ω
1. In
w only varies below −1. The
Received 21 November 2005, Revised 23 January 2006 Supported by National Natural Science Foundation of China (90403023, 10375087) * 1) E-mail: liyh1218@
Key words scalar field, dark energy, vector field, equation of state
1
Introduction
The recent observations
[1—3]
parameter w for the general k-essence models[8] also fails to cross w = −1. suggest that the Uni-
[5] [4]
2
The evolution equation of universe in new model
Recently some dynamical dark energy models
with the property of w crossing −1 have been proposed
(Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China) (Graduate University of Chinese Academy of Sciences, Beijing 100049, China)
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Abstract We propose a dark energy model with single dynamical scalar field and fixed background vector field, in which the parameter w can cross −1 during the evolution of the universe. It is found that in certain cases w can cross −1 and transition from decelerating to accelerating occur at z ≈ 0.2 and z ≈ 1.7 respectively, which is consistent with the observations.
30 2006
6 6 HIGH ENERGY PHYSICS AND NUCLEAR PHYSICS
Vol. 30, No. 6 Jun., 2006
Scalar-Field Model for Dark Energy with a Fixed Background Vector Field *
LI Yan-Heng1)
verse consists of dark energy (73%), dark matter (23%) and baryon matter (4%). What the dark energy becomes one of the essential questions in theoretical physics and cosmology. Although the simplest form of dark energy — the cosmological constant — fits the observation is evidence
eff LT = LEH + Lm + Lφ , [3]
only interested in the late stage of the evolution of the universe, all matter can be regarded as dust, whose
3 energy density satisfies ρ = ρ0 a3 0 /a . Then, the Fried-
where Aµ is the fixed background covariant vector. In Ref. [14], it has been shown that due to the existence of a fixed background vector field, the parameter w of dark energy for a flat universe may cross w = −1 in
mann equation reads H2 + (2) Let ϕ = b0 = h= 8πG φ, 3 8πG A0 , 3 H0 H . H0 χ= ˙ 8πG φ , 3 H0 8πGV (φ) , 2 3 H0 (12) k 8πG 1 ˙ 2 ˙ ρ0 a3 0 = φ − φA + v ( φ ) + . (11) 0 a2 3 2 a3
[9—14]
quite well, which has the ef-
. In the present letter, we shall further
[14]
fective equation of state p = −ρ, or w = −1, there to show that dark energy might evolve from ω > −1 in the past to ω < −1 today. It seems that the dynamics model with a varying w is preferable rather than the simple constant. Various models have been proposed in order to describe the evolving dark energy. Most of them are characterized by a scalar field, which grew in part out of the inflation scenario in part from ideas from particle physics. Unfortunately, most of these dynamical dark energy models cannot explain the fact w crossing −1. For example, w in the quintessence models is always evolving in the range of −1 the phantom models
522 — 525
6
523
the evolution of the universe. Since the cosmological observations prefer that the universe is closed , we will study the model in a closed universe. The present letter is to focus on the problem. Consider the Lagrangian
2
v (ϕ) =
(3)
Eq. (10) and Eq. (11) can be simplified as dimensionless ordinary differential equations dϕ χ = − , dz h(1 + z ) 3(χ − b0 ) v ′ (ϕ) dχ = + , dz 1+z h(1 + z ) and h2 = 1 2 χ − b0 χ + v (ϕ) + 2 k (1 + z )2 ΩM0 (1 + z )3 − 2 2 . a 0 H0 χ2 − 2χb0 − 2v (ϕ) . χ2 − 2χb0 + 2v (ϕ) (13) (14)
[7] [6]
study the scalar-field model for dark energy with a fixed background vector field . In this model the dark energy is effectively described by a dynamical scalar field coupled to a fixed background vector field with a constant norm and constant zeroth-component in a comoving system, which describes some unknown effects of the universe. Namely, the dark energy is supposed to be governed by the Lagrangian, 1 eff Lφ = g µν φ,µ φ,ν − g µν φ,µ Aν − V (φ). 2 (1)
(15)
In terms of new variables, Eq. (7) becomes w= (16)
For a closed universe the present values which serve as the initial values of integration are taken as ω0 = −1.33, ΩM0 = 0.275, 1 ΩDE0 := ( χ2 − b0 χ + v )|z=0 = 0.745, 2 (χ2 − 2b0 χ)|z=0 = −0.246, v |z=0 = 0.868 concordant with the observation data
where LEH and Lm are the Lagrangian for gravity and for matter, including bayonic matter and dark matter. Suppose the universe is homogenous and isotropic, namely satisfying the cosmological principle as usual, so that the Robertson-Walker metric ds2 = dt2 − a2 (t) can be used. The stress-energy tensor for the homogenous and isotropic scalar field is ˙ 2 δ 0 δ 0 − 2φA ˙ 0 δ0 δ0 − Tµν = φ µ ν µ ν 1 ˙ 2 − 2φA ˙ 0 − 2V (φ)). gµν (φ (4) 2 It is equivalent to perfect fluid with energy density 1 ˙2 ˙ ρφ = φ − φA0 + V (φ) (5) 2 and pressure 1 ˙2 ˙ pφ = φ − φA0 − V (φ), (6) 2 respectively. The effective equation of state for the dark energy is p = wρ with ˙ 2 − 2φA ˙ 0 − 2V (φ) φ w= . 2 ˙ ˙ 0 + 2V (φ) φ − 2φA Remember that the energy density ρφ seen that for A0 > 0, −1, φ ˙ 2A0 or φ ˙ w < −1, 2A > φ ˙ > 0, 0 for A0 < 0, −1, φ ˙ 0 or φ ˙ w < −1, 0 > φ ˙ > 2A0 . (7) 0. It can be dr 2 + r2 dΩ2 1 − kr2