Isotropic LQC and LQC--inspired Models with a massless scalar field as Generalised Brans--Dicke theories

We explore whether generalised Brans – Dicke theories, which have a scalar field $\mathrm{\Phi }$ and a function $\omega (\mathrm{\Phi })$, can be the effective actions leading to the effective equations of motion of the LQC and the LQC–inspired models, which have a massless scalar field $\sigma$ and a function $f(m)$. We find that this is possible for isotropic cosmology. We relate the pairs $(\sigma ,f)$ and $(\mathrm{\Phi },\omega )$ and, using examples, illustrate these relations. We find that near the bounce of the LQC evolutions for which $f(m) = sin ; m$,thecorrespondingfield$\Phi \to 0$andthefunction$\omega(\Phi) \propto \Phi^2 ;$. We also find that the class of generalised Brans – Dicke theories, which we had found earlier to lead to non singular isotropic evolutions, may be written as an LQC–inspired model. The relations found here in the isotropic cases do not apply to the anisotropic cases, which perhaps require more general effective actions.