Nonsingular bouncing cosmologies in light of BICEP2

Yi-Fu Cai,1, ∗ Jerome Quintin,1, † Emmanuel N. Saridakis,2, 3, ‡ and Edward Wilson-Ewing4, §

1Department of Physics, McGill University, Montréal, QC, H3A 2T8, Canada2Physics Division, National Technical University of Athens, 15780 Zografou Campus, Athens, Greece3Instituto de F́ısica, Pontificia Universidad de Católica de Valparáıso, Casilla 4950, Valparáıso, Chile

4Department of Physics and Astronomy, Louisiana State University, Baton Rouge, 70803 USA

We confront various nonsingular bouncing cosmologies with the recently released BICEP2 dataand investigate the observational constraints on their parameter space. In particular, within thecontext of the effective field approach, we analyze the constraints on the matter bounce curvatonscenario with a light scalar field, and the new matter bounce cosmology model in which the universesuccessively experiences a period of matter contraction and an ekpyrotic phase. Additionally, weconsider three nonsingular bouncing cosmologies obtained in the framework of modified gravitytheories, namely the Hořava-Lifshitz bounce model, the f(T ) bounce model, and loop quantumcosmology.

PACS numbers: 98.80.Cq

I. INTRODUCTION

Very recently, the BICEP2 collaboration announcedthe detection of primordial B-mode polarization in thecosmic microwave background (CMB), claiming an in-direct observation of gravitational waves. This result,if confirmed by other collaborations and future obser-vations, will be of major significance for cosmology andtheoretical physics in general. In particular, the BICEP2team found a tensor-to-scalar ratio [1]

r = 0.20+0.07−0.05, (1)

at the 1σ confidence level for the ΛCDM scenario. Al-though there remains the possibility that the observedB-mode polarization could be partially caused by othersources [2–4], it is indeed highly probable that the ob-served B-mode polarization in the CMB is due at leastin part to gravitational waves, remnants of the primordialuniverse.

The relic gravitational waves generated in the veryearly universe is a generic prediction in modern cos-mology [5, 6]. Inflation is one of several cosmologicalparadigms that predicts a roughly scale-invariant spec-trum of primordial gravitational waves [6–8]. The sameprediction was also made by string gas cosmology [9–12]and the matter bounce scenario [13–15]. (Note that thespecific predictions of r and the tilt of the tensor powerspectrum can be used in order to differentiate betweenthese cosmologies.) So far, a lot of the theoretical analy-ses of the observational data have been in the context ofinflation (see, for instance, [16–34]).

In this present work, we are interested in exploringthe consequences of the BICEP2 results in the frame-

∗Electronic address: yifucai@physics.mcgill.ca†Electronic address: jquintin@physics.mcgill.ca‡Electronic address: Emmanuel Saridakis@baylor.edu§Electronic address: wilson-ewing@phys.lsu.edu

work of bouncing cosmological models. In particular,we desire to study the production of primordial grav-itational waves in various bouncing scenarios, in boththe settings of effective field theory and modified gravity.First, we show that the tensor-to-scalar ratio parameterobtained in a large class of nonsingular bouncing modelsis predicted to be quite large compared with the obser-vation. Second, in some explicit models this value canbe suppressed due to the nontrivial physics of the bounc-ing phase, namely, the matter bounce curvaton [35] andthe new matter bounce cosmology [36–38]. Additionally,for bounce models where the fluid that dominates thecontracting phase has a small sound velocity, primor-dial gravitational waves can be generated with very lowamplitudes [39]. We show that the current Planck andBICEP2 data constrain the energy scale at which thebounce occurs as well as the slope of the Hubble rateduring the bouncing phase in these specific models.

The paper is organized as follows. In Sec. II, we fo-cus on matter bounce cosmologies from the effective fieldtheory perspective. In particular, we explore the mat-ter bounce curvaton scenario [35] and the new matterbounce cosmology [37]. In Sec. III, we explore anotheravenue for obtaining nonsingular bouncing cosmologies,that is modifying gravity. We comment on the statusof the matter bounce scenario in Hořava-Lifshitz gravity[40], in f(T ) gravity [41], and in loop quantum cosmology[42]. We conclude with a discussion in Sec. IV.

II. MATTER BOUNCE COSMOLOGY

As an alternative to inflation, the matter bounce cos-mology can also give rise to scale-invariant power spec-tra for primordial density fluctuations and tensor pertur-bations [13, 14]. In the context of the original matterbounce cosmology, both the scalar and tensor modes ofprimordial perturbations grow at the same rate in thecontracting phase before the bounce. As a result, thisnaturally leads to a large amplitude of primordial tensor

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fluctuations [15], greater than the observational upperbound. However, an important issue that has to be addi-tionally incorporated in these calculations is how the per-turbations pass through the bouncing phase, which candrastically decrease the tensor-to-scalar ratio. One ex-ample is the matter bounce in loop quantum cosmology,where the tensor-to-scalar ratio is suppressed by quan-tum gravity effects during the bounce [43]. Also, for someparameter choices in the new matter bounce model, r canbe suppressed by a small sound speed of the matter fluid[37].

In this section, we focus on two particularly interestingmatter bounce cosmological models in the effective fieldtheory setting. First, we consider the matter bounce cur-vaton model, in which the primordial curvature pertur-bation can be generated from the conversion of entropyfluctuations seeded by a second scalar field [35]. Sec-ondly, we investigate the new matter bounce cosmology,where the primordial curvature perturbations can achievea gravitational amplification during the bounce [36].

A. The matter bounce curvaton model

The matter bounce curvaton model was originallystudied to examine whether the bouncing solution of theuniverse is stable against possible entropy fluctuations[35] and particle creation [44]. In a toy model studied in[35], a massless entropy field χ is introduced such thatit couples to the bounce field φ via the interaction termg2φ2χ2. The entropy field evolves as a tracking solutionin the matter contracting phase and its field fluctuationsare nearly scale-invariant provided the coupling parame-ter g2 is sufficiently small. The amplitude of this mode iscomparable to the tensor modes and scales as the abso-lute value of the Hubble parameter H before the bounce.Afterwards, the universe enters the bouncing phase andthe kinetic term of the entropy field varies rapidly in thevicinity of the bounce. As in the perturbation equation ofmotion this term effectively contributes a tachyonic-likemass, a controlled amplification of the entropy modes canbe achieved in the bouncing phase. Since this term doesnot appear in the equation of motion for tensor pertur-bations, the amplitude of primordial gravitational waveis conserved through this phase. After the bounce, theentropy modes will be transferred into curvature pertur-bations, and this increases the amplitude of the power-spectrum of the primordial density fluctuations. An im-portant consequence of this mechanism is its suppressionof the tensor-to-scalar ratio.

In this subsection, we briefly review the analysis of [35],in light of the BICEP2 results. In the simplest version ofthe matter bounce curvaton mechanism, there are onlythree significant model parameters, namely the couplingparameter g2, the slope parameter of the bouncing phaseΥ (which is defined by H ≡ Υt around the bounce), andthe maximal value of the Hubble parameter HB . Thevalue of HB is associated with the mass of the bounce

field m through the following relation

HB '4m

3π. (2)

The propagation of primordial gravitational waves de-pends only on the evolution of the scale factor, and it ispossible to calculate the power spectrum for primordialgravitational waves in this scenario1

PT =2H2m

9π2M2p, (3)

from which we see that the amplitude is determined solelyby the maximal Hubble scale Hm. However, the am-plitude of the entropy fluctuations is increased duringthe bounce. Since tensor perturbations do not couple toscalar perturbations, the entropy perturbations do not af-fect the power spectrum of gravitational waves, whereasthe entropy modes are amplified and act as a sourcefor curvature perturbations. This asymmetry leads toa smaller tensor-to-scalar ratio of

r ' 35F2

, (4)

where the amplification factor is given by

F ' e√y(2+y)+ 3√

2sinh−1(

2√

y

3 ) , with y ≡ m2

Υ, (5)

and Υ is the slope parameter of the bouncing phase asdefined before Eq. (2). Since the exponent in the aboveequation is approximately linear in y in the regime ofinterest, we see that the tensor-to-scalar ratio can begreatly suppressed for large values of y, that is for largem or small Υ. Also, we see that r will reach a maxi-mal value in the massless limit or in the limit where thebounce is instantaneous (i.e., Υ → ∞), in which caseentropy perturbations are not enhanced.

We recall that, according to the latest observation ofthe CMB (Planck+WP), the amplitude of the powerspectrum of primordial curvature perturbations is con-strained to be [45]

ln(1010As) = 3.089+0.024−0.027 (1σ CL) , (6)

at the pivot scale k = 0.002 Mpc−1. Moreover, the re-cently released BICEP2 data indicate that [1]

r = 0.20+0.07−0.05 (1σ CL) . (7)

By making use of the above data, we performed a nu-merical estimate and derived the constraint on the modelparameters Υ and m shown in Fig. 1. From the result,we find that the mass scale m and the slope parameter

1 Mp ≡ 1/√8πG is the reduced Planck mass.

3

3× 10−4 4× 10−4m/Mp

10−7

6× 10−8

2× 10−7

3× 10−74× 10−7

Υ/M

2 p

FIG. 1: Constraints on the mass parameter m and the slopeparameter Υ of the bounce phase from the measurements ofPlanck and BICEP2 in the matter bounce curvaton scenario.The blue bands show the 1σ and 2σ confidence intervals of thetensor-to-scalar ratio and the red bands show the confidenceintervals of the amplitude of the power spectrum of curvatureperturbations.

Υ appearing in the matter bounce curvaton model haveto be in the following ranges

2.5× 10−4 . m/Mp . 4.5× 10−4 , (8)7.0× 10−8 . Υ/M2p . 3.5× 10−7 , (9)

respectively. The resulting constraints suggest that if theuniverse has experienced a nonsingular matter bouncecurvaton, then the energy scale of the bounce should beof the order of the GUT scale with a smooth and slowbouncing process.

B. New matter bounce cosmology

In the new matter bounce scenario, as first developedin [36], the universe starts with a matter-dominated pe-riod of contraction and evolves into an ekpyrotic phasebefore the bounce. This scenario combines the ad-vantages of the matter bounce cosmology, which givesrise to scale-invariant primordial power spectra, andthe ekpyrotic universe, which strongly dilutes primordialanisotropies [46]. The model can be implemented by in-troducing two scalar fields, as analyzed in the context ofeffective field theory [37].

In the effective model of the two field matter bounce[37], one scalar field is introduced to drive the matter-dominated contracting phase and the other is responsiblefor the ekpyrotic phase of contraction and the nonsingu-lar bounce. Therefore, similar to the matter bounce cur-vaton scenario, there also exist curvature perturbationsand entropy fluctuations during the matter-dominatedcontracting phase. However, the main difference betweenthese two models is that, in the present case, the entropy

modes have already been converted into curvature per-turbation when the universe enters the ekpyrotic phasebefore the bounce, while in the matter bounce curvatonmechanism, this process occurs after the bounce.

In this model, when the universe evolves into thebouncing phase, the kinetic term of the scalar field thattriggers the bounce could vary rapidly which is similar tothe analysis of the matter bounce curvaton mechanism.This process can also effectively lead to a tachyonic-likemass for curvature perturbations, and therefore, the cor-responding amplitude can be amplified. For the samereason as the matter bounce curvaton mechanism, thiseffect only works on the scalar sector. Correspondingly,the tensor-to-scalar ratio is suppressed when primordialperturbations pass through the bouncing phase in thenew matter bounce cosmology. We would like to pointout that this effect is model dependent, namely, it couldbe secondary if the kinetic term of the background scalarevolves very smoothly compared to the bounce phase[47, 48].

Following [37], one can write the expression of thepower spectrum for primordial tensor fluctuations as

PT 'F2ψγ2ψH2E

16π2(2q − 3)2M2p, (10)

with

γψ '1

2(1− 3q),

Fψ ' exp[2√

ΥtB+ +2

3Υ3/2t3B+

], (11)

up to leading order. In the above expression, HE is thevalue of the Hubble rate at the beginning of the ekpyroticphase and q is an ekpyrotic parameter which is much lessthan unity. Note that we have assumed that the bouncingphase is nearly symmetric around the bounce point withthe values of the scale factor before and after the bouncebeing comparable. We denote the time at the end of thebounce phase by tB+.

At leading order, the power spectrum for curvaturefluctuations is given by

Pζ 'F2ζH2Ea2E8π2M4p

γ2ζm2 |Uζ |2 , (12)

with γζ ' γψ and

Uζ = −(25 + 49q)iHE24m

− 27q24

,

Fζ ' e2√

2+ΥT 2(

tB+T

)+

2(2+3ΥT2+Υ2T4)

3√

2+ΥT2

(t3B+

T3

). (13)

Similarly to HE, we introduced aE, which is the value ofthe scale factor at the beginning of the ekpyrotic phase(in the pre-bounce branch of the universe). Also, we in-troduced the mass m of the scalar field responsible for thephase of matter contraction. We also note the presence

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1.82 1.83 1.84 1.85 1.86 1.87 1.88 1.89 1.90

tB+/T

4× 10−7

5× 10−7

6× 10−7

7× 10−7Υ/M

2 p

FIG. 2: Constraints on the dimensionless duration parametertB+/T and the slope parameter Υ of the bounce phase fromthe measurements of Planck and BICEP2 in the new matterbounce cosmology. The value of the Hubble parameter at thebeginning of the ekpyrotic phase is fixed to beHE/Mp = 10

−7.As in Fig. 1, the blue and red bands show the confidenceintervals of the tensor-to-scalar ratio and of the amplitude ofthe power spectrum of curvature perturbations, respectively.

of the variable T , which comes into play in the evolutionof the bounce field (see [37] for more details). From eqs.(10) and (12), the tensor-to-scalar ratio in this modelthen takes the form of

r ≡ PTPζ'

F2ψM2p

2(2q − 3)2F2ζ a2Em2∣∣∣U (k)ζ ∣∣∣2 . (14)

Similarly to the previous subsection, we perform a nu-merical estimate to investigate the consequences of theobservational constraints on the parameter space of thenew matter bounce scenario. Note that, although themodel under consideration involves a series of parame-ters, there are three main parameters that are most sen-sitive to observational constraints, i.e., the slope param-eter Υ, the Hubble rate at the beginning of the ekpy-rotic phase HE, and the dimensionless duration parame-ter tB+/T of the bouncing phase.

We first look at the correlation between Υ and tB+/T ,with the numerical result shown in Fig. 2. One can readthat the slope parameter Υ and the dimensionless dura-tion parameter tB+/T are slightly negatively correlated.This implies that one expects either a slow bounce witha long duration or a fast bounce with a short duration.However, it is easy to find that the constraint on the di-mensionless duration parameter tB+/T is very tight witha value slightly less than 2. Therefore, it is importantto examine whether the model predictions accommodatewith observations by fixing the parameter tB+/T .

Then, we analyze the correlation between Υ and HEafter setting tB+/T = 1.86. The allowed parameter spaceis depicted by the intersection of the blue and red bandsshown in Fig. 3. From the result, we find that the Hubble

10−7 10−6

HE/Mp

10−6

4× 10−7

5× 10−7

6× 10−7

7× 10−78× 10−79× 10−7

Υ/M

2 p

FIG. 3: Constraints on the Hubble parameter at the begin-ning of the ekpyrotic phase HE and the slope parameter Υof the bounce phase from the measurements of Planck andBICEP2 in the new matter bounce cosmology. The dimen-sionless bounce time duration is fixed to be tB+/T = 1.86. Asin Figs. 1 and 2, the blue and red bands show the confidenceintervals of r and of the amplitude of Pζ , respectively.

scale HE and the slope parameter Υ introduced in thenew matter bounce cosmology are constrained to be inthe following ranges

1.9× 10−8 . HE/Mp . 1.9× 10−6 , (15)4.9× 10−7 . Υ/M2p . 8.5× 10−7 , (16)

respectively. One can easily see that the constraints onthe slope parameter Υ in the new matter bounce cosmol-ogy and in the matter bounce curvaton scenario are inthe same ballpark, i.e., Υ ∼ O(10−7). For the new mat-ter bounce cosmology, if we assume that the bounce oc-curs at the GUT scale, then the duration of the bouncingphase is roughly O(104) Planck times. We also note thatthe amplitude of the Hubble scale HE in the new mat-ter bounce cosmology is much lower than the GUT scale.This allows for a long enough ekpyrotic contracting phasethat can suppress the unwanted primordial anisotropiesas addressed in [46].

In summary, from the analysis of the matter bouncecurvaton and the new matter bounce cosmology sce-narios, we can conclude that, in general, a nonsingu-lar bouncing cosmology has to experience the bouncingphase smoothly for it to agree with latest observationaldata. In other words, the Hubble parameter cannot growtoo fast during the bounce phase since the constraintsthat we find favor a small value of Υ. Depending on thedetailed bounce mechanism, the observed amplitude ofthe spectra of the CMB fluctuations may be determinedby the bounce scale or the value of the Hubble parame-ter at the moment when primordial perturbations werefrozen at super-Hubble scales. For the matter bouncecurvaton, the mass scale of the bounce is of the order ofO(10−4)Mp which is close to or slightly lower than the

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GUT scale (O(1016) GeV). On the other hand, for thenew matter bounce cosmology, due to the introductionof the ekpyrotic phase, it is the Hubble parameter HE atthe onset of the ekpyrotic phase that determines the am-plitude of the primordial spectrum of the perturbations,and it must be much lower than the GUT scale in orderto agree with observations. These interesting results en-courage further study of bouncing cosmologies followingthe effective field approach.

III. IMPLICATIONS FOR MODIFIED-GRAVITYBOUNCING COSMOLOGY

In the previous section, we performed numerical com-putations to constrain two representative bounce cos-mologies that are described by the effective field ap-proach. It is interesting to extend the analysis to bounc-ing cosmology models where the bounce occurs due tomodified-gravity theories. In the following, we shall fo-cus on three specific models. The first one is to obtainthe matter bounce solution in the framework of a non-relativistic modification to Einstein gravity, namely theHořava-Lifshitz bounce model [40, 49]. The second is torealize the nonsingular bounce by virtue of torsion grav-ity, i.e., the f(T ) bounce model [41]. And the third is astudy of the new matter bounce cosmology in the settingof loop quantum cosmology [42].

A. Matter bounce in Hořava-Lifshitz gravity

Hořava-Lifshitz gravity is argued to be a potentiallyUV complete theory for quantizing the graviton, and ithas important implications in the physics of the veryearly universe. In particular, a nonsingular bouncingsolution can be achieved in this theory when a non-vanishing spatial curvature term is taken into account[50, 51]. In this case, the higher order spatial derivativeterms of the gravity Lagrangian can effectively contributea stiff fluid with negative energy, which can trigger thenonsingular bounce as well as suppressing unwanted pri-mordial anisotropies. Thus, the bouncing solutions ob-tained in this picture are marginally stable against theBKL instability.

As was shown in [40], if the contracting phase is dom-inated by a pressure-less matter fluid, Hořava-Lifshitzgravity can provide a realization of the matter bouncescenario. Moreover, for primordial perturbations inthe infrared limit, the corresponding power spectrumfor both the scalar and tensor modes are almost scale-invariant [52]. However, the paradigm derived in thisframework belongs to the traditional matter bounce cos-mology, and so the amplitude of the tensor power spec-trum is of the same order as the scalar power spectrum.In this regard, the corresponding tensor-to-scalar ratio istoo large to explain the latest cosmological observations.To address this issue, one needs to enhance the amplitude

of the curvature perturbations generated in the contract-ing phase in the infrared limit, for example by applyingthe matter bounce curvaton mechanism.

B. The f(T ) matter bounce cosmology

We briefly describe the realization of the matterbounce in the f(T ) gravitational modifications of generalrelativity. The f(T ) gravity theory is a generalizationof the formalism of the teleparallel equivalent of generalrelativity [53–55], in which one uses the curvature-lessWeitzenböck connection instead of the torsion-less Levi-Civita one, and thus all of the gravitational informationis included in the torsion tensor.

We use the vierbeins eA(xµ) (Greek indices run over

the coordinate space-time and capital Latin indices runover the tangent space-time) as the dynamical field, re-lated to the metric through gµν(x) = ηAB e

Aµ (x) e

Bν (x),

with ηAB = diag(1,−1,−1,−1). Thus, the torsion ten-sor is Tλµν = e

λA (∂µe

Aν − ∂νeAµ ), and the torsion scalar is

given by

T =1

4TµνλTµνλ +

1

2TµνλTλνµ − T νµν Tλλµ . (17)

Thus, inspired by the f(R) modifications of Einstein-Hilbert action, one can construct the f(T ) modifiedgravity by taking the gravity action to be an arbitraryfunction of the torsion scalar through S =

∫d4x e[T +

f(T )]/16πG. One interesting feature of the f(T ) gravityis that the null energy condition can be effectively vio-lated, and thus nonsingular bouncing solutions are pos-sible. In particular, it has been shown that the matterbounce cosmology can be achieved by reconstructing theform of f(T ) under specific parameterizations of the scalefactor [41].

In this model the power spectrum of primordial curva-ture perturbations is also scale-invariant if the contract-ing branch is matter-dominated. Its form is given by

Pζ =H2m

48π2M2p, (18)

where Hm is the maximal value of the Hubble parameterthroughout the whole evolution [41] (and thus its defini-tion is the same as that introduced in the matter bouncecurvaton scenario).

Now we investigate the evolution of primordial tensorfluctuations in the f(T ) matter bounce. The perturba-tion equation for the tensor modes can be expressed as[54](

ḧij + 3Hḣij −∇2

a2hij

)− 12HḢf,TT

1 + f,Tḣij = 0 , (19)

where the tensor modes are transverse and traceless. Itis interesting to note that the last term appearing in Eq.(19) plays a role of an effective “mass” for the tensor

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modes which may affect their amplitudes along the cos-mic evolution. However, as it was pointed out in [41], theeffect brought by f,TT is negligible since f(T ) is approx-imately a linear function of T in the matter contractingphase. Hence, for primordial tensor fluctuations at largelength scales, although the power spectrum is also scale-invariant, the amplitude takes of the form of

PT =H2m

2π2M2p. (20)

Thus, this result is already ruled out by the present obser-vations unless one introduces some mechanism to mag-nify the amplitude of scalar-type metric perturbations.This issue can be resolved by the matter bounce curva-ton mechanism. Doing so, the tensor-to-scalar ratio inour model can be suppressed by the kinetic amplificationfactor in the bouncing phase as described in the previoussection.

C. Loop quantum cosmology

The realization of bouncing cosmologies becomes verynatural in the frame of loop quantum cosmology (LQC)since the classical big-bang singularity is generically re-placed by a quantum bounce when the space-time cur-vature of the universe is of the order of the Planck scale[56, 57]. Several cosmological models have been studiedin the context of LQC, including inflation [58–60], thematter bounce [43], and the ekpyrotic scenario [61]. Notehowever that anisotropies are generically expected to be-come important near the bounce point —with the excep-tion of the ekpyrotic scenario— and, while the bounce isrobust in the presence of anisotropies [62, 63], the anal-ysis of the cosmological perturbations becomes consider-ably more complex when anisotropies are important.

There are two realizations of the matter bounce sce-nario that have been studied so far in LQC: the “pure”matter bounce model, where the dynamics are matter-dominated at all times, including the bounce, andthe new matter bounce model (also called the matter-ekpyrotic bounce) where the space-time is matter-dominated at the beginning of the contracting phase,while an ekpyrotic scalar field dominates the dynamicsduring the end of the contracting era and also the bounce.

In LQC, the dynamics of cosmological perturbationsare given by the effective equation of motion for theMukhanov-Sasaki variables that include quantum gravityeffects,

vik′′

+

(c2sk

2 − z′′i

zi

)vik = 0 , (21)

where k labels the Fourier modes and the indexi = {S, T} denotes the scalar and tensor modes respec-tively. The detailed forms of the sound speed parametercs and the coefficient zi for holonomy-corrected LQC arec2s = 1−2ρ/ρc, z2T = a2/c2s, and z2S = a2(ρ+P )/H2. Here

ρc ∼M4p is the critical energy density of LQC where thebounce occurs. These are the equations of motion thatwere used to determine the observational predictions ofthe pure matter bounce and the matter-ekpyrotic bouncemodels in LQC.

In the pure matter bounce model in LQC, tensor per-turbations are strongly suppressed during the bounce dueto the quantum gravity modification of zT and this givesa predicted tensor-to-scalar ratio of r ∼ O(10−3), wellbelow the signal detected by BICEP2 [43]. In addition,the amplitude of the spectrum of scalar perturbations isof the order of ρc/M

4p , and therefore in order to match

observations, it is necessary for ρc to be several ordersof magnitude below the Planck energy density. This isproblematic as heuristic arguments relating LQC and thefull theory of loop quantum gravity indicate that ρc isexpected to be at most one or two orders of magnitudebelow ρPl.

This last problem is avoided in the new matter bouncemodel for the following reason: when the universe evolvesinto the ekpyrotic phase, all the perturbation modes atsuper-Hubble scales freeze and thus the amplitude of theperturbations are entirely determined by the value ofthe Hubble parameter at the beginning of the ekpyroticphase, HE [42]. Because of this, the observed amplitudeof the scalar perturbations determines HE, not ρc. In ad-dition, the ekpyrotic phase also dilutes the anisotropiesbefore the bounce occurs and hence the BKL instabilityis avoided in this model. In order to determine how therecent results of the BICEP2 collaboration constrain thematter-ekpyrotic bounce in LQC, it is necessary to deter-mine the amplitude of the primordial tensor fluctuationsin this scenario.

The dynamics of scalar perturbations in the LQCmatter-ekpyrotic bounce model have been studied in de-tail in [42], and this analysis is easy to extend to tensorperturbations as their evolution is given by a very similardifferential equation as seen in Eq. (21). Due to the factthat their equations of motion are very similar at timeswell before the bounce, the amplitude of the spectra ofthe scalar and tensor modes are of the same order, andit is easy to check that if the ekpyrotic scalar field dom-inates the dynamics during the bounce,both the scalarand tensor modes evolve trivially through the bounce(note that this is very different to what happens if thematter field dominates the dynamics during the bounce).The result is that, as in other matter-ekpyrotic bouncemodels without entropy perturbations, the resulting am-plitude of the tensor perturbations is significantly largerthan for the scalar perturbations and therefore this par-ticular model is ruled out by observations.

However, if there is more than one matter field thenentropy perturbations may become important, and theyhave been neglected in the above analysis. As explainedin Sec. II, entropy perturbations can significantly in-crease the amplitude of scalar perturbations, while notaffecting the dynamics of tensor perturbations in anyway, thus decreasing the tensor-to-scalar ratio. There-

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fore, for the matter-ekpyrotic bounce model to be viablein LQC, it will be necessary to include entropy pertur-bations in some manner, perhaps as is done in the newmatter bounce model presented in Sec. II B.

Finally, it is possible (at least for the flat FLRW space-time) to express LQC as a teleparallel theory, which leadsto slightly different equations of motion for cosmolog-ical perturbations [64]. In this setting, as there existsolutions with a large range of tensor-to-scalar ratios,r ∈ [0.1243, 13.4375] [65], it is possible to obtain a valueof r that is compatible with the results of the BICEP2collaboration.

IV. CONCLUSION

In this work, we confronted various bouncing cosmolo-gies with the recently released BICEP2 data. In par-ticular, we analyzed two scenarios in the effective fieldtheory framework, namely the matter bounce curvatonscenario and new matter bounce cosmology, and threemodified gravity theories, namely Hořava-Lifshitz grav-ity, the f(T ) theories, and loop quantum cosmology. Inall of these models, we showed their capability of gener-ating primordial gravitational waves.

Since matter bounce models typically produce a largeamount of primordial tensor fluctuations, specific mech-anisms for their suppression are needed. In the mat-ter bounce curvaton scenario, introducing an extra scalarcoupled to the bouncing field induces a controllable am-plification of the entropy modes during the bouncingphase, and since these modes will be transferred into cur-vature perturbations the resulting tensor-to-scalar ratiois suppressed to a value in agreement with the observa-tions of the BICEP2 collaboration.

Another possibility, called the new matter bounce cos-mology, is to have two scalar fields, one driving the mat-ter contracting phase and the other driving the ekpyroticcontraction and the nonsingular bounce. Thus, the en-tropy modes are converted into curvature perturbationswhen the universe enters the ekpyrotic phase before thebounce, and the resulting tensor-to-scalar ratio is againsuppressed to observed values.

Furthermore, in both of these models we used the BI-CEP2 and the Planck results in order to constrain thefree parameters in these models, namely the energy scaleof the bounce, the slope of the Hubble rate during thebouncing phase, or the Hubble rate at the beginningof the ekpyrotic-dominated phase for the new matterbounce cosmology.

Finally, we considered bouncing cosmologies in theframework of modified gravity. In particular, in both

the Hořava-Lifshitz bounce model and the f(T ) gravitybounce, we have argued that the presence of a curva-ton field may suppress the tensor-to-scalar ratio to itsobserved values. We leave the detailed analysis of thistopic for a follow-up study.

In loop quantum cosmology, two realizations of thematter bounce have been studied. In the simplest mat-ter bounce model where there is only one matter field,the amplitude of the tensor perturbations is significantlydiminished during the bounce due to quantum grav-ity effects; this process predicts a very small value ofr ∼ O(10−3), well below the value observed by BICEP2.The other model that has been studied is the new matterbounce scenario, which in the absence of entropy pertur-bations predicts a large amplitude for the tensor per-turbations (in this case quantum gravity effects do notmodify the spectrum during the bounce). Therefore, forthe new matter bounce scenario in LQC to be viable,it is also necessary to include entropy perturbations inorder to lower the value of r to a value in agreementwith the results of BICEP2. Also, as can be seen here,the dominant field during the bounce significantly affectshow the value of r changes during the bounce and there-fore it seems likely that by carefully choosing this field,it may be possible to obtain a tensor-to-scalar ratio inagreement with observations. We leave this possibilityfor future work.

In summary, the predictions of the matter bounce cos-mologies where entropy perturbations significantly in-crease the amplitude of scalar perturbations remain con-sistent with observations, and thus these models are goodalternatives to inflation.

Acknowledgments

We are indebted to Robert Brandenberger and Jean-Luc Lehners for valuable comments. We also thank Tao-tao Qiu for helpful discussions in the initial stages of theproject. The work of YFC and JQ is supported in part byNSERC and by funds from the Canada Research Chairprogram. The research of ENS is implemented within theframework of the Action “Supporting Postdoctoral Re-searchers” of the Operational Program “Education andLifelong Learning” (Actions Beneficiary: General Secre-tariat for Research and Technology), and is co-financedby the European Social Fund (ESF) and the Greek State.The work of EWE is supported in part by a grant fromthe John Templeton Foundation. The opinions expressedin this publication are those of the authors and do notnecessarily reflect the views of the John Templeton Foun-dation.

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I IntroductionII Matter Bounce CosmologyA The matter bounce curvaton modelB New matter bounce cosmology

III Implications for modified-gravity bouncing cosmologyA Matter bounce in Horava-Lifshitz gravityB The f(T) matter bounce cosmologyC Loop quantum cosmology

IV Conclusion Acknowledgments References