  We consider the massless Sine-Gordon model in de Sitter spacetime, in the
regime $\beta^2 < 4 \pi$ and using the framework of perturbative algebraic
quantum field theory. We show that a Fock space representation exists for the
free massless field, but that the natural one-parameter family of vacuum-like
states breaks the de Sitter boost symmetries. We prove convergence of the
perturbative series for the S matrix in this representation, and construct the
interacting Haag-Kastler net of local algebras from the relative S matrices. We
show that the net fulfills isotony, locality and de Sitter covariance (in the
algebraic adiabatic limit), even though the states that we consider are not
invariant. We furthermore prove convergence of the perturbative series for the
interacting field and the vertex operators, and verify that the interacting
equation of motion holds.
