Table Of ContentBrane-bulk energy exchange : a model with the present universe as a global attractor
Georgios Kofinas1∗, Grigorios Panotopoulos2† and Theodore N. Tomaras2,3‡
1Departament de F´ısica Fonamental, Universitat de Barcelona, Diagonal 647, 08028 Barcelona, Spain
2Department of Physics and Institute of Plasma Physics,
University of Crete, 71003 Heraklion, Greece and
3Foundation of Research and Technology, Hellas, 71110 Heraklion, Greece
(Dated: February 1, 2008)
Theroleofbrane-bulkenergyexchangeandofaninducedgravitytermonasinglebraneworld of
negative tension and vanishing effective cosmological constant is studied. It is shown that for the
physically interesting cases of dust and radiation a unique global attractor which can realize our
present universe (accelerating and 0<Ωm0<1) exists for a wide range of the parameters of the
model. ForΩm0=0.3,independentlyoftheotherparameters, themodelpredictsthat theequation
6 of state for the dark energy today is wDE,0=−1.4, while Ωm0=0.03 leads to wDE,0=−1.03. In
0 addition, duringits evolution, wDE crosses the wDE=−1line to smaller values.
0
2
I. INTRODUCTION conservation equation of matter. In 4-dimensional the-
n
ories, an accelerating late time cosmological phase char-
a
J In cosmologies where the present universe is realized acterized by a frozen ratio of dark matter/dark energy
appears in coupled dark energy scenarios [1] as a result
6 as a finite point during the cosmic evolution, the answer
1 to the coincidence question “why it is that today Ω of the interaction of the dark matter with other energy-
m0
momentum components, such as scalar fields. In higher
andΩDE,0 areofthesameorderofmagnitude”,relieson
2 dimensional theories, where the universe is represented
appropriatechoiceofinitial conditions. Bycontrast,ina
v as a 3-brane, this violation could be the result of energy
scenarioinwhichthepresentuniverseisinitsasymptotic
7
exchange between the brane and the bulk. In particular
0 era(closetoafixedpoint)the answerto the aboveques-
infivedimensions,auniversewithfixedpointscharacter-
2 tion reduces to an appropriate choice of the parameters
ized by Ω =0, q <0 was realized in [2] in the context
0 of the model. However, this latter situation is not easily m∗ ∗
6
1 realized if today’s universe is accelerating,because: oftheRandall-Sundrumbraneworldscenariowithenergy
5 influx fromthe bulk. However,these fixed points cannot
Iftheenergydensityofaperfectfluidwithequationof
0 represent the present universe, since they have Ω >2.
state w> 1/3 of any cosmological system is conserved, m∗
h/ all fixed p−oints of the system with Ω =0 are decelerat- In this paper we present a brane-bulk energy exchange
t m6 model with induced gravity whose global attractor can
- ing.
p represent today’s universe.
Indeed, with ρ the energy density of the perfect fluid
e Let us consider an arbitrary cosmology in the form
withconservationequationρ˙+3(1+w)Hρ=0,theHubble
h (1). Instances of such cosmologies arise in braneworld
v: equationofanarbitrarycosmologycanbe writteninthe modelsorintheorieswithmodified4-dimensionalactions
form
i leadingtoH2=f(ρ),orincosmologieswhereρDE isdue
X
to additional fields. Assuming that as a result of some
r H2 =2γ(ρ+ρDE), (1) interaction ρ is not conserved, it will satisfy an equation
a
of the form
where γ = 4πG /3. Then, the equation governing
N
ρ can always be brought into the form ρ˙ +3(1+ ρ˙+3(1+w)Hρ= T. (2)
DE DE
−
w )Hρ =0, where w is time-dependent and dis-
DE DE DE
Then,theequationgoverningρ canalwaysbebrought
tiguishes one model from the other. It can be easily DE
into the form
seen that d(Ω /Ω )/dlna = 3(Ω /Ω )(w w)
m DE m DE DE
and 2q=1+3(wΩ +w Ω ), where Ω =2γρ/−H2,
m DE DE m ρ˙ +3(1+w )Hρ =T, (3)
Ω =2γρ /H2 and q= a¨/aH2. At the fixed point DE DE DE
DE DE
−
(denoted by ∗) d(Ωm/ΩDE)/dlna=0. For Ωm∗6=0 one where wDE is time and model dependent. Whenever a
obtains wDE∗=w, and 2q∗=1+3w>0. fixed point of the system satisfies
Thus, independently of the cosmological model, the
H T =0 , ρ˙ =ρ˙ =0, (4)
only way our accelerating universe with Ω =0 can be ∗ ∗ DE
m∗ 6
6
closetoalatetimefixedpointisbyviolatingthestandard
one obtains
1+w
w = 1 . (5)
DE∗ − − Ω−1 1
m∗
∗kofinas@ffn.ub.es −
†[email protected] Equation (5) is model-independent, in the sense that it
‡[email protected] doesnotdependontheformofT orthefunctionwDE(t).
2
For Ω <1 equation (5) gives w < 1. Specifically, TA is any possible additional energy-momentum in
for wm=∗0 and Ω =Ω =0.D3Eo∗ne −obtains w = thCe|mbu,Blk, the brane matter content TA consists of a
m∗ CDM DE∗ C|m,b
1.4, while for Ω =Ω =0.03, w = 1.03. perfect fluid with energy density ρ and pressure p, while
m∗ bar DE∗
− −
The cosmology discussed in the present paper has a thecontributionsarisingfromthescalarcurvatureofthe
global attractor of the form (4), (5) [3]. Moreover, the brane are given by
universe during its evolution crosses the w = 1 bar-
DE − 6m2 a˙2 kn2 δ(y)
rier from higher values. This behavior is favored by T0 = + (17)
several recent model-independent [4] as well as model- 0|ind n2 (cid:16)a2 a2 (cid:17) b
dependent [5, 6, 7, 8] analyses of the astronomical data. Ti = 2m2 a˙2 2a˙n˙ + 2a¨ + kn2 δiδ(y). (18)
j|ind n2 (cid:16)a2 − an a a2 (cid:17) j b
AssumingaZ symmetryaroundthebrane,thesingu-
II. THE MODEL 2
lar part of equations (12) gives the matching conditions
We consider the model described by the gravitational
a′ ρ+V r a˙2 kn2
brane-bulk action [9] o+ = + c o+ o (19)
aobo −12M3 2n2o(cid:16)a2o a2o (cid:17)
S = d5x√ g(M3R Λ)+ d4x√ h(m2Rˆ V), (6)
Z − − Z − −
n′ 2ρ+3p V r 2a¨ a˙2 2a˙ n˙ kn2
where R,Rˆ are the Ricci scalars of the bulk metric o+ = − + c o o o o o (20)
g and the induced metric h =g n n respec- nobo 12M3 2n2o(cid:16) ao −a2o− aono − a2o (cid:17)
AB AB AB A B
tively (nA is the unit vector normal to−the brane and (the subscript o denotes the value on the brane), while
A,B = 0,1,2,3,5). The bulk cosmological constant is fromthe 05,55 components ofequations (12) we obtain
Λ/2M3 < 0, the brane tension is V, and the induced-
gravity crossoverscale is rc=m2/M3. n′oa˙o + a′ob˙o a˙′o = T05 (21)
We assume the cosmologicalbulk ansatz noao aobo − ao 6M3
ds2 = n(t,y)2dt2+a(t,y)2γ dxidxj+b(t,y)2dy2, (7)
ij
− a′ a′ n′ b2 a¨ a˙ a˙ n˙ kb2 T Λb2
where γ is a maximally symmetric 3-dimensional met- o o + o o o + o o o o= 55− o,
ric, paraijmetrized by the spatial curvature k = 1,0,1. ao(cid:16)ao no(cid:17)−n2ohao ao(cid:16)ao−no(cid:17)i− a2o 6M3
− (22)
The non-zero components of the five-dimensional Ein-
whereT ,T arethe 05and55componentsofT
stein tensor are 05 55 AC|m,B
evaluated on the brane. Substituting the expressions
a˙ a˙ b˙ n2 a′′ a′ a′ b′ kn2 (19), (20) in equations (21), (22), we obtain the semi-
G =3 + + + (8)
00 na(cid:16)a b(cid:17)− b2h a a(cid:16)a − b(cid:17)i a2 o conservation law and the Raychaudhuri equation
a2 a′ a′ 2n′ b′ n′ 2a′ 2a′′ n′′
Gij=b2γijna(cid:16)a + n (cid:17)− b(cid:16)n + a (cid:17)+ a + n o ρ˙+3aa˙o(ρ+p)=−2bn2oT50 (23)
a2 a˙ 2n˙ a˙ 2a¨ b˙ n˙ 2a˙ ¨b o o
+ γ + kγ (9)
n2 ijna(cid:16) n −a(cid:17)− a b(cid:16)n− a (cid:17)−bo− ij k r2(ρ+3p 2V) r2(ρ+3p 2V)(ρ+V)
G05=3 n′a˙ + a′b˙ a˙′ (10) (cid:16)Ho2+a2o(cid:17)h1− c 24m−2 i+ c 14−4m4
(cid:16)n a a b − a(cid:17) H˙ r2 k r2(ρ+V) Λ T5
a′ a′ n′ b2 a¨ a˙ a˙ n˙ kb2 + o+H2 1 c H2+ + c = − 5,(24)
G55=3na(cid:16)a + n(cid:17)− n2ha + a(cid:16)a−n(cid:17)i− a2 o,(11) (cid:16)no o(cid:17)h − 2 (cid:16) o a2o(cid:17) 12m2 i 6M3
whereprimesindicatederivativeswithrespecttoy,while where Ho = a˙o/aono is the Hubble parameter of the
dots derivatives with respect to t. The five-dimensional brane. One can easily check that in the limit m 0,
→
Einstein equations take the usual form equation (24) reduces to the corresponding second order
equation of the model without Rˆ [2]. Energy exchange
1
GAC = 2M3TAC|tot, (12) between the brane and the bulk has also been investi-
gated in [10, 11, 12].
where
Since only the 55 component of T enters equa-
AC m,B
|
TA =TA +TA +TA +TA +TA (13) tion (24), one can derive a cosmological system that is
C|tot C|v,B C|m,B C|v,b C|m,b C|ind largely independent of the bulk dynamics, if at the po-
is the total energy-momentum tensor,
sition of the brane the contribution of this component
TA =diag( Λ, Λ, Λ, Λ, Λ) (14) relative to the bulk vacuum energy is much less impor-
C|v,B − − − − − tantthanthe branematterrelativeto the branevacuum
δ(y)
TA =diag( V, V, V, V,0) (15) energy, or schematically
C|v,b − − − − b
δ(y) T5 ρ
TA =diag( ρ,p,p,p,0) . (16) 5 . (25)
C|m,b − b (cid:12) Λ (cid:12)≪(cid:12)V (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
3
Then,for Λ notmuchlargerthantheRandall-Sundrum (30), which simplify to (we omit the subscript o in the
value V2/|12|M3, the term T5 in equation (24) can be following)
5
ignored. Alternatively, the term T5 can be ignored in
5
equation (24) if simply ρ˙+3(1+w)Hρ= T (35)
−
k
H2 =µ+2γρ β λ+24γρ+χ (36)
T55 1. (26) ± p − a2
(cid:12) Λ (cid:12)≪ 1
(cid:12) (cid:12) χ˙ +4Hχ=24γT 1 λ+24γρ+χ , (37)
(cid:12) (cid:12) (cid:16) ± 6βp (cid:17)
Notethatrelations(25)and(26)areonlyboundarycon-
ditionsforT5,whichinarealisticdescriptionintermsof while the second order equation (24) for the scale factor
5
bulkfieldswillbetranslatedintoboundaryconditionson becomes
thesefields. Inthespecialcasewhere(25),(26)arevalid
a¨ λ+6(1 3w)γρ
throughout the bulk, the latter remains unperturbed by =µ (1+3w)γρ β − . (38)
the exchange of energy with the brane. a − ± √λ+24γρ+χ
Onecannowcheckthatafirstintegralofequation(24)
Finally,settingψ √λ+24γρ+χ,equations(36),(37),
is ≡
(38) take the form
2H2 ρ+V 6 3k ρ+V k 2
Ho4− 3o(cid:16) 2m2 +rc2−a2o(cid:17)+(cid:16)6m2 −a2o(cid:17) + H2 =µ+2γρ±βψ− ak2 (39)
4 Λ k χ λ+6(1 3w)γρ 2γT
+ =0, (27) ψ˙ +2H ψ − = (40)
rc2(cid:16)12M3−a2o(cid:17)− 3rc2 (cid:16) − ψ (cid:17) ± β
a¨ λ+6(1 3w)γρ
with χ satisfying =µ (1+3w)γρ β − . (41)
a − ± ψ
χ˙ +4n H χ= rc2n2oT H2 ρ+V + k , (28) Throughout, we will assume T(ρ)=Aρν, with ν > 0, A
o o m2bo (cid:16) o− 6m2 a2o(cid:17) constant parameters [2, 13]. Notice that the system of
equations(35)-(37)hastheinflux-outflowsymmetryT
and T=2T50 is the discontinuity across the brane of the T,H H,t t. ForT =0 the systemreduces→to
05componentofthebulkenergy-momentumtensor. The −the cosm→ol−ogy stu→die−d in [14].
solution of (27) for Ho is Wewillbe referringtotheupper (lower) solutionas
±
BranchA (Branch B). We shall be interested in a model
H2 = ρ+V + 2 k 1 2(ρ+V)+12 Λ +χ 12, that reduces to the Randall-Sundrum vacuum in the ab-
o 6m2 rc2−a2o ± √3rch m2 rc2−M3 i sence of matter, i.e. it has vanishing effective cosmo-
(29) logical constant. This is achieved for µ= β√λ, which,
∓
and equation (28) becomes giventhatm2V+12M6isnegative(positive)forbranches
A (B), is equivalent to the fine-tuning Λ= V2/12M3.
2n2T r 2(ρ+V) 12 Λ 1 NoticethatforBranchA,V isnecessarilyne−gative. Cos-
χ˙+4n H χ= o 1 c + +χ 2 .
o o m2bon ±2√3h m2 rc2−M3 i o mologieswithnegativebranetensionintheinducedgrav-
(30) ity scenario have also been discussed in [15].
At this point we find it convenient to employ a coor- Consider the case k = 0. The system possesses the
dinate frame in whichbo=no=1 in the aboveequations. obvious fixed point (ρ∗,H∗,ψ∗) = (0,0,√λ). However,
This canbe achievedby usingGauss normalcoordinates forsgn(H)T <0therearenon-trivialfixedpoints,which
with b(t,z)=1, and by going to the temporal gauge on are found by setting ρ˙ = ψ˙ = 0 in equations (35), (40).
the brane with n =1. It is also convenient to define the For w 1/3 these are:
o ≤
parameters
2T(ρ )2
∗
=2µ+(1 3w)γρ
2V 12 Λ 9(1+w)2ρ2 − ∗
λ = + (31) ∗
mV2 rc22− M3 ±p9(1+w)2γ2ρ2∗+4β2[λ+6(1−3w)γρ∗] (42)
µ = + (32) T(ρ∗)
6m2 r2 H∗ = (43)
c −3(1+w)ρ
∗
1
γ = (33) 3(1+w)
12m2 ψ∗2± β γρ∗ψ∗−[λ+6(1−3w)γρ∗]=0. (44)
1
β = . (34)
√3r Equation (41) gives
c
For a perfect fluid on the brane with equation of state a¨ T(ρ∗)2
= , (45)
p = wρ our system is described by equations (23), (29), (cid:16)a(cid:17)∗ 9(1+w)2ρ2∗
4
which is positive, and also, it has the same form (as a to use (dimensionless) flatness parameters such that the
function of ρ ) as in the absence of Rˆ. The deceleration state space is compact [20]. Defining
∗
parameter is found to have the value
2γρ βψ H
q∗ =−1, (46) ωm= D2 , ωψ = D2 , Z = D , (49)
whichmeansH˙ =0. Furthermore,atthis fixedpointwe
∗
find
where D= H2 µ, we obtain the equations
−
2γρ 18(1+w)2 p
Ω ∗ = γρ3−2ν. (47)
m∗ ≡ H∗2 A2 ∗ ωm+ωψ =1 (50)
oEnqeuarotiootnf(o4r2e)a,cwhhbernaenxcphressedintermsofΩm∗,hasonly ωm′ =ωmh(1+3w)(ωm−1)Z− Aµ (cid:16)|µ2|ωγm(cid:17)ν−1(1−Z2)32−ν
| |
ρ = β 6(1−3w)β±√λ(1−3w−4Ω−m1∗). (48) 2Z(1 Z2)1−Zp2−3(1−3w)β2µ−1ωm (51)
∗ 2γ (2Ω−m1∗+1+3w)(Ω−m1∗−1) − − 1−ωm i
1 Z2 3(1 3w)β2µ−1ω
However,itcanbeseenfrom(48)thatfor 1 w 1/3 Z′=(1 Z2) (1 Z2) − − − m 1
and Ωm∗ <1 the BranchB is inconsistent−wit≤h equ≤ation − h − 1−ωm −
1+3w
(42). On the contrary, Branch A with 1 w 1/3 ω ,(52)
and Ω < 1 is consistent for 0 < 6(1 −3w)≤β+√≤λ(1 − 2 mi
m∗
− −
3w 4Ω−1)<3 4(1 3w)2β2 (1+w)2λ. Thus,since we
are−intemre∗sted inpreali−zing the−present universe as a fixed with ′=d/dτ=D−1d/dt. Note that 1 Z 1, while
− ≤ ≤
point, Branch B should be rejected, and from now on both ω’s satisfy 0 ω 1. The deceleration parameter
≤ ≤
we will only consider Branch A. So, we have seen until is given by
now that for negative brane tension, we can have a fixed
point of our model with acceleration and 0 < Ωm∗ < 1. 1 1+3w ω Z2 3(1 3w)β2µ−1ω
Ttahinisedbeihnatvhioerciosnqtuexatlitoaftitvheelymdoiffdeelrepnrtesfreonmtedthieno[2n]e(ofobr- q=Z2h 2 ωm−(1−Z2) m− − 1−−ωm (5m3i)
−1/3≤ w ≤ 1/3), where for positive brane tension we andH′ = HZ(q+1). Thesystemofequations(51)-(52)
have Ω > 2, while for negative brane tension the uni- −
m∗ inheritsfromequations(35)-(37)thesymmetryA A,
versenecessarilyexhibiteddeceleration;therefore,inthat →−
Z Z, τ τ. The system written in the new
modeltheideathatthepresentuniverseisclosetoafixed → − → −
variablescontainsonlythreeparameters. However,going
point could not be realized.
backtothephysicalquantitiesH,ρonewillneedspecific
Concerning the negative brane tension the following
values of two more parameters.
remarks are in order: (a) In the conventional, non-
Itisobviousthatthepointswith Z =1haveH = .
supersymmetric setting, it is well known that a negative | | ∞
Therefore,from(39)itarisesthattheinfinitedensityρ=
tension brane with or without induced gravity is accom-
big bang (big crunch) singularity, when it appears, is
panied by tachyonic bulk gravitational modes [16]; how- ∞
representedbyoneofthepointswithZ=1(Z= 1). The
ever, including the Gauss-Bonnet corrections relevant at points with ω =1, Z =1,0 have ω′ = , Z′−= and
high-energies,thetachyonicmodescanbecompletelyre- m | |6 m ∞ ∞
finite ρ, H; for w 1/3, one has in addition a¨/a=+ ,
moved for a suitable range of the parameters [17]. (b) ≤ ∞
i.e. divergent 4D curvature scalar on the brane.
Asshownin[18],insupersymmetrictheories,spacetimes
withtwobranesofoppositetensionarestable;inparticu- The system possesses, generically, the fixed point
lar, there is no instability due to expanding “balooning” (a) (ωm∗,ωψ∗,Z∗) = (0,1,0), which corresponds to the
modes on the negative brane. It is, however, unclear fixed point (ρ ,H ,ψ ) = (0,0,√λ) discussed above.
∗ ∗ ∗
what happens in models with supersymmetry unbroken For ν 3/2 there are in addition the fixed points
≤
in the bulk but softly broken on the brane. (c) Finally, (b) (ω ,ω ,Z ) = (0,1,1) and (c) (ω ,ω ,Z ) =
m∗ ψ∗ ∗ m∗ ψ∗ ∗
it has been shown [19] that with appropriate choice of (0,1, 1). All these critical points are either non-
−
boundary conditions, both at the linearized level as well hyperbolic, or their characteristic matrix is not defined
asinthefulltheory,thegravitationalpotentialofamass at all; thus, their stability cannot be studied by first
onanegativetensionbranehasthecorrect1/rattractive order perturbation analysis. In cases like these, one
behaviour. can find non-conventional behaviors (such as saddle-
nodes and cusps [21]) of the flow-chart near the criti-
cal points. There are two more candidate fixed points
III. CRITICAL POINT ANALYSIS
(d) (ω ,ω ,Z ) = (1,0,1) and (e) (ω ,ω ,Z ) =
m∗ ψ∗ ∗ m∗ ψ∗ ∗
(1,0, 1), whose existence cannot be confirmed directly
−
We shallrestrictourselvesto the flat casek=0. In or- from the dynamical system, since they make equations
der tostudy the dynamicsofthe system,it is convenient (51), (52) undetermined. Apart from the above, there
5
are other critical points given by in the region of the big bang/big crunch singularity one
obtains a(t) √ǫt, ρ(t) t−2, as in the standard radia-
A µω ν−1 3(1+w)Z ∼ ∼
| | m∗ = ∗ (54) tion dominated big-bang scenario. This means that for
|µ|(cid:16) 2γ (cid:17) −(1−Z∗2)32−ν ν < 3/2 the energy exchange has no observable effects
p 6β2 close to the big bang/big crunch singularity.
(1+3w)ω2 +(1 3w)1 (1 Z2) ω 2[1 (1 Z2)2]
m∗ − h − µ − ∗ i m∗− − − ∗
Ζ
1
=0. (55)
0.75
They exist only for AZ <0 and correspond to the ones
∗
givenbyequations(42)-(44). Forthe physicallyinterest-
0.5
ingcasew=0withinfluxwescannedtheparameterspace
and were convinced that for ν=3/2 there is always only 0.25
onefixedpoint; forν<3/2this6 isanattractor(A),while ω
for ν>3/2 this is a saddle (S). For w=0, ν=3/2 there 0 m
is either one fixed point (attractor) or no fixed points,
depending on the parameters. For the other character- -0.25
istic value w=1/3, we concluded that for ν<3/2 there
-0.5
is only one fixed point (attractor), for ν>2 there is only
one fixed point (saddle), while for 3/2<ν<2 there are
-0.75
either two fixed points (one attractor and one saddle)
or no fixed points at all, depending on the parameters.
0.2 0.4 0.6 0.8 1
For w=1/3, ν=3/2 there is either one fixed point (at-
tractor) or no fixed points. Finally, for w=1/3, ν =2
FIG. 1: Influx, w=0, ν<3/2. The arrows show the direc-
thereiseitheronefixedpoint(saddle)ornofixedpoints.
tion of increasing cosmic time. The dotted line corresponds
Theseresultswereobtainednumericallyforawiderange to wDE =−1. The region inside (outside) the dashed line
of parameters and are summarized in Tables 1 and 2. corresponds to acceleration (deceleration). The region with
Z>0 represents expansion, while Z<0 represents collapse.
The present universe is supposed to be close to the global
ν <3/2 ν =3/2 ν >3/2
attractor.
No. of F.P. 1 0 or 1 1
Nature A A S
Table 1: The fixed points for w=0, influx
Ζ 1
ν<3/2 ν=3/2 3/2<ν<2 ν=2 ν>2 0.75
No. of F.P. 1 0 or 1 0 or 2 0 or 1 1
Nature A A A,S S S 0.5
Table 2: The fixed points for w=1/3, influx
0.25
ω
m
The approachtoanattractordescribedby the linearap- 0
proximation of (51)-(52) is exponential in τ and takes
infinite time τ for the universe to reach it. Given that -0.25
nearthisfixedpointtherelationbetweenthecosmictime
-0.5
t and the time τ is linear, we conclude that it also takes
infinite cosmic time to reach the attractor.
-0.75
Defining ǫ=sgn(H), we see from (51)-(52) that the
lines Z = ǫ (ν 3/2), ω = 0 are orbits of the sys-
≤ m 0.2 0.4 0.6 0.8 1
tem. Furthermore, the family of solutions with Z ǫ
and dZ/dω =Z′/ω′ 0 is approximately described≈for
ν < 3/2 bymω′ = ǫ(m1≈+3w)ω (ω 1), and thus, they FIG. 2: Outflow, w=1/3, ν <3/2. The arrows show the
m m m− direction of increasing cosmic time. The region inside (out-
move away from the point (ωm∗,Z∗)=(1,1), while they side) the dashed line corresponds to acceleration (decelera-
approach the point (ωm∗,Z∗)=(1, 1). In addition, the tion). The region with Z > 0 represents expansion, while
solution of this equation is ωm=[1−+ceǫ(1+3w)τ]−1, with Z<0represents collapse.
c > 0 an integration constant. Using this solution in
equation H′/H = Z(q+1) we find that for w = 1/3, Since our proposalrelies onthe existence ofanattrac-
−
H/Ho = √ωm/(1 ωm), where Ho is another integra- tor, we shall restrict ourselves to the case ν<3/2. It is
−
tion constant. Then, the equation for ωm(t) becomes convenient to discuss the four possible cases separately:
dω /dt= 2ǫω H2ω µ(1 ω )2, and can be inte- (i) w =0 with influx. The generic behavior of the so-
m − m o m− − m
grated giving t aps a function of ω or H. Therefore, lutions of equations (51)-(52) is shown in Figure 1. We
m
6
see that all the expanding solutions approach the global with negative tension, zero effective cosmological con-
attractor. Furthermore, there is a class of collapsing so- stant,andinthepresenceoftheinducedcurvaturescalar
lutions which bounce to expanding ones. Finally, there term in the action. Adopting the physically motivated
are solutions which collapse all during their lifetime to ρν power-law form for the energy transfer and assuming
a state with finite ρ and H. The physically interest- a cosmologicalconstantin the bulk, an autonomous sys-
ing solutions are those in the upper part of the diagram temofequationswasisolated. Inthisscenario,the“dark
emanating from the big bang (ω,Z) (1,1). These solu- energy” is a result of the geometry and the brane-bulk
≈
tionsstartwithaperiodofdeceleration. Thesubsequent energy exchange. The negative tension of the brane is
evolution depends on the value of 3β2/µ, which deter- necessary in order to realize the present universe (accel-
| |
minestherelativepositionofthedashedanddottedlines. erating with 0<Ω <1) as being close to a future fixed
m0
Specifically,for3β2/µ >1(the caseofFigure1)onedis- point of the evolution equations. We studied the possi-
| |
tinguishes two possible classes of universe evolution. In ble cosmologies using bounded normalized variables and
thefirst,theuniversecrossesthedashedlineenteringthe the corresponding global phase portraits were obtained.
accelerationerastillwithw > 1,andfinallyitcrosses By studying the number and nature of the fixed points
DE
−
the dotted line to w < 1 approaching the attractor. we demonstrated numericaly that our present universe
DE
−
Inthesecond,whileinthedecelerationera,itfirstcrosses can be easily realized as a late-time fixed point of the
the dotted line to w < 1, and then the dashed line evolution. Thisprovidesanalternativeanswertothe co-
DE
entering the eternally acce−lerating era. For 3β2/µ 1, incidence problem in cosmology, which does not require
| |≤
the dotted line lies above the dashed line, and, conse- specific fine-tuning of the initial data. Furthermore, the
quently, only the second class of trajectories exists. To equation of state for the dark energy at the attractor is
connect with the discussion in the introduction, notice uniquely specified by the value Ω . Remarkably, for
m0
that the Friedmann equation (39) can be written in the Ω =0.3, one obtains w = 1.4, independently of
m0 DE,0
−
form(1)withdarkenergyρ =(βψ+µ)/2γ. Using(40), theotherparameters,whilefortheothersuggestivevalue
DE
the equation for ρ takes the form (3) with Ω =0.03, w = 1.03. In the past, the function
DE m0 DE,0
−
w crosses the line w = 1 to larger values.
DE DE
1 β2ω (1 Z2) −
w = − 2Z2 ω 1 6(1 3w) m − .
DE 3(1−ωm)h − m− − − µ Z2−ωm i It would be interesting to investigate if the above par-
(56)
tialsuccessofthepresentscenariopersistsafteronetries
The global attractor (42)-(44) satisfies relations (4) and
tofitthe supernovadataandthedetailedCMB spectum
consequently, w evolves to the value w given by
DE DE∗ [22]. Of course, the nature of the content of the bulk
(5). As for the bouncing solutions, they approach the
andofthemechanismofenergyexchangewiththebrane
attractor after they cross the line Z2=ω , where w
m DE is another crucial open question, which we hope to deal
jumps from + to ; however, the evolution of the
∞ −∞ with in a future publication.
observable quantities is regular.
(ii) w=0 withoutflow. Thegenericbehaviorinthis case
is obtained from Figure 1 by the substitution Z Z
→−
and τ τ, which reflects the diagram with respect to
→−
the ω axis and converts attractors to repelers.
m
(iii) w=1/3 with outflow. Figure 2 depicts the flow dia-
gramofthiscase. Eventhoughinthecaseofradiationin
general w > 1/3 from equation (56), there are both
DE
−
accelerationanddecelerationregions. Furthermore,from
equation (5) it is Ω >1.
m∗
(iv) w=1/3 with influx. This arises like in (ii) by reflec-
tion of Figure 2 and resembles Figure 1.
Acknowlegements. SupportedinpartbytheEUgrant
MRTN-CT-2004-512194. The work of G.K. is also sup-
IV. CONCLUSIONS
portedbytheEuropeanCommissionMarie-CurieFellow-
ship under contract MEIF-CT-2004-501432. G.P. is also
In this work, we studied the role of brane-bulk en- supported by the Greek Ministry of Education research
ergy exchange on the cosmological evolution of a brane program“Herakleitos”.
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