Holographic Complexity in dS$_{d+1}$

TL;DR

This work extends holographic complexity to de Sitter spacetimes by evaluating CV, CA, and CV2.0 proposals in $dS_{d+1}$. All three observables share a universal pattern: a hyperfast growth and a finite-time divergence at a critical time, which is regulated by a geometric cutoff near the future timelike boundary, yielding subsequent linear growth whose rate is set by the cutoff. The analysis highlights the role of the de Sitter entropy $N$ as a prefactor and connects the late-time behavior to horizon-based holographic descriptions, with implications for dS/CFT and the interpretation of complexity in cosmological settings. The results suggest a robust, regulator-dependent linear growth regime and motivate further exploration of complexity in more general cosmological backgrounds and potential topological or JT-gravity extensions.

Abstract

We study the CV, CA, and CV2.0 approaches to holographic complexity in $(d+1)$-dimensional de Sitter spacetime. We find that holographic complexity and corresponding growth rate presents universal behaviour for all three approaches. In particular, the holographic complexity exhibits `hyperfast' growth [arXiv:2109.14104] and appears to diverge with a universal power law at a (finite) critical time. We introduce a cutoff surface to regulate this divergence, and the subsequent growth of the holographic complexity is linear in time.

Figures (18)

• Figure 1: The Penrose diagram for dS$_{d+1}$ spacetime consists of four quadrants I, II, III, IV. The left/right gray region presents the static patch covered by $(t,r)$ coordinates. The dashed purple curves are referred to as the spacelike hypersurface with a constant radial coordinate, i.e.,$r=\rho L=\text{constant}$. The future and past infinity located at $r=\infty$ are denoted by blue lines and labeled by $i^+, i^-$, respectively.
• Figure 2: Extremal surfaces joining the left and right stretched horizons at the boundary time $\tau$ are denoted by the black curves.
• Figure 3: The orange square region denotes the WdW patch at a boundary time $t_{\textrm{\tiny R}}=\tau L =- t_{\textrm{\tiny L}}$. The purple dashed curves represent the stretch horizon at $r=\rho L$. Left: The WdW patch at $\tau=0$. Right: The WdW patch at the critical time $\tau=\tau_{\infty}$ where the future tip $F$ touches $i^+$. The shadowed blue region presents the WdW patch at $\tau=0$ with exactly anchoring the boundary at the south and north poles ( i.e.,$\rho=0$).
• Figure 4: The time evolution of tip radii $r_\pm$ of WdW patch with $\rho =1/2$. Hence from eq. (\ref{['eq:rpm']}), $r_+$ diverges at $\tau=\tau_{\infty}\simeq 0.5493$, and $r_-$ diverges at $\tau=-\tau_{\infty}$.
• Figure 5: The time evolution of holographic complexity and its growth rate from CV2.0 with $\rho=1/2$.