Gravity model behaving as a dark energy source
Introduction
The ambiguous dark energy (DE) has been regarded as one of the most tantalizing topics in cosmology since its profound discovery in 1998 (Riess, et al., 1998, Perlmutter, et al., 1999). It is considered to be the reason behind the late time expanding universe at an expedited rate due to its huge negative pressure with repulsive gravitation. It is uniformly permeated throughout the space and vary slowly or almost consistent with time (Chan, 2015, Peebles, Ratra, 2003, Carroll, 2001, Carroll). Cosmologists all over the map have conducted a series of studies with the aim of hunting its origin and are still scrabbling for a perfect answer. Some worth mentioning such studies that have not escaped our notice in the recent years are briefly discussed below.
In Singh and Kar (2019), the authors assert that emergent D-instanton might indicate us a hint to the root of DE. A cosmological model associated with an antineutrino star is constructed by Neiser (2020) in order to search the origin of DE. In Dikshit (2019), the author presents an explanation for DE with pure quantum mechanical method. In Huterer and Shafer (2017), the investigation of the twenty years old history of DE and the current status can be seen. The authors in Wang et al. (2018) study the evolution of the DE using a non-parametric Bayesian approach in the light of the latest observation. In Capolupo (2018), the author claims that vacuum condensate can provide us the origin of DE. According to Josset et al. (2017), DE is originated from the violation of energy conservation. A unified dark fluid is obtained as a source of DE by Tripathy et al. (2015). The presence of particle with imaginary energy density can lead us to the source of DE (Chan, 2015a). The explanation of a physical mechanism as a source of DE is presented by Gontijo (2012). Lastly, in Alexander et al. (2010), DE evolves as a result of the condensation of fermions formed during the early evolution.
It is an obvious fact that the universe is dominated by the cryptic DE with negative pressure and positive energy density (Carroll, 2001, Law, 2020, Singh, Singh, 2019, Agrawal, Obied, Steinhardt, Vafa, 2018, Ray, Mukhopadhyay, Rahaman, Sarkar, 2013, Straumann, 2007, de Araujo, 2005, Wu, Yu, 2005). This qualifies DE a completely irony of nature as the dominating component is also the least explored. As against the positive energy density condition, it is fascinating to see many authors introducing the concept of the possibility of negative energy density (NED) with convincig arguments in support. In Ijjas and Steinhardt (2019), the authors discuss NED where models evolve with a bounce. The authors continued that there might be bounces in the future too. The discussion of negative vacuum energy density in Rainbow Gravity can be seen in Wong et al. (2019). In Nemiroffa et al. (2015), we can witness, under certain conditions, a repelling negative gravitational pressure with NED. Further, we can find a repelling negative phantom energy with NED. In Fay (2014), the author claims that the universe evolves by inflation when the coupled fluid has NED in the initial epoch. An accelerating universe with NED is studied by Sawicki and Vikman (2013). In Macorra and German (2004), we can find an explaination of energy density with negative value with equation of state parameter (EoS) . The author in Carroll (2001a)predicts that NED is possible only if the DE is in the form of vacuum energy. In Huang (1990), the investigation of models which evolved with NED in the infinite past can be found. According to Parker and Fulling (1973), the introduction of quantized matter field with NED to energy momentum tensor might by pass cosmological singularity. Besides defying the energy conditions of GR, NED also disobeys the second law of thermodynamics (Hawking and Ellis, 1973). However, the condition should be solely obeyed on a large scale or on a mean calculation, thereby neglecting the probable violation on a small scale or for a short duration, in relativity(Fewster, Graham, Olum, 2003, Visser, Barcelo, Pfenning, Ford, Helfer, 1998, Helfer, 1998, Ford, Roman, 1996, Roman, 1986, Epstein, Glaser, Jaffe). Hence, in the initial epoch, if there were circumstance of defiance for a short duration measured against the present age of Gyr estimated by the latest Planck 2018 result (Planck collabration, 2019), it will remain as an important part in the course of evolution.
In the present cosmology, authors prefer to opt alternate or modified theories of gravity in order to precisely understand the underlying mechanism of the late time expedited expansion of the universe. One such well appreciated modified theory is the gravity introduced by Harko et al. (2011) in which the gravitational Lagrangian is represented by an arbitrary function of the Ricci scalar and the trace of the energy-momentum tensor. In the past few years, this theory has captivated many cosmologists and theoretical physicists as it presents a natural gravitational substitute to DE (Chirde and Shekh, 2019). Recently, Myrzakulov (2020) studies the theory and predicts the conditions to obtain expanding universe in the absence of any dark component. In Sahoo et al. (2020), the authors investigate a mixture of barotropic fluid and DE in gravity where the model evolves from the Einstein static era and approaches CDM. In Singh and Singh (2019b), the study of cosmological dynamics of DE within the theory can be seen. Pawar et al. (2019)study a modified holographic Ricci DE model in the theory obtaining a singularity free model. The authors in Zia et al. (2018) investigate gravity discussing future singularities in DE dominated universe. In Srivastava and Singh (2018), we can find a discussion of new holographic DE model in gravity thereby obtaining CDM in the late times. In Fayaz et al. (2016), the examination of ghost DE model within the theory can be seen, predicting model behaving as phantom or quintessence like nature. The investigation of cosmological models within the theory without DE is observed in Sun and Huang (2016). The authors in Mishra et al. (2016b)and Singh and Kumar (2016) study the relation of the theory with DE. Houndjo and Piattella (2012) present a reconstruction of the theory from holographic DE. The study of cosmological model in gravity obtaining DE induced cosmic acceleration can be seen in Mishra et al. (2016a). Zubair et al. (2016) discuss Bianchi space-time within the theory with time-dependent deceleration parameter. Ahmed et al. (2016) investigate a model in which the cosmological constant is considered as a function of . The authors in Rao and Rao (2015)discuss a higher dimensional anisotropic DE model within the theory obtaining the EoS parameter . Jamil et al. (2012)construct models within the theory asserting that dust fluid leads to CDM. Houndjo (2012)predicts a model in gravity that transit from matter dominated to accelerating phase. From these worth appreciating studies, it won’t be a wrong guess to sum up that there must be some sort of hidden correspondences between the pair of DE and gravity theory. Consequently, in this work, we will try to find out if gravity theory itself behaves as a DE source.
The possibility of space-time possessing with more than 4D has fascinated many authors. In the recent years, there has been a trend of preferring higher dimensional space-time to study cosmology. Higher dimensional model was introduced by Kaluza (1921) and Klein (1926) in an effort to unify gravity with electromagnetism. Higher dimensional model can be regarded as a tool to illustrate the late time expedited expanding paradigm (Banik and Bhuyan, 2017). Investigation of higher dimensional space-time can be regarded as a task of paramount importance as the universe might have come across a higher dimensional era during the initial epoch (Singh et al., 2004). Marciano (1984) asserts that the detection of a time varying fundamental constant can possibly show us the proof for the extra dimension. According to Alvax and Gavela (1983) and Guth (1981), the extra dimension generates a huge amount of entropy which gives possible solutions to the flatness and horizon problem. Since we are living in a 4D space-time, the hidden extra dimension in 5D is highly likely to be associated with the invisible DM and DE (Chakraborty and Debnath, 2010).
Keeping in mind the above notable works by different authors, we have analysed a spherically symmetric metric in 5D setting within the framework of gravity with a focus to predict a possible source of DE. Here, we observe the field equations with due consideration of reasonable cosmological assumptions within the limit of the present cosmological scenario. The paper has been structured into sections. In Sect. 2, the field equations of the theory are discussed. In Sect. 23, in addition to obtaining the solutions of the field equations, the cosmological parameters are also solved. In Sect. 34, the physical and kinematical aspects of our model are discussed with graphs. Considering everything, a closing remark is presented in Sect. 45.
Section snippets
The field equations of gravity theory
The action of gravity theory is given bywhere is an arbitrary function of the Ricci scalar and the trace of the energy-momentum tensor of matter defined by Koivisto (2006) as
Here, the matter Lagrangian density is assumed to rely solely on so that we obtain
The action is varied w.r.t. the metric tensor and hence, the field equations of gravity are given by
Formulation of the problem and solutions
The five-dimensional spherically symmetric metric is given bywhere and are cosmic scale factors.
Now, using co-moving co-ordinates, the surviving field equations are obtained as followswhere an overhead dot indicates differentiation w.r.t. .
From Eqs. (12) and (13), the expressions for the cosmic scale factors are obtained aswhere
Discussions
For convenience sake and to obtain realistic results, specific values of the arbitrary constants involved are chosen i.e., and and . The graphs of the cosmological parameters w.r.t. cosmic time are presented with the detailed discussion in view of the latest observations.
Fig. 1 and Fig. 2 can be regarded as the perfect evidences for the present spatial expansion at an expedited rate. When and other related parameters are constants (), implying that the model
Conclusions
Within the framework of gravity, we have analysed a spherically symmetric space-time in 5D setting. We have obtained an isotropic model universe undergoing super-exponential expansion. The variation of the pressure and energy density with cosmic time are analysed when and . In both the cases, it is fascinating to see that our gravity model behaves as a DE (vacuum energy) model. The model is free from an initial singularity. The model expands with a slow and
CRediT authorship contribution statement
Pheiroijam Suranjoy Singh: Writing - original draft, Software, Investigation, Visualization, Investigation, Data curation, Formal analysis, Conceptualization. Kangujam Priyokumar Singh: Supervision, Validation, Resources, Methodology, Writing - review & editing.
Declaration of Competing Interest
None.
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