Table Of ContentG-Bounce Inflation: Towards Nonsingular Inflation Cosmology with Galileon Field
Taotao Qiu1,2∗ and Yu-Tong Wang3†
1 Institute of Astrophysics, Central China Normal University, Wuhan 430079, China
2 State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,
Chinese Academy of Sciences, Beijing 100190, China and
3 School of Physics, University of Chinese Academy of Sciences, Beijing 100049, China
Westudyanonsingularbounceinflationmodel,whichcandrivetheearlyuniversefromacontract-
ingphase,bounceintoanordinaryinflationaryphase,followedbythereheatingprocess. Besidesthe
bouncethatavoidedtheBig-Bang singularitywhichappearsinthestandardcosmological scenario,
5
1 wemakeuseoftheHorndeskytheoryand design thekineticandpotential forms of thelagrangian,
0 sothatneitherofthetwobigproblemsinbouncingcosmology,namelytheghostandtheanisotropy
2 problems, will appear. The cosmological perturbations can be generated either in the contracting
phase or in the inflationary phase, where in the latter the power spectrum will be scale-invariant
r
p and fit the observational data, while in the former the perturbations will have nontrivial features
A thatwill betested bythelarge scale structureexperiments. Wealso fitourmodel totheCMB TT
power spectrum.
9
2
]
O I. INTRODUCTION
C
. The standard model of cosmology regards inflation [1–6] as an important period in the early universe. As a super-
h
fast expansion after the Big Bang, the inflation can solve a series of cosmologicalproblems such as horizon, flatness,
p
- monopole and so on, as well as give rise to scale-invariant scalar perturbations that can fit with the data. However,
o traditional inflation scenario still needs to be improved, because of the so-called “singularity problem” which was
r
t proposedbyHawkingetal. intheirearlyworkofsingularitytheorem[7,8]. Accordingtotheirprooftothis theorem,
s
if we track back inflation to its very beginning, we will generally meet the singularity of the early universe (the Big-
a
[ Bang singularity). At the singularity,everything blows up and one can not get controlof the universe under classical
description. Since the singularityoccursbefore the onsetofinflation,it canhardly be solvedwithin inflationscenario
2
itself. This motivates us to find alternative theories in pre-inflation era.
v
Phenomenologically, there might be quite a few evolutions that can be set in front of inflation in order to get
8
6 rid of this problem. As an example, the universe may undergo a contracting phase where the scale factor a(t)
5 shrinks initially and then, by some mechanism, “bounces” into an expanding one [9]. The whole process can be done
3 non-singularly if at the bouncing point a (t) = 0 [10]. The bouncing scenario has many interesting properties, for
B
0 example,the Big-Bangpuzzlessuchashorizon6problemandflatnessproblemcanbe solvedevenincontractingphase,
.
1 andscale-invariantprimordialperturbationscanbegenerated,etc[11]. Moreover,suchnon-singularscenariocanalso
0 be non-trivially extended to the cyclic universe [12, 13].
5 In bouncing inflation scenario however, there will be two more latent problems. One of them is the so-called
1 “ghostinstability”. This problemclaimsthat fieldtheorymodelsthatviolatesthe Null EnergyCondition(NEC)will
:
v generally give rise to “ghost” degrees of freedom, which causes the instability. The problems is originated from the
i “Ostradski problem” [14] in field theory, and the first cosmological application of this problem was on dark energy
X
models[15](seealso[16],whichdiscussedfromquantumaspect). Since anonsingularbounceviolatesNEC,normally
r
theghostwillappear. However,recentlyaninteresting“Galileon”theories[17,18]hasbeenshowntobeabletoavoid
a
such ghost degree of freedom while violating NEC (see pioneering work on such kind of models in [19]). The reason
is that although Galileons have higher derivative operators that can violate NEC, due to the delicate design of the
Lagrangian, there will only be one dynamical degree of freedom, which can be made normal. The other degrees of
freedomisnon-dynamical,thuswillnotleadtoanyinstabilities. Therefore,inthispaperwewillmakeuseofGalileon
theory to build our model. Note that pure Galileon bounce models has also been proposedin the literatures [20, 21],
which can avoid ghost instability on NEC violation.
The otherproblemisknownas“anisotropyproblem”[22],whichwillimposeadditionalconstraintonthe evolution
of the universe before the bounce. An exact isotropic universe at the initial time needs somehow fine-tuning, so in
realistic models of the early universe, certain amount of anisotropies will exist. In pure inflation scenario, this will
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected]
2
not be a problem because the anisotropies will decay in expanding universe, and will eventually be diluted away by
inflation to get an isotropic universe. In bounce cosmology however, the anisotropies will grow in contracting phase,
and if the growthof the backgroundenergy density does not exceed that of anisotropies,the latter will dominate the
universetomakeitatotallyanisotropicone,andprobablynobouncewilloccurandtheuniversewillneverenterinto
an expanding phase, which shows contradiction to our observations. By calculation, we can know that the growth of
anisotropies scales as a−6, and to have growth of background faster than this speed one must have the equation of
state (EoS) larger than unity, which gives constraint to the model building. In fact, a large EoS can be obtained by
requiringa negativepotential. Thoughcurrentlysucha potentialis only for phenomenogicallyuse, itcanactually be
reduced from more fundamental theories such as Ekpyrotic theories.
Thephenomenologyofbounceinflationscenariowasfirststudiedin[23],whereitwasshownthatsuchascenariocan
notonlyobtainscale-invariantscalarperturbationsthatcanfitwith the data,the perturbationsgeneratedbefore the
bounce can also give rise to tilted spectrum, which can explain the suppressionof CMB TT spectrum at large scales.
In the second paper of Ref. [11], a bounce inflation model was realized, but the ghost and anisotropy problems has
notbeenaddressed. In[24],asketchplotofhealthybounceinflationmodelhasbeendrawnwiththe waysofavoiding
thoseproblemsmentionedabove,andin[25], the1st-yearPLANCKdatahasbeenusedtoconstrainthis scenario. In
[26](alsosee[27]), itwasfoundthatthe bounce inflationmaygeneratealargecircularlypolarizedgravitationalwave
on large scale, which will lead to the enhancement of TB and EB-mode correlations on corresponding scale, see [28]
for recent study on large scale power deficits on the CMB fluctuations along with various pre-inflationary models.
In this paper, we aimed at building a realistic bounce inflation model, and fit with the newest PLANCK data.
Following [24] and also the above arguments, we make use of the Galileon theory the avoid ghosts, and introduce
negative potential to keep anisotropies away during the evolution. Moreover,another technical point is how to unify
a negative potential with a positive potential which is used to drive a period of inflation, and in the following we
will show that, with the help of some “shape functions”, one can naturally match the evolutions of contracting and
expanding phases together. In the case of model building the shape functions are shown to be very useful, although
its fundamental origination needs to be more explored.
The rest of the paper will be organized as follows: in Sec. II we set up our model, and show by numerics that it
can give us non-singular bounce followed by a period of inflation, and the ghost and anisotropy problems will not
appear. In Sec. III we calculate the perturbations generated in our model, and show that for perturbations of small
wavelengths,nearlyscale-invariancewillberealizedsimilartoinflationcase,whileforthoseoflongwavelengths,there
will be a blue-tilt which can explain the low-l suppression in CMB TT spectrum. in Sec. IV we will fit our model
with the newly released PLANCK data. Sec. V comes as a conclusion.
II. THE GALILEON BOUNCING INFLATION MODEL
Ashasbeendemonstratedintheintroduction,inordertoavoidghostinstability,itisusefultomakeuseofGalileon
theory to build up the bounce inflation model. The most generalized Lagrangianof Galileon is constructed as [18]:
5
= ,
i
L L
i=2
X
= K(φ,X) ,
2
L
= G (φ,X)✷φ ,
3 3
L −
= G (φ,X)R+G (✷φ)2 ( φ)2 ,
4 4 4X µ ν
L − ∇ ∇
h G i
= G (φ,X)G µ νφ 5,X (✷φ)3 3(✷φ)( φ)2+2( φ)3 ,
5 5 µν µ ν µ ν
L ∇ ∇ − 6 − ∇ ∇ ∇ ∇
h i
where X ∂ φ∂µφ/2 is the canonical kinetic term of φ, ✷φ gµν φ, R is the Ricci scalar and G
µ µ ν µν
≡ − ≡ ∇ ∇ ≡
R g R/2 is the Einstein tensor. However, in order to make our model simpler, we will only use and
µν µν 2 3
− L L
instead of all the ’s. To be explicit, we take the Lagrangianto be:
i
L
=k(φ)X +t(φ)X2 V(φ) G(X,φ)✷φ , (1)
L − −
where the last term, which we called the “G-term”, contains the second derivative of the scalar field φ which will be
essential in violating NEC and trigger the bounce. Note that for the kinetic part k(φ)X +t(φ)X2, when k(φ) = 1,
t(φ) = 0, it will reduce to trivial kinetic term X, and when k(φ) = 1, t(φ) = 1 it reduces to the form of “ghost
−
condensate”. We will see in the following that kinetic part of this form is useful in matching bounce with inflation,
3
which was first proposed for ghost-free bounce by Cai et al [29], and later connected with Supergravity in [30].
Following the Lagrangian(1), one can get the equation of motion for φ as:
[k(φ)+6t(φ)X +6G Hφ˙ +6HG φ˙ 2(G +G X)]φ¨+3H[k(φ)+2t(φ)X 2(G G X)]φ˙
X XX φ Xφ φ Xφ
− − −
+[2k (φ)+4t (φ)X +6G (H˙ +3H2) 2G ]X k (φ) t (φ)X2+V (φ)=0 , (2)
φ φ X φφ φ φ φ
− − −
and the energy density and pressure are
ρ=k(φ)X +3t(φ)X2+3G Hφ˙3 2G X +V(φ) , P =k(φ)X +t(φ)X2 2(G +G φ¨)X V(φ) (3)
X φ φ X
− − −
respectively.
We can furtherly determine the functions in Lagrangian (1) with the evolutions we want to get. First of all, an
anisotropy-freecontractionphasewhichrequirestheEoSlargerthanunitygenerallyleadstoanegativepotential,the
simplest Lagrangianof which is:
con =X Vcon(φ) , (4)
L −
where Vcon(φ)<0. Meanwhile, the inflationary expansion phase after the bounce will have a positive flat potential:
inf =X Vinf(φ) (5)
L −
where Vinf(φ) > 0, Vinf/(HVinf) 1. Therefore, we will choose functions k(φ), t(φ), G(X,φ) and V(φ) so that
φ ≪
they can approach Lagrangians (4) and (5) in the limits of past and future, respectively. As a simple example, we
choose forms of k(φ), t(φ) and G(X,φ) as:
2k 1 t 1 γX
0 0
k(φ)=1 , t(φ)= , G(X,φ)= . (6)
− [1+2κ (φ/M )2]2 M4[1+2κ (φ/M )2]2 M3[1+2κ (φ/M )2]2
1 p p 2 p p 2 p
and the potential is:
φ φ φ2
V(φ)=[1 tanh(λ )]Vcon(φ)+[1+tanh(λ )]Vinf(φ) , Vcon(φ)= V ecφ/Mp ,Vinf(φ)=Λ4(1 )2 , (7)
− 1M 2M − 0 −v2
p p
with k , t , γ, κ , κ , λ , λ , V , c, Λ andv areallfree parameters. Note thatVcon(φ) is inspiredfromthe Ekpyrotic
0 0 1 2 1 2 0
potential [21, 31, 32], and Vinf(φ) is the well-known spontaneously symmetry breaking potential [33] which fits well
with the PLANCK data [34]. So the choice of this potential is not only simple, but may have a strong backgroundof
fundamental theories.
From Eqs. (6) and (2) one can see that, in the (φ/M ) 1 limit, we have
p
≫
k(φ)=1 , t(φ)=G(X,φ)=0 , V(φ)=Vcon(φ) , (8)
while in the (φ/M ) 1 limit, we have:
p
≪−
k(φ)=1 , t(φ)=G(X,φ)=0 , V(φ)=Vinf(φ) , (9)
whichcangiveus(4)and(5)respectively. Aroundφ=0however,t(φ), G(X,φ)andthesecondtermofk(φ)becomes
large,sothelineartermofX willflipitssign,while the X2-termandG-termwillbe dominate. Thenforthe solution
where φ increasesmonotonically(whichis a verynaturalsolution), one will getthe anisotropy-freecontractionin the
past and inflation in the future, between which the bounce is triggered around φ=0.
Furthermore,we wouldlike to emphasize that by having the shapes of k(φ), t(φ) and G(X,φ), one cansuccessfully
control each term such that the higher derivative term only dominates for a while in order to trigger the bounce,
while disappearsbefore andafter that, notdisturbing the evolutionbefore and after the bounce. In fact, if we do not
controlthehigherderivativeterm,itwillusuallybecomemoreandmoreimportantafterdominance,leadingtoavery
fast evolution of the universe instead of a modest one, or even the “big-rip” singularity. Moreover, using tanh-like
shape function, we can naturally connect Vcon(φ) and Vinf(φ) so that the field φ can evolve naturally from a lower
potential to a higher one, leading to inflation. Therefore we can see that, shape functions are actually very useful in
constructing models. We also show the shape functions and potential explicitly by numerics in Fig. 1 and 2. For
pioneer works of using this shape functions to model building, see [29, 30] for ghost-free bounce and see [35] for dark
energy models.
4
k@ΦD t@ΦD X-1G@Φ,XD
1.0 5 0.0010
0.8 4 0.0008
0.6
3 0.0006
0.4
2 0.0004
0.2
1 0.0002
Φ
-3 -2 -1 1 2 3
-0.2 -3 -2 -1 1 2 3 Φ -3 -2 -1 1 2 3 Φ
FIG.1: Plotsof functionsk(φ),t(φ)andX−1G(X,φ) inEq. (6)w.r.t. thescalar, φ. Notethat thelast oneisalso function of
φ only. The parameters are chosen as k0 =0.6, κ1 =15, t0 =5, κ2 =10, γ =1 103. In such achoice, all thethree functions
×
havenontrivial value only around the bouncingpoint, which is useful to trigger thebounce.
V@ΦD
1.5´10-7
1.´10-7
5.´10-8
Φ
-5 5 10 15
-5.´10-8
-1.´10-7
FIG. 2: Plots of potential V(φ) in Eq. (2) w.r.t. the scalar, φ. The parameters are chosen as λ1 = λ2 = 10, V0 = 0.7Mp4,
c = √20, Λ 1.5 10−2Mp, v = 10Mp. From the plot we can see that the shape of the potential shares the same shape of
Ekpyroticpo≈tentia×lVcon(φ) fornegative φ while thatof symmetry breakinginflaton potential Vinf(φ)for positive φ. Wewill
see below that φ=0 is almost the division of contracting and expandingphases.
Now let’s go to a little bit more detail about the field evolution both before and after the bounce, which actually
become very simple. In the contracting phase before the bounce when Eq. (8) applies, the equation of motion of φ
becomes:
c
φ¨+3Hφ˙ V ecφ/Mp =0 . (10)
0
− M
p
The energy density and pressure of φ thus becomes
1 1
ρcon X +Vcon = φ˙2 V ecφ/Mp , pcon X Vcon = φ˙2+V ecφ/Mp , (11)
0 0
≃ 2 − ≃ − 2
while the EoS and the slow-rollparameter of the universe are
Pcon 4V0ecφ/Mp 3 6V0ecφ/Mp
w 1+ , ǫ (1+w )=3+ . (12)
c ≡ ρcon ≃ φ˙2 2V0ecφ/Mp c ≡ 2 c φ˙2 2V0ecφ/Mp
− −
Since φ˙2 > 2V0ecφ/Mp due to the requirement of positivity of the energy density, wc is obviously larger than unity,
and the growthof the energydensity ρ a−3(1+wc) will be largerthan the anisotropy,avoidingthe dominance of the
∝
latter. So the negative potential is rather useful in avoiding the cosmic anisotropy problem.
Setting the initial values of φ be negative, and once φ approaches 0 with certain amount of velocity φ˙, the higher
derivative terms is turned on, which is responsible to trigger the bounce. After bounce however, as Eq. (9) applies,
these terms decay away again, and one will have:
φ φ2
φ¨+3Hφ˙ +2Λ4 ( 1)=0 , (13)
v2 v2 −
5
which givesthe solutionofinflation. Accordingto the potential, one canexpressthe slow-rollparameterǫ as wellas
e
the e-folding number N of the inflation as:
e
M2 V 2 8M2φ2 φf V φ2 v2 φf
p φ p
ǫ (φ) = , N dφ= lnφ , (14)
e ≃ 2 V (φ2 v2)2 e ≃ V 8 − 4
(cid:18) (cid:19) − Zφi (cid:18) φ(cid:19) (cid:18) (cid:19)(cid:12)φi
(cid:12)
(cid:12)
By requiring ǫe(φf)=1 and Ne(φ)=60, the field values at starting and ending points of infla(cid:12)tion can be calculated
as:
φ 1.9M , φ 11.66M . (15)
i p f p
≈ ≈
One can see again that the φ has monotonically increasing evolution. The potential has valley with a minimum at
v =13M ,andwhenφfalls downthe valley,itwillbecomeoscillatingrapidly,whichreheatsouruniverse. The whole
p
evolution of the φ, φ˙, H and w are sketched in Fig. 3.
Φ'@tD
Φ@tD 0.0020
0.4
0.0015
0.2
0.0010
t
-2000 -1000 1000 2000
-0.2 0.0005
-0.4
t
-2000 -1000 0 1000 2000
H@tD 15w@tD 15w@tD
0.00020
10 10
0.00015
0.00010 5 5
0.00005
t t
t -2000 -1000 1000 2000 100000 200000 300000 400000 500000
-1000 -500 500 1000 1500 2000
-0.00005 -5 -5
-0.00010
-10 -10
FIG. 3: Top row: Plots of scalar φ and its velocity φ˙ w.r.t. cosmic time t. During the evolution, φ goes from negative to
positivevalues,andascanbeseenbelow,theswitchingpointisnearlythebouncingpoint. Thevelocitypeaksatthebouncing
point,anddecaytowardsbothpastandfuturedirections. Bottom row: Plot ofHubbleparameterH (left)andEoS w (middle
andright). Thehubbleparametertransitsfrom negativetopositivevaluesatthebouncingpoint,andwill approachtocertain
constant positive value, indicating a de-Sitter (inflationary) expansion after the bounce. The middle plot is the EoS near the
bounce region. From the plot we see that while before bounce it approach to value larger than unity, it goes down to minus
infinityatthebouncepointandapproachesto 1afterthebounce. TherightplotistheEoSduringwholeevolution,andsince
−
theinflationary period is verylong, thefeaturesnear thebounceisnot viewable. Nevertheless, wecan see that afterinflation,
EoS shows some oscillating behavior, referring to thereheating process.
III. PERTURBATIONS
A. scalar perturbations
Inthis section,we analyzethe perturbationsthatcouldbe generatedinthis model. Followingthe generalformulae
for calculating perturbations of Galileon models in [36, 37], the 2-nd order perturbed lagrangian based on (1) turns
out to be:
1 Q
(2) = dηd3xa2 ζ′2 c2(∂ζ)2 , (16)
S 2 c2 − s
Z sh i
6
where
2M4X 2G2 X2
Q = p [k(φ)+2t(φ)X +2(G +G X)φ¨+4HG φ˙ X ] , (17)
(M2H G Xφ˙)2 X XX X − M2
p − X p
c2 = (Mp2H −GXXφ˙)2[k(φ)+6t(φ)X +6H(G +G X)φ˙ + 6G2XX2]−1Q . (18)
s 2M4X X XX M2
p p
Here ζ is the variable of adiabatic perturbation, while c2 is the sound speed squared of ζ, representing how fast the
s
perturbations can propagate. From the quantum field theories, the ghost-free condition of ζ is Q/c2 > 0. Moreover,
s
the gradient stability requires 0 c2 1. From Eqs. (17) and (18) one can see that, for most of the evolution when
approximations (8) and (9) are r≤eacshe≤d, we have Q 2M2ǫ or 2M2ǫ and c2 1. However, for the region near the
≃ p c p e s ≃
bounce, evolution are very complicated, and numerical calculations are required. Here we numerically plot Q/c2 and
s
c2 in Fig. 4.
s
Q
50cs2 cs2
1.0
40
0.8
30 0.6
20 0.4
0.2
10
t
-2000 -1000 1000 2000
t
-2000 -1000 1000 2000
-0.2
-10
FIG. 4: Plots of functions Q/c2 and c2 during the whole evolution. The parameters are chosen as before. We can see that
s s
during the whole evolution Q/c2 >0, indicating that we don’t have any ghost instability in our model. In regions away from
s
the bounce, Q/c2 reduces to constant proportional to the slow-roll parameter. c2 is trivially equal to unity in regions away
s s
from the bounce, howeveraround bouncingpoint, it goes down to below zero, causing short period of gradient instability.
From the Fig. 4 we can see that, in the whole evolution (including the bounce) we have positive Q/c2, indicating
s
thatourmodeliscompletelyghost-free. However,duringthebounceregion,thenumericsshowthatthereisainstant
period of negative c2, which may induce fast increase of the perturbations. We will later show what sequent it will
s
cause to have negative c2 and how it can be ameliorated. In fact, the same issue also appears in the model given by
s
[29, 30]. Let’s first write down the equation of motion for ζ according to lagrangian(16):
z′′
u′′+c2 2u u=0 , (19)
s∇ − z
where
a√Q
u zζ , z , (20)
≡ ≡ c
s
and “” and “ ” denotes derivatives with respect to conformal time η a−1(t)dt and space x respectively. When
transf′ormed in∇to momentum space, one has u(x,t) u (t) for each wa≡venumber k, and 2 can be substituted into
k
k2. Accordingtothenumericalresultinthelastse→ction,wecanplottheeRvolutionsoffluc∇tuationmodesfordifferent
−
wavelengths λ = a/k as well as the Hubble horizon of our model as in Fig. 5. From the sketch plot one can
phys
see that, these fluctuation modes can been divided into two parts, one is of small k’s (green, cyan and blue), while
the other is of large k’s (navy and purple). The former modes are generated in the contracting phase, and will exit
the horizon before bouncing, while the latter ones are generated in expanding phase, and will exit the horizon after
bouncing. As will be shown very soon, they will have very different evolution and k-dependence. We remark the
dividingmode betweenthe twoask (the blueone),the wavelengthofwhichisjustequaltohorizonatthebeginning
0
of the bounce region.
7
FIG. 5: The sketch plot of the horizon of our model (red) and the fluctuation modes that cross the horizon before or during
inflation period (from green to purple). Since before the bounce the universe evolves slowly with a small value of Hubble
parameter, the Hubble horizon is large at most of the time, with a sharp damp near the bounce. However, when bounce
happens, it goes to infinity as H = 0. At the inflation era when H becomes nearly a constant, the horizon also behaves as
a constant. The fluctuations with k k0 exit the horizon at contracting phase, thus will stay outside the horizon all over
≤
the inflation region. Those with k >k0 exits the horizon at inflationary era, and will stay inside the horizon till the universe
expands,and exit thehorizon at somewhere during theinflation era.
1. initial condition
Since during the whole evolution, the background is different at each stage, and will lead to different evolution of
perturbation. Inthefollowingsubsections,wewillanalyzeevolutionineachstageseparately,andjointthemtogether
using matching conditions to get the final power spectrum.
At far past, everymode is inside the horizon. In this region,we solvethe Eq. (19) and taking the k2 z′′/z limit,
≫
and obtain the solution:
u =AeikR−η∞csdη′ . (21)
k
Asweknowthatincontractingphaseexceptnearthebouncingphase,Qandc canbeapproximatedas1. Moreover,
s
the coefficient A can be normalized by requiring a quantum origin of the perturbations [38]. We assume that the
perturbations starts from Bunch-Davies vacuum as usual, and from the quantum condition, we have A = (2k)−1/2.
So the initial condition for u is:
1
u = eikη . (22)
k
√2k
2. contracting phase
Inthe mostpartof contractingphase,the modelwill be governedby the potentialtermwhile the higherderivative
termsareseverelysuppressed. Inthiscase,theapproximation(8)applies,andfrom(17)and(18)onehasQ 2M2ǫ
≃ p c
andc2 1. Moreover,iftheuniverseevolvesaccordingtothe scalingsolutionofEq. (10)withthe EoSbeing as(12),
s ≃
the scale factor of the universe in contracting phase can be parameterized as
acon(η) (η η˜B−)ǫc1−1 , (23)
∼ −
where η˜ η [(ǫ 1) ]−1, and η and are the conformaltime and conformal Hubble parameterat
B− B− c B− B− B−
≡ − − H H
the end point of the contracting phase, respectively. The perturbation equation (19) then becomes
2 ǫ u
u′′+k2u − c =0 . (24)
k − (ǫ 1)2(η η˜ )2
c B−
− −
The solution of the above equation is Hankal function:
u = η η˜ [c (k)J (k η η˜ )+c (k)J (k η η˜ )] , (25)
k | − B−| 1 ν− | − B−| 2 −ν− | − B−|
with the coefficients c (k) andpc (k) to be determined by the initial conditions (22). Here
1 2
3 ǫ
c
ν = − . (26)
−
2(ǫ 1)
c
−
8
In the region far away from the bounce where all the fluctuation modes are deep inside the horizon, one have
k2 z′′/z , and the solution (25) can be approximated as:
≫| |
2 2
u c (k) cos(k η η˜ ) , c (k) sin(k η η˜ ) ,
k 1 B− 2 B−
≃ πk | − | πk | − |
r r
1 1
c1(k) 1 η η˜B− 1−ǫc c2(k) 1 η η˜B− 1−ǫc
ζ − cos(k η η˜ ) , − sin(k η η˜ ) ,(27)
k B− B−
≃ a πǫ k η η˜ | − | a πǫ k η η˜ | − |
B− r c (cid:18) B−− B−(cid:19) B− r c (cid:18) B−− B−(cid:19)
In the region near the bounce, the horizon shrinks rapidly to a minimum value. In this case, some fluctuation modes
withsmallk,namelyk2 z′′/z willexitthehorizonandbecomesuperhorizonmodesinthe contractingphase. The
≪| |
approximate behavior of these modes according to the solution (25) will be:
c1(k) 1 c2(k) ǫc−2
uk (k η η˜B− )ǫc−1 , (k η η˜B− )ǫc−1 ,
≃ 2ν−Γ(ν−+1)√k | − | 2−ν−Γ( ν−+1)√k | − |
−
1 1
ζk ≃ c21ν(−kΓ)[((νǫ−c−+11))H√B2−ǫc]a1B−ǫ−c k−2(ǫǫcc−−31) , 2−c2ν(−kΓ)[((ǫcν−−+1)1H)B√−2]ǫ1c−aǫBc−k−2(ǫǫcc−−31)(k|η−η˜B−|)ǫǫcc−−13 , (28)
−
while the large k mode still remains the same as (27).
The subhorizon solution in the contracting phase, which describes the modes deep inside the horizon, can be
connected to the initial condition (22) to find out the coefficients c (k) and c (k). Using (22) and (27) we will get:
1 2
√π √π
c (k)= , c (k)=i . (29)
1 2
2 2
One can see that while subhorizon modes shows oscillating behavior, the superhorizon modes splits into two
branches, one keeps constant and the other is decaying. Assuming the two modes are generated equally, one could
easily understood that for modes that exits horizon at contracting phase, the constant mode will dominate over
the decaying one, and will be inherited by the perturbations through the bounce and in expanding phase. We will
investigate the following evolution of perturbations in the coming sections.
3. through the bounce
After the Hubble parameter of the universe reaches the minimum value, it will no longer evolve as an monotonic
function which keepsnegative,but ratherturns aroundtowardspositive values. This will cause the universecontract
more and more slowly, while finally it stops contracting and becomes expanding, triggering the bounce. The Hubble
parameter will go from its minimum value to its maximum value, the process during which we call as a bouncing
phase. During the bouncing phase, the Hubble parameter is increasing rather than decreasing, and shape functions
such as k(φ), t(φ) and G(X,φ), as well as Q and c2, will have nontrivial features, which will make Eq. (19) hard to
s
solve. For sake of analyticity, we assume that the Hubble parameter increases as:
H α(t t ) , (30)
B
≃ −
when t is the time point when the bounce happens. Then we have
B
1
a a eα(t−tB)2/2 a [1+ α(t t )2] (31)
B B B
≃ ≃ 2 −
for the scale factor. Here a refers to the scale factor at t . From the numerical calculation,it can be identified that
B B
around the bouncing point the first two terms of the Lagrangian(1) will be dominant, so one roughly have:
2M4k(φ)X k(φ)
Q p , c2 , (32)
≃ (M2H G Xφ˙)2 s ≃ 6t(φ)X
p − X
respectively. Using Friedmann equations, one can furtherly simplify c2 to be
s
11+2α(t t )2
c2 − B , (33)
s ≃−21+3α(t t )2
B
−
9
andnearthebouncingpointwherethet-dependenttermscanbeviewedashigherorder. Moreover,nearthebouncing
point, the hubble parameter will be small compared to the term G Xφ˙ in the dominator of Q (at bouncing point
X
H =0), where G is obtained from G(X,φ) in (6). From (6), one can notice that in our model G =M (γ/t )t(φ),
X X p 0
so Q can be expressed as
2M4k(φ)X M2t2k(φ)X 3t2X
Q p p 0 0 . (34)
≃ G2 X2φ˙2 ≃ γ2t(φ)2X3 ≃− γ2α
X
It is convenient to approximately parameterize X as function of t [29]:
X X e−χ(t−tB)2 , (35)
B
≃
where χ is the parameter and X is the value of X at the bouncing point. From the numerical calculation we find
B
that the parametrization can efficiently mimic the evolution of X near the bouncing point.
It is convenient to transfer everything into conformal coordinate where the time is denoted by the conformal time
η. From the definition of η one can see that for leading order η η =a−1(t t ), where η is the conformal time
− B B − B B
at the bouncing point. Then from Eq. (32) one can express Q and c2 w.r.t. η approximately as:
s
3t2X 11+2αa2(η η )2
Q 0 Be−χa2B(η−ηB)2 , c2 B − B , (36)
≃− γ2α s ≃−21+3αa2(η η )2
B − B
and the parameter z turns out to be:
z =a√csQ ≃aBs6tγ202XαBea22B(α−χ)(η−ηB)2 . (37)
ThoughQ/c2 >0insuresthereisnoghostinstability,c2 <0leadstoagradientinstability,whichmainlyaffectsthe
s s
small-scale perturbation modes. These small-scale modes are within the horizon, i.e. the wavelength λ< ,
B− B+
H H
(noting that the superhorizon modes are hardly affected, see [30].), in which case the perturbation modes present
quantum-behavior. At classical level, it still remains an open issue of how to estimate the effect of this gradient
instability. Here, we will not involve this issue in this paper, which is beyond our scope.
However,the nasty instability might be avoided if we expect some additional corrections in the high energy region
such as around the bounce [42]. For example, we phenomenologically assume that there is an additional term in the
action of gravity such as:
1
∆S(2) = dηd3xa2ξ(t)R(3) , (38)
2 s
Z
the physics of which is yet to be known, where the 3-metric R(3) can be viewed as the scalar component of potential
s
term of the 4D gravity. In order not to affect gravity at other regions of evolution, ξ(t) is required to go close to zero
at past or future limit but reach its extreme value at the bounce. Since as a spatial curvature, R(3) (∂ζ)2 only, so
s
∼
this termwill do nothing to the backgroundevolutionas wellas the time derivative ofζ, but will addanother spatial
derivative term of ζ such as
1 2M2ξ˙(t)
dηd3xa2 Qξ(t)+ p (∂ζ)2 , (39)
− 2Z Mp2H −GXXφ˙!
so adding this term, the effective sound speed squared becomes:
2M2ξ˙(t)
c¯2 =c2 1+ξ(t)+ p . (40)
s s" Q(Mp2H −GXXφ˙)#
In order to make c¯2 >0 around the bounce, we only need
s
2M2ξ˙(t)
p
1+ξ(t)+ <0 , (41)
Q(M2H G Xφ˙)
p − X
whichcanbeeasilysatisfiedbychoosingproperformofξ(t). As anexample,weparameterizeξ(t)asξ(t)=ξ0e−λξt2,
and plot the evolutionof c¯2(t) in Fig. 6. From the plot we can see that now the sound speed squaredwill be positive
s
10
cs2
0.5
0.4
0.3
0.2
0.1
t
-774.2 -774.1 -774.0 -773.9 -773.8 -773.7 -773.6
FIG. 6: Plot of the new sound speed squared under correction (38) during bouncing phase. With the correction, the sound
speed squared can be kept above zero, avoiding the gradient instability. Various examples are shown with ξ0 set to be 1
−
(blue), 1.5 (pink)and 2(gold). At the edge of bounceregion where thecorrection term starts to decay,it will bematched
to value−s of c2 in contrac−ting/expanding region, which is unity.
s
all the time, avoidingthe gradientinstability aroundthe bounce, while reduce to the one in contracting phase due to
the quick decay of ξ(t) far from the bounce.
The EoM of perturbation now becomes
u′′+ c¯2k2 (α χ)a2 u =0 , (42)
k s − − B k
with a positive c¯2. As explicit examples, we se(cid:0)t ξ = 1, 1.5 an(cid:1)d 2 while λ = 1, and find that c¯2 can be easily
s 0 − − − ξ s
kept above zero (the minimum required value of ξ is 1, which gives c¯2 0). The solution is:
0 − s ≃
u = d cos[l(η η )]+d sin[l(η η )] (43)
k 1 B 2 B
− −
foranyk,wherel2 =c¯2k2+(α χ)a2. Ingeneral,wehaveχ>αduringbouncingphase,andthisisthe requirement
s − B
of kinetic dominance in order to get a bounce: φ¨>Hφ˙. This relation can also be numerically checked in our model.
The solution during bouncing phase should be matched onto the solution in contracting phase via the matching
conditions. WeapplytheHwang-Vishniac(Deruelle-Mukhanov)matchingconditions[39,40]thatζ anditsderivative
k
ζ′ should be matched at the pivot point η (see also [41] for matching conditions in Galileon theories.). Using the
k B−
solutions (27), (29) and (43) we have:
2
d = sin[c¯ k(η η )] c cos[k(η η˜ )] c sin[k(η η˜ )]
1 sπǫcc¯2sk( s B − B− 2 B−− B− − 1 B−− B−
(cid:16) (cid:17)
+c¯ cos[c¯ k(η η )] c cos[k(η η˜ )]+c sin[k(η η˜ )] ,
s s B B− 1 B− B− 2 B− B−
− − − )
(cid:16) (cid:17)
2
d = cos[c¯ k(η η )] c cos[k(η η˜ )] c sin[k(η η˜ )]
2 sπǫcc¯2sk( s B − B− 2 B−− B− − 1 B−− B−
(cid:16) (cid:17)
c¯ sin[c¯ k(η η )] c cos[k(η η˜ )]+c sin[k(η η˜ )] (44)
s s B B− 1 B− B− 2 B− B−
− − − − )
(cid:16) (cid:17)
for subhorizon modes while with (28), (29) and (43) we have:
1
d1 = [2(ǫνc−−Γ(11)+HBν−−)]√1−ǫǫcc k−12+−11+ǫc cos[l(−ηB+ηB−)]c1 ,
1
d2 = [2(ǫνc−−Γ(11)+HBν−−)]√1−ǫǫcc k−12+−11+ǫc sin[l(−ηB+ηB−)]c1 (45)
for superhorizon modes.
