CC BY
33
UDC 530.12:531.551
A. V Timoshkin
DARK ENERGY wITH SPECIAL FORM INHOMOGENEOUS EQUATION OF STATE
The four-dimensional flat Friedman universe filled with ideal fluid with a nonlinear inhomogeneous equation of state depending on time is considered. The equations of motion are solved. It is shown that in some cases there appears a quasi-periodical universe. On the other hand, such dark fluid models may also describe quintessence-like cosmic acceleration. The appearance of future singularities resulting from various choices for the input parameters is discussed.
Key words: dark energy, cosmological singularity, cosmic acceleration.
1. Introduction
As is known, the present universe is subject to acceleration, which can be explained in terms of an ideal fluid (dark energy) weakly interacting with usual matter. Among the different possible models that have been considered in the literature, one that has a good probability to reflect what may be going on there, is a model in which dark energy is described by some rather complicated ideal fluid with an uncommon equation of state (EoS). Very general dark fluid models can be described by means of an inhomogeneous EoS [1-4]. However, it is worth stressing that term inhomogene-ous does not mean that we consider an inhomogeneous universe (so that the RW metric may still be retained) nor that we consider the possibility that dark energy may cluster in the nonlinear regime of perturbation. Some particular examples of such kind of equations have been considered in [5-11]. Moreover, a dark energy fluid obeying a time-depending EoS [6, 12] may also be successfully used with the purpose to mimic the classical string landscape picture [13], which is very interesting as regard establishing a connection with a different fundamental approach.
In this paper we shall investigate a specific model for a dark fluid with a non-linear EoS, which seems to be particularly natural and exhibit quite nice properties.
In this sense it can be considered as a conviniently simple paradigm for that class of models.
2. Inhomogeneous equation of state for the universe and its solutions
Let us assume that our universe can be conveniently described by a field giving rise to an ideal fluid (dark energy) that obeys a non-linear inhomogeneous EoS depending on time [14]:
p = w(f)p + /(p) + A(t), (1)
where w (t) and A (t) depend on time t, and where f(p) is an arbitrary function in general. Let us write down the law of energy conservation:
p + 3H(p + p) = 0 (2)
and the corresponding FRW equation for spatially flat FRW universe:
-H2 = p.
where p is the energy density, p is the pressure.
a
H = — is the Hubble parameter, a (t) is the scale factor
a
of the three-dimensional flat Friedman universe, X2 = 8nG with Newton’s gravitational constant G.
Taking into account (1)-(3), we obtain the gravitational equation of motion:
1
p + V3k p2 [ (w (t) +1 )p + f (p) + A(t)] = 0. (4)
First of all, we assume here, that the function f (p) = 0 and that both the function w (t) and A (t) depend linearly on time:
w( t) = a1t + b, (5)
A(t) = ct + d , (6)
where a1,b, c, d are some constants.
This kind of behaviour may be consequence of a modification of gravity (see [1, 13]).
Note that special version of our model introduced in [15] was considered recently in [16].
Let us assume, for simplicity that a1 = 0, b ^ 1. For the energy density we obtain:
p(5) =-----------------7~------------
3 X2 52 (1+b )2
1+-
2
(7)
Y
Hubble’s parameter takes the form: 2
3^(1 + b)
1+
(8)
Y
where y = -^- and E = — y2 (ct + .
3cx s 4X V '
d
(3)
If ^ —> 0 ^t ^—J, the functions H (%) and p(£,) simultaneously approach zero. For b > -1 when
p > 0 the energy density grows, and there occurs expansion of the universe. In the case when ^>^2, p <0, a < 0, there occurs compression of the universe. The derivative H > 0 if $1 <^2 and we obtain the phantom phase of the universe. When and 5>5 2 the derivative H < 0 we obtain non-phantom phase.
But if b < -1 then with 5<5 2 the energy density diminishes, p <0 and a < 0, and there occurs compression of the universe. When 5>52, P > 0, there occurs expansion of the universe.
A graphical representation of H (%) is given in Fig. 1.
Since d > 0, we have H < 0, and we get the nonphantom phase of the universe. The functions p (t) and H (t) change quasi-periodically (Fig. 2). Cosmological singularities appear also quasi-periodically. The energy density and the Hubble parameter simultaneously approach infinity. That is, we have singularity of the type Big Rip [17]. Note that such singularities can be of very different types [18].
X / d -3 X a y n(+1)3
3 V b +1 Z 2 3 n (b+1)5
(12)
Fig. 1. Hubble parameter H (^), t>1 = 0.757y3, ^2 = 1.272y
Assume now that w(t~) = a1t + b and A(t) = d . Then the energy density takes the form:
P(t ) =
z
-11% (it + b + ^
Qi\ 3
(9)
Fig. 2. Hubble parameter H = H (t)
Let us finally consider the case of the nonlinear EoS with function A (t) = 0 and parameter w (t) to be linearly dependent on time (5); while for the function f (p) we choose it to be f (p) = Ayjp , where A is a parameter.
The solution of (4) looks like
and the Hubble parameter becomes
Id (t + b + ^2
P(t ) =
1
H (t ) = * . rd
3 ,J(t + b +1)
Ce
Axt
ax
2 A
(13)
-^a^ + b + l)2
(10)
and the Hubble parameter is H (t) = - X
where Zv = CjJv + C2Yv, i.e. the general solution of
Bessel’s equation, C1, C2 are arbitrary constants, and
b +1 t >--------.
The derivative equals
d X2
^ Axt
Ce 2 X
(14)
6
f ] 2
1 + 3
Z 2
V 3 )
(11)
where C is an integration constant. We shall investigate, for further simplicity, the case when a1 = 0. Then the EoS acquires the form
P(t) = bP(t) + A/p . (15)
The Hubble parameter is H ()= ?, — ■ (16)
AX't b + 1
A
2
c
a
a
2
e
and the derivatives of H(t) and p (t) correspondingly become
H (1 ) = •
^3 A ^A%t
—AXe
tAx1 b +1
Y
(17)
A
3^3
P(t ) = ~
A 'yA%t
— e 2
X______
(18)
, 2
A
The scale factor is given by the expression
Ax-t b +1
A
Sx{b+i)
The derivative of scale factor is
i(t ) = ,
A 4Ax-t b+1V3^1)
A
2
-ln
(19)
(20) b +1
and we see that a(t) = 0 for t = tx =- r ,
\3%A A
while the second derivative of scale factor is
A
o -----1
b+1
a e
S — Aft ^
1 Axe 2
(21)
2 2
and we also have a (t) = 0 for t = t2 = ^— ln^—•
. y/3%A yj3%A
The simultaneous divergence of p (t) and H (t) appears at t1. However, the scale factor a (t1) = 0, which
is finite. That is, we have singularity of the type III.
In the case t > t, a simple study of the Hubble parameter sigh shows (H > 0) that one has an expanding universe and the contrast (H < 0) (the contracting universe) when t < t1. If A > 0 and b > -1 the expanding universe is always in the non-phantom phase (H < 0).
Provided A > 0 and b > -1 for values of t < t2 it turns out that the first and second derivatives of the scale factor are both positive (i.e., the universe expands with acceleration), while for t > t2 the first derivative is positive but the second derivative is negative (in this case it is still expanding but the expansion is decelerated). This is a very interesting transition.
3. Summary
In this work we have studied a model of the universe in which there is a nonlinear inhomogeneous EoS with a linear dependence on time. The consequences of various choices of parameters in the linear functions are examined. The presence of a linear inho-mogeneous term in the linear EoS leads either to compression of the universe in the evolution process or to quasi-periodical change in the energy density and in the Hubble parameter, and also to the quasi-periodical appearance of singularities. In the absence inhomoge-neous term in the nonlinear EoS it is possible to have a description of the evolutionary transition according to the values of the scale factor a (t) and its first and second derivatives, which characterize different types of accelerating and decelerating expansions of the universe. Adding a cosmological constant-like term to the EoS led us to the conclusion that a future singularity forms, in the sense that the energy density and the Hubble parameter simultaneously approach infinity. We have considered a pure dark energy model.
Acknowledgements. We thank professor Sergey Odintsov for very valuable discussions.
References
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Tomsk State Pedagogical University.
Ul. Kievskaya, 60, Tomsk, Russia, 634061.
E-mail: timoshkinAV@tspu.edu.ru
Received 14.03.2011.
А. В. Тимошкин
ТЕМНАЯ эНЕРГИЯ СО СПЕЦИАЛЬНОЙ ФОРМОЙ НЕОДНОРОДНОГО УРАВНЕНИЯ СОСТОЯНИЯ
Рассматривается пространственно-плоская Вселенная Фридмана, заполненная идеальной жидкостью с нелинейным неоднородным уравнением состояния. Получены решения гравитационного уравнения движения. Показано, что в некоторых случаях возникает квазипериодическая Вселенная. С другой стороны, такая модель темной жидкости может также описывать космическое расширение с ускорением (квинтэссенцию). Обсуждается возможность появления будущих сингулярностей в зависимости от выбора ряда значений параметров.
Ключевые слова: темная энергия, космологическая сингулярность, космическое ускорение.
Тимошкин А. В., кандидат физико-математических наук.
Томский государственный педагогический университет.
Ул. Киевская, 6О, Томск, Россия, 634О61.
E-mail: timoshkinAV@tspu.edu.ru
