content_image/369966.jpg content_image/369967.jpg content_image/369968.jpg content_image/369969.jpg content_image/369970.jpg content_image/369964.jpg content_image/369965.jpg content_image/369962.jpg
with the solution 

where 

and P is taken as a free parameter. We can assign any real value to P . For the results presented in this work, we used a negative value. This negative sign is necessarily to overcome the sign from the integral limits. The geometric and thermodynamic quantities of the Universe read 

<img src='content_image/369966.jpg'>

<img src='content_image/369967.jpg'>

<img src='content_image/369968.jpg'>

<img src='content_image/369969.jpg'>

<img src='content_image/369970.jpg'>

<img src='content_image/369964.jpg'>

<img src='content_image/369965.jpg'>

a$_{0}$ is an arbitrary constant of the integration. The sign of q indicates whether the Universe decelerates (positive) or accelerates (negative). q can also be given as a function of the ther- modynamic, gravitational and cosmological quantities q ( t ) = [ ρ ( t )+3 p ( t )+3Π( t )] / 2 ρ ( t ) [33]. 

## IV. DE SITTER UNIVERSE 

Besides the approximation in g ( z ), previous solution apparently depends on the free parameter P . In this section, we suggest a particular solution to overcome P . Eq. (12) can easily be obtained by assuming that H doesn’t depend one t , i.e, de Sitter Universe. With a simple calculation, we get an estimation for H 

<img src='content_image/369962.jpg'>

