Cosmological tests of sudden future singularities

0.26 MB·English

by  Tomasz Denkiewicz

Cosmological tests of sudden future singularities.
Tomasz Denkiewicz∗ and Mariusz P. Da¸browski†
Institute of Physics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland and
Copernicus Center for Interdisciplinary Studies, S lawkowska 17, 31-016 Krak´ow, Poland
Hoda Ghodsi‡ and Martin A. Hendry§
SUPA, School of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, UK
(Dated: February 14, 2012)
Wediscusscombinedconstraints,comingfromthecosmicmicrowavebackgroundshiftparameter
2 R,baryonacousticoscillations(BAO)distanceparameterA,andfromthelatesttypeIasupernovae
1 data,imposedoncosmological modelswhichallowsuddenfuturesingularitiesofpressure. Weshow
0 that due to their weakness such sudden singularities may happen in the very near future and that
2 at present they can mimic standard dark energy models.

b
PACSnumbers: 98.80.Es;98.80.Cq;04.20.Dw
e
F
3 I. INTRODUCTION gularities may be investigated by using the higher-order
1 characteristicsofthe expansionofthe universeknownas
Suddenfuturesingularitiesofpressure[1](SFSortype statefinders [19].
]
O II singularities [2, 3]) are some exotic-type singularities Sudden future singularity models result in leaving no
which are allowed to occur in the universe. The in- restrictionsonanequationofstatep=p(̺)ofthecosmo-
C
spiration for investigating these new types of singular- logicalmatter, which allows the unconstrainedevolution
.
h ity was the observation of high-redshift Type Ia super- of the energy density andpressure. The nature of a sud-
p novae (SNIa) which over the past decade has provided denfuturesingularityisdifferentfromthatofastandard
- strong evidence that the expansion of the universe is ac- big-bang singularity and also from an exotic big-rip sin-
o
r celerating [5] and, in particular, the further extension of gularityinthatitdoesnotexhibitgeodesicincompletness
st these datasets [6, 7] which remain consistent with the andthecosmicevolutionmayeventuallybeextendedbe-
a dark energy existing in the form of phantom [8] energy. yondit[12,13]. The onlyphysicalcharacteristicofthese
[ Phantom-driven dark energy leads to a big-rip singu- singularities is a momentarily infinite peak of the tidal
2 larity (BR or type I according to [2]) in which all the forces in the universe. In generalized sudden future sin-
v matterisdissociatedbythephantom-drivendarkenergy gularitymodels this peak may alsoappear in the deriva-
1 which acts as antigravityin a large and a dense universe tives of the tidal forces [1]. It is interesting to note that
6 [9]. This behavior is of course different from the stan- these singularities are in a way similar to yet another
6
dard picture of cosmic evolution which allows big-bang type, which were termed finite density singularities [14].
6
(BB) or big-crunch (BC) standard types of singularities The crucial difference is that finite density singularities
.
1 only. Besides, phantom energy violates all the energy occurassingularitiesinspaceratherthanintime,which
0 conditions: the null (̺c2 +p 0), weak (̺c2 0 and means that evenat the presentmoment of cosmic evolu-
2 ̺c2+p 0), strong (̺c2+p ≥0 and ̺c2+3p ≥0), and tiontheycouldexistsomewhereintheUniverse[15]. We
1 ≥ ≥ ≥
dominant energy (̺c2 0, ̺c2 p ̺c2) (here c is will not discuss in detail these finite density singularities
:
v the speed of light, ̺ - ≥the m−ass de≤nsity≤in kg m−3, and in this paper since they basically appear in cosmologi-
i
X p - the pressure). Apart from big-rip and sudden future cal models without homogeneity. On the other hand, it
singularitiestherearenumerousothertypessuchas: gen- is worth mentioning that the sudden future singularities
r
a eralized sudden future singularities (GSFS), finite scale are quite generic, since they may arise in both homoge-
factor singularities (FSF or type III) which were tested neous[1]andinhomogeneous[16]modelsoftheuniverse.
against observations in [10], big-separation singularities In Ref. [17] we discussed the constraints imposed on
(BS or type IV) and w-singularities (type V) [11]. They SFS models which came from SNIa data. We showed
arecharacterizedbyviolationofallorsomeoftheenergy that those data were consistent with the occurrence of
conditionswhichresultsinablow-upofallorsomeofthe a sudden future singularity in the very near future –
appropriate physical quantities such as: the scale factor, as little as 8.7 million years from now. In Ref. [18]
the energy density, the pressure, and the barotropic in- we extended our investigation to confront SFS models
dex (for a review see Ref. [3]). In fact, all of these sin- withothercosmologicaldata,fromthecosmicmicrowave
background(CMB)andbaryonacousticoscillations. We
demonstratedthattheclassofSFSmodelsintroducedin
∗ [email protected] [1] was not compatible with current observations, in the
† [email protected] particular case where the asymptotic behaviour of the
‡ [email protected] scale factor close to the BB singularity mimics a dust-
§ [email protected] filledEinsteindeSitteruniverse. Inthispaperweextend
2
our investigation further, confronting current cosmolog- where
ical data with SFS models that simulate the behaviour
c2
of more general flat, barotropic fluid models. The paper
w (t)= [2q(t) 1] , (II.6)
s
is organized as follows. In Section II we present sudden 3 −
future singularities in a framework which is appropriate
and q(t) = a¨a/a˙2 is the deceleration parameter. The
to further discussion. In Sections III-V we discuss cur-
−
index‘s’hasbeenattachedtomarkthefactthatwehave
rent observational data on supernovae, shift parameter
a type of matter which is of sudden future singularity
and baryon acoustic oscillations in the universe. In Sec-
origin and this may play the role of the dark energy. In
tion VI we compare these observational data with our
thelimitδ 0wehaveq(t)=(1 m)/mwhichfordust
SFS models and use them to determine constraints on
→ −
(i.e. for m =2/3) gives q(t) = 1/2, as required. In fact,
the model parameters and present our conclusions.
the relations (II.5) and (II.6) remain of the same form
if the time t is replaced by the scaled time to a sudden
II. THE MODELS singularity y.
ThestandardFriedmannlimit(i.e. modelswithoutan
SFS) of (II.4) is achieved when δ 0; hence δ becomes
Inorderto obtainasuddenfuture singularityconsider →
the“non-standardicity”parameterofSFSmodels. Addi-
the simple framework of an Einstein-Friedmann cosmol-
tionally, notwithstanding Ref. [1] and in agreementwith
ogy governedby the standard field equations
the field equations (II.1)-(II.2), we assume that δ can be
3 a˙2 kc2 both positive and negative leading to a deceleration or
̺(t)= + , (II.1)
8πG a2 a2 an acceleration (cf. (II.4)) of the universe, respectively.
(cid:18) (cid:19)
c2 a¨ a˙2 kc2 It is important to our discussion that the asymp-
p(t)= 2 + + , (II.2) totic behaviour of the scale factor (II.4) close to the
−8πG a a2 a2
(cid:18) (cid:19) BB singularity at t = 0 is given by a simple power-
where the energy-momentum conservation law law a = ym, simulating the behaviour of flat k = 0
BB
barotropic fluid models with m = 2/[3(w +1)] . This
a˙ p(t)
̺˙(t)= 3 ̺(t)+ , (II.3) allows us to preserve all the standard observed charac-
− a c2 teristics of early universe cosmology – such as the cos-
(cid:18) (cid:19)
mic microwave background, density perturbations, nu-
is trivially fulfilled due to the Bianchiidentity. Here a
≡ cleosynthesis etc. – provided we choose an appropri-
a(t)isthescalefactor,thedotmeansthederivativewith
ate value of m. On the other hand, close to an SFS
respect to time t, G is the gravitational constant, and
the asymptotic behaviour of the scale factor is non-
the curvature index k =0, 1. What is crucialto obtain
± standard, a (y) = a [1 m(1 δ)(1 y) ], showing
a sudden future singularityis that in the generalcase no SFS s − − −
that a = a for t = t (i.e. y = 1) at the SFS. No-
link between the energy density and pressure (i.e. the SFS s s
ticethatonedoesnotviolatetheenergyconditionsifthe
equationof state) is specified (but see below for the case
parameter m lies in the range
of the flat models).
From equations (II.1)-(II.2) one can easily see that a
0<m 1 (w 1/3), (II.7)
pressuresingularityp occurswhentheacceleration ≤ ≥−
→∞
a¨ , no matter that the value of the energy density
→−∞ This range of values is, in fact, equivalent to a standard
̺ andthe scalefactora(t)areregular. Since inthatcase
(neither quintessence-like nor phantom-like)evolutionof
p > ̺, it is clear that the dominant energy condition
| | the universe. However, with no adverse impact on the
is violated. This condition can be achieved if the scale
fieldequations(II.1)-(II.2),onecouldalsoextendtheval-
factor takes the form [1, 4, 17]
ues of m to lie in the complementary ranges [12] m > 1
a(y)=a [δ+(1 δ)ym δ(1 y)n] , y t/t (i.e. 1 < w < 1/3) for quintessence, and m < 0 (i.e.
s − − − ≡ (IsI.4) w < −1)forphan−tommodels,althoughtheserangesmay
−
withtheappropriatechoiceoftheconstantsδ,t ,a ,m,n. lead to violation of the strong and weak energy condi-
s s
First publication of the eq.(II.4) was in [4] where it was tions, respectively.
usedtoshowthatcloseduniversesobeyingenergycondi- A special case in which sudden future singularities al-
tions did not necessarilyrecollapse. Notice that in equa- lowforanexplicitequationofstateistheanti-Chaplygin
tion (II.4) a has the unit of length and all the terms gas model [21]
s
(including y) in the bracket are dimensionless. before
A
the expansion max ie a sudden singularity as I termed p(t)= (A 0) , (II.8)
it later. It is interesting to note that for a flat (k = 0) ̺(t) ≥
Friedmann model we have an explicit relation between
andthepressuresingularity,knowninthiscaseastheso-
the pressureandthe energydensity,thoughwithatime-
called ‘Big-Brake’, is achieved for ̺ 0 and so p .
dependent barotropic index, in the form
→ →∞
Here we can easily check that one is able to obtain an
p (t)=w (t)̺ (t) , (II.5) anti-Chaplygin gas behavior for the scale factor (II.4).
s s s
3
In order to prove that we need simply to consider the onecanwritedownthe flatFriedmannequation(II.1)in
first time derivative of (II.4) the form
m(1 δ) n a˙2 8πG
a˙(t)=as t− ym−1+δt (1−y)n−1 . (II.9) H2(t)= a2 = 3 ̺s(t) (II.20)
(cid:20) s s (cid:21)
t
Wsoethreaqtuwireeha˙av→e a0,cwonhdicithiocnortrheastpoenitdhsertom̺ →00oartδy =11. =H02Ωs0exp(cid:20)−3Zt0 [w(t′)+1]H(t′)dt′(cid:21) ,
→ →
In fact, these conditions are almost equivalent since or
lim a(y)=a [1 δ(1 y)n] , (II.10)
s y
m→0 − −
H(y)=H Ω exp 3 [w(y′)+1]H(y′)dy′ .
δli→m1a(y)=as[1−(1−y)n] . (II.11) 0s s0 (cid:20)− Zy0 (cid:21)
(II.21)
It follows that (II.10) is more general since additionally
In this paper we consider three observational con-
it does not restrict δl, and moreover it has a standard
straints on SFS models: luminosity distance moduli to
Friedmann limit δ 0, though a static one. It is useful
SNIa, the CMB shift parameter, which is a scaled dis-
→
to calculate from (II.10) that (cf. (III.4))
tance to the last scattering surface of the cosmic mi-
nδ(1 y)n−1 crowave background, and the distance parameter con-
H(y)= − , (II.12) strained by baryon acoustic oscillations.
[1 δ(1 y)n]
− −
(n 1) 1
q(y)= − δ . (II.13)
nδ (1 y)n − III. SUPERNOVAE
(cid:20) − (cid:21)
Using the conservation law (II.3) together with the
barotropic relation (II.5) and (II.6) (which holds for flat We proceed within the framework of Friedmann cos-
k =0 models only) we may write mology, and consider an observer located at r = 0 at
coordinate time t = t . The observer receives a light
0
d̺s = 2[q(t)+1]a˙dt , (II.14) ray emitted at r = r1 at coordinate time t = t1 and,
̺s − a according to (II.4), with redshift given by
which integrates to give a(t ) δ+(1 δ)ym δ(1 y )n
1+z = 0 = − 0 − − 0 , (III.1)
t a(t ) δ+(1 δ)ym δ(1 y )n
̺ (t)=̺ exp 2 [q(t′)+1]H(t′)dt′ (II.15) 1 − 1 − − 1
s 0s
−
(cid:20) Zt0 (cid:21) where y = y(t ) and y = y(t ). We then have a stan-
t 0 0 1 1
=̺ exp 3 [w(t′)+1]H(t′)dt′ .(II.16) dard null geodesic equation
0s
−
(cid:20) Zt0 (cid:21)
r1 dr t0 cdt
The following regimes of the relations (II.15) and = , (III.2)
(II.16) are in order. Firstly, the regime of the early uni- Z0 √1−kr2 Zt1 a(t)
verse can be recovered by taking t t (which is equiv-
s
≪ withthescalefactora(t)givenby(II.4). ForaflatFried-
alent to δ 0) one has
→ mann model we can write down the radial coordinate to
̺ /̺ =(y /y)2 t−2 . (II.17) an observer in any of the forms
s 0s 0
∝
Secondly, the near-to-SFS limit can be obtained by tak- t0 cdt y0 (ct )dy t0 cda a0 cda
s
r = = = =
ing t∝ts which gives 1 Zt1 a(t) Zy1 a(y) Zt1 a˙a Za1 Ha2
2 z cdz c z dz
̺s(t)=̺0s(cid:20)((11−−αα))ttss++ααtt0(cid:21) ∝̺0st2 , (II.18) =Z0 H(z)a0 = H0a0 Z0 E(z) , (III.3)
where we have used that y = t/t and so dy = dt/t
with α = m(1 δ). In fact, one can see from (II.18) s s
− which implies a′(y) da(y)/dy = t da(t)/dt = t a˙(t),
that the contribution from the ‘SFS-driven’ matter is a′(y )=t a˙(t ) and ≡ s s
more and more important and starts dominating from 0 s 0
the current moment of the evolution, i.e., t < t implies
0 1 a′(y ) 1 a′(y)
̺ansd<t̺→s0,tts=imtp0liiemsp̺lsie=s̺̺s0=st2s̺(0tss,−t>αtts0+imαptl0i)e−s2̺.s >̺s0, H0 ≡H(t0)= ts a00 = tsH(y0) ; H(y)≡ a((IyI)I.4.)
Defining the density parameter
In (III.3) the transition from the integral of da to the
8πG integral of dz was given by the application of the defini-
Ω = ̺ , (II.19)
s0 3H2 s0 tionofredshift (III.1). Besides,due a lackofananalytic
0
4
form for the equation of state for SFS models the func- where
tion E(z) can only be given by a formula which involves
tdec dt
an integral over z, as follows
r =a S(r )=a S c , (IV.3)
s dec s dec s
a(t)
H2(z) z 2q(z′) 1 (cid:18)Z0 (cid:19)
E2(z)= H2 =Ωδ0(1+z)3exp dz′ 1+z−′ , withcsbeingthesoundvelocityandtheangulardiameter
0 (cid:20)Z0 (cid:21) distance reads as
(III.5)
in(III.3)withΩδ0beingthedensityparameterofsudden- t0 dt
future-singularity-driven dark energy [3]. It is easy to dA =adecS(rdec)=adecS c (IV.4)
a(t)
notice that in the limit δ 0 and m = 2/3 one has (cid:18)Ztdec (cid:19)
that E(z) = Ω (1 + z)3,→as for the standard matter
δ0 where c is the velocity of light and S(r)=r for k =0.
dominated case. Let us recall that the standard formula
Following [25] we can write, using IV.1 and IV.2 as
forthemodelswhichinclude thedarkenergycomponent
follows
Ω reads as [25, 26]
w0
r d′ (z′ )
= s A dec , (IV.5)
E2(z)=Ωm0(1+z)3+Ωw0(1+z)3(w+1)+Ωk0(1+z)2 , R rs′ dA(zdec)
(III.6)
whereΩ isthecurvaturecomponentandp=w̺. How- which, by assuming that at decoupling the amount of
k0
ever,inourfurthercalculationswewillnotbeexpressing radiationwasthesameinboththeflatreferencestandard
E(z) in either forms (III.5) or (III.6), using an explicit CDM model and in our SFS model (which we assume to
form of a(y) as in (III.3) instead. bejustthesameasastandardmatter-radiationmodelof
The luminosity distance to e.g. a supernova observed the early universe, since SFS models do not change the
at redshift, z, is given by evolution there) we have that
r 1
d (z)=(1+z)a(t )r (III.7) s
L 0 1 = . (IV.6)
r′ √Ω
s m0
where r is given in one of the forms (III.3). For our
1
Onthe otherhand,fora referencestandardCDMmodel
calculations we have applied the following expression
y0 dy 2ca′ a′
dL(z)=(1+z)a(y0)ctsZy1 a(y) . (III.8) d′A = a0Hde0c  Ω′r0+1−sΩ′r0+ ad0ec
p
The distance modulus is 2ca′  
= decf(Ω′ ,a′ ) , (IV.7)
a H r0 dec
0 0
µ(z)=5log d (z)+25. (III.9)
10 L
while for our SFS model the angular diameter distance
We compared the luminosity distance predicted for SFS is given by
models with SNIa data using the SCP Union2 [23]
d =a r (IV.8)
dataset,consistingof557supernovae,whichisthelargest A dec dec
compilation published to date.
with r givenby (III.3) takenatdecoupling. Using the
dec
above, we may write that for our SFS models the shift
parameter is
IV. SHIFT PARAMETER
2c 2ct
s
= = , (IV.9)
The standard formula for the CMB shift parameter is R H a √Ω r a′(y )√Ω r
0 0 m0 dec 0 m0 dec
given by [24, 25]:
where we have assumed that the function f(Ω′ ,a′ ) is
r0 dec
l′TT approximately unity [25].
= 1 , (IV.1)
R lTT Finally, the rescaled shift parameter is
1
H a z dz
where lTT is the temperature perturbation CMB spec- ¯ = Ω 0 0r = Ω
1 R m0 c dec m0 E(z)
trum multipole of the first acoustic peak in the model Z0
under consideration and l′TT corresponds to a reference p y0 dpy
1 = Ω a′(y ) , (IV.10)
flatstandardColdDarkMatter(CDM)model. Themul- m0 0 a(y)
Zydec
tipole number is relatedto anangularscale of the sound p
horizon r at decoupling by [26, 27] whereinordertoobtainthelastexpressionwehaveused
s
(III.4). TheWMAPdatagives =1.70 0.03[28]. Note
R ±
r 1 that, as we discussed in [18], this value is not directly
s
θ = , (IV.2)
1 d ∝ l “observed” but is derived from the CMB data assuming
A 1
5
a specific class of cosmological model; hence one must where l and ∆θ are the linear and angular size of an
be careful in employing it to test our SFS model. How- object,anditsline-of-sightextent,∆r,usingtheredshift
ever,as we showedin [18], the effective equation of state distance
for our SFS model displays general similarity to that of
c∆t ct ∆y
the concordancemodelandonthis basiswe considerthe ∆r = = s (V.2)
use of the “observed” value of the shift parameter to be a(t) a(y)
appropriate for our SFS model too.
(seee.g. Ref. [25]). Asaresultonecandefinethevolume
Moreoverin [18] we also consideredthe acoustic scale,
distance, D , as
l , and how its incorporation into the analysis could re- V
a
sultinparameterconstraintswhichgaveacloserapproxi-
D3 =r2∆r , (V.3)
mationtothoseobtainedfromafittothefullCMBpower V
spectrum. However,uponvaryingtheparametermfrom
so that using (III.3), one has
theEinstein-deSittervalueof2/3consideredin[18],itis
notclearthatthe useofthe acousticscaleremainsvalid. 1
To see this we recall that the acoustic scale is defined in y0 ctsdy 2 cts∆y 3
D = (V.4)
terms of two distinct distance scales: it is proportional V a(y) a(y)
"(cid:18)Zy1 (cid:19) (cid:18) (cid:19)#
to the ratio of the angular diameter distance, d , and
A 1
the sound horizon, rs, both evaluated at the redshift of c z dz 2 c ∆z 3
= (.V.5)
recombination, z :
CMB a H E(z) a H E(z)
"(cid:18) 0 0 Z0 (cid:19) (cid:18) 0 0 (cid:19)#
d (z )
l =π A CMB , (IV.11) Ref. [30]uses the acousticpeakscalefor 46748luminous
a
r (z )
s CMB red galaxies selected from the Sloan Digital Sky Survey
to measure D (∆z =z =0.35)=1370 64 Mpc.
where the comoving sound horizon is defined as: V BAO ±
Usually, it is more convenient to work with a dimen-
∞ c dz′ 1 sionless quantity which for our SFS model (II.4) is
rs = Es(z′) with cs = (I,V.12) obtained multiplyiAng (V.5) by √Ωm0(a0H0)/(czBAO) to
ZzCMB 3(1+Rba) obtain
q
wwpoihtteherneRtibEal=(zd)3iffi1=5c0u0lHtΩy(bzhin)2/(cHTa2C0r.M7rkyaBnin)d−g4co,suTtiCstMhtBehel=sp2ca.e7elcKdu.loaNftoioswonutfnhoder A=Ω10/m2E(zBAO)−1/3(cid:20)zB1AO Z0z1 Ed(zz)(cid:21)2/3 , (V.6)
a
the m = 2/3 case arises from the sound speed calcula- or,usingthedefinitionofE(z)givenbythefirstequality
6
tion. Previously in the m = 2/3 case we could interpret in (III.5), together with (III.4),
the sound speed as being set at the time of recombina-
tion in the “standard early universe”. The value of the √Ω a′(y ) a(y ) 31 1 y0 dy 23
m0 0 BAO
aScFoSusutnicivsecraseledwueoutlodtshteillSFinSgmenoedreal’lsbdeiffdeirffeenrtenextpfaonrsoiounr A= [a′(yBAO)]31 (cid:20) a(y0) (cid:21) (cid:20)zBAO ZyBAO a(y)((cid:21)V.7)
history and hence different angular diameter distance-
The same result can be obtained from (V.4) if we com-
redshift relation. In the case of m = 2/3, however, the
pute the difference ∆y =y y and a(y)=a(y ) in
6 0 BAO 0
abovesimplificationisnolongervalidandboth thesound −
(V.2), i.e.,
horizonand angulardiameter distance will in generalbe
different for our SFS universe. In particular computing 1
the soundspeedin the m=2/3case wouldrequiremore y0 yBAO y0 dy 2 3
6 DV =cts − (V.8)
detailed modelling of the SFS fluid in the early universe " a(y0) (cid:18)ZyBAO a(y)(cid:19) #
which lies beyond the scope of this paper. Hence we do
not consider the acoustic scale further in this paper. and use the fact that ct = c/H a′(y )/a(y ). Following
s 0 0 0
Ref. [30] the parameter should have the value
A
V. BARYON ACOUSTIC OSCILLATIONS n −0.35
=0.469 0.017 , (V.9)
A 0.98 ±
(cid:16) (cid:17)
TheAlcock-Paczyn´skieffect[29]statesthatoneisable
where n is the spectral index (now taken to be 0.96).
tocalculatethedistortionofasphericalobjectinthesky ∼
without knowing its true size. This canbe done by mea-
suring its transverse extent using the angular diameter
VI. RESULTS AND CONCLUSIONS
distance, r
l We used a Bayesian framework to confront our SFS
r = , (V.1) model with the cosmological observations discussed in
∆θ
6
the previous sections. For each cosmological probe we mimics a dust-filled Einstein de Sitter universe. This of
took the likelihood function to be Gaussian in form, i.e. courseis not surprisingsince as noted inSection I it was
the conclusion we reached in [18]. On the other hand
1
p(dataΘ) exp( χ2), (VI.1) it is also clear from Figs. 1 and 2 that there is a sig-
| ∝ −2 nificant region of parameter space where the marginal
posterior distributions of the SFS model parameters ap-
where Θ denotes the parameters of the SFS model and
pear to overlap with each other – i.e. one can identify
“data” denotes generically the observed data for one of
the three cosmological probes. For the SNIa data χ2 sets of model parameters which occupy the 68.3% credi-
ble region for all of the marginal posterior distributions
takes the form
shown. Thisdoesnotinitselfguaranteethatthesesetsof
N (µ (z ) µ (z ))2 modelparametersgiveanacceptablefittotheSFSmodel
χ2SN = obs σi2−+σp2red i , (VI.2) sincethemarginalposteriordistributionsareprojections
i=1 i int of the fully, four-dimensional joint posterior: the appar-
X
entoverlapofthetwo-dimensionalmarginaldistributions
where σ is the quoted observational error on the ith
i for eachpair ofparameters,for example, couldin princi-
Union2 SNIa and σ is the SNIa intrinsic scatter. Fol-
int ple be merely a projection effect. We have checked this
lowing [23] we take σ = 0.15. For the CMB shift pa-
int rigorously,however,via the followingapproach. We con-
rameter χ2 takes the form
sidered each of the three cosmological probes separately
( 1.70)2 and produced MCMC chains that explored the full four-
χ2R = R0−.032 , (VI.3) dimensional SFS model parameter space for each probe.
WethenidentifiedtheMCMCchainpointsthatsampled
while for the BAO distance parameter it takes the form e.g. the 68.3% credible region for the SNIa chain and
verified (via direct computation of the likelihood values)
( 0.469)2
χ2 = A− . (VI.4) that there was significantoverlapin four dimensions be-
A 0.0172 tweenthesepointsandthecorrespondingcredibleregions
of both other probes. Thus we established that the ap-
Since our three cosmological probes are mutually inde-
parentoverlapofthe marginalposteriordistributionsfor
pendent, their joint likelihood function is given by the
the joint likelihood of all three cosmological probes was
product of their individual likelihoods, and thus takes
not merely a projection effect.
the form
It is clear from this analysis, therefore, that the
1
p(all dataΘ) exp( χ2 ), (VI.5) SFS model under consideration is compatible with cur-
| ∝ −2 TOT
rent cosmological data, based on the three cosmological
where χ2 =χ2 +χ2 +χ2. probes that we considered. In particular, provided that
TOT SN R A
m 0.72 we see that we can identify ranges for the
We used Bayes’ theorem and a Markov Chain Monte
≥
other parameters for which the predicted values of the
Carlo (MCMC) approach [20] to obtain posterior prob-
cosmological probes are in good agreement with current
ability distributions for the SFS model parameters: n,
observations. As the value of m increases towards unity
m, δ and y = t /t , where t is the present age of the
0 0 s 0
the credible regions for the parameters δ and n extend
universe. We employed the Metropolis-Hastings algo-
to fill their allowed prior ranges, while the credible re-
rithm, adopting uniform priors for each of the param-
gion for y is pushed towards lower values – i.e. the SFS
eters: δ ( 30,0), n (1,2), m (0,3), y (0,1). 0
0
∈ − ∈ ∈ ∈ modelisstillcompatiblewithcurrentdatabuttheepoch
In Fig. 1 we present contour plots showing the joint
ofthesuddenfuturesingularityispushedfurtherintothe
marginal posterior distribution for each pair of SFS
future. Interestingly, however, we see that as m 0.72
modelparameters. Eachsub-panelshowsthreecontours,
→
the credible regions for n and δ become much narrower,
denoting 68.3%, 95.4% and 99.73% (from light gray to
constrained to lie ever closer to n = 2 and δ = 0, while
dark grey respectively) credible regions.
thecredibleregionfory tendstowardsunity,sothatthe
InFig. 2wealsoshowmarginalposteriordistributions 0
SFS may happen in the very near future.
for each of the model parameters individually, again de-
termined via Markov Chain Monte Carlo. It is immedi- Thus we conclude that our SFS models are compat-
atelyclearfromFigs. 1and2thatthecurrentcosmologi- ible with current observations provided the parameter
caldatastronglyexcludeourSFSmodelform=2/3(in- m, which characterizes the near-to-big-bang evolution
dicatedbythedashedhorizontalandverticallinesinthe of the scale factor, is at least 0.72. This excludes the
relevantsub-panelsofeachfigure). Inthe leftmost panel Einstein de Sitter dust solution, and requires a form
of Fig. 2, for example, we see that the marginal pos- of matter which has slightly negative pressure (since
terior distribution for m is strongly peaked between 0.7 m = 2/3(w+1), so that w 0.083). Note also that,
≤ −
and 1.0, but is negligible outside the range (0.7,1.0). In in order to match the current observations, the value of
other words our SFS model is incompatible with current the “non-standardicity” parameter δ is seen necessarily
observationsin the particular case where the asymptotic to be negative. This naturally eliminates the validity of
behaviour of the scale factor close to the BB singularity anti-Chaplygin gas models [21] given in a special form
7
FIG. 1. Marginalized contours for pairs of parameters are plotted. There are three levels of credible region shown: 68.3%,
95.4%, 99.73% (from light grey to dark grey respectively) calculated from A, R, and SN Ia jointly
FIG. 2. Unnormalized probability densities for all parameters, calculated for A, R, and SN Ia jointly.
by (II.11) which require δ 1 (but it does not elimi- VII. ACKNOWLEDGEMENTS
→
natethemodelsintheformof(II.10)despitetheir static
Friedmann limit and also other types of anti-Chaplygin
gas models studied in Ref. [22]). T.D. and M.P.D. acknowledge the support of the Na-
tional Science Center grant No N N202 3269 40. We
thankJohnBarrow,LeonardoFernandez-Jambrina,Las-
In conclusion, similarly as in our previous paper [17] zlo Gergely and Sergei Odintsov for discussions.
where we applied supernovae data only, here we have Partofthesimulationsreportedinthisworkwereper-
shown that a sudden future singularity may happen in formed using the HPC cluster HAL9000 of the Comput-
the near future of the universe. ing Centreofthe Faculty ofMathematics andPhysicsat
8
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