Viability of the Matter Bounce Scenario
VViability of the Matter Bounce Scenario
Jaume de Haro ∗ and Jaume Amor´os † Departament de Matem`atica Aplicada I, Universitat Polit`ecnica de Catalunya,Diagonal 647, 08028 Barcelona, Spain
Abstract
It is shown that teleparallel F ( T ) theories of gravity combined with LoopQuantum Cosmology support a Matter Bounce Scenario which is an altern-ative to the inflation scenario in the Big Bang paradigm. It is checked thatthese bouncing models provide theoretical data that fits well with the currentobservational data, allowing the viability of the Matter Bounce Scenario. It is well-known that inflation suffers from several problems (see [1] for a reviewabout these problems), like the initial singularity which is usually not addressed,or the fine-tuning of the degree of flatness required for the potential in order toachieve successful inflation [2].In order to avoid these problems, an alternative scenario to the inflationaryparadigm, called
Matter Bounce Scenario (MBS), has been developed in order toexplain the evolution of our Universe (see [3]). Essentially, it depicts at very earlytimes a matter dominated Universe in a contracting phase, that evolves towardsthe bounce and afterwards enters an expanding phase. This model, like inflation,solves the horizon problem that appears in General Relativity (GR) and improvesthe flatness problem in GR (where spatial flatness is an unstable fixed point andfine tuning of initial conditions is required), because the contribution of the spatialcurvature decreases in the contracting phase at the same rate as it increases in theexpanding one (see for instance [4]).The aim of our work is to construct viable bouncing cosmologies where thematter part of the Lagrangian is composed of a single scalar field and, therefore,have to go beyond General Relativity, since GR forbids bounces when one dealswith a single field. Hence, theories such as holonomy corrected Loop QuantumCosmology (LQC) [5], where a big bounce appears owing to the discrete structure ∗ E-mail: [email protected] † E-mail: [email protected] a r X i v : . [ g r- q c ] N ov f space-time [6] or teleparalellism [7] must be taken into account. When dealingwith these theories, in order to obtain a theoretical value of the spectral index and itsrunning that may fit well with current experimental data, a quasi-matter dominatedregime in the contracting phase termed by the condition (cid:12)(cid:12)(cid:12) w ≡ Pρ (cid:12)(cid:12)(cid:12) (cid:28) , where P and ρ are respectively the pressure and the energy density of the Universe, has tobe introduced [8].Since in Matter Bounce Scenario the number of e-folds before the end of thequasi-matter domination regime can be relatively small, the horizon problem doesnot exist in bouncing cosmologies and the flatness problem is neutralized [4]. Thisargues for the viability of such models, making it possible that for certain mat-ter bounce scenarios the forecast values of the spectral index and of the runningparameter agree well with the most accurate current observations.In contrast, in slow roll inflation one must consider the running of the spectralindex corresponding to N e-folds before the end of the inflation, which in general,is of the order of N − . This value turns out to be very small, when one substitutesfor N the minimum number of e-folds which are needed to solve the horizon andflatness problem in inflationary cosmology (the usual accepted value is N > ),as compared with its corresponding observational value − . ± . comingfrom the most recent Planck data [9]. This shows that these slow roll models areless favored by observations.The units used in the paper are: (cid:126) = c = 8 πG = 1 . F ( T ) gravity in flat FLRW geometry Teleparallel theories are based in the
Weitzenb¨ock space-time . This space is R ,with a Lorentz metric, in which a global, orthonormal basis of its tangent bundlegiven by four vector fields { e i } has been selected, that is, they satisfy g ( e i , e j ) = η ij with η = diag ( − , , , . The Weitzenb¨ock connection ∇ is defined byimposing that the basis vectors e i be absolutely parallel, i.e. that ∇ e i = 0 .The Weitzenb¨ock connection is compatible with the metric g , and it has zerocurvature because of the global parallel transport defined by the basis { e i } . Theinformation of the Weitzenb¨ock connection is carried by its torsion, and its basicinvariant is the scalar torsion T . The connection, and its torsion, depend on thechoice of orthonormal basis { e i } , but if one adopts the flat Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) metric and selects as orthonormal basis { e = ∂ , e = a ∂ , e = a ∂ , e = a ∂ } , then the scalar torsion is T = − H , (1)where H = ˙ aa is the Hubble parameter, and this identity is invariant with respectto local Lorentz transformations that only depend on the time, i.e. of the form (cid:101) e i = Λ ki ( t ) e k (see [10, 11]). 2ith the above choice of orthonormal fields, the Lagrangian of the F ( T ) the-ory of gravity is L T = V ( F ( T ) + L M ) , (2)where V = a is the volume of the Universe, and L M is the matter Lagrangiandensity.The Hamiltonian of the system is H T = (cid:18) T dF ( T ) d T − F ( T ) + ρ (cid:19) V , (3)where ρ is the energy density. Imposing the Hamiltonian constrain H T = 0 leadsto the modified Friedmann equation ρ = − dF ( T ) d T T + F ( T ) ≡ G ( T ) (4)which, as T = − H , defines a curve in the plane ( H, ρ ) .Equation (4) may be inverted, so a curve of the form ρ = G ( T ) defines an F ( T ) theory with F ( T ) = − √−T (cid:90) G ( T ) T √−T d T . (5)To produce a cyclically evolving Universe, let us take the F ( T ) theory arisingfrom the ellipse that defines the holonomy corrected Friedmann equation in LoopQuantum Cosmology H = ρ (cid:18) − ρρ c (cid:19) , (6)where ρ c is the so-called critical density .To obtain a parametrization of the form ρ = G ( T ) , the curve has to be split intwo branches ρ = G ± ( T ) = ρ c (cid:32) ± (cid:115) T ρ c (cid:33) , (7)where the branch ρ = G − ( T ) corresponds to ˙ H < and ρ = G + ( T ) is the branchwith ˙ H > . Applying Eq. (5) to these branches produces the model ([12, 13, 14]) F ± ( T ) = ± (cid:114) − T ρ c (cid:32)(cid:115) − T ρ c (cid:33) + G ± ( T ) . (8)3 Matter Bounce Scenario
Matter Bounce Scenarios (see [3] for a recent review) are essentially characterizedby the Universe being nearly matter dominated at very early times in the contract-ing phase (to obtain an approximately scale invariant power spectrum) and evolvingtowards a bounce where all the parts of the Universe become in causal contact [12],solving the horizon problem, to enter into a expanding regime, where it matches thebehavior of the standard hot Friedmann Universe. They constitute an alternative tothe inflationary paradigm.According to the current observational data, in order to obtain a viable MBSmodel, the bouncing model has to satisfy some conditions that we have summar-ized as follows:1. The latest Planck data constrain the value of the spectral index for scalarperturbations and its running, namely n s and α s , to . ± . and − . ± . respectively [9]. The analysis of these parameters providedby Planck makes no slow roll approximation (in fact, the determination ofcosmological parameters from the first year WMAP observations was doneconsidering the Λ CDM model [15]), which means that the parameters n s and α s could be used to test bouncing models. On the other hand, it is well-known that the ways to obtain a nearly scale invariant power spectrum ofperturbations with running are either a quasi de Sitter phase in the expand-ing phase or a nearly matter domination phase at early times, in the contract-ing phase [16]. Then, since for the MBS one has n s = 1 , if one wants toimprove the model to match correctly with this observational data, one hasto consider, at early times in the contracting phase, a quasi-matter domina-tion period characterized by the condition (cid:12)(cid:12)(cid:12) w ≡ Pρ (cid:12)(cid:12)(cid:12) (cid:28) , being P and ρ thepressure and the energy density of the Universe.2. The Universe has to reheat creating light particles that will thermalize match-ing with a hot Friedmann Universe. Reheating could be produced due to thegravitational particle creation in an expanding Universe [17]. In this case,an abrupt phase transition (a non adiabatic transition) is needed in order toobtain sufficient particle creation that thermalizes producing a reheating tem-perature that fits well with current observations. This method was used in thecontext of inflation in [18, 19], where a sudden phase transition from a quaside Sitter phase to a radiation domination or a quintessence phase was as-sumed in the expanding regime. It is shown in [20] that gravitational particleproduction could be applied to the MBS, assuming a phase transition fromthe matter domination to an ekpyrotic phase in the contracting regime, andobtaining a reheating temperature compatible with current data.3. Studies of distant type Ia supernovae ([21] and others) provide strong evid-ence that our Universe is expanding in an accelerating way. A viable model4ust take into account this current acceleration, which could be incorpor-ated, in the simplest case, with a cosmological constant, or by quintessencemodels [22]. There are other ways to implement the current cosmic acceler-ation, for example using F ( R ) gravity (see for instance [23]), but the currentmodels that provide this behavior are very complicated, and the main object-ive in MBS is to present the simplest viable models.4. The data of the seven-year survey WMAP ([24]) constrains the value of thepower spectrum for scalar perturbations to be P S ( k ) ∼ = 2 × − . Thenumerical results (analytical ones will be impossible to obtain) calculatedwith bouncing models have to match with that experimental data.5. The constrain of the tensor/scalar ratio provided by WMAP and Planck pro-jects ( r ≤ . ) is obtained indirectly assuming the consistency slow rollrelation r = 16 (cid:15) (where (cid:15) = − ˙ HH ∼ = (cid:16) V ϕ V (cid:17) is the main slow roll para-meter) [25], because gravitational waves are not longer detected by thoseprojects. This means that, the slow roll inflationary models must satisfy thisconstrain, but not the bouncing ones, where there is not any consistency re-lation. This point is very important because some very complicated mechan-isms are sometimes implemented to the MBS in order to enhance the powerspectrum of scalar perturbation to achieve the observational bound providedby Planck [26]. In fact, in matter bounce scenario, to check if the mod-els provide a viable value of the tensor/scalar ratio, first of all gravitationalwaves must be clearly detected in order to determine the observed value ofthis ratio. The authors hope that more accurate unified Planck-BICEP2 data(the B2P collaboration), which is going to be issued soon, may adress thispoint. In contrast, as we have pointed out in (i), the spectral index of scalarperturbations and its running could be calculated independently of the theory,which means that in order to check bouncing models, while in the absenceof evidence of gravitational waves, one has to work in the space ( n s , α s ) . The Mukhanov-Sasaki equations (see [27] for a deduction of these equations inGR) for F ( T ) gravity and LQC are given by [28, 29] ζ (cid:48)(cid:48) S ( T ) − c s ∇ ζ S ( T ) + Z (cid:48) S ( T ) Z S ( T ) ζ (cid:48) S ( T ) = 0 , (9)where ζ S and ζ T denote the amplitude for scalar and tensor perturbations.In F ( T ) gravity one has Z S = a | Ω | ˙ ϕ c s H , Z T = a c s | Ω | , c s = | Ω | arcsin (cid:16) (cid:113) ρ c H (cid:17) (cid:113) ρ c H , with
Ω = 1 − ρρ c . (10)5n contrast, for LQC, Z S = a ˙ ϕ H , Z T = a Ω , c s = Ω . (11)The power spectrum for scalar perturbations is given by [30] P S ( k ) = 3 ρ c ρ pl (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∞−∞ Z − S ( η ) dη (cid:12)(cid:12)(cid:12)(cid:12) , (12)where, in order to obtain this formula, the scale factor a ( t ) ∼ = ( ρ c t ) / at earlytimes has been used. In the particular case of an exactly matter dominated uni-verse during all the background evolution, i.e., when a ( t ) = ( ρ c t + 1) / forteleparalell F ( T ) gravity one has P S ( k ) = ρ c ρ pl C , [31] where C = 1 − + − ... = 0 . ... is the Catalan’s constant, and for holonomy corrected LQC P S ( k ) = π ρ c ρ pl [32].The ratio of tensor to scalar perturbations in MBS is given by r = 83 (cid:32) (cid:82) ∞−∞ Z − T ( η ) dη (cid:82) ∞−∞ Z − S ( η ) dη (cid:33) , (13)where the factor 8 appears due to the two polarizations of the gravitational wavesand to the renormalization with respect to a canonical field [33].The spectral index for scalar perturbations and its running are calculated in [8]given n s − w, α s = − δ , (14)where the parameters w and δ , calculated in the quasi-matter domination, as afunctions of the potential are w ∼ = 13 (cid:18) V ϕ V (cid:19) − , δ ∼ = − (cid:18) V ϕ V (cid:19) ϕ . (15) ( n s , α s ) In slow-roll inflation, for the general models (monomial, natural, hilltop and plat-eau potentials), − n s is of the order N − , while the running parameter is of order N − and, consequently, one has α s ∼ (1 − n s ) , which in most cases is incom-patible with Planck and WMAP data, because the observed value of the running isnot small enough [34, 35].Thus, the observation of a large negative running implies that any inflationaryphase requires multiple fields or the breakdown of slow roll. Following this secondpath, in [35] the authors consider the break of the slow-roll approximation for ashort while, due to the inclusion of a quickly oscillating term in the potential. As a6onsequence, the theoretical value of the running parameter gets larger and couldmatch well with observational data.In contrast, in MBS the situation is completely different. For example, in [8]dealing with a perfect fluid whose Equation of State (EoS) is parametrized by thenumber of e-folds before the end of the quasi-matter domination period, namely N , the authors have shown that the theoretical values of the spectral index of scalarperturbations and its running fit well with their corresponding observational data.To be more precise, for the EoS P = β ( N +1) α ρ , ( α > , β < ) the followingrelation α s = 2 αN + 1 ( n s − (16)is obtained, which is perfectly compatible with the experimental data. In fact,for instance, if one takes α = 2 and N = 12 (note that in bouncing cosmologies alarge number of e-folds is not required, because the horizon problem does not exist,since at the bounce all parts of the Universe are already in causal contact, and alsothe flatness problem gets improved [4]), one obtains, for n s = 0 . ± . ,the following value for the running parameter: α s = − . ± . , which iscompatible with the Planck data. Effectively, for these values of α and N one gets n s − β ∼ = 0 . β , which is indeed compatible with its observed value, bychoosing β ∼ = − . Acknowledgements
The authors would like to thank Professor Sergei D. Odintsov for his valuableand useful comments. This investigation has been supported in part by MINECO(Spain), project MTM2011-27739-C04-01, MTM2012-38122-C03-01.
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