Can quantum effects due to a massless conformally coupled field avoid gravitational singularities?

Using quantum corrections from massless fields conformally coupled to gravity, we study the possibility of avoiding singularities that appear in the flat Friedmann–Robertson–Walker model. We assume that the universe contains a barotropic perfect fluid with the state equation $p=\omega\rho$, where $p$ is the pressure and $\rho$ is the energy density. We study the dynamics of the model for all values of the parameter $\omega$ and also for all values of the conformal anomaly coefficients $\alpha$ and $\beta$. We show that singularities can be avoided only in the case where $\alpha>0$ and $\beta<0$. To obtain an expanding Friedmann universe at late times with $\omega>-1$ (only a one-parameter family of solutions, but no a general solution, has this behavior at late times), the initial conditions of the nonsingular solutions at early times must be chosen very exactly. These nonsingular solutions consist of a general solution (a two-parameter family) exiting the contracting de Sitter phase and a one-parameter family exiting the contracting Friedmann phase. On the other hand, for $\omega<-1$ (a phantom field), the problem of avoiding singularities is more involved because if we consider an expanding Friedmann phase at early times, then in addition to fine-tuning the initial conditions, we must also fine-tune the parameters $\alpha$ and $\beta$ to obtain a behavior without future singularities: only a one-parameter family of solutions follows a contracting Friedmann phase at late times, and only a particular solution behaves like a contracting de Sitter universe. The other solutions have future singularities.