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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: [email protected]†Electronic
address: [email protected]‡Electronic address: Emmanuel
[email protected]§Electronic address:
[email protected]
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.
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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-
-
7
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