aa r X i v : . [ a s t r o - ph ] M a y A parametric model for dark energy
E. M. Barboza Jr. ∗ and J. S. Alcaniz , † Observat´orio Nacional, 20921-400, Rio de Janeiro - RJ, Brasil and Instituto Nacional de Pesquisas Espaciais/CRN, 59076-740, Natal – RN, Brasil (Dated: February 2, 2018)Determining the mechanism behind the current cosmic acceleration constitutes a major questionnowadays in theoretical physics. If the dark energy route is taken, this problem may potentiallybring to light new insights not only in Cosmology but also in high energy physics theories. Followingthis approach, we explore in this paper some cosmological consequences of a new time-dependentparameterization for the dark energy equation of state (EoS), which is a well behaved function ofthe redshift z over the entire cosmological evolution, i.e., z ∈ [ − , ∞ ). This parameterization allowsus to divide the parametric plane ( w , w ) in defined regions associated to distinct classes of darkenergy models that can be confirmed or excluded from a confrontation with current observationaldata. A statistical analysis involving the most recent observations from type Ia supernovae, baryonacoustic oscillation peak, Cosmic Microwave Background shift parameter and Hubble evolution H ( z )is performed to check the observational viability of the EoS parameterization here proposed. PACS numbers: 98.80.CqKeywords:
I. INTRODUCTION
Over the last decade, a considerable number of highquality observational data have transformed radically thefield of cosmology. Results from distance measurementsof type Ia supernovae (SNe Ia) [1, 2] combined with Cos-mic Microwave Background (CMB) observations [3] andthe Large-Scale Structure (LSS) data [4, 5] seem to in-dicate that the simple picture provided by the standardcold dark matter scenario is not enough. These obser-vations are usually explained by introducing a new hy-pothetical energy component with negative pressure, theso-called dark energy or quintessence , usually character-ized by the equation of state (EoS) parameter w ≡ p/ρ ,i.e., the ratio between the dark energy pressure to itsenergy density (see, e.g., [6] for some recent reviews).Among the many candidates for dark energy, the en-ergy density associated with the quantum vacuum or thecosmological constant (Λ) emerges as the simplest andthe most natural possibility. However, this interpreta-tion of the cosmological term brings to light an unsettledsituation in the Particle Physics/Cosmology interface, inwhich the cosmological upper bound ( ρ Λ < ∼ − GeV )differs from theoretical expectations ( ρ Λ < ∼ GeV )by more than 100 orders of magnitude [7]. Thus, al-though Λ may be able to explain the majority of obser-vations available so far, if dark energy is in fact associatedwith the vacuum energy density, we should look for anexplanation for this enormous discrepancy between the-ory and observation.In this regard, many proposals have appeared in the lit-erature trying to solve this problem, but so far no reason- ∗ E-mail: [email protected] † E-mail: [email protected] able explanation, if there is one, was obtained. Thus, de-spite the beauty and simplicity of the cosmological term,other proposals, even if not so attractive, should be ex-plored. In the context of the General Theory of Relativ-ity, for instance, models with a time-varying cosmologicalterm [8], irreversible processes (e.g., cosmological mattercreation [9]), barotropic fluids (e.g., Chapligyn Gas [10])and, dynamical scalar fields Φ (e.g., quintessence [11],phantom fields [12] and quintom [13]), are some of thosealternatives .Phenomenologically, it is usual to explore some pos-sible time-dependent parameterizations to describe thedark energy EoS. In this concern, Taylor series-like pa-rameterizations w ( z ) = X n =0 w n x n ( z ) , where w n ’s are constants to be fixed by observations and x n ( z )’s are functions of redshift, are among the mostcommonly discussed and, depending on the allowed val-ues for w n ’s, they may have quintessence ( − ≤ w ( z ) ≤
1) and phantom fields ( w ( z ) < −
1) as special cases.Among the parameterizations based on series expansionwe can quote the following first order expansions: w ( z ) = w + w z (redshift) [16] w + w z/ (1 + z ) (scale factor) [17] w + w ln(1 + z ) (logarithmic) [18] Out of the context of the General Relativity, some other attrac-tive approaches to the dark energy problem, such as brane-worldmodels [14] and f(R) derived cosmologies [15] have also been re-cently explored.
The first parameterization represents a good fit for lowredshifts, but presents a problematic behavior for highredshifts. For example, it fails to explain the estimatedages of high- z objects [19]. The second one solves thisproblem, since w ( z ) is a well behaved function for z ≫ z at low redshifts. Thelatter was built empirically to adjust some quintessencemodels at z < ∼
4. It is worth mentioning that it is dif-ficult to obtain the above parameterizations from scalarfield dynamics since they are not limited functions, i.e.,the EoS parameter does not lie in the interval defined by w = ˙Φ / − V (Φ)˙Φ / V (Φ) , where V (Φ) is the field potential. Thisamounts to saying that when extended to the entire his-tory of the universe, z ∈ [ − , ∞ ), the three parameteriza-tions above are divergent functions of the redshift . How-ever, since the dark energy dominance is a very recentphenomena, this particular aspect is not usually takenas important because it is always possible to obtain aquintessence-like behavior as a particular approximationwhen z is not too large. Even so, one can suspect thatthe information that can be obtained about dark energyfrom these parameterizations may be compromised. Infact, to avoid the ambiguities and uncertainty that can becontained in these parameterizations, it is desirable a pa-rameterization that can be extended to entire expansionhistory of the Universe, so that the constraints obtainedfrom the scalar field behavior can also be applied.In this paper, in order extend the range of applicabilityof the dark energy EoS, we study some cosmological con-sequences of a new phenomenological parameterizationfor this quantity. We discuss the classification of this ex-pression in the parametric space w − w , and explore itsmain observational features. We test the viability of thisnew dark energy scenario from the most recent distancemeasurements from type Ia supernovae (SNe Ia), mea-surements of the baryonic acoustic oscillations (BAO),the shift parameter of the cosmic microwave backgroundand measurements of the rate of the cosmic expansion H ( z ). II. PARAMETERIZATION
In this paper, we will consider the following parame-terization for the dark energy EoS: w ( z ) = w + w z (1 + z )1 + z , (1)where w is the EoS value at present time (the subscriptand superscript zero denotes the present value of a quan-tity) and w = dw/dz | z =0 gives a measure of how time-dependent is the dark energy EoS. This parameterization As an example, the scale factor parameterization above blows upexponentially in the future as z → − w > has the same linear behavior in z at low redshifts pre-sented by the parameterizations discussed above and hasthe advantage of being a limited function of z throughoutthe entire history of the Universe.For a flat Friedmann-Robertson-Walker universe, it isstraightforward to show from the continuity equation foreach component, ˙ ρ i = 3 ρ i ˙ z [1 + w i ( z )] / (1 + z ), that thedark energy density ρ X for parameterization (1) evolvesas f ( z ) ≡ ρ X ρ X = (1 + z ) w ) (1 + z ) w / . (2)Note that, if Eq. (1) should be always valid, the param-eters w and w should be constrained by w + w < , (3)so that the dark energy is always subdominant at z ≫ H ≡ (cid:16) ˙ aa (cid:17) = H [Ω m (1 + z ) + (1 − Ω m ) f ( z )] , (4)where a is the cosmological scale factor, H is the Hubbleparameter and Ω i = ρ i /ρ c ( ρ c = 3 H / π G ) is densityparameter of the i th-component ( m ≡ baryonic + darkmatter).The deceleration parameter is given by q ( z ) = − ¨ aa H = 12 [1 + 3 w ( z ) Ω X ] . (5)By solving the above equation for w ( z ) at z = 0 we obtain w = 2 q − − Ω m ) or w < − − Ω m ) , (6)since q <
0, as indicated by current observations [1].Note that, for Ω m = 0 . ± .
04 [3], it is possible obtaina model-independent bound on the current value of thedark energy EoS, i.e., w < − . A. The w − w plane By differentiating Eq. (1) with respect to z , we findthat w ( z ) has absolute extremes at z ± = 1 ± √ w − = w ( z − ) = w − . w and w + = w ( z + ) = w + 1 . w . For w > < w − is a minimum (maximum) and w + is a maximum (mini-mum). Since quintessence and phantom scalar fields EoSare limited by − ≤ w ( z ) ≤ w ( z ) < −
1, respec-tively, the region occupied in the ( w , w ) plane by thesefields can be easily determined by imposing that the max-imum and minimum of w ( z ) satisfy these limits. Thus,for quintessence fields we find the following bounds − ≤ w − . w and w +1 . w ≤ w > , -1.5 -1.0 -0.5 0.0-2-1012 w w QuintessencePhantom Forbidden RegionDecelerated
FIG. 1: The ( w , w ) parametric space for Parameterization(1). The forbidden region represents the constraint (3) whilethe decelerating region is limited by the upper bound (6) withΩ m = 0 .
27, i.e., w < − .
43. The blanc regions indicate mod-els that at some point of the cosmic evolution, z ∈ [ − , ∞ ),have switched or will switch from quintessence to phantombehaviors or vice-versa. and − ≤ w +1 . w and w − . w ≤ w < . whereas for phantom fields we obtain w < − (1 + w ) / .
21 (if w > , and w > (1 + w ) / .
21 (if w < . Figure 1 shows different classes of dark energy mod-els in the ( w , w ) plane that arises from parameteriza-tion (1). The forbidden region represents the constraint(3) while the decelerating region is limited by the upperbound (6) with Ω m = 0 .
27, i.e., w < − .
43. The blankregions indicate models that at some point of the cos-mic evolution, z ∈ [ − , ∞ ), have switched or will switchfrom quintessence to phantom behaviors or vice-versa.Clearly, Parameterization (1) provides a simple way toclassify distinct DE models in the ( w , w ) plane. Thisis a direct consequence from the fact that (1) is a wellbehaved bounded function along the entire history of theUniverse. The standard ΛCDM scenario corresponds tothe intersection point between quintessence ( w > − w < − III. OBSERVATIONAL CONSTRAINTS
In the previous Section, we have defined the regionsoccupied by different classes of dark energy models de-rived from parameterization (1) in the plane ( w , w ).Now, we will test the viability of these scenarios by usingthe most recent cosmological data, namely, 115 distancemeasurements of SNe Ia from the Supernova Legacy Sur-vey (SNLS) [2], measurements of the baryonic acousticoscillations (BAO) from SDSS [5], the CMB shift param-eter as given by the WMAP team [3] and estimates ofthe Hubble parameter H ( z ) obtained from ages of high- z galaxies [21] (for more details on the statistical analysisdiscussed below we refer the reader to Ref. [22]). A. SNe Ia observations
The predicted distance modulus for a supernova at red-shift z , given a set of parameters P , is µ p ( z | P ) = m − M = 5 log d L + 25 , (7)where m and M are, respectively, the apparent and ab-solute magnitudes, and d L stands for the luminosity dis-tance (in units of megaparsecs), d L ( z ; P ) = (1 + z ) Z z dz ′ H ( z ′ ; P ) , (8)where H ( z ; P ) is given by Eq. (4).We estimated the best fit to the set of parameters s byusing a χ statistics, with χ SNe = N X i =1 (cid:2) µ ip ( z | P ) − µ io ( z ) (cid:3) σ i , (9)where µ ip ( z | P ) is given by Eq. (7), µ io ( z ) is the extinc-tion corrected distance modulus for a given SNe Ia at z i ,and σ i is the uncertainty in the individual distance mod-uli. Since we use in our analyses the SNLS collaborationsample (see [2] for details), N = 115. B. Baryonic acoustic oscillations
As well known, the acoustic peaks in the cosmic mi-crowave background (CMB) anisotropy power spectrumis an efficient way for determining cosmological param-eters (e.g., [3]). Because the acoustic oscillations in therelativistic plasma of the early universe will also be im-printed on to the late-time power spectrum of the non-relativistic matter [23], the acoustic signatures in thelarge-scale clustering of galaxies yield additional tests forcosmology.In particular, the characteristic and reasonably sharplength scale measured at a wide range of redshifts pro-vides an estimate of the distance-redshift relation, which -1.5 -1.0 -0.5 0.0-3-2-1012 w w QuintessencePhantom Forbidden RegionDecelerated -1.5 -1.0 -0.5 0.0-3-2-1012 w w Forbidden RegionDeceleratedPhantom
FIG. 2: The results of our statistical analyses.
Left:
The 68.3%, 95.4%, and 99.7% confidence contours for Parameterization(1) arising from SNLS SNe Ia, SDSS BAO, WMAP CMB shift parameter and H ( z ) data. Right:
The same as in the previousPanel for the scale factor parameterization. is a geometric complement to the usual luminosity-distance from SNe Ia. Using a large spectroscopic sampleof 46,748 luminous, red galaxies covering 3816 square de-grees out to a redshift of z = 0 .
47 from the Sloan DigitalSky Survey, Eisenstein et al. [5] have successfully foundthe peaks, described by the A -parameter, i.e., A ≡ p Ω m h H z ∗ H ( z ∗ ; P ) Z z ∗ H dzH ( z ; P ) i / , (10)where z ∗ = 0 .
35 is the redshift at which the acoustic scalehas been measured.
C. CMB shift parameter
The shift parameter R which determines the wholeshift of the CMB angular power spectrum is given by[24] R ≡ p Ω m Z z l s H dzH ( z ; P ) , (11)where the z l s = 1089 is the redshift of the last scatteringsurface, and the current estimated value for this quantityis R obs = 1 . ± .
03 [25].
D. Hubble Expansion
In our joint analysis we also use 9 determinations ofthe Hubble parameter as a function of redshift, as given in Ref. [21]. These determinations, based on differen-tial age method, relates the Hubble parameter H ( z ) di-rectly to measurable quantity dt/dz and can be achievedfrom the recently released sample of old passive galax-ies from Gemini Deep Deep Survey (GDDS) [26] andarchival data [27]. The use of these data to constraincosmological models is interesting because, differently ofluminosity distance measures, the Hubble parameter isnot integrated over (see [21] for more details). To per-form this test we minimize the quantity χ H = X i =1 (cid:2) H i ( z | P ) − H iobs ( z ) (cid:3) σ i , (12)where the predicted Hubble evolution for parameteriza-tion (1) is given by Eq. (4). E. Analyses and Discussions
1. Parameterization (1)
Figure (2) shows the main results of our analyses. Inorder to compare the theoretical frame with the observa-tional constraints discussed above the three dimensionalparameter space (Ω m , w , w ) has been projected into theplane ( w , w ). In Fig. (2a) we show confidence inter-vals (68.3%, 95.4% and 99.7%) in this parametric space( w , w ) for parameterization (1). The best-fit values forthese parameters are w = − .
11 and w = 0 .
43 whereasat 68.3% (c.l.) they lie, respectively, in the intervals − . ≤ w ≤ − .
86 and − . ≤ w ≤ .
91. Notethat no DE model is preferred or ruled out by observa-tions, although the largest portion of the confidence con-tours lies into the blanc region (indicating models thathave switched or eventually will switch from quintessenceto phantom behaviors or vice-versa). At 99% C.L. wealso have 0 . ≤ Ω m ≤ .
33 so that the decelerated re-gion is limited, in accordance with (6), by the constraint w < − .
42 and the possibility of a decelerated universetoday is almost completely excluded. For the best-fit val-ues discussed above, the transition redshift z t , at whichthe Universe switches from deceleration to acceleration,occurs at z t ≃ .
2. Scale factor Parameterization
For the sake of comparison, we employ the same analy-sis to scale factor parameterization w + w z/ (1 + z ) [17].Note that this parameterization has an absolute extremein w ∞ = w ( z = ∞ ) = w + w . For w > w ∞ is amaximum whereas for w < w ∞ < − w >
0, whereas similarconstraints cannot be obtained for the quintessence case.The region occupied by phantom fields in the contextof this parameterization and the confidence intervals forthe statistical analysis discussed above are displayed inthe Fig. 3. At 68.3% c.l., we found w = − . +0 . − . , w = 0 . +0 . − . and Ω m = 0 . ± .
03. As can be seen,at this confidence level, the possibility of an early darkenergy dominance is not completely excluded.Finally, we also note that for this parameterization,the continuity equation solves as f SF ( z ) = (1 + z ) w + w ) e − w z/ (1+ z ) (13)so that it must also be submitted to the constraint (3).Therefore, when z → − a → ∞ ), f SF ( z ) blows upif w > f ( z ), given by Eq. (1), blows up if w < −
1. Thus, the roles of the parameters w and w are inverted in these scenarios, in the sense that whilefor Parameterization (1) the fate of the Universe is dic-tated by the equilibrium part ( w ), for the scale factorparameterization the future of the Universe is driven bythe time-dependent term w . IV. FINAL REMARKS
The recent cosmic expansion history has the poten-tial to greatly extend our physical understanding of theUniverse. This in turn is closed related to the originand nature of the mechanism behind the current cosmicacceleration, for instance, if it is associated with a newcomponent of energy or large-scale modifications of grav-ity.In this paper, by assuming a hypothetical componentof dark energy as the fuel that drives the acceleration ofthe Universe, we have proposed and studied some the-oretical and observational aspects of a new parameter-ization for this dark energy EoS, as given by Eq. (1).This parameterization is a well-behaved, bounded func-tion of the redshift throughout the entire cosmic evolu-tion, which allows us to study the effects of a time vary-ing EoS component to the distant future of the Universeat z = − w − w ) andstudied their theoretical and observational consequences.In order to check the observational viability of the phe-nomenological scenario proposed here, we also have per-formed a joint statistical analysis involving some of themost recent cosmological measurements of SNe Ia, BAOpeak, CMB shift parameter and the Hubble expansion H ( z ). From a pure observational perspective, we haveshown that both quintessence and phantom behavioursare fully acceptable regimes, although the largest por-tion of the confidence contours arising from these obser-vations lies in the region of models that have crossed orwill eventually cross these regimes at some point of thecosmic evolution. Acknowledgments
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