Statefinder hierarchy model for the Barrow holographic dark energy
Introduction
The cosmological findings (Riess, et al., 1998, Perlmutter, et al., 1999, Spergel, et al., 2003, Tegmark, et al., 2004) have indicated that there is an accelerated expansion. The responsible cause behind this accelerated expansion is a miscellaneous element having exotic negative pressure termed as dark energy (DE). In this field, an enormous DE models are suggested for the conceivable existence of DE such as CDM models, holographic dark energy (HDE) models (Li, Miao, Wei, Cai, 2008, Gao, Wu, Chen, Shen, 2009, Zhang, 2010, Zhang, Zhao, Cui, Zhang, 2014, Ratra, Peebles, 1988), and the scalar-field models (Peebles, Ratra, 1988, Zhang, 2007, Zhang, Zhang, Liu, 2008, Zhang, 2006, Bousso, 2000, De Felice, Tsujikawa, 2010), etc. In the direction of modified gravity, various efforts have been made, for instance, the theories (De Felice, Tsujikawa, 2010, Nojiri, Odintsov, 2011, Paul, 2019), the Dvali–Gabadadze–Porrati (DGP) braneworld model (Dvali et al., 2000) etc. Amidst other prevailing theoretical perspectives, the least complicated amongst all is the CDM model which is in good agreement with observations. The CDM model normally suffers from many theoretical problems such as fine tuning and coincidence problems (Copeland, Sami, Tsujikawa, 2006, Sahni, Starobinsky, 2006, Kamionkowski, Li, Li, Wang, Wang, 2011).
For clarifying the accelerating expansion of universe, the dark energy has been considered as most encouraging factor (Copeland, Sami, Tsujikawa, 2006, Li, Li, Wang, Wang, 2011, Li, Li, Wang, Wang, 2013, Frieman, Turner, Huterer, 2008). A large number of models have been proposed on idea of dark energy, but DE is still assumed as mysterious causes (Caldwell, Kamionkowski, Weinberg, 2003, Feng, Wang, Zhang, 2005, Wei, Cai, Zeng, 2005, Xia, Zhang, 2007, Wang, Zhang, 2008). Fundamentally, DE problem might be an issue of quantum gravity. It is normally accepted that the holographic principle (HP) (t’Hooft, Susskind, 1995) is one of the fundamental principles in quantum gravity. In 2004, Li proposed the (HDE) model (Li, 2004), which is the primary DE model inspired by the HP. This model is in generally excellent concurrence with recent observational data (Huang, Li, Li, Wang, 2009, Wang, Li, Li, 2010, Li, Li, Wang, Zhang, Huang, Li, 2011, Cui, Xu, Zhang, Zhang, 2015, He, Zhang, 2017, Wang, H., Li, 2017, Pradhan, Dixit, Singhal, 2019). Recently, Nojiri et al. (2020) proposed a HDE model in which they have established the unification of holographic inflation with holographic dark energy.
Motivated by the holographic fundamentals and utilizing different framework entropies, some new types of DE models were suggested such as, the Tsallis holographic dark energy (THDE) Tavayef et al. (2018); Dixit et al. (2019), the Tsallis agegraphic dark energy (TADE) Zadeh et al. (2019) the Renyi holographic dark energy (RHDE) Moradpour et al. (2018); Dixit et al. (2020),and the Sharma–Mittal holographic dark energy (SMHDE) model (Jahromi, 2018, Dubey, Sharma, Pradhan, 2021). Recently many authors shows a great interest in HDE models and explored in different context (Nojiri, Odintsov, Saridakis, 2019, Huang, Huang, Chen, Zhang, Tu, 2019, Ghaffari, Moradpour, Lobo, Graaa, Bezerra, 2018, Saridakis, Bamba, Myrzakulov, Anagnostopoulos, 2018, Ghaffari, Moradpour, Bezerra, Graaa, Lobo, 2019, Srivastava, Sharma, Pradhan, 2019).
Now a day our main concern is the CDM instead of DE models, because our observational analysis are based on CDM. The models generally deal with two key parameters: dimensionless HDE parameter and fractional density of cosmological constant. Since they decide the dynamical nature of DE and the predetermination of universe. For the evaluation of the various DE models, two diagnostic tools are generally used. The first one is the state finder hierarchy (Sahni, Saini, Starobinsky, Alam, 2003, Arabsalmani, Sahni, 2011), which is a geometrical diagnostic tool and it is model independent. The second parameter is the fractional growth parameter (FGP) (Acquaviva, Hajian, Spergel, Das, 2008, Acquaviva, Gawiser, 2010), which presents an independent scale diagnosis of the universe’s growth history. Additionally, a variation of the state finder hierarchy along with FGP, referred to as composite null diagnosis (CND) Arabsalmani and Sahni (2011) is often used to diagnose DE models.
In this way a clearer diagnosis called as state finder hierarchy (SFH) has been recently implemented in Yu et al. (2013). The diagnostic and statefinders are connected to the derivative of scale factor and the expansion rate . Now the composite null diagnosis (CND) is a helpful and beneficial technique to the state finder progressive system, where the fractional growth parameter associated with the structure growth rate (Acquaviva, Hajian, Spergel, Das, 2008, Acquaviva, Gawiser, 2010). In this model (Saridakis, 2020a), the researchers developed Barrow holographic energy. Here the authors utilized the holographic principle in a cosmological structure and Barrow entropy rather then the popular Bekenstein-Hawking. BHDE is also an engaging and thought-provoking optional framework for quantitatively describing DE.
Barrow (2020) has recently found the possibility that the surface of the black hole could have a complex structure down to arbitrarily tiny due to quantum-gravitational effects. The above potential impacts of the quantum-gravitational space time form on the horizon region would therefore prompt another black hole entropy relation, the basic concept of black hole thermodynamics. In particularHere and stand for the normal horizon area and the Planck area respectively. The new exponent is the quantum-gravitational deformation with bound as (Saridakis, Pradhan, Dixit, Bhardwaj, 2021, Barrow, 2020, Saridakis, Basilakas, Anagnostopoulos, Basilakos, Saridakis, Barrow, Basilakos, Saridakis, 2021). The value gives to the most complex and fractal structure, while relates to the easiest horizon structure. Here as a special case the standard Bekenstein–Hawking entropy is reestablished and the scenario of BHDE has been developed. The inequality , is given by the standard HDE, where stands for the horizon length under the assumption (Wang et al., 2017b). The Barrow holographic dark energy models have been explored and discussed by various researchers (Srivastava, Sharma, Mamon, Paliathanasis, Saha, Abreu, Neto, 2020, Abreu, Neto, 2020) in various other contexts. The above relationship results in some fascinating holographic and cosmological setup results (t’Hooft, Zadeh, Sheykhi, Moradpour, 2019). It should be noted that the above relationship offers the usual HDE for the special case of , i.e. . Consequently, the BHDE is definitely a more general paradigm than the standard HDE scenario. We are concentrating here on the general () scenario. We consider Hubble horizon (HH) as an IR cut-off (). The energy density of BHDE is expressed aswhere is an unknown parameter.
In the current research article, we are considering the fundamental geometry of the universe to be a spatially flat, homogeneous and isotropic space time. The Hubble horizon has been regarded as an effective IR cut-off to describe the continuing accelerated expansion of the universe (Saridakis, Pradhan, Dixit, Bhardwaj, 2021). The objective of this research is to apply the diagnostic tools to differentiate between the BHDE models with various other values of . Here, we use the diagnostic tool of the state finder hierarchy for BHDE that achieves the value of the model and demonstrates consistency of the model for proper estimation of the parameters. It is worth mentioning here that we have also demonstrated DE’s physical scenario by taking Barrow exponent . We arrange the manuscript as follows: The BHDE model suggested in Saridakis (2020a) and the basic field equations are added in Section 2. The state finder hierarchy are discussed in Section 3. We explore the fractional state finder diagnostic in Section 4. Growth rate perturbations are discussed in Section 5. Section 6 contains the conclusive statements, discussion and discourse.
Section snippets
Basic field equations
The construction of modified Friedman equations with the application of gravity-thermodynamic conjecture is discussed in this section with the use of entropy from Barrow (Saridakis, 2020b). The metric reads as for the flat FRW Universewhere is the cosmic time and is the dimensionless scale factor normalized to unity at the present time i.e. .
The Friedmann first field equation for BHDE is given as :where and read as
Statefinder hierarchy
Statefinder hierarchy (Zhou, Wang, 2016, Zhao, Wang, 2018, Wang, Steinhardt, 1998) is an effective diagnosis of geometry. It is appropriated to discriminate, various DE models from the CDM model by utilizes the higher order derivative of the scale factor. In this case, the scale factor a(t) is the main dynamical variable. Here we will be focused on the late time evolution of the universe. Now we assume the Taylor expansion of the scale factor around the current age (Yu et al., 2013):
Fractional Statefinder
The statefinder hierarchy equations produce a progression of diagnostic for model with . By using the relationship , valid in CDM. It is also possible to write Statefinders in an alternative form as:
etc
It is clearly visible that in both of the methods yield identical results for : =1.
However for other DE models, it is proposed that the two alternative definitions, of the state finder, , would provide
Growth rate of perturbations
We use the ”parametrized post-Friedmann” theoretical framework for the estimation of growth rate perturbations for the interaction of DE to acquire the values for the models numerically.
The perturbation growth rate parameter (Acquaviva, Hajian, Spergel, Das, 2008, Acquaviva, Gawiser, 2010) is given bywhere represents the structure’s growth rate. Here, , with matter density perturbation, energy density.
If the matter density perturbation is
Conclusion
In this manuscript we have discussed the BHDE by utilizing the Barrow entropy, rather than the standard Bekenstein-Hawking. Our main focus in the manuscript is to diagnose BHDE model by the State finder hierarchy and composite null diagnostic (CND). The diagnostic tools which are frequently apply to test the DE models: (a) state finder hierarchy (b) fractional growth parameter . Additionally the CND pair, are also considered to diagnosing DE models. So, the major concern of the work
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The authors are thankful to Dr. Kasturi Sinha Ray (GLA University) for her help in preparing the manuscript. The authors also thank to the anonymous referee for his/her constructive comments which helped to improve the quality of paper in present form.
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