Reconciling tensor and scalar observables in G-inflation

The simple $m^2\phi^2$ potential as an inflationary model is coming under increasing tension with limits on the tensor-to-scalar ratio $r$ and measurements of the scalar spectral index $n_s$. Cubic Galileon interactions in the context of the Horndeski action can potentially reconcile the observables. However, we show that this cannot be achieved with only a constant Galileon mass scale because the interactions turn off too slowly, leading also to gradient instabilities after inflation ends. Allowing for a more rapid transition can reconcile the observables but moderately breaks the slow-roll approximation leading to a relatively large and negative running of the tilt $\alpha_s$ that can be of order $n_s-1$. We show that the observables on CMB and large scale structure scales can be predicted accurately using the optimized slow-roll approach instead of the traditional slow-roll expansion. Upper limits on $|\alpha_s|$ place a lower bound of $r\gtrsim 0.005$ and conversely a given $r$ places a lower bound on $|\alpha_s|$, both of which are potentially observable with next generation CMB and large scale structure surveys.