content_image/328277.jpg content_image/328276.jpg content_image/328275.jpg content_image/328274.jpg content_image/328273.jpg content_image/328270.jpg content_image/328271.jpg content_image/328269.jpg content_image/328268.jpg
We can rewrite Eq. (13) in the form 

where 

By introducing a new independent variable z = ∫ F$_{1}$ ( H ) dH , we obtain d Ω 

with g ( z ) is defined parametrically as, 

<img src='content_image/328277.jpg'>

<img src='content_image/328276.jpg'>

<img src='content_image/328275.jpg'>

<img src='content_image/328274.jpg'>

As shown in Appendix B, g ( z ) can be approximated as a simple function of z 

<img src='content_image/328273.jpg'>

where C is a constant. We proceed with this approximation to get solvable differential equa- tions. Keeping the parametric solution of g ( z ), Eq. (50), results in much more complicated differential equations. This would be the subject of a future work. 

From the definitions of Ω and z we have 

Analogous to the solution of reduced Abel type canonical equation, 

<img src='content_image/328270.jpg'>

<img src='content_image/328271.jpg'>

<img src='content_image/328269.jpg'>

(see Appendix C) we obtain the relation Ω = z/ P . Therefore, from Eqs. (18) and (19) we obtain the following first order differential equation for Hubble parameter H , 

<img src='content_image/328268.jpg'>

