Inhomogenized sudden future singularities

0.13 MB·English

by  Mariusz P. Dcabrowski

Inhomogenized sudden future singularities
∗
Mariusz P. Da¸browski
Institute of Physics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland
(Dated: February 7, 2008)
We find that sudden future singularities of pressure may also appear in spatially inhomogeneous
Stephanimodelsoftheuniverse. Theyaretemporalpressuresingularitiesandmayappearindepen-
dentlyof thespatial finitedensity singularities already known toexist in these models. It is shown
that the main advantage of thehomogeneous sudden future singularities which is thefulfillment of
thestrong and weak energy conditions may not be thecase in the inhomogeneous case.
PACSnumbers: 98.80.Hw
I. INTRODUCTION conditions violation,whichmainly refersto the factthat
5 they admit FD singularities.
0 The SF singularities appear in the simple framework
It has been shown by Barrow [1] that for Friedmann
0
of Friedmann cosmology with the assumption that the
2 cosmological models which do not admit an equation of
energy-momentum is conserved, so that one can write
state which links the energy density ̺ and the pressure
n the energy density and pressure as follows (following [1]
p, a sudden future (SF) singularity of pressure may ap-
a
we assume 8πG=c=1, k=0, 1)
J pear,evenforthematterfulfillingthestrongenergycon- ±
9 dition ̺ > 0,̺ + 3p > 0, though violating the domi- a˙2 k
1 nant energy condition ̺ > 0, ̺ < p < ̺ [2]. This is ̺ = 3 + , (1)
− (cid:18)a2 a2(cid:19)
in contrast to the most of the current observational dis-

3 cussion in cosmology, mainly devoted to determining of a¨ a˙2 k
v p = 2 + + . (2)
the barotropic index w in a barotropic equation of state −(cid:18) a a2 a2(cid:19)
3
p=w̺, whichtightly constraintstheenergydensity and
3
the pressure [3]. On the other hand, the observational From (2) one is able to notice that the singularity of
0
0 data interpreted by such an equation of state cannot ex- pressure p occurs, when the accelerationa¨ .
→∞ →−∞
1 clude a possibility of barotropic phantom cosmological This can be achieved for the scale factor
4 models [4]. These models violate null energy condition q n
t t
0 ̺+p>0,andconsequentlyalltheremainingenergycon- a(t)=A+(a A) A 1 , (3)
s
c/ ditions [5]. Besides, phantommodels allowfor a Big-Rip − (cid:18)ts(cid:19) − (cid:18) − ts(cid:19)
q (BR) curvature singularity, which appears as a result of
where a a(t ) with t being the SF singularity time
- havingthe infinite values ofthe scalefactor a(t) atfinite s ≡ s s
r and A,q,n= const. It is obvious from (3) that a(0)=0
g future. This is in opposition to a curvature Big-Bang
andsoatzerooftime aBBsingularitydevelops. Forthe
: (BB) singularity which takes place in the limit a 0.
v → sake of further considerations it is useful to write down
i The common feature of BB and BR singularities is the derivatives of the scale factor (3), i.e.,
X
that both ̺ and p blow up equally. This is not the case
ar with a SF singularity for which a blow up occurs only t q−1 n t n−1
for the pressure p, but not for the energy density ̺. It is a˙ = qts(as A) +A 1 , (4)
− (cid:18)t (cid:19) t (cid:18) − t (cid:19)
s s s
interesting that SF singularities are similar to those ap-
pearing in spatially inhomogeneous Stephani models of a¨ = q(q 1)t2(a A) t q−2
the universe [6], in which they were termed finite den- − s s− (cid:18)t (cid:19)
s
sity (FD) singularities [7, 8]. However, FD singularities n−2
n(n 1) t
occur as singularities in spatial coordinates rather than A − 1 . (5)
− t2 (cid:18) − t (cid:19)
in time (as SF singularities do), which means that even s s
at the present moment of the evolution they may exist
The main point is that the evolution of the Universe, as
somewhere in the Universe [9, 10, 11]. In this letter we
described by the scale factor (3), begins with the stan-
show that sudden future (SF) singularities (as temporal
dard BB singularity at t = 0, and finishes at SF singu-
singularities) can be inhomogenized in the sense, that
larity at t=t , provided we choose
s
they may appear in spatially inhomogeneous models of
the universe, independently of the spatial finite density 1<n<2, 0<q 1 . (6)
≤
(FD)singularitiesallowedinthesemodels. Wealsoshow
that the inhomogeneous Stephani models lead to energy Forthese values ofn andq, the scale factor(3) vanishes,
and its derivatives (4)-(5) diverge at t = 0, leading to a
divergenceof̺andpin(1)-(2)(BBsingularity). Onthe
otherhand,thescalefactor(3)anditsfirstderivative(4)
∗Electronicaddress: [email protected] remain constant, while its second derivative (5) diverge,
2
leadingtoadivergenceofpressurein(2)only,withfinite with α= const. In fact, the limit α 0 gives the Fried-
→
energy density (1). This behaviour means, for example, mann models (cf. the discussion of the conditions to
thatpositivecurvature(k =+1)Friedmannmodelsmay derivesuchalimitin[8]). Inserting(11)into(10)weget
not recollapse to a second BB singularity – instead they
a˙2(t) α
terminate in a SF singularity [12].
̺(t) = 3 , (12)
(cid:20)a2(t) − a(t)(cid:21)
a¨ a˙2 α 1 a˙2
II. INHOMOGENEIZED SUDDEN FUTURE p(t,r) = 2 +2 αr2 2 2a¨ α (13.)
SINGULARITIES − a − a2 a − 4 (cid:18) a − − (cid:19)
From (13) one can see, that p , when acceleration
→ ∞
Now, let us consider inhomogeneous Stephani mod- a¨ for an arbitrary value of the radial coordinate
→ −∞
els. They appear as the only conformally flat perfect- r. We can then say that we generalized SF singulari-
fluid models which can be embedded in a 5-dimensional ties (given, for example, by the scale factor (3)) onto an
flat space [6, 13]. Their metric in the spherically sym- inhomogeneous model of the universe.
metric case reads as (notice that we have introduced a However, it is very interesting to notice that in such
Friedmann-like time coordinate which eliminated one of a generalization not only SF singularities appear, but
the functions of time in the originalStephani metric [8]) alsoFDsingularitiesarepossiblefortheradialcoordinate
r2 . These seem to be far awayfrom us, and so not
ds2 = a2 a2 V · 2dt2 ver→yh∞armful,sincer2 →∞definesanantipodalcenterof
− a˙2V2 (cid:20)(cid:18)a(cid:19)(cid:21) symmetryinthesphericallysymmetricStephanimodels.
a2 The advantage of these FD singularities is that they are
+ dr2 + r2 dθ2 + sin2θdϕ2 , (7) able to drivethe currentaccelerationofthe universe[10,
V2
(cid:2) (cid:0) (cid:1)(cid:3) 17, 18].
where TheprocedureofinhomogenizingSFsingularitiesmay
1 be extended into the general Stephani models for which
V(t,r)=1+ k(t)r2 , (8)
4 there is no spacetime symmetry at all, and so they are
completely inhomogeneous. The most general Stephani
·
and (...) ∂/∂t. The function a(t) plays the role
≡ metric in cartesian coordinates (x,y,z) [6, 13] reads as
of a generalized scale factor, k(t) has the meaning of a
time-dependent ”curvature index”, and r is the radial a2 a2 V · 2
coordinate. ds2 = dt2
− a˙2V2 (cid:20)(cid:18)a(cid:19)(cid:21)
TheiranalogytoSFsingularitymodelsisthatthey do
not admit any equation of state linking p to ̺ through- + a2 dx2 +dy2 + dz2 , (14)
out the whole evolution of the universe, although at any V2
(cid:2) (cid:3)
givenmomentofthe evolution,suchanequationofstate
where
(varying from one spacelike hypersurface to the other)
can be admitted. An analytic equation of state can also V(t,x,y,z)= (15)
be admitted at any fixed subspace with constant radial 1
coordinater,butnotglobally[8]. Forthesakeofsimplic- 1+ k(t) [x x0(t)]2+[y y0(t)]2+[z z0(t)]2 ,
4 n − − − o
ity,firstthesphericallysymmetricmodelswillbeconsid-
ered(note that in [8, 9] a time coordinatet analogousto and x0,y0,z0 are arbitrary functions of time. Now the
the cosmic time in Friedmann models was marked by τ, general expression for the pressure is (the expression for
which had nothing to do with a commonconformaltime the energy density (9) remains the same)
coordinate). The energy density and pressure are given
V(t,x,y,z)
by [8]
p(t,x,y,z)= 3C2(t) + 2C(t)C˙(t)h a(t) i· . (16)
a˙2(t) k(t) − V(t,x,y,z)
̺(t) = 3C2(t) 3 + , (9) a(t)
≡ (cid:20)a2(t) a2(t)(cid:21) h i
Inserting the time derivative of (9) and the function
V(t,r)
p(t,r) = 3C2(t) + 2C(t)C˙(t)h a(t) i· , (10) V(t,x,y,z) from (15) into (16) gives
− V(t,r)
a˙2 k
a(t)
h i p(t,x,y,z)= 3 3 (17)
− a2 − a2
andgeneralizetherelations(1)and(2)toinhomogeneous
V(t,x,y,z)
moWdeelsn.owshowthatitispossibletoextendSFsingulari- +aa˙ (cid:20)2aa¨ −2aa˙22 + a12 (cid:16)k˙aa˙ −2k(cid:17)(cid:21) hV(ta,x(,ty),z)i· .
tiesintoinhomogeneousmodels. Following[8]weassume a(t)
h i
that the functions k(t) and a(t) are related by
It is easy to notice that SF singularity p appears
· →±∞
k(t)= αa(t) , (11) with (3) for a¨ , if (V/a)/(V/a) is regular and the
− →−∞
3
signofthepressuredependsonthesignsofbotha˙/aand α > 0 [18] (the spatial acceleration scalar reads as u˙ =
(V/a)/(V/a)·. This provesthat we caninhomogenize SF 2αr –lowerpressureregionsareawayfromthecenter).
−
singularities for a Stephani model with no symmetry. This means that the strong and weak energy conditions
In fact, SF singularities appear independently of FD are not necessarily fulfilled for the models with α > 0.
singularities whenever a¨ and the blow-up of p is In particular, they cannot be fulfilled at an antipodal
guaranteed by the involv→em−en∞t of the time derivative of centerofsymmetryatr2 ,unlessa 0(whereBig-
→∞ →
the function C(t) in (10). BangsingularityappearsandsoBBandFDsingularities
Thatmakesacomplimentarygeneralizationtotheone coincide - see Section III).
which extends SF singularities into the theories with On the other hand, the first part of the dominant en-
actions being arbitrary analytic functions of the Ricci ergy condition may not be violated if the contribution
scalar and into anisotropic (but homogeneous) models from the last term with a¨ in (20), which includes r2,
[14, 15, 16]. does not overweighthe first one, i.e., when
It appears that the main motivation for studying SF
1 α
singularities [1] was the fact that, unlike phantom mod- < r2 . (23)
els[4],they obey the strongandweakenergyconditions, a 4
thoughthey donotobeythedominantenergycondition.
This should be appended by the condition (19) which is
Inthis paperwe raisethe pointthatthe questionofpos-
equivalent to (22), i.e., the dominant energy condition is
sible violation of the energy conditions for the inhomog-
fulfilled if
enized SF singularity models is a bit more complex than
for the homogeneous ones. From (10) and (13) for the 1 α 1
< r2 < . (24)
strong, weak and dominant energy conditions to be ful-
a 4 a
filled we have:
This is obviouslya contradictionwhichmeans that,sim-
a¨ α
̺+3p = 6 +3 ilarly as in the isotropicFriedmann models, SF singular-
− a a ities violate the dominant energy condition.
3 a˙2 Such a violation of the dominant energy condition
αr2 2 2a¨ α >0 , (18)
− 4 (cid:18) a − − (cid:19) also appears in M-theory-motivated ekpyrotic models in
a¨ a˙2 α which p ̺ during recollapse [19].
≫
̺+p = 2 +2 Let us now discuss the problem of the possible energy
− a a2 − a
conditionsviolationinthegeneralStephanimodel. Using
1 a˙2
αr2 2 2a¨ α >0 , (19) (9) and (17) we get for the strong, weak, and dominant
− 4 (cid:18) a − − (cid:19) energy conditions
a¨ a˙2 α
̺ p = 2 +4 5 a˙2 k
− a a2 − a ̺+3p= 6 6 + (25)
1 a˙2 − a2 − a2
+ αr2 2 2a¨ α >0 . (20)
4 (cid:18) a − − (cid:19) 3a˙ 2a¨ 2a˙2 + 1 k˙a 2k hV(ta,x(t,)y,z)i· >0 ,
In fact, the dominant energy condition requires fulfilling a(cid:20) a − a2 a2 (cid:16) a˙ − (cid:17)(cid:21) V(t,x,y,z)
a(t)
both(19)and(20). Noticethatinviewof(9)the energy h i
̺+p= (26)
densityininhomogeneousStephanimodelsisalwayspos-
itive, i.e., V(t,x,y,z)
2a˙ 2a¨ 2a˙2 + 1 k˙a 2k h a(t) i· >0 ,
̺>0 . (21) a(cid:20) a − a2 a2 (cid:16) a˙ − (cid:17)(cid:21) V(t,x,y,z)
a(t)
h i
This means that the strong and weak energy conditions a˙2 k
̺ p=6 +6 (27)
are still not violated if a¨ , in analogy to homoge- − a2 a2
→ −∞
neous models, provided V(t,x,y,z)
1 > αr2 , (22) −aa˙ (cid:20)2aa¨ −2aa˙22 + a12 (cid:16)k˙aa˙ −2k(cid:17)(cid:21) hV(ta,x(,ty),z)i· >0 .
a 4 a(t)
h i
sincethefirsttermwitha¨in(18)and(19)mustdominate Obviously,thedominantenergyconditionrequiresfulfill-
the second (remember that a¨ ). Notice that the ing both (26) and (27). Before we go any further, using
→ −∞
equality 1/a =αr2/4 may lead to a pressure singularity (15), we note that
avoidance in (13). Assuming that the generalized scale
factor a(t) > 0, we conclude from (22) that the strong V(t,x,y,z)
= (28)
andweakenergyconditions arealwaysfulfilled, if α<0. a(t)
However,anacceleratedexpansionforanobserveratthe 1 1k
center of symmetry at r = 0 can only be achieved, if + (x x0)2+(y y0)2+(z z0)2 ,
a 4a h − − − i
4
·
V(t,x,y,z) Under the choice of (11), we have (k/a)· =0, and so the
= (29)
(cid:20) a(t) (cid:21) singularities appear at r2 . In general, it may not
→ ∞
beso,whichwasexplicitlyshownin[8]. Forexample,by
a˙ 1 a˙
−a2 + 4a(cid:18)k˙ −ka(cid:19)h(x−x0)2+(y−y0)2+(z−z0)2i choosing
k a(t)=αt2+βt+γ , (33)
[(x x )x˙ +(y y )y˙ +(z z )z˙ ] .
0 0 0 0 0 0
−2a − − − k(t)=1 a˙2 = 4αa(t)+∆ , (34)
− −
It is important to notice that the ratio of (28) and (29)
with
which appears in the conditions (25), (26), and (27) al-
lows to cancel 1/a from both the numerator and the de- ∆=4αγ+1 β2 , (35)
nominator. Apart from that, one is able to take a˙/a out −
in(29)andcancelitwiththesametermstandinginfront the FD singularities appear for [8]
ofthelasttermintheseconditions. Thisbasicallymeans
that, bearing in mind the fact that a¨ , the strong r =2/√ ∆ . (36)
and weak energy conditions are fulfille→d−pr∞ovided one of | | −
the expressions Of course the condition (11) is obtained in the limit
∆ 0 from (34) which moves FD singularities to an
V 1+ 1k (x x )2+(y y )2+(z z )2 ,(30) ant→ipodal center of symmetry at r2 . Having cho-
1 ≡ 4 h − 0 − 0 − 0 i sen γ = 0,∆ = 1 β2 in (33) (Mo→del∞I of Ref. [10],
V 1 k˙a k (x x )2+(y y )2+(z z )2 called Da¸browski m−odel in Ref. [11]) the energy density
2 0 0 0
≡ 4(cid:16) a˙ − (cid:17)h − − − i and pressure are then given by
ka
[(x x0)x˙0+(y y0)y˙0+(z z0)z˙0] 1,(31) 1
− a˙ − − − − ̺ = 3 , (37)
t2(αt+β)2
is negative. It is clear that for k(t) > 0 (30) is always
1+2αt(αt+β)r2
positive,so that(31) mustnecessarilybe negativeandit p = . (38)
− t2(αt+β)2
certainly does, at least in some regions of space. On the
other hand, for k(t) < 0 (30) can be both positive and
For the strong, weak and dominant energy conditions to
negativewhich requires(31)to be negativeand positive,
be fulfilled, respectively, we get the requirements
respectively. In conclusion, similarly as in the spheri-
cally symmetric case, the strong and weak energy condi- r2
tions may be violated for inhomogenized SF singularities ̺+3p = 6α >0 , (39)
− t(αt+β)
in Stephani models. This is different from what we have
1 αt(αt+β)r2
for isotropic SF singularities. Finally, one can easily no-
̺+p = 2 − >0 , (40)
ticethatinordertofulfillthedominantenergycondition t2(αt+β)2
theratioof(30)to(31)shouldbesimultaneouslypositive 2+αt(αt+β)r2
̺ p = 2 >0 . (41)
andnegative,whichis a contradiction. This means that, − t2(αt+β)2
like in the isotropic case, a general Stephani model al-
lows SF singularities which always violate the dominant If α > 0 (acceleration [18]), then the strong energy con-
energy condition. In conclusion, one can say that the dition is violated if
problem of the energy conditions violation by SF singu-
laritiesintheinhomogeneousmodelsismorecomplicated t(αt+β)>0 , (42)
than in the isotropic ones and so the results based only
on the isotropic models cannot be trusted. and this may happen independently of the radial coor-
dinate r. The weak energy condition is violated for the
domain of space in which
III. INHOMOGENEOUS FINITE DENSITY
SINGULARITIES 1
>αt(αt+β) . (43)
r2
In the context of temporalSF singularities of pressure With the strong energy condition violated, this gives a
we will further briefly discuss the occurrence of spatial weak energy condition violationonly in some spatial do-
FD singularities of pressure in the Stephani models and mainsincetheright-handsideof(43)hasapositivevalue,
possible energy conditions violation. From (10) we can but including the center of symmetry at r = 0. On the
see that FD singularities appear whenever the radial co- otherhand,forthedeceleratedexpansion,α<0,andthe
ordinate right-handside of (43) has a negative value, so the weak
· energy condition is violated everywhere in the universe
1
r2 =−4(cid:0)ka(cid:1)· . (32) n(io.et.,vfioorlaatleldvaalnudesαof>r)0. (Ifactcheelesrtartoinogn)e,ntehrgenycaognadinititohneries
a
(cid:0) (cid:1)
5
is aweakenergyconditionviolationforallvalues ofr. If avoided, if
the strong energy condition is violated, and α <0, then
theweakenergyconditionisviolatedonlyinsomespatial
∆>0 . (47)
domainwhich, however,includes the centerofsymmetry
at r =0. The dominant energy condition is violated for
1 1
< αt(αt+β) , (44) IV. CONCLUSIONS
r2 −2
which, combined with (43), gives In conclusion, we have shown that one is able to spa-
tially inhomogenize sudden future (SF) singularities in
1 1
αt(αt+β)< < αt(αt+β) . (45) the sense that these singularities do appear in inhomo-
r2 −2 geneous models of the universe. However, despite ho-
mogeneous SF singularities, they may violate the strong
This last condition can only be fulfilled either if α < 0
and weak energy conditions in some regions of space, al-
and t(αt+β)>0, or if α>0 and t(αt+β)<0. Then,
though they share the dominant energy condition viola-
atleastfortheseparticularclassofStephanimodels,FD
tionwithhomogeneousmodels. Itshowsthatmakingany
singularities may lead to a violation of the energy con-
important conclusions about physics, on the basis of the
ditions in a similar way as BR singularities in phantom
isotropicmodels only,maybe misleadingandshouldnot
cosmology do.
be trusted. A possible violation of all the energy condi-
An interesting problem is a possible avoidance of SF
tionsbyinhomogenizedSFsingularitiesissimilartowhat
andFDsingularitiesintheuniverse. SFsingularitiescan
happenstoBig-Ripsingularitiesinphantomcosmologies.
easily be avoided by imposing an analytic form of the
equation of state p = p(̺). Even without this assump- Besides, we have noticed that, apart from sudden fu-
tion, some other ways of their avoidance by introduc- ture singularities, the inhomogenized models also admit
ing quadratic in Ricci curvature scalar terms [16], or by finite density (FD) singularities which are spatial rather
quantum effects [20], are possible. On the other hand, a than temporal. In relation to this we have discussed an
necessaryconditiontoavoidFDsingularitiesinStephani example of an inhomogeneous model with spatial finite
models comes from (10) and reads as densitysingularitiesofpressureandstudiedthe domains
of its energy conditions violation.
1 · V. ACKNOWLEDGEMENTS
(cid:0)ka(cid:1)· >0 . (46)
a
(cid:0) (cid:1) I acknowledge the support from the Polish Research
In our special model (33)-(34) they can be simply Committee (KBN) grant No 2P03B 090 23.
[1] J.D. Barrow, Class. Quantum Grav. 21, L79 (2004). 67 (1998).
[2] K.Lake, e-print: gr-qc/0405055. [11] R.K. Barrett and C.A. Clarkson, Class. Quantum Grav.
[3] S. Perlmutter et al., Astroph. J. 517, (1999) 565; A. G. 17, 5047 (2000).
Riess et al. Astron. J. 116 (1998) 1009; A.G. Riess et [12] J.D. Barrow, G. Galloway, and F.J.T. Tipler, Mon. Not.
al., Astroph. J. 560 (2001), 49; astro-ph/0207097; J.L. R. Astron. Soc. 223 (1986), 835.
Tonry et al., Astroph. J. 594 (2003), 1; M. Tegmark et [13] A.Krasin´ski,InhomogeneousCosmologicalModels,Cam-
al.,Phys.Rev. D 69 (2004), 103501. bridge University Press (1997).
[4] R.R.Caldwell, Phys.Lett.B 545, 23(2002); S.Hannes- [14] J.D. Barrow, e-print: gr-qc/0409062.
tad and E. M¨orstell, Phys. Rev. D 66, 063508 (2002); [15] J.D. Barrow, and Ch. Tsagas, e-print: gr-qc/0411045.
M.P. Da¸browski, T. Stachowiak and M. Szydl owski, [16] M. Abdalla, S. Nojiri, and S. Odintsov, e-print:
Phys. Rev. D 68, 103519 (2003); P.H. Frampton, Phys. hep-th/0409198.
Lett.B562(2003),139;S.NojiriandD.Odintsov,Phys. [17] J. Stelmach and I. Jakacka, Class. Quantum Grav. 18
Lett. B 562 (2003), 147; J.E. Lidsey, Phys. Rev. D 70 (2001), 1.
(2004), 041302. [18] M.P. Da¸browski, Gravit. & Cosmology 8 (2002), 190.
[5] Hawking, S.W., Ellis, G.F.R., The Large-scale Structure [19] J. Khoury et al, Phys. Rev. D 64 (2001), 123522; P.J.
of Space-time, Cambridge Univ.Press (1999) . Steinhardt and N. Turok, Phys. Rev. D 65 (2002),
[6] H.Stephani, Commun.Math. Phys. 4 (1967), 137. 126003; J. Khoury, P.J. Steinhardt and N. Turok, Phys.
[7] R.A.Sussmann,Journ.Math.Phys.28,1118(1987);29, Rev. Lett.92 (2004), 031302.
945 (1988); 29, 1177 (1988). [20] S. Nojiri and S.D. Odintsov, e-print: hep-th/0405078;
[8] M.P. Da¸browski, Journ. Math. Phys. 34, 1447 (1993). hep-th/0408170.
[9] M.P. Da¸browski, Astroph.J. 447, 43 (1995).
[10] M.P.Da¸browskiandM.A.Hendry,Astroph.Journ.498,