Status on Bidimensional Dark Energy Parameterizations Using ... - arXiv

Status on bidimensional dark energy parameterizations using SNe Ia JLA and BAO datasets Celia Escamilla-Rivera1, 2, ∗ 1

arXiv:1605.02702v1 [astro-ph.CO] 9 May 2016

Departamento de F´ısica, Universidade Federal do Esp´ırito Santo, Av. Fernando Ferrari 514, Vit´ oria - ES, Brazil 2 Mesoamerican Centre for Theoretical Physics (ICTP regional headquarters in Central America, the Caribbean and Mexico), Universidad Aut´ onoma de Chiapas, Carretera Zapata Km. 4, Real del Bosque (Ter´ an), 29040, Tuxtla Guti´errez, Chiapas, M´exico. Using current observations forecast type Ia supernovae (SNe Ia) Joint Lightcurve Analysis (JLA) binned and baryon acoustic oscillations (BAO), in this paper we investigate six bidimensional dark energy parameterizations in order to explore which has more constraining power. Our results indicate that for Taylor series-like parameterizations at second order in redshift z, the tension (σ-distance) between these data sets seems to be reduced and their behaviour are 0 in the radiation era. However, this ansatz has some corrections when radiation becomes important [20].

and this condition is always satisfied when zc = 0.5, i.e. ac = 2/3. Arguably, the choice zc ≈ 0.5 offers a good

To perform the cosmological test we will employ a binned sample from the most recent SNe Ia catalog avail-

where w1 is called bending parameter and characterized the redshift where an approximately constant EoS turns over to a different behaviour. Using Eq.(22) in Eq.(5) we obtain the following evolution h 3 1+ 1+w

E(z)2 = Ωm (1 + z)3 + (1 − Ωm )(1 + z)

IV.

STATISTICAL ANALYSIS

A.

Analysis using SNe Ia data

w0 1 ln (1+z)

i

4 The χ2 function for the BAO data is defined as: able: the JLA (Joint Lightcurve Analysis) [10]. This compilation shows the same trend as using the full cataT χ2BAO (θ) = XTBAO C−1 (31) log itself, for this reason we will use this binned sample BAO XBAO , which can be found in the above reference and explicitly where XBAO is given as in [21]. This dataset consist of NJLA = 31 SN events distributed over the redshift interval 0.01 < z < 1.3. The   rs (zd ) statistical analysis of the JLA binned data can be done DV (0.106,Ωm ;w0 ,w1 ) − 0.336   rs (zd ) using the definition of the modulus distance: ) − 0.1126  , (32) XBAO =  DV (0.35,Ω m ;w0 ,w1 rs (zd )   ˆ z ) − 0.07315 DV (0.57,Ωm ;w0 ,w1 d˜ z E −1 (˜ z , Ωm ; w0 , w1 ) + µ0 , µ(zi , µ0 ) = 5 log10 (1 + z) 0 and (24) C−1 (33) BAO = diag(4444, 215156, 721487), where (w0 , w1 ) are the free parameters of the model. The best fits will be obtained by minimizing the quantity χ2SNJLA

=

N JLA X i=1

2

[µ(zi , Ωm ; µ0 , w0 , w1 ) − µobs (zi )] , (25) 2 σµ,i

2 where the σµ,i are the measurements variances.

B.

Analysis using BAO data

The compilation of BAO observations used in this paper is presented in [22–24]. Some require quantities are defined via the ratio rs (zd ) dz ≡ , DV (z)

(26)

where rs (zd ) is the comoving sound horizon at the baryon dragging epoch ˆ ∞ c cs (z) rs (zd ) = dz , (27) H0 zd E(z) with c the light velocity, cs the sound speed and zd the drag epoch redshift. By definition the dilation scale DV (z) is  2 DV (z, Ωm ; w0 , w1 ) = (1 + z)2 DA

cz H(z, Ωm ; w0 , w1 )

1/3 ,

(28) where DA is the angular diameter distance: ˆ z 1 c d˜ z DA (z, Ωm ; w0 , w1 ) = . (29) 1 + z 0 H(˜ z , Ωm ; w0 , w1 ) Through the comoving sound horizon, the distance ratio dz is related to the expansion parameter h (defined . such that H = 100h) and the physical densities Ωm and Ωb . Specifically, we have  rs (zd ) = 153.5

Ωb h2 0.02273

−0.134 

Ωm h2 0.1326

−0.255 Mpc,

(30) with Ωm = 0.295 ± 0.304 and Ωb = 0.045 ± 0.00054 [10].

C.

Discussion

We will employ the maximum likelihood method in order to determine the best fit values of the parameters w0 and w1 for the six parameterizations described. The ΛCDM case can be set with Om as an independent parameter and compute its best fit. The total likelihood for joint data analysis is expressed as the sum of each dataset, i.e., χ2total = χ2SNe IaJLA + χ2BAO .

(34)

Finally, to compare results and test the tension among datasets, we compute the so called σ-distance, dσ , i.e. the distance in units of σ between the best fit points of the SNe Ia, BAO and the total compilation SNe Ia + BAO and the best fit points of each parameterization in comparison to the ΛCDM model. Following [25], the σ-distance is calculated by solving √ 1 − Γ(1, |∆χ2σ /2|)/Γ(1) = erf(dσ / 2). (35) For homogeneity and consistency our ‘ruler’ is in every case the total χ2 function Eq.(34), and our prescription is the following [26]: if we want to calculate the tension between SNe Ia and SNe Ia+BAO and the best fit parameters ([w0 ,w1 ]) then the previous ∆χ2σ will be defined as χ2tot ([w0 , w1 ]SNeIa+BAO ) − χ2tot ([w0 , w1 ]SNeIa ); other cases follow this recipe. Looking our results regarding the σ-distances in Tables I- II we can verify that the tension between compilations seems to be reduced when we use the parameterizations that correspond to Taylor series-like of second order in z, as the BA and JBP models. These two parameterizations shows the minor σ-distance and are more compatible with ΛCDM than the other parameterizations (see Figures 1-2). Is important to address that this tension effect can change depending of the priors Ωm and Ωb as it was showed in [26], where the influence of these values combined with their systematic error can increase (if Ωm and Ωb decrease) or decrease (if Ωm and Ωb grows) the tension effect, but even with these variations, the tension remains reduced for the BA and JBP parameterizations

5 in comparison to Linear, CPL, LC and WP parameterizations. Furthermore, we found in our analysis that using the latest values over these priors [10, 24] in comparison to those reported by the Planck mission [27], the tension in each parameterization is reduced to < 1σ using the JLA binned data and the total compilation JLA + BAO. The LC case exclude this observation for the total sample. From Tables I- II we also notice that the best fits obtained by using the BA and JBP parameterizations are in better agreement with ΛCDM, around 1% of difference than using either of the other cases as we can see from the value of their σ-distances. In our bidimensional analysis, we observe that the confidence error over the second free parameter w1 are much larger in JBP parameterization in comparison to BA parameterization.

V.

CONCLUSIONS

dimensional ΛCDM model. All of them were tested using current observations as SNe Ia JLA binned and BAO data sets, together with the combinations of both (SNe Ia JLA + BAO). Furthermore, after performing the statistical tests, using the σ-distance recipe, our results indicate that for Taylor series-like parameterizations at second order in z (as BA and JBP models), the tension between these data sets are reduced for both parameterizations and their behaviour are 2 w0 = −0.827 ± 0.276, w1 = −2.529 ± 22.057 1.20 w0 = −1.046 ± 0.048, w1 = −0.171 ± 3.308 0.82 w0 = −0.889 ± 0.083, w1 = −0.610 ± 0.580 1.24 w0 = −1.013 ± 0.028, w1 = −0.149 ± 0.202 0.85

TABLE II: Dark energy parameterizations with best fits parameters using BAO and the JLA binned + BAO compilation data. In 3-5-columns it is show the σ-distance in comparison to ΛCDM model.

[26] C. Escamilla-Rivera, R. Lazkoz, V. Salzano and I. Sendra, JCAP 1109, 003 (2011) doi:10.1088/14757516/2011/09/003 [arXiv:1103.2386 [astro-ph.CO]]. [27] C. Burigana, C. Destri, H. J. de Vega, A. Gruppuso,

N. Mandolesi, P. Natoli and N. G. Sanchez, Astrophys. J. 724, 588 (2010) doi:10.1088/0004-637X/724/1/588 [arXiv:1003.6108 [astro-ph.CO]].