Astrophys Space Sci (2011) 336:315–325
DOI 10.1007/s10509-011-0870-z
INVITEDREVIEW
Modified Chaplygin gas and solvable F-essence cosmologies
Mubasher Jamil · Yerlan Myrzakulov · Olga Razina ·
Ratbay Myrzakulov
Received: 3 July 2011 / Accepted: 19 September 2011 / Published online: 8 October
2011 © Springer Science+Business Media B.V. 2011
Abstract The Modified Chaplygin Gas (MCG) model belongs to the class of a unified models of dark energy and
dark matter. In this paper, we have modeled MCG in the
frame-work of f-essence cosmology. By constructing an
equation connecting the MCG and the f-essence, we solve
it to ob-tain explicitly the pressure and energy density of
MCG. As special cases, we obtain both positive and
negative pressure solutions for suitable choices of free
parameters. We also calculate the state parameter which
describes the phantom crossing.
Department of Physics, California State University,
Fresno, CA 93740, USA
e-mail: rmyrzakulov@csufresno.edu
(WMAP) satellite results which gathered data about the cosmic microwave background radiation, such a cosmic acceleration is produced by a so-called dark energy (DE) (Sher-win
et al. 2011). Such a new element of the universe, capa-ble of
accelerating, must, in accordance with the Friedman equation,
have a pressure less than minus one third of the energy
density. The notion of a cosmic medium possessing negative
pressure is not new in cosmology. The very first proposal for
dark energy is the cosmological constant de-noted by _.
Originally proposed by Einstein to construct a theoretical
model of the universe such that the spherical configuration of
matter in the universe is balanced with the negative pressure
of cosmological constant. Thereby creat-ing a static universe,
in the conformity of the notions held by many scientists of
that time. The origin of cosmic accel-eration in recent time
needs _ but it faces serious problems of fine tuning (the value
of _ is several orders of magnitude larger then estimated from
the empirical results) and cosmic coincidence problem (the
energy density of matter and dark energy component are
approximately same in our presence). Then theorists looked
for other candidates of dark energy. There has been a wide
variety of theoretical models of dark energy constructed in the
literature including quintessence, phantom energy, Chaplygin
gas, tachyon and dilaton dark energy, etc., see Copeland et al.
(2006) for further details. The quintessence and phantom
energy models are based on spatially homogeneous and time
dependent scalar fields, in conformity of the cosmological
principle. The Lagrangians for the quintessence (phantom
energy) has positive (nega-tive) kinetic energy. Where
quintessence faces still fine tun-ing problem of the parameters
of its potential, while the phantom energy gives very esoteric
possibilities of Big Rip and black hole evaporations (Jamil et
al. 2008).
R. Myrzakulov
e-mail: rmyrzakulov@gmail.com
Quintessence as a model of dark energy relies on the
suitable choice of the potential function or the potential
Keywords Modified Chaplygin gas · K-essence
· F-essence · G-essence
1 Introduction
Several complementary cosmological observations guide us
that our Universe is experiencing an accelerated expansion in
the current era (Perlmutter et al. 1999; Riess et al. 1998).
From the results of Wilkinson Microwave Anisotropy Probe
M. Jamil
Center for Advanced Mathematics and Physics, National
University of Sciences and Technology, Islamabad,
Pakistan e-mail: mjamil@camp.nust.edu.pk
Y. Myrzakulov · O. Razina · R. Myrzakulov
Eurasian International Center for Theoretical Physics,
Eurasian National University, Astana 010008, Kazakhstan
R. Myrzakulov ( )
316
energy of scalar fields. It is also possible that the cosmic
acceleration could appear due to modification of the kinetic energy of the scalar fields. Such modifications are
termed non-canonical. The kinetically driven cosmic acceleration was originally proposed as a model for inflation,
namely k-inflation (Armendariz-Picon et al. 1999), and
then as a model for dark energy, namely k-essence
(Armendariz-Picon et al. 2000, 2001; Chiba et al. 2000;
Chiba 2002; Scherrer 2004; Yang and Gao 2011; De Putter
and Lin-der 2007; Capozziello et al. 2010; Karami et al.
2011; Malekjani et al. 2010; Adabi et al. 2011; Farooq et
al. 2010). This model is free from fine-tuning and
anthropic argu-ments. K-essence has been proposed as a
possible means of explaining the coincidence problem of
the Universe be-ginning to accelerate only at the present
epoch (Malquarti et al. 2003). Instead, k-essence is based
on the idea of a dy-namical attractor solution which causes
it to act as a cos-mological constant only at the onset of
matter-domination. Consequently, k-essence overtakes the
matter density and induces cosmic acceleration at about the
present epoch. In some models of k-essence, the cosmic
acceleration contin-ues forever while in others, it continues
for a finite duration (Armendariz-Picon et al. 2000).
In the last years, the k-essence model has received much
attention. It is still worth investigating in a systematic way
the possible cosmological behavior of the k-essence. Quite
recently, a model named g-essence is proposed (Yerzhanov
et al. 2010) which is a more generalized version of kessence. In fact, the g-essence contains, as particular cases,
two important models: k-essence and f-essence. Note that
f-essence is the fermionic counterpart of k-essence.
To our knowledge, in the literature there are relatively few
works on dark energy models with fermionic fields. However,
in the recent years several approaches were made to explain
the accelerated expansion by taking fermionic fields as the
gravitational sources of energy (see e.g. Ribas et al. 2005;
Samojeden et al. 2009, 2010; Myrzakulov 2010; Tsyba et al.
2011; Yerzhanov et al. 2011; Ribas and Kre-mer 2010; Cai
and Wang 2008; Wang et al. 2010; Ribas et al. 2008; Rakhi et
al. 2009, 2010; Chimento et al. 2008; Anischenko et al. 2009;
Saha 2001, 2004, 2006a, 2006b; Saha and Shikin 1997; Vakili
and Sepangi 2008; Wei 2011; Dereli et al. 2001; Balantekin
and Dereli 2007; Armendariz-Picon and Greene 2003). In
particular, it was shown that the fermionic field plays very
important role in: (i) isotropiza-tion of initially anisotropic
spacetime; (ii) formation of sin-gularity free cosmological
solutions; (iii) explaining late-time acceleration.
A very appealing proposal to describe the dark sector are
the so-called unified models. The prototype of such model is
the Chaplygin gas. In the unified models, dark energy and
dark matter are described by a single fluid, which behaves as
ordinary matter in the past, and as a cosmological constant
Astrophys Space Sci (2011) 336:315–325
term in the future. In this sense, it interpolates the different
periods of evolution of the Universe, including the present
state of accelerated expansion. The Chaplygin gas model
leads to very good results when confronted with the observational data of supernova type Ia. Concerning the matter
power spectrum data, the statistic analysis leads to results
competitive with the _CDM model, but the unified (called
quartessence) scenario must be imposed from the beginning.
It means that the only pressureless component is the usual
baryonic one, otherwise there is a conflict between the constraints obtained from the matter power spectrum and the
supernova tests. Note that many variations of the Chaply-gin
gas model have been proposed in the literature. One of them is
the so-called Modified Chaplygin Gas model. It is important
that the MCG model belongs to the class of a uni-fied models
of dark energy and dark matter. In this context, it is important
to study the relation between MCG and the other unified
models of dark energy and dark matter. For example, in
Myrzakulov et al. (2011) relationship between MCG and kessence was established. In this paper, we have modeled
MCG in the framework of f-essence cosmology. By
constructing an equation connecting the MCG and the fessence, we solve it to obtain explicitly the pressure and
energy density of MCG. As special cases, we obtain both
positive and negative pressure solutions for suitable choices of
free parameters.
This paper is organized as follows. In Sect. 2, we introduce the F-essence formalism. In Sect. 3, we briefly discuss Modified Chaplygin gas and its connection with the fessence. In Sect. 4, we construct a governing differential
equation of our model and solve it for several special cases
and a general case. Conclusion is presented in the last section. At last, in the Appendix we give the derivation of the
equations of motion of g-essence, k-essence and f-essence.
2 F-essence
Let us briefly present some basics of f-essence. Its action
has the form (Myrzakulov 2010; Tsyba et al. 2011)
d 4x √
S
ψ) ,
−g R + 2K(Y, ψ,
=
ψ
ψ†
(1)
¯
0
γ
denote the fermion field and its
adjoint, respectively, the dagger represents complex conwhere ψ and ¯ =
jugation and R is the Ricci scalar. The fermionic fields are
treated here as classically commuting fields (see e.g. Ribas
et al. 2005; Cai and Wang 2008; Wang et al. 2010; Rakhi
et al. 2010; Chimento et al. 2008; Saha 2004; Vak-ili and
Sepangi 2008; Wei 2011; Dereli et al. 2001; Bal-antekin
and Dereli 2007; Armendariz-Picon and Greene 2003).
What we mean by a classical fermionic field is a
Astrophys Space Sci (2011) 336:315–325
317
set of four complex-valued space-time functions that transform according to the spinor representation of the Lorenz
and
group. The existence of such fields is crucial in our work
since in spite of fact that fermions are described by quantized fermionic fields which do not have a classical limit,
we assume such classical fields exist and use them as matter
fields coupled to gravity. A possible justification for the existence of classical fermionic fields is given in the appendix
of reference Armendariz-Picon and Greene (2003). So for
a more extensive and physically detailed discussion of the
properties of such classical fermionic fields we refer to this
fundamental work (Armendariz-Picon and Greene 2003).
Furthermore, K is the Lagrangian density of the fermionic
field, the canonical kinetic term has the form
1 i ψ _μD
Y
=2
ψ
¯
μ
(D
μ
μ
¯
trices satisfying the Clifford algebra
μν
= 2g ,
_μψ,
∂ μψ
=
Dμ ψ
−
¯
ψ _μ.
∂ μψ
=
¯
+
a
(4)
¯
with _
ρ
(5)
μδ
We work with the Friedmann-Robertson-Walker (FRW)
spacetime given by
2
2
2
2
(6)
ds = −dt + a dx + dy + dz .
For this metric, the vierbein is chosen to be
ea
μ
=
diag(1, 1/a, 1/a, 1/a),
,
γ
= I0
,
(11)
are Pauli matrices having
the following form
σ
1
01
= 10
σ
,
2
0 −i ,
=
σ
3
1 0
= 0 −1
i0
. (12)
2
2
K
H
− ρ = 0,
˙
+
Y
˙
3H 2
+ p = 0,
0.5(3H K
ψ
+
KY
(13)
ψ
˙Y
+
0. ( HK
iγ 0K
K )ψ
Y
Y
−
KY )ψ
+
˙
iKψ γ
¯+
0
ψ
¯=
0
,
(14)
(15)
0,
(16)
=
(17)
where
denoting the Christoffel symbols.
2
−σ 0
k
0I
5
ρ˙ + 3H (ρ + p) = 0,
1
ρ
ρ b σ δ
4 gρσ _μδ − eb ∂μeδ _ _ ,
2
where I = diag(1, 1) and the σ
k
k
γ =
,
¯+
Above, the fermionic connection _μ is defined by
_μ = −
0 σ
k
0 −I
=
γ
3H
μ
denotes the tetrad or “vierbein” while the covariant derivatives are given by
Dμ ψ
I0
0
(3)
where the braces denote the anti-commutation relation. e
ψ).
We are ready to write the equations of f-essence (in detail, the derivation of these equation we will present in the
Appendix). Here we write the final form of these equations.
We have
μ
μ a
Moreover, _ = ea γ are the generalized Dirac-Pauli ma-
μ ν
γ ,γ
0
ψγ
0ψ
=
¯
˙− ¯
Finally, we note that the gamma matrices we write in the
Dirac basis that is as
(2)
ψ)_ ψ .
−
0.5i(ψγ
Y
eμ
a
=
p = K,
ρ = Y KY − K,
(18)
are the energy density and pressure of the fermionic field.
If K = Y − V , then from the system (13)–(18) we get the corresponding equations of the Einstein-Dirac model. At last,
we note that f-essence is the particular case of g-essence
for which the energy density and pressure are given by
ρ = 2XKX + Y KY − K, p = K,
diag(1, a, a, a).
where
(7)
X
=
0.5φ
2
(19)
is the canonical kinetic term for the scalar
˙
field φ (see the Appendix).
μ
The Dirac matrices of curved spacetime _ are
0
5
0 j
=γ ,_ = a
√
3 Modified Chaplygin gas and f-essence
−1 j
γ ,
0 1 2 3
(8)
5
= −i −g_ _ _ _ = γ ,
_0 = γ 0,
j
_j = aγ , (j = 1, 2, 3).
(9)
Hence we get
j 0
_0 = 0, _j = 0.5aγ˙ γ
(10)
In the cosmological context, the Chaplygin gas was first
suggested as an alternative to quintessence and was demon-
strated to have an increasing _ (cosmological constant) behavior for the evolution of the universe (Kamenshchik et al.
2001; Bento et al. 2002). The EoS of the MCG dark energy
model was proposed by Benaoum (2002) as an exotic fluid
which could explain the cosmic accelerated expansion. Later
318
Astrophys Space Sci (2011) 336:315–325
on, it was shown that the EoS of MCG is valid from the radiation era to _CDM model (Debnath et al. 2004). The MCG
parameters α and B have been constrained by the cosmic
microwave background CMB data (Liu and Li 2005). The
stable scaling solutions (attractor) of the Freidmann equa-tion
have been obtained in Jamil and Rashid (2008, 2009). The
MCG is given by Benaoum (2002)
ρα
da . The last equation has the following solution
Y
Y0e
p+ρ
(27)
,
da
=
where Y0 is an integration constant. We can rewrite this expression as
pζ
Y0e
p+ρ
(28)
,
dζ
=
,
(20)
where A and B are positive constants and 0 ≤ α ≤ 1. In the light
of 3-year WMAP and the SDSS data, the MCG best fits the
data by choosing A = −0.085 and α = 1.724 (Lu et al. 2008).
The dynamical attractor for the MCG exists at ω = −1 (where p
= ωρ), hence MCG can do the phan-tom crossing from ω > −1
to ω < −1, independent to the choice of initial conditions (Jing
et al. 2008). A general-ization of MCG was proposed in
n
Debnath (2007) by tak-ing B ≡ B(a) = B0a , where n and B0 are
constants. The MCG is the generalization of generalized
α
Chaplygin gas p = −B/ρ (Barreiro and Sen 2004; Carturan and
Finelli 2003) with the addition of a barotropic term. Using
(17) and (20), we can show that the MCG energy density and
pressure are given by Benaoum (2002)
−3(1+α)(1+A)
where ζ = Ca
. Then the corresponding expres-sions for the
energy density and pressure take the form
1
ρ = [D + ζ ]
B(1 + A)−1 + Ca−3(1+α)(1+A)
1+α
1+α
(29)
,
α
−
p = A(D + ζ ) − B [D + ζ ]
with D = B(1 + A)
−1
p + ρ = (1 + A)ζ (D + ζ )
pζ = A(D + ζ )
. Then we have
−
α
1+α
1+α
(21)
p = AB(1 + A)−1 + ACa−3(1+α)(1+A)
− 1
α
(Aζ
− D)(D + ζ )
−
α
1+α
−1.
(32)
Hence we get
p
ζ
,
(31)
,
α
α
−
(30)
1+α
+
1
ρ=
dp
pa
Y
B
p = Aρ −
where pa =
p
(1+α)A+α
ρ
dζ = ln ζ
(D +ζ )
(1+α)(1+A)
+
Finally we have
−
α
1+α
.
(33)
α
− B B(1 + A)−1 + Ca−3(1+α)(1+A)
(1+α)A+α
−
1+α
,
(22)
where C is a constant of integration. This constant of integration C can be found from the condition that the fluid has
vanishing pressure p = 0 when a = a0:
C = B A(1 + A) −1a03(1+α)(1+A).
For the modified Chaplygin gas case, the EoS parameter is
ω =ACa−3(1+α)(1+A) − B(1 + A)−1
B(1
1
A)
3(1
Ca
+ − +
From (18) we get
−
+
.
α)(1
(23)
B)
+
Y = Y0 ζ
α
(D +
(1+α)(1+A)
ζ)
−
1+α
.
4 Solvable f-essence cosmologies
In this section, we consider some solvable f-essence
models related with the MCG given by (20).
4.1 The case: B = 0
In this case the EoS takes the form
dK
ρ=
− K,
d ln Y
(24)
(25)
p = K.
Hence follows that
dp
ρ+p=
dp
da
da
p
+
−3(1+A)
p = Aρ0a
pa
=
where ρ evolves like
ρ
,
, ρ0 = const
(36)
so that
so that we have
d ln Y
(35)
p = Aρ,
ρ = ρ0a
= da d ln Y
d ln Y
(34)
(26)
−3(1+A)
.
(37)
On the other hand, (24)–(25) for (35) give
1+A
ρ=FY
A
, F = const.
(38)
Astrophys Space Sci (2011) 336:315–325
319
Fig. 1 Evolution of kinetic energy Y as a function of redshift z
Fig. 2 Evolution of Y against z from (47). Other parameters are fixed
from (40). Other parameters are fixed at Y0 = 2 and A = −1
at C = 0.5, B = 0.4, W = 0.3 and α = 1.2
In this case, the pressure is
sure:
(39)
.
ρ
1
Y0a−3 ,
A
=
Y0
−
ρ 1+A F
=
0
(40)
.
We get the scale factor as
−1
Y
a(t) =
=−
Y
(41)
.
The behavior of (40) against redshift a
−1
in Fig. 1 where we see that this evolution is of power-law
− 1 = z is plotted
1+α
−
B 1+α 1
(45)
,
+α
(W Y )
α
α
1
α
(46)
+ ,
where W = const. The solution for Y is determined from
(43) and (45):
=
3A
Y0
α
α
1
A
1+A
1 − (W Y )
=
p
A
1
+
Comparison of (36) and (38) yields
Y
1
B
1+A
A
p ≡ K = AF Y
W
−1
α
C
C
1+α
Ba3(1+α) −
+
α
1+α
.
(47)
The behavior of (47) is shown in Fig. 2, where we see
that the kinetic energy of the f-essence increases and then
stays constant at higher redshifts. Note that this conclusion
depends crucially on the chosen values of free parameters.
form.
4.3 The general case
4.2 The case: A = 0
In this section, we consider the general case when A _= 0,
Ignoring the barotropic term in MCG, we have
=_
0. So in this case we must solve the following system
B
B
p= −
ρ
α
(42)
.
ρ = Y KY − K,
(48)
B
It is called the generalized Chaplygin gas (GCG) (Kamenshchik et al. 2001). Recently it is proposed using perturbative analysis and power spectrum observational data that the
MCG model is not a successful candidate for the cosmic
medium unless A
= 0, i.e. the usual GCG model is favored
(Fabris et al. 2011). As well-known, the corresponding en-
K = Aρ −
ergy density and pressure are given by
p = Aρ −
1+α
or
ρ = YpY − p,
(50)
B
ρ
α
(43)
,
α
p = −B B + Ca−3(1+α)
ρ
−
1+α
,
where C is a constant of integration. From (34), (43) and
(44) we get the expressions for the energy density and pres-
(51)
.
Solving (48) for K , we arrive at
1
ρ = B + Ca−3(1+α)
(49)
ρα
(44)
p ≡ K = EY + Y
(52)
where E
if ρ =
(ψ, ψ)
Y 2 dY,
is an integration constant. Note that
(ψ, ψ)
V
¯
then from (52), it follows that K = EY − V
¯
320
Astrophys Space Sci (2011) 336:315–325
Consider some particular solutions of this equation. If α =
i.e. the purely Dirac case. Equations (51) and (52) give
0, then we have
ρ
EY + Y
B
2
dY = Aρ −
α
1
(1 + A)ρ
ρ
α
+
− B = 0,
α(1 + A)
=
α(1
A
(55)
.
A)
+
−B
− B.
The corresponding EoS parameter is given by
−1
ω
ln(C Y ) .
1
= − −1
p = ln(C1Y )
(54)
where W is a constant and
n
(66)
and for the pressure
n(1α)
n(1+α)
− (W Y )
+
B
ρ = ln(C1Y ) ,
(53)
ρα ,
which has the following solution:
Y
+
(67)
(68)
Second example is α = −1. Then for the energy density and
From (55) it follows that A and α are related by
pressure we obtain
(n − 1)α
A
or α
1)α
= − (n
.
nA
= − (n
n
1)(1
(56)
B
A)
C2Y 1+B
ρ
−
+
−
+
The search of the analytical solutions of (54) is a tough job.
(69)
const),
(C2
=
and
=
So let us find some particular solutions for some values of n.
B
p = −(1 + B)C2Y
1+B
(70)
.
4.3.1 Example 1: n = 0
The corresponding EoS parameter is given by
It follows from (55) that this case (n = 0) realized as α = 0
ω
n = 0 and (48)–(49) take the form
=−
1
(71)
B.
−
4.3.2 Example 2: n
or A = −1. (1) Let us first consider the case α = 0. Then
1
=
ρ = Y KY − K,
(57)
If n = 1 then from (56), it follows that A = 0. This case was
K = Aρ − B
(58)
considered in Sect. 4.2, so we omit that here.
(59)
4.3.3 Example 3: n = 2
Now we consider the case when n
and (54) becomes
α
1
(1 + A)ρ
1+A
− (W Y )
+
A
− B = 0.
related by
Hence we write
ρ = (1 + A)
−
1
1+A
B
+
(W Y )
(60)
A
and for the pressure
A)−
(1
p
1
B
.
1+A
A(W Y )
A
=
A
−
parameter is
+
2A
or
2
α
=− 1
1 α 2(1
(1 + A)ρ
+
+
A
1
A
+
− (W Y )
− α2
1
(62)
.
α
α)
ρ
2(1 α)
+
1
(
α)
2 +
1 − B(1 + α) (W Y )
× 1∓
(2) Now we consider the case when A = −1. Then n = 0
1
1 α
+
α
.
The pressure is given by
ρ = Y KY − K,
α
−
K
= − ρ − Bρ .
Hence we get
(63)
(64)
= ln(C1Y )
(73)
α
+
+A
and (48)–(49) take the form
αB ln ρ − (1 + α) ρ
− B = 0.
1
2
−1 1+α
(72)
.
It has the solution
ρ = (W Y )
A)
B + (W Y )
=−α
(61)
+
ω
A
2. In this case A and α
The equation for ρ (54) takes the form
=
+
− +
The corresponding equation of state (EoS)
given by
B(1
α
=
−B
(65)
.
α(W Y )
p= −
α
+
× 1∓
−
α
2
2
1
1
+
α
2
1
2(1+α)
1 − B(1 + α) (W Y )
α
1+α
(74)
Astrophys Space Sci (2011) 336:315–325
321
4.3.4 Example 4: n = 0.5
If n = 0.5 then A and α satisfy the relation
α
A
A = − α 1 or α = 1 A .
−
+
Equation (54) becomes
1 α
(77)
0.5(1 α)
(1+α)
+ − (W Y )
+ − B = 0.
α ρ
This equation has the following solution
(1 + A)ρ
Fig. 3 For n = 2, the EoS parameter w− is plotted against the ki-netic
energy Y , while other parameters are fixed at B = 0.1, W = 0.2,
1−
2
1
ρ
=
(78)
2
(W Y )
α
α = 1 .5
2
α
1 ±
4B
1+ 1
1
−
−
(W Y )
α
α
+α
1+α
.
(79)
The pressure is given by
1
p
1−α
α(W Y ) α
=−
α
−
1
2
2
4B
× 1± 1+1
1
B(W Y )−
1
(W Y )−
α
−
2
4B
× 1±
α = 1 .5
2
+
(W Y )−
α
α
−
1−
1
(W Y )
−
+α
.
(80)
α
−
(W Y )
1 +1
× 1∓
−
1+α
2
.
(81)
α
−
In Fig. 5, we have plotted the EoS parameter against the
α
2(1+α)
2α
1+α
α
4B
2
−
α
1+α
α
1
1
1+1
1
−
The corresponding EoS parameter reads
ω∓ = −
− B(W Y )
+α
1−α
−
Fig. 4 For n = 2, the EoS parameter w+ is plotted against the ki-netic
energy Y , while other parameters are fixed at B = 0.1, W = 0.2,
1+α
α
α
1+α
α
α
× 1∓
1 − B(1 + α) (W Y )
.
(75)
The corresponding EoS parameter is
ω∓ = −
α
α
+
2 − B(W Y )
−
1
2(1+α)
1
α
2
+
α
4.3.5 Example 5: n = −1
2(1+α)
α
× 1∓
1 − B(1 + α) (W Y )
kinetic term. Here we chose α < 0 which corresponds to the
polytropic term added in the barotropic equation of state.
Such a f-essence polytropic EoS can also cause the superacceleration.
.
(76)
In Figs. 3 and 4, we have plotted the EoS parameter and it
is shown that subnegative values of ω are permissible in
our model. This corresponds to f-essence MCG behaving
as phantom energy which causes super-accelerated
expansion (Debnath et al. 2004).
Our next example is n = −1. Then A and α satisfy the relation
2α
A= −
2α
1
or α =
−
Equation (54) becomes
(1 + A)ρ
1+α
− (W Y )
−
1+αα
A
2(1
+
A)
.
ρ−(1+α) − B = 0.
(82)
(83)
322
Astrophys Space Sci (2011) 336:315–325
Fig. 5 For n = 0.5, the EoS parameter w− is plotted against the kinetic
Fig. 6 For n = −1, the EoS parameter w+ is plotted against the kinetic
energy Y , while other parameters are fixed at B = −0.1, W = 0.2,
energy Y , while other parameters are fixed at B = −1, W = 2, α = 1.5
α = −1.5
It has the solution
B 2 + 4(1 + A)(W Y )−
B±
ρ=
2(1
+
The pressure is given by
2
B± B
p=A
+
−B
(84)
.
−
1
1+α
1+α
α
A)
B2 + 4(1 + A)(W Y )−
B±
1+α
α
A)
+ 4(1 + A)(W Y )
2(1
1
1+α
2(1
1+αα
A)
−
+
The corresponding EoS parameter is given by
ω∓ = −
2α
2α
−
B
α
1+α
.
(85)
1
2
B −
netic energy Y , while other parameters are fixed at B = −3, W = 0.8,
α = −1.5
1+α
1 + 2 (W Y ) α
× B∓
Fig. 7 For n = −1, the EoS parameter w+ is plotted against the ki-
1 (W Y )
−
.
1+αα
(86)
−
In Figs. 6 and 7, we plotted the above state parameter
against kinetic energy. We choose positive and negative
2α
values of α for the sake of completeness. It is apparent that
the state parameter achieves subnegative values showing
the viability of our dark energy model.
5 Conclusion
In summary, we have modeled modified Chaplygin gas in fessence cosmology. The use of MCG is useful as a tool of
explaining dark energy and dark matter in a unified man-ner,
while f-essence cosmology essentially suitable to de-scribe
cosmic acceleration only at present time. Thus the
correspondence of MCG with f-essence is useful in learn-ing
how these two models are connected to each other. We
studied this link by constructing a differential equation (54)
connecting the MCG and the f-essence. We solved it to obtain explicitly the pressure and energy density of MCG.
We observed that f-essence MCG has one additional free
pa-rameter namely W along with A, B , α. As special cases,
we obtain both positive (37), (80) and negative (46), (68),
(72) pressure solutions for suitable choices of free
parameters. The negative pressure solution is essentially
useful for cos-mic expansion with acceleration. Prior to
this, we studied the model with barotropic and generalized
Chaplygin gas equation of states.
The present work is concerned only with the correspondence of MCG with the f-essence. However it would be more
interesting to study the dynamical features of this model. For
instance, it is interesting to check whether such a model is
useful to model inflation at earlier cosmic epoch and cosmic
acceleration at the present time. Next it will be important to
compare this model with the observational data and constrain
its free parameters, particularly W . Also ob-servational
constraints on the state parameter ω will indicate if it
corresponds to cosmological constant, quintessence or
phantom energy. We finish our work in the hope that merits
Astrophys Space Sci (2011) 336:315–325
323
and demerits of f-essence will be clear in the future investigations.
One of old methods for finding exact solutions of nonlinear differential equations is considered. Modifications of
the method are discussed. Application of the method is
illus-trated for finding exact solutions of the Fisher
equation and nonlinear ordinary differential equation of the
seven order. It is shown that the method is one of the most
effective ap-proaches for finding exact solutions of
nonlinear differential equations. Merits and demerits of the
method are discussed. [3] arXiv:1108.3296 [pdf, other].
Finally, we note that this paper is the logical continuation of Myrzakulov et al. (2011), where the relation
between k-essence and MCG was studied. The action of kessence reads as
S = d4 x
√
(87)
R + 2K(X, φ) ,
−g
Acknowledgements We would like to thank the anonymous refer-ees for
providing us with constructive comments and suggestions to improve this
work. One of the authors (R.M.) thanks D. Singleton and Department of
Physics, California State University Fresno for their hospitality during his
one year visit (October, 2010 – October, 2011).
Appendix: The derivation of the equations of motion of
g-essence, k-essence and f-essence
In this Appendix we would like to present the derivation of
the equations of motion for the f-essence action (2.1) that
is the system (13)–(17). But, as f-essence is the exact
particular case of g-essence, we consider more general case
and give the derivation of the equations of motion for gessence. Let us consider the following action of g-essence
S
√
d 4x
=
where the kinetic term X for the scalar field φ (for the FRW
metric) reads as
φ
2
˙
.
(88)
The corresponding equations for the FRW metric look like
X = 0.5
3H − ρ = 0,
2
H2
H
˙
K
+
+3
φ
p
0,
=
3H K )φ
(K
X
¨
˙
+
(89)
(90)
X+
X
˙
0,
K
φ=
−
ρ˙ + 3H (ρ + p) = 0,
(91)
(92)
where the energy density and pressure of the scalar field are
(98)
2K(X, Y, φ, ψ, ψ) ,
−[ +
¯]
where R is the scalar curvature, X is the kinetic term for the
scalar field φ, Y is the kinetic term for the fermionic field ψ and
K is some function (Lagrangian) of its arguments. In the case
of the FRW metric
2
2
ds = −dt + a
2
R
g
2
2
2
2
dx + dy + dz ,
(99)
R, X and Y have the form
a˙ 2
6 a¨
R
=
(100)
(101)
,
a2
a+
0.5φ2,
X
Y =
0
ψγ ψ ,
˙
0.5i ψγ 0ψ
(102)
˙
¯
given by
ρ = 2XKX − K, p = K.
(93)
In this case, the common system of equations for the kessence and MCG has the form
ρ = 2XpX − p,
ρ
α
.
(1 + A)ρ
+
− (W X)
2α
ρ
+
− B = 0,
2 2
2aa¨ + a˙ + a K = 0.
(104)
Now by varying the above Lagrangian (103) with respect
to the scalar field φ we obtain its equation of motion as
(96)
A
K
φ
3a˙ K φ
Kφ
˙ ˙
where W = W (φ) and
n = α(1 + A)
(103)
(95)
n(1 α)
n(1+α)
L = −2 3aa˙2 − a3K .
Variation of Lagrangian (103) with respect to a yields the
equation of motion of the scale factor
The compatibility condition for the equations (94)–(95) is
given by Myrzakulov et al. (2011)
1 α
¯
Lagrangian in the mini-superspace { } a, φ, ψ, ψ
(94)
B
p = Aρ −
−
=
¯
˙
respectively. Substituting (100)–(102) into (98) and integrating over the spatial dimensions, we are led to an effective
0.
K
(105)
˙
¨ + X+
X
− φ=
a
At least, the variation of Lagrangian (103) with respect to
X
.
(97)
α(1 A)
+
+
In Myrzakulov et al. (2011), different type solvable kessence cosmologies compatible with the MCG model are
ψ, ψ that is the corresponding Euler-Lagrangian equations
¯
for the fermionic fields give
Y
found for the different values of n.
1.5a˙ K γ 0ψ
γ 0ψ
K
0.5K γ
˙
Y
˙
+
a
+
0
ψ
iK
Y
−
0,
ψ
¯
=
(106)
324
K
Y
Astrophys Space Sci (2011) 336:315–325
ψγ
0
1.5a˙ K Y ψγ
˙
+
¯
0
0.5K Y ψγ
¯
a
+
˙
0
¯
iK
+
0.
ψ
(107)
References
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Also, we have the “zero-energy” condition given by
L aa
L
φ
φ
L
ψ
˙
˙
˙˙+
+
ψ
Lψ
¯
˙
+
0
L
˙
ψ
˙
˙
−
¯
=
which yields the constraint equation
−2 2
−
3a a
2XKX
+
˙
+
Y KY
−
K
=
0.
(109)
Collecting all derived (104)–(107) and (109) and rewriting
them using the Hubble parameter H = (ln a)t , we come to
the following closed system of equations of g-essence (for
the FRW metric case):
2
3H − ρ = 0,
H
H
2 ˙ + 3
K
φ
X
(110)
2
+
(K
¨+
0.
K Y ψ
˙
=
+
˙X
3H K )φ
Y
( HK
Y
¯+
iγ 0 K
0,
(113)
0,
(114)
ψ
−
¯
=
iK γ 0
K )ψ
+ ˙
(112)
0,
˙Y
+
( HK
53
φ=
K )ψ
Y
53
0.
ψ
˙
K
˙−
X
+
K
(111)
0,
p
Y
ψ
=
¯+
ρ˙ + 3H (ρ + p) = 0.
(115)
Here
ρ = 2XKX + Y KY − K,
(116)
p=K
are the energy density and pressure of g-essence. It is clear
that these expressions for the energy density and pressure
represent the components of the energy-momentum tensor
of g-essence:
T00 = 2XKX + Y KY − K,
T11 = T22 = T33 = −K.
(117)
Also we note that g-essence admits two particular cases (reductions): k-essence and f-essence. In fact, let ψ = 0. Then
Y = 0 and the system (110)–(115) becomes
2
3H
H
2
˙
− ρ = 0,
+ 3
φ
H2
+
p
(K
¨
KX +
˙X
=
(118)
(119)
0,
φ
K
˙
+ 3H KX )
−
φ=0
,
ρ˙ + 3H (ρ + p) = 0,
(120)
(121)
where
ρ = 2XKX − K,
p = K.
(122)
It is the system of equations of k-essence. Now let us consider the case when φ = 0 that is the purely fermionic case.
Then X = 0, K = K(Y,
ψ, ψ) and the system (110)–(115)
¯
takes the form of the equations of f-essence that is (13)–(17).
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