Constraining the Dynamical Dark Energy Parameters

8/13/2019 Constraining the Dynamical Dark Energy Parameters

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Prepared for submission to JCAP

Constraining the dynamical darkenergy parameters: Planck-2013 vsWMAP9

B. Novosyadlyj, a O. Sergijenko, a R. Durrer, b V. Pelykh c

a Astronomical Observatory of Ivan Franko National University of Lviv,Kyryla i Methodia str., 8, Lviv, 79005, Ukraine

bUniversité de Genève, Département de Physique Théorique and CAP,24 quai Ernest-Ansermet, CH-1211 Genève 4, Switzerland

cYa. S. Pidstryhach Institute for Applied Problems of Mechanics and Mathematics,Naukova str., 3-b, Lviv, 79060, UkraineE-mail: [email protected] , [email protected] ,[email protected] , [email protected]

Abstract. We determine the best-t values and condence limits for dynamical dark en-ergy parameters together with other cosmological parameters by Markov chain Monte Carlotechnique on the base of different datasets which include WMAP9 and Planck-2013 resultson CMB anisotropy, BAO distance ratios from recent galaxy surveys, magnitude-redshift re-lations for distant SNe Ia from SNLS3 and Union2.1 samples and the HST determination of the Hubble constant. It is shown that the most precice determination of cosmological param-eters with the narrowest condence limits is obtained for the Planck+HST+BAO+SNLS3dataset. The best-t values and 2 σ condence limits for cosmological parameters in thiscase are Ωde = 0 .718 ± 0.022, w0 = − 1.15+0 .14

− 0.16 , c2a = − 1.15+0 .02

− 0.46 , Ωbh2 = 0 .0220 ± 0.0005,Ωcdm h2 = 0 .121 ± 0.004, h = 0 .713 ± 0.027, ns = 0.958+0 .014

− 0.010 , As = (2 .215+0 .093− 0.101 ) · 10− 9,

τ rei = 0 .093+0 .022− 0.028 . For this dataset, the ΛCDM model is just outside the 2 σ condenceregime, while for the dataset WMAP9+HST+BAO+SNLS3 the ΛCDM model is only 1σaway from the best t. The tension in the determination of some cosmological parameters onthe basis of the two CMB datasets WMAP9 and Planck-2013 is highlighted.

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Contents

1 Introduction 1

2 Scalar eld model of dark energy and cosmological background 1

3 Observational data and method 5

4 Results and discussion 6

5 Conclusion 12

1 Introduction

The year 2013 is epochal for cosmology owing to the publication of observational data from twospace observatories, the nal WMAP results (at the end of 2012) [1, 2] and rst cosmologicalPlanck results [ 3–5]. WMAP has started the precision epoch of cosmology and Planck has agood chance to improve it considerably. Cosmologists are trying to use these data to answerimportant questions especially about the dark sector of the Universe. In this paper we studythe nature of dark energy.

Most observational data obtained so far can not distinguish between a cosmologicalconstant and quintessence/phantom dark energy at a statistically signicant level (see e. g.books [6–8], special issue of the journal General Relativity and Gravitation [ 9] and citationstherein): the marginalized 1σ range of value of equation of state (EoS) parameter wde asrule covers the comparable area on both sides of wΛ = − 1. We have recently analyzed [ 10]the arbitrating power of available and expected observational data in distinction betweenquintessence and phantom types of dark energy and have studied prospects to improve thissituation. The data on CMB temperature anisotropy from the space observatory Planckand the parameters of baryon acoustic oscillations (BAO) extracted from advanced galaxiessurveys seemed to be most promising.

One of the models of dark energy, suitable for the problem of recognizing of its type, isa scalar eld with wde < − 1/ 3 that lls the Universe almost uniformly. One of the simplestscalar eld models of dynamical dark energy that can be quintessence, vacuum or phantomis the scalar eld with barotropic EoS [ 11–13]. The existence of analytical solutions for theevolution of such a scalar eld, their regularity and applicability for any epoch in the pastas well as in the future make it a useful model to investigate the general properties of darkenergy, especially to establish its type – quintessence, vacuum or phantom.

The goal of this paper is the estimation of parameters of scalar eld dark energy jointlywith other cosmological parameters on the base of different datasets including either WMAP9or Planck CMB anisotropy measurements, as well as the last data releases on BAO and SNeIa magnitude-redshift relations.

2 Scalar eld model of dark energy and cosmological background

We assume the standard paradigm of inationary cosmology: the Universe is lled withbaryonic (b) matter, cold dark matter (cdm), dark energy (de), neutrinos ( ν ) and cosmic

– 1 –



microwave background (CMB) radiation (r); its structure is formed by gravitational insta-bility from nearly scale invariant primordial perturbations generated during ination. Thetheory of post-ination evolution of the Universe including the dynamics of expansion, cos-mological nucleosynthesis and recombination, formation of CMB anisotropies and large scale

structure is well elaborated [ 6–8, 14–17] and implemented numerically [18–25] for computa-tion of theoretical predictions with subpercent accuracy. The model of dark energy has to bespecied.

We assume a scalar eld model of minimally coupled dark energy which describes equallywell quintessence and phantom dark energy with constant or variable EoS parameters. Thereexist different methods for specifying a scalar eld which can mimic these properties (seebooks cited above). In this paper we consider the scalar eld model with generalized linearbarotropic EoS pde = c2

a ρde + C , where c2a ≡ ˙ pde / ρde and C are arbitrary constants which

dene the dynamical properties of the scalar eld on a cosmological background, which isassumed to be spatially at, homogeneous and isotropic with Friedmann-Robertson-Walker(FRW) metric ds2 = c2dt2 − a2(t)δ αβ dxα dxβ , where a(t) is the scale factor normalized to 1

today and we set c = 1 . Here and below () ≡ d/dt . The differential form of energy-momentumconservation law for such model of dark energy has analytical solution, which for density ρdeand EoS parameter wde ≡ pde /ρ de are given by [11, 13]:

ρde = ρ(0)de

(1 + w0)a − 3(1+ c2a ) + c2

a − w0

1 + c2a

, (2.1)

wde = (1 + c2

a )(1 + w0)1 + w0 − (w0 − c2

a )a3(1+ c2a ) − 1, (2.2)

where ρ(0)de and w0 are the density of dark energy and EoS parameter at the current epoch.

Clearly ρde and pde are analytical functions of a for any values of the constants c2a andw0. The constant C from generalized linear barotropic EoS is connected with others byrelation C = ρ(0)

de (w0 − c2a ). As the Universe expands the energy density of such scalar eld

monotonically decreases with a when w0 > − 1 (dρde /da < 0) and increases with a whenw0 < − 1 (dρde /da < 0). The rst case corresponds to quintessence, in the second case it isphantom dark energy. For some combinations of values of c2

a and w0 the energy density ρdechanges sign from positive to negative if c2

a < w0 and w0 > − 1, and from negative to positiveif c2

a > w0 and w0 < − 1. The EoS parameter at the moment a(ρde =0) has a discontinuityof the second kind (see for details [ 11, 13]). The constants c2

a and w0 dening the type andthe dynamics of the scalar eld are the parameters of our dark energy model which must bedetermined jointly with other cosmological parameter. Both have a clear physical meaning:w0 is the EoS parameter wde at the current epoch, c2a is asymptotic value of the EoS parameterwde at early times ( a → 0) for c2

a > − 1 and in the far future ( a → ∞ ) for c2a < − 1. The

asymptotic value of wde in the opposite time direction is − 1 in both cases. Equations ( 2.1)-(2.2) illustrate also that the scalar eld becomes Λ or vacuum-like, pde = − ρde = const whenw0 or c2

a or both are equal to − 1. If w0 = c2a = − 1, we have dark energy with a constant EoS

parameter, wde =const. If w0 = − 1, the value of c2a is undetermined.

The following subclass of our dark energy model leads to ρtot < 0 at some time in thepast: w0 < − 1, c2

a > w0. In this paper we exclude this possibility. The further analysisinvolves 3 subclasses of models without peculiarities in the past: i) w0 > − 1, c2

a > − 1;ii) w0 > − 1, c2

a < − 1; iii) w0 < − 1, c2a < − 1, c2

a < w0. The rst one corresponds to

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quintessence, the third to phantom, while the second to the model which mimics Λ at thebeginning of expansion, is quintessence in past but will become phantom in future.

The Friedmann equations for our model of the Universe are

H ≡ aa = H 0 Ωr a

−

4 + Ω m a−

3 + Ω def (a), (2.3)

q ≡ −aaa2 =

12

2Ωr a − 4 + Ω m a − 3 + (1 + 3 wde )Ωde f (a)Ωr a − 4 + Ω m a − 3 + Ω de f (a)

, (2.4)

where f (a) = ρde /ρ (0)de , H 0 is Hubble constant and Ωr ≡ ρ(0)

r /ρ (0)tot , Ωm ≡ ρ(0)

m /ρ (0)tot , Ωde ≡

ρ(0)de /ρ (0)

tot are the dimensionless density parameters for radiation, matter and dark energycomponents correspondingly at the current epoch ( ρ(0)

tot ≡ ρ(0)r + ρ(0)

m + ρ(0)de ). The matter density

parameter is the sum of cold dark matter, baryons and active neutrinos, Ωm ≡ Ωcdm +Ω b+Ω ν .We follow [5], which includes a minimal-mass normal hierarchy for the neutrino masses: asingle massive eigenstate with mν = 0 .06 eV. It gives very small contributions into current

matter density component, Ων ≈ m ν / 93.04h2 eV ≈ 0.0006/h 2 , where h ≡ H 0/ 100km/s ·Mpc.The current density of thermal radiation (CMB) is also very small, Ωr = 16πGa SB T 40 / 3H 20 =2.49 · 10− 5h − 2, where T 0 is the current CMB temperature assumed here and below to be2.7255K. The rst Friedmann equation ( 2.3) today yields the constraint Ωr + Ω m + Ω de = 1(vanishing curvature).

The scalar eld causes accelerated expansion when |(1 + 3 wde )Ωde f (a)| > Ωm a − 3 anddenes the future of the Universe. Integrating ( 2.3) over a we can obtain the a − t relationfor the model,

t(a) = a

0

da ′

a ′ H (a ′ ). (2.5)

The results for nine sets of values w0 and c2a and xed all other cosmological parameters

at the values ( H 0 = 70 km/s ·Mpc, Ωm = 0 .3, Ωde = 0 .7) are shown in Fig. 1. One seesthat a scalar eld with barotropic EoS can describe most possible scenarios of the post-ination dynamics of expansion which are predicted by modern cosmology: 1) decelerated,accelerated, eternal exponential expansion (eternal late de Sitter ination); 2) decelerated,accelerated, eternal power law expansion (eternal late quasi de Sitter ination); 3) decelerated,accelerated, decelerated expansion, turn around, collapse, Big Crunch (BC) singularity ( a →0) is reached within nite time; 4) decelerated, accelerated, superfast expansion, Big Rip(BR) singularity ( a → ∞ ) is reached within nite time. The time of BC singularity canbe estimated as twice the turn around time, tB C = 2 t ta , and tta is the integral ( 2.5) withupper limit ata ≈ [(1 + w0)/ (w0 − c2

a )]1/ 3(1+ c2a ) [11, 12]. Among the models in Fig. 1 only

model 4 with quintessence behavior (dash-three dotted line) has a BC singularity in 170 Gyrs(a ta ≈ 9.8). The BR singularity is reached for all models with a phantom scalar eld withinnite time estimated as [ 13]

tBR − t0 ≈ 23

1H 0

1|1 + c2

a | 1 + c2a

(1 + w0)Ωde. (2.6)

For the phantom models 6 to 9 with scalar eld in Fig. 1 it will be reached approximately in25, 35, 80 and 500 Gyrs correspondingly.

The three parameters of our scalar eld with barotropic EoS, Ωde , w0 and c2a , are suf-

cient to describe the different evolutionary tracks of the homogeneous Universe, but not

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Figure 1 . Dependences of scale factors on time, a(t), for cosmological models with quintessencescalar eld as dark energy with w0 = − 0.8 and c2

a = 0 .0 (dotted line, 1), − 0.7 (dashed line, 2), − 0.8(dash-dotted line, 3), − 0.9 (dash-three-dotted line, 4) and with phantom dark energy with w0 = − 1.2and c2

a = − 2.0 (dotted line, 6), − 1.5 (dashed line, 7), − 1.1 (dash-dotted line, 8) and w0 = − 1.05 andc

2a = − 1.01 (dash-three-dotted line, 9). For the ΛCDM model a(t) is shown by thick solid line (5). In

all models Ωm = 0 .3, Ωde = 0 .7, H 0 = 70 km/s ·Mpc. The right panel corresponds to the lower leftcorner of the left panel. In the left panel at t > 25 Gyrs the lines are ordered as (from top to bottom):6, 7, 8, 9, 5, 1, 2, 3, 4; in the right panel they are in order: 1, 2, 3, 4, 5, 9, 8, 7, 6.

determine the scalar eld evolution φ. Once functional form of the scalar eld LagrangianLde (X, U ), where X (φ) = φ2/ 2 is kinetic term and U (φ) is potential, is dened, then thepotential U (φ) and eld variable φ(a) can be reconstructed (see [11, 13] for details).

Also, to analyze the gravitational instability of scalar eld in the context of the formationof structure of the Universe we need to know the effective sound speed c2s = δpde /δρ de . For

a given Lagrangian it is easily computed, since c2s = L,X / (L ,X + 2X L ,XX ), where L,X ≡

∂ L/∂X . For example, for a canonical Lagrangian Lde = ± X − U (“+ ” for quintessence, and “− ” for phantom) the effective sound speed is equal to the speed of light.

We assume also that large scale structure of the Universe is formed from Gaussian,adiabatic scalar perturbations generated in the early Universe. The initial power spectrum of density perturbations of all components is a power-law, P i (k) = As kn s , where A s and n s arethe amplitude and the spectral index ( k is wave number) which must be determined jointlywith other cosmological parameters. The scalar eld is perturbed too, the system of lineardifferential equations for the evolution of quintessence and phantom scalar eld perturbationsand their numerical solutions are studied in Refs. [ 11–13]. The main conclusions are asfollows: (i) the amplitude of scalar eld density perturbations at any epoch depends stronglyon the parameters of the barotropic scalar eld Ωde , w0, c2

a and c2s ; (ii) although the density

perturbations of dark energy at the current epoch are signicantly smaller than matter densityperturbations, they leave noticeable imprints in the matter power spectrum, which can beused to constrain the scalar eld parameters.

The assumed cosmological model contains 7 free parameters: w0, c2a , Ωb, Ωcdm , H 0, A s ,

n s , which are subject of joint determination from observational data. Ωde is determined bythe constraint Ωde + Ωm = 1 .

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3 Observational data and method

Using expressions ( 2.1)-(2.3) we can compute the “luminosity distance - redshift“ or “angulardiameter distance - redshift” relations to constrain the above mentioned parameters by com-

parison with the corresponding data on standard candles (supernovae type Ia, γ -ray bursts orother) and standard rulers (positions of the CMB acoustic peaks, baryon acoustic oscillations,X-ray gas in clusters or other). The linear power spectrum of matter density perturbationscan be computed by numerical integration of the linearized Einstein-Boltzmann equations[14–17] using publicly available code CAMB [ 21, 22] with the corresponding modications forthe barotropic scalar eld as dark energy,

P lin (k) = P i (k)T 2m (k; Ωb, Ωcdm , Ωde , w0, c2a ) .

Here T m (k; Ωb, Ωcdm , Ωde , w0 , c2a ) is the transfer function of matter density perturbations which

depends also on the parameters listed after the semicolon. We use the modied CAMB codealso to compute the angular power spectra of CMB temperature anisotropies C T T

ℓ for compar-ison with WMAP9 and Planck data to constrain the cosmological and DE parameters men-tioned above. The calculation of CMB anisotropies requires also the knowledge of reionizationhistory of the Universe, which depends on complicated non-linear physics of star formation.It is parameterized by the value of optical depth from the current epoch to decoupling causedby Thomson scattering. It is denoted by τ rei and added to the list of parameters tted to thedata. The complete list of parameters which we will determine contains 9 parameters

Θk : Ωde , w0, c2a , Ωb, Ωcdm , H 0, As , ns , τ rei ,

8 of which are free since Ωb + Ω cdm + Ω de + Ω ν = 1 for the spatially at cosmological modelconsidered here. To determine their best-t values and condence ranges we use the following

observational data:

1. CMB temperature uctuations angular power spectra from WMAP9 [ 1] and Planck2013 results [4];

2. Hubble constant measurements from Hubble Space Telescope (HST) [31];

3. BAO data from the galaxy surveys SDSS DR7 [27], SDSS DR9 [26], 6dF [28];

4. Supernovae Ia luminosity distances from SNLS3 compilation [ 29] and Union2.1 [30]compilations.

Throughout the paper we will use the combination of the Planck temperature powerspectrum with the WMAP9 polarization likelihood [1] and will refer to this CMB data com-bination as Planck (see for details [ 4]). It is obvious that WMAP9 includes these data also.

The best-t parameters correspond to the model with maximal likelihood function

L(x ; Θk ) = exp −12

(x i − x thi )C ij (x j − x th

j ) . (3.1)

The condence ranges correspond to the marginalized posterior functions

P (Θ k ; x ) = L(x ; Θk ) p(Θk )

g(x ) , (3.2)

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− 1 . 6 − 1 . 2 − 0 . 8 − 0 . 4

c

2

a

− 1 . 6

− 1 . 2

− 0 . 8

w

0

0 . 4 5 0 . 6 0 0 . 7 5

Ω

d e

− 1 . 6

− 1 . 2

− 0 . 8

− 0 . 4

c

2

a

− 1 . 6 − 1 . 2 − 0 . 8

w

0

− 1 . 6 − 1 . 2 − 0 . 8 − 0 . 4

c

2

a

− 1 . 6

− 1 . 2

− 0 . 8

w

0

0 . 5 0 . 6 0 . 7 0 . 8

Ω

d e

− 1 . 6

− 1 . 2

− 0 . 8

− 0 . 4

c

2

a

− 1 . 6 − 1 . 2 − 0 . 8

w

0

− 1 . 2 − 0 . 8 − 0 . 4

c

2

a

− 1 . 2 5

− 1 . 0 0

− 0 . 7 5

w

0

0 . 6 8 0 . 7 2 0 . 7 6

Ω

d e

− 1 . 2

− 0 . 8

− 0 . 4

c

2

a

− 1 . 2 5 − 1 . 0 0 − 0 . 7 5

w

0

− 1 . 5 0 − 1 . 2 5 − 1 . 0 0 − 0 . 7 5

c

2

a

− 1 . 4

− 1 . 2

− 1 . 0

w

0

0 . 6 7 5 0 . 7 0 0 0 . 7 2 5 0 . 7 5 0

Ω

d e

− 1 . 5 0

− 1 . 2 5

− 1 . 0 0

− 0 . 7 5

c

2

a

− 1 . 4 − 1 . 2 − 1 . 0

w

0

Figure 2 . One-dimensional marginalized posteriors (solid lines) and mean likelihoods (dotted lines)for Ωde , w0 and c2

a (top subpanels in each gure); color subpanels show two-dimensional mean likeli-hood distributions in the planes Ωde − w0 , Ωde − c2

a and w0 − c2a , where solid lines show the 1σ and

2σ condence contours. Left plots are for datasets including WMAP9 and right ones are for datasets

including Planck. Top plots are for CMB alone data, bottom ones for CMB+HST+BAO data.

functions are similar. 2 σ condence ranges for h and τ rei determined from WMAP9 andPlanck data are wide, about 30% of the mean value. The dark energy parameters are ratherbadly constrained by CMB data alone, the well-known Ωde − wde degeneracy is illustrated bygure 2: the marginalized likelihood ( 3.1) and posterior ( 3.2) functions strongly diverge andhave different shapes and extrema. For example, the maximum of the marginalized likelihoodfunction for w0 in the case of Planck CMB data lies in the phantom range ( ≈ − 1.23), whilethe maximum of the marginalized posterior function lies in the quintessence range ( ≈ − 0.96).In the case of WMAP9 data both lie in the quintessence range, ≈ − 0.81 and ≈ − 0.75 cor-

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mentioned above for baryon and dark matter density parameters remains present but issomewhat less severe: the best-t values for WMAP9+HST+BAO dataset are at the limitsof the 2σ condence ranges for Planck+HST+BAO dataset.

The SNe Ia magnitude-redshift relation is the key observational evidence for the ex-

istence of dark energy at very high condence level, but it is not sufficiently accurate andstatistically complete to discriminate between different types of dark energy, quintessence orphantom for example. Moreover, some tension exists between distance moduli obtained us-ing different lightcurve tters applied to the same SNe Ia (for example, SALT2 [34] andMLCS2k2 [35]). This has already been highlighted and analyzed in [ 36, 37], but up tonow we have no decisive arguments to favor one of the proposed lightcurve tters. InRefs.[13, 38] it was shown that the dataset with SNe Ia distance moduli determined withthe MLCS2k2 tter prefers quintessence dark energy, while the one with SALT2 appliedto the same supernovae prefers phantom dark energy. To avoid this ambiguity we use thehigh-quality joint sample of 472 SNe Ia compiled by [ 29] and denoted here as SNLS3. Forthese supernovae the updated versions of two independent light curve tters, SiFTO [39] and

SALT2 [34], have been used for the distance estimations. We denote the two datasets in-cluding this compilation as WMAP9+HST+BAO+SNLS3 and Planck+HST+BAO+SNLS3.To evaluate the reliability of the parameters and their condence intervals based on thesedatasets we use also other homogeneous sample of distance module - redshift measurementsfor 580 SNe Ia from the Union2.1 compilation. We denote the datasets with these supernovaas WMAP9+HST+BAO+Union2.1 and Planck+HST+BAO+Union2.1. The results of theMCMC determination of cosmological parameters from these four datasets are presented inthe table 2 and gure 3. We denote the sets of best-t parameters Ωde , w0 , c2

a , Ωbh2, Ωcdm h2,h, n s , As and τ rei by p 1, p 2, p 3 and p 4 for the WMAP9 + HST + BAO + SNLS3, WMAP9+ HST + BAO + SN Union2.1, Planck + HST + BAO + SNLS3 and Planck + HST + BAO+ SN Union2.1 datasets correspondingly.

Let us rst compare the cosmological parameters ( Ωbh2, Ωcdm h2, ns , As , h, τ rei ) fromthe datasets with SNLS3 and Union2.1 SNe Ia (2, 3 vs 4, 5 and 6, 7 vs 8, 9 columns of table2). One nds that for the same non-SN Ia data (CMB+HST+BAO), the best-t and meanvalues of these parameters are practically identical. The condence limits are different onlyfor the Hubble parameter h and the optical depth to reionization τ rei : they are narrower indetermination with SNLS3 moduli distances than with Union2.1. No tension between SNLS3and Union2.1 data in the determination of the best parameters is found.

Next we compare the results for the parameters Ωbh2, Ωcdm h2 , ns , As , τ rei from theWMAP9+HST+BAO+SNe Ia and Planck+HST+BAO+SNe Ia datasets (SNe Ia here de-notes either SNLS3 or Union2.1). Clearly, the SNe Ia data does not inuence results of theseparameter in a signicant way: this follows from the comparison of columns 6 and 7 of table 1with columns 2-5 s of table 2 as well as columns 8 and 9 columns of table 1 with columns 6-9of table 2. We also note that inclusion of SNe Ia data reduces slightly the Planck-WMAP9tension mentioned above: best-t values of baryon and dark matter density parameters de-termined from WMAP9+HST+BAO+SNe Ia are outside the 1 σ condence limits but wellinside the 2 σ range from the Planck+HST+BAO+SNe Ia datasets. The Hubble parame-ter is determined reliably by the datasets including supernovae: the 2 σ limits consist now4.0%, 4.4%, 3.8% and 4.2% of best-t values of h determined from the WMAP9+...+SNLS3,WMAP9+...+Union2.1, Planck+...+SNLS3, Planck+...+Union2.1 dataset correspondingly.The dataset Planck+HST+BAO+SNLS3 is the most self-consistent and the most accurate.

Let us now return to the determination of the dynamical dark energy parameters, see

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Figure 4 . Theoretical predictions of models with parameters p i from table 2 versus observational datafor the CMB power spectrum of temperature uctuations (left top panel), BAO’s (right top panel),SNe Ia distance moduli (left bottom panel) and the power spectrum of matter density perturbationsdeduced from luminous red galaxies (right bottom panel).

Fig. 3. First of all we note that all datasets with SNe Ia moduli distance - redshift relationsprefer phantom dark energy. In the case of WMAP9+...+SNLS3 and WMAP9+...+Union2.1datasets the phantom divide line is within the 1 σ condence limits of w0. In the case of Planck+...+Union2.1 dataset it is within the 2 σ and for the Planck+HST+BAO+SNLS3dataset it is outside the 2 σ condence limits for w0. Adding the SNe Ia data improves thedetermination of all the dark energy parameters. This follows from a comparison of theresults presented in tables 1-2 and gures 2-3. The 2σ condence ranges of Ωde , w0 andc2

a are narrowest for the dataset Planck+HST+BAO+SNLS3: 3.1%, 12% and 20% of theirbest-t values correspondingly. For WMAP9+HST+BAO+SNLS3 the ranges are 3.1%, 14%

and 25.6%.In gure 4 we compare the theoretical predictions of models with the parameters p i from

table 2 with observational data on the CMB power spectrum of temperature uctuations[1, 4], on BAO’s [26–28], on SNe Ia distance moduli [30] and on the power spectrum of matter density perturbations [ 40] deduced from luminous red galaxies. All models matchthe observational data well (the lines are superimposed), despite the somewhat differentparameters of dynamical dark energy. It is interesting to study the difference of the darkenergy dynamics and the expansion history of the Universe for these cases.

The evolution of dark energy EoS parameter wde (a) and dark energy density in units of the current total one, ρde (a)/ρ (0)

tot , as well as the rate of expansion of the Universe H (a) and

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Figure 5 . Left panels: the evolution of the EoS parameter (top) and the dark energy density (bottom)with the parameters from table 2. Right panels: the evolution of the Hubble parameter (top) and thedeceleration parameter (bottom) of the Universe for the cosmological models with best-t parametersp i . The symbols show the observational data on H (a) from [41].

deceleration parameter q (a) in the cosmological models with sets of best-t parameters p iare presented in gure 5. The dynamics of expansion of the Universe in the past ( a < 1) ispractically indistinguishable for all models while in the future ( a > 1) it will be quite different.The data on H (a) [41] are well matched by the models. The dark energy parameters weresignicantly different in the past and will be so in the future. In the models with p 1, p 2, p 3and p 4 parameters the Big Rip singularity ( 2.6) is reached in 73.4, 55.0, 72.4 and 27.8 Gyrscorrespondingly.

The results presented in tables 1 and 2 are in agreement with other determinations, inparticular with [ 2, 5, 42, 43]. Small differences between best-t values of some parametersare due to i) the statistical nature of MCMC technique, ii) the difference of the dark energymodels and iii) differences in the sets of observational data and priors. Our best determination(p 3) gives Ωm = 0 .281 ± 0.012 for matter density parameter at 1 σ and wde = − 1.169 ± 0.069for dark energy EoS parameter at current epoch. This means that dark energy EoS parameterdiffers from the cosmological constant value of − 1 by more than 2 σ, this is in agreement withsimilar study in [43], which uses the combination of the 1.5 year Pan-STARRS1 supernovaeIa measurements with Planck+HST+BAO and ’excludes’ the ΛCDM model of dark energyin a at Universe at the level of 2.4 σ (with Ωm = 0 .277 ± 0.012, wde = − 1.186 ± 0.076).

5 Conclusion

We have determined the best-t values and the condence limits for parameters of cosmolog-ical models with dynamical dark energy using the MCMC technique on the basis of differentdatasets, which include the results from the nal WMAP data release and the Planck-2013data, the type Ia supernovae samples SNLS3 and Union2.1, the updated BAO measurements

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together with the HST prior on the Hubble constant. The results, presented in tables 1 and2, and gures 2 and 3, can be summarized as follows:

1) In the class of spatially at models of the Universe both WMAP and Planck dataalone prefer dark energy density dominated models at a very high level of condence. In

the class of dynamical dark energy models WMAP9 data alone prefer quintessence, but Λand phantom models are within 1 σ condence limits of wde . The Planck 2013 data alone, incontrary, prefer phantom models of dark energy, but the condence level of this preference islow, Λ and quintessence models are within the 1 σ condence limits for wde . The condencelimits of the dark energy parameters are narrower for the Planck data than for WMAP9.

2) Adding HST and BAO data to WMAP9 and Planck 2013 data improves the ac-curacy of the determinations of Ωbh2, Ωcdm h2, ns and As : their 2σ condence limits arethen 4.0%, 5.1%, 2.3% and 6.0% of the corresponding best-t values for WMAP9 data and2.3%, 3.3%, 1.3% and 5.0% for the Planck 2013 results. Both WMAP9 and Planck datatogether with HST+BAO prefer a phantom scalar eld model of dark energy. In the caseof WMAP9+HST+BAO, the ΛCDM model and quintessence dark energy ( w0 ≥ − 1) are

inside the 1 σ condence limits of w0. For Planck+HST+BAO the Λ model is outside the 1 σcondence limits of w0 but still inside the 2 σ range. For the Planck+HST+BAO dataset thecondence limits for the dynamical dark energy parameters, Ωde , w0 and c2

a are signicantlynarrower and constitute 4.0%, 18.0% and 16.8% of the best-t values accordingly.

3) Adding nally supernova data, the SNe Ia samples SNLS3 or Union2.1, to WMAP9+HST+BAO or Planck+HST+BAO increases the precision of the Hubble constant and of the dynamical dark energy parameters. In all combinations of WMAP9+HST+BAO andPlanck+HST+BAO datasets with SNLS3 and Union2.1, the phantom scalar eld model of dark energy is preferred. The most reliable determination of cosmological and dynamicaldark energy parameters is obtained from the Planck+HST+BAO+SNLS3 dataset. The best-t values of the parameters and their 2 σ condence limits are: Ωde = 0 .718 ± 0.022, w0 =

− 1.15+0 .14− 0.16 , c2a = − 1.15+0 .02− 0.46 , Ωbh2 = 0 .0220 ± 0.0005, Ωcdm h2 = 0 .121 ± 0.004, h = 0 .713 ±0.027, ns = 0 .958+0 .014

− 0.010 , As = (2 .215+0 .093− 0.101 ) · 10− 9 , τ rei = 0 .093+0 .022

− 0.028 . The ΛCDM modelis disfavored by this dataset at 2 σ condence. The dataset WMAP9+HST+BAO+SNLS3disfavors the Λ-model only at 1 σ.

4) The results presented in the tables 1 and 2 highlight a tension between WMAP9 andPlanck-2013: the best-t and mean values of the baryon and dark matter density parametersas well as the spectral index determined from datasets including WMAP9 are outside the 1 σlimits of the corresponding values determined from datasets with Planck 2013.

The CMB and matter density perturbations power spectra, the BAO distance ratios,the SN Ia distance moduli and the Hubble parameter at different redshifts computed for thecosmological models with best-t parameters p i (table 2) match well both, observational datawhich have been used in the MCMC search procedure and the data which have not been usedhere.

Acknowledgments

This work was supported by the project of Ministry of Education and Science of Ukraine(state registration number 0113U003059), research program “Scientic cosmic research” of the National Academy of Sciences of Ukraine (state registration number 0113U002301) andthe SCOPES project No. IZ73Z0128040 of the Swiss National Science Foundation. Authorsalso acknowledge the use of CAMB and CosmoMC packages.

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Constraining the Dynamical Dark Energy Parameters

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