Imprints of an extended Chevallier-Polarski-Linder parametrization on the large scales

Supriya Pan, ∗ Weiqiang Yang, † and Andronikos Paliathanasis 4, ‡ Department of Mathematics, Presidency University, 86/1 College Street, Kolkata 700073, India Department of Physics, Liaoning Normal University, Dalian, 116029, P. R. China Instituto de Ciencias F́ısicas y Matemáticas, Universidad Austral de Chile, Valdivia, Chile Institute of Systems Science, Durban University of Technology, PO Box 1334, Durban 4000, Republic of South Africa


INTRODUCTION
According to the theory of general relativity, one possible way to describe the recent observational evidences is to introduce the dark energy, a hypothetical fluid with large negative pressure [1]. However, apart from this negativity condition on the pressure of dark energy, no one knows what exactly this particular fluid is. The simplest explanation to the dark energy theory comes through the introduction of positive cosmological constant, Λ, which does not evolve with the time. But, the cosmological constant already suffers from two major problems, one which is recognized as the cosmological constant problem and the other is the cosmic coincidence problem. Thus, although as stated by a series of observational data, the Λ-cosmology is an elegant version to model the recent observational features of the universe, the problems associated with the above motivate us to think of the scenarios beyond the standard Λ-cosmology paradigm.
The simplest extension to Λ-cosmology is the w xcosmology in which w x is the dark energy equation-ofstate quantified as the ratio of pressure to its density, mathematically which is w x = p x /ρ x . One can identify that p x and ρ x are respectively the pressure and energy density of the dark energy fluid. The equation-of-state w x being −1 recovers the Λ-cosmology. In general one can assume w x ( = −1) to be either time independent or dependent while the latter scenario is the most general one. Thus, in the present work we shall focus on the alternative cosmologies to the Λ-cosmology in which the dark energy equation-of-state is evolving with the expansion of the universe.
The parametrization of w x could be any function of the redshift z or the scale factor a(t) of the Friedmann-Lemaître-Robertson-Walker universe; note that, 1 + z = a 0 /a(t), where a 0 is the present value of the scale factor in this universe. Thus, since w x ≡ w x (z) ≡ w x (a) could be any arbitrary function of the redshift or the scale factor, therefore, in principle this gives us a complete freedom to pick up any particular model of interest and test it with the observational data in order to see whether that model is able to correctly describe the evolution of the universe. In fact one can realize that the introduction of the dark energy equation-of-state is a reverse mechanism to probe the expansion history of the universe. Going back to literature, one can find that this particular area of cosmology has been investigated well both at the level of background and perturbations where various parametrizations for w x were introduced earlier [2][3][4][5][6][7][8][9][10][11][12] and later . Precisely, the dark energy parametrization with only a single free parameter, with two free parameters, with three free parameters and finally with more than three parameters have been rigorously studied by various investigators.
The aim of the present work is slightly different. Here, we are considering an exponential dark energy parametrization that in its first order approximation around z = 0 recovers the CPL parametrization, and further we allow its higher order corrections in order to investigate how such extended corrections affect the evolution of the universe both at the level of background and perturbations. More specifically, we consider upto the third order expansion of the exponential dark energy model. We remark that in general every analytic function for the equation-of-state parameter around the z = 0 describes the CPL parametrization in the first correction; however, while we want to assume a general Taylor expansion of an analytic function f (a) around a = 1, i.e. f (a) = ∞ i=0 w i (a − 1) i , every new term which is introduced in the correction provides a new degree of freedom, a free parameter, in the model. Consequently, the models will have different degrees of freedom and they will not be in comparison. Hence, special relations amount the constants w i should be considered, and for our analysis we assume that w 0 is free while w j = w1 j! , which j = 0, in which f (a) is now the exponential function. However, by this approach we will get a remarkable information on how the nonlinear terms in the parametrizations of the equation-of-state affect the viability of the model in higher-redshifts.
The work has been organized in the following way. In section 2 we introduce the models for w x (z) and describe the general equations at the level of background and perturbations. In section 3 we describe the observational data and the statistical analysis that are used to constrain the models. After that in section 4 we describe the observational constraints extracted from the models using the astronomical data described in section 3. Then in section 5 we compute the evidences of the dark energy parametrizations through the MCEvidence. Finally, we close the work in section 6 with a brief summary of everything.

BASIC EQUATIONS AND THE MODELS
Considering a spatially flat Friedmann-Lemaître-Robertson-Walker line element is the expansion scale factor of the universe), in the context of the Einstein gravity, we assume that (i) matter is minimally coupled to gravity, (ii) there is no interaction between any two fluids under consideration and (iii) all the fluids satisfy barotropic equation of state, i.e., p i = w i ρ i in which w i being the barotropic state parameter for the i-th fluid having (ρ i , p i ) as its the energy density and pressure, respectively. Precisely, we consider that the total energy density of the universe is, ρ tot = ρ r + ρ b + ρ c + ρ x and the total pressure thus becomes p tot = p r + p b + p c + p x . Here, the subscripts r, b, c and x respectively stands for radiation, baryons, cold dark matter and dark energy. Thus, the barotropic indices are, w r = 1/3, w b = w c = 0 and we assume w x to be dynamical. The Einstein's field equations for the above FLRW universe can be written down as in which an overhead dot represents the cosmic time differentiation and H ≡ȧ/a is the Hubble rate of this universe. Now, using (1) and (1) (or alternatively the Bianchi's identity), one can find the balance equatioṅ Now, since as we assumed that we don't have any interaction between any two fluids of the universe, thus, they should satisfy their own conservation equation leading tȯ from which using the relation between pressure and energy density for the radiation, baryons, and cold (pressureless-) dark matter, one can find that Here, ρ i0 is the present value of ρ i . And finally, the evolution of the dark energy fluid can be given by, where ρ x0 being the current value of ρ x and a 0 is the present value of the scale factor that we set to be unity (a 0 = 1) without any loss of generality. We further note that the scale factor is related to the redhisft that we shall frequently use hereafter via 1 + z = a 0 /a = 1/a. Thus, once the dark energy equation of state is prescribed, the evolution of the dark energy density can be found. As we discussed above, we consider that the dark energy fluid follows a general parametrization in the following way: where w 0 is the present value of the dark energy equation of state, that means, w x (z = 0) = w 0 and w a is another free parameter. The model (6) is very interesting by its construction since one can easily recognize that it could return a number of interesting parametrization that includes the classic Chevallier-Polarski-Linder parametrization w x (z) = w 0 + w a z/(1 + z) if we take the first approximation of the exponential function in (6). We expand the exponential function of (6) upto its first, second and third order corrections leading to the following class of dark energy parametrization: and for convenience we call the dark energy parametrization of equations (7), (8) and (9) as "Extension 1" (Ext1 in short), "Extension 2" (Ext2 in short) and "Extension 3" (Ext 3 in short), respectively. At the end of this section, we would like to present the qualitative features of the present dark energy parametrizations in terms of the evolution of their equations of state and the deceleration parameters. In order to do so, we assume three different values of w 0 , namely, w 0 = −0.95, w 0 = −1 and w 0 = −1.1 and in each case we consider various values of w a to understand how the curves behave with the increasing of the w a parameter. In Fig. 1 we show the evolution of the dark energy parameterizations (6), (7), (8) and (9)  We then plot the evolution of the deceleration parameter for all the DE parametrizations, namely, (6), (7), (8) and (9). Here we have considered three fixed values of w 0 , namely, w 0 = −0.95, −1, −1.1 but in each case we have assumed different values of w a similar to what we have shown in Fig. 1, Fig. 2 and Fig. 3. Finally, we depict the evolution of the deceleration parameter in  (Fig. 4, Fig. 5 and Fig. 6) representing the evolution of the deceleration parameters are same, as one can see that irrespective of the values of w 0 , a fine transition from the past decelerating phase to the current accelerating one is observed, however, the impacts of w a should be discussed. For that reason, we only consider Fig. 4 because the other figures lead to same conclusion. From Fig. 4, we see that for negative values of w a the transition redshifts are shifting towards higher redshifts (although mild) while for positive values of w a , we see the reverse, that means the transition redshifts are shifting towards lower values of the redhift.
Overall, we find that the models at the level of background do not exhibit any deviations from one another. This is not surprising because the deviations between the cosmological models are usually reflected from their analysis at the level of perturbations. In what follows we shall consider the perturbation equations for all the DE parametrizations in this work.
We start with the following metric which is the perturbed form of the FLRW line element: Here, η denotes the conformal time; δ ij , h ij are the unperturbed and the perturbative metric tensors, respectively. Now, considering the perturbed Einstein's field equations, for a mode with wave-number k one can write down [43][44][45]: where δ i = δρ i /ρ i is the density perturbation for the ith fluid; the prime associated to any quantity denotes the derivatives with respect to conformal time; H = a /a is the conformal Hubble parameter; θ i ≡ ik j v j is the divergence of the i-th fluid velocity; h = h j j , is the trace of the metric perturbations h ij ; σ i denotes the anisotropic stress related to the i-th fluid. Let us also note that c 2 a,i =ṗ i /ρ i , is the adiabatic speed of sound of the i-th fluid which can also be written in terms of other physical quantities as c 2 where we fix the sound speed c 2 s = δp i /δρ i to be unity. Finally, we also note that we have neglected the anisotropic stress from the system for simplicity.

OBSERVATIONAL DATA
For the convenience of the reader and for our presentation we provide the details of the observational data used to constrain the dynamical dark energy parametrization and also the methodology.
• Cosmic microwave background observations: the cosmic microwave background (CMB) observations are one of the powerful data to probe the nature of dark energy. Here we use the CMB from Planck 2015 [46,47]. The high-temperature and polar-ization data as well as the low-temperature and polarization data from Planck 2015 (precisely the dataset: Planck TT, TE, EE + lowTEB) [46,47] have been considered.
• Supernovae Type Ia: We also use latest released Pantheon sample [52] from the Supernovae Type Ia.  • Hubble parameter measurements: Finally, we use the Hubble parameter measurements from the Cosmic Chronometers (CC) [53].
Now we come to the technical part of the statistical analysis. Thus, we have performed the fitting analysis using the modified version of cosmomc [54,55], an efficient markov chain monte carlo package equipped with a convergence diagnostic given by the Gelman and Rubin statistics [56]. This package includes the sup-port for the Planck 2015 likelihood code [47] (see http: //cosmologist.info/cosmomc/). In Table I we have shown the flat priors on the model parameters that have been used during the observational analysis. Perhaps it might be important to mention here that in the present analysis we have used Planck 2015 likelihood [47] instead of Planck 2018 likelihood (although the cosmological parameters from Planck 2018 are already available [57]) because Planck 2018 likelihood code is not public yet. However, it will be worth to run the same codes that we  use for the present models but with the new Planck 2018 likeliood which will enable us to understand any effective changes in the cosmological parameters and consequently more stringent constraints on them as well.

OBSERVATIONAL CONSTRAINTS AND THE ANALYSIS
In this section we describe the observational constraints on all the dark energy parametrization, namely the general parametrization of eqn. (6), Extension 1 or the CPL parametrization of eqn. (7), Extension 2 of eqn. (8) and extension 3 of eqn. (9) using various astronomical datasets summarized in section 3. In particular, we focus on the two key parameters of the dark energy parametrization, namely, w 0 and w a in order to investigate the qualitative changes in the parametrization as long as nonlinear terms are considered. In what follows we describe the observational constraints extracted from each dark energy scenario.
Let us first focus on the general dark energy parametrization given in equation (6). We have constrained this dark energy scenario using different observational combinations, the results of which are summarized in Table II. From Table II, one can see that the best constraints on the model parameters are achieved for the combinations CMB+BAO+Pantheon and CMB+BAO+Pantheon+CC since the addition of Pantheon and CC to the combination CMB+BAO significantly decrease the error bars on the model parameters. We find that for the general parametrization, the mean value of the dark energy equation of state at present, i.e., w 0 is always in the quintessential regime while looking at the exact estimations on w 0 for CMB+BAO+Pantheon (w 0 = −0.963 +0.060 −0.082 at 68% CL) and for CMB+BAO+Pantheon+CC (w 0 = −0.933 +0.071 −0.070 at 68% CL), it is also clear that w 0 could cross the w 0 = −1 boundary, but of course marginally. Additionally, we find that the remaining key parameter w a may assume nonnull values, however, w a = 0 is allowed within 68% CL of course. For a better understanding of all the parameters of this model, in Fig. 7, we have shown the one dimensional posterior distributions for some selected parameters of this model as well as the two dimensional contour plots for various combinations of the parameters. From Fig. 7, one can see that all the parameters shown in this figure are correlated with each other. Specifically, we find a strong correlation between w 0 , w a and H 0 .
We now consider the first extension of the general parametrization (6) that leads to the well known CPL parametrization of (7). The results of this parametrization are also extracted using the same observational datasets applied to the general DE parametrization which can be found from Table III. One can easily see that the conclusions on w 0 and w 0 remain same similar to what we have found in the general parametrization (6). So, effectively we see that the first approximation (7) of the original parametrization (6) returns similar fit to the original parametrization (6). Similarly, for this parametrization we plot Fig. 8 containing the one dimensional marginalized posterior distributions as well as the two dimensional contour plots at 68% and 95% CL.
Then we move to the observational constraints of the next parametrization given in eqn. 8. The results for this parametrization are shown in Table IV and in Fig. 9 we have shown the graphical variations of the model parameters. We find that this parametrization behaves similarly   to the previous two parametrization, that means concerning the free parameters w 0 , w a , we have exactly similar conclusion as observed with model (6) and model (7). Finally, we focus on the last parametrization of this series, namely Extension 3 shown in eqn. (9). We have summarized the results in Table V and in Fig. 10 we have shown the corresponding graphical variations of the model parameters. Looking at all the parameters, it is clear that except w a , the other parameters have similar constraints as already found in models (6), (7), (8). For the w a parameter, we see that for the last two combinations (the CMB data alone provide higher values, similar to other models as well), the mean values of w a as well as the error bars are significantly reduced compared to the previous models. In fact for the last combination that means for CMB+BAO+Pantheon+CC, the estimated value of w a is very very close to zero which means that the dynamical nature of the dark energy parametrization is very weak, however, the statistical estimation also offers its non-null value finding, w a = −0.094 +0.089+0.158 −0.074−0.163 (68% CL, CMB+BAO+Pantheon+CC). The difference in the results can be found from Fig. 11 which clearly shows that for this parametrization, the w 0 − w a plane is much reduced compared to others.
Thus, based on the analyses presented above one can see that as long as we consider the higher order corrections in the generalized parametrization (6), the parameter w a quantifying the dynamical nature of the dark energy parametrization becomes weak. For a detailed understanding, we refer to Table VI. Now we investigate how the present dark energy parametrization, namely, the new dark energy parameterization in eqn. (6), and its extensions in equations (7), (8) and (9) affect the temperature anisotropy in the cosmic microwave background spectra as well as in the matter power spectra. Such an investigation is important because this certainly enables us to understand how the higher order extensions of the original dark energy parametrization (6) is important in the context of structure formation. Thus, keeping these all in mind, in    V: Observational constraints on the dark energy parametrization, namely, the Ext3 of (9) using various observational datasets. We note that H0 is in the units of km/Mpc/sec and Ωm0 is the present value of Ωm = Ω b + Ωc. Fig. 12, one can easily find that as long as w a increases, the higher order corrections of the generalized model (6), that means model in eqn. (9) gets differentiated from the remaining models. While on the other hand, for the negative values of w a (corresponding to the lower graphs of Fig. 12), we don't find any kind of differences between the models. In a similar way, in Fig. 13 we have shown plots representing the matter power spectra for positive and negative values of w a with fixed w 0 = −0.95. The upper graphs of Fig. 13 correspond to w a > 0 while the lower graphs correspond to w a < 0. From this figure, we have similar observation as already noticed in Fig. 12.

BAYESIAN EVIDENCE
A general and natural question that we will be looking for in this section is that, how the models are efficient analyses of the models presented in this work the best constraint on w 0 is around that value. compared to the standard ΛCDM cosmology. Thus, we need a statistical comparison between all four dynamical DE parametrizations where the base model will be fixed as ΛCDM. This statsitical comparison comes through the Bayesian evidence. Here we apply publicly available code MCEvidence [58,59] 2 to compute the evidences of the models. The use of MCEvidence is very easy since the code only needs the MCMC chains used to extract the free parameters of the DE parametrizations.
While dealing with Bayesian analysis we need the posterior probability of the model parameters (denoted by θ), given a specific observational data (x) with any prior information for a model (M ). Following Bayes theorem, one can write, where p(x|θ, M ) is the likelihood as a function of θ and π(θ|M ) refers to the prior information. Here, the quantity p(x|M ) appearing in the denominator of (12) is the Bayesian evidence that we actually need for the model     comparison. Now, for two cosmological models M i , M j where M j is acting as the reference model 3 , the posterior probability is, in which B ij = p(x|Mi) p(x|Mj ) , is the Bayes factor of the model M i relative to M j . And based on the values of B ij (alternatively, ln B ij ) we quantify the observational support of the underlying model M i relative to M j . The quantification is done through the widely accepted Jeffreys scales [60] (see Table VII). We also note that the negative values of ln B ij indicate that the reference model (M j ) is preferred over the underlying model (M i ).
In Table VIII we have shown the values of ln B ij computed for all DE parametrizations considering all the datasets. We find that the values of ln B ij are all negative indicating that ΛCDM is always preferred and this is true for all the observational datasets.   . We note that the negative value of ln Bij means that ΛCDM is preferred.

CONCLUDING REMARKS
The dark energy, a hypothetical fluid in Einstein gravity is the main concern of this work. This dark energy, as examined by many investigators since the year 1998, could be anything obeying only one condition that the pressure of the fluid should be negative. Thereafter, a cluster of dark energy models have been introduced and confronted with the observational data, see [1] to get an overview of the models.
Among them an interesting construction of the dark energy models comes through the equation of state of dark energy, w x = p x /ρ x which in principle is the function of the underlying cosmological time parameter, usu-ally the function of the redshift. Technically, there is no such restriction to pick up any specific functional form for w x , however, the viability of the model is only tested through the observational data and its effects on the large scale structure of the universe indeed. According to the investigations performed in the last couple of years, the Chevallier-Polarski-Linder parametrization is a feasible and well functioning dark energy parametrization with the observational data. The present work is motivated in the same direction whilst we have investigated something different as follows.
We have introduced a new dark energy parametrization (6) having a novel feature. The model recovers the well known CPL parametrization in its first order Taylor series expansion around z = 0. Thus, the model actually presents a generalized version of the CPL parametrization. Since the model is a nonlinear generalized version of the CPL model, thus, a natural inquiry one may ask for is, how its higher order corrections are important for the expansion history of the universe, and moreover, how the higher order corrections could affect the evolution of the universe at the level of background and perturbations. In order to investigate these issues, we have considered the generalized model (6) together with its first, second and third order Taylor approximations around the present cosmic epoch z = 0, given in equations (7), (8) and (9). Since the original model (6) contains only two free parameters w 0 (current value of the dark energy equation of state) and w a (parameter quantifying the dynamical nature of the DE), thus its extensions contain the same free parameters. We then constrain all the models using a class of astronomical data, such as CMB, BAO, Pantheon from SNIa and the Hubble parameter measurements (summarized in section 3).
The observational constraints are summarized in Table II (for eqn. (6)), Table III (for eqn. (7)), Table IV (for eqn. (8)), Table V (for eqn. (9)) and the graphical variations of the model parameters are also shown in Fig. 7, Fig. 8, Fig. 9 and Fig. 10, respectively for the general, Ext1, Ext2, and Ext3. From the analyses, it is clear that up to Ext2, the cosmological parameters assume similar constraints while the third order correction (Ext3 of eqn. (9)) presents slightly different results compared to others (see Fig. 11). Precisely, for the third order correction (eqn. (9)), we find that the parameter w a quantifying the dynamical nature of the model is very weak (≡ w a ∼ 0) at least for the last two combinations (i.e., CMB+BAO+Pantheon and CMB+BAO+Pantheon+CC). We mention that for the CMB+BAO combination, the error bars of the parameters for all the models are significantly large compared to others (hence weakly constrained) and moreover, one can notice that for this particular dataset the parameters w 0 and w a are strongly correlated to each each other (as seen from Fig. 10). Such a disparity has also been reflected from the analysis at large scale structure. Looking at Fig. 12 and Fig. 13, one can see that as soon as we allow the higher order corrections, the model (Ext3) is distinguished from the others and the dynamical nature of the model becomes insignificant.
Finally, we perform the Bayesian analysis using the MCEvidence and compared the models with respect to the standard ΛCDM reference scenario. Our analysis reveals that ΛCDM is indeed favored over all the dynamical DE models. This is an expected result because the parameters space for all the dynamical DE models are of eight dimensional while the ΛCDM has only six parameters.
Last but not least, we would like to comment that the model (6), so far we are aware of the literature, is a new one in the field of dark energy which naturally recovers CPL parametrization in its first order approximation and sounds good with the Bayesian evidence. Therefore, a number of investigations can be performed in various contexts. We hope to address some of them in near future.