Reconstructing the inflationary $f(R)$ from observations
  The BICEP2 collaboration has recently released data showing that the
scalar-to-tensor ratio $r$ is much larger than expected. The immediate
consequence, in the context of $f(R)$ gravity, is that the Starobinsky model of
inflation is ruled out since it predicts a value of $r$ much smaller than what
is observed. Of course, the BICEP2 data need verification, especially from
Planck with which there is some tension, therefore any conclusion seems
premature. However, it is interesting to ask what would be the functional form
of $f(R)$ in the case when the value of $r$ is different from the one predicted
by the Starobinsky model. In this paper, we show how to determine the form of
$f(R)$, once the slow-roll parameters are known with some accuracy. The
striking result is that, for given values of the scalar spectral index $n_{S}$
and $r$, the effective Lagrangian has the form $f(R)=R^{\zeta}$, where
$\zeta=2-\varepsilon$ and $|\varepsilon|\ll 1$. Therefore, it appears that the
inflationary phase of the Universe is best described by a $R^{2}$ theory, with
a small deviation that, as we show, can be obtained by quantum corrections.
