for which one has the condition f f R G f = F, ξ R G R, G =0.. In this case, perturbations of F and ξ are not independent. For generic backgrounds, this is the case if L = F φr + ξφg V φ. Below, let us consider special cases in which fr, G isafunctionintheformof fr + λg withλ being a constant. In this case, δξ = ξ F δf. generally holds. Therefore, one can eliminate δξ from the perturbation action so that the number of even-parity degrees of freedom is reduced to two. Then, the no-ghost conditions for even-parity modes reduce to only two conditions, det A = f + f F f f J f ξ + f F Ff + ξ f f F f ξ 3f F Ff 8f f ξf F f f ξf f 6f f ξ + f Ff ξ > 0,.3 A = f F f J Ff + ξ f f f 3 ξ F f 3 + ξ f γ f 6 F f ξ γ f 7 f ξ γ f 6 f 3 f ξ Ff F f 6f f ξ Ff F f f 3 f f 3 F F f f ξ Ff F f 3 3f f ξ Ff F f 3 f 3 f F f F ξ f f 5 f F f F f f f f F F 8f 3 f F f ξ f 6f 3 f 3 ξ FF f f f ξ f F f 6f f ξ 3 Ff f + ξ 3 f γ F f 7 6f f F f F ξ f 8f ξ3 F f f 3 f ξ 3 f F f 5 8f f 3 ξ 3 F f 3 6f f ξ 3 Fγ f f f ξ3 γ F f 6 8f f ξ3 Fγ f 5 f f ξ 3 γ F f 5 + f ξ f F f 6 γ f f 3 f 3 F F ξ f f ξ F f f 3 6f ξ3 Ff f 6 γ 3f 5 F f γ F f 3f f f F γ F 6f 3 F f γ ξ f f ξ FF f f 3 f f ξ F f γ f f ξ F f 5 γ +8f f F γ ξ f +f f F γ ξ f 3 f ξ3 f γ F f 6 6f 3 f ξ Ff F f 3 γ 8f f F f γ F ξ f 0f f ξ Ff F f γ ξ f 5 f 3 f F f γ F ξ f 3 +f 5 f F γ F ξ f f f ξ FF f γ 3f f ξ f γ f 5 F 8f 3 F f γ F ξ f 3 f ξ Ff γ F f 5 f ξ Ff γ F f f 3 f ξ Fγ F f f 3 ξ Ff γ F f 0f 3 f ξ Fγ F f 3 f f ξ FF f 3 γ 3f f ξ3 Ff γ f 5 +f 5 f F γ F ξ f 8f ξ Ff γ F f 5 3f 3 f F f γ ξ f 3 3f 5 f F f γ F f > 0,. Downloaded from by guest on December 08 where J has been defined in Eq It is interesting to note that for these theories the reduction of the degrees of freedom does not occur as the Kasner-type background approaches the FLRW one, since det A does not either vanish nor diverge in the

10 5 A. De Felice and T. Tanaka FLRW limit. In fact, for the limiting FLRW case, we have det A = 9 F +H ξ a + γ A,.5 A = H F +H ξ F +H ξ F F +HF +H ξ,.6 which are both positive when F +H ξ >0 as well as the odd-parity mode. A famous example is fr gravity. The result for this particular case is obtained by simply setting ξ = 0 in the above equations. More explicitly, we have det A = 3F f + f [Ff f +f f F + Ff f +γ f F f + f F ],.7 A = F f f + f f + f f f f +3f f γ f +3f f γ f +3f γ f [Ff f +f f F + Ff f +γ f F f..8 + f F ] In this case, the sound speed is easy to evaluate, and one can verify that all modes propagate at the speed of light..3. Ghost crossing For the particular models discussed in this section, all modes become ghost when F +H ξ crosses 0. It might be interesting to study if this ghost crossing i.e., the time when any of the perturbation modes becomes a ghost can appear without the background reaching a spacetime singularity. An example that shows ghost crossing can be easily constructed in the FLRW model. For definiteness, we consider models specified by fr, G = fr + λg. In this case, the action can be written with the aid of one auxiliary field as S = M P d x g [ R + λ GF V F ]..9 6π Here, ξ in the above calculation is to be identified with λf. Hence, the ghost crossing occurs when F +λhf = 0. As long as the function V F can be chosen freely, the only nontrivial equation of motion is +λh F H 8λ Ḣ +λh F +ḢF =0..0 As a simple example, we assume power law expansion, a t n. Then, F is solved as F = w F n ++ n +0n +, n + n +0n + ; n + ; t n λ +w t n+3 F n +5 n +0n +, n +5+ n +0n + ; n +5 ; t n λ,. Downloaded from by guest on December 08 with w and w being constants. By choosing w and w appropriately, we can make models that pass through F +λh F = 0. For example, if we set n =andw =0,

11 Inevitable Ghost and the Degrees of Freedom in fr, G Gravity 53 we have F +λh F = w 3t +6λt + 56λ 768λ.. For negative λ, this crosses 0 when t = 6λ/3 or 6λ. Since the values of R and G are respectively given by 6nn R = t, n n3 G = t,.3 they are both regular at the ghost crossing points. Here we discussed the simple FLRW case, but the ghost crossing without having a singularity in the background geometry generically occurs even if we consider Kasner-type backgrounds. Finally, we want to remark that the ghost crossing does not in general coincide with the phantom crossing, that is the point where the effective dark energy equation of state parameter crosses. In fact, since we considered here only vacuum solutions, then we have that w DE p DE /ρ DE = w eff 3Ḣ/H,wherewe defined ρ DE and p DE by using the field equations of motion, namely, 3H =8πGρ DE and Ḣ = πg p DE + ρ DE. Since in our example n =,wehaveh =/t, then w DE = 3, which implies that the ghost crossing occurs whereas the phantom one never does. 5. Conclusions We studied the behavior of the perturbation for general modifications of gravity whose Lagrangian consists of a general function of R, the Ricci scalar, and of G, the Gauss-Bonnet scalar, on Kasner-type backgrounds. We have shown that the existence of ghost is inevitable for generic models. However, the ghost mode can decouple from the physical spectrum in the FLRW limit. The kinetic term of this mode vanishes in this limit, which does not mean strong coupling here because the mass term does not vanish. Hence, such cosmological models may have a chance to survive as an effective theory that describes only a small deviation from the FLRW universe and/or the modifications of gravity tend to vanish at early times. We also found that the modified dispersion relation of the kind ω k first discussed in Ref. 8 is due to integrating out the ghost mode in the FLRW limit. Also, we presented special cases that avoid the presence of the ghost. In this case, one degree of freedom is absent from the beginning since the action satisfies a special relation, and there is no reduction of degrees of freedom in the FLRW limit. If we require this condition to be satisfied for rather generic backgrounds, the form of the Lagrangian is restricted to L = F φr + ξφg V φ. For such models, the Downloaded from by guest on December 08 Also, other Riemann curvature invariants are finite for the power law solution a t n. For example, also the Kretschmann scalar is proportional to t. An example is given by the Lagrangian L = FR + ξf G 3 F /ξ, where ξf = F p F λ 6 /λ, which admits the solution f t, f t,andf = λ t + λ. The ghost crossing occurs at t = 3 λ 5 /λ,withλ > 0andλ < 0, whereas the singularity is in t =0.

12 5 A. De Felice and T. Tanaka ghost crossing, where the sign of the kinetic term flips for some modes, can happen. We have presented an example of ghost crossing within FLRW models without the background reaching a classical singularity. Acknowledgements The discussion during the workshops YITP-T and YITP-T-0-0 at the Yukawa Institute was very useful to complete this work. We thank Prof. Jiro Soda for useful comments. The work of A. D. F. was supported by the Grant-in-Aid for Scientific Research Fund of the JSPS No T. T. is supported by the JSPS through Grant No We also acknowledge the support of the Grant-in-Aid for the Global COE Program The Next Generation of Physics, Spun from Universality and Emergence and the Grant-in-Aid for Scientific Research on Innovative Area No. 006 from the Ministry of Education, Culture, Sports, Science and Technology MEXT of Japan. References A. De Felice and S. Tsujikawa, Living Rev. Relativity 3 00, 3, arxiv: A. A. Starobinsky, Phys. Lett. B 9 980, S.Capozziello,Int.J.Mod.Phys.D 00, 83. S. Capozziello, S. Carloni and A. Troisi, Recent Res. Dev. Astron. Astrophys. 003, 65. S.Capozziello,V.F.Cardone,S.CarloniandA.Troisi,Int.J.Mod.Phys.D 003, 969. S. M. Carroll, V. Duvvuri, M. Trodden and M. S. Turner, Phys. Rev. D 70 00, L. Amendola, R. Gannouji, D. Polarski and S. Tsujikawa, Phys. Rev. D , B.LiandJ.D.Barrow,Phys.Rev.D75 007, W. Hu and I. Sawicki, Phys. Rev. D , A. A. Starobinsky, JETP Lett , 57. S. A. Appleby and R. A. Battye, Phys. Lett. B , 7. S. Tsujikawa, Phys. Rev. D , E. V. Linder, arxiv: L. Amendola and S. Tsujikawa, Phys. Lett. B , 5. 5 S. M. Carroll, I. Sawicki, A. Silvestri and M. Trodden, New J. Phys , 33. S. Carloni, P. K. S. Dunsby and A. Troisi, Phys. Rev. D , 00. Y. S. Song, W. Hu and I. Sawicki, Phys. Rev. D , 000. I. Sawicki and W. Hu, Phys. Rev. D , 750. R. Bean, D. Bernat, L. Pogosian, A. Silvestri and M. Trodden, Phys. Rev. D , T.Faulkner,M.Tegmark,E.F.BunnandY.Mao,Phys.Rev.D76 007, L. Pogosian and A. Silvestri, Phys. Rev. D , S. Nojiri, S. D. Odintsov and M. Sasaki, Phys. Rev. D 7 005, T. Koivisto and D. F. Mota, Phys. Lett. B 6 007, 0; Phys. Rev. D , S. Kawai, M. a. Sakagami and J. Soda, Phys. Lett. B , 8. S. Nojiri and S. D. Odintsov, Phys. Lett. B ,. A. De Felice and S. Tsujikawa, Phys. Lett. B ,. T. P. Sotiriou, arxiv: K. Uddin, J. E. Lidsey and R. Tavakol, Gen. Relat. Gravit. 009, 75. G. Cognola, E. Elizalde, S. Nojiri, S. D. Odintsov and S. Zerbini, Phys. Rev. D , B.Li,J.D.BarrowandD.F.Mota,Phys.Rev.D76 007, 007. S. Y. Zhou, E. J. Copeland and P. M. Saffin, J. Cosmol. Astropart. Phys , 009. Downloaded from by guest on December 08

13 Inevitable Ghost and the Degrees of Freedom in fr, G Gravity 55 A. De Felice and S. Tsujikawa, Phys. Rev. D , A. De Felice, D. F. Mota and S. Tsujikawa, Phys. Rev. D 8 00, 0353, arxiv: S. M. Carroll, A. De Felice, V. Duvvuri, D. A. Easson, M. Trodden and M. S. Turner, Phys.Rev.D7 005, G. Calcagni, S. Tsujikawa and M. Sami, Class. Quantum Grav. 005, A. De Felice and T. Suyama, J. Cosmol. Astropart. Phys , 03; Phys. Rev. D , 08353, arxiv: A. De Felice, J. M. Gerard and T. Suyama, arxiv: K. Bamba, S. D. Odintsov, L. Sebastiani and S. Zerbini, Eur. Phys. J. C 67 00, 95, arxiv: O. Gorbunova and L. Sebastiani, arxiv: E. Elizalde, R. Myrzakulov, V. V. Obukhov and D. Saez-Gomez, Class. Quantum Gravit. 7 00, , arxiv: K. S. Stelle, Gen. Relat. Gravit , 353. N. H. Barth and S. M. Christensen, Phys. Rev. D 8 983, 876. N. Boulanger, T. Damour, L. Gualtieri and M. Henneaux, Nucl. Phys. B , 7, hep-th/ A. De Felice, M. Hindmarsh and M. Trodden, J. Cosmol. Astropart. Phys , 005. G. Calcagni, B. de Carlos and A. De Felice, Nucl. Phys. B , 0. J. O Hanlon, Phys. Rev. Lett. 9 97, 37. P. Teyssandier and P. Tourrenc, J. Math. Phys. 983, 793. T. Chiba, Phys. Lett. B ,. J. M. Bardeen, Phys. Rev. D 980, 88. H. Kodama and M. Sasaki, Prog. Theor. Phys. Suppl. No ,. V.F.Mukhanov,H.A.FeldmanandR.H.Brandenberger,Phys.Rep.5 99, 03. Downloaded from by guest on December 08