Holographic Complexity and de Sitter Space

TL;DR

This work investigates holographic complexity via the complexity=volume (CV) proposal in two-dimensional dilaton-gravity flow geometries that asymptotically approach AdS but possess either a de Sitter or a black hole interior. By deriving a general geodesic formalism for arbitrary $f(r)$, the authors compute the length of spacelike geodesics anchored on opposite AdS boundaries and study their time dependence across AdS$_2$, AdS$_2$ black hole, and dS$_2$ spacetimes, as well as flow geometries that interpolate between these interiors. They find that AdS$_2$ black holes exhibit the familiar linear growth of extremal volumes at late times, while geometries with dS interiors generally yield finite-length geodesics only for times of order the inverse temperature and do not show linear growth, with many geodesics ceasing to exist beyond a finite time window. In contrast, AdS-to-AdS flow geometries restore linear growth at late times, illustrating a qualitative sensitivity of the CV observable to interior geometry. The results illuminate how holographic complexity behaves in cosmological-horizon contexts and suggest exploring alternative complexity constructions (e.g., CV restricted to behind-the-horizon regions or the complexity=action proposal) to better capture dS holography and scrambling in these settings.

Abstract

We compute the length of spacelike geodesics anchored at opposite sides of certain double-sided flow geometries in two dimensions. These geometries are asymptotically anti-de Sitter but they admit either a de Sitter or a black hole event horizon in the interior. While in the geometries with black hole horizons, the geodesic length always exhibit linear growth at late times, in the flow geometries with de Sitter horizons, geodesics with finite length only exist for short times of the order of the inverse temperature and they do not exhibit linear growth. We comment on the implications of these results towards understanding the holographic proposal for quantum complexity and the holographic nature of the de Sitter horizon.

Figures (14)

• Figure 1: Penrose diagram for AdS$_2$. The boundary $r=R_b$, corresponding to some fixed value of the dilaton $\phi=\phi_b$, is indicated by a dashed black line. The times $t_L=t_R$ run upwards along both boundaries. The axis of changing $u_{L/R},v_{L/R}$ are indicated in the figure. We have also illustrated a geodesic with turning point $r_t$ (see below).
• Figure 2: Penrose diagram for global AdS$_2$. The geodesics in blue connect equal times on the two boundaries. The black dashed line is the cutoff surface $r=R_b \gg 1$.
• Figure 3: Volume and its time derivative as a function of the boundary time $t$ with $R_b = 100$.
• Figure 4: Penrose diagram for the AdS$_2$ black hole and the geodesics in blue connecting equal times at the two boundaries. $R_b=10$ is the black dashed line.
• Figure 5: Penrose diagram for (half of) dS$_2$ and the geodesics in blue connecting $t_R=t_L=0$. $P$ runs from $-\infty$ to $\infty$ and in these limits, the geodesics become (almost everywhere) null.