arxiv: 2604.04601 · v1 · submitted 2026-04-06 · 🌀 gr-qc · hep-th

Recognition: no theorem link

Cosmic Inflation From Regular Black Holes

Kensuke Sueto , Riku Yoshimoto , Pablo A. Cano

Authors on Pith no claims yet

Pith reviewed 2026-05-10 20:07 UTC · model grok-4.3

classification 🌀 gr-qc hep-th

keywords braneworld cosmologyquasi-topological gravityregular black holescosmic inflationhigher-curvature termsmodified Friedmann equationsde Sitter phase

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The pith

A brane in a regular black hole bulk undergoes de Sitter inflation set only by the higher-curvature scale at small radii.

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

The paper shows that braneworlds in quasi-topological gravity with regular black hole solutions in the bulk enter an early de Sitter phase whose Hubble scale is fixed solely by the length √α of the higher-derivative corrections. Standard Einstein-gravity Friedmann dynamics are recovered once the scale factor becomes large. The duration of this inflationary stage is estimated universally by the ratio of the bulk black-hole radius to √α and is independent of the matter living on the brane. This supplies a geometric mechanism that automatically evades the trans-Planckian density problem while still matching late-time cosmology.

Core claim

In the small scale factor regime, the brane generically approaches a de Sitter phase characterized solely by the length scale √α of the higher-derivative terms, while the standard Einstein-gravity braneworld dynamics is recovered in the low-energy regime. A universal estimate for the number of e-folds is given in terms of the ratio r_g/√α between the black-hole scale and the new-physics scale; the inflationary regime is fully independent of the brane matter content.

What carries the argument

The Z2-symmetric junction conditions that relate the extrinsic curvature of an FLRW brane to the bulk curvature, yielding modified Friedmann equations whose small-a limit is de Sitter controlled by √α.

Load-bearing premise

The bulk must admit regular black-hole solutions and the brane must be Z2-symmetric with FLRW geometry moving inside a static spherically symmetric bulk metric.

What would settle it

A direct integration of the modified Friedmann equations for any explicit regular bulk solution (Dymnikova or Hayward) that fails to produce an early de Sitter phase whose duration scales with r_g/√α.

read the original abstract

We study braneworld cosmology in quasi-topological gravity (QTG) with an infinite tower of higher-curvature terms, focusing on the case in which the bulk admits regular black hole solutions. We derive the $\mathbb{Z}_2$-symmetric junction conditions for a FLRW brane moving in a static, spherically symmetric bulk geometry, and obtain the corresponding modified Friedmann equations for the scale factor. We prove that, in the small scale factor regime, the brane generically approaches a de Sitter phase characterized solely by the length scale $\sqrt{\alpha}$ of the higher-derivative terms, while the standard Einstein-gravity braneworld dynamics is recovered in the low-energy regime. We further provide a universal estimate for the number of e-folds of the de Sitter phase in terms of the ratio between the black hole scale and the scale of new physics $r_g/\sqrt{\alpha}$. The inflationary regime is fully independent of the brane matter content and hence avoids the problem of trans-Planckian matter densities. Numerical integrations for explicit regular bulk solutions (Dymnikova-like and Hayward black holes) confirm these estimates and illustrate how the bulk black hole sector controls the onset and termination of inflation. This framework leverages the powerful properties of QTGs, defined only in $D\ge 5$, to study consequences for a four-dimensional universe.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Regular black holes in a quasi-topological gravity bulk drive a matter-independent de Sitter phase on the brane at small scale factors, with e-folds set by r_g over sqrt(alpha).

read the letter

The central result is that Z2-symmetric junction conditions for an FLRW brane in a static spherical bulk yield modified Friedmann equations whose small-a limit is de Sitter, fixed only by the higher-curvature scale sqrt(alpha), while low-energy behavior recovers standard Einstein gravity. The number of e-folds is estimated directly from the ratio r_g/sqrt(alpha) and checked numerically on Dymnikova-like and Hayward bulk solutions. This is new as a concrete link between regular bulk black holes and early-universe expansion that avoids trans-Planckian densities on the brane. The derivation of the junction conditions and the analytic proof for the attractor are the parts that hold up cleanly. The numerics confirm the estimate for the two explicit cases without obvious fitting. The main soft spot is that everything rests on the existence of those regular black-hole solutions in the QTG bulk; the paper assumes they are available and works out the consequences, but does not explore how generic the solutions are or what happens if the bulk metric deviates slightly. The e-fold count still requires choosing the ratio of scales to be large enough, which is transparent but leaves alpha as a free parameter. No internal contradictions appear in the limit-taking or the independence from brane matter. This is for people working on braneworlds, higher-curvature gravity, or geometric models of inflation. A reader who already knows the junction-condition machinery will get the most out of the explicit attractor and the numerical illustrations. The work is grounded enough in derivations and checks to deserve a serious referee rather than a desk reject, though the review should focus on the stability of the de Sitter phase and the range of valid bulk solutions.

Referee Report

2 major / 2 minor

Summary. The paper studies braneworld cosmology in quasi-topological gravity (QTG) with an infinite tower of higher-curvature terms, assuming the bulk admits regular black hole solutions. It derives the Z2-symmetric junction conditions for a FLRW brane moving in a static spherically symmetric bulk and obtains the corresponding modified Friedmann equations. The central claim is an analytic proof that, in the small scale factor regime, the brane generically approaches a de Sitter phase determined solely by the higher-derivative length scale √α, independent of brane matter content, while recovering standard Einstein-gravity braneworld dynamics at low energies. A universal estimate for the number of e-folds is given in terms of the ratio r_g/√α, with numerical integrations for Dymnikova-like and Hayward black holes confirming the estimates and illustrating control by the bulk black hole sector.

Significance. If the derivations hold, this provides a mechanism for inflation sourced by the bulk geometry in higher-curvature gravity rather than brane matter, avoiding trans-Planckian densities. The independence from brane matter content and the control of the inflationary duration by the ratio r_g/√α constitute a potentially falsifiable prediction. The leverage of QTG properties (defined only in D≥5) for 4D cosmology, together with the analytic proof and explicit numerical confirmations for specific regular black holes, strengthens the result.

major comments (2)

[§3] §3 (derivation of modified Friedmann equations from Z2 junction conditions): the explicit form of the equations obtained after imposing the junction conditions should be displayed, as the small-a de Sitter attractor and its independence from matter terms follow directly from them; without the explicit expressions it is difficult to verify that the higher-derivative contributions dominate exactly as claimed.
[§4] §4 (small scale factor regime and e-fold estimate): the universal e-fold estimate is expressed in terms of r_g/√α, where r_g is set by the bulk black-hole solution; the paper should clarify whether r_g is fixed by the choice of regular black-hole metric or remains a free parameter, since this affects how predictive the duration of inflation is.

minor comments (2)

[Abstract] The abstract states that the inflationary regime is 'fully independent of the brane matter content'; this should be cross-referenced to the precise limit in which matter terms drop out of the modified Friedmann equation.
[Numerical section] Numerical integrations are mentioned for Dymnikova-like and Hayward black holes; a brief statement of the integration tolerances or convergence checks used would strengthen the confirmation of the analytic estimates.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript, positive assessment of the results, and constructive comments. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [§3] §3 (derivation of modified Friedmann equations from Z2 junction conditions): the explicit form of the equations obtained after imposing the junction conditions should be displayed, as the small-a de Sitter attractor and its independence from matter terms follow directly from them; without the explicit expressions it is difficult to verify that the higher-derivative contributions dominate exactly as claimed.

Authors: We agree that the explicit form of the modified Friedmann equations, obtained after imposing the Z2-symmetric junction conditions, should be displayed explicitly in §3. This will make it straightforward for readers to see how the higher-derivative terms dominate at small scale factor, yielding the de Sitter attractor independent of brane matter. In the revised manuscript we will insert these explicit expressions immediately following the derivation of the junction conditions. revision: yes
Referee: [§4] §4 (small scale factor regime and e-fold estimate): the universal e-fold estimate is expressed in terms of r_g/√α, where r_g is set by the bulk black-hole solution; the paper should clarify whether r_g is fixed by the choice of regular black-hole metric or remains a free parameter, since this affects how predictive the duration of inflation is.

Authors: We thank the referee for this clarification request. In the framework, r_g is fixed by the choice of regular black-hole metric in the bulk (e.g., the specific Dymnikova-like or Hayward solution), rather than being an arbitrary free parameter. Different regular black-hole solutions yield different values of r_g, which in turn control the duration of the inflationary phase via the ratio r_g/√α. We will revise §4 to state this explicitly, noting that the model’s predictivity for the number of e-folds is tied to the bulk geometry once a regular black-hole solution is selected, while still allowing a range of outcomes across viable bulk solutions. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives Z2-symmetric junction conditions from the braneworld embedding in a static spherically symmetric regular black hole bulk in quasi-topological gravity, obtains the modified Friedmann equations, and proves the small-scale-factor de Sitter attractor fixed solely by the input scale √α. The universal e-fold estimate follows directly from integrating those equations in terms of the ratio r_g/√α (both parameters of the setup, not fitted to inflation data). All steps are explicit consequences of the junction conditions and bulk geometry assumptions without self-definition, parameter fitting renamed as prediction, or load-bearing self-citation chains. The independence from brane matter content is shown by direct substitution into the derived equations. The framework is self-contained against external benchmarks such as the stated Z2 symmetry and regular BH solutions.

Axiom & Free-Parameter Ledger

2 free parameters · 3 axioms · 0 invented entities

The framework rests on standard assumptions of braneworld models plus the existence of regular black hole solutions in QTG; no new entities are postulated beyond the higher-curvature scale alpha.

free parameters (2)

alpha
Length scale of higher-derivative terms that sets the de Sitter Hubble rate in the small-scale regime.
r_g
Characteristic scale of the bulk regular black hole used to estimate the number of inflationary e-folds.

axioms (3)

domain assumption Z2 symmetry of the brane in the bulk
Invoked to derive the junction conditions for the FLRW brane.
domain assumption Bulk geometry is static and spherically symmetric
Required for the regular black hole solutions and the motion of the brane.
domain assumption Quasi-topological gravity admits an infinite tower of higher-curvature terms
Foundation of the bulk theory in D >= 5.

pith-pipeline@v0.9.0 · 5543 in / 1650 out tokens · 90635 ms · 2026-05-10T20:07:20.096515+00:00 · methodology

discussion (0)

Forward citations

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Reference graph

Works this paper leans on

67 extracted references · 56 canonical work pages · cited by 2 Pith papers · 4 internal anchors

[1]

Penrose,Gravitational collapse and space-time singularities,Phys

R. Penrose,Gravitational collapse and space-time singularities,Phys. Rev. Lett.14(1965) 57–59

1965
[2]

S. W. Hawking and R. Penrose,The Singularities of gravitational collapse and cosmology,Proc. Roy. Soc. Lond. A314(1970) 529–548

1970
[3]

Modified Gravity and Cosmology

T. Clifton, P. G. Ferreira, A. Padilla and C. Skordis,Modified Gravity and Cosmology,Phys. Rept.513(2012) 1–189, [1106.2476]

work page internal anchor Pith review arXiv 2012
[4]

Extended Theories of Gravity

S. Capozziello and M. De Laurentis,Extended Theories of Gravity,Phys. Rept.509(2011) 167–321, [1108.6266]

work page Pith review arXiv 2011
[5]

A. H. Guth,The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems,Phys. Rev. D23(1981) 347–356

1981
[6]

A. D. Linde,A New Inflationary Universe Scenario: A Possible Solution of the Horizon, Flatness, Homogeneity, Isotropy and Primordial Monopole Problems,Phys. Lett. B108(1982) 389–393

1982
[7]

Albrecht and P

A. Albrecht and P. J. Steinhardt,Cosmology for Grand Unified Theories with Radiatively Induced Symmetry Breaking,Phys. Rev. Lett.48(1982) 1220–1223

1982
[8]

A. D. Linde,Chaotic Inflation,Phys. Lett. B129(1983) 177–181

1983
[9]

B. A. Bassett, S. Tsujikawa and D. Wands,Inflation dynamics and reheating,Rev. Mod. Phys. 78(2006) 537–589, [astro-ph/0507632]. – 19 –

work page Pith review arXiv 2006
[10]

A. D. Linde,Inflationary Cosmology,Lect. Notes Phys.738(2008) 1–54, [0705.0164]

work page Pith review arXiv 2008
[11]

A. A. Starobinsky,A New Type of Isotropic Cosmological Models Without Singularity,Phys. Lett. B91(1980) 99–102

1980
[12]

A. L. Berkin and K.-i. Maeda,Effects of R**3 and R box R terms on R**2 inflation,Phys. Lett. B245(1990) 348–354

1990
[13]

Biswas, A.S

T. Biswas, A. S. Koshelev, A. Mazumdar and S. Y. Vernov,Stable bounce and inflation in non-local higher derivative cosmology,JCAP08(2012) 024, [1206.6374]

work page arXiv 2012
[14]

Bonanno and A

A. Bonanno and A. Platania,Asymptotically safe inflation from quadratic gravity,Phys. Lett. B 750(2015) 638–642, [1507.03375]

work page arXiv 2015
[15]

A. S. Koshelev, L. Modesto, L. Rachwal and A. A. Starobinsky,Occurrence of exactR2 inflation in non-local UV-complete gravity,JHEP11(2016) 067, [1604.03127]

work page arXiv 2016
[16]

A. R. Romero Castellanos, F. Sobreira, I. L. Shapiro and A. A. Starobinsky,On higher derivative corrections to theR+R 2 inflationary model,JCAP12(2018) 007, [1810.07787]

work page arXiv 2018
[17]

Carloni, J

S. Carloni, J. L. Rosa and J. P. S. Lemos,Cosmology off(R,□R)gravity,Phys. Rev. D99 (2019) 104001, [1808.07316]

work page arXiv 2019
[18]

P. A. Cano, K. Fransen and T. Hertog,Novel higher-curvature variations ofR2 inflation,Phys. Rev. D103(2021) 103531, [2011.13933]

work page arXiv 2021
[19]

Bianchi and M

E. Bianchi and M. Gamonal,Precision predictions of Starobinsky inflation with self-consistent Weyl-squared corrections,Phys. Rev. D112(2025) 124006, [2506.10081]

work page arXiv 2025
[20]

Bueno and P

P. Bueno and P. A. Cano,Einsteinian cubic gravity,Phys. Rev. D94(2016) 104005, [1607.06463]

work page arXiv 2016
[21]

R. A. Hennigar, D. Kubizňák and R. B. Mann,Generalized quasitopological gravity,Phys. Rev. D95(2017) 104042, [1703.01631]

work page arXiv 2017
[22]

Bueno and P

P. Bueno and P. A. Cano,On black holes in higher-derivative gravities,Class. Quant. Grav.34 (2017) 175008, [1703.04625]

work page arXiv 2017
[23]

Ahmed, R

J. Ahmed, R. A. Hennigar, R. B. Mann and M. Mir,Quintessential Quartic Quasi-topological Quartet,JHEP05(2017) 134, [1703.11007]

work page arXiv 2017
[24]

Bueno, P

P. Bueno, P. A. Cano and R. A. Hennigar,(Generalized) quasi-topological gravities at all orders, Class. Quant. Grav.37(2020) 015002, [1909.07983]

work page arXiv 2020
[25]

Bueno, P

P. Bueno, P. A. Cano, R. A. Hennigar, M. Lu and J. Moreno,Generalized quasi-topological gravities: the whole shebang,Class. Quant. Grav.40(2023) 015004, [2203.05589]

work page arXiv 2023
[26]

Bueno, P.A

P. Bueno, P. A. Cano, J. Moreno and Á. Murcia,All higher-curvature gravities as Generalized quasi-topological gravities,JHEP11(2019) 062, [1906.00987]

work page arXiv 2019
[27]

Arciniega, J

G. Arciniega, J. D. Edelstein and L. G. Jaime,Towards geometric inflation: the cubic case, Phys. Lett. B802(2020) 135272, [1810.08166]

work page arXiv 2020
[28]

Arciniega, P

G. Arciniega, P. Bueno, P. A. Cano, J. D. Edelstein, R. A. Hennigar and L. G. Jaime, Geometric Inflation,Phys. Lett. B802(2020) 135242, [1812.11187]

work page arXiv 2020
[29]

Cisterna, N

A. Cisterna, N. Grandi and J. Oliva,On four-dimensional Einsteinian gravity, quasitopological gravity, cosmology and black holes,Phys. Lett. B805(2020) 135435, [1811.06523]. – 20 –

work page arXiv 2020
[30]

Cosmological higher-curvature gravities

J. Moreno and A. J. Murcia,Cosmological higher-curvature gravities,Class. Quant. Grav.41 (2024) 135017, [2311.12104]

work page internal anchor Pith review Pith/arXiv arXiv 2024
[31]

J. D. Edelstein, D. Vázquez Rodríguez and A. Vilar López,Aspects of Geometric Inflation, JCAP12(2020) 040, [2006.10007]

work page arXiv 2020
[32]

Bueno, R

P. Bueno, R. A. Hennigar and Á. J. Murcia,Birkhoff implies Quasi-topological,2510.25823

work page arXiv
[33]

Bueno, P.A

P. Bueno, P. A. Cano and R. A. Hennigar,Regular black holes from pure gravity,Phys. Lett. B 861(2025) 139260, [2403.04827]

work page arXiv 2025
[34]

Bueno, P

P. Bueno, P. A. Cano, R. A. Hennigar and Á. J. Murcia,Dynamical Formation of Regular Black Holes,Phys. Rev. Lett.134(2025) 181401, [2412.02742]

work page arXiv 2025
[35]

Bueno, P

P. Bueno, P. A. Cano, R. A. Hennigar and Á. J. Murcia,Regular black holes from thin-shell collapse,Phys. Rev. D111(2025) 104009, [2412.02740]

work page arXiv 2025
[36]

Bueno, P

P. Bueno, P. A. Cano, R. A. Hennigar, Á. J. Murcia and A. Vicente-Cano,Regular black holes from Oppenheimer-Snyder collapse,Phys. Rev. D112(2025) 064039, [2505.09680]

work page arXiv 2025
[37]

R. A. Konoplya and A. Zhidenko,Dymnikova black hole from an infinite tower of higher-curvature corrections,Phys. Lett. B856(2024) 138945, [2404.09063]

work page arXiv 2024
[38]

Aguayo, L

M. Aguayo, L. Gajardo, N. Grandi, J. Moreno, J. Oliva and M. Reyes,Holographic explorations of regular black holes in pure gravity,JHEP09(2025) 030, [2505.11736]

work page arXiv 2025
[39]

P. G. S. Fernandes,Singularity resolution and inflation from an infinite tower of regularized curvature corrections,Phys. Rev. D112(2025) 084028, [2504.07692]

work page arXiv 2025
[40]

P. G. S. Fernandes,Regular BTZ black holes from an infinite tower of corrections,Phys. Lett. B 868(2025) 139772, [2504.08565]

work page arXiv 2025
[42]

J. P. Arbelaez,Grey-body factors of higher dimensional regular black holes in quasi-topological theories,2601.22340

work page arXiv
[43]

Coll´ eaux, S

A. Colléaux, S. Chinaglia and S. Zerbini,Nonpolynomial Lagrangian approach to regular black holes,Int. J. Mod. Phys. D27(2018) 1830002, [1712.03730]

work page arXiv 2018
[44]

Bueno, P

P. Bueno, P. A. Cano, R. A. Hennigar and Á. J. Murcia,Regular black hole formation in four-dimensional nonpolynomial gravities,Phys. Rev. D113(2026) 024019, [2509.19016]

work page arXiv 2026
[45]

Borissova and R

J. Borissova and R. Carballo-Rubio,Regular black holes from pure gravity in four dimensions, 2602.16773

work page arXiv
[46]

All $2D$ generalised dilaton theories from $d\geq 4$ gravities

J. Borissova,All2Dgeneralised dilaton theories fromd≥4gravities,2603.06786

work page internal anchor Pith review Pith/arXiv arXiv
[47]

Borissova and J

J. Borissova and J. Magueijo,Modified Friedmann equations and non-singular cosmologies in d= 4non-polynomial quasi-topological gravities,2603.17654

work page arXiv
[48]

A Large Mass Hierarchy from a Small Extra Dimension

L. Randall and R. Sundrum,A Large mass hierarchy from a small extra dimension,Phys. Rev. Lett.83(1999) 3370–3373, [hep-ph/9905221]

work page internal anchor Pith review arXiv 1999
[49]

An Alternative to Compactification

L. Randall and R. Sundrum,An Alternative to compactification,Phys. Rev. Lett.83(1999) 4690–4693, [hep-th/9906064]. – 21 –

work page Pith review arXiv 1999
[50]

Binetruy, C

P. Binetruy, C. Deffayet and D. Langlois,Nonconventional cosmology from a brane universe, Nucl. Phys. B565(2000) 269–287, [hep-th/9905012]

work page arXiv 2000
[51]

Ida,Brane world cosmology,JHEP09(2000) 014, [gr-qc/9912002]

D. Ida,Brane world cosmology,JHEP09(2000) 014, [gr-qc/9912002]

work page arXiv 2000
[52]

The Einstein Equations on the 3-Brane World

T. Shiromizu, K.-i. Maeda and M. Sasaki,The Einstein equation on the 3-brane world,Phys. Rev. D62(2000) 024012, [gr-qc/9910076]

work page Pith review arXiv 2000
[53]

Heterotic and Type I String Dynamics from Eleven Dimensions

P. Horava and E. Witten,Heterotic and Type I string dynamics from eleven dimensions,Nucl. Phys. B460(1996) 506–524, [hep-th/9510209]

work page Pith review arXiv 1996
[54]

G. W. Horndeski,Second-order scalar-tensor field equations in a four-dimensional space,Int. J. Theor. Phys.10(1974) 363–384

1974
[55]

Kunstatter, H

G. Kunstatter, H. Maeda and T. Taves,New 2D dilaton gravity for nonsingular black holes, Class. Quant. Grav.33(2016) 105005, [1509.06746]

work page arXiv 2016
[56]

Dymnikova,Vacuum nonsingular black hole,Gen

I. Dymnikova,Vacuum nonsingular black hole,Gen. Rel. Grav.24(1992) 235–242

1992
[57]

S. A. Hayward,Formation and evaporation of regular black holes,Phys. Rev. Lett.96(2006) 031103, [gr-qc/0506126]

work page Pith review arXiv 2006
[58]

R. A. Hennigar, D. Kubizňák, S. Murk and I. Soranidis,Thermodynamics of regular black holes in anti-de Sitter space,JHEP11(2025) 121, [2505.11623]

work page arXiv 2025
[59]

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
[60]

J. M. M. Senovilla,Junction Conditions for General Gravitational Theories,2603.04645

work page arXiv
[61]

Padilla and V

A. Padilla and V. Sivanesan,Boundary Terms and Junction Conditions for Generalized Scalar-Tensor Theories,JHEP08(2012) 122, [1206.1258]

work page arXiv 2012
[62]

Maartens and K

R. Maartens and K. Koyama,Brane-World Gravity,Living Rev. Rel.13(2010) 5, [1004.3962]

work page arXiv 2010
[63]

Kofinas, R

G. Kofinas, R. Maartens and E. Papantonopoulos,Brane cosmology with curvature corrections, JHEP10(2003) 066, [hep-th/0307138]

work page arXiv 2003
[64]

Davis, S

A.-C. Davis, S. C. Davis, W. B. Perkins and I. R. Vernon,Brane world phenomenology and the Z(2) symmetry,Phys. Lett. B504(2001) 254–261, [hep-ph/0008132]

work page arXiv 2001
[65]

L. A. Gergely,Generalized Friedmann branes,Phys. Rev. D68(2003) 124011, [gr-qc/0308072]

work page arXiv 2003
[66]

Padilla,Cosmic acceleration from asymmetric branes,Class

A. Padilla,Cosmic acceleration from asymmetric branes,Class. Quant. Grav.22(2005) 681–694, [hep-th/0406157]

work page arXiv 2005
[67]

Konya,Gauss-Bonnet brane-world cosmology without Z(2)-symmetry,Class

K. Konya,Gauss-Bonnet brane-world cosmology without Z(2)-symmetry,Class. Quant. Grav. 24(2007) 2761–2772, [gr-qc/0605119]

work page arXiv 2007
[68]

Yamauchi and M

D. Yamauchi and M. Sasaki,Brane world in arbitrary dimensions without Z(2) symmetry,Prog. Theor. Phys.118(2007) 245–256, [0705.2443]. – 22 –

work page arXiv 2007