arXiv:1806.03538v1 [astro-ph.CO] 9 Jun 2018
Prepared for submission to JCAP
Constraints on barotropic dark energy models by a new phenomenological q(z) parameterization. Jaime Román-Garza,a,b,1 Tomás Verdugo,b Juan Magañac Verónica Mottac a Facultad
de Ciencias Físico Matemáticas, Universidad Autónoma de Nuevo León, San Nicolás de los Garza, México b Instituto de Astronomía, Universidad Nacional Autónoma de México, Apdo. postal 106, CP 22800, Ensenada, B.C, México c Instituto de Física y Astronomía, Universidad de Valparaíso, Avenida Gran Bretaña 1111, Valparaíso, Chile E-mail:
[email protected],
[email protected],
[email protected],
[email protected]
Abstract. In this paper, we propose a new phenomenological two parameter parametrization of q(z) to constrain barotropic dark energy models by considering a spatially flat FRW universe, neglecting the radiation component, and reconstructing the effective equation of state (EoS). This two free-parameter EoS reconstruction shows a non-monotonic behavior, pointing to a more general fitting for the scalar field models, like thawing and freezing models. We constrain the q(z) free parameters using the observational data of the Hubble parameter obtained from cosmic chronometers, the joint-light-analysis type Ia Supernovae sample and a joint analysis from these data. We obtain a value of q(z) today, q0 = −0.48+0.10 −0.11, and a transition redshift, zt = 0.71+0.12 (when the Universe change from an decelerated phase to an −0.12 accelerated one). The effective EoS reconstruction and the ω 0 -ω plane analysis pointed out a quintom dark energy, which is consistent with a non parametric EoS reconstruction, reported by other authors, and using the latest cosmological observations.
Keywords: dark energy theory, cosmological parameters
1
Corresponding author.
Contents 1 Introduction
1
2 Theoretical framework 2.1 Proposed parameterization for the deceleration parameter 2.2 The effective Equation of State
2 2 3
3 Observational data and methodology 3.1 Observational Hubble Data from cosmic chronometers. 3.2 Type Ia Supernovae 3.3 Fitting the data
5 5 6 7
4 Dynamical Dark Energy 4.1 The resulting EoS 4.2 Discriminating dark energy models
8 8 9
5 Summary
11
A The behavior of Ωm (z) and the singularities of ω
13
1
Introduction
Several cosmological observations point out that the Universe experiments a late-time acceleration [1]. This feature was evidenced for the first time by the observations of distant Type Ia Supernovae[(SNIa) 2, 3] and is one of the major puzzles in modern cosmology. In general, there are two ways to explain this mysterious cosmic phase: i) to postulate a fluid with negative pressure, the so-called dark energy (DE) into the canonical Einsteinian general relativity theory or ii) to modify the gravity laws. Between these two branches, numerous models have been proposed. Most of them can explain a wide range of the cosmological observations and distinguish among them is not a trivial problem. Despite of this, one simple model has been established as the standard one, the one with a cosmological constant associated to the quantum vacuum fluctuations. Nevertheless, the model have theoretical problems [4] which motivates further studies of alternative models [5]. For instance, some consider a dynamical DE involving scalar fields, for instance, quintessence [6–8], phantom [9, 10], quintom [11], and k-essence fields [12, 13]. Since the equation of state (EoS) of a dynamical DE evolves with time, it can be parameterized by a function of the scale factor (redshift, as proposed by [14, 15]) and explore its cosmological behavior. To study these models, the standard way is to calculate (in a background Cosmology) the Friedmann and Raychaudhuri equations to constrain their free parameters, related to the dynamics of the components of the Universe. Another model-independent approach is to investigate the cosmographic parameters that characterize the kinematics of the cosmic expansion. The advantage of this procedure is that the only assumption is the Cosmological Principle, i.e. an homogeneous and isotropic Universe, and it is not necessary to know its composition. Indeed, it is very common to consider the Hubble parameter, H ≡ a/a, ˙ and
–1–
the deceleration parameter, q(a) ≡ −¨ aa/a˙ 2 1 . However, other parameters such jerk and snap, which are higher order derivatives of the scale factor a, can be considered[e.g., 17]. By probing the cosmographic parameters using cosmological data, it is possible to associate them to a given dynamical DE entity and reconstruct its features as well as the Universe dynamics. In this vein, several authors have proposed a number of functions to parameterize the deceleration parameter q(z) (see for example [17–22] for recent studies) and reconstructed the features of any kind of dark energy. The motivation of the present work is to propose a new parameterization of the deceleration parameter as function of redshift, based only in the cosmological principle. The ansatz is a continuous and differentiable function that is valid from the matter domination epoch till the near future. We constrain the q(z) free parameter by performing a Bayesian analysis employing the latest compilation of observational Hubble data (hereinafter OHD) from cosmic chronometers and Type Ia Supernova. Using the mean value parameters, we reconstruct an effective equation of state associated to the dynamical dark energy. The paper is organized as follows. In Sec. 2 we state the theoretical framework and present the parametric equation of the deceleration parameter. Section 3 provides a description of the data sets and the methodology used to constrain the parameters of the deceleration parameter. The Sec. 4 presents the obtained EoS, and the tools to discriminate between different DE models. Finally, in Sec. 5 the remarks and conclusions are presented.
2
Theoretical framework
2.1
Proposed parameterization for the deceleration parameter
The deceleration parameter as function of H(z) is H˙ q =− 1+ 2 H
! ,
(2.1)
if q > 0 the Universe is at a decelerated phase, otherwise q < 0 corresponds to a accelerated phase. By integrating the eq. (2.1), the Hubble parameter can be written as: Z
z
H(z) = H0 exp 0
1 + q(z 0 ) 0 dz , 1 + z0
(2.2)
where H0 is the Hubble parameter at the present epoch and z = (1/a) − 1 is the redshift. The OHD suggest that q < 0 at the present epoch and q > 0 during an early epoch when the matter dominates as shown in ref. [23, 24]. The structure formation at this early epoch is explained by a decelerated phase, so the value of the deceleration parameter transit from positive in past to negative at the present. The parameterization of the deceleration parameter is a useful method to reconstruct cosmological parameters and constrain the dynamical evolution of the universe in a general scheme [25]. There are several parameterizations for q(z) reported in the literature, see refs. [17–23, 25–30]. We propose a new one as follows 2
2
q(z) = q1 + (q0 − q1 )(z + 1)ezc −(z+zc ) , 1
(2.3)
Alan Sandage claimed that the cosmic expansion can be determined by these two parameters at z=0 [16]
–2–
0.6 0.0 0.4
−0.2 −0.4
ω(z)
q(z)
0.2
0.0
−0.2
−0.8 −1.0 −1.2
−0.4
−1.4
−0.6 0.0
−0.6
0.5
1.0
z
1.5
2.0
2.5
0.0
0.5
1.0
z
1.5
2.0
2.5
Figure 1: Left panel.- Functional form of the proposed q(z) given by equation (2.3) for different (q0 ,zc ) values. Notice that both an accelerated and decelerated stage at z = 0 are allowed. Right panel.- Functional form of ω(z) calculated through equation(2.9) where, q0 and q1 are the values for the deceleration parameter at the present epoch, and at high redshift, respectively. We set q1 = 0.5 to consider the matter-dominated epoch of the Universe. The characteristic redshift, zc , is a free parameter related to the transition redshift, zt , i.e. the redshift at which the Universe underwent a transition from deceleration to an acceleration phase. This well behaved parametrization (figure 1) can reproduce a soft step transition, as well as changes in concavity in the deceleration parameter (notice that both an accelerated and decelerated stage at z = 0 are allowed), and facilitates the analytical reconstruction of other cosmological parameters like H(z). Note how combinations of q0 and zc can yield the same transition redshift. Substituting the equation (2.3) into the equation(2.2), we obtain the analytical expression for the Hubble parameter in terms of z: H(z) = H0 (z + 1)q1 +1 eξ[erf(z+zc )−erf(zc )] ,
(2.4)
√ 2 where ξ = ( π/2)(q0 − q1 )ezc , and erf(x) is the error function of x. This is the expression that is fitted to the data. 2.2
The effective Equation of State
With the metric for a spatially flat FRW space-time ds2 = −dt2 + a2 (t){dr2 + r2 dΩ2 },
(2.5)
considering a space-time filled with a non-relativistic component ρm and a barotropic fluid with an effective density ρeff and an effective pressure peff , the Einstein field equations in units of 8πG = c = 1 are obtained following ref. [4] as
3H 2 = ρm + ρeff , 2H˙ + 3H = −peff , 2
–3–
(2.6) (2.7)
and the effective EoS is written as ω=
peff . ρeff
(2.8)
Substituting (2.6) and (2.7) in (2.8), the EoS in terms of q(z) and H(z) is obtained following ref. [22] as ω(z) =
2 3
q(z) −
1 2
1 − Ωm,0 (1 + z)3
H0 H(z)
2 .
(2.9)
By substituting equations (2.3) and (2.4) in equation (2.9), we obtain the expression
ω(z) =
2 (q0 − q1 )(z + 1)exp(zc2 − (z + zc )2 ) , 3 1 − Ωm,0 (1 + z)1−q1 exp(−2ξ(erf(z + zc ) − erf(zc )))
(2.10)
The right panel of figure 1 shows how the EoS changes for different values of q0 and zc . The reconstruction of ω(z) yield distinct DE behaviors when the barotropic fluid is associated to a minimally coupled scalar field: quintessence (−1 ≤ ω(z) ≤ 1), phantom (ω(z) < −1) or quintom (where the DE component moves across the quintessence and phantom regions through two scalar fields)[see 4, and references therein]. In contrast with some ω(z) parameterizations analyzed in the literature [14, 15, 31–36], our EoS concavity changes from low to high z values if there is at least one inflexion point at z > 0.
0.2
q0
0.4 0.6 0.8 0.9
zc
0.6 0.3 0.0 0.675 0.700 0.725 0.750
h
0.8
0.6
q0
0.4
0.2
0.0
0.3
zc
0.6
0.9
Figure 2: 1D marginalized posterior distributions and the 2D 68%, 95%, and 99.7% confidence levels for the h, q0 and zc parameters for the joint analysis. The green star represents the joint analysis mean values.
–4–
3
Observational data and methodology
In this section we introduce the cosmological data and the methodology used to constraint the q(z) free parameters of the equation (2.3). 3.1
Observational Hubble Data from cosmic chronometers.
Several authors have shown that the OHD can be used to constrain cosmological parameters. There are two techniques to measure the cosmic expansion at different redshifts: using the baryon acoustic oscillation analysis or applying the differential age technique (DA) in cosmic chronometers, i.e. passive-early-type galaxies. This last method was proposed by [37] and measures H(z) using the following relation for two early-type galaxies separated by a small redshift interval ∆z 1 dz H(z) = − , (3.1) 1 + z dt where dz/dt is measured by estimating the differential age ∆t with the 4000Å break (D4000) feature in their spectra. We employ the latest OHD obtained from DA, which contains 31 data points covering 0 < z < 1.97, compiled by [38] and references therein. The figure-of-merit for the OHD is written as 31 X [H(zi ) − HDA (zi )]2 χ2OHD = , (3.2) 2 σ H i i=1 where H(zi ) is the theoretical Hubble parameter, HDA (zi ) is the observational one at redshift zi , and σHi its uncertainty.
0.6
0.4
250
0.2
q(z)
H(z)
200
150
0.0
−0.2 100 −0.4 50 −0.6 0.0
0.5
1.0
z
1.5
2.0
2.5
0.0
0.5
1.0
z
1.5
2.0
2.5
Figure 3: Fit to OHD (left panel) and the reconstructed q(z) (right panel) using the joint analysis constraints. The dashed and red shadow regions show the 1σ confidence limits estimated from a MCMC analysis. The black dashed line in the right panel represents the transitional redshift, zt , for the joint analysis mean values.
–5–
3.2
Type Ia Supernovae
The standard test to investigate the accelerating expansion is employing the observations of type Ia Supernovae at high redshifts. We use one of the latest compilations presented by [39], the so-called joint-light-curve-analysis (JLA) sample, that contains 740 points spanning a redshift range 0.01 < z < 1.2. The figure-of-merit for the JLA data is given by χ2JLA = (ˆ µ − µqz )† C−1 µ − µqz ), η (ˆ
(3.3)
where µqz = 5 log10 (dL /10pc) is the theoretical distance modulus for the q(z) parameterization and dL is the luminosity distance given by z
Z dL = c(1 + z) 0
dz0 . H(z 0 )
(3.4)
The observational distance modulus, µ ˆ, for the the JLA data reads as µ ˆ = m?b − (MB − α × X1 + β × C) ,
(3.5)
where m?b corresponds to the observed peak magnitude, α, and β are nuisance parameters. The X1 and C variables describe the time stretching of the light-curve and the Supernova color at maximum brightness respectively. The absolute magnitude MB is related to the host stellar mass, Mstellar , by the step function: MB =
if Mstellar < 1010 M , otherwise.
MB1 + ∆M
MB1
(3.6)
Finally, Cη is the covariance matrix2 of µ ˆ provided by [39], which takes into account several statistical and systematic errors in the SNIa data.
0.9 0.8 0.7 0.6
mz ϕz 1 Ωm (z) 5 Ωϕ (z) Ω ( )
0.5
Ω ( )
0.4 0.3 0.2 0.1 0.0
0.5
1.0
z
1.5
2.0
2.5
Figure 4: The matter and DE density parameter of the Universe and its ratio using the joint analysis mean values. The dashed regions show the 1σ confidence limits estimated from a MCMC analysis. 2
available at http://supernovae.in2p3.fr/sdss_snls_jla/ReadMe.html
–6–
3.3
Fitting the data
To estimate the values of the parameters q0 and zc from equation (2.3), a Markov Chain Monte Carlo (MCMC) Bayesian statistical analysis is performed using the Affine-Invariant MCMC Ensemble sampler from the emcee Python module [40]. The computations are running with 1500 steps to stabilize the estimations (burn-in phase), and 5000 MCMC steps using 600 walkers. We assume the following flat priors: h ∈ [0, 1], q0 ∈ [−1, 1], zc ∈ [0, 1], Mb1 ∈ [−20.0, −18.0], ∆M ∈ [−0.1, 0.1], α ∈ [0.0, 0.2], β ∈ [0.0, 4.0]. To assess the convergence of our analysis, a Gelman-Rubin test is employed. In figure 2 we show the results obtained for the joint analysis. The goodness of the fit is quantified by a total χ2 defined as: χ2T = χ2OHD + χ2JLA ,
(3.7)
where χ2OHD , and χ2JLA are calculated using equation(3.2) and equation(3.3). Thus, a joint Gaussian likelihood can be expressed as: Ljoint ∝ exp(−χ2T /2).
(3.8)
Where Ljoint is the product of the likelihood functions of each data set. The mean values of the fit are presented in table 1. In the left panel of figure 3 we show the OHD along with the function given by equation (2.4) using the mean values obtained from the fitting. In the right panel of the same figure is the reconstructed deceleration parameter.
0.0
ω dω/dz
2.0
−0.2 1.5
ω(z)
−0.4 1.0
−0.6
0.5
−0.8
0.0
−1.0
−0.5
−1.2
−1.0
−1.4 0.0
0.5
1.0
z
1.5
2.0
2.5
0.0
0.5
1.0
z
1.5
2.0
2.5
Figure 5: The ω reconstruction using the joint analysis mean values and its functional form (left panel). The horizontal black dashed line represent the phantom divide. The right panel shows the effective EoS (cyan) and its derivative with respect to z (magenta). The regions delimited by dashed lines show the 1σ confidence limits estimated from a MCMC analysis.
Along with the narrow constraints obtained with the joint analysis (see figure 2), we note an anticorrelation between zc and q0 parameters. This degeneracy has a mathematical origin, thus, as q0 becomes more negative, the transition redshift zt is greater (see figure. 1), which in turn increases zc . A numerical analysis of the roots of q(z) allowed us to estimate the value of the transition redshift, zt = 0.71+0.12 −0.12, for the joint dataset. This result is consistent with values reported in literature [17, 27, 30, 41, 42] that indicate the Universe passed from a decelerated phase to an accelerated one at z ≈ 0.7. The right panel of figure 3 illustrates
–7–
the reconstructed q(z) for the joint analysis constraints. Note that q0 = −0.48+0.10 −0.11, and the reconstructed Ωm (z) are in agreement with the dynamics of the standard cosmological model. The matter component is dominant with respect to the dark energy component for high redshift values, the opposite occurs at late times (see figure 4).
1.2 1.0 0.8
zc
0.6 0.4 0.2 0.0 −0.2 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1
0.0
q0
Figure 6: Decision regions in the parameter space and the 68%, 95%, 99.7% confidence levels for the q0 and zc parameters for the joint analysis constraints. The classification of the EoS (depending on the given ω value) is represented in different regions: green for quintesence models; purple for quintom models; red for EoS with singular points. The black star represents the joint analysis mean values for zc and q0 .
4 4.1
Dynamical Dark Energy The resulting EoS
The figure 5, left panel, presents the EoS constructed from the equation (2.10) using the parameter mean values and Ωm,0 = 0.27. Namely, we plot
ω(z) =
+0.20 2 +0.20 2 2 (−0.48+0.10 −0.11 − 0.5)(z + 1)exp((0.50−0.19 ) − (z + 0.50−0.19 ) ) . +0.20 3 1 − 0.3(1 + z)0.5 exp(−2ξ(erf(z + 0.50+0.20 −0.19 ) − erf(0.50−0.19 )))
(4.1)
Although the above equation is a well-behaved function, the denominator may be zero, leading to a singularity in the EoS (see next section). A way to overcome this problem is studying the derivative of the EoS [43]. From eq. (2.9), it is straightforward to show that
–8–
1 dω(z) 2 (1 − Ωm (z))(q(z)( z+1 − 2(z + zc )) + q1 (2(z + zc ) − = dz 3 (1 + z)(1 − Ωm (z))2
1 1+z ))
+ 3(q(z) − 12 )Ωm (z)
.
(4.2)
The equation ω and the derivative dω/dz are shown in the right panel of figure 5. Note that around z ≈ 1 the EoS changes concavity (inflexion point), producing a maximum in dω/dz. Besides, the first derivative of ω with respect z gives a value, as shown in figure 5, of dω/dz|z=0 = −0.97+0.37 −0.37, consistent with [41].
4.2
Discriminating dark energy models
The nature of DE is connected to the characteristics of the EoS. The reconstruction of the EoS by equation (2.9) may have singular points on its domain, i.e. it might diverge, which occurs when the denominator is equal to zero. To find the singular points we consider the next equation:
1 − Ωm,0 (1 + z)
3
H0 H(z)
2 = 0,
(4.3)
which can be written as: 1 − Ωm (z) = 0.
(4.4)
We expect Ωm (z) to be a monotonically increasing function from the present (at z = 0), to a matter dominated epoch when q(z) → q1 = 1/2 (see appendix A). As equation (2.3) asymptotically tends to q(z) ∼ q1 as z → ∞ [44], our EoS reconstruction is valid only from today to an epoch of the Universe when matter dominates. In future works, we expect to study a more general parameterization of the deceleration by using q1 as free parameter. The condition given by equation (4.4) is satisfied for z > 0. As comment before, the EoS is valid too in a matter dominated epoch, i.e., z >> 1, let us assume for simplicity that z → ∞. Thus, by substituting the equation (2.4) into equation (A.2), the limit for Ωm (z) at such epoch is:
lim Ωm (z) = Ωm,0 exp [2ξ(erf(zc ) − 1)] .
z→∞
–9–
(4.5)
4
P hantom
Quintessence
3 2
ω′
1
3 1− ω 2ω
(1 −
+ 3(1
ω )(
1+
ω)
(1 + ω )
ω)
0.2ω(1 + ω)
0 −3 (1
−1
−
ω)
(1
+
−2
ω)
−3 −4 −1.4
−1.2
−1.0
−0.8
ω
−0.6
−0.4
−0.2
0.0
Figure 7: Discrimination regions for quintessence (thawing and freezing behavior) and phantom models in the ω 0 -ω plane. The red dashed lines represent the bounds for the thawing discrimination region, where ω 0 = 3(1 + ω) is the upper bound and ω 0 = (1 + ω) is the lower bound [45]. The black dashed lines in the quintessence region (ω > −1) are the bounds for freezing models, where ω 0 = 0.2ω(1 + ω) is the upper bound, and ω 0 = −3(1 − ω)(1 + ω) is the lower bound [see 46]. In the phantom region (ω < −1), 0 ω = 3ω(1 − ω)(1 + ω)/(1 − 2ω) is the upper bound, and ω = −1 is the lower bound [see 47]. In shades of blue are the 68%, 95%, 99.7% confidence levels for the reconstruction of ω and ω 0 , and the orange line is the mean value of these reconstructions. The orange square is the value at redshift z = 0. Considering that the reconstruction of Ωm (z) for our model is a monotonic increasing function for z ≥ 0 (see appendix A), for given a pair of fixed q0 and zc there exist a real positive value of the redshift z for which equation (2.9) will contain a singular point if Ωm,0 exp [2ξ(erf(zc ) − 1)] > 1.
(4.6)
Figure 6 illustrates the q0 − zc region bounded for this inequality, showing two regions of interest: the quintessence, and quintom DE area. If ω(z) ∈ [−1, 1], the bartropic fluid can be represented with a minimally coupled scalar field, known as quintessence DE model, but if ω(z) < −1 the behavior of the fluid is represented as a phantom DE [4]. Since in our proposed EoS (see equation (2.9)) does not exist an evident restriction for its codomain, it is important to know whether the reconstruction go through the phantom divide, defined as ω = −1. If the EoS cross the phantom divide, the fluid behavior can be represented by a combination of a negative-kinetic and a normal scalar field, known as quintom DE [48]. Notice that our joint analysis mean values for q0 and zc rely on the quintom region.
– 10 –
Table 1: Mean values for the model parameters (h, q0 , zc ) derived from OHD and SN Ia measurements. Data set OHD (CC) SNIa (JLA) Joint (CC+JLA)
χ2min
h
q0
zc
Mb1
δM
α
β
15.08 682.85 700.63
0.726+0.015 −0.015 0.729+0.18 −0.18 0.713+0.01 −0.01
−0.74+0.18 −0.14 −0.42+0.10 −0.12 −0.48+0.10 −0.11
0.75+0.16 −0.23 0.45+0.30 −0.27 0.50+0.20 −0.19
— −18.95+0.48 −0.65 −19.01+0.04 −0.04
— −0.06+0.02 −0.01 −0.06+0.02 −0.01
— 0.14+0.006 −0.006 0.14+0.006 −0.006
— 3.10+0.08 −0.07 3.11+0.08 −0.08
Quintessence models can be classified by the behavior of the potential associated with the scalar field. The two categories are thawing models and freezing (tracking) models (see [31, 49] and references therein). In the thawing models, the scalar field is frozen at early times due to the Hubble parameter damping3 , while at late times the friction term becomes subdominant. The ω(z) is a decreasing function that asymptotically reaches the cosmological constant EoS (i.e. ω ≈ −1) at early times. In the freezing models, the scalar potential is steep enough at early times to develop the kinetic term, while at late times it becomes shallower allowing the slowing down of the scalar field. The ω(z) is an increasing function that tends to the Cosmological Constant EoS at late times. An effective tool to discriminate between these models is the ω 0 -ω plane, where ω 0 = dω/dlna [45]; since different models are bounded by different regions [45–47]. A phantom DE can be represented by a scalar field minimally coupled to gravity with a non-canonical negative-kinetic energy term, and whose energy density grows with time. Thus, the tracking behavior of a phantom model can be depicted in the ω 0 -ω plane [47]. Because in the quintom models the evolution equations of the negative-kinetic and the normal scalar fields are independent [50], the potential behavior can be classified by the quintessence and phantom discrimination regions obtained separately. Figure 7 shows the discrimination regions for quintessence (thawing and freezing behavior) and phantom models in the ω 0 -ω plane. The thawing discrimination region is delimited between ω 0 = 1 + ω (lower bound) and ω 0 = 3(1 + ω) (upper bound) [45]. The freezing quintessence limits are provided by ω 0 = 0.2ω(1 + ω) (upper bound) and ω 0 = −3(1 − ω)(1 + ω) (lower bound) [46, 47]. The upper bound for phantom region is ω 0 = 3ω(1 − ω)(1 + ω)/(1 − 2ω) [47]. Notice that our joint constraints on the q(z) parameters crosses both the quintessence and phantom regions, hence, confirming that our results are consistent with a quintom dark energy.
5
Summary
There are several ways to approach the dynamical evolution of the Universe with the aim of describing the late and early epoch expansion. Many models of DE, such as normal and negative-kinetic scalar field models, are represented by a barotropic fluid. Recent observations indicate a transition between a decelerated and an accelerated phase of the cosmic expansion, from a matter dominated epoch to recent times, respectively. In this work we proposed a new phenomenological well-behaved parametrization of the deceleration parameter, equation (2.3), to approach the accelerated evolution of the cosmic expansion, and to reconstruct the effective EoS of DE. We performed a MCMC Bayesian analysis of the reconstruction of the Hubble parameter, +0.01 equation (2.4), using OHD and JLA datasets. We obtain q0 = −0.48+0.10 −0.11, h = 0.713−0.01, and +0.12 zt = 0.71−0.12 , which are consistent with values reported by other authors. The reconstruction 3
Indicates that the dynamics of the scalar field is governed by the Klein-Gordon equation
– 11 –
of the EoS (see figure 6) using these values cross the phantom divide, rejecting quintessence DE models. Our result points to a quintom DE, and it is consistent with a non parametric reconstruction of the EoS using the latest observations (see ref. [51]) within the range of validity of the equation (2.9). The behavior of the two free-parameter reconstruction of the EoS (equation (2.10)) is a more general expression, including both the thawing or freezing scalar field models. Indeed, the functional form of ω does not impose an apriori category of scalar field model for its entire domain. Furthermore, the discrimination analysis we presented in figure 7 is also consistent with a quintom DE model. The confidence contours for ω 0 vs. ω, are not subsets of a single model region within the regions delimited by thawing and freezing models. This is a complex behavior of our two free parameter reconstruction of the EoS, in contrast to the parameterizations analyzed in ref. [31]. In a future work, we plan to extend the study presented here, and analyze the consequences of the cosmic expansion in a early epoch by setting q1 as a free parameter, and its repercussions on the behavior of the effective EoS.
– 12 –
A
The behavior of Ωm (z) and the singularities of ω
By considering the definition of the matter density in terms of z: Ωm (z) =
ρm (z) , 3H 2 (s)
(A.1)
where ρm (z) = 3H02 Ωm,0 (1 + z)3 , equation (A.1) can be rewritten as H0 2 3 . Ωm (z) = Ωm,0 (1 + z) H(z)
(A.2)
Let us calculate the first derivative of Ωm (z) with respect of z dΩm (z) = Ωm,0 dz from equation (2.2)
dH(z) dz
3
H0 H(z)
2
H02 dH(z) (1 + z) − 2 (1 + z)3 H(z)3 dz 2
! ,
(A.3)
= H(z) 1+q(z) 1+z , simplifying equation (A.3)
dΩm (z) = Ωm,0 (1 + z)2 dz
H0 H(z)
2 (1 − 2q(z)),
(A.4)
By the reconstruction of the Hubble parameter using the joint dataset, H(z) > 0 and q(z) < 1/2 for z ≥ 0, see figure 3. Introducing both considerations in equation (A.4), we obtain dΩm (z) >0 dz
∀z ≥ 0,
(A.5)
for our model. Thus, in this case, Ωm (z) is a monotonic increasing function for all z ≥ 0. Given equation (4.5) and Ωm (0) = Ωm,0 < 1 [41], then the codomain of this function is delimited: Ωm (z) ∈ [ Ωm,0 , Ωm,0 exp(2ξ(erf(zc ) − 1)) )
∀z ≥ 0.
(A.6)
Let us consider the next cases: • If Ωm,0 exp(2ξ(erf(zc ) − 1)) < 1: ⇒Ωm (z) < 1
⇒1 − Ωm (z) > 0
∀z ≥ 0 ∀z ≥ 0
(A.7) (A.8)
then, there is not a value of z ≥ 0 such that ωeff (z) diverges. • If Ωm,0 exp(2ξ(erf(zc ) − 1)) = 1: ⇒Ωm (z) ∼ 1
as z → ∞
⇒ωeff (z) → ∞
as z → ∞
then, ωeff (z) diverges as z → ∞.
– 13 –
(A.9) (A.10)
• If Ωm,0 exp(2ξ(erf(zc ) − 1)) > 1: Because the codomain of Ωm (z) is delimited as equation (A.6), 1 ∈ [ Ωm,0 , Ωm,0 exp(2ξ(erf(zc ) − 1)) ),
(A.11)
then there is a value z 0 > 0 such that Ωm (z 0 ) = 1 ⇒1 − Ωm (z 0 ) = 0
where z 0 > 0.
(A.12)
Therefore, the last case gives the condition to have a singular point of ωeff .
Acknowledgments The authors thank Luis Ureña for his thoughtful comments. This research has been carried out thanks to PROGRAMA UNAM-DGAPA-PAPIIT IA102517. J.M. acknowledges support from CONICYT/FONDECYT 3160674 and thanks the hospitality of the staff of IA-Ensenada where part of this work was done.
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