Pith. sign in

REVIEW 4 major objections 5 minor 107 references

Timelike subregion complexity in de Sitter is real and finite, grows exponentially at early durations, and diverges hyperfast near a maximal duration unless a black hole horizon intervenes.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-03 02:20 UTC pith:V4NG7MYR

load-bearing objection The pure dS calculation is solid and worth reading, but the SdS Case 2 headline is contradicted by the paper's own Appendix B, so the abstract and Section 5.2 need correction before this can be trusted. the 4 major comments →

arxiv 2607.29662 v1 pith:V4NG7MYR submitted 2026-07-31 hep-th

Holographic timelike complexity for de Sitter

Arpan Bhattacharyya , Md Khalid Hossain , Sanhita Parihar , Shubho R. Roy This is my paper
classification hep-th
keywords timelike subregion complexityholographic complexityde Sitter static patchvolume complexitySchwarzschild-de Sittertimelike entanglement entropyextremal surfacesstretched horizon
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends the holographic 'complexity equals volume' idea to timelike subregions of the stretched horizon in de Sitter static-patch holography. It argues that this timelike subregion complexity is a genuine observable: real, positive, and free of ultraviolet divergences, unlike timelike entanglement entropy, which picks up an imaginary part. In pure de Sitter it grows as e^{2T} for short subregion durations and diverges as (T_max - T)^{-(d-2)} (logarithmically in d=3) near a maximal duration T_max set by the stretched-horizon cutoff. For Schwarzschild–de Sitter with the stretched horizon near the cosmological horizon the same qualitative behavior persists with mass-dependent exponents; with the stretched horizon near the black hole horizon, the finite-time blow-up disappears and late-time growth becomes a power law. If correct, this gives a boundary-anchored, causally well-defined measure of complexity that can probe the black hole interior.

Core claim

On the paper's own terms, the discovery is that timelike-subregion volume complexity is computable and well-defined in de Sitter static-patch holography. For a boundary subregion of duration 2T on the stretched horizon, the complexity is the bulk volume bounded by two spacelike extremal surfaces anchored at the subregion's endpoints and a timelike extremal surface joining their turning points. In pure dS_3 this volume is analytic, real and positive; in higher dimensions it is numerical, checked in Kruskal and Eddington–Finkelstein coordinates. The observable reproduces the early exponential and late hyperfast growth of spacelike volume complexity, with the divergence exponent lowered by one,

What carries the argument

The construction has three parts: (i) the stretched-horizon prescription, which places the holographic boundary just inside the cosmological horizon and regulates the maximum boundary duration as T_max = (1/2) ln((2-epsilon)/epsilon); (ii) the timelike-subregion complexity prescription, in which two spacelike extremal surfaces anchored at the past and future endpoints and one timelike extremal surface joining their turning points bound a bulk volume; and (iii) the reflection-symmetry assumption that the turning points lie on the t=0 slice (U=V in Kruskal coordinates), which fixes all boundary conditions. In pure dS_3 the extremal surfaces are known analytically; for higher dimensions and for

Load-bearing premise

The entire calculation assumes that the turning point of every spacelike extremal surface lies exactly on the t=0 slice (U=V), argued from the left-right symmetry of the extended Penrose diagram but not independently derived for finite boundary subregions; if the turning point is off t=0, every boundary condition, volume, and late-time scaling in the paper changes.

What would settle it

For the Schwarzschild–de Sitter Case-2 geometry, numerically integrate the extremal-surface equations without imposing U=V (or v(r*) = r̃(r*)) at the turning point; if the resulting late-time complexity growth differs from the claimed power-law/linear regime, the symmetry assumption is the point of failure. Separately, a late-time run with epsilon and T-max approached to high precision can settle whether C_s grows like r_min^{d-1} T or like a genuinely nonlinear power of T, since the paper currently asserts both.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If the proposal is correct, timelike subregion complexity is a real, positive, finite holographic quantity in de Sitter, with no ultraviolet divergence, in contrast to timelike entanglement entropy which is complex.
  • In pure de Sitter, the observable grows as e^{2T} at short durations and diverges as (T_max - T)^{-(d-2)} near the maximal duration, with T_max identical to the critical time of full-volume complexity.
  • For Schwarzschild–de Sitter with the stretched horizon near the cosmological horizon, the same exponential-then-hyperfast pattern holds, with the black-hole mass increasing T_max and modifying the divergence exponent.
  • With the stretched horizon near the black hole horizon, timelike complexity does not diverge at finite time; its long-duration growth is a power law (per the paper's abstract and discussion) or linear with coefficient r_min^{d-1} (per the appendix), indicating the black hole interior rather than the expanding cosmological region.
  • The growth-rate pattern — e^{T} for full volume complexity and e^{2T} for timelike subregion complexity — suggests the rate tracks the codimension of the boundary anchoring surface, giving a new organizational rule for holographic complexity.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A testable extension: compute timelike subregion complexity for rotating or charged de Sitter black holes; if the Case-2 softening persists, the black-hole-horizon effect is generic rather than a quirk of Schwarzschild–de Sitter.
  • The e^{2T} early growth matches the behavior expected of Krylov-type operator growth, so timelike subregion complexity may be a more direct holographic proxy for operator growth in the static patch than full volume complexity.
  • The paper's Case-2 late-time result is stated in two ways: the abstract and discussion describe nonlinear growth, while the appendix derives C_s ~ r_min^{d-1} T, linear in T. Since these are not the same statement, a high-precision late-time numerical check is needed to decide which behavior the extremal-surface construction actually produces.
  • If the turning-point-at-t=0 symmetry is relaxed, the entire Case-2 late-time scaling would shift; testing this assumption in the numerical integration would reveal whether the claimed black-hole-horizon softening is robust.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper applies the recently proposed holographic timelike subregion volume complexity to de Sitter spacetime in the static patch holography framework. For pure dS it computes analytic dS3 expressions for the spacelike and timelike extremal surfaces, the timelike entanglement entropy, and the complexity, then extends the computation numerically to d=4,5. For Schwarzschild–de Sitter, it treats two stretched-horizon positions: near the cosmological horizon (Case 1) and near the black hole horizon (Case 2). The headline claims are that pure dS timelike complexity grows as e^{2T} for short durations and diverges hyperfastly near a maximal duration, while SdS Case 2 replaces the hyperfast growth by a 'nonlinear growth regime' for long durations.

Significance. If established, this would be a useful extension of timelike complexity to cosmological horizons, showing that the construction yields a finite, real, positive measure in dS and that the late-time divergence is dimension-dependent. The dS3 analytic solution, the numerical cross-checks in Kruskal and Eddington–Finkelstein coordinates (Tables 3–6, Appendix A), and the absence of fitted constants in the extremization are genuine strengths. However, the central Case 2 claim is internally inconsistent with the paper's own Appendix B, and the pure-dS late-time divergence formulae contradict each other for d=3. These issues must be resolved before the paper's main conclusions can be accepted.

major comments (4)
  1. [Abstract; §5.2; Appendix B] The SdS Case 2 claim of 'nonlinear growth' is contradicted by Appendix B. Eq. (B.6) gives C_s ∼ r_min^{d−1} T, which is linear in T for fixed r_min, while the sentence immediately after (B.6) states that this 'does not show linear growth with time.' Section 5.2 and the Discussion instead describe 'nonlinear growth (mix of powers)' and a 'power law increase with the boundary subregion duration,' without any fitted exponent. Both statements cannot be correct. The authors must determine whether Eq. (B.6) or the numerical reading of Fig. 17 is right; if the former, the abstract and §5.2 must be revised to report linear growth; if the latter, the asymptotic reduction in (B.4)–(B.6) contains an error.
  2. [Eqs. (1.2), (1.3), (3.40), (4.13)] The pure-dS late-time divergence is stated inconsistently. Eq. (1.2) and Eq. (3.40) give C_late ∼ ln[(T_max−T)^{−1}] for d=3, while Eq. (1.3) and Eq. (4.13) give C_late ∼ (T_max−T)^{−(d−2)}, which for d=3 is (T_max−T)^{−1}, not logarithmic. Section 4 and Fig. 8 apply the power-law form to d=3,4,5. This is a direct contradiction in a central quantitative claim. If logarithmic divergence is special to d=3, the general formula must state d>3 and explain how the d=3 limit arises; if the power law is correct, the §3.2 analytic result needs re-examination.
  3. [§5.1, Eq. (5.16); §5.2, Eqs. (5.29), (5.35)] The SdS computation relies on the assumption that the turning point of the spacelike extremal surface lies on the t=0 slice, and that the timelike connecting curve is u(r)=−r̃(r) or v(r)=r̃(r). This is asserted via 'insight based on symmetry' (§5.1) and 'we can argue from the symmetry of SdS geometry' (§5.2), rather than derived from the extremization conditions with finite boundary subregions. For pure dS this condition follows from the U↔V reflection symmetry of the maximal-radius point; for the infinitely extended SdS Penrose diagram and a finite timelike boundary interval it is not automatic. Since this assumption fixes the boundary conditions for all computed surfaces and volumes in Cases 1 and 2, a proper justification is load-bearing. The authors should either prove the t=0 turning point from the equations of motion or state the precise symmetry and why it survives the choice of a
  4. [§5.2, Fig. 17] The evidence for 'nonlinear (polynomial)' growth is not quantified. Fig. 17 plots ln C_s versus T, not versus ln T, and shows only two mass values with no error bars, no fitted exponent, and no alternative linear fit. Given the conflict with Appendix B, a quantitative late-time analysis is necessary: for example, extract d ln C_s/d ln T at large T, or directly test C_s = a T + b and compare with a power law. The present plot does not by itself support a 'mix of powers' claim.
minor comments (5)
  1. [Eq. (3.36)] The expression contains 'ln ln' where a single logarithm is clearly intended; the formula should read C_s = (4c/3) Tϵ + (4c/3) ln(...).
  2. [Fig. 13 caption] The caption mentions 'AdS radius L' in a de Sitter context; this should be 'de Sitter radius L'.
  3. [Eq. (5.21)] The exponent is written as '(d−2)−#g(μ)' with the symbol '#' unexplained and the function g(μ) not defined. Please define g and specify the range of T_max−T over which this fit is valid.
  4. [§2] The notation V_{S^{d−1}} is used without definition; it should be identified as the volume of the unit (d−1)-sphere.
  5. [References] Several references use incomplete entries, e.g., [1] is listed with only an arXiv number and no year, and [82] repeats the arXiv identifier. Please normalize the bibliography.

Circularity Check

0 steps flagged

No significant circularity: the central derivation is self-contained; the noted issues are internal consistency and unproven symmetry assumptions, not circular reasoning.

full rationale

The central derivation chain is self-contained. For pure dS, the d=3 case is solved analytically (Eq. 3.15 and Eq. 3.35), and the higher-dimensional cases use numerical solutions of Eq. 4.5, validated against the analytic d=3 results in Tables 3-5 and against independent Eddington-Finkelstein numerics in Appendix A. The SdS analysis uses the conserved charge from Eq. 5.12/5.26 with boundary conditions; no fitted parameter is later renamed as a prediction. The stretched-horizon cutoff epsilon and mass parameter mu are physical inputs, not parameters fitted to the output. Self-citations (e.g., [82], [103], [104], [106], [107], [109]) appear as comparisons or outlook examples; for instance, "Contrast this with volume complexity growth in global de Sitter space... [82]" is a side remark, not a load-bearing premise. The paper is benchmarked against external work [2], [90] and its own analytic/numeric cross-checks, so the computation does not reduce to its inputs by construction. Two issues are correctness risks rather than circularity. First, Appendix B's late-time analysis gives Eq. (B.6), "Cs ∼ rd−1 min T", i.e., linear growth, yet the next sentence says "the timelike complexity does not show linear growth with time," and Sections 5.2/6 claim "nonlinear growth (mix of powers)". Second, the pure-dS d=3 divergence is stated logarithmically in Eq. (1.2) and Eq. (3.40), but as a power law (Tmax−T)^{-(d−2)} in Eq. (1.3) and Section 4; for d=3 these cannot both hold. Also, the SdS turning-point condition is invoked "based on symmetry" (Secs. 5.1-5.2) without an independent derivation, an assumption that may affect the results but is not circular.

Axiom & Free-Parameter Ledger

3 free parameters · 4 axioms · 0 invented entities

No new particles, forces, or conserved quantities are invented. The paper relies on two conjectural frameworks (static patch holography and the timelike-complexity prescription) as axioms, plus a symmetry assumption for turning points in SdS. The only hand-chosen numerical inputs are the stretched-horizon cutoff ϵ, the SdS mass parameter µ, and the de Sitter radius L, none of which are fitted to reproduce the claimed scaling laws.

free parameters (3)
  • ϵ (stretched-horizon cutoff) = 10^{-7} (pure dS and SdS Case 1 plots), 10^{-5} (SdS Case 2), 10^{-3} (some comparisons)
    Distance of the stretched horizon inside the cosmological (or outside the black hole) horizon. It is a physical regulator in static-patch holography, but its numerical value is chosen by hand and all T_max values and scaling exponents depend on it.
  • µ (SdS mass parameter) = 0.1, 0.11, 0.2, 0.3
    Black hole mass parameter chosen for representative numerics; the mass-dependent functions g(µ) and the mass dependence of T_max and the early-time exponent are inferred from these specific values.
  • L (dS radius) = 1 (pure dS), 3 (SdS numerics)
    Overall length scale; setting L=1 is a unit choice, but L=3 is chosen for numerical convenience in the SdS sections and affects the mass bound and the plotted behavior.
axioms (4)
  • domain assumption Timelike subregion complexity is the volume of the bulk region bounded by the spacelike and timelike extremal surfaces anchored on the timelike boundary subregion (the Alishahiha prescription, adapted to dS by joining turning points with the U=V curve).
    Adopted from ref. [1]; the paper extends it to dS in Section 3.2 (Eq. 3.34, Fig. 6) and Sections 5.1–5.2. The construction is used without an independent derivation.
  • domain assumption Static patch holography: the dual field theory lives on the stretched horizon at r=1−ϵ and the standard dictionaries S=A/4G_N and CV=V/G_N apply.
    Taken from refs. [2, 40, 41, 63]; all quantities are computed under this dictionary.
  • domain assumption The classical extremization (Euler–Lagrange equations (3.14), (4.5), (5.12), (5.27)) selects the correct codimension-two surface, and the branch choices (spacelike vs timelike) are the ones dictated by the timelike-entanglement proposals [65, 66].
    The sign choices in Sections 3.1.1, 3.1.2, 5.1, and 5.2 follow prior conventions; the authors do not prove uniqueness of the extremal configuration for finite subregions.
  • domain assumption In SdS, the turning point of the spacelike extremal surface lies on the t=0 slice by reflection symmetry of the infinitely extended Penrose diagram (leading to Eq. (5.16) and Eq. (5.29)).
    Invoked in Sections 5.1 and 5.2 to fix boundary conditions for the numerical surfaces and the timelike branch. If this symmetry fails for finite subregion duration, the computed surfaces and volumes change.

pith-pipeline@v1.3.0-daily-deepseek · 30782 in / 16283 out tokens · 152175 ms · 2026-08-03T02:20:41.166577+00:00 · methodology

0 comments
read the original abstract

We investigate the recent proposal of holographic volume complexity for timelike subregions \cite{Alishahiha:2025xml} in the framework of static patch holography for de Sitter spacetime. Using the stretched-horizon prescription, we compute the timelike subregion complexity as a function of the subregion duration for pure de Sitter and Schwarzschild de Sitter geometries. In pure de Sitter spacetime, the timelike subregion complexity displays exponential growth for short durations, and hyperfast growth near a maximal duration, paralleling the features of spacelike volume complexity \cite{Jorstad:2022mls}. For Schwarzschild de Sitter, when the stretched horizon is near the cosmological horizon, the behavior broadly remains similar to pure de Sitter. However, when the stretched horizon is near the black hole horizon, the hyperfast growth for long durations is replaced by nonlinear growth regime. Along the way, we also compute the corresponding timelike holographic entanglement entropy for de Sitter and Schwarzschild de Sitter.

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

107 extracted references · 98 linked inside Pith

  1. [1]

    Alishahiha,Timelike Holographic Complexity,2510.25700

    M. Alishahiha,Timelike Holographic Complexity,2510.25700

    Pith/arXiv arXiv

  2. [2]

    Jørstad, R.C

    E. Jørstad, R.C. Myers and S.-M. Ruan,Holographic complexity in dS d+1,JHEP05(2022) 119 [2202.10684]

    Pith/arXiv arXiv 2022

  3. [3]

    Bekenstein,Bekenstein-Hawking entropy,Scholarpedia3(2008) 7375

    J. Bekenstein,Bekenstein-Hawking entropy,Scholarpedia3(2008) 7375

    2008

  4. [4]

    Hawking,Particle Creation by Black Holes,Commun

    S.W. Hawking,Particle Creation by Black Holes,Commun. Math. Phys.43(1975) 199

    1975

  5. [5]

    ’t Hooft,Dimensional reduction in quantum gravity,Conf

    G. ’t Hooft,Dimensional reduction in quantum gravity,Conf. Proc. C930308(1993) 284 [gr-qc/9310026]

    Pith/arXiv arXiv 1993

  6. [6]

    Susskind,The World as a hologram,J

    L. Susskind,The World as a hologram,J. Math. Phys.36(1995) 6377 [hep-th/9409089]

    Pith/arXiv arXiv 1995

  7. [7]

    Bousso,A Covariant entropy conjecture,JHEP07(1999) 004 [hep-th/9905177]

    R. Bousso,A Covariant entropy conjecture,JHEP07(1999) 004 [hep-th/9905177]

    Pith/arXiv arXiv 1999

  8. [8]

    Bousso,Holography in general space-times,JHEP06(1999) 028 [hep-th/9906022]

    R. Bousso,Holography in general space-times,JHEP06(1999) 028 [hep-th/9906022]

    Pith/arXiv arXiv 1999

  9. [9]

    Maldacena,The LargeNlimit of superconformal field theories and supergravity,Adv

    J.M. Maldacena,The LargeNlimit of superconformal field theories and supergravity,Adv. Theor. Math. Phys.2(1998) 231 [hep-th/9711200]

    Pith/arXiv arXiv 1998

  10. [10]

    Gubser, I.R

    S.S. Gubser, I.R. Klebanov and A.M. Polyakov,Gauge theory correlators from noncritical string theory,Phys. Lett. B428(1998) 105 [hep-th/9802109]

    Pith/arXiv arXiv 1998

  11. [11]

    Witten,Anti de Sitter space and holography,Adv

    E. Witten,Anti de Sitter space and holography,Adv. Theor. Math. Phys.2(1998) 253 [hep-th/9802150]

    Pith/arXiv arXiv 1998

  12. [12]

    Aharony, S.S

    O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri and Y. Oz,Large N field theories, string theory and gravity,Phys. Rept.323(2000) 183 [hep-th/9905111]

    Pith/arXiv arXiv 2000

  13. [13]

    El-Showk and K

    S. El-Showk and K. Papadodimas,Emergent Spacetime and Holographic CFTs,JHEP10 (2012) 106 [1101.4163]

    Pith/arXiv arXiv 2012

  14. [14]

    Pasterski, S.-H

    S. Pasterski, S.-H. Shao and A. Strominger,Flat Space Amplitudes and Conformal Symmetry of the Celestial Sphere,Phys. Rev. D96(2017) 065026 [1701.00049]

    Pith/arXiv arXiv 2017

  15. [15]

    Pasterski, M

    S. Pasterski, M. Pate and A.-M. Raclariu,Celestial Holography, inSnowmass 2021, 11, 2021 [2111.11392]

    Pith/arXiv arXiv 2021

  16. [16]

    Duval, G.W

    C. Duval, G.W. Gibbons and P.A. Horvathy,Conformal Carroll groups and BMS symmetry,Class. Quant. Grav.31(2014) 092001 [1402.5894]

    Pith/arXiv arXiv 2014

  17. [17]

    Bagchi, P

    A. Bagchi, P. Dhivakar and S. Dutta,Holography in flat spacetimes: the case for Carroll, JHEP08(2024) 144 [2311.11246]

    Pith/arXiv arXiv 2024

  18. [18]

    Ruzziconi,Carrollian physics and holography,Phys

    R. Ruzziconi,Carrollian physics and holography,Phys. Rept.1182(2026) 1 [2602.02644]

    arXiv 2026

  19. [19]

    Banks, W

    T. Banks, W. Fischler, S.H. Shenker and L. Susskind,M theory as a matrix model: A conjecture,Phys. Rev. D55(1997) 5112 [hep-th/9610043]. [20]Supernova Search Teamcollaboration,Observational evidence from supernovae for an accelerating universe and a cosmological constant,Astron. J.116(1998) 1009 [astro-ph/9805201]. [21]Supernova Cosmology Projectcollabora...

    Pith/arXiv arXiv 1997

  20. [22]

    Gibbons and S.W

    G.W. Gibbons and S.W. Hawking,Cosmological Event Horizons, Thermodynamics, and Particle Creation,Phys. Rev. D15(1977) 2738. – 40 –

    1977

  21. [23]

    Spradlin, A

    M. Spradlin, A. Strominger and A. Volovich,Les Houches lectures on de Sitter space, inLes Houches Summer School: Session 76: Euro Summer School on Unity of Fundamental Physics: Gravity, Gauge Theory and Strings, pp. 423–453, 10, 2001 [hep-th/0110007]

    Pith/arXiv arXiv 2001

  22. [24]

    Anninos,De Sitter Musings,Int

    D. Anninos,De Sitter Musings,Int. J. Mod. Phys. A27(2012) 1230013 [1205.3855]

    Pith/arXiv arXiv 2012

  23. [25]

    Galante,Modave lectures on de Sitter space & holography,PoSModave2022(2023) 003 [2306.10141]

    D.A. Galante,Modave lectures on de Sitter space & holography,PoSModave2022(2023) 003 [2306.10141]

    Pith/arXiv arXiv 2023

  24. [26]

    Strominger,The dS / CFT correspondence,JHEP10(2001) 034 [hep-th/0106113]

    A. Strominger,The dS / CFT correspondence,JHEP10(2001) 034 [hep-th/0106113]

    Pith/arXiv arXiv 2001

  25. [27]

    Witten,Quantum gravity in de Sitter space, inStrings 2001: International Conference, 6, 2001 [hep-th/0106109]

    E. Witten,Quantum gravity in de Sitter space, inStrings 2001: International Conference, 6, 2001 [hep-th/0106109]

    Pith/arXiv arXiv 2001

  26. [28]

    Maldacena,Non-Gaussian features of primordial fluctuations in single field inflationary models,JHEP05(2003) 013 [astro-ph/0210603]

    J.M. Maldacena,Non-Gaussian features of primordial fluctuations in single field inflationary models,JHEP05(2003) 013 [astro-ph/0210603]

    Pith/arXiv arXiv 2003

  27. [29]

    Chowdhury, V

    C. Chowdhury, V. Godet, O. Papadoulaki and S. Raju,Holography from the Wheeler-DeWitt equation,JHEP03(2022) 019 [2107.14802]

    Pith/arXiv arXiv 2022

  28. [30]

    Chakraborty, J

    T. Chakraborty, J. Chakravarty, V. Godet, P. Paul and S. Raju,The Hilbert space of de Sitter quantum gravity,JHEP01(2024) 132 [2303.16315]

    Pith/arXiv arXiv 2024

  29. [31]

    Dine, J.A.P

    M. Dine, J.A.P. Law-Smith, S. Sun, D. Wood and Y. Yu,Obstacles to Constructing de Sitter Space in String Theory,JHEP02(2021) 050 [2008.12399]

    Pith/arXiv arXiv 2021

  30. [32]

    Maldacena and C

    J.M. Maldacena and C. Nunez,Supergravity description of field theories on curved manifolds and a no go theorem,Int. J. Mod. Phys. A16(2001) 822 [hep-th/0007018]

    Pith/arXiv arXiv 2001

  31. [33]

    Robbins and S

    D. Robbins and S. Sethi,A Barren landscape?,Phys. Rev. D71(2005) 046008 [hep-th/0405011]

    Pith/arXiv arXiv 2005

  32. [34]

    Green, E.J

    S.R. Green, E.J. Martinec, C. Quigley and S. Sethi,Constraints on String Cosmology, Class. Quant. Grav.29(2012) 075006 [1110.0545]

    Pith/arXiv arXiv 2012

  33. [35]

    Kutasov, T

    D. Kutasov, T. Maxfield, I. Melnikov and S. Sethi,Constraining de Sitter Space in String Theory,Phys. Rev. Lett.115(2015) 071305 [1504.00056]

    Pith/arXiv arXiv 2015

  34. [36]

    Sethi,Supersymmetry Breaking by Fluxes,JHEP10(2018) 022 [1709.03554]

    S. Sethi,Supersymmetry Breaking by Fluxes,JHEP10(2018) 022 [1709.03554]

    Pith/arXiv arXiv 2018

  35. [37]

    Russo and P.K

    J.G. Russo and P.K. Townsend,Time-dependent compactification to de Sitter space: a no-go theorem,JHEP06(2019) 097 [1904.11967]

    Pith/arXiv arXiv 2019

  36. [38]

    Obied, H

    G. Obied, H. Ooguri, L. Spodyneiko and C. Vafa,De Sitter Space and the Swampland, 1806.08362

    Pith/arXiv arXiv

  37. [39]

    Ooguri, E

    H. Ooguri, E. Palti, G. Shiu and C. Vafa,Distance and de Sitter Conjectures on the Swampland,Phys. Lett. B788(2019) 180 [1810.05506]

    Pith/arXiv arXiv 2019

  38. [40]

    Susskind,De Sitter Holography: Fluctuations, Anomalous Symmetry, and Wormholes, Universe7(2021) 464 [2106.03964]

    L. Susskind,De Sitter Holography: Fluctuations, Anomalous Symmetry, and Wormholes, Universe7(2021) 464 [2106.03964]

    Pith/arXiv arXiv 2021

  39. [41]

    Susskind,Entanglement and Chaos in De Sitter Space Holography: An SYK Example, JHAP1(2021) 1 [2109.14104]

    L. Susskind,Entanglement and Chaos in De Sitter Space Holography: An SYK Example, JHAP1(2021) 1 [2109.14104]

    Pith/arXiv arXiv 2021

  40. [42]

    Anninos, S.A

    D. Anninos, S.A. Hartnoll and D.M. Hofman,Static Patch Solipsism: Conformal Symmetry of the de Sitter Worldline,Class. Quant. Grav.29(2012) 075002 [1109.4942]

    Pith/arXiv arXiv 2012

  41. [43]

    Susskind,De Sitter Space, Double-Scaled SYK, and the Separation of Scales in the Semiclassical Limit,JHAP5(2025) 1 [2209.09999]

    L. Susskind,De Sitter Space, Double-Scaled SYK, and the Separation of Scales in the Semiclassical Limit,JHAP5(2025) 1 [2209.09999]. – 41 –

    Pith/arXiv arXiv 2025

  42. [44]

    Narovlansky and H

    V. Narovlansky and H. Verlinde,Double-scaled SYK and de Sitter holography,JHEP05 (2025) 032 [2310.16994]

    Pith/arXiv arXiv 2025

  43. [45]

    Verlinde and M

    H. Verlinde and M. Zhang,SYK correlators from 2D Liouville-de Sitter gravity,JHEP05 (2025) 053 [2402.02584]

    Pith/arXiv arXiv 2025

  44. [46]

    Lewkowycz, J

    A. Lewkowycz, J. Liu, E. Silverstein and G. Torroba,T Tand EE, with implications for (A)dS subregion encodings,JHEP04(2020) 152 [1909.13808]

    arXiv 2020

  45. [47]

    Coleman, E.A

    E. Coleman, E.A. Mazenc, V. Shyam, E. Silverstein, R.M. Soni, G. Torroba et al.,De Sitter microstates from T T+Λ 2 and the Hawking-Page transition,JHEP07(2022) 140 [2110.14670]

    Pith/arXiv arXiv 2022

  46. [48]

    Ryu and T

    S. Ryu and T. Takayanagi,Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett.96(2006) 181602 [hep-th/0603001]

    Pith/arXiv arXiv 2006

  47. [49]

    Ryu and T

    S. Ryu and T. Takayanagi,Aspects of Holographic Entanglement Entropy,JHEP08(2006) 045 [hep-th/0605073]

    Pith/arXiv arXiv 2006

  48. [50]

    Maldacena and L

    J. Maldacena and L. Susskind,Cool horizons for entangled black holes,Fortsch. Phys.61 (2013) 781 [1306.0533]

    Pith/arXiv arXiv 2013

  49. [51]

    Faulkner, M

    T. Faulkner, M. Guica, T. Hartman, R.C. Myers and M. Van Raamsdonk,Gravitation from Entanglement in Holographic CFTs,JHEP03(2014) 051 [1312.7856]

    Pith/arXiv arXiv 2014

  50. [52]

    Engelhardt and A.C

    N. Engelhardt and A.C. Wall,Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime,JHEP01(2015) 073 [1408.3203]

    Pith/arXiv arXiv 2015

  51. [53]

    Penington,Entanglement Wedge Reconstruction and the Information Paradox,JHEP09 (2020) 002 [1905.08255]

    G. Penington,Entanglement Wedge Reconstruction and the Information Paradox,JHEP09 (2020) 002 [1905.08255]

    Pith/arXiv arXiv 2020

  52. [54]

    Almheiri, T

    A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini,Replica Wormholes and the Entropy of Hawking Radiation,JHEP05(2020) 013 [1911.12333]

    Pith/arXiv arXiv 2020

  53. [55]

    Penington, S.H

    G. Penington, S.H. Shenker, D. Stanford and Z. Yang,Replica wormholes and the black hole interior,JHEP03(2022) 205 [1911.11977]

    Pith/arXiv arXiv 2022

  54. [56]

    Susskind,Computational Complexity and Black Hole Horizons,Fortsch

    L. Susskind,Computational Complexity and Black Hole Horizons,Fortsch. Phys.64(2016) 24 [1403.5695]

    Pith/arXiv arXiv 2016

  55. [57]

    Belin, R.C

    A. Belin, R.C. Myers, S.-M. Ruan, G. S´ arosi and A.J. Speranza,Does Complexity Equal Anything?,Phys. Rev. Lett.128(2022) 081602 [2111.02429]

    Pith/arXiv arXiv 2022

  56. [58]

    Belin, R.C

    A. Belin, R.C. Myers, S.-M. Ruan, G. S´ arosi and A.J. Speranza,Complexity equals anything II,JHEP01(2023) 154 [2210.09647]

    Pith/arXiv arXiv 2023

  57. [59]

    Hawking, J.M

    S. Hawking, J.M. Maldacena and A. Strominger,de Sitter entropy, quantum entanglement and AdS / CFT,JHEP05(2001) 001 [hep-th/0002145]

    Pith/arXiv arXiv 2001

  58. [60]

    Narayan,On extremal surfaces and de Sitter entropy,Phys

    K. Narayan,On extremal surfaces and de Sitter entropy,Phys. Lett. B779(2018) 214 [1711.01107]

    Pith/arXiv arXiv 2018

  59. [61]

    Narayan,de Sitter space, extremal surfaces, and time entanglement,Phys

    K. Narayan,de Sitter space, extremal surfaces, and time entanglement,Phys. Rev. D107 (2023) 126004 [2210.12963]

    Pith/arXiv arXiv 2023

  60. [62]

    Narayan,Further remarks on de Sitter space, extremal surfaces, and time entanglement, Phys

    K. Narayan,Further remarks on de Sitter space, extremal surfaces, and time entanglement, Phys. Rev. D109(2024) 086009 [2310.00320]

    Pith/arXiv arXiv 2024

  61. [63]

    Shaghoulian and L

    E. Shaghoulian and L. Susskind,Entanglement in De Sitter space,JHEP08(2022) 198 [2201.03603]. – 42 –

    Pith/arXiv arXiv 2022

  62. [64]

    Shaghoulian,The central dogma and cosmological horizons,JHEP01(2022) 132 [2110.13210]

    E. Shaghoulian,The central dogma and cosmological horizons,JHEP01(2022) 132 [2110.13210]

    Pith/arXiv arXiv 2022

  63. [65]

    K. Doi, J. Harper, A. Mollabashi, T. Takayanagi and Y. Taki,Pseudoentropy in dS/CFT and Timelike Entanglement Entropy,Phys. Rev. Lett.130(2023) 031601 [2210.09457]

    Pith/arXiv arXiv 2023

  64. [66]

    K. Doi, J. Harper, A. Mollabashi, T. Takayanagi and Y. Taki,Timelike entanglement entropy,JHEP05(2023) 052 [2302.11695]

    Pith/arXiv arXiv 2023

  65. [67]

    Heller, F

    M.P. Heller, F. Ori and A. Serantes,Geometric Interpretation of Timelike Entanglement Entropy,Phys. Rev. Lett.134(2025) 131601 [2408.15752]

    Pith/arXiv arXiv 2025

  66. [68]

    Li, Z.-Q

    Z. Li, Z.-Q. Xiao and R.-Q. Yang,On holographic time-like entanglement entropy,JHEP04 (2023) 004 [2211.14883]

    Pith/arXiv arXiv 2023

  67. [69]

    Chu and H

    C.-S. Chu and H. Parihar,Time-like entanglement entropy in AdS/BCFT,JHEP06(2023) 173 [2304.10907]

    Pith/arXiv arXiv 2023

  68. [70]

    Jiang, P

    X. Jiang, P. Wang, H. Wu and H. Yang,Timelike entanglement entropy and TT¯ deformation,Phys. Rev. D108(2023) 046004 [2302.13872]

    Pith/arXiv arXiv 2023

  69. [71]

    Afrasiar, J.K

    M. Afrasiar, J.K. Basak and D. Giataganas,Holographic timelike entanglement entropy in non-relativistic theories,JHEP05(2025) 205 [2411.18514]

    Pith/arXiv arXiv 2025

  70. [72]

    Afrasiar, J.K

    M. Afrasiar, J.K. Basak and D. Giataganas,Timelike entanglement entropy and phase transitions in non-conformal theories,JHEP07(2024) 243 [2404.01393]

    Pith/arXiv arXiv 2024

  71. [73]

    Chu and H

    C.-S. Chu and H. Parihar,Timelike entanglement entropy with gravitational anomalies, JHEP08(2025) 038 [2504.19694]

    Pith/arXiv arXiv 2025

  72. [74]

    Nunez and D

    C. Nunez and D. Roychowdhury,Timelike entanglement entropy: A top-down approach, Phys. Rev. D112(2025) 026030 [2505.20388]

    Pith/arXiv arXiv 2025

  73. [75]

    Narayan,de Sitter extremal surfaces, time contours, complexifications and pseudo-entropies,2604.00108

    K. Narayan,de Sitter extremal surfaces, time contours, complexifications and pseudo-entropies,2604.00108

    arXiv

  74. [76]

    Jiang, P

    X. Jiang, P. Wang, H. Wu and H. Yang,Timelike entanglement entropy in dS 3/CFT2, JHEP08(2023) 216 [2304.10376]

    Pith/arXiv arXiv 2023

  75. [77]

    A. Das, S. Sachdeva and D. Sarkar,Bulk reconstruction using timelike entanglement in (A)dS,Phys. Rev. D109(2024) 066007 [2312.16056]

    Pith/arXiv arXiv 2024

  76. [78]

    Li, M.-H

    G.-Y. Li, M.-H. Xiao, S. He and J.-R. Sun,Entanglement first law for timelike entanglement entropy and linearized Einstein’s equation,JHEP06(2026) 144 [2511.17098]

    Pith/arXiv arXiv 2026

  77. [79]

    Prihadi, M.A.R

    H.L. Prihadi, M.A.R. Al-Faritsi, R.R. Firdaus, F. Khairunnisa, Y.P. Sarwono and F.P. Zen, Holographic timelike entanglement and subregion complexity with scalar hair,JHEP04 (2026) 174 [2601.18310]

    Pith/arXiv arXiv 2026

  78. [80]

    Pal and Y

    J. Pal and Y. Shi,Holographic Timelike Entanglement and Subregion Complexity in Localized AdS3*S3*T4 Black Holes,2607.11641

    Pith/arXiv arXiv

  79. [81]

    Li and R.-Q

    Z.-H. Li and R.-Q. Yang,Linear Growth of Holographic Time-like Entanglement Entropy and Kasner exponents,2606.21079

    Pith/arXiv arXiv

  80. [82]

    Parihar and S.R

    S. Parihar and S.R. Roy,Holographic complexity of conformal fields in global de Sitter spacetime. arXiv:2604.21408 [hep-th],2604.21408

    Pith/arXiv arXiv

Showing first 80 references.