UV stable, Lorentz-violating dark energy with transient phantom era
UV stable, Lorentz-violating dark energy with transient phantom era
Maxim Libanov and Valery Rubakov
Institute for Nuclear Research of the Russian Academy of Sciences,
60th October Anniversary Prospect, 7a, Moscow, 117312, Russia
Eleftherios Papantonopoulos
Department of Physics, National Technical University of Athens,
Zografou Campus GR 157 73, Athens, Greece
M. Sami
Centre for Theoretical Physics, Jamia Millia, New Delhi-110025, India
Shinji Tsujikawa
Department of Physics, Gunma National College of Technology, Gunma 371-8530, Japan
Phantom fields with negative kinetic energy are often plagued by the vacuum quantum instability
in the ultraviolet region. We present a Lorentz-violating dark energy model free from this problem
and show that the crossing of the cosmological constant boundary w = −1 to the phantom equation
of state is realized before reaching a de Sitter attractor. Another interesting feature is a peculiar time-
dependence of the effective Newton’s constant; the magnitude of this effect is naturally small but may
be close to experimental limits. We also derive momentum scales of instabilities at which tachyons
or ghosts appear in the infrared region around the present Hubble scale and clarify the conditions
under which tachyonic instabilities do not spoil homogeneity of the present/future Universe.
I. INTRODUCTION
The compilations of various observational data show that the Universe has entered the stage of an accelerated
expansion around the redshift z ∼ 1 [1, 2, 3, 4, 5, 6]. The equation of state (EOS) parameter w of Dark Energy (DE)
responsible for the acceleration of the Universe has been constrained to be close to w = −1. However, the phantom
EOS (w < −1) is still allowed by observations and even favored by some analyses of the data [7]. It is also possible
that the EOS of DE crossed the cosmological constant boundary (w = −1) in relatively near past [8].
The presence of the phantom corresponds to the violation of weak energy condition, the property which is generally
difficult to accommodate within the framework of field theory. The simplest model which realizes the phantom EOS is
provided by a minimally coupled scalar field with a negative kinetic term [9, 10] (see also Refs. [11, 12]). The negative
kinetic energy is generally problematic because it leads to a quantum instability of the vacuum in the ultraviolet
(UV) region [10, 13, 14, 15, 16, 17]: the vacuum is unstable against the catastrophic particle production of ghosts and
normal (positive energy) fields.
There have been a number of attempts to realize the phantom EOS without having the pathological behaviour in
the UV region. One example is scalar-tensor gravity in which a scalar field φ with a positive kinetic term is coupled to
Ricci scalar R [18, 19]. This coupling leads to the modification of gravitational constant, but it was shown in Ref. [20]
that there are some parameter regions in which a phantom effective EOS is achieved without violating local gravity
constraints in the present Universe.
Another example is provided by the so-called modified gravity, including f(R) gravity models [21] and the Gauss-
Bonnet (GB) models [22]. In f(R) models it is possible to obtain a strongly phantom effective EOS, but in that case
the preceding matter epoch is practically absent [23]. For GB DE models it was shown in Ref. [24] that the crossing of
the cosmological constant boundary, w = −1, is possible, but local gravity experiments place rather strong constraints
on the effective GB energy fraction [25]. In addition, tensor perturbations are typically plagued by instabilities in
the UV region if the GB term is responsible for the accelerated expansion of the Universe [26]. Thus, it is generally
not so easy to construct viable modified gravity models that realize the phantom effective EOS without violating
cosmological and local gravity constraints.
The third example is the Dvali-Gabadadze-Porrati (DGP) braneworld model [27] and its extension [28] with a GB
term in the bulk, which allow for the possibility to have w < −1 [29, 30]. However, it was shown in Ref. [31] that the
DGP model contains a ghost mode, which casts doubts on the viability of the self-accelerating solution.
While the above models more or less correspond to the modification of gravity, it was recently shown that in the
Einstein gravity in a Lorentz-violating background the phantom EOS can be achieved without any inconsistency in
the UV region [32, 33]. In particular, in the model of Ref. [33] Lorentz invariance is broken in the presence of a vector
field Bµ which has two-derivative kinetic terms similar to those given in Ref. [34]. The effect of the Lorentz violation
http://arxiv.org/abs/0704.1848v2
is quantified by a parameter Ξ ≡ BµBµ/M2, where M is an UV cut-off scale. In analogy to Ref. [35] the vector field
also has one-derivative coupling ǫ∂µΦB
µ with a scalar field Φ, where ǫ is a small parameter that characterizes an IR
scale. In the UV region, where the spatial momentum p is much larger than ǫ, ghosts, tachyons and super-luminal
modes are not present. Meanwhile tachyons or ghosts can appear in the IR region p <∼ ǫ. This is not problematic
provided that ǫ is close to the present Hubble scale.
In this paper we apply this Lorentz-violating model to dark energy and study the cosmological dynamics in detail
in the presence of mass terms in the potential, V = 1
m2Φ2 − 1
M2X2 (where X2 = BµB
µ). We show that the
model has a de Sitter attractor responsible for the late-time acceleration. At early times DE naturally has normal
EOS with w > −1, while the phantom EOS can be realized between the matter-dominated era and the final de Sitter
epoch. We clarify the conditions under which the cosmological constant boundary crossing to the phantom region
occurs. Interestingly, in a range of parameters this crossing takes place at the epoch when Ωm ∼ ΩDE thus making
the crossing potentially observable.
Another interesting feature of our model is the time-dependence of the effective Newton’s constant. It is naturally
weak, but may well be comparable with current experimental limits. Moreover, the effective Newton’s constant G∗(t)
has a peculiar behaviour correlated with the deviation of w from −1.
We also derive momentum scales of instabilities of perturbations, first in Minkowski spacetime. This is the extension
of the work [33] that mainly focused on the case of massless scalar (m = 0). We show that in the UV region (p ≫ ǫ)
the model does not have any unhealthy states such as ghosts, tachyons or super-luminal modes. In the IR region
(p <∼ ǫ) tachyons or ghosts appear, depending on the momentum. Finally, we study the evolution of perturbations
in the cosmological background and estimate the amplitude of perturbations amplified by the tachyonic instability
around the scale of the present Hubble radius. The perturbations remain to be smaller than the background fields
under certain restriction on the model parameters.
This paper is organized as follows. In Sec. II we present our Lorentz-violating model and derive basic equations
describing spatially flat Friedmann–Robertson–Walker cosmology in the presence of DE, radiation and non-relativistic
matter. In Sec. III the cosmological dynamics is discussed in detail analytically and numerically with an emphasis
on the occurrence of a phantom phase before reaching a de Sitter attractor. The time-dependence of the effective
gravitational constant is also considered. In Sec. IV we study the Minkowski spectrum of field perturbations and
clarify the properties of tachyons and ghosts in the IR region. We then discuss the tachyonic amplification of field
perturbations around the present Hubble scale in the cosmological background. We summarize our results in Sec. V.
Appendix A contains the derivation of the effective “Newtonian gravitational constant” in our model. In Appendix
B we derive the fixed points of the system by rewriting the equations in autonomous form. We analyse the stability
of the fixed points and show analytically that the cosmological evolution proceeds from radiation-dominated stage
through matter-dominated stage to the final de Sitter regime.
II. LORENTZ-VIOLATING MODEL
We study a 4-dimensional Lorentz-violating model whose Lagrangian density includes a vector field Bµ and a scalar
field Φ:
L = −1
α(Ξ)gνλDµBνD
µBλ +
β(Ξ)DµBνD
µΦ + ǫ∂µΦB
µ − V (B,Φ) , (1)
where Ξ = BµB
µ/M2 with M being an UV cut-off scale of the effective theory. The dimensionless parameters α
and β are the functions of Ξ, and ǫ is a free positive parameter that characterizes an IR scale. The first two terms
in (1) are familiar in two-derivative theory [34], whereas the one-derivative term ǫ∂µΦB
µ is introduced following the
approach of Ref. [35].
We study dynamics of flat Friedmann-Robertson-Walker (FRW) Universe
ds2 = N 2(t)dt2 − a2(t)dx2 , (2)
where N (t) is a Lapse function and a(t) is a scale factor. In the case of spatially homogeneous fields with Bi = 0
(i = 1, 2, 3), the Lagrangian (1) reads
−gL = γ
2 − 3α
2 + ǫa3φ̇X − a3NV (X,φ) , (3)
where X = B0/N , φ is the homogeneous part of the field Φ and
γ(X) =
M2β(X)− α(X) . (4)
Hereafter we study the case in which the following condition holds
α > γ > 0 .
This is required to avoid a superluminal propagation in Minkowski spacetime [33], as we will see later. Throughout
this paper we assume that α and γ are of order unity.
For fixed X , the second term in the Lagrangian (3) has precisely the form of the Einstein–Hilbert action specified
to the flat FRW metric. Hence, it leads to the change of the “cosmological” effective Planck mass [33]
m2pl,cosm = m
pl + 4παX
2 . (5)
Another effective Planck mass mpl,Newton determines the strength of gravitational interactions at distances much
shorter than the cosmological scale; in general, these two effective Planck masses are different [19, 35, 36]. We show
in Appendix A that the “Newtonian” Planck mass in our model is given by
m2pl,Newton = m
pl − 4παX2 . (6)
Both effective Planck masses depend on time via X = X(t). Since the time-dependent terms in (5) and (6) differ by
sign only, it will be sufficient to study one of these effective masses. In what follows we concentrate on the “Newtonian”
mass (6) for definiteness.
In this paper we focus on the case in which the potential V takes a separable form:
V = W (φ) + U(X) . (7)
We take into account the contributions of non-relativistic matter and radiation whose energy densities ρm and ρr,
respectively, satisfy
ρ̇m + 3Hρm = 0 , (8)
ρ̇r + 4Hρr = 0 . (9)
The energy density of the fields is derived by taking the derivative with respect to N of the action S =
ρ = − 1
Ẋ2 − 3α
H2X2 +
φ̇2 + V . (10)
We set N = 1 for the rest of this paper.
The Friedmann equation is given by
γẊ2 − 3α
H2X2 +
φ̇2 +W (φ) + U(X) + ρm + ρr
, (11)
where κ2 = 8π/m2pl. The equations of motion for the homogeneous fields φ and χ are
Ẍ + 3HẊ
γ,XẊ
2 − 3
2X2 − 3αH2X + ǫφ̇ = U,X , (12)
−(φ̈+ 3Hφ̇)− ǫ(Ẋ + 3HX) = W,φ , (13)
where γ,X = dγ/dX , etc. Taking the time-derivative of Eq. (11) and using Eqs. (12) and (13), we obtain
Ḣ = −κ
ρ+ p+ ρm +
where
ρ+ p = ǫφ̇X + αḢX2 + 2αHXẊ + γẊ2 + φ̇2 + α,XHX
2Ẋ . (14)
In what follows we assume for simplicity that α and γ are constants, i.e., α,X = γ,X = 0.
Following Ref. [33] we consider the simplest potential for the fields,
W (φ) =
m2φ2 , U(X) = −1
M2X2 , (15)
which allows for a possibility to realize a phantom phase.
III. DYNAMICS OF DARK ENERGY
One way to analyse the cosmological dynamics in our model is to make use of the autonomous equations, the
techniques widely used in the context of dark energy studies [6, 38, 39]. This approach is presented in Appendix B,
where we analytically confirm that our model can lead to the sequence of radiation, matter and accelerated epochs.
Also, in Appendix B we derive the conditions under which the de Sitter solution given below is an attractor. Here we
first present a simpler analysis based on the slow-roll approximation. Then we give numerical solutions to eqs. (8),
(9), (11), (12), (13), exhibiting transient phantom behaviour, and study their dependence on various parameters of
our model, including the initial values of the fields.
A. Final and initial stages
One immediate point to note is that in the absence of radiation and matter, the system of equations (11), (12),
(13) has a de Sitter solution, H = const, for which φ and X are also independent of time, provided that
. (16)
Indeed, for constant H,φ and X eqs. (11), (12), (13) reduce to a simple algebraic system
H2X2 − M
3αH2 = M2 ,
−3ǫHX = m2φ . (17)
Once the inequality (16) is satisfied, this system has a solution
Mmplǫ√
3ǫ2/m2 − 2α
XA = −
3ǫ2/m2 − 2α
. (18)
We will see in what follows, and elaborate in Appendix B, that in a range of parameters this solution is an attractor
which corresponds to the de Sitter phase in asymptotic future (hence the notation). In order to use this for dark
energy we require that the mass scale M is of the order of the present Hubble parameter H0. Then the Newtonian
effective Planck mass, Eq. (6), is given by
m2pl,Newton = m
3ǫ2/m2 − 2α
. (19)
In order that the change of the Planck mass be small, we impose the condition
αm . (20)
It is worth noting that under this condition, the contribution of the field φ in the energy density dominates in the
de Sitter regime,
φ2A ≫
X2A =
X2A . (21)
Thus, as the system approaches the de Sitter attractor, the total energy density in the Universe becomes determined
by the scalar field energy density.
Another point to note is that at early times (at the radiation-dominated epoch already), when the Hubble parameter
is large enough, the term (−3αH2X) in Eq. (12) drives the field X to zero, the relevant time being of the order of
the Hubble time. Soon after that the field φ obeys the usual scalar field equation in the expanding Universe, so the
Hubble friction freezes this field out. Thus, the initial data for the interesting part of the DE evolution are
Xi = 0 ,
φi = const . (22)
The value of φi is a free parameter of the cosmological evolution in our model. Since at early times the field X is
close to zero, its effect on the evolution of the field φ is negligible. The field φ slowly rolls down its potential, and its
energy density dominates over that of X . Therefore, EOS for DE at early times is normal, w > −1, with w being
close to −1. We refer to this regime as quintessence stage. As we will see below, in a range of parameters, the system
eventually crosses the cosmological constant boundary w = −1 and passes through a transient phantom phase before
reaching the de Sitter asymptotics (18).
B. Slow roll phantom regime
The approach to the de Sitter solution (18) occurs in the slow roll regime. To see how this happens, we truncate
Eqs. (12) and (13) to
ǫφ̇− 3αH2X = U,X , (23)
−3ǫHX = W,φ . (24)
This truncation is legitimate provided that in addition to the usual slow-roll conditions φ̈ ≪ Hφ̇ and Ẍ ≪ HẊ, the
following conditions are satisfied:
φ̇ ≪ ǫX , (25)
ǫφ̇X ≪ V , (26)
Ẋ ≪ HX . (27)
[When writing inequalities, we always mean the absolute values of the quantities.] Note that we do not impose the
condition ǫφ̇ ≫ 3αH2X unlike in Ref. [33], since the term 3αH2X is not necessarily negligible relative to the term
U,X in Eq. (23).
From Eq. (23) we obtain
X = − ǫφ̇
, (28)
where
ξ ≡ 1− 3αH
. (29)
Note that ξ may be considered as a measure of the deviation from the de Sitter regime (18).
Substituting Eq. (28) into Eq. (24) we get the following equation
3Hφ̇ = ξW̃,φ , (30)
where
W̃ (φ) ≡ m
Equation (30) shows that the field φ rolls up the potential W (φ) for ξ > 0, i.e., for
. (31)
This is the region in which the phantom equation of state (w < −1) is realized; indeed, Eq. (14) gives ρ+ p ≈ ǫφ̇X =
ξXU,X ≡ −ξM2X2. Another way to understand the phantom behaviour is to notice that when the system approaches
the de Sitter regime, the field φ dominates the energy density, see Eq. (21), so the energy density increases as the
field φ rolls up.
Let us find out whether the slow roll conditions (25), (26) and (27) are indeed satisfied. Making use of Eq. (28) we
obtain that the condition (25) is equivalent to
ǫ2 ≫ ξM2 , (32)
while using Eqs. (28) and (30) we rewrite the condition (26) as
ǫ2 ≫ ξm
. (33)
The second inequality ensures also the validity of the relation (27); this can be seen by taking the time derivative
of Eq. (24). The latter two inequalities are automatically valid at small ξ, that is near the de Sitter solution (18).
We conclude that the approach to the de Sitter solution indeed occurs in the slow roll regime, and that the phantom
phase is indeed realised provided that the relation (31) holds. Our analysis implies also that the de Sitter solution
(18) is an attractor: the Hubble parameter slowly increases towards its de Sitter value, ξ decreases, and the dynamics
gets frozen as ξ → 0.
Since the field φ dominates the energy density at the phantom slow roll stage, the condition (31) takes a simple
φ < φA =
MmPl√
, (34)
where we made use of (20). The latter relation translates into the range of initial conditions which eventually lead to
the transient phantom behaviour,
φi <∼ φA . (35)
Indeed, during the radiation- and matter-dominated stages the field φ remains almost constant, and at the quintessence
stage it also does not roll down much.
Recalling again that φ dominates the energy density, we rewrite the inequality (33) as εs ≪ 1, where
ξ . (36)
The parameter εs may be viewed as the slow roll parameter for the field φ. Indeed, one observes that
= εs , (37)
which, together with Eq. (30), justifies this interpretation.
From Eqs. (10), (14), (28) and (30) it follows that during the slow roll phantom stage, the EOM parameter of DE
is given by
w = −1− εs . (38)
Hence, the appreciable deviation from w = −1 occurs when εs is not much smaller than unity, i.e., when φi is
appreciably smaller than φA.
In the next section we confirm these expectations by numerical analysis, and also show explicitly that in a range of
parameters, the cosmological evolution proceeds from radiation-dominated to matter-dominated epoch, and then to
the slow roll phantom stage, before finally ending up in the de Sitter regime (18).
C. Numerical solutions
In our numerical analysis we choose initial conditions Ẋ = φ̇ = X = 0 with nonzero values of φ, ρm and ρr. This
choice corresponds to the initial data (22). We have also tried many other initial conditions and found that the results
are not sensitive to the initial values of X , Ẋ and φ̇, in accord with the discussion in the end of Sec. III A.
In Fig. 1 we plot the cosmological evolution for the model parameters α = 1, γ = 1/2, ǫ/m = 3, M/m = 1 and the
initial value φi = 0.5φA. We find it convenient to present the plots in terms of the variable
N = ln a . (39)
Figure 1 clearly shows that the sequence of radiation, matter and de Sitter epochs can be achieved together with the
w = −1 crossing. The DE EOS parameter w is nearly a constant, w ≃ −1, during the radiation and matter epochs
because the fields are almost frozen. At the transition era from matter domination to DE domination, w begins
to grow because the kinetic energies of the fields become important; this is the quintessence phase. However, the
system soon enters the phantom phase during which the field φ rolls up the potential. Hence the equation of state w
crosses the cosmological constant boundary w = −1 and reaches a minimum value wmin < −1. The solution finally
approaches the de Sitter regime (18) from the phantom side. Of some interest is also the EOS parameter of the entire
system,
weff =
. (40)
It is seen from Fig. 1, this parameter also becomes smaller than −1 soon after w = −1 crossing.
- 1 . 5
- 1 . 0
- 0 . 5 0
0 . 0
0 . 5 0
1 . 0
1 . 5
0 5 1 0 1 5
FIG. 1: Cosmological evolution for the model parameters α = 1, γ = 1/2, ǫ/m = 3 and M/m = 1. We choose initial conditions
X = Ẋ = φ̇ = 0, φi = 0.5φA, and Ωr,i = 0.99, Ωm,i = 0.01. Shown is the evolution of ΩDE, Ωm, Ωr , w and weff as functions
of N ≡ ln a. Note that the present epoch corresponds to Ωm = 0.3 and ΩDE = 0.7, which is denoted by a vertical line. After
the cosmological constant boundary crossing, the DE EOS parameter w reaches a minimum wmin = −1.19 and then increases
towards the de Sitter value w = −1 from the phantom side.
The pattern shown in Fig. 1 is generic in our model, provided that its parameters and initial data obey m/ǫ ≪ 1,
M/ǫ ≪ 1 and φi < φA (in fact, the inequalities here need not be strong). The strengths of the effects depend, of
course, on the values of these parameters. In particular, the minimum value wmin is related to the slow-roll parameter
εs, in accord with Eq. (38). If the field φ evolves very slowly, one has εs ≪ 1, so wmin is close to −1. On the contrary,
the appreciable deviation from w = −1 occurs if εs is not very much smaller than unity.
FIG. 2: The minimum value of the EOS parameter w of DE, its value w0 at the present epoch (Ωm = 0.3, ΩDE = 0.7) and its
maximum value as functions of m/ǫ for M/ǫ = 1/30 and the initial value φi = 0.5φA.
Since the field φ is practically frozen during the radiation- and matter-dominated epochs, and evolves rather slowly
later on, the dependence of the cosmological evolution on the parameters of the model and on the initial value φi can
be understood, at qualitative level, by inspecting Eq. (36). For the qualitative discussion of the evolution well before
the asymptotic de Sitter regime sets in, the parameter ξ in (36) may be set equal to 1, while the value of φ may be set
equal to its initial value φi. Then Eq. (36) implies that with other parameters and φi fixed, for smaller m/ǫ one gets
smaller absolute value of εs at its minimum, leading to the value of wmin closer to −1. This is shown in Fig. 2. From
Fig. 1 it is clear, however, that the minimum of w occurs after the present epoch (ΩDE ≃ 0.7 and Ωm ≃ 0.3). Again,
this is a rather generic feature of our model. Therefore, instead of wmin, more interesting quantities are the present
value w0 of the DE EOS parameter and also its maximum value before the cosmological constant boundary crossing.
These quantities are also given in Fig. 2. Overall, the behaviour shown in Fig. 1 is more pronounced at larger m/ǫ,
once other parameters of solutions are kept fixed.
According to Eq. (36), the initial value of the field φ is also important to determine the amplitude of εs and hence
w: a smaller φi results in a stronger deviation of wmin from −1. The present value w0 also becomes more negative
(stronger deviating from −1), while the cosmological constant boundary crossing occurs earlier. From the numerical
analysis we find that at relatively large values of m/ǫ the increase of M has the opposite effect, while at smaller m/ǫ
the effects due to the variation of M are small. These properties are illustrated in Figs. 3 and 4.
As discussed above, the phantom phase occurs only if the initial value of φ obeys (35), otherwise the de Sitter
attractor is approached from the quintessence side, w > −1. This is illustrated in Fig. 5.
- 2 . 0
- 1 . 8
- 1 . 6
- 1 . 4
- 1 . 2
- 1 . 0
- 0 . 8 0
0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8
 m/e=1/3, M/e=1/3
 m/e=1/3, M/e=1/30
 m/e=1/10, M/e=1/3
 m/e=1/10, M/e=1/30
FIG. 3: The dependence of the present value w0 (i.e., w at Ωm = 0.3, ΩDE = 0.7) on the initial value of φ, for different sets of
the model parameters.
- 1 . 0
- 0 . 5 0
0 . 0
0 . 5 0
1 . 0
1 . 5
2 . 0
0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8
 m/e=1/3, M/e=1/3
 m/e=1/3, M/e=1/30
 m/e=1/10, M/e=1/3
 m/e=1/10, M/e=1/30
FIG. 4: The dependence of the redshift (from the present epoch, Ωm = 0.3, ΩDE = 0.7) of the cosmological constant boundary
crossing, w = −1, on the initial value of φ.
- 1 . 5
- 1 . 0
- 0 . 5 0
0 . 0
0 . 5 0
1 . 0
1 . 5
0 5 1 0 1 5 2 0
FIG. 5: Cosmological evolution for the model parameters α = 1, γ = 1/2, ǫ/m = 3 and M/m = 0.1. We choose initial
conditions X = Ẋ = φ̇ = 0, φi = 3φA, and Ωr,i = 0.99, Ωm,i = 0.01. The present epoch corresponds to ΩDE = 0.7 and
Ωm = 0.3, which is denoted by a vertical line. In this case the cosmological constant boundary crossing is not realized because
φ remains always larger than φA.
At the end of this section we discuss the variation of the effective gravitational constant. According to Eq. (6), the
effective Newton’s constant that determines the interaction deep inside the horizon scale is given by
G∗ = G(1 − 4πGαX2)−1 . (41)
Its variation in time is conveniently expressed in terms of the following quantity
d lnG∗
d ln a
≡ Ġ∗
8παXẊ
H(m2pl − 4παX2)
. (42)
The typical experimental and observational constraints on the variation of G∗ in the present Universe are given by
|Ġ∗/G∗| <∼ 10−12 yr−1 [37], which translates into the condition
Ġ∗/G∗
<∼ 10
−2H0 . (43)
- 0 . 0 1 0
0 . 0
0 . 0 1 0
0 . 0 2 0
0 . 0 3 0
0 . 0 4 0
0 5 1 0 1 5 2 0
( a )
( b )
FIG. 6: The evolution of the variation of the effective gravitational constant d lnG∗/d ln a as function of N = ln a and redshift
z. The cases (a) and (b) correspond to the model parameters and initial conditions given of Figs. 1 and 5, respectively. The
black points represent the values at the present epoch (Ωm ≃ 0.3).
FIG. 7: The maximum value of d lnG∗/d ln a as a function of m/ǫ. Other parameters and initial conditions are the same as in
Fig. 2.
In Fig. 6 we plot the evolution of the quantity d lnG∗/d lna for the model parameters and initial conditions given in
Figs. 1 and 5. At the present epoch (Ωm ≃ 0.3) we obtain the values Ġ∗/G∗ = 3.5× 10−2H−10 and 2.4× 10−2H
0 for
these two cases, respectively. Comparing Fig. 6 with Figs. 1 and 5 one observes that the variation of the gravitational
constant is correlated in time with the deviation of w from −1. This is clear from Eq. (41) too: the gravitational
constant varies when the field X changes in time, while the latter occurs during the transition from the matter
dominated stage to the final de Sitter attractor. It is precisely at this transition stage that w substantially deviates
from −1.
Figure 7 shows the maximum value of d lnG∗/d lna as a function of m/ǫ. Again, the variation of the effective
Newton’s constant is more pronounced at larger m/ǫ. This means that it is correlated with the amplitude of the
deviation of w from −1. The dependence of d lnG∗/d lna on M and on the initial value of φ is rather weak.
It is worth pointing out that as long as the deviation of the EOS from w = −1 is not so significant, the models
satisfy the constraint (43), and also that our model suggests that the variation of G∗ is close to the present upper
bound on Ġ∗/G∗.
IV. MOMENTUM SCALES OF INSTABILITIES
In this section the momentum scales of instabilities are present in our model. We first study dispersion relations in
the Minkowski space-time and then proceed to those in the FRW space-time. We wish to clarify the conditions under
which a tachyon or a superluminal mode appears by considering dispersion relations. We also evaluate the energy of
the modes to find out a ghost state.
A. Minkowski spectrum
Let us consider the perturbations for the fields,
B0 = X + b0 , Bi = bi , Φ = φ+ ϕ . (44)
The quadratic Lagrangian for perturbations, following from the general expression (1), is
Lb0,bi,ϕ =
∂µb0∂
µb0 +
∂µbi∂
µbi +
µϕ+ ǫ∂0ϕb0 − ǫ∂iϕbi
2 , (45)
where
m20 = U,XX , m
1 = −
, m2ϕ = W,φφ . (46)
For our model (15) one has −m20 = m21 = M2 and m2ϕ = m2. In what follows we concentrate on this case, and assume
the following relations, see (20) and (32),
ǫ ≫ M . (47)
Varying the Lagrangian (45) with respect to bi, b0 and ϕ, we obtain the equations for the field perturbations. In
order to find the spectrum of the system we write the solutions in the form b0 = b̃0e
= b̃0e
i(ωt−p·r), bi = b̃ie
and ϕ = ϕ̃eipµx
. The transverse mode of the vector field Bi has the dispersion relation
ω20 = p
. (48)
The three scalar modes b̃i = (pi/p)b̃L, b̃0 and ϕ̃ satisfy the following equations
ω2 − p2 − M
b̃L + i
pϕ̃ = 0 , (49)
ω2 − p2 + M
b̃0 − i
ωϕ̃ = 0 , (50)
ω2 − p2 −m2
ϕ̃− iǫωb̃0 − iǫpb̃L = 0 . (51)
Expressing b̃L and b̃0 in terms of ϕ̃ from Eqs. (49), (50) and plugging them into Eq. (51), we find that the eigenfre-
quencies corresponding to three mixed states satisfy
(z −m2)
z − M
− ǫ2z
= 0 , (52)
where
z ≡ ω2 − p2 . (53)
The spectrum in the case m = 0 was studied in Ref. [33]. Our purpose here is to extend the analysis to the case of
non-zero m. Denoting the solutions of Eq. (52) as z1, z2 and z3, we obtain the relation
z1z2z3 = −
. (54)
Once the conditions (47) are satisfied, then one can show that if the relation
z1 < z2 < z3 (55)
holds at some momentum, then the inequality (55) is satisfied for all momenta.
In the limits p → ∞ and p → 0, we obtain the following dispersion relations, respectively.
• (A) UV limit (p → ∞)
ω1 = p−
+O(M2/p) , (56)
ω2 = p+
2p3ǫ2(α+ γ)
+O(1/p5) ,
ω3 = p+
+O(M2/p) .
We see that ω1 < ω2 < ω3 and z1 < 0, z2,3 > 0. In all three cases the group velocities ∂ωi/∂p are less than 1,
so neither mode is superluminal at high three-momenta, provided that α > γ. The two-derivative terms in the
Lagrangian (45) dominate in the UV limit, so there are neither ghosts nor tachyons in this limit.
• (B) IR limit (p → 0)
ω1 = −
1 +O(m2/ǫ2,M2/ǫ2)
, (57)
+m2 +O(M2) .
We see again that ω1 < ω2 < ω3 and z1 < 0, z2,3 > 0. This means that using the property (55) we can identify
the modes: the first one has the behaviour (56) and (57), and so on.
It follows from Eq. (54) that zi never vanish. In fact the coefficients in Eq. (52) are regular at all momenta, so zi
are regular as well. Therefore, zi never change signs and hence z1 < 0, z2,3 > 0 for all momenta. This means, in
particular, that the second and third modes never become tachyonic.
Let us discuss the dangerous mode with the dispersion relation ω = ω1(p) in some detail. The expression for the
fields in each mode is
b̃L,i = −iǫp
γzi +M
· Ci ,
b̃0,i = iǫω(αzi −M2) · Ci ,
ϕ̃i = (γzi +M
2)(αzi −M2) · Ci , (58)
where Ci are the normalization factors. Setting ω
2 = 0 in Eq. (52), we obtain three critical momenta
p21,2 =
ǫ2 −M2
−m2 ±
ǫ2 −M2
 , (59)
p23 =
. (60)
Under the conditions (47), the critical momenta p21,2 are approximately given by
p21 ≃
ǫ2 −M2
−m2 , p22 ≃
, (61)
so that p21 > p
3 > p
2 > 0. The tachyonic mode (ω
1 < 0) is present for 0 < p
2 < p22 and p
3 < p
2 < p21.
In order to find whether there are ghosts we calculate the energy of the modes (58),
Ei(p) = 2ω
2|Ci|2
αǫ2p2(γzi +M
2)2 + ǫ2(γp2 −M2)(αzi −M2)2 + (γzi +M2)2(αzi −M2)2
. (62)
For the modes with ω2,3 we have E2,3(p) > 0. For the mode with ω1 the energy is equal to zero at p = p3 = M/
While ω21 > 0 for p
2 < p23 ≡ M2/γ, the energy E1(p) changes its sign at this momentum. Thus the mode with ω1 is a
ghost for p2 < M2/γ.
We summarize the properties of the dangerous mode as follows:
• (i) p2 > (ǫ2 −M2)/α−m2: healthy
• (ii) M2/γ < p2 < (ǫ2 −M2)/α−m2: tachyon
• (iii) m2M2/ǫ2 < p2 < M2/γ: ghost, but not tachyon
• (iv) 0 < p2 < m2M2/ǫ2: tachyon.
Unlike the case m = 0 [33] the tachyon is present in the deep IR region (iv).
To end up the discussion of the modes in Minkowski space-time, we give the expressions for the minimum values of
ω2 in the tachyonic regions,
• (ii):
ω2min = −
4α(α+ γ)
at p2 =
γ + 2α
γ + α
. (63)
• (iv):
ω2min = −
at p2 = 0 .
Note that |ω2min| is relatively large in the region (ii), so this region is the most problematic.
B. Evolution of perturbations in cosmological background
Finally we discuss the evolution of field perturbations in the FRW background (2). In the cosmological context
the physical momentum p is related to the comoving momentum k as p = k/a. Once the parameters of the model
and initial data are such that the cosmological boundary crossing occurs, the present epoch (ΩDE ≃ 0.7) typically
corresponds to the phantom region. From Fig. 1 one can see that the Hubble parameter does not change much during
the transition from the phantom epoch to the final de Sitter era. Hence the present value of the Hubble parameter (H0)
is of the same order as the value H = M/
3α in the de Sitter asymptotics. This means that the value p3 = M/
is of the same order as H0 provided that γ and α are of order unity.
The tachyon appears when the momentum p = k/a of the dangerous mode becomes smaller than
(ǫ2 −M2)/α−m2 and temporally disappears when the mode crosses the value M/√γ. Hence this instability
is present for the modes which are inside the Hubble radius and satisfy M/
γ < p <
(ǫ2 −M2)/α−m2, but it
is absent for the modes deep inside the Hubble radius, satisfying p >
(ǫ2 −M2)/α−m2. After the Hubble radius
crossing (k = aH), the tachyonic instability disappears in the momentum region m2M2/ǫ2 < p2 < M2/γ, but the
tachyon appears again for p2 < m2M2/ǫ2. Note that the ghost existing at m2M2/ǫ2 < p2 < M2/γ is a not a problem
because of its low energy [13, 40].
In what follows we discuss the evolution of field perturbations in the two tachyonic regimes. Before doing that it is
instructive to study the high-momentum regime that sets the initial data for the tachyonic evolution.
1. p2 ≫ (ǫ2 −M2)/α−m2
We denote the overall amplitude of the dangerous mode as ϕ. Since the modes are deep inside the Hubble radius
(k/a ≫ H) in the regime we discuss here, the field perturbation χ approximately satisfies
χ+ k2χ ≃ 0 , (64)
where χ = aϕ and η is conformal time defined by η =
a−1dt. Taking the asymptotic Minkowski vacuum state,
χ = e−ikη/
2k, the squared amplitude of the field perturbation ϕ is given by [41]
(2π)3
|ϕ|2 =
. (65)
Since the maximum momentum at which the tachyon appears is k/a ≃ ǫ/
α, one has the following estimate for the
amplitude of the field perturbation at the beginning of the tachyonic instability,
. (66)
As usual, this amplitude characterizes the contribution of a logarithmic interval of momenta into 〈ϕ2(x)〉.
2. M2/γ < p2 < (ǫ2 −M2)/α−m2
This interval of momenta is dangerous, as the perturbations undergo the tachyonic amplification. Since the modes
are still inside the Hubble radius, one can neglect the gravitational effects on the “frequency” ω when estimating the
growth of field perturbations. By the time the tachyonic amplification ends up, the amplitude of field perturbations
is estimated as
ϕ ≃ ϕi exp
|ω1|dt
= ϕi exp
. (67)
Recall that p1 ≃ ǫ/
α and p3 = M/
γ. The largest value of |ω21 | is approximately given by (63). Substituting this
value into Eq. (67) and recalling that the background changes slowly (H ≃ const), one finds that the amplitude of
the field perturbation after exit from the tachyonic regime is of order
ϕ ≃ ǫ
α(γ + α)
, (68)
where we used Eq. (66).
Recall now that H is of the same order as M/
3α during the phantom phase. Hence the large ratio ǫ/M leads to
a strong amplification of field perturbations. From (18), the homogeneous field φ at the phantom and de Sitter phase
is estimated as φ ≃
. The requirement that the perturbation ϕ is smaller than the background field φ leads
to the constraint
α(γ + α)
. (69)
As an example, in the case m = H0 = 10
−42 GeV, M =
3αH0, α = 1 and γ = 1/2, we obtain the constraint
ǫ/M <∼ 70. As long as α and γ are of order one, the ratio ǫ/M should not be too much larger than unity.
3. 0 < p2 < m2M2/ǫ2
After the Hubble radius crossing, the effect of the cosmic expansion can no longer be neglected when estimating
the “frequencies” of the field perturbations. Since there are no tachyonic instabilities for m2M2/ǫ2 < p2 < M2/γ,
we consider the evolution of perturbations in the region 0 < p2 < m2M2/ǫ2. In Ref. [33] the equations for the field
perturbations were derived in the slow-rolling background under the condition p2 ≪ M2,m2. The equation for the
perturbation χ = aϕ is approximately given by
k2 − 1
− a2m
χ = 0 . (70)
Here we neglected the contribution of metric perturbations on the r.h.s. of this equation. Note that metric pertur-
bations works as a back reaction effect after the field perturbation is sufficiently amplified. The growth rate of the
perturbation χ is mainly determined by the terms in the parenthesis of Eq. (70) rather than the backreaction of metric
perturbations.
The last term corresponds to the tachyonic mass term, which already appeared in Minkowski spacetime [see Eq. (57)].
Since the term (d2a/dη2)/a is of order a2H2, one can estimate the ratio of the tachyonic mass relative to this
gravitational term:
δ ≡ a
2m2M2/ǫ2
(d2a/dη2)/a
. (71)
If we use the de Sitter value H = M/
3α, this ratio is estimated as δ = 3αm2/ǫ2 ≪ 1. Hence the gravitational term
(d2a/dη2)/a dominates over the tachyonic mass.
In the de Sitter background with a = −1/(Hη) the approximate solutions to Eq. (70) can be obtained by setting
χ = −(C/H)η−1+δ̃. One finds that δ̃ = −m2M2/(3H2ǫ2) for the growing solution, thereby giving
ϕ = Cη
3H2ǫ2 ∝ aδ/3 . (72)
In Ref. [33] it was shown that the physical temporal component of the vector field perturbations evolves as b0/a ∝ aδ/3,
whereas the physical spatial component of the vector field decreases as Bi/a ∝ a−1+δ/3. The growth rate of ϕ and
b0/a is small due to the condition δ ≪ 1. So, the second tachyonic instability is harmless for the past and present
cosmological evolution. However, we notice that since the de Sitter solution is a late-time attractor, the perturbations
ϕ and b0/a become larger than the homogeneous background fields in the distant future. At this stage we expect that
the contribution of metric perturbations can be also important.
V. CONCLUSIONS
In this paper we have studied the dynamics of dark energy in a Lorentz-violating model with the action given in
(1). The model involves a vector field Bµ and a scalar field Φ with mass terms M and m, respectively. The presence
of the one-derivative term ǫ∂µΦB
µ leads to an interesting dynamics at the IR scales larger than ǫ−1. The phantom
equation of state can be realized without having ghosts, tachyons or superluminal modes in the UV region.
We have taken into account the contributions of radiation and non-relativistic matter and studied the cosmological
evolution of the system. Interestingly, there exists a de Sitter attractor solution that can be used for the late-time
acceleration. The phantom regime is not an attractor, but we have found that in a range of parameters, the phantom
stage occurs during the transition from the matter epoch to the final de Sitter attractor. As is seen, e.g., in Fig. 1
the equation of state parameter w of dark energy crosses the cosmological constant boundary towards the phantom
region. We clarified the conditions under which the w = −1 crossing is realized together with the existence of the
stable de Sitter solution.
In the model studied in this paper, the effective Newton’s constant is time-dependent. We have found, however,
that this dependence is typically mild, though for interesting values of parameters it is close to the experimental
bounds.
We have also considered the field perturbations in Minkowski spacetime and obtained the momentum scales of
instabilities present in the IR region (p <∼ ǫ). We have found that either tachyons or ghosts appear for the spatial
momenta p smaller than
(ǫ2 −M2)/α−m2, while in the UV region there are no unhealthy modes. In the cosmo-
logical context the presence of tachyons at the IR scales leads to the amplification of large-scale field perturbations
whose wavelengths are roughly comparable to the present Hubble radius. There are two tachyonic regions of spatial
momenta in this model: (a) one is sub-horizon and its momenta are characterized by M2/γ < p2 < (ǫ2−M2)/α−m2;
(b) another is super-horizon and has 0 < p2 < m2M2/ǫ2. In the region (a) we derived the condition under which the
perturbations always remain smaller than the homogenous fields, see Eq. (69). While the existence of the phantom
phase requires that ǫ > M , the condition (69) shows that ǫ cannot be very much larger than M . Thus the allowed
range of ǫ is constrained to be relatively narrow. In the tachyonic region (b) the growth of the perturbations is
estimated as ϕ ∝ aαm2/ǫ2 . Since the growth rate is suppressed by the factor m2/ǫ2, this effect is negligible in the past
and at present, though the inhomogeneities can start to dominate over the homogenous fields in the distant future.
There are several issues yet to be understood. The presence of the tachyonic instability on sub-horizon scales may
lead to the variation of the gravitational potential, which can be an additional source of the late-time integrated
Saches-Wolfe effect on the CMB power spectrum. Another property of this model is the peculiar time-dependence of
the effective Newton’s constant, which may result in interesting phenomenology.
The model studied in this paper is likely to belong to a wider class of Lorentz-violating theories exhibiting the
phantom behaviour (see Refs. [42] for a number of Lorentz-violating models). It would be interesting to understand
how generic are the features we found in our particular model — late-time de Sitter attractor, transient phantom stage,
time-dependent Newton’s constant, sub-horizon tachyons, super-horizon ghosts, etc. One more direction is to modify
our model in such a way that it would be capable of describing inflationary epoch rather than the late-time acceleration.
Since it is known that the spectra of scalar and tensor perturbations produced during the phantom inflationary phase
are typically blue-tilted [43], this model may give rise to some distinct features in the CMB spectrum.
ACKNOWLEDGEMENTS
This work is supported by RFBR grant (M. L. and V.R., 05-02-17363-a), grant of the President of Russian Federation
(M. L. and V.R., NS-7293.2006.2), INTAS grant (M. L., YSF 04-83-3015), grant of Dynasty Foundation awarded by
the Scientific Board of ICFPM (M.L.), the European Union through the Marie Curie Research and Training Network
UniverseNet (E. P., MRTN-CT-2006-035863) and JSPS (S. T., No. 30318802).
Appendix A
We are going to find the effective Newton’s constant that determines the strength of gravitational interactions at
distances shorter than all scales present in our model, including ǫ−1, M−1, m−1 as well as the Hubble distance. To
this end, we neglect the last two terms in the action (1), and also neglect the time dependence of the background fields
φ and X . We also neglect the space-time curvature of the Universe, and therefore consider our model in Minkowski
space-time.
Let us impose the gauge h0i = 0, where hµν is the metric perturbation about the Minkowski background. Then the
quadratic Lagrangian for perturbations of metric, vector and scalar fields is readily calculated,
ḃi +
X∂ih00
∂ibj −
Xḣij
ḃ0 +
Xḣ00
∂ib0 +
Xḣ00
ϕ̇2 − (∂iϕ)2
. (73)
where X2 = B20 is the background value. Clearly the scalar field ϕ decouples in our approximation, so we will not
consider it in what follows.
By varying the quadratic action with respect to h00 and hij , one obtains (00)- and (ij)-components of the linearized
energy-momentum tensor for perturbations (note that T µν = −2δS/δhµν). Specifying further to scalar perturbations
with bi = ∂ibL and choosing conformal Newtonian gauge, h00 = 2Φ, hij = −2Ψδij, one obtains (we keep the standard
notation for the Newtonian potential, even though the same notation was used for the original scalar field in the main
text)
T 00 = αX(X∆Φ−∆ḃL) + γX (b0 +XΦ) ,
T ij = αX∂i∂j ḃL − δijαX2Ψ̈ , (74)
where ∆ = ∂i∂i and = ∂
0 −∆. The field equations for b0 and bL in the absence of sources for these fields read
(b0 +XΦ) = 0 ,
− bL +X(Φ̇− Ψ̇) = 0 . (75)
Now, the longitudinal (proportional to ∂i∂j) part of the (ij)-component of the Einstein equations, in the absence of
external anisotropic stresses, gives
Φ + Ψ = 8πGαXḃL , (76)
while the trace part and (00)-component are
∆(Φ + Ψ) = −4πGαX2Ψ̈− 4πGpext , (77)
−∆Ψ = 4πG(αX2∆Φ− αX∆ ˙bL) + 4πGρext , (78)
where ρext and pext are energy density and pressure of an external source.
For time-independent, pressureless source it is consistent to take all perturbations independent of time and set
bL = 0. Then one finds, as usual, Ψ = −Φ and obtains the following equation for the Newtonian potential,
(1 − 4πGαX2)∆Φ = 4πGρext . (79)
Thus, the effective Newton’s constant in the background field X is
G∗ = G(1 − 4πGαX2)−1 . (80)
This means that the effective Planck mass entering the Newton’s law is given by (6).
Appendix B
A. Autonomous equations
Let us define the following dimensionless variables which are convenient for studying the dynamical system [6, 38]:
, x2 =
, x3 =
, x4 =
, x5 =
, x6 =
. (81)
Then we obtain the following autonomous equations
x′1 = −3x1 −
x2x5 +
5 − x1
, (82)
x′2 = −3x2 −
x1x5 − 3
x4x5 −
x3x5 − x2
, (83)
x′3 =
x2x5 − x3
, (84)
x′4 =
x1 , (85)
x′5 = −x5
, (86)
x′6 = −2x6 − x6
, (87)
1 + x21 + x
2 − x23 + x24(3α+ x25) + x26/3 + 2(ǫ/M)x2x4x5 + 4(α/
γ)x1x4
1 + 3αx24
where prime denotes the derivative with respect to
N ≡ ln(a) .
Equation (11) gives the constraint
= 1− x21 − x22 − x23 + x24
3α+ x25
− x26 . (88)
Note that the above equations are invariant under the simultaneous change of the signs of φ and X . Hence it is not
restrictive to study the case of positive φ. Note also that we study the case of an expanding Universe with H > 0.
B. Fixed points
By setting x′i = 0 one formally finds the following six fixed points:
• (A) de Sitter (i): (x1, x2, x3, x4, x5, x6) =
0, 0, ǫ
3ǫ2/m2−2α
,− 1√
3(3ǫ2/m2−2α)
3α, 0
• (B) de Sitter (ii): (x1, x2, x3, x4, x5, x6) = (0, 0, const, 0, 0, 0) ,
• (C) matter: (x1, x2, x3, x4, x5, x6) = (0, 0, 0, 0, 0, 0) ,
• (D) radiation: (x1, x2, x3, x4, x5, x6) = (0, 0, 0, 0, 0, 1) ,
• (E1) kinetic point (i): (x1, x2, x3, x4, x5, x6) = (0, 1, 0, 0, 0, 0) ,
• (E2) kinetic point (ii): (x1, x2, x3, x4, x5, x6) = (0,−1, 0, 0, 0, 0) .
The fixed point (A) is precisely the de Sitter solution (18) that we discussed in Sec. III A. We will comment on its
stability shortly.
The point (B) is also in some sense a de Sitter point. It exists even in the absence of the field X and satisfies
the relation 3H2 = κ2W (φ). To reach the solution (B), the Hubble parameter needs to increase towards infinity
(M/H → 0), and the field φ needs to diverge as well.
The point (C) corresponds to matter-dominated era satisfying Ωm = 1 and weff = 0, whereas the point (D) describes
radiation-dominated epoch with Ωr = 1 and weff = 1/3.
The points (E1) and (E2) are kinetic solutions satisfying ΩDE = 1 and weff = 1. These solutions are used neither
for dark energy nor for radiation/matter dominated epochs.
A cosmologically viable trajectory starts from the radiation point (D), connects to the matter solution (C) and
finally approaches the de Sitter point (A). [Note that the initial data (22) indeed correspond to x1, x2, x3, x4, x5 → 0
as t → 0.] To see that this sequence of events is indeed possible, let us study the stability of the fixed points against
perturbations.
Let us consider linear perturbations δxi. By perturbing Eqs. (82)-(86) we obtain
δx′1 =
−3− H
− c1x1
δx1 +
x5 − c2x1
δx2 − c3x1δx3
x25 −
− c4x1
δx4 +
x4x5 − c5x1
δx5 − c6x1δx6 , (89)
δx′2 = −
x5 + c1x2
δx1 −
3 + c2x2 +
δx2 −
x5 + c3x2
x5 + c4x2
δx4 −
x1 + 3
x3 + c5x2
δx5 − c6x2δx6 , (90)
δx′3 = −c1x3δx1 +
x5 − c2x3
δx2 − c3x3δx3 − c4x3δx4 +
x2 − c5x3
δx5 − c6x3δx6 , (91)
δx′4 =
δx1 , (92)
δx′5 = −c1x5δx1 − c2x5δx2 − c3x5δx3 − c4x5δx4 −
+ c5x5
δx5 − c6x5δx6 , (93)
δx′6 = −c1x6δx1 − c2x6δx2 − c3x6δx3 − c4x6δx4 − c5x6δx5 −
+ c6x6
δx6 , (94)
where δ(H ′/H) =
i=1 ciδxi with
c1 = −
3x1 + 6(α/
1 + 3αx24
x4 , c2 = −
3x2 + 3(ǫ/M)x4x5
1 + 3αx24
, c3 =
1 + 3αx24
c4 = −
3x4(3α+ x
5) + 3(ǫ/M)x2x5 + 6(α/
1 + 3αx24
− 6αx4
1 + 3αx24
c5 = −
3x24x5 + 3(ǫ/M)x2x4
1 + 3αx24
, c6 = −
1 + 3αx24
. (95)
The stability of fixed points can be analyzed by considering eigenvalues of the 6 × 6 matrix M for perturbations
along the lines of Ref. [6, 38].
The stability of the de Sitter point (A) is important for having the late-time accelerated epoch. This depends upon
the two ratios ǫ/m and M/m once the parameters α and γ are fixed. When α = 1 and γ = 1/2, for example, the
parameter range of ǫ/m is determined by the ratio M/m. We find that the point (A) is a stable attractor if the
following conditions hold:
• (i) When M/m = 0.1, ǫ/m > 0.817,
• (ii) When M/m = 1, ǫ/m > 1.35,
• (iii) When M/m = 10, ǫ/m > 3.52,
When ǫ ≫ m, the stability of the point (A) is ensured automatically unless the ratio M/m is too much larger than
unity. In view of (20) the case ǫ ≫ m is of particular interest.
For another de Sitter point (B), the eigenvalues are
− 3,−3,−3
9− 12α
, 0,−1/2 , (96)
This means that this point is marginally stable. The zero eigenvalue comes from the perturbation equation for δx5.
If H continues to increase toward the solution (B), this eigenvalue actually obtains a small negative value, as can be
seen from Eq. (93). Thus in such a case the point (B) is stable. However, we know that the phantom phase is realized
only for finite field values bounded by φA, see Eq. (34). Hence it is not possible that the actual solutions approach
the point (B) with infinite H and φ.
The matter point (C) has the eigenvalues
3/2,−3/2,−4
9− 48α
, 0,−1/2 , (97)
which shows that the matter era corresponds to a saddle point with one positive eigenvalue. Hence the solutions
eventually repel away from this fixed point even if they temporarily approach it.
The radiation point (D) has the eigenvalues
1− 12α
,−1, 0, 2, 1 . (98)
One finds that the radiation epoch corresponds to a saddle point with two positive eigenvalues.
The kinetic points (E1) and (E2) have the eigenvalues
3, 3, 1, 0,
i , (99)
which shows that they are unstable.
The above stability analysis shows that the sequence of radiation, matter and de Sitter epochs can indeed be
realized.
[1] A. G. Riess et al., Astron. J. 116, 1009 (1998); Astron. J. 117, 707 (1999); S. Perlmutter et al., Astrophys. J. 517, 565
(1999); P. Astier et al., Astron. Astrophys. 447, 31 (2006); W. M. Wood-Vasey et al., arXiv:astro-ph/0701041.
http://arxiv.org/abs/astro-ph/0701041
[2] D. N. Spergel et al. [WMAP Collaboration], Astrophys. J. Suppl. 148, 175 (2003); D. N. Spergel et al.,
arXiv:astro-ph/0603449.
[3] U. Seljak et al. [SDSS Collaboration], Phys. Rev. D 71, 103515 (2005); M. Tegmark et al., Phys. Rev. D 74, 123507 (2006).
[4] D. J. Eisenstein et al. [SDSS Collaboration], Astrophys. J. 633, 560 (2005).
[5] V. Sahni and A. A. Starobinsky, Int. J. Mod. Phys. D 9, 373 (2000); V. Sahni, Lect. Notes Phys. 653, 141 (2004)
[arXiv:astro-ph/0403324]; S. M. Carroll, Living Rev. Rel. 4, 1 (2001); T. Padmanabhan, Phys. Rept. 380, 235 (2003);
P. J. E. Peebles and B. Ratra, Rev. Mod. Phys. 75, 559 (2003); V. Sahni and A. Starobinsky, Int. J. Mod. Phys. D 15
2105 (2006); S. Nojiri and S. D. Odintsov, Int. J. Geom. Meth. Mod. Phys. 4, 115 (2007).
[6] E. J. Copeland, M. Sami and S. Tsujikawa, Int. J. Mod. Phys. D 15, 1753 (2006).
[7] A. Melchiorri, L. Mersini-Houghton, C. J. Odman and M. Trodden, Phys. Rev. D 68, 043509 (2003); J. Weller and
A. M. Lewis, Mon. Not. Roy. Astron. Soc. 346, 987 (2003); B. A. Bassett, P. S. Corasaniti and M. Kunz, Astrophys. J.
617, L1 (2004); U. Seljak, A. Slosar and P. McDonald, JCAP 0610, 014 (2006).
[8] U. Alam, V. Sahni, T. D. Saini and A. A. Starobinsky, Mon. Not. Roy. Astron. Soc. 354, 275 (2004); R. Lazkoz, S. Nesseris
and L. Perivolaropoulos, JCAP 0511, 010 (2005); J. Q. Xia, G. B. Zhao, B. Feng, H. Li and X. Zhang, Phys. Rev. D 73,
063521 (2006); S. Nesseris and L. Perivolaropoulos, JCAP 0701, 018 (2007); U. Alam, V. Sahni and A. A. Starobinsky,
JCAP 0702, 011 (2007); G. B. Zhao, J. Q. Xia, B. Feng and X. Zhang, arXiv:astro-ph/0603621.
[9] R. R. Caldwell, Phys. Lett. B 545, 23 (2002); R. R. Caldwell, M. Kamionkowski and N. N. Weinberg, Phys. Rev. Lett.
91, 071301 (2003).
[10] S. M. Carroll, M. Hoffman and M. Trodden, Phys. Rev. D 68, 023509 (2003).
[11] P. Singh, M. Sami and N. Dadhich, Phys. Rev. D 68 023522 (2003); S. Nojiri and S. D. Odintsov, Phys. Lett. B 562,
147 (2003); Phys. Lett. B 565, 1 (2003); J. G. Hao and X. Z. Li, Phys. Rev. D 67, 107303 (2003); P. F. Gonzalez-Diaz,
Phys. Rev. D 68, 021303 (2003); L. P. Chimento and R. Lazkoz, Phys. Rev. Lett. 91, 211301 (2003); M. P. Dabrowski,
T. Stachowiak and M. Szydlowski, Phys. Rev. D 68, 103519 (2003); L. P. Chimento and R. Lazkoz, Phys. Rev. Lett.
91, 211301 (2003); S. Tsujikawa, Class. Quant. Grav. 20, 1991 (2003); M. Sami and A. Toporensky, Mod. Phys. Lett.
A 19, 1509 (2004); E. Elizalde, S. Nojiri and S. D. Odintsov, Phys. Rev. D 70, 043539 (2004); H. Stefancic, Phys. Lett.
B 586, 5 (2004); V. B. Johri, Phys. Rev. D 70, 041303 (2004); Z. K. Guo, Y. S. Piao and Y. Z. Zhang, Phys. Lett.
B 594, 247 (2004); J. M. Aguirregabiria, L. P. Chimento and R. Lazkoz, Phys. Rev. D 70, 023509 (2004); S. Nojiri,
S. D. Odintsov and S. Tsujikawa, Phys. Rev. D 71, 063004 (2005); S. Nojiri and S. D. Odintsov, Phys. Rev. D 72, 023003
(2005); L. Perivolaropoulos, Phys. Rev. D 71, 063503 (2005); T. Chiba, JCAP 0503, 008 (2005); M. Bouhmadi-Lopez and
J. A. Jimenez Madrid, JCAP 0505, 005 (2005); V. Faraoni, Class. Quant. Grav. 22, 3235 (2005); L. P. Chimento, Phys.
Lett. B 633, 9 (2006); L. P. Chimento and D. Pavon, Phys. Rev. D 73, 063511 (2006); O. Hrycyna and M. Szydlowski,
arXiv:0704.1651 [hep-th].
[12] A. Vikman, Phys. Rev. D 71, 023515 (2005); Z. K. Guo, Y. S. Piao, X. M. Zhang and Y. Z. Zhang, Phys. Lett. B 608, 177
(2005); W. Hu, Phys. Rev. D 71, 047301 (2005); R. R. Caldwell and M. Doran, Phys. Rev. D 72, 043527 (2005); H. Wei,
R. G. Cai and D. F. Zeng, Class. Quant. Grav. 22, 3189 (2005); G. B. Zhao, J. Q. Xia, M. Li, B. Feng and X. Zhang, Phys.
Rev. D 72, 123515 (2005); S. Tsujikawa, Phys. Rev. D 72, 083512 (2005); I. Y. Aref’eva, A. S. Koshelev and S. Y. Vernov,
Phys. Rev. D 72, 064017 (2005); B. McInnes, Nucl. Phys. B 718, 55 (2005); L. P. Chimento, R. Lazkoz, R. Maartens
and I. Quiros, JCAP 0609, 004 (2006); R. Lazkoz and G. Leon, Phys. Lett. B 638, 303 (2006); X. F. Zhang and T. Qiu,
Phys. Lett. B 642, 187 (2006); W. Zhao, Phys. Rev. D 73, 123509 (2006); H. Mohseni Sadjadi and M. Alimohammadi,
Phys. Rev. D 74, 043506 (2006); Z. K. Guo, Y. S. Piao, X. Zhang and Y. Z. Zhang, Phys. Rev. D 74, 127304 (2006);
I. Y. Aref’eva and A. S. Koshelev, JHEP 0702, 041 (2007); Y. f. Cai, M. z. Li, J. X. Lu, Y. S. Piao, T. t. Qiu and
X. m. Zhang, arXiv:hep-th/0701016; Y. F. Cai, T. Qiu, Y. S. Piao, M. Li and X. Zhang, arXiv:0704.1090 [gr-qc].
[13] J. M. Cline, S. Jeon and G. D. Moore, Phys. Rev. D 70, 043543 (2004).
[14] N. Arkani-Hamed, H. C. Cheng, M. A. Luty and S. Mukohyama, JHEP 0405, 074 (2004).
[15] F. Piazza and S. Tsujikawa, JCAP 0407, 004 (2004).
[16] R. V. Buniy and S. D. H. Hsu, Phys. Lett. B 632, 543 (2006).
[17] S. Dubovsky, T. Gregoire, A. Nicolis and R. Rattazzi, JHEP 0603, 025 (2006).
[18] J. P. Uzan, Phys. Rev. D 59, 123510 (1999); L. Amendola, Phys. Rev. D 60, 043501 (1999); T. Chiba, Phys. Rev. D 60,
083508 (1999); N. Bartolo and M. Pietroni, Phys. Rev. D 61, 023518 (2000); F. Perrotta, C. Baccigalupi and S. Matarrese,
Phys. Rev. D 61, 023507 (2000); A. Riazuelo and J. P. Uzan, Phys. Rev. D 66, 023525 (2002); D. F. Torres, Phys. Rev.
D 66, 043522 (2002); L. Perivolaropoulos, JCAP 0510, 001 (2005); M. X. Luo and Q. P. Su, Phys. Lett. B 626, 7 (2005);
J. Martin, C. Schimd and J. P. Uzan, Phys. Rev. Lett. 96, 061303 (2006).
[19] B. Boisseau, G. Esposito-Farese, D. Polarski and A. A. Starobinsky, Phys. Rev. Lett. 85, 2236 (2000).
[20] R. Gannouji, D. Polarski, A. Ranquet and A. A. Starobinsky, JCAP 0609, 016 (2006).
[21] S. Capozziello, V. F. Cardone, S. Carloni and A. Troisi, Int. J. Mod. Phys. D 12, 1969 (2003); S. M. Carroll, V. Duvvuri,
M. Trodden and M. S. Turner, Phys. Rev. D 70, 043528 (2004).
[22] S. Nojiri, S. D. Odintsov and M. Sasaki, Phys. Rev. D 71, 123509 (2005); M. Sami, A. Toporensky, P. V. Tretjakov and
S. Tsujikawa, Phys. Lett. B 619, 193 (2005); G. Calcagni, S. Tsujikawa and M. Sami, Class. Quant. Grav. 22, 3977 (2005);
B. M. N. Carter and I. P. Neupane, JCAP 0606, 004 (2006).
[23] L. Amendola, R. Gannouji, D. Polarski and S. Tsujikawa, Phys. Rev. D 75, 083504 (2007). See also L. Amendola, D. Polarski
and S. Tsujikawa, Phys. Rev. Lett. 98, 131302 (2007); arXiv:astro-ph/0605384.
[24] T. Koivisto and D. F. Mota, Phys. Lett. B 644, 104 (2007); S. Tsujikawa and M. Sami, JCAP 0701, 006 (2007); T. Koivisto
and D. F. Mota, Phys. Rev. D 75, 023518 (2007); B. M. Leith and I. P. Neupane, arXiv:hep-th/0702002.
[25] L. Amendola, C. Charmousis and S. C. Davis, JCAP 0612, 020 (2006).
http://arxiv.org/abs/astro-ph/0603449
http://arxiv.org/abs/astro-ph/0403324
http://arxiv.org/abs/astro-ph/0603621
http://arxiv.org/abs/0704.1651
http://arxiv.org/abs/hep-th/0701016
http://arxiv.org/abs/0704.1090
http://arxiv.org/abs/astro-ph/0605384
http://arxiv.org/abs/hep-th/0702002
[26] A. De Felice, M. Hindmarsh and M. Trodden, JCAP 0608, 005 (2006); G. Calcagni, B. de Carlos and A. De Felice, Nucl.
Phys. B 752, 404 (2006); Z. K. Guo, N. Ohta and S. Tsujikawa, Phys. Rev. D 75, 023520 (2007).
[27] G. R. Dvali, G. Gabadadze and M. Porrati, Phys. Lett. B 485, 208 (2000).
[28] G. Kofinas, R. Maartens and E. Papantonopoulos, JHEP 0310 (2003) 066 [arXiv:hep-th/0307138].
[29] V. Sahni and Y. Shtanov, JCAP 0311, 014 (2003).
[30] A. Lue and G. D. Starkman, Phys. Rev. D 70, 101501 (2004).
[31] D. Gorbunov, K. Koyama and S. Sibiryakov, Phys. Rev. D 73, 044016 (2006).
[32] L. Senatore, Phys. Rev. D 71, 043512 (2005); P. Creminelli, M. A. Luty, A. Nicolis and L. Senatore, JHEP 0612, 080
(2006).
[33] V. A. Rubakov, Theor. Math. Phys. 149, 1651 (2006) [arXiv:hep-th/0604153].
[34] B. M. Gripaios, JHEP 0410, 069 (2004).
[35] M. V. Libanov and V. A. Rubakov, JHEP 0508, 001 (2005).
[36] S. M. Carroll and E. A. Lim, Phys. Rev. D 70, 123525 (2004).
[37] J. P. Uzan, Rev. Mod. Phys. 75, 403 (2003).
[38] E. J. Copeland, A. R. Liddle and D. Wands, Phys. Rev. D 57, 4686 (1998).
[39] B. Gumjudpai, T. Naskar, M. Sami and S. Tsujikawa, JCAP 0506, 007 (2005).
[40] S. L. Dubovsky, JHEP 0410, 076 (2004).
[41] B. A. Bassett, S. Tsujikawa and D. Wands, Rev. Mod. Phys. 78, 537 (2006).
[42] J. M. Cline and L. Valcarcel, JHEP 0403, 032 (2004); N. Arkani-Hamed, H. C. Cheng, M. Luty and J. Thaler, JHEP 0507,
029 (2005); M. V. Libanov and V. A. Rubakov, JCAP 0509, 005 (2005); Phys. Rev. D 72, 123503 (2005); S. M. Carroll
and J. Shu, Phys. Rev. D 73, 103515 (2006); H. C. Cheng, M. A. Luty, S. Mukohyama and J. Thaler, JHEP 0605, 076
(2006); S. Kanno and J. Soda, Phys. Rev. D 74, 063505 (2006); P. G. Ferreira, B. M. Gripaios, R. Saffari and T. G. Zlosnik,
Phys. Rev. D 75, 044014 (2007).
[43] M. Baldi, F. Finelli and S. Matarrese, Phys. Rev. D 72, 083504 (2005).
http://arxiv.org/abs/hep-th/0307138
http://arxiv.org/abs/hep-th/0604153
	Introduction
	Lorentz-violating model
	Dynamics of dark energy
	Final and initial stages
	Slow roll phantom regime
	Numerical solutions
	Momentum scales of instabilities
	Minkowski spectrum
	Evolution of perturbations in cosmological background
	p2 (2-M2)/-m2
	M2/<p2<(2-M2)/-m2
	0<p2<m2M2/2
	Conclusions
	ACKNOWLEDGEMENTS
	Appendix A
	Appendix B
	Autonomous equations
	Fixed points
	References
