The Cosmological Switchback Effect II

Holography in de Sitter space

• Static patch holography places the dual quantum theory on a timelike stretched horizon just inside the cosmological horizon, rather than on the spacelike boundaries used in dS/CFT.This framework supplies a bulk-time interpretation for the dual theory and supports links to operator-algebra and SYK constructions.Source: §1 · p. 3

Holographic complexity in de Sitter space

• De Sitter holographic complexity differs from AdS through hyperfast growth, with a divergent rate at finite boundary time caused by contributions from timelike infinities I±.Some codimension-one CAny observables and observables spanning cosmological and black-hole regions are exceptions.Source: §1 · pp. 4–5

Novelties and main results

• CV and CA exhibit an approximately constant plateau before hyperfast growth, extending the switchback effect beyond CV2.0 to finite-time shockwaves in generic dimensions d ≥2.The analysis targets Schwarzschild-de Sitter geometries with a shockwave inserted at finite boundary time.Source: §1 · p. 5
• The plateau duration increases when the shockwave is inserted earlier and asymptotically approaches a linear dependence on the insertion time, shifted by the scrambling time t∗.The scrambling time is analyzed using a double-scaling limit with ε →0 and ρ →1 while (1−ρ)/ε remains fixed.Source: §1 · p. 5
• The scrambling time has logarithmic dependence on shockwave energy ε and inverse dependence on the Hawking temperature, features associated with chaotic systems.The analytic expression also depends on horizon quantities before and after the shockwave.Source: §1 · p. 6
• The scrambling time has a novel logarithmic dependence log(1 −ρ) on stretched-horizon location and increases as the stretched horizon moves closer to the cosmological horizon.ρ determines the stretched-horizon location; when ρ →1, it approaches the cosmological horizon.Source: §1 · pp. 5–6
• The universal scrambling expression applies to CV, CV2.0, and CA despite their different geometric constructions.The paper relates this universality to de Sitter causal structure: after matter insertion, the Penrose diagram grows taller and the static patches become causally connected.Source: §1 · pp. 4–6

Outline

• The study evaluates CA through the WDW patch and CV through maximal-volume surfaces after reviewing shockwave geometries, causal structure, and stretched-horizon definitions.The paper includes analytic calculations, numerical examples, and separate CA and CV analyses.Source: §1 · pp. 4, 7

2.1.1 General dimensions

• The SdS geometry has black hole and cosmological horizons, with the static patch lying between them and the cosmological and black hole patches distinguished causally.For m ∈(0, mcr), the horizons satisfy rh < rc; the general geometry is analytically extended using EF and Kruskal coordinates.Source: §2.1.1 · pp. 8–9
• For generic dimensions and nonzero subcritical mass, the two horizons have unequal temperatures, so the background is generally out of thermal equilibrium.The exceptions include empty de Sitter space, three dimensions, and the Nariai limit.Source: §2.1.1 · p. 8

2.1.2 Three dimensions

• Three-dimensional SdS has a single cosmological horizon and is geometrically related to empty dS by a quotient producing a conical deficit at the origin.Its Penrose diagram matches empty dS, while the angular periodicity changes through the deficit angle αdef = 2π (1 −a).Source: §2.1.2 · p. 10

2.2 Perturbation with shockwaves

• A spherically symmetric null shockwave transitions the geometry from mass m1 before insertion to mass m2 afterward, with the null energy condition requiring a mass decrease.Consequently, rc1 ≤rc2 and rh1 ≥rh2, so the cosmological horizon expands while the black hole horizon contracts.Source: §2.2 · pp. 11–12

2.3.1 Definition in asymptotic dS geometries

• The stretched horizon is a timelike constant-radius surface inside the cosmological horizon where the putative dual quantum system is defined and boundary times are anchored.Its radius rst = (1 −ρ)rh + ρrc interpolates between the black hole and cosmological horizons, with ρ →1 approaching the latter.Source: §2.3.1 · pp. 12–13
• The analysis restricts geometric observables to the symmetric region of the cosmological patch between two stretched horizons, omitting the black hole patch.The left and right stretched horizons define boundary times tL and tR, which can be synchronized by a connecting spacelike surface.Source: §2.3.1 · p. 13

2.3.2 Definition in the presence of shockwaves

• After a shockwave, stretched horizons can be prescribed by constant redshift, continuous time across the shock, or constant proper acceleration.The paper adopts constant redshift for concreteness, while noting that all prescriptions yield qualitatively similar results.Source: §2.3.2 · p. 15
• All stretched-horizon prescriptions approach fixed radial locations in the far past and future and admit limits in which the horizons approach the corresponding cosmological horizons.The subsequent analysis focuses on tR ≥ −tw, where the time-dependent horizon shape is simpler and more relevant to geometric observables.Source: §2.3.2 · pp. 15–16

3.1 WDW patch

• The WDW patch requires future and past radial cutoffs because CA divergences occur when its null-boundary joints reach timelike infinities, and divergences return as δ → 0.The patch is defined as the bulk domain of dependence of a spacelike surface attached to the stretched horizons, following the treatment in.Source: §3.1 · p. 16

3.1.2 Universal critical times of the WDW patch

• Universal critical times tc1, tc2, and tc3 occur when WDW-patch joints or the position rb intersect the future or past cutoff surfaces.Their ordering can depend on the shockwave insertion time for tc0 and tc1, whereas the other critical times have definite order; tc1 and tc2 control the switchback analysis.Source: §3.1.2 · pp. 18–19, Figure 7 · p. 18

3.2 General computation of the action

• The CA calculation focuses on t ≥ tc0 and the intermediate regime tc1 ≤ t < tc2, where the WDW vertices avoid the asymptotic cutoff surfaces and complexity develops a plateau.The remaining time regimes are treated in appendix A, while special WDW-patch configurations occur only during this intermediate evolution stage.Source: §3.1.2 · p. 18, §3.2 · p. 23, Figure 10 · p. 23

3.1.3 Special configurations of the WDW patch

• Special WDW-patch configurations arise because the shockwave brings the two static patches into causal contact, with cosmological horizons satisfying rc1 ≤ rc2.These configurations occur when one or both WDW joints move inside the cosmological horizon; for sufficiently large tw, the associated fig. 9 configuration always exists and is crucial for the switchback effect.Source: §3.1.3 · pp. 19, 21

3.2 General computation of the action

• The on-shell action includes bulk, Gibbons–Hawking–York, null-boundary, joint, and null counterterm contributions, with affine parametrization making the null-boundary term vanish.The counterterm restores reparametrization invariance and introduces the length scale ℓct.Source: §3.2 · pp. 21–22

3.2.1 Bulk term

• Because the Einstein–Hilbert Lagrangian is constant, the bulk action is directly proportional to CV2.0, while special joint configurations require modified rates and additional GHY terms.The standard bulk-rate expression applies throughout tc1 ≤ t < tc2 except when both joints lie inside the cosmological horizon, as in fig. 9.Source: §3.2.1 · p. 24, §3.2.2 · p. 25

3.2.3 Hyperfast growth

• CA’s rate diverges at critical times, with hyperfast growth at t_c2 and an oppositely signed divergence as t approaches t_c1 from above.At t_c2, null-boundary counterterms leave a residual divergence after bulk–joint cancellation; the same critical time occurs for CV2.0.Source: §3.2.3 · p. 26

3.3 Explicit examples

• The CA examples show plateaus, extended by special WDW-patch configurations, with kinks at stretched-horizon crossings and possible negative complexity for later shock insertion.The standard configuration resembles empty de Sitter, while the special configuration lengthens the plateau; the rate is generally discontinuous at t_c1 and t_c2 but can be made continuous by tuning ℓ_ct.Source: §3.3 · pp. 26–28

3.4 Cosmological switchback effect (action)

• The action results identify the cosmological switchback effect with causal contact between otherwise spacelike-separated de Sitter static patches caused by shockwaves.This causal connection permits communication between the stretched horizons and accompanies the special WDW-patch configuration.Source: §3.4 · p. 28, §3.4.2 · p. 32

3.4.1 Plateau of complexity

• The CA plateau has the same duration as CV2.0 and grows when the shockwave is inserted earlier, asymptotically approaching a linear regime governed by a scrambling time.The plateau begins at an ε-independent constant and exhibits the switchback effect through the scrambling-time shortening of its duration.Source: §3.4.1 · pp. 28–30

3.4.2 Complexity of formation

• For shocks inserted far in the past, CA complexity of formation grows linearly with a scrambling-time delay, matching CV2.0’s switchback time while differing in overall prefactor.Unlike CV2.0, complexity of formation is always negative for sufficiently large ρ; CA itself can also become negative after the WDW joints move behind the stretched horizon.Source: §3.4.2 · pp. 31–32

4.1 General analysis

• CV is computed by decomposing the maximal volume across two Schwarzschild–de Sitter regions and matching conserved momenta across the shockwave.The symmetric boundary-time problem can be parametrized by fixing P1 rather than independently fixing tL and tR.Source: §4 · p. 33, §4.1 · pp. 33–34

4.2 Time evolution of the volume

• P1 > 0 sends the maximal surface through the future exterior, P1 < 0 through the past exterior, and P1 = 0 across the bifurcation surface.A surface through the past exterior necessarily has a turning point in region 1.Source: §4.2 · p. 34
• At early and late times the maximal surface follows configurations (a) and (c), while configuration (b) appears at intermediate times when tw ≫ L.Configuration (b) depends on the geometry parameters and occurs near t ≃ 0 for sufficiently early shock insertion.Source: §4.2 · p. 35, Figure 17 · p. 35

4.2.1 Hyperfast growth

• CV and its rate diverge when a turning point reaches timelike infinity, producing hyperfast growth at finite critical boundary times.The divergence follows from the rmax → ∞ limit and the associated |P_i| → ∞ behavior.Source: §4.2.1 · pp. 36–37

4.3 Explicit examples

• For larger tw, configuration (b) appears at intermediate times and lengthens the plateau because the positive null-energy pulse makes the de Sitter Penrose diagram taller.This behavior parallels the special WDW-patch configurations found for CV2.0 and CA.Source: §4.3 · pp. 37–39
• Numerical examples show a plateau with approximately constant complexity between past and future divergences, and the plateau lasts longer when the shockwave is inserted earlier.The numerical study uses d = 2, restricts to t ≥ −2tw, and focuses on the interval where turning points do not reach timelike infinity.Source: §4.3 · pp. 37, 39

4.4 Cosmological switchback effect (volume)

• The plateau begins when configuration (a) reaches past timelike infinity and ends when configuration (c) reaches future timelike infinity.These endpoints define the critical times tc1 and tc2 used to determine the plateau duration.Source: §4.4 · pp. 39–40
• CV has the same plateau duration and scrambling time as CV2.0 and CA because its critical-time definitions coincide with those of the WDW patch in the rmax → ∞ limit.The maximal-surface shockwave intersections approach the corresponding WDW-patch intersections as |P_i| → ∞.Source: §4.4 · pp. 39–40

5.1 Conclusions

• CV, CV2.0, and CA all exhibit an approximately constant plateau near tL = tR = 0, whose duration grows for earlier shockwave insertion and approaches linear growth with a universal scrambling-time delay.The complexity of formation also grows linearly with shockwave insertion time in the CV2.0 and CA cases.Source: §5.1 · p. 40
• A precise numerical analysis of the complexity-of-formation phenomenon in the CV case remains outstanding.The authors expect the same phenomenon but defer its verification because of the difficulty of the numerical computation.Source: §5.1 · p. 40
• The beginning of hyperfast growth is always delayed, including the limit where the shockwave is sent along the cosmological horizon.For a special shockwave energy, the scrambling time matches the result obtained in reference.Source: §5.1 · p. 41
• The scrambling time scales as t∗∼βc1 log(1/GN), indicating fast scrambling analogous to black holes in AdS and agreeing with other de Sitter correlator diagnostics.For perturbations changing the energy by a few thermal quanta, the entropy variation satisfies ∆Sc1 ∼d and the scrambling time simplifies further.Source: §5.1 · p. 41
• The switchback effect arises because positive-energy perturbations make the de Sitter cosmological horizon grow, bringing the left and right static patches into causal contact and allowing communication between stretched horizons.The relevant plateau configurations differ between codimension-zero proposals and CV, but both depend on the horizon relation rc1 ≤ rc2.Source: §5.1 · pp. 41–42
• The CA observable can become negative when the shockwave is inserted far in the past and the stretched horizon approaches the cosmological horizon.The authors leave quantum corrections and the interpretation of the de Sitter reference state as possible explanations for this puzzle.Source: §5.1 · p. 42

5.2 Future developments

• The paper proposes future tests involving circuit models, explicit dual quantum theories, more general CAny observables, conformal boundary conditions, centaur geometries, earlier times, two dimensions, and de Sitter causality.These directions include checking whether the scrambling-time formula remains universal beyond the studied CV, CV2.0, and CA observables.Source: §5.2 · pp. 43–44

• The appendix organizes the complexity=action calculation into an intermediate-time regime and other evolution regimes treated separately.Source: A Details for complexity=action · pp. 44–45

• At fixed t ∈[tc1, tc2], the calculation uses the WDW-patch configuration shown in fig. 10(b).The same computation applies to the shapes in fig. 8 except case 9, where the top and bottom joints move behind the stretched horizons.Source: §A.1 · p. 45

Codimension-one boundaries

• The standard configuration has no GHY terms, and affine parametrization of the null generators sets I_N = 0.Source: §A.1.1 · p. 45

Joint terms

• The null normals are assigned positive normalization constants, with additional conditions enforcing affine parametrization across the shockwave.The conditions arise by requiring selected null-null joints at r_s and r_b to vanish.Source: §A.1.1 · p. 45

Standard configuration of the WDW patch

• The shockwave contributes a null surface, while its apparent diagrammatic jump does not represent a discontinuity in the bulk WDW region.Source: §A.1.1 · p. 46

Total boundary term

• The calculation combines nonvanishing joint contributions, null-boundary counterterms, and all boundary terms into the total result.Source: §A.1.1 · pp. 46–47
• The final boundary expression is independent of null-normal normalization, leaving the counterterm scale ℓct as the only arbitrary scale.Source: §A.1.1 · p. 47

• The special configuration requires a separate analysis before evaluating its boundary contributions.Source: §A.1.2 · p. 47

GHY term

• The calculation includes timelike stretched-horizon boundaries and corresponding GHY contributions.Source: §A.1.2 · p. 47

Joint terms

• The WDW patch has four null-boundary parts, whose normal one-forms determine the joint terms.Source: §A.1.2 · p. 48

Total boundary term

• Summing the GHY, joint, and counterterm pieces yields the total boundary term for this configuration.Source: §A.1.2 · p. 48

Counterterm on null boundaries

• The null-boundary expansion is used to compute the counterterm contribution.Source: §A.1.2 · p. 48

• The remaining regimes are organized by the intervals tc0 ≤ t < tc1, tc2 ≤ t < tc3, and t ≥ tc3.Source: §A.2 · p. 49
• The total complexity=action observable is continuous, while its rate can jump at critical times tc1 and tc2 because of additional exterior GHY terms.Source: §A.2 · p. 50
• For late times, the rate is obtained by differentiating the complexity=action expression and evaluating the asymptotic geometric data.Source: §A.2 · p. 50
• At late times, the behavior coincides with empty de Sitter space and is controlled by the asymptotic structure near timelike infinity, independently of the black hole.Source: §A.2 · p. 50

A.3 Linear approximation for the complexity of formation

• The approximation applies for a shockwave inserted in the far past, tw ≫ L, under the stated double-scaling limit.Source: §A.3 · p. 50
• In the far-past limit, the special positions approach rc2, rc1, and rst, simplifying the WDW-patch geometry.Source: §A.3 · p. 50

A.3.1 Three dimensions

• In three dimensions, the complete complexity=action calculation can be performed analytically, with the bulk term inherited from the earlier CV2.0 computation.Source: §A.3.1 · p. 50
• The three-dimensional boundary contribution is expanded using the factor a from eq. (2.11), then combined with the bulk term to obtain formula (3.34a).Source: §A.3.1 · pp. 50–51

A.3.2 Higher dimensions

• In higher dimensions, no closed expression is available, but the double-scaling limit permits a linear expansion in tw.Source: §A.3.2 · p. 51
• The higher-dimensional bulk term follows from the computation in, while tortoise-coordinate manipulations provide the remaining simplification.Source: §A.3.2 · p. 51
• Combining the higher-dimensional intermediate expressions yields formula (3.35a).Source: §A.3.2 · p. 51

• Six maximal-surface configurations are possible, distinguished by turning-point number and passage through the future or past cosmological-horizon exterior.Source: B Details for complexity=volume · p. 51
• Case A passes through the past IP1, has P1 < 0, and turns in IP1 but not in BH2.Source: B Details for complexity=volume · p. 51
• Case B passes through the future IP1 with P1 > 0 and has no turning points.Source: B Details for complexity=volume · p. 51
• Case C passes through the future IP1 with P1 > 0 and turns in IP2 but not in IP1.Source: B Details for complexity=volume · p. 51
• Here IP denotes the inflationary patch of Schwarzschild–de Sitter space outside the cosmological horizon.Source: B Details for complexity=volume · p. 51
• Numerical evolution shows that cases A–C are the only configurations realized during the time evolution.Source: Figure 20 · p. 52

B.1 General strategy

• Boundary times are related to conserved momenta by integrating the extremal surface from the left to the right stretched horizon and summing the segment contributions.The final relation uses the coordinate continuity rules across horizons and the shockwave, together with the boundary-time relation.Source: §B.1 · pp. 52–53
• The null-coordinate conventions depend on whether the radial coordinate increases or decreases toward the right, and the coordinates used in all quadrants are specified in Figure 21.The sign choice is tied to the direction of radial evolution along each surface segment.Source: §B.1 · p. 53, Figure 21 · p. 53
• The integration begins with the continuous v coordinate from the left stretched horizon to the shockwave, with turning-point segments split when present.Case A uses a special combination of u and v integrations, while the right side uses u, which remains continuous there.Source: §B.1 · p. 53
• For equal boundary times and fixed P1, the procedure solves for P2 and the shock intersection rsh, then integrates Ri[Pi, r] to obtain the maximal volume.This determines the time dependence of the extremal surfaces.Source: §B.1 · p. 53

B.2 Analysis of the cases for the extremal surfaces

• Numerically, only configurations A–C occur during the evolution, so the subsequent analysis reports these cases.The cases follow the integration rules established in appendix B.1.Source: §B.2 · p. 54

Case A

• Case A has a shock intersection rsh ≤ rc1; the left surface passes through a turning point, while the right side is integrated using uR across the future horizon.The maximal volume is assembled from the corresponding left and right radial integrals.Source: §B.2 · p. 54

Case B

• Case B has no turning point on either side, with rsh ∈ [rc1, rc2] or rsh > rc2; both possibilities are treated formally together.The possible configurations are depicted in Figure 22.Source: §B.2 · p. 54, Figure 22 · p. 55
• Case B’s maximal-surface volume is constructed from the radial integrals over the relevant regions after integrating from the left stretched horizon without a turning point.The displayed configurations distinguish the possible shockwave locations within this common treatment.Source: §B.2 · p. 54

Case C

• Case C requires rsh ≥ rc2 for a turning point after the shockwave, and the right-side uR integration is split around that turning point.The total volume is obtained by summing the resulting region-2 contributions and integrating along three extremal-surface portions.Source: §B.2 · p. 55

B.3 Rate of growth of the volume

• The calculation starts from the volume expression for the relevant configuration and rewrites it with the identities before differentiating in time.The resulting terms include the right-side momentum multiplied by the shifted boundary time.Source: §B.3 · pp. 55–56
• The rate of growth is evaluated using case C as a representative configuration because the calculation yields a universal result.The derivation introduces identities relating R, its momentum derivative, and null-coordinate derivatives.Source: §B.3 · pp. 55–56
• Continuity of the null-coordinate derivative across the shockwave supplies the identity connecting the two sides’ radial functions and the shock position.This relation is used to simplify the time derivative of the volume.Source: §B.3 · p. 56
• After applying the case identities and relating boundary times to conserved momenta, all terms in parentheses vanish, producing the simplified growth-rate expression.The same result can be shown to hold for any extremal-surface shape.Source: §B.3 · p. 56