This paper explains how to make quantum field operators in de Sitter space into fully gravitational observables, working to the first nontrivial order in Newton’s constant G. "Gravitational dressing" means adding the appropriate gravitational field to an operator so it is invariant under the gauge symmetry of gravity. Doing this in de Sitter space is nontrivial because that spacetime has no spatial infinity, so familiar tricks used in flat or anti-de Sitter space do not straightaway apply.

The authors first show a simple necessary condition for an underlying field-theory operator to be dressable: it must be invariant under the symmetries of de Sitter space. This requirement is related to a phenomenon known as linearization instability, which says that some perturbative solutions are not allowed unless they obey certain global constraints. If an operator meets the de Sitter invariance condition, then a gravitational dressing can be built by enforcing that the dressed operator commute with the gravitational constraints. Those constraints generate diffeomorphisms, the coordinate changes that are the gauge symmetry of general relativity.

Concretely, the construction is perturbative. The authors expand the metric around the de Sitter background using a standard ADM time slicing and treat the metric perturbations order by order in G. The problem of finding a dressing reduces to solving a Green function problem: one finds specific Green functions on a chosen time-symmetric spatial slice and uses them to build the metric perturbation that must be added to the original operator so it commutes with the constraints.

To make the method explicit, they give an example of an operator that creates a two-particle state of scalar fields and satisfies the de Sitter invariance condition. In the limit where the particles are very heavy, the particles sit at opposite points on the spatial slice (they are antipodal). The authors construct the dressing for this operator and check it by computing correlators involving the dressed state. Those correlators reproduce, at linear order in G, the gravitational field known as the Schwarzschild–de Sitter solution, the expected leading gravitational effect of two masses in de Sitter space.