Bouncing solutions from generalized EoS

0.57 MB·

by  F. Contreras

Bouncing solutions from generalized EoS
F. Contreras,1 N. Cruz,2, and G. Palma2
∗
1Departamento de Matema´ticas, Universidad de Santiago de Chile, Casilla 307, Santiago, Chile
2Departamento de F´ısica, Universidad de Santiago, Casilla 307, Santiago, Chile
Wepresentanexact analytical bouncingsolution for aclosed universefilled with onlyoneexotic
fluid with negative pressure, obeying a Generalized Equations of State (GEoS) of the form P(ρ)=
Aρ+Bρλ, where A, B and λ are constants. In our solution A= −1/3 and λ =1/2 and B <0 is
keptasafreeparameter. Forparticularvaluesoftheinitialconditions, weobtain thatoursolution
obeys Null Energy Condition (NEC), which allows us to reinterpret the matter source as that of a
realscalarfield,φ,withapositivekineticenergyandapotentialV(φ). Wecomputenumericallythe
scalarfieldasafunctionoftimeaswellasitspotentialV(φ),andfindananalyticalfunctionforthe
potential that fits very accurately with the numerical results obtained. The shape of this potential
7
canbewelldescribedbyaGaussian-typeoffunction,andhence,thereisnospontaneoussymmetry
1
0 minimum of V(φ). We further show that the bouncing scenario is structurally stable under small
2 variations of theparameter A, such that a family of bouncing solutions can be find numerically, in
a small vicinity of thevalue A=−1/3.
n
a
PACSnumbers: 98.80.-k,98.80.Jk,04.20.-q
J

2
1 I. INTRODUCTION phantom epoch. For example, at early times and aim-

ing to extend the range of known inflationary behaviors,
]
c Non singular cosmologies such as the described by an Barrow [5] assumed a GEoS with A = 1 and B > 0,
−
q emergentor bouncing universe havebeen studied during which correspondsto the standardEoSof a perfect fluid
r- the last decades as alternative scenarios to the inflation- p = (B 1)ρ when λ = 1. A non singular flat universe
−
g aryparadigm,whichisthemostacceptedonetodescribe was found for the case λ = 1/2 and B < 0, represent-
[ the early universe [1], [2]. Nevertheless, in inflation the ing an emergent cosmological solution. It is interesting
1 problem of the initial singularity still remains [3]. On tomentionthatthedoubledexponentialbehaviorofthis
v the other hand, the scale-invariant spectrum of cosmo- solution was previously found for a bulk viscous source
8 logical perturbations can be obtained in most inflation- in the presence of an effective cosmological constant [6].
3 ary models and one natural question is if in these non This is a consequence of the inclusion of bulk viscosity
4 singular scenarios an scale-invariant spectrum can also in the Eckarts theory, which leads to a viscous pressure
3
.0 beIonbttahienedca.se of bouncing models the universe has fΠoromf tξh=e fξo0rρmλ.−O3tξhHer,ewmheerrgeeξntisflaatsssuomluetdiounssuwaellryeifnoutnhde
1 emergedfromacosmologicalbounce,wherethescalefac- by Mukherjee et al [7] for A> 1 and B >0.
0 −
7 tor takesa non-zerominimum value sothere is no initial TheGEoSrepresentedinEq.(1)canalsobeseenasthe
1 singularity. Bouncing universes have been investigated sum of the standard linear EoS p=Aρ and a polytropic
: in a wide variety of frameworks, which includes among EoS with the polytropic exponent λ =n/(n+1), where
v
i others,higherordertheoriesofgravity,scalar-tensorthe- n is the polytropic index. Non singular inflationary sce-
X ories and braneworlds. See [4] for a detailed discussions narioswereinvestigatedin[8]takenparticularvalues for
r of different approaches to obtain bouncing solutions. A, B, and n>0.
a
Inthispaperouraimwastofindbouncingsolutionsfor
In the study of late time evolution of the universe, it
a universe only filled with one exotic fluid with negative
hasbeenalsoassumedGEoSofthetypegivenbyEq.(1),
pressure,obeyingaGEoS.Awidevarietyofcosmological
motivated by the fact that the constraints from the ob-
modelshavebeeninvestigatedconsideringaGEoSofthe
servational data implies ω 1 for the EoS of the dark
form ≈−
energycomponent,ifitisruledbyabarotropicEoS.Nev-
P(ρ)=Aρ+Bρλ, (1) ertheless, the values ω < 1, corresponding to a phan-
−
tom fluid, or ω > 1, corresponding to quintessence can
−
where A, B and λ are constants. In the framework of not be discarded. Within a phenomenological approach
general relativity the inclusion of Eq.(1) has been used to phantom fluids, a GEoS of the form p = ρ f(ρ),
− −
to describe the behavior of the cosmic fluid components with f(ρ) > 0, was proposed in [9]. To overcome the
at early and late times, as well as the possible present hydrodynamic instability of a fluid with an EoS p=wρ,
with w = const < 0, a general linear EoS of the form
p = A(ρ ρ ) was postulated in [11], being A and ρ
0 0
−
constant and free parameters. This EoS corresponds to
∗e-mail: [email protected] the particular choice λ=0 and B =Aρ0 and was inves-
tigated as a dark fluid filling the universe. A bouncing behavior and how this allow to describe the matter con-
solutionwasobtainedwhen1+A<0andAρ <0. Fora tent by a usual real scalar field with a potential. In sec-
0
Bianchi-Icosmology,theinclusionofaperfectfluidobey- tion III we evaluate numerically the scalar field and its
ing a GEoS with λ = 2 leads to a great suppression on potential associated to our exact solution. We also find
the anisotropies in the contracting phase of a bouncing an analytical expression for this potential that fits the
cosmology [12]. numericalresultsfound. InsectionIV weinvestigatethe
The case with A= 1 and λ=1/2 was considered in stabilityofthebouncingsolutionwhentheGEoSismod-
−
[9] and [10]. In both works the cosmologicalsolutions of ifieddisturbingtheparameterAbyansmallquantity. So
dark energy models with this fluid was analyzed, focus- inthiscase,wemakeastudyofstructuralstabilityunder
ing in the future expansion of the universe. A late time small variations of the parameter A. Finally, in section
behavior of a universe filled with a dark energy compo- Vwediscusssomefeaturesandtheirfurtherpossibleap-
nent with an EoS given by Eq.(1) has been investigated plications to suitable bouncing models.
in [13], [14],wheretheallowedvaluesofthe parameters
A and γ were constrainedusing H(z)-z data, a modelin-
dependent BAO peak parameter and cosmic parameter II. EXACT BOUNCING SOLUTION FROM
(WMAP7 data). GEOS
Also theoretical studies like the so called running vac-
uumenergyinQFT(see [15])givesrisetoacosmological In what follows we will discuss an analytical solution for
constant with a dynamical evolution during the cosmic a closed universe found in [16], for the case in which the
time, which allows to conclude that GEoS of the type parameterAtakesthevalue 1/3andλ=1/2inEq.(1).
−
of Eq.(1) could also effectively represent these scenarios As we discuss bellow the ω = p/ρ parameter can repre-
under some specific assumptions. sent phantom and quintessence fluids, depending on the
In this work we use a rather conservative setup in- initial conditions. This exact solution describe a bounc-
troducing a positive curvature and the particular values ing universe, assuming that one fluids with a GEoS is
A= 1/3andλ=1/2,lettingB <0asafreeparameter present in the early universe.
−
of the model. With this election the strong energy con- For a universe with positive curvature (k = 1), the
dition is violated, which is a condition to have bouncing equation of constraint of the Friedmann equations is
solutions, but NEC holds, and thus our particular GEoS given by
hasaparameterω thatevolveswiththecosmictime,but
lies in the range of quintessence fluids, for some choice a˙ 2 3
ρ=3 + , (2)
of the initial conditions, except for t , where the a a2
→ ±∞ (cid:18) (cid:19)
fluid behaves like a cosmlogical constant. These partic-
ular values of A and λ allow to find an exact analytical and the equation of continuity by
bouncing solution for the scale factor.
Reinterpreting the matter source in terms of a real ρ˙+3H(ρ+p)=0. (3)
scalar field, we can compute numerically the scalar field
2
anditspotential. Wealsofoundananalyticalexpression Using the change of variable s = the GEoS of
for this potential that fit very accurately the numerical −B√3
solutions, with a coefficient of determination (r2) equal Eq. (1) can be rewritten as
to r2 =0.99999.
1 2
We also study the robustness of the bouncing solution p(ρ)= ρ ρ1/2. (4)
−3 − s√3
when the GEoS is modified by including a perturbative
term in the standard linear coefficient A. We find that
Solvingtheaboveequationsonefindsthefollowingso-
under reasonableconstraintsonthe perturbativeparam-
lution
eter, the solution is analytic in ǫ and the first order cor-
rection allows to extend the behavior of the bouncing t t
0
a(t)=s cosh − c , (5)
solution beyond the value 1/3, for which an explicit s −
analyticsolutionwasfound.−Theperturbativeexpansion (cid:20) (cid:18) (cid:19) (cid:21)
leadsaswelltoconcludethatthepropertiesofthescalar where t and c are integration constants. The bouncing
0
potential (shape and minimum) proposed as source for solutionisobtainedwhens>0. Thissolutionrepresents
the effective equation of state are stable, provided ǫ re- a universe expanding exponentially for t ( , ).
∈ −∞ ∞
mains small enough. The scale factor takes a minimum value a(t = t ) =
0
This paper is organized as follows. In section II we s(1 c). The positivity of the scale factor constraints c
−
presenttheparticularconsideredGEoSandshowthean- to be in the following range ( ,1). Before to express
−∞
alytical bouncing solution found and their main proper- c in terms of the initial energy density we evaluate H,
ties. Inparticular,wepresenttheevolutionoftheparam- H˙ and H¨ using the Eq.(5). Their expressions are the
eter ω with the cosmic time, discussingits quintessential following:
and in our case the Eq.(6) with c = 0 takes the similar
form:
sinh(u)
H(u)= , (6)
s(cosh(u) c)
1 t t
− 0
H(t)= tanh − . (14)
1 ccosh(u)
H˙(u)= [s(c−osh(u) c)]2, (7) s (cid:18) s (cid:19)
−
r 1 Inthe next subsectionwe giveaninterpretationofthe
sinh(u) 2 c2 +sinh(2u) +
− 3 s used GEoS in terms of well known fluids.
H¨(u)= (cid:18) (cid:19), (8)
− (cid:0) [s(co(cid:1)sh(u) c)]3
−
where
A. Fluid sources of the bouncing solution
t t
0
u= − . (9)
s We can obtain the energy density as a function of the
scale factor if we replace the Eq.(5) in the Eq.(10)
For c < 0 the Hubble parameter is a strictly increas-
ing function, so there are no critical points and we have
2
H(t ) = 1 and H(t ) = 1, then for late 1 c
→ −∞ −s → ∞ s ρ(a)=3 + . (15)
times this solution behaves like a de Sitter universe. s a
(cid:18) (cid:19)
It is straightforwardto evaluate the energy density as
a function of the cosmic time using Eq.(2) and the ex-
IntroducingEq.(15)inEq.(4)weobtainthefluidpres-
pression for a(t) and H(t) given by Eq.(5) and Eq.(6),
sure as a function of the scale factor
respectively. The expression for the energy density is
then given by
2
1 c 2 1 c
p(a)= + + . (16)
3cosh2(u) − s a − s s a
ρ= . (10) (cid:18) (cid:19) (cid:18) (cid:19)
[s(cosh(u) c)]2
−
Expanding the terms of the above both expressions
Using the initial conditions a(t ) = a in (5) and
0 0 yields:
ρ(t )=ρ in (10) we obtain that
0 0
a =s(1 c), 3=a2ρ . (11) 3 6c 3c2
0 − 0 0 ρ(a)= s2 + sa + a2 =ρ1+ρ2+ρ3, (17)
One dimensional restoration lead to the Eq.(11) takes 3 4c c2
the following form: p(a)= =p +p +p . (18)
−s2 − sa − a2 1 2 3
v2 v2 8πG Comparing each terms of the expansions our fluid
a =s(1 c) , = ρ a2, (12)
0 − R √8πG R2 3 0 0 can be seen as the sum of three fluids with the EoS
0 0
given by ω = p /ρ = 1, ω = p /ρ = 2/3 and
1 1 1 2 2 2
where v is the speed of light, R0 is the radius of curva- ω3 = p3/ρ3 = 1/3, respe−ctively. So the first−fluid cor-
ture, G is the gravitational contant and ρ is the energy responds to a −cosmological constant, the second one is
density. Because the model considers the universe with
a quintessence and the last corresponds to a fluid which
curvature positive, the radius the curvature R0 also is a drivesanexpandinguniversewithzeroacceleration. No-
free parameter.
tice that in the above descomposition the EoS of each
A very special situation occurs for c = 0 or equiva- fluid is constant. Therefore, each ω is independent of
i
lently ρ = 3/s2 and a =s, because the energy density
0 0 the parameters s, t and c.
0
preserves this constant value during all the cosmic evo-
Lets us evalue the EoS, ω = p/ρ, for this fluid, which
lution. It means that for a closed universe with the EoS
in terms of the cosmic time takes the expression
that we are considering, the universe expand with accel-
eration but the energy density remains constant, like in
2c
de Sitter solution for a closed universe. Note that the ω = 1+ . (19)
Hubble parameter for the de Sitter solution is given by − 3cosh t−st0
(cid:0) (cid:1)
λ λ
H(t)= tanh (t t ) , (13) The lowerplotin Fig. 1. depicted the behaviorofthis
0
r3 r3 − ! parameter ω.
where u is given by the Eq.(9). Integrating the Eq.(22),
-0.6
the function φ is obtained as:
c = 0.5
-0.7
c = -0.5
-0.8
iu c iu 2
-0.9 φ(u)= i√8c F ,√2 + Π , ,√2 ,
− 2 1 c 2 1 c
ω -1 (cid:20) (cid:18) (cid:19) − (cid:18) − (2(cid:19)4(cid:21))
-1.1 where i is the imaginary unit, F is the elliptic integral
-1.2 ofthe first kind andΠ is the elliptic integralof the third
-1.3 kind. Both are defined in [20].
The function dφ in Eq.(22) is a continuous function.
-1.4-10 -8 -6 -4 -2 0t 2 4 6 8 10 Therefore, this fduunction have a real primitive function,
but this can’t be represented by elementary functions.
FIG. 1: Plot of the parameter ω given by Eq.(19), for the Thus the imaginary value in φ(u) in the Eq.(24) is only
parameter values c=0.5, s=1 and t0 =0. a artifice of the representation of the function.
In order to numerically obtain φ(u) from the above
equations,wehaveusedstandardintegrationsubroutines
from Matlab, imposing for consistence the initial condi-
Note that the EoS becomes like a cosmological con-
tionφ(0)=0. TheresultisdisplayedinFig. 2,wherefor
stant ω 1 for t . For the case with c = 0.5
→ − → ±∞ comparisonwehavealsoplotteditsMaclaurinexpansion
the fluid ruled by the EoS given in Eq.(19) behaves like
up to order 14th, whose coefficients are obtained in Ap-
quintessenceforthelapseassociatedatthetimeofbounc-
pendix A. In addition, in this appendix the convergence
ing. For the case c= 0.5 the EoS behaves like a phan-
− radiusofthis seriesshowntobe arccos(c). Aremarkable
tom fluid within the period associated to the bouncing.
agreement among both results is found, within the com-
Wewillfocusontheparticularinterval1>c>0because
monrangeofu,whichrepresentsaseveretestofaccuracy
in this casethe GEoSleads to a quintessence-type ofbe-
to the numerical solution.
havior. With c within this range, the matter content of
the universe can be described by a real scalar field with
a lagrangianminimally coupled to gravity given by 4
3
1 Maclaurin up to order 14
= φ,µφ,µ V(φ). (20) 2 Numerical data
L 2 −
1
In order to reinterpret the matter source as that of a φ 0
scalar field, we will evaluate the scalar field φ and the
-1
potential V(φ) in the next section.
-2
-3
-4
III. COMPUTATION OF THE SCALAR FIELD -50 -40 -30 -20 -10 0 10 20 30 40 50
u
AND ITS POTENTIAL
FIG.2: Plotofφ(u)usingtheMaclaurinseriesobtainedfrom
Ifweconsiderthemattercontentoftheuniversemodeled
Eq.(A14)anditsnumericalsolution fortheparametervalues
by a perfect fluid, then the density and pressure in term
c=0.5 and s=1.
of the scalar field are given by
Moreover, we have compared the results for φ(u) ob-
1 1
ρ= φ˙2+V(φ), p= φ˙2 V(φ). tained by the Maclaurin series with the one obtained by
2 2 − (21)
thenumericalintegrationbycomputingthePearson’sco-
efficient r =0.999985,which allows to use the numerical
Using the Eq.(4) and the Eq.(10) in the Eq.(21) we solution beyond the convergence radius of the series ex-
obtain: pansion.
WehavealsocomputednumericallyV(φ)fromEq.(23)
as well as its Maclaurin expansion using the expression
dφ 2ccosh(u) deduced in Appendix A (see the Eq.(A21)). Similarly
= , (22)
du ± cosh(u) c to the analysis explained above, we have measured the
p − degree of agreementamong both methods by computing
cosh(u)(2cosh(u) c)
V(u)= − , (23) the Pearson’s coefficient, which in this case is 1 (r =1).
[s(cosh(u) c)]2 Both results are displayed in Fig. 3.
−
0.99981, for the fit parameters θ 3.5892 and θ
1 2
10 ≈ ≈
0.42631.
Maclaurin up to order 14
9 Wehavealsofittedthenumericaldataforthefieldφ(u)
Numerical data
8 to the function θ1arctan(θ2u), where θ1 and θ2 are the
fit parameters, but the quality of the fit was not quite
7
comparable to the one obtained by using the analytic
V
6 form given by Eq.(25).
5 Because of the shape of the scalar potential V(φ), we
have used a Gaussian as a trial function :
4
3 V(φ)=σ1 exp σ2 φ2 +σ3, (26)
-4 -3 -2 -1 0 1 2 3 4 · − ·
φ
whereσ ,σ andσ arefitpa(cid:0)rameters(cid:1). Theresultofthis
1 2 3
fit is displayed in Fig. 5.
FIG. 3: Plot of V(φ) using the Maclaurin series obtained
in the Appendix A Eq.(A21), as well as by the numerical
integration from Eq.(23), for theparticular parameter values
c=0.5 and s=1. 10 Gaussian
9 Numerical data
8
In the next subsecction we will find analytical expres-
7
sions for the field φ and its potencial V by performing V
high accuracy fits. 6
5
4
A. Analytical representation of φ(u) and V(φ)
3
-4 -3 -2 -1 0 1 2 3 4
φ
In order to characterize analytically the shape of the
scalarpotential, andeventuallytocompareit withother FIG.5: PlotofthenumericaldataforV(φ)andtheGaussian
quintaessence potentials, we performa fit of the numeri- fit definedby Eq.(26), for thevalues c=0.5 and s=1.
cal data for φ(u) using the function tanh(x),
AsitisshowninFig. 5,theGaussianfunctionfitsvery
φ(u)=θ tanh(θ u), (25)
1 2 accuratelythenumericaldata,moreoverthecoefficientof
· ·
determination r2 = 0.99965 for the fit parameters σ
1
whereθ andθ areparemeterspositivesoffit. Thefit ≈
1 2 7.2164, σ 0.32953and σ 2.896.
2 3
of field φ can be observed in the Fig. 4. ≈ ≈
DuetotheboundeddomainofthepotentialV(φ), one
has to modify the Gaussian such that it falls off to zero
as the field φ approaches the limits φ . We have
4 ± max
fulfilled this constraint by introducing the modified trial
3 θ1· tanh(θ2· u) function:
2 Numerical data
1
σ φ2
φ 0 V(φ)=(V V )exp σ φ2 2 +V
max− min − 1 − φ2 φ2 min
-1 (cid:20) max− (cid:21)
(27)
-2 where σ and σ are the fit parameters. φ turns out
1 2 max
-3 to be φ 3.6009. V and V are the maximum
max max min
≈
-4 and minimum values of the potential V(φ). They can
-50 -40 -30 -20 -10 0 10 20 30 40 50
u explicitlybe obtainedasfollows: insertingEqs. (15)and
(16) into Eq.(21) one obtains:
FIG. 4: Plot of φ(u) obtained from its numerical solution as
well as from the fit given by Eq.(25) for the values c = 0.5
2c 2c2 3 5c 2c2
and s=1. φ˙(a)= + , V(a)= + + . (28)
sa a2 s2 sa a2
Sincetheconstantscandsarepositive,themaximum
The fit represented by Eq.(25) is of high quality as (minimum)ofV is obtainedwhenaisaminimum(max-
the corresponding coefficient of determination is r2 = imum). But from Eq.(5) the maximum and minimum
values of a(t) are and s(1 c) respectively. Inserting
∞ − 4
these values into Eq.(28) we obtain:
3 Numerical solution
2 Exact solution
1
3 c 3
Vmax = s2(1− c)2, Vmin = s2. (29) φ 0
− -1
-2
TheresultofthefitofV(φ) usingthefunctiondefined -3
in Eq.(27) is shown in Fig. 6. -4
-50 -40 -30 -20 -10 0 10 20 30 40 50
t
FIG. 7: Plot of φ as a function of time obtained numeri-
10
Modified Gaussian cally from Eq.(28) using the fitted expression for V given by
9 Numerical data Eq.(27), as well as from the exact solution of Eq.(5), for the
8 parameter values c=0.5 and s=1.
7
V
6
5
30
4 Numerical solution
25 Exact solution
3
-4 -3 -2 -1 0 1 2 3 4
φ 20
a15
FIG. 6: Plot of V(φ) obtained from the numerical solution
and by the fit defined by Eq.(27), for the values c=0.5 and 10
s=1.
5
0
-4 -3 -2 -1 0 1 2 3 4
The modified Gaussian function fits remarkable well
t
the numerical data for V(φ) as the coefficient of deter-
mination is r2 = 0.99999 for the values σ1 = 0.29243 FIG. 8: Plot of a(t) obtained of Eq.(28) used V given by
and σ =0.32674. Consideringthe constrainton the do- Eq.(27), as well as from the exact solution of Eq.(5), for the
2
mainofφandthehigheraccuracyofthismodifiedGaus- parameter values c=0.5 and s=1.
sian function, we conclude that the expression given by
Eq.(27) is the faithfuliest representation of V(φ).
IV. STRUCTURAL STABILITY OF THE GEOS
AND PREVALENCE OF THE BOUNCING
SOLUTION
B. Analysis of the modified Gaussian potential
Since and exact bouncing solution is obtained for the
very particular value A = 1/3, it is an open question
−
As a consistency check it is possible to start with the whethervariationsoftheparameterAleadstostructural
expression of Eq.(27) for the potential V(φ) and solve stabilityofthefieldequations,orinotherwordswhether
numericallythe setofequations(28). By usingstandard the system still have bouncing solutions when the GEoS
integrationsubroutines this numericalstrategy allows to is modified by disturbing the parameter A by A A+
→
obtainthe functions φ(t) and a(t), whose results are dis- ǫ in the Eq.(1). We will explore this issue taking into
played in Figs. 7 and 8, where for comparison, we have account a GEoS of the form
included the exact solutions given by Eqs. (24) and (5)
respectively. 1 2
p= +ǫ ρ ρ1/2, (30)
Both figures show a remarkable agreement among the (cid:18)−3 (cid:19) − s√3
numerical solutions and the exact expressions, which
is quantified by the coefficients of determination r2 = whereǫ 1/3. We study firstperturbationsonthe ex-
0.99999forthe scalarfieldφ(t), andr2 =0.99962forthe act solut≪ion| fou|nd, driven by the initial GEoS of Eq.(4).
scale factor a(t). We shall consider the following perturbations on the pa-
rameters a, ρ and P we can obtain a system to first order in ǫ, which allows
to find a and ρ . The system is:
(1) (1)
a(u)=a (u)+ǫa (u)+O(ǫ2),
(0) (1)
ρ(u)=ρ (u)+ǫρ (u)+O(ǫ2), (31)
(0) (1) a (u)=A(u)a (u)+B(u)ρ (u),
′1 1 1
p(u)=p (u)+ǫp (u)+O(ǫ2),
(0) (1) ρ (u)=C(u)a (u)+D(u)a (u)+E(u)ρ (u)+F(u),
′1 ′1 1 1
(34)
where for the case ǫ = 0 the solutions are given by the
where the “prime” ( ′ ) denotes the derivative with re-
Eq.(5) in the following form
spect to u, and the cofficients A(u), B(u), C(u), D(u),
E(u), F(u) are obtain in the Appendix B (see the
a (u)=s(cosh(u) c), Eqs.(B4) and (B7)).
(0)
−
3(cosh(u))2 Solving the system using the given initial condition,
ρ (u)= ,
(0) s2(cosh(u) c)2 (32)
−
1 2
p (u)= ρ ρ1/2.
(0) −3 (0)− s√3 (0)
a(u=0)=a (0), a˙(u=0)=a˙ (0), (35)
(0) (0)
Then using Eq.(2) and the continuity equation
we obtain that the first order contribution for the scale
a˙
ρ˙ = 3 (ρ+P), (33) factor, the energy density and the pressure are given by
− a
3s 2carctan(N)
a = 1+cosh(u)+cln[M] sinh(u) u+ ,
(1) 2 − − √1 c2
(cid:26) (cid:20) − (cid:21)(cid:27)
9cosh(u) 2c2sinh(u)arctan(N)
ρ(1) = s2(cosh(u) c)3 cosh(u)−cosh2(u)+cusinh(u)+ √1 c2 −ccosh(u)ln[M] , (36)
− (cid:26) − (cid:27)
ρ ρ cosh(u) c (1+c)tanh(u/2)
(1) (1)
p =ρ , where M = − and N = .
(1) (0)− 3 − s√3ρ1/2 1 c √1 c2
(0) − −
Now,weanalyzethefunctionsa andρ ofaboveequa- than original. Finally, if we capare magnitudes of the
1 1
tion. Let us consider the first two nonzero terms of the order u2 for a and a , and if in addition we include the
0 1
MaclaurinexpansionofEq.(32),whichrepresentsa ,and term ǫ in a , we obtain that
0 1
the first nonzero nonzero of the Maclaurin expansion of
the expression for a , which is given by Eq.(36). We
1 s 3s 3ǫ
obtain that > ǫ c<1 | |. (38)
2 | | −4(1 c) ⇔ − 2
su2 3su2 Therefo(cid:12)(cid:12)(cid:12)re,(cid:12)(cid:12)(cid:12)we wi(cid:12)(cid:12)(cid:12)(cid:12)ll have−a bo(cid:12)(cid:12)(cid:12)(cid:12)uncing behavior for u≈0
a (u)=s(1 c)+ , a (u)= . (37) foranycoupleofcostantscandǫthatsatisfytheEq.(38).
0 1
− 2 −4(1 c)
− Note that for any c between (0,1)alwaysexist one ǫ>0
and ǫ 1 which satisfy the above equation.
Note that a1(0) = 0 and that the sign of a1 is always No|w|,≪let3’s analyze ρ of Eq.(36). Evaluating the first
negative contrary to a that is positive. For this reason 1
0 nonzerotermsofMaclaurinseriesofρ andρ ,weobtain
for ǫ > 0 the scale factor of the bouncing universe in 0 1
a neighborhood of u = 0 grows lesser than the original
solutionandforǫ<0the scalefactorgrowsfastherthan 3 3cu2 9(1 2c)u2
ρ = , ρ = − . (39)
theoriginalone. ThisbehaviorcanbeseenfromEq.(30), 0 s2(1 c)2−s2(1 c)3 1 −2s2(1 c)4
since for ǫ > 0 the GEoS is the same that the original − − −
GEoSplusthetermǫρ. Therefore,theexpectedbehavior Note that a (0) = ρ (0) = 0 and also that the sign
1 1
of the new scale factor should be less pronunced that of ρ is negative for c < 0.5 and positive for c > 0.5.
1
the scale factor of the original solution, because the new In the case c = 0.5 the first ρ = 0. Thus, for any
1
quintessence fluid have one state parameter ω greater ǫ 1, ρ will dominate over ρ for u 0. Now, if we
| | ≪ 3 0 1 ≈
compare magnitudes of the order u2 for ρ and ρ , and orderarequitesimilartothe obtainedwiththe exactso-
0 1
if inadditionwe include the termǫin ρ , weobtainthat lution. Moreover,ifwe changethe parametersc ands in
1
for c=0.5 theirrespectivedomainwewillgetuniverseskeepingthe
6
same shape of the bouncing but with different growths.
We evaluate numerically the first order perturbation
3c 9(1 2c) 2c(1 c)
> ǫ − − > ǫ. (40) of the field φ, which we denoted by φ1(u). Its behavior
s2(1 c)3 | | 2s2(1 c)4 ⇔ 31 2c | |
(cid:12) − (cid:12) (cid:12) − (cid:12) | − | is showed in the Fig. 11.
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12)This equat(cid:12)ion te(cid:12)ll us what v(cid:12)alues of c and ǫ lead ρ
(cid:12) (cid:12) (cid:12) (cid:12) 0
dominates over ρ1 for u 0. 5
≈
Thebehavioroforiginalscalefactorandtheperturbed φ
(0)
ones are shown in Fig. 9. The ǫ that we consider in the φ +ǫ φ
(0) (1)
graphics corresponds to a 45% of 1/3 and c=0.5. With
φ -ǫ φ
(0) (1)
these c and ǫ we obtain that the Eq.(38) is satisfied and
therefore, there is a bouncing behavior. φ 0
18
16 a(0)
14 a +ǫ a -5
(0) (1) -50 -40 -30 -20 -10 0 10 20 30 40 50
u
12 a -ǫ a
(0) (1)
10
a FIG.11: Plotofthescalarfieldφanditsfirstordercorrection,
8
for the parameter values ǫ=0.15, c=0.5 and s=1.
6
4
Likethe scalefactorandthe energydensity,we obtain
2
that the first order perturbed field φ preserve the
0 1
-3 -2 -1 u0 1 2 3 shape of unperturbed solution φ0. The coefficient of
determination is r2 = 0.99999, for φ and φ +ǫφ , and
0 0 1
r2 =0.99998 for ρ and ρ ǫρ .
FIG. 9: Plot of thescale factor and its first order correction, 0 0 1
−
obtained from Eq.(36), for the values ǫ = 0.15, c = 0.5 and
s=1. To obtain the potential V1 we exapand the function
V(φ) in ǫ. From this expansion we obtain the following
Thecoefficientofdeterminationisr2 =0.94759,fora expression
0
anda +ǫa , andr2 =0.99875,fora anda ǫa . The
0 1 0 0 1
−
energy density ρ given by Eq.(36) is shown in Fig. 10.
1 φ
V(φ)=V (φ )+ǫ V (u) (1) +V (φ ) +O(ǫ2).
(0) 0 " (′0) · φ′(0) (1) (0) #
12 (41)
ρ
(0) The potential V(φ) is plotted in the Fig. 12.
10
ρ +ǫ ρ
(0) (1)
8 ρ -ǫ ρ
(0) (1)
11
ρ 6 10 V(0)
9 V +ǫ V
4 (0) (1)
8 V -ǫ V
(0) (1)
2 φ) 7
(
V 6
0
-8 -6 -4 -2 u0 2 4 6 8 5
4
FIG.10: Plotofenergydensityρanditsfirstordercorrection 3
obtained from Eq.(36), for the parameter values ǫ = 0.15, 2
-4 -3 -2 -1 0 1 2 3 4
c=0.5 and s=1. φ
(0)
Thecoefficientofdeterminationisr2 =0.99980,forρ FIG.12: PlotofthepotentialV(φ)anditsfirstordercorrec-
0
and ρ +ǫρ and r2 = 0.98566, for ρ and ρ ǫρ . We tion, for the parameter values ǫ=0.01, c=0.5 and s=1.
0 1 0 0 1
−
cannote thatdespite ofthe high value ofǫ,the behavior
ofscalefactorandoftheenergydensityperturbedtofirst We obtain a r2 = 0.99975 for V and V +ǫV and a
0 0 1
r2 =0.99975,for V and V ǫV . der of one up to 10 4 and 10 5 respectively (see Eqs.
0 0 1 − −
−
(25) and (27). The shape of this potential can be very
precisely described by a Gaussian-type of function that
The perturbation of the GEoS in the parameter A has a bounded domain given by the condition φmin <
around the vaue 1/3 leads also to bouncing solutions φ < φmax. With this exact analytical expression for the
−
whosebehaviorinthescalefactorandintheenergyden- scalar potential we evaluated numerically the scale fac-
sity are quite similar than the exact analytical solution tor, founding a curve very close to the exact analytical
found in this work. This also applied for the scenario in bouncing solution already found. This represents a very
which a scalar field describe the matter content of the astringent test on the accuracy of the numerical method
universe. used to compute the scalar potential V(φ).
Wehavealsostudiedthestructuralstabilityofthean-
alyticalbouncingsolutionwhentheGEoSismodifiedby
including a perturbative term in the standard linear co-
V. CONCLUSIONS
efficientA(seeEq.(30)). Forsufficientlysmallǫ,the sce-
nario predicted by the analytic solution is still preserved
The study of GEoS has been very important in the
asthescalefactoraandthe density ρbehavequitesimi-
exploration of new scenarios in the very early phases of
lartotheunperturbedsolutions(seeEqs. (38)and(40)).
the universe like inflation, or bouncing universe theories
The shape of the scalar potential -introduced as a possi-
in which there are no initial singularities.
ble source to generate the effective GeoS- also confirms
Inthisworkwehavefoundandexactanalyticalbounc-
the unperturbed scenario: the absence of a spontaneous
ing solution for a closed universe filled with one fluid
symmetry breaking minimum, for ǫ small enough within
which obeys a GEoS of the form p = 1/3ρ+Bρ1/2,
− the validity range of the first order approximation.
where B < 0 is a free parameter. We have chosen the
In summary, the exact analytical bouncing solution
initial conditions that allow no violation of NEC, which
can be extended to a vicinity of A = 1/3, confirm-
leads the parameter ω to evolve with the cosmic time −
ing a bouncing scenario beyond the particular value re-
in the domain of quintessence. For t , ω 1
→ ±∞ → − quired for the exact solution. Moreover, an analytical
the fluid behaves like a cosmological constant. We have
quintessence potential has been found by using a high
also shown that the well known de Sitter solution with
accuracy fit to the numerical data. A scalar field theory
positive curvature is obtained as a particular case of our
minimally coupled to gravity and ruled by this potential
exact analytical bouncing solution.
leads to bouncing solutions for closed universes, which
Anotherinterestingfeatureofourresultisthepossibi-
does not present spontaneous symmetry breaking.
lity to interpret the fluid ruled by the GEoS of Eq.(1) in
The analytic scalar potential found in this article can
termsofknownfluids. Infact,expandingtheexpressions
further be used to study other interesting issues associ-
for the pressure, p, and the energy density, ρ, in terms
ated, like the consequences of considering perturbations
of the scale factor and inserting them into the GEoS, we
inthebackgroundmetricinthetrivialminimumandab-
obtainedthatthe mattercontentcanbe seenasthe sum
sence of spontaneous symmetry breaking .
of the contributions coming from three fluids: a cosmo-
logical constant, quintessence (ω = 2/3) and the cor-
−
responding fluid which arises from the particular value
ω = 1/3. Acknowledgements
−
Since the investigated fluid behaves effectively like
quintessence, it is possible to reinterpret the matter This work was supported by CONICYT through Grant
source in terms of an ordinary scalar field φ, minimally FONDECYTN01140238(NC).GPacknowledgespartial
coupledtogravitywith apositive kinetic termanda po- funding by Dicyt-Usach Grant N0 041531PA.
tential V(φ).
WehavesolvedforφandV(φ)thesetofcoupledequa-
tions(21)byusingMaclaurinexpansionsupto14thorder
and, in paralell as an accuracy test, we have computed Appendix A: Maclaurin coefficients of φ and V(φ)
themnumericallyfromthe implicitEqs.(22)and(23). A
remarkableagreementwasfoundbycomparingthescalar We will find the coefficients of Maclaurin series of the
fieldφandthepotentialV(φ)obtainedbybothmethods, functions φ(u) and V(φ) and we will compute their con-
whose precision is characterized by coefficients of deter- vergence radius. These formal expansions up to order
minationr2 =0.99997andr2 =1respectively,(seeFigs. 14th are required in Figs. 2 and 3. We will firstly find
(2) and (3)), which holds in the common range. the Maclaurinseriesofφ(u), andthenwe willinvertthis
Performing a high accuracy fit we have found an an- seriestoobtaintheexpansionforu(φ). Theseexpressions
alytical expression for the field and its scalar potential will allow us to obtain the Maclaurin series of V(φ). In
with coefficients of determination typically of the or- fact, starting with the definitions of the coefficients α ,
n
βn, and cn through the formal expansions ij−=11(k−j+1) .
Using Lheibniz’s rule for thie derivative of a product
Q
φ(u)= ∞n=0αnun, V(u)= ∞n=0cnun,
(A1)
V(φ)=P∞n=0βnφn. P (AB)(k)(y)= k k A(k i)(y)B(i)(y), (A6)
−
i
we will procePedfinding the βn coefficients in three steps: Xi=0(cid:18) (cid:19)
• First, we find the coefficients αn of the Maclaurin swioitnhsA(y) = √y and B(y) = (y−c)−1, and the expres-
seriesofφ(u), usingits exactanalyticalexpression.
Second,wefindthecoefficientsc oftheMaclaurin
• series of V(u) by using Eq.(23).n A(n)(y)= (−1)n+21n(2n−3)!!y21−n, (A7)
B(n)(y)=n!( 1)n(y c) n 1.
third, we find the coefficients βn of the Maclaurin − − − −
•
series of V(φ) solving the implicit expression:
we obtain
V(φ)= ∞ cnun(φ) (A2) F(k)(cosh(0))= k k! (2k−2i−3)!! (−1)k+1 .
(k i)! · 2k i (1 c)i+1
nX=0 i=0 − − −
X
(A8)
where u(φ) should be computed by inverting the expan- InordertoobtainthefunctionsU (0)weusetheiden-
k
sion for φ(u) (see Eq. (A1)). We begin by finding αn. tities
TofindtheMaclaurinseriesofφ(u)weintegrateEq.(22),
which gives
cosh2k(u)=2−2k ik=−012 2ik cosh(2(k−i)u)+ 2kk ,
φ(u)=±√2cZ cposcho(ush)(−u)cdu+s1, (A3) cosh2k−1(u)=22−h2PkPik=−01(cid:0)(cid:0)2(cid:1)ki−1(cid:1)cosh((2k−2i−(cid:0)1()(cid:1)Aui)9.)
From which we obtain for l 1 and n 1
with s1 being an integration constant. Now we need the ≥ ≥
√cosh(u)
Maclaurinseriesoftheintegrand ,whichcanbe
cosh(u) c (2n 1)
derived by using the identity − coshl(u) − (0)=0,
(cid:16) (cid:17)
(2n)
f(n)(u)= n Uk(u)F(k)(y), n 1, (A4) cosh2l(u) (0)=2−2l il−=102 2il (2(l−i))2n,
k! ≥ (cid:16) (cid:17) (2n) P (cid:0) (cid:1)
kX=1 cosh2l−1(u) (0)=22−2l il−=10 2l−i1 (2l−2i−1)2n.
where: (cid:16) (cid:17) (A10)
U (u)= k (−1)i+1yi 1(u) yk+1 i(u) (n)(u) Moreover, using Eq.(A8), EPq.(A1(cid:0)0) an(cid:1)d Eq.(A5), we
k i=1 (i 1)! − − × find
PΠij−=11(k+−1−j) , andh(cid:0)f(u)=(F (cid:1)◦y)(u).i
(cid:2) (cid:3) f(m)(0) =f(2n)(0)= 2n ( 1)k+1U (0)δ
Note that each Uk also depends of n. If we consider k=1 − k k
wFh(yic)h=is py√−ryecciasneldyψth(eu)in=tegcroasnhd(uo)f,Ewqe.(oAb3t)a.in that f(u) = nk=1U2k−P1(0)δ2k−1− nk=1U2k(0)δ2k,
P P
(A11)
uaNtinowg,ureaetmup=lac0i,nwgeyo=btcaoinsh(u) into Eq.(A4) and eval- twhheeerexpreδsksi=on ki=0 ((2kk−−i2)!i2−k3−)i!!(1−c)−i−1,whichleadsto
P
n
U (0) n
f(n)(0)= k F(k)(cosh(0)), (A5) f(2n)(0)= (U (0)δ U (0)δ ), (A12)
k! 2k 1 2k 1 2k 2k
k=1 − − −
X k=1
X
where:
where n 1, and
Uk(0)=Pki=1 (−(i−1)1i+)!1 (cid:20)(cid:16)coshk+1−i(u)(cid:17)(n)(0)(cid:21)× U2k(0)=≥Pki=1n(2ηi−i,k2)!hQj2=i−12(2k−j+1)i