Impacts of different SNLS3 light-curve fitters on cosmological consequences of interacting dark energy models

1.1 MB·English

by  Shuang Wang

Astronomy&Astrophysicsmanuscriptno. light-curve-fitters (cid:13)cESO2016
May24,2016
Impacts of different SNLS3 light-curve fitters on cosmological
consequences of interacting dark energy models
ShuangWang1,YazhouHu1,2,3,MiaoLi1,andNanLi1,2,3
1 SchoolofPhysicsandAstronomy,SunYat-SenUniversity,Guangzhou510275,P.R.China
e-mail:[email protected],[email protected], [email protected], [email protected]
2 KavliInstituteforTheoreticalPhysicsChina,ChineseAcademyofSciences,Beijing100190,P.R.China
3 StateKeyLaboratoryofTheoreticalPhysics,InstituteofTheoreticalPhysics,ChineseAcademyofSciences,Beijing100190,P.R.
6 China
1
0 May24,2016
2

y ABSTRACT
a
M Weexplorethecosmologicalconsequencesofinteractingdarkenergy(IDE)modelsusingtheSNLS3supernovasamples. Inpartic-
ular,wefocusontheimpactsofdifferentSNLS3light-curvefitters(LCF)(correspondingto“SALT2”,“SiFTO”,and“Combined”
3 sample). Firstly,makinguseofthethreeSNLS3datasets,aswellasthePlanckdistancepriorsdataandthegalaxyclusteringdata,
2 weconstraintheparameterspacesofthreeIDEmodels. Then,westudythecosmicevolutionsofHubbleparameterH(z),decelera-
tiondiagramq(z),statefinderhierarchyS(1)(z)andS(1)(z),andcheckwhetherornotthesedarkenergydiagnosiscandistinguishthe
] differencesamongtheresultsofdifferent3SNLS3LCF4.Atlast,weperformhighredshiftcosmicagetestusingthreeoldhighredshift
O objects(OHRO),andexplorethefateoftheUniverse.Wefindthat,theimpactsofdifferentSNLS3LCFarerathersmall,andcannot
C bedistinguishedbyusingH(z),q(z),S(1)(z),S(1)(z),andtheagedataofOHRO.Inaddition,weinfer,fromthecurrentobservations,
3 4
howfarwearefromacosmicdoomsdayintheworstcase,andfindthatthe“Combined”samplealwaysgivesthelargest2σlower
.
h limitofthetimeintervalbetween“bigrip”andtoday,whiletheresultsgivenbythe“SALT2”andthe“SiFTO”samplearecloseto
p eachother. Theseconclusionsareinsensitivetoaspecificformofdarksectorinteraction. Ourmethodcanbeusedtodistinguishthe
- differencesamongvariouscosmologicalobservations.
o
r Keywords. cosmology:darkenergy,observations,cosmologicalparameters
t
s
a
[
1. Introduction and “Combined”, which consists of 472 SN. It should be men-

4 tioned that, in the cosmology fits, the SNLS group treated two
vTypeIasupernovaisasub-categoryofcataclysmicvariablestars important quantities, stretch-luminosity parameter α and color-
2thatresultsfromtheviolentexplosionofawhitedwarfstarina luminosityparameterβofSNeIa,asfreemodelparameters.No-
6binarysystem(Hillebrandt,&Niemeyer2000). Nowithasbe- ticethatαandβareparametersfortheluminositystandardiza-
9comeoneofthemostpowerfulprobestoilluminatethemystery
tion. Inaddition,theintrinsicscatterσ arefixedtoensurethat
6ofcosmicacceleration(Riessetal.1998;Perlmutteretal.1999), χ2/dof =1. int
0which may be due to an unknown energy component that can
1.produceananti-gravitationaleffect,i.e.,darkenergy(DE),ora thesAysmteomstatcircituicnaclecrthaainllteinesgeofotfySpNeIcaossumpeorlnoogvyaies(tShNeecoIna)tr.oOlnoef
0modification of general relativity, i.e., modified gravity (MG).
ofthemostimportantfactorsisthepotentialSNevolution. For
51 Along with the rapid progress of supernova (SN) cosmology,
examples, previousstudiesontheUnion2.1Mohlabeng&Ral-
1several high quality SN datasets were released in recent years,
ston(2013)andthePan-STARRS1datasetsScolnicetal.(2014)
:such as “SNLS” Astier et al. (2006), “Union” Kowalski et al.
v allindicatedthatβshouldevolvealongwithredshiftz. Besides,
i(2008), “Constitution” Hicken et al. (2009), “SDSS” Kessler itwasfoundthattheintrinsicscatterσ hasthehintofredshift-
Xet al. (2009), “Union2” Amanullah et al. (2010), “Union2.1” dependencethatwillsignificantlyaffecitnttheresultsofcosmology
rSuzukietal.(2012),and“Pan-STARRS1”Scolnicetal.(2014).
a fits (Marriner et al. 2011). In addition to the SN evolution, an-
In 2010, the Supernova Legacy Survey (SNLS) group released
otherimportantfactoristhechoiceofLCF(Kessleretal.2009).
their three years data Guy et al. (2010). Soon after, combining
Forinstance,ithasbeenprovedthatevenforthesameSNsam-
theseSNsampleswithvariouslow-ztomid-zsamplesandmak-
ples,using“MLCS2k2”(Jhaetal.2007)and“SALT2”(Guyet
inguseofthreedifferentlight-curvefitters(LCF)2,Conleyetal. al.2007)LCFwillleadtocompletelydifferentfittingresultsfor
presentedthreeSNLS3datasetsConleyetal.(2011):“SALT2”,
various cosmological models (Sollerman et al. 2009; Sanchez
whichconsistsof473SN;“SiFTO”,whichconsistsof468SN;
etal.2009;Pigozzoetal.2011;Smale&Wiltshire2011;Ben-
gochea2011;Bengochea&DeRossi2014).
1 Forrecentreviews,seeFrieman,Turner,&Huterer(2008);Caldwell
Oneofthepresentauthorshasalsodoneaseriesofresearch
&Kamionkowski(2009);Uzan(2010);Wang(2010);Li,Li,Wang,&
Wang(2011,2013);Weinbergetal.(2013). workstostudythesystematicuncertaintiesofSNeIa. Byusing
2 ALCFisamethodtomodelthelightcurvesofSN;byusingitone theSNLS3dataset,wefoundthatαisstillconsistentwithacon-
canestimatethedistanceinformationofeachSN. stant, but β evolves along with z at very high confidence level
Articlenumber,page1of12
(CL) Wang & Wang (2013a). Soon after, we showed that this 2.1. TheoreticalModels
resulthassignificanteffectsontheparameterestimationofstan-
This work is a sequel to the previous studies of our group, and
dardcosmology(Wang,Li&Zhang2014),andtheintroduction
thus we study the same IDE models considered in (Wang, et
ofatime-varyingβcanreducethetensionbetweenSNeIaand
al. 2014). It must be mentioned that, here we consider a flat
othercosmologicalobservations. Moreover,weprovedthatour
Universe and treat the present fractional density of radiation as
conclusionholdstrueforvariousDEandMGmodels(Wang,et
a model parameter, which is a little different from the case of
al. 2014; Wang, Wang & Zhang 2014; Wang, et al. 2015). Be-
(Wang,etal.2014). InaflatUniverse3,theFriedmannequation
sides,theevolutionofαhasalsobeenfoundfortheJLAsampls
(Li et al. 2016). In addition, by using three different SNLS3 is
LCF,including“SALT2”,“SiFTO”and“Combined”,webriefly 3M2H2 =ρ +ρ +ρ +ρ , (1)
discussed the effects of different LCF on the parameter estima- pl c de r b
tion of the Λ-cold-dark-matter (ΛCDM) model Wang & Wang where H ≡ a˙/a is the Hubble parameter, a = (1+z)−1 is the
(2013a) and the holographic dark energy (HDE) model (Wang, scale factor of the Universe (we take today’s scale factor a =
0
et al. 2015). It must be emphasized that in these two papers,
1), the dot denotes the derivative with respect to cosmic time
onlythefittingresultsgivenbydifferentSNLS3LCFareshown, t, M2 = (8πG)−1 is the reduced Planck mass, G is Newtonian
and the corresponding cosmological consequences are not dis- p
gravitationalconstant,ρ ,ρ ,ρ andρ aretheenergydensities
cussed. Therefore,theimpactsofdifferentLCFarenotstudied c de r b
of CDM, DE, radiation and baryon, respectively. The reduced
in details in our previous works. The main scientific objective HubbleparameterE(z)≡ H(z)/H satisfies
0
of the current work is presenting a comprehensive and system-
aticinvestigationontheimpactsofdifferentSNLS3LCF.Todo E2 =Ω ρc +Ω ρde +Ω ρr +Ω ρb , (2)
this,boththecosmology-fitresultsandthecosmologicalconse- c0ρ de0ρ r0ρ b0ρ
c0 de0 r0 b0
quenceresultsareshowninthiswork.
Asmentionedabove,theeffectsofdifferentSNLS3LCFon where Ωc0, Ωde0, Ωr0 and Ωb0 are the present fractional den-
theΛCDMmodelhasbeenbrieflydiscussedin(Wang&Wang sities of CDM, DE, radiation and baryon, respectively. As
mentioned above, we treat Ω as a model parameter in this
2013a). To obtain new scientific results, new elements need to r0
be taken into account. Since the interaction between different work. In addition, ρr = ρr0(1 + z)4, ρb = ρb0(1 + z)3. Since
Ω =1−Ω −Ω −Ω ,Ω isnotanindependentparame-
componentswidelyexistinnature,andtheintroductionofain- de0 c0 b0 r0 de0
ter.
teractionbetweenDEandCDMcanprovideanintriguingmech-
In an IDE scenario, the energy conservation equations of
anismtosolvethe“cosmiccoincidenceproblem”(Guo,Ohta,&
CDMandDEsatisfy
Tsujikawa2007;Lietal.2009a,b;Heetal.2010)andalleviate
the“cosmicageproblem”(Wang&Zhang2008;Wang,Li&Li ρ˙ +3Hρ = Q, (3)
c c
2010;Cui&Zhang2010;Duran&Pavon2011),itisinteresting
to study the effect of different LCF on interacting dark energy ρ˙de+3H(ρde+pde)=−Q, (4)
(IDE) model. Here we adopt the w-cold-dark-matter (wCDM) wherep =wρ isthepressureofDE,Qistheinteractionterm,
de de
model with a direct non-gravitational interaction between dark
which describes the energy transfer between CDM and DE. So
sectors. It must be emphasized that, if our conclusions are de-
far, the microscopic origin of interaction between dark sectors
pendent on the specific form of dark sector interaction, these
is still a puzzle. To study the issue of interaction, one needs to
conclusions will not be reliable at all. So in this work, three
write down the possible forms of Q by hand. In this work we
kinds of interaction terms are taken into account to ensure that
considerthefollowingthreecases:
our study is insensitive to a specific interaction form. In addi-
tion,tomakeacomparison,wealsoconsiderthecaseofwCDM Q =3γHρ , (5)
1 c
modelwithoutinteractionterm. Q =3γHρ , (6)
2 de
According to the previous studies on the potential SN evo- ρ ρ
lution, we adopt a constant α and a linear β(z) = β0 + β1z in Q3 =3γHρ c+dρe , (7)
thiswork. MakinguseofthethreeSNLS3datasets,aswellas c de
thelatestPlanckdistancepriordata(Wang&Wang2013b),the whereγisadimensionlessparameterdescribingthestrengthof
galaxyclustering(GC)dataextractedfromSloandigitalskysur- interaction,γ>0meansthatenergytransfersfromDEtoCDM,
vey (SDSS) data release 7 (DR7) (Hemantha, Wang & Chuang γ<0impliesthatenergytransfersfromCDMtoDE,andγ=0
2014)anddatarelease9(DR9)(Wang 2014),weconstrainthe denotesthecasewithoutdarksectorinteraction,i.e. thewCDM
parameterspacesofthewCDMmodelandthethreeIDEmodels, model. Notice that these three kinds of interaction form have
andinvestigatetheimpactsofdifferentSNLS3LCFonthecos- been widely studied in the literature (Guo, Ohta, & Tsujikawa
mologyfits. Moreover,basedonthefittingresults,westudythe 2007;Lietal.2009b;Heetal.2010;Li&Zhang2014).Forsim-
possibilityofdistinguishingtheimpactsofdifferentLCFbyus- plicity,hereafterwecallthemIwCDM1model,IwCDM2model,
ingvariousDEdiagnosistoolsandcosmicagedata,anddiscuss andIwCDM3model,respectively.
thepossiblefateoftheUniverse. ForthewCDMmodel,wehave
WepresentourmethodinSection2,ourresultsinSection3,
andsummarizeandconcludeinSection4. E(z)=(cid:16)Ω (1+z)4+(Ω +Ω )(1+z)3+Ω (1+z)3(1+w)(cid:17)1/2. (8)
r0 b0 c0 de0
FortheIwCDM1model,thesolutionsofEqs. (3)and(4)are
2. Methodology
ρ =ρ (1+z)3(1−γ), (9)
Inthissection,firstlywereviewthetheoreticalframeworkofthe c c0
IDEmodels,thenwebrieflydescribetheobservationaldataused 3 Theassumptionofflatnessismotivatedbytheinflationscenario.For
in the present work, and finally we introduce the background adetaileddiscussionoftheeffectsofspatialcurvature,see(Clarkson,
knowledgeaboutDEdiagnosisandcosmicage. Cortes,&Bassett2007)
Articlenumber,page2of12
Wangelal.:ImpactsofdifferentSNLS3LCFoncosmologicalconsequencesofIDEmodels
2.2.1. SNeIaData
ρ = γρc0 (cid:0)(1+z)3(1+w)−(1+z)3(1−γ)(cid:1)+ρ (1+z)3(1+w). (10) As mentioned above, we use all the three SNLS3 data sets. In
de w+γ de0 the following, we briefly introduce how to include these three
datasetsintotheχ2analysis.
Adoptingaconstantαandalinearβ(z) = β +β z,thepre-
Thenwehave 0 1
dictedmagnitudeofanSNbecomes
(cid:16)
E(z)= Ωr0(1+z)4+Ωb0(1+z)3+Ωde0(1+z)3(1+w) mmod =5log10DL(z)−α(s−1)+β(z)C+M. (19)
+Ωc0(cid:0)wγ+γ(1+z)3(1+w)+ ww+γ(1+z)3(1−γ)(cid:1)(cid:17)1/2. (11) TheluminositydistanceDL(z)isdefinedas
D (z)≡ H (1+z )r(z), (20)
L 0 hel
FortheIwCDM2model,thesolutionsofEqs. (3)and(4)are wherezandzhelaretheCMBrestframeandheliocentricredshifts
ofSN,and
ρde =ρde0(1+z)3(1+w+γ), (12) r(z)= H−1(cid:90) z dz(cid:48) . (21)
0 E(z(cid:48))
0
Here sandCarestretchmeasureandcolormeasurefortheSN
γρ γρ lightcurve,Misaparameterrepresentingsomecombinationof
ρ =ρ (1+z)3+ de0(1+z)3− de0(1+z)3(1+w+γ). (13)
c c0 w+γ w+γ SNabsolutemagnitudeMandHubbleconstantH0.
Forasetof N SNewithcorrelatederrors,theχ2 functionis
givenby
Thenweget
χ2 =∆mT ·C−1·∆m, (22)
(cid:16) SN
E(z)= Ω (1+z)4+(Ω +Ω )(1+z)3
r0 c0 b0 where∆m≡m −m isavectorwithN components,andm
B mod B
+Ω (cid:0) γ (1+z)3+ w (1+z)3(1+w+γ)(cid:1)(cid:17)1/2. (14) is the rest-frame peak B-band magnitude of the SN. The total
de0 w+γ w+γ covariancematrixCcanbewrittenasConleyetal.(2011)
C=D +C +C . (23)
stat stat sys
For the IwCDM3 model, Eqs. (3) and (4) still have analytical
HereD denotesthediagonalpartofthestatisticaluncertainty,
solutions stat
C and C denote the statistical and systematic covariance
stat sys
ρc =ρc0(1+z)3(cid:0)ρ ρ+c0ρ + ρ ρ+deρ0 (1+z)3(w+γ)(cid:1)−wγ+γ, (15) manacterimceast,rrixesCpe,csteiveeClyo.nFleoyrtehteald.e(t2a0il1s1o)f.constructingthecovari-
c0 de0 c0 de0
It must be emphasized that, in order to include host-galaxy
information in the cosmological fits, Conley et al. split the
ρde =ρde0(1+z)3(1+w+γ)(cid:0)ρ ρ+c0ρ SanNdLmS3adseamMplteobbaeseddiffoenrehnotsfto-rgathlaexytwsoteslalamrpmleasssCoantl1ey01e0tMa(cid:12)l.,
c0 de0
+ρ ρ+deρ0 (1+z)3(w+γ)(cid:1)−wγ+γ. (16) (S2N0L11S)3. Sdaotath.erMeoarreeotvweor,vCalounelseyofeMt al(.i.er.emMo1veadndMM12a)nfdorMthe2
c0 de0 from cosmology-fits by analytically marginalizing over them
(for more details, see the appendix C of Conley et al. (2011)).
Thenweobtain
In the present work, we just follow the recipe of Conley et al.
E(z)=(cid:0)Ω (1+z)4+Ω (1+z)3+Ω C(z)(1+z)3 (2011),anddonottreatMasmodelparameter.
r0 b0 c0
+Ω C(z)(1+z)3(1+w+γ)(cid:1)1/2. (17)
de0
2.2.2. CMBData
where
For CMB data, we use the distance priors data extracted from
Planckfirstdatarelease(Wang&Wang2013b)4. CMBgiveus
Ω Ω
C(z)=(cid:0)Ω +c0Ω + Ω +deΩ0 (1+z)3(w+γ)(cid:1)−wγ+γ. (18) 4 Itshouldbementionedthat,inthispaperwejustusethepurelyge-
c0 de0 c0 de0
ometricmeasurementsofCMB,i.e. thedistancepriordata. Thereare
someothermethodsofusingCMBdata. Forexamples, theobserved
Foreachmodel,theexpressionofE(z)willbeusedtocalculate
positionofthefirstpeakoftheCMBanisotropiesspectrumcanbeused
theobservationalquantitiesappearinginthenextsubsection.
toperformcosmology-fits(Carneiroetal.2008;Pigozzoetal.2011).
In addition, the CMB full data can also be used to constrain cosmo-
logicalmodelsviatheglobalfittechnique. Tomakeacomparison,we
2.2. ObservationalData
constraintheparameterspacesofIwCDM1modelbyusingthesethree
Inthissubsection,wedescribetheobservationaldata. Herewe methods of using CMB data. We find that the differences among the
fitting results given by different method are very small (For example,
use the SNe Ia, CMB, and GC data, which is same with our
thedifferencesonDEEoSwareonlytheorderof1%).Inotherwords,
previouspaper(Wang,Li&Zhang2014). Itshouldbeempha-
OurresultsareinsensitivetothemethodofusingCMBdata.Thiscon-
sizedthat,inthisworkweuseallthethreeSNLS3datasets(i.e.
clusionalsoholdstrueforotherDEmodels(suchasHDEmodel(Liet
“SALT2”, “SiFTO”, and “Combined” sample), while only the
al.2013)),showingthattheCMBdatacannotputstrictconstraintson
“combined”dataareusedin(Wang,Li&Zhang2014). inaddi- thepropertiesofDE.SincethemainpurposeofusingCMBdataisto
tion,theGCdataarealsoupdatedinthiswork,comparedtothat putstrictconstraintsonΩ andΩ ,wethinkthattheuseofthePlanck
c0 b0
usedin(Wang,Li&Zhang2014). distancepriorissufficientenoughforourwork.
Articlenumber,page3of12
the comoving distance to the photon-decoupling surface r(z )5, &Chuang2014),usingthetwo-dimensionalmatterpowerspec-
∗
and the comoving sound horizon at photon-decoupling epoch trumofSDSSDR7samples,Hemantha,Wang,andChuanggot
r (z ). WangandMukherjeeshowedthattheCMBshiftparam-
s ∗ H(z=0.35)r (z )/c = 0.0431±0.0018,
eters(Wang&Mukherjee 2007) s d
D (z=0.35)/r (z ) = 6.48±0.25. (30)
A s d
l ≡ πr(z )/r (z ),
a ∗ s ∗
(cid:113) In a similar work (Wang 2014), using the anisotropic two-
R ≡ Ω H2r(z )/c, (24)
m 0 ∗ dimensionalgalaxycorrelationfunctionofSDSSDR9samples,
Wangobtained
togetherwithω ≡Ω h2,provideanefficientsummaryofCMB
b b
data as far as dark energy constraints go. The comoving sound H(z=0.57)r (z )/c = 0.0444±0.0019,
s d
horizonisgivenby(Wang&Wang2013b) D (z=0.57)/r (z ) = 9.01±0.23. (31)
A s d
rs(z)=cH0−1(cid:90)0a (cid:113)3(1+Rbdaa(cid:48)(cid:48))a(cid:48)4E2(z(cid:48)), (25) χGG2CC2d,awtaitahrezGinCc1lu=de0d.3i5naonudr azGnCal2ys=is0b.y57a,dtdointgheχχG22Co=f aχG2gCiv1e+n
model. Notethat
where a is the scale factor of the Universe, Rb = χ2 =∆q (cid:104)C−1 (q,q )(cid:105)∆q , ∆q =q −qdata, (32)
31500Ω h2(T /2.7K)−4,andT =2.7255K. GCi i GCi i j j i i i
b cmb cmb
UsingthePlanck+lensing+WPdata,themeanvaluesand1σ whereq = H(z )r (z )/c,q = D (z )/r (z ),andi = 1,2.
1 GCi s d 2 A GCi s d
errorsof{la,R,ωb}areobtained(Wang&Wang2013b), BasedonRefs. (Hemantha,Wang&Chuang2014)and(Wang
(cid:104)l (cid:105)=301.57,σ(l )=0.18, 2014),wehave
a a
(cid:104)R(cid:105)=1.7407,σ(R)=0.0094, (cid:32) 0.00000324 −0.00010728 (cid:33)
C = , (33)
(cid:104)ω (cid:105)=0.02228,σ(ω )=0.00030. (26) GC1 −0.00010728 0.0625
b b
Defining p = l (z ), p = R(z ), and p = ω , the normalized
1 a ∗ 2 ∗ 3 b (cid:32) (cid:33)
covariancematrixNormCovCMB(pi,pj)canbewrittenas(Wang C = 0.00000361 0.0000176111 . (34)
&Wang2013b) GC2 0.0000176111 0.0529
 1.0000 0.5250 −0.4235 
 0.5250 1.0000 −0.6925 . (27) 2.2.4. Totalχ2 Function
−0.4235 −0.6925 1.0000
Nowthetotalχ2functionis
Then,thecovariancematrixfor(l ,R,ω )isgivenby
a b
χ2 =χ2 +χ2 +χ2 . (35)
Cov (p,p )=σ(p)σ(p )NormCov (p,p ), (28) SN CMB GC
CMB i j i j CMB i j
We perform an MCMC likelihood analysis using the “Cos-
wherei, j = 1,2,3. TheCMBdataareincludedinouranalysis moMC”package(Lewis&Bridle2002).
byaddingthefollowingtermtotheχ2function:
(cid:104) (cid:105)
χ2 =∆p Cov−1 (p,p ) ∆p , ∆p = p −pdata, (29) 2.3. DarkEnergyDiagnosisandCosmicAge
CMB i CMB i j j i i i
Sincepreviousworkspointedoutthatthedifferencesamongthe
where pdiataarethemeanvaluesfromEq. (26). cosmological results given by the three different SNLS3 LCF
arerathersmall,weneedmoretoolstodistinguishtheeffectsof
2.2.3. GCData differentLCF.Inthepresentwork,weusetheHubbleparameter
H(z), the deceleration parameter q(z), the statefinder hierarchy
To improve the cosmological constraints, we also use the GC S(1)(z) and S(1)(z) (Arabsalmani & Sahni 2009; Zhang, Cui &
dataextractedfromSDSSsamples. In(Chuang&Wang 2013), 3 4
Zhang 2014), and the cosmic age t(z) as our diagnosis tools to
Chuang and Wang measured the Hubble parameter H(z) and distinguishtheeffectsofdifferentLCF.Thissubsectionconsists
the angular-diameter distance D (z) separately 6 by scaling the
A of two parts. Firstly, we introduce the tools of DE diagnosis,
model galaxy two-point correlation function to match the ob-
includingtheHubbleparameterH(z),thedecelerationparameter
served galaxy two-point correlation function. Since the scal-
q(z),andthestatefinderhierarchy{S(1),S(1)}. Then,wediscuss
ing is measured by marginalizing over the shape of the model 3 4
theissuesaboutcosmicage,includingthehigh-redshiftcosmic
correlation function, the measured H(z) and D (z) are model-
A agetestandthefateoftheUniverse.
independent and can be used to constrain any cosmological
LetusintroducetheDEdiagnosistoolsfirst.Thescalefactor
model(Chuang&Wang 2013).
oftheUniverseacanbeTaylorexpandedaroundtoday’scosmic
It should be mentioned that, compared with using H(z) and
aget asfollows:
D (z),usingH(z)r (z )/candD (z)/r (z )7cangivebettercon- 0
A s d A s d
straints on various cosmological models. In (Hemantha, Wang (cid:88)∞ A
a(t)=1+ n[H (t−t )]n, (36)
5 In this work, we adopt the result of z given in (Hu & Sugiyama n! 0 0
∗ n=1
1996).
6 Angular-diameterdistanceDA(z) = cH0−1r(z)/(1+z),wherecisthe where
speedoflight.
7 r (z )isthesoundhorizonatthedragepoch,wherer (z)isgivenby a(t)(n)
s d s A = , n∈ N, (37)
Eq.(25,zdisgivenin(Eisenstein&Hu 1998). n a(t)Hn
Articlenumber,page4of12
Wangelal.:ImpactsofdifferentSNLS3LCFoncosmologicalconsequencesofIDEmodels
witha(t)(n) =dna(t)/dtn.TheHubbleparameterH(z)containthe will use these three age data to test the wCDM model and the
information of the first derivative of a(t). Based on the Baryon threeIDEmodels.
AcousticOscillations(BAO)measurementsfromtheSDSSdata AnotherinterestingtopicisthefateoftheUniverse. Thefu-
release9anddatarelease11,Samushiaetal.gaveH ≡ H(z= tureoftheUniversedependsonthepropertyofDE.IftheUni-
0.57
0.57) = 92.4 ± 4.5km/s/Mpc Samushia et al. (2013), while verse is dominated by a quintessence Caldwell, Dave & Stein-
Delubacetal.obtainedH ≡ H(z=2.34)=222±7km/s/Mpc hardt(1998);Zlatev,Wang&Steinhardt(1999)oracosmologi-
2.34
Delubac et al. (2014). These two H(z) data points will be used calconstant,theexpansionoftheUniversewillcontinueforever.
tocomparethetheoreticalpredictionsofthewCDMmodeland If the Universe is dominated by a phantom Caldwell (2002);
thethreeIDEmodels. Inaddition,thedecelerationparameterq Caldwell,Kamionkowski&Weinberg(2003),eventuallythere-
isgivenby pulsivegravityofDEwillbecomelargeenoughtotearapartall
thestructures,andtheUniversewillfinallyencounteradooms-
q=−A =− a¨ , (38) day,i.e. theso-called“bigrip”(BR).Setting x=−ln(1+z),we
2 aH2 cangetthetimeintervalbetweenaBRandtoday
whichcontainstheinformationofthesecondderivativesofa(t). (cid:90) ∞ dx
For the ΛCDM model, A2|ΛCDM = 1 − 23Ωm, A3|ΛCDM = 1, tBR−t0 = H(x), (46)
A4|ΛCDM =1−322Ωm. Thestatefinderhierarchy,Sn,isdefinedas 0
Arabsalmani&Sahni(2009):
wheret denotesthetimeofBR.Itisobviousthat, foraUni-
BR
verse dominated by a quintessence or a cosmological constant,
S = A + 3Ω , (39)
2 2 2 m this integration is infinity; and for a Universe dominated by a
S3 = A3, (40) phantom, this integration is convergence. We would like to in-
S = A + 32Ω , (41) fer, from the current observational data, how far we are from a
4 4 2 m
cosmicdoomsdayintheworstcase.Sointhisworkwecalculate
Thereasonforthisredefinitionistopegthestatefinderatunity the2σlowerlimitsoft −t forallthefourDEmodels.
BR 0
forΛCDMduringthecosmicexpansion,
Sn|ΛCDM =1. (42) 3. Result
This equation defines a series of null diagnostics for ΛCDM In this section, firstly we show the cosmology-fit results of the
when n ≥ 3. By using this diagnostic, we can easily distin- wCDM model given by various SNLS3 samples without and
guish the ΛCDM model from other DE models. Because of withsystematicuncertainties,nextwepresentthefittingresults
Ωm|ΛCDM = 23(1 + q), when n ≥ 3, statefinder hierarchy can ofthethreeIDEmodels,thenweshowthecosmicevolutionsof
berewrittenas: Hubbleparameter H(z),decelerationparameterq(z),statefinder
hierarchyS(1)(z)andS(1)(z)accordingtothefittingresultsofthe
S(1) = A , (43) 3 4
3 3 IDEmodels,finallyweperformthehigh-redshiftcosmicagetest
S(1) = A +3(1+q), (44) and discuss the fate of the Universe. Since both three SNLS3
4 4 datasets and a time-varying β are considered at the same time,
wherethesuperscript(1)istodiscriminatebetweenS(1)andS . alltheresultsofthisworkarenewcomparedwiththeprevious
n n
Inthispaper,weusethestatefinderhierarchyS(1)(z)andS(1)(z) studies.
3 4
asourdiagnosistoolstodistinguishtheeffectsofdifferentLCF.
Now,letusturntotheissuesaboutcosmicage. Theageof 3.1. CosmologyFits
theUniverseatredshiftzisgivenby
In table 1, we present the fitting results of the wCDM model
(cid:90) ∞ dz˜ given by various SNLS3 samples without and with systematic
t(z)= . (45)
(1+z˜)H(z˜) uncertainties. The first row of table 1 shows the fitting results
z
for the case of only considering constant β and statistical un-
In history, the cosmic age problem played an important role in certainties. From this row we find that the best-fit values of
thecosmologyAlcaniz&Lima(1999);Lanetal.(2010);Ben- Ω and w given by the Combined sample are in between the
m0
galyJr. etal.(2014);Liu&Zhang(2014)). Obviously,theUni- best-fit results given by the SALT2 sample and by the SiFTO
verse cannot be younger than its constituents. In other words, sample. ThisresultisconsistentwiththeresultofConleyetal.
theageoftheuniverseatanyhighredshiftzcannotbeyounger (2011). The second row of table 1 shows the fitting results for
than its constituents at the same redshift. There are some old thecaseofconsideringconstantβandstatistical+systematicun-
high redshift objects (OHRO) considered extensively in the lit- certainties. From thisrow wecan see that, once the systematic
erature. For instance, the 3.5Gyr old galaxy LBDS 53W091 at uncertaintiesofSNLS3samplesaretakenintoaccount,thebest-
redshift z = 1.55 Dunlop et al. (1996), the 4.0Gyr old galaxy fit values of Ω and w given by the Combined sample are not
m0
LBDS 53W069 at redshift z = 1.43 Dunlop (1999), and the in between the best-fit results given by the SALT2 sample and
old quasar APM 08279+5255, whose age is estimated to be by the SiFTO sample any more. Therefore, the reason causing
2.0 Gyr, at redshift z = 3.91 Hasinger, Schartel, & Komossa thisstrangephenomenonisduetothesystematicuncertaintiesof
(2002). Intheliterature,theagedataofthesethreeOHRO(i.e. SNLS3samples. Tofurtherstudythisissue,inthethirdrowof
t ≡ t(z = 1.43) = 4.0Gyr, t ≡ t(z = 1.55) = 3.5Gyr and table1,wepresentthefittingresultsforthecaseofconsidering
1.43 1.55
t ≡ t(z = 3.91) = 2.0Gyr)havebeenextensivelyusedtotest linear β and statistical+systematic uncertainties. We find that,
3.91
variouscosmologicalmodels(seee.g. Alcaniz, Lima&Cunha afterconsideringtheevolutionofβ,thebest-fitvalueofwgiven
(2003);Wei&Zhang(2007);Wang&Zhang(2008);Wang,Li by the Combined sample is in between the results given by the
&Li(2010);Yan, Liu&Wei(2014)). Inthepresentwork, we SALT2sampleandbytheSiFTOsample,whilethedifferences
Articlenumber,page5of12
Table1.FittingresultsforthewCDMmodelgivenbyvariousSNLS3sampleswithoutandwithsystematicuncertainties.Boththebest-fitvalues
andthe1σerrorsofΩ andwarelisted. “StatOnly”and“StatPlusSys”representtheSNdataonlyincludingstatisticaluncertaintiesandthe
m0
SNdataincludingbothstatisticalandsystematicuncertainties,respectively. Moreover,forcomparison,boththecasesofconstantβandlinearβ
aretakenintoaccount.
Combined SALT2 SiFTO
Ω 0.188+0.079 0.216+0.069 0.178+0.105
Constantβ(StatOnly) m0 −0.059 −0.047 −0.141
w −0.88+0.14 −0.93+0.14 −0.84+0.19
−0.11 −0.12 −0.22
Ω 0.167+0.084 0.236+0.083 0.220+0.083
Constantβ(StatPlusSys) m0 −0.070 −0.060 −0.061
w −0.88+0.18 −0.99+0.19 −1.01+0.19
−0.11 −0.16 −0.15
Ω 0.171+0.085 0.186+0.095 0.21+0.11
linearβ(StatPlusSys) m0 −0.070 −0.13 −0.11
w −0.88+0.18 −0.856+0.22 −0.96+0.26
−0.12 −0.096 −0.11
Table 2. Fitting results for the wCDM model and the three IDE models, where both the best-fit values and the 1σ errors of various pa-
rameters are listed. “Combined”, “SALT2” and “SiFTO” represent the SN(Combined)+CMB+GC, the SN(SALT2)+CMB+GC and the
SN(SiFTO)+CMB+GCdata,respectively.
wCDM IwCDM1 IwCDM2 IwCDM3
Parm Combined SALT2 SiFTO Combined SALT2 SiFTO Combined SALT2 SiFTO Combined SALT2 SiFTO
α 1.417+0.068 1.576+0.135 1.360+0.049 1.406+0.080 1.597+0.110 1.345+0.066 1.430+0.051 1.581+0.124 1.358+0.053 1.414+0.065 1.602+0.110 1.357+0.053
ΩΩΩββbcr01000 15000.....412003144009050−0+−+−+−+−0000000008.........0215700004788870000199618922 23000.....072002234081750−+−0+−+−+−0000000008.........1216500003229151000501000921 15000.....412008634008650−+−+−+−+−00000000008.........0227600003455711000669171721 01400.....5920009340327600−+−+−+−+−0000000008.........0226600005528721000980851722 23000.....062005934006850−+−0+−+−+−0000000008.........1115400004389860000418559922 01500.....4120089340898500−+−+−+−+−0000000008.........0226600006336880100455659022 01500.....402006234098950−1+−+−+−+−0000000000.........0226600004854480000046699821 23000.....082000034025550−+−+−0+−+−0000000008.........1215600007226021000457792621 15000.....4120068340237501−+−+−+−+−0000000000.........0226600000483781000586230922 01400.....582002944067150−1+−+−+−+−0000000000.........0127500002699910100283678022 23000.....072005034027650−+−0+−+−+−0000000009.........1126500007471001000398110722 15000.....4220071340586500−+−+−+−+−0000000009.........0226600001455571000496260721
γ .......... .......... .......... 0.0008+0.0020 −0.0001+0.00260.0001+0.0024 0.0024+0.0049 0.0008+0.0052 0.0013+0.0047 0.0017+0.0145 0.0039+0.0094 −0.0002+0.0136
w −1.051+0.049 −1.066+0.053 −1.068+0.047 −1.039−+00..0005207 −1.062+0−.00.508024−1.066−+00..0005285 −1.038−+00..0005683 −1.062−+00..0006651 −1.050−+00..0004688 −1.043−+00..0016050 −1.055−+00..0015445 −1.067+0−.00.600119
h 0.701+0−.001.0241 0.706+0−.001.0143 0.707+0−.001.0040 0.698+0−.001.0856 0.704+0−.001.0848 0.733+0−.001.0751 0.701+0−.001.0146 0.708+0−.000.0951 0.705+0−.001.0468 0.700+0−.001.0345 0.705+0−.001.0258 0.707+0−.001.0053
−0.013 −0.014 −0.015 −0.018 −0.019 −0.017 −0.014 −0.016 −0.013 −0.012 −0.014 −0.014
amongthebest-fitvaluesofΩ givenbythethreeSNLS3sam- best-fitpointsgivenbythethreedatasetscorrespondtoaphan-
m0
plesareeffectivelyreduced. Theseresultsimplytheimportance tom,bothphantom,quintessenceandcosmologicalconstantare
of considering β’s evolution in the cosmology-fits. Therefore, consistentwiththecurrentcosmologicalobservationsat2σCL.
fromnowonweonlyconsiderthecaseoflinearβ. This means that the current observational data are still too lim-
itedtoindicatethenatureofDE.
Intable2,byadoptingalinearβ,wegivethefittingresultsof
thewCDMmodelandthethreeIDEmodels. Fromthistablewe
seethat,forallthefourDEmodels,βsignificantlydeviatesfrom 3.2. HubbleParameter,DecelerationParameterand
aconstant,consistentwiththeresultsofWang,etal.(2014); in StatefinderHierarchy
addition,thereisnoevidencefortheexistenceofdarksectorin-
teraction. Although the best-fit results of w given by the three The1σconfidenceregionsofHubbleparameterH(z)atredshift
LCF are always less than −1, w = −1 is still consistent with region [0,4] for the wCDM model and the three IDE models
the current cosmological observations at 2σ CL. Moreover, we areplottedinFig. 2, wherethetwo H(z)datapoints, H0.57 and
check the impacts of different SNLS3 LCF on parameter esti- H2.34,arealsomarkedbydiamondswitherrorbarsforcompar-
mationandfindthatforallthefourDEmodels: (1)The“Com- ison. Wefind thatthedatapoint H0.57 canbeeasilyaccommo-
bined”samplealwaysgivesthelargestw;inaddition,thevalues dated in the wCDM model and the three IDE models, but the
ofwgivenbythe“SALT2”andthe“SIFTO”samplearecloseto datapointH2.34 significantlydeviatesfromthe1σregionsofall
eachother. (2)TheeffectsofdifferentLCFonotherparameters thefourDEmodels. Inotherwords,themeasurementofH2.34is
are negligible. It is clear that these results are insensitive to a intensionwithothercosmologicalobservationsandthisresultis
specificDEmodel. consistentwiththeconclusionofSahni,Shafieloo&Starobinsky
(2014);Huetal.(2014). Inaddition,the1σconfidenceregions
In Fig. 1, we plot the probability contours at the 1σ and of H(z) given by different LCF are almost overlap; this means
2σ CL in the γ-w plane, for the three IDE models. The best-
that using H(z) diagram is almost impossible to distinguish the
fit values of {γ,w} of the “Combined”, the “SALT2” and the differencesamongdifferentSNLS3LCF.
“SiFTO” data are marked as a black square, a red triangle and
InFig. 3,weplotthe1σconfidenceregionsofdeceleration
a green diamond, respectively. For comparison, the fixed point
parameterq(z)atredshiftregion[0,4],forthewCDMmodeland
{γ,w} = {0,−1} for the ΛCDM model is also marked as a blue
thethreeIDEmodels. Again,weseethatthe1σconfidencere-
round dot. A most obvious feature of this figure is that, for all gions of q(z) given by different LCF are almost overlap. This
the IDE models, the fixed point {0,−1} of the ΛCDM model is impliesthatusingq(z)diagramalsohasgreatdifficultytodistin-
outside the 1σ contours given by the three data sets; however, guishthedifferencesamongdifferentSNLS3LCF.
theΛCDMmodelisstillconsistentwiththeobservationaldata
In Fig. 4, we plot the 1σ confidence regions of S(1)(z) at
at 2σ CL. Moreover, according to the evolution behaviors of 3
redshift region [0,4], for the wCDM model and the three IDE
ρ (see Eqs. 10, 12 and 16) at z → −1, we divide these γ-w
de models. Fromthisfigureweseethat,the1σconfidenceregions
planes into two regions: the region above the dividing line de-
notesaquintessencedominatedUniverse(withoutbigrip), and of S3(1)(z) given by the three SNLS3 LCF are almost overlap at
the region below the dividing line represents a phantom domi- high redshift. Although there is an separating trend for S(1)(z)
3
natedUniverse(withbigrip). Wecanseethat,althoughallthe when z → 0, most parts of these three 1σ regions are overlap
Articlenumber,page6of12
Wangelal.:ImpactsofdifferentSNLS3LCFoncosmologicalconsequencesofIDEmodels
IIwwCCDDMM11 IIwwCCDDMM22
−0.9 −0.9
NNoo BBiigg RRiipp
−0.95 NNoo BBiigg RRiipp −0.95
−1 −1
BBiigg RRiipp
ww−1.05 ww−1.05
BBiigg RRiipp
−1.1 −1.1
SALT2
SALT2 SiFTO
−1.15 SCioFmTObined −1.15 CΛoCmDMbined
ΛCDM
−1.2 −1.2
−0.006−0.004−0.002 0 0.002 0.004 0.006 0.008 −0.02 −0.01 0 0.01 0.02
γ γ
IIwwCCDDMM33
NNoo BBiigg RRiipp
−0.9
−1
ww
−1.1
SALT2
SiFTO
Combined
−1.2 BBiigg RRiipp ΛCDM
−0.03 −0.02 −0.01 0 0.01 0.02 0.03
γ
Fig.1. (coloronline). Probabilitycontoursatthe1σand2σCLintheγ-wplane, fortheIwCDM1(upperleftpanel), theIwCDM2(upper
rightpanel)andtheIwCDM3(lowerpanel)model.“Combined”(blacksolidlines),“SALT2”(reddashedlines),and“SiFTO”(greendash-dotted
lines)denotetheresultsgivenbytheSN(Combined)+CMB+GC,theSN(SALT2)+CMB+GC,andtheSN(SiFTO)+CMB+GCdata,respectively.
Furthermore,thebest-fitvaluesof{γ,w}ofthe“Combined”,the“SALT2”andthe“SiFTO”dataaremarkedasablacksquare,aredtriangleand
agreendiamond,respectively. Tomakeacomparison,thefixedpoint{γ,w}={0,−1}fortheΛCDMmodelarealsomarkedasabluerounddot.
Thegraysolidlinedividesthepanelintotworegions:theregionabovethedividinglinedenotesaquintessencedominatedUniverse(withoutbig
rip),andtheregionbelowthedividinglinerepresentsaphantomdominatedUniverse(withbigrip).
evenatcurrentepoch. Thismeansthat,althoughS(1)(z)isabet- andt canbeeasilyaccommodatedinallthefourDEmodels,
3 1.55
ter tool than H(z) and q(z), it still have difficulty to distinguish butthepositionoft issignificantlyhigherthanthe2σupper
3.91
the effects of different SNLS3 LCF. Moreover, our conclusion boundsofallthefourDEmodels. Inotherwords,theexistence
holdstrueforallthefourDEmodels,showingthatthisconclu- of the old quasar APM 08279+5255 still can not be explained
sionisinsensitivetoaspecificinteractionform. in the frame of IDE model. This result is consistent with the
In Fig. 5, we plot the 1σ confidence regions of S(1)(z) at conclusionsofpreviousstudiesAlcaniz,Lima&Cunha(2003);
redshift region [0,4], for the wCDM model and the thr4ee IDE Wei & Zhang (2007); Wang & Zhang (2008); Wang, Zhang &
models. Fromthisfigureweseethatmostofthe1σconfidence Xia (2008); Wang, Li & Li (2010); Yan, Liu & Wei (2014). In
regionsofS(1)(z)givenbythethreeLCFareoverlap.Thismeans addition,the2σregionsoft(z)givenbydifferentLCFarealmost
thattheeffec4tsofdifferentLCFcannotbedistinguishedbyusing overlap, showing that the impacts of different SNLS3 LCF can
notbedistinguishedbyusingtheagedataofOHRO.
thestatefinderS(1)(z)either. Again,weseethatthisconclusion
4 We want to infer how far we are from a cosmic doomsday
isinsensitivetoaspecificinteractionform.
in the worst case. So in Fig. 7, we plot the 2σ lower limits of
In conclusion, we find that the differences given by
thetimeintervalt−t betweenafuturemomentandtoday, for
“SALT2”,“SiFTO”,and“Combined”LCFarerathersmalland 0
thewCDMmodelandthethreeIDEmodels. Moreover,the2σ
cannotbedistinguishedbyusingH(z),q(z),S(1)(z)andS(1)(z).
This result is quite different from the case of “3MLCS2k24” Jha lower limit values of tBR −t0 given by the three LCF are also
markedonthisfigure. 8. Alltheevolutioncurvesoft−t tend
et al. (2007) and “SALT2” Guy et al. (2007), where using tothecorrespondingconvergencevaluesat−ln(1+z)(cid:39)200. The
“MLCS2k2” and “SALT2” LCF will give completely different
most important factor of determining t −t is the EoS w. In
cosmologicalconstraintsforvariousmodelsBengochea(2011); BR 0
a phantom dominated Universe, a smaller w corresponds to a
Bengochea&DeRossi(2014).
largerincreasingrateofρ ; thismeansthatallthegravitation-
de
allyboundstructureswillbetornapartinashortertime,andthe
Universe will encounter a cosmic doomsday in a shorter time,
3.3. CosmicAgeandFateofTheUniverse
too. Among the three SNLS3 LCF data, the “Combined” sam-
The2σconfidenceregionsofcosmicaget(z)atredshiftregion plealwaysgivesthelargestw,andthusgivesthelargestvalueof
[0,4]forthewCDMmodelandthethreeIDEmodelsareplotted
inFig. 6,wherethethreet(z)datapoints,t1.43,t1.55andt3.91,are 8 The2σupperlimitsoftBR−t0isinfinite;inotherwords,theUniverse
alsomarkedbySquaresforcomparison. Wefindthatbotht1.43 willexpandeternally
Articlenumber,page7of12
wCDM IwCDM1
500 500
168 168
450 166 450 166
164 164
400 400
162 162
350 160 350 160
158 158
z)300 1516.50 1.52 1.54 1.56 1.58 1.60 z)300 1516.50 1.52 1.54 1.56 1.58 1.60
H( H(
250 250
200 H2.34 = 222±7.0 km/s/Mpc 200 H2.34 = 222±7.0 km/s/Mpc
150 150
SALT2(1σ) SALT2(1σ)
100 SiFTO(1σ) 100 SiFTO(1σ)
H0.57 = 92.4±4.5 km/s/Mpc Combined(1σ) H0.57 = 92.4±4.5 km/s/Mpc Combined(1σ)
50 50
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
z z
IwCDM2 IwCDM3
500 500
168 168
450 166 450 166
164 164
400 400
162 162
350 160 350 160
158 158
z)300 1516.50 1.52 1.54 1.56 1.58 1.60 z)300 1516.50 1.52 1.54 1.56 1.58 1.60
H( H(
250 250
200 H2.34 = 222±7.0 km/s/Mpc 200 H2.34 = 222±7.0 km/s/Mpc
150 150
SALT2(1σ) SALT2(1σ)
100 SiFTO(1σ) 100 SiFTO(1σ)
H0.57 = 92.4±4.5 km/s/Mpc Combined(1σ) H0.57 = 92.4±4.5 km/s/Mpc Combined(1σ)
50 50
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
z z
Fig. 2. (color online). The 1σ confidence regions of Hubble parameter H(z) at redshift region [0,4], for the wCDM (upper left panel), the
IwCDM1 (upper right panel), the IwCDM2 (lower left panel) and the IwCDM3 (lower right panel) model, where the data points of H and
0.57
H arealsomarkedbydiamondswitherrorbarsforcomparison. “Combined”(grayfilledregions),“SALT2”(bluesolidlines),and“SiFTO”
2.34
(purple dashed lines) denote the results given by the SN(Combined)+CMB+GC, the SN(SALT2)+CMB+GC, and the SN(SiFTO)+CMB+GC
data,respectively.
t −t . Inaddition,thevaluesofwgivenbythe“SALT2”and rameterspacesofthewCDMmodelandthethreeIDEmodels.
BR 0
the“SIFTO”sampleareclosetoeachother,andthusthevalues According tothe results of cosmology-fits, we have plotted the
oft −t givenbythesetwosamplesareclosetoeachother. cosmic evolutions of Hubble parameter H(z), deceleration pa-
BR 0
rameter q(z), statefinder hierarchy S(1)(z) and S(1)(z), and have
3 4
checkedwhetherornottheseDEdiagnosescandistinguishthe
4. SummaryandDiscussion differencesamongtheresultsofdifferentLCF.Furthermore,we
As is well known, different LCF will yield different SN sam- haveperformedhigh-redshiftcosmicagetestusingthreeOHRO,
ple. In 2011, based on three different LCF, the SNLS3 group andhaveexploredthefateoftheUniverse.
Conley et al. (2011) released three kinds of SN samples, i.e.,
WefindthatforthewCDMandallthethreeIDEmodels:(1)
“SALT2”, “SiFTO” and “Combined”. So far, only the SNLS3
the“Combined”samplealwaysgivesthelargestw;inaddition,
“Combined” sample is studied extensively, both the “SALT2”
thevaluesofwgivenbythe“SALT2”andthe“SIFTO”sample
andthe“SiFTO”datasetsareseldomtakenintoaccountinthe are close to each other. (see table 2 and Fig. 1); (2) the effects
literature. Moreover, the cosmological consequences given by ofdifferentSNLS3LCFonotherparametersarenegligible(see
thesethreeSNLS3LCFarenotdiscussedbefore. Therefore,the table2). Besides,wefindthattheΛCDMmodelisinconsistent
impacts of different SNLS3 LCF have not been studied in de-
with the three SNLS3 samples at 1σ CL, but is still consistent
tailinthepast. Themainaimofthepresentworkispresenting
withtheobservationaldataat2σCL.
acomprehensiveandsystematicinvestigationontheimpactsof
differentSNLS3LCF. Moreover,wefindthattheimpactsofdifferentSNLS3LCF
Since the interaction between different components widely arerathersmallandcannotbedistinguishedbyusingH(z)(see
exist in nature, and the introduction of a interaction between Fig. 2),q(z)(seeFig. 3),S(1)(z),(seeFig. 4),S(1)(z)(seeFig.5),
3 4
DE and CDM can provide an intriguing mechanism to solve andt(z)diagram(seeFig. 6). Thisresultisquitedifferentfrom
the“cosmiccoincidenceproblem”andalleviatethe“cosmicage thecaseof“MLCS2k2”Jhaetal.(2007)and“SALT2”Guyetal.
problem”, here we adopt the wCDM model with a dark sector (2007), where using “MLCS2k2” and “SALT2” LCF will give
interaction. To ensure that our study is insensitive to a specific completely different cosmological constraints for various mod-
darksectorinteraction,threekindsofinteractiontermsaretaken elsBengochea(2011);Bengochea&DeRossi(2014). Inaddi-
into account in this work. In addition, to make a comparison, tion, we infer how far we are from a cosmic doomsday in the
wealsoconsiderthecaseofwCDMmodelwithoutdarksector worst case, and find that the “Combined” sample always gives
interaction. thelargest2σlowerlimitoft −t ,whiletheresultsgivenby
BR 0
WehaveusedthethreeSNLS3datasets, aswellastheob- the “SALT2” and the “SiFTO” sample are close to each other
servationaldatafromtheCMBandtheGC,toconstrainthepa- (seeFig. 7).
Articlenumber,page8of12
Wangelal.:ImpactsofdifferentSNLS3LCFoncosmologicalconsequencesofIDEmodels
wCDM IwCDM1
0.6 0.6
0.4 0.4
0.2 0.14 0.2 0.14
0.13 0.13
0.0 0.12 0.0 0.12
0.11 0.11
q(z) 0.2 00..1009 q(z) 0.2 00..1009
0.08 0.08
0.4 0.07 0.4 0.07
0.86 0.88 0.90 0.92 0.94 0.96 0.86 0.88 0.90 0.92 0.94 0.96
0.6 0.6
SALT2(1σ) SALT2(1σ)
0.8 SiFTO(1σ) 0.8 SiFTO(1σ)
Combined(1σ) Combined(1σ)
1.0 1.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
z z
IwCDM2 IwCDM3
0.6 0.6
0.4 0.4
0.2 0.14 0.2 0.14
0.13 0.13
0.0 0.12 0.0 0.12
0.11 0.11
q(z) 0.2 00..1009 q(z) 0.2 00..1009
0.08 0.08
0.4 0.07 0.4 0.07
0.86 0.88 0.90 0.92 0.94 0.96 0.86 0.88 0.90 0.92 0.94 0.96
0.6 0.6
SALT2(1σ) SALT2(1σ)
0.8 SiFTO(1σ) 0.8 SiFTO(1σ)
Combined(1σ) Combined(1σ)
1.0 1.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
z z
Fig. 3. (color online). The 1σ confidence regions of deceleration parameter q(z) at redshift region [0,4] for the wCDM (upper left
panel), the IwCDM1 (upper right panel), the IwCDM2 (lower left panel) and the IwCDM3 (lower right panel) model. “Combined” (gray
filledregions), “SALT2”(bluesolidlines), and“SiFTO”(purpledashedlines)denotetheresultsgivenbytheSN(Combined)+CMB+GC,the
SN(SALT2)+CMB+GC,andtheSN(SiFTO)+CMB+GCdata,respectively.
wCDM IwCDM1
1.5 1.5
SALT2(1σ) SALT2(1σ)
SiFTO(1σ) SiFTO(1σ)
1.4 Combined(1σ) 1.4 Combined(1σ)
1.3 1.3
(1)(z)S31.2 (1)(z)s31.2
1.1 1.1
1.0 1.0
0.9 0.9
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
z z
IwCDM2 IwCDM3
1.5 1.5
SALT2(1σ) SALT2(1σ)
SiFTO(1σ) SiFTO(1σ)
1.4 Combined(1σ) 1.4 Combined(1σ)
1.3 1.3
z) z)
(1)(S31.2 (1)(S31.2
1.1 1.1
1.0 1.0
0.9 0.9
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
z z
Fig. 4. (color online). The 1σ confidence regions of S(1)(z) at redshift region [0,4] for the wCDM (upper left panel), the IwCDM1 (upper
3
right panel), the IwCDM2 (lower left panel) and the IwCDM3 (lower right panel) model. “Combined” (gray filled regions), “SALT2” (blue
solidlines),and“SiFTO”(purpledashedlines)denotetheresultsgivenbytheSN(Combined)+CMB+GC,theSN(SALT2)+CMB+GC,andthe
SN(SiFTO)+CMB+GCdata,respectively.
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wCDM IwCDM1
1.5 1.5
SALT2(1σ) SALT2(1σ)
SiFTO(1σ) SiFTO(1σ)
1.4 Combined(1σ) 1.4 Combined(1σ)
1.3 1.3
z) z)
(1)(S41.2 (1)(S41.2
1.1 1.1
1.0 1.0
0.9 0.9
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
z z
IwCDM2 IwCDM3
1.5 1.6
SALT2(1σ) SALT2(1σ)
SiFTO(1σ) 1.5 SiFTO(1σ)
1.4 Combined(1σ) Combined(1σ)
1.4
1.3
1.3
z) z)
(1)(S41.2 (1)(S41.2
1.1
1.1
1.0
1.0
0.9
0.9 0.8
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
z z
Fig. 5. (color online). The 1σ confidence regions of S(1)(z) at redshift region [0,4] for the wCDM (upper left panel), the IwCDM1 (upper
4
right panel), the IwCDM2 (lower left panel) and the IwCDM3 (lower right panel) model. “Combined” (gray filled regions), “SALT2” (blue
solidlines),and“SiFTO”(purpledashedlines)denotetheresultsgivenbytheSN(Combined)+CMB+GC,theSN(SALT2)+CMB+GC,andthe
SN(SiFTO)+CMB+GCdata,respectively.
wCDM IwCDM1
14
2.70 SALT2(2σ) 14 2.70 SALT2(2σ)
12 2.65 SCioFmTbOin(2eσd)(2σ) 12 2.65 SCioFmTbOin(2eσd)(2σ)
2.60 2.60
10
2.55 10 2.55
8 2.50 2.50
z) 2.50 2.52 2.54 2.56 2.58 2.60 z) 8 2.50 2.52 2.54 2.56 2.58 2.60
t( t(
6
6
t1.43 = 4.0 Gyr t1.43 = 4.0 Gyr
4 4
t3.91 = 2.0 Gyr t3.91 = 2.0 Gyr
2 t1.55 = 3.5 Gyr 2 t1.55 = 3.5 Gyr
0 0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
z z
IwCDM2 IwCDM3
14 2.70 SALT2(2σ) 14 2.70 SALT2(2σ)
SiFTO(2σ) SiFTO(2σ)
12 2.65 Combined(2σ) 12 2.65 Combined(2σ)
2.60 2.60
10 2.55 10 2.55
2.50 2.50
z) 8 2.50 2.52 2.54 2.56 2.58 2.60 z) 8 2.50 2.52 2.54 2.56 2.58 2.60
t( t(
6 6
t1.43 = 4.0 Gyr t1.43 = 4.0 Gyr
4 4
t3.91 = 2.0 Gyr t3.91 = 2.0 Gyr
2 t1.55 = 3.5 Gyr 2 t1.55 = 3.5 Gyr
0 0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
z z
Fig.6. (coloronline). The2σconfidenceregionsofcosmicaget(z)atredshiftregion[0,4],forthewCDM(upperleftpanel),theIwCDM1
(upperrightpanel),theIwCDM2(lowerleftpanel)andtheIwCDM3(lowerrightpanel)model.Threet(z)datapoints,t ,t andt ,arealso
1.43 1.55 3.91
markedbySquaresforcomparison. “Combined”(grayfilledregions),“SALT2”(bluesolidlines),and“SiFTO”(purpledashedlines)denotethe
resultsgivenbytheSN(Combined)+CMB+GC,theSN(SALT2)+CMB+GC,andtheSN(SiFTO)+CMB+GCdata,respectively.
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