Table Of ContentSingularities around w = 1
−
Leonardo Fernández-Jambrina
ETSINavales,ArcodelaVictorias/n,E-28040-Madrid,Spain
5
Abstract. In this talk we would like to analyse the appearanceof singularities in FLRW cosmo-
1
logicalmodelswhichevolveclosetow= 1,wherewisthebarotropicindexoftheuniverse.We
0 −
relatesmalltermsincosmologicaltimearoundw= 1withthecorrespondentscalefactorofthe
2 −
universeandcheckfortheformationofsingularities.
n
a Keywords: Singularities,cosmology,BigRip,geodesiccompleteness
J PACS: 98.80.-k,04.20.Dw
6
2
INTRODUCTION
]
c
q
Accelerated expansion of our universe has been established observationally in the last
-
r decade from several sources of information. Even more, the barotropic index w of the
g
[ equation of state of the content of the universe, that is the ratio between pressure and
energydensity,hasbeencheckedtobecloseto 1,valuewhichhasbeendubbedasthe
2
−
v phantomdivide.Thisvaluecorrespondstoacosmologicalconstantasenergycontentof
0 theuniverse.
6
The fact of the accelerated expansion of the universe has lead to several scenarios
9
3 for a singular fate of the universe, different from eternal expansion or Big Crunch. A
0 classification of such singularities is provided in [1]. Among these new scenarios one
.
1 can find Big Rip singularities [2], sudden singularities [3], Big Freeze singularities [4],
0
w-singularities[5]ordirectionalsingularities[6].
5
1 Some of then however have been shown to be not strong enough [7] to mean the
: actual end of the universe. For instance, sudden singularities [8] and w-singularities [9]
v
i are weak singularities. They have also been studied within the framework of modified
X
theoriesofgravitation[10].
r
a The main idea behind the previous analysis is to assume that the scale factor of the
universemaybewrittenasgeneralisedpowerexpansionofthetimecoordinate[11]and
thenanalysethegeodesiccompleteness[12]oftheFLRWspacetimeendowedwithsuch
scalefactor.
With these ideas in mind, we would like to analyse the behaviour of cosmological
models in the vicinity of w= 1. We shall show that there are scenarios which fall out
−
ofthepreviousclassificationand aretoberegarded separately.
PERTURBED BAROTROPIC INDEX
Let us assume that the barotropic index behaves approximately as a cosmological con-
stantatpresent time,thatis,
w= 1+h(t), h(t) 1. (1)
− | |≪
Itisformallypossibletogetanexplicitexpressionforthescalefactora(t)=ef(t) for
suchbarotropicindex,
p 1 2aa¨ 2 dt
w= = f(t)= +C.
r −3−3a˙2 ⇒ 3 h(t)dt+k
Z
The constant C is irrelevant since it amounts to aR change of scale or time origin,
a(t) eCa(t).Sincethedensityoftheenergy contentis
→
2
a˙ 2
r = = f˙2 √r = ,
a ⇒ 3 h+k
(cid:18) (cid:19)
wemayuseittofix theconstantk. R
Forinstance, takingtheorigin inthe nearfutureand assuminga power-lawdeviation
fromunity,h(t)=a ( t)p, p>0 sothatw(0)= 1,
− −
2(p+1) 1
r (t)= .
− 3a k+( t)p+1
−
p
Two differentcases arisein thesemodels:
• k=0: √r hassimplepolesoutoft =0 andr (t t0)−2, butw= 1 isregular.
6 ∼ − −
• k=0: Singulardensityand scalefactorat w= 1. Namingb =2(1+p)/3a p,
−
2
4 p+1 1 21+p
r (t)= , f(t)= ( t) p, a(t)=e b /( t)p.
9 a t2p+2 −3 a p − − − −
(cid:18) (cid:19)
Weareinterestedinthelattermodels,whichhaveasingulardensityatthetimewhen
w = 1 is reached. These models have a non-analytical scale factor with an essential
singu−larity at t = 0 and the energy density blow up as 1/t2p+2 instead of t 2, which
−
is the usual power for divergencies with analytical scale factors. Two possibilitiesarise
dependingon thesignoftheconstanta :
• Fora >0, wehaveaGreatCrunch:a(t) 0.
→
• Fora <0, wehaveaGreatRip:a(t) ¥ .
→
GEODESICS NEAR w = 1
−
Equationsforcausal geodesicsparametrised bypropertimet ,
dt = g dxidxj,
ij
−
q
inaflat FLRW modelreduce to
dt P2 dr P
= d + , = ,
dt s a2(t) dt ±a2(t)
whereP isaconstantofmotionand d =0 fornulland d =1for timelikegeodesics.
Thecaseofnullgeodesicsis simplersincetheequationsmaybeintegrated,
0
t =P 1 a(t)dt.
−
t
Z0
ThismeansittakesaninfinitepropertimetoGreatRip(a(0)=¥ ),sothissingularity
isnotaccesiblealonglightlikegeodesics.
On thecontrary, fortimelikegeodesics,
0 dt
t = ,
t 1+P2a 2(t)
Z0 −
allgeodesicsreach w= 1 in finiteproppertime.
−
Hence w = 1 becomes singular but for null geodesics. This sort of behaviour is
−
similarto theoneinthevicinityofaBig Rip singularity.
STRONG SINGULARITIES
In spite of the singular character of w= 1, it could happen that the singularity could
−
be not strong enough to be the end of the universe, since extended objects might avoid
beingdisruptedbytidalforces on crossingthesingularity.
ThisconceptofstrengthofsingularitieswasfirstcoinedbyEllisandSchmidt[12]and
it was related to the cosmic censorship conjecture later on. It was further developed by
otherauthors. For instance, for Tipler[13] a singularityis strong if thevolumespanned
by three Jacobi fields orthogonal to the velocity of the geodesic tends to zero as it
approaches its end. Królak [14] suggests a less restrictive criterion, requiring just the
derivativeofthevolumewithrespect to propertimetobenegative.
Since these definitions involve calculations with Jacobi fields, checking the strength
of singularities may be cumbersome. Fortunately, necessary and sufficient conditions
[15] have been derived involving just integrals of some components of the curvature
tensoralongincompletegeodesics.
Forinstance,followingTipler’sdefinition,asingularityisstrongalonganullgeodesic
ofvelocityu ifand onlyiftheintegral
t t
dt ′dt R uiuj
′ ′′ ij
0 0
Z Z
divergesas t tendsto t , wheret isthepropertimeassignedto thesingularity.
0 0
For Królak’s criterion the necessary and sufficient condition is the divergence of the
integral
t
dt R uiuj.
′ ij
0
Z
ast tends tot .
0
Such calculationsaresimpleinourcase andallowustoconcludethatsingularitiesat
w = 1 are strong according to both criteria. The same result is obtained for timelike
−
geodesics.
CONCLUSIONS
Wehaveshownthattherearetwopossiblebehavioursformodelswithabarotropicindex
closeto w= 1:
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• Regularcrossingofw= 1.
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• Essentialsingularityatw= 1:Great Crunch /Rip.
−
Forthelattermodelstheenergydensityblowsupatthesingularityanditsdivergence
is worse than t 2. The essential singularities are strong, though null geodesics never
−
reach theGreat Rip. Moredetailsand references may befoundelsewhere[16].
ACKNOWLEDGMENTS
The author wishes to thank the University of the Basque Country for their hospitality
andfacilitiesto carry out thiswork.
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