Cosmic slowing down of acceleration for several dark energy parametrizations
aa r X i v : . [ a s t r o - ph . C O ] S e p Prepared for submission to JCAP
Cosmic slowing down of accelerationfor several dark energyparametrizations
Juan Maga˜na a V´ıctor H. C´ardenas a Ver´onica Motta a a Instituto de F´ısica y Astronom´ıa, Facultad de Ciencias, Universidad de Valpara´ıso, Avda.Gran Breta˜na 1111, Valpara´ıso, Chile.E-mail: [email protected], [email protected], [email protected]
Abstract.
We further investigate slowing down of acceleration of the universe scenariofor five parametrizations of the equation of state of dark energy using four sets of Type Iasupernovae data. In a maximal probability analysis we also use the baryon acoustic oscillationand cosmic microwave background observations. We found the low redshift transition of thedeceleration parameter appears, independently of the parametrization, using supernovae dataalone except for the Union 2.1 sample. This feature disappears once we combine the TypeIa supernovae data with high redshift data. We conclude that the rapid variation of thedeceleration parameter is independent of the parametrization. We also found more evidencefor a tension among the supernovae samples, as well as for the low and high redshift data.
Keywords: dark energy, Cosmology ontents
The observations from Type Ia supernova (SNIa) were the first clear evidence of the acce-lerated expansion of the Universe [1, 2]. The hypothesis of an expanding universe is currentlysupported by a wide range of large-scale measurements [3–5]. To explain this accelerationwithin general relativity, it is necessary to introduce an exotic component, named dark energy(DE). Current observations point towards a cosmological constant Λ [6–8]; nevertheless, itcould be necessary to consider dynamical DE models where the equation of state (EoS)parameter, w , is a function of the scale factor a (or redshift z ), i.e. w = P/ρ = w ( a ), where P and ρ are the pressure and energy density of DE respectively [9–12]. One popular ansatzfor this dynamical EoS is the Chevallier-Polarski-Linder (CPL) parametrization [13, 14] givenby w ( a ) = w + (1 − a ) w , where w is the present value of the EoS and w is the derivativewith respect to scale factor. Recently, [15] found a cosmic slowing down of the acceleration atlow redshifts using the CPL model with the Constitution SNIa sample in combination withbaryon acoustic oscillation (BAO) data. Nevertheless, this apparent behavior disappearswhen the cosmic microwave background (CMB) data are added in the analysis, indicating atension between low and high redshift measurements. Similarly, using the Union2 SNIa set,[16] it was found the same behavior at low redshift. The authors of [17] found the tensionbetween the sets at low redshifts and at high redshifts can be ameliorated incorporating acurvature term as a free parameter in the analysis.The slowing down scenario using only supernovas can be considered a sort of an artifactfrom the data. However, recent studies have been able to demonstrate the same trend usingusing 42 measurements of gas mass fraction in galaxy clusters [18]. They found the sametrend as using SNIa, i.e. the acceleration of the universe has already reached its maximumat z ∼ . § § § We consider a flat Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) universe whose DE hasa dynamical EoS w ( z ). The dimensionless Hubble parameter E ( z ) for this universe is givenby E ( z ) = H ( z ) /H = Ω m (1 + z ) + Ω r (1 + z ) + Ω de X ( z ) , (2.1)where Ω m is the density parameter for matter. The density parameter for radiation isΩ r = 2 . × − h − (1 + 0 . N eff ), where h = H /
100 kms − Mpc − and the numberof relativistic species is set to N eff = 3 .
04 [19]. The density parameter for DE is written asΩ de = 1 − Ω m − Ω r and the function X ( z ) reads as X ( z ) = ρ de ( z ) ρ de (0) = exp (cid:18) Z z w ( z )1 + z dz (cid:19) , (2.2)where ρ de ( z ) is the energy density of DE at redshift z , and ρ de (0) its present value. Thecomoving distance from the observer to redshift z is given by r ( z ) = cH Z z dz ′ E ( z ′ ) . (2.3)The deceleration parameter q(z) is defined as q ( z ) = − ¨ a ( z ) a ( z )˙ a ( z ) , (2.4)where a is the scale factor of the Universe. Using eq. (2.1), this expression can be rewrittenas q ( z ) = − (1 + z ) E ( z ) dE ( z ) dz − . (2.5)In the following section, we study five different parametrizations of w ( z ) and for eachone we shall use expression (2.5) to reconstruct the deceleration parameter. Because we do not known what physics is behind the DE, one way to explore models goingbeyond the cosmological constant is by using an explicit parametrization for the EoS pa-rameter w ( z ). Actually, the apparent cosmic slowing down of acceleration was found firstusing an explicit parametrization of w ( z ). This low redshift feature of the reconstructed q ( z ), remains after using different data sets by applying the CPL parametrization [15, 18].In the [20], the author adopted a different approach focusing his study not in a w ( z ) trialfunction, but in a simple (blind analysis) interpolation for the DE density X ( z ). However,– 2 –e found the same low redshift transition for the reconstructed q ( z ). This means that bothare valid complementary methods which enable us to explore evolution in the DE density,and consequently, the cosmic acceleration.In this work, although we do not expect that a phenomenological specific parametriza-tion for w ( z ) will reproduce the eventual real behavior of q ( z ), we investigate through severalparametrizations of the DE EoS whether the cosmic slowing down of acceleration is a realfeature of the Universe at low redshifts. We choose parametrizations with two independentfree parameters, thus allowing for direct statistical model comparisons. In the following weintroduce the parametrizations to be tested by cosmological data. Jassal-Bagla-Padmanabhan (JBP) [21, 22] propose an EoS of DE as function of redshift inthe following way w ( z ) = w + w z (1 + z ) , (3.1)where w = w (0), w = w ′ (0), and w ( ∞ ) = w . In addition, the EoS parameter has thesame value at present and early epochs and performs a rapid variation around z = 1 withamplitude w + w /
4. For the JBP parametrization (3.1), the function (2.2) reads as X ( z ) = (1 + z ) w ) exp (cid:18) w z (1 + z ) (cid:19) , (3.2)and E ( z ) is completely determined. Another interesting parametrization was proposed by Feng, Shen, Li and Li (FSLL) [23] toovercome the divergence of the CPL model when z → −
1. They propose the two followingrelations: FSLL I : w ( z ) = w + w z z , FSLL II : w ( z ) = w + w z z , (3.3)where for both models w = w (0) and w = w ′ (0). For FSLL I, w ( ∞ ) = w and at lowredshift it reduces to the linear form w ( z ) ≈ w + w z , while for FSLL II w ( ∞ ) = w + w and at low redshifts it yields w ( z ) ≈ w + w z . For the parametrizations (3.3), the functions(2.2) are given by X ( z ) ± = (1 + z ) w ) exp (cid:20) ± w (cid:21) (cid:0) z (cid:1) w / (1 + z ) ∓ w / , (3.4)where X + and X − correspond to FSLL I and FSLL II respectively. A generalization of the CPL model consists in an expansion in powers of (1 + z ) [24]. One ofthese polynomial parametrizations was proposed by Sendra and Lazkoz (SL) [25] w ( z ) = − c (cid:18) z z (cid:19) + c (cid:18) z z (cid:19) , (3.5)– 3 –here c = (16 w − w . + 7) / c = − w + (9 w . − / w and w . are the values ofthe equation of state at z = 0 and z = 0 . w . improved the situation of the CPL model, which suffers from a significant correlationbetween the parameters w and w [26]. The proposed function is: X ( z ) = (1 + z ) − w +9 w ) / exp (cid:20) z ( w (52 z + 40) − w (5 z + 4) + 7 z + 4)8(1 + z ) (cid:21) . (3.6)Notice the ΛCDM model is recovered for w = w . = − We also consider the Barbosa-Alcaniz parametrization [27], where: w ( z ) = w + w z (1 + z )1 + z , (3.7)where w = w (0), w = w ′ (0), w ( ∞ ) = w + w , and at low redshift it reduces to the linearform w ( z ) ≈ w + w z . This ansatz is a well behaved function of the redshift z over the range z ∈ [ − , ∞ ). The DE density in this case evolves as X ( z ) = (1 + z ) w ) (1 + z ) w / . (3.8)As a closing for this section, we have to mention that an important condition that agiven parametrization must fulfill, is that the DE density has to be lower that the matterenergy density in the past. This is easily checked using the explicit expressions we havederived, finding that all of our best fit functions satisfies this requirement. The maximal probability analysis we perform to constrain the parameters of the differentparametrizations considers cosmological observations at different redshifts: Type Ia super-novae data, the acoustic peaks of BAO and the WMAP 9-yr distance posterior. In all ouranalysis, we assume a flat geometry, and a fixed reduced Hubble constant h = 0 .
697 [3], thusleaving Ω m , w o , and w ( w . for the SL model) as the only free parameters of the problem.For the supernova data we use the Constitution (C) set consisting of 397 SNIa points [28]covering a redshift range 0 . < z < . . < z < .
12 [29], and Union 2 . . < z < .
41 [30]. We also use the sample presented by [31] consisting in586 SNIa in the redshift range 0 . − . µ obs ( z ) and itserror σ µ . Theoretically, the distance modulus is computed as µ ( z ) = 5 log [ d L ( z ) / Mpc ] + 25 , (4.1)which is a function of the cosmology through the luminosity distance (measured in Mpc) d L ( z ) = (1 + z ) r ( z ) , (4.2)– 4 –alid for a flat universe with r ( z ) given by Eq. (2.3). We fit the SNIa with the cosmologicalmodel by minimizing the χ value defined as χ SNIa = N X i =1 [ µ ( z i ) − µ obs ( z i )] σ µ i . (4.3)We also consider the data from BAO and CMB. The BAO measurements considered in ouranalysis are the following: three acoustic parameter measurements at z = 0 .
44, 0 .
6, and 0 . z = 0 .
2, and 0 . z = 0 .
106 from the Six-degree-Field Galaxy Survey (6dFGS) [35]. Thus, the total χ for allthe BAO data sets can be written as χ BAO = χ W iggleZ + χ SDSS + χ dF GS (see [18] for moredetails to construct χ BAO ). The CMB information considered is derived from the 9 yearsWMAP data [3]. In our analysis we compare the measurements of the acoustic scale, theshift parameter and the redshift of decoupling with the theoretical ones (see [18] for detailsto construct χ CMB ). We constrain the cosmological parameters for two cases: using only theType Ia supernovae data, where the χ T ot = χ SNIa ; and using all data (SNIa+BAO+CMB), where χ T ot = χ SNIa + χ BAO + χ CMB . We minimize these χ T ot functions with respect tothe parameters { Ω m , w , w ( w . ) } to compute the best estimated values and their errors.Our results from the Bayesian analysis for the models are given in Table 1. In Fig.1 we show the evolution of the deceleration parameter as function of redshift for the JBPparametrization using the best fit values for these data sets. Notice that using the C set - - - - - zq H z L JBP - - - - - zq H z L JBP
Figure 1 . Evolution of the deceleration parameter q ( z ) vs z for the JBP parametrization. The leftpanel shows the reconstructed q ( z ) using the best fit obtained from Constitution (dotted line), Union 2(dot-dashed line), Union 2 . q ( z ) using the best fit obtained from C+BAO+CMB (dotted line), U2+BAO+CMB(dotted-dash line), U21+BAO+CMB (dashed line) and LU+BAO+CMB (solid line). Notice thecosmic slowing down of the acceleration at z ∼ the Universe pass through a maximum of acceleration at z ∼ . z ∼ . z ∼ .
2) sets. This feature is preserved when propagating theerror at 1 σ in the best fit parameter. Therefore, there is a statistically significant evidence ofthe cosmic slowing down of the acceleration at low redshifts. Nevertheless this behavior doesnot occur using the U2.1 data. When adding the information from BAO and CMB data,the Universe has a transition from a decelerated phase to an accelerated phase at z ∼ .
7– 5 –for all SNIa samples). Our analysis strongly suggests a tension between low-redshifts andhigh-redshifts measurements. This results is in agreement with the previous works usingSNIa data and information from galaxy clusters [15–18].In Fig. 2 we show the reconstructed q ( z ) as function of z for the FSLL I model. Noticethat the slowing down of the acceleration occurs at z ∼ . z ∼ . z ∼ .
25 using C,U2 an LU samples respectively. For the FSLL II parametrization, the cosmic decelerationoccurs at z ∼ . z ∼ . z ∼ . q ( z ) has another transition at z ∼ .
07 for U2 and LU sets, and the Universebegins an accelerated phase. However, this oscillating behavior is not statistically significantwhen propagating the error at 1 σ in the best fit parameters. Using the U2.1 SNIa samplethere is no evidence for the deceleration of the Universe at low redshift. As is shown in Figure4 the slowing down of the acceleration for the SL w ( z ) function occurs at z ∼ . z ∼ . z ∼ .
23 using C, U2 and LU samples respectively. Finally, in Fig. 5 we show that theUniverse has a maximum of acceleration and it begins to decelerate at z ∼ . z ∼ . z ∼ .
25 using the same SNIa samples respectively. - - - - - zq H z L FSLL I - - - - - zq H z L FSLL I
Figure 2 . We show the slope of the q ( z ) as in Fig. 1 for the FSLL I parametrization. Therefore, our analysis shows significant evidence for a cosmic slowing down of the ac-celeration at low redshifts for dynamical DE models with EoS varying with redshift usingConstitution, Union 2 and LOSS-Union SNIa sets. Interestingly, this feature disappears whenusing the Union 2.1 sample, which suggests an important tension between this particular dataset and the other SNIa samples. Moreover, when the analysis is performed using the combi-nation of SNIa data with the BAO and CMB observations, the scenario of the deceleration ofthe Universe’s expansion at low redshift disappears. Actually, we obtain w (0) ≈ − σ forall parametrizations in agreement with a cosmological constant. Moreover, we found that theUniverse has a transition from a decelerated phase to an accelerated phase at z ∼ z ∼ . z ∼ .
75 and z ∼ . w ( z ) on the results and final conclusions. Recently [20] found aconnection between this low redshift transition of q ( z ) with the DE density decreasing withincreasing redshift. Therefore, it would be important to check such a behavior. In Figure6 we show the slope of the X ( z ) function for all the five parametrizations using the best– 6 – .0 0.5 1.0 1.5 2.0 - - - - - zq H z L FSLL II - - - - - zq H z L FSLL II
Figure 3 . We show the slope of the q ( z ) as in Fig. 1 for the FSLL II parametrization. - - - - - zq H z L SL - - - - - zq H z L SL Figure 4 . Idem as Fig. 1 for the SL parametrization. - - - - - zq H z L BA - - - - - zq H z L BA Figure 5 . We show the slope of the q ( z ) as in Fig. 1 for the BA parametrization. fit parameters estimated from the LOSS-Union sample. The low redshift data suggest theDE density is in fact increasing with time. This same behavior occurs using the constraintsderived from the other Type Ia supernovae data. As it was mentioned in [20] this particularresults is in agreement with the recent BAO DR11 measurements [36] where a tension betweenBAO data and CMB was informed. This tension reveals that, in order to accommodate thesenew data assuming a flat universe with dark matter and DE, we have to assume the DE– 7 –ensity decreases with increasing redshift, and it reaches a negative value at z ≃ . ρ de ( z = 2 . ρ de ( z = 0) = − . ± . , (4.4)in agreement with what we have found in this work. Similar conclusions are obtained in [46]based on the same data from BAO. zX H z L LOSS - Union zX H z L LU + BAO + CMB
Figure 6 . Evolution of the function X ( z ) vs z for all parametrizations. The left panel shows thereconstructed X ( z ) using the best fit obtained from LOSS-Union for the JBP (solid line), FSLL I(dotted line), FSLL II (dashed), SL (dot-dashed) and, BA (long dashed) parametrizations. The rightpanel shows the reconstructed X ( z ) using the best fit obtained from LOSS-Union+BAO+CMB forthe five parametrizations. Notice the low redshift data suggest the DE density is in fact increasingwith time. This same behavior occurs using the constraints derived from the other Type Ia supernovaedata. To discern which model is the preferred one by observations we only compare the χ min ,given in Table 1, among data sets. Notice there is no significant differences among thesevalues for the parametrizations and, it is difficult to distinguish which model is the favoredby the data. Therefore, any parametrization could be a plausible model of dynamical darkenergy. In this paper, we have studied five alternative models for dark energy with an equation of statevarying with redshift. We put constraints on the parameters for these parametrizations usingfour different SNIa samples. We reconstructed the cosmological evolution of the decelerationparameter, and found that using the best fit obtained with Constitution, Union 2 and LOSS-Union samples, the Universe reaches a maximum of acceleration at low redshift ( z ∼ . w CDM with w =constant, using the same data,prefers a value w < − w ( z )in any of the five parametrizations studied, implies the deceleration parameter has alreadyreached its maximum and its evolving towards lower acceleration regimes. Nevertheless, thiscosmic slowing down of the acceleration disappears using the constraints derived from theUnion 2.1 SNIa set. This suggests a tension among the different SNIa compilations. Thistension is very intriguing for Union 2 and Union 2.1 because they differ only in 24 data pointsand in the lightcurve fitter SALT for Union 2 and its improved version SALT2 for Union 2.1.– 8 –imilar tension between these Union 2.1 and Union 2 as well as other recent SNIa sampleswas also found by [37–39]. Additionally, Union 2.1 and LOSS-Union compilations are verysimilar and both of them use the lightcurve fitter SALT2 trained on data from low redshiftsSNIa. However, they predict a reconstructed deceleration parameter considerably different.Therefore, we conclude the cosmic slowing down of the acceleration does not depend on theparametrization of the equation of state of dark energy and it could be due to several factors.For instance, systematic errors in the measurements from SNIa (such as peculiar motions[40, 41]), the presence of a Hubble bubble, anisotropic cosmological models, to mention someof them (see [18] for a discussion about this point). As was also discussed in [20], this lowredshift transition of q ( z ) was first found by the authors in [42], in the context of a Lemaitre-Tolman-Bondi inhomogeneous models. In that work, and also in recent ones ([43, 44]), theauthors derived an effective deceleration parameter for void models, indicating that such abehavior of q ( z ) may be considered as a signature for the existence of voids.Nevertheless,these results is in contrast with those obtained when a q ( z ) parametrization is considered[45]. On the other hand, when adding the BAO and CMB measurements, the cosmic slow-ing down of the acceleration disappears for all parametrizations and they could mimic thecosmological constant at the present epoch. Therefore, we also confirmed a tension betweenthe cosmological constraints obtained from low and high redshift data.In agreement with the results of the recent BAO DR11 measurements [36] – where atension between low to high redshift observational probes was detected – and also from thestudy in [20] – where the low redshift transition of q ( z ) was demonstrated to be linked witha DE density that decrease with increasing redshift – we have also found evidence for DEdensity evolution. In fact, using the best fit parameters found for each parametrization, wecan directly plot the DE density expressions (3.2), (3.4), (3.6) and (3.8), obtaining for eachcase the same result found first in [20]: the data suggest the DE density is increasing withepoch. The astonishing agreement between our approach based on using low redshift data(as SNIa, f gas , and others) with the result obtained from BAO measurements [36], make astrong case for DE evolution, as it was recently highlighted in [46]. Acknowledgments
We thank the anonymous referee for thoughtful suggestions. We wish to acknowledge usefuldiscussion with Diego Pav´on and Gael Fo¨ex. J.M. acknowledges support from ESO Comit´eMixto, and Gemini 32130024. V.M. acknowledges support from FONDECYT 1120741, andECOS-CONICYT C12U02. V.C. acknowledges support from FONDECYT Grant 1110230and DIUV 13/2009,
References [1] A. G. Riess et al. , Observational evidence from supernovae for an accelerating universe and acosmological constant , Astron. J. (1998) 1009, [astro-ph/9805201].[2] S. Perlmutter et al. , Measurements of Omega and Lambda from 42 high redshift supernovae ,Astrophys. J. (1999) 565, [astro-ph/9812133].[3] G. Hinshaw, D. Larson, E. Komatsu, D. N. Spergel, C. L. Bennett, J. Dunkley, M. R. Noltaand M. Halpern et al. [WMAP Collaboration],
Nine-Year Wilkinson Microwave AnisotropyProbe (WMAP) Observations: Cosmological Parameter Results , Astrophys. J. Suppl. 208(2013), 19, [arXiv:1212.5226]. – 9 –
4] P. A. R. Ade et al. , Planck 2013 results. XVI. Cosmological parameters , 2013,[arXiv:1303.5076].[5] T. M. Davis,
Cosmological constraints on dark energy , Gen. Rel. Grav. 46 (2014)arXiv:1404.7266 [astro-ph.CO].[6] P. Serra et al. , No evidence for dark energy dynamics from a global analysis of cosmologicaldata , Phys. Rev. D. 80 (2009) 121302, [arXiv:0908.3186].[7] R. Nair, J. Sanjay,
Is dark energy evolving? , JCAP 02 (2013) 049, [arXiv:1212.6644].[8] S. Postnikov, M. G. Dainotti, X. Hernandez and S. Capozziello,
Nonparametric Study of theEvolution of the Cosmological Equation of State with SNeIa, BAO, and High-redshift GRBs ,Astrophys. J. (2014) 126 [arXiv:1401.2939 [astro-ph.CO]].[9] C. Wetterich,
Cosmology and the Fate of Dilatation Symmetry , Nucl. Phys. B (1988) 668.[10] S. Weinberg,
The cosmological constant problem , Rev. Mod. Phys. 61 (1989) 1.[11] E. J. Copeland, M. Sami, S. Tsujikawa,
Dynamics of dark energy , Int. J. Mod. Phys. D 15(2006), 1753, [arXiv:hep-th/0603057].[12] J. Alberto Vazquez, M. Bridges, M. P. Hobson and A. N. Lasenby,
Reconstruction of the DarkEnergy equation of state , JCAP (2012) 020 [arXiv:1205.0847 [astro-ph.CO]].[13] M. Chevallier, D. Polarski,
Accelerating Universes with Scaling Dark Matter , Int. J. Mod. Phys.D , 10 (2001), 213, [gr-qc/0009008].[14] E. V. Linder,
Mapping the Dark Energy Equation of State , Phys. Rev. Lett. , 90 (2003),091301, [astro-ph/0311403].[15] A. Shafieloo, V. Sahni and A. A. Starobinsky,
Is cosmic acceleration slowing down? , Phys. Rev.D. 80 (2009), 101301, [arXiv:0903.5141].[16] Z. Li, P. Wu, H. Yu,
Examining the cosmic acceleration with the latest Union2 supernova data
Phys. Lett. B 695 (2011) 1, arXiv:1011.1982 [gr-qc].[17] V.H. C´ardenas, M. Rivera,
The role of curvature in the slowing down acceleration scenario ,Phys. Lett. B , 710 (2012), 251, [arXiv:1203.0984].[18] V.H. C´ardenas, C. Bernal, A. Bonilla,
Cosmic slowing down of acceleration using f gas , Mon.Non. Roy. Astron. Soc. , 433 (2013), 3534, [arXiv:1306.0779].[19] E. Komatsu et al. [WMAP Collaboration], Seven-Year Wilkinson Microwave Anisotropy Probe(WMAP) Observations: Cosmological Interpretation , Astrophys. J. Suppl. (2011) 18[arXiv:1001.4538 [astro-ph.CO]].[20] V. H. Cardenas,
Exploring dark energy density evolution in light of recent data , 2014,arXiv:1405.5116 [astro-ph.CO].[21] H. K. Jassal, J. S. Bagla and T. Padmanabhan,
WMAP constraints on low redshift evolution ofdark energy , Mon. Not. Roy. Astron. Soc. (2005) L11 [astro-ph/0404378].[22] H. K. Jassal, J. S. Bagla and T. Padmanabhan,
Observational constraints on low redshiftevolution of dark energy: How consistent are different observations? , Phys. Rev. D (2005)103503 [astro-ph/0506748].[23] C. -J. Feng, X. -Y. Shen, P. Li and X. -Z. Li, A New Class of Parametrization for Dark Energywithout Divergence , JCAP (2012) 023 [arXiv:1206.0063 [astro-ph.CO]].[24] J. Weller and A. Albrecht,
Future supernovae observations as a probe of dark energy , Phys.Rev. D (2002) 103512 [astro-ph/0106079].[25] I. Sendra and R. Lazkoz, SN and BAO constraints on (new) polynomial dark energyparametrizations: current results and forecasts , Mon. Not. Roy. Astron. Soc. (2012) 776[arXiv:1105.4943 [astro-ph.CO]]. – 10 –
26] Y. Wang,
Figure of Merit for Dark Energy Constraints from Current Observational Data ,Phys. Rev. D (2008) 123525 [arXiv:0803.4295 [astro-ph]].[27] E. M. Barboza, Jr. and J. S. Alcaniz, A parametric model for dark energy , Phys. Lett. B (2008) 415, [arXiv:0805.1713 [astro-ph]].[28] M. Hicken et al.,
Improved Dark Energy Constraints from 100 New CfA Supernova Type IaLight Curves , Astrophys. J. (2009) 1097 [arXiv:0901.4804 [astro-ph.CO]].[29] R. Amanullah et al. , Spectra and Light Curves of Six Type Ia Supernovae at 0.511 ¡ z ¡ 1.12and the Union2 Compilation , Astrophys. J. (2010) 712 [arXiv:1004.1711 [astro-ph.CO]].[30] N. Susuki et al. , The Hubble Space Telescope Cluster Supernova Survey. V. Improving theDark-energy Constraints above z > and Building an Early-type-hosted Supernova Sample Astrophys. J. (2012) 85 [arXiv:1105.3470].[31] M. Ganeshalingam, W. Li and A. V. Filippenko,
Constraints on dark energy with the LOSS SNIa sample , Mon. Not. Roy. Astron. Soc. (2013) 2240 arXiv:1307.0824 [astro-ph.CO].[32] M. Ganeshalingam et al. , Results of the Lick Observatory Supernova Search Follow-upPhotometry Program: BVRI Light Curves of 165 Type Ia Supernovae , Astrophys. J. Supp. (2010) 418.[33] C. Blake, E. Kazin, F. Beutler, T. Davis, D. Parkinson, S. Brough, M. Colless andC. Contreras et al. , The WiggleZ Dark Energy Survey: mapping the distance-redshift relationwith baryon acoustic oscillations , Mon. Not. Roy. Astron. Soc. (2011) 1707[arXiv:1108.2635 [astro-ph.CO]].[34] W. J. Percival et al. , Baryon Acoustic Oscillations in the Sloan Digital Sky Survey Data Release7Galaxy Sample , 2010, Mon. Not. Roy. Astron. Soc. 401 (2010), 2148, [arXiv:0907.1660].[35] F. Beutler, C. Blake, M. Colless, D. H. Jones, L. Staveley-Smith, L. Campbell, Q. Parker andW. Saunders et al. , The 6dF Galaxy Survey: Baryon Acoustic Oscillations and the Local HubbleConstant , Mon. Not. Roy. Astron. Soc. (2011) 3017 [arXiv:1106.3366 [astro-ph.CO]].[36] T. Delubac et al. [BOSS Collaboration],
Baryon Acoustic Oscillations in the Ly α forest ofBOSS DR11 quasars , arXiv:1404.1801 [astro-ph.CO].[37] D. L. Shafer and D. Huterer, Chasing the phantom: A closer look at Type Ia supernovae and thedark energy equation of state , Phys. Rev. D (2014) 063510 [arXiv:1312.1688 [astro-ph.CO]].[38] J.-F. Zhang, Y.-H. Li, X. Zhang, A global fit study on the new agegraphic dark energy model ,Eur. Phys. J. C., 73 (2013) 2280, [arXiv:1212.0300].[39] X. Yang, F. Y. Wang, Z. Chu,
Searching for a preferred direction with Union2.1 data , Mon.Non. Roy. Astron. Soc. , 437 (2014) 1840, [arXiv:1310.5211].[40] L. Hui and P. B. Greene,
Correlated Fluctuations in Luminosity Distance and the (Surprising)Importance of Peculiar Motion in Supernova Surveys , Phys. Rev. D (2006) 123526[astro-ph/0512159].[41] C. Gordon, K. Land and A. Slosar, Cosmological Constraints from Type Ia Supernovae PeculiarVelocity Measurements , Phys. Rev. Lett. (2007) 081301 [arXiv:0705.1718 [astro-ph]].[42] R. A. Vanderveld, E. E. Flanagan and I. Wasserman, Mimicking dark energy withLemaitre-Tolman-Bondi models: Weak central singularities and critical points , Phys. Rev. D (2006) 023506, [astro-ph/0602476].[43] S. February, J. Larena, M. Smith and C. Clarkson, Rendering dark energy void
Mon. Not. Roy.Astron. Soc. (2010) 2231, [arXiv:0909.1479].[44] G. R. Bengochea and M. E. De Rossi,
Dependence on supernovae light-curve processing in voidmodels , Phys. Lett. B (2014) 258, [arXiv:1402.3167 [astro-ph.CO]]. – 11 –
45] S. del Campo, I. Duran, R. Herrera and D. Pavon,
Three thermodynamically-basedparameterizations of the deceleration parameter , Phys. Rev. D (2012) 083509[arXiv:1209.3415 [gr-qc]].[46] V. Sahni, A. Shafieloo and A. A. Starobinsky, Model independent evidence for dark energyevolution from Baryon Acoustic Oscillations , arXiv:1406.2209 [astro-ph.CO]. – 12 –ata Set χ min /d.o.f. Ω m w w ( w . )JBP parametrizationConstitution 461 . /
394 0 . ± . − . ± . − . ± . . /
403 0 . ± . − . ± . − . ± . . /
554 0 . ± . − . ± . − . ± . . /
563 0 . ± . − . ± .
164 0 . ± . . /
577 0 . ± . − . ± . − . ± . . /
586 0 . ± . − . ± .
156 0 . ± . . /
583 0 . ± . − . ± . − . ± . . /
592 0 . ± . − . ± . − . ± . . /
394 0 . ± . − . ± . − . ± . . /
403 0 . ± . − . ± . − . ± . . /
554 0 . ± . − . ± . − . ± . . /
563 0 . ± . − . ± .
124 0 . ± . . /
577 0 . ± . − . ± .
630 0 . ± . . /
586 0 . ± . − . ± .
119 0 . ± . . /
583 0 . ± . − . ± . − . ± . . /
592 0 . ± . − . ± . − . ± . . /
394 0 . ± . − . ± . − . ± . . /
403 0 . ± . − . ± . − . ± . . /
554 0 . ± . − . ± . − . ± . . /
563 0 . ± . − . ± .
066 0 . ± . . /
577 0 . ± . − . ± .
648 0 . ± . . /
586 0 . ± . − . ± .
065 0 . ± . . /
583 0 . ± . − . ± . − . ± . . /
592 0 . ± . − . ± .
055 0 . ± . . /
394 0 . ± . − . ± . − . ± . . /
403 0 . ± . − . ± . − . ± . . /
554 0 . ± . − . ± . − . ± . . /
563 0 . ± . − . ± . − . ± . . /
577 0 . ± . − . ± . − . ± . . /
586 0 . ± . − . ± . − . ± . . /
583 0 . ± . − . ± . − . ± . . /
592 0 . ± . − . ± . − . ± . ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± Table 1 . Best fits for the free parameters using several data sets for the parametrizations of the EoSof dark energy.. Best fits for the free parameters using several data sets for the parametrizations of the EoSof dark energy.