  We study the Cauchy problem for the Einstein-Boltzmann system with soft
potentials in a cosmological setting. We assume the Bianchi I symmetry to
describe a spatially homogeneous, but anisotropic universe and consider a
cosmological constant $ \Lambda > 0 $ to describe an accelerated expansion of
the universe. For the Boltzmann equation we introduce a new weight function and
apply the method of Illner and Shinbrot to obtain the future global existence
of spatially homogeneous, small solutions. For the Einstein equations we assume
that the initial value of the Hubble variable is close to $ ( \Lambda / 3 )^{ 1
/ 2 } $. We obtain the future global existence and asymptotic behavior of
spatially homogeneous solutions to the Einstein-Boltzmann system with soft
potentials.
