Inflation in an effective gravitational model & asymptotic safety

We consider an inflationary model motivated by quantum effects of gravitational and matter fields near the Planck scale. Our Lagrangian is a re-summed version of the effective Lagrangian recently obtained by Demmel, Saueressig and Zanusso~\cite{Demmel:2015oqa} in the context of gravity as an asymptotically safe theory. It represents a refined Starobinsky model, ${\cal L}_{\rm eff}=M_{\rm P}^2 R/2 + (a/2)R^2/[1+b\ln(R/\mu^2)]$, where $R$ is the Ricci scalar, $a$ and $b$ are constants and $\mu$ is an energy scale. By implementing the COBE normalisation and the Planck constraint on the scalar spectrum, we show that increasing $b$ leads to an increased value of both the scalar spectral index $n_s$ and the tensor-to-scalar ratio $r$. Requiring $n_s$ to be consistent with the Planck collaboration upper limit, we find that $r$ can be as large as $r\simeq 0.01$, the value possibly measurable by Stage IV CMB ground experiments and certainly from future dedicated space missions. The predicted running of the scalar spectral index $\alpha=d n_s/d\ln(k)$ is still of the order $-5\times 10^{-4}$ (as in the Starobinsky model), about one order of magnitude smaller than the current observational bound.