Brans-Dicke theory of gravity with torsion: A possible solution of the ω -problem
BBrans-Dicke theory of gravity with torsion: A possible solution of the ω -problem Yu-Huei Wu a,b ∗ and Chih-Hung Wang b † a Center for Mathematics and Theoretical Physics, National Central University, Chungli 320, Taiwan, R.O.C. b Department of Physics, National Central University, Chungli 320, Taiwan, R.O.C. (Dated: November 1, 2018)We study the Brans-Dicke theory of gravity in Riemann-Cartan space-times, and obtain general torsion solu-tions, which are completely determined by Brans-Dicke scalar field Φ , in the false vacuum energy dominatedepoch. The substitution of the torsion solutions back to our action gives the original Brans-Dicke action with Φ -dependent Brans-Dicke parameter ω (Φ) . The evolution of ω (Φ) during the inflation is studied and it is foundthat ω approaches to infinity at the end of inflation. This may solve the ω -problem in the extended inflationmodel. PACS numbers: 98.80.Cq, 98.80.Jk
I. INTRODUCTION
The discovery of spin-1/2 fermions indicates that matterfields have microscopic structure, which can be character-ized by the spin angular momentum 3-forms S ab . Since gen-eral relativity (GR) is established (by hypothesis) in pseudo-Riemannian (i.e. torsion free) space-times, the gravitatingsources are solely described by the stress-energy tensor, andintrinsic spin does not play any role in the conservation lawsof angular momentum. Hence, GR lacks of a description ofspin-orbital coupling. This unsatisfactory situation actuallyhappens in any theory of gravitation established in pseudo-Riemannian space-time, and may be resolved when we extendrelativistic theories of gravity to Riemann-Cartan (RC) space-time. Several well-kown theories of gravitation, e.g. Poincarégauge theory of gravity (PGT) (see a review article [1]), arebuilt in RC space-times. Moreover, it was discovered that aLorentz gauge-covariant form of the Brans-Dicke (BD) the-ory of gravity yields torsion fields determined by the gradientof the BD scalar field Φ [2, 3].In RC space-times the concept of the metric g and metric-compatible connection ∇ are fundamentally independent, sothe associated independent variables of gravitational fields areorthonormal co-frames { e a } and connection 1-forms { ω ab } [4]. The intrinsic spin quantities S ab become the sourcesof torsion fields, and the Bianchi identities yield the conser-vation law of orbital angular momentum and intrinsic spin.Since torsion and the intrinsic spin have direct interactions,spin-polarized bodies can be used to detect torsion directlyin the laboratory (see a review article [5]). Up to now, thereis no experimental evidence showing the existence of torsionfield, so the constraints on torsion-spin coupling turn out tobe extremely small [5, 6]. The result is not surprising sinceit is very difficult to produce a significant magnitude of theintrinsic spin in the laboratory. However, we expect that in-trinsic spin should generate observable torsion effects in theearly Universe. Refs [7, 8] discovered that the spin- + torsion(sometimes called scalar torsion) mode in PGT [9] can nat- ∗ Electronic address: [email protected] † Electronic address: [email protected] urally explain the current acceleration of our Universe with-out introducing the dark energy. Moreover, we found that thequadratic curvature terms in RC space-times can generate apower-law inflation without introducing inflaton fields [10].However, this inflation model is not satisfactory since it re-quires some fine-tuning in the parameters.The old inflation model was originally proposed to solvehorizon and flatness problems by considering the Universe toundergo a first-order phase transition [11]. Cosmic inflation isdriven by the false vacuum energy of an inflaton field σ and issupposed to end by bubble nucleations. It was soon realizedthat inflation will never come to an end because of the small-ness of the (dimensionless) bubble nucleation rate (cid:15) ≡ λ/H ,where λ is the number of bubbles nucleated per unit time perunit 3-volume and H = ˙ a/a is the Hubble parameter duringinflation [12]. If we assume that bubble nucleation is domi-nated by quantum-mechanical tunneling, λ can be expressedas λ = Ae − S E in the semiclassical limit [13–16]. The pref-actor A is equal to T c times terms expected to be of orderunity and S E is the Euclidean action of the bounce solution,where T c denotes the critical temperature of the phase transi-tion. Since the scale factor a ( t ) is exponentially expanding, a ∝ e Ht , and H is a constant during inflation, we obtain that (cid:15) is a constant. In [12], Guth and Weinberg proved that if (cid:15) > (cid:15) cr ≈ . , the system of bubbles will percolate at somefinite time. However, a direct calculation of (cid:15) from effectivepotentials of some specific models shows that the value of (cid:15) is quite small ( − is even plausible) [12]. It turns out thatinflation never comes to the end in the old inflation model, andthis is called the "graceful exit" problem.La and Steinhardt [17] discovered that the "graceful exit"problem can naturally be resolved in the BD theory of gravity[18], and they called this model the extended inflation. Themain feature of the extended inflation is that a ( t ) has a power-law solution a ∝ t ω + instead of exponential expansion,where ω denotes the dimensionless BD parameter, so that (cid:15) ( t ) ∝ t is now time-dependent and monotonically increaseswith respect to time. It means that (cid:15) can be very small inthe beginning of inflation and then grows to the critical value (cid:15) cr , where the system of bubbles will percolate. Althoughthe Universe can exit from false-vacuum energy dominationin the extended inflationary scenario, it was soon realized thatin order to satisfy the nearly isotropic spectrum of the cosmic a r X i v : . [ g r- q c ] N ov microwave background radiation (CMB), the constraint on abubble-size distribution requires ω < [19–21]. However,the current solar system observation of Cassini spacecraft re-quires that ω must exceed [22, 23]. Apparently, theconstraint of the bubble-size distribution is conflict with thesolar system observations. This is called the ω -problem.There are several phenomenological approaches to solvethe ω -problem by assuming ω (Φ) to be a function of Φ [25] oradding a potential V (Φ) in the BD action [20]. In [25], ω (Φ) is taken to be ω (Φ) = ω + ω m ( Φ M Pl ) m , which increasesmonotonically during the evolution of Φ . Here, M P l denotesthe Planck mass and ω m must be assigned a huge value tosatisfy the solar system observations. When Φ approaches to M P l in the post-inflationary state, ω m dominates, and the so-lution of a ( t ) gives a ( t ) ∝ exp( α t f ) , where α is a positiveconstant and f = m m +1 . This solution is called the inter-mediate inflation. A further investigation of the bubble-sizedistribution in this intermediate inflation gives a constraint on ω and m [26]. Roughly speaking, it requires that the transi-tion between ω and ω m must be rapid, which corresponds torequiring a large value of m .The Einstein-Cartan theory is a natural generalization ofGR to RC space-times, and its torsion fields are completelydetermined by the distribution of the intrinsic spin S ab . If themagnitude of S ab is too small to observe, there is no differencebetween the Einstein-Cartan theory and GR. However, this isnot true in the BD theory with torsion, since torsion fields willbe produced not only by the intrinsic spin but also by the gra-dient of BD scalar field Φ [2]. Here, the torsion field generatedby Φ is called BD torsion field. Hence both S ab and Φ becomethe sources of torsion fields. In this paper, we find that the BDtorsion field will contribute to ω and obtain the explicit formof ω (Φ) . Moreover, we show that the ω -problem can naturallybe resolved in the BD theory with torsion, instead of usingphenomenological approaches. ω (Φ) only contains one phys-ical parameter a , and the present value of Φ , which is equalto M P l , will yield a ≈ M P l (see Sec. III). Hence, ω (Φ) actually does not contain any free parameter. To understandwhether this extended inflation model with torsion can satisfythe CMB anisotropic observation, it requires a further investi-gation on the cosmological perturbation in RC space-times.This paper is outlined as below. In Sec. II, we generalizethe BD theory to RC space-times. The main difference be-tween our work and Dereli & Tucker’s work [2] is that ourBD action with torsion includes three irreducible pieces ofquadratic torsion, which were not considered in [2]. Thesequadratic torsion terms may be associated to kinematic en-ergy of orthonormal co-frames e a . Adding these terms doesnot spoil the field equations as the second-order differentialequations. Without introducing any symmetry of space-time,we obtain a general torsion solution completely determinedby Φ , where the Lagrangian of matter is assumed to be thepotential U ( σ ) of the inflaton field. In Sec. II A, we substitutethe torsion solution back to our original action, and obtain aneffective action, which is equivalent to the original BD theoryexcept that ω (Φ) now is a function of Φ instead of the dimen-sionless parameter. In Sec. III, we study field equations in thehomogeneous and isotropic Universe, and obtain analytic and numerical solutions of a ( t ) and Φ( t ) during the inflation. Sec.IV gives a discussion and conclusion. In this paper, we usethe units c = (cid:126) = 1 and πG = M − P l [27].
II. BRANS-DICKE THEORY OF GRAVITY WITHTORSION
Mach’s principle is a fundamental principle to explain theorigin of inertia [28]. In attempting to incorporate Mach’sprinciple, the BD theory introduces an inertial field Φ whichplays the role of the gravitational constant G and is deter-mined by the matter field distributions. So the gravitationalfields are described by the metric g and the BD scalar field Φ ,which has the dimension [Φ] = [ M ] . The BD theory startsfrom the following action: ˆ S BD (Φ , e a ; Ψ) = (cid:90) Φ2 ˆ R ab ∧ ∗ ( e a ∧ e b ) − ω d Φ ∧ ∗ d Φ + ˆ L M (Ψ) , (1)where ˆ R ab are Riemann curvature 2-forms, ω is the BD di-mensionless parameter, and ∗ is the Hodge map associated to4-dimensional metric g . ˆ L M denotes the Lagrangian 4-formof matter fields Ψ , where the minimal coupling of gravita-tional fields is assumed, so there is no direct interaction be-tween Ψ and Φ . An important feature of BD theory is thatwhen ω approaches to infinity, the field equations of Φ yieldsthat Φ becomes a constant Φ . Hence, the BD theory willrecover to GR in the limit of ω → ∞ .The most natural generalization of the BD theory to RCspace-times is to start from the following action: S BD (Φ , e a , ω ab ; Ψ) = (cid:90) Φ2 R ab ∧ ∗ ( e a ∧ e b )+ (cid:88) n =1 a n n T a ∧ ∗ n T a − ω d Φ ∧ ∗ d Φ + L M (Ψ) , (2)where R ab are curvature 2-forms defined by R ab = dω ab + ω ac ∧ ω cb , and n T a denotes three irreducible pieces of torsion2-forms T a defined by T a = de a + ω ac ∧ e c . One may noticethat the orthonormal co-frames e a and the connection 1-forms ω ab become independent variables. The trace vector torsion T a (scalar torsion) and the axial torsion T a (pseudo-scalartorsion) can be expressed as T a = −
13 ( i p T p ) ∧ e a , T a = 13 i a ( e p ∧ T p ) , (3)where i a denotes the interior derivative, and the tensor part oftorsion T a is defined by T a ≡ T a − T a − T a . Clearly,the dimensions of three parameters a n are the same and equalto [ a n ] = [ M ] . In Eq. (2), we also assume the minimalcoupling of the matter fields Ψ and the gravitational fields, so Φ does not appear in the Lagrangian L M . Generally speak-ing, the difference between ˆ L M and L M occurs when Ψ has adirect interaction with ω ab in the Lagrangian 4-forms, and ac-cording to the standard model of particle physics, only spin- / fermions will directly couple to the connection 1-formsin the action. This is the reason why fermions become thesources of torsion in RC space-times. In Eq. (2), the firstterm on the right-hand side shows that Φ couples to the fullscalar curvature instead of the Riemannian scalar curvature, soone can expect that Φ will generate torsion through these cou-pling. Furthermore, the field equations remain second-orderdifferential equations when adding three irreducible quadratictorsion terms. In this paper, we will concentrate on the evolu-tion of Φ and the gravitational fields at the inflationary epoch,when is dominated by the false vacuum energy, so we put L M = − U ( σ ) ∗ , where the potential U ( σ ) of the inflaton σ is constant during inflation.Since the field equation δS BD δω ab = 0 yields the algebraicequations for torsion 2-forms T a , we first solve these equa-tions and then obtain general solutions T a = e a ∧ d Φ2(Φ − a ) with T a = T a = 0 . One should notice that, to obtain T a = T a = 0 , we have excluded degenerate situations where Φ = a or Φ = 2 a . It is clear that the coupling of Φ and thescalar curvature will only produce scalar torsion T a . When a = 0 , we return to the result in [2]. The substitution of thetorsion solutions n T a back to the remaining field equations,i.e. δS BD δe c = 0 and δS BD δ Φ = 0 , yields Φ2 R ab ∧ ∗ e abc + a { ( i c ∗ T p ) ∧ T p − i c T p ∧ ∗ T p (4) +2 D ∗ T c } = − ω
2Φ ( i c d Φ ∗ d Φ + d Φ ∧ i c ∗ d Φ) + U ∗ e c ,d ∗ d Φ = − Φ2 ω R ab ∧ ∗ e ab + 12Φ d Φ ∧ d ∗ Φ , (5)where the BD torsion T a is T a = T a = e a ∧ d Φ2(Φ − a ) . (6) e a ··· bc ··· d ≡ e a ∧ · · · ∧ e b ∧ e c ∧ · · · ∧ e d and D denotescovariant exterior derivative [29]. Since T a and T a vanish,we expect that Eq. (4) does not contain the parameters a and a . It is clear that Eqs. (4) and (5) are the second-orderdifferential equations for e a and Φ . In Subsection II A, wewill show that the effects of the BD torsion can actually becombined with ω to form an effective BD "parameter" ω (Φ) ,which is now a function of Φ . More specifically, Eqs. (4)and (5) are equivalent to original BD equations in Riemannianspace-time with the effective BD parameter ω (Φ) . A. The effective action
In order to compare Eqs. (4) and (5) to the original BDequations, we should decompose the curvature 2-forms R ab into Riemannian curvature 2-forms ˆ R ab and torsion parts. Thefirst step is to decompose ω ab into the connection 1-forms ˆ ω ab associated with the Levi-Civita connection and the con-torsion 1-forms K ab , which are defined by [29] ˆ ω ab = 12 ( e p i a i b d e p + i b d e a − i a d e b ) ,K ab = 12 ( − e p i a i b d T p − i b d T a + i a d T b ) . (7)Substituting ω ab = ˆ ω ab + K ab into the definition of R ab yields R ab = ˆ R ab + ˆ DK ab + K ac ∧ K cb , (8)where ˆ D is the covariant exterior derivative associated to ˆ ω ab .Using Eq. (6), we obtain K ab = 12(Φ − a ) ( e a i b d Φ − e b i a d Φ) . (9)The substitution of Eqs. (8) and (9) into Eq. (2) gives theeffective BD action: S BD ( e a , Φ; σ ) = (cid:90) Φ2 ˆ R ab ∧ ∗ e ab − ω (Φ)2Φ d Φ ∧ ∗ d Φ+ U ( σ ) ∗ d B , (10)where the effective BD parameter is ω (Φ) = ω + 3Φ2( a − Φ) , (11)and d B denotes the boundary term. If a = 0 , we obtain ω = ω − , which agrees with the result in [2]. It is notdifficult to verify that the field equations obtained by varying S BD with respect to e a and Φ are equivalent to Eqs. (4) and(5) with the substitution of Eqs. (8)-(9).Before we present a detail study of the evolution of Φ( t ) and the scale factor a ( t ) during the inflation, we can first ex-amine the behavior of ω (Φ) . Eq. (11) indicates that when Φ (cid:28) a , we have ω ≈ ω . Moreover, when Φ approaches to a , ω (Φ) will then approach to infinity. It is clear that ω (Φ) monotonically increases with respect to the growth of Φ . Inorder to satisfy the constraint ω > of the current so-lar system observation [22], it requires that Φ should be veryclose to a at present time. In Sec. III, we apply analytic andnumerical approaches to study the evolution of Φ( t ) duringthe inflation, which gives that Φ is asymptotically approach to a in the post-inflationary stage. Furthermore, the equationsof motion indicate that Φ continuously approaches to a in theradiation and matter domination epochs, so this result can beused to explain why ω (Φ) is so large at the present time. III. EQUATIONS OF MOTION IN ROBERTSON-WALKERSPACE-TIMES
Although our Universe is apparently inhomogenous andanisotropic in small scales (e.g. galactic scale), the astro-physical observations strongly support the homogeneity andisotropy of our observable Universe in the cosmological scale.It allows us to assume that our observable Universe exists 3-dimensional space-like hypersurfaces, which are maximallysymmetric 3-spaces [30]. The assumption of homogeneityand isotropy in RC space-times gives e = dt, e α = a ( t )(1 − kr ) dx α , (12) T = 0 , T α = f ( t ) e α ∧ e + h ( t ) ∗ ( e ∧ e α ) , (13)and Φ = Φ( t ) , where k = {− , , } denotes the constantcurvature of 3-dimensional spaces and r ≡ √ x α x α . It is con-venient to introduce a dimensionless scalar field χ defined by χ ≡ Φ a . The substitution of Eq. (13) into Eq. (6) yields f ( t ) = ˙ χ χ − , (14)and h ( t ) = 0 . Moreover, the substitution of Eqs. (12)-(14)into Eqs. (4)-(5) yields H = − ka − H ˙ χχ + ˙ χ χ (1 − χ ) + ω (cid:18) ˙ χχ (cid:19) + M F χa , (15) (cid:18) ω + 32(1 − χ ) (cid:19) ( ¨ χ + 3 H ˙ χ ) = − χ − χ ) + 2 M F a , (16)where H ≡ ˙ aa and U ≡ M F [31]. M F denotes the falsevacuum energy, which may be around the GUT energy scale Gev. We should now try to determine the energy scale of a . Since χ ( t ) at present time t P is extremely close to 1, i.e. Φ( t P ) ≈ a , and Φ( t P ) should be normalized to (8 πG ) − ,we obtain that a ≈ M P l .Eqs. (15)-(16) describe the evolution of a ( t ) and χ ( t ) dur-ing the inflation. We first observe that Eq. (16) has a veryinteresting feature. On the right-hand side of Eq. (16), thefirst term is definitely negative and is proportional to ˙ χ , soone may identify it as frictional force. Moreover, the secondterm is a definitely positive constant, so it can be consideredas a constant external force supplying χ with kinetic energy.If we consider χ (cid:28) at the beginning of inflation, the firstterm can actually be neglected so the false vacuum energy willdrive χ to have positive velocity and acceleration. It meansthat χ ( t ) grows with positive acceleration. However, when χ is approaching to , the frictional force cannot be neglectedanymore. So one can expect that χ ( t ) will evolve from accel-erating phase to decelerating phase. In Subsection III A, weobtain analytic solutions of Eqs. (15) and (16) in the earlystage of inflation, where χ (cid:28) , and in the post-inflationarystage, where χ ≈ . In Subsection III B, we use numericalcalculations to illustrate our analytic studies. A. Analytic solutions
We first study the early stage of inflation. When χ (cid:28) ,it yields that ω = ω . So Eqs. (15) and (16) return to theequations of motion in the extended inflation [17], and we thenobtain the power-law solutions a ( t ) = a B (cid:16) γα t (cid:17) ω + , (17) χ ( t ) = χ B (cid:16) γα t (cid:17) , (18) with f ( t ) = − χ B γα (cid:0) γα t (cid:1) , where α = (2 ω +3)(6 ω + 5) and γ = M F a χ B . Here, a B and χ B denote theinitial values of a ( t ) and χ ( t ) . Eq. (18) indicates that χ has aconstant acceleration. If ω > , we obtain a power-law in-flation, which yields a time-dependent bubble nucleation rate (cid:15) ( t ) . As mentioned in [17], the initial bubble nucleation rate (cid:15) B can be small and then grows to a critical value (cid:15) cr , wherethe system of bubbles will percolate at some finite time. Itmeans that (cid:15) will reach (cid:15) cr in the post-inflationary stage. Theconstraint of the bubble-size distribution required ω < inthe extended inflationary model [19, 20], so we may require ω ( χ ) < in this power-law inflationary stage. Moreover,since ω ( χ ) becomes large in the post-inflationary stage, weshould restrict the e-folding number N ( t ) ≡ ln a end a ( t ) to beless than at the post-inflationary epoch in order not to pro-duce a large- ω , scale-invariant bubble spectrum. In [26], Lid-dle and Wands analyzed the intermediate inflationary model,which has ω (Φ) = ω + ω m ( Φ M Pl ) m , and obtained a con-straint on ω and m . They concluded that the choice of ω ( χ ) must exhibit a prolonged flat region and only increase rapidlyonce χ approaches to . It corresponds to choose a large m . Inthis extended inflation model with torsion, we find that χ − χ ) changes very rapidly when χ approaches to and only be-comes significant when χ is extremely close to . So the con-straint of the bubble-size distribution can be achieved in thisinflationary model by requiring ω ≤ . A more detailedstudy of the bubble spectrum in this extended inflation modelwith torsion may lower the upper bound of ω .In the post-inflationary stage, we try to find an attractor so-lution, which asymptotically approaches to . More precisely,the solution satisfies lim t →∞ χ = 1 and lim t →∞ d n χdt n = 0 , ∀ n ≥ . When χ ≈ , the Eqs. (15)-(16) become H = M F χa and ¨ χ + 3 H ˙ χ = − ˙ χ − χ ) + 4 M F (1 − χ )3 a , (19)which yields an approximate analytic solution a ( t ) ∝ sinh βt, χ ( t ) ≈ tanh βt, (20)with f ( t ) ≈ − β tanh βt , where β = M F √ a . We see that χ asymptotically approaches to , and a ( t ) becomes nearly ex-ponential expansion in this post-inflationary stage. It meansthat ω ( χ ) will grows to a large value at the end of inflation.After the end of inflation, the Universe may be thermalized bybubble collisions and returns to radiation domination, so thefalse vacuum energy in Eq. (19) should be replaced by ρ − p ,which is zero at radiation domination. From Eq. (19), one canactually see that either in the radiation- or matter-dominatedera, the matter fields ( ρ − p ) do not affect the evolution of χ since they all multiply a very small value − χ . It turns outthat χ will continuously approach to at the radiation- andmatter-domainated epochs and it naturally gives an extremelylarge value of ω at the present matter-dominated epoch, whichsatisfies the solar system observations. In Subsection III B, weapply numerical calculations to study the evolution of χ and a during the inflation. B. Numerical demonstration
In this subsection, we use a numerical method to demonstrate our analytic study. In the numerical calculation, we normalize a = 1 and choose M F = 10 − [32]. Moreover, we set the BD parameter ω = 16 , and the initial values are chosen as a B = 1 , χ B = 10 − and ˙ χ B = 2 × − . In Fig. 1, one can clearly see that χ ( T ) is proportional to T at early stage of inflation,which agrees with our analytic solution, and will then pass a critical point ( T ≈ ), where its acceleration ¨ χ ( T ) becomesdeceleration. In the post-inflationary stage ( T > ), both the velocity ˙ χ ( T ) and acceleration ¨ χ ( T ) approach to zero, so ityields that χ asymptotically approaches to , which also agrees with our analytic result. In the ln ω − T diagram, we observethat ω is nearly ω and grows rapidly to a large value at the post-inflationary stage. FIG. 1: Evolution of a ( T ) , χ ( T ) and ω during the inflation, where T is a dimensionless time-parameter normalized by T ≡ − M F t . (1)The top left-panel indicates ln a − T diagram and the top right-panel denotes χ ( T ) − T diagram. (2) The bottom left-panel indicates theevolution of the velocity of χ and the bottom right-panel shows the evolution of ln ω with respect to T . lna Χ ! T " d Χ ! dT ln Ω IV. CONCLUSION AND DISCUSSION
We study the BD theory with torsion and obtain a generaltorsion solution, which is completely determined by the di-mensionless BD scalar field χ . We further discover that tor-sion fields will contribute to the BD parameter ω to form aneffective BD parameter ω ( χ ) = ω + χ − χ ) . In the extendedinflation model, the constraint of bubble-size distribution re-quires ω < . However, the current solar system observa-tion requires ω > . This apparent conflict is called the ω -problem. We study the evolution of χ ( t ) and a ( t ) duringthe inflation, i.e false vacuum energy domination, and showthat ω ( χ ) is approximate to ω during the inflation and willrapidly transit to a large value in the post-inflationary stage.Moreover, since χ continuously approaches to during theradiation- and matter-dominated epochs, ω ( χ ) will becomeextremely large at present time, which naturally explains thesolar system observations.In this work, we solve the ω -problem by generalizing theBD theory to RC space-times. The next question is to un-derstand whether this extended inflation model with torsionwill satisfy CMB anisotropic observations. In particular, thesuperhorizon-scale anisotropic spectrum ( l < modes) of CMB, which contains the information of primordial quantumfluctuations, has been used to test and constraint the inflationmodels. From WMAP 7-year data [33], the best-fit cosmolog-ical parameters give a spectral index of density perturbation n ≈ . , which is nearly scale-invariant. Refs. [34, 35] indi-cate that the extended inflation with ω < yields n < . ,which is inconsistent with the WMAP 7-year data. Further-more, the scalar-tensor ratio in the extended inflation with ω < is too large to satisfy the same data [35]. In orderto answer whether this extended inflation model with torsionsatisfies the CMB anisotropic spectrum, further study on cos-mological perturbation in RC spacetimes will be followed. Acknowledgements
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