On solving dynamical equations in general homogeneous isotropic cosmologies with scalaron

摘要: We study general dynamical equations describing homogeneous isotropic cosmologies coupled to a scalaron $\psi$. For flat ($k=0$), we analyze in detail the gauge-independent equation differential, $\chi(\alpha)\equiv\psi^\prime(\alpha)$, of map metric $\alpha$ field $\psi$, which is main mathematical characteristic locally defining `portrait' cosmology `$\alpha$-version'. In `$\psi$-version', similar for differential inverse map, $\bar{\chi}(\psi)\equiv \chi^{-1}(\alpha)$, can be solved asymptotically or some `integrable' potentials $v(\psi)$. case, $\bar{\chi}(\psi)$ and $\chi(\alpha)$ satisfy first-order depending only on logarithmic derivative potential. Once know analytic solution one these $\chi$-functions, explicitly derive all characteristics cosmological model. $\alpha$-version, whole system integrable $k\neq 0$ with any `$\alpha$-potential', $\bar{v}(\alpha)\equiv v[\psi(\alpha)]$, replacing There no priori relation between two before deriving $\chi$ $\bar{\chi}$, implicitly depend potential itself, but relations pictures found by asymptotic expansions inflationary perturbation theory. Explicit applications results more rigorous treatment chaotic inflation models their comparison ekpyrotic-bouncing ones are outlined frame our `$\alpha$-formulation' cosmologies. particular, establish an expansion $\chi$. When conditions satisfied obeys certain boundary (initial) condition, get standard parameters, higher-order corrections.