Holographic complexity of the extended Schwarzschild-de Sitter space

TL;DR

This work analyzes holographic complexity in an extended Schwarzschild-de Sitter spacetime under static patch holography, studying a broad set of complexity probes anchored to various stretched horizons. By systematically evaluating codimension-zero (WDW patch, CV2.0, CA) and codimension-one (CV, CAny) proposals across multiple horizon configurations (SdS^n), it reveals a rich pattern: hyperfast growth dominates when observables lie purely in the cosmological patch, linear late-time growth emerges for black-hole patches, and mixed configurations can lead to time-independent behavior for several proposals. A key result is that the location of the stretched horizon crucially determines whether complexity grows, remains constant, or diverges, with codimension-one observables (CAny) offering a route to sustained linear growth even in some mixed patches. The findings have implications for understanding holographic descriptions of de Sitter cosmologies and multiverse scenarios, and point to future work on dynamical shocks, lower-dimensional reductions, and quantum corrections in static patch holography.

Abstract

According to static patch holography, de Sitter space admits a unitary quantum description in terms of a dual theory living on the stretched horizon, that is a timelike surface close to the cosmological horizon. In this manuscript, we compute several holographic complexity conjectures in a periodic extension of the Schwarzschild-de Sitter black hole. We consider multiple configurations of the stretched horizons to which geometric objects are anchored. The holographic complexity proposals admit a hyperfast growth when the gravitational observables only lie in the cosmological patch, except for a class of complexity=anything observables that admit a linear growth. All the complexity conjectures present a linear increase when restricted to the black hole patch, similar to the AdS case. When both the black hole and the cosmological regions are probed, codimension-zero proposals are time-independent, while codimension-one proposals can have non-trivial evolution with linear increase at late times. As a byproduct of our analysis, we find that codimension-one spacelike surfaces are highly constrained in Schwarzschild-de Sitter space. Therefore, different locations of the stretched horizon give rise to different behaviours of the complexity conjectures.

Figures (35)

• Figure 1: SdS$_{d+1}^n$ space (illustrated for $n=2$) where $r_{\rm st}^{\rm (h)}$, $r_{\rm st}^{\rm (c)}$ (both in green) denote the stretched horizons close to the black hole and the cosmological horizon, respectively. $r_h$ and $r_c$ denote the black hole and cosmological horizon radii. The purple line region near $\mathcal{I}^+$ indicates a spacelike slice where an observer could collect information encoded in the inflating region.
• Figure 2: Summary of the possible configurations for the stretched horizons. Holographic observables are located between the stretched horizons.
• Figure 3: Penrose diagram of SdS$_{d+1}$ space in dimensions $d\geq 3,$ in the regime where the mass parameter satisfies $\mu \in (0, \mu_{N}).$$r_h$ denotes the black hole horizon and $r_c$ the cosmological horizon. We display the orientation of the null coordinates in eqs. \ref{['eq:general_null_coordinates']} and \ref{['eq:Kruskal_coord_sec2']}, where the subscripts $\lbrace b,c \rbrace$ denote black hole and cosmological patches, respectively.
• Figure 4: Prescriptions for the stretched horizons (in green) in a holographic setting. (a) The cosmological stretched horizons are placed in consecutive static patches. Geometric quantities anchored to the cosmological stretched horizons explore the region beyond the cosmological horizon. (b) The stretched horizons are placed in consecutive black hole patches. Geometric quantities anchored to the black hole stretched horizon explore the region behind the black hole horizon. The gray arrows represent the orientation of the Killing vector $\partial_t$.
• Figure 6: Novel prescriptions for the stretched horizons in a holographic setting with two copies of SdS space. (a) The cosmological stretched horizons (in green) are located in different copies of the static patch. (b) The stretched horizons (in green) are located in different copies of the black hole patch. Gray arrows represent the orientation of the Killing vector $\partial_t$.