The new form of the equation of state for dark energy ?uid and accelerating universe
Shin’ichi Nojiri1, ? and Sergei D. Odintsov?2, ?
2 1 Department of Physics, Nagoya University, Nagoya 464-8602. Japan Instituci` Catalana de Recerca i Estudis Avan?ats (ICREA) and Institut de Ciencies de l’Espai (IEEC-CSIC), o c Campus UAB, Facultat de Ciencies, Torre C5-Par-2a pl, E-08193 Bellaterra (Barcelona), Spain
arXiv:hep-th/0606025v2 29 Jun 2006
We suggest to generalize the dark energy equation of state (EoS) by introduction the relaxation equation for pressure which is equivalent to consideration of the inhomogeneous EoS cosmic ?uid which often appears as the e?ective model from strings/brane-worlds. As another, more wide generalization we discuss the inhomogeneous EoS which contains derivatives of pressure. For several explicit examples motivated by the analogy with classical mechanics the accelerating FRW cosmology is constructed. It turns out to be the asymptotically de Sitter or oscillating universe with possible transition from deceleration to acceleration phase. The coupling of dark energy with matter in accelerating FRW universe is considered, it is shown to be consistent with constrained (or inhomogeneous) EoS.
PACS numbers: 11.25.-w, 95.36.+x, 98.80.-k
The number of attempts is aimed to the resolution of dark energy problem (for recent review, see [1, 2, 3]) which is considered as the most fundamental one in modern cosmology. Among the di?erent descriptions of late-time universe the easiest one is phenomenological approach where it is assumed that universe is ?lled with mysterious cosmic ?uid of some sort. One can mention imperfect ?uids , general equation of state (EoS) ?uid where pressure is some (power law) function of energy-density , ?uids with inhomogeneous equation of state , where EoS with time-dependent bulk viscosity is the particular case , coupled ?uids [8, 9], etc. The EoS ?uid description may be even equivalent to modi?ed gravity approach as is shown in . As it has been recently discussed in  it is not easy to construct the dark energy model which describes the universe acceleration and on the same time keep untouched the radiation/matter dominated epochs with subsequent transition from deceleration to acceleration. In order to minimize the dark energy e?ect at intermediate epoch one may speculate about sudden appearence of dark energy around the deceleration-acceleration transition point. In other words, one may suppose that EoS
of DE ?uid is of the form p = θ(t ? td )wρ where td is transition time and w is DE EoS parameter. Before transition point, DE plays a role of usual dust which changes EoS by some unknown scenario. In the similar way, one can generalize other cosmic ?uids with more complicated EoS. This introduces the idea of structure/form changing EoS in di?erent epochs. The simplest example of such cosmic ?uid is oscillating dark energy [12, 13]. Finally, the reason why cosmic ?uid still escapes of direct observations could be that it has completely unexpected properties, for instance, in EoS picture. In the present letter the new form of dark energy EoS is considered. As the ?rst step, we introduce the relaxation equation for pressure (the analog of energy conservation law for energy-density). It is then shown that such constrained EoS is equivalent to usual but inhomogeneous EoS which is known to be the e?ective description for brane-worlds or modi?ed gravity . The generalized inhomogeneous EoS which contains time derivatives of pressure is introduced. The number of examples for such EoS cosmic ?uids is presented and the corresponding FRW cosmologies are described. It is shown that cosmic speed-up in the examples under consideration corresponds to the asymptotically de Sitter or the oscillating universe where accelerating/decelerating epochs repeat with possibility to cross the phantom barrier or to make transition from deceleration to acceleration. In all cases, dark energy EoS parameter is close to ?1, being within the observational bounds to it. It is demonstrated that the inclusion of matter may be consistent with constrained EoS.
at Lab. Fundam. Study, Tomsk State Pedagogical University, Tomsk ? Electronic address: firstname.lastname@example.org ? Electronic address: email@example.com
II. CONSTRAINED EQUATION OF STATE
Let us discuss the possible modi?cation of the equation of state in such a way that it would change its structure/form during the universe evolution. Consider the balance equation for the energy (energy conservation law) ρ + 3H(ρ + P ) = 0 . ˙ It can be represented as a relaxation equation 1 ˙ Ψ = ? (Ψ ? Ψ0 ) , τ (2) (1)
water drop. At the point of phase transition, since the system becomes unstable, the pressure may be governed by a equation like (4). We prefer to measure the relaxation time τ in terms of Hubble function H, i.e., consider τ H = ξ = const. In this case it is convenient to use a new ˙ variable x ≡ a(t) . In terms of x the expression τ P
dP ˙ τ P = ξx dx
and one obtains the pair of relaxation type equations for ρ and P 1 dρ x + ρ = ?P , 3 dx dP + P = f (ρ, x) . ξx dx (6) (7)
where Ψ coincides with ρ, relaxation time τ is τ = 1 3H and stationary (or equilibrium) value of ρ is ρ0 ≡ Ψ0 = ?P . To formulate consistently this equation we need, as usual, the equation of state (EoS). The standard barotropic EoS is P = P (ρ), providing the equation for ρ only: 1 ρ = ?(ρ + P (ρ)) . ˙ 3H (3)
Extracting P from the ?rst equation and inserting it to the second one we obtain the second order, master equation for the energy density ρ x2 dρ d2 ρ +x dx2 dx 4+ 1 ξ 3 + [ρ + f (ρ, x)] = 0 ξ (8)
Let us conjecture now that cosmological ?uid is described by di?erent EoS at di?erent epochs. In other words, to describe the transition from one epoch in cosmological evolution to another we try to introduce the transition from one EoS to another, or in simplest form to modify EoS to permit the presence of pressure derivatives. The simplest way is to introduce the relaxation equation for pressure ˙ τ P + P = f (ρ, a(t)) . (4)
This is new, dynamical equation to energy-density which is compatible with energy conservation law. As the explicit example, let the function f be of the form (of course, more complicated choices may be considered) f (ρ, x) = ?ρ + γ(x)(ρ ? ρc ) , (9)
where ρc = const is some critical value of the energy density, and γ(x) = γ0 + αxm . When ρc = 0 and α = 0 we recover the standard linear EoS P = (γ0 ? 1)ρ . (10)
When τ = 0 and f (ρ, a(t)) = P (ρ) we recover the standard EoS. Such an equation may be considered as some (dynamical) constraint to usual EoS. Of course, the physical sense of such equation (unlike to energy conservation law) is not clear at the moment although some explanations are given in the next section. In daily life, however, there could occur similar phenomena where the time change of the presure depends on the density. For example, consider the water. There is a pressure in the steam, which is the gas of water. When the density increases, the molecules of water make drops of water, like fog. The pressure of the drops could be neglected. At high density, the total pressure could decrease. The equation (4) seems to express such a process. Then if dark energy consists of particles or some objects with internal structure, there may occur the phase transition like that between steam and
The equation for ρ can be reduced to the Bessel equation and the solution is of the form a a ρ = ρc + xσ C1 Jν (?xλ ) + C2 J?ν (?xλ ) . Here σ = ?2 ? m 1 ? , 2 ξ λ= m , 2 a = αm/2 . (12) ? (11)
Using (6), we ?nd P = ?ρc ? σ + 1 xσ C1 Jν (?xλ ) + C2 J?ν (?xλ ) a a 3 λ? a ? xσ+λ C1 Jν?1 (? xλ ) ? Jν+1 (?xλ ) a a 6 a a +C2 J?ν?1 (?xλ ) ? J?ν+1 (? xλ ) . (13)
3 When x is large, ρ behaves as an oscillating function ρ ? ρc + 2ν + 1 2 σ?λ/2 C1 cos axλ ? ? x π π? a 4 ?2ν + 1 +C2 cos axλ ? ? π . (14) 4 By using the conservation law (1), one can rewrite (17) in a form similar to (4): ˙ τP + P = g (ρ, a) + τ H ?3 + ?g (ρ, a) a ?a . ?g (ρ, a) (ρ + g (ρ, a)) ?ρ (18)
Since σ ? λ/2 = ?2 ? (3/4)m ? 1/ξ, if we naturally assume that m and ξ should be positive, the second term damps with oscillation. Then ρ goes to a constant ρ → ρc . On the other hand, for large x, P behaves as P ? ?ρc ?
When other contributions to the energy density can be neglected, the ?rst FRW equation looks as 3 2 H =ρ. κ2 Then Eq.(18) can be rewritten as ˙ τP + P = g (ρ, a) + τ κ + ?g (ρ, a) a . ρ 3 ?3 ?g (ρ, a) (ρ + g (ρ, a)) ?ρ (20) (19)
2ν ? 1 2 σ+λ/2 C1 cos axλ ? ? x π π? a 4 ?2ν + 1 +C2 cos axλ + ? π . (15) 4 λ? a 3
Since σ + λ/2 = ?2 ? m/4 ? 1/ξ, when m and ξ are positive, the second term damps with oscillation again and P goes to a constant P → ?ρc . Then the e?ective EoS parameter w ≡ P/ρ goes to ?1, which corresponds to a cosmological constant. The Bessel function is quasi-oscillating and we obtain an in?nite number of epochs, in which ρ, P , H and a are also quasi-oscillating. In other words we have an in?nite number of points in which the deceleration replaces the acceleration and vice-versa. The presence of ρc can guarantee that ρ is positive, thus, H 2 is also positive. Nevertheless, P can change its sign, and this phenomenon can mimic the dark energy e?ect. When ρc = 0, α = 0 the equation becomes of the Euler type, and the solution is also very simple.
III. THE RELATION WITH STANDARD EQUATION OF STATE.
By comparing (20) with (4), we may identify f (ρ, a) = g (ρ, a) + τ κ + ?g (ρ, a) a . ρ 3 ?3 ?g (ρ, a) (ρ + g (ρ, a)) ?ρ (21)
This shows the relation between standard (generally speaking, inhomogeneous EoS) and relaxation equation for pressure.
IV. GENERALIZED INHOMOGENEOUS EQUATION OF STATE
We now consider the relation with the standard EoS. Let us start with the scale factor dependent EoS: P = g (ρ, a) . Then we have d et/τ P dt 1 ˙ = P +P τ 1 ?g (ρ, a) ?g (ρ, a) = g (ρ, a) + ρ+ ˙ aH . (17) τ ?ρ ?a e?t/τ (16)
As it was indicated above, there is a possibility ˙ that the EoS contains P or even higher time derivatives of pressure. More generally, the EoS could de˙ pend on H or H (inhomogeneous EoS ) like ˙ ˙ U (ρ, P, P , H, H) = 0 . (22)
Note that many e?ective dark energy models like brane-worlds, modi?ed gravity and string compacti?cations have such a form ( for very recent example compatible with observational data, see  and references therein). As particular example, one may consider ˙ ˙ ˙ U (ρ, P, P , H, H) = P + ˙ H ? 3H H (ρ + P )+ W (ρ) . 3H (23)
4 Here W (ρ) is a proper function of the energy density ρ. Using the energy conservation law (1), one gets ρ = W (ρ) . ¨ (24) which oscillates around we? = ?1 as in . As an another example, we consider the EoS ˙ P ? 3H (ρ + P ) = U (H) . (34)
If ρ is regarded as a coordinate, Eq.(24) has a form of Newtonian equation of motion of the classical particle with the “force” W . For example, if W is a constant W = w0 , we ?nd ρ behaves as a coordinate of the massive particle in the uniform gravity: w0 2 ρ= (t ? t0 ) + c0 . (25) 2 Here t0 and c0 are constants of the integration. As an another example, we may consider the case of the harmonic oscillator: W (ρ) = ?ω 2 (ρ ? ρ0 ) . ρ = ρ0 + A sin (ωt + α) . (26)
Here U (H) is a proper function of the Hubble rate H. Then by using (1), one arrives at ˙ ρ + P = U (H) . ˙ In a simplest case, U (H) = 0, it follows ρ+P =c (c : constant) . (36) (35)
When the other contributions to the energy density and pressure are neglected, because of (30), we ?nd ˙ H is constant and H =? κ2 c t. 2 (37)
Then an oscillating energy density follows [12, 13]: (27)
If other contributions to the energy density may be neglected, by using the ?rst FRW equation (19), we ?nd the behavior of the Hubble rate, for (25), κ H= √ 3 w0 (t ? t0 )2 + c0 . 2 (28)
As an another case, we may consider U (H) = 2ω 2 H . κ2 (38)
Here ω is a constant. Then combining (35), (30), and (38), we ?nd ¨ H = ?ω 2 H , (39)
˙ ˙ As H < 0 when t < t0 , and H > 0 when t > t0 , there is a transition from non-phantom era to phantom one at t = t0 . For (27), we have oscillating H: κ H= √ 3 ρ0 + A sin (ωt + α) . (29)
which is the equation typical for the harmonic oscillator in classical mechanics. Hence, the oscillating Hubble rate is obtained H = H0 sin (ωt + α) . (40)
When we neglect the other contributions to the energy density and pressure, we also have 2 ˙ ? 2H = ρ + p . κ (30)
Combining (30) with (19), one may de?ne the e?ective EoS parameter we? by we? ≡ ?1 ? Hence, for (28) we? = ?1 ? √ 3κ t ? t0
Here H0 and α are constants of the integration. Thus, we demonstrated that inhomogeneous generalized EoS (linear in the pressure derivative) leads to the interesting accelerating (often oscillating) latetime universe.
V. THE EQUATION OF STATE QUADRATIC ON THE PRESSURE DERIVATIVE
˙ 2H . 3H 2
(t ? t0 )2 + c0
which surely crosses we? = ?1 when t = t0 . On the other hand, for (29), one gets 2Aω cos (ωt + α) we? = ?1 ? √ , 3/2 3κ (ρ0 + A sin (ωt + α)) (33)
In this section, as an immediate generalization, ˙ the case that the EoS is not linear on P but quadratic is considered. Let the equation to pressure and its derivatives looks like an energy in the classical mechanics: E= 1 ˙2 P + V (P ) . 2 (41)
5 Here E is a constant but as it is an analogue of the energy, it is denoted as E. We should note that E does not correspond to real energy in universe. This may be also considered as implicit form of EoS. First example is V (P ) = aP , ? (42) Then we have P = A sin (ωt + α) E = A2 ω 2 . (50) which goes to ?1 when t goes to in?nity. Hence, the emerging universe seems to be the asymptotically de Sitter one. Second example is V (P ) = 1 2 2 ω P . 2 (49)
with a constant a. Then by the analogy with the ? classical mechanics, we ?nd 1 P = ? at2 + v0 t + p0 , ? 2 1 2 E = v0 + ap0 . ? 2
Here v0 and p0 are constants. In case that other contributions to the total energy density are large, as in the early universe, the Hubble rate H could not be so rapidly changed. Then we may assume that the Hubble rate H could be almost constant H = H0 . Using (1), one obtains ρ = ρ0 e?3H0 t 2 2t t2 a ? ? + ? 3 2 2 27H0 9H0 3H0 p0 t 1 + . ?v0 ? 2 + 3H 9H0 3H0 0
where A and α are constants. Then in the case that other contribution to the total energy density is large, as in the early universe, the Hubble rate H could be almost constant H = H0 , we ?nd ρ = ρ0 e?3H0 t A (3H0 sin (ωt + α) ? 2 9H0 + ω 2 ?ω cos (ωt + α)) ,
with a constant ρ0 . This corresponds to de Sitter universe. On the other hand, when other contributions to the total energy density can be neglected, as in the late-time universe, by using (45), one gets (44) d2 a3/2 3 + κ2 A sin (ωt + α) a3/2 = 0 . dt2 4 By de?ning a new variable s π s ≡ ωt + α + , 2 one obtains a kind of Mathieu equation: d2 a3/2 3κ2 + cos s a3/2 , ds2 4ω 2 whose solution is given by 0=
Here ρ0 is a constant. The explicit form of (inhomogeneous) EoS may be found combining two above equations. On the other hand, we may also consider the case that the other contributions to the energy density and pressure are neglected as in late-time or future unverse. Then deleting ρ from (1) and (19), we have 3 κ2 ˙ H + H2 + P = 0 . 2 2 For (43), Eq.(45) admits the solution H = h0 t + h1 , when a = 3h0 , v0 = ?3h0 h1 , ? p0 = ?h0 ? h2 . 1 (46)
cn cos(nt) +
sn sin(nt) .
Here the coe?cients cn and sn are given by recursively solving the following equations: c1 = 0 q ?n cn + (cn?1 + cn+1 ) = 0 (n ≥ 1) , 2 2 s 1 = s , s2 = ? s , q q 2 ?n sn + (sn?1 ? sn+1 ) = 0 (n ≥ 2) , 2 3κ2 . (56) q≡ 4ω 2
c0 = c ,
For (46), the e?ective EoS parameter we? de?ned by (31) has the following form: we? = ?1 ? 2h0 3 (h0 t + h1 )
6 Hence, a has a periodicity 1/ω. In the expression (55), a is not always positive. Then physically the regions where a3/2 is not negative could be allowed and the points a = 0 could correspond to Big Bang/Big Crunch/Big Rip. We should note that the expressions of ρ0 in (44) and (51) are not always positive. Then only the period(s) where ρ0 is positive could be allowed in the real universe.
VI. A. COUPLING WITH THE MATTER
which gives, as well-known, H?
2 3(1+wm )
On the other hand if σ ? λν < ?3(1 + wm ) < 0, Hubble rate is H=κ which gives H=
2 ? σ?λν
a ? 2
No direct interaction between dark energy and matter
Let us now include the matter. For simplicity, we consider the matter with constant EoS parameter wm so that the matter energy density ρm is given by a(t) a(t0 )
By comparing (65) with (63) or (11), it follows that the e?ective EoS parameter is given by we? = ?1 ? σ ? λν . 3 (66)
ρ = ρ0
For the model in (23), solving (24), we ?nd the t-dependence of ρ. Then the FRW equation gives 3 2 H = ρ(t) + ρ0 κ2 a(t) a(t0 )
In case of (11), the total energy density is given by
a a ρtot = ρc +xσ C1 Jν (?xλ ) + C2 J?ν (?xλ ) +ρ0 x?3(1+wm ) ,For the case (25), when t is large enough, the second (58) term in the r.h.s. of (67) could be neglected and we and the Hubble rate H is given by will obtain (28). If c0 in (25) is small enough, when t ? t0 , the second term in (67) could be dominant 1 σ λ λ and we may obtain (63). For the case (29), in the H = κ ρc + x C1 Jν (? x ) + C2 J?ν (?x ) a a 3 early universe, where a is small, the second term in 1/2 (67) could be dominant and one obtains (63), again. . (59) +ρ0 x?3(1+wm ) Especially for the dust wm = 0, we ?nd H ? 2/3 , t 2 that is, a ? t 3 . In the late time universe, the ?rst In future, x becomes large, then the Hubble rate H term could be dominant and one gets (29). goes to a constant (with oscillations): Three years WMAP data are recently analyzed in Ref., which shows that the combined analysis ρ0 H →κ , (60) of WMAP with supernova Legacy survey (SNLS) 3 constrains the dark energy equation of state wDE pushing it towards the cosmological constant. The which tells we? → ?1. On the other hand, in the marginalized best ?t values of the equation of state early universe, x should be small. Hence, one ?nds parameter at 68% con?dance level are given by ν ?ν ?1.14 ≤ wDE ≤ ?0.93. In case of a prior that axλ ? axλ ? 1 ρc + xσ C1 + C2 H = κ universe is ?at, the combined data gives ?1.06 ≤ 3 2 2 wDE ≤ ?0.90. 1/2 In our models, as shown in (16), (17), (48), and +ρ0 x?3(1+wm ) . (61) (60), the e?ective EoS parameter is we? ? ?1 and there is no contradiction with the above WMAP If ?3(1 + wm) < σ ? λ, the contribution from matter data. We should also note that when mater is coubecomes dominant and Hubble rate is pled, we ?nd we? ? wm in the early universe, as in (63). Thus when wm < ?1/3, there should occur ρ0 1/2 ?3(1+wm )/2 x , (62) H=κ the transition from deceleration to acceleration. 3
B. Dark energy interacting with matter
Note that Eq.(8) is also modi?ed: it now contains the inhomogeneous terms: d2 ρ 1 3 dρ 4+ + [ρ + f (ρ, x)] +x dx2 dx ξ ξ dS(x) 3 + S(x) = 3x dx ξ d = 3x1?1/ξ x1/ξ S(x) . (78) dx x2 Let a (special) solution of (8) be ρ = ρs (x). Then in case of (9) with γ(x) = γ0 + αxm , the general solution corresponding to (11) is given by a a ρ = ρs (x) + xσ C1 Jν (?xλ ) + C2 J?ν (?xλ ) . (79) where ρc should be included in ρs (x). It is also noted that the initial conditions are relevant to determine C1 and C2 but irrelevant for ρs (x). As an example, we can ?nd ρs (x) = ρc + ρ0 xη , with constants ρ0 and η when eq(a) = eq0 η 2 ? 3η/4 + η/ξ + 3γ0 /ξ a?η aη+3(1+wm ) ρ0 0 ? ρm0 (η + 1/ξ) (η + 3(1 + wm )) + ?
?m?η m+η+3(1+wm ) (3α/ξ)a0 a (m + η + 1/ξ) (m + η + 3(1 + wm ))
Generally speaking, the matter interacts with the dark energy. In such a case, the total energy density ρtot consists of the contributions from the dark energy and the matter: ρtot = ρ + ρm . If we de?ne, however, the matter energy density ρm properly, we can also de?ne the matter pressure pm and the dark energy pressure p by pm ≡ ?ρm + ρ ˙ , 3H P ≡ Ptot ? Pm . (68)
Here Ptot is the total pressure. Hence, the matter and dark energy satisfy the energy conservation laws separately, ρ + 3H (ρ + P ) = 0 . ˙ (69) In case, however, that the EoS parameter wm for the matter is almost constant, one may write the conservation law as ρm + 3H (1 + wm ) ρm = Q , ˙ and therefore for the dark energy ρ + 3H (ρ + P ) = ?Q , ˙ (71) (70) ρm + 3H (ρm + Pm ) = 0 , ˙
so that the total energy density and the pressure satisfy the conservation law: ρtot + 3H (ρtot + Ptot ) ρm = 0 . ˙ (72)
In (70), Q expresses the shift from the constant EoS parameter case. As an example, we consider the case that Q is given by a function q = q(a) as Q = Haq (a)ρm . Combining (73) with (70), one gets ρm = ρm0 a?3(1+wm ) eq(a) . (74)
3s0 a0 a?1/ξ+3(1+wm ) . ρm0 (?1/ξ + 3(1 + wm ))
which gives S = ρ0 3 + η 2 ? 3η/4 + η/ξ + 3γ0 /ξ xη η + 1/ξ (82)
Here ρm0 is a constant of the integration. Hence, the conservation law (1) is modi?ed, through (71) as ρ + 3H (ρ + P ) = ?ρm0 Ha ˙ and (6) is also modi?ed as 1 dρ x + ρ = ?P ? S(x) . 3 dx Here S(x) ≡ ? ρm0 ?(2+3wm ) ′ (a(t0 )x) q (a(t0 )x) eq(a(t0 )x) . 3 (77) (76)
?(2+3wm ) ′
3αxm+η + s0 x?1/ξ . (m + η + 1/ξ) ξ
In (81) and (82), q0 , s0 are constants and a0 ≡ a(t0 ). In case that η+3(1+wm ), m+η+3(1+wm), ?1/ξ+3(1+wm) < 0 , (83) we ?nd eq(a) → eq0 when a becomes large, that is, in the late time universe. Thus, ρm → ρm0 eq0 a?(2+3wm ) . Furthermore if η < 0, we ?nd ρs → ρ0 , that is, H goes to a constant, which may lead to the asymptotically deSitter space. Clearly, for more complicated coupling Q, more sophisticated accelerating cosmology may be constructed.
In summary, we discussed the constrained EoS for cosmic ?uid where the relaxation equation for pressure is introduced. It is shown that such EoS is equivalent to usual inhomogeneous EoS  which contains scale factor dependent terms. Subsequently, the generalized inhomogeneous EoS with time derivatives of pressure is presented. For the number of explicit examples, the accelerating dark energy cosmology as follows from such EoS cosmic ?uid is constructed. It turns out to be the asymptotically de Sitter universe or oscillating universe with long accelerating phase and transtion from deceleration to acceleration. The consistent coupling of such constrained EoS dark ?uid with matter is discussed. It is shown that emerging FRW cosmology may be consistent with three years WMAP data. Of course, there are many ways to generalize the EoS for cosmic ?uid and to investigate the corresponding impact of such generalization to dark cosmos. The physics behind such generalization remains to be quite obscure (as dark energy itself and its sudden appearence). At best, this may be
considered as some phenomenological approximation. Nevertheless, having in mind, that most of modern attempts to understand dark energy including strings/M-theory, brane-worlds, modi?ed gravity, etc lead to e?ective description in terms of cosmic ?uid with unusual form of EoS, it turns out to be extremely powerful approach. From another side, the reconstruction of the cosmic ?uid EoS may be done for any given cosmology compatible with observational data which may ?nally select the true dark energy theory.
We are very grateful to A. Balakin for stimulating discussions and participation at the early stage of this work. The research by SDO was supported in part by LRSS project n4489.2006.02 (Russia), by RFBR grant 06-01-00609 (Russia), by project FIS2005-01181 (MEC, Spain) and by the project 2005SGR00790 (AGAUR,Catalunya, Spain) and the research by S.N. was supported in part by YITP computer facilities.
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