Department of Physics, Taraba State University, P. M. B. 1167, Jalingo, Taraba State, Nigeria



In a bid to resolve lingering problems in modern Cosmology, Cosmologist are turning to models in which Physical constants of nature are not real constants but vary as function of the scale factor of the universe. In this work, a cosmological model is developed by incorporating Ghost Dark Energy model with Barrow's ansatz for varying Gravitational Constant. A model obtained is free from initial Big Bang Singularity and Inflation Problem. A test of Causality showed that the model depends of the constant m and is classically stable at  


In
The simplest model of DE is the cosmological constant, which is a key ingredient in the ΛCDM model. Although the ΛCDM model is consistent very well with all observational data, it faces the finetuning problem
A dynamical dark energy model, known as Veneziano ghost dark energy (GDE) has been introduced in the late 70s by Veneziano [10]. This has signiﬁcant nontrivial physical properties for the expanding universe or in spacetime having nontrivial topological formation. The existence of Veneziano GDE is supposed to resolve the coincidence problem
According to
The Newtonian gravitational constant G occurs in the source term of Einstein’s field equation of the general theory of relativity, which is a fundamental equation for developing every model of cosmology. In Einstein’s field equation, G acts as a coupling constant between the geometry of spacetime and matter. In quantum mechanics, G is essential in the definition of the Planck constant
The structure of this paper is as follows: In section I, we gave a brief introduction, section II contents the ghost dark energy model. The evolution of the Universe studied in section III. We tested the classical causality in section IV and conclude in section V.
In this work, we shall use the action of variable speed of light given by:
)(1)
Where λ= cosmological constant which serves as vacuum energy, = matter Lagrange density, R = Ricci scalar, G = Newton’s gravitational constants, and The determinant of the metric tensor. Varying the action for the metric and ignoring surface terms leads to
(2)
Where = Einstein tensor = Energy momentum tensor. The Greek indices run from 0 – 3. In the cosmological context, the Friedmann Robertson Walker metric is:
(3)
Where
t is the comoving time and = 0, 1, 1 representing flat, closed and open universe respectively. The Friedmann equations are;
(4)
(5)
Here we will consider the energy density of ghost dark energy given by:
(6)
And Chaplygin gas equation of state (EoS)
(7)
We shall use the conservation equation given by
(8)
For flat FRW universe . The conservation equation can be rewritten as
(9)
For varying, we shall assume the Barrow ansatz
(10)
We shall use this to solve for the scale factor and use the scale factor to determine physical properties of cosmic importance such as deceleration parameter equation of state parameter e.t.c
Evolution of the Scale Factor
For this model, the scale factor of the universe for a flat spacetime is computed using the conservation given by
(11)
Taking the time derivative of equation (10) and substituting it into Equation (11) we obtain:
(12)
Where and for dust particles. The solution to (12) is
(13)
is a constant, indicating an empty universe.
Now applying the energy density for the Ghost Dark Energy model (6) and the Chaplygin gas E0S (7) into (11) we obtained a differential equation of the form:
The solution of the above equation is :
Where C is a constant. A simple case is when , the scale factor becomes:
The evolution of the scale factor is shown in figure 1 below. It is observed that the model is free from the initial Big Bang singularity and the universe ends in a big rip. The figure also shows that there was no inflation as alleged by the standard model
The Hubble parameter is :
From figure 2, it is observed that the Hubble parameter diverges at the beginning and end of the universe. Such a universe starts with a positive deceleration parameter indicating deceleration expansion and transits to a negative deceleration parameter indicating accelerating expansion phase
The deceleration parameter is
(19)
The universe is decelerating when
For any cosmological model to survive, it is established that the speed of sound cannot exceed the local speed of light, cs≤ 1. The second requirement for stability is that the square of the speed of sound must be positive, i.e., c2s>0. In case the model is classically we obtain:
(20)
A requirement of the classical stability is and the causality is . Hence, the best value of , for both stability conditions are
Cosmic models in which physical constants of nature are not constant but vary with time havebeen of interest in recent years. Indeed, the speed of light in our universe which has gone throughvarious phase transition due to various content might not have been a constant, especially at theearly stage. The Newtonian gravitational constant which acts as a coupling term in the Einstein Field Equation used here varies as a function of the scale factor of the universe to obtain a solution to the Friedmann equation that governs the dynamics of the universe from the very beginning, through various phase transitions to the present epoch.
The value of the scale factor obtained indicates that the size of the universe was never zero. By incorporating Ghost dark energy and variable gravitational constant, a model is presented free from the initial big bang singularity and inflation. The model further revealed that the Hubble parameter diverges at the beginning and end of the universe. Such a universe starts with a positive deceleration parameter indicating deceleration expansion and transits to a negative deceleration parameter indicating accelerating expansion phase. Furthermore, studying the stability of this model it is observed that it depends on the constant m and is classically stable at .
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