  We investigate a recent proposal for a distinguished vacuum state of a free
scalar quantum field in an arbitrarily curved spacetime, known as the
Sorkin-Johnston (SJ) vacuum, by applying it to de Sitter space. We derive the
associated two-point functions on both the global and Poincar\'e (cosmological)
patches in general d+1 dimensions. In all cases where it is defined, the SJ
vacuum belongs to the family of de Sitter invariant alpha-vacua. We obtain
different states depending on the spacetime dimension, mass of the scalar
field, and whether the state is evaluated on the global or Poincar\'e patch. We
find that the SJ vacuum agrees with the Euclidean/Bunch-Davies state for heavy
("principal series") fields on the global patch in even spacetime dimensions.
We also compute the SJ vacuum on a causal set corresponding to a causal diamond
in 1+1 dimensional de Sitter space. Our simulations show that the mean of the
SJ two-point function on the causal set agrees well with its expected continuum
counterpart.
