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arXiv:1109.1961v1 [gr-qc] 9 Sep 2011
On coincidence problem and attractor solutions in
ELKO dark energy model
H. Mohseni Sadjadi∗
Department of Physics, University of Tehran,
P. O. B. 14395-547, Tehran 14399-55961, Iran
September 12, 2011
Abstract
We study the critical points of a Universe dominated by ELKO
spinor field dark energy and a barotropic matter in an almost general
case. The coincidence problem and attractor solutions are discussed
and it is shown the coincidence problem can not be alleviated in this
model.
PACS numbers: 95.36.+x, 95.35.+d, 98.80.-k
1Introduction
To describe the present accelerated expansion of the universe [1], many
models have been considered. In dark energy models, almost 70% of our
universe is assumed to be filled with a smooth unknown matter with negative
pressure known as dark energy. Although the first candidates proposed
for dark energy were exotic dynamical scalar field such as quintessence or
phantom, but, spinor dark energy model has also attracted some attentions
recently [2].
In [3], a class of non standard spinors, constructed in momentum space
from the eigenspinors of the charge conjugation operator, known as ELKO
spinor(Eigenspinoren des LadungsKonjugationsOperators), satisfying (CPT)2=
−1, was introduced. Cosmological consequences of this model in a spatially
flat Friedmann Robertson Walker (FRW) space-time were studied in [4].
However viable models must be consistent with astrophysical data. These
data indicate that, despite the expansion of the universe, the ratio of matter
to dark energy density is of order r :=ρm
dark energy density, ρd, is of the same order of the matter density, ρm, is
known as the coincidence problem [5].
ρd≃ O(1). This problem, i.e. why
∗mohsenisad@ut.ac.ir
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In [6], it was found that, for some special potentials and also for some
special interactions between (dark) matter and dark energy, the coincidence
problem cannot be solved in ELKO dark energy model. In this paper, by
considering a general dark energy potential and also a general interaction
between dark matter and dark energy, it is shown that, in principle, the
coincidence problem cannot be alleviated in this model.
We use the units ? = c = 1.
2Attractor solutions in ELKO cosmology
We consider the spatially flat FRW space-time
ds2= dt2− a2(t)(dx2+ dy2+ dz2),
(1)
where a(t) is the scale factor. In this background by writing the dark spinor
as ψ = φλ, where λ is a constant spinor, the energy momentum tensor of
dark energy sector may be derived as (the derivation of energy momentum
tensor of spinors in terms of the field φ can be found in [4], in details)
T0
0
=
ρd=1
2
˙φ2+ V +3
8H2φ2
Ti
j
=
−δi
jPd= δi
j
?3
8H2φ2+ V −1
2
˙φ2+1
4
˙Hφ2+1
2φ˙φH
?
,
(2)
where V = V (φ2) is the potential. ρd, and Pdare the energy density and
pressure of dark energy respectively. In the absence of interaction, the con-
tinuity equation for the dark sector
˙ ρd+ 3H(Pd+ ρd) = 0,
(3)
implies
¨φ + 3H˙φ +dV
dφ−3
4H2φ = 0.
(4)
Now let us consider a FRW Universe dominated by ELKO dark energy
and a barotropic matter ρmwhose the pressure is Pm= wmρm. The equation
of state parameter of the matter, wm, is assumed to be non negative wm≥ 0,
e.g. for cold dark matter we have wm= 0. The continuity equations for
dark energy and matter component in the presence of the interaction source
C become
˙ ρd+ 3H(Pd+ ρd) = −C
˙ ρm+ 3Hγρm= C,
(5)
where γ := wm+ 1. The Friedmann equation is given by
H2=
1
3M2
p
(ρd+ ρm),
(6)
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which can be rewritten as
(1 −
1
8M2
p
φ2)H2=
1
3M2
p
(ρm+1
2
˙φ2+ V ).
(7)
Mpis the reduced Planck mass. So the effective gravitational coupling con-
stant is modified in this theory. For (ρm+1
|φ| < 2√2Mp.
Raychaudhuri equation reads:
1
8M2
p
2M2
2˙φ2+ V ) > 0, we must have
(1 −
φ2)˙H = −
1
p
(γρm+˙φ2−1
2Hφ˙φ).
(8)
The scalar field, φ, satisfies the classical equation of motion:
˙φ
?
¨φ + 3H˙φ + V,φ−3
4H2φ
?
= −C.
(9)
The equation of state parameter of the Universe defined by w =Pd+Pm
ρd+ρmis
given by w = −1 +2
To study the cosmological dynamics of this model, we define dimension-
less variables [7]
√V
√3MpH, z =
3ω, where ω = −
˙H
H2.
x =
˙φ
√6MpH, y =
√ρm
√3MpH, u =
φ
Mp
√8.
(10)
Hence
(1 − u2)ω = 3x2+ 1.5γz2−
√3xu.
(11)
By the assumption that the potential is only a function of u, and by
defining
f(u) =MpV,φ
V
,
(12)
where V,φ=dV
dφ, we find out the autonomous system of differential equations
√3
2u −
?
(ω −3
√3
2x,
where, prime denotes derivatives with respect to the e-folding time N = lna,
and
C
√6MpH2 ˙φ
C
2√3MpH2√ρm
x′
=(ω − 3)x +
3
2xf + ω)y
?
3
2y2f − C1
y′
=(
z′
=
2γ)z + C2
u′
=
(13)
C1
=
C2
=
=x
zC1.
(14)
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C is taken to be a function of x,z,u. Note that x,y,z,u are not independent
and are constraint to the Friedmann equation :
x2+ y2+ z2+ u2= 1.
(15)
Most generally, critical points of the autonomous system (13) denoted
with {¯ x, ¯ y, ¯ z, ¯ u} can be arranged as follows:
I: {¯ x = 0, ¯ y = 0, ¯ z = 0, ¯ u = 0} which is in contradiction with (15), and
then is ruled out.
II:{¯ x = 0, ¯ y = 0, ¯ z ?= 0, ¯ u = 0}. From (15) we have ¯ z2= 1. (13) implies
¯ C2= 0 and¯ C1= 0, where bar denotes the value at the critical point.
III:{¯ x = 0, ¯ y = 0, ¯ z = 0, ¯ u ?= 0}. From (15) we have ¯ u2= 1, (13) gives
¯ C2= 0 and¯ C1= ±
IV:{¯ x = 0, ¯ y = 0, ¯ z ?= 0, ¯ u ?= 0}. In this case, ¯ u2+¯ z2= 1, and¯ C1=
(11) gives ¯ ω =3
It is obvious that in the absence of interaction critical points III and IV
do not exist.
V:{¯ x = 0, ¯ y ?= 0, ¯ z = 0, ¯ u = 0}, ¯ y2= 1. From (13) we have¯ C1= −
¯ C2= 0, and (11) gives ¯ ω = 0.
VI:{¯ x = 0, ¯ y ?= 0, ¯ z ?= 0, ¯ u = 0}, ¯ z2+ ¯ y2= 1. From (13) we obtain ¯ ω = 0
which using ¯ ω = 1.5γ¯ z2, results in γ = 0. In this case,¯ C1= −
¯ C2= 0.
In the absence of interaction, critical points V and VI exist only for
potential satisfying¯f(u) = f(¯ u) = 0. E.g. for the potential V ∝ exp(λφ),
we have¯f(u) = λMP, and these critical points do not exist when C = 0.
VII: {¯ x = 0, ¯ y ?= 0, ¯ z = 0, ¯ u ?= 0}, ¯ u2+ ¯ y2= 1. From (13) we obtain
¯ C2= 0, ¯ ω = 0 and¯ C1=
2¯ u −
VIII: {¯ x = 0, ¯ y ?= 0, ¯ z ?= 0, ¯ u ?= 0}. ¯ ω¯ y = 0 gives ¯ ω = 0. We obtain
also ¯ C2 =
2¯ u −
¯ C2= 0 which results in γ = 0. This could be obtained in another way : as
¯ ω =3γ
2
In the absence of interaction, critical points VII and VIII exist only for
potential satisfying ¯ u =√2¯ y2¯f.
To study the stability of the system around the above points, we consider
small perturbation around these critical points, {¯ x, ¯ y, ¯ z, ¯ u} → {¯ x + δx, ¯ y +
δy, ¯ z + δz, ¯ u + δu}. If the real part of all the eigenvalues of M defined by
√3
2.
√3
2¯ u.
2γ. We have also¯ C2= 0.
?
3
2¯f,
?
3
2¯f¯ y2, and
√3
?
3
2¯ y2¯f.
3
2γ¯ z, and ¯ C1 =
√3
?
3
2¯ y2¯f. Using C2 =
x
zC1, we obtain
¯ z2
1−¯ u2, ¯ ω = 0 implies γ = 0.
d
dN
δx
δy
δu
= M
δx
δy
δu
(16)
are negative at a critical point, the system has stable attractor solution. In
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our model
M =
¯ ω − 3 −¯C1,x
3
2¯ y¯f −
√3
2
−√6¯ y¯f −¯ y¯C1,z
¯ ω −3γ¯ y2
0
¯ z
√3
2−
?
1−¯ u2(2¯ ω¯ u − 3γ¯ u)
0
3
2¯ y2¯ df
du−¯C1,u−¯ u¯C1,z
¯ z
?
¯ y
1−¯ u2
√3¯ u
1−¯ u2
¯ y
.
(17)
As we are interested to study the coincidence problem, among the situ-
ations I-VIII, we need only consider cases where r is of order O(1) or more
precisely:
r :=ρm
ρd
In the cases III, V, and VII, we have ¯ r = 0, and in the case II, ¯ r → ∞.
Therefore among all the possible critical points II −V III, only IV, VI, and
VIII may be consistent with ¯ r ∼ O(1).
In VI, and VIII, we have γ = 0 which implies wm= −1, in contradiction
with our assumption that the universe is dominated by ELKO dark energy
and a matter with non-negative pressure. So, finally, we are left only with
the case IV. In this case
=
z2
1 − z2≃3
7.
(18)
MIV =
3
2(γ − 2) −¯C1,x
0
√3
2
0
3
2γ
0
√3
2−¯C1,u−¯ u¯C1,z
0
0
¯ z
.
(19)
One of the eigenvalues of MIV is λ =3γ
situation the system is not stable and there is no scaling attractor. Besides
for an accelerated expanding universe we must have w < −1
ω < 1, but in IV, we have ¯ ω =
2γ > 1 which, in contradiction with the
nowadays accelerated expansion of the Universe, describes a decelerated
expanding Universe.
2which is positive, so even in this
3, implying
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3Summary
Using Friedmann and Raychaudhury equations we obtained an autonomous
dynamical system describing the behavior of a spatially flat FRW Universe
dominated by ELKO non standard spinor dark energy, interacting with a
barotropic matter. We did not restrict the problem to special potentials or
interactions. The crirtical points and attractor solutions of the problem were
studied. The coincidence problem was discussed in this framework and it
was found that there is no stable solution which can alleviate the coincidence
problem.
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