A class of transient acceleration models consistent with Big Bang ...

arXiv:1312.3962v1 [astro-ph.CO] 13 Dec 2013

A class of transient acceleration models consistent with Big Bang Cosmology Tianlong Zu ∗, Jiewen Chen †, Yang Zhang ‡ Key Laboratory for Researches in Galaxies and Cosmology, Department of Astronomy, University of Science and Technology of China, Hefei, Anhui, 230026, China

Abstract Is it possible that the current cosmic accelerating expansion will turn into a decelerating one? Can this transition be realized by some viable theoretical model that is consistent with the standard Big Bang cosmology? We study a class of phenomenological models of a transient acceleration, based on a dynamical dark energy with a very general form of equation of state pde = αρde − βρm de . It mimics the cosmological constant ρde → const for small scale factor a, and behaves as a barotropic gas with ρde → a−3(α+1) with α ≥ 0 for large a. The cosmic evolution of four models in the class has been examined in details, and all yields a smooth transient acceleration. Depending on the specific model, the future universe may be dominated either by the dark energy or by the matter. In two models, the dynamical dark energy can be explicitly realized by a scalar field with an analytical potential V (φ). Moreover, the statistical analysis shows that the models can be as robust as ΛCDM in confronting the observational data of SN Ia, CMB, and BAO. As improvements over the previous studies, our models overcome the over-abundance problem of dark energy during early eras, and satisfy the constraints on the dark energy from WMAP observations of CMB.

Key words: cosmology, dark energy, transient acceleration. 1.Introduction Cosmological observations, such as the SN Ia [1, 2] and CMB anisotropies [3] , have indicated that the universe is now in an accelerating expansion. Interpreting in the framework of general relativity, the acceleration can be attributed to the existence of some dark energy, which currently dominates the total cosmic energy in the Universe. There have been a number of candidates for the dark energy, such as the cosmological constant Λ, various scalar field dynamical dark energy models, [4, 5, 6, 7, 8, 9], Chaplygin gas model [10], Quantum Yang-Mills condensate models [11, 12, 13], etc. So far, there is no observational evidences whether the current acceleration is eternal or transient. In an eternally accelerating universe, there is an event horizon, and the S-matrix in the string theory will be ill-defined [14, 15, 16]. Recently, the analysis of combined data of SN Ia+BAO+CMB [17] seems to indicate that the acceleration may be slowing down. More data are needed for the issue [18]. Thus the possibility of return to decelerating expansion in the future has been explored. There have been various models based on different possible ∗

[email protected] [email protected] ‡ [email protected] †

1

mechanisms [19, 20, 21, 22, 23, 24, 25, 26]. In particular, a scalar field with an exponential potential as dark energy in Ref.[21], and two coupled scalar fields as dark energy in [27] are studied respectively, and, for certain range of parameters, a transient acceleration occurs in these scalar models. Refs.[28, 29, 30] consider possible interaction between matter and dark energy that can lead to a transient accelerating expansion. A coupling between Chaplygin gas and scalar field is studied in [31]. Based certain ansatzs on the dark energy density to achieve a return to deceleration, some scalar field models are proposed, which have a potential of exponential type with a quadratic dependence on the scalar field [32, 33, 34]. However, as has been checked [35], when extended back to the earlier stages, these exponential scalar dark energy would be dominant over the matter component, jeopardizing the standard Big Bang cosmological scenario. To handle this over-abundance problem within the whole class of scalar field models, one might directly design some special form of scalar potential and go ahead by trial and error to see if it works. This is essentially the method in Refs.[32, 33, 34] which has not worked for their chosen form potential. Moreover, there are infinite number of possible forms of scalar potentials, and it is not practical to try each of them. Viewing this, we adopt a parameterization approach instead. That is, we take some simple form of dynamic dark energy density ρde (t) which is subdominant to the matter density ρm (t) during the early stage of the cosmic expansion. If this works, it will automatically overcome the over-abundance problem and provide a possible model of transient acceleration. In this paper, we will specifically work with those ρde , which mimics the cosmological constant for a small scale factor a, and behaves, for large a, like a barotropic gas with ρde → a−3(α+1) with α ≥ 0. In certain cases, an explicit expression of scalar potential V (φ) is obtained analytically. Depending on specific models in the class, the future universe may be dominated either by dark energy or by the matter. The interesting characteristic of our simple scalar models is that the dark energy will always be subdominant to the matter when extended to earlier stages, allowing for a transient acceleration within the framework of the standard Big Bang cosmology. 2. Models The spacetime background is the homogeneous and isotropic flat FRW metric: ds2 = −dt2 + a2 (t)(dx2 + dy 2 + dz 2 ).

(1)

We will set a = 1 as the current value. The dynamical expansion of spacetime is determined by the Friedmann equation: 8πG a˙ (ρde + ρm ), (2) ( )2 = a 3 a ¨ 4πG =− (ρm + ρde + 3pde ), (3) a 3 where ρde is the dark energy density to be discussed in the following, ρm = Ωm ρc a−3 is the matter density, and Ωm + Ωde = 1. First, we consider a dynamical dark energy density as a function of the scale factor a: ρde (t) = Ωde ρc

1+r , 1 + ra(t)3

(4)

where Ωde is the current dark energy fraction, ρc is the critical density, and r is a dimensionless parameter of the model. It behaves as ρde ∝ Ωde ρc (1 + r) for a3 ≪ 1/r, and ρde ∝ a−3 for 2

a3 ≫ 1/r, similar to that of the matter component. But we do not regard ρde as the matter. For simplicity, we do not include the coupling between the dark energy and the matter. By the equation of energy conservation a dρdade + 3(ρde + pde ) = 0, the pressure of the dark energy is given by ρ2de (t) (1 + r) . (5) = − pde (t) = −Ωde ρc (1 + ra(t)3 )2 Ωde ρc (1 + r) The equation of state of dark energy is wde = pde /ρde = − Ωde ρρcde(1+r) . Eq.(5) turns out to be similar to a model p ∝ −ρ−α , α < −1 in Ref.[36], but is different from the generalized Chaplygin gas model p ∝ −ρ−α , α ≥ −1 [10]. From Eq.(2) and (3) follows the deceleration parameter explicitly q=−

3Ωde (1 + r)a3 a ¨a 1 − . = (a) ˙ 2 2 2(1 + ra3 )[Ωm + (r + Ωde )a3 ]

(6)

We plot q as a function of a in Fig. 1. Here Ωde = 0.74 and Ωm = 0.26 are taken for concreteness. It is seen that q → 12 as its asymptotic values in both the limits a → 0 and a → ∞. Therefore, this model predicts a decelerating expansion for both the past and for the future, and, in the interval a ∼ (0.5, 5), q < 0, the current acceleration is transient, as shown in Fig 1 for various 3r < Ωde , the acceleration is values of r. In fact, for a finite value of r > 0 and a constraint 4r+1 transient and is always followed by a deceleration expansion. By checking through calculation, a larger Ωde yields an earlier entry into the current acceleration, and a larger value of parameter r yields an earlier return to deceleration. In the limiting case r = 0, the model reduces to the ΛCDM model. On the other hand, for r 6= 0, the model predicts a dark energy fraction ∼ Ωde (1+r) for the early stage. However, based on the observations of CMB, WMAP7 has given an error band ±4% for ΩΛ for the ΛCDM model [3]. To be consistent with the observations, we impose an upper limit r ≃ 0.04 for our model. That is, for the parameter r ∼ (0, 0.04), our model is within the constraint from the the observations by WMAP. In Fig. 2 we plot the ratio ρde /ρm as a function of a. At early times when a ≪ 1, ρde /ρm → 0, the dark energy is comparatively small and the universe is matter dominated, as desired. In the remote future when a ≫ 1, the ratio ρde /ρm → (1 + r)ΩΛ /rΩm > 1, asymptotically approaching a constant. Thus the universe in the future is dominated by ρde , which is decreasing as ∝ a(t)−3 , and a(t) ∝ t2/3 , expanding like the matter-dominated. The dark energy in this model can also be explicitly realized by a scalar field φ. For simplicity, we do not consider the effects of matter component [37]. We start with the Lagrangian of the scalar field: 1 (7) L = φ˙ 2 − V (φ), 2 where V (φ) is the potential to be determined. The energy density and the pressure are ρφ = dρφ 1 ˙2 1 ˙2 2 φ + V (φ) and pφ = 2 φ − V (φ), and satisfy the conservation equation a da + 3(ρφ + pφ ) = 0, which can be written as the scalar field evolution equation: 1 dρφ . φ˙ 2 = − a 3 da

(8)

Now require that ρφ behaves as ρde in Eq(4): ρφ = Ωde ρc

3

1+r . 1 + ra3

(9)

140

0.6

0.4

120

0.2

100 m

-0.2

80

r=0.0215

/

r=0.0215

de

q(a)

0.0

r=0.0175

-0.4

60 40

r=0.0175 -0.6

20

-0.8

0

-1.0 0

1

2

3

4

5

6

0

1

2

3

4

5

Scale factor a

Scale factor a

Figure 2: The ratio ρde /ρm in the model of Eq.(5). ρde will still dominate over ρm in future.

Figure 1: The deceleration parameter q in Eq.(6).

Using Eq(2) without the matter, Eq.(8) can be written as r r 8πG dφ ra =± . 3 da 1 + ra3

(10)

By integrating, one obtains √

√ √ ra3 + 1 + ra3 . 6πG(φ − φ0 ) = ± ln p 3 p ra0 + 1 + ra30

(11)

where a0 and φ0 are constants. For simplicity, we set φ0 = 0, a0 = 0. Then Eq(11) shrinks to: i h√ p √ (12) 6πGφ = ± ln ra3 + 1 + ra3 . Taking the + sign yileds

6

√ √ √ 2 ra3 = e 6πGφ − e− 6πGφ .

(13)

  2 + ra3 1 V = Ωde ρc (1 + r) . 2 (1 + ra3 )2

(14)

By V = (ρφ − pφ )/2, one has

Substituting Eq(13) into Eq(14), we finally obtain V in terms of the field φ: " # 4 1 √ √ √ + √ V (φ) = 2Ωde ρc (1 + r) . (e 6πGφ + e− 6πGφ )2 (e 6πGφ + e− 6πGφ )4

(15)

Note that V (φ) in Eq.(15) is proportional to the factor Ωde ρc (1 + r). In Fig. 3 the re-scaled potential V (φ)/Ωde ρc (1 + r) is plotted. It is worthy noticing that the same V (φ) holds if “ − ” sign is taken in Eq.(12). Indeed, the resulting Lagrangian L is symmetric under φ → −φ. √ √ 6πGφ −2 , a simple exponential function When 6πGφ ≫ 1 the potential reduces to V (φ) ∝ e of φ, which is similar to what Ratra and Peebles [4] used for dark energy and dark matter in a different context. Notice that, the expression of Eq.(15) is a combination of the exponential 4

1.0

0.6

V( ) / (

de

c

(1+r))

0.8

0.4

0.2

0.0

0

1

2

(6

3

G)

4

1/2

Figure 3: The re-scaled potential V (φ)/Ωde ρc (1 + r) is shown. functions of φ, but differs from the exponential potential with a quadratic φ2 in Refs. [32, 33]. In particular, the profile of our V (φ) is more slope around φ = 0 than that in Refs. [32, 33]. To examine the observational viability of this simple model of transient acceleration, we perform a joint analysis involving the data of SN Ia, CMB, and BAO. We use the distance modulus µobs (zi ) data of 557 SNIa [38], the shift parameter of CMB by the WMAP observations [39], and the BAO measurement from the Sloan Digital Sky Survey (SDSS) [40]. We shall follow the calculational routine in Refs.[12, 13, 41]. Assuming that these three sets data of observations are mutual independent and that the measurement errors for each set are Gaussian, the likelihood function is then of the form 2 L ∝ e−χ /2 (16) with χ2 = χ2SN + χ2BAO + χ2CM B . χ2SN

=

557 X [µobs (zi ) − µth (zi )]2

σi2

i=1

(17) ,

(18)

χ2BAO =

(A − Aobs )2 , 2 σA

(19)

χ2CM B =

(R − Robs )2 . 2 σR

(20)

The detailed specifications of these formulae have been given in Ref.[13]. Variations of values of the model parameters yield respective values of χ2 . For demonstration, we take the model of r = 0.02 and let Ωm vary. The resulting χ2 (Ωm ) is plotted (in dash) in Fig.4. To compare with the standard ΛCDM, we also plot the case of r = 0 (corresponding to ΛCDM). The minimal is χ2 = 542.91 at Ωm = 0.268 for the model of r = 0.02, whereas ΛCDM has χ2 = 542.88 at Ωm = 0.271. Its corresponding likelihood L is 0.99 times that of ΛCDM. Thus the joint analysis 5

tells that the model of r = 0.02 is quite close to ΛCDM in confronting the observational data, and is robust in providing a transient acceleration. 546.5

CDM r=0.02

546.0

r=4.28 r'=-4.25 545.5

2

545.0

544.5

544.0

543.5

543.0

0.25

0.26

0.27

0.28

0.29

m

Figure 4: The χ2 of the transient acceleration models based upon the joint data of SNIa, BAO and CMB. The dashed line is of the first mode with r = 0.02 in Eq.(4). The dotted line is of the second model in Eq.(21) with r = 4.28 and r′ = −4.25. For comparison, χ2 of ΛCDM (r = 0) is also shown in the solid line.

The above model can be extended so that the universe in future is dominated by the matter ρm , while retaining the return to deceleration. We consider a dark energy density ρde = Ωde ρc

1 + r + r′ , 1 + ra3 + r ′ aǫ(a)

(21)

where r and r ′ are constant parameters, ǫ(a) is some function of the scale factor a. By the energy conservation, one obtains pde = −

θ ρ2 , Ωde ρc (1 + r + r ′ ) de

(22)

where

d 1 θ ≡ 1 + r ′ aǫ(a) − r ′ a aǫ(a) . 3 da Then the deceleration parameter is q=

1 3 Ωde (1 + r + r ′ )θa3 . − 2 2 (1 + ra3 + r ′ aǫ(a) )[Ωm + (r + (1 + r ′ )Ωde )a3 + Ωm r ′ aǫ(a) ]

(23)

(24)

This model admits a return to deceleration as long as ǫ(a) > 3 for small a, and ǫ(a) < 3 for large a. For instance, taking ǫ(a) = (2a + 4)/(a + 1), one has ǫ(a) → 2 for a → ∞, and ǫ(a) → 4 ′ de 1+r+r →Ω as a → ∞. A future matter domination is achieved if for a → 0, and the ratio ρρde Ωm r m 1+r+r ′ r ρde ρm →

Ωm . To be specific, we take r = 5.02, r ′ = −5.00 for Ωde = 0.74, Ωm = 0.26, yielding < Ω de 0.704 in the future. Fig. 5 and Fig. 6 demonstrate q and ρρde , respectively. It is also m

6

5

1.5

r=4.28, r'=-4.25

r=4.28,

r'=-4.25

4

1.0

m

3 de

q(a)

/

0.5

2

0.0

1 -0.5

0.0

0 0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

0

1

2

3

4

5

6

scale factor a

Scale factor a

Figure 5: The deceleration parameter q in Eq.(24) in the second model.

Figure 6: The ratio ρde /ρm of Eq.(21) in the second model. ρm will dominate over ρde in future.

checked that larger values of r or r ′ yield an earlier return to the deceleration expansion. To be consistent with the error band ±4% for ΩΛ by WMAP7 [3], the allowed value of the parameters should be constrained to r + r ′ < 0.04. In confronting the joint data of SNIa, BAO and CMB, the χ2 of this mode with r = 4.28 and r ′ = −4.25 is also obtained and shown (in dots) in Fig.4. Its minimum is χ2 = 543.60 at Ωm = 0.276. Its corresponding likelihood L is 0.69 times that of ΛCDM. Thus, this second model is also close to ΛCDM by statistical analysis, although its is not as good as the first model. In the above two models, the dark energy have a negative pressure pde < 0. In fact, the first model can be generalized so that a positive pressure pde > 0 for a ≫ 1 can be achieved. Consider 1+r , (25) ρde = Ωde ρc 1 + ra3(α+1) where α is a positive constant. In the limit α → 0, this reduces to Eq.(4) of the first model. From the energy conservation, the pressure is given by 1+α ρ2 , (26) pde = αρde − Ωde ρc (1 + r) de α consisting of two terms. When a ≫ 1, one has ρde ≪ Ωde ρc (1 + r) 1+α , and pde ≃ αρde , which is positive. From Eqs.(25) and (26) follows the deceleration parameter

q=

1 3 Ωde (1 + r)(1 − αra3α+3 )a3 − , 2 2 (1 + ra3α+3 )[Ωm (1 + ra3α+3 ) + Ωde (1 + r)a3 ]

(27)

which is shown in Fig. 7 for α = 1/3 and r = 0.0287. When a ≫ 1, the matter will dominate, similar to the second model. This dark energy can also be realized by a scalar field φ. By calculation, we obtain √ √ p 2 ra3(1+α) = e 6πG(1+α)φ − e− 6πG(1+α)φ , (28) and the potential

# 1+α 1−α √ √ √ . +4 √ V (φ) = 2Ωde ρc (1 + r) (e 6πG(1+α)φ + e− 6πG(1+α)φ )2 (e 6πG(1+α)φ + e− 6πG(1+α)φ )4 (29) "

7

1.0

r=0.0287,

=1/3

q(a)

0.5

0.0

-0.5

0

1

2

3

4

5

6

Scale factor a

Figure 7: The deceleration parameter q in Eq.(27) in the third model. In the special case α = 0, Eq.(29) reduces to Eq.(15) of the first model. In fact, Eq.(26) can be further extended into the following most general form: pde = αρde − βρm de ,

(30)

where α, β > 0 and m > 1 are constant. By the energy conservation, Eq.(30) yields ρde = Ωde ρc (

1 1+r m−1 , ) 1 + ra3(α+1)(m−1)

(31)

, and σ is an integral constant and can be fixed by the initial where the parameter r ≡ σ(α+1) β condition ρde |a=1 = Ωde ρc . The deceleration parameter is q(a) =

1 3 Ωde (1 − αra3(α+1)(m−1) )a3 . − 1 2 2 (1 + ra3(α+1)(m−1) )[Ωde a3 + Ωm ( 1+ra3(α+1)(m−1) ) m−1 ] 1+r

(32)

When m = 2, this general model reduces to the third model, and reduces to the first model if further α = 0. In the limit ρde ≪ (α/β)1/m−1 , Eq.(30) reduces to pde ≃ αρde , a barotropic gas. The general model in Eq.(31) has the following asymptotic behavior: in the limit a → 0, 1/(m−1) like the cosmological constant, and, in the limit a ≫ 1, ρ −3(1+α) , ρde → ( α+1 de → a β ) which mimics a barotropic gas. As we have checked by detailed computations, shown in Fig. 8 for α = 1/3 and m = 2.5, the general model also admits a transient accelerating expansion, in which the matter will dominate for a ≫ 1, similar to the second and the third models. Ref.[42] also discussed a special case of α = −1 of Eq.(30) in a different context. 3 Conclusion and Discussion We have demonstrated, by explicit model constructions, that the current cosmic acceleration driven by some dynamical dark energy can be transient, and can transit smoothly into redeceleration. Four specific models have been examined in details, each being a special case of the most general one with the equation of state: pde = αρde − βρm de . In all the models, the 8

1.0

r=0.03,

=1/3,

m=2.5

q(a)

0.5

0.0

-0.5

0

1

2

3

4

Scale factor a

Figure 8: The deceleration parameter q in Eq.(31) in the general model. dark energy behaves like a barotropic gas with ρde → a−3(α+1) with α ≥ 0 for a ≫ 1, and the total cosmic energy can be dominated either by ρm as in the last three models, or by ρde as in the first model. Our dark energy models can be realized by a scalar field φ. In two cases, by analytical integration, we have also obtained the explicit expressions of scalar field potential V (φ), which√is as simple as a combination of the exponential functions of φ, and approaches to V (φ) ∝ e−2 6πGφ for a ≫ 1. This function V (φ) differs from the previous exponential type of potential in literature. The interesting feature of these models is that the dark energy density will always ρde → const, subdominant to the matter during earlier stages. This is an improvement over the previous models of transient acceleration that employed the exponential type of scalar field potentials. Moreover, in all our models the fraction of dark energy at a ≪ 1 is very close to the value of the cosmological constant ΩΛ in ΛCDM, and is within the error band from WMAP observations. Besides, the joint likelihood analysis also shows that the transient acceleration models can be as robust as ΛCDM in confronting the observational data of SN Ia, CMB, and BAO. Therefore, our models can be further incorporated into the framework of the standard Big Bang cosmology to achieve the possible transient acceleration. Acknowledgements Y. Zhang’s research work has been supported by the CNSF No.11073018, 11275187, SRFDP, and CAS.

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