The string swampland constraints require multi-field inflation
  An important unsolved problem that affects practically all attempts to
connect string theory to cosmology and phenomenology is how to distinguish
effective field theories belonging to the string landscape from those that are
not consistent with a quantum theory of gravity at high energies (the "string
swampland"). It was recently proposed that potentials of the string landscape
must satisfy at least two conditions, the "swampland criteria", that severely
restrict the types of cosmological dynamics they can sustain. The first
criterion states that the (multi-field) effective field theory description is
only valid over a field displacement $\Delta \phi \leq \Delta \sim \mathcal
O(1)$ (in units where the Planck mass is 1), measured as a distance in the
target space geometry. A second, more recent, criterion asserts that, whenever
the potential $V$ is positive, its slope must be bounded from below, and
suggests $|\nabla V| / V \geq c \sim \mathcal O(1)$. A recent analysis
concluded that these two conditions taken together practically rule out
slow-roll models of inflation. In this note we show that the two conditions
rule out inflationary backgrounds that follow geodesic trajectories in field
space, but not those following curved, non-geodesic, trajectories (which are
parametrized by a non-vanishing bending rate $\Omega$ of the multi-field
trajectory). We derive a universal lower bound on $\Omega$ (relative to the
Hubble parameter $H$) as a function of $\Delta, c$ and the number of efolds
$N_e$, assumed to be at least of order 60. If later studies confirm $c$ and
$\Delta$ to be strictly $\mathcal O(1)$, the bound implies strong turns with
$\Omega / H \geq 3 N_e \sim 180$. Slow-roll inflation in the landscape is not
ruled out, but it is strongly multi-field.
