Tracing the high energy theory of gravity: an introduction to Palatini
  inflation
  We present an introduction to cosmic inflation in the context of Palatini
gravity, which is an interesting alternative to the usual metric theory of
gravity. In the latter case only the metric $g_{\mu\nu}$ determines the
geometry of space-time, whereas in the former case both the metric and the
space-time connection $\Gamma^\lambda_{\mu\nu}$ are a priori independent
variables - a choice which can lead to a theory of gravity different from the
metric one. In scenarios where the field(s) responsible for cosmic inflation
are coupled non-minimally to gravity or the gravitational sector is otherwise
extended, assumptions of the underlying gravitational degrees of freedom can
have a big impact on the observational consequences of inflation. We
demonstrate this explicitly by reviewing several interesting and well-motivated
scenarios including Higgs inflation, $R^2$ inflation, and $\xi$-attractor
models. We also discuss some prospects for future research and argue why
$r=10^{-3}$ is a particularly important goal for future missions that search
for signatures of primordial gravitational waves.
