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The Cosmological Switchback Effect

Stefano Baiguera, Rotem Berman, Shira Chapman, Robert C. Myers

arXiv:2304.15008v2hep-thgr-qc

TL;DR

The paper asks whether holographic complexity can characterize a proposed de Sitter dual theory. It studies shockwave-induced changes in spacetime volume and finds delayed growth like the AdS switchback effect, with a different geometric origin.

  • Problem

    The paper investigates whether holographic complexity can characterize de Sitter holography, where cosmological horizons are observer-dependent.

    Source: §1 · pp. 3–4

  • Method

    The paper studies how shockwaves affect the spacetime volume behind the de Sitter horizon, using the CV2.0 complexity proposal.

    Source: §1 · p. 6

  • Results

    Shockwaves produce delayed growth of the dS volume, with a scrambling time proportional to the inverse cosmological-horizon temperature and logarithmic in the stretched-horizon parameter (1−ρ).

    Source: §1 · p. 6, §7.1 · p. 61

  • Takeaways & Limitations

    The dS volume exhibits a switchback-like delay, although its geometric origin differs from the AdS case.

    Source: §7.1 · p. 61

    The leading scrambling-time contribution can vanish for specific choices of the stretched-horizon factor, whose meaning for the dual theory remains unclear.

    Source: §7.1 · p. 62

Abstract

from arXiv · show

The volume behind the black hole horizon was suggested as a holographic dual for the quantum computational complexity of the boundary state in AdS/CFT. This identification is strongly motivated by the switchback effect: a characteristic delay of complexity growth in reaction to an inserted perturbation, modelled as a shockwave in the bulk. Recent proposals of de Sitter (dS) holography suggest that a dual theory could be living on a stretched horizon near the cosmological horizon. We study how the spacetime volume behind the cosmological horizon in Schwarzschild-dS space reacts to the insertion of shockwaves in an attempt to characterize the properties of this dual theory. We demonstrate that a switchback effect can be observed in dS space. That is, the growth of complexity is delayed in reaction to a perturbation. This delay is longer for earlier shocks and depends on a scrambling time which is logarithmic in the strength of the shockwave and proportional to the inverse temperature of the cosmological dS horizon. This behavior is very similar to what happens for AdS black holes, albeit the geometric origin of the effect is different.

1 Introduction

The introduction motivates extending holographic complexity to de Sitter space within static patch holography. It proposes testing this setting with shockwaves and reports delayed volume growth resembling the AdS switchback effect, though with a different geometric origin.

1 Introduction

  • In static patch holography, the cosmological horizon is proposed to encode a finite-dimensional quantum system in a maximally mixed state on a slice just inside the horizon.Cosmological horizons are observer-dependent, so the observer’s causal patch matters to this proposal.Source: §1 · p. 4
  • The paper extends complexity studies to dS by focusing on CV2.0, which relates complexity to spacetime volume and is described as technically easier to compute than CA or CV.The introduction presents CV and CA as alternative gravity-side proposals for holographic complexity.Source: §1 · p. 5
  • In dS, complexity-related geometric regions can diverge near timelike infinity; regulating those regions recovers linear late-time growth.The stretched horizon is proposed as the location of the holographic degrees of freedom.Source: §1 · pp. 5–6
  • The AdS switchback effect is a delay in complexity growth after a shockwave, attributed in the boundary picture to cancellations of gates that commute with the perturbation.This effect motivates the dS shockwave test.Source: §1 · p. 6
  • Shockwaves delay the growth of the dS spacetime volume, producing a switchback effect similar to AdS black holes but with a different geometric origin.The paper studies this response using the CV2.0 proposal.Source: §1 · pp. 6–7
  • The proposed geometric explanation in dS involves positive-energy insertions changing the Penrose diagram and advancing null rays across the shockwave.This can bring the stretched horizons on opposite sides into causal contact, which the paper connects to special WDW-patch configurations and scrambling time.Source: §1 · p. 7

2 Geometric preliminaries

The section introduces empty de Sitter and Schwarzschild–de Sitter geometries and describes shockwave transitions between them. It outlines the null-energy-condition consequences: positive-energy shocks decrease the black-hole mass and enlarge the cosmological horizon.

2.3 Null energy condition

  • The null energy condition requires the black-hole mass to decrease after a shockwave.The section frames this as a consistency condition on the transition between geometries.Source: §2 · p. 8, §2.3 · p. 18

2.1.1 Empty de Sitter space

  • Empty de Sitter space has a static patch bounded by a cosmological horizon at r = L, with a conjectured dual theory living on its stretched horizon.The stretched horizons' time coordinates are identified with the dual theory's boundary times.Source: §2.1.1 · pp. 10–11

2.1.2 Schwarzschild-de Sitter black hole

  • In d = 2, Schwarzschild–de Sitter has only a cosmological horizon, while for d ≥ 3 it generally has both black-hole and cosmological horizons.In four dimensions, increasing the mass enlarges the black-hole horizon and shrinks the cosmological horizon until they meet in the Nariai limit.Source: §2.1.2 · pp. 11, 13, 14

2.2 Perturbation with shockwaves

  • A spherically symmetric null-fluid shock shifts the mass parameter and both horizon locations; under the null energy condition, the cosmological horizon moves outward.The geometry can transition from a black-hole solution to empty de Sitter space.Source: §2.2 · pp. 15–17

2.3 Null energy condition

  • The null energy condition implies m1 ≥ m2, so positive-energy shockwaves decrease the black hole’s mass.The only non-vanishing stress-tensor component is Tuu.Source: §2.3 · p. 18
  • Because lower-mass Schwarzschild–de Sitter black holes have larger cosmological horizons, the constraint is equivalently rC1 ≤ rC2.Source: §2.3 · p. 19

3 Stretched horizon in the presence of shockwaves

The section examines how to define the stretched horizon in shockwave geometries, comparing a constant-redshift prescription tied to a fixed-radius observer with a continuous-time prescription. It describes how the horizon behaves during shock crossing and in the far past and future under these definitions.

3 Stretched horizon in the presence of shockwaves

  • A fixed-radius stretched horizon cannot approach the larger post-shock cosmological horizon, so the authors impose constant redshift between the horizon and a fixed-radius observer.For SdS3, the observer is placed at the north pole; the prescription keeps cosmological redshift constant through time.Source: §3 · p. 19
  • Different local and non-local prescriptions yield qualitatively similar results and coincide for light shocks, with the scrambling-time behavior requiring only a fixed post-shock radius.Alternatives include fixing the local acceleration magnitude or requiring continuity of the time coordinate; the prescriptions share fixed-radius regions in the far past and after the shock and a limit in which both stretched horizons approach their respective cosmological horizons.Source: §3 · p. 20, §3.1 · p. 20
  • When the shock propagates along the cosmological horizon (tw → ∞), that horizon does not change and the stretched horizon retains the same form throughout.Source: §3 · p. 20

3.1 Constant redshift prescription

  • The constant-redshift condition makes the early and late stretched horizons approach their respective cosmological horizons together, but equal ρ values are not general.The equality holds for transitions between SdS3 black holes with an observer at Ro = 0; it fails when Ro ≠ 0 or in higher dimensions.Source: §3.1 · pp. 21, 24
  • During the interval when emitted light rays cross the shock, the stretched horizon becomes time-dependent, then returns to a constant-radius surface that jumps from its prior value.It is time-independent in the far past and future; the SdS4 solution illustrates the intermediate evolution.Source: §3.1 · pp. 21, 26, Figure 8 · p. 26
  • The discontinuity decreases as ρ approaches 1, shock strength ε approaches 0, or the insertion becomes earlier (tw → ∞).Source: §3.1 · p. 26
  • Under the constant-redshift prescription, the stretched horizon jumps at the shock—initially toward smaller r, then toward larger r as ρ_1^st increases—and the pre- and post-shock horizons converge toward the cosmological horizon as ρ_1^st → 1.The jump is easier to observe in three dimensions; its qualitative behavior is similar in higher dimensions.Source: §3.1 · p. 27

3.2 Continuous time prescription

  • The continuous-time prescription keeps the time coordinate continuous while allowing a radial-coordinate jump; as the pre-shock stretched horizon approaches r_C1, the post-shock horizon approaches the larger r_C2.In SdS3, both horizons approach the corresponding cosmological horizon as ρ → 1 and the pole r = 0 as ρ → 0; higher-dimensional solutions are numerical. The prescription is technically more efficient and permits an analytic scrambling-time expression.Source: §3.2 · pp. 28–29

4 WDW patch

This section sets up the WDW-patch calculation for the CV2.0 proposal and tracks its evolution after shockwave insertion, including how its joints can move beyond the cosmological horizon and behind stretched horizons. It gives an analytic account of special-position motion in three dimensions and a qualitative numerical account in higher dimensions.

4.1 Regularization

  • Near-timelike-infinity cutoff surfaces regularize the divergent WDW-patch volume; the past cutoff avoids the shockwave and needs no shockwave-specific prescription.The paper keeps the two cutoff locations distinct in general calculations.Source: §4.1 · pp. 29–30

4.2 Time evolution of the WDW patch

  • Top and bottom WDW-patch joints, usually behind the horizon, can move outside the cosmological horizon in the dS shockwave geometry.The patch’s special positions are the shockwave intersections and its top and bottom joints.Source: §4.2 · p. 31

4.3 Critical times of the WDW patch

  • Universal critical-time regimes occur for all asymptotically dS metrics, but outside-horizon configurations depend on shock parameters and occur for sufficiently early shocks.The standard times satisfy tc1 ≤ tc2 ≤ tc3, while tc0 and tc1 change ordering as the insertion approaches tw = 0; the analysis focuses on early shocks tw ≫ L and times t > tc0.Source: §4.3 · pp. 32, 34, Table 1 · p. 34

4.4 Special configurations of the WDW patch

  • As rs increases into [rC1, rC2], it shifts to the top quadrant, while the bottom joint moves inside the cosmological horizon; the configurations arise from shockwave time advance.For sufficiently early shock insertion, the special outside-horizon configurations always occur.Source: §4.4 · p. 35
  • When rb enters [rC1, rC2], the top joint moves outside the cosmological horizon; depending on shock parameters, this regime can overlap with the rs configuration.The overlap occurs for tc,s ≤ t ≤ tc,b when tc,s ≤ tc,b.Source: §4.4 · p. 36

4.4.1 Joints behind the stretched horizon

  • The joints can pass behind the stretched horizons only during the special interval when both shockwave intersections lie between the cosmological horizons; this configuration exists for sufficiently large tw.Its critical-time ordering relative to other transitions is not generally fixed.Source: §4.4.1 · pp. 36–38, Table 2 · p. 38

4.5 Explicit cases

  • The analytic 3D treatment assumes t ≥ t_c0; earlier times are left for future study.Source: §4.5 · p. 39
  • After shockwave insertion, the 3D positions r_s and r_b increase monotonically with boundary time: r_s moves toward r_C2, while r_b moves from r_C1 toward future timelike infinity.Figure 14 illustrates the evolution for two choices of ε and t_w in the symmetric configuration.Source: §4.5 · p. 39, Figure 14 · p. 40
  • When r_s or r_b enters [r_C1, r_C2], the WDW joints r_m1 and r_m2 move outside the corresponding cosmological horizon, so the position functions must be defined by cases.Source: §4.5 · p. 39
  • In higher dimensions, the special-position evolution cannot be solved analytically over the full time range, but numerical checks show behavior qualitatively similar to the 3D plots.Source: §4.5 · p. 40

5 Complexity=Volume 2.0

The section applies CV2.0 to the regularized WDW spacetime volume in shockwave-perturbed asymptotically dS geometries, deriving complexity across patch regimes and its growth rate.

5.1 General strategy

  • The intermediate-regime calculation splits the WDW patch into seven regions and provides a plateau; other cases follow by changing integration endpoints.With fixed cosmological redshift, the pre-shock stretched horizon becomes time-dependent, making earlier-time calculations more complicated and deferred.Source: §5 · p. 41, §5.1 · p. 42, Figure 16 · p. 42
  • The complexity rate diverges when WDW vertices reach timelike infinity, motivating cutoff surfaces; at r_s = r_C1, the rate remains finite.At that limit, r_m1 also equals r_C1.Source: §5.1 · p. 43

5.2 CV2.0 with the joints behind the stretched horizons

  • For joints behind the stretched horizons, exchanging integration endpoints requires reversing the overall sign to keep CV2.0 positive.This expression follows from a limiting procedure applied to the intermediate-regime result.Source: §5.2 · pp. 44–45

5.3 Integrated Complexity=Volume 2.0

  • The integrated expressions cover four post-insertion time regimes, with the exceptional joint-behind-horizon configuration retaining the same formal functional dependence.The other regimes are obtained as limits of the intermediate regime.Source: §5.3 · pp. 45–46

5.4 Complexity rate of change

  • With finite δ, late-time growth is constant and matches empty dS without a shockwave; δ→0 instead gives positive divergence.The finite-regulator rate is the sum of the rates in the two geometries.Source: §5.4 · p. 47

6 Explicit results for the Complexity=Volume 2.0

The CV2.0 analysis examines the intermediate complexity plateau in Schwarzschild–de Sitter shockwave geometries, including how its duration and evolution depend on shockwave and stretched-horizon parameters across three and four dimensions. It develops a scrambling-time description of the plateau, including its universal pattern in the relevant limit, the higher-dimensional double-scaling limit, and horizon-based interpretations for light black holes.

6.1 Three-dimensional SdS space

  • In SdS3, complexity has a plateau between critical times tc1 and tc2, then grows linearly at late times; it remains continuous across transitions.The plateau can become arbitrarily long for early shocks. Complexity of formation also has a plateau and kink, followed by delayed linear growth proportional to (tw − t∗).Source: §6.1 · pp. 48–50

Complexity rate of change.

  • The complexity rate is continuous across transitions, approximately vanishes during the plateau, then decreases toward its constant late-time value, as in AdS black holes.In SdS, the plateau arises because the stretched horizon keeps the WDW joints from reaching the past and future singularities simultaneously; the cancellation occurs while the patch avoids the singularities.Source: Complexity rate of change · p. 50, §6.1.1 · p. 51

6.1.1 Plateau of complexity

  • In SdS3, the plateau lasts longer when the stretched horizon approaches the cosmological horizon, the shock is inserted earlier, or the perturbation is heavier.A kink appears when WDW joints move behind the stretched horizon; complexity vanishes there because the WDW-patch volume vanishes. If tc1 < tc0, the plateau begins at shock insertion, and complexity remains very small in the additional behind-horizon regime.Source: §6.1 · p. 49, §6.1.1 · pp. 52–53
  • For tw ≫ L, the SdS3 plateau duration enters a linear regime delayed by a scrambling time that decreases for larger perturbations and has a logarithmic divergence whose prefactor is proportional to T1.The leading scrambling-time behavior matches three-dimensional AdS-Vaidya when the shockwave energy is ∆E = 2T1.Source: §6.1.1 · pp. 53, 55, 56

6.2 Four-dimensional SdS space

  • Numerical integration shows that SdS4 also has an intermediate plateau, with time evolution qualitatively similar to SdS3 despite the different Penrose diagram.The analysis of higher-dimensional SdS geometries is numerical rather than closed-form.Source: §6.2 · p. 57

6.2.1 Plateau of complexity

  • For SdS4 at tw ≫ L, the plateau is described by an accurate interpolation when tc0 ≤ tc1, and the small-ε scrambling time is expanded to leading logarithmic order.The stated plateau-duration equation applies only when tc0 ≤ tc1.Source: §6.2.1 · p. 58
  • In three dimensions, the continuous-time and constant-redshift prescriptions give the same result in the relevant ε → 0 limit, though the latter computation is harder.The plateau-duration figure plots dependence on shock insertion time and marks the 4(t_w − t*) reference curves.Source: §6.2.1 · p. 58, Figure 25 · p. 59
  • In the relevant limit, the scrambling time follows a universal pattern common to asymptotically de Sitter spacetimes, with T_c1 ≈ T_c2 at first order in ε when ε ≪ 1.In the double-scaling limit, the logarithm's argument also acquires a factor of 1 − ρ.Source: §6.2.1 · p. 59

6.3 Higher dimensions

  • For higher dimensions, the scrambling-time description is more cumbersome generally but simplifies in the double-scaling limit ε → 0, ρ → 1 with (1 − ρ)/ε fixed.Source: §6.3 · p. 59
  • For light black holes, positive t* requires T_h1^(d−2)(1 − ρ) > ε, while equivalent scrambling-time expressions emphasize either the black hole or cosmological horizon.The passage notes that these expressions are equivalent despite their different physical interpretations.Source: §6.3 · p. 60

7 Discussion

The discussion finds a de Sitter complexity plateau whose geometric origin differs from AdS and summarizes scrambling-time behavior in a small-black-hole regime. It also describes shockwave-induced causal contact and identifies open directions for testing and extending these results.

7.1 Summary of results

  • A dS complexity plateau near tL=tR=0 lasts for a scrambling time, but arises before the WDW joints reach infinities, unlike AdS’s behind-singularity regime.In SdS, the stretched horizon’s inclusion distinguishes the plateau’s geometric origin.Source: §7.1 · p. 61
  • In the small-black-hole regime, scrambling time scales inversely with cosmological-horizon temperature and logarithmically with the stretched-horizon location parameter (1−ρ).Equivalent expressions emphasize either the cosmological and stretched horizons or the black-hole horizon; small black holes have Sh1≪SC1 but Th1≫TC1.Source: §7.1 · p. 61
  • For d≥3, the complexity-of-formation derivation matches the other scrambling-time expressions in the small-black-hole limit.The match holds in the double-scaling limit with the simplifying small-black-hole limit.Source: §7.1 · pp. 61–62
  • The WDW volume grows at late times and decreases at early times linearly, with rates governed by regulator placement after the corresponding joints reach timelike infinity.For small black holes, the late-time rate matches the empty-dS result; the authors expect a full nonperturbative description to reflect finite Hilbert-space dimension.Source: §7.1 · pp. 62–63
  • Positive-energy shocks can create exceptional WDW configurations by bringing the left and right causal patches into contact, so they no longer bipartition a global time slice.For sufficiently late insertion times, the special configurations appear; the causal patches therefore do not correspond to two halves of an entangled state.Source: §7.1 · pp. 64–65

7.2 Future directions

  • Future work includes the pre-shock, time-dependent stretched horizon, other complexity conjectures, and circuit tests of whether perturbations reproduce the holographic behavior.The proposed static-patch dual’s dictionary remains unclear.Source: §7.2 · p. 65
  • Other proposed directions include studying shockwaves in dS2, general-relativistic explanations for shared AdS/dS behavior, centaur geometries, and localized perturbations.The centaur comparison could help distinguish universal features of complexity.Source: §7.2 · p. 66

A Comparison with reference [1]

The appendix compares this paper’s conventions and critical-time analysis with reference, finding agreement after accounting for differing stretched-horizon prescriptions and limit orders.

A Comparison with reference [1]

  • The stretched-horizon prescriptions differ: this work ties the horizon to the corresponding cosmological horizon and makes it time-independent after shock crossing, unlike reference.Reference instead varies the stretched-horizon location over r_st ∈ [0, ℓ], independently of the cosmological horizon.Source: A Comparison with reference [1] · pp. 67–68
  • Boundary times run along the stretched horizons here but are measured at the poles in reference, producing a constant shift in boundary times.The appendix describes the shift as minor.Source: A Comparison with reference [1] · p. 68
  • The critical time for the WDW patch’s top joint to reach future timelike infinity matches the corresponding time in reference after accounting for conventions.The appendix’s table summarizes where each paper states its conventions.Source: A Comparison with reference [1] · p. 68, Table 3 · p. 68
  • The analyses use different orders of limits for critical times, but correspond to similar formulas after the shockwave energy is chosen appropriately.That choice cancels the linear dependence on t_w and the finite part, recovering the result in reference.Source: A Comparison with reference [1] · p. 69

B Ping-pong prescription for the constant redshift

The constant-redshift prescription has ambiguities tied to observer location and light-ray orientation; this appendix changes the boundary conditions used to locate the stretched horizon.

B Ping-pong prescription for the constant redshift

  • Observer location and ingoing versus outgoing light rays make the constant-redshift prescription ambiguous, so this appendix changes its boundary conditions.The authors plan to discuss the resulting discrepancies and similarities with the earlier analysis.Source: B Ping-pong prescription for the… · p. 69

B.1 Details of the prescription

The ping-pong prescription recursively reconstructs the stretched and auxiliary horizons by exchanging light-ray emitter and receiver roles while imposing constant redshift and continuity across the shock.

B.1 Details of the prescription

  • The procedure alternates ingoing and outgoing light rays to reconstruct the stretched and auxiliary horizons, repeatedly exchanging emitter and receiver roles.The schematic is for a transition between two SdS3 black holes; the authors describe higher dimensions as technically harder but conceptually similar.Source: §B.1 · pp. 69–71
  • The ping-pong prescription requires both horizons to remain continuous across the shock and the redshift to remain constant, which uniquely determines their backward-time trajectories.The construction begins from time-independent curves in the far future and evolves toward the past.Source: §B.1 · pp. 70–71
  • Requiring continuity throughout the evolution proves too strong and yields a stretched-horizon solution inconsistent with expectations, so the authors later allow a jump at the shock.The relaxation is studied in section B.3.Source: §B.1 · p. 71

B.2 Implementation of the procedure

The implementation reconstructs the horizons backward in time, alternating null-ray and redshift conditions to solve for their trajectories and relate their time coordinates.

B.2 Implementation of the procedure

  • The first stretched-horizon segment is determined from constant-redshift conditions and null rays crossing the shock, with continuity imposed at the shock.The associated critical times mark rays received at the shock and emitted precisely from it.Source: §B.2 · pp. 71–72
  • Each recursive auxiliary-horizon step uses outgoing constant-u rays and redshift to solve for the trajectory and relate auxiliary time to the initial stretched-horizon time.The procedure differentiates the time relation, solves a redshift differential equation, then substitutes the result back to recover the time mapping.Source: §B.2 · pp. 72–73
  • Each subsequent stretched-horizon segment uses ingoing constant-v rays and redshift, with continuity linking it to the preceding segment.The reconstructed quantities remain parameterized by time along the initial stretched-horizon curve.Source: §B.2 · pp. 73–74

B.3 Results

Numerical construction of the stretched and auxiliary horizons yields distinct outcomes depending on the shock-crossing boundary condition. Tuning its discontinuity can produce horizons that remain approximately at constant radius in the far past.

B.3 Results

  • The continuous construction makes both horizons reach the north pole at finite but different times.The plotted solution imposes continuity across the shockwave.Source: §B.3 · p. 74, Figure 28 · p. 74
  • Allowing a discontinuity δ > 0 in the stretched-horizon boundary condition addresses the failure of the initial curves to reach past infinity.The modification was also motivated by the possibility of approaching a constant-radius trajectory in the far past.Source: §B.3 · p. 74
  • Three regimes occur around δcr: below it the horizons reach the north pole, above it they approach the past cosmological horizon, and near it they can remain nearly constant in radius.For the plotted parameters, δcr ≃ 0.000605; fine-tuning can yield a solution reaching the pole at past null infinity.Source: §B.3 · p. 75, Figure 29 · p. 75, Figure 30 · p. 76
  • Fine-tuning δ can satisfy the constant-redshift prescription's far-past requirement, while time independence after the shock is also a universal construction requirement.Source: §B.3 · p. 75

C Complexity of formation for shockwaves in SdS

The section defines complexity of formation for shockwave-perturbed SdS geometries and studies its time dependence. In three dimensions its large-insertion-time behavior tracks the plateau scrambling time, and under stated assumptions the matching extends to higher dimensions.

C Complexity of formation for shockwaves in SdS

C.1 General setting

  • For d ≥ 2, the formation calculation uses the WDW-patch positions at tL = tR = 0 and must account for both ordinary and behind-stretched-horizon joint configurations.At sufficiently large tw, the joints reach the stretched horizon; the expressions described are valid in any d ≥ 2.Source: §C.1 · pp. 77–78

C.2 Three-dimensional SdS space

  • In SdS3, formation complexity is smooth at small tw until a kink where rs = rb and the WDW-patch spacetime volume vanishes.When ε = 0, the complexity is independent of tw, consistent with the absence of a shockwave.Source: §C.2 · p. 78
  • For tw ≫ L, the SdS3 analytic expression resembles the plateau result; as ε → 0, its leading logarithmic divergence matches the plateau scrambling-time behavior.The comparison assumes shockwave energy proportional to a few thermal quanta and identifies the pre-shock black-hole Hawking temperature.Source: §C.2 · pp. 79–80

C.3 Higher-dimensional SdS space

  • Higher-dimensional formation complexity is difficult to compute explicitly because the tortoise coordinate and integrals can resist closed forms.Source: §C.3 · p. 80
  • The higher-dimensional analytic scrambling-time result assumes tw ≫ L, a double-scaling limit, and small black holes with rh,i/rC,i ≪ 1.Source: §C.3 · p. 80
  • In the stated higher-dimensional regime, formation and plateau scrambling times match for large insertion times, with similar universal behavior reported for AdS.The comparison is valid for small black-hole horizons and the assumptions specified for the higher-dimensional analysis.Source: §C.3 · pp. 80–81
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