Route to Lambda in conformally coupled phantom cosmology
Orest Hrycyna∗
Department of Theoretical Physics, Faculty of Philosophy,
The John Paul II Catholic University of Lublin, Al. Rac lawickie 14, 20-950 Lublin, Poland and
Astronomical Observatory, Jagiellonian University, Orla 171, 30-244 Kraków, Poland
Marek Szyd lowski†
Astronomical Observatory, Jagiellonian University, Orla 171, 30-244 Kraków, Poland and
Mark Kac Complex Systems Research Centre, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland
In this letter we investigate acceleration in the flat cosmological model with a conformally coupled
phantom field and we show that acceleration is its generic feature. We reduce the dynamics of the
model to a 3-dimensional dynamical system and analyze it on a invariant 2-dimensional submanifold.
Then the concordance FRW model with the cosmological constant Λ is a global attractor situated
on a 2-dimensional invariant space. We also study the behaviour near this attractor, which can
be approximated by the dynamics of the linearized part of the system. We demonstrate that
trajectories of the conformally coupled phantom scalar field with a simple quadratic potential crosses
the cosmological constant barrier infinitely many times in the phase space. The universal behaviour
of the scalar field and its potential is also calculated. We conclude that the phantom scalar field
conformally coupled to gravity gives a natural dynamical mechanism of concentration of the equation
of state coefficient around the magical value weff = −1. We demonstrate route to Lambda through
the infinite times crossing the weff = −1 phantom divide.
PACS numbers: 98.80.Bp, 98.80.Cq, 11.15.Ex
At present the scalar fields play a crucial role in modern cosmology. In an inflationary scenario they generate an
exponential rate of evolution of the universe as well as density fluctuations due to vacuum energy. The Lagrangian
for a phantom scalar field on the background of the Friedmann-Robertson-Walker (FRW) universe is assumed in the
[gµν∂µψ∂νψ + ξRψ
2 − 2U(ψ)], (1)
where gµν is the metric of the spacetime manifold, ψ = ψ(t), t is the cosmological time, R = R(g) is the Ricci scalar
for the spacetime metricg, ξ is a coupling constant which assumes zero for a scalar field minimally coupled to gravity
and 1/6 for a conformally coupled scalar field, U(ψ) is a potential of the scalar field.
The minimally coupled slowly evolving scalar fields with a potential function U(ψ) are good candidates for a
description of dark energy. In this model, called quintessence [1, 2], the energy density and pressure from the scalar
field are ρψ = −1/2ψ̇2 +U(ψ), pψ = −1/2ψ̇2−U(ψ). From recent studies of observational constraints we obtain that
wψ ≡ pψ/ρψ < −0.55 [3]. This model has been also extended to the case of a complex scalar field [4, 5].
Observations of distant supernovae support the cosmological constant term which corresponds to the case ψ̇ ≃ 0.
Then we obtain that wψ = −1. But there emerge two problems in this context. Namely, the fine tuning and the cosmic
coincidence problems. The first problem comes from the quantum field theory where the vacuum expectation value
is of 123 orders of magnitude larger than the observed value of 10−47GeV4. The lack of a fundamental mechanism
which sets the cosmological constant almost zero is called the cosmological constant problem. The second problem
called “cosmic conundrum” is a question why the energy densities of both dark energy and dark matter are nearly
equal at the present epoch.
One of the solutions to this problem offers the idea of quintessence, which is a version of the time varying cosmological
constant conception. Quintessence solves the first problem through the decaying Λ term from the beginning of the
Universe to a small value observed at the present epoch. Also the ratio of energy density of this field to the matter
density increases slowly during the expansion of the Universe because the specific feature of this model is the variation
of the coefficient of the equation of state with respect to time. The quintessence models [2, 6] describe the dark energy
with the time varying equation of state for which wX > −1, but recently quintessence models have been extended to
the phantom quintessence models with wX < −1. In this class of models the weak energy condition is violated and
∗Electronic address: hrycyna@kul.lublin.pl
†Electronic address: uoszydlo@cyf-kr.edu.pl
http://arxiv.org/abs/0704.1651v3
mailto:hrycyna@kul.lublin.pl
mailto:uoszydlo@cyf-kr.edu.pl
such a theoretical possibility is realized by a scalar field with a switched sign in the kinetic term ψ̇2 → −ψ̇2 [7, 8, 9].
From theoretical point of view it is necessary to explore different evolutional scenarios for dark energy which provide
a simple and natural transition to wX = −1. The methods of dynamical systems with notion of attractor (a limit
set with an open inset) offers the possibility of description of transition trajectories to the regime with wX = −1.
Moreover they demonstrate whether this mechanism is generic.
Inflation and quintessence with non-minimal coupling constant are studied in the context of formulation of necessary
conditions for the acceleration of the universe [10] (see also [11, 12]). We can find two important arguments which
favour the choice of conformal coupling over ξ 6= 1/6. The first, equation for the massless scalar field is conformally
invariant [13, 14]. The second argument is that if the scalar field satisfy Klein-Gordon equation in the curved space
then ψ does not violate the equivalence principle, and ξ is forced to assume the value 1/6 [15].
While recent astronomical observations give support that the equation of state parameter for dark energy is close to
constant value −1 they do not give a “corridor” around this value. Moreover, Alam et al. [16] pointed out that evolving
state parameter is favoured over constant wX = −1. The first step in the direction of description of the dynamics of
the dark energy seems to be investigation of the system with evolving dark energy in the close neighbourhood of the
value wX = −1. For this aim we linearize dynamical system at this critical point and then describe the system in a
good approximation (following the Hartman-Grobman theorem [17]) by its linearized part.
Other dark energy models like the Chaplygin gas model [18, 19, 20], [9, and references therein] and the model with
tachyonic matter can also be interpreted in terms of a scalar field with some form of a potential function.
Recent applications of the Bayesian framework [21, 22, 23, 24, 25] of model selection to the broad class of cosmo-
logical models with acceleration indicate that a posteriori probability for the ΛCDM model is 96%. Therefore the
explanation why the current universe is such close to the ΛCDM model seems to be a major challenge for modern
theoretical cosmology.
In this letter we present the simplest mechanism of concentration around wX = −1 basing on the influence of a
single scalar field conformally coupled to gravity acting in the radiation epoch. Phantom cosmology non-minimally
coupled to the Ricci scalar was explored in the context of superquintesence (wX < −1) by Faraoni [26, 27] and there
was pointed out that the superacceleration regime can be achieved by the conformally coupled scalar field in contrast
to the minimally coupled scalar field.
Let us consider the flat FRW model which contains a negative kinetic scalar field conformally coupled to gravity
(ξ = 1/6) (phantom) with the potential function U(ψ). For the simplicity of presentation we assume U(ψ) ∝ ψ2.
In this model the phantom scalar field is coupled to gravity via the term ξRψ2. We consider massive scalar fields
(for recent discussion of cosmological implications of massive and massless scalar fields see [28]). The dynamics of a
non-minimally coupled scalar field for some self-interacting potential U(ψ) and for an arbitrary ξ is equivalent to the
action of the phantom scalar field (which behaves like a perfect fluid) with energy density ρψ and pressure pψ [29]
ρψ = −
ψ̇2 + U(ψ) − 3ξH2ψ2 − 3ξH(ψ2)̇, (2)
pψ = −
ψ̇2 − U(ψ) + ξ
2H(ψ2)̇ + (ψ2)̈
2Ḣ + 3H2
ψ2, (3)
where the conservation condition ρ̇ψ = −3H(ρψ + pψ) gives rise to the equation of motion for the field
ψ̈ + 3Hψ̇ + ξRψ2 − U ′(ψ) = 0, (4)
where R = 6
Ḣ + 2H2
is the Ricci scalar.
Let us assume that both the homogeneous scalar field ψ(t) and the potential U(ψ) depend on time through the
scale factor, i.e.
ψ(t) = ψ(a(t)), U(ψ) = U(ψ(a)); (5)
then due to this simplified assumption the coefficient of the equation of state wψ is parameterized by the scale factor
wψ = wψ(a), pψ = wψ(a)ρψ(a), (6)
ψ′2H2a2 − U(ψ) + ξ
2(ψ2)′H2a+ (ψ2 )̈
Ḣ + 3H2
ψ′2H2a2 + U(ψ) − 3ξH2ψ2 − 3ξ(ψ2)′H2a
where prime denotes the differentiation with respect to the scale factor.
We assume the flat model with the FRW geometry, i.e., the line element has the form
ds2 = −dt2 + a2(t)[dr2 + r2(dθ2 + sin2 θdϕ2)], (8)
where 0 ≤ ϕ ≤ 2π, 0 ≤ θ ≤ π and 0 ≤ r ≤ ∞ are comoving coordinates, t stands for the cosmological time. It is also
assumed that a source of gravity is the phantom scalar field ψ with the conformal coupling to gravity ξ = 1/6. The
dynamics is governed by the action
m2pR+ (g
µνψµψν +
Rψ2 − 2U(ψ))
where m2p = (8πG)
−1; for simplicity and without lost of generality we assume 4πG/3 = 1 and U(ψ) is the scalar field
potential
U(ψ) =
m2ψ2. (10)
After dropping the full derivatives with respect to time, rescaling phantom field ψ → φ = ψa and the time variable
to the conformal time dt = adη we obtain the energy conservation condition
a′2 +
φ′2 −
m2a2φ2 = ρr,0 (11)
where ρr,0 is constant corresponding to the radiation in the model. The equations of motion are
a′′ = m2aφ2,
φ′′ = m2a2φ
where a prime denotes the differentiation with respect to the conformal time dt = adη and m2 > 0.
From the energy conservation condition we have
a′2 =
ρr,0 +
m2a2φ2
1 + φ̇2
and now from the equations of motion (12) we receive
(ρr,0 +
m2a2φ2)φ̈+
m2aφ(1 + φ̇2)(φφ̇ − a) = 0. (14)
The effective equation of state parameter is
weff =
ρφ + ρr
, (15)
for our model this parameter reduces to
weff = −
φ′2 + 1
m2a2φ2 − ρr,0
φ′2 + 1
m2a2φ2 + ρr,0
where a prime denotes the differentiation with respect to the conformal time and finally taking into account equa-
tion (13) we have
weff = −
φ̇2 +
m2a2φ2 − ρr,0
m2a2φ2 + ρr,0
(1 + φ̇2)
. (17)
For a, φ≫ ρr,0 this equation reduces to
weff = −
(2φ̇2 + 1), (18)
and it is clear that for any value of φ̇ weff is always negative.
To analyze equation (14) we reintroduce the original phantom field variable ψ = φ
and da/a = d ln a. Now equation
(14) reads
(ψ′′ + ψ′) +
m2ψ(1 + (ψ′ + ψ)2)(ψ(ψ′ + ψ) − 1) = 0 (19)
where a prime now denotes the differentiation with respect to a natural logarithm of the scale factor. Introducing
new variables y = ψ′ and ρr = ρr,0a
−4 we can represent this equation as an autonomous dynamical system
ψ′ = y
y′ = −y −
(ψ(y + ψ) − 1)(1 + (y + ψ)2) (20)
ρ′r = −4ρr.
There are the two critical points in the phase space (ψ, y, ρr), namely ψ = ±1, y = 0, ρr = 0. The linearization
matrix reads
0 1 0
−2(1 + ψ2) −1 − (1 + ψ2) 2
(ψ2 − 1)(1 + ψ2)
0 0 −4
y=0,ρr=0
0 1 0
−4 −3 0
0 0 −4
y=0,ρr=0,ψ=±1
. (21)
The eigenvalues for this matrix are λ1,2 =
(−3 ± i
7) and λ3 = −4.
To find a global phase portrait it is necessary to study the system in the neighbourhood of the critical points which
correspond, from the physical point of view, stationary states (or asymptotic solutions). Then the Hartman-Grobman
theorem guaranties us that the linearized system at this point is a well approximation of the nonlinear system. First,
we must note that ρr = 0 is in the invariant submanifold of the 3-dimensional nonlinear system. It is also useful to
calculate the eigenvectors for any eigenvalue. We obtain following eigenvectors
v1,2 =
 , v3 =
 . (22)
They are helpful in construction of the exact solution of the linearized system
~x(t) = ~x(0) exp t
0 1 0
−4 −3 0
0 0 −4
0 − 3
0 1 0
1 0 0
e−4t 0 0
t cos
t −e− 32 t sin
t sin
t cos
0 0 1
0 1 0
where x = ψ − ψ0, y = ψ′ − ψ′0, z = ρr − ρ0 and x0, y0, z0 are initial conditions and we have substituted ln a = t.
If we consider linearized system on the invariant stable submanifold z = 0, it is easy to find the exact solution. If
we return to the original variables ψ, ψ′, then ψ(ln a) is the solution of the linear equation
(ψ − ψ0)′′ + 3(ψ − ψ0)′ + 4(ψ − ψ0) = 0, (24)
i. e.,
(ψ − ψ0) = C1 exp
+ C2 exp
(φ− φ0) = C1a−
2 cos
+ C2a
2 sin
. (26)
Because of the lack of alternatives to the mysterious cosmological constant [21, 22] we allow that energy might vary
in time following assumed a priori parameterization of w(z). In the popular parameterization [30, 31, 32, 33] appears
a free function in most scenarios which is a source of difficulties in constraining parameters by observations. However
most parameterizations of the dark energy equation of state cannot reflect real dynamics of cosmological models with
dark energy. The assumed form of w(z) can be incompatible with the w(z) obtained from the underlying dynamics
of the cosmological model. For example some of parameters can be determined from the dynamics which can be
crucial in testing and selection of cosmological models [21]. Our point of view is to obtain the form of w(z) specific
for given class of cosmological models from dynamics of this models and apply it in further analysis both theoretical
and empirical. In practice we put the cosmological model in the form of the dynamical system and linearize it around
the neighbourhood of the present epoch to find the exact formula of w(z). For the phantom scalar field model this
incompability manifests by the presence of a focus type critical point (therefore damping oscillations) in the phase
space rather than a stable node (Fig. 1 and its 3D version Fig. 2).
The properties of the minimally coupled phantom field in the FRW cosmology using the phase portrait have been
investigated by Singh et al. [34] (see also [35] for more recent studies). Authors showed the existence of the deSitter
attractor and damped oscillations (the ratio of the kinetic to the potential energy |T/U | to oscillate to zero).
We can also express weff in these new variables
weff = −
(ψ + ψ′)2 −
ρr − 12m
1 + (ψ + ψ′)2
w′eff =
dweff
d ln a
(ψ + ψ′)(ψ′ + ψ′′) + ρrm
2 + ψ
(ρr +
m2ψ2)2
1 + (ψ + ψ′)2
. (28)
Recently Caldwell and Linder [36] have discussed dynamics of quintessence models of dark energy in terms of w−w′
phase variables, where w′ was the differentiation with respect to the logarithm of the scale factor. These methods
were extended to the phantom and quintom models of dark energy [37, 38]. Guo et al. [38] examined the two-field
quintom models as the illustration of the simplest model of transition across the wX = −1 barrier. The interesting
mechanism of acceleration with a periodic crossing of the w = −1 barrier have been recently discussed in the context
of the cubic superstring field theory [39]. In the model under consideration we obtain this effect but trajectories cross
the barrier infinitely many times. The main advantage of the discovered road to Λ is that it takes place in the simple
flat FRW model with the quadratic potential of the scalar field.
It is easy to check that at the critical points weff = −1 and dweffd lna = 0. Since these points are sinks there is infinite
many crossings of weff = −1 during the evolution.
The methods of the Lyapunov function are useful in discussion of stability of the critical point of the non-linear
system. The stability of any hyperbolic critical point of dynamical system is determined by the signs of the real parts
of the eigenvalues λi of the Jacobi matrix. A hyperbolic critical point is asymptotically stable iff real λi < 0 ∀i, if x0
is a sink. The hyperbolic critical point is unstable iff it is either a source or a saddle. The method of the Lyapunov
function is especially useful in deciding the stability of a non-hyperbolic critical points [17, p.129]. The construction
of the Lyapunov function was used by [40] for demonstration that periodic behaviour of a single scalar field is not
possible for minimally coupled phantom scalar field (see also [41]).
The quantity w′eff in terms of weff and (lnψ)
′ reads
w′eff = −(1 − 3weff)(1 + weff +
(lnψ)′). (29)
It is interesting that equation (29) can be solved in terms of w̄(a) – the mean of the equation of the state parameter
in the logarithmic scale defined by Rahvar and Movahed [42] as
w̄(a) =
w(a′)d(ln a′)
d(ln a′)
, (30)
namely:
w(a) =
a3(1+w̄(a))ψ2. (31)
They argued that this phenomenological parameterization removes the fine tuning of dark energy and ρX/ρm ∝
a−3w̄(a) approaches a unity at the early universe. Note that in w̄(a) = −1 that
w(z) + 1 =
(1 − ψ2), (32)
FIG. 1: The phase portrait (weff, w
eff) of the investigated model on the submanifold ρr = 0. This figure illustrates the evolution
of the dark energy equation of the state parameter as a function of redshift for different initial conditions. In all cases trajectories
cross the boundary line weff = −1 infinite many times but this state also represents the global attractor.
where
ψ = ψ0 + (1 + z)
C1 cos(
ln(1 + z)) − C2 sin(
ln(1 + z))
. (33)
In Fig. 3 we present the relation w(z) for different values of parameters ψ0 = ±1, C1 and C2.
In this letter we regarded the phantom scalar field conformally coupled to gravity in the context of the problem of
acceleration of the Universe. We applied the methods of dynamical systems and the Hartman-Grobman theorem to find
universal behaviour at the late times – damping oscillations around weff = −1. We argued that most parameterizations
of the dark energy, such as linear evolution of w(z) in redshift or the scale factor, cannot reflect realistic physical
models because of the presence of non-hyperbolic critical point of a focus type on the phase plane (w,w′). We
suggested a parameterization of a type
wX(z) = −1 + (1 + z)3
C1 cos(ln(1 + z)) + C2 sin(ln(1 + z))
which parameterizes damping oscillations around wX = −1 “phantom divide”, and finally, with the help of this
formula one can simply calculate energy density for dark energy ρX
ρX = ρX,0 exp
(1 + z)3
A sin(ln(1 + z)) +B cos(ln(1 + z))
. (35)
Acknowledgments
The work of M.S. has been supported by the Marie Curie Actions Transfer of Knowledge project COCOS (contract
MTKD-CT-2004-517186).
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FIG. 3: The relation w(z) for different values of the parameters: C1 = 0.01, C2 = 0 dash-dotted line (red el. version); C1 = 0,
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	Acknowledgments
	References

  In this letter we investigate acceleration in the flat cosmological model
with a conformally coupled phantom field and we show that acceleration is its
generic feature. We reduce the dynamics of the model to a 3-dimensional
dynamical system and analyze it on a invariant 2-dimensional submanifold. Then
the concordance FRW model with the cosmological constant $\Lambda$ is a global
attractor situated on a 2-dimensional invariant space. We also study the
behaviour near this attractor, which can be approximated by the dynamics of the
linearized part of the system. We demonstrate that trajectories of the
conformally coupled phantom scalar field with a simple quadratic potential
crosses the cosmological constant barrier infinitely many times in the phase
space. The universal behaviour of the scalar field and its potential is also
calculated. We conclude that the phantom scalar field conformally coupled to
gravity gives a natural dynamical mechanism of concentration of the equation of
state coefficient around the magical value $w_{\text{eff}}=-1$. We demonstrate
route to Lambda through the infinite times crossing the $w_{\text{eff}}=-1$
phantom divide.

Route to Lambda in conformally coupled phantom cosmology
Orest Hrycyna∗
Department of Theoretical Physics, Faculty of Philosophy,
The John Paul II Catholic University of Lublin, Al. Rac lawickie 14, 20-950 Lublin, Poland and
Astronomical Observatory, Jagiellonian University, Orla 171, 30-244 Kraków, Poland
Marek Szyd lowski†
Astronomical Observatory, Jagiellonian University, Orla 171, 30-244 Kraków, Poland and
Mark Kac Complex Systems Research Centre, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland
In this letter we investigate acceleration in the flat cosmological model with a conformally coupled
phantom field and we show that acceleration is its generic feature. We reduce the dynamics of the
model to a 3-dimensional dynamical system and analyze it on a invariant 2-dimensional submanifold.
Then the concordance FRW model with the cosmological constant Λ is a global attractor situated
on a 2-dimensional invariant space. We also study the behaviour near this attractor, which can
be approximated by the dynamics of the linearized part of the system. We demonstrate that
trajectories of the conformally coupled phantom scalar field with a simple quadratic potential crosses
the cosmological constant barrier infinitely many times in the phase space. The universal behaviour
of the scalar field and its potential is also calculated. We conclude that the phantom scalar field
conformally coupled to gravity gives a natural dynamical mechanism of concentration of the equation
of state coefficient around the magical value weff = −1. We demonstrate route to Lambda through
the infinite times crossing the weff = −1 phantom divide.
PACS numbers: 98.80.Bp, 98.80.Cq, 11.15.Ex
At present the scalar fields play a crucial role in modern cosmology. In an inflationary scenario they generate an
exponential rate of evolution of the universe as well as density fluctuations due to vacuum energy. The Lagrangian
for a phantom scalar field on the background of the Friedmann-Robertson-Walker (FRW) universe is assumed in the
[gµν∂µψ∂νψ + ξRψ
2 − 2U(ψ)], (1)
where gµν is the metric of the spacetime manifold, ψ = ψ(t), t is the cosmological time, R = R(g) is the Ricci scalar
for the spacetime metricg, ξ is a coupling constant which assumes zero for a scalar field minimally coupled to gravity
and 1/6 for a conformally coupled scalar field, U(ψ) is a potential of the scalar field.
The minimally coupled slowly evolving scalar fields with a potential function U(ψ) are good candidates for a
description of dark energy. In this model, called quintessence [1, 2], the energy density and pressure from the scalar
field are ρψ = −1/2ψ̇2 +U(ψ), pψ = −1/2ψ̇2−U(ψ). From recent studies of observational constraints we obtain that
wψ ≡ pψ/ρψ < −0.55 [3]. This model has been also extended to the case of a complex scalar field [4, 5].
Observations of distant supernovae support the cosmological constant term which corresponds to the case ψ̇ ≃ 0.
Then we obtain that wψ = −1. But there emerge two problems in this context. Namely, the fine tuning and the cosmic
coincidence problems. The first problem comes from the quantum field theory where the vacuum expectation value
is of 123 orders of magnitude larger than the observed value of 10−47GeV4. The lack of a fundamental mechanism
which sets the cosmological constant almost zero is called the cosmological constant problem. The second problem
called “cosmic conundrum” is a question why the energy densities of both dark energy and dark matter are nearly
equal at the present epoch.
One of the solutions to this problem offers the idea of quintessence, which is a version of the time varying cosmological
constant conception. Quintessence solves the first problem through the decaying Λ term from the beginning of the
Universe to a small value observed at the present epoch. Also the ratio of energy density of this field to the matter
density increases slowly during the expansion of the Universe because the specific feature of this model is the variation
of the coefficient of the equation of state with respect to time. The quintessence models [2, 6] describe the dark energy
with the time varying equation of state for which wX > −1, but recently quintessence models have been extended to
the phantom quintessence models with wX < −1. In this class of models the weak energy condition is violated and
∗Electronic address: hrycyna@kul.lublin.pl
†Electronic address: uoszydlo@cyf-kr.edu.pl
http://arxiv.org/abs/0704.1651v3
mailto:hrycyna@kul.lublin.pl
mailto:uoszydlo@cyf-kr.edu.pl
such a theoretical possibility is realized by a scalar field with a switched sign in the kinetic term ψ̇2 → −ψ̇2 [7, 8, 9].
From theoretical point of view it is necessary to explore different evolutional scenarios for dark energy which provide
a simple and natural transition to wX = −1. The methods of dynamical systems with notion of attractor (a limit
set with an open inset) offers the possibility of description of transition trajectories to the regime with wX = −1.
Moreover they demonstrate whether this mechanism is generic.
Inflation and quintessence with non-minimal coupling constant are studied in the context of formulation of necessary
conditions for the acceleration of the universe [10] (see also [11, 12]). We can find two important arguments which
favour the choice of conformal coupling over ξ 6= 1/6. The first, equation for the massless scalar field is conformally
invariant [13, 14]. The second argument is that if the scalar field satisfy Klein-Gordon equation in the curved space
then ψ does not violate the equivalence principle, and ξ is forced to assume the value 1/6 [15].
While recent astronomical observations give support that the equation of state parameter for dark energy is close to
constant value −1 they do not give a “corridor” around this value. Moreover, Alam et al. [16] pointed out that evolving
state parameter is favoured over constant wX = −1. The first step in the direction of description of the dynamics of
the dark energy seems to be investigation of the system with evolving dark energy in the close neighbourhood of the
value wX = −1. For this aim we linearize dynamical system at this critical point and then describe the system in a
good approximation (following the Hartman-Grobman theorem [17]) by its linearized part.
Other dark energy models like the Chaplygin gas model [18, 19, 20], [9, and references therein] and the model with
tachyonic matter can also be interpreted in terms of a scalar field with some form of a potential function.
Recent applications of the Bayesian framework [21, 22, 23, 24, 25] of model selection to the broad class of cosmo-
logical models with acceleration indicate that a posteriori probability for the ΛCDM model is 96%. Therefore the
explanation why the current universe is such close to the ΛCDM model seems to be a major challenge for modern
theoretical cosmology.
In this letter we present the simplest mechanism of concentration around wX = −1 basing on the influence of a
single scalar field conformally coupled to gravity acting in the radiation epoch. Phantom cosmology non-minimally
coupled to the Ricci scalar was explored in the context of superquintesence (wX < −1) by Faraoni [26, 27] and there
was pointed out that the superacceleration regime can be achieved by the conformally coupled scalar field in contrast
to the minimally coupled scalar field.
Let us consider the flat FRW model which contains a negative kinetic scalar field conformally coupled to gravity
(ξ = 1/6) (phantom) with the potential function U(ψ). For the simplicity of presentation we assume U(ψ) ∝ ψ2.
In this model the phantom scalar field is coupled to gravity via the term ξRψ2. We consider massive scalar fields
(for recent discussion of cosmological implications of massive and massless scalar fields see [28]). The dynamics of a
non-minimally coupled scalar field for some self-interacting potential U(ψ) and for an arbitrary ξ is equivalent to the
action of the phantom scalar field (which behaves like a perfect fluid) with energy density ρψ and pressure pψ [29]
ρψ = −
ψ̇2 + U(ψ) − 3ξH2ψ2 − 3ξH(ψ2)̇, (2)
pψ = −
ψ̇2 − U(ψ) + ξ
2H(ψ2)̇ + (ψ2)̈
2Ḣ + 3H2
ψ2, (3)
where the conservation condition ρ̇ψ = −3H(ρψ + pψ) gives rise to the equation of motion for the field
ψ̈ + 3Hψ̇ + ξRψ2 − U ′(ψ) = 0, (4)
where R = 6
Ḣ + 2H2
is the Ricci scalar.
Let us assume that both the homogeneous scalar field ψ(t) and the potential U(ψ) depend on time through the
scale factor, i.e.
ψ(t) = ψ(a(t)), U(ψ) = U(ψ(a)); (5)
then due to this simplified assumption the coefficient of the equation of state wψ is parameterized by the scale factor
wψ = wψ(a), pψ = wψ(a)ρψ(a), (6)
ψ′2H2a2 − U(ψ) + ξ
2(ψ2)′H2a+ (ψ2 )̈
Ḣ + 3H2
ψ′2H2a2 + U(ψ) − 3ξH2ψ2 − 3ξ(ψ2)′H2a
where prime denotes the differentiation with respect to the scale factor.
We assume the flat model with the FRW geometry, i.e., the line element has the form
ds2 = −dt2 + a2(t)[dr2 + r2(dθ2 + sin2 θdϕ2)], (8)
where 0 ≤ ϕ ≤ 2π, 0 ≤ θ ≤ π and 0 ≤ r ≤ ∞ are comoving coordinates, t stands for the cosmological time. It is also
assumed that a source of gravity is the phantom scalar field ψ with the conformal coupling to gravity ξ = 1/6. The
dynamics is governed by the action
m2pR+ (g
µνψµψν +
Rψ2 − 2U(ψ))
where m2p = (8πG)
−1; for simplicity and without lost of generality we assume 4πG/3 = 1 and U(ψ) is the scalar field
potential
U(ψ) =
m2ψ2. (10)
After dropping the full derivatives with respect to time, rescaling phantom field ψ → φ = ψa and the time variable
to the conformal time dt = adη we obtain the energy conservation condition
a′2 +
φ′2 −
m2a2φ2 = ρr,0 (11)
where ρr,0 is constant corresponding to the radiation in the model. The equations of motion are
a′′ = m2aφ2,
φ′′ = m2a2φ
where a prime denotes the differentiation with respect to the conformal time dt = adη and m2 > 0.
From the energy conservation condition we have
a′2 =
ρr,0 +
m2a2φ2
1 + φ̇2
and now from the equations of motion (12) we receive
(ρr,0 +
m2a2φ2)φ̈+
m2aφ(1 + φ̇2)(φφ̇ − a) = 0. (14)
The effective equation of state parameter is
weff =
ρφ + ρr
, (15)
for our model this parameter reduces to
weff = −
φ′2 + 1
m2a2φ2 − ρr,0
φ′2 + 1
m2a2φ2 + ρr,0
where a prime denotes the differentiation with respect to the conformal time and finally taking into account equa-
tion (13) we have
weff = −
φ̇2 +
m2a2φ2 − ρr,0
m2a2φ2 + ρr,0
(1 + φ̇2)
. (17)
For a, φ≫ ρr,0 this equation reduces to
weff = −
(2φ̇2 + 1), (18)
and it is clear that for any value of φ̇ weff is always negative.
To analyze equation (14) we reintroduce the original phantom field variable ψ = φ
and da/a = d ln a. Now equation
(14) reads
(ψ′′ + ψ′) +
m2ψ(1 + (ψ′ + ψ)2)(ψ(ψ′ + ψ) − 1) = 0 (19)
where a prime now denotes the differentiation with respect to a natural logarithm of the scale factor. Introducing
new variables y = ψ′ and ρr = ρr,0a
−4 we can represent this equation as an autonomous dynamical system
ψ′ = y
y′ = −y −
(ψ(y + ψ) − 1)(1 + (y + ψ)2) (20)
ρ′r = −4ρr.
There are the two critical points in the phase space (ψ, y, ρr), namely ψ = ±1, y = 0, ρr = 0. The linearization
matrix reads
0 1 0
−2(1 + ψ2) −1 − (1 + ψ2) 2
(ψ2 − 1)(1 + ψ2)
0 0 −4
y=0,ρr=0
0 1 0
−4 −3 0
0 0 −4
y=0,ρr=0,ψ=±1
. (21)
The eigenvalues for this matrix are λ1,2 =
(−3 ± i
7) and λ3 = −4.
To find a global phase portrait it is necessary to study the system in the neighbourhood of the critical points which
correspond, from the physical point of view, stationary states (or asymptotic solutions). Then the Hartman-Grobman
theorem guaranties us that the linearized system at this point is a well approximation of the nonlinear system. First,
we must note that ρr = 0 is in the invariant submanifold of the 3-dimensional nonlinear system. It is also useful to
calculate the eigenvectors for any eigenvalue. We obtain following eigenvectors
v1,2 =
 , v3 =
 . (22)
They are helpful in construction of the exact solution of the linearized system
~x(t) = ~x(0) exp t
0 1 0
−4 −3 0
0 0 −4
0 − 3
0 1 0
1 0 0
e−4t 0 0
t cos
t −e− 32 t sin
t sin
t cos
0 0 1
0 1 0
where x = ψ − ψ0, y = ψ′ − ψ′0, z = ρr − ρ0 and x0, y0, z0 are initial conditions and we have substituted ln a = t.
If we consider linearized system on the invariant stable submanifold z = 0, it is easy to find the exact solution. If
we return to the original variables ψ, ψ′, then ψ(ln a) is the solution of the linear equation
(ψ − ψ0)′′ + 3(ψ − ψ0)′ + 4(ψ − ψ0) = 0, (24)
i. e.,
(ψ − ψ0) = C1 exp
+ C2 exp
(φ− φ0) = C1a−
2 cos
+ C2a
2 sin
. (26)
Because of the lack of alternatives to the mysterious cosmological constant [21, 22] we allow that energy might vary
in time following assumed a priori parameterization of w(z). In the popular parameterization [30, 31, 32, 33] appears
a free function in most scenarios which is a source of difficulties in constraining parameters by observations. However
most parameterizations of the dark energy equation of state cannot reflect real dynamics of cosmological models with
dark energy. The assumed form of w(z) can be incompatible with the w(z) obtained from the underlying dynamics
of the cosmological model. For example some of parameters can be determined from the dynamics which can be
crucial in testing and selection of cosmological models [21]. Our point of view is to obtain the form of w(z) specific
for given class of cosmological models from dynamics of this models and apply it in further analysis both theoretical
and empirical. In practice we put the cosmological model in the form of the dynamical system and linearize it around
the neighbourhood of the present epoch to find the exact formula of w(z). For the phantom scalar field model this
incompability manifests by the presence of a focus type critical point (therefore damping oscillations) in the phase
space rather than a stable node (Fig. 1 and its 3D version Fig. 2).
The properties of the minimally coupled phantom field in the FRW cosmology using the phase portrait have been
investigated by Singh et al. [34] (see also [35] for more recent studies). Authors showed the existence of the deSitter
attractor and damped oscillations (the ratio of the kinetic to the potential energy |T/U | to oscillate to zero).
We can also express weff in these new variables
weff = −
(ψ + ψ′)2 −
ρr − 12m
1 + (ψ + ψ′)2
w′eff =
dweff
d ln a
(ψ + ψ′)(ψ′ + ψ′′) + ρrm
2 + ψ
(ρr +
m2ψ2)2
1 + (ψ + ψ′)2
. (28)
Recently Caldwell and Linder [36] have discussed dynamics of quintessence models of dark energy in terms of w−w′
phase variables, where w′ was the differentiation with respect to the logarithm of the scale factor. These methods
were extended to the phantom and quintom models of dark energy [37, 38]. Guo et al. [38] examined the two-field
quintom models as the illustration of the simplest model of transition across the wX = −1 barrier. The interesting
mechanism of acceleration with a periodic crossing of the w = −1 barrier have been recently discussed in the context
of the cubic superstring field theory [39]. In the model under consideration we obtain this effect but trajectories cross
the barrier infinitely many times. The main advantage of the discovered road to Λ is that it takes place in the simple
flat FRW model with the quadratic potential of the scalar field.
It is easy to check that at the critical points weff = −1 and dweffd lna = 0. Since these points are sinks there is infinite
many crossings of weff = −1 during the evolution.
The methods of the Lyapunov function are useful in discussion of stability of the critical point of the non-linear
system. The stability of any hyperbolic critical point of dynamical system is determined by the signs of the real parts
of the eigenvalues λi of the Jacobi matrix. A hyperbolic critical point is asymptotically stable iff real λi < 0 ∀i, if x0
is a sink. The hyperbolic critical point is unstable iff it is either a source or a saddle. The method of the Lyapunov
function is especially useful in deciding the stability of a non-hyperbolic critical points [17, p.129]. The construction
of the Lyapunov function was used by [40] for demonstration that periodic behaviour of a single scalar field is not
possible for minimally coupled phantom scalar field (see also [41]).
The quantity w′eff in terms of weff and (lnψ)
′ reads
w′eff = −(1 − 3weff)(1 + weff +
(lnψ)′). (29)
It is interesting that equation (29) can be solved in terms of w̄(a) – the mean of the equation of the state parameter
in the logarithmic scale defined by Rahvar and Movahed [42] as
w̄(a) =
w(a′)d(ln a′)
d(ln a′)
, (30)
namely:
w(a) =
a3(1+w̄(a))ψ2. (31)
They argued that this phenomenological parameterization removes the fine tuning of dark energy and ρX/ρm ∝
a−3w̄(a) approaches a unity at the early universe. Note that in w̄(a) = −1 that
w(z) + 1 =
(1 − ψ2), (32)
FIG. 1: The phase portrait (weff, w
eff) of the investigated model on the submanifold ρr = 0. This figure illustrates the evolution
of the dark energy equation of the state parameter as a function of redshift for different initial conditions. In all cases trajectories
cross the boundary line weff = −1 infinite many times but this state also represents the global attractor.
where
ψ = ψ0 + (1 + z)
C1 cos(
ln(1 + z)) − C2 sin(
ln(1 + z))
. (33)
In Fig. 3 we present the relation w(z) for different values of parameters ψ0 = ±1, C1 and C2.
In this letter we regarded the phantom scalar field conformally coupled to gravity in the context of the problem of
acceleration of the Universe. We applied the methods of dynamical systems and the Hartman-Grobman theorem to find
universal behaviour at the late times – damping oscillations around weff = −1. We argued that most parameterizations
of the dark energy, such as linear evolution of w(z) in redshift or the scale factor, cannot reflect realistic physical
models because of the presence of non-hyperbolic critical point of a focus type on the phase plane (w,w′). We
suggested a parameterization of a type
wX(z) = −1 + (1 + z)3
C1 cos(ln(1 + z)) + C2 sin(ln(1 + z))
which parameterizes damping oscillations around wX = −1 “phantom divide”, and finally, with the help of this
formula one can simply calculate energy density for dark energy ρX
ρX = ρX,0 exp
(1 + z)3
A sin(ln(1 + z)) +B cos(ln(1 + z))
. (35)
Acknowledgments
The work of M.S. has been supported by the Marie Curie Actions Transfer of Knowledge project COCOS (contract
MTKD-CT-2004-517186).
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FIG. 3: The relation w(z) for different values of the parameters: C1 = 0.01, C2 = 0 dash-dotted line (red el. version); C1 = 0,
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