Deducing cosmological observables from the $S$ matrix

We study one loop quantum gravitational corrections to the long range force induced by the exchange of a massless scalar between two massive scalars. The various diagrams contributing to the flat space $S$ matrix are evaluated in a general covariant gauge, and we show that dependence on the gauge parameters cancels at a point considerably before forming the full $S$ matrix, which is unobservable in cosmology. It is possible to interpret our computation as a solution to the effective field equations—which could be done even in cosmology—but taking account of quantum gravitational corrections from the source and from the observer.

Figure 1

The left diagram shows how the self-mass contributes to the amputated $4\text{−}\psi$ vertex function. The diagram on the right shows how graviton correlations between the two vertices contribute. Solid lines represent the massless scalar, wavy lines represent the graviton, and dashed lines stand for the massive scalar. These graphs have the same topology as Bjerrum-Bohr’s Diagrams 8 and 4, respectively [31].

Figure 2

These diagrams show correlations between the source (primed) or observer (unprimed) and the opposite vertex. Solid lines represent the massless scalar, wavy lines represent the graviton, and dashed lines represent the massive scalar. These graphs have the same topology as Bjerrum-Bohr’s Diagram 3 [31].

Figure 3

These diagrams show correlations between one of the vertices and the massless scalar force carrier. Solid lines represent the massless scalar, wavy lines represent the graviton, and dashed lines stand for the massive scalar. These graphs have the same topology as Bjerrum-Bohr’s Diagram 7 [31].

Figure 4

These diagrams show correlations between the source (primed) and the observer (unprimed). Solid lines represent the massless scalar, wavy lines represent the graviton, and dashed lines represent the massive scalar. These graphs have the same topology as Bjerrum-Bohr’s Diagram 2 [31].

Figure 5

These diagrams show correlations between the source (primed) or observer (unprimed) and the massless scalar force carrier. Solid lines represent the massless scalar, wavy lines represent the graviton, and dashed lines represent the massive scalar. These graphs have the same topology as Bjerrum-Bohr’s Diagram 6 [31].