Abstract
The paper is devoted to the study of cosmological models with time-varying cosmological term () in the presence of creation field in the framework of Bianchi type III space-time. To obtain deterministic model of the universe, we have assumed , where is the scale factor, for steady state cosmology and creation field, and shear () is proportion to expansion () which leads to , where and are the metric potentials to explain small anisotropic behaviour of the universe and its isotropy. To obtain the results in terms of cosmic time , we have assumed . The model satisfies conservation equations, and creation field increases with time. The present model is free from singularity, has particle horizon, and provides a natural explanation for inflationary scenario and isotropization. Creation field and Einstein field equations are derived using principle of least action and Lagrangian formulation of variable cosmological term. For illustrative purposes, evolutionary behaviour of some cosmological parameters are shown graphically. The other physical aspects like accelerating behaviour of the model are also discussed. Thus, the model represents not only expanding universe but also accelerating which matches with the results of present-day observations.
1. Introduction
The investigations deal with the physical process and use a model of the universe which is called big bang model, but the big bang models face problems such as the following: (i) singularity problem, (ii) the conservation of energy is violated, (iii) it leads to a very small particle horizon in early epoch of the universe, (iv) it fails to provide a consistent scenario that explains the origin and evolution of the universe, and (v) it has flatness problem.
If a model successfully explains the creation of positive energy matter without violating the conservation of energy, then it becomes necessary to have a negative energy mode that provides a natural way for creation of matter. Hoyle and Narlikar [1] adopted a field theoretic approach for creation of matter by introducing a massless and chargeless scalar field to discuss creation field (-field) cosmology. There is no big bang-type singularity in -field cosmology. The creation field cosmological models create more interest in the study because these models solve a number of outstanding problems of big bang cosmology like singularity, horizon, and flatness problems of the universe [2]. These problems could be solved by (i) quantum gravity models and (ii) inflationary cosmological models. But we do not have complete quantum theory of gravity, and inflationary models do not solve the problem of singularity for creation. Therefore, the -field cosmology with negative energy field is justified as introduced by Hoyle and Narlikar [1].
Frieman et al. [3] have pointed out that we have entered an era dominated by dark energy (cosmological term). A wide range of observations suggest that cosmological term () is the most favoured candidate of dark energy representing energy density of vacuum. In Einstein’s field equations, the nontrivial role of vacuum generates a cosmological constant () term which leads to inflationary scenario [4]. The present-day observations of smallness of cosmological constant () support to assume that the cosmological constant is time dependent. Gibbons and Hawking [5] have investigated that cosmological models with positive cosmological constant lead to de-Sitter space-time asymptotically. Therefore, the cosmological models linking the variation of cosmological constant and having the form of Einstein field equations unchanged have been studied by many authors, viz., [6–10].
In the late eighties, astronomical observations revealed that the predictions of FRW models do not always meet our requirements as was believed earlier (Smoot et al. [11]). Therefore, spatially homogeneous and anisotropic Bianchi space-times (I-IX) have been investigated to study the universe in its early stages of evolution. Recent cosmological observations support the existence of anisotropic universe that approaches to isotropic phase (Land and Maguejo [12]). Among these, Bianchi type III space-time is simple generalisation of Bianchi type I space-time which plays a significant role for the study of the universe.
Recently, Bali [13] investigated creation field cosmological model for dust distribution and time-dependent cosmological term in Bianchi type II space-time.
In this paper, we have investigated creation field cosmological model for dust distribution and cosmological term in the framework of Bianchi type III space-time and discussed inflationary scenario, horizon problem, inflationary parameters, and isotropisation. The model fulfills the requirements as per astronomical observations.
2. Metric and Field Equations
We consider Bianchi type III space-time in orthogonal form represented by the line element where , , and are the metric potential functions of alone.
By the introduction of creation field (), the modified Einstein field equation is given by Hoyle and Narlikar [1] as
The energy momentum tensor for perfect fluid distribution and for creation field are given by where is the coupling constant between matter and creation field and . We note that for . Thus, -field has negative energy density that produces a repulsive gravitational effect. This repulsive force drives the expansion of the universe. The variation of (creation field) gives the source equation in the form where is the number of net creation events per unit proper 4-volume.
3. The Lagrangian Formulation of Einstein Field Equation with Term
The Lagrangian density () is defined by Moffat [14] as where where is the variable cosmological constant treated as dynamical field, is the Lagrangian multiplier field, is observed for velocity along a world line in a space-time, is a constant, and is a matter of Lagrangian density. A variation of with respect to and yields the equation of motion as
Variation of with respect to and using (8) gives
The cosmological constant enters through the vacuum energy density.
has small Gyr. By the introduction of -field given by Hoyle and Narlikar [1], the Einstein field equation leads to equations (2) with (3) and (4).
The coordinates are assumed to be comoving so that and .
The modified Einstein’s field equation (2) for dust distribution and the space-time (1) leads to where suffix 4 after , , and indicates partial derivative with respect to cosmic time .
Following Hoyle and Narlikar [1], we have used (dust fluid distribution). The source equation of -field leads to for large . Thus, . Equations (12)–(15) lead to
Equation (16) leads to where is a constant. For deterministic models of the universe, we assume two conditions: (i)Shear () α expansion (θ) as investigated by Thorne [15](ii) as investigated by Hoyle et al. [16] for steady state and creation field cosmology, being the scale factor
The assumption is given by Thorne [15] as per astronomical observations. Thorne [15] has investigated that the current observations of velocity redshift relation for extragalactic sources suggest that the Hubble expansion of the universe is isotropic to within ~30%. Also, redshift studies place the limit in the neighbourhood of our galaxy. By contrast, cosmic microwave background measured isotropy is within 3%, at least around the celestial equator as mentioned by Partridge and Wilkinson [17]. In our model, which is very small in large-scale structure of the universe. The shear () and expansion () for metric (1) are given by
Using , we have where is a constant. The spatial volume is given as and .
Using (21), (23), and (25) in (17), we have where .
To get deterministic solution, we assume
Thus, .
Now, equation (26) leads to
Thus, we have which leads to where is a constant of integration.
To get deterministic model of the universe in terms of cosmic time , we assume and .
Thus, equation (30) for and leads to which leads to
That is,
Therefore, we have
That is, where , , and is a constant of integration.
We can choose without loss of generality.
Thus, metric (1) leads to
4. Physical and Geometrical Aspects
The matter density (), spatial volume (), shear (), expansion (), Hubble parameter (), the cosmological term (), and deceleration parameter () for model (36) are given by
which agrees with the result as investigated by Thorne [15].
The reality condition leads to
The creation rate is determined by the conservation equation which leads to where .
Thus, we have which leads to
Thus, , where .
Therefore, and which matches with the result as investigated by Hoyle and Narlikar [1] in creation field cosmology.
In anisotropic models, the directional Hubble parameters , , and are always different. We define average Hubble parameter and calculate which agrees with the result as investigated by Thorne [15]. Crawford and Aguiar [18] have shown that if the parameter is almost zero, where is the shear and is the anisotropic parameter, then we find isotropic phase (Figures 1, 2, 3 and 4).
5. Discussion and Conclusion
The spatial volume for model (36) increases with time exponentially, thus representing inflationary universe. The matter density is positive which satisfies reality condition. The model also satisfies the condition which is very small, as pointed out by Crawford and Aguiar [18]. Thus, the model leads to isotropic phase. The cosmological term . The shear () is 0 at the initial stage and at the late time; i.e., at , it leads to 1. But % which is small and agrees with the result as investigated by Thorne [15]. We have studied the matter dominated stage for creation field cosmology. But the creation always exists in the beginning and end. This is also applicable to radiation dominated stage, i.e., . The deceleration parameter indicates that the model shows accelerating universe throughout. The cosmological parameter () helps to solve the problems of steady state theory as mentioned by Hoyle et al. [16]. The unit of time in the figure is indicated in gigayear. The model represents expanding and accelerating universe which agrees with the result obtained by Riess et al. [19] and Perlmutter el al. [20] as deceleration parameter .
The coordinate distance to the horizon is the maximum distance a null ray could have travelled at time starting from the infinite past. Thus,
We can extend the proper time to in the past because of the nonsingular nature of model (36). Therefore,
Taking , . which shows that model (36) has particle horizon.
Data Availability
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Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This study was supported by the Wellcome Trust, the European Commission, the Ministry of Science and Technology in Taiwan, and the National Earth System Science Data Sharing Infrastructure in China. A list of policies of some funders and other organisations is available in FAIRsharing.