Complex geodesics in de Sitter space

TL;DR

This work resolves a tension in the geodesic approximation for scalar two-point functions in de Sitter space by showing that complex geodesics, revealed via Euclidean continuation to the sphere, reproduce the correct large-mass behavior even when no real geodesics connect the points. It computes one-loop corrections around Euclidean geodesic saddles and demonstrates that, upon analytic continuation, both Euclidean saddles can contribute to the Lorentzian correlator in the spacelike regime, while only the shorter saddle dominates otherwise. The results hold in any dimension and for arbitrary point configurations, and they offer insights for de Sitter holography, including static patch and boundary interpretations. The analysis also highlights how complexified geometries encode physical information about correlations in de Sitter space, with potential implications for non-Hermitian holographic models and the role of complex surfaces in cosmological holography.

Abstract

The two-point function of a free massive scalar field on a fixed background can be evaluated in the large mass limit by using a semiclassical geodesic approximation. In de Sitter space, however, this poses a puzzle. Certain spacelike separated points are not connected by real geodesics despite the corresponding two-point function in the Bunch-Davies state being non-vanishing. We resolve this puzzle by considering complex geodesics after analytically continuing to the sphere. We compute one-loop corrections to the correlator and discuss the implications of our results to de Sitter holography.

Figures (5)

• Figure 1: Penrose diagram for (half of) dS$_2$ with geodesics in blue. The horizons are drawn in red, and past and future infinity are in green. In dashed black, we also plot the position of stretched horizons.
• Figure 2: Upon analytic continuation to the Euclidean sphere, we look for two different types of geodesics, depending on the type of correlator under consideration. The geodesic with shorter length is shown in red, while the one going on the other side of the great circle is shown in black.
• Figure 3: (a) Penrose diagram showing the geodesics (in blue) between two stretched horizons (in dashed black). The position of the stretched horizon is set to $r_{st}=2/3$. (b) Exact (dashed red) and large mass (solid blue) correlator for two symmetric points on opposite stretched horizons as a function of the static patch time. At early times we observe a decay of the correlation function while at later times we observe an oscillatory behaviour. We multiply by $e^{m\pi}$ in the inset to make the oscillations apparent. In the plot, $m=20$ and $r_{st}=0.99$, so $t_c \sim 2.65$. Given the stretched horizon is very close to the actual horizon, at $t\sim0$, the two points become very close to each other, and so the approximation breaks down.
• Figure 4: Density plot of $|G(P_{X,Y})|$ for $d=4$ and $m=25$. In the large-$m$ limit zeroes (blue dots) accumulate near $P_{X,Y}=-1$.
• Figure 5: Illustration of the null rays involved in fixing the critical static time $t_c$, after which symmetric geodesics between the two stretched horizons do not exist anymore.