Fermi-Bounce Cosmology and Scale-Invariant Power Spectrum

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7-9-2018

Stephon Alexander Dartmouth College

Cosimo Bambi Fudan University

Antonino Marcianò Fudan University

Leonardo Modesto Fudan University

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Dartmouth Digital Commons Citation Alexander, Stephon; Bambi, Cosimo; Marcianò, Antonino; and Modesto, Leonardo, "Fermi-Bounce Cosmology and Scale-Invariant Power Spectrum" (2018). Open Dartmouth: Published works by Dartmouth faculty. 2698. https://digitalcommons.dartmouth.edu/facoa/2698

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For more information, please contact [email protected]. arXiv:1402.5880v2 [gr-qc] 13 Nov 2014 h ubesaeaesaeivrati h cl factor scale the if scale-invariant as evolves are cross Hubble-scale con- that the the perturbations During gauge-invariant phase, phase. and tracting dominated invariant epoch matter inflationary scale contracting the a in the generated between spectrum authors power “duality” a These in a universe. spectrum demonstrated power contracting pos- scale-invariant is dominated it a matter that generate shown was to it sible 5], [4, Wands by dependently phases expanding mode in [4]. and issues contracting to the due between mainly matching models, these of adia- phase number obtain contracting a the to in in difficult [4]. fluctuations proven invariant scenarios has scale it Pre-Big-Bang batic hand, and other [3] the Ekpy- Gas On as String such [2], phase contracting rotic of a context with the models in nearly bouncing attempted the observed was in invariance the Scale fluctuations CMBR. predict adiabatic the must of spectrum universe behind scale-invariant early assumptions theory the the singularity successful of of a Bang However, all Big theorem. or Hawking-Penrose the one avoid obviating to cosmologies by proposed bouncing non-singular been condi- years have energy the Gen- null Over the by obeys described tion. matter is if and space-time Relativity eral if singu- unavoidable initial is the that larity states curvature theorem all Their in invariants[1]. Big singularities from Standard suffers Friedmann–Lemaˆıtre– cosmology the Bang the demon- of times, metric Penrose (FLRW) initial and Robertson–Walker at Hawking that of strated work pioneering The ∗ § ‡ † [email protected] [email protected] [email protected] [email protected] npoern ok yBadnegr iel n in- and Finelli Brandenberger, by works pioneering In 2 etrfrFedTer n atcePyis&Dprmn of Department & Physics Particle and Theory Field for Center a tpo Alexander, Stephon ( emoi ed.TeuulBgBn iglrt saoddt avoided si is ghost-free is singularity theory Bang Our Big fermions. the usual non-tr from The a contribution using cosmology fields. bouncing fermionic non-singular a develop We ASnmes 88.q 88.s 46.p 04.62.+v 04.60.Pp, 98.80.Es, 98.80.Cq, inflation numbers: the PACS choice to suitable alternative by possible recovered a be providing scale thus can a that mass, phase show contracting We ter g the coupling. kinetic for in non-minimal no sector a topological has with fields a which Dirac plus torsion, relativity to general equivalent standard is bounce the t ) ∼ .INTRODUCTION I. em-onecsooyadsaeivratpower-spectru invariant scale and cosmology Fermi-bounce ( − t ) 2 / 3 1 utemr,i h oneis bounce the if Furthermore, . etrfrCsi rgn n eateto hsc n Astr and Physics of Department and Origins Cosmic for Center atot olg,Hnvr e aphr 35,USA 03755, Hampshire New Hanover, College, Dartmouth 1, ∗ oioBambi, Cosimo Dtd uy9 2018) 9, July (Dated: 2, † noioMarcianò,Antonino tagtowrl ne rmtersl fBrandenberger of the may result con- we the that as the from Indeed, time infer in scale-invariant. straightforwardly first be fermions can the phase of for tracting fluctuation show quantum we adiabatic Moreover, ature. utn onei o-iglrpoie htanisotropic sub-dominant that is provided stress non-singular is bounce sulting e h remneutost aeangtv energy negative a have like redshifts to that equations density Friedman the fies aisyedasaeivratpwrspectrum power these dy- scale-invariant that whose a show interactions yield We four-fermion namics [17]). generate with Ref. terms endowed two instance [14– term for Refs gravitational (see non- in topological a torsion analyzed a both (as and propose fermions we 16]) work, of this coupling In minimal vi- theory studied. parity torsion-free a been a of has in role self-interaction the 4-fermion [13], olating Ref. a In yields interaction bounce. four-fermion non-singular induced torsion a where de assumed. a was was and expansion interaction perturbation background scalar four-fermion Sitter differs the the work computing in where Our neglected [10], field. Dirac Ref. previously class the wide from of been a of potentials has perspective generic the research of from [10] of Ref. in line addressed scenario This four-fermion bounce the on matter interaction. specifically, a fermions, present Dirac [2, we on fields paper based scalar this In fundamental on 6–9]. based mostly proposed, phase. expanding the matched in be modes scale-invariant can to modes scale-invariant the non-singular, 1 2 ouin 1] hc o utbecoc fsm paramete some form. a of black never choice as may of suitable findings theory fate a these the the for used for which us [19], consequences of solutions discuss some to [18], point contributi paper starting a companion into topologica a term term. In the surface interaction a four-fermion turns from the background action to Holst torsion the the in term of presence The utemr,tefu-emo urn est modi- density current four-fermion the Furthermore, nti okw olwteprpcietknb 1,12] [11, by taken perspective the follow we work this In been since has scenarios bounce matter of handful A hsc,FdnUiest,203 hnhi China Shanghai, 200433 University, Fudan Physics, aiy n emoi atrdsrbdby described matter fermionic and ravity, c h emoi prtrta generates that operator fermionic the nce r scenario. ary ffrinnme est n bare and density number fermion of s s h hsclsse osssof consists system physical The ms. va opigo eea eaiiyto relativity general of coupling ivial ak oangtv nrydensity energy negative a to hanks nain oe-pcrmgenerated power-spectrum invariant 2, ‡ n enroModesto Leonardo and 2 ossetwt rvosliter- previous with consistent , onomy, ∼ a ( t ) 6 eso httere- the that show We . m 2, § 1 . sof rs hole on l 2 and Finelli, since the bounce is non-singular our scale- in which α R is the so called non-minimal coupling pa- invariant curvature perturbation induced by the fermion rameter. The∈ Einstein-Cartan action can be found if we quantum fluctuations will enter the expanding phase as consider SECH=SGR +SDirac and α=γ, with a term that a scale invariant fluctuation. An advantage of the mech- reduces to the Nieh-Yan invariant [16] when the second anism shown in this paper is that it does not require any Cartan structure equation holds. From the point of view fundamental scalar field. The fermionic field is sufficient of the Holst action (1), minimal coupling is recovered in to account for both the bounce and the generation of the limit α . Constraints on α and γ can be de- nearly scale-invariant scalar perturbations. rived from the→ four ±∞ fermion axial-current Lagrangian (7), The following is an outline of the paper. In Section II based on measurements of lepton-quark contact interac- we introduce the theoretical framework, and cite the rel- tions [22, 23], but these are not at all stringent. evant works in the literature. In Section III we address The covariant derivative for Dirac spinors is defined the consequences of the model for the Matter-Bounce sce- to be ∂ + 1 ωIJ γ γ , while the field-strength ∇µ ≡ µ 4 µ [I J] nario. In Section IV we address the cosmological per- of the Lorentz connection is obtained from [ µ, ν ] = 1 IJ ∇ ∇ turbations induced by the fermionic field. In Section V 4 Fµν γ[I γJ]. Because of the presence of fermions, a tor- we discuss consistency with experimental data. In Sec- sional part of the connection enters the non-minimal tion VI we provide some concluding remarks and mention ECH action. Nevertheless, the latter can be integrated works in progress. out of the theory through the Cartan equation, which is found by varying the total action with respect to IJ the connection ωµ . We provide the usual definition of II. THE THEORY IJ the contortion tensor, denoted as Cµ and defined by J ( µ µ)VI = C VJ , where µ is the covariant deriva- In what follows we provide our theoretical framework and ∇ −∇ µ I ∇ tive compatible with the tetrad eI and V a vector in the conventions following the formalism of Refs. [11, 14, 15, µ J internale space. The Cartan equatione then relates the con- 17, 20, 21]. We start by considering a generalization of tortion tensor CIJ to the fermionic currents and tetrad: the Einstein-Hilbert action with a topological term: this µ is the Holst action for gravity in the Palatini formalism µ κ γ e C = βǫ J L 2θ η J , which allows us to couple gravity to chiral fermions. We I µJK 4 γ2 +1 IJKL − I[J K] then couple this theory to a Dirac field ψ, whose com- L L J = ψγ γ5ψ , (3) T 0 plex conjugate reads ψ = (ψ∗) γ . The action for the fermionic field is cast in terms of the Dirac matrices, γI where the coefficients are functions of the free parameters with I = 0,..., 3 and γ5, expressed in the Dirac-Pauli within the non-minimal ECH theory, β = γ+1/α and θ = 1 γ/α. Thanks to (3) the non-minimal ECH action can basis. The action for pure gravity can be cast in terms − I J I be completely recast in terms of the metric compatible of the gravitational field gµν = eµeν ηIJ , where eµ is µ connection, as a sum of the Einstein-Hilbert action and the tetrad/frame field (with inverse eI and determinant IJ the Dirac action. The latter is now written in terms of e), and the Lorentz connection ωµ (whose curvature is IJ IJ IJ metric compatible variables, and now includes a novel Fµν =2∂[µω ν] + [ωµ,ων ] ). The action for the fermion interaction term that captures the new physics within 0 fields involves the spinors ψ and ψ = ψ† γ . the non-minimal ECH theory SECH. The theory then The total action is the sum of the Einstein-Cartan- becomes: Holst (ECH) action plus the non-minimal covariant Dirac action 3. The ECH action is (see [18]), SECH = SGR + SDirac + SInt , (4) where the Einstein-Hilbert action is expressed in terms IJ IJ 1 4 µ ν IJ KL of the mixed-indices Riemann tensor Rµν = Fµν [ω(e)] SHolst = d x e eI eJ P KLFµν (ω) , (1) 2κ M | | 1 Z S = d4x e eµeν RIJ , (5) where κ = 8πG is the reduced Planck length square GR 2κ | | I J µν e N ZM and the operator P IJ = δ[I δJ] ǫIJ /(2γ), ǫ KL K L KL IJKL the Dirac action SDirac on curved space-time reads being the Levi-Civita symbol, is defined− in terms of the 1 4 I µ Barbero–Immirzi parameter γ, and can be inverted for SDirac = d x e ψγ e ı µψ mψψ + h.c.,(6) 2 1 4 I γ = 1. The Dirac action is S = d x e , 2 M | | ∇ − Dirac 2 Dirac Z where6 − | |L R and the interacting term is: e 1 I µ ı Dirac = ψγ eI 1 γ5 ı µψ mψψ + h.c.,(2) 4 L M L 2 − α ∇ − SInt = ξκ d x e J J ηLM , (7) − | | h i ZM where we define the coefficient ξ as a function of the 3 Notice that, in absence of the gravitational Holst topological fundamental parameters of the theory, term, the whole theory provided with torsion and minimally- coupled fermions is referred to in the literature as the Einstein- 3 γ2 2 1 ξ := 1+ . (8) Cartan-Sciama-Kibble theory. See e.g. Refs. [20]. 16γ2 +1 αγ − α2 3

In what will follow it is useful to compute the energy- in which ⋆ denotes complex conjugation. It is easy to ⋆ momentum tensor, derive the equation of motion for the bilinear ψ0 ψ0,

d ⋆ ⋆ ψ0 ψ0 +3 H ψ0 ψ0 =0 , (12) fer 1 I dt Tµν = ψγI e(µı ν)ψ + h.c. gµν fer. (9) 4 ∇ − L which yields the familiar expression in terms of a constant In canonical quantume field theory, spinors are initial density n0 operator-valued fields ψˆ that act on a definite Hilbert n ψ⋆ψ 0 . (13) space. We can express a classical spinor as the expec- 0 0 ∼ a3 tation value of the spinor operator on an appropriate Using the solutions of (12) the Friedmann equation quantum state s , such that ψ = s ψˆ s , which is a becomes, complex number.| i The observable bilinearh | | thati will enter the classical equation will be evaluated on such a quan- κ2 n2 mκ n H2 = ξ 0 + 0 . (14) tum state, and their renormalized value will be obtained 3 a6 3 a3 by subtracting the vacuum expectation value, namely ... s ... s 0 ... 0 . We see from (14) that the four-fermion term has the h iren ≡ h | | i − h | | i 1 The Dirac equation on a curved background for the crucial a6 redshifting which will control the bounce. We interacting system is found to be4 now consider a contracting scale factor and immediately recognize that the bounce is due to the vanishing of the total energy density. As we approach the would be sin- I µ I gularity (the scale-factor approaching zero), the nega- γ eI ı µψ mψ =2ξκ(ψψ + ψγ5ψγ5 + ψγI ψγ )ψ. (10) ∇ − tive energy (for ξ < 0) four-fermion term dominates and e drives the Hubble parameter to zero, resulting in a non- III. NON-SINGULAR BOUNCE singular bounce. At the bounce we will have to obtain the initial value of ˙ In previous bouncing models, the issue of the robustness H which we can get from the second Friedmann equation, of the singularity avoidance depends on whether quantum corrections (i.e. curvature or matter) were under- 2 a¨ 1 n0 n control at the bounce [4]. The advantage of our model is H˙ H2 = = mκ +4 ξκ2 0 . (15) − a −6 a3 a6 that torsion in this scheme, which is responsible for the bounce, has no kinetic term (i.e. it is an auxiliary field) At the bounce, t = t0, the vanishing of H = H0 and will not experience any quantum corrections as we in (14), the scale factor approaches a constant, a0 = approach the bounce. ( ξκn /m)1/3. For negative values of the ξ parameter, − 0 We would like to find self-consistent initial values of the scale factor a0 reaches its minimum, as from (15) one the fermionic densities so as to not spoil isotropy of our 2 finds that H˙ 0 = m /(3ξ). Notice that both the bilinear FLRW space-time. First we cast our metric eI in FLRW − µ ψψ and the field ψ reach their maxima at t0: although I I form, which in the comoving gauge reads e0 = δ0 and the effective potential in Sfer is unbounded in ψ when ξ I I ej = δj a(t). Homogeneity and isotropy on spatial hyper- is negative, the gravitational bounce prevents the clas- surfaces demand a vanishing fermionic current. sically unbounded energy spectrum from taking infinite As for the Dirac field components, the vanishing of the negative values. It is then straightforward to find the de- spatial fermionic current yields [10] ψ = (ψ0, 0 , 0 , 0). In terministic evolution of the scale factor that leads to the the comoving gauge, the only non-vanishing spin connec- bounce: IJK IJ µ tion components for ω = ωµ eK are ω0ij = ωi0j = 1 − 3 Hδij, where the Hubble parameter is defined by H = 3mκn0 2 κn0 − a = (t t0) ξ . (16) a/a˙ and “˙” represents the time-derivative. This implies 4 − − m = ∂ and = ∂ + aH/2δ diag(σj , σj ), where σj 0 0 i i ij This solution can be shown to be stable under perturba- denotes∇ Pauli∇ matrices. The Dirac equation− then follows tions to the fermionic matter field if the anisotropic and e e3 inhomogeneous contribution to the energy density, which ψ˙ + H ψ +ı (m+4κ ξ ψ⋆ψ )ψ =0, (11) 0 2 0 0 0 0 reads

Tr[γiγj] ρ˜ 2 ψψ δψδψ , (17) ∼ Mp h i 4 To further simplify the four-fermion term, we have used the Pauli-Fierz identity is subdominant with respect to the isotropic contribution in the right hand side of (14). The criterion to have a I 2 2 I (ψγ5γ ψ)(ψγ5γI ψ) = (ψψ) + (ψγ5ψ) + (ψγ ψ)(ψγI ψ) . subdominant contribution is δψδψ /M 2 <

We show in Fig. 1 the range of values in the (γ, α) This finally provides the expression for the power- parameter-space for which ξ is negative, and the matter- spectrum bounce happens. The bounce takes place when the in- 3 3 teraction energy of the fermion fields provide a nega- ma (t) 2ξκn0 k ~ ~ (k) − 2 vh(t, k) vh(t, k) , (23) P ∼ 4mn0 4π tive contribution that violates the null energy condi- h tion, as shown for non-conventional fermion fields in X Refs. [10, 24, 25]. where the mode function vh is obtained by expanding the quantum fluctuations of the spinor: d3k IV. COSMOLOGICAL CURVATURE δψ = (24) PERTURBATIONS (2π)3/2 × Xh Z ı~k ~x ı~k ~x uh(t,~k)a(~k,h)e · + vh(t,~k)b†(~k,h)e− · . In this section, we demonstrate that fermions can induce scale-invariant adiabatic perturbations. We show this in We now proceed to evaluate the solution of the mode a way similar to the treatment of scalar fields, in that function vh of the spinor perturbation. Using the back- we solve the mode functions for the fermionic perturba- ground solution of ψ, and a conformal rescaling, ψ = tions and evaluate its contribution to the gauge invariant a3/2ψ, we obtain the equation of motion for the spinor curvature perturbation. Following [10], we introduce a perturbation e quantity that is conserved on large scales and can be related to CMBR temperature fluctuations, analogous to the Bardeen variable5, i.e. 2ξκn ıγµ∂ ma(η) 0 δψ =0 . (25) µ − − a2(η) δρ ζ = . (19) ρ + p We can now solve the Dirac equationf (25) in terms of the following mode6 functions: After some algebra and the use of the Pauli-Fierz identity, we obtain 1 ˜ 1 ~ ~ ζ = m + ξκψψ δψ ψ + ψ δψ + f h = [˜uL,h(k, η)+˜uR,h(k, η)] , L ± √2 mψψ +2ξκJLJ n 1 ~ ~ ξκ ψγ ψ(δψγ ψ + ψγ δψ)+ψγLψ(δψγ ψ + ψγ δψ) . g˜ h = [˜vL,h(k, η)+˜vR,h(k, η)] . (26) 5 5 5 L L ± √2 h io We can further simplify ζ by using the background solu- Equation(25) can be expressed in terms of f h: tions for the spinors: ± ˜ 2 2 2 m a′ ˜ f ′′h + k +m a + ıma′ +2ξκn0 ı f h =0, ζ = f(t)(δψψ + ψδψ) , ± a − a3 ± (1 ξκψψ/m) (27) with f(t) − . (20) ≃ ψψ where h denotes helicity and the and ′ denotes deriva- ± The power spectrum (k) is implicitly defined in terms tive with respect to conformal time. An identical system P of the equal-time two-point function of ζ: of coupled equations is recovered forg ˜h. Rescaling (27) 2 2 by κ and then taking the limits η0 <<κ and κm << 1, dk sin kr 2 2 2 ζ(t, ~x)ζ(t, ~x + ~r) = (k) , (21) provided that also κ m <<η0 is fulfilled, equation (27) h i k kr P reduces to Z 2 where the expectation value is taken in the vacuum state ˜ 2 ν 1 ˜ f ′′h+ k − f h =0 , (28) and defined by a 0 = b 0 = 0. Using the simplified ± − 4 η2 ± expression for ζ in| termsi of| iδψ and the background Dirac field, the two point function becomes

6 Given a unit eigenspinor ξh, the helical components of the mode ψψ functions are ζ(t, ~x)ζ(t, ~x) = f 2(t) δψ δψ . (22) h i 4 h i ~ ~ ~ u˜L,h(k, η) u˜(t, k) = u˜h(t, k)= ~ ξh , u˜R,h(k, η) ! Xh Xh 5 The cosmological models whose parameters α and γ encode a ~ ~ ~ v˜R,h(k, η) negative ξ may have application as alternative model of Inﬂation. v˜(t, k)= v˜h(t, k)= ~ ξh . v˜L,h(k, η) ! Below we study the perturbation variable sourced by fermionic Xh Xh matter. 5

5 ξ > 0 ξ > 0

ξ < 0 α 0 ξ < 0

-5 ξ > 0 ξ > 0

-5 0 5 γ

FIG. 1. Sign of ξ on the plane (γ, α). Natural values of the fundamental parameters are allowed in order to obtain ξ < 0. where the parameter ν is related to the fermion coupling which is evaluated at the end of the matter contracting parameter ξ by phase tE, at which the scale factor takes the value aE. The resulting expression would then be ν2 =1 8ξ . (29) − mH2 E , (32) An equation identical to (28) is then found forg ˜ h; their PS ≃ 32 n solutions have been extensively studied in the literature,± 0 and for non densitized components read where η = 2/(a )=2/H has been applied. No- E EHE E tice that the time tE marks the moment at which per- m πkη turbations become constant, throughout the rest of the f h(k, η)= − Z ν ( kη) , (30) ± k 8a3(η) | | − primordial epoch, until they reenter the Hubble radius. r s

Z ν denoting the Bessel functions labeled by the parameter| | ν . In the contracting epoch and on sub-horizon V. CONSISTENCY WITH OBSERVATIONS | | scales, when kη >> 1, both f h(k, η) and g h(k, η) the − ± ± fermionic perturbations are oscillatory and suppressed by An exact scale-invariance of the power spectrum would a factor a3/2(η). In this limit, the factor m/k in (30) immediately constrain the parameter ξ to take the value determines the quantum vacuum initial conditions [32]. For super-Hubble perturbations, i.e. p kη << 1, 3 − ξ = . (33) the solutions of (28) are Bessel functions, Z ν | | 8 ν /2 1/2 | | ≃ Γ( ν )( kη)−| | − , in which Γ( ν ) denotes the Eu- But observed deviations from scale invariance, namely | | − 7 | | ler function . We now see that ν, which is related to ns = 0.960 0.007 [33], as parametrized from (31) four-fermion coupling parameter, determines a scale in- through the relation± variant power-spectrum. Far away from the bounce, the power-spectrum (23) can be evaluated to be d ln (k) n 1 P =2 ν , (34) s − ≡ d ln k − | | 2 2 m k Γ( ν ) ν (k) | | | | kη −| | . requires a slightly diﬀerent value for ξ, i.e. P ≃ 16 n0 | | 1 (3 n )2 A value of ν = 2 then ensures scale-invariance, provid- ξ = − − s 0.395 0.004 , (35) ing the expression| | for the power-spectrum 8 ≃− ± m once we have taken into account (29). (k) 2 , (31) P ≃ 16 n0η Notice that the value of ξ consistent with the CMB (35) will restrict the bare parameters in our theory (γ and α) to one-parameter family of theories. Finally, the 1 choice ξ 4 10− is also clearly consistent with parti- | |≃ · 7 For any value of ν, this provides perturbations δψ(η, ~x) and cle physics data, given the lack of a stringent constraint δψ′(η, ~x) which decrease during the expanding phase of the uni- coming from the lepton-quark contact interactions. Mea- verse. surements constrain ξ <1032 [22, 23], which in turn may | | 6

2 2 allow a region of natural values for the parameters en- tently with the conditions η0<<κ and κm <<1 previously tering the non-minimal Einstein-Cartan-Hilbert theory required for scale-invariance. Therefore we conclude that resulting from (1) and (2). if the BICEP2 detection is confirmed in the future, then Recently, there has been much discussion about the our specific model could be ruled out8. possible detection of primordial gravitational-waves by the BICEP2 collaboration [34]. The result has since been questioned in the literature by a few studies (see e.g. VI. SUMMARY AND CONCLUSION [35] and [36]), which point out possible flaws in the data analysis. It has been shown that a proper dust profile might still account for all or most of the signal of the When general covariance accommodates non-minimal primordial gravitational waves [33]. coupling in the fermionic sector, a four-fermion inter- With a view towards more detailed data analyses to action modifies the cosmological evolution to yield a be delivered by BICEP2 and other collaborations, it is bounce. In this work we have demonstrated that the sensible in this work to show the derivation of the phe- same fermions that regulate the singularity also gener- nomenological parameter r, which accounts for the ra- ate scale-invariant quantum fluctuations in the contract- tio between the primordial gravitational waves’ power- ing phase. Using the arguments of Brandenberger and spectrum and the scalar perturbations power-spectrum. Finelli, we can easily match these fermionic perturba- This can be achieved, recalling that at the perturbative tions to the scale invariant modes in the expanding phase. level both the scalar and the tensor metric fluctuations The bounce is non-singular because the torsion, which is can be treated linearly and as uncoupled degrees of free- responsible for the bounce, does not receive quantum cor- dom. Thus the derivation of the primordial gravitational rections. The gravitational wave power spectrum and the waves’ power-spectrum will be immediately achieved fol- resulting tensor to scalar ratio have been derived. In a lowing the standard procedure outlined in [37], which is future paper, it will be interesting to compute corrections specialized to general matter-bounce scenarios (see e.g. to the tensor to scalar ratio due to the coupling of the [38]). The primordial gravitational power-spectrum is: gravitons to the fermions. Furthermore, in order to fully understand the gener- 1 H2 ation of scale-invariant scalar perturbations, it is essen- = E , (36) T 2 2 tial to address the mechanism here discussed in terms of P ϑ Mp the canonical Mukhanov-Sasaki variables, to which both where ϑ =8π(2q 3)(1 3q) (the coefficient q is a back- matter and metric perturbations contribute. In a forth- − − ground parameter associated with the contracting phase coming paper [41] some of us are considering how to re- and typically required to be less than unity), and the co- cover those variables by looking at the second order ac- moving Hubble parameter HE is evaluated at the end of tion of the theory. In this context, it is interesting to matter contracting phase, just before the phase transi- notice that while matter perturbations are derived from tion to the bounce. The maximal amplitude of the Hub- the relevant fermionic bilinears, which behave as scalar ble rate can then be evaluated from requiring the scale and vector fields, perturbations of the fermionic bilinears m factor to be of order √ξ , once the Universe has been as- must be expressed in terms of the fundamental fermionic sumed to evolve through the bounce. fields, the dynamics of which is dictated by first order Most of the bouncing models that generate adiabatic differential equations. This feature is at the origin of the fluctuations in the contraction phase (before the bounce), very different behavior of fermionic matter perturbations including the Ekpyrotic scenario, would be disfavored or relative to scalar field perturbations. eventually ruled out if the claim by the BICEP2 collaboration [34] on the detection of B-modes coming from primordial gravitational waves, and the related value of the ACKNOWLEDGMENTS tensor to scalar ratio r 0.2, is confirmed. The model in this paper, consisting≃ of only one fermionic species, We thank the Referees for their valuable remarks and would then suffer a similar fate, as the theoretical value suggestions. We dedicate this paper to Leon Cooper, for r consistent with the mass parameter of the model is whose work continues to inspire and challenge us. SA found to be too large. Indeed, it follows from (32) and was supported by the Department of Energy Grant (36) that de-sc0010386. This work was supported by the NSFC 32 n grant No. 11305038, the Shanghai Municipal Education r 0 , (37) 2 2 Commission grant for Innovative Programs No. 14ZZ001, ≃ ϑ mMp the Thousand Young Talents Program, and Fudan Uni- which can not match experimental constraints consis- versity.

8 Nevertheless, the introduction of a second fermionic species in the a smaller value of r. This latter argument has been explored in analysis can account for a new degree of freedom able to match [39]. 7

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