Inflationary universe from higher-derivative quantum gravity
aa r X i v : . [ g r- q c ] A p r Inflationary universe from higher-derivative quantum gravity
R. Myrzakulov , S. D. Odintsov , , and L. Sebastiani Department of General & Theoretical Physics and Eurasian Center for Theoretical Physics,Eurasian National University, Astana 010008, Kazakhstan Consejo Superior de Investigaciones Cient´ıficas,ICE/CSIC-IEEC, Campus UAB,Facultat de Ci`encies, Torre C5-Parell-2a pl,E-08193 Bellaterra (Barcelona), Spain Instituci´o Catalana de Recerca i Estudis Avan¸cats (ICREA), Barcelona, Spain Inst.of Physics, Kazan Federal Univer.,420008 Kazan and Tomsk State Pedagogical Univer.,634061 Tomsk, Russia
We consider higher-derivative quantum gravity where renormalization group improved effectiveaction beyond one-loop approximation is derived. Using this effective action, the quantum-correctedFRW equations are analyzed. De Sitter universe solution is found. It is demonstrated that such deSitter inflationary universe is instable. The slow-roll inflationary parameters are calculated. Thecontribution of renormalization group improved Gauss-Bonnet term to quantum-corrected FRWequations as well as to instability of de Sitter universe is estimated. It is demonstrated that in thiscase the spectral index and tensor-to-scalar ratio are consistent with Planck data.
PACS numbers: 98.80.Cq, 12.60.-i, 04.50.Kd, 95.36.+x
I. INTRODUCTION
Recent more precise observational WMAP data [1] as well as corrected Planck constraints [2] increased the interestto the theoretical models for inflationary universe. There is large variety of the inflationary models (for review, see,for instance, Refs. [3]) which may comply with observational data, at least, up to some extent (see also Ref. [4] aboutBICEP experiment).In fact, during last years there was much activity in the account of quantum effects of General Relativity in theconstruction of inflationary universe (for the introduction and review, see Ref. [5]). Furthermore, recent study [6]indicates that quantum effects of specific models of (non-renormalizable) higher-derivative F ( R )-gravity may giveconsistent inflation which complies with Planck data. The next natural step is extension of quantum-correctedinflationary scenario for multiplicatively-renormalizable higher derivative gravity (for a general review, see Ref. [7]).The very interesting attempt in this direction has been recently made in Ref. [8]. Note that being multiplicatively-renormalizable one, higher-derivative quantum gravity is based on the use of higher-derivative propagator. As a result,such theory eventually leads to problem with unitarity what is related with well-known Ostrogradski instability ofhigher-derivative theories. In fact, there were made some attempts to resolve this problem with the proposal thatunitarity maybe restored at the non-perturbative level. However, there is no complete proof of non-perturbativerestoration of unitarity. Hence, so far this theory maybe considered as effective theory teaching us different generalaspects of quantum gravity.The purpose of the current work is the study of the inflationary universe in general higher-derivative quantum grav-ity [7]. Making use the fact that one-loop beta-functions of such theory are well-known and their asymptotically freeregime is well investigated, we apply renormalization group (RG) considerations to get RG improved effective actionin general higher-derivative gravity. This technique is well-developed in quantum field theory in curved spacetime [10].It permits to get the effective action beyond one-loop approximation, making sum of all leading logs of the theory.The paper is organized as follows. In Section , we present the renormalization-group improved effective action ofmultiplicatively-renormalizable higher-derivative gravity. In order to do so, the one-loop effective coupling constantsare used. Subsequently, the quantum-corrected equations of motion are derived on the flat Friedmann-Robertson-Walker space-time. In Section , using the asymptotic behaviour of the gravitational running constants, de Sitterinflationary universe is constructed. The asymptotically-free regime is discussed in detail. Section is devoted tothe study of the dynamics of such quantum-corrected inflation. It is shown that de Sitter space is unstable and canlead to a large amount of inflation. Slow-roll conditions are discussed and the expressions for slow-roll parameters arefound. In Section , we consider the contribution from total derivative and surface terms (topological Gauss-Bonnetterm and dalambertian of the curvature) to RG improved effective action. It is demonstrated that with these termsthe spectral index can be compatible with Planck data. Conclusions and final remarks are given in Section . II. RENORMALIZATION-GROUP IMPROVED EFFECTIVE ACTION ANDQUANTUM-CORRECTED FRW EQUATIONS
In this section we start from the general action of higher-derivative gravity which is known to be multiplicatively-renormalizable theory (see Ref. [7] for general introduction and review). The starting action has the following form : I = Z M d x √− g (cid:18) Rκ − Λ + aR µν R µν + bR + cR µνξσ R µνξσ + d (cid:3) R (cid:19) , (II.1)where g is the determinant of the metric tensor g µν , M is the space-time manifold, R , R µν , R µνξσ are the Ricciscalar, the Ricci tensor and the Riemann tensor, respectively, and (cid:3) ≡ g µν ∇ µ ∇ ν is the covariant d’Alembertian, ∇ µ being the covariant derivative operator associated with the metric g µν . Here, κ >
0, Λ , a , b , c and d are constantswhich characterize the gravitational interaction. The above lagrangian contains some terms not important in fourdimensions. First of all, we note that (cid:3) R is a surface term which does not give any contribution to the dynamicalequations. Second, we have R µν R µν = C − G R , R µνξσ R µνξσ = 2 C − G + R , (II.2)where G and C are the Gauss-Bonnet term and the “square” of the Weyl tensor, G = R − R µν R µν + R µνξσ R µνσξ , C = 13 R − R µν R µν + R ξσµν R ξσµν . (II.3)The Gauss-Bonnet term is a topological invariant in four dimensions, and we can drop it from the action. Thus, wecan rewrite the higher derivative terms with the help of the Weyl squared tensor.Let us express the constants which appear in the starting action in terms of more convenient coupling constantswhich stress that the theory under consideration is asymptotically-free one. In order to do it, we follow the notationsof Ref. [7]. To take into account quantum gravity effects we use the renormalization-group (RG) improved effectiveaction. The calculation of RG improved effective action has been developed in multiplicatively-renormalizable quantumfield theory in curved spacetime. In general terms, this technique is described in detail in Refs. [7, 10]. Recently,RG improved scalar potential in curved spacetime has been applied in the study of inflation [11]. In the simplestversion [10], RG improved effective action follows from the solution of RG equation applied to complete effectiveaction of the multiplicatively renormalizable theory. The final result is very simple: one has to replace constants inthe classical action by one-loop effective coupling constants where corresponding RG parameter is defined as log termof chacteristic mass scale in the theory.Applying the above considerations to higher-derivative quantum gravity, one can get RG improved effective actionas the following: I = Z M d √− g (cid:20) Rκ ( t ′ ) − ω ( t ′ )3 λ ( t ′ ) R + 1 λ ( t ′ ) C − Λ( t ′ ) (cid:21) . (II.4)The effective coupling constants λ ≡ λ ( t ′ ), ω ≡ ω ( t ′ ), κ ≡ κ ( t ′ ) and Λ ≡ Λ( t ′ ) obey to the one-loop RG equations [12] dλdt ′ = − β λ ≡ − (cid:18) (cid:19) λ , (II.5) dωdt ′ = − λ ( ωβ + β ) ≡ − λ (cid:18) ω + 18310 ω + 512 (cid:19) , (II.6) dκ dt ′ = κ γ ≡ κ λ (cid:18) ω − − ω (cid:19) , (II.7) d Λ dt ′ = β ( κ ) − γ Λ( t ′ ) ≡ λ ( κ ) (cid:18)
52 + 18 ω (cid:19) + λ Λ (cid:18)
283 + 13 ω (cid:19) . (II.8) Note that higher-derivative theory of the type of (II.1) as well as other higher-derivative modified gravities may even pass solar systemtests, for instance, due to chameleon scenario [9] and so on.
Note that κ ( t ′ ) is positive defined, and in general λ ( t ′ ) and Λ( t ′ ) are also positive defined to have a positive con-tribution to the Weyl tensor and a positive effective cosmological constant in the action; on the other hand, ω ( t ′ ) isexpected to be negative to have a positive R -term. In the above expressions, β , , and γ correspond to [7] β = 13310 , β = 103 ω +5 ω + 512 , β = λ (cid:18) ω (cid:19) + λ (cid:0) κ (cid:1) Λ (cid:18) ω + 15 − ω (cid:19) , γ = λ (cid:18) ω − − ω (cid:19) . (II.9)The RG parameter t ′ is given by t ′ = t ′ (cid:20) RR (cid:21) , (II.10)where t ′ > R is the mass scale for the Ricciscalar. We set R as the value of the Ricci scalar in the current nearly de Sitter universe ( R = 4Λ, Λ being thecosmological constant), such that t ′ ( R = R ) = 0 today, while in the past 0 < t ′ ( R < R ). Note that the de Sittersolution of the current accelerated expansion is a final attractor of Friedmann universe.For Eq. (II.5) we also have the explicit solution λ ( t ′ ) = λ (0)1 + λ (0) β t ′ , (II.11)where λ (0) is the integration constant corresponding to the value of λ at t ′ = 0, namely λ ( t = t ) ≡ λ ( R = R ) = λ (0).One important remark is in order: when we introduce the effective running constants in (II.1), we also get acontribution from the Gauss-Bonnet and (cid:3) R in RG improved effective action, since it is not more possible to writethe Gauss-Bonnet term like a total derivative and (cid:3) R in terms of a flux in three dimensions. This fact will bediscussed in below, but for the moment we work with the simplified action.Let us consider the flat Friedmann-Robertson-Walker (FRW) space-time, whose general form is given by ds = − N ( t ) dt + a ( t ) ( dx + dy + dz ) , (II.12)where a ≡ a ( t ) is the scale factor depending on the cosmological time t and N ≡ N ( t ) is an arbitrary lapse function,which describes the gauge freedom associated with the reparametrization invariance of the action. For the abovemetric, the Ricci scalar and the square of the Weyl tensor read R = 1 N " (cid:18) ˙ aa (cid:19) + 6 (cid:18) ¨ aa (cid:19) − ˙ NN ! (cid:18) ˙ aa (cid:19) , C = 0 , (II.13)where the dot denotes the derivative with respect to the cosmological time t . The fact that the Weyl tensor is zeroon the general form of the metric indicates that its contribution to the action and therefore to the derivation of thefield equations of the theory is null. In fact one can write on FRW background δI C = 1 λ ( t ′ ) δ (cid:0) √− gC (cid:1) + (cid:0) √− gC (cid:1) δ (cid:18) λ ( t ′ ) (cid:19) = 1 λ ( t ′ ) δ (cid:0) √− gC (cid:1) , (II.14)but 1 λ ( t ′ ) δ (cid:0) √− gC (cid:1) = 0 , (II.15)and it is well known that the square of the Weyl tensor does not enter in the Friedmann-like equations.To derive the equations of motion (EOMs), we will use a method based on the Lagrangian multiplayer [13–16]. Ifwe plug the expression for the Ricci scalar (II.13) into the action (II.4), we get higher derivative lagrangian theory.In order to derive a standard (first order) lagrangian theory, we introduce a Lagrangian multiplier ξ as [13, 14], I = Z M d √− g " Rκ ( t ′ ) − ω ( t ′ )3 λ ( t ′ ) R − Λ( t ) − ξ " R − N " (cid:18) ˙ aa (cid:19) + 6 (cid:18) ¨ aa (cid:19) − ˙ NN ! (cid:18) ˙ aa (cid:19) , (II.16)where we have taken into account (II.13). By making the derivation with respect to R , one finds ξ = − R ω ( t ′ )3 λ ( t ′ ) + 1 κ − ∆( t ′ ) dt ′ dR , (II.17)where ∆( t ′ ) = (cid:20) R ( κ ( t ′ )) dκ ( t ′ ) dt ′ + R ddt ′ (cid:18) ω ( t ′ )3 λ ( t ′ ) (cid:19) + d Λ( t ′ ) dt ′ (cid:21) , (II.18)since it is understood that the functions κ ( t ′ ) , Λ( t ′ ) , λ ( t ′ ) and ω ( t ′ ) depend on R throught t ′ as in Eq. (II.10).Therefore, by substituting (II.17) and making an integration by parts one obtains the (standard) Lagrangian L ( a, ˙ a, N, R, ˙ R ) = − N a Λ( t ′ ) − a aκ ( t ′ ) N + 6 ˙ aa ˙( κ ( t ′ )) N ( κ ( t ′ )) + ω ( t ′ )3 λ ( t ′ ) a N " R + 12 RN ˙ a a + 12 ˙ a ˙ RaN + ddt ′ (cid:20) ω ( t ′ )3 λ ( t ′ ) (cid:21) (cid:18) dt ′ dR ˙ R (cid:19) Ra ˙ aN + 6 a N (cid:18) R a a N (cid:19) ∆( t ′ ) dt ′ dR + 6 ˙ a (cid:18) a N (cid:19) " d ∆( t ′ ) dt ′ (cid:18) dt ′ dR (cid:19) + ∆( t ′ ) d t ′ dR ˙ R . (II.19)If we derive this Lagrangian with respect to N ( t ) and therefore we choose the gauge N ( t ) = 1, we get0 = − a Λ( t ′ ) + 6 ˙ a aκ ( t ′ ) − aa ˙( κ ( t ′ ))( κ ( t ′ )) + ω ( t ′ )3 λ ( t ′ ) a " R − R ˙ a a −
12 ˙ a ˙ Ra − Ra ˙ a ddt ′ (cid:18) ω ( t ′ )3 λ ( t ′ ) (cid:19) (cid:18) dt ′ dR ˙ R (cid:19) +6 a (cid:18) R − ˙ a a (cid:19) ∆( t ′ ) dt ′ dR − aa " d ∆( t ′ ) dt ′ (cid:18) dt ′ dR (cid:19) + ∆( t ′ ) d t ′ dR ˙ R . (II.20)The variation with respect to a ( t ) leads to0 = − a Λ( t ′ ) + 6 κ ( t ′ ) (cid:0) ˙ a + 2¨ aa (cid:1) + 6 κ ( t ′ ) a ˙( κ ( t ′ )) ( κ ( t ′ )) − aa ˙( κ ( t ′ )) κ ( t ′ ) − a ( ¨ κ ( t ′ )) κ ( t ′ ) ! + ω ( t ′ ) λ ( t ′ ) (cid:16) R a − R ˙ a − R ˙ aa − R ¨ aa − Ra (cid:17) − ddt (cid:18) ω ( t ′ ) λ ( t ′ ) (cid:19) h ˙ Ra + Ra ˙ a i − d dt (cid:18) ω ( t ′ ) λ ( t ′ ) (cid:19) Ra + (cid:0) a R − a − a ¨ a (cid:1) ∆( t ′ ) dt ′ dR − (cid:16) a ˙ a ˙ R + 6 a ¨ R (cid:17) " d ∆( t ′ ) dt ′ (cid:18) dt ′ dR (cid:19) + ∆( t ′ ) d t ′ dR − a ˙ R " d ∆( t ′ ) dt ′ (cid:18) dt ′ dR (cid:19) + 3 d ∆( t ′ ) dt ′ (cid:18) dt ′ dR (cid:19) d t ′ dR + ∆( t ′ ) d t ′ dR , (II.21)where we have set N ( t ) = 1 again and d/dt ≡ ˙ R ( dt ′ /dR ) d/dt ′ . Finally, the variation of the Lagrangian with respectto R , remembering that t ′ is a function of R , returns to be the expression in (II.13), and by putting N ( t ) = 1 we have R = 6 (cid:18) ˙ aa (cid:19) + 6 (cid:18) ¨ aa (cid:19) . (II.22)We obtained a system of three second order equations (II.20)–(II.22), where one is redundant (in the absence of mattercontributions), namely it can be derived from the other two.Eq.(II.20) and Eq. (II.22) can be rewritten as0 = − Λ( t ′ ) + 6 H κ ( t ′ ) − H ( κ ( t ′ )) dκ ( t ′ ) dt ′ t ′ ˙ RR ! + ω ( t ′ )3 λ ( t ′ ) h R ˙ H − H ˙ R i − H ddt ′ (cid:18) ω ( t ′ )3 λ ( t ′ ) (cid:19) (cid:16) ˙ Rt ′ (cid:17) +6 (cid:16) H + ˙ H (cid:17) ∆( t ′ ) t ′ R − H " d ∆( t ′ ) dt ′ (cid:18) t ′ R (cid:19) − ∆( t ′ ) t ′ R ˙ R , (II.23) R = 12 H + 6 ˙ H , (II.24)where we have introduced the Hubble parameter H = ˙ a/a and we have used (II.10) to write dt ′ /dR = t ′ /R . In thefollowing expression, we explicit develop Eq. (II.23) in terms of the functions λ ( t ′ ) , ω ( t ′ ) , κ ( t ′ ) and Λ( t ′ ) by usingthe set of equations (II.5)–(II.8) and Eq. (II.24) for the Ricci scalar,0 = 12 ω (cid:16) − H ˙ H − H ¨ H + ˙ H (cid:17) λ − Hλt ′ (cid:0) ω − ω − (cid:1) (4 H ˙ H + ¨ H )2 κ ω (cid:16) H + ˙ H (cid:17) + 6 H κ − t ′ κ ω (cid:16) H + ˙ H (cid:17) (cid:18) H (4 H ˙ H + ¨ H ) (cid:18) λt ′ (cid:18) κ ω (4 ω + 3)(2 ω (100 ω + 549) + 25) (cid:16) H + ˙ H (cid:17) − κ λω (cid:16) H ( ω ( ω (20 ω (100 ω + 409) − H ( ω ( ω (20 ω (100 ω + 409) − κ Λ(4 ω (1616 ω − − (cid:1) − λ ( ω (2 ω (4 ω (50 ω + 97) − − − (cid:1) − κ ω (cid:16) H + ˙ H (cid:17)(cid:16) κ ω (4 ω (2 ω + 3) + 1) (cid:16) H + ˙ H (cid:17) + λ (cid:0) − ω + 26 ω + 3 (cid:1)(cid:17)(cid:17) + 15 ω (cid:16) H + 7 H ˙ H + H ¨ H + ˙ H (cid:17)(cid:18) κ ω (4 ω (2 ω + 3) + 1) (cid:16) H + ˙ H (cid:17) + 4 κ λω (cid:16) H (cid:0) − ω + 26 ω + 3 (cid:1) + ˙ H (6(13 − ω ) ω + 9) − κ Λ(28 ω + 1) (cid:1) − λ (cid:0) ω + 1 (cid:1)(cid:1)(cid:1) + 10 Ht (cid:0) ω + 12 ω + 1 (cid:1) (4 H ˙ H + ¨ H ) − Λ . (II.25)Here, λ ≡ λ ( t ′ ), ω ≡ ω ( t ′ ), κ ≡ κ ( t ′ ) and Λ ≡ Λ( t ′ ). One should remember that t ′ is related to R as in Eq. (II.10),and only λ ( t ′ ) is given by (II.11). Note that the above approach suggests the consistent way to account for quantumeffects of higher-derivative gravity. Note also that different approach to take into account such quantum effects at theinflationary universe was developed in Ref. [8].On the de Sitter solution R dS = 12 H , where H dS is a constant, the system is simplified as0 = 6 H κ − t ′ κ ) ω (cid:0) H ( κ ) ω (4 ω (2 ω + 3) + 1) + 4 κ λω (cid:0) H (cid:0) − ω + 26 ω + 3 (cid:1) − κ Λ(28 ω + 1) (cid:1) − λ (cid:0) ω + 1 (cid:1)(cid:1) − Λ , (II.26)where the functions λ , ω , κ and Λ are assumed to be constant and H ≡ H dS .Hence, we obtained consistent system of quantum-corrected FRW equations from RG improved effective actioncorresponding to higher-derivative quantum gravity. III. ASYMPTOTIC BEHAVIOUR OF THE EFFECTIVE COUPLING CONSTANTS AND DE SITTERSOLUTION FOR INFLATION
In order to solve the system (II.25), we need to investigate the asymptotic behaviour of the implicitly-given effectivecoupling constants ω ( t ′ ) , κ ( t ′ ) , Λ( t ′ ), when t ′ → ∞ , namely at the high curvature limit ( R → ∞ ) describing inflation(see (II.10)). Eq. (II.6) has two fixed points at ω ≃ − . , ω ≃ − . , (III.1)and the analysis of the solution around the fixed points ω ( t ′ ) = ω , + δω ( t ′ ), with | δω ( t ′ ) | ≪
1, leads to dω ( t ′ ) dt ′ ≃ − λ ( t ′ ) (cid:18) ω + 18310 (cid:19) | ω , δω ( t ′ ) − λ ( t ′ ) β (cid:18) dt ′ dω ( t ′ ) (cid:19) (cid:18) ω + 18310 ω + 512 (cid:19) | ω , δω ( t ′ )= − λ ( t ′ ) (cid:18) ω + 1585 (cid:19) | ω , δω ( t ′ ) , (III.2)such that, ω ( t ′ ) = ω , + c (1 + λ (0) β t ′ ) q , q = 1 β (cid:18) ω + 1585 (cid:19) | ω , , | c | ≪ , (III.3)where c is a constant and we have introduced λ ( t ′ ) as in (II.11). We immediatly see that q ≃ .
37 for ω renderingthe solution stable when t ′ → ∞ , but for ω one gets q ≃ − .
37 and the solution is unstable when t ′ → ∞ . Thus,we expect that for large values of t ′ the function ω ( t ′ ) tends to the attractor ω . Since between ω and ω thederivative dω ( t ′ ) /dt ′ with 0 < λ ( t ′ ) is positive, ω ( t ′ ) grows up with t ′ and approaches to ω being ω ( t ′ ) < ω . When ω < ω ( t ′ ) < ω we may estimate from (III.2), dω ( t ′ ) dt ′ = − λ ( t ′ )2 (cid:18) (cid:19) ( ω − ω ) δω ( t ′ ) . (III.4)Therefore, the solution (III.3) is rewritten as (see third Ref. in [12]), ω ( t ′ ) = ω + c (1 + λ (0) β t ) p , p = (cid:18) (cid:19) ( ω − ω ) β ≃ . , | c | ≪ . (III.5)Note that related study for the behaviour of above dimensionless coupling constants in relation with dimensionaltransmutation is given in Ref. [17].In order to study the behaviour of κ ( t ′ ) and Λ( t ′ ), we introduce˜Λ( t ′ ) = ( κ ( t ′ )) Λ( t ′ ) , (III.6)and Eq. (II.8) with Eq. (II.7) lead to d ˜Λ( t ′ ) dt ′ = β ≡ λ ( t ′ ) (cid:18) ω ( t ′ ) (cid:19) + λ ( t ′ )˜Λ( t ′ ) (cid:18) ω ( t ′ ) + 5 − ω ( t ′ ) (cid:19) . (III.7)In the asymptotic limit ω ( t ′ ) ≃ ω we get˜Λ = − λ (0)(1 + 20 ω )4 ω (1 + λ (0) β t ′ )( − ω + 6 β ω + 40 ω ) + ˜Λ (1 + λ (0) β t ′ ) W/β , W = 203 ω + 5 − ω = 13 . . (III.8)As a consequence, ˜Λ( t ′ ) ≃ ˜Λ (1 + λ (0) β t ′ ) W/β , (III.9)where the constant ˜Λ is assumed to be positive. On the other side, from Eq. (II.7) we have at ω ( t ′ ) ≃ ω , κ ( t ′ ) ≃ κ (1 + λ (0) β t ′ ) Z/β , Z = (cid:18) ω − − ω (cid:19) ≃ . , (III.10)such that finally Λ( t ′ ) ≃ ˜Λ ( κ ) (1 + λ (0) β t ′ ) X/β , X = ( W − Z ) ≃ − . . (III.11)Let us summarize the results. From the investigation of the asymptotic region, we can derive the effective runningcoupling constants of the model (II.4) as λ ( t ′ ) = λ (0)(1 + λ (0) β t ′ ) , ω ≃ ω + c (1 + λ (0) β t ′ ) . , κ ( t ′ ) ≃ κ (1+ λ (0) β t ′ ) . , Λ( t ′ ) ≃ Λ λ (0) β t ′ ) . . (III.12)Here, Λ = ˜Λ / ( κ ) and | c | ≪ | ω | , and we will omit its contribution at large t ′ . One remark is in order. In principlethese expressions correspond to the behaviour of the coupling constants in the high energy limit, when t ′ → ∞ and R ≪ R , R being the Ricci scalar at the present time, and they are valid as soon as ω ( t ′ ) is close to ω . However,we may assume that the structure of the coupling constants keeps the same form at every epoch, since in fact outof inflation the curvature of the universe drastically decreases, t ′ →
1, and the coupling constants are expected tobe constant: in fact, we can consider ω ( t ′ ) sufficiently close to − ω at every time, namely we will not consider theadditional corrections at small curvature. In particulary, at the present de Sitter epoch with R = R and t ′ = 0 (seeEq. (II.10) and the comment below) we must find κ ( t ′ ) ≡ κ = 16 πM P l , Λ( t ′ ) ≡ Λ = 2Λ , (III.13)where M P l is the Planck mass and Λ is the cosmological constant, which is much smaller than the curvature at theinflation scale. By considering λ (0) of the order of the unit to avoid the R -correction at the present epoch, at thetime of inflation one can put Λ( t ′ ) = 0.Let us assume that R = R dS describes the curvature of (de Sitter) inflation. Since it must be R ≪ R dS ≡ H ,where R = 4Λ, one has log (cid:20) R dS R (cid:21) = log (cid:2) H κ (cid:3) − log (cid:20) Λ3 κ (cid:21) ≃ − log (cid:20) Λ3 κ (cid:21) . (III.14)Thus, from (II.10) we get t ′ ≃ − t ′ log (cid:20) Λ3 κ (cid:21) , ≪ t ′ , (III.15)namely t ′ expresses the rate of the curvature of the current universe with respect to the Planck mass on logaritmscale: this approximation is valid as soon as R dS is near to M P l during inflation, where “near” is understood as “withrespect to the cosmological constant scale”. In fact, the solution of Eq. (II.26) depends on the value of today λ (0),which fixes the bound of inflation. From (II.26), we derive the following solution, H κ ≃ . t ′ ( λ (0) t ′ ) . ≡ . t ′ . ( λ (0)) . (cid:2) − log (cid:2) Λ3 κ (cid:3)(cid:3) . , (III.16)where we have taken into account that 1 ≪ t ′ . If we use the recent cosmological data [1] for the evaluation of Λ inPlanck units (see also Ref. [18]), Λ κ ≃ . × − , (III.17)and we set for simplicity t ′ = 1, we finally obtain H κ ≃ × − λ (0) . . (III.18)For example, for λ (0) = 1, we have − ω λ (0) (cid:0) κ (cid:1) R ≃ . × − R ≪ R , . × − M P l ≃ (cid:18) Λ3 (cid:19) ≪ H ≃ . × − M P l . (III.19)The first condition guarantees that at the present epoch the R -contribution to the action (II.4) is negligible withrespect to the Hilbert-Einstein term R /κ , where R = 4Λ. The second condition shows that de Sitter solution ofinflation takes place at very high curvature near to the Planck scale, such that the approximation (III.14) is wellsatisfied. We also note that during inflation Rκ ( t ′ ) ≃ . × − M P l ≪ − ω ( t ′ )3 λ ( t ′ ) R ≃ . × − M P l , (III.20)and the second term in (II.4) is dominant with respect to the Hilbert-Einstein contribution at the early universe,thanks to the fact that the running constant κ ( t ′ ) increases back into the past. IV. DYNAMICS OF INFLATION
In this section, we would like to analyze the behaviour of the model (II.4) at high curvature, when the de Sittersolution describing inflation (III.16) takes place. First of all, in order to have the exit from inflation, one must showthat the solution is unstable. Hence, we can try to describe the inflation in terms of e -folds number and slow-rollparameters. A. Instability of de Sitter universe
Let us consider the following form of Hubble parameter which is used in Eq. (II.25), H = H dS + δH ( t ) , | δH ( t ) | ≪ , (IV.1)where δH ( t ) is the perturbation with respect to de Sitter inflation. By making use of Eq. (II.26) and (III.12)–(III.13)with c , Λ = 0 in Eq. (II.25), and by multiplying it by κ , one has at the first order in δH ( t ) ≡ δH ,0 = ( κ ˙ δH ) " t ′ ( H dS κ ) (cid:18) . − . t ′ t ′ (cid:19) + 0 . t ′ + 0 . t ′ t ′ ( H dS κ ) ( λ (0) t ′ ) . + 0 . t ′ − . t ′ t ′ ( λ (0) t ′ ) . ! + 19 . t ′ ( H dS κ ) + ( κ ¨ δH ) t ′ ( H dS κ ) (cid:20) t ′ ( H dS κ ) (cid:0) . t ′ + t ′ (11 . t ′ − . t ′ ) (cid:1) − . t ′ t ′ ( H dS κ ) ( λ (0) t ′ ) . + 0 . t ′ ( λ (0) t ′ ) . + 0 . t ′ t ′ ( H dS κ ) ( λ (0) t ′ ) . + 2 × − t ′ t ′ ( λ (0) t ′ ) . + ( H dS κ ) δH (cid:20) . λ (0) t ′ ) . + 0 . λ (0) t ′ ( λ (0) t ′ ) . − . t ′ ( H dS κ ) (cid:21) . (IV.2)If we assume 1 ≪ ( H dS κ ) t ′ . , (IV.3)the above expression is simplified as D δH + t ′ [19 . H dS κ )( κ ˙ δH ) + 6 . κ ¨ δH )] ≃ , (IV.4)where D = (cid:18) . λ (0) t ′ ) . − . t ′ ( H dS κ ) (cid:19) . (IV.5)Thus, the solution of the equation reads δH = h ± exp [ A ± t ] , A ± = " H dS − ± s − . D ( H dS κ ) t ′ ! , | h ± | ≪ , (IV.6)where h ± are the integration constants corresponding to plus and minus signs inside A ± . By choosing the sign plusin (IV.6), the solution is unstable under the condition D < . (IV.7)We would like to note that if we ignore the contribution from δH in (IV.4), we get − ω λ h ( − H ) ˙ δH ( t ) − δH ( t ) i ≃ , (IV.8)which is the equation for perturbation around the de Sitter solution in pure R -theory with Lagrangian L = − ( ω/ (3 λ )) R , ω/ λ being constant. From this equation is not possible to know if the solution is stable or not,since δH mainly goes like δH ∼ const in the time and even a small contribution from the coefficient in front of δH ( t ) could make the solution unstable, such that a further analysis is required. In particular, the fact that thecoefficient in front to R is not a constant contributes to the instability of the solution, since for the Lagrangian L = − ( ω ( t ′ ) / (3 λ ( t ′ ))) R we get the equation − ω λ h ( − H ) ˙ δH − δH i + (24 H dS ) (6 H dS ) ddR (cid:18) ω ( t ′ )3 λ ( t ′ ) (cid:19) δH ≃ , (IV.9)where we have omitted the additional contributions to ˙ δH , ¨ δH . The term related to δH corresponds to the last termof D in (IV.5), and, if it is dominant, it makes the solution (IV.6) unstable.Let us discuss the conditions (IV.3) and (IV.7). If0 . t ( λ (0) t ′ ) . < ( H dS κ ) , (IV.10)both of the conditions are well satisfied and by taking into account de Sitter solution (III.16) we see that this formulaholds always true and it is independendent on the bound of inflation encoded in λ (0)! It means, that de Sitter solutionis unstable with D ≃ − . λ (0) t ′ ) . , (IV.11)where we have used (III.16). Moreover, A + ≃ . H dS t ′ t ′ , A − ≃ − H dS , (IV.12)where D has been considered very small. For example, by setting H dS κ with (III.16)–(III.17) and by putting t ′ = 1and λ (0) = 1, one derives δH = h − e − × − M Pl t + h + e . × − M Pl t . (IV.13)During inflation, as soon as t ≪ /A + , avoiding the contribution of h − which quickly disappears, one may estimate δH ≃ h + , ˙ δH ≃ h + A + , ¨ δH ≃ h + A , (IV.14)where A + is the instability parameter. The duration of inflation ∆ t is of the order of magnitude∆ t ∼ A + , (IV.15)but may continue after the linear approximation of the perturbation. In the case of (IV.13) one has∆ t ∼ × M P l . (IV.16)The inflation solves the problems of initial conditions of the Friedmann universe (horizon and velocities problems), if˙ a i / ˙ a < − , where ˙ a i , ˙ a are the time derivatives of the scale factor at the Big Bang and today, respectively, and10 − is the estimated value of the inhomogeneity (anisotropy) in our universe. Since at decelerating universe ˙ a ( t )decreases by a factor 10 , it is required that ˙ a i / ˙ a f < − , with a i the scale factor at the beginning of inflation and a f the scale factor at the end of inflation. If inflation is governed by a (quasi) de Sitter solution where a ( t ) = exp ( H dS t ),we introduce the number of e -folds N as N = ln (cid:18) a f a i (cid:19) ≡ Z t f t i H ( t ) dt , (IV.17)and inflation is viable if N >
76, but the spectrum of fluctuations of CMB say that it is enough
N ≃
55 to havethermalization of observable universe. In our case, N ≃ H dS ∆ t ∼ H dS A + ≃ . (cid:18) t ′ t ′ (cid:19) , (IV.18)due to the fact that the Hubble parameter is almost a constant during inflation. In order to obtain a viable inflationit must be 61 < (cid:18) t ′ t ′ (cid:19) . (IV.19)It means, from (II.10) and (III.17), 3 . R × < R , (IV.20)and this condition is always satisfied for realistic inflation. For the case of (III.18), where the Hubble parameterduring inflation is 117 times larger than today and whose duration of inflation is given by (IV.16), we get N ∼ , (IV.21)and it is guaranteed the thermalization of a portion of universe much larger with respect to the observed one.It is clear that a large e -folds number, which corresponds to a huge amount of inflation, may be related to thefact that the universe remains extremely close to the de Sitter space-time during inflation. In fact, even if, withoutadditional data about the decay of the primordial accelerated expansion (the so called “false vacuum”), we cannotpose any upper limit to the e -folds number and we could expect that the homogeneity and isotropy continue for somedistance beyond our observable universe, the primordial perturbations at the end of inflation depend on the e -folds.As a consequence, as we will see in the next subsection, a large e -folds could generate wrong predictions for thespectral index. In the last part of the work we will find how it is possible to make inflation shorter according with acorrect prediction of such index.0 B. Slow-roll parameters and spectral index
During the inflation the Hubble parameter must slowly decrease and the following approximations must be meet, | ˙ HH | ≪ , | ¨ HH ˙ H | ≪ . (IV.22)Thus, one introduces the slow-roll parameters ǫ = − ˙ HH , η = − ˙ HH − ¨ H H ˙ H ≡ ǫ − ǫH ˙ ǫ , (IV.23)whose magnitude must be small during inflation and ˙ H is assumed to be negative. In particular, since the accelerationis expressed as ¨ aa = ˙ H + H , (IV.24)we see that the universe expands in accelerated way as soon as ǫ <
1. By integrating the formula for the (positiveand almost constant) ǫ parameter in (IV.23) we also get H ( t ) = 1 ǫ ( t dS + t ) , t dS ≃ ǫH dS , (IV.25)where t dS is a positive time parameter and when the time increases the Hubble parameter decreases. In the limit t/t dS ≪
1, one has H ( t ) ≃ H dS − H ǫt , (IV.26)and by taking into account (IV.14) we get ǫ ≃ ( − h + ) A + ( H dS ) = 0 . (cid:18) t ′ t ′ (cid:19) ( − h + ) H dS , (IV.27)where h + < A + is given by (V.19). This relation is consistent with a direct evaluation of the slow-roll parameter ǫ (IV.23) in the slow-roll limit (IV.22) of the equation of motion (II.25),0 = 2 λ ǫ (cid:2) H κ ω (4 ω + 3)(2 ω (100 ω + 549) + 25) + 2 κ λ Λ ω (4(355 − ω ) ω + 45) − λ ( ω (2 ω (4 ω (50 ω + 97) − − − (cid:3) + 720 κ ω (cid:0) H κ ωǫ + 6 H λ − κ λ Λ (cid:1) +15 λω (cid:0) − H κ ω (4 ω (2 ω + 3) + 1)(8 ǫ + 1) + 4 κ λω (cid:0) H (4 ω − ω + 1)(7 ǫ + 2) + 2 κ Λ(28 ω + 1) (cid:1) + λ (cid:0) ω + 3 (cid:1)(cid:1) . (IV.28)By using (III.12)–(III.13) with c = Λ = 0, one obtains the solution ǫ ≃ − × − t ′ t ′ ( λ (0) t ′ ) . − . λ (0) t ′ ( Hκ ) ( λ (0) t ′ ) . − . Hκ ) ( λ (0) t ′ ) . + 7 . t ( Hκ ) − . t ′ ) t ′ ( λ (0) t ′ ) . + . t ′ ) ( Hκ ) t ′ + t ′ ( Hκ ) (cid:16) . λ (0)( λ (0) t ′ ) . − .
056 ( Hκ ) (cid:17) − . t ′ ( Hκ ) , (IV.29)and under the condition (IV.3) we derive ǫ ≃ . t ′ ( λ (0) t ′ ) . ( Hκ ) − . t ′ . (IV.30)By expanding H ( t ) around de Sitter solution (III.16) we finally get ǫ ≃ − . t ′ ( λ (0) t ′ ) . ( H dS κ ) κ δH = 0 . t ′ ( λ (0) t ′ ) . ( H dS κ ) ǫκ A + , (IV.31)where Eqs. (IV.14) and (IV.27) are considered: the equation is well satisfied by using (III.16) again and (V.19). Thus,the ǫ slow-roll parameter is related to the (initial) amplitude of perturbation and by using (IV.18) one may estimate ǫ ≃ ( − h + ) A + ( H dS ) ∼ ( − h + )( H dS ) N . (IV.32)1Moreover, for the η slow-roll parameter in (IV.23) with (IV.14) one has η ≃ − A + H dS ≃ − . t ′ t ′ ∼ N . (IV.33)Both of the paramter ǫ , | η | (IV.32)–(IV.33) are very small during inflation and the slow-roll approximations (IV.22)hold true. We also note that, since | h + | ≪ H dS , ǫ ≪ | η | , (IV.34)like in other scalar tensor theories for inflation, where usually ǫ ∼ /N , as in (IV.32) if we consider ( − h + ) /H dS ∼ /N .Given the slow-roll parameters, one can evaluate the universe anisotropy coming from inflation by introducing thespectral indexes. To be specific, the amplitude of the primordial scalar power spectrum reads∆ R = κ H π ǫ , (IV.35)and for slow-roll inflation the spectral index n s and the tensor-to-scalar ratio are given by n s = 1 − η , r = 48 ǫ , (IV.36)where we use the results for modified gravity [19]. The last Planck data [1] constrain these quantities as n s = 0 . ± . , r < . . (IV.37)For our model one has the scalar power spectrum∆ R ≃ . H dS κ ) (cid:18) t ′ t ′ (cid:19) ( − κ h + ) − , (IV.38)and the spectral index and the tensor-to-scalar ratio, n s = 1 − A + H dS ∼ − N , r = 48 A H ( − h + ) H ≪ N , (IV.39)where we have used (IV.34). We see that the tensor-to-scalar ratio can satisfy the Planck results, being the e -folds ofrealistic inflation quite large. On the other side, in order to find the spectral index n s in agreement with the Planckdata (IV.37), we must require 21 < A + H dS (cid:18) = 2 . (cid:18) t ′ t ′ (cid:19)(cid:19) < , (IV.40)Since A + /H dS depends on the ratio between the curvature of the universe at the time of inflation and the curvatureof today universe, it results particulary high and does not satisfy this condition, contributing to render near to onethe spectral index n s of the model. For example, in the case of (III.18) where the Hubble parameter during inflationis 117 times larger than today and the e -folds N ∼ H dS / ( A + ) ≃
339 as in (IV.21), n s ≃ . , r ≃ . − h + ) H . (IV.41)Since ( − h + /H dS ) ≪
1, the tensor-to-scalar ratio is much smaller than 0 .
11, but the spectral index does not satisfy thePlanck data. This should be compared with analysis of inflationary parameters for general F ( R )-theory in fluid-likepresentation [20] which maybe consistent with Planck data.The large e -folds number and the n s spectral index too close to one are consequences of the small value of A + (V.19),which depends on d ( ω ( t ′ ) / λ ( t ′ )) /dt ′ , as we explained under (IV.9). In particulary, the fact that d ( ω ( t ′ ) / λ ( t ′ )) /dt ′ = − β /
3, where β is given in (II.9), such that β ≪
1, makes this term too small compared with the coefficients infront of ¨ δH ( t ) , ˙ δH ( t ) in the equation for perturbation (IV.8). In the next section, we suggest a possible solution ofthe problem returning to the general action (II.4) with the Gauss-Bonnet and (cid:3) R terms which have been omitted inthe above study.2 V. THE ACCOUNT OF GAUSS-BONNET AND (cid:3) R TERMS AND SPECTRAL INDEX
As it was mentioned in second section, to construct the Lagrangian of higher-derivative gravity, also the Gauss-Bonnet and the (cid:3) R terms must be taken into account. They may give a non-zero contribution to the dynamicalequations if the coefficients in front of them are not constant but depend on the curvature. This is precisely whathappens when one solves RG equation and gets RG improved effective action. In the first part of this work we did notconsider such contributions. Let us analyze their role on the dynamics of the inflation induced by higher-derivativequantum gravity. Let us consider the following additional piece to the action (II.4), I G , (cid:3) R = − Z M d x √− g [ γ ( t ′ ) G − ζ ( t ′ ) (cid:3) R ] , (V.1)where G is given by (II.3) and γ ( t ′ ) , ζ ( t ′ ) are effective coupling constants depending on t ′ (II.10) and therefore on R .We assume γ ( t ′ ) = γ (1 + c t ′ ) , ζ ( t ′ ) = ζ (1 + c t ′ ) , (V.2)where γ , ζ are generic constants and c , are numerical coefficients whose explicit values are not necessary in thebelow analysis. As it is explained in review [7] this is result of one-loop quantum calculation of these terms (vacuumpolarization). For recent discussion of contribution of GB term in higher-derivative gravity, see Ref. [21]. Actually, thecalculation of surface terms may be done in less/more than four dimensions, with subsequent dimensional continuation.Hence, when t ′ ≪
1, at the low curvature limit, γ ( t ′ ) , ζ ( t ′ ) tend to constants, the derivatives do not diverge and(V.1) turns out to be zero: on the other side, when 1 ≪ t ′ , at the high curvature limit, they give a significativecontribution to the dynamical equations of motion. The Gauss-Bonnet represents a new curvature invariant. OnFRW metric it (II.12) reads G = 24 ˙ a a N (cid:16) ¨ aN − ˙ a ˙ N (cid:17) . (V.3)Adding to the Lagrangian (II.16) the piece (V.1), we make an integration by parts with respect to (cid:3) R , where (cid:3) R = ( √− g ) − ∂ µ ( g µν √− g∂ ν R ) ≡ − ( √− g ) − ∂ t ( √− g∂ t R ), and introduce a new Lagrangian multiplier σ for theGauss-Bonnet term [15], such that I G , (cid:3) R = − Z M d x √− g (cid:20) γ ( t ′ ) G + σ (cid:20) G −
24 ˙ a a N (cid:16) ¨ aN − ˙ a ˙ N (cid:17)(cid:21) − (cid:18) dζdt ′ dt ′ dA ˙ A (cid:19)(cid:21) , σ = − γ ( t ′ ) , (V.4)Here the second expression has been derived from the variation with respect to G and A ≡ A ( N, ˙ N , a, ˙ a ) is the explicitform of the Ricci scalar as a function of the metric (II.13), A ( N, ˙ N , a, ˙ a ) = 1 N " (cid:18) ˙ aa (cid:19) + 6 (cid:18) ¨ aa (cid:19) − ˙ NN ! (cid:18) ˙ aa (cid:19) . (V.5)Thus, ∆( t ′ ) in (II.17) reads∆( t ′ ) = (cid:20) R ( κ ( t ′ )) dκ ( t ′ ) dt ′ + R ddt ′ (cid:18) ω ( t ′ )3 λ ( t ′ ) (cid:19) + d Λ( t ′ ) dt ′ + dγ ( t ′ ) dt ′ G (cid:21) , (V.6)and the additional piece to the Lagrangian (II.19) results to be L G , (cid:3) R ( N, ˙ N , ¨ N, a, ˙ a, ¨ a, R, ˙ R ) = 6 ˙ a (cid:18) a N (cid:19) (cid:20) dγ ( t ′ ) dt ′ dt ′ dR ˙ G (cid:21) + 8 ˙ a N dγ ( t ′ ) dt ′ dt ′ dR ˙ R + ( N a ) (cid:18) dζdt ′ dt ′ dA ˙ A (cid:19) , (V.7)where the first piece comes from the integration by parts of the second derivative metric functions of the Ricci scalar,the second term comes from the ones of the Gauss-Bonnet and the last piece corresponds to (cid:3) R -term. Note thatnow the Lagrangian depends on the higher derivatives of the metric due to the introduction of ˙ A . Equation (II.23),3in the gauge N = 1, is derived as0 = − Λ( t ′ ) + 6 H κ ( t ′ ) − H ( κ ( t ′ )) dκ ( t ′ ) dt ′ t ′ ˙ RR ! + ω ( t ′ )3 λ ( t ′ ) h R ˙ H − H ˙ R i − H ddt ′ (cid:18) ω ( t ′ )3 λ ( t ′ ) (cid:19) (cid:16) ˙ Rt ′ (cid:17) +6 (cid:16) H + ˙ H (cid:17) ∆( t ′ ) t ′ R − H " d ∆( t ′ ) dt ′ (cid:18) t ′ R (cid:19) − ∆( t ′ ) t ′ R ˙ R − H dγ ( t ′ ) dt ′ t ′ ˙ RR − H " dγ ( t ′ ) dt ′ t ′ ˙ GR − A ˙ R − B ˙ R R + 6 ddt h A (cid:16) H + 3 ˙ H (cid:17) ˙ R + B H ˙ R i + 18 H h A (cid:16) H + 3 ˙ H (cid:17) ˙ R + B H ˙ R i − (cid:16) H + ˙ H (cid:17) A H ˙ R − H ddt (cid:16) A H ˙ R (cid:17) − d dt (cid:16) A H ˙ R (cid:17) , (V.8)where A = (cid:18) dζ ( t ′ ) dt ′ t ′ R (cid:19) , B = " d ζ ( t ′ ) dt ′ (cid:18) t ′ R (cid:19) − dζ ( t ′ ) dt ′ t ′ R , (V.9)and the Ricci scalar R is given by (II.24). The derivative of the Lagrangian with respect to the Gauss-Bonnet leadsto the Ricci scalar in (II.13), and the derivative with respect to the Ricci scalar leads to the Gauss-Bonnet one in(V.3), which reads in the gauge N = 1, G = 24 H (cid:16) H + ˙ H (cid:17) . (V.10)On de Sitter solution R dS = 12 H , G dS = 24 H , H dS being constant, equation (II.26) is corrected as0 = 6 H κ − t ′ κ ) ω (cid:0) H ( κ ) ω (4 ω (2 ω + 3) + 1) + 4 κ λω (cid:0) H (cid:0) − ω + 26 ω + 3 (cid:1) − κ Λ(28 ω + 1) (cid:1) − λ (cid:0) ω + 1 (cid:1)(cid:1) − Λ + 12 H dγdt ′ t ′ , (V.11)where the functions λ , ω , κ , Λ and γ , dγ/dt ′ are constants in the time. By using (III.12)–(III.13) with c = Λ = 0,and 1 ≪ t ′ , we obtain the solution H κ ≃ . . − . dγ/dt ′ )) t ′ ( λ (0) t ′ ) . , dγdt ′ < , (V.12)where | dγ/dt ′ | ≪ t ′ is used and we require that such a derivative is negative ( γ c < γ t ′ ( t ′ ), de Sitter solution depends on the current value of λ ( t ′ = 0) = λ (0). Obviously, the (cid:3) R -term does notgive any contribution to the de Sitter solution. By using again the parametrization (III.12)–(III.13) with c = Λ = 0,and therefore by multiplying (V.8) by κ , and by perturbationg it with respect to de Sitter solution (V.12) as in(IV.1), we get κ t ′ ( H dS κ ) h κ ¨ δH (cid:0) t ′ ( H dS κ ) (cid:0) . t ′ + t ′ t ′ ( − γ t ′ ( t ′ ) + 18 ζ t ′ ( t ′ ) − γ t ′ t ′ ( t ′ ) t ′ + 11 . − . t ′ (cid:1) − . t ′ t ′ ( H dS κ ) ( λ (0) t ′ ) − . + 0 . t ′ ( λ (0) t ′ ) − . + 0 . t ′ t ′ ( H dS κ ) ( λ (0) t ′ ) − . +2 × − t ′ t ′ ( λ (0) t ′ ) − . (cid:1) + ( H dS κ ) ˙ δH (cid:16) t ′ ( H dS κ ) (cid:0) . t ′ + t ′ t ′ ( − γ t ′ ( t ′ ) + 72 ζ t ′ ( t ′ ) − γ t ′ t ′ ( t ′ ) t ′ + 34 . − . t ′ (cid:1) − . t ′ t ′ ( H dS κ ) ( λ (0) t ′ ) − . + 0 . t ′ ( λ (0) t ′ ) − . +0 . t ′ t ′ ( H dS κ ) ( λ (0) t ′ ) − . + 0 . t ′ t ′ ( λ (0) t ′ ) − . (cid:17)i + ( H dS κ ) δH (cid:20) . λ (0) t ′ ) . + t ′ (cid:18) . λ (0)( λ (0) t ′ ) . + ( H dS κ ) (48 γ t ′ ( t ′ ) − . (cid:19)(cid:21) = 0 , H ( t ) = H dS + δH ( t ) , | δH ( t ) | ≪ , (V.13)where we introduced the notation γ t ′ ( t ′ ) ≡ dγ ( t ′ ) dt ′ , γ t ′ t ′ ( t ′ ) ≡ d γ ( t ′ ) dt ′ , ζ t ′ ( t ′ ) ≡ dζ ( t ′ ) dt ′ . (V.14)4If one assumes (IV.3) and takes into account that | γ t ′ ( t ′ ) | , | ζ t ′ ( t ′ ) | ≪ t ′ and | γ t ′ t ′ ( t ′ ) | ≪
1, this expression is simplifiedas ˜ D δH + t ′ [19 . H dS κ )( κ ˙ δH ) + 6 . κ ¨ δH )] ≃ , (V.15)where ˜ D = (cid:20) . λ (0) t ′ ) . − (30 . − γ t ′ ( t ′ )) t ′ ( H dS κ ) (cid:21) . (V.16)Thus, the solution of the above differential equation reads δH = h ± exp h ˜ A ± t i , ˜ A ± = H dS − ± s − .
627 ˜ D ( H dS κ ) t ′ , | h ± | ≪ , (V.17)where h ± are the integration constants corresponding to the signs: plus and minus inside ˜ A ± . The solution is unstableif ˜ D <
0, namely 0 . λ (0) t ′ ) . < [30 . − γ t ′ ( t ′ )] t ′ ( H dS κ ) , (V.18)and, by using (V.12), one sees that this inequality is always satisfied independently on the value of γ t ′ ( t ′ ). As aconsequence, also (IV.3) that we have used to derive (V.15) is verified and it is interesting to note that ˜ D evaluatedwith respect to de Sitter solution (V.12) is equal to D in (IV.11) evaluated with respect to de Sitter solution (III.16),from which we can understand that Gauss-Bonnet term contribution to the stability of de Sitter solution behaves likethe one of a R -term (see (IV.8)–(IV.9) and related comment). By using (V.12) one gets˜ A + ≃ × − H dS t ′ t ′ (22085 . − . γ t ′ ( t ′ )) , ˜ A − ≃ − H dS , (V.19)where ˜ D is taken to be small. Thanks to the presence of the Gauss-Bonnet term in the action, the instabilityparameter ˜ A + can be increased with respect to the case considered before. Let us introduce our Ansatz (V.2). Weobtain H κ ≃ . . − . γ c ] t ′ ( λ (0) t ′ ) . , γ c < , (V.20)˜ A + ≃ × − H dS t ′ t ′ (22085 . − . γ c ) , ˜ A − ≃ − H dS . (V.21)As a consequence, the instability parameter ˜ A + is larger than A + in the absence of Gauss-Bonnet correction if γ c is negative, namely, by taking 0 < c and γ <
0, the Gauss-Bonnet contribution to the action is positive (see (V.1)):the analysis of inflation is similar to the previous case, but the e -folds and therefore the spectral index n s are smaller.To be specific, the η slow-roll parameter (IV.33) and the spectral index n s in (IV.36) read η ≃ − × − t ′ (22085 . − . γ c ) t ′ , n s ≃ − × − t ′ (22085 . − . γ c ) t ′ , (V.22)since we can still use (IV.34). The spectral index n s is consistent with Planck data (IV.37) if450 < t ′ t ′ (22085 . − . γ c ) < . (V.23)If we set λ (0) = t = 1 and take (III.15) together with (III.17), we get from (V.23), − . < γ c < − . . (V.24)For example, for c = 1 and γ = − n s ≃ . , (V.25)5which is in agreement with the Planck data (IV.37). The de Sitter solution results to be H ≃ . × − M P l ,and inflation takes place near to the Planck scale, such that (III.15) is valid. In this kind of model, as we noted in § IV B, the e-folds N ∼ / (1 − n s ), and in the present case we have N ∼
60: this is an order of magnitude/lowerbound of the e -folds which permits the thermalization of observable universe (the acceleration finishes when ǫ = 1,and therefore the exact amount of inflation depends on the initial amplitude | h + | as in (IV.32)). Thanks to theGauss-Bonnet contribution in the action, we can see that the value of the e -folds has considerably decreased (see forexample (IV.21)), rendering correct the prediction of the spectral index. In the present example, a viable inflation isobtained for 1 ≪ t ′ /t ′ , which is always true due to the large curvature scale of inflation.We have demonstrated that the contribution from RG improved Gauss-Bonnet term can modify the instability ofde Sitter solution describing inflation given a viable spectral index. In our derivation, we have taken into accountalso the (cid:3) R contribution, but, due to the Ansatz (V.2), it disappears. However, we furnished the formalism to treatthe Lagrangian (V.1) with generic coefficients: if they grow up in the early-time universe, they modify the dynamicsof inflation and can lead to a model compatible with the Planck data.As a final result of the work, we are able to present the very general quantum-corrected Lagrangian constructed withsecond degree corrections to the Einstein gravity: I = Z M d √− g (cid:20) Rκ ( t ′ ) − ω ( t ′ )3 λ ( t ′ ) R + 1 λ ( t ′ ) C − γ ( t ′ ) G + ζ ( t ′ ) (cid:3) R − Λ( t ′ ) (cid:21) , t ′ = t ′ (cid:20) RR (cid:21) , (V.26)where t is a number and R = 4Λ is the curvature of today universe, Λ being the cosmological constant. The one-looprunning coupling constants λ ( t ′ ) , ω ( t ′ ) , κ ( t ′ ) , Λ( t ′ ) , γ ( t ′ ) and ζ ( t ′ ) are found from higher-derivative quantum gravity.They can be written as λ ( t ′ ) = λ (0)(1 + λ (0)(133 / t ′ ) , ω ( t ′ ) = ω , κ ( t ′ ) = κ (1 + λ (0)(133 / t ′ ) . , Λ( t ′ ) = Λ (1 + λ (0)(133 / t ′ ) . , (V.27)with ω = − . κ = 16 π/M P l , Λ = 2Λ. The expressions for ω ( t ′ ) , κ ( t ′ ) and Λ( t ′ ) are derived by investigating theasymptotic behaviour of the running constants at high curvature. However, the derivatives of the coupling constantsobey to a set of RG equations that we have taken into account in our analysis. The form of γ ( t ′ ) and ζ ( t ′ ) is given by γ ( t ′ ) = γ (1 + c t ′ ) , ζ ( t ′ ) = ζ (1 + c t ′ ) , c γ < , (V.28) γ , ζ and c , constants. Finally, λ (0) is a number related to the bound of inflation. At small curvature ( t ′ ≪ I = Z M d √− g (cid:20) Rκ + 0 . λ (0) R + 1 λ (0) C − (cid:21) , t ′ = t ′ (cid:20) RR (cid:21) , (V.29)and the contributions of Gauss-Bonnet and (cid:3) R -terms disappear when the coefficients become constant.Inflation is described at high curvature for 1 ≪ t ′ , near to the Planck mass. The model possesses a de Sitter solutionwhich depends on λ (0). This solution is always unstable and the model exits from inflation. It is possible to calculatethe behaviour of perturbations and show that the slow-roll conditions of inflation are satisfied with the ǫ slow-rollparameter much smaller than the η slow-roll parameter. The amount of inflation ( e -folds) is sufficiently large, thetensor-to-scalar ratio r is very close to zero and, due to the contribution of the RG improved Gauss-Bonnet term inthe action, the spectral index n s satisfies the Planck data. The RG improved (cid:3) R -term does not play any importantrole in the dynamics of inflation.After inflation, the reheating process with the particle production must take place to recover the FRW universe.These processes occur when the curvature (Ricci scalar) oscillates and eventually in the presence of the interactionbetween the gravity and matter quantum fields. At the end of inflation t ′ → R of Einstein’s gravity (on FRW metric the square of Weyl tensor gives a zero contribution):this model has been well-investigated in the literature and it has been demonstrated that it is compatible with thereheating scenario. VI. DISCUSSION
In this work we investigated the inflationary universe taking into account quantum gravity effects in frames of RGimproved effective action of higher-derivative quantum gravity. The effective coupling constants in higher-derivative6quantum gravity obey to a set of one-loop RG equations found in Refs. [12] and may show the asymptotically-freebehaviour. These one-loop RG equations which define the effective coupling constants are used to derive quantum-corrected dynamical FRW equations. In order to find the explicit form of the running coupling constants, their(asymptotically free) behaviour at high energy scale is used.The model possesses a de Sitter solution at high curvature to describe expanding inflationary universe. The boundof de Sitter solution depends on the value of the running constant of R -term today. We have demonstrated thatde Sitter solution is always unstable and takes place near to the Planck scale. Thus, it is possible to evaluate theinstability parameter of the model and the amplitude of perturbations. The slow-roll conditions are well satisfied, andthe η slow-roll parameter is much larger than the ǫ slow-roll parameter: their behaviour with respect to the e -folds N seems to be the same of the ones in scalar-tensor theories (see review [22]) for inflation ( ǫ ∼ /N and | η | ∼ /N ).The amount of inflation of the model is sufficiently large, the tensor-to-scalar ratio r is very close to zero. However,in order to have the correct spectral index n s compatible with the Planck data it is necessary to take into accountthe contribution of RG improved Gauss-Bonnet term in the action. Note that other RG-improved surface term ( (cid:3) R )does not play any important role during inflation. At low energy, the effective running constants become constantand we recover the Friedmann universe.It would be very interesting to compare the inflationary predictions (including the exit and reheating) of higher-derivative quantum gravity with those of Einstein quantum gravity in more detail. This will be considered elsewhere. Acknoweledgments
The research by SDO has been supported in part by MINECO(SPAIN), projects FIS2010-15640 and FIS2013-44881and by the Russ. Government Program of Competitive Growth of Kazan Federal University. [1] G. Hinshaw et al. [WMAP Collaboration], Astrophys. J. Suppl. (2013) 19 [arXiv:1212.5226 [astro-ph.CO]]; E. Komatsu et al. [WMAP Collaboration], Astrophys. J. Suppl. (2009) 330 [arXiv:0803.0547 [astro-ph]];[2] P. A. R. Ade et al. [Planck Collaboration], Astron. Astrophys. (2014) A22 [arXiv:1303.5082 [astro-ph.CO]].[3] V. Mukhanov, “Physical foundations of cosmology,” Cambridge, UK: Univ. Pr. (2005) 421 p; D. S. Gorbunov andV. A. Rubakov, “Introduction to the theory of the early universe: Cosmological perturbations and inflationary theory,”Hackensack, USA: World Scientific (2011) 489 p.[4] P. A. R. Ade et al. [BICEP2 Collaboration], Phys. Rev. Lett. (2014) 241101 [arXiv:1403.3985 [astro-ph.CO]].[5] R. P. Woodard, arXiv:1407.4748 [gr-qc]; M. G. Romania, N. C. Tsamis and R. P. Woodard, Lect. Notes Phys. (2013)375 [arXiv:1204.6558 [gr-qc]].[6] M. Rinaldi, G. Cognola, L. Vanzo and S. Zerbini, arXiv:1410.0631 [gr-qc]; K. Bamba, G. Cognola, S. D. Odintsovand S. Zerbini, Phys. Rev. D (2014) 023525 [arXiv:1404.4311 [gr-qc]]; B. J. Broy, F. G. Pedro and A. Westphal,arXiv:1411.6010 [hep-th]; K. Bamba and S. D. Odintsov, Symmetry (2015) 220 [arXiv:1503.00442 [hep-th]].[7] I. L. Buchbinder, S. D. Odintsov and I. L. Shapiro, Effective action in quantum gravity, Bristol, UK: IOP (1992) 413 p;Riv. Nuovo Cim. (1989) 1[8] A. Salvio and A. Strumia, JHEP , 080 (2014) [arXiv:1403.4226 [hep-ph]].[9] J. Khoury and A. Weltman, Phys. Rev. D , 044026 (2004) [astro-ph/0309411].[10] E. Elizalde and S. D. Odintsov, Phys. Lett. B (1993) 240; Phys. Lett. B (1994) 199; Z. Phys. C (1994) 699[hep-th/9401057]; S. D. Odintsov, Fortsch. Phys. (1991) 621.[11] A. De Simone, M. P. Hertzberg and F. Wilczek, Phys. Lett. B (2009) 1 [arXiv:0812.4946 [hep-ph]]; H. M. Lee, Phys.Lett. B (2013) 198 [arXiv:1301.1787 [hep-ph]]; G. Barenboim, E. J. Chun and H. M. Lee, Phys. Lett. B (2014) 81[arXiv:1309.1695 [hep-ph]]; N. Okada and Q. Shafi, arXiv:1311.0921 [hep-ph]; M. Herranen, T. Markkanen, S. Nurmi andA. Rajantie, Phys. Rev. Lett. (2014) 21, 211102 [arXiv:1407.3141 [hep-ph]]; T. Inagaki et al. , arXiv:1408.1270 [gr-qc];E. Elizalde, S. D. Odintsov, E. O. Pozdeeva and S. Y. Vernov, Phys. Rev. D (2014) 084001 [arXiv:1408.1285 [hep-th]];Y. Hamada, H. Kawai and K. y. Oda, JHEP (2014) 026 [arXiv:1404.6141 [hep-ph]; H. J. He and Z. Z. Xianyu, JCAP (2014) 019 [arXiv:1405.7331 [hep-ph]].[12] K. S. Stelle, Phys. Rev. D (1977) 953; E. S. Fradkin and A. A. Tseytlin, Nucl. Phys. B (1982) 469; I. G. Avramidiand A. O. Barvinsky, Phys. Lett. B (1985) 269.[13] A. Vilenkin, Phys. Rev. D 32 (1985) 2511.[14] S. Capozziello, Int. J. Mod. Phys. D114483 (2002).[15] G. Cognola, M. Gastaldi and S. Zerbini, Int. J. Theor. Phys. , 898 (2008) [gr-qc/0701138].[16] G. Cognola, L. Sebastiani and S. Zerbini, arXiv:1006.1586 [gr-qc].[17] M. B. Einhorn and D. R. T. Jones, arXiv:1410.8513 [hep-th].[18] J. D. Barrow and D. J. Shaw, Gen. Rel. Grav. , 2555 (2011) [Int. J. Mod. Phys. D , 2875 (2011)] [arXiv:1105.3105[gr-qc]]. [19] Hwang, J.-C., and Noh, H., Phys. Lett. B, 506, 13–19, (2001); H. Noh and J. c. Hwang, Phys. Lett. B , 231 (2001)[astro-ph/0107069].[20] K. Bamba, S. Nojiri, S. D. Odintsov and D. Saez-Gomez, arXiv:1410.3993 [hep-th].[21] M. B. Einhorn et al.et al.