Loop cosmological implications of a nonminimally coupled scalar field

Nonminimal actions with matter represented by a scalar field coupled to gravity are considered in the context of a homogeneous and isotropic universe. The coupling is of the form $-\frac{1}{2}\xi {\varphi }^{2}R$. The possibility of successful inflation is investigated taking into account features of loop cosmology. For that end a conformal transformation is performed. That brings the theory into the standard minimally coupled form (Einstein frame) with some effective field and its potential. Both analytical and numerical estimates show that a negative coupling constant is preferable for successful inflation. Moreover, provided fixed initial conditions, larger $|\xi |$ leads to a greater number of $e$-folds. The latter is obtained for a reasonable range of initial conditions and the coupling parameter and indicates a possibility for successful inflation.

Figure 1

The effective potential for $\sigma <0$ (upper curve) and $\sigma >0$ (lower curve). The solid curve represents the original quadratic potential. The argument of the potential is $x\equiv \sqrt{|\sigma |}\varphi$ for $\sigma \ne 0$ and $x=\varphi$ for $\sigma =0$.

Figure 2

Maximum change of the inflaton as a function of the initial field momentum for $\sigma =0$ and $p_0=p_{*}$. The analytical curve is the Lambert function defined in (59), whereas the numerical points are obtained as solutions to the equations of motion (42, 43, 44).

Figure 3

The spectrum of the geometrical density operator $D(q)$ introduced in Eq. (38).

Figure 4

The change of the inflaton during the classical phase as function of its initial momentum for different values of $\sigma$ (measured in the units of ${\ell }_{\mathrm{P}}^2$).