  The relation between different notions measuring proximity to $\ell_1$ and
distortability of a Banach space is studied. The main result states that a
Banach space, whose all subspaces have Bourgain $\ell_1$ index greater than
$\omega^\alpha$, $\alpha<\omega_1$, contains either an arbitrary distortable
subspace or an $\ell_1^\alpha$-asymptotic subspace.
