Pith. sign in

REVIEW 3 major objections 5 minor 1 cited by

Entanglement Entropy and Complexity of Multicomponent Universe from Holography

T0 review · 3 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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

Pith's one-line read The paper shows that a black brane with p-brane gas supports two-component FLRW universes, and derives holographic entanglement entropy and complexity whose leading time scalings are set by the dominant matter component — linear early, powe

desk verdict Two-component holographic cosmology with plausible scalings undone by a load-bearing RT area functional that does not match the stated metric. read the letter →

arxiv 2601.05628 v2 pith:FWM2OHHY submitted 2026-01-09 hep-th

Ritam Mahanta , Gopinath Guin , Souvik Paul , Sunandan Gangopadhyay This is my paper · ORCID
classification hep-th
keywords holographicentanglemententropycomplexitybraneworldcosmologyIsraeljunctionconditionRyu-Takayanagiformulap-branegasFLRWuniversetwo-componentmatter
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Most holographic cosmologies studied so far put a single matter component on the brane. This paper tries to move to a more realistic two-component universe by adding p-brane gas to the bulk AdS black brane, yielding a lapse function whose two terms act as radiation plus dark matter (p=1) or radiation plus exotic matter (p=2). Using the second Israel junction condition to turn brane motion into cosmic time, and the Ryu-Takayanagi and Complexity=Volume 2.0 prescriptions on a circular region, the paper derives the time-dependent holographic entanglement entropy and subregion complexity. The leading scalings are linear in cosmic time during the radiation era, τ^{4/3} entropy and τ^2 complexity in the radiation-dark-matter era, and τ^2 entropy and τ^3 complexity in the radiation-exotic-matter era — always the scaling of the dominant component. A sympathetic reader would care because these mixed-component results are a step toward holographic information measures that match the known thermal history of our universe.

What carries the argument

The central object is the blackening (lapse) function f(r)=1−ρ_p/r^{4−p}−m/r^4 of a five-dimensional AdS black brane, where the Schwarzschild-like −m/r^4 piece supplies radiation and the p-brane gas piece supplies a second component. The second Israel junction condition (K_{MN}=−(T/3)h_{MN} plus its trace) converts the brane's radial motion into a scale factor and yields differential equations for r(τ); these are solved perturbatively in the early and late times. With the brane position in hand, the Ryu-Takayanagi formula computes entanglement entropy from the minimal area of a circular region, and the Complexity=Volume (2.0) conjecture computes subregion complexity from the volume underneat

What would settle it

Recompute the induced metric determinant on a static slice of ds²=(R²/z²)(−f dt²+dz²/f+dx²); if the resulting area functional is √(1+z'^2/f) rather than √(1+z'^2)/√f, the Euler-Lagrange equations (4.15)-(4.16) and the perturbative area integrals would change, so the printed quantitative expressions would fail for the stated geometry. Alternatively, solve the full RT surface numerically for f=1−ρ/r^3−m/r^4 and check whether the late-time entropy exponent is exactly 4/3.

Watch

Extended reading notes

Core claim

The paper's central claim is that the black brane metric with a p-brane gas, f(r)=1−ρ_p/r^{4−p}−m/r^4, is enough to describe a four-dimensional FLRW universe on the brane that contains two coexisting matter sources: the m/r^4 term radiates, and the ρ_p term acts as dark matter for p=1 and exotic matter for p=2. Solving the second Israel junction condition with this lapse gives the brane position z̄(τ) in early and late time regimes. Substituting those positions into the perturbatively solved Ryu-Takayanagi surface and into the volume under that surface yields the time dependence: for radiation plus dark matter, entanglement entropy goes as τ early and τ^{4/3} late, while complexity goes as τ

Load-bearing premise

The computation assumes the area functional in eq. (4.14) is the correct RT integrand for the stated black brane; if the printed radical placement is wrong, the entropy and complexity are computed for a different geometry and only the qualitative scalings survive.

Editorial extensions

If this is right

  • For radiation and dark matter, holographic entanglement entropy grows as τ in the radiation era and as τ^{4/3} in the matter era; since the physical length grows as τ^{2/3}, this satisfies the expected area law.
  • For radiation and exotic matter, the same entropy grows as τ in the radiation era and as τ^2 later, again an area law in the physical length (L∝τ).
  • Volume complexity grows as τ^2 for radiation plus dark matter and as τ^3 for radiation plus exotic matter in the late era, matching volume-law growth in physical size.
  • Subleading terms in both quantities contain the density of the subdominant component, so the holographic information measures are sensitive to both matter components, not just the dominant one.
  • The early-time linear growth of both measures coincides with radiation domination and matches the transition from radiation to matter seen in the universe's thermal history.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • One could turn the result around: the late-time growth exponent of holographic entanglement entropy (e.g., 4/3 vs 2) is set by the dominant component's equation of state, so in a larger family of p-brane universes it could serve as a holographic probe of matter content.
  • The same two-term lapse construction should extend to other coexisting combinations, such as adding curvature or a cosmological constant as further terms, producing richer power laws; the paper only works out p=1 and p=2.
  • A direct check against an exact, non-perturbative integration of the RT surface for these lapses would test whether the leading exponents survive beyond the small-density expansion used here.
  • For cosmology, the cleanest observable signature would be a late-time crossover in entanglement entropy growth when the universe transitions from radiation to matter domination, analogous to the shift in the scale factor's power law.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. This paper extends earlier single-component holographic braneworld calculations to a universe with two coexisting matter components. The setup is a five-dimensional AdS black brane in Randall-Sundrum-type braneworld cosmology, with the bulk modified by a uniform gas of p-branes. The resulting blackening factor is f(r)=1 - m/r^4 - rho_p/r^{4-p}. Using the second Israel junction condition at the critical brane tension, the authors derive the time-dependent brane position for p=1 (radiation plus matter) and p=2 (radiation plus exotic matter), obtaining z ~ const - const*tau at early times and z ~ tau^{-2/3} or z ~ tau^{-1} at late times. They then use the Ryu-Takayanagi formula and the CV-2.0 volume conjecture to compute the holographic entanglement entropy and subregion complexity for a spherical region on the brane, obtaining S_EE ~ tau and C_V ~ tau at early times, and S_EE ~ tau^{4/3}, C_V ~ tau^2 for radiation+matter or S_EE ~ tau^2, C_V ~ tau^3 for radiation+exotic matter at late times. The paper argues that these scalings are consistent with radiation domination in the early universe and matter/exotic domination at late times.

Significance. If the computations were correct, the paper would provide a useful multi-component generalization of previous holographic entanglement/complexity results for FLRW braneworlds. The leading time scalings are physically well motivated and are robust: they follow from the brane-position power laws and the leading AdS area/volume terms, so the qualitative statement that the dominant component controls the leading scaling of both information measures is attractive and likely survives a corrected derivation. The paper also has the virtue of presenting explicit junction-condition derivations and perturbative solutions. However, in the present form the central quantitative results are not reliably derived from the stated bulk geometry because of the area and volume integrand inconsistencies detailed below. The manuscript also frames the two-component Friedmann behavior as an output when it is structurally an input of the model. These issues are local and fixable in principle, so the correct level of revision is major rather than rejection.

major comments (3)
  1. [Sec. 4.2, Eq. (4.14); Sec. 4.3, Eq. (4.30)] The area functional as displayed is not the induced area of a static surface in the bulk metric (3.8). With z=1/r and t=const, z=z(u), the induced metric is ds_ind^2 = R^2 z^{-2}[(1 + z'^2/f) du^2 + u^2 dOmega_2^2], so the RT area integrand is u^2 z^{-3} sqrt(1 + z'^2/f). Equations (4.14) and (4.30) print u^2 z^{-3} sqrt(1+z'^2)/sqrt(f), which agrees with the correct integrand only when f=1. Since f_{m,r}=1-tilde r z^3 - tilde m z^4 and f_{r,e}=1-tilde chi z^2 - tilde m z^4, all perturbative profiles z1,z2,z3, the areas (4.26), (4.41), and hence the HEE expressions (4.28)-(4.29), (4.43)-(4.44) are computed for a different, unstated geometry. If the intended radical in the original is actually sqrt(1 + z'^2/f), that convention must be stated explicitly; the displayed Euler-Lagrange equations (4.15)-(4.16) are not the EL equations for the induced metric, so the derivation as written remain
  2. [Sec. 5.2, Eq. (5.9); Sec. 5.3, Eq. (5.14)] The volume integrand used for CV-2.0 complexity is also inconsistent with the metric (3.8). For a static slice, the four-dimensional spatial volume element is R^4 z^{-4} f^{-1/2} dz d^3x, not R^4 z^{-4} sqrt(f). Equations (5.9) and (5.14) write dz/z^4 sqrt(f(z(u))), which is the inverse of the correct factor. The unperturbed f=1 limit is unaffected, but the tilde-m, tilde-r, tilde-chi corrections in (5.10)-(5.11) and (5.15)-(5.16), and consequently the early/late complexity expressions (5.12)-(5.13) and (5.17)-(5.18), do not follow from the stated geometry. The leading late-time volume-law scalings (tau^2 and tau^3) are dominated by the brane-position power laws and the leading AdS term, so they may survive, but the subleading terms and coefficients are unsupported.
  3. [Sec. 3, Eqs. (3.19)-(3.20); Sec. 6] The claim that the model 'supports a universe with two-component matter sources' and that the results are 'consistent with the thermal history' is, as it stands, partly circular. The Friedmann-type ODE (dr/dtau)^2 = tilde m/r^2 + tilde r/r for p=1, and (dr/dtau)^2 = tilde m/r^2 + chi for p=2, is obtained by choosing the p-brane gas source and the critical tension; the radiation and matter/exotic terms are put into the bulk metric (3.13) by construction. The paper does not derive this Friedmann behavior from the holographic setup; it uses it as an input to find the brane position. The new content is the subsequent computation of entanglement entropy and complexity in these backgrounds, not the derivation of the two-component cosmological history. The abstract and conclusion should be reframed accordingly.
minor comments (5)
  1. [Eq. (3.19)] The notation in the last term is ambiguous: 'r/(R^2 r)' should presumably read tilde r/r with tilde r = rho_p/R^6. Please define all rescaled parameters consistently.
  2. [Sec. 4.1, text after Eq. (4.7)] The statement 'inside the subsystem, on the brane, l^2 + \bar z^2 is always much less than u^2' is reversed; since 0≤u≤l, one has l^2+\bar z^2 ≥ u^2. This likely should be 'much greater than' or refer to the late-time limit.
  3. [Sec. 2 and throughout] There are numerous typos and spelling errors: 'Platini' for 'Palatini', 'sclaes' for 'scales', 'time-dependnet' for 'time-dependent', 'presciption' for 'prescription', etc. A careful proofread is needed.
  4. [Sec. 4.1, Eq. (4.11)] The notation r_0 is introduced without definition; earlier the initial brane position is r_i. Please use one symbol.
  5. [Sec. 1] The paper itself notes in the Introduction that RT is only a leading-order approximation to HRT for time-dependent backgrounds. Since all entropy results are obtained using static RT surfaces, this caveat should be repeated in the Conclusion so that the reader does not overinterpret the exact time-dependent scalings.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the derivation is self-contained given the stated bulk action, though the RT area functional in Eq. (4.14) has a separate technical inconsistency.

full rationale

The claimed derivation chain is not circular. The bulk action (3.1) with the p-brane gas determines the lapse function f(r) in Eq. (3.13); the Israel junction condition in Eq. (2.37) then yields the brane-position ODE (3.19)-(3.20) containing the same matter/radiation parameters. The RT area and CV-2.0 volume are subsequently evaluated on a profile z(u) with the boundary condition z(l)=zbar(τ). No entanglement-entropy or complexity datum is used to fix m, r-tilde, or chi-tilde, and the early/late time scalings are consequences of integrating the ODE and of evaluating the area/volume functionals, not assumptions inserted into the target quantities. The only self-citation, [2], supplies the perturbative solution technique; it is not a load-bearing uniqueness claim, and the perturbative method is a standard, checkable expansion. The two-component cosmology is indeed encoded in the bulk construction by design, but that is model input, not a circular prediction. The more serious issue is a technical error: for metric (3.8), the induced area integrand is sqrt(1+z'^2/f), not sqrt(1+z'^2)/sqrt(f), so the quantitative EE/complexity expressions are not derived from the stated geometry. That undermines correctness but is an internal inconsistency, not a self-referential reduction, and therefore does not raise the circularity score above the minor self-citation level.

Assumptions & free parameters 5 free parameters · 9 assumptions · 0 invented entities

No new entities are postulated: the black brane, p-brane gas, and the 'exotic matter' fluid are all inherited from [1,2,84] (the p=2 exotic component is a placeholder for a non-standard fluid, not a new particle or field). The model carries two matter-sector inputs (m, rho_p/chi) plus a tuned tension and an integration constant, none fit to observation. The load-bearing assumptions are the radiation/matter dictionary, the leading-order validity of RT, and the unstated perturbative regime.

free parameters (5)
  • m (ADM mass of the bulk black brane; m_tilde = m/R^8)
    Section 3, eqs. (3.12)-(3.13). Governs the radiation component and the early-time brane dynamics (sqrt(m_tilde) appears in eqs. (3.23)-(3.24) and in all early-time HEE/HSC results). Input of the model, not fit to data.
  • rho_p (p-brane gas density; rho_tilde = rho_p/R^6 for p=1)
    Section 3, eq. (3.13). Controls the late-time matter-dominated evolution z ~ tau^{-2/3} and the tau^{4/3} entropy scaling. Input of the model, not fit to data.
  • chi (exotic matter density; chi_tilde = chi/R^4 for p=2)
    Section 4.3, eq. (4.31). Constant-density term in the junction condition, giving z ~ tau^{-1} and the tau^2/tau^3 late-time scalings. No independent determination.
  • T (brane tension) = T_c = 6/R (critical value)
    Section 3, eqs. (3.19)-(3.20). Chosen to cancel the (T^2/36 - 1/R^2)r^2 vacuum term so matter terms survive; in the pure-AdS sector H = sqrt(T^2/36 - 1/R^2) is instead a free parameter.
  • r_i (initial brane position; z_i = 1/r_i)
    Integration constant of the junction-condition ODE (eqs. (3.23), (3.30), etc.). Appears in every final HEE/HSC expression (e.g., eqs. (4.28), (4.43), (5.12)); no observational determination.
assumptions (9)
  • domain assumption RS-II braneworld embedding with Z2 symmetry: K^+ = -K^- on the brane
    Section 2, eqs. (2.21)-(2.26). All junction-condition results and hence the brane Friedmann equation assume this mirror symmetry and the Israel thin-shell framework.
  • standard math AdS black brane ansatz ds^2 = r^2/R^2(-f dt^2 + dx^2) + R^2/(r^2 f) dr^2 solves the backreacted bulk EOM
    Section 3, eq. (3.8). Standard ansatz for a homogeneous 5D solution; assumes no bulk fields besides the p-brane gas and the negative cosmological constant.
  • ad hoc to paper p-branes are uniformly distributed in the transverse (4-p)-dimensional space, with constant number density n_p
    Section 3, eqs. (3.3)-(3.4). Needed so the p-brane worldvolume dimension matches the bulk dimension and the averaged stress tensor (3.6) takes the stated form; no derivation is given.
  • domain assumption p-brane gas stress tensor (3.6) with (p-1)/3 average spatial extension
    Section 3, eq. (3.6), taken from Park [84]. The identification of the resulting lapse terms with matter/exotic-matter densities inherits this prior result without re-derivation.
  • domain assumption Dictionary: Schwarzschild term m/r^4 ↔ radiation; p-brane term rho_p/r^{4-p} ↔ dark matter (p=1) / exotic matter (p=2) on the brane
    Section 3, eq. (3.13) and surrounding text; inherited from [1,2,84]. All physical conclusions about 'radiation early, matter late' depend on this dictionary, which is not independently established in this paper.
  • ad hoc to paper Critical brane tension choice T = T_c = 6/R in the matter-sector calculations
    Section 3, eqs. (3.19)-(3.20). Standard RS fine-tuning, but a free choice; in the pure-AdS case H = sqrt(T^2/36 - 1/R^2) is an equally free parameter.
  • domain assumption RT formula applied to the time-dependent brane is valid in the UV leading order
    Sections 4 (intro) and 6. The paper acknowledges HRT is the correct prescription for a non-static background and uses RT because it 'matches the results of the HRT formalism in the leading order' — an admitted approximation, not a derivation.
  • ad hoc to paper Perturbative truncation: drop O(m^2), O(rho^2), O(m rho), O(chi^2) corrections to the EL equations and areas
    Sections 4.2-4.3, eqs. (4.15)-(4.17) and (4.32)-(4.34). No bound is given on the actual expansion parameters m l^4, rho l^3, chi l^2, which grow with subsystem size l; subleading-time scalings are drawn from this truncated series.
  • domain assumption Complexity = Volume 2.0 conjecture (C = V/(8 pi R G)) is the correct holographic complexity prescription
    Section 5, eq. (5.2), citing Stanford-Susskind [109] and Alishahiha [110]. All complexity results depend on this unproven conjecture; the paper does not compare with CA or CV-1.0.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Entanglement Entropy and Complexity of Multicomponent Universe from Holography." pith.science (2026). https://pith.science/paper/FWM2OHHY

@misc{pith2026260105628,
  author       = {Pith},
  title        = {Pith review of: Entanglement Entropy and Complexity of Multicomponent Universe from Holography},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FWM2OHHY}},
  note         = {Machine review of arXiv:2601.05628}
}
abstract

Recent studies in \cite{Park:2020jio,Paul:2025gpk} have calculated various holographic information-theoretic quantities of the four-dimensional FLRW universe for different matter-dominated eras using the braneworld model of cosmology. These studies are done for a single matter component, which is a good toy model for understanding the entanglement properties of the universe. However, for a more realistic model, one should consider a scenario where our universe has coexisting matter components like radiation-dark matter or radiation-exotic matter, etc. In this work, we have presented a systematic way to study various holographic information-theoretic quantities, namely, entanglement entropy and complexity, of the FLRW universe in the presence of coexisting matter components. We have shown that the black brane geometry in the presence of $p$-brane gas indeed supports the existence of a universe with two-component matter sources. The second Israel junction condition, along with the Ryu-Takayanagi formula, is used to compute the time-dependent holographic entanglement entropy of the universe with coexisting radiation-dark matter and radiation-exotic matter. The expression of the time-dependent volume complexity is also evaluated in these scenarios. For both universes, these information-theoretic quantities show a clear radiation dependence in the early time and matter and exotic matter dominance in the late time, which is consistent with the thermal history of the universe \cite{WMAP:2010qai,WMAP:2010sfg,Planck:2014loa,Planck:2018vyg}.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. One-point holographic correlator in the expanding universe

    hep-th 2026-07 conditional novelty 5.0 of 10

    In the RS-II braneworld with p-brane gas, the holographic thermal one-point function of heavy operators inherits its time dependence from the brane motion: late-time power-law decays τ^{-Δ/2}, τ^{-2Δ/3}, τ^{-Δ} for ra...

Reference graph

Works this paper leans on

112 extracted references · 95 linked inside Pith · cited by 1 Pith paper

  1. [1]

    Park,Time Evolution of Entanglement Entropy in Holographic FLRW Cosmologies,Phys

    C. Park,Time Evolution of Entanglement Entropy in Holographic FLRW Cosmologies,Phys. Rev. D101(2020) 126006 [2004.08020]

    arXiv 2020

  2. [2]

    S. Paul, G. Guin and S. Gangopadhyay,Holographic entanglement entropy and complexity for the cosmological braneworld model,JHEP08(2025) 164 [2505.11553]

    arXiv 2025

  3. [3]

    Von Neumann,Mathematische grundlagen der quantenmechanik, vol

    J. Von Neumann,Mathematische grundlagen der quantenmechanik, vol. 38, Springer-Verlag (2013)

    2013

  4. [4]

    Calabrese and J.L

    P. Calabrese and J.L. Cardy,Entanglement entropy and quantum field theory,J. Stat. Mech. 0406(2004) P06002 [hep-th/0405152]

    arXiv 2004

  5. [5]

    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]

    arXiv 1998

  6. [6]

    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]

    arXiv 1998

  7. [7]

    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]

    arXiv 1998

  8. [8]

    S. Paul, S. Gangopadhyay and A. Saha,Gauss–Bonnet AdS planar and spherical black hole thermodynamics and holography,Class. Quant. Grav.41(2024) 235010 [2403.07543]

    arXiv 2024

Show all 112 references
  1. [9]

    Gregory, R

    R. Gregory, R. Whisker, K. Beckwith and C. Done,Observing braneworld black holes,JCAP 10(2004) 013 [hep-th/0406252]

    2004 arXiv

  2. [10]

    Tanaka,Classical black hole evaporation in Randall-Sundrum infinite brane world,Prog

    T. Tanaka,Classical black hole evaporation in Randall-Sundrum infinite brane world,Prog. Theor. Phys. Suppl.148(2003) 307 [gr-qc/0203082]. – 29 –

    2003 arXiv

  3. [11]

    Emparan, A

    R. Emparan, A. Fabbri and N. Kaloper,Quantum black holes as holograms in AdS brane worlds,JHEP08(2002) 043 [hep-th/0206155]

    2002 arXiv

  4. [12]

    Bak,Dual of big-bang and big-crunch,Phys

    D. Bak,Dual of big-bang and big-crunch,Phys. Rev. D75(2007) 026003 [hep-th/0603080]

    2007 arXiv

  5. [13]

    Rovelli and T

    C. Rovelli and T. Thiemann,The Immirzi parameter in quantum general relativity,Phys. Rev. D57(1998) 1009 [gr-qc/9705059]

    1998 arXiv

  6. [14]

    Silva,A note on the AdS/CFT correspondence and the nature of spacetime in quantum gravity,Nucl

    C. Silva,A note on the AdS/CFT correspondence and the nature of spacetime in quantum gravity,Nucl. Phys. B998(2024) 116402 [2312.05260]

    2024 arXiv

  7. [15]

    Engelhardt and G.T

    N. Engelhardt and G.T. Horowitz,Holographic Consequences of a No Transmission Principle, Phys. Rev. D93(2016) 026005 [1509.07509]

    2016 arXiv

  8. [16]

    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]

    2006 arXiv

  9. [17]

    Ryu and T

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

    2006 arXiv

  10. [18]

    Nishioka, S

    T. Nishioka, S. Ryu and T. Takayanagi,Holographic Entanglement Entropy: An Overview,J. Phys. A42(2009) 504008 [0905.0932]

    2009 arXiv

  11. [19]

    Takayanagi and K

    T. Takayanagi and K. Umemoto,Entanglement of purification through holographic duality, Nature Phys.14(2018) 573 [1708.09393]

    2018 arXiv

  12. [20]

    P. Jain, V. Malvimat, S. Mondal and G. Sengupta,Holographic entanglement negativity conjecture for adjacent intervals inAdS3/CF T2,Phys. Lett. B793(2019) 104 [1707.08293]

    2019 arXiv

  13. [21]

    Ghasemi, A

    M. Ghasemi, A. Naseh and R. Pirmoradian,Odd entanglement entropy and logarithmic negativity for thermofield double states,JHEP10(2021) 128 [2106.15451]

    2021 arXiv

  14. [22]

    Jokela and A

    N. Jokela and A. Pönni,Notes on entanglement wedge cross sections,JHEP07(2019) 087 [1904.09582]

    2019 arXiv

  15. [23]

    Cáceres, J

    E. Cáceres, J. Couch, S. Eccles and W. Fischler,Holographic Purification Complexity,Phys. Rev. D99(2019) 086016 [1811.10650]

    2019 arXiv

  16. [24]

    Mishra and H

    R. Mishra and H. Singh,Entanglement asymmetry for boosted black branes and the bound,Int. J. Mod. Phys. A32(2017) 1750091 [1603.06058]

    2017 arXiv

  17. [25]

    Karar, R

    S. Karar, R. Mishra and S. Gangopadhyay,Holographic complexity of boosted black brane and Fisher information,Phys. Rev. D100(2019) 026006 [1904.13090]

    2019 arXiv

  18. [26]

    Chowdhury, A

    A.R. Chowdhury, A. Saha and S. Gangopadhyay,Entanglement wedge cross-section for noncommutative Yang-Mills theory,JHEP02(2022) 192 [2106.04562]

    2022 arXiv

  19. [27]

    Chowdhury Roy, A

    A. Chowdhury Roy, A. Saha and S. Gangopadhyay,Mixed state information theoretic measures in boosted black brane,Annals Phys.452(2023) 169270 [2204.08012]

    2023 arXiv

  20. [28]

    Nguyen, T

    P. Nguyen, T. Devakul, M.G. Halbasch, M.P. Zaletel and B. Swingle,Entanglement of purification: from spin chains to holography,JHEP01(2018) 098 [1709.07424]

    2018 arXiv

  21. [29]

    Kusuki, J

    Y. Kusuki, J. Kudler-Flam and S. Ryu,Derivation of holographic negativity in AdS3/CFT2, Phys. Rev. Lett.123(2019) 131603 [1907.07824]

    2019 arXiv

  22. [30]

    Chaturvedi, V

    P. Chaturvedi, V. Malvimat and G. Sengupta,Holographic Quantum Entanglement Negativity, JHEP05(2018) 172 [1609.06609]

    2018 arXiv

  23. [31]

    Saha and S

    A. Saha and S. Gangopadhyay,Holographic study of entanglement and complexity for mixed states,Phys. Rev. D103(2021) 086002 [2101.00887]

    2021 arXiv

  24. [32]

    Csaki, M

    C. Csaki, M. Reece and J. Terning,The AdS/QCD Correspondence: Still Undelivered,JHEP 05(2009) 067 [0811.3001]. – 30 –

    2009 arXiv

  25. [33]

    Andreev and V.I

    O. Andreev and V.I. Zakharov,Heavy-quark potentials and AdS/QCD,Phys. Rev. D74 (2006) 025023 [hep-ph/0604204]

    2006 arXiv

  26. [34]

    Karch, E

    A. Karch, E. Katz, D.T. Son and M.A. Stephanov,Linear confinement and AdS/QCD,Phys. Rev. D74(2006) 015005 [hep-ph/0602229]

    2006 arXiv

  27. [35]

    Kruczenski, L.A

    M. Kruczenski, L.A. Pando Zayas, J. Sonnenschein and D. Vaman,Regge trajectories for mesons in the holographic dual of large-N(c) QCD,JHEP06(2005) 046 [hep-th/0410035]

    2005 arXiv

  28. [36]

    Erlich, E

    J. Erlich, E. Katz, D.T. Son and M.A. Stephanov,QCD and a holographic model of hadrons, Phys. Rev. Lett.95(2005) 261602 [hep-ph/0501128]

    2005 arXiv

  29. [37]

    Panero,Thermodynamics of the QCD plasma and the large-N limit,Phys

    M. Panero,Thermodynamics of the QCD plasma and the large-N limit,Phys. Rev. Lett.103 (2009) 232001 [0907.3719]

    2009 arXiv

  30. [38]

    Hartnoll, C.P

    S.A. Hartnoll, C.P. Herzog and G.T. Horowitz,Holographic Superconductors,JHEP12(2008) 015 [0810.1563]

    2008 arXiv

  31. [39]

    Hartnoll, C.P

    S.A. Hartnoll, C.P. Herzog and G.T. Horowitz,Building a Holographic Superconductor,Phys. Rev. Lett.101(2008) 031601 [0803.3295]

    2008 arXiv

  32. [40]

    Herzog,Lectures on Holographic Superfluidity and Superconductivity,J

    C.P. Herzog,Lectures on Holographic Superfluidity and Superconductivity,J. Phys. A42 (2009) 343001 [0904.1975]

    2009 arXiv

  33. [41]

    Gangopadhyay and D

    S. Gangopadhyay and D. Roychowdhury,Analytic study of properties of holographic superconductors in Born-Infeld electrodynamics,JHEP05(2012) 002 [1201.6520]

    2012 arXiv

  34. [42]

    Li, R.-G

    H.-F. Li, R.-G. Cai and H.-Q. Zhang,Analytical Studies on Holographic Superconductors in Gauss-Bonnet Gravity,JHEP04(2011) 028 [1103.2833]

    2011 arXiv

  35. [43]

    Horowitz,Introduction to Holographic Superconductors,Lect

    G.T. Horowitz,Introduction to Holographic Superconductors,Lect. Notes Phys.828(2011) 313 [1002.1722]

    2011 arXiv

  36. [44]

    S. Paul, A. Roy Chowdhury, A. Saha and S. Gangopadhyay,Information theoretic measures for Lifshitz system,JHEP10(2024) 033 [2408.03670]

    2024 arXiv

  37. [45]

    Paul and S

    S. Paul and S. Gangopadhyay,Noncommutative p-wave holographic superconductors,Class. Quant. Grav.42(2025) 185016 [2502.08275]

    2025

  38. [46]

    McFadden and K

    P. McFadden and K. Skenderis,Holography for Cosmology,Phys. Rev. D81(2010) 021301 [0907.5542]

    2010 arXiv

  39. [47]

    Banks and W

    T. Banks and W. Fischler,An Holographic cosmology,hep-th/0111142

    arXiv

  40. [48]

    Bak and S.-J

    D. Bak and S.-J. Rey,Holographic principle and string cosmology,Class. Quant. Grav.17 (2000) L1 [hep-th/9811008]

    2000 arXiv

  41. [49]

    Hertog,Course 8 - holographic cosmology, inParticle Physics and Cosmology: The Fabric of Spacetime, F

    T. Hertog,Course 8 - holographic cosmology, inParticle Physics and Cosmology: The Fabric of Spacetime, F. Bernardeau, C. Grojean and J. Dalibard, eds., vol. 86 ofLes Houches, pp. 397–410, Elsevier (2007), DOI

    2007

  42. [50]

    S. Lepe, J. Saavedra and F. Pena,Holographic Cosmological Models on the Braneworld,Phys. Lett. B671(2009) 323 [0806.0981]

    2009 arXiv

  43. [51]

    Nastase and K

    H. Nastase and K. Skenderis,Holography for the very early Universe and the classic puzzles of Hot Big Bang cosmology,Phys. Rev. D101(2020) 021901 [1904.05821]

    2020 arXiv

  44. [52]

    Waddell,Bottom-up holographic models for cosmology,JHEP09(2022) 176 [2203.03096]

    C. Waddell,Bottom-up holographic models for cosmology,JHEP09(2022) 176 [2203.03096]

    2022 arXiv

  45. [53]

    Betzios and O

    P. Betzios and O. Papadoulaki,Brane Cosmology and the self-tuning of the cosmological constant in the presence of bulk black holes,Eur. Phys. J. C80(2020) 660 [2003.05767]

    2020 arXiv

  46. [54]

    Nielsen and I.L

    M.A. Nielsen and I.L. Chuang,Quantum computation and quantum information, Cambridge university press (2010). – 31 –

    2010

  47. [55]

    Baiguera, V

    S. Baiguera, V. Balasubramanian, P. Caputa, S. Chapman, J. Haferkamp, M.P. Heller et al., Quantum complexity in gravity, quantum field theory, and quantum information science,Phys. Rept.1159(2026) 1 [2503.10753]

    2026

  48. [56]

    Susskind,Computational Complexity and Black Hole Horizons,Fortsch

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

    2016 arXiv

  49. [57]

    Susskind,Three Lectures on Complexity and Black Holes, SpringerBriefs in Physics, Springer, 10, 2018, DOI [1810.11563]

    L. Susskind,Three Lectures on Complexity and Black Holes, SpringerBriefs in Physics, Springer, 10, 2018, DOI [1810.11563]

    2018 arXiv

  50. [58]

    Susskind,Entanglement is not enough,Fortsch

    L. Susskind,Entanglement is not enough,Fortsch. Phys.64(2016) 49 [1411.0690]

    2016 arXiv

  51. [59]

    Friedman,On the Curvature of space,Z

    A. Friedman,On the Curvature of space,Z. Phys.10(1922) 377

    1922

  52. [60]

    Friedmann,On the Possibility of a world with constant negative curvature of space,Z

    A. Friedmann,On the Possibility of a world with constant negative curvature of space,Z. Phys.21(1924) 326

    1924

  53. [61]

    Lemaitre,A Homogeneous Universe of Constant Mass and Increasing Radius accounting for the Radial Velocity of Extra-galactic Nebulæ,Mon

    G. Lemaitre,A Homogeneous Universe of Constant Mass and Increasing Radius accounting for the Radial Velocity of Extra-galactic Nebulæ,Mon. Not. Roy. Astron. Soc.91(1931) 483

    1931

  54. [62]

    Lemaitre,The expanding universe,Annales Soc

    G. Lemaitre,The expanding universe,Annales Soc. Sci. Bruxelles A53(1933) 51

    1933

  55. [63]

    Robertson,Kinematics and World-Structure

    H.P. Robertson,Kinematics and World-Structure. 2,Astrophys. J.83(1935) 187

    1935

  56. [64]

    Robertson,Kinematics and World-Structure,Astrophys

    H.P. Robertson,Kinematics and World-Structure,Astrophys. J.82(1935) 284

    1935

  57. [65]

    Walker,On Milne’s Theory of World-Structure,Proc

    A.G. Walker,On Milne’s Theory of World-Structure,Proc. Lond. Math. Soc. s2-42(1937) 90

    1937

  58. [66]

    Hubeny, M

    V.E. Hubeny, M. Rangamani and T. Takayanagi,A Covariant holographic entanglement entropy proposal,JHEP07(2007) 062 [0705.0016]

    2007 arXiv

  59. [67]

    Randall and R

    L. Randall and R. Sundrum,A Large mass hierarchy from a small extra dimension,Phys. Rev. Lett.83(1999) 3370 [hep-ph/9905221]

    1999 arXiv

  60. [68]

    Randall and R

    L. Randall and R. Sundrum,An Alternative to compactification,Phys. Rev. Lett.83(1999) 4690 [hep-th/9906064]

    1999 arXiv

  61. [69]

    Chamblin, S.W

    A. Chamblin, S.W. Hawking and H.S. Reall,Brane world black holes,Phys. Rev. D61(2000) 065007 [hep-th/9909205]

    2000 arXiv

  62. [70]

    Chamblin and H.S

    H.A. Chamblin and H.S. Reall,Dynamic dilatonic domain walls,Nucl. Phys. B562(1999) 133 [hep-th/9903225]

    1999 arXiv

  63. [71]

    Brax and C

    P. Brax and C. van de Bruck,Cosmology and brane worlds: A Review,Class. Quant. Grav.20 (2003) R201 [hep-th/0303095]

    2003 arXiv

  64. [72]

    P. Brax, C. van de Bruck and A.-C. Davis,Brane world cosmology,Rept. Prog. Phys.67 (2004) 2183 [hep-th/0404011]

    2004 arXiv

  65. [73]

    Flanagan, S.H.H

    E.E. Flanagan, S.H.H. Tye and I. Wasserman,A Cosmology of the brane world,Phys. Rev. D 62(2000) 024011 [hep-ph/9909373]

    2000 arXiv

  66. [74]

    Coley,Dynamics of brane world cosmological models,Phys

    A.A. Coley,Dynamics of brane world cosmological models,Phys. Rev. D66(2002) 023512 [hep-th/0110049]

    2002 arXiv

  67. [75]

    Kraus,Dynamics of anti-de Sitter domain walls,JHEP12(1999) 011 [hep-th/9910149]

    P. Kraus,Dynamics of anti-de Sitter domain walls,JHEP12(1999) 011 [hep-th/9910149]

    1999 arXiv

  68. [76]

    Park and S.-J

    C. Park and S.-J. Sin,Moving domain walls in AdS(5) and graceful exit from inflation,Phys. Lett. B485(2000) 239 [hep-th/0005013]

    2000 arXiv

  69. [77]

    Papantonopoulos and V

    E. Papantonopoulos and V. Zamarias,AdS/CFT correspondence and the reheating of the brane-universe,JHEP10(2004) 051 [hep-th/0408227]

    2004 arXiv

  70. [78]

    Okuyama and K.-i

    N. Okuyama and K.-i. Maeda,Domain wall dynamics in brane world and non-singular cosmological models,Phys. Rev. D70(2004) 064030 [hep-th/0405077]. – 32 –

    2004 arXiv

  71. [79]

    Yoshiguchi and K

    H. Yoshiguchi and K. Koyama,Bulk gravitational field and dark radiation on the brane in dilatonic brane world,Phys. Rev. D70(2004) 043513 [hep-th/0403097]

    2004 arXiv

  72. [80]

    Chang and I.G

    C.-J. Chang and I.G. Moss,Brane inflation with dark reheating,Phys. Lett. B607(2005) 214 [hep-ph/0411021]

    2005 arXiv

  73. [81]

    H. Geng, A. Karch, C. Perez-Pardavila, S. Raju, L. Randall, M. Riojas et al.,Information Transfer with a Gravitating Bath,SciPost Phys.10(2021) 103 [2012.04671]

    2021 arXiv

  74. [82]

    H. Geng, Y. Nomura and H.-Y. Sun,Information paradox and its resolution in de Sitter holography,Phys. Rev. D103(2021) 126004 [2103.07477]

    2021 arXiv

  75. [83]

    H. Geng, A. Karch, C. Perez-Pardavila, L. Randall, M. Riojas, S. Shashi et al.,Constraining braneworlds with entanglement entropy,SciPost Phys.15(2023) 199 [2306.15672]

    2023 arXiv

  76. [84]

    Park,Holographic time-dependent entanglement entropy in p-brane gas geometries,Phys

    C. Park,Holographic time-dependent entanglement entropy in p-brane gas geometries,Phys. Lett. B838(2023) 137672 [2106.05500]

    2023 arXiv

  77. [85]

    Iwashita, T

    Y. Iwashita, T. Kobayashi, T. Shiromizu and H. Yoshino,Holographic entanglement entropy of de Sitter braneworld,Phys. Rev. D74(2006) 064027 [hep-th/0606027]

    2006 arXiv

  78. [86]

    Kushihara, K

    K. Kushihara, K. Izumi and T. Shiromizu,Holographic entanglement entropy of a de Sitter braneworld with Lovelock terms,PTEP2021(2021) 043E01 [2102.12597]

    2021 arXiv

  79. [87]

    Feng, B.-M

    S. Feng, B.-M. Gu and F.-W. Shu,Quantum gravity induced entanglement of masses with extra dimensions,Eur. Phys. J. C84(2024) 59 [2307.11391]

    2024 arXiv

  80. [88]

    D. Basu, A. Chandra and H. Chourasiya,Reflected entropy and islands in a braneworld cosmology,2503.17819

    arXiv

  81. [89]

    Bhattacharya, A

    A. Bhattacharya, A. Bhattacharyya and A.K. Patra,Holographic complexity of Jackiw-Teitelboim gravity from Karch-Randall braneworld,JHEP07(2023) 060 [2304.09909]

    2023 arXiv

  82. [90]

    Narayan, H.K

    K. Narayan, H.K. Saini and G. Yadav,Cosmological singularities, holographic complexity and entanglement,JHEP07(2024) 125 [2404.00761]

    2024 arXiv

  83. [91]

    Jiang, X.-L

    X.-Y. Jiang, X.-L. Huang and S.-M. Wu,Cosmological entanglement of initial multipartite states,Eur. Phys. J. C85(2025) 851

    2025

  84. [92]

    Noumi, F

    T. Noumi, F. Sano and Y.-k. Suzuki,Holographic entanglement entropy in the FLRW universe,JHEP08(2025) 115 [2504.10457]

    2025 arXiv

  85. [93]

    Giantsos and N

    V. Giantsos and N. Tetradis,Entanglement entropy in a four-dimensional cosmological background,Phys. Lett. B833(2022) 137331 [2203.06699]

    2022 arXiv

  86. [94]

    Giataganas and N

    D. Giataganas and N. Tetradis,Entanglement entropy in FRW backgrounds,Phys. Lett. B 820(2021) 136493 [2105.12614]

    2021 arXiv

  87. [95]

    Carrillo-Gonzalez, K

    M. Carrillo-Gonzalez, K. Hinterbichler, J. Stokes and M. Trodden,Holographic 2-Point Functions in the Pseudo-Conformal Universe,Phys. Rev. D102(2020) 126009 [2009.07289]

    2020 arXiv

  88. [96]

    Engelhardt, T

    N. Engelhardt, T. Hertog and G.T. Horowitz,Holographic Signatures of Cosmological Singularities,Phys. Rev. Lett.113(2014) 121602 [1404.2309]

    2014 arXiv

  89. [97]

    Marcori and T.S

    O.H. Marcori and T.S. Pereira,Two-point Correlation Functions in Inhomogeneous and Anisotropic Cosmologies,JCAP02(2017) 032 [1612.01994]

    2017 arXiv

  90. [98]

    Chatterjee, S.P

    S. Chatterjee, S.P. Chowdhury, S. Mukherji and Y.K. Srivastava,Nonvacuum AdS cosmology and comments on gauge theory correlator,Phys. Rev. D95(2017) 046011 [1608.08401]

    2017 arXiv

  91. [99]

    Banerjee, S

    S. Banerjee, S. Bhowmick, S. Chatterjee and S. Mukherji,A note on AdS cosmology and gauge theory correlator,JHEP06(2015) 043 [1501.06317]

    2015 arXiv

  92. [100]

    C. Park, H. Kim and K. Cho,Correlation functions in expanding universes,Modern Physics Letters A0(0) 2550233 [https://doi.org/10.1142/S0217732325502335]. – 33 – [101]WMAPcollaboration,Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Sky Maps, Systematic Err...

    2011 arXiv

  93. [103]

    Gibbons and S.W

    G.W. Gibbons and S.W. Hawking,Action Integrals and Partition Functions in Quantum Gravity,Phys. Rev. D15(1977) 2752

    1977

  94. [104]

    Hawking and G.T

    S.W. Hawking and G.T. Horowitz,The Gravitational Hamiltonian, action, entropy and surface terms,Class. Quant. Grav.13(1996) 1487 [gr-qc/9501014]

    1996 arXiv

  95. [105]

    York, Jr.,Role of conformal three geometry in the dynamics of gravitation,Phys

    J.W. York, Jr.,Role of conformal three geometry in the dynamics of gravitation,Phys. Rev. Lett.28(1972) 1082

    1972

  96. [106]

    Deruelle, N

    N. Deruelle, N. Merino and R. Olea,Einstein-Gauss-Bonnet theory of gravity: The Gauss-Bonnet-Katz boundary term,Phys. Rev. D97(2018) 104009 [1709.06478]

    2018 arXiv

  97. [107]

    Israel,Singular hypersurfaces and thin shells in general relativity,Nuovo Cim

    W. Israel,Singular hypersurfaces and thin shells in general relativity,Nuovo Cim. B44S10 (1966) 1

    1966

  98. [108]

    Arnowitt, S

    R.L. Arnowitt, S. Deser and C.W. Misner,Dynamical Structure and Definition of Energy in General Relativity,Phys. Rev.116(1959) 1322

    1959

  99. [109]

    Stanford and L

    D. Stanford and L. Susskind,Complexity and Shock Wave Geometries,Phys. Rev. D90(2014) 126007 [1406.2678]

    2014 arXiv

  100. [110]

    Alishahiha,Holographic Complexity,Phys

    M. Alishahiha,Holographic Complexity,Phys. Rev. D92(2015) 126009 [1509.06614]

    2015 arXiv

  101. [111]

    Alishahiha, K

    M. Alishahiha, K. Babaei Velni and M.R. Mohammadi Mozaffar,Black hole subregion action and complexity,Phys. Rev. D99(2019) 126016 [1809.06031]

    2019 arXiv

  102. [112]

    Brown, D.A

    A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle and Y. Zhao,Complexity, action, and black holes,Phys. Rev. D93(2016) 086006 [1512.04993]

    2016 arXiv

  103. [113]

    Brown, D.A

    A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle and Y. Zhao,Holographic Complexity Equals Bulk Action?,Phys. Rev. Lett.116(2016) 191301 [1509.07876]

    2016 arXiv

  104. [114]

    K. Goto, H. Marrochio, R.C. Myers, L. Queimada and B. Yoshida,Holographic Complexity Equals Which Action?,JHEP02(2019) 160 [1901.00014]. – 34 –

    2019 arXiv

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.