Perturbations in Regularized Lovelock Gravity
aa r X i v : . [ g r- q c ] A p r Perturbations in Regularized Lovelock Gravity
Alessandro Casalino
1, 2, ∗ and Lorenzo Sebastiani
3, 4, † Dipartimento di Fisica, Universit`a di Trento,Via Sommarive 14, I-38123 Povo (TN), Italy Trento Institute for Fundamental Physics and Applications (TIFPA)-INFN,Via Sommarive 14, I-38123 Povo (TN), Italy Istituto Nazionale di Fisica Nucleare, Sezione di Pisa, Italy Dipartimento di Fisica, Universit´a di Pisa,Largo B. Pontecorvo 3, 56127 Pisa, Italy
Abstract
In this paper we study the perturbation theory of the recently proposed Regularized LovelockGravity [1], on the curved Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) space-time. We providethe first order perturbation equations both in the scalar and tensor sector in the presence of anadditional minimally coupled scalar field. A general expression for the velocity of gravitationalwaves at the generic order is inferred. Moreover, we apply the results on the study of slow-rollinflation on flat FLRW background, at second order in the Regularized Lovelock Gravity, i.e. inthe presence of the Gauss-Bonnet correction only. We derive the power spectra of scalar and tensorperturbations, and provide the equations for the inflation observable quantities: the spectral indicesand the tensor-to-scalar spectra ratio. ∗ [email protected] † [email protected] . INTRODUCTION The Lovelock theorem [2] provides the most general gravitational theory, described bythe Lovelock-Lanczos action, leading to second order field equations in d dimensions. Infour dimension, since higher orders Lovelock terms contribute to the field equations withidentically vanishing contributions, the theory reduces to General Relativity (GR) with acosmological constant [3].There are different ways to circumvent the Lovelock theorem in the metric formalism,without adding degrees of freedom to the action. For instance, one method considers theoriesleading to second order field equations only for specific classes of metric fields. This is usuallyexploited in Quasi-Topological gravity theories [4, 5] and Non-polynomial gravity theories[6–8].In this paper we will consider a different approach, firstly proposed in the four dimensionalcase by Tomozawa [9]. Consider the following regularized Einstein-Gauss-Bonnet theory S = Z d d x √− g (cid:18) M P R + αd − G (cid:19) , (1)where G ≡ R − R µν R µν + R µνρσ R µνρσ and M p ≡ / πG . Tomozawa proved that, ifthe associated equations of motion are evaluated considering a static spherically symmetricansatz, we can set the dimension to d = 4, preserving the contribution coming from theGauss-Bonnet term. The resulting theory admits a black hole solution with a repulsivegravitational force near the time-like singularity r →
0. Note that the same solution hasbeen found in different contexts motivated by quantum corrections to gravity. See for in-stance the Hoˇrava-Lifshitz gravity case [10], and the semi-classical Einstein equations withconformal anomaly [11]. The geodesic structure, as well as its quasi-normal modes and sta-bility, have been investigated recently in, respectively, [12] and [13]. The charged case wasinvestigated in [11] and more recently in [14]. The Anti de Sitter (AdS) black holes and theirthermodynamics were found and discussed in [15, 16]. Finally, a rotating generalization hasbeen studied in [17, 18]. Other works on black holes in the context of this theory can befound in [19–34].The same theory has been also studied by Cognola et al. [35]. They showed that the en-tropy of the black hole is logarithmic in its area, which provides a motivation for interpretingthese terms as quantum corrections. In fact, such a behaviour of the entropy typically arises2nce the quantum effects are taken into account [36, 37]. Finally they proved that the flatFLRW sector of this theory is well-defined in the limit d →
4. An analogous investigationin the case of spatially curved FLRW sector has also been recently studied [38].Glavan and Lin further extended the applicability of this theory to first order in pertur-bation theory around (A)dS vacua. In fact, they showed that this theory only contains thedegrees of freedom of a massless graviton, as in GR [1], which is a very crucial result to assessthe viability of this theory, as it indicates that it is not plagued by ghost-like instabilities.Although this is not a sufficient proof, it provides a strong indication that the regularizationof Lovelock-Lanczos gravity is a well-defined procedure.Recently, a discussion arose about the well-posedness of this regularization. See forinstance [39, 40]. In fact, at least for FLRW or SSS metric fields, is always possible toextract the dimensional factor from the equations of motion. In other words, for FLRW orSSS metric fields g ρσ , is always possible to write the equations of motion in the form G µν [ g ρσ ] = ( d − D ) H µν [ g ρσ ] , (2)where H µν is a well-defined covariant tensor, and the limit d → D is possible for any D .This, for a generic metric, is not possible, creating questions on the well-posedness of thewhole regularization procedure. However, in this paper we limit our discussion to the FLRWmetric, which is not affected by this problem, and use the theory with its regularization asan effective theory with a prescription to obtain the equations of motion with non trivialcorrections with respect to General Relativity. This has also been noted by [41]. On thesubject, see also [42].Finally, a reformulation of Regularized Lovelock theory with a sub-class of Horndeskigravity has been recently proposed [43, 44].In this paper we will extend the results obtained by Glavan and Lin [1], and by Casalinoet al [38]. In particular we will study the first order perturbation theory in the case ofcurved FLRW with an additional minimally coupled scalar field, providing the perturbationsequation both in the scalar and tensor sector. In particular, the equations are derived at thirdorder in Regularized Lovelock Gravity corrections, and then we infer a generalization forany Lovelock order. We will then derive the formalism of first order perturbation theory forinflation in the reduced case of regularized Gauss-Bonnet theory (second order of RegularizedLovelock Gravity), which is the second order in Regularized Lovelock Gravity, in a flat3LRW.The paper is structured as follows. In Sec. II we provide a brief introduction to theRegularized Lovelock Gravity theory, showing the associated regularization and providingthe general equations of motion. In Sec. III we show the background results on a spatiallycurved FLRW background. In the following sections Sec. IV and V, we derive respectivelythe tensor and scalar perturbation equations. We also provide the equations for the tensorand scalar waves velocity. In Sec.VI we study the corrections to slow-roll scalar field inflationgiven by Lovelock theory. Finally, in Sec. VII we draw the conclusions.In this paper we follow the convention c = 1. Moreover, we define the Planck mass as M p ≡ / πG N . II. REGULARIZED LOVELOCK GRAVITY
In this section we review the Regularized Lovelock Gravity theory. We consider thefollowing d − dimensional action S = Z d d x √− g " t X p =0 α p L p ( R, R µν , R µνρσ ) + S ϕ , (3)where the first part is the Lovelock–Lanczos action, while the second S ϕ part is the con-tribution from a scalar field which is minimally coupled to the metric. The parameter t isthe order of the Lovelock gravity, α p the coupling constants and L p are curvature tensorfunctions, defined as L p = 12 p δ µ ν ...µ p ν p σ ρ ...σ p ρ p p Y r =1 R µ r ν r σ r ρ r , (4)where δ µ ν ...µ p ν p σ ρ ...σ p ρ p is the generalized Kronecker delta. Explicitly, the definition of these func-tions up to the third order in Lovelock gravity are L = 1 , (5) L = R , (6) L = R − R µν R µν + R µνρσ R µνρσ ≡ G , (7) L = R − RR µν R µν + 16 R µν R µρ R νρ + 24 R µν R ρσ R µρνσ + 3 RR µνρσ R µνρσ − R µν R µρσκ R νρσκ + 4 R µνρσ R µνηζ R ρσηζ − R µρνσ R µ νη ζ R ρησζ . (8)4oreover, we define the scalar field part of the action as S ϕ = Z d d x √− g (cid:20) − ∇ µ ϕ ∇ µ ϕ − V ( ϕ ) (cid:21) , (9)where the potential V ( ϕ ) is a scalar function of the field ϕ . Note that, since the scalar fieldis minimally coupled to the metric, the contribution from S ϕ to the equations of motion willbe a simple addition.The equations of motion can be computed varying the action in Eq. (3) respectively withrespect to the metric g µν and ϕ . We obtain G µν = t X p =0 α p G ( p ) µν = 12 (cid:20) ( ∇ µ ϕ ) ( ∇ ν ϕ ) − g µν ( ∇ ρ ϕ ) ( ∇ ρ ϕ ) − g µν V ( ϕ ) (cid:21) , (10) ∇ µ ∇ µ ϕ − V ( ϕ ) = 0 , (11)where G ( p ) αβ (cid:2) g µν (cid:3) = g αγ √− g δ [ √− g L p ] δg γβ = − p +1 δ αµ ν ...µ p ν p βσ ρ ...σ p ρ p p Y r =1 R µ r ν r σ r ρ r . (12)Since in the equations of motion (12) we have the totally anti-symmetric Kronecker delta,the d ≤ p contributions identically vanish. To circumvent this problem, we can implementthe aforementioned regularization by re-scaling the coupling constants α p , and then by fixingthe dimension of the manifold. With this procedure we obtain non-vanishing contributionsto the equations of motion coming from generic Lovelock gravity orders. Therefore, wedefine the following coupling constants α = M d − P α = αd − α = βd − , (13)where the d − α ∝ Λ = 0.In the following we will consider these equations of motion on a Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) manifold.
III. BACKGROUND ON FLRW
In this section we review some results of Lovelock gravity on a FLRW manifold. Weconsider the metric ds = − dt + a ( t ) g ( d − ij ( ~r ) dx i dx j , (14)5here g ( d − ij is the spatially curved d − κ ,written in some coordinates x i . Using the equations of motion (10) and (11), we obtain theequations of motion F ( J ) = (cid:20) ˙ ϕ V ( ϕ ) (cid:21) , (15)( Q − J ) F ′ ( J ) = − ˙ ϕ , (16)¨ ϕ + ( d − H ˙ ϕ + V ( ϕ ) = 0 , (17)where we define the functions J and Q as J ≡ H + κa and Q ≡ H + ˙ H , (18)and function F reads F ( J ) ≡ ∞ X p =1 " p Y s =1 ( d − s ) α p J p , (19)and F ′ ( J ) ≡ dF ( J ) dJ . (20)In particular, at cubic order in Lovelock gravity ( t = 3), in the limit d →
4, we obtain J (cid:18) αM P J + 4 βM P J (cid:19) = 13 M P (cid:20) ˙ ϕ V ( ϕ ) (cid:21) , (21) (cid:0) Q − J (cid:1) (cid:18) αM p J + 12 βM p J (cid:19) = − ˙ ϕ M p , (22)¨ ϕ + 3 H ˙ ϕ + V ( ϕ ) = 0 . (23) IV. TENSOR PERTURBATIONS
In this section we present the first order tensor perturbation equations in a curved FLRWspace. We consider the following line element ds = − dt + a ( t ) h g ( d − ij ( ~r ) + h ij ( d − ( t, ~r ) i dx i dx j , (24)where h is a spatial tensor whose value is always less than unity. The following results areshown in the limit d →
4. 6n order to obtain the tensor waves perturbation equation, we perform the computationon the i − j Einstein equation with i = j . We obtain¨ h ij + 3 H (cid:20) M P (cid:0) α + 6 βJ (cid:1) ( Q − J ) (cid:21) ˙ h ij + (cid:20) M P (cid:0) α + 6 βJ (cid:1) ( Q − J ) (cid:21) (cid:18) κa − ∂ l ∂ l a (cid:19) h ij = 0 , (25)where Q − J = ˙ H − κ/a by definition, and Γ is a dimensionless factor defined asΓ ≡ M P (cid:0) αJ + 6 βJ (cid:1) . (26)Therefore we can define the velocity of gravity waves as c T = 1 + 8Γ M P (cid:0) α + 6 βJ (cid:1) ( Q − J ) , (27)from which we can see that the addition of the regularized Gauss-Bonnet adds a correctionto the General Relativity result c T = 1.We can also infer a generalization of the tensor perturbation equations (25) to a genericorder t in the Lovelock series as¨ h ij + 3 H (cid:18) t (cid:19) ˙ h ij + (1 + Θ t ) (cid:18) κa − ∂ l ∂ l a (cid:19) h ij = 0 , (28)where we define the adimensional coefficientsΘ t ≡ Q − J Γ t M P p = t X p =2 p p ! α p J p − (29)and Γ t ≡ M P p = t X p =1 p ! α p J p − = 1 + 2 M P p = t X p =2 p ! α p J p − . (30)Note that when t = 1, which is the General Relativity case, Γ t = 1 and Θ t = 0.Finally, we can write the velocity of gravity waves up to order t in the Lovelock series c T = 1 + Θ t . (31)From this result is possible to gather information about the value of the coupling con-stants. In fact, although the value is a combination of the terms in Θ t coming from differentorders of Lovelock gravity up to order t , we have stringent constraint on the value of Θ t itself, which should be approximately null to respect the experimental constraint c T ∼ = 1[45]. 7 . SCALAR PERTURBATIONS In this section we present the first order scalar perturbation equations in a curved FLRWspace, in the case t = 3. We consider the following line element ds = − [1 + 2Ψ( t, ~r )] dt + a ( t ) [1 − t, ~r )] g ( d − ij dx i dx j , (32)where Ψ and Φ are the Newtonian potentials, whose absolute values is always smaller thanunity. The following results are shown in limit d → t − t equation3 H ˙Φ − κa Φ + (cid:18) Q + 2 J − κa (cid:19) Ψ − ∂ l ∂ l Φ a = − M P (cid:18) ˙ ϕδ ˙ ϕ + dVdϕ δϕ (cid:19) , (33)the t − i equations (cid:16) ˙Φ + H Ψ (cid:17) = ˙ ϕ M P δϕ , (34)the traceless part of the i − j equation ∂ l ∂ l Ψ − (1 + Θ ) ∂ l ∂ l Φ = 0 , (35)and finally the trace of the i − j equation¨Φ + 3 H (cid:18) (cid:19) ˙Φ + H ˙Ψ − κa (1 + Θ ) Φ + (cid:2) J + Q + H (1 + Θ ) (cid:3) Ψ −− (1 + Θ ) ∂ l ∂ l Φ3 a + ∂ l ∂ l Ψ3 a = 12Γ M P (cid:18) ˙ ϕδ ˙ ϕ − dVdϕ δϕ (cid:19) . (36)In the above equations we neglect an overall factor of Γ in all the equations. Moreover, thedefinition of Θ is given by Eq. (29) evaluated for t = 3Θ ≡ M P (cid:0) α + 6 βJ (cid:1) (cid:0) Q − J (cid:1) . (37)Note that we can infer a generalization of these equations to any order t of Lovelock gravity,substituting Γ with Γ t and Ω with Ω t . For completeness we also write the Klein-Gordonequation for the scalar field, δ ¨ ϕ + 3 Hδ ˙ ϕ + d Vdϕ δϕ − ∂ l ∂ l ϕa = ˙ ϕ (cid:16) (cid:17) − dVdϕ Ψ , (38)which is not affected by the Lovelock term.We can also obtain an equation for the second derivative of the potential Φ only. Fromnow we will consider the case t = 2 for simplicity. We choose to start from the t − t equation833). From the derivative of the t − i equation (34) we can find the term ˙ ϕδ ˙ ϕ which we cansubstituted in the t − t equation. Then from the traceless part of the i − j equation we canfind the relation between Φ and Ψ which readsΨ = Φ (cid:20) α ( Q − J )Γ M P (cid:21) . (39)Substituting also this, and using the background equations, we obtain¨Φ+ H (cid:20) α ( Q − J ) M P Γ (cid:21) ˙Φ+ (cid:26) Q − J ) − κa + 8 α Γ M P h ( Q − J ) + (cid:16) Q − κa (cid:17) ˙ H + H ¨ H i(cid:27) Φ − ∂ l ∂ l Φ a == − δϕM P Γ (cid:18) H ˙ ϕ + dVdϕ (cid:19) . (40)We see that the velocity (the coefficient of the gradient of Φ) is equal to the speed of light.Therefore, the velocity is not modified with respect to the case of General Relativity by theaddition of the regularized Gauss-Bonnet term. VI. INFLATION
In this section we apply the Regularized Lovelock theory to inflation, in the flat case k = 0 and at second order in Lovelock series, i.e. β = 0. We split the discussion in twoparts, respectively the scalar and tensor sectors. Our derivation resembles the one in [46]. A. Scalar waves during inflation
By introducing u = s Γ − ˙ H Φ , (41)we can rewrite Eq. (40) in the flat case as¨ u + H ˙ u + " H − H H − H ¨ H H + ... H H − α ˙ HM P Γ H − H Γ ! u − ∂ l ∂ l ua = 0 , (42)where we have used (34) with (39).We can rewrite the above equation in terms of the conformal time dη = dt/a as u ′′ a − ∂ l ∂ l ua + (cid:20) H H ′ a − HH ′′ aH ′ − H ′′ a H ′ + H ′′′ a H ′ − αH ′ M p Γ a (cid:18) H − H ′ a Γ (cid:19)(cid:21) u = 0 , (43)9hich can be further simplified in the form u ′′ − ∂ l ∂ l u − θ ′′ θ u = 0 , (44)where θ = 1 a r aH − H ′ √ Γ . (45)In the short-wavelenght limit | θ ′′ /θ | ≪ k , k being the Fourier vector norm associated tothe ~r vector, we obtain the solution u ( η ) = C e ± kη , (46)where C is an integration constant.On the other hand, in the long-wavelength limit | θ ′′ /θ | ≫ k , we have u ( η ) ≃ C θ + C θ Z dηθ , (47)where the C mode can be absorbed in the integral and Z dηθ = (cid:18) aH − Z a dη (cid:19) − αH M P (cid:20) aH − H Z H a dη (cid:21) . (48)When the slow roll approximation is valid the above result can be approximated as Z dηθ ≃ − Γ H ′ H , (49)such that one finally obtains for the long-wavelength limit, u ( η ) = C p − H ′ /aH Γ , (50)where C is an integration constant. Thus, the factor Γ contains the correction given by theRegularized Lovelock Gravity to the result of General Relativity.We also introduce a canonical quantization variable, called v , which should be derivedfrom a well-posed Lagrangian derivation. This permits to normalize the vacuum quantumfluctuations generated at the beginning of inflation. We can infer the action for cosmologicalperturbations from the equations of motion. In particular, Eq. (44) is a direct consequenceof the two relations ∂ l ∂ l u = z (cid:16) vz (cid:17) ′ , v = θ (cid:16) uθ (cid:17) ′ , (51)where z ≡ /θ . For comparison with the set of field equations presented in the Sec. V, weshould consider v = a [ δϕ + ( aϕ ′ /a ′ )Φ] / p M P .10he quadratic action for perturbations in terms of v is S v = 12 Z dη d x (cid:18) v ′ + v∂ l ∂ l v + z ′′ z v (cid:19) , (52)with the associated equation of motion, v ′′ − ∂ l ∂ l v − z ′′ z v = 0 . (53)After decomposing v in Fourier modes v k and introducing the Bunch-Davies vacuum stateas the boundary condition of the solution, such that v k ( η i ) = 1 / √ k and v ′ k ( η i ) = i √ k at thereference (conformal) time η i , we can use the Eq.s (46) and (51) to derive the correspondingmodes u k in the short-wavelength limit. We obtain u k ( η ) = − ik / e ik ( η − η i ) . (54)After a Fourier mode enters the long-wavelength regime, its evolution is described by Eq.(47), namely u k ( η ) ≃ A k θ Z dηθ == A k √ Γ r a − H ′ (cid:20)(cid:18) − Ha Z a dη (cid:19) − αH M P (cid:18) − aH Z H a dη (cid:19)(cid:21) , (55)where the amplitude A k can be fixed by comparing (54) and (55) at the time when theFourier mode crosses the Hubble horizon. Thus, using the slow-roll approximation derivedin (50) for the modes u k , after the identification of C = A k we find A k ≃ − ik / H p − H ′ /a ! k ≃ Ha . (56)Thus, taking into account (41), the scale-invariant power spectrum of long-wavelength scalarperturbations is δ = 18 π M P (cid:18) − H ′ a Γ (cid:19) | u k | k == 18 π M P (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) aH − H ′ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) k ≃ Ha (cid:20)(cid:18) − Ha Z a dη (cid:19) − αH M P (cid:18) − aH Z H a dη (cid:19)(cid:21) , (57)which is valid for perturbations with wavelength λ ≡ π/k > / ( Ha ). Note that, contrarilyto the case of General Relativity, the scalar perturbations are not completely frozen duringthe radiation dominated era before they re-enter the horizon. In fact, a time dependence11s preserved, as we can see from the following result. Using a scale factor time dependence a ( t ) ∝ t / , i.e. the scale factor time evolution in a radiation dominated period, we obtainthe following δ = 18 π M P (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) aH − H ′ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) k ≃ Ha (cid:18) − αH M P (cid:19) (cid:18) αH M P (cid:19) − , (58)and only if α = 0 we recover the constant result of General Relativity. B. Gravitational waves during inflation
In this section we study the tensor perturbations described by Eq. (25) in the context ofinflation. Also in this case, we consider β = 0 and k = 0. After the decomposition of h ij inFourier modes h k , and introducing a the new variable y k ≡ Γ a h k , (59)we obtain, in terms of the conformal time, y ′′ k − (cid:18) αH ′ Γ aM P (cid:19) ∂ l ∂ l y k −− (cid:20) a H + H ′ + αM P Γ (cid:18) H ′ H a − HH ′′ − H ′ Γ (cid:19)(cid:21) y k = 0 . (60)Note that this equation of motion can be obtained from a well posed action derivation.On the de Sitter space-time with H constant and a = − / ( Hη ) we simply have y ′′ k − ∂ l ∂ l y k − η y k = 0 . (61)The above equation form coincides with the one of General Relativity. In fact the Lovelockcorrections vanish as they depend on the derivatives of H , which are null on de Sitter.Considering the standard initial conditions given by y k ( η i ) = 1 / √ k and y k ( η i ) = i √ k with k | η i | ≫
1, we obtain the solution y k = 1 √ k (cid:18) ikη (cid:19) e ik ( η − η i ) , (62)such that the power spectrum of tensor perturbations is given by δ h = 1 π M | y k | k a Γ = 1 π M H Γ (1 + k η ) , ka ≫ Hηη i . (63)12n the other hand, long-wavelength tensor mode perturbations with λ a = 2 πa/k ≫ ( η i /η ) /H have a flat spectrum with an amplitude proportional to H/ Γ. Since during infla-tion H changes slowly, the power spectrum of long-wavelength tensor perturbations is fixedat the time when these perturbations cross the horizon, namely δ h = 1 π M H Γ | k ≃ Ha . (64)We note that the tensor perturbations power spectrum gains some corrections from theRegularized Lovelock Gravity through the factor Γ. C. Spectral indices and tensor-to-scalar power spectra ratio
In this subsection we show the equations for the spectral indices of scalar and tensorperturbations, and for the tensor-to-scalar power spectra ratio. The results are given interms of the cosmological time.The expression for the spectral index of the scalar perturbations is derived from Eq. (58).Using the definition, we obtain n s − ≡ dδ d log k = 4 ˙ HH − ¨ HH ˙ H − α ˙ HM P Γ | k ≃ Ha , (65)which can be checked against the experimental results [47] once the scalar field potential isfixed. The spectral index for the tensor perturbations is defined as n T ≡ dδ T d log k = 2 ˙ HH − α ˙ HM P Γ | k ≃ Ha , (66)where we used Eq. (64).Finally, the tensor-to-scalar spectra ratio results to be r ≡ δ δ h = 16 − ˙ HH ! | k ≃ Ha , (67)where the two polarizations of the tensor modes are taken into account.Therefore, the spectral indices results reduce to the ones of General Relativity in the limit α →
0, while the form of the tensor-to-scalar spectra ratio is not affected by RegularizedLovelock Gravity corrections. 13
II. CONCLUSIONS
The Regularized Lovelock Gravity theory permits to circumvent the Lovelock theorem.In this theory, the action is formulated in a generic dimensions and by adding a particularcombination of the curvature tensors, given by Lovelock theorem, to the Hilbert-Einsteinaction of General Relativity. Then, by using the dimensional regularization and by takingthe limit d → c T = 1 of General Relativity. These corrections can be arbitrarily small as they dependexplicitly on the action parameter α . Moreover, we were able to infer an expression for thevelocity of gravitational waves at a generic order, showing that higher order Lovelock termsintroduce additional corrections. On the other hand, we found that the velocity of scalarperturbations corresponds to the velocity of light at any order in Lovelock gravity.In the second part of our work, we studied the theory of cosmological perturbations ina slow-roll scalar field inflation, on the flat FLRW background. We found that the spectralindices gain some corrections with respect to General Relativity results. On the contrary,the tensor-to-scalar spectra ratio assumes the same form. Acknowledgments
This work has been partially performed using the software xAct [48] and xPand [49]. A. C.acknowledges the financial support of the Italian Ministry of Instruction, University and Re-search (MIUR) for his Doctoral studies. The authors thank Aimeric Coll´eaux, Massimiliano14inaldi, Silvia Vicentini and Sergio Zerbini for the useful discussions. [1] D. Glavan and C. Lin, Phys. Rev. Lett. (1971) 498.[3] D. Lovelock, J. Math. Phys. (1972) 874.[4] R. C. Myers and B. Robinson, JHEP (2010) 1008.[5] A. Cisterna, L. Guajardo, M. Hassane, J. Oliva, JHEP (2017).[6] S. Deser, O. Sarioglu, B. Tekin, Gen.Rel.Grav. (2008).[7] E. Bellini, R. Di Criscienzo, L. Sebastiani, S. Zerbini, Entropy (2010).[8] A. Coll´eaux, PhD thesis, Trento University (2019).[9] Y. Tomozawa, “Quantum corrections to gravity,” arXiv:1107.1424 [gr-qc].[10] A. Kehagias, K. Sfetsos, Phys. Lett . B (2009).[11] R. G. Cai, L. M. Cao and N. Ohta, JHEP (2010) 082.[12] M. Guo and P. C. Li, “The innermost stable circular orbit and shadow in the novel 4 D Einstein-Gauss-Bonnet gravity,” arXiv:2003.02523 [gr-qc].[13] R. A. Konoplya, A. F. Zinhailoa, “Quasinormal modes, stability and shadows of a black holein the novel 4D Einstein-Gauss-Bonnet gravity,” arXiv:2003.01188 [gr-qc].[14] P. G. S. Fernandes, “Charged Black Holes in AdS Spaces in 4D Einstein Gauss-Bonnet Grav-ity,” arXiv:2003.05491 [gr-qc].[15] R-G. Cai, Phys. Lett. B (2014) 183-189.[16] K. Hegde, A. N. Kumara, C. L. A. Rizwan, K. M. Ajith, Md S. Ali, “Thermodynamics,Phase Transition and Joule Thomson Expansion of novel 4-D Gauss Bonnet AdS Black Hole,”arXiv:2003.08778 [gr-qc].[17] S-W. Wei, Y-X. Liu, “Testing the nature of Gauss-Bonnet gravity by four-dimensional rotatingblack holeshadow,” arXiv:2003.07769 [gr-qc].[18] R. Kumar, S. G. Ghosh, “Rotating black holes in the novel 4D Einstein-Gauss-Bonnet gravity,”arXiv:2003.08927 [gr-qc].[19] M. Cuyubamba, “Stability of asymptotically de Sitter and anti-de Sitter black holes in 4 D regularized Einstein-Gauss-Bonnet theory,” arXiv:2004.09025 [gr-qc].
20] S. Yang, J. Wan, J. Chen, J. Yang and Y. Wang, “Weak cosmic censorship conjecture forthe novel 4 D charged Einstein-Gauss-Bonnet black hole with test scalar field and particle,”arXiv:2004.07934 [gr-qc].[21] A. Naveena Kumara, C. A. Rizwan, K. Hegde, M. S. Ali and A. K. M, “Rotating 4D Gauss-Bonnet black hole as particle accelerator,” arXiv:2004.04521 [gr-qc].[22] C. Zhang, S. Zhang, P. Li and M. Guo, “Superradiance and stability of the novel 4D chargedEinstein-Gauss-Bonnet black hole,” arXiv:2004.03141 [gr-qc].[23] M. Heydari-Fard, M. Heydari-Fard and H. Sepangi, “Bending of light in novel 4 D Gauss-Bonnet-de Sitter black holes by Rindler-Ishak method,” arXiv:2004.02140 [gr-qc].[24] C. Liu, T. Zhu and Q. Wu, “Thin Accretion Disk around a four-dimensional Einstein-Gauss-Bonnet Black Hole,” arXiv:2004.01662 [gr-qc].[25] A. Kumar and S. G. Ghosh, “Hayward black holes in the novel 4 D Einstein-Gauss-Bonnetgravity,” arXiv:2004.01131 [gr-qc].[26] S. U. Islam, R. Kumar and S. G. Ghosh, “Gravitational lensing by black holes in 4 D Einstein-Gauss-Bonnet gravity,” arXiv:2004.01038 [gr-qc].[27] D. V. Singh, S. G. Ghosh and S. D. Maharaj, “Clouds of string in the novel 4 D Einstein-Gauss-Bonnet black holes,” arXiv:2003.14136 [gr-qc].[28] S. Wei and Y. Liu, “Extended thermodynamics and microstructures of four-dimensionalcharged Gauss-Bonnet black hole in AdS space,” arXiv:2003.14275 [gr-qc].[29] C. Zhang, P. Li and M. Guo, “Greybody factor and power spectra of the Hawking radiationin the novel 4 D Einstein-Gauss-Bonnet de-Sitter gravity,” arXiv:2003.13068 [hep-th].[30] A. Kumar and R. Kumar, “Bardeen black holes in the novel 4 D Einstein-Gauss-Bonnet grav-ity,” arXiv:2003.13104 [gr-qc].[31] R. Konoplya and A. Zhidenko, “(In)stability of black holes in the 4D Einstein-Gauss-Bonnetand Einstein-Lovelock gravities,” arXiv:2003.12492 [gr-qc].[32] R. Kumar and S. G. Ghosh, arXiv:2003.08927 [gr-qc].[33] R. Konoplya and A. Zhidenko, Phys. Rev. D (2020), 084038doi:10.1103/PhysRevD.101.084038 arXiv:2003.07788 [gr-qc].[34] S. Wei and Y. Liu, “Testing the nature of Gauss-Bonnet gravity by four-dimensional rotatingblack hole shadow,” arXiv:2003.07769 [gr-qc].[35] G. Cognola, R. Myrzakulov, L. Sebastiani and S. Zerbini, Phys. Rev. D (2013) no.2, 024006.
36] S. Carlip, Class. Quant. Grav (2000) 4175-4186.[37] J. Engle, K. Noui, A. Perez, Phys. Rev. Lett. (2010) 031302.[38] A. Casalino, A. Colleaux, M. Rinaldi and S. Vicentini, “Regularized Lovelock gravity,”arXiv:2003.07068 [gr-qc].[39] M. Gurses, T. C. Sisman and B. Tekin, “Is there a novel Einstein-Gauss-Bonnet theory infour dimensions?,” arXiv:2004.03390 [gr-qc].[40] W. Ai, “A note on the novel 4D Einstein-Gauss-Bonnet gravity,” arXiv:2004.02858 [gr-qc].[41] D. Malafarina, B. Toshmatov and N. Dadhich, “Dust collapse in 4D Einstein-Gauss-Bonnetgravity,” arXiv:2004.07089 [gr-qc].[42] S. Mahapatra, “A note on the total action of 4 D Gauss-Bonnet theory,” arXiv:2004.09214[gr-qc].[43] H. Lu and Y. Pang, “Horndeski Gravity as D → et al. [LIGO Scientific and Virgo], Phys. Rev. Lett. (2017) no.16, 161101.[46] Physical Foundations of Cosmology, V. Mukhanov (Munich U. ), 2005, 421p.[47] N. Aghanim et al. [Planck], “Planck 2018 results. VI. Cosmological parameters,”arXiv:1807.06209 [astro-ph.CO].[48] J. M. Mart´ın-Garc´ıa et. al., “xAct: Efficient tensor computer algebra for Mathematica,” url: http://xact.es/ .[49] C. Pitrou, X. Roy and O. Umeh, Class. Quant. Grav. (2013) 165002.(2013) 165002.