Inflation and Scale-invariant $R^2$-Gravity
  In scale-invariant models of fundamental physics, mass scales are generated
by spontaneous symmetry breaking. In this work, we study inflation in
scale-invariant $R^2$ gravity, in which the Planck mass is generated by a
scalar field, which is responsible for spontaneous breaking of scale--symmetry.
If the self-interactions of the scalar field are non-zero, a cosmological
constant is generated, which can be potentially quite large. To avoid
fine-tuning at late times, we introduce another scalar field which drives the
classical cosmological constant to zero during inflation. Working in the
Einstein-frame, we find that due to a conserved Noether current the
corresponding three-field inflationary model (consisting of the two scalar
fields plus the scalaron) becomes effectively a two-field model. The prize to
be paid for introducing the field which cancels the classical cosmological
constant at the end of inflation is that the running of the spectral index and
the running of the running can be quite large due to entropy perturbations
during inflation, making the model testable with future cosmological
experiments.
