Nov 7, 2009 - As for the explanation of the SNeIa data, phantom is ... One of the most interesting features of phantom models is that they allow for a Big-Rip (BR) .... 3v. + 12A2 Ëu2 ,. (22) where u is the cyclic variable. The conserved current ...
Noether symmetry approach in phantom quintessence cosmology S. Capozziello, E. Piedipalumbo, C. Rubano, and P. Scudellaro Dipartimento di Scienze Fisiche, Università di Napoli “ Federico II” and INFN Sez. di Napoli,
arXiv:0908.2362v2 [astro-ph.CO] 7 Nov 2009
Complesso Universitario di Monte S. Angelo, Via Cinthia, Edificio N’, 80126 Napoli, Italy (Dated: November 8, 2009)
Abstract In the framework of phantom quintessence cosmology, we use the Noether Symmetry Approach to obtain general exact solutions for the cosmological equations. This result is achieved by the quintessential (phantom) potential determined by the existence of the symmetry itself. A comparison between the theoretical model and observations is worked out. In particular, we use type Ia supernovae and large scale structure parameters determined from the 2-degree Field Galaxy Redshift Survey (2dFGRS)and from the Wide part of the VIMOS-VLT Deep Survey (VVDS). It turns out that the model is compatible with the presently available observational data. Moreover we extend the approach to include radiation. We show that it is compatible with data derived from recombination and it seems that quintessence do not affect nucleosynthesis results. PACS numbers: 04.50.+h, 04.80.Cc, 98.80.-k, 11.25.-w, 95.36.+x
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I.
INTRODUCTION
Recent analysis of the three year WMAP data [1, 2, 3] provides no indication of any significant deviations from Gaussianity and adiabaticity of the CMBR power spectrum and therefore suggests that the Universe is spatially flat to within the limits of observational accuracy. Further, the combined analysis of the three-year WMAP data with the Supernova Legacy Survey (SNLS), in [1], constrains the equation of state wde , corresponding to almost 74% of dark energy present in the currently accelerating Universe, to be very close to that of the cosmological constant value. The marginalized best fit values of the equation of state parameter gave −1.14 ≤ wde ≤ −0.93 at 68% confidence level. Thus, it was realized that a viable cosmological model should admit a dynamical equation of state that might have crossed the phantom value w = −1, in the recent epoch of cosmological evolution. Phantom fluid was first investigated in the current cosmological context by Caldwell [4], who also suggested the name referring to the fact that phantom (or ghost) must possess negative energy which leads to instabilities on both classical and quantum level [5, 6]. Since it violates the energy conditions, it also could put in doubt the pillars of general relativity and cosmology such as: the positive mass theorems, the laws of black hole thermodynamics, the cosmic censorship, and causality [7, 8]. On the other hand, phantom becomes a real challenge for the theory, if its support from the supernovae Ia-Type (SNeIa) data is really so firm. From the theoretical point of view, a release of the assumption of an analytic equation of state which relates energy density and pressure and does not lead to energy conditions violation (except for the dominant one) may also be useful [9]. As for the explanation of the SNeIa data, phantom is also useful in killing the doubled positive pressure contribution in several braneworld models [10]. Phantom type of matter was also implicitly suggested in cosmological models with a particle production [11], in higher-order theories of gravity models [12], Brans-Dicke models, in nonminimally coupled scalar field theories [13, 14], in "mirage cosmology" of the braneworld scenario [15], and in kinematically-driven quintessence (k-essence) models [16, 17], for example. Such phantom models have well-known problems but, nevertheless, have also been widely studied as potential dark energy candidates, and actually the interest in phantom fields has grown vastly during the last years and various aspects of phantom models have been investigated [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35]. One of the most interesting features of phantom models is that they allow for a Big-Rip (BR) curvature singularity, which appears as a result of having the infinite values of the scale factor
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a(t) → ∞ at a finite future. However, as it was already mentioned, the evidence for phantom from observations is mainly based on the assumption of the barotropic equation of state which tightly constraints the energy density and the pressure. It is puzzling [9] that for Friedmann cosmological models, which do not admit an equation of state which links the energy density ̺ and the pressure p, a sudden future singularity of pressure may appear. This is a singularity of pressure only, with finite energy density which has an interesting analogy with singularities which appear in some inhomogeneous models of the Universe [36, 37]. Recently, phantom cosmologies which lead to a quadratic polynomial in canonical Friedmann equation have been investigated [38], showing that interesting dualities exist between phantom and ordinary matter models which are similar to dualities in superstring cosmologies [39, 40]. These dualities were generalized to non-flat and scalar field models [41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53], brane models [54], and are also related to ekpyrotic models [55, 56]. Furthermore, some theoretical studies have been devoted to shed light on phantom dark energy within the quantum gravity framework, since, despite the lack of such a theory at present, we can still make some attempts to probe the nature of dark energy according to some of its basic principles [57]. Finally, phantom cosmology can provides the opportunity to "connect" the phantom driven (low energy meV scale) dark energy phase to the (high energy GUT scale) inflationary era. This is possible because the energy density increases in phantom cosmology. Concrete models in this sense have been recently elaborated with some interesting results [58]. In this paper, we want to investigate if the existence of phantom fields can be connected to Noether symmetries. Such an issue becomes recently extremely important due to the fact that several phenomenological models have been constructed but, some of them, have no self-consistent theoretical foundation. The idea to derive the equation of state from symmetries is not new [59] and recently has been applied to dark energy [60]. From a mathematical point of view, the general consideration is that symmetries greatly aid in finding exact solutions [59, 61]. Besides, due to the Noether theorem, symmetries are always related to conserved quantities which, in any case, can be considered as conserved "charges". Specifically, the form of the self-interacting scalar-field potential is "selected" by the existence of a symmetry and then the dynamics can be controlled. The equation of state, being related to the form of scalar-field potential, is determined as well. However, the symmetry criterion is not the only that can be invoked to discriminate physically consistent models but it could be considered a very straightforward one since, as we will see be3
low, it allows also to achieve exact solutions. In Sect.II, we actually show that phantom fields come out by requiring the existence of Noether symmetry to the Lagrangian describing a standard single scalar field quintessential cosmological model: we show that it allows a phantom dark energy field, and also provides an explicit form for the (phantom) self-interaction potential. Sect. III studies how this gives rise to exact and general solutions. Also extending the approach to include radiation, we show that it is also compatible with the post recombination observational data and that quintessence does not influence the results of nucleosynthesis. In Sect IV, we work out a comparison between the theoretical solution and observational dataset, as the publicly available data on SNeIa, the parameters of large scale structure determined from the 2-degree Field Galaxy Redshift Survey (2dFGRS)and from the Wide part of the VIMOS-VLT Deep Survey (VVDS). In Sect.V, we discuss the presented results and draw conclusions.
II.
THE NOETHER SYMMETRY APPROACH
The Noether Symmetry Approach has revealed a useful tool in order to find out exact solutions, in particular in cosmology [61, 62, 63, 64]. The existence of the Noether symmetry allows to reduce the dynamical system that, in most of cases, results integrable. It is interesting to note that the self-interacting potentials of the scalar field [64], the couplings [61] or the overall theory [63], if related to a symmetry (i.e. a conserved quantity) have a physical meaning. In this sense, the Noether Symmetry Approach is also a physical criterion to select reliable models (see [63] for a discussion). In the present case, let us consider a matter–dominated model in homogeneous and isotropic cosmology with signature (−, +, +, +) for the metric, with a single scalar field, φ, minimally coupled to the gravity. It turns out that the point-like Lagrangian action takes the form ! φ˙ 2 2 3 L = 3a˙a − a ǫ − V(φ) + Da−3(γ−1) 2
(1)
where a is the scale factor and the constant D is a constant defined in such a way that the matter density ρm is expressed as ρm = D(ao /a)3γ , where 1 ≤ γ ≤ 2. For the moment, we will limit our analysis to γ = 1, corresponding to cosmological dust. The value of the constant ǫ discriminates between standard and phantom quintessence fields: in the former case, it is ǫ = 1; in the latter, it
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is ǫ = −1. The effective pressure and energy density of the φ-field are given by 1 pφ = ǫ φ˙ 2 − V(φ) , 2
(2)
1 ρφ = ǫ φ˙ 2 + V(φ) . 2
(3)
These two expressions define an effective equation of state wφ =
pφ , which drives the behavior of ρφ
the model. The field equations are a¨ 1 2 + H 2 + ǫ φ˙ 2 − V(φ) = 0, a 2
(4)
a¨ + 3H φ˙ + ǫV ′ (φ) = 0, a
(5)
3H 2 = ρφ + ρm ,
(6)
where prime denotes derivative with respect to φ, while dot denotes derivative with respect to time. The Noether theorem states that, if there exists a vector field X, for which the Lie derivative of a given Lagrangian L vanishes i.e. LX L = 0, the Lagrangian admits a Noether symmetry and thus yields a conserved current [61]. In the Lagrangian under consideration, the configuration space ˙ Hence the infinitesimal is M = {a, φ} and the corresponding tangent space is T M = {a, φ, a˙ , φ}. generator of the Noether symmetry is X=α
∂ ∂ ∂ ∂ + β + α˙ + β˙ , ∂a ∂φ ∂˙a ∂φ˙
(7)
where α and β are both functions of a and φ and ∂α ∂α a˙ + φ˙ ∂a ∂φ ∂β ∂β ˙ β˙ ≡ a˙ + φ. ∂a ∂φ α˙ ≡
(8) (9)
The Cartan one–form is
θL =
∂L ∂L da + dφ. ∂˙a ∂φ˙
(10)
The constant of motion Q = iX θL is given by Q = α(a, φ)
∂L ∂L + β(a, φ) . ∂˙a ∂φ˙
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(11)
If we demand the existence of a Noether symmetry, LX L = 0, we get the following equations, ∂α =0 ∂a ∂β ∂α − ǫa2 =0 6 ∂φ ∂a ∂β =0 3α + 2ǫa ∂φ ′ 3V(φ)α + aV (φ)aβ = 0 α + 2a
(12) (13) (14) (15)
We have now to look for conditions on the integrability of this set of equations, limiting ourselves to the phantom case (1.e., ǫ = −1), since the standard case has been already investigated in [65, 66]. It is possible to assume that α and β are separable (and non–null), i.e. α(a, φ) = A1 (a)B1 (φ), β(a, φ) = A2 (a)B2 (φ).
(16)
This is not true in general but, in such a case, it is straightforward to achieve a solution for the system (12-15). It is q 1 2
III.
2A cos , √ a q √ −2 6A sin 21 32 φ β= , √ a a r 2 1 3 V(φ) = V0 sin φ 2 2 α=
which selects the Noether symmetry.
3 φ 2
(17)
(18) (19)
SOLUTIONS FROM NEW COORDINATES AND LAGRANGIAN
Once that X is found, it is then possible to find a change of variables {a, φ} → {u, v}, such that one of them (say u, for example) is cyclic for the Lagrangian L in Eq. (1), and the transformed Lagrangian produces a reduced dynamical system which is generally solvable. Solving the system of equations iX du = 1 and iX dv = 0 (where iX du and iX dv are the contractions between the vector field X and the differential forms du and dv, respectively), we obtain: 1
a = (v + 9A2 u2 ) 3 p (3Au) φ = 2 2/3 arccos √ . v + 9A2 u2 6
(20) (21)
Under this transformation, the Lagrangian takes the suitable form L = D + vV0 +
v˙ 2 + 12A2 u˙ 2 , 3v
(22)
where u is the cyclic variable. The conserved current gives Q=
∂L = 24A2 u˙ = B, ∂˙u
(23)
which can be trivially integrated to obtain u(t) = Bt + C. We use now the energy condition E L = 0 to find v. We obtain the following differential equation ˙ 2 − 3v(t) D + V0 v(t) − 12A2 u(t) ˙ 2 = 0. v(t)
(24)
It is a first order equation which for , i.e. can be factorized into the form (p − F 1 ) (p − F 2 ) = 0,
(25)
˙ and F i = F i (t, v). We are then left with solving two first-degree equations p = being p = v(t) F i (t, v). Writing the solutions to these first-degree equations as Gi (t, v) = 0 the general solution to Eq. (24) is given by the product G1 (t, v)G2 (t, v) = 0. It turns out that1: √ 2 p exp (− 3V0 t) 2 2 . 3V t) + 48A B V − 4DV v(t) = exp ( 0 0 0 16V02 The substitution of the functions a = a(u, v) and φ = φ(u, v) into Eqs. (20,21) yields 2 31 √ √ 2 exp − 3V t exp 3V t + 48ω V − 4DV 0 0 0 0 2 2 a(t) = 9ω t + 16V02 r 3ABt 2 , arccos r φ(t) = 2 √ √ 3 2 9ω2 t2 + (exp − 3V0 t)(exp 3V0 t+48ω2 V0 −4DV0 )
(26)
(27)
(28)
16V02
where we have defined ω = AB. Setting a(0) = 0, we can construct a relation among the integra-
tion constants ω, D and V0 : actually it turns out that D =
1+48ω2 V0 . 4V0
To determine the integration
constant ω, we set the present time t0 = 1. This fixes the time-scale according to the (formally unknown) age of the Universe. That is to say that we are using the age of the Universe, t0 , as a unit of time. We then set a0 = a(1) = 1, to obtain q √ 8 + V12 − cosh 3V0 0 ω= . V02 1
In the following we can set C = 0 in u(t) = Bt + C without loosing generality.
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(29)
The two conditions specified above allow one to express all the basic cosmological parameters in terms of V0 , the constant that determines the scale of the potential. It is not directly measurable. However we can also strongly constrain its range of variability through its relation with the Hubble constant: actually because of our choice of time unit, the expansion rate H(t) is dimensionless, so b0 = H(t0 ) is clearly of order 1 and not (numerically) the same as the H0 that our Hubble constant H that is usually measured in kms−1Mpc−1 . Actually, we can consider the relation h = 9.9
b0 H , τ
(30)
b0 fixes only the where, as usual, h = H0 /100 and τ is the age of the Universe in Gy. We see that H product hτ. In particular, following e.g. [1], we can assume that τ = 13.73+0.16 −0.15 , thus we get h
