Tracing primordial black holes in nonsingular bouncing cosmology

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Tracing primordial black holes in nonsingular bouncing

cosmology

Jie-Wen Chen, Hao-Lan Xu, Yi-Fu Cai

To cite this version:

Jie-Wen Chen, Hao-Lan Xu, Yi-Fu Cai. Tracing primordial black holes in nonsingular bouncing

cosmology. Physics Letters B, Elsevier, 2017, 111, �10.1016/j.physletb.2017.03.036�. �hal-01494881�

Contents lists available atScienceDirect

Physics

Letters

B

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Tracing

primordial

black

holes

in

nonsingular

bouncing

cosmology

Jie-Wen Chen

a

,

Hao-Lan Xu

a

,

b

,

Yi-Fu Cai

a

a_{CASKeyLaboratoryforResearchesinGalaxiesandCosmology,DepartmentofAstronomy,UniversityofScienceandTechnologyofChina,Hefei,Anhui230026,}

China

b_{Institutd’AstrophysiquedeParis,UMR7095-CNRS,UniversitéPierreetMarieCurie,98bisboulevardArago,75014Paris,France}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received18September2016

Receivedinrevisedform18March2017 Accepted18March2017

Availableonlinexxxx Editor:M.Trodden

Weinthispaperinvestigatetheformationandevolutionofprimordialblackholes(PBHs)innonsingular bouncingcosmologies.WediscusstheformationofPBHinthecontractingphaseandcalculatethePBH abundanceasafunctionofthesoundspeedand Hubbleparameter.Afterwards,bytakingintoaccount the subsequent PBH evolution during the bouncing phase, wederive the densityof PBHs and their Hawking radiation. Ouranalysis shows that nonsingular bounce modelscan be constrained from the backreactionofPBHs.

©2017PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction

The matter bounce scenario [1–3] is one type of nonsingular bounce cosmology [4–9], which isoften viewed asan important alternativetothestandardinflationaryparadigm[10–13].By sug-gestingthattheuniversewasinitiallyinacontractingphase dom-inated by dust-like fluid (with a vanishing equation-of-state pa-rameterw

=

0), thenexperiencedaphase ofnonsingularbounce, andafterwardsenteredaregularphaseofthermalexpansion.The matter bouncecosmology can solve thehorizon problemas suc-cessfully asinflation andmatch with the observed hot big bang historysmoothly.Basedonprimordialfluctuationsgenerated dur-ingmattercontractingandtheirevolutionthroughthenonsingular bounce,onecanobtainascaleinvariantpowerspectrumof cosmo-logicalperturbations.Unlikeinflation,thematter bouncescenario doesnotneed a strongconstrain onthe flatness ofthepotential oftheprimordialscalarfieldthatdrivestheevolutionofthe back-groundspacetime [14,15].Also,thisscenariocan avoidtheinitial singularityproblemandthetrans-Planckianproblem,whichexists ininflationaryandhotbigbangcosmologies[16,17].

The aforementioned scenario has been extensively studied in theliterature, suchasthe quintombounce [18,19],the Lee–Wick bounce[20],theHorava–Lifshitzgravitybounce[21–23],the f

(

T

)

teleparallelbounce[24–26],theghostcondensatebounce[27],the Galileonbounce[28,29],thematter-ekpyroticbounce[30–32],the fermionicbounce [33,34],etc. (see,e.g. Refs. [1,35] forrecent

re-E-mailaddresses:chjw@mail.ustc.edu.cn(J.-W. Chen),xhl1995@mail.ustc.edu.cn (H.-L. Xu),yifucai@ustc.edu.cn(Y.-F. Cai).

views).Ingeneral,itwasdemonstratedthatonlengthscaleslarger thanthetimescaleofthebouncingphase,boththeamplitudeand the shapeofthe powerspectrum ofprimordialcurvature pertur-bations can remain unchanged through the bouncing point due to a no-go theorem [36,37]. A challenge that the matter bounce cosmology has to address is how to obtain a slightlyred tilt on the nearly scaleinvariant primordialpower spectrum.To address thisissue,ageneralizedmatterbouncescenario,whichisdubbed asthe



-Cold-Dark-Matter(



CDM)bounce,wasproposed in[38] andpredictedan observational signature ofa positiverunning of thescalarspectralindex[39,40].

As a candidate describing the very early universe,the matter bouncescenarioisexpectedtobeconsistentwithcurrent cosmo-logicalobservationsandtobedistinguishablefromthe experimen-tal predictions ofcosmic inflation aswell asother paradigms[7, 41].Meanwhile,apossibleprobeofprimordialblackholes(PBHs) mayofferapromising observationalapproachto distinguish vari-ousparadigmsoftheveryearlyuniverse[42,43].PBHscouldform at very early times of the universe, where a large amplitude of density perturbations wouldhave obtained. Correspondingly, the formationprocessandtheabundanceofPBHsstronglydependon thoseearlyuniversemodels,inwhichfluctuationsofmatterfields areresponsibleforsuchlargeamplitudesofdensityperturbations [44].

In the literature, most of attentions were paid on the com-putation of PBH predictions from the inflationary paradigm (for instancesee[45–49]),whilesofar,onlyafewworksaddressedthe PBHformationinabouncingscenario[50,51].Furthermore,those studiesofPBHsinabouncingscenariohavenotyetbeendiscussed in detail, for specific cosmological paradigms orbeen applied to

http://dx.doi.org/10.1016/j.physletb.2017.03.036

JID:PLB AID:32717 /SCO Doctopic: Astrophysics and Cosmology [m5Gv1.3; v1.208; Prn:22/03/2017; 11:43] P.2 (1-8) 2 J.-W. Chen et al. / Physics Letters B•••(••••)•••–•••

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falsify various early universe cosmologies, especially the matter bouncescenario.Inthecontextofmatterbouncecosmology,there areseveraldifferencesonthecomputationofthePBHabundance comparing with that in an expanding universe. First, comparing withinflation wheretheprimordialfluctuationsbecomefrozenat the momentof theHubble exit,those primordialfluctuationson matterfieldsinbouncecosmologywouldcontinuetoincreaseafter theHubbleexitduringthecontractingphaseuntiltheuniverse ar-riveatthebouncingphase[9,20],andthecontractingphasewould yielda differentinitial condition forthePBH formationand evo-lution. Second, once thesePBHs have formed, the contractionof spacetimecouldalsocompress andenlargetheprimordialmatter density,thuschangethePBHhorizonradiuswhichthen canlead toeffectsontheirevolution.

In this paper, we perform a detailed survey on the PBH for-mation and evolution in the background of the matter bounce cosmology. In Section 2, we briefly introduce the matter bounce scenarioanddescribetheformationofthepowerspectrumof pri-mordial curvature perturbation in an almost model-independent framework.InSection3,aphysicalpictureofthePBHformationin thecontractingbackgroundispresented.Afteraprocessofdetailed calculations,thethresholdforformingPBHsandthecorresponding massfractionare provided.InSection 4,wediscusstheevolution ofPBHsinthe bouncingphase bytakingintoaccount theeffects arisenfrom the contraction of the background andthe Hawking radiation.InSection 5, wesummarize ourresultsanddiscusson someoutlookofthePBHphysicswithinthenonsingularbouncing cosmology.

2. Nonsingularbounce

Nonsingular bounce can be achieved in various theoretical models,namely,tomodifythegravitationalsectorbeyondEinstein, toutilizematter fieldsviolating theNullEnergy Condition(NEC), orinthebackgroundofnon-flatgeometries(seee.g.[52,53]).Itis interesting tonoticethat, ingeneral, onlength scales largerthan thetimescaleofthenonsingularbouncingphase,primordial cos-mologicalperturbationsremainalmostunchangedthroughoutthe bounce[36,37].Inthisregard,one expectsthat theeffectivefield theoryapproachshouldbeefficienttodescribetheinformationof anonsingularbouncemodelatbackgroundandperturbationlevel. Recently,itwasfoundin[30]thatanonsingularbouncemodelcan beachievedunderthehelp ofscalarfieldwithaHorndeski-type, non-standard kinetic term and a negative exponential potential. Withinthismodelconstruction,the mattercontractingphasecan be obtained directlyby including the dust-like fluid orinvolving asecond matterfield[31].Notethat, inthepresentstudywe as-sume that the effective field approach of bouncing cosmology is validthrough thewhole evolutionwithout modifications to Gen-eralRelativity.

2.1. Themodel

Itturnsoutthat,underthedescriptionoftheeffectivefield the-oryapproach,thebackgrounddynamicsofthenonsingular bounc-ing cosmology can be roughly separated into three phases: the matter-dominated contraction, the non-singular bounce, and the thermal expansion. We consider a simple model starting with a matter contracting phase (t

<

t₋) from an initial time tinitial

→

−∞

,andthen enteringinto anonsingularbouncing phaseatt₋, whichlaststillt₊.Afterthebounceendsatt₊,theuniversebegins thehotbigbangexpansion,whichisinaccordancetothecurrent observations.

The evolution of the matter bounce cosmology in each stage canbeapproximatelydescribedasfollows.

(i)Inthemattercontractingphase,thescalefactoroftheuniverse shrinksas a

(

t

)

=

a₋



t

− ˜

t₋ t₋

− ˜

t₋



2/3

,

(2.1)

where a₋ is the scale factor at time t₋, and

˜

t₋ is related to the Hubble parameter att₋ via the relationt₋

− ˜

t₋

=

_3H2

−. The

equation-of-stateparameterduringthisphaseis w

=

0,whichcan berealizedinmanyways,suchasbycolddust,bymassivefieldor bythegravitysectorinvolvingnon-minimalcouplings.We param-eterizethesedifferentmechanismsbyintroducingthesoundspeed

cs,whichcanaffectthepropagationofprimordialperturbationsin

thegradientterms.

(ii)Inthenonsingularbouncingphase,thescalefactorofthe uni-versecanbeapproximatelydescribedas[30]

a

(

t

)

=

aBe

ϒt2

2

_,

_(2.2)

wherethecoefficientaB isthescalefactorexactlyatthebouncing

point, andfromEq.(2.2) one obtainsa₋

=

aBexp

[ϒ

t₋2

/

2

]

.

ϒ

is a

modelparameterdescribingtheslopeofHubbleparametertotime duringthebouncingphase,as:

H

(

t

)

= ϒ

t

.

(2.3)

Itcanbeseenthatt

=

0 correspondstothebouncingpointwhen theuniversestopsthecontractionandstarts theexpansion.Thus, it canbe foundthat t₋

=

H₋

/ϒ

,andthe valueof H vanishesat

t

=

0 whichisatthebouncingpoint.InsertingEqs.(2.2)and(2.3) intotheFriedmannequation,onecanseethatthenullenergy con-dition

ρ

+

p

>

0 isviolated around the bouncing point.It is the negative pressure that avoids the singularity and drives the uni-versetoevolvefromacontractingphasetoanexpandingphase. (iii)Intheeraofradiation-dominatedexpansion,wehave

a

(

t

)

=

a₊



t

− ˜

t₊ t₊

− ˜

t₊



1/2

,

(2.4) where t₊

=

H₊

/ϒ

,t₊

− ˜

t₊

=

_2H1 + and a+

=

aBe ϒt2₊ 2 _{. In present}

analysiswehaveadoptedtheassumptionthattheheatingprocess happensinstantlyafterthebounce(see[54,55]forrelevant analy-ses).

Fromsuchparameterizations,thematterbouncecosmologycan beapproximatelydescribed bymodelparameters H₋,H₊,

ϒ

,and also cs if perturbations are taken into account. In Fig. 1 we

de-picttheevolutionofcomovingHubblelength

|

H

−1

|

= |

aH

|

−1,and one can read that one Fourier mode of cosmological perturba-tioninthematterbouncecosmologycouldexittheHubbleradius duringthecontractingphase,andthenenterandre-exitthe Hub-bleradiusduringthebouncingphase,andeventuallyre-enterthe Hubble radius againin the classical Big Bang era. Note that the changeoftheHubbleradiusinthevicinityofthebouncing point canbeverylarge.Intheliterature,observationalconstraintsupon the bounce cosmology can be derived fromvarious cosmological experiments such as thecosmic microwave background [41] and primordial magneticfields[56].Inthiswork we willprovide the independentconstraintsonthemodelparametersfromPBHs.

2.2. Curvatureperturbationduringmattercontractingphase

Duringthemattercontractingphase,equationofmotionforthe curvatureperturbationcanbeexpressedas

v_k

+ (

c2_sk2

−

z

1 66 2 67 3 68 4 69 5 70 6 71 7 72 8 73 9 74 10 75 11 76 12 77 13 78 14 79 15 80 16 81 17 82 18 83 19 84 20 85 21 86 22 87 23 88 24 89 25 90 26 91 27 92 28 93 29 94 30 95 31 96 32 97 33 98 34 99 35 100 36 101 37 102 38 103 39 104 40 105 41 106 42 107 43 108 44 109 45 110 46 111 47 112 48 113 49 114 50 115 51 116 52 117 53 118 54 119 55 120 56 121 57 122 58 123 59 124 60 125 61 126 62 127 63 128 64 129 65 130

Fig. 1. Sketches ofthe evolutionsofthe comoving Hubbleradius (greencurve)

|H|−1_{= |aH|}−1 _{andaneffectivelengthscalefromthecoefficient|z}_/z|−1/2 _(red dashed)wherez=a√ρ+p

H ,inthenonsingularbouncecosmology.Theblackcurveis thecomovinglengthforwavenumberk.Thehorizontalaxiscorrespondstothe co-movingtimewhichisdefinedbyη≡dt

a.(Forinterpretationofthereferencesto colorinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

where vk

=

z

R

k is the Mukhanov–Sasaki variable with

R

k

be-ingthecomovingcurvatureperturbation, z

=

a

√

ρ+p

csH reliesonthe

detailed evolution of the background dynamics, and the prime represents the derivative with respect to the comoving time

η

. Assuming that primordial perturbations originated from vacuum fluctuations at initial times,1 one can derive the solution of Eq. (2.5) vk

=



π

(

−

η

)

2 H (1) 3/2

[

csk

(

−

η

)

] ,

(2.6)

whereH(₃1_/)₂ isthe 3₂-orderHankelfunctionofthefirstkind. From Eq.(2.6),thepowerspectrumof

R

kisthengivenby



2_R

≡

k 3 2

π

2





vk z





2

=

c2sk3

(

−

η

)

24

π

Mp2a2





H(₃1_/)₂

(

−

csk

η

)





2

,

(2.7)

during the matter contracting phase. At large scales csk

 |

H|

,



2_R



a

3 −H2₋

48π2_csM2 pa3

is almost scale independent, and

R

k





a3 −H2 − 24k3_csM2 p

a−3/2_{. This result is different from that in an}

expand-ing spacetime, in which

R

k is time independent at large scales.

Moreover, atsmall scales csk

|

H|

, curvature perturbation is of

quantumfluctuationwith



2

R



a 3 −H2 − 12π2_csM2 pa3



csk H

2 .

3. PBHformationduringmattercontractingphase

PBHsoriginatefromthecollapsedover-denseregionsseededby cosmologicalperturbationsinearlyuniverse.Whenan over-dense region starts to form a black hole (BH),its radius R isexpected tobelargerthan theJeansradius RJ

≡

c₂s

π ¯ ρ

=

2 3

π

cs

|

H− 1

_|

_and

smallerthan theHubblelength

|

H−1

_|

_{[42,44,48,58,59],which}

im-plies,



2 3

π

cs

|

H

−1

_{| ≤}

_R

_{≤ |}

_H−1

_{| .}

_(3.1)

Forscalessmallerthan RJ,thepressuregradientforcewould

pre-ventthe collapsingoftheover-dense region. Itis clearthat RJ

≥

1 _{Notethatthevacuumstate isnotnecessarilythe onlychoice oftheinitial} conditionforprimordialcurvatureperturbations,namely,theymayalsoarisefrom fluctuationsofathermalensemble[20,57].

|

H−1

_|

_{when c}

s

≥

0

.

39,andhence,therewouldbe noPBHinthis

case.

We note that the PBH formation in the contracting phase is differentfromthatinapurelyexpandinguniverse.Inthe expand-ing universe,thePBH formationwouldbeginonly whenthesize ofaregionreenterstheHubblescale.However,inthecontracting phase,when thesize oftheregion exitstheJeansscale, thePBH formationwouldhavestartedifthisregionisdenseenough.Thus, we wouldliketofocusonthe fluctuationswithin theJeansscale insteadoftheHubblescaleinthecontractinguniverse.

3.1. PBHmassfraction

The background dust-like fluid inside the over-dense regions may collapse into PBHs, andthe abundance ofPBHs is depicted bythemassfraction,whichinthemattercontractingphaseis

β

=

ρ

PBH

ρ

PBH

+

ρ

bg

,

(3.2)

where

ρ

PBH denotes the density of PBHs and

ρ

bg the density of

the remnant dust-like fluid. In general, PBHs andremnant dust-likefluidtogetherdrivetheevolutionofbackground

ρ

PBH

+

ρbg

=

3M2

pH2. Notethat thisrelation only holds when the subsequent

evolutions, includingthe accretionandHawking radiation,of the formedPBHsarenotconsidered.

The mass fraction can be obtained from the Press–Schechter theory[60](see[42,44,48]also):

β(

t

)

=

δm



δc 2

√

2

π σ

exp

(

−

δ

2 2

σ

2

)

d

δ

=

erfc



δc

√

2

σ

(

t

)



−

erfc



δm

√

2

σ

(

t

)



,

(3.3)

where

δ

≡

δ_ρρ_¯ is thefractional densityfluctuations,

σ

≡



δ

2

_is

themeanmassfluctuationattheJeansscale,whichcanbederived fromthepowerspectrumofcurvatureperturbation



2_R,

δ

c isthe

threshold ofthe PBH formation, andthe upper limit

δ

m ensures

thatthePBHsarenolargerthantheHubblescale[42,44,48].Note thatEq.(3.3)isbasedontheassumptionthat

δ

obeystheGaussian distributionN

[

0

,

σ

]

.

Thevalueofthreshold

δ

c isdeterminedasfollows.Considering

asphericalover-denseregionwitharadius R,thespaceinsidethe regionsatisfiesFriedmannequation[61,62]

(

H

+ δ

H

)

2

=

H2

(

1

+ δ) −

δ

K

a2

=

H

2

₍

₁

_{+ ˜δ) ,}

_(3.4)

whereH istheHubbleparameterofthebackground,a isthescale factorinsidetheover-dense region,

δ

H and

δ

K aretheperturbed Hubbleparameterandcurvaturerespectively,and

˜δ ≡ δ−

_a2δK_H2. Itis

noticedthat H

<

0 foracontractingphase,and

δ

H

<

0 intheover denseregion.Theouterregion(

>

R)isthoughttobeunperturbed forsimplicity.WeassumethatwhentheregioncollapsestoaBH, itssurface hasanadditionalphysicalspeed v

=

1 withrespectto the conformally staticbackground, i.e.collapsingat the speed of light.Inthecomovingslicing,itiswrittenas

δ

H

·

R

= −

1

.

(3.5)

InsertingEq.(3.4)intotheaboveequation,oneobtainsthe thresh-oldforanarbitraryscale R

˜δ

c

= δ

c

−

δ

Kc a2_H2

=

1 H2_R2

+

2

|

H

|

R

.

(3.6)

OntheJeansscale RJ,onehas

˜δ

c

=

√ 6 πcs

+

3 2π2_c2 s .

JID:PLB AID:32717 /SCO Doctopic: Astrophysics and Cosmology [m5Gv1.3; v1.208; Prn:22/03/2017; 11:43] P.4 (1-8) 4 J.-W. Chen et al. / Physics Letters B•••(••••)•••–•••

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ThemassofBHwitharadiusR,inthecontractingbackground, canbedeterminedby M

(

R

)

=

4

π

3 R 3

_ρ

_¯



₁

_{+ ˜δ}

c

=

R 2G

(

1

+ |

H

|

R

)

2

_.

_(3.7)

Notethat,both thedensityandcurvaturefluctuationscould con-tribute to the BH mass, as shownin [48]. We mention that the above descriptionofBH isarough estimate, andhence,Eq.(3.7) slightlydiffersfromtheMisner–SharpmassM

=

_2GR .Theaccurate description ofthe BH in thematter contractingphase should be based onthe Tolman–Bondi–Lemaitre metric [63–65],which will beaddressedinourfollow-upstudy.Inpresentanalysis, however, we note that at small scale R

 |

H−1

|

the above model natu-rallyrecoversthesolutionofaSchwarzschildBH M

=

_2GR inaflat spacetime,whichindicatesthatthisestimateremainsreliable.

Toderivethevaluesof

δ

c and

σ

,oneneeds toknowthe

rela-tions amongthevariables

δ

,

δ

K and

R

k.It isconvenient totake

thecomovinggauge,inwhich

H2

δ

=

2 3

∇

2

₍

_x

_,

_t

_{) ,}

δ

K a2

= −

2 3

∇

2

_R

₍

_x

_,

_t

_{) ,}

_(3.8)

where

(

x

,

t

)

istheBardeenpotentialand

R(

x

,

t

)

isthecurvature perturbation[45,66,67].TherelationbetweentheirFourier compo-nents



kand

R

kisgivenby[45,67]

−(

1

+

w

)

R

k

=

5

+

3w

3



k

+

2

˙

k

3H

,

(3.9)

wherethe dot denotes the derivative withrespect tothe cosmic timet.Thesolutionduring thematter contractingera

R

k

∝

a−3/2

is



k

= −

3

2

R

k

,

(3.10)

differingfrom



k

= −

3₅

R

k inthematterdominantexpandingera.

Onealsoobtains

δ

k

=

k2

a2_H2

R

k

,

(3.11)

where

δ

k istheFouriercomponentofthedensityperturbation

δ

.

From (3.6) and(3.8),one also has

δ

=

_2a32δK_H2

=

3

˜δ

.Therefore,the

thresholdis

δ

c

=

3 √ 6 πcs

+

9 2π2_c2 s .

Thevalue oftheupperlimit

δ

m forthe Jeansscaleisfixed as

follows.If

δ

= δ

m,thefluidinsidetheHubbleradiuswouldforma

BH,2whichwouldhave

4

π

3 R 3 J

ρ

¯



1

+ ˜δ

m

+

4

π

3

ρ

¯

(

|

H −3

_{| −}

_R3 J

)

=

2

|

H−1

|

G

,

(3.12)

where the first term on the left side is the mass of the over-dense region and the second term is the mass outside the RJ

butinside the Hubbleradius. As a result, we obtain

δ

m

=

3

˜δ

m

=

9



2 3

π

cs



−3 .

FromEq.(3.11),onecangetthepowerspectrumofthedensity perturbation



2_δ

=

k 3 2

π

2

|δ

k

|

2

₌

k4 a4_H4



2 R

.

(3.13) Accordingly,

σ

isdeterminedby

2 _{Inthiscase,thePBHisformedbyanover-densecorewithradius R} J,anda shellofthebackgroundfluidwithintheradiusfromRJ to|H−1|.

Fig. 2. β(t−)varyingfromthe modelparameters H− and cs.Thevalue ofβ is illustratedbythecolor,decreasingfromredtopurple.Theisolineofβ(t−)=0.1 is shown.(Forinterpretationofthereferencestocolorinthisfigurelegend,thereader isreferredtothewebversionofthisarticle.)

σ

2

=



dk k



2 δW



kRJ a



=

kJ



0 dk k



2 δ

,

(3.14) whereW



kRJ a

isthewindowfunction.Forsimplicity,wetake

W



kRJ a



=



1 k

≤

kJ 0 k

>

kJ

,

andthenobtain

σ

=

3 4c2s



a3₋H2₋ 48

π

2_c sM2pa3

.

(3.15)

Therefore,thePBHmassfractionby(3.3)is

β(

t

)



erfc



(

48c3s/2

+

9

.

355c1s/2

)



_M p

|

H₋

|



_a

₍

_t

₎

a₋

3/2



−

erfc



10

.

9427 c1s/2



_Mp

|

H₋

|



_a

₍

_t

₎

a₋

3/2



,

(3.16)

for cs

<

0

.

39;and

β(

t

)

=

0 for cs

≥

0

.

39.It can beseen that the

value of

β

increasesasa

(

t

)

becomes smalleralongwiththe con-tractingandreachesitsmaximumvalue attheendofthe matter contractingphaset₋.

From the above expression, one can read that the maximal valueof

β

isunity,whichcorrespondstothecasethattheuniverse isdominatedbyblackholes.Accordingly,weexpectthat

β



1 so that theuniverseis dominatedbythe backgrounddust-likefluid. Bynumericallysolving Eq.(3.16),we thereforederive the param-eter spaceof

|

H₋

|

andcs foreach fixed value of

β

asshownin

Fig. 2.From thisfigure,onecanreadthatthenon-vanishingmass fraction

β(

t₋

)

,whencs

<

0

.

39,issmallin boththevery low

en-ergyregime

|

H₋

|



Mp andveryhighone

|

H−

|

Mp.Inthehigh

energy regime, the density fluctuation is generally large

σ

1 according to Eq. (3.15), but

β(

t₋

)

is smalldue to the constraint that PBHs ought to be within the cosmic apparent horizon. The constraint of

β

from today’s observation could be loose. For in-stance, if one assumes that PBHs constitute the totality of dark matter in the universe, then the value of

β

could be of order

1 66 2 67 3 68 4 69 5 70 6 71 7 72 8 73 9 74 10 75 11 76 12 77 13 78 14 79 15 80 16 81 17 82 18 83 19 84 20 85 21 86 22 87 23 88 24 89 25 90 26 91 27 92 28 93 29 94 30 95 31 96 32 97 33 98 34 99 35 100 36 101 37 102 38 103 39 104 40 105 41 106 42 107 43 108 44 109 45 110 46 111 47 112 48 113 49 114 50 115 51 116 52 117 53 118 54 119 55 120 56 121 57 122 58 123 59 124 60 125 61 126 62 127 63 128 64 129 65 130

O

(

0

.

1

)

at most which might have occurred at the moment of matter-radiationequality (see forexample[68] for related analy-ses).Therefore,ifoneconsidersthecaseof

β



0

.

1 andcs

=

10−4

as a specific example, then Eq. (3.16) yields

|

H₋

|



10−1Mp or

|

H₋

|



104_M

p.Note that,

|

H−

|



10−1Mp can beeasily satisfied,

but

|

H₋

|



104_M

p isdisfavoredfrommodelconstruction.

4. EvolutionofPBHinbouncingphase

Itis importantto pointout that the subsequentevolutions of PBHs after their formation are not considered in the above sec-tion. Associatedwith the evolutions, including the accretionand the evaporation due to Hawking radiation, one can obtain con-straintsuponbouncingcosmology.Thisisthemainsubjectinthis section.Forsimplicity,weonlyconsiderthePBHevolutionsduring thebouncingphase.

4.1.ThegrowthofPBHmass

Atfirst,wecalculatethegrowthofPBHby accretingthemass aroundin the contractingbackground.Forsimplicity, we still as-sumethatthespaceoutsidetheBHhorizonisunperturbed Fried-mannuniverseandthematterwhichflowsintothehorizonisdue to the cosmic contracting. For a BH withmass M and radius R,

themass increasedequals to that flowinginto thehorizon, with thespeed

|

H

|

R,

˙

M

=

4

π

R2

ρ

¯

|

H

|

R

.

(4.1)

We shall investigate the evolution in the contracting era of the bouncing phase t₋

<

t

<

0.During this stage, the Hubble length diverges quickly (see Fig. 1), and all PBHs can be treated as SchwarzschildBHs and R

=

2G M, accordingto thearguments in thelastsection.Therefore,Eq.(4.1)reducesto

˙

R

= −

3R3H3

= −

3R3

ϒ

3t3

,

(4.2)

andthesolutionis

R

(

t

)

=









₁ 1 R2_(t−)

+

32

ϒ

3

(

t4

−

H4−

/ϒ

4

)

.

(4.3)

Itis obvious that,the BH radius increaseswitht approaching to thebouncingpoint.Eq.(4.3)alsogivesaconstraintasfollows,

R2

(

t₋

)

≤

2

ϒ

3H4₋

.

(4.4)

Forthe models violating the constraints above, the massof PBH willgrowtoinfinitybeforethebouncingpoint.Accordingto(3.1), onecanreadthatR

(

t₋

)

≥

2 3

π

cs

|

H−

1

−

|

,andhence,theabove

con-straintyieldsexplicitlythat3

ϒ

≥

c2_s

π

2H2₋

,

(4.5)

forcs

≤

0

.

39.Eq.(4.5)impliesthatthebouncingphaseisexpected

to proceed rapidly so that the mass ofPBHs wouldnot become divergent andtheuniverse is safefromcollapsing intothe black holecompletely.

3 _{Wenotethat,infact,ifc}

sisnotverysmall,theradiusofBHsuffersan in-creasingafterthebouncingbegins,butthiseffecthasbeenignoredinthepresent considerationforsimplicity.

4.2. Backreactionandtheoreticalconstraints

The bouncing phase is driven by the background fluid which violates NEC in the frame of general relativity. From the Fried-mann equations not considering PBHs, the density and pressure of the background fluid at the bouncing point are

ρ

bg

=

0 and pbg

= −

2M2p

ϒ

respectively. To include the contribution of

back-reactionfromPBHs,theFriedmannequationatthebouncingpoint canbewrittenas

¨

a a

= −

1 6M2p

(

ρ

BR

−

6M2p

ϒ) ,

(4.6)

where

ρ

BR

=

ρPBH

+

ρ

γ

+

3pγ istheeffectivedensityofthe

back-reaction,

ρ

PBH is thedensity ofPBH,

ρ

γ and pγ are the density

andpressureofHawkingradiationrespectively.Forsimplicity,we assumeallHawkingradiationparticlesareultra-relativistic, which satisfytherelationpγ

=

ρ

γ

/

3.Ifthebackreactionsneutralizethe

negative pressure of background

ρ

BR

−

6M2p

ϒ >

0, the universe

would not expand any more, as mentioned in Section 2. In the following,wewillconstrainthemodelthroughPBHs.

RecallthattheenergydensityofPBHsis

ρ

PBH

=



M

L3

=

4

π

M2_p



R

L3

,

(4.7)

whichdependsonthemeanPBHmass



M

orthemeanPBH ra-dius



R

,andthemeanPBHseparationL.Atthebeginningofthe bouncingphaset₋,thePBHdensityisknown

ρ

PBH(t₋

)

= β(

ρ

PBH

+

ρ

bg)

=

3M2pH−2

β(

t−

) ,

(4.8)

accordingto Eq. (3.2), wherethe evolution ofthe earlierformed PBHsisnotconsidered.If



M

atthemomentt₋isknown,onecan obtainthemeanPBHseparationL

(

t₋

)

fromEqs.(4.7)and(4.8).To obtain



M

(

t₋

)

, one should first calculated the mass function of PBHs.HereweassumethatallPBHshavethesamemass,whichis asimplificationgenerallyused[49],andallPBHsare oftheJeans scale



R

(

t₋

)



2 3

π

cs

|

H−

1

−

|

.Therefore,themeanseparationatt−

is L

(

t₋

)

=



R

(

t−

)

c2 s

β

1 3

.

(4.9)

Atthepresentinvestigation,wehaveneglectedthenewlyformed PBHs during the bouncingphase. Therefore, themean separation follows L

(

t

)

∝

a

(

t

)

, andaccordingly, L

(

0

)



L

(

t₋

)

e−

H2₋

2ϒ. Notethat

the PBH evaporationdue toHawking radiation has beenstudied clearlyin[43].Followingthis,onecanestimatethelifetimeofPBH tobe

τ



13

.

7 Gyr



M 5

×

1014_g



3



5

×

103M 3 M4p

.

(4.10)

We have adopted the approximation that for massive PBHs

τ

>

|

t₋

|

, or

ϒ >

H4

−

/(

108c3sMp2

)

, the Hawking radiation is

sub-dominant. A closer analysis combing both the Hawking radia-tion andthe growth ofPBHs will be addressedin our follow-up study.Asaresult,theevolutionofthePBHsfollowsEq.(4.3),and

theirsizeapproximatelytakes R

(

0

)



R

(

t₋

)



ϒ ϒ−c2sπ2H−2



1/2

atthe bouncingpoint.Otherwise,PBHwouldhaveevaporatedcompletely beforetheuniversereachesthebouncingpointwithR

(

0

)

=

0.Asa result,wecanestimatetheenergydensityofPBHsatthebouncing pointas

JID:PLB AID:32717 /SCO Doctopic: Astrophysics and Cosmology [m5Gv1.3; v1.208; Prn:22/03/2017; 11:43] P.6 (1-8) 6 J.-W. Chen et al. / Physics Letters B•••(••••)•••–•••

1 66 2 67 3 68 4 69 5 70 6 71 7 72 8 73 9 74 10 75 11 76 12 77 13 78 14 79 15 80 16 81 17 82 18 83 19 84 20 85 21 86 22 87 23 88 24 89 25 90 26 91 27 92 28 93 29 94 30 95 31 96 32 97 33 98 34 99 35 100 36 101 37 102 38 103 39 104 40 105 41 106 42 107 43 108 44 109 45 110 46 111 47 112 48 113 49 114 50 115 51 116 52 117 53 118 54 119 55 120 56 121 57 122 58 123 59 124 60 125 61 126 62 127 63 128 64 129 65 130

ρ

PBH(0

)

=

4

π

M 2 pR

(

0

)

L

(

0

)

3

=

⎧

⎪

⎨

⎪

⎩

6M2 pH2− π

β(

t−

)

ϒ ϒ−c2 sπ2H2−e 3H2₋ 2ϒ

_{, ϒ >}

H 4 − 108_c3 sM2p 0

,

ϒ <

H4− 108_c3 sM2p

.

(4.11)

Similarly,we analyzethebackreaction duetotheHawking radia-tion.For

ϒ >

H

4 − 108_c3 sM2p

,theHawking radiationcanbe ignored and

ρ

γ

(

0

)

=

0. In the limit that PBHs finish their evaporation att−, one has

ρ

γ

(

t−

)

=

ρPBH

(

t₋

)

andthen

ρ

γ

∝

a−4.In the limit that

theeffectiveevaporationhappensatthebouncing point,

ρ

γ

(

0

)

=

ρPBH

(

0

)

,and

ρ

PBH

∝

a−3 beforeitsevaporation.Therefore,onehas

ρ

γ

(

0

)

=

⎧

⎪

⎨

⎪

⎩

0

,

ϒ >

H4− 108_c3 sM2p

ρ

PBH(t₋

)



e H2₋ 2ϒ

n

, ϒ <

H4− 108_c3 sM2p

,

(4.12)

where

ρ

PBH

(

t−

)

is given by Eq. (4.8) andn is a parameter that describes the moment of the complete evaporation of PBH. For instance,n

=

4 correspondstothelimitthat theevaporation hap-pens att₋; and, n

=

3 corresponds to thislimit at thebouncing point.

Inserting Eqs. (4.11) and (4.12) into the Friedmann equation (4.6),inordernottoneutralizethenegativepressurep

¯

= −

2M2p

ϒ

,

thefollowingconstraintshouldbesatisfied

ρ

BR

| ¯

p

|

=

⎧

⎪

⎨

⎪

⎩

β(t−)H2 − πϒ

ϒ ϒ−c2 sπ2H−2 e 3H2₋ 2ϒ

_<

₁

_{, ϒ >}

H 4 − 108_c3 sM2p β(t−)H2₋ ϒ



e H2₋ 2ϒ

n

<

1

,

ϒ <

H 4 − 108_c3 sM2p

.

(4.13)

The resultingconstraints are numericallyshownin Fig. 3forthe model with cs

=

10−3. It is seen that the lines of

ϒ

=

c2s

π

2H2−

and

ϒ

=

H

4 − 108_c3 sM2p

,which representthelimit caseof(4.5) andthe boundary betweenthe PBH growthand evaporationrespectively, play important roles in the constraints.The two lines shape the forbiddenparameterspaceinwhich PBHgrowsinto infinity. Fur-thermore, two lines intersect at the characteristic energy scale

H2

−



109c5sMp2.AtthelowenergyH2−



109c5sMp2,

ϒ

=

c2s

π

2H2−is

theasymptoteforallisolinesof

ρ

BR

/

| ¯

p

|

.Therefore,theconstraint

ρBR

/

| ¯

p

|

<

1 reducestoEq.(4.5)atthelowenergy.

4.3. Estimatesofobservationalconstraints

Afterthe bouncingphase, PBHs will evolve inthe regular ex-pandinguniverse,andhencecanbeconstrainedbytoday’s obser-vations.Inthissubsection,webrieflyestimatetheconstraintsupon thebouncingmodelsthoughtheobservationallimitsofPBHs.

FormassivePBHswithaninitial massM

>

1015 g, theycould surviveuntiltoday andtheir densityscales asa−3 _{approximately}

intheexpandinguniverse[43].Asaresult,thecurrentPBH(M

>

1015g)densityparametercanbeexpressedas



PBH



ρ

PBH(0

)

3Mp2H20



ab a0



3

,

wherea0andH0denotethescalefactorandHubbleparameterof

today,abisthescaleofbounce,and

ρ

PBH

(

0

)

isthePBHdensityat

thebouncingpoint.SincethePBHsmaycontributetopartofdark matter, oneimmediatelyhasthe roughconstraint



PBH

< 

DM



0

.

25 andthuscangetabound

Fig. 3. ThebackreactionρBR/| ¯p|varyingfrommodelparametersH−andϒ,with

cs=10−3andn=4 taken.ThevalueofρBR/| ¯p|isillustratedbycolor,increasing frompurpletored.ThewhitethickcurveistheisolineρBR/| ¯p|=1,andonlythe parameterspaceoverthiscurveisallowed.Thelightbluecurveistheboundary betweenPBHgrowthandevaporationϒ=₁₀8Hc4−3

sM2p,theareaovertheboundaryis

thePBHgrowingregion,andthatundertheboundaryistheevaporationregion. ThegrayregionistheforbiddenparameterspaceduetoEq.(4.5),andtheupper boundaryofthisregionisϒ=c2

sπ2H2− illustratedbygreencurve. (For

interpre-tationofthereferencestocolorinthisfigurelegend,thereaderisreferredtothe webversionofthisarticle.)

ρ

PBH(0

) <

3M_p2H2₀

DM



ab a0



₋3

.

(4.14)

For the light PBHs with M

<

1015 g, they have been completely evaporatedtoday.Therefore,thedensityparameteroftheHawking radiation



HR isapproximatelyestimatedas

HR



ρ

PBH

(

0

)

3Mp2H20



ar a0



4



_a b ar



3

>

ρ

PBH

(

0

)

3M2pH02



ab a0



4

,

(4.15)

wherear isthescalefactorwhenPBHscompletetheevaporation,

and ab

<

ar

<

a0.



HR is not allowed to be larger than the total

densityoftoday’sradiation



r

∼

10−4,measuredby theCMB

ex-periments.Asaconsequence,onegets

ρ

PBH(0

) <

3M_p2H2₀

r



ab a0



−4

.

(4.16)

InFig. 4,we providetheconstraintson

ρ

PBH

(

0

)

along withab

/

a0

fromthe bounds(4.14)and(4.16),respectively. ThePBHmass M

anddensity

ρ

PBH

(

0

)

arefunctionsofcs,H−and

ϒ

,whichare pro-videdintheprevioussubsection.Asaresult,wefindthatthe non-singularbouncemodelscanbeconstrainedbyobservationsdueto (4.14) and(4.16). We note that these constraintsare quite loose. Morestringentconstraintscanbeobtainedbytakingthe observa-tional boundsof



PBH and



HR fromthe CMB, gamma-rayburst

andBBNetc.[43].

5. Conclusionandoutlook

In this paper, we have investigated the formation and evolu-tionofPBHsinthematterbouncecosmology.Firstly,wedescribed

1 66 2 67 3 68 4 69 5 70 6 71 7 72 8 73 9 74 10 75 11 76 12 77 13 78 14 79 15 80 16 81 17 82 18 83 19 84 20 85 21 86 22 87 23 88 24 89 25 90 26 91 27 92 28 93 29 94 30 95 31 96 32 97 33 98 34 99 35 100 36 101 37 102 38 103 39 104 40 105 41 106 42 107 43 108 44 109 45 110 46 111 47 112 48 113 49 114 50 115 51 116 52 117 53 118 54 119 55 120 56 121 57 122 58 123 59 124 60 125 61 126 62 127 63 128 64 129 65 130

Fig. 4. TheconstraintsonρPBH(0)alongwithab/a0.Theblueshadowedregionis givenby(4.14)forPBHswithM>1015 _{g,andtheredshadowedregionisgiven} by(4.16)forM<1015_{g.(Forinterpretationofthereferencestocolorinthisfigure} legend,thereaderisreferredtothewebversionofthisarticle.)

thegeneralmatterbouncemodelsbysomeparameterslikecs,H−

and

ϒ

.Thecomovingcurvatureperturbation

R

kisalsocalculated

duringthemattercontractingphase,whichseeds thePBH forma-tion.ThenwehadadiscussionabouttheconditionofPBH forma-tion in the contracting background, which is different from that in the expanding universe. Bytaking a simple collapsingmodel, thethresholdofthe densityfluctuation forforminga PBHis de-rived.Furthermore,inthecomovinggauge,thedensityfluctuation anditsthresholdareallrelatedtothecurvatureperturbation

R

k.

Therefore,wecancalculatethemassfractionofPBHs

β

inthe con-tractingphase fromthePress–Schechtertheory,andconstrainthe bouncingmodelsfromPBHs.PBHformationdependsonthemodel parameters cs and H−. Forinstance, PBHs can formin the con-tractingphaseonlyifcs

≤

0

.

39,sincethe BHsare notallowed to

belargerthanthecosmicapparent horizon.Whencs

≤

0

.

39,

β

is

smallandthemodelissafefor

|

H₋

|



Mpor

|

H−

|

Mp,butonly

thelowenergyregimeisfavoredfromthemodelconstruction. ThesubsequentevolutionofPBHsinthebouncingphaseisalso investigated,withthe PBH accretionandHawking radiation con-sidered. The growth behavior of PBH yields a constraint to the model

ϒ

≥

c2_s

π

2_H2

−,incasethat themassofPBHsgrowto

infin-itybeforethebouncingpoint.Moreover,thebackreactionofPBH anditsHawkingradiationiscalculatedtoconstrainmodels,in or-der not to neutralize the negative pressure pbg

= −

2M2p

ϒ

, since

insuch casetheuniversecannot expandagain.Theconstraint re-duces to

ϒ

≥

c2_s

π

2_H2

− at the low energy scale H2−



109c5sM2p.

Afterwards,aroughconstraintofbouncingmodelthoughthePBH observationsisgiven,andtheconstraintisstringentonlywhen

ϒ

isslightlylargerthanc2s

π

2H2−.Amorepreciseanalysistakeninto

accountboththePBHgrowthandtheHawkingevaporationaswell asthedetailedconstraintsfromcosmologicalobservationswillbe addressedinourfollow-upworks.

We note that while our paper was being prepared, an inde-pendentworkwasbeingcarriedoutbyanothergroup[59],which exploressimilarfeaturesofBHformationinbouncingcosmology.

Acknowledgements

Weare gratefultoR. Brandenberger,J.Martin,P. Peter, T.Qiu, J. Quintin,D.-G.Wang,Z.Wang,X.-Y.YangandY.Zhangfor valu-able comments. We especially thank the anonymous Referee for verydetailedandconstructivesuggestions.YFCandJWCare sup-ported in part by the Chinese National Youth Thousand Talents Program (Grant No. KJ2030220006), by the USTC start-up fund-ing(GrantNo.KY2030000049)andbytheNationalNaturalScience

FoundationofChina(GrantNos.11421303,11653002).HLXis sup-portedin partby the Fund forFostering Talentsin BasicScience of the National Natural Science Foundation of China (Grant No. J1310021)andinpartbytheYanJiciTalentStudentsProgram.HLX alsothanksJ.MartinandP.Peterfortheirwarmesthospitality dur-ing visiting theInstitut d’Astrophysique de Pariswhen thiswork wasfinalized.Partofnumericalcomputationsareoperatedonthe computerclusterLINDAintheparticlecosmologygroupatUSTC.

References

[1] R.H.Brandenberger,arXiv:1206.4196[astro-ph.CO].

[2] D.Wands,Phys.Rev.D60(1999)023507,arXiv:gr-qc/9809062.

[3] F. Finelli, R. Brandenberger, Phys. Rev.D 65 (2002) 103522, arXiv:hep-th/ 0112249.

[4] R.H.Brandenberger,Int.J.Mod.Phys.Conf.Ser.01(2011)67,arXiv:0902.4731 [hep-th].

[5] M.Novello,S.E.P.Bergliaffa,Phys.Rep.463(2008)127,arXiv:0802.1634 [astro-ph].

[6] J.L.Lehners,Phys.Rep.465(2008)223,arXiv:0806.1245[astro-ph].

[7] Y.F.Cai,Sci.China,Phys.Mech.Astron.57(2014)1414,arXiv:1405.1369 [hep-th].

[8] D.Battefeld,P.Peter,Phys.Rep.571(2015)1,arXiv:1406.2790[astro-ph.CO]. [9] R.Brandenberger,P.Peter,arXiv:1603.05834[hep-th].

[10] A.H.Guth,Phys.Rev.D23(1981)347. [11] A.A.Starobinsky,Phys.Lett.B91(1980)99. [12] K.Sato,Mon.Not.R.Astron.Soc.195(1981)467. [13] L.Z.Fang,Phys.Lett.B95(1980)154.

[14] F.C.Adams,K.Freese,A.H.Guth,Phys.Rev.D43(1991)965.

[15] E.J.Copeland,A.R.Liddle,D.H.Lyth,E.D.Stewart,D.Wands,Phys.Rev.D49 (1994)6410,arXiv:astro-ph/9401011.

[16] A.Borde,A.Vilenkin,Phys.Rev.Lett.72(1994)3305,arXiv:gr-qc/9312022. [17] J.Martin,R.H. Brandenberger,Phys. Rev.D63(2001) 123501,arXiv:hep-th/

0005209.

[18] Y.F.Cai,T.Qiu,Y.S.Piao,M.Li,X.Zhang,JHEP0710(2007)071,arXiv:0704.1090 [gr-qc].

[19] Y.F.Cai,T.Qiu,R.Brandenberger,Y.S.Piao,X.Zhang,JCAP 0803(2008)013, arXiv:0711.2187[hep-th];

Y.F.Cai,X.Zhang,JCAP0906(2009)003,arXiv:0808.2551[astro-ph]. [20] Y.F.Cai,T.T.Qiu,R.Brandenberger,X.M.Zhang,Phys.Rev.D80(2009)023511,

arXiv:0810.4677[hep-th].

[21] R.Brandenberger,Phys.Rev.D80(2009)043516,arXiv:0904.2835[hep-th]. [22] Y.F.Cai,E.N.Saridakis,JCAP0910(2009)020,arXiv:0906.1789[hep-th]. [23] X.Gao,Y.Wang,W.Xue,R.Brandenberger,JCAP1002(2010)020,arXiv:0911.

3196[hep-th].

[24] Y.F.Cai,S.H.Chen,J.B.Dent,S.Dutta,E.N.Saridakis,Class.QuantumGravity28 (2011)215011,arXiv:1104.4349[astro-ph.CO].

[25] J.deHaro,J.Amoros,Phys.Rev.Lett.110(2013)071104,arXiv:1211.5336 [gr-qc].

[26] Y.F. Cai,S. Capozziello,M. DeLaurentis,E.N. Saridakis,Rep.Prog. Phys. 79 (2016)106901,arXiv:1511.07586[gr-qc].

[27] C.Lin,R.H.Brandenberger,L.PerreaultLevasseur,JCAP1104(2011)019,arXiv: 1007.2654[hep-th].

[28] T.Qiu,J.Evslin,Y.F.Cai,M.Li,X.Zhang,JCAP1110(2011)036,arXiv:1108.0593 [hep-th].

[29] D.A.Easson,I.Sawicki,A.Vikman,JCAP1111(2011)021,arXiv:1109.1047 [hep-th].

[30] Y.F.Cai,D.A.Easson,R.Brandenberger,JCAP1208(2012)020,arXiv:1206.2382 [hep-th].

[31] Y.F.Cai,E.McDonough,F.Duplessis,R.H.Brandenberger,JCAP1310(2013)024, arXiv:1305.5259[hep-th].

[32] Y.F.Cai,E.Wilson-Ewing,JCAP1403(2014)026,arXiv:1402.3009[gr-qc]. [33] S.Alexander,C.Bambi,A.Marciano,L.Modesto,Phys.Rev.D90(2014)123510,

arXiv:1402.5880[gr-qc].

[34] S.Alexander,Y.F.Cai,A.Marciano,Phys.Lett.B745(2015)97,arXiv:1406.1456 [gr-qc].

[35] R.H.Brandenberger,AIPConf.Proc.1268(2010)3,arXiv:1003.1745[hep-th]. [36] J.Quintin,Z.Sherkatghanad,Y.F.Cai,R.H.Brandenberger,Phys.Rev.D92(2015)

063532,arXiv:1508.04141[hep-th].

[37] L.Battarra,M.Koehn,J.L.Lehners,B.A.Ovrut,JCAP1407(2014) 007,arXiv: 1404.5067[hep-th].

[38] Y.F.Cai,E.Wilson-Ewing,JCAP1503(2015)006,arXiv:1412.2914[gr-qc]. [39] Y.F.Cai,F.Duplessis,D.A.Easson,D.G.Wang,Phys.Rev.D93(2016)043546,

arXiv:1512.08979[astro-ph.CO].

[40] Y.F. Cai,A.Marciano,D.G. Wang,E.Wilson-Ewing,arXiv:1610.00938 [astro-ph.CO].

JID:PLB AID:32717 /SCO Doctopic: Astrophysics and Cosmology [m5Gv1.3; v1.208; Prn:22/03/2017; 11:43] P.8 (1-8) 8 J.-W. Chen et al. / Physics Letters B•••(••••)•••–•••

1 66 2 67 3 68 4 69 5 70 6 71 7 72 8 73 9 74 10 75 11 76 12 77 13 78 14 79 15 80 16 81 17 82 18 83 19 84 20 85 21 86 22 87 23 88 24 89 25 90 26 91 27 92 28 93 29 94 30 95 31 96 32 97 33 98 34 99 35 100 36 101 37 102 38 103 39 104 40 105 41 106 42 107 43 108 44 109 45 110 46 111 47 112 48 113 49 114 50 115 51 116 52 117 53 118 54 119 55 120 56 121 57 122 58 123 59 124 60 125 61 126 62 127 63 128 64 129 65 130

[41] Y.F. Cai,J. Quintin, E.N.Saridakis, E.Wilson-Ewing,JCAP 1407 (2014) 033, arXiv:1404.4364[astro-ph.CO].

[42] B.J.Carr,Astrophys.J.201(1975)1.

[43] B.J.Carr,K.Kohri,Y.Sendouda,J.Yokoyama,Phys.Rev.D81(2010)104019, arXiv:0912.5297[astro-ph.CO].

[44] B.J.Carr,S.W.Hawking,Mon.Not.R.Astron.Soc.168(1974)399.

[45] D.H.Lyth,K.A.Malik,M.Sasaki,I.Zaballa,JCAP0601(2006)011, arXiv:astro-ph/0510647.

[46] I.Zaballa,A.M.Green,K.A.Malik,M.Sasaki,JCAP0703(2007)010, arXiv:astro-ph/0612379.

[47] M.Kopp,S.Hofmann,J.Weller,Phys.Rev.D83(2011)124025,arXiv:1012.4369 [astro-ph.CO].

[48] T.Harada,C.M.Yoo,K.Kohri,Phys.Rev.D88(2013)084051,Erratum:Phys. Rev.D89(2014)029903,arXiv:1309.4201[astro-ph.CO].

[49] A.S.Josan,A.M.Green,K.A.Malik,Phys.Rev.D79(2009)103520,arXiv:0903. 3184[astro-ph.CO].

[50] B.J.Carr,A.A.Coley,Int.J.Mod.Phys.D20(2011)2733,arXiv:1104.3796 [astro-ph.CO].

[51] B.J.Carr,SpringerProc.Phys.170(2016)23,arXiv:1402.1437[gr-qc]. [52] J.Martin,P.Peter,Phys.Rev.D68(2003)103517,arXiv:hep-th/0307077. [53] D.M.Solomons,P.Dunsby,G.Ellis,Class.Quantum Gravity23(2006)6585,

arXiv:gr-qc/0103087.

[54] J.Quintin,Y.F.Cai,R.H.Brandenberger,Phys.Rev.D90(2014)063507,arXiv: 1406.6049[gr-qc].

[55] J.deHaro,Y.F.Cai,Gen.Relativ.Gravit.47(2015)95,arXiv:1502.03230[gr-qc]. [56] P.Qian,Y.F.Cai,D.A.Easson,Z.K.Guo,arXiv:1607.06578[gr-qc].

[57] Y.F.Cai,W.Xue,R.Brandenberger,X.m.Zhang,JCAP0906(2009)037,arXiv: 0903.4938[hep-th].

[58] C.Gao,X.Chen,Y.G.Shen,V.Faraoni,Phys.Rev.D84(2011)104047,arXiv: 1110.6708[gr-qc].

[59] J.Quintin,R.H.Brandenberger,JCAP1611 (11)(2016)029,arXiv:1609.02556 [astro-ph.CO].

[60] W.H.Press,P.Schechter,Astrophys.J.187(1974)425. [61] J.E.Gunn,J.R.GottIII,Astrophys.J.176(1972)1. [62] P.J.E.Peebles,Astrophys.J.147(1967)859. [63] R.C.Tolman,Proc.Natl.Acad.Sci.USA20(1934)410. [64] H.Bondi,Mon.Not.Astron.Soc.107(1947)343. [65] G.Lemaitre,Ann.Soc.Sci.Brux.IA53(1933)51. [66] D.H.Lyth,Phys.Rev.D31(1985)1792.

[67] A.R.Liddle,D.H.Lyth,CosmologicalInflationandLargeScaleStructure,Univ. Pr.,Cambridge,UK,2000,400 pp.