I'm studying a Friedmann-Lemaitre-Robertson-Walker (FLRW) cosmology model with a simple scalar field as source (no dust-like matter, no radiation, no cosmological constant). For the moment, the field is a Klein-Gordon (KG) scalar only, of potential $\mathcal{V}(\phi) = \frac{1}{2} \; m^2 \, \phi^2$ (later, I'll generalize it to the quartic potential used in inflation theory).
The equations that I need to solve are these (FLRW equations, and the KG equation) : \begin{gather} \frac{\dot{a}^2}{a^2} + \frac{k}{a^2} = \frac{8 \pi G}{3} \; \rho,\tag{1} \\[12pt] \frac{\ddot{a}}{a} = -\, \frac{4 \pi G}{3} (\rho + 3 p), \tag{2} \\[12pt] \ddot{\phi} + 3 \frac{\dot{a}}{a} \; \dot{\phi} + \mathcal{V}^{\, \prime} = 0. \tag{3} \end{gather} The variables are $a(t)$, $\phi(t)$. The scalar field density and pressure are these : \begin{gather} \rho = \frac{1}{2} \; \dot{\phi}^2 + \mathcal{V}(\phi), \tag{4} \\[12pt] p = \frac{1}{2} \; \dot{\phi}^2 - \mathcal{V}(\phi). \tag{5} \end{gather} The initial conditions are $\phi(0) = \phi_0$ and $\partial_t \, \phi \, |_{t = 0} = \dot{\phi}_0$ (two arbitrary real numbers) and the mass parameter $m$ (an arbitrary positive number). For the scale factor, I set $a(0) = 1$. Since the expansion rate at $t = 0$ is $H_0 \equiv \frac{\dot{a}}{a} |_{t=0}$ (the Hubble constant), its inverse could be used as an unit of time. So I set the initial condition $\dot{a}(0) = 1$.
I now could numerically solve equations (2) and (3) using $\phi_0$, $\dot{\phi}_0$ and $m$ as arbitrary input, and draw nice graphics of the evolution of $a(t)$ and $\phi(t)$.
Now the question is what should be the proper variables to define the phase-space of the scale factor and of the field? Currently, I'm using these: \begin{equation}\tag{6} \big(\, a, \; \dot{a} \big), \qquad \big( \, \phi, \; \dot{\phi} \big). \end{equation} I suspect it should be something more complicated. From a lagrangian that I'm not writing here ($p_q = \frac{\partial L}{\partial \dot{q}}$ is the canonical momentum associated to the $q$ variable), I suspect the proper variables to be used in a phase-space diagram are these instead: \begin{equation}\tag{7} \big(\, a, \; -\, 6 \, a \, \dot{a} \big), \qquad \big( \, \phi, \; a^3 \, \dot{\phi} \big). \end{equation} But when I plot the graphics of those variables, I'm getting weirdly deformed patterns that are hard to rescale. Apparently, variables (6) are giving better results (nicer to look at). I need advices on this.
EDIT: About the FLRW cosmology of a simple KG scalar field, is there any papers that discuss the results (from numerical integration)? I would like to compare my results with something, since I never saw any discussion of this model in all the General Relativity books that I have (except the usual inflation scenarios with slow roll approximations or other variations...).