Anisotropy in a Nonsingular Bounce

0.81 MB·English

by  Yi-Fu Cai

Anisotropy in a Nonsingular Bounce
Yi-Fu Cai,1,∗ Robert Brandenberger,1,† and Patrick Peter2,‡
1Department of Physics, McGill University, Montréal, QC H3A 2T8, Canada
2GRεCO – Institut d’Astrophysique de Paris, UMR7095 CNRS,
Université Pierre & Marie Curie, 98 bis boulevard Arago, 75014 Paris, France
(Dated: January 29, 2013)
Followingrecentclaimsrelativetothequestionoflargeanisotropyproductioninregularbouncing
scenarios, we study the evolution of such anisotropies in a model where an Ekpyrotic phase of
contractionisfollowedbydominationofaGalileon-typeLagrangianwhichgeneratesanon-singular
bounce. We show that the anisotropies decrease during the phase of Ekpyrotic contraction (as
expected)andthattheycanbeconstrainedtoremainsmallduringthenon-singularbouncephase(a
non-trivialresult). Specifically,wederivethee-foldingnumberofthephaseofEkpyroticcontraction
which leads to a present-day anisotropy in agreement with current observational bounds.
3 PACSnumbers: 98.80.-k,98.80.Cq
1
0
2 I. INTRODUCTION ferentiated from inflation is the shape of the bispectrum
[12].
n
a
Bouncingcosmologies, however, faceseveralproblems.
J The idea of a non-singular bouncing universe as an al-
ternative to the singular Big Bang paradigm has been Firstofall,newphysicsisrequiredtoprovidethebounce
7
(a pure GR bounce sourced by a regular fluid also leads
2 attractive on philosophical grounds for a long time.
to dynamical instabilities [13], while with a scalar field
More recently [1, 2] it has been realized that a non-
] singular bouncing cosmology with a matter-dominated a positive spatial curvature [14] or non-canonical kinetic
c term [15] is required). Such new physics can either come
q initialphaseofcontraction-a“matterbounce” cosmology
fromintroducingnewformsofmattersuchasphantomor
- -mayprovideanalternativetotheinflationaryparadigm
r quintom fields [16], ghost condensates [17, 18], Galileons
of cosmological structure formation (see e.g. [3] for re-
g
[19], S-branes [20] or effective string theory actions [21].
[ cent reviews). Since the curvature perturbation ζ grows
It can also arise from corrections to Einstein gravity, as
on super-Hubble scales in the contracting phase, fluctu-
2 e.g. in the non-singular universe construction of [22],
ations on long wavelengths get boosted by a larger fac-
v specifically applied to the bouncing scenario in [23], in
tor than those on small scales. In a matter-dominated
3
the ghost-free higher derivative gravity model of [24], in
0 phase of contraction the relative enhancement factor is
termsoftorsiongravity[25],orinHořava-Lifshitzgravity
7 preciselysuchastotransformaninitialvacuumspectrum
[26]. Finally, a bouncing phase can also originate from
4 into a scale-invariant one. Studies in many non-singular
. models [4] have shown that the scale-invariance of the pure quantum cosmological effects [27]; those also pro-
1
0 spectrum of ζ1 before the bounce is maintained through videanaturalframeworkinwhichadust-dominatedcon-
traction is easily implemented to yield a scale-invariant
3 the bounce on length scales which are larger than the
1 duration of the bouncing phase (the phase in which the spectrum of perturbation [28].
v: violation of the usual energy conditions is localized)2. A more serious problem for bouncing cosmologies is
i Hence, the matter bounce scenario leads to the same
X thefactthatthecontractingphaseisunstabletovarious
shape of the spectrum of primordial curvature fluctua-
effects. As discussed e.g. in [29], many matter bounce
r tions on super-Hubble scales as inflationary cosmology.
a models are unstable to the addition of radiation. This
Starting with vacuum initial conditions, the fluctuations
arises since the energy density of radiation scales as a−4,
are also Gaussian and coherent. A specific prediction of
where a(t) is the cosmological scale factor, whereas the
the matter bounce with which the scenario can be dif-
density of matter grows as a−3. In some models (e.g.
[16]) this will lead to a “Big Crunch” singularity instead
of a smooth bounce. Some constructions (e.g. those of
[18, 22], [20] and [30]) are free from this problem. How-
∗Electronicaddress: [email protected]
ever, a problem for all constructions mentioned so far is
†Electronicaddress: [email protected]
‡Electronicaddress: [email protected] the instability to the growth of anisotropic stress, whose
1 The variable ζ is the curvature fluctuation in comoving gauge. associated energy density grows as a−6. This is the fa-
It is proportional to the canonical fluctuation variable v - the mous BKL instability of a contracting universe [31].
Sasaki-Mukhanov variable [5, 6] - in terms of which the action
forcosmologicalfluctuationshascanonicalform. Seee.g. [7]for Asolutionofthisanisotropyproblemcanberealizedin
a comprehensive review and [8, 9] for introductory overviews of
theEkpyroticscenario[32],inwhichitispostulatedthat
thetheoryofcosmologicalperturbations.
2 Note that there are counterexamples as discussed in [10] and a matter field with equation of state parameter w (cid:29) 1
morerecentlyin[11]. (where w =p/ρ is the ratio of pressure p to energy den-
2
sity ρ) is dominant in the contracting phase3. Such an not apply to all non-singular bouncing cosmologies.
equationofstatecanberealizedbytreatingthedominant Theoutlineofthispaperisasfollows. Inthenextsec-
form of matter as a scalar field with negative exponen- tion we review the bounce model introduced in [38] and
tial potential. Since the energy density of the dominant derive the resulting equations of motion for a homoge-
matterthenscaleswitha−q with q (cid:29)6, anisotropiesbe- neous but anisotropic universe. In Section 3 we analyt-
come negligible and the BKL instability is avoided [37]4. ically study the background dynamics in each phase of
In a recent paper [38], a subset of the present authors the cosmological evolution from the initial matter phase
introduced a scalar field with an Ekpyrotic potential to ofcontractionthroughtheEkpyroticphasetothebounc-
constructamatterbouncescenariowhichisfreefromthe ing phase and the subsequent fast-roll expanding period.
BKL instability problem. Specifically, we determine the decay or growth rates of
The Ekpyrotic scenario in its original formulation [32] the anisotropy parameter in each phase. In Section 4 we
involves a singular bounce. In addition, the curvature solve the dynamical system numerically and present our
spectrumof ζ isan n =3spectrumratherthan ascale- final results. We close with a general discussion.
S
invariant n =1 one [39–42]. Hence, without non-trivial A word on notation: We define the reduced Planck
S (cid:112)
matching of ζ across the bounce, one cannot obtain a mass by M = 1/ 8πG where G is Newton’s gravi-
Pl N N
scale-invariant spectrum at late time5. To solve this tational constant. The sign of the metric is taken to be
problem,anewandnon-singularversionoftheEkpyrotic (+,−,−,−). Note that we take the value of the mean
scenario [46] was proposed in which a second scalar field scalefactoratthebouncepointtobea =1throughout
B
is introduced which does not influence the background the paper.
dynamics but develops a scale-invariant spectrum which
starts out as an isocurvature mode but which is trans-
ferred to the adiabatic mode during the evolution. The II. A NONSINGULAR BOUNCE MODEL
second field can also be given a “ghost condensate” La-
grangian [47] in which case it mediates a non-singular We consider a nonsingular bounce model in which the
bounce. However, as has been pointed out in [48], in universe is filled with two matter components, a cosmic
this “New Ekpyrotic” scenario the anisotropies which are scalar field φ and a generic matter fluid, as proposed in
highly suppressed during the contracting phase again Ref. [38] (which, in turn, is based on the theory devel-
raise their head and lead to a BKL instability. oped in [49]). The Lagrangian of φ is given by
Inourpreviouswork[38],wearguedqualitativelythat
in the model we considered the anisotropies remained L[φ(x)]=K(φ,X)+G(φ,X)(cid:3)φ, (1)
negligibly small during the bouncing phase. The reason
forthedifferencecomparedtowhathappensinthemodel where K and G are functions of φ and its canonical ki-
of [46] is that in our model the kinetic condensate which netic term
grows as the bounce is approached does not need to de-
1
crease again by the time of the bounce point. This leads X ≡ ∂ φ∂µφ, (2)
2 µ
toashorterbouncetimescaleandtodifferentdynamics.
In this paper we carefully study the development of while the other kinetic terms of φ include the operator
anisotropies in the bouncing cosmology with an Ekpy-
rotic phase of contraction introduced in [38]. We work (cid:3)φ≡gµν∇ ∇ φ. (3)
µ ν
in the context of a homogeneous but anisotropic Bianchi
cosmology in which the scale factors in each spatial di- Variation of the above scalar field Lagrangian mini-
mension evolve independently. We are able to show that mally coupled to Einstein gravity leads to the following
no BKL type instability develops, in agreement with corresponding energy momentum tensor
what the study of [38] indicated. Our work thus shows
that the arguments against non-singular (as opposed to Tφ = (−K+2XG +G ∇ X∇σφ)g
µν ,φ ,X σ µν
singular) bouncing cosmologies put forwards in [48] do +(K +G (cid:3)φ−2G )∇ φ∇ φ
,X ,X ,φ µ ν
−G (∇ X∇ φ+∇ X∇ φ), (4)
,X µ ν ν µ
in which we use the notation that F and F denote
,φ ,X
3 There are other approaches to address the anisotropy prob-
derivatives of whatever functional F(φ,X) may be with
lem. For example, nonlinear matter terms may smooth out the
respect to φ and X, respectively.
anisotropies[33]. AddingquadraticR Rαβ termstothegrav-
αβ
itationalactioncanalsopreventtheBKLinstability[34]. For the model under consideration we choose:
4 Note,however,thatincludinganisotropicpressuresmayreintro-
duceinstabilitiestowardsanisotropygeneration[35]. K(φ,X)=M2 [1−g(φ)]X+βX2−V(φ), (5)
5 However, the spectrum of the Bardeen potential Φ is scale- Pl
invariant[43],and,asarguedin[10]andshownexplicitlyinsome
where we introduce a positive-definite parameter β so
examples [44, 45], it is this spectrum which may pass through
that the kinetic term is bounded from below at high en-
the bounce, thus yielding a scale-invariant spectrum of curva-
turefluctuationsatlatetimes. ergy scales. Note that the first term of K involves M2
Pl
3
since in the present paper we adopt the convention that
the scalar field φ is dimensionless.
The function g(φ) is chosen such that a phase of ghost
condensationonlyoccursduringashorttimewhenφap-
proachesφ=0. Thisrequiresthedimensionlessfunction
g to be smaller than unity when |φ|(cid:29)1 but larger than
unity when φ approaches the origin. To obtain a nonsin-
gular bounce, we must make an explicit choice of g as a
function of φ. We want g to be negligible when |φ|(cid:29)1.
In order to obtain a violation of the Null Energy Con-
dition after the termination of the Ekpyrotic contracting
phase, g must become the dominant coefficient in the
quadratic kinetic term when φ approaches 0. Thus, we
suggest its form to be
2g
g(φ)= 0 , (6)
(cid:113) (cid:113)
e− p2φ+ebg p2φ
where g is a positive constant defined as the value of g
0 Figure1: Modelfunctionsg(φ)andV(φ)asgivenbyEqs.(6)
at the moment when φ=0, and is required to be larger
and(7),withbackgroundparameterstakenasforthefollow-
than unity, g >1.
0 ing evolution figures, namely as in Eqs. (65) and (66).
Wehavealsointroducedanon-trivialpotentialV forφ.
This potential is chosen such that Ekpyrotic contraction
is possible. It is well known that the homogeneous tra- function of only X:
jectoryofascalarfieldcanbeanattractorsolutionwhen
its potential is an exponential function. One example is G(X)=γX, (8)
inflationaryexpansionoftheuniverseinapositive-valued
exponentialpotential,andtheotheroneistheEkpyrotic where γ is a positive-definite number.
model in which the homogeneous field trajectory for a We now turn to the study of the cosmology of this
negative exponential potential is an attractor in a con- model. In order to characterize a homogeneous but
tracting universe. For a phase of Ekpyrotic contraction, anisotropicuniverse,wetakethemetrictobeoftheform
we take the form of the potential to be
(cid:88)
ds2 =dt2−a2(t) e2θi(t)σiσi, (9)
2V
V(φ)=− 0 , (7) i
(cid:113) (cid:113)
e− q2φ+ebV q2φ where t is cosmic time, σi are linearly independent at
all points in space-time and form a three dimensional
whereV isapositiveconstantwithdimensionof(mass)4.
0 homogeneous space.
Thusthepotentialisalwaysnegativeandasymptotically
In the case of a Ricci flat space, one can consider the
approaches zero when |φ|(cid:29)1. Ignoring the second term projection σi = dxi and thus the metric is of Bianchi
of the denominator, this potential reduces to the form
type-I form. The factor a(t) can be viewed as the mean
used in the Ekpyrotic scenario [32]. Both functions g(φ) scale factor of this universe, and the functions eθi(t) de-
and V(φ) are shown on Fig. 1 with the parameters used
scribe the correction of anisotropies to the scale factor.
in the later parts of this work.
Sincethevaluesofscalefactorscanbere-scaledarbitrar-
The term G(φ,X) is a Galileon type6 operator which
ily, one can impose an additional constraint
is consistent with the fact that the Lagrangian contains
higher order derivative terms in φ, but the equation of (cid:88)
θ =0. (10)
i
motionremainsasecondorderdifferentialequation. Phe-
i
nomenologically, there are few requirements on the ex-
plicit form of G(φ,X). We introduce this operator since Then,onecanimmediatelydefineameanHubbleparam-
we expect that it can be used to stabilize the gradi- eter as follows,
ent term of cosmological perturbations, which requires
a˙
thatthesoundspeedparameterbehavessmoothlyandis
H ≡ , (11)
positive-definitethroughoutmostofthebackgroundevo- a
lution. For simplicity, we will choose G to be a simple
and the individual Hubble parameters along spatial di-
rections are given by,
6 See[36]foradiscussionofGalileontypeLagrangians. Hi ≡ ae1θi ddt(cid:0)aeθi(cid:1)=H +θ˙i, (no sum) (12)
4
where the overdot denotes the derivative with respect to valuesofφ˙. However,thesetermswillbecomeimportant
cosmic time t. during the bouncing phase where φ˙ reaches a maximal
Since we are interested in studying anisotropies rather value. For simplicity, in the following we will consider
thaninhomogeneitieswecantreatthematterfieldstobe matter fluid is cold and thus w =0.
m
homogeneous,whichimpliesφisonlyafunctionofcosmic Finally, we can write down Einstein equations in this
time. Thus, the kinetic terms of the homogeneous scalar background, given by
field background become
(cid:18) (cid:19)
R
M2 R − g =Tφ +Tm. (19)
X = 1φ˙2, Pl µν 2 µν µν µν
2
(cid:3)φ = φ¨+3Hφ˙, (13) Once expanded in components, this tensor equation
yields the effective Friedmann equations,
so that, for this background, the energy density of the
scalar field is H2 = ρT + 1(cid:88)θ˙2, (20)
3M2 6 i
1 3 Pl i
ρφ = 2MP2l(1−g)φ˙2+ 4βφ˙4+3γHφ˙3+V(φ), (14) H˙ = −ρT +pT − 1(cid:88)θ˙2, (21)
2M2 2 i
and the pressure is Pl i
where ρ and p represent the total energy density and
p = 1M2(1−g)φ˙2+ 1βφ˙4−γφ˙2φ¨−V(φ), (15) pressureTin theTBianchi type-I universe, i.e., the sum of
φ 2 Pl 4
the contributions of the scalar field and the fluid.
as follows by computing the diagonal components of the Moreover, combining the spatial component of Ein-
stress-energy tensor (4). stein equation with the constraint equation (10) yields
Additionally, the matter fluid contributes its own en-
ergy density ρm and pressure pm, and usually they are θ¨i+3Hθ˙i =0, (22)
associated with a constant equation-of-state parameter
w = p /ρ . Namely, for normal radiation, w = 1, from which it follows that
m m m m 3
whTiloedfoerrivneortmhealemquaatttieorn, wofmm=ot0io.n for φ, one can either θ˙i(t)=Mθ,iaa3(3Bt), (23)
vary the Lagrangian with respect to φ or, equivalently,
require that the covariant derivative of its stress-energy where a is the mean scale factor of the universe at the
B
tensor vanishes. This yields bouncing point. The coefficients M are integral con-
θ,i
stants with a dimension of mass. According to the con-
Pφ¨+Dφ˙+V =0, (16)
,φ straint equation (10), one can read off that
where we have introduced (cid:88)
M =0. (24)
θ,i
P = (1−g)M2 +6γHφ˙+3βφ˙2+ 3γ2 φ˙4, (17) i
Pl 2M2
Pl Plugging Eq. (23) into Eq. (20) shows that one can
(cid:18) (cid:19)
1 introduce an effective energy density of anisotropy
D = 3(1−g)M2H + 9γH2− M2g φ˙+3βHφ˙2
Pl 2 Pl ,φ
−3(1−g)γφ˙3− 9γ2Hφ˙4 − 3βγφ˙5 ρθ ≡ M2P2l (cid:88)θ˙i2 ∝a−6, (25)
2 2M2 2M2 i
Pl Pl
−3G (cid:88)θ˙2φ˙− 3G,X(ρ +p )φ˙. (18) whose evolution as 1/a6 implies an effective equation-of-
2 ,X i 2M2 m m state parameter equal to wθ =1. We see that this effec-
i Pl tiveenergydensityincreasesfasterthanthatofpressure-
From Eq. (16), it is clear that the function P determines less matter or radiation in a contraction universe. This
the positivity of the kinetic term of the scalar field and is the source of the BKL instability of the contracting
thus can be used to determine whether the model con- phase of many bouncing cosmologies.
tainsaghostornotattheperturbativelevel;thefunction
D on the other hand, represents an effective damping
term. By keeping the first terms of the expressions of III. BACKGROUND EVOLUTION
P and D and setting g = 0, one can recover the stan-
dard Klein-Gordon equation in the FRW background. The initial conditions of our model are chosen (as in
Neglectingtheothertermsisagoodapproximationwhen [38])suchthatwestartinacontractingphasedominated
the velocity of φ is sub-Planckian. Note that the friction byregularmatter. SincetheenergydensityoftheEkpy-
term D contains the contributions from anisotropic fac- roticscalarfieldφgrowsfasterthanthatofregularmat-
tors and matter fluid, which can be suppressed for small ter, φ will at some time begin to dominate the energy
5
density. At this point, the Ekpyrotic phase of contrac- Integrating the above yields the following expressions
tion begins, and lasts until the nonsingular bounce in- for the anisotropy factors
terval begins (this is the phase where the new physics
(cid:90) t 2a3M t −t˜ 2a3M
effectsdominate),followedbyaperiodoffast-rollexpan- θ = θ˙ (t(cid:48))dt(cid:48) =− B θ,i E E =− B θ,iH(t).
sion, which in turn ends at a transition to the expansion i i 3a3H t−t˜ 3a3H2
−∞ E E E E E
of Standard Big Bang cosmology. We choose the initial (30)
conditions for the density of regular matter and for the From Eq. (30), we can observe that the anisotropy fac-
value of φ such that the temperature at which the Ekpy- tors θ approach zero in the limit of t → −∞. As the
i
roticphasebeginsishigherthanthatatthetimeofequal universe contracts with a matter-dominated equation of
matter and radiation in the Standard Big Bang expand- state, the absolute value of H increases and thus θ be-
i
ing phase. In this way, we ensure that initial vacuum comes larger as well.
perturbations develop into a scale-invariant spectrum of Before the Ekpyrotic phase, the universe is dominated
cosmologicalfluctuationsonallscaleswhicharecurrently by the pressureless matter fluid and the energy density
probed. of normal matter fluid evolves as ρ (cid:39) 3M2H2. How-
m
Pl
In the following we study the evolution of the ever, Eq. (29) shows that the effective energy density of
anisotropy in each of the periods of cosmological evo- anisotropies evolves as ρθ ∼ H4 and will therefore come
lution mentioned above. to dominate. This is nothing but another way of stating
the BKL instability in a contracting matter-dominated
universe. In the following we will see that the Ekpyrotic
phase of contraction cures this problem. However, in or-
A. Matter contraction
der to be able to enter this phase, the initial anisotropy
cannot be too large for the model not to break down al-
We start by considering the period when the universe together. The requirement is that ρ is less than ρ at
θ m
is dominated by a pressureless matter fluid, i.e., when the transition time t , which implies
the background equation of state parameter is roughly E
w =0. In this phase, the mean scale factor evolves as (cid:88) a6
M2 (cid:46)6H2 E , (31)
θ,i Ea6
(cid:18) t−t˜ (cid:19)2/3 i B
a(t)(cid:39)a E , (26) a requirement which we implement in our numerical dis-
E t −t˜
E E cussion below.
Let us now move on to the discussion of the evolution
wheret denotesthefinalmomentofmattercontraction
E of the anisotropy factors during the Ekpyrotic phase of
and the beginning of the Ekpyrotic phase, and a is the
E contraction.
value of the mean scale factor at the time t . In the
E
above, t˜ is an integration constant which is introduced
E
tomatchthemeanHubbleparametercontinuouslyatthe
B. Ekpyrotic contraction
time t , i.e.,
E
2 We assume a homogeneous scalar field φ which is ini-
t˜ (cid:39)t − . (27)
tially placed in the regime of φ (cid:28) −1 in the phase of
E E 3H
E mattercontraction. Inthiscase,theLagrangianforφap-
Correspondingly,themeanHubbleparameterduringthe proachestheconventionalcanonicalformandthusyields
period of matter contraction is given by an attractor solution which is given by
(cid:114)q (cid:20)2V (t−t˜ )2(cid:21)
2 φ(t)(cid:39)− ln 0 B− , (32)
H(t)= 3(t−t˜ ). (28) 2 q(1−3q)MP2l
E
where t˜ is an integration constant which is chosen in
Following Eq. (23), one can immediately write down B−
order that the mean Hubble parameter at the end of the
the time derivatives of the anisotropy factors as follows
phase of Ekpyrotic contraction matches with the one at
the beginning of the bouncing phase. This attractor so-
a3H2(t)
θ˙ (t)=M B , (29) lution corresponds to an effective equation of state
i θ,i a3H2
E E 2
w (cid:39)−1+ . (33)
where H is the value of mean Hubble parameter at the 3q
E
moment t . Note that the dimensional parameters of
E During the phase of Ekpyrotic contraction, the mean
anisotropy Mθ,i are therefore related to the values of θ˙i scale factor evolves as
at the moment t ; a smooth bounce will thus take place
only if these valEues satisfy some constraints which we (cid:18) t−t˜ (cid:19)q
a(t)(cid:39)a B− , (34)
derive below. B− t −t˜
B− B−
6
(cid:112) (cid:112)
wherea isthevalueofmeanscalefactoratthetimet φ ∼− p/2ln(2g )andφ ∼ p/2ln(2g )/b (assum-
B− B− − 0 + 0 g
which corresponds to the end of Ekpyrotic contraction ing one term in the denominator of g(φ) dominates over
and the beginning of the bouncing phase. Therefore, the the other at each transition time), the value of the func-
mean Hubble parameter is given by tiong(φ)becomeslargerthanunityandthustheuniverse
enters a ghost condensate state. The occurrence of the
q
H(t)(cid:39) , (35) ghost condensate naturally yields a short period of null
t−t˜
B− energy condition violation and this in turn gives rise to
where,inordertomakeH(t)continuousatthetimet , a nonsingular bounce.
B−
one must set As shown in Ref. [38], we have two useful parameteri-
q zationstodescribetheevolutionofthescalefactorinthe
t˜ =t − . (36) bounce phase. One is the linear parametrization of the
B− B− H
B− mean Hubble parameter
Additionally, we require the mean scale factor to evolve
smoothly and continuously at the time t . This leads to H(t)(cid:39)Υt, (40)
E
the relation
(cid:18)H (cid:19)q and the other is the evolution of the background scalar
a (cid:39)a B− . (37)
E B− HE φ˙(t)(cid:39)φ˙ e−t2/T2, (41)
B
Giventhesepreliminaries,wefindthatthetimederiva-
tives of the anisotropy factors evolve as wherethecoefficientΥissetbythedetailedmicrophysics
of the bounce. The coefficient T can be determined by
θ˙ (t) (cid:39) M a3B (cid:18) H (cid:19)3q, (38) matching the detailed evolution of the scalar field at the
i θ,ia3 H beginningortheendofthebouncingphase,whichwillbe
B− E
addressed in next subsection. Thus, during the bounce
andthustheeffectiveenergydensityofanisotropyevolves
the mean scale factor evolves as
asρ ∼H6q. Hence,whereasρ stillincreasesduringthe
θ θ
phase of Ekpyrotic contraction, the growth rate is much a(t)(cid:39)a e12Υt2. (42)
slower than in the matter-dominated phase (given the B
small value of q). In particular, the energy density in φ
Notethatanonsingularbouncerequiresthatthetotal
increases much faster than that due to the anisotropies:
energy density vanishes at the bounce point. The total
theBKLinstabilityistamedinthismannerbytheEkpy-
energy density includes the contributions from the mat-
rotic contraction phase.
ter fields and the anisotropy factors. This leads to the
Integrating the above expressions for θ˙i we obtain following result for the value of φ˙
B
(cid:90) t
θi(t) = θi,E + θ˙i(t(cid:48))dt(cid:48) (g −1)M2 (cid:34) (cid:115) 12β(V +ρ +ρ )(cid:35)
t φ˙2 (cid:39) 0 Pl 1+ 1+ 0 m θ
(2−3q)aE3M (cid:34) 3q (cid:18)H (cid:19)1−3q(cid:35) B 3β (g0−1)2MP4l
θ,i
(cid:39) B −1+ E . 2(g −1)
3(1−3q)a3H 2−3q H (cid:39) 0 M2, (43)
E E 3β Pl
(39)
From the expression (39) we see that the absolute values where we have made use of approximations that ρm and
of βi are monotonically increasing when cosmic time t ρθ are much less than V0 and V0 (cid:28) MP4l in the second
evolvesfromt totheonsetofthebouncingphaseattime line. These approximations must be valid for the model
t . HoweverE, they very rapidly converge to their limit- toholdsincebothρm andρθ aregreatlydilutedinEkpy-
inB−g values, and since any constant part of the anisotropy rotic phase and V0 is the maximal absolute value of the
factors can be absorbed in a rescaling of the spatial co- potential of φ which, according to the observational con-
ordinates, it turns out that the actual anisotropy, effec- straintfromtheamplitudeofcosmologicalperturbations,
tively measured as the distance to an effective FLRW must be far below the Planck scale.
space, is effectively decreasing as the amplitude of the Now we calculate the anisotropy factors in the bounc-
mean Hubble parameter increases: the Ekpyrotic phase ing phase. We first study their time derivatives, which
thus provides a simple solution to the BKL instability are given by
growth in a contracting phase.
θ˙i(t)(cid:39)Mθ,ie−32Υt2. (44)
C. Bounce phase Thus we see that the integration constants M can
θ,i
be interpreted as the values of θ˙ at the bounce point
i
In our model the scalar field evolves monotonically t = 0, and they are the maximal values θ˙ will ever
B i
from φ (cid:28) −1 to φ (cid:29) 1. For values of φ between take throughout the whole cosmic evolution. It is again
7
easy to perform the integrals and obtain the anisotropy phase we find the following approximate solution for the
factors: evolution of φ:
θi(t)(cid:39)θi,B− +Mθ,i(cid:114)6πΥ Erf(cid:32)(cid:114)32Υt(cid:33)(cid:12)(cid:12)(cid:12)(cid:12)t , (45) φ˙(t)(cid:39)φ˙B+aa33B(+t) (cid:39)φ˙Be−t2B+/T2HH(t), (52)
(cid:12)t B+
B−
where we have applied (41) in the second equality. This
where Erf is the Gauss error function. Around the
implies that
bouncepoint,wecanperformaTaylorexpansionof (45)
uptoleadingorderandthusobtainthefollowingapprox- M2 M2φ˙2 H2
imate expression, ρ (cid:39) Plφ˙2 (cid:39) Pl B . (53)
(cid:0) (cid:1) φ 2 2e2t2B+/T2 HB2+
θ (t)(cid:39)θ +M t−t , (46)
i i,B− θ,i B− Moreover, the Friedmann equation requires that ρ (cid:39)
φ
which is a linear function of cosmic time. We find that 3M2H2 in the fast-roll phase, so that T2 is given by
Pl
thegrowthoftheanisotropyatlinearorderonlydepends
on the coefficients M . However, at nonlinear order in 2H2
θ,i T2 (cid:39) B+ . (54)
cosmic time, the growth would also depend on the slope (cid:34)M2(g −1)(cid:35)
0
of the bounce phase characterized by the parameter Υ. Υ2ln Pl
9βH2
Finally, we have the relations B+
Υt2 Duringthefast-rollexpansionepoch, onecansolvefor
aB− (cid:39) aBe2 B−, (47) the time derivatives of the anisotropy factors
H (cid:39) Υt , (48)
B− B−
a3H(t)
by matching the mean scale factor and the mean Hubble θ˙i(t)(cid:39)Mθ,ia3B H , (55)
parameter at the beginning of the bouncing phase. B+ B+
from which it follows that the effective energy density
of anisotropy evolves as ρ = ρ H2/H2 , and thus
D. Fast-roll expanding phase θ θ,B+ B+
its ratio relative to the total energy density does not
change(recallthattheabsolutevalueofthisratioisalso
After the bounce, the universe enters the expanding very small because of the decrease of the shear contri-
phase, which typically begins with a period of fast-roll bution during the Ekpyrotic phase of contraction dis-
expansion for the background during which the universe cussed above). On the other hand, the matter fluid,
isstilldominatedbythescalarfieldφ. Duringthisstage, having w = 0, sees its energy density going as ρ =
m m
themotionofφisdominatedbyitskinetictermwhilethe ρ H/H ,andthereforeitscontributiontotheback-
m,B+ B+
potentialisnegligible. Thus,thebackgroundequationof groundwillcatchupwiththatofthescalarfieldandthe
state parameter is w (cid:39) 1. Correspondingly, the mean anisotropy,bringingthefastrollphasetoanendandini-
scale factor evolves as tiating the transition to the usual expansion history of
Standard Big Bang cosmology.
(cid:18) t−t˜ (cid:19)1/3
The anisotropy factors in the fast roll phase are given
a(t)(cid:39)a B+ , (49)
B+ t −t˜ by
B+ B+
where t represents the end of the bounce phase and a3M (cid:18) t−t˜ (cid:19)
B+ θ (t)(cid:39)θ + B θ,i ln B+ , (56)
thebeginningofthefast-rollperiod,andaB+ isthevalue i i,B+ 3a3 H t −t˜
of the mean scale factor at that moment. Then one can B+ B+ B+ B+
write down the mean Hubble parameter in the fast-roll so they are safely growing only logarithmically during
phase this phase.
Toconcludethissection,itisclearthattheanisotropy
1
H(t)(cid:39) , (50) contributioncaneasilybemadetoremainsmallthrough-
3(t−t˜ )
out the entire evolution, i.e. including the matter con-
B+
traction and the bounce phases. We now turn to the
and the continuity of the mean Hubble parameter at t
B+ actualconstraintsthatshouldbeimposedontheparam-
yields
eters of our model.
1
t˜ =t − . (51)
B+ B+ 3H
B+ E. Theoretical constraints
Recall that, in Eq. (41), we made use of a Gaussian
parametrizationofthescalarfieldevolutioninthebounce Inthissubsection,westudythetheoreticalconstraints
phase, with characteristic timescale T. In the fast roll on the scenario of anisotropic nonsingular bounce. It is
8
convenient to introduce an effective e-folding number of we expect this may impose a much more stringent con-
the Ekpyrotic phase as follows straint on the anisotropy parameters. For that, one
couldrequirethattheanisotropycontributionbesmaller
(cid:18)a H (cid:19) (cid:18)H (cid:19) that the observed level of perturbations, namely Ω ∼
N ≡ln B− B− =(1−q)ln B− , (57) pert
E a H H 10−10.
E E E
ItisinterestingtonotethatincreasingtheEkpyrotice-
which characterizes the variation of the mean value of
foldnumberto,say,N ∼15whilekeepingq ∼0.1,leads
the conformal Hubble scale aH during this period. to relation (31) to yieEld Ω ∼ 10−10 ∼ Ω2 (cid:28) Ω .
θ pert pert
Recall that the anisotropy is not allowed to dominate
This means that after a sufficiently long Ekpyrotic con-
before the onset of the Ekpyrotic phase. This is the con-
traction,theanisotropycontributionistotallynegligible,
straint (31). Applying (37) and (57), we can further
even compared with the amplitude of primordial pertur-
derive the following relation for the anisotropy at the
bations. At this particular level however, the anisotropy
bounce time:
contribution could be comparable to second order in
terms of the primordial curvature perturbation expan-
(cid:88) Mθ2,i (cid:39)e−2(1−3q)NE/(1−q), (58) sion, and thus could contribute to the primordial non-
6H2
Gaussianities of local shape with f ∼O(1).
i B− NL
If one wants to make successful contact with late time
which is, as expected, exponentially suppressed by the
cosmology,thereisasecondconstraintthatshouldbeim-
numberofe-foldingsofEkpyroticcontraction. Notethat
plemented on the model, namely that the fast roll phase
one can introduce a relative density parameter to de-
endsbeforethetimeofBigBangNucleosynthesis(BBN).
scribe the contribution of the anisotropies as follows
Roughly speaking, the energy density of regular matter
ρ doesnotchangemuchduringthephaseofEkpyroticcon-
Ωθ ≡ 3M2θH2. (59) traction (for small values of q). On the other hand, the
Pl density of φ grows rapidly. In the fast roll phase of ex-
pansion, the decrease in the density of regular matter is
This parameter must be less than the amplitude of the
nolongernegligible. Hence,theenergydensityofmatter
observed anisotropies in the cosmic microwave back-
ground (CMB), which is of order Ωpert ∼ O(10−5), will be much lower at the time tF when ρm(tF)=ρφ(tF)
than at the time t when Ekpyrotic contraction begins.
hereby defining Ωpert as the amount of energy density in E
In fact, it is straightforward to derive that
perturbations,asgivenbyCMBobservations. Thisleads
to the following constraint on the number of e-foldings
H (cid:46)|H |e−(1−3q)NE/(1−q), (62)
F E
(cid:88)6MHθ,2i2 <Ωpert, (60) showing that the value of HF should be much less than
i B− |H |. The Hubble rate H is associated with the ini-
E F
tial temperature T when the expansion begins to follow
where we have made use of (55). Provided that this F
the Standard Big Bang evolution (it is the equivalent of
constraint is satisfied, the current anisotropy will be be-
the temperature of reheating in inflationary cosmology).
low the current upper bound provided that it was ini-
Specifically, the relation is
tiallysmallenoughnottopreventtheonsetofEkpyrotic
contraction. Combining this inequality and the approxi-
g1/2πT2
mate relation (58), we can obtain a lower bound on the s
H (cid:39) F, (63)
e-folding number of the Ekpyrotic phase F 9.5M
Pl
(cid:18) (cid:19)
1−q 1 where g is the effective partible number for radiation.
N > ln . (61) s
E 2(1−3q) Ω As a consequence, in analogy with inflationary cosmol-
pert
ogy, the constraint (62) leads to an upper bound on the
Note that the nonsingular bounce model studied in the effective “reheating” temperature:
present paper belongs to “fast bounce” category and the
absolute values of H and H are of the same order. (cid:32) (cid:33)1
B− B− 3M |H | 2
Tofhtuhse,wbeoucnacne.simApslayncoenxasimdeprleH, Bif−wteocbheotohseeeqn=erg0y.1scaanlde TF (cid:46) Pgl1/2B− e−(2−3q)NE/[2(1−q)] (64)
s
Ω = 10−5, then the relation (61) yields N > 7.4.
pert E
This result implies that the Ekpyrotic phase is rather in our nonsingular bounce model. From the BBN con-
efficientatloweringthecontributionofanisotropiessince straint, we find that the lower limit of the “reheating”
a mere few e-folds of Ekpyrotic contraction are enough temperature is of the order O(MeV). If we consider this
to damp any reasonable initial anisotropy to negligible lower bound and take g ∼ 100, N ∼ 30 and q ∼ 0.1,
s E
level. then we find that H > 10−17M which can easily be
B+ Pl
Note that the e-folding number N can also be con- implementedinthemodel,asweshallseeinthefollowing
E
strained on observable scales of CMB experiments, and numerical calculations.
9
F. Numerical estimates

Toillustratethatanonsingularbouncecanbeachieved 1 .5 x 1 0 -4 5.00x10-5
in our model, we numerically solved the background 2.50x10-5
1 .0 x 1 0 -4 0.00
equations of motion. Expressing all relevant functions
-2.50x10-5
andparametersinthecorrespondingunitsofthereduced 5 .0 x 1 0 -5 -5.00x10-5
Planck mass M , we set -0.08 -0.04 0.00 0.04 0.08
Pl 0 .0
H
V0 =10−7, g0 =1.1, β =5, γ =10−3, -5 .0 x 1 0 -5 H
H
b =5, b =0.5, p=0.01, q =0.1 (65) 1
V g -1 .0 x 1 0 -4 H 2
H
3
to illustrate the calculations.
-1 .5 x 1 0 -4
Moreover, we consider the following parameters of the
matter fluid and the anisotropy -4 .0 x 1 0 4 -2 .0 x 1 0 4 0 .0 2 .0 x 1 0 4 4 .0 x 1 0 4
t
ρ =2.8×10−10, M =2.2×10−6,
m,B θ,1
M =3.4×10−6, M =−5.6×10−6, (66) Figure2: TimeevolutionoftheHubbleparametersH (black
θ,2 θ,3
line)andH (reddashed,bluedottedandmagentadot-dashed
i
and choose as the initial conditions for the scalar field linesfortheHubbleexpansionratesalongthex1,x2,andx3
the following: axes, respectively), in units of the reduced Planck mass M ,
Pl
with background parameters given by Eqs. (65) and (66),
φ =−2, φ˙ =7.8×10−6. (67) and initial conditions as in (67). The main plot shows that
ini ini
a nonsingular bounce occurs, and that the time scale of the
The actual computation also requires the initial value bounceisshort(itisa“fastbounce” model). Theinnerinsert
of the mean Hubble parameter, which is determined by showsablowupofthesmoothHubbleparametersduringthe
bouncephase: thiszoomed-inviewoftheHubbleparameters
imposing the Hamiltonian constraint equation. Figs. 2
around the bounce point shows that the Hubble rates vanish
and 3 show the evolution of the Hubble parameters and
atdifferenttimes,sothatthescalefactorsbounceatdifferent
“effective” energy densities for matter components and
times as well.
for the anisotropy, respectively.
From Fig. 2, one can see that Hubble parameters
alongallspatialcoordinatesevolvesmoothlythroughthe
bouncing point with an approximate dependence on cos- tE tB tF
mictimewhichislinear. Themaximalvalueofthemean r f
Hubble parameter, which we denote as the bounce scale 1 0 -8 r m
H , is mainly determined by the value of the potential r q1
paBrameterV . Specifically,H isoforderO(10−4M )in r q2
our numerica0l result. We alsoBnote that the bouncePsloc- 1 0 -10 r q3
curringinthethreespatialdirectionsdonotoccuratex- r
actly the same moment – a consequence of the existence 1 0 -12
of anisotropy. This could leave a smoking gun signature
fordetectingnonsingularbouncecosmologyinhighaccu-
1 0 -14
racy CMB experiments since the difference in the times
of the bounces along various spatial coordinates would
-3 .0 x 1 0 4 0 .0 3 .0 x 1 0 4 6 .0 x 1 0 4 9 .0 x 1 0 4
affect the ultraviolet (UV) modes of primordial pertur-
t
bations passing through the bouncing phase. We leave
this issue for a forthcoming investigation.
From Fig. 3, one can easily see that the universe in Figure3: Timeevolutionofthe“Effective” energydensitiesof
ourmodelexperiencesfourphases,whicharemattercon- the scalar field ρφ (full black line), the matter fluid ρm (dot-
traction,Ekpyroticcontraction,thebounce,andfast-roll dashedgreenline)andtheanisotropyfactorsρθi (reddashed,
bluedottedandmagentadot-dashedlines),withsameinitial
expansion, in turn. At the beginning, the universe is
conditions and background parameters are in Fig. 2.
dominated by the matter fluid. At some point (time t )
E
during the phase of contraction, the contribution of the
scalar field becomes dominant, and the universe enters
the Ekpyrotic phase. Note that the effective energy den- bounce, the scalar field φ enters a fast roll phase with
sity of anisotropies grows faster than ρ but slower than an effective equation of state equal to unity. As a con-
m
ρφ during the matter contraction phase. Thus, if ρθi sequence, the energy densities ρφ and ρθi dilute at the
does not dominate over the background before t , it will same rate, and finally the matter fluid catches up with
never become dominant throughout the whole evEolution, the density of φ at the time t .
F
as already discussed in the previous sections. After the Fig. 4 shows the evolution of the anisotropy factors
10
t t t t t t
E B F 1 0 0 E B F
2 .5 0 x1 0 -6
1 0 -1
0 .0 0 W f
q. -2 .5 0 x1 0 -6 qq..12 W 11 00 --32 WW mq
-5 .0 0 x1 0 -6 q.3 1 0 -4
0 .1 4 .0 x1 0 -6 s 1
q 0 .0 qq12 s 2 .0 x100.0-6 ss 23
-0 .1 q3 -2 .0 x1 0 -6
-0 .2 -4 .0 x1 0 -6
-3 .0 x1 0 4 0 .0 3 .0 x1 0 4 6 .0 x1 0 4 9 .0 x1 0 4 -3 .0 x1 0 4 0 .0 3 .0 x1 0 4 6 .0 x1 0 4 9 .0 x1 0 4
t t
Figure 4: Time evolution of the anisotropy factors θ (lower Figure 5: Time evolution of the density parameters Ω , Ω ,
i φ m
panel)andtheirtimederivativesθ˙i (upperpanel),withsame and Ωθ (upper panel), and of the shear function σi (lower
initial conditions and background parameters are in Fig. 2. panel),withsameinitialconditionsandbackgroundparame-
The anisotropies increase during the contracting phase but ters are in Fig. 2.
rapidly approach constant values in the following expanding
phase.
Indeed, the parameters chosen do not satisfy the bounds
imposed by CMB observations, so the effects of setting
θi and their time derivatives. Although the anisotropy non vanishing initial anisotropies can be enlarged so as
functions grow during the contraction, they evolve to- to be visible at all. Assuming parameter values taking
wards constant values in the expanding epoch. There- into account the experimental constraints would lead to
fore,afterthetimet ,onecanrescaleallscalefactorsby figures exactly similar to those obtained for a regular
F
absorbing the asymptotic factors in a redefinition of the isotropic bouncing model.
coordinates, and we finally get an isotropic universe. It
implies that at the level of homogeneous cosmology the
anisotropies do not destabilize our nonsingular bounce IV. CONCLUSION AND DISCUSSION
model. This can also be read from the upper panel of
Fig. 4 which shows that θ˙ approach zero after a suffi-
i We have studied homogeneous but anisotropic cos-
ciently long period of expansion.
mological solutions of the Galileon type action with an
In order to better characterize the anisotropy quanti-
Ekpyrotic scalar field potential, as originally introduced
tatively, we can define the so-called shear parameters
in [38]. The model was constructed to yield non-singular
σ ≡θ˙ e2θi, (68) isotropicbouncingcosmologicalsolutions. Ingeneral,the
i i
contracting phase of a bouncing cosmology suffers from
and the density parameters a BKL instability against the growth of anisotropies. By
ρ studying the solutions of Bianchi-type homogeneous so-
Ω ≡ I , (69)
(cid:80) lutions in which the scale factors in different spatial di-
I ρ
I I mensions are allowed to be independent we have stud-
where the subscript “I” represents φ, m and θ, respec- ied the evolution of the anisotropies in our model. We
tively. Fig. 5 shows the numerical solution we obtained haveshownthattheanisotropiesaredampedinthephase
fortheirtimedevelopment. Theshearfunctionsincrease of Ekpyrotic contraction, as was already conjectured on
up to their maximal values at the bounce point, after qualitativegroundsin[38]. Theyremainsmallduringthe
which they rapidly decrease to end up vanishingly small non-singular bounce phase, and are further suppressed
when the universe connects with the Standard Big Bang once space begins to expand again. This (counter) ex-
evolution. Fromtheevolutionofthedensityparameters, ample shows that a no-go theorem concerning smooth
weseethatthecontributionoftheanisotropyonlygrows bounces cannot be formulated, at least in terms of BKL
relative to the dominant density in the phase of matter instabilities.
contraction, but it then rapidly decreases in the Ekpy- It would be of interest to study the evolution of lin-
rotic phase and in the fast roll phase. earized inhomogeneities in our anisotropic background
Note that all numerical calculations shown here are and to see if there are any late time signatures of
meanttoillustratethediscussionoftheprevioussections. the initial (pre-Ekpyrotic) contracting phase when the