$\phi^2$ Inflation at its Endpoint

In the simplest inflationary model $V=\frac12 m^2\phi^2$, we provide a prediction accurate up to $1\%$ for the spectral index $n_s$ and the tensor-to-scalar ratio $r$ assuming instantaneous reheating and a standard thermal history: $n_s = 0.9668\pm0.0003$ and $r=0.131\pm 0.001$. This represents the simplest and most informative point in the $(n_s,r)$ plane. The result is independent of the details of reheating (or preheating) provided the conversion to radiation is sufficiently fast. A slower reheating or a modified post-inflationary evolution push towards smaller $n_s$ (and larger $r$), so that our prediction corresponds to the maximum $n_s$ (and minimum $r$) for the quadratic potential. We also derive similar results for a general $V \propto \phi^p$ potential.