Bouncing cosmologies via modified gravity in the ADM formalism: Application to Loop Quantum Cosmology
BBouncing cosmologies via modified gravity in the ADM formalism: Application toLoop Quantum Cosmology
Jaume de Haro ∗
1, 2 and Jaume Amor´os † Departament de Matem`atiques, Universitat Polit`ecnica de Catalunya, Colom 11, 08222 Terrassa, Spain Departament de Matem`atiques, Universitat Polit`ecnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain
We consider the ADM formalism as a tool to build bouncing cosmologies. In this approach, thefoliation of the spacetime has to be fixed in order to go beyond General Relativity modifying thegravitational sector. Once a preferred slicing, which we choose based on the matter content ofthe universe following the spirit of Weyl’s postulate, has been fixed, f theories depending on theextrinsic and intrinsic curvature of the slicing are covariant for all the reference frames preserving thefoliation, i.e., the constraint and dynamical equations have the same form for all these observers.Moreover, choosing multivalued f functions, bouncing backgrounds emerge in a natural way. Infact, the simplest is the one corresponding to holonomy corrected Loop Quantum Cosmology. Thefinal goal of this work is to provide the equations of perturbations which, unlike the full equations,become gauge invariant in this theory, and apply them to the so-called matter bounce scenario. PACS numbers: 04.20.Fy, 04.50.Kd, 98.80.Jk.
Keywords: Modified gravity, ADM formalism, Loop Quantum Cosmology, Weyl postulate.
1. INTRODUCTION
Bouncing cosmologies (see the reviews [1–5]) seem to be the most natural alternative to the inflationary paradigm[6–8], although in some papers [9–12] it seems to be only an implementation of it, introducing a pre-inflationarydynamics in order to remove the initial singularity. However, it is not an easy task to build up this kind of scenariosin the framework of the General Relativity (GR) [13–17], or to go beyond GR using modified theories such as f ( R )gravity (see for example [18–20]), modified Gauss-Bonnet gravity [21–23] or Born-Infeld inspired gravity theories[24, 25]. A simple non-singular bounce, which replaces the Big Bang singularity, emerges from holonomy correctedLoop Quantum Cosmology (LQC) [26–35].The LQC theory, in the flat Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) spacetime, containing only holonomycorrections (not inverse-volume effects) can be built as follows: since the gravitational part of the classical Hamiltoniancontains the Ashtekar connection, namely c = β ˙ a , where β is the Immirzi parameter [36] and a the scale factor, whichdoes not have a well defined quantum operator (see for instance [37, 38]), so one has to express it in terms of holonomiesin order to obtain a quantity able to be quantized which captures the underlying loop quantum dynamics (see forinstance [39]). At effective level, as has been shown in [40, 41], this is equivalent to replacing the Ashtekar connectionby the function sin( µ c ) /µ , where µ is a parameter with dimensions of length. Then, with the new holonomy correctedHamiltonian constraint one obtains the modified LQC-Friedmann equation, which in the phase space ( H, ρ ) representsan ellipse [42], where H is the Hubble parameter and ρ the energy density, and thus, contrarily to GR where thiscurve is an unbounded parabola, singularities such as the big bang, big rip [43] or little rip [44] are directly removed.Given the success of the effective theory for homogeneous and isotropic space-times, it is natural to try the sameapproach for cosmological perturbations in LQC. If one simply replaces the Ashtekar connection everywhere it appearsin the Hamiltonian constraint by a sinusoidal function c → sin( nµ c ) nµ where n is a natural number [45], the result is atheory with anomalies in its constraint algebra. A more careful treatment that gives an anomaly-free theory in thelongitudinal gauge has been found [46], and this has since been extended to the gauge-invariant case by allowing fornew terms in the holonomy modified Hamiltonian that vanish in the classical limit [47–50]). The result is a modifiedMukhanov-Sasaki equation that can be used to study the dynamics of cosmological perturbations in LQC. ∗ E-mail: [email protected] † E-mail: [email protected] a r X i v : . [ g r- q c ] F e b On the other hand, it is well-known that the background of holonomy corrected LQC in flat FLRW spacetime can beeasily mimicked in modified gravity if one has an scalar field that only depends on the square of the Hubble parameter[51–53]. This scalar appears in teleparallelism, where the spacetime is equipped by the Weitzenb¨ock connection [54]and an orthonormal basis parallelaly transported is chosen in the tangent bundle (see for example [55, 56]). For aFLRW geometry, using synchronous co-moving coordinates, the scalar named torsion and denoted by T , is given by T = − H [57, 58]. Then, the f ( T ) theory that leads to the same ellipse as holonomy corrected LQC could benamed teleparallell LQC [61]. Dealing with perturbations, the perturbed equations in teleparallelism [59, 60] havebeen applied to this particular function in [62, 63] obtaining a different kind of equation, which differs from the onesof holonomy corrected LQC starting with the velocity of the speed of sound. Unfortunately, besides avoiding theusual Levi-Civita connection, the scalar torsion is not local Lorentz invariant, that is, a local Lorentz transformationof the orthonormal basis could lead to a new orthogonal basis, in which the value of the scalar torsion is differentform the one obtained using the first basis [64, 65]. Thus, given a torsion, a f ( T ) theory becomes preferred frame,which is an undesirable feature. Effectively, let { e ν } ν =0 ,.., and { ¯ e ν } ν =0 ,.., be two orthonormal basis in the tangentbundle parallelally transported, and let T and ¯ T be their corresponding torsions. Since its difference is a divergence,when f is the identity their corresponding actions are the same, which will not happen for a general function f .In the same way as in teleleparallelism, in the Arnowitt-Deser-Misner (ADM) formalism of GR [66, 67], where aglobally hyperbolic spacetime could be foliated by three-dimensional space-like sub-manifolds parameterized by a time and the connection is the usual Levi-Civita one, the extrinsic curvature scalar, namely I , is also equal to − H inthe flat FLRW spacetime using synchronous co-moving coordinates. Since the scalar curvature of the full manifold,namely R , is equal to R + I plus a divergence, where R is the intrinsic curvature, i.e., the scalar curvature of thesub-manifolds, on can built some classes of F ( R , I ) gravitational theories, leading to different backgrounds and oneof them will be the same as in holonomy corrected LQC. In this way, the main goal of this work is to provide thegravitational equations in a particular case, namely F ( R , I ) = R + f ( I ), its perturbed equations and, as a particularcase, to calculate explicitly them for the f theory that leads to the same background as LQC. At this point, it isimportant to emphasize that the theory is gauge dependent, in the sense that, it depends on the foliation chosen or,equivalently, on the choice of the normal vector, because only the sum of the intrinsic and extrinsic curvature is thesame, up a divergence, for all the slicing. So, when the gauge is fixed, the theory becomes covariant only for thecoordinate systems preserving the foliation, i.e., the equations only will become with the same aspect in these referenceframes. Since cosmology deals with the evolution of the universe, in order to have a well defined Cauchy problem[67, 68] one has to fix precisely a foliation, and of course there are some gauges such as the so-called “geodesicslicing” [69], where the lapse function is chosen to be 1, the constant main curvature (the trace of the extrinsiccurvature tensor) slicing [70], the maximal slicing [71], which corresponds to the vanishing of the mean curvature ofthe hyper-surfaces Σ t , or the harmonic slicing [72], where the t coordinate is required to be harmonic with respectto the D’Alembertian. In this work, as in the Chapter 4 of [73], and following the spirit of Weyl’s principle (see [74]for a historical and critical review), we argue that there is a preferred 4-velocity field in the whole spacetime, whichgenerates a preferred non-crossing family of world-lines. Therefore, taking the parameter t as the proper time of thesepreferred observers one obtains a preferred synchronous gauge , or choosing the time t defining the so-called co-movingslicing , i.e., the slicing orthogonal to the world-lines [75], which, as we will see, is the best choice when the universe isfilled by an scalar field, one obtains two preferred different 3 + 1 splittings of the spacetime. Fortunately, and it is anice feature of the theory, the equations of perturbations are gauge invariant, i.e., slicing independent, provided thatthe background, which is the FLRW spacetime, has been split as in GR, using co-moving coordinates.The manuscript is organized as follows: In section II we review the ADM formalism and, for a given foliation,we generalize GR to a R + f ( I ) gravitational theory, obtaining the particular f that leads to the same backgroundas holonomy corrected LQC. At the end of the Section, the gravitational equations that generalize those of GR arederived. Section III is devoted to the study of scalar perturbations in this theory, comparing them with those of theteleparallel f ( T ) gravity, and also applying them to LQC. In section IV we do the same as in section III but for tensorperturbations, and finally, in section V we apply our previously obtained results to the matter bounce scenario inLQC.The units used throughout the paper are (cid:126) = c = 1, where M pl = √ πG is the reduced Planck’s mass.
2. ADM FORMALISM REVISITED
In order to deal with bouncing cosmologies without initial or final singularities we assume that the spacetime isglobally hyperbolic, i.e., it admits a three-dimensional space-like Cauchy surface Σ, four-dimensional Lorentz manifold M with metric g (see [76] or the Chapter 3 of [67] for a review of these concepts), and thus, following the Geroch’ssplitting theorem [77], topologically the spacetime is homeomorphic to Σ × R and thus it admits a foliation or slicing (Σ t ) t ∈ R where for t ∈ R Σ t ≡ { p ∈ M : ˆ t ( p ) = t } , (1)being ˆ t : M → R a scalar field defined in the full spacetime.As in the Arnowitt-Deser-Misner (ADM) formulation of GR [66], we make the decomposition ∂ t = N n + N where N is the so-called lapse function, N is a vector belonging in the tangent space of Σ t named the shift vector and n isa vector perpendicular to Σ t , satisfying g ( n , n ) = −
1. Now, let { ∂ j } j =1 , , be a basis of the tangent space of Σ t , thenthe metric, in coordinates, is given by ds = g µν dx µ dx ν = − N dt + γ ij ( dx i + N i dt )( dx j + N j dt ) , (2)where γ ij = g ( ∂ i , ∂ j ) are the entries of the induced metric in Σ t , which we denote by γ , and N i are the coordinatesof the vector N in the basis { ∂ j } j =1 , , , i.e., N = N j ∂ j .An important ingredient of this formulation is the extrinsic curvature tensor which, although defined with a negativesign in some books, in this work is given by K ( u , v ) = g ( ∇ u v , n ) , (3)where we have denoted by ∇ the Levi-Civita connection, and u , v are vectors of the tangent space of Σ t .Let R be the intrinsic curvature, i.e., the scalar curvature of Σ t , and I = K ij K ij − ( T r ( K )) , where T r ( K ) = K ii is the trace of the extrinsic curvature tensor. Then, the scalar curvature of the whole spacetime M is related withthese quantities via the equality [67] R = R + I − ∇ i v i , (4)where ∇ i v i , is the divergence of the vector field v = ∇ n n − ∇ i n i n .Therefore, the gravitational part of the Hilbert-Einstein action could be written as S HE ; grav = M pl (cid:90) M √− gR = M pl (cid:90) t t (cid:26)(cid:90) Σ t N √ γ ( R + I ) dx (cid:27) dt, (5)which could be generalized, as a F gravitational theory, taking the action S F ; grav = M pl (cid:90) t t (cid:26)(cid:90) Σ t N √ γF ( R , I ) dx (cid:27) dt. (6)Here, it is important to stress that this generalization depends on the chosen foliation. Effectively, for two differentslicing, namely Σ t and ¯Σ ¯ t , the difference between R + I and ¯ R + ¯ I is a divergence, so, in GR, the action is the same,but when one considers a F ( R , I ) theory the action is different from one to other slicing. For this reason, as we haveexplained in the Introduction, we have to fix a foliation Σ t (the gauge) as in the Cauchy problem, and thus, oncethe foliation has been chosen, namely Σ t , the action is covariant for all the coordinate systems where the foliation ¯Σ ¯ t coincides with the chosen one, what always happens for transformations of the form ¯ t = g ( t ), where g is an arbitraryfunction independent of the spatial coordinates.To make physical sense of the theory, the fixed slicing has to be preferred in some sense, as it could be the frame atrest with respect the Cosmic Microwave Background. To justify our choice, we first remember the Weyl’s postulate,which asserts: ” The world-lines of galaxies (on average) form a bundle of non-intersecting geodesics orthogonal to aseries of space-like hyper-surfaces. This series of hyper-surfaces allows for a common cosmic time and the space-likehyper-surfaces are the surfaces of simultaneity with respect to this cosmic time ” [78]. So, the idea behind the Weyl’spostulate is to build up a co-moving reference frame in which the constituents of the universe are at rest (on average)relative to the co-moving coordinates. However, the basis of Weyl postulate (e.g. non-intersecting (on average) world-lines of galaxies) seems questionable if there exists some period in cosmic history when the universe is not filled bydust, for example in the radiation period when the word ”galaxies” has to be changed by relativistic gas particles, orearlier.To overcome the difficulties of Weyl’s postulate and develop it, we follow the Ellis proposal [79] (see also [80]), basedon an specific choice of preferred fundamental world-lines whose 4-velocity, which following the spirit of the Weyl’spostulate has to be intrinsically determined by the constituents of the universe, is the time-like eigenvector of thestress tensor, i.e., satisfyes T νµ u µ = λu ν . For realistic matter, it was shown (see page 89 −
90 of [76]) that such familyexists and is unique due to the weak energy condition: T ( v , v ) ≥ v . What is importantis that if this is a 4-velocity field defined on the whole manifold it implies that its world-lines never cross due to theunicity of the solutions of a differential equation, and thus one has a 3 + 1 splitting of the spacetime. Finally, the timecould be the corresponding proper time of these world-lines defining a preferred synchronous gauge or the co-movingslicing defined as the foliation Σ t perpendicular to the world-lines. In fact, as we will show, when the he universeis filled by an scalar field φ , the later prescription is the better one because in this case the slices coincide with thehyper-surfaces φ = t = constant , and thus, the matter component of the universe is not perturbed along a slice.Note also that, this generalization is completely different from the standard modified F ( R ) gravity, where themodified Friedmann equation is a dynamical equation which depends on the second derivative of the Hubble parameter(see for instance the reviews [81, 82]), because in our theory, as we will immediately see, the modified Friedmannequation is a constraint between de Hubble parameter, the energy density and the scale factor, that does not containhigher order derivatives.As an special case in cosmology, assuming the isotropy and homogeneity of the universe, the Friedman-Lemaˆıtre-Robertson-Walker (FLRW) geometry has to be used, whose line element for co-moving observers is given by ds = − N ( t ) dt + a ( t ) d Σ t . (7)Now we consider the stress tensor for a perfect fluid T νµ = ( ρ + P ) u µ u ν + P δ νµ where P is the pressure and u isthe 4-velocity of the observers. Since T νµ u µ = − pu ν , this shows that the 4-velocity is the time-like eigenvector of thestress tensor which leads to the preferred foliation in the FLRW geometry. In the same way, for an scalar field thestress tensor is T νµ = ∂ µ φ∂ ν φ − (cid:0) g ( ∇ φ, ∇ φ ) + V ( φ ) (cid:1) δ νµ where ∇ φ is the gradient of the scalar field. Provided thatthe gradient was time-like, the 4-velocity u = − ∇ φ √ − g ( ∇ φ, ∇ φ ) is an eigenvector, which for the flat FLRW geometry, hasthe form u = ( ˙ φN | ˙ φ | , , ,
0) and thus leads to the same foliation as the co-moving observers. In fact, when N = 1, t isthe proper time of the observers and also the slice t = constant are perpendicular to the world-lines of the observers,that is, for the FLRW spacetime with N = 1, the co-moving and synchronous slicing coincide.Note that, if one takes another foliation ¯Σ ¯ t with ¯ t = g ( t, x ) and g a function depending of the spatial coordinates x , quantities such as the energy density, the intrinsic or extrinsic curvatures which are constant in every slice, becomenon-homogeneous in the new slices ¯ t = constant , and thus, introduce fictitious perturbations when the new slice is aconsequence of a simple coordinate perturbation (see the beginning of section 7 in [83]).Dealing with this preferred slicing, the extrinsic and intrinsic curvatures are: I = − H N , R = 6 ka , (8)where k = 0 , , − t , respectively.In that case the total action would be S F ; tot = (cid:90) t t N V L grav dt + (cid:90) t t N V L m dt, (9)where the volume is V = a , the gravitational piece of the Lagrangian is L grav = M pl F ( R , I ), and the matter partfor a barotropic fluid is L m = − ρ ( t ) [84, 85], and L m = P = N ˙ φ − V ( φ ) for a scalar field minimally coupled withgravity, where P is the pressure. As a function of the volume and its derivative one has I = − V N V , R = 6 kV / . (10)Since the action does not depend on the derivative of the lapse function, the Euler-Lagrange equation becomes ∂ N ( N ( L grav + L m )) = 0 , (11)which leads to the well-known Hamiltonian constraint, or equivalently, for N = 1 to the modified Friedmann equation − I F I + F = 2 ρM pl ⇐⇒ H F I + 16 F = ρ M pl , (12)where we have denote by F I the partial derivative of F with respect to the extrinsic curvature.We see that this is a constraint that defines a surface in the space ( a, H, ρ ), and, in particular, for the flat FLRWmetric a curve in the plane ( H, ρ ).The dynamical equation is obtained from the first principle of thermodynamics d ( ρV ) = − P dV obtaining theconservation equation ˙ ρ = − H ( ρ + P ). Note also that, when F ( R , I ) = R + I one obtains the usual Friedmannequation H + ka = ρ M pl . (13)Finally, dealing with the flat FLRW geometry, since R = 0, we will use the notation f ( I ) ≡ F (0 , I ). Then, givena curve g ( I ) = ρM pl in the plane ( I , ρ ) we can obtain the corresponding f theory solving the first order differentialequation (12), where ρ must be replaced by g ( I ). Using the variation of constants method for first order differentialequations, the result is given by: f ( I ) = − √−I (cid:90) g ( I ) I√−I d I . (14) Dealing with the flat FLRW geometry with N = 1, i.e., working, as usual, in co-moving coordinates, one has R = 0.Then, redefining the Hubble parameter as ¯ H = √ ρ c M pl H where ρ c is a constant with units of energy density, inorder that ¯ H has the same units as ρ , in the plane ( ¯ H, ρ ), the simplest closed curve is a circle, and since the energydensity has to be positive and the Hubble parameter has to be zero when the energy density vanishes, we mustchoose a circle centered at (0 , ρ c ) with radius ρ c , that is, ¯ H + (cid:0) ρ − ρ c (cid:1) = ρ c , which in the plane ( H, ρ ) depictsthe well-known ellipse (see Fig. 1) that characterizes the holonomy corrected Friedmann equation in Loop QuantumCosmology (LQC) (see for instance [27, 28, 86]) H ρ c M pl + (cid:0) ρ − ρ c (cid:1) ρ c = 1 ⇐⇒ H = ρ M pl (cid:18) − ρρ c (cid:19) . (15)The meaning of ρ c , which is called the critical density, is the maximum value of the energy density. Since thisequation could be written as ρ = ρ c ± (cid:115) I M pl ρ c , (16)where the sign − correspond to the lower branch of the ellipse, i.e., to values of ρ between 0 and ρ c / ρ c / ρ c , one can see that the energy density is bi-valued as a functionof I , which, taking into account that mathematically the square root is bi-valued, we will write as ρ = ρ c − (cid:115) I M pl ρ c , (17)where we use the convention that the sign positive of the square root corresponds to the lower branch and the negativeto the upper one, i.e., ρ = ρ c (cid:18) − (cid:113) I M pl ρ c (cid:19) in the lower branch and ρ = ρ c (cid:18) (cid:113) I M pl ρ c (cid:19) in the upperone. -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 H (a) Hamiltonian constraint or equivalently holonomycorrected Friedmann equation in LQC (units: ρ c = M Pl = 1).The arrow shows that the dynamics is ant-clockwise fornon-phantom matter. -0.6 -0.4 -0.2 0 0.2 0.4 0.6 I (b) Hamiltonian constraint in the plane ( I , ρ ). FIG. 1: Curve depicting the holonomy corrected Friedmann equation.Then, to depict this constraint we will need bi-valued functions f , which could easily be obtained integrating theequation (14), [51–53, 86]: f ( I ) = ρ c M pl (cid:16) − (cid:112) − s − s arcsin s (cid:17) , (18)where s ≡ (cid:113) − I M pl ρ c = (cid:113) H M pl ρ c , belongs in the interval [0 , f in the upper and lower branch. Here, we take two differentprescriptions to define the bi-valued function whose picture appears in Fig. 2:1. Prescription 1:
We choose the sign of the square root positive in the lower branch and negative in the upperone and, arcsin s ≡ (cid:82) s √ − ¯ s d ¯ s where the criterium for the sign of the square root is the same.2. Prescription 2:
We choose the sign of the square root positive in the lower branch and negative in the upperone and, arcsin s ≡ (cid:82) s √ − ¯ s d ¯ s in the lower branch and arcsin s ≡ (cid:82) s √ − ¯ s d ¯ s + π in the upper one, with thesame criterium for the sign of the square root.The advantage of the second prescription is that it ensures that the function f is uni-valued when both branchesmatch, i.e. at s = 1, and in this prescription the function arcsin s is always positive due to the π added in upperbranch. Moreover, as we will show dealing with perturbations, the square of the velocity of sound changes from oneprescription to the other.However, in spite of the authors of [51, 52, 86] arguing that the equation (18) gives the f theory that mimics LQC,they only use f as a one-valued function, meaning that they are only considering the lower branch of the ellipse. Weinsist in this essential point, to obtain the equivalent background to LQC one needs a bi-valued function.Some important final remarks are in order:1. As has been pointed out in [90], the bi-valued functions √ − s and arcsin s could be avoided, allowing thevariable s to belongs in the interval [ − ,
1] (note that by definition it belongs in [0 , s = sin(2 ψ ) with ψ ∈ [0 , π ]. Then, the ellipse is parametrized in the following way H = (cid:115) ρ c M pl sin(2 ψ ) , ρ = ρ c sin ψ, (19) s -1-0.500.511.522.53 f lower branch upper branch (a) Prescription 1. s -1-0.500.511.52 f lower branch upper branch (b) Prescription 2. FIG. 2: Bivalued function f for the upper (green) and lower (red) branches.and the total Hamiltoninan H tot ≡ ˙ V p V + ˙ φp φ − L tot = − ρ c M pl (1 − (cid:112) − s ) + ρ, (20)where p V = ∂ L tot ∂ ˙ V and p φ = ∂ L tot ∂ ˙ φ becomes H tot = − ρ c M pl sin ψ + ρ, (21)which coincides with the usual Hamiltonian in LQC.2. At low energy densities a viable theory must coincide with General Relativity, so in the flat FLRW geometryit will be H = ρ M pl , meaning that for H = 0 the energy density vanishes. On the other hand, if one has abounce, at the bounce the energy density does not vanish when the Hubble parameter is zero. This means that,in a theory with bounces where the modified Friedmann equation is a constraint relating only H and ρ , theenergy density as a function of the Hubble parameter has to be a multi-valued function, and thus, multi-valuedfunctions f are needed to obtain the corresponding modified Friedmann equation.3. Bouncing cosmologies mimicking LQC could also be obtained, in a gauge invariant, approach named F ( R, T )gravity [87], where T denotes the trace of the the stress tensor. In this theory, on the contrary of the oneproposed in this manuscript, it is the matter sector which is modified. Dealing with the particular case F ( R, T ) = R + M pl f ( T ), the modified Friedmann equation, for co-moving observers in the flat FLRW spacetime, becomes[88] 3 H = 1 M pl (cid:20) (1 + f (cid:48) ( T )) ρ + f (cid:48) ( T ) L m − f ( T )2 (cid:21) . (22)Then, assuming that the universe is filled by an scalar field (this is necessary in order to deal with primordialperturbations) on has L m = P , but in order that the right hand side of (22) only depends of T , one has toassume that the field mimics a perfect fluid with a constant Equation of State parameter w ≡ Pρ . To simplify,we assume that the scalar field mimics a dust fluid ( w = 0 = ⇒ T = − ρ ), obtaining the equation3 H = − M pl (cid:20) (1 + f (cid:48) ( T )) T + f ( T )2 (cid:21) . (23)To obtain the uni-valued function f leading to the same Friedmann equation as in holonomy corrected LQC,one only has to solve the first order differential equation(1 + f (cid:48) ( T )) T + f ( T )2 = T (cid:18) Tρ c (cid:19) , (24)whose solution is f ( T ) = ρ c T .Unfortunately, apart from other difficulties that leads this approach unviable [89], the continuity equation differsform the usual one, namely ˙ ρ = − Hρ for a pressureless fluid. For f ( T ) = ρ c T , one has˙ ρ = − Hρ − ρρ c (cid:18) − ρ ρ c (cid:19) , (25)which is singular at ρ = ρ c /
2, i.e., when both branches of the ellipse matches. So, only few unnatural anunrealistic bouncing solutions could be obtained (see Section 4 of [88]).4. Recently, the function (18) has been used in the context of mimetic gravity [90–92] and also as a function of theCarminati-MacLenaghan scalars [93], to reproduce, in a covariant way, the same background as in holonomycorrected LQC. A perturbed theory of this approach has not been devoloped yet, although it deserve futureinvestigation. F theory To get the gravitational equation we start with the total action S F,tot = (cid:90) t t (cid:40)(cid:90) Σ t N √ γ (cid:32) M pl F ( R , I ) + L matt (cid:33) d x (cid:41) dt, (26)where we assume the universe is filled by an scalar field minimally coupled with gravity, whose Lagrangian is L matt = (cid:18) − φ µ φ µ − V ( φ ) (cid:19) = (cid:32) ˙ φ N − N i N ˙ φφ i − (cid:18) γ ij − N i N j N (cid:19) φ i φ j − V ( φ ) (cid:33) . (27)Then, to obtain the constraints and dynamical equations we have to perform the variation with respect the variables N, N i , γ ij and with respect the scalar field φ .Taking into account that the components of the extrinsic curvature could be written as follows [67] K ij = 12 N ( D i N j + D j N i − ˙ γ ij ) , (28)where D is the induced Levi-Civita connection in the space tangent of Σ t , that the variation of the intrinsic curvatureis δ R = R ij δγ ij + D k (cid:0) γ ij δ Γ kij − γ kj δ Γ iij (cid:1) , (29)where R ij = R ic ( ∂ i , ∂ j ) are the entries of the Ricci tensor and Γ kij are the Christoffel symbols, and using the formula γ km ( D i δγ mj + D j δγ mi − D m δγ ij ) = 2 δ Γ kij , (30)and the Gauss divergence theorem one gets (see [94], to our knowledge the first instance where these equations werederived): F − I F I = − M pl ∂ N ( N L matt ) , (31) D j ( F I π ji ) = − NM pl ∂ N i L matt , (32) − ( D j D i − δ ji D )( F R N ) + N G jF,i + 2
N F I ( K mj K im − T r ( K ) K ji ) − √ γ γ ki ∂ t ( √ γF I π kj ) − D k ( F I ( π ki N j + π kj N i − π ji N k )) = NM pl T ji , (33)where π ji are the entries of the tensor π ≡ K − T r ( K ) γ , D = D k D k is the three-dimensional Laplace-Beltramioperator, G F = − F γ + F R R ic is the three-dimensional modified Einstein tensor, and T ji = √ γ γ mj ∂ γ im ( √ γ L matt )are the entries of the three-dimensional stress-energy tensor.The equations (31) and (32) are the well-known Hamiltonian and diffeomorphism constraints, and the equation(33) is the dynamical one.For the scalar field, the conservation equation is −∇ φ + V φ = 0, where ∇ = ∇ i ∇ i is the Laplace-Beltrami operatorin the full manifold M , which in the ADM formalism becomes1 √ γN ∂ t (cid:18) √ γN ˙ φ (cid:19) − N D i (cid:0) N φ i (cid:1) − N D i (cid:18) N i N ( ˙ φ − N j φ j ) (cid:19) + V φ = 0 . (34)Finally, once these equations are obtained one can deal with the Cauchy initial value problem as in GR, i.e., givena fixed foliation, in our case the preferred one, one has to choose a slice at some time t , namely Σ, which is assumedto be a Cauchy hyper-surface), and we have to consider on it the set of variables ( γ, K, φ, ˙ φ ) where γ is the metric and K the extrinsic curvature tensor on Σ, which has to satisfy the Hamiltonian and Diffeomorphism constraints. Then,to solve the dynamical equations one has to follow, as in GR, some of the schemes proposed in numerical relativity (see [95, 96] for reviews).
3. SCALAR PERTURBATIONS
In the previous Section we have shown the lack of covariance of our approach. Nevertheless, our theory is gaugeinvariant at the perturbation level. To prove it, we consider the general line element containing scalar perturbations[75] ds = − (1 + 2Ψ) dt + 2 a∂ i Bdtdx i + a [(1 − δ ij − ∂ ij E − h ij ] dx i dx j , (35)where h is a symmetric, traceless and transverse tensor: h ii = ∂ i h ij = 0, and Ψ, Φ and E are scalars [83].In order to deal with scalar perturbations, since there are two degrees of freedom we choose the so-called Newtonianor Longitudinal gauge, where the two metric variables E and B vanishes. Then, we obtain the metric ds = − (1 + 2Ψ) dt + a (1 − δ ij dx i dx j . (36) Remark 3.1
In General Relativity, i.e., when F ( R , I ) = R + I and the matter part of the Lagrangian is depictedby an scalar field, one has Φ = Ψ [83], but in general, for a F theory, these functions are different (see section of[97]). Fortunately, as we will immediately show, if one chooses F ( R , I ) = R + f ( I ) the equality holds. Due to thissimplification in the theory, we will only consider this kind of F models. Since in this gauge one has N = (1 + 2Ψ), γ ij = a (1 − δ ij and N i = 0, the extrinsic curvature is given by I = − H (1 − − ˙Φ) (1 − (1 + 2Ψ) ≈ − H + 12 H (Ψ H + ˙Φ) , (37)where the symbol ≈ means: up to first order.On the other hand, since the Christoffel symbols satisfyΓ kij = − − (cid:0) δ ki Φ j + δ kj Φ i − δ ij δ mk Φ m (cid:1) , (38)0the entries of the Ricci tensor will be R ij = ∂ k Γ kij − ∂ j Γ kik + Γ kij Γ lkl − Γ lik Γ klj ≈ ∂ k Γ kij − ∂ j Γ kik = Φ ij + δ ij ∆Φ , (39)and the scalar curvature will become R ≈ a ∆Φ, where ∆ = (cid:80) i =1 ∂ ii is the standard three-dimensional Laplacian.Now, in the linear approximation, we will use the following notation: I = I + I , with I = − H , I = 12 H (Ψ H + ˙Φ) , (40) φ = φ ( t ) + δφ = ⇒ V ( φ ) = V + V φ, δφ, with V = V ( φ ) , V φ, = V φ ( φ ) . (41) g = f − I f I , g = g ( I ) and g I , = ∂ I g ( I ) ≡ g I ( I ) , (42)where the meaning of g is that the equation g = ρ M pl , where ρ is the non-perturbed energy density, is the non-perturbed hamiltonian constraint, i.e., the modified Friedmann equation.For i (cid:54) = j , the equation (33) becomes − D j D i ( F R N ) + N G jF,i = 0 , (43)which for our choice F ( R , I ) = R + f ( I ) (essential to obtain the desired result), becomes − D j D i ( N ) + N R ji = 0 = ⇒ ∂ ij (Φ − Ψ) = 0 , (44)which leads to the solution Φ = Ψ. Then, the perturbed equations become:1 a ∆Φ + 3 H g I , Φ + 3 Hg I , ˙Φ = 12 M pl (cid:16) ˙ φ ( ˙ δφ − ˙ φ Φ) + V φ, δφ (cid:17) , (45) − g I , ( H Φ + ˙Φ) = 12 M pl ˙ φ δφ, (46) − g I , ¨Φ − (4 Hg I , + ˙ g I , ) ˙Φ − ((3 H + 2 ˙ H ) g I , + H ˙ g I , )Φ = 12 M pl (cid:16) ˙ φ ( ˙ δφ − ˙ φ Φ) − V φ, δφ (cid:17) . (47)Combining the equations and using the cosmic time one gets the dynamical equation for the Newtonian potentialthe following equation, which coincides with the one of GR (see equation (6.48) of [97]) when one takes f ( I ) = I ,Φ (cid:48)(cid:48) − c S ∆Φ + 2 (cid:18) H − (cid:18) φ (cid:48)(cid:48) φ (cid:48) + (cid:15) (cid:19)(cid:19) Φ (cid:48) + 2 (cid:18) H (cid:48) − H (cid:18) φ (cid:48)(cid:48) φ (cid:48) + (cid:15) (cid:19)(cid:19) Φ = 0 , (48)where the square of the velocity of sound for scalar perturbations is c S = − g I , and we have introduced the notation (cid:15) = − g (cid:48)I , g I , .Now introducing, as in standard cosmology, (see for example formulae (8.56)-(8.58) of [83]) the gauge invariantMukhanov-Sasaki (M-S) variables v S = a (cid:18) δφ + φ (cid:48) H Φ (cid:19) , z S = aφ (cid:48) H , (49)the equation (48) becomes, in the Fourier space, the following gauge invariant M-S equation v (cid:48)(cid:48) S,k + (cid:18) c S k − z (cid:48)(cid:48) S z S (cid:19) v S,k = 0 . (50)Some important final remarks are in order:11. If one changes I by T , where T is the torsion that appears when one uses the Weitzenb¨ock connection insteadof the Levi-Civita one, these equations are practically the same as the ones that appear in teleparallel f ( T )gravity [59]. The difference appears in equation (45), where in f ( T ) gravity the coefficient that multiplies theLaplacian is f T a , and it is very important because the velocity of sound changes from one theory to the other.Effectively, in this theory the equation of the Newtonian potential is the same as (48) but with a velocity ofsound given by ˜ c S,s = − f I , g I , . Moreover, in teleparallel f ( T ) gravity the variables v S and z S has to be [61, 62] v S = a (cid:112) | f I , | (cid:18) δϕ + φ (cid:48) H Φ (cid:19) , z S = aφ (cid:48) (cid:112) | f I , |H , (51)in order to recover the M-S equation (50), but replacing c S by ˜ c S .2. Applying the theory to the particular model given in (18), we will obtain a theory, which we name Extrinsiccurvature LQC , where − g I , = , with Ω ≡ − ρρ c , and thus, c S = Ω and (cid:15) = Ω (cid:48) . Then, the equations (48) and(50) are identical to the ones obtained in holonomy corrected LQC [47], which from our point of view supportsthe consistency of our approach to obtain bouncing cosmologies via modified gravity in the ADM formalism.3. On the contrary, in teleparallel LQC, since f I , = 1 s arcsin s, (52)where s = (cid:113) − I M pl ρ c , the square of the velocity of the sound will be ˜ c S = Ω s arcsin s , and contrary to holonomycorrected and extrinsic curvature LQC, is always positive if one takes the first prescription below formula (18),but taking the second one, ˜ c S is positive in the lower branch, negative in the upper one and unbounded at thebounce.
4. TENSOR PERTURBATIONS
For metric perturbations the line element, in the preferred foliation, is ds = − dt + a ( δ ij − h ij ) dx i dx j , (53)where, as we have already introduced in the previous section, h is a symmetric, traceless and transverse tensor: h ii = ∂ i h ij = 0. Due to these properties one has √ γ ≈ a , T r ( K ) ≈ − H and I ∼ = − H . Then, the equation (33)reduces to 2 f I , ( K mj K mi + 3 HK ji ) + G jF,i − a γ ki ∂ t ( a f I , π kj ) = 0 . (54)And taking into account that R ji ≈ − a ∆ h ji and R ≈
0, the equation for the gravitational waves becomes f I , ¨ h ji + 3 Hf I , ˙ h ji − a ∆ h ji + ˙ f I , ˙ h ji = 0 . (55)On can see that, this equation differs from the one obtained in f ( T ) gravity, in the coefficient that multiplies theLaplacian.Introducing the variable v T = hz T with z T = aM pl (cid:112) | f I , | and h denoting h ji , one obtains, in Fourier space, theM-S equation for tensor perturbations v (cid:48)(cid:48) T,k + (cid:18) c T k − z (cid:48)(cid:48) T z T (cid:19) v T,k = 0 , (56)where now the square of the velocity of the sound is equal to c T = f I , .The variables v , z T are the same used en teleparallel f ( T ) gravity, and the M-S for gravitational waves only differsin the square of the velocity of sound, which in teleparallelism is always 1.2Now, if we consider the function that leads to the same background as holonomy corrected LQC, one has f I , = s arcsin s, what implies that, if one takes the first prescription below (18) the square of the velocity of the sound c T = f I , is positive in the lower branch of the ellipse and negative and the upper one, as in holonomy corrected LQC,although it is different because in LQC for both kind of perturbations, scalar and tensors, one has c T = Ω. Moreover,in extrinsic curvature LQC following this prescription, the square of the velocity is discontinuous when the branchesmatch. This does not happen if one chooses the second prescription where c T is always positive and continuous.However, the main difference with holonomy corrected LQC appears in the definition of z T , because in the formercase one has [48] z T = aM pl √ c T = aM pl √ Ω , which becomes complex in the upper branch, being an exotic feature ofholonomy corrected LQC, and reducing the ratio of tensor to scalar perturbations to negligible values in the matterbounce scenario (see next section) [62, 98]. This does not happen in theories such as GR, F ( R ) or F ( T ) gravity, andneither does it in our approach defining z T = aM pl √ | c T | = aM pl (cid:113) | arcsin s | s . But, if one chooses z T = aM pl √ c T = aM pl (cid:113) arcsin ss (which could be done because the equation (56) is the same for both definitions of z T ), for the first prescription of f , since the square of the velocity of sound becomes negative in the upper branch, z T will be complex in the upperbranch as in holonomy corrected LQC. This will reduce the tensor/scalar ratio to negligible values in the case of thematter bounce scenario, which does not happen with the second prescription, because in that case c T = s arcsin s isalways positive.Summing up, the M-S for scalar and tensor perturbation in Extrinsic curvature LQC are: v (cid:48)(cid:48) S,k + (cid:18) c S k − z (cid:48)(cid:48) S z S (cid:19) v S,k = 0 , v (cid:48)(cid:48)
T,k + (cid:18) c T k − z (cid:48)(cid:48) T z T (cid:19) v T,k = 0 , (57)where, if one considers the model that leads to the same background as holonomy corrected LQC, i.e., given by thefunction (18) and we use the second prescription, the square of the velocity of sound is c S = Ω = 1 − ρρ c for scalarperturbations and c T = s arcsin s for the tensor ones. Then, since in holonomy corrected LQC one has c S = c T = Ωwe see that both approaches coincide for scalar perturbations, but for tensor ones they differ in the square of thevelocity of sound, where in holonomy corrected LQC the sign of the square of the velocity of sound changes its signpassing from one branch of the ellipse to the other one, but in Extrinsic curvature LQC it is always positive andin fact, for tensor perturbations, as we already explained, our approach is closer to F ( T ) gravity. In conclusion,Extrinsic curvature LQC leads to the same scalar perturbation equations that holonomy corrected LQC, and veryclose equations to F ( T ) gravity for tensor perturbations.
5. THE MATTER BOUNCE SCENARIO IN THEORIES WITH THE SAME BACKGROUND AS LQC
In this scenario the universe is filled by dust matter, whose background in LQC, can be mimicked by a scalar fieldwhose potential is [99] V ( φ ) = 2 ρ c e − √ φ Mpl (cid:18) e − √ φ Mpl (cid:19) , (58)where φ is the homogeneous part of the scalar field, i.e., the unperturbed part. The conservation equation is givenby ¨ φ + 3 H ± ˙ φ + V φ = 0 , (59)where H ± = ± (cid:114) ρ M pl (cid:16) − ρ ρ c (cid:17) are the values of the Hubble parameter in the contracting and expanding phase, and ρ is the unperturbed energy density ρ = ˙ φ + V ( φ ). Thus, for each of the infinite solutions of (59) one obtains adifferent background H ( t ). On the other hand, there is an analytic solution given by φ ( t ) = 2 M pl √ (cid:32)(cid:115) ρ c t M pl + (cid:115) ρ c t M pl + 1 (cid:33) , (60)3 -10 -5 0 5 10-1.5-1-0.500.511.5 (a) Phase portrait of the system, with curves bouncing from H <
H > φ >
0: theunstable and stable varieties for
H >
H <
H >
H < -10 -5 0 5 10t-0.3-0.2-0.100.10.20.3 H (b) H ( t ) for the orbits in Fig. 3a bouncing left of the stable variety (red), between the stable and unstable varieties (blue),right of the unstable variety (purple), and for the analytic orbit (black). FIG. 3: The extrinsic curvature LQC model: possible backgrounds (black: analytic solution; red, blue, purple:qualitatively different alternatives).which leads to the following analytic background H ( t ) = ρ c t ρ c t + M pl . (61)The other solutions have to be calculated numerically, and lead to different backgrounds (see Figures 3a, 3b, alsothe figures 8 and 9 of [100] and the corresponding explanation).4Dealing with the power spectrum, for scalar perturbations it is given by the formula [75] P S ( k ) = k π | ζ k | , (62)where, as we have already explained in Section III, ζ k ≡ v k z S is the curvature fluctuation in co-moving coordinates. Inthe case of the matter bounce scenario, taking at early times the scale factor a ( t ) = (cid:16) ρ c t M pl (cid:17) / , one has [63] P S ( k ) = 3 ρ c ρ pl (cid:18)(cid:90) ∞−∞ a ( t ) z S ( t ) dt (cid:19) , (63)and for tensor perturbations P T ( k ) = ρ c ρ pl (cid:18)(cid:90) ∞−∞ a ( t ) z T ( t ) dt (cid:19) , (64)where ρ pl ≡ π M pl is Planck’s energy density.Therefore, the ratio of tensor to scalar perturbations is r = (cid:16)(cid:82) ∞−∞ a ( t ) z T ( t ) dt (cid:17) (cid:16)(cid:82) ∞−∞ a ( t ) z S ( t ) dt (cid:17) . (65)Using the analytic solution (60), and the previous notation Ω = 1 − ρρ c and s = (cid:113) H M pl ρ c , one obtains for the firstprescription of f :1. Holonomy corrected LQC: In this case one has z S = a ˙ φ H and z T = aM pl √ Ω , which leads to P S ( k ) = π ρ c ρ pl (see [98])and P T ( k ) = 0. Thus, r = 0.2. Teleparallel LQC: In this case one has z S = a ˙ φ H (cid:113) s | arcsin s | and z T = aM pl (cid:113) | arcsin s | s , which leads to P S ( k ) = C ρ c ρ pl , where C = (cid:80) ∞ n =1 ( − n (2 n +1) ∼ = 0 . ... is Catalan’s constant, and P T ( k ) = Si ( π ) ρ c ρ pl where Si ( x ) = (cid:82) x zz dz is the Sine integral function, and thus, r = 3 (cid:16) Si ( π ) C (cid:17) ∼ = 6 . z S = a ˙ φ H and z T = aM pl (cid:113) | arcsin s | s , which leads to P S ( k ) = π ρ c ρ pl and P T ( k ) = Si ( π ) ρ c ρ pl , and thus, r = π Si ( π ) ∼ = 9 . r ≤ .
12 provided by the joint analysis of BICEP2/Keck Array and Planck teams [101]. Indeed, the authors’ numericalcomputations, illustrated in Figure 4, show that for backgrounds bouncing at φ M pl ∈ [ − . , − . ∪ [1 . , .
20] thetensor/scalar ratio r is below 0.12, while the scalar power spectrum P S ( k ) is still finite, requiring a fine-tuning of thevalue of the scalar field at the bounce point. The values of P S ( k ) in the admissible domain [ − . , − . ∪ [1 . , . φ M pl = − . , .
20, and descend to 16 ρ c ρ pl at theinner bounds φ M pl = − . , .
16. Consequently, since in Extrinsic curvature LQC, the critical energy density is aparameter of the theory, in order to match with the observational data P S ∼ = 2 × − [102] we have to choose ρ c ∼ − ρ pl and fine tune the value of the backgrounds at the bouncing time in order to satisfy that φ M pl (atbounce) was extremely close to − .
20 or 1 .
16. However in holonomy corrected LQC the critical energy density hasa determined value obtained relating the black hole entropy in LQC with the Bekenstein-Hawking entropy formula[103], which is approximately ρ c ∼ = 0 . ρ pl disagreeing with the value obtained in the matter bounce scenario.Finally, note that using the second prescription (see below (18)) to define the bi-valued function f , scalar pertur-bations in teleparallel LQC are not well defined because the power spectrum of scalar perturbations diverges, this5 -1.5 -1 -0.5 0 0.5 1 1.50246810 r ( p r e sc r i p t i on ) (a) For prescription 1 of f . -1.5 -1 -0.5 0 0.5 1 1.5012345 r ( p r e sc r i p t i on ) (b) For prescription 2 of f . FIG. 4: Tensor/scalar ratio r for backgrounds solution to Extrinsic curvature LQC Eq. (59). The backgrounds areindexed according to the value of φ at bounce. The Planck-BICEP2 bound r ≤ .
12 is indicated.is the reason why the first prescription is used in [62, 63]. On the contrary, in extrinsic curvature LQC both kindsof perturbations are well defined: the scalar ones are independent of the prescription and for tensor ones, in bothprescriptions, the power spectrum is of the same order. In fact, for the analytic solution (60) we have obtained P T ( k ) = Si ( π ) ρ c ρ pl , leading to a tensor/scalar ratio equal to r = π Si ( π ) ∼ = 4 .
1. Figure 4 illustrates the comparisonof the values of the tensor/scalar ratio r for both prescriptions of f ; the backgrounds for which r ≤ .
12 turn out tobe almost exactly the same.
As we have seen above when calculating the power spectrum, dealing with the LQC background, it is of the order ρ c ρ pl , then to match with the observational value P S ( k ) ∼ × − , one has to take ρ c ∼ − ρ pl . This is not aproblem neither for teleparallel LQC nor for our extrinsic curvature LQC because in both theories the critical densityis only a parameter, however this is in contradiction with the value of the the critical energy density in holonomycorrected LQC which has the determined value ρ c ∼ = 0 . ρ pl .In the same way, the matter bounce scenario leads to a flat power spectrum, i.e., the spectral index of scalarperturbations is exactly equal to 1, which does not agree with the recent observational results provided by the jointanalysis of BICEP2/Keck Array and Planck teams [101]. This problem could easily overpassed assuming that theuniverse is not exactly matter dominated, rather with an Equation of State P = − (cid:15)ρ with (cid:15) ≈ .
003 [98]. However,the problem is not solved at all because, as we have already shown, in order to have a tensor/scalar ratio satisfyingthe constraint r ≤ .
12 we have to fine tune the value of the scalar field at the bounce. For this reason, a moresophisticated improvement seems necessary. One way comes from the so-called matter-ekpyrotic bounce scenariointroduced in [104], developed in [105, 106], and studied for the particular of the LQC background in [100, 107].In this new scenario, the universe evolves at very early times in the contracting phase from a matter dominationregime to an ekpyrotic one, which leads, in the case of the LQC background with the model proposed in [107], to apower spectrum of scalar perturbations of the order H E /M pl , where H E is the value of the Hubble parameter at thephase transition from matter domination to ekpyrotic phase. Moreover, in the same framework of LQC, the improvedmodel studied in [100] leads to the same power spectrum of scalar perturbations as in [107], with a spectral index andrunning entering in its observational 1-dimensional marginalized domain at 2 σ C.L..On the other hand, apart from solving the well-known Belinsky-Khalatnikov-Lifshitz (BKL) instability: the efectiveenergy density of primordial anisotropy scales as a − in the contracting phase [108], the more important thing is thatthe phase transition provides the mechanism to create enough particles, via gravitational particle production, to6reheat and thermalize the universe in order to math with the hot Friedmann universe [109, 110].
6. CONCLUSIONS
We have presented a theory based in the ADM formalism where, given a pre-determined slicing, the Lagrangian ofthe gravitation sector is proportional to the intrinsic curvature plus a function f depending on the extrinsic one. Thea priori slicing is the price one has to pay in order to generalize, using the ADM formalism, GR to an f -theory, andsimilarly in teleparallelism where a fixed orthonormal basis in the tangent bundle has to be chosen, or in holonomycorrected LQC where the Asthekar connection is replaced by a convenient sinus function.In the work we have tried to justify the choice of a preferred foliation based on an improvement of Weyl’s postulate,where the 4-velocity of the preferred observers is the time-like eigenvector of the stress tensor, which goes precisely inthe spirit of Weyl’s postulate, where the matter content of the universe must fix a preferred frame. Fortunately, wehave shown that the equations dealing with perturbations are gauge invariant, as in General Relativity, teleparallelismor holohomy corrected LQC.At the level of the background, i.e., working in the flat FLRW spacetime with the usual co-moving frame, we showthat in order to have a bounce, the use of multivalued functions f is mandatory , and the simplest one is the one thatleads to the holonomy corrected Friedmann equation in LQC. For a general metric, using the variational principle wehave obtained, in the case of a general f -theory, the Hamiltonian and diffeomorphism constraints and the dynamicalequations, which have been used to calculate the equations of perturbations in this theory.We have compared them with the ones obtained in teleparallel f ( T ) gravity, being T the torsion provided by theWeitzenb¨ok connection, showing that for tensor perturbations the M-S equation coincides, and for scalar perturbationsit only differs in the square of the velocity of sound. Moreover, for the particular case of the f which leads to thesame background of LQC, we have compared them with the perturbed equations of the holonomy corrected LQC,obtaining that for scalar perturbations, the M-S equation is the same, but for tensor perturbations it is completelydifferent.Finally, we have applied the above results to the matter bounce scenario in LQC, calculating the power spectrum forscalar and tensor perturbations, and showing numerically that, by choosing carefully some backgrounds, the ratio oftensor to scalar perturbations, namely r , satisfies the observational constraint r ≤ .
12 provided by the joint analysisof BICEP2/Keck Array and Planck teams [101].
Acknowledgments
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