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In this paper we have performed a comparative study of different types of oscillating dark energy models using the Metropolis algorithm of MCMC. Eight different oscillating parameterization being examined herein which have demonstrated considerable ability to fit the overall cosmological observations including Pantheon sample of SnIa, Baryon Acoustic Oscillations, Cosmic Chronometer Hubble data and distance priors of Planck CMB. In order to compare the consistency of these models with observations, we have used both Akaike and Deviation information criteria. Although, the values of Akaike information criteria for different models indicate that there is no support for oscillating DE models, Deviation information criteria showed that there is significant support for some of these models. Our results showed that these models are capable of solving cosmic coincidence problem and alleviating the Hubble tension. Comparing the $H_0$ values obtained for different oscillating scenarios, with that of $\Lambda$CDM, we observe that our oscillating models led to $\bar{H_0}=69.78$ which is $0.29$ greater than $H_{0,\Lambda}$ and thus reduce the Hubble tension. Among all of models, Model(1) with $H_0=70.00 \pm 0.71$ is the most capable of alleviating the $H_0$ tension. Furthermore, we examined our models assuming $H_0=73.0 \pm 1.4$ from SHoES measurements. We find that adding this data point to our data combination, led to a $\Delta \bar{H_0}=0.95$ increase in $H_0$ value for different models., Comment: 11 pages, 2 tables and 4 figures, Submitted in APJ

We investigate the dark energy phenomenology in an extended parameter space where we allow the curvature density of our universe as a free-to-vary parameter. The inclusion of the curvature density parameter is motivated from the recently released observational evidences indicating the closed universe model at many standard deviations. Here we assume that the dark energy equation-of-state follows the PADE approximation, a generalized parametrization that may recover a variety of existing dark energy models. Considering three distinct PADE parametrizations, labeled as PADE-I, SPADE-I and PADE-II, we first constrain the cosmological scenarios driven by them using the joint analyses of a series of recently available cosmological probes, namely, Pantheon sample of Supernovae Type Ia, baryon acoustic oscillations, big bang nucleosynthesis, Hubble parameter measurements from cosmic chronometers, cosmic microwave background distance priors from Planck 2018 and then we include the future Gravitational Waves standard sirens (GWSS) data from the Einstein telescope with the combined analyses of these current cosmological probes. We find that the current cosmological probes indicate a very strong evidence of a dynamical dark energy at more than 99\% CL in both PADE-I, and PADE-II, but no significant evidence for the non-flat universe is found in any of these parametrizations. Interestingly, when the future GWSS data from the Einstein telescope are included with the standard cosmological probes an evidence of a non-flat universe is found in all three parametrizations together with a very strong preference of a dynamical dark energy at more than 99\% CL in both PADE-I, and PADE-II. Although from the information criteria analysis, namely, AIC, BIC, DIC, the non-flat $\Lambda$-Cold Dark Matter model remains the best choice, however, in the light of DIC, PADE parametrizations are still appealing., Comment: 21 pages, 10 captioned figures, 6 tables; revised version; to appear in JCAP

We perform a comparative study of different types of dynamical dark energy models (DDES) using the cosmographic method. Among the models being examined herein we have the Running Vacuum models (RVMs), which have demonstrated considerable ability to fit the overall cosmological data at a level comparable to the standard cosmological model, $\Lambda$CDM, and capable of alleviating the $\sigma_8$ and $H_0$ tensions. At the same time we address a variety of Holographic dark energy models (HDEs) with different options for the time (redshift)-varying model parameter $c=c(z)$. We deal with the HDEs under the double assumption of fixed and evolving holographic length scale and assess which one is better. Both types of DDEs (RVMs and HDEs) are confronted with the most robust cosmographic data available, namely the Pantheon sample of supernovae (SnIa), the baryonic acoustic oscillation data (BAOs) extracted from measurement of the power spectrum and bispectrum of the BOSS data release, and the cosmic chronometer measurements of the Hubble rate (CCHs) at different redshifts obtained from spectroscopic observations of passively evolving galaxies. Using these data samples we assess the viability of the mentioned DDEs and compare them with the concordance $\Lambda$CDM model. From cosmographic analysis we conclude that the RVMs fare comparably well to the $\Lambda$CDM, a fact which adds up more credit to their sound phenomenological status. In contrast, while some of the HDEs are favored using the current Hubble horizon as fixed holographic length, they become highly unfavoured in the more realistic case when the holographic length is dynamical and evolves as the Hubble horizon., Comment: 15 pages, 3 figures, 7 tables, Accepted for publication in EPJC

Cosmographic approach, a Taylor expansion of the Hubble function, has been used as a model-independent method to investigate the evolution of the universe in the presence of cosmological data. Apart from possible technical problems like the radius of convergence, there is an ongoing debates about the tensions appear when one investigates some high redshift cosmological data. In this work, we consider two common data sets namely SNIa (Pantheon sample) and the Hubble data to investigate advantages and disadvantages of the cosmographic approach. To do this, we obtain the evolution of cosmographic functions using cosmographic method as well as two other well known model-independent approaches namely, the Gaussian process and the Genetic algorithm. We also assume $\Lambda$CDM model as concordance model to compare the results of mentioned approaches. Our results indicate that the results of cosmography comparing with the other approaches, are not exact enough. Considering the Hubble data which is less certain, the results of $q_0$ and $j_0$ obtained in cosmography, provides a tension at more than $3\sigma$ away from the best result of $\Lambda$CDM. Assuming both of data samples in different approaches we show that the cosmographic approach, because of providing some biased results, is not the best approach for reconstruction of cosmographic functions, especially at higher redshifts., Comment: 12 pages, 4 figures, 2 tables, Accepted for publication in APJ

In this work we study different types of dark energy (DE) models in the framework of the cosmographic approach, with emphasis on the Running Vacuum models (RVMs). We assess their viability using different information criteria and compare them with the so-called Ghost DE models (GDEs) as well as with the concordance $\Lambda$CDM model. We use the Hubble diagrams for Pantheon SnIa, quasars (QSOs), gamma-ray bursts (GRBs) as well as the data on baryonic acoustic oscillations (BAOs) in four different combinations. Upon minimizing the $\chi^2$ function of the distance modulus in the context of the Markov Chain Monte Carlo method (MCMC), we put constraints on the current values of the standard cosmographic parameters in a model-independent way. It turns out that, in the absence of BAOs data, the various DE models generally exhibit cosmographic tensions with the observations at the highest redshifts (namely with the QSOs and GRBs data). However, if we include the robust observations from BAOs to our cosmographic sample, the $\Lambda$CDM and RVMs are clearly favored against the GDEs. Finally, judging from the perspective of the deviance information criterion (DIC), which enables us to compare models making use of the Markov chains of the MCMC method, we conclude that the RVMs are the preferred kind of DE models. We find it remarkable that these models, which had been previously shown to be capable of alleviating the $\sigma_8$ and $H_0$ tensions, appear now also as the most successful ones at the level of the cosmographic analysis., Comment: 16 pages, 2 figures, Accepted for publication in MNRAS

There are several methods for model selection in cosmology which have at least two major goals, that of finding the correct model or predicting well. In this work we discuss through a study of well-known model selection methods like Akaike information criterion (AIC), Bayesian information criterion (BIC), deviance information criterion (DIC) and Bayesian evidence, how these different goals are pursued in each paradigm. We also apply another method for model selection which less seen in cosmological literature, the Cross-validation method. Using these methods we will compare two different scenarios in cosmology, $\Lambda$CDM model and dynamical dark energy. We show that each of the methods tends to different results in model selection. While BIC and Bayesian evidence overrule the dynamical dark energy scenarios with 2 or 3 extra degree of freedom, the DIC and cross-validation method prefer these dynamical models to $\Lambda$CDM model. Assuming the numerical results of different analysis and combining cosmological and statistical aspects of the subject, we propose cross-validation as an interesting method for model selection in cosmology that can lead to different results in comparison with usual methods of model selection., Comment: 12 pages, 3 figures, PUBLISHED IN EPJP. arXiv admin note: text overlap with arXiv:1510.00547 by other authors

In the context of cosmography approach and using the data of Hubble diagram for supernovae, quasars and gamma-ray bursts, we study some DE parametrizations and also the concordance $\Lambda$CDM universe. Using the different combinations of data sample including ({\it i}) supernovae (Pantheon), ({\it ii}) Pantheon + quasars and ({\it iii}) Pantheon + quasars + gamma-ray bursts and applying the minimization of $\chi^2$ function of distance modulus of data samples in the context of Markov Chain Monte Carlo method, we first obtain the constrained values of the cosmographic parameters in model independent cosmography scenario. We then investigate our analysis, for different concordance $\Lambda$CDM cosmology, $w$CDM, CPL and Pade parametrizations. Comparing the numerical values of the cosmographic parameters obtained for DE scenarios with those of the model independent method, we show that the concordance $\Lambda$CDM model has a serious tension when we involve the quasars and gamma-ray bursts data in our analysis. While the high redshift quasars and gamma-ray bursts can falsify the concordance model, our results of cosmography approach indicate that the other DE parametrizations are still consistent with these observations., Comment: Accepted in ApJ

In this work we examine the possibility that the dark energy (DE) density, $\rho_{de}$ can be dynamical and appear as a power series expansion of the Hubble rate (and its derivatives), i.e.$\rho_{de}(H,\dot{H},...)$. For the present universe, however, only the terms $H$, $\dot{H}$ and $H^2$ can be relevant, together with an additive constant term. We fit these models to the current cosmological data on the main observables SNIa+$H(z)$+BAO+LSS+CMB+BBN. Our analysis involves both the background as well as the cosmic perturbation equations. The latter include, apart from the matter density perturbations, also the DE density perturbations. We assume that matter and dynamical DE are separately self-conserved. As a result the equation of state of the DE becomes a nontrivial function of the cosmological redshift, $w_D=w_D(z)$. The particular subset of DE models of this type having no additive constant term in $\rho_{de}$ include the so-called entropic-force and QCD-ghost DE models, as well as the pure linear model $\rho_{de} \sim H$ all of which are strongly disfavored in our fitting analysis. In contrast, the models that include the additive term plus one or both of the dynamical components $\dot{H}$ and $H^2$ appear more favored than the $\Lambda$CDM. In particular, the dynamical DE models provide a value of $\sigma_8\simeq 0.74-0.77$ which is substantially lower than that of the $\Lambda$CDM and hence more in accordance with the observations. This helps to significantly reduce the $\sigma_8$-tension in the structure formation data. At the same time the predicted value for $H_0$ is in between the local and Planck measurements, thus helping to alleviate this tension as well., Comment: 20 pages, 6 figures and 8 tables, Accepted in Phys. Rev. D

We study the imprints on the formation of cosmic structures of a particular class of dark energy parameterizations dubbed PADE parameterization. Here we investigate how dark energy can affect the growth of large scale structures of the universe in framework of spherical collapse model. The dynamics of the spherical collapse of a dark matter halo depends on the properties of the dark energy model. We show that the properties of spherical collapse scenario are directly affected by the evolution of dark energy. We obtain the main parameters of spherical collapse for two different DE parameterizations in two different approaches: first the homogeneous DE approach, in which dark energy does not exhibit fluctuations on cluster scales and the other, clustered DE scenario in which, dark energy components inside the overdense region collapses similar to dark matter. Using the Sheth-Tormen mass function, we investigate the abundance of virialized halos in the framework of PADE parameterizations. Specifically, the present analysis shows that the number count of dark matter halos depends on the evolution of DE parameterizations and clustering properties of dark energy. Also we show that perturbations in phantom DE components enhance the growth of matter perturbations. This result were obtained in the literature for dark energy parameterizations that are phantom at all redshifts. But we obtained same results for dark energy parameterizations which are in phantom regime and enter in quintessence region at relatively low redshifts. We also show that in parameterizations under study, low mass halos were formed before massive halos., Comment: 12 pages, 7 figures, 2 tables

In this study we combine the background and the growth rate data in order to study the ability of the two oscillating dark energy parameterizations, to fit the observational data. Using the likelihood and MCMC method we try to explore the posterior space and put constraints on the free parameters of the models. Based on the values of the well known Akaike and Bayesian information criteria we find that both of oscillating dark energy models considered in this work are disfavored by the combined (background+growth rate) data. Although using the expansion data we can not reject oscillating dark energy models, the combined analysis provides strong evidences against these models., Comment: 10 pages, 4 figures, Accepted in MNRAS

In this work we investigate the holographic dark energy models with slowly time-varying model parameter defined based on the current Hubble horizon length scale. While the previous studies on the three popular holographic dark energy models defined based on the future event horizon, Ricci scale and Granda-Oliveros IR cutoffs showed that these models cannot fit the observational data [1], in this work we show that the holographic dark energy models with time-varying model parameter defined on the current Hubble radius are well favored by observations. Using the standard $\chi^2$ minimization in the context of Markov Chain Monte Carlo method, we compare the ability of holographic dark energy models with time-varying $c^2$ parameter constructed on the current Hubble length scale against different sets of observational data namely expansion data, growth rate data and expansion+growth rate data respectively. Based on the values of Akaike and Bayesian information criteria, we find that these types of holographic dark energy models are well fitted to both expansion and growth rate observations as equal to $\Lambda$CDM cosmology. We also put constraints on the cosmological parameters and show that the transition epoch form early decelerated to current accelerated expansion calculated in holographic dark energy models with time-varying model parameter defined on the Hubble length is consistent with observations., Comment: 12 pages, 4 figures, 6 tables, Accepted in PRD

We put constraints on dark energy properties using the PADE parameterisation, and compare it to the same constraints using Chevalier-Polarski-Linder (CPL) and $\Lambda$CDM, at both the background and the perturbation levels. The dark energy equation of state parameter of the models is derived following the mathematical treatment of PADE expansion. Unlike CPL parameterisation, the PADE approximation provides different forms of the equation of state parameter which avoid the divergence in the far future. Initially, we perform a likelihood analysis in order to put constraints on the model parameters using solely background expansion data and we find that all parameterisations are consistent with each other. Then, combining the expansion and the growth rate data we test the viability of PADE parameterisations and compare them with CPL and $\Lambda$CDM models respectively. Specifically, we find that the growth rate of the current PADE parameterisations is lower than $\Lambda$CDM model at low redshifts, while the differences among the models are negligible at high redshifts. In this context, we provide for the first time growth index of linear matter perturbations in PADE cosmologies. Considering that dark energy is homogeneous we recover the well known asymptotic value of the growth index, namely $\gamma_{\infty}=\frac{3(w_{\infty}-1)}{6w_{\infty}-5}$, while in the case of clustered dark energy we obtain $\gamma_{\infty}\simeq \frac{3w_{\infty}(3w_{\infty}-5)}{(6w_{\infty}-5)(3w_{\infty}-1)}$. Finally, we generalize the growth index analysis in the case where $\gamma$ is allowed to vary with redshift and we find that the form of $\gamma(z)$ in PADE parameterisation extends that of the CPL and $\Lambda$CDM cosmologies respectively., Comment: 15 pages, 6 figures and 5 tables. Accepted in ApJ