capacity.

With the use of Eqs. (\[8\]), (\[13\]) and (\[tau\]), respectively, we obtain the following equation describing the cosmological evolution of the Hubble function $H$ $$\begin{aligned}
\label{init}
\ddot H + \frac{3}{2} [1+(1-r) \gamma] H\dot H + \frac{1}{\alpha}\dot H - (1+r) H^{-1} \dot H^2 + \frac{9}{4}(\gamma -2) H^3 +
\frac{3}{2}\frac{\gamma}{\alpha} H^2 &=& 0.\end{aligned}$$

An approximate solution {#approx}
=======================

We introduce the transformation $u=\dot{H}$, so that Eq. (\[init\]) is transformed into a first order ordinary differential equation, $$\label{init2}
u\frac{du}{dH}-(1+r)H^{-1}u^{2}+\left(\frac{3}{2}[1+(1-r)\gamma ]H+\alpha
^{-1}\right) u+\frac{9}{4}\frac{1}{(\gamma)}H^{3}+\frac{3}{2}\frac{\gamma}{\alpha}
H^{2}=0.$$ We can rewrite Eq. (\[init2\]) in the form $$\label{OmegH1}
\Omega \frac{d\Omega }{dH} = F_1(H)\Omega + F_0(H),$$ where $$\begin{aligned}
\Omega &=& u \; E \;\; = u\; \exp\left(-\int \frac{1+r}{H} dH\right), \nonumber \\
F_1(H) &=& -\left( \frac{3}{2} [1+(1-r)\gamma]H + \frac{1}{\alpha}\right)E, \nonumber \\
F_0(H) &=& -\left(\frac{9}{4}(\gamma-2) H^3 + \frac{3}{2} \frac{\gamma}{\alpha} H^2\right)E^2. \nonumber\end{aligned}$$ By introducing a new independent variable $z=\int F_1(H)\,dH$, we obtain $$\Omega \frac{d\Omega}{dz} - \Omega = g(z),$$ with $g(z)$ is defined parametrically as, $$\label{gofzz1}
g(z) = \frac{F_0}{F_1}.$$ As shown in Appendix B, $g(z)$ can be approximated as a simple function of $z$ $$g(z)\approx {\cal C}\; z,$$ where ${\cal C}$ is a constant. We proceed with this approximation to get solvable differential equations. Keeping the parametric solution of $g(z)$, Eq. (\[fullgofz\]), results 