Emergent Universe from A Composition of Matter, Exotic Matter and Dark Energy

0.18 MB·English

by  B. C. Paul

Mon.Not.R.Astron.Soc.000,1–??(2008) Printed10January2011 (MNLATEXstylefilev2.2)
Emergent Universe from A Composition of Matter, Exotic
Matter and Dark Energy
1 B. C. Paul,1,3 ⋆ S. Ghose,1,3 † P. Thakur2, 3 ‡
1
1Physics Department, North Bengal University
0
Dist. : Darjeeling, Pin : 734 013, West Bengal, India
2
2Physics Department, Alipurduar College

n Dist. : Jalpaiguri, Pin : 736122, West Bengal, India
a 3 IUCAA Reference Centre, Physics Department
J North Bengal University

7

]
O
C
h. ABSTRACT
p
- A specific class of flat Emergent Universe (EU) is considered and its viability is
o
testedinviewoftherecentobservations.ModelparametersareconstrainedfromStern
r
t data for Hubble Parameter and Redshift (H(z) vs. z) and from a model independent
s
measurementofBAOpeakparameter.ItisnotedthatacompositionofExoticmatter,
a
[ dust and dark energy,capable of producing an EU,can not be ruledout with present
data.Evolutionofotherrelevantcosmologicalparameters,viz.densityparameter(Ω),

1 effective equation of state (EOS) parameter (ω ) are also shown.
eff
v
0 Key words: Emergent Universe, CosmologicalParameters,Observations.
6
3
1
.
1
1 INTRODUCTION scribes the matter in theuniverse. An alternative approach
0
is also followed where gravitational sector of the Einstein
1 Itis knownfrom observational cosmology that ouruniverse
field equation is modified by including higher order terms
1 is passing through a phase of acceleration. Unfortunately,
: in the Einstein-Hilbert action (Sotiriou 2007). To address
v the present phase of acceleration of the universe is not
the present accelerating phase of the universe once again
i clearly understood. Standard Big Bang cosmology with
X attemptsaremadewheretheorieswithamodificationofthe
perfect fluid assumption fails to accommodate the observa-
gravitational sector taking into account higher order terms
r tional fact. However, an accelerating universe is permitted
a thatarerelevantatthepresentenergyscaleareconsidered.
if a small cosmological constant (Λ) be included in the
There are other approaches generally adopted considering
Einstein’s gravity . There is, however, no satisfactory
modificationofthemattersectorbyincludingverydifferent
theory that explains the origin of Λ which is required to
kind of matter known as exotic matter namely, Chaplygin
be unusually small. Moreover, Standard Big Bang model
gas and its variations (Bento, Bertolami & Sen 2002;
withoutacosmological constantisinevitablypleaguedwith
Bilic, Tupper& Viollier 2001), models consisting one or
a time like singularity in the past. The Big Bang model is
morescalarfieldandtachyonfields(Lyth2003).Whilemost
also found to be entangled with some of the observational
of these models address dark energy part of the universe,
features which do not have explantion in the framework of
other models based on non-equilibrium thermodynamics
perfect fluid model. Consequently an inflationary epoch in
and Boltzmann formulation, which do not require any
the early universe is required (Guth 1981) to resolve the
dark energy (Zimdahl et al. 2001; Balakin et al. 2003;
outstandinng issues in cosmology. It is not yet understood
Lima, Silva & Santos2008),arealsoconsidered suitablefor
when and how the universe entered the phase. However,
describinglateuniverse.Aviablecosmologicalmodelshould
the concept of inflation is taken up to build a consistent
accommodate an inflationary phase in the early universe
scenario of the ealy universe. Inflation may be realized in
withasuitable accelerating phaseatlate time.Aninterest-
a semiclassical theory of gravity where one requires an
ing area of cosmology is to consider models which are free
additional inputs like existence of a scalar field which de-
from the initial singularity also. Emergent Universe (EU)
scenario is one of the well known choices in this field. EU
modelsareproposedindifferentframeworklikeBrans-Dicke
⋆ Electronicmail:[email protected] theory (delCampo, Herrera & Labrana 2007), brane world
† Electronicmail:[email protected] cosmology (Banerjee, Bandyopadhyay& Chakraborty
‡ Electronicmail:prasenjit [email protected]
2 B. C. Paul, S. Ghose and P. Thakur
2008;Beesham, Chervon & Maharaj 2009;Debnath2008),
Table 1.SternData(H(z)vs.z)
Gauss-Bonnet modified gravity (Paul & Ghose 2009), loop
quantum cosmology (Mulryne, Tavakol, Lidsey & Ellis
zData H(z) σ
2005) and standard General Relativity (GR)
(Mukherjee et al. 2006). Some of these models are im- 0.00 73 ±8.0
plemented in a closed universe (Ellis, Murugan & Tsgas 0.10 69 ±12.0
0.17 83 ±8.0
2004)whileothersinaflatuniverse(Mukherjee et al.2006).
0.27 77 ±14.0
If EU be developed in a consistent way it might solve some
0.40 95 ±17.4
of the well known conceptual problems not understood
0.48 90 ±60.0
in the Big-Bang model. An interesting class of EU model
0.88 97 ±40.4
in the standard GR framework has been obtained by 0.90 117 ±23.0
Mukherjee et al. (2006) considering a non-linear equation 1.30 168 ±17.4
of state in a flat universe. The EU model evolves from a 1.43 177 ±18.2
static phase in the infinite past into an inflationary phase 1.53 140 ±14.0
andfinallyitadmitsanacceleratingphaseatlatetime.The 1.75 202 ±40.4
universe is free from initial singularity and large enough
to begin with so as to avoid quantum gravity effects. The
relevant parameters in our model. The paper is organized
non-linear equation of state is the input of the model
as follows : in section 2 field equations for the model are
which permits different composition of matter in addition
discussed, in section 3 and 4 we the model is constrained
to normal matter as cosmic fluid. The model has been
with Stern data and Stern+BAO data respectively. Finally
explored in a flat universe as such universe is supported by
in section 5 thefindingsare summarized with a discussion.
recent observations. The EOS considered in obtaining EU
model by Mukherjee et al. (2006) is
1
p=Aρ Bρ2, (1) 2 FIELD EQUATIONS
−
where A and B are unknown parameters with B > 0 al- We consider Robertson-Walker(RW) metric which is given
ways. Different values of A and B corresponds to different by:
composition of matters in the EU model. In the literature
dr2
(Fabris et al.2007),similarkindofnon-linearEOShasbeen ds2= dt2+a2(t) +r2(dθ2+sin2θdφ2) (2)
considered as a double component dark energy model and − 1 kr2
(cid:20) − (cid:21)
analyzed to obtain acceptable values of model parameters.
wherek=0,+1( 1) isthecurvatureparameterin thespa-
The EOS given by eq. (1) is a special form of a more gen- −
tial section representing flat or closed (open) universe and
eral EOS, p = Aρ Bρα; which permits Chaplygin gas as
− a(t)isthescalefactor oftheuniverse,r,θ,φarethedimen-
a special case (with α < 0) (Bento, Bertolami & Sen 2002;
sionless comoving co-ordinates. The Einstein field equation
Bilic, Tupper& Viollier 2001). Chaplygin gas is considered
is
widely in recent times to build a consistent cosmological
1
model. It interpolates between a matter dominated phase R g R=8πGT (3)
µν − 2 µν µν
andadeSitterphase.LatervariousmodifiedformsofChap-
lygingaswereproposed(Liu et al.2005)totrackcosmolog- where Rµν, R and Tµν represent Ricci tensor, Ricci scalar
ical evolution. For example models like Modified Chaply- andenergymomentumtensorrespectively.UsingRWmetric
gin gas interpolates between radiative era and ΛCDM era. in Einstein field equation we obtain time-time component
Fabris et al.(2007)showedintheirworkthatsuchinterpola- which is ghiven by
tionispermissibleevenwithα>0andastringspecificcon- a˙ k
figurationmaybephenomenologicallyrealizedwithanEOS 3 a + a2 =ρ (4)
considered by Mukherjee et al. (2006). Recently using eq. (cid:16) (cid:17)
where we consider natural units i.e., c = 1, 8πG = 1.
(1) for an EU model proposed by Mukherjee et al. (2006),
Another field equation is the energy conservation equation
we determined various constraints that are imposed on the
which is given by
EOSparametersfromobservationaldatanamely,SNIadata,
BAO peak parameter measurement and CMB shift pa- dρ
+3H(ρ+p)=0, (5)
rametermeasurement(Paul, Thakur& Ghose2010).Itwas dt
noted that an EU model is permitted with A < 0. It is where p, ρ and H are respectively pressure, energy density,
found that the possibility of A = 0 case is also permitted Hubbleparameter H = a˙ TheHubbleparameter(H)can
when we probe the contour diagram of A B plane with be expressed in terms ofaredshift parameter (z) which is
−
95%confidence.ThecaseA=0correspondstoacomposi- given by (cid:0) (cid:1)
tion of dust, exotic matter and dark energy in the universe
1 dz
which is certainly worth exploring. In this paper a specific H(z)= . (6)
−1+z dt
EUmodelistakenupwherethematterenergycontentofthe
universe comprises of dust, exotic matter and dark energy. Since the components of matter and dark energy (exotic
Using Stern data (Table. 1), the admissibility of model pa- matter) are conserved separately, we may use energy con-
rametersaredeterminedfromH(z)vs.z (Stern et al.2006) servation equation together with EOS given by eq. (1) to
and using measurement of model independent BAO peak determinetheenergydensitywhichisobtainedonintegrat-
parameter . We also plotted evolution of cosmologically ing eq.(5) :
A
Emergent Universe from A Matter-Energy Composition 3
1.0 1.0
H(z)-z Data (Stern) Sterrn+BAO data
0.8 0.8
0.6 0.6
K K
0.4 0.4
0.2 0.2
0.0 0.0
0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0
B B
Figure 1. (Colour Online)Constraints from Stern Data Figure 2. (Colour Online)Constraints from joint analysis with
(H(z)vs.z)68.3%(Solid)95%(Dotted)and99.7%(Dashing)con- Stern (H(z)-z) data and BAO peak parameter measurement for
tours.Thebestfitpointisshown(0.0122,−0.0823). 68.3%(Solid) 95% (Dashed) and 99.7% (Outermost) confidence
levelareshowninthefigurealongwiththebestfitvalue(0.0094,-
0.1573)
2
B 1 K
ρ = + , (7)
emu 1+A A+1 3(A+1)
(cid:20) a 2 (cid:21) energy.Inthiscaseeq.(10)canberepresentedinfunctional
where K is an integration constant and for a consistent form given by
formulation of EU it is required to be positive definite
H2(H ,B,K,z)=H2E2(B,K,z), (11)
(Mukherjee et al. 2006). It is evident that the energy den- 0 0
sity ρ contains three terms corresponding to three different where
composition of fluids:
E(B,K,z)2= Ωconst+Ω1(1+z)32 +Ω2(1+z)3 . (12)
ρ(z)= B 2+ 2BK (1+z)32(A+1)+ (cid:16) (cid:17)
A+1 (A+1)2 In the present case Ωconst denotes a constant density pa-
(cid:16) (cid:17) K 2 rameterwhichcorrespondstoenergydensitydescribedbya
(1+z)3(A+1) (8) cosmological constant Λ. We denote the above density pa-
A+1
(cid:16) (cid:17) rameterbyΩΛ.Ω1 correspondstosomeexoticmatterwhich
In the above the first term is a constant which may be
we denote by Ω . Ω corresponds to a dust like fluid which
e 2
consideredtodescribeenergydensitycorrespondingtodark
we denoteby Ω . Usingthe abovein eq. (12) we obtain :
d
energy.In a simpler form eq. (8) can be written as:
ρ(z)=ρconst+ρ1(1+z)32(A+1)+ρ2(1+z)3(A+1) (9) E(B,K,z)2= ΩΛ+Ωe(1+z)32 +Ωd(1+z)3 . (13)
(cid:16) (cid:17)
we denote ρ = B 2, ρ = 2BK and ρ = K 2 Theabovefunctionswillbeusedinthenextsectionforanal-
const A+1 1 (A+1)2 2 A+1 ysis with observational results and to determine the model
denoteenergydensitiesfordifferentfluidcomponentsatthe
(cid:0) (cid:1) (cid:0) (cid:1) parameters.
presentepochamongwhichρ denotestheconstantcom-
const
ponent. We note that present value of densities depend on
both B and K. The Einstein field equation given by eq.(4) 3.1 Analysis with Stern (H(z)vs.z) data
can berewritten for a flat universe (k=0) as :
In thissection we defineχ2 function as :
Hwh(ezr)e2Ω==H02ρ(cid:16)Ωsdcoennsott+edΩe1n(s1it+ypz)ar2(aAm3+e1)te+rsΩfo2r(1co+rrzes)p3(oAn+d1i)n(cid:17)g(10) χ2stern(H0,A,B,K,z)= (H(H0,B,K,σzz2)−Hobs(z))2(14)
fluid and ρρc= 3Ho2 here is thecritical density. whereHobs(z)istheobseXrvedHubbleparameteratredshift
c 8πG zandσz istheerrorassociatedwiththatparticularobserva-
tion. The present day Hubbleparameter (H ) is a nuisance
0
parameterhere.Theobjectiveoftheanalysisistodetermine
3 CONSTRAINTS ON MODEL PARAMETERS the constraints imposed on the model parameters namely,
FROM OBSERVATIONAL DATA B and K from the observational input. So we can safely
marginalize over H , defininga function
0
In this section we consider an EU model implemented in
L(A,B,K,z)= Exp χ2(H0,A,B,K,z) P(H )dH
a flat universe using EOS given by eq.(1). A special case − 2 0 0
A=0istakenupheretoexploreEUscenariowithadefinite where P(H0) repRresenths a prior distribuition function. Here
compositionofmatternamely,dust,exoticmatteranddark we consider a Gaussian Prior with H = 72 8. In the
0
±
4 B. C. Paul, S. Ghose and P. Thakur
theoretical model it is demanded that the model parame- 0.6
ters should satisfy the inequalities (i) B > 0,(ii) K > 0.
Therefore, themodel parameters obtained from thebest fit
0.5
analysis with observational data are determined in thethe-
oretical parameter space. The best fit values obtained for
the parameters here are: B = 0.2615 and K = 0.4742 to- 0.4
gether with χ2 = 1.02593 ( per degree of freedom). The
min
plotsof68.3%,95%and99.7%confidencelevelcontoursare
shown infig.2.Thefollowing rangeof valuesare permitted Wd0.3
: 0.003 < B < 0.5996 and 0.303 < K < 0.63 within 68.3%
confidencelevel.
0.2
0.1
3.2 Analysis with Stern+BAO data
ForaflatuniverseBAOpeakparametermaybedefinedasin
0.0
a low redshift region such as 0<z <0.35 (Eisenstein et al. 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
2005): We
= √Ωm 0z1 Ed(zz) 2/3 (15) 9F9i.g7u%re(ou3t.er()CcoolnofiudrenOcnelilneev)e6l8c.o3n%tou(irnsnienr)Ω,d9−5%Ωe(pmlaidndel.e) and
A E(z1)1/3 R z1 !
whereΩ isthetotaldensityparameterformattercontent 0.0
m
ofuniverse.Onehastoconsideraconstantω(EOSparame-
ter).Evenifω isnotstrictlyconstant,itisquitereasonable
to take a constant ω value over a small redshift interval. -0.2
It would not be strictly the value of ω at z = 0 but rather
someaveragevalueintheregion0<z<0.35.Hereweusea
techniqueadoptedbyEisenstein et al.(2005)toexplorethe -0.4
parameter which is independent of dark energy model.
The value oAf for a flat universe is = 0.469 0.017 as Ωeff
A A ±
measured in Eisenstein et al. (2005) using SDSS data. We
-0.6
define χ2 = (A−0.469)2. For a joint analysis scheme we
BAO (0.017)2
consider χ2 =χ2 +χ2 . Thebest fit valuesfoundin
tot stern BAO
the joint analysis are : B = 0.2599 and K = 0.4751 along -0.8
with a χ2 =1.1681 (per degree of freedom). Contours of
min
68.3%, 95% and 99.7% confidence level are shown in fig. 2.
Here we found that the range of values permitted within -1.0
68.3% confidence is a bit elevated : 0.009 <B <0.606 and 0 2 4 6 8 10
0.3126<K <0.6268. z
Figure4.(ColourOnline)Evolutionofωeff withbestfitvalues
andvalueswithindifferentconfidence level
4 RELEVANT COSMOLOGICAL 5 DISCUSSIONS
PARAMETERS
In this paper considering a very specific model of flat EU,
The range of permitted values of model paramters B and wedeterminetheobservationalconstraintsonthemodelpa-
K are determined above. In this section we determine the rameters. For this recent observational data namely, Stern
variation of both the density parameter and the effective data, measurement of BAO peak parameter are used. The
equation of state. Notethatthemodel isan asymptotically specific form of EOS given by eq. (1) to obtain EU sce-
de Sitter model and a late time phase of acceleration is as- nario in a flat universe is employed here for the purpose.
sured. 68.3%, 95% and 99.7% confidence level contours in We set A = 0 in the eq. (1) to begin with. A equal
Ω Ω plane are shown in fig. 3. It is found that the best to zero represents a universe with a composition of ex-
d e
−
fit value (Ω +Ω = 0.3374) permits Ω = 0.6626. It is otic matter only. This kind of EOS has been considered in
d e Λ
also noted that the generally predicted values Ω 0.72 Nojiri, Odinstov & Tsujikawa (2005). In our previous work
Λ
≈
and Ω 0.04 are permitted here within 68.3% confidence (Paul, Thakur& Ghose2010)onEUmodelitisnotedthat
d
≈
level. Wealso plot theevolution of theeffective EOS in fig. asmall nonzerovalueofA(althoughzeroisnotruledout)
4. As expected it remains negative throughout. EU, as we is permitted. As a result the analysis was done with non
havenoted,isanasymptotically deSitteruniverseandit is zero A. In this paper since we are interested in a specific
evidentfrom fig.5that ρdecreases toaverysmall valueat composition of matterenergy contentof theuniversecorre-
late universe. spondingtoA=0anlysisiscarriedoutforEOSgivenby(1)
Emergent Universe from A Matter-Energy Composition 5
0 100 Bento M. C., Bertolami O. & Sen A.A., 2002, Phys. Rev.
D,66, 043507
Bilic N.,TupperG. B.& Viollier R.D.., 2001, Phys.Lett.
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0 100
Komatsu E. et al., 2010, preprint (arXiv: 1001.4538)
aHtL
Kowalaski M. et al., 2008, Astrophys. J.,686,749,
Figure 5. (Colour Online) Evolution of the matter-energy den- preprint(arXiv:astro-ph/0804.4142)
sityinaasymptoticallydeSitteruniverse Lima J. A. S., Jesus J. F., Oliveira F. A.,2009, preprint
(arXiv:0911.5727)
Lima J. A. S., Silva F. E. & Santos R. C.,2008, Class.
with A = 0 only. As suggested by (Mukherjee et al. 2006),
Quant.Grav., 25, 205006
itcorrespondstothecontentoftheuniversewhichisacom-
Liu D-Jand Li X-Z,2005, preprint(astro-ph/0501115)
position of dark energy, dust and exotic fluid¿ The above
Lue A.,2002, preprint(hep-th/0208169)
composition is reasonable to obtain a viable scenario of the
Lyth D.H., 2003, preprint(hep-th/0311040)
universe considering the observational facts. It seems that
Mukherjee S. et al., 2006, Class. Quant.Grav., 23, 6927
the exotic part of the EOS may also contributein the bud-
MulryneD.J.,TravakolR.,LidseyJ.E.&Ellis G.F.R.,
getofdarkenergycontentoftheuniverse.Wefoundthatthe
2005, Phys.Rev.D, 71, 123512
observationally favoured amount of dark energy present in
NojiriS.,OdintsovS.D.&TsujikawaS.,2005,Phys.Rev.D,
universe today Ω 0.72 is permitted in our model within
Λ
≈ 71, 063004
68.3% confidence level. However, it may be mentioned here
PaulB.C.andGhoseS.,2009,preprint(arXiv:0809.4131.)
that the model may be extended even if A << 1 and so
| | Paul B. C., Thakur P. & Ghose S., 2010, Mont. Not. R.
that we can write A+1 1. We found that a composi-
≈ Astron.Soc., 407, 415
tionofdust,exoticmatteranddarkenergymayproducean
Sotiriou T. P., 2007, preprint(arXiv:0710.4438v1)
EUmodelwithintheframeworkofEinstein’sgravitywitha
Stern D. et al., 2006, preprint( arXiv:0907.3149v1 )
non-linearequationofstate.Itisalsonotedthatthiskindof
Zimdahl W. et al.,2001, Phys. Rev.D., 64, 3501
modelcanaccommodatemanyothercomposition ofmatter
energy dependingon thevalueof A.Theviability for those
will be taken up elsewhere.
ACKNOWLEDGMENTS
BCPandPTwouldliketothankIUCAAReference Centre,
Physics Department, N.B.U for extending the facilities of
research work. SG would like to thank CSIR for awarding
Senior Research Fellowship. BCP would like to thank Uni-
versity Grants Commission, New Delhi for Minor Research
Project (No. 36-365/2008 SR). SG would like to thank Dr.
A.A.SenforvaluablediscussionsandforhospitalityatCen-
tre for Theoretical Physics, JMI, New Delhi.
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