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Statefinder diagnostic is a useful method which can differ one dark energy model from the others. The Statefinder pair ${r, s}$ is algebraically related to the equation of state of dark energy and its first time derivative. We apply in this paper this method to the dilaton dark energy model based on Weyl-Scaled induced gravitational theory. We investigate the effect of the coupling between matter and dilaton when the potential of dilaton field is taken as the Mexican hat form. We find that the evolving trajectory of our model in the $r-s$ diagram is quite different from those of other dark energy models.
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Using a new method--statefinder diagnostic which can differ one dark energy model from the others, we investigate in this letter the dynamics of Born-Infeld(B-I) type dark energy model. The evolutive trajectory of B-I type dark energy with Mexican hat potential model with respect to $e-folding$ time $N$ is shown in the $r(s)$ diagram. When the parameter of noncanonical kinetic energy term $etato0$ or kinetic energy $dot{phi}^2to0$, B-I type dark energy(K-essence) model reduces to Quintessence model or $Lambda$CDM model corresponding to the statefinder pair ${r, s}$=${1, 0}$ respectively. As a result, the the evolutive trajectory of our model in the $r(s)$ diagram in Mexican hat potential is quite different from those of other dark energy models.
$Om$ diagnostic can differentiate between different models of dark energy without the accurate current value of matter density. We apply this geometric diagnostic to dilaton dark energy(DDE) model and differentiate DDE model from LCDM. We also investigate the influence of coupled parameter $alpha$ on the evolutive behavior of $Om$ with respect to redshift $z$. According to the numerical result of $Om$, we get the current value of equation of state $omega_{sigma0}$=-0.952 which fits the WMAP5+BAO+SN very well.
We apply in this paper the statefinder parameters to the interacting phantom energy with dark matter. There are two kinds of scaling solutions in this model. It is found that the evolving trajectories of these two scaling solutions in the statefinder parameter plane are quite different, and that are also different from the statefinder diagnostic of other dark energy models.
We have analyzed the Barrow holographic dark energy (BHDE) in the framework of the flat FLRW Universe by considering the various estimations of Barrow exponent $triangle$. Here we define BHDE, by applying the usual holographic principle at a cosmological system, for utilizing the Barrow entropy rather than the standard Bekenstein-Hawking. To understand the recent accelerated expansion of the universe, considering the Hubble horizon as the IR cut-off. The cosmological parameters, especially the density parameter ($Omega_{_D}$), the equation of the state parameter ($omega_{_D}$), energy density ($rho_{_{D}}$) and the deceleration parameter($q$) are studied in this manuscript and found the satisfactory behaviors. Moreover, we additionally focus on the two geometric diagnostics, the statefinder $(r,s)$ and $O_{m}(z)$ to discriminant BHDE model from the $Lambda CDM$ model. Here we determined and plotted the trajectories of evolution for statefinder $(r, s)$, $(r,q)$ and $O_{m}(z)$ diagnostic plane to understand the geometrical behavior of the BHDE model by utilizing Planck 2018 observational information. Finally, we have explored the new Barrow exponent $triangle$, which strongly affects the dark energy equation of state that can lead it to lie in the quintessence regime, phantom regime, and exhibits the phantom-divide line during the cosmological evolution.
In this work we study a non-flat Friedmann-Robertson-Walker universe filled with a pressure-less dark matter (DM) and Barrow holographic dark energy (BHDE) whose IR cutoff is the apparent horizon. Among various DE models, (BHDE) model shows the dynamical enthusiasm to discuss the universes transition phase. According to the new research, the universe transitioned smoothly from a decelerated to an accelerated period of expansion in the recent past. We exhibit that the development of $q$ relies upon the type of spatial curvature. Here we study the equation of state (EoS) parameter for the BHDE model to determine the cosmological evolution for the non-flat universe. The (EoS) parameter and the deceleration parameter (DP) shows a satisfactory behaviour, it does not cross the the phantom line. We also plot the statefinder diagram to characterize the properties of the BHDE model by taking distinct values of barrow exponent $triangle$. Moreover, we likewise noticed the BHDE model in the $(omega_{D}-omega_{D}^{})$ plane, which can furnish us with a valuable, powerful finding to the mathematical determination of the statefinder. In the statefinder trajectory, this model was found to be able to reach the $Lambda CDM$ fixed point.
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