Article
On coincidence problem in ELKO dark energy model
General Relativity and Gravitation (Impact Factor: 1.77). 09/2011; 44(9). DOI: 10.1007/s107140121392x
Source: arXiv
Fulltext
Available from: Hossein Mohseni Sadjadi, Aug 19, 2014arXiv:1109.1961v1 [grqc] 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 143995596 1, 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
1 Introduction
To describe the present accelerated expansion of the universe [1], many
models have been considered. In dark energy m odels, almost 70% of our
universe is assumed to be ﬁlled w ith 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, sp inor dark energy mod el 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 (CP T )
2
=
−1, was introduced. Cosmological consequences of this model in a spatially
ﬂat Friedmann Robertson Walker (FRW) spacetime 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 den sity is of order r :=
ρ
m
ρ
d
≃ O(1). This problem, i.e. why
dark energy density, ρ
d
, is of the same order of the matter density, ρ
m
, is
known as the coincidence problem [5].
∗
mohsenisad@ut.ac.ir
1
Page 1
In [6], it was found th at, 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 th is model.
We use the units ~ = c = 1.
2 Attractor solutions in ELKO cosmology
We consider the spatially ﬂat FRW spacetime
ds
2
= dt
2
− a
2
(t)(dx
2
+ dy
2
+ dz
2
), (1)
where a(t) is the scale factor. In this background by writing the dark spinor
as ψ = φλ, wh er e λ is a constant spinor, th e energy momentum tensor of
dark energy sector may be derived as (the derivation of energy momentum
tensor of spinors in term s of the ﬁeld φ can be found in [4], in details)
T
0
0
= ρ
d
=
1
2
˙
φ
2
+ V +
3
8
H
2
φ
2
T
i
j
= −δ
i
j
P
d
= δ
i
j
3
8
H
2
φ
2
+ V −
1
2
˙
φ
2
+
1
4
˙
Hφ
2
+
1
2
φ
˙
φH
, (2)
where V = V (φ
2
) is the potential. ρ
d
, and P
d
are the energy density and
pressure of dark energy respectively. In the absen ce of interaction, the con
tinuity equation for th e dark sector
˙ρ
d
+ 3H(P
d
+ ρ
d
) = 0, (3)
implies
¨
φ + 3H
˙
φ +
dV
dφ
−
3
4
H
2
φ = 0. (4)
Now let us consider a FRW Universe dominated by E LKO dark energy
and a barotropic matter ρ
m
whose the pressure is P
m
= w
m
ρ
m
. The equation
of state parameter of the matter, w
m
, is assumed to be non negative w
m
≥ 0,
e.g. for cold dark matter we have w
m
= 0. The continuity equations for
dark energy and matter component in the presence of the interaction source
C become
˙ρ
d
+ 3H(P
d
+ ρ
d
) = −C
˙ρ
m
+ 3Hγρ
m
= C, (5)
where γ := w
m
+ 1. The Friedmann equation is given by
H
2
=
1
3M
2
p
(ρ
d
+ ρ
m
), (6)
2
Page 2
which can be rewritten as
(1 −
1
8M
2
p
φ
2
)H
2
=
1
3M
2
p
(ρ
m
+
1
2
˙
φ
2
+ V ). (7)
M
p
is the reduced Planck mass. So the eﬀective gravitational coupling con
stant is modiﬁed in this theory. For (ρ
m
+
1
2
˙
φ
2
+ V ) > 0, we must have
φ < 2
√
2M
p
.
Raychaudhuri equation reads:
(1 −
1
8M
2
p
φ
2
)
˙
H = −
1
2M
2
p
(γρ
m
+
˙
φ
2
−
1
2
Hφ
˙
φ). (8)
The scalar ﬁeld, φ, satisﬁes the classical equation of motion:
˙
φ
¨
φ + 3H
˙
φ + V
,φ
−
3
4
H
2
φ
= −C. (9)
The equation of state parameter of the Universe deﬁned by w =
P
d
+P
m
ρ
d
+ρ
m
is
given by w = −1 +
2
3
ω, where ω = −
˙
H
H
2
.
To study the cosmological dynamics of this model, we deﬁne dimension
less variables [7]
x =
˙
φ
√
6M
p
H
, y =
√
V
√
3M
p
H
, z =
√
ρ
m
√
3M
p
H
, u =
φ
M
p
√
8
. (10)
Hence
(1 − u
2
)ω = 3x
2
+ 1.5γz
2
−
√
3xu. (11)
By the assumption that the potential is only a f unction of u, and by
deﬁning
f (u) =
M
p
V
,φ
V
, (12)
where V
,φ
=
dV
dφ
, we ﬁnd out the autonomous system of diﬀerential equations
x
′
= (ω − 3)x +
√
3
2
u −
r
3
2
y
2
f − C
1
y
′
= (
r
3
2
xf + ω)y
z
′
= (ω −
3
2
γ)z + C
2
u
′
=
√
3
2
x, (13)
where, prime denotes derivatives with respect to the efolding time N = ln a,
and
C
1
=
C
√
6M
p
H
2
˙
φ
C
2
=
C
2
√
3M
p
H
2
√
ρ
m
=
x
z
C
1
. (14)
3
Page 3
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 :
x
2
+ y
2
+ z
2
+ u
2
= 1. (15)
Most generally, cr itical points of the autonomous system (13) denoted
with {¯x, ¯y, ¯z, ¯u} can be arran ged 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 6= 0, ¯u = 0}. From (15) we have ¯z
2
= 1. (13) implies
¯
C
2
= 0 and
¯
C
1
= 0, where bar denotes the value at th e critical point.
III :{¯x = 0, ¯y = 0, ¯z = 0, ¯u 6= 0}. From (15) we have ¯u
2
= 1, (13) gives
¯
C
2
= 0 and
¯
C
1
= ±
√
3
2
.
IV:{¯x = 0, ¯y = 0, ¯z 6= 0, ¯u 6= 0}. In this case, ¯u
2
+ ¯z
2
= 1, and
¯
C
1
=
√
3
2
¯u.
(11) gives ¯ω =
3
2
γ. We have also
¯
C
2
= 0.
It is obvious that in the absence of interaction critical points III and IV
do not exist.
V:{¯x = 0, ¯y 6= 0, ¯z = 0, ¯u = 0}, ¯y
2
= 1. Fr om (13) we h ave
¯
C
1
= −
q
3
2
¯
f,
¯
C
2
= 0, and (11) gives ¯ω = 0.
VI:{¯x = 0, ¯y 6= 0, ¯z 6= 0, ¯u = 0}, ¯z
2
+ ¯y
2
= 1. From (13) we obtain ¯ω = 0
which using ¯ω = 1.5γ¯z
2
, results in γ = 0. In this case,
¯
C
1
= −
q
3
2
¯
f ¯y
2
, and
¯
C
2
= 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) = λM
P
, and these critical points do not exist when C = 0.
VII: {¯x = 0, ¯y 6= 0, ¯z = 0, ¯u 6= 0}, ¯u
2
+ ¯y
2
= 1. From (13) we obtain
¯
C
2
= 0, ¯ω = 0 and
¯
C
1
=
√
3
2
¯u −
q
3
2
¯y
2
¯
f .
VIII: {¯x = 0, ¯y 6= 0, ¯z 6= 0, ¯u 6= 0}. ¯ω¯y = 0 gives ¯ω = 0. We obtain
also
¯
C
2
=
3
2
γ ¯z, and
¯
C
1
=
√
3
2
¯u −
q
3
2
¯y
2
¯
f. Using C
2
=
x
z
C
1
, we obtain
¯
C
2
= 0 which results in γ = 0. This could be obtained in another way : as
¯ω =
3γ
2
¯z
2
1−¯u
2
, ¯ω = 0 implies γ = 0.
In the absence of interaction, critical points VII and VIII exist only for
potential satisfying ¯u =
√
2¯y
2
¯
f.
To study the stability of the system arou nd 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
d
dN
δx
δy
δu
= M
δx
δy
δu
(16)
are negative at a critical point, the system has stable attractor solution. In
4
Page 4
our model
M =
¯ω − 3 −
¯
C
1,x
−
√
6¯y
¯
f −
¯y
¯
C
1,z
¯z
√
3
2
−
q
3
2
¯y
2
¯
df
du
−
¯
C
1,u
−
¯u
¯
C
1,z
¯z
q
3
2
¯y
¯
f −
¯y
1−¯u
2
√
3¯u ¯ω −
3γ ¯y
2
1−¯u
2
¯y
1−¯u
2
(2¯ω ¯u − 3γ ¯u)
√
3
2
0 0
.
(17)
As we are interested to study the coincidence problem, among the situ
ations IVIII, we need only consid er cases where r is of order O(1) or more
precisely:
r :=
ρ
m
ρ
d
=
z
2
1 − z
2
≃
3
7
. (18)
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 w
m
= −1, in contradiction
with our assumption th at the un iverse 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
M
IV
=
3
2
(γ − 2) −
¯
C
1,x
0
√
3
2
−
¯
C
1,u
−
¯u
¯
C
1,z
¯z
0
3
2
γ 0
√
3
2
0 0
. (19)
One of the eigenvalues of M
IV
is λ =
3γ
2
which is positive, so even in this
situation the sy stem is not stable and there is no scaling attractor. Besides
for an accelerated expanding universe we must have w < −
1
3
, implying
ω < 1, but in IV, we have ¯ω =
3
2
γ > 1 which, in contradiction with the
nowadays accelerated expansion of the Un iverse, describes a decelerated
expanding Universe.
3 Summary
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 cr irtical points and attractor solutions of the problem were
studied. The coincidence problem was discussed in this framework and it
was foun d that there is no stable solution which can alleviate the coincidence
problem.
5
Page 5
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 "In this work we have developed a new approach to study the stability of a system composed by an ELKO field interacting with DM, which could give some contribution in order to alleviate the cosmic coincidence problem. Since that recent works44454647 have not found stable points for such system for different dynamic variables and interactions terms, we are led to believe (without demonstration) that the system ELKODM does not allow stable points. Based on these results, we have supposed an additional constraint to the dynamical system, namely that the potential of the ELKO field is related to a constant parameter δ, then we have analysed the stability conditions for such new system. "
[Show abstract] [Hide abstract] ABSTRACT: In this work it has been developed a new approach to study the stability of a system composed by an ELKO field interacting with dark matter, which could give some contribution in order to alleviate the cosmic coincidence problem. It is assumed that the potential that characterizes the ELKO field is not specified, but it is related to a constant parameter $\delta$. The strength of the interaction between the matter and the ELKO field is characterized by a constant parameter $\beta$ and it is also assumed that both the ELKO field as the matter energy density are related to their pressures by equations of state parameters $\omega_\phi$ and $\omega_m$, respectively. The system of equations is analysed by a dynamical system approach. It was found out the conditions of stability between the parameters $\delta$ and $\beta$ in order to have stable fixed points for the system for different values of the equation of state parameters $\omega_\phi$ and $\omega_m$, and the results are presented in form of tables. The possibility of decay of Elko field into dark matter or vice verse can be read directly from the tables, since the parameters $\delta$ and $\beta$ satisfy some inequalities. This opens the possibility to constrain the potential in order to have a stable system for different interactions terms between the Elko field and dark matter. The cosmic coincidence problem can be alleviated for some specific relations between the parameters of the model. 
Article: Dynamics of Teleparallel Dark Energy
[Show abstract] [Hide abstract] ABSTRACT: Recently, Geng et al. proposed to allow a nonminimal coupling between quintessence and gravity in the framework of teleparallel gravity, motivated by the similar one in the framework of General Relativity (GR). They found that this nonminimally coupled quintessence in the framework of teleparallel gravity has a richer structure, and named it "teleparallel dark energy". In the present work, we note that there might be a deep and unknown connection between teleparallel dark energy and Elko spinor dark energy. Motivated by this observation and the previous results of Elko spinor dark energy, we try to study the dynamics of teleparallel dark energy. We find that there exist only some darkenergydominated de Sitter attractors. Unfortunately, no scaling attractor has been found, even when we allow the possible interaction between teleparallel dark energy and matter. However, we note that $w$ at the critical points is in agreement with observations (in particular, the fact that $w=1$ independently of $\xi$ is a great advantage).  [Show abstract] [Hide abstract] ABSTRACT: In this work, we have studied the BransDicke (BD) cosmology in anisotropic models. We present three dimensional dynamical system describing the evolution of anisotropic models containing perfect fluid and BD scalar field with selfinteracting potential. The relevant equations have been transformed into the dynamical system. The critical points and the corresponding eigen values have been found in radiation, dust, dark energy, $\Lambda$CDM and phantom phases of the universe. The natures and the stability around the critical points have also been investigated.