Higgs-$R^2$ inflation -- full slow-roll study at tree-level
  We consider Higgs inflation with an $\alpha R^2$ term. It adds a new scalar
degree of freedom, which leads to a two-field model of inflation. We do a
complete slow-roll analysis of the three-dimensional parameter space of the
$R^2$ coefficient $\alpha$, the non-minimal coupling $\xi$ and the Higgs
self-coupling $\lambda$. We find three classes of inflationary solutions, but
only pure $R^2$ and attractor solutions fit observations. We find that pure
Higgs inflation is impossible when the $R^2$ term is present regardless of how
small $\alpha$ is. However, we can have Higgs-like inflation, where the
amplitude of the perturbations does not depend on $\alpha$ and the predictions
as a function of e-folds are the same as in Higgs inflation, although the
inflationary trajectory is curved in field space. The spectral index is $0.939
< n_R < 0.967$, and constraining it to the observed range, the tensor-to-scalar
ratio varies from $3.8\times10^{-3}$ to the maximum allowed by observations,
$0.079$. Observational constraints on isocurvature perturbations contribute to
these limits, whereas non-Gaussianity is automatically in the range allowed by
observations.
