In this talk we review the appearance of new types of singularities (big rip,
sudden singularities...) in FLRW cosmological models that have arisen on
considering explanations for accelerated expansion of our universe.Comment: 3 pages, ws-procs975x65.cls to appear in Proceedings of 12th Marcel
  Grossmann Meeting, Pari

Fernandez-Jambrina, L.

Lazkoz, R.

English

arXiv

In this talk we review the appearance of new types of singularities (big rip, sudden singularities. . . ) in FLRW cosmological models that have arisen on considering explanations for accelerated expansion of our univers

Fernández Jambrina, Leonardo

Lazkoz, Ruth

Archivo Digital UPM

Cosmological singularities in FLRW spacetimes

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COSMOLOGICAL SINGULARITIES IN FLRW SPACETIMES 
L. FERNÁNDEZ-JAMBRINA 
ETSI Navales, Universidad Politécnica de Madrid 
Arco de la Victoria s/n, E-28040-Madrid, Spain 
* E-mail: leonardo.fernandez@upm.es 
www. etsin. upm. es/ilfj. htm, 
R. LAZKOZ 
Física Teórica, Facultad de Ciencia y Tecnología, 
Universidad del País Vasco, Apdo. 644> E-48080 Bilbao, Spain 
E-mail: ruth.lazkoz@ehu.es 
tp.lc.ehu.es/RLS.html 
In this talk we review the appearance of new types of singularities (big rip, sudden singu-
larices. . . ) in FLRW cosmological models that have arisen on considering explanations 
for accelerated expansión of our universe. 
Keywords: Cosmological singularities, dark energy, geodesics. 
1. Introduction 
During the last decade the observational evidence in favour of accelerated expan-
sión of our universe has increased considerably.1 In order to cope with this fact two 
lines of development have been proposed: either general relativity is to be modi-
fied2 or our universe is pervaded by an exotic source named dark energy,3 which is 
responsible for the accelerated expansión. Violation of classical energy conditions 
by this fluid has made feasible a number of new scenarios4 for the final fate of our 
universe (big rip,5 sudden singularities,6 big brake,7 big freeze,8 inaccesible singu-
larities,9 directional singularities,10 w-singularities,11 in braneworld models12 . . .), 
which had not been considered previously. In this talk we deal with the appearance 
and strength of these singular behaviours in FLRW cosmological spacetimes from 
the point of view of geodesic incompleteness. 
2. Behaviour of singularities 
We choose coordinates for which the FLRW metric is written as 
ds2 = -dt2+a2(t)\f2(r)dr2+r2dQ2}, f2(r) = \ , , A; = 0, ± 1 , 
1 — krz 
and we assume that the scale factor a(t) admits a generalised power expansión13 in 
the vicinity of an event at to, 
a(t) = c0\t-t0\ri0+c1\t - t i l " 1 + ••• , 
where r/o < rr}\ < • • •, CQ > 0. Models with oscillatory behaviour14 might drop out 
of this scheme. At lowest order three different qualitative behaviours arise: 
• r/o > 0: a vanishing scale factor at to means a Big Bang or Big Crunch. 
• -/yo = 0: a finite scale factor at to rneans that either a(t) is analytical and the 
event is regular or on the other hand a weak or sudden singularity appears. 
• r/o < 0: a diverging scale factor at to rneans a Big Rip singularity. 
Geodesics are parametrised by their proper time T, ( Í (T) , r(r), 0(T) , </>(T)) , 
a2(í) ' r a2(í) / ( r) ' 
may be simplified in terms of a constant of geodesic motion P and 5 = 0,1 for nuil 
and timelike geodesics. The nuil case S = 0 is straightforwardly solved 
a(t)t' =P => / a(í') di' = P ( T - T0) , 
and the geodesic is complete if and only if the scale factor is an integrable function. 
Dealing with finite bodies, a singularity is strong15 if tidal forces are capable of 
disrupt them. Following Tipler, a singularity is strong if their volume tends to zero. 
Folllowing Królak, just a negative derivative of the volume is required. The results 
are consigned in Table l.16 Conformal diagrams for these models can be found in.17 
Table 1. Strength of singularities in FLRW models. 
?7o Í?I k CQ Tipler Królak 
( -oo,0) 
0 
0 
(0,1) 
1 
(»?o,°o) 
(0,1) 
[l,oo) 
(770,00) 
( l ,oo) 
0 ,±1 
0, ± 1 
0, ± 1 
0, ± 1 
0,1 
(0 ,00 
(0 ,00 
(0 ,00 
(0 ,00 
(0 ,00 
Strong 
Weak 
Weak 
Strong 
Strong 
Strong 
Strong 
Weak 
Strong 
Strong 
1 (l,oo) - 1 (0, 1)U(1,00) Strong Strong 
1 (1,3) - 1 1 Weak Strong 
1 [3,oo) - 1 1 Weak Weak 
(l,oo) (r/o, 00) 0, ±1 (0,oo) Strong Strong 
3. Conclusions 
We have obtained a classification of singular events in FLRW models in terms of 
their strength and the exponents of an expansión of the scale factor around an event 
at to- In models with 770 < — 1, nuil geodesics avoid the Big Rip singularity. The 
only models which allow regular behaviour cióse to to are those with 770 = 0 (de 
Sitter, sudden...) and with rjo = 1, k = — 1, CQ = 1 (Milne). 
Acknowledgements 
R.L. is supported by the University of the Basque Country through research grant 
GIU06/37 and by the Spanish Ministry of Education and Culture through research 
grant FIS2007-61800. 
(
' = V 
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Cosmological singularities in FLRW spacetimes

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