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REVIEW 3 major objections 4 minor 39 references

Unitary evolution and cosmic acceleration in Loop Quantum Cosmology

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Pith's one-line read For flat FLRW loop quantum cosmology with weight $\lambda$ between the Euclidean and Lorentzian constraints, unitary evolution holds for all real $\lambda$: directly for $\lambda\le0$, and through a $U(1)$ family of self-adjoint…

desk verdict Clean math for the soluble LQC model, but the observational claims outrun the proof. read the letter →

arxiv 2412.07916 v2 pith:X3IVTTD7 submitted 2024-12-10 gr-qc

Omar Gallegos , Tonatiuh Matos , Hugo A. Morales-Técotl This is my paper · ORCID
classification gr-qc MSC 81Q1083C4583F05 PACS 04.60.Pp98.80.Qc
keywords loopquantumcosmologyself-adjointextensionsdeficiencyindicesunitaryevolutioncosmologicalconstantFLRWuniverseLorentziantermHamiltonianconstraint
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

This paper asks whether the flat FLRW loop quantum cosmology model with an arbitrary weight parameter $\lambda$ between the Euclidean and Lorentzian parts of the Hamiltonian constraint admits unitary evolution. It shows that for $\lambda\le0$ the gravitational constraint operator is essentially self-adjoint, so unitary evolution is automatic, while for $\lambda>0$ the operator is not self-adjoint but possesses a one-parameter family of self-adjoint extensions labeled by $\beta\in[0,\pi)$. The paper implements the extensions in an explicit propagator. This matters because the positive weight needed to reproduce the observed cosmological constant in earlier work must be unitary to be physical.

What carries the argument

The load-bearing object is the gravitational constraint operator in its soluble differential form, obtained from the LQC difference operator by a change of representation: $\hat\Theta_{\lambda,g}=12\pi G\gamma^2[\lambda(\sin b\,\partial_b)^2-\xi_\lambda(\sin 2b\,\partial_b)^2]$, with $\xi_\lambda=(1+\lambda\gamma^2)/(4\gamma^2)$. Further $x$-transformations reduce it to $-\partial_x^2$ for $\lambda\le0$ and to a sign-changing second-order operator $-12\pi G\,\mathrm{sgn}(|x|-x_0)\partial_x^2$ for $\lambda>0$. The sign change is the source of the nontrivial gluing condition at $x=\pm\pi/2$, and the freedom in that condition is precisely the $U(1)$ family of self-adjoint extensions parameterized by $\beta$.

What would settle it

Compute the deficiency indices of the genuine difference operator in Eq. (20) on the discrete LQC Hilbert space for a positive weight such as $\lambda=1$. If the indices are not $n_+=n_-=1$, or if the required gluing condition depends on the lattice spacing in a way that the differential proxy in Eq. (24) does not capture, then the claimed $U(1)$ classification belongs to the soluble approximation rather than to the physical model.

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Extended reading notes

Core claim

The central claim, on the paper's own terms, is that the soluble flat FLRW LQC Hamiltonian with weight $\lambda$ has a complete unitary dynamics for every real $\lambda$. Using the deficiency index method on the differential form of the constraint, the operator is essentially self-adjoint for $\lambda\le0$; for $\lambda>0$ the deficiency indices are $(1,1)$, giving a $U(1)$ family of self-adjoint extensions. The extensions are encoded as a gluing condition for the wave function at $x=\pm\pi/2$, parametrized by $\beta$, and are built into the propagator kernel $K_{\lambda>0,\beta}$. Known cases sit inside this family: $\lambda=-1/\gamma^2$ and $\lambda=0$ are essentially self-adjoint, while $\lambda=1$ requires extensions.

Load-bearing premise

The entire self-adjointness analysis is done on a smooth differential operator that replaces the actual discrete difference operator of the model, and the paper does not prove that this replacement preserves the deficiency indices.

Editorial extensions

If this is right

  • Every real weight $\lambda$ yields a unitary evolution for the flat FLRW loop quantum cosmology model, either essentially self-adjoint for $\lambda\le0$ or through a chosen self-adjoint extension for $\lambda>0$.
  • The observationally relevant weight $\lambda_0\sim10^{-122}$ that reproduces the measured cosmological constant now has a unitary implementation, but only after fixing one of the extensions $\beta$.
  • The propagator depends on the extension for all $\beta\neq\pi/4$, so the choice of extension affects the evolution of generic states, not just the spectrum.
  • Earlier results are recovered as special cases: $\lambda=-1/\gamma^2$ and $\lambda=0$ need no extensions, while $\lambda=1$ requires the extension family already found in the emergent de Sitter studies.
  • For $\lambda>0$ the gravitational spectrum remains continuous for every extension, so a bouncing cosmology with an accelerating late-time phase is compatible with unitarity for any $\beta$.

Reading between the lines

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

  • If the differential approximation is faithful, the model with the observed positive weight still has no unique quantum dynamics: the extension label $\beta$ must be fixed by an additional physical criterion, such as a boundary condition at the bounce or a semiclassical selection rule.
  • The $\pi(1-\tan\beta)$ shift in the propagator pole suggests that different extensions alter the interference of late-time wave packets; a semiclassical analysis of the effective dynamics could turn the extension label into a testable prediction.
  • Applying the same deficiency index calculation directly to the original difference operator, or extending it to spatially curved models, would settle whether the $U(1)$ classification is a property of loop quantum cosmology or of its soluble differential approximation.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The manuscript studies a flat FLRW loop quantum cosmology model whose Hamiltonian constraint contains Euclidean and Lorentzian terms weighted by a parameter λ. Working in what it calls the soluble form of the model, it passes from the LQC difference operator to a differential operator and then to an x-representation. For λ≤0 the resulting operator is claimed to be essentially self-adjoint, while for λ>0 the paper finds a one-parameter family of self-adjoint extensions labelled by β∈[0,π), implements the extensions in eigenfunctions and a propagator, and concludes that positive values of λ, including the observationally motivated λ0∼10^{-122}, require self-adjoint extensions for unitary evolution.

Significance. If the differential proxy is a faithful representation of the polymer difference operator, the paper provides a useful and explicit extension classification that unifies previous results for λ=−1/γ², λ=0 and λ=1, and it supplies a concrete propagator implementation of the self-adjoint extensions. The deficiency-index computations are transparent, the boundary-condition translation is explicit, and the propagator formulas are welcome additions to the LQC literature. The significance is conditional, however, because the central classification is proven for the soluble continuum differential operator and the manuscript does not establish that the original discrete LQC operator has the same deficiency indices.

major comments (3)
  1. [Sec. III, Eqs. (20) and (24)] The decisive step from the LQC difference operator (20) on the superselected sector to the differential operator (24) is asserted rather than proved. Equation (22) defines the transform with a factor 1/√|v|, and Eq. (23) gives only the formal action of the basic operators, but no domain, measure, or unitary equivalence is specified that would allow one to conclude that the deficiency indices of (20) equal those of (26) and (34). The text itself says the soluble representation is adopted 'for convenience' in Sec. III and calls the model 'soluble' in Sec. VI, which confirms that the analysis is for the differential proxy. Hence the abstract's statement that for positive λ self-adjoint extensions 'are required' and are 'mandatory' to encompass observations is strictly a claim about the soluble model unless a bridge theorem is supplied. The authors should either prove that the transformation preserves the self-adjointness classification, or explicitly restrict the central claims, the abstract, and the observational conclusion to the soluble model.
  2. [Sec. V, Eqs. (53)--(55)] The closed-form propagator (55) is obtained by inserting the large-eigenvalue approximation (54) for ϕ(β,k) into the integral (53), which is integrated over all k>0. For k near zero the correction O(e^{−kπ}) is not small, so (55) is not an exact evaluation of (53). This does not affect the extension classification, but it does affect the paper's propagator claim. The authors should either evaluate (53) with the full transcendental relation (48) or state clearly that (55) is a large-k approximation and discuss its regime of validity.
  3. [Sec. IV, Eq. (45)] The reparametrization from α to β is described by 'β∈[0,π), tan(β)≥0, which is bijective in Uα'. These conditions are mutually inconsistent: tanβ is not nonnegative throughout [0,π), and the right-hand side of Eq. (45) is not sign-definite as α varies. Since β labels the extensions and enters the eigenfunctions (47) and the propagator (55), the parametrization should be stated consistently, for example β∈[0,π) with tanβ taking all real values, or with an explicitly restricted range that matches the sign of the right-hand side.
minor comments (4)
  1. [Throughout] There are several typographical errors, including 'wight' in Sec. VI, 'Lorentizian' in Sec. III, and 'deficit' in Sec. IV; these should be corrected.
  2. [Sec. V, paragraph after Eq. (55)] The text says extensions with β∈(0,π/4)∪(π/4,π] affect evolution, but earlier β is defined in [0,π); the endpoint convention should be made consistent.
  3. [Sec. IV, Eq. (41)] The domain D is written as L²(R_Bohr,dµ_H), but after the x-representation the relevant Hilbert space is L²(R,dx); please clarify which representation is being used.
  4. [Sec. V, Eq. (51)] The normalization constant ζ=4/√|k| and the measure in the k-integral are not specified, which makes it difficult for the reader to verify the closed form (52); a brief derivation or measure statement would help.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the self-adjointness classification is a genuine operator analysis of the stated soluble model, with no fitted parameters and no load-bearing self-citations.

full rationale

The paper's central claim—that the gravitational constraint is essentially self-adjoint for lambda <= 0 and admits a U(1) family of self-adjoint extensions for lambda > 0—is derived within the paper from the explicitly stated differential operators (26) and (34) using von Neumann's deficiency-indices method. The deficiency solutions are exhibited (Eqs. (36) and (38)), so the classification is not assumed from the input; it is a computed property of the operator. The observed cosmological constant enters only through the externally fixed value lambda_0 ~ 10^-122 taken from Ref. [22]; the paper does not fit alpha or beta to observations, nor does it use unitarity to reverse-engineer lambda or beta. Thus no fitted-input-called-prediction pattern occurs. The cited prior works [13,23,24] are by independent authors (Pawlowski/Ashtekar, Assanioussi et al.), and the relevant calculations are reproduced in the text rather than merely cited; these are not self-citations of Gallegos, Matos, and Morales-Tecotl, so no self-citation is load-bearing. The paper explicitly labels the model as 'soluble' and the b-representation as adopted 'for convenience,' and in Sec. VI states 'We adopted for simplicity the soluble model LQC,' so the gap between the discrete difference operator (20) and the differential proxy (24) is an openly declared modeling limitation rather than a hidden circular reduction. The large-k approximation used in deriving the propagator (55) is an approximation inside an integral, not a circular identification of a prediction with an input. Overall, the derivation chain is self-contained for the model it analyzes; the main caveat is scope, not circularity.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new physical entities. Its theoretical input is the weight parameter lambda, which is carried over from previous work, and the extension parameter beta, which is a mathematical label for the U(1) family. The main burden is the unproven equivalence between the differential proxy and the actual LQC difference operator.

free parameters (2)
  • lambda = positive, specifically lambda0 ~ 10^-122 in [22]
    The weight parameter between Euclidean and Lorentzian terms is free in the model. The paper does not fit it, but imports the observed-cosmological-constant value from Zhang, Long, and Ma. The positive regime is where self-adjoint extensions are needed.
  • beta = free in [0, pi)
    The self-adjoint extension label is undetermined by the model. The paper notes beta = pi/4 gives no effect on the propagator at the displayed level, but does not derive or select a physical value.
assumptions (4)
  • domain assumption The Thiemann regularization plus a 'convenient approximation' yields the soluble differential operator of Eq. (24).
    This is the core modeling step: the discrete difference operator on the Bohr Hilbert space is replaced by a differential operator on L^2(R). The paper does not prove that this proxy preserves the self-adjointness properties (deficiency indices) of the original LQC operator.
  • domain assumption The scalar field phi serves as a global clock and the physical Hilbert space is selected by the positive part of the gravitational operator.
    The paper assumes the standard 'deparametrized' LQC treatment where phi is a clock and the square root of the constraint generates evolution in phi. This is a standard LQC procedure but is an interpretive assumption about time.
  • standard math The Barbero-Immirzi parameter gamma = 0.2375 is taken from black hole entropy calculations.
    This is a standard input in LQC, not derived in the paper. It is used in the definitions of the variables and in the harmonic approximation.
  • standard math The deficiency index theorem (von Neumann) is applied to the formal differential expression on the whole real line with the given domain.
    This is standard functional analysis, used without proof.

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Cite this review

Pith. "Pith review of Unitary evolution and cosmic acceleration in Loop Quantum Cosmology." pith.science (2026). https://pith.science/paper/X3IVTTD7

@misc{pith2026241207916,
  author       = {Pith},
  title        = {Pith review of: Unitary evolution and cosmic acceleration in Loop Quantum Cosmology},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X3IVTTD7}},
  note         = {Machine review of arXiv:2412.07916}
}
read the original abstract

Loop quantum cosmology was shown to interpolate between de Sitter and FLRW Universe phases through a bounce by including Euclidean and Lorentzian terms of the Hamiltonian constraint with weight one -that corresponding to classical General Relativity. Unitary evolution required self-adjoint extensions of the constraint and a Planckian cosmological constant was obtained. Independent work took a positive weight to get a cosmological constant with the observed value, without considering unitarity. In this work we address the unitary evolution of the model for arbitrary weight. For non positive weight parameter unitary holds but for positive values self-adjoint extensions are required. To encompass observations the extensions here provided are mandatory. These are implemented in a propagator. Finally, we discuss our results and perspectives.

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Works this paper leans on

39 extracted references · 21 canonical work pages

  1. [1]

    Aghanim et al

    N. Aghanim et al. (Planck), Planck 2018 re- sults. I. Overview and the cosmological legacy of Planck, Astron. Astrophys. 641, A1 (2020), arXiv:1807.06205 [astro-ph.CO]

    arXiv 2020

  2. [2]

    Aghanim et al

    N. Aghanim et al. (Planck), Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys. 641, A6 (2020), [Erra- tum: Astron.Astrophys. 652, C4 (2021)], arXiv:1807.06209 [astro-ph.CO]

    arXiv 2020

  3. [3]

    A. G. Adame et al. (DESI), DESI 2024 VI: Cosmological Constraints from the Measurements of Baryon Acoustic Oscillations, (2024), arXiv:2404.03002 [astro-ph.CO]

    arXiv 2024

  4. [4]

    Weinberg, The Cosmological Constant Problem, Rev

    S. Weinberg, The Cosmological Constant Problem, Rev. Mod. Phys. 61, 1 (1989). 7

    work page 1989

  5. [5]

    Rovelli, Quantum gravity , Cambridge Monographs on Mathematical Physics (Univ

    C. Rovelli, Quantum gravity , Cambridge Monographs on Mathematical Physics (Univ. Pr., Cambridge, UK, 2004)

    2004

  6. [6]

    Thiemann, Modern Canonical Quantum General Relativity , Cambridge Monographs on Mathematical Physics (Cam- bridge University Press, 2007)

    T. Thiemann, Modern Canonical Quantum General Relativity , Cambridge Monographs on Mathematical Physics (Cam- bridge University Press, 2007)

    2007

  7. [7]

    H. A. Morales-T´ ecotl, S. Rastgoo, and J. C. Ru- elas, Effective dynamics of the Schwarzschild black hole interior with inverse triad cor- rections, Annals Phys. 426, 168401 (2021), arXiv:1806.05795 [gr-qc]

    work page Pith review arXiv 2021

  8. [8]

    Bojowald, Absence of singularity in loop quan- tum cosmology, Phys

    M. Bojowald, Absence of singularity in loop quan- tum cosmology, Phys. Rev. Lett. 86, 5227 (2001), arXiv:gr-qc/0102069

    arXiv 2001

Show all 39 references
  1. [9]

    Bojowald, Loop quantum cosmology, Living Rev

    M. Bojowald, Loop quantum cosmology, Living Rev. Rel. 8, 11 (2005), arXiv:gr-qc/0601085

    2005 arXiv

  2. [10]

    Ashtekar, M

    A. Ashtekar, M. Bojowald, and J. Lewandowski, Mathematical structure of loop quantum cos- mology, Adv. Theor. Math. Phys. 7, 233 (2003), arXiv:gr-qc/0304074

    2003 arXiv

  3. [11]

    Ashtekar, T

    A. Ashtekar, T. Pawlowski, P. Singh, and K. Vander- sloot, Loop quantum cosmology of k=1 FRW models, Phys. Rev. D 75, 024035 (2007), arXiv:gr-qc/0612104

    2007 arXiv

  4. [12]

    Ashtekar, T

    A. Ashtekar, T. Pawlowski, and P. Singh, Quan- tum Nature of the Big Bang: Improved dynamics, Phys. Rev. D 74, 084003 (2006), arXiv:gr-qc/0607039

    2006 arXiv

  5. [13]

    Pawlowski and A

    T. Pawlowski and A. Ashtekar, Positive cos- mological constant in loop quantum cos- mology, Phys. Rev. D 85, 064001 (2012), arXiv:1112.0360 [gr-qc]

    2012 arXiv

  6. [14]

    Kaminski and J

    W. Kaminski and J. Lewandowski, The Flat FRW model in LQC: The Self- adjointness, Class. Quant. Grav. 25, 035001 (2008), arXiv:0709.3120 [gr-qc]

    2008 arXiv

  7. [15]

    Kaminski and T

    W. Kaminski and T. Pawlowski, The LQC evolution operator of FRW universe with positive cosmo- logical constant, Phys. Rev. D 81, 024014 (2010), arXiv:0912.0162 [gr-qc]

    2010 arXiv

  8. [16]

    Bentivegna and T

    E. Bentivegna and T. Pawlowski, Anti-deSitter univers e dynamics in LQC, Phys. Rev. D 77, 124025 (2008), arXiv:0803.4446 [gr-qc]

    2008 arXiv

  9. [17]

    Martin-Benito, G

    M. Martin-Benito, G. A. M. Marugan, and J. Olmedo, Further Improvements in the Understanding of Isotropic Loop Quantum Cosmology, Phys. Rev. D 80, 104015 (2009), arXiv:0909.2829 [gr-qc]

    2009 arXiv

  10. [18]

    Ashtekar, A

    A. Ashtekar, A. Corichi, and P. Singh, Ro- bustness of key features of loop quantum cosmology, Phys. Rev. D 77, 024046 (2008), arXiv:0710.3565 [gr-qc]

    2008 arXiv

  11. [19]

    Szulc, W

    L. Szulc, W. Kaminski, and J. Lewandowski, Closed FRW model in Loop Quantum Cos- mology, Class. Quant. Grav. 24, 2621 (2007), arXiv:gr-qc/0612101

    2007 arXiv

  12. [20]

    Szulc, Open FRW model in Loop Quantum Cosmology, Class

    L. Szulc, Open FRW model in Loop Quantum Cosmology, Class. Quant. Grav. 24, 6191 (2007), arXiv:0707.1816 [gr-qc]

    2007 arXiv

  13. [21]

    J. Yang, Y. Ding, and Y. Ma, Alternative quantization of the Hamiltonian in loop quantum cosmology II: In- cluding the Lorentz term, Phys. Lett. B 682, 1 (2009), arXiv:0904.4379 [gr-qc]

    2009 arXiv

  14. [22]

    Zhang, G

    X. Zhang, G. Long, and Y. Ma, Loop quantum gravity and cosmological con- stant, Phys. Lett. B 823, 136770 (2021), arXiv:2101.07527 [gr-qc]

    2021 arXiv

  15. [23]

    Assanioussi, A

    M. Assanioussi, A. Dapor, K. Liegener, and T. Paw/suppress lowski, Emergent de Sitter Epoch of the Quantum Cosmos from Loop Quantum Cos- mology, Phys. Rev. Lett. 121, 081303 (2018), arXiv:1801.00768 [gr-qc]

    2018 arXiv

  16. [24]

    Assanioussi, A

    M. Assanioussi, A. Dapor, K. Liegener, and T. Paw/suppress lowski, Emergent de Sitter epoch of the Loop Quantum Cosmos: a detailed analysis, Phys. Rev. D 100, 084003 (2019), arXiv:1906.05315 [gr-qc]

    2019 arXiv

  17. [25]

    J. Yang, C. Zhang, and X. Zhang, Alternative k=-1 loop quantum cosmology, Phys. Rev. D 107, 046012 (2023), arXiv:2212.05748 [gr-qc]

    2023 arXiv

  18. [26]

    K. A. Meissner, Black hole entropy in loop quan- tum gravity, Class. Quant. Grav. 21, 5245 (2004), arXiv:gr-qc/0407052

    2004 arXiv

  19. [27]

    Ghosh and P

    A. Ghosh and P. Mitra, A Bound on the log correction to the black hole area law, Phys. Rev. D 71, 027502 (2005), arXiv:gr-qc/0401070

    2005 arXiv

  20. [28]

    Rovelli, Black hole entropy from loop quantum gravity, Phys

    C. Rovelli, Black hole entropy from loop quantum gravity, Phys. Rev. Lett. 77, 3288 (1996), arXiv:gr-qc/9603063

    1996 arXiv

  21. [29]

    Ashtekar, J

    A. Ashtekar, J. Baez, A. Corichi, and K. Kras- nov, Quantum geometry and black hole entropy, Phys. Rev. Lett. 80, 904 (1998), arXiv:gr-qc/9710007

    1998 arXiv

  22. [30]

    Reed and B

    M. Reed and B. Simon, Methods of Modern Mathematical Physics. I. Functional analysis (1972)

    1972

  23. [31]

    Reed and B

    M. Reed and B. Simon, Methods of Modern Mathematical Physics. 2. Fourier Analysis, Self-adjointness (1975)

    1975

  24. [32]

    Simon, The classical moment problem as a self-adjoint finite difference operator, Adv

    B. Simon, The classical moment problem as a self-adjoint finite difference operator, Adv. Math , 82 (1998)

    1998

  25. [33]

    Kato, Perturbation Theory for Linear Operators (Springer, Berlin, Heidelberg, 1995)

    T. Kato, Perturbation Theory for Linear Operators (Springer, Berlin, Heidelberg, 1995)

    1995

  26. [34]

    Gitman, I

    D. Gitman, I. Tyutin, and B. Voronov, Self-adjoint Ex- tensions in Quantum Mechanics: General Theory and Applications to Schr¨ odinger and Dirac Equations with Singular Potentials (Birkh¨ auser, 2012)

    2012

  27. [35]

    Kowalczyk and T

    M. Kowalczyk and T. Paw/suppress lowski, Regulariza- tions and quantum dynamics in loop quan- tum cosmology, Phys. Rev. D 108, 086010 (2023), arXiv:2212.12527 [gr-qc]

    2023 arXiv

  28. [36]

    Ashtekar, M

    A. Ashtekar, M. Campiglia, and A. Henderson, Path Integrals and the WKB approximation in Loop Quantum Cosmology, Phys. Rev. D 82, 124043 (2010), arXiv:1011.1024 [gr-qc]

    2010 arXiv

  29. [37]

    Flores-Gonz´ alez, H

    E. Flores-Gonz´ alez, H. A. Morales-T´ ecotl, and J. D. Reyes, Propagators in Polymer Quan- tum Mechanics, Annals Phys. 336, 394 (2013), arXiv:1302.1906 [math-ph]

    2013 arXiv

  30. [38]

    L. Qin, G. Deng, and Y.-G. Ma, Path integrals and alternative effective dynamics in loop quan- tum cosmology, Commun. Theor. Phys. 57, 326 (2012), arXiv:1206.1131 [gr-qc]

    2012 arXiv

  31. [39]

    Huang, Y

    H. Huang, Y. Ma, and L. Qin, Path Inte- gral and Effective Hamiltonian in Loop Quan- tum Cosmology, Gen. Rel. Grav. 45, 1191 (2013), arXiv:1102.4755 [gr-qc]

    2013 arXiv

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