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Reconstruction of $5D$ Cosmological Models From Equation of State of Dark Energy


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IJMPD

International Journal of Modern Physics D c World Scienti?c Publishing Company

arXiv:astro-ph/0510673v1 24 Oct 2005

Reconstruction of 5D Cosmological Models From Equation of State of Dark Energy

Lixin Xu? Hongya Liu? Chengwu Zhang , , Department of Physics, Dalian University of Technology, Dalian, 116024, P. R. China

Received Day Month Year Revised Day Month Year We consider a class of ?ve-dimensional cosmological solutions which contains two arbitrary function ?(t) and ν(t). We found that the arbitrary function ?(t) contained in the solutions can be rewritten in terms of the redshift z as a new arbitrary function f (z). We further showed that this new arbitrary function f (z) could be solved out for four known parameterized equations of state of dark energy. Then the 5D models can be reconstructed and the evolution of the density and deceleration parameters of the universe can be determined. Keywords: Kaluza-Klein theory; cosmology

1. Introduction Recent observations of high redshift Type Ia supernovae reveal that our universe is undergoing an accelerated expansion rather than decelerated expansion 1,2,3 . Meanwhile, the discovery of Cosmic Microwave Background (CMB) anisotropy on degree scales together with the galaxy redshift surveys indicate ?total ? 1 4 and ?m ? 1/ 3. All these results strongly suggest that the universe is permeated smoothly by ’dark energy’, which violates the strong energy condition with negative pressure and causes the expansion rate of the universe accelerating. The dark energy and accelerating universe have been discussed extensively from di?erent points of view 5,6,7 . In principle, a natural candidate for dark energy could be a small cosmological constant. However, there exist serious theoretical problems: ?ne tuning and coincidence problems. To overcome the coincidence problem, some self-interact scaler ?elds φ with an equation of state (EOS) wφ = pφ /ρφ were introduced dubbed quintessence, where wφ is time varying and negative. Generally, the potentials of the scalar ?eld should be determined from the underlying physical theory, such as Supergravity, Superstring/M-theory etc.. However, from the phenomenal level, one can also design many kinds of potentials to solve the concrete problems 5,8 . Once
? lxxu@dl.cn ? Corresponding

author: hyliu@dlut.edu.cn 1

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the potentials are given, EOS wφ of dark energy can be found. On the other hand, the potential can also be reconstructed from a given EOS wφ 9 . That is, the forms of the scalar potential can be determined from observational data. Although there are many kinds of models for dark energy, one still knew little about it’s properties. And, one also needs some mechanism to distinguish these di?erent models. Therefore, one may wish to use the model independent method to study the universe without specifying a particular model for dark energy. That is, we can use observational data to parameterize the EOS of dark energy, and then to study the evolution of the universe directly. The idea that our world may have more than four dimensions is due to Kaluza 10 , who uni?ed Einstein’s theory of General Relativity with Maxwell’s theory of Electromagnetism in a 5D manifold. In 1926, Klein reconsidered Kaluza’s idea and treated the extra dimension as a compact small circle topologically 11 . Afterwards, the Kaluza-Klein idea has been studied extensively from di?erent points of view. Among them, a kind of theory called Space-Time-Matter (STM) theory, is designed to incorporate the geometry and matter by Wesson and his collaborators (for review, please see 12 and references therein). In STM theory, our 4D world is a hypersurface embedded in a 5D Ricci ?at (RAB = 0) manifold, and all the matter in our 4D world are induced from the extra dimension. This theory is supported by Campbell’s theorem 13 which says that any analytical solution of Einstein ?eld equation of N dimensions can be locally embedded in a Ricci-?at manifold of (N + 1) dimensions. Since the matter are induced from the extra dimension, this theory is also called induced matter theory. Within the framework of STM theory, a cosmological solution is presented in 14 in which it was shown that the universe is characterized by having a big bounce instead of a big bang. It was also shown that both the radiation and matter dominated cosmological models could be recovered from the solution. Further studies of this solution include the embedding to brane models 15 , the isometry with 5D black holes 16 , the big bounce singularity 17 , and the dark energy models 18,20 . The purpose of this paper is to study the acceleration of the 5D solution. The solution contains two arbitrary functions ?(t) and ν(t). We will show in Section 2 that one of these two arbitrary functions, ?(t), plays a similar role as the potential V (φ) in quintessence or phantom dark energy models. This enable us to study the evolution of the 5D universe in a model independent way. We will reconstruct the evolution of the 5D universe by using four known parameterized methods. Section 3 is a short discussion.

2. Dark energy in a class of ?ve-dimensional cosmological model The 5D cosmological solution was originally given by Liu and Mashhoon in 1995 . Then, in 2001, Liu and Wesson 14 restudied the solution and showed that it describes a cosmological model with a big bounce as opposed to a big bang. The
19

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5D metric of this solution reads dS 2 = B 2 dt2 ? A2 where d?2 ≡ dθ2 + sin2 θdφ2 and A2 = ?2 + k y 2 + 2νy + B= ˙ A 1 ?A ≡ . ? ?t ? ν2 + K , ?2 + k (2) dr2 + r2 d?2 1 ? kr2 ? dy 2 (1)

Here ? = ?(t) and ν = ν(t) are two arbitrary functions of t, k is the 3D curvature index (k = ±1, 0), and K is a constant. This solution satis?es the 5D vacuum equation RAB = 0. So, the three invariants are I1 ≡ R = 0, I2 ≡ RAB RAB = 0, 72K 2 . (3) I3 = RABCD RABCD = A8 The invariant I3 in Eq. (3) shows that K determines the curvature of the 5D manifold. Using the 4D part of the 5D metric (1) to calculate the 4D Einstein tensor, one obtains 3 ?2 + k (4) 0 G0 = , A2 2?? ?2 + k ˙ (4) 1 . (4) + G1 = (4) G2 =(4) G3 = 2 3 ˙ A2 AA In Ref. 20 , the induced matter was set to contain three components: dark matter, radiation and x-matter. In this paper, we assume, for simplicity, the induced matter to contain two parts: cold dark matter (CDM) ρcd and dark energy (DE) ρde . So, we have 3 ?2 + k = ρcd + ρde , A2 2?? ?2 + k ˙ = ? (pcd + pde ) , (5) + ˙ A2 AA where pcd = 0, pde = wde ρde . (6)

From Eqs.(5) and (6), one obtains the EOS of the dark energy wde = ˙ 2 ??/ AA + ?2 + k A2 ˙ pde =? , 2 + k)/ A2 ? ρ ?3 ρde 3 (? cd0 A (7)

and the dimensionless density parameters ρcd0 ρcd = , ?cd = ρcd + ρde 3 (?2 + k) A ?de = 1 ? ?cd .

(8) (9)

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where ρcd0 = ρcd0 A3 (?cd0 and A0 denote the current density of CDM and scale ? 0 ρ factor at present time, respectively (The subscript 0 denotes value at present time), and ?cd and ?de are dimensionless density parameters of CDM and DE, respectively. The Hubble parameter and deceleration parameter should be given as 14 , 20 , ˙ ? A = AB A d2 A q (t, y) ≡ ?A 2 dτ H≡ (10) dA dτ
2

=?

A? ˙ , ˙ ?A

(11)

from which we see that ? /? > 0 represents an accelerating universe, ? /? < 0 rep˙ ˙ resents a decelerating universe. So the function ?(t) plays a crucial role in de?ning the properties of the universe at late time. In this paper, we consider the spatially ?at k = 0 cosmological model. From equations (7)-(11), it is easy to see that these equations do not contain the second arbitrary ν(t) explicitly. So if we use the relation A0 /A = 1 + z (12)

and de?ne ?2 ?2 = f (z) with f (0) ≡ 1, then these equations (7)-(11) can be z 0 expressed in terms of redshift z as wde = ? ?cd
2

1 + (1 + z) d ln f (z) /dz , 3 ? 3?cd = ?cd0 (1 + z) f (z) ,
2 H0 (1 2 ?1

(13) (14) (15) (16) (17)

?de = 1 ? ?cd , H = + z) f (z) , 1 + 3?de wde (1 + z) d ln f (z) q= =? . 2 2 dz

Note that in the 5D bounce model the scale factor A reaches a nonzero minimum at t = tb where tb is the bouncing time. From Eq. (12), this tb corresponds to a maximum redshift zb . Therefore, the relation (12) and all the equations after it only valid in the range z < zb (i.e., after the bounce). Now let us consider equation (13) which is a ?rst order ordinary di?erential equation of the function f (z) w.r.t. redshift z. This equation could be integrated if the form of wde (z) is given. It was shown in 9 that the scalar potentials can be constructed from a given EOS of dark energy wφ . Following this spirit, we can also reconstruct the forms of function f (z) from a given concrete form of wde (z). And once the function f (z) is constructed, the evolution of the universe can be determined. At this point, we say that the cosmological models are reconstructed. Following Ref.9 , we consider the following four cases to reconstruct the forms of f (z).

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Case I: wde = w0 (Ref. 21 ) For this case, wde is a constant and we ?nd that Eq. (13) can be integrated, giving 1 . (18) f (z) = (1 + z) (?cd0 + ?de0 (1 + z)3w0 ) Then using this f (z) in (14), (15) and (17), we obtain ?cd , ?de and ?q expressed in terms of z alone. The evolutions of the dimensionless energy density parameters ?cd and ?de , EOS of dark energy ωde , and the deceleration parameter q are plotted in Fig. (1).

1
cd

0.5 0 -0.5 -1 -1.5 wde

q
de

wde w0 2.5 5 7.5 10 z 12.5 15 17.5 20

Fig. 1. Case I: The evolution of the dimensionless density parameters ?cd , ?de , and deceleration parameter q, EOS of dark energy wde versus redshift z, where ?cd0 = 0.25, ?de0 = 0.75, and ω0 = ?1.14.

Case II: wde = w0 + w1 z (Ref. giving f (z) =

22

) For this case Eq. (13) can also be integrated,

?cd0 (1 + z)3w1

(1 + z)3w1 ?1 . + ?de0 (1 + z)3w0 exp(3w1 z)

(19)

Using this f (z) in (14), (15) and (17), we obtain the expressions of the evolutions of the dimensionless energy density parameters ?cd and ?de , EOS of dark energy ωde , and the deceleration parameter q, and we plot them in Fig. (2). z Case III: wde = w0 + w1 1+z (Ref. 23,24 ) Similar as in case I and case II, for this case we have 1 . (20) f (z) = (1 + z) ?cd0 + ?de0 (1 + z)3w0 +3w1 exp(? 3w1 z ) 1+z

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1
cd

0.5 0 -0.5 -1 -1.5 wde

q
de

wde w0 w1 z 2.5 5 7.5 10 z 12.5 15 17.5 20

Fig. 2. Case II: The evolution of the dimensionless density parameters ?cd , ?de , and deceleration parameter q, EOS of dark energy wde versus redshift z, where ?cd0 = 0.25, ?de0 = 0.75, ω0 = ?1.14 and ω1 = 0.03.

We plot the evolutions of the dimensionless energy density parameters ?cd and ?de , EOS of dark energy ωde and deceleration parameter q in Fig. (3). Case IV: wde = w0 + w1 ln(1 + z) (Ref. 25 ) For this case f (z) is also integrable and we ?nd f (z) = 1 (1 + z) ?cd0 + ?de0 (1 + z)3w0 exp( 3w1 ln(1+z) ) 2
2

.

(21)

We plot the evolutions of the dimensionless energy density parameters ?cd and ?de , EOS of dark energy ωde and deceleration parameter q in Fig. (4). We see that in Case I the EOS of dark energy wde is assumed to be a constant w0 during the whole evolution of the universe. In the other three cases, wde deviates from w0 in di?erent ways as the redshift z increases. This causes the density and deceleration parameters ?cd , ?de and q deviate from those in Case I explicitly at higher redshift. It is expected that these deviations may become very large at very large redshift. In further studies we are going to use more observational dada such as those from the SNe Ia data to constrain the parameters in the expression of wde . Here, in this paper, we just want to use above four cases to illustrate how to reconstruct function f (z) from a given EOS of dark energy, and the results show that our procedure works.

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1
cd

0.5 0 -0.5 -1 -1.5 wde

q
de

wde w0 w1 2.5 5 7.5 10 z 12.5 15

z 1 z 20

17.5

Fig. 3. Case III: The evolution of the dimensionless density parameters ?cd , ?de , and deceleration parameter q, EOS of dark energy wde versus redshift z, where ?cd0 = 0.25, ?de0 = 0.75, ω0 = ?1.14 and ω1 = 1.1.

3. Discussion The 5D cosmological solution presented by Liu, Mashhoon and Wesson in 19 and14 is rich in mathematics because it contains two arbitrary functions ?(t) and ν(t). However, this also brings us a problem: how to determine the two arbitrary functions. In this paper we ?nd that one of the two functions, ?(t), plays a similar role as the potential V (φ) in the quintessence and phantom dark energy models. Meanwhile, another arbitrary function ν(t) seems do not a?ect the densities and the EOS of dark energy in an explicit way. This reminds us of a similar situation happened in the 4D general relativity where one can study the cosmic evolution of dark energy (as well as other densities) just from a given parameterized EOS of dark energy without knowing it’s explicit form and without knowing the explicit form of the scale factor a(t). Following this kind of model independent methods we have used the relation (12) and successfully derived the di?erential equation (13) which governs the function f (z). Furthermore, for four known EOS of dark energy, we have successfully integrated Eq. (13) and plotted the evolutions of the density and deceleration parameters ?cd , ?de and q. In this sense, we say that we have reconstructed the 5D solution. However, we should also say that this kind of reconstruction is not complete. As we mentioned in Section 2 that the relation A(z) = A0 (1 + z)?1 in (12) does not cover the whole 5D manifold of the bounce solution; it can only cover ”half” of it, i.e., that half after the bounce for z < zb .

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1
cd

0.5 0 -0.5 -1 -1.5

q
de

wde

wde w0 w1 ln 1 z 2.5 5 7.5 10 z 12.5 15 17.5 20

Fig. 4. Case IV: The evolution of the dimensionless density parameters ?cd , ?de , and deceleration parameter q, EOS of dark energy wde versus redshift z, where ?cd0 = 0.25, ?de0 = 0.75, ω0 = ?1.14 and ω1 = 0.3.

There is actually a one to two correspondence between A and t for the whole curve of A in the bounce model; while the relation A(z) = A0 (1 + z)?1 is just a one to one correspondence between A and z. On the other hand, even if in the 4D general relativity, the scale factor a(t) is not easy to be integrated out analytically if the cosmic matter contains more than one di?erent components of matter. In our 5D case, the scale factor A(t, y) might be more complicated than a(t). To study the evolution of the whole bounce model, one may need look for other method such as the numerical simulation which beyond the scope of this paper. 4. Acknowledgments This work was supported by NSF (10273004) and NBRP (2003CB716300) of P. R. China. References
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