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arxiv: 2505.14813 · v3 · submitted 2025-05-20 · 🌀 gr-qc

Stable initial conditions and analytical investigations of cosmological perturbations in a modified loop quantum cosmology

Rui Pan , Jamal Saeed , Anzhong Wang This is my paper

Pith reviewed 2026-05-22 13:35 UTC · model grok-4.3

classification 🌀 gr-qc
keywords cosmological perturbationsloop quantum cosmologyinitial conditionsBirrell-Davies stateuniform asymptotic approximationmLQC-Iparticle creationmode functions
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0 comments X

The pith

A stable initial state in the remote contracting phase minimizes particle creation and diagonalizes the Hamiltonian for perturbations in modified loop quantum cosmology.

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

The paper examines cosmological perturbations in the mLQC-I model of loop quantum cosmology. It applies the Birrell-Davies method to select an initial state during the remote contracting phase. This state proves stable and reduces particle production even when some modes remain outside the Hubble horizon and have not reached adiabatic vacuum conditions. The authors then use the uniform asymptotic approximation to derive first-order analytic solutions for the mode functions, expressed with Airy functions or cylindrical functions depending on wavenumber. These solutions fix the two integration constants directly from the chosen initial state.

Core claim

In the mLQC-I model, an initial state selected via the Birrell-Davies method in the remote contracting phase is stable, minimizes particle creations, and diagonalizes the Hamiltonian, even for modes outside the Hubble horizon and not yet in their adiabatic states. Using the uniform asymptotic approximation method, first-order approximate solutions of the mode function are obtained in terms of Airy functions or cylindrical functions of the first or second kind, with the two integration constants uniquely fixed by the initial state.

What carries the argument

The Birrell-Davies method for selecting the initial quantum state in the remote contracting phase, which minimizes particle creation and diagonalizes the Hamiltonian despite modified background dynamics.

If this is right

  • The initial state permits consistent quantization of perturbations across the quantum bounce without requiring all modes to be adiabatic at early times.
  • Analytic expressions for the mode functions enable direct computation of primordial power spectra without full numerical integration of the perturbation equations.
  • Particle creation is suppressed, which reduces the backreaction on the background and yields cleaner predictions for observable fluctuations.
  • The same initial-state choice and approximation technique can be applied to other modified loop quantum cosmology models with altered effective potentials.

Where Pith is reading between the lines

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

  • This approach may allow pre-bounce initial conditions to be set in a wider range of quantum cosmology models without assuming adiabatic vacuum states for all modes.
  • If the minimized particle creation persists, it could reduce uncertainties in matching perturbation spectra to late-time observations such as CMB anisotropies.
  • The method suggests a route to parameter-free predictions for the tensor-to-scalar ratio in bouncing cosmologies once the background is fixed.

Load-bearing premise

The Birrell-Davies method for selecting the initial state remains valid when applied to the modified background dynamics of the mLQC-I model where quantum corrections alter the Hubble parameter and the effective potential for perturbations.

What would settle it

Evolve the proposed initial state numerically through the bounce in mLQC-I and check whether the resulting power spectrum matches the analytic mode-function predictions to within the expected approximation error.

Figures

Figures reproduced from arXiv: 2505.14813 by Anzhong Wang, Jamal Saeed, Rui Pan.

Figure 1
Figure 1. Figure 1: FIG. 1: The Penrose diagram of the de Sitter spacetime. The spacelike horizontal line AB [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: The equation of state for the Starobinsky potential, [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: In this figure, we compare the relative magnitudes of the effective potential and [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: In this figure, we show the range near the bounce in which the magnitude of the [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: (a) Schematic plot of the quantity [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: The function [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: The function [PITH_FULL_IMAGE:figures/full_fig_p021_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: The numerical and UAA solutions with [PITH_FULL_IMAGE:figures/full_fig_p023_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: The numerical and UAA solutions with [PITH_FULL_IMAGE:figures/full_fig_p023_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: The numerical and UAA solutions with [PITH_FULL_IMAGE:figures/full_fig_p024_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13: The four regions, I - IV for [PITH_FULL_IMAGE:figures/full_fig_p025_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14: The function [PITH_FULL_IMAGE:figures/full_fig_p025_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15: The numerical and UAA solutions with [PITH_FULL_IMAGE:figures/full_fig_p030_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16: The numerical and UAA solutions with [PITH_FULL_IMAGE:figures/full_fig_p031_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17: The four regions, I - IV for [PITH_FULL_IMAGE:figures/full_fig_p032_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18: The numerical and UAA solutions with [PITH_FULL_IMAGE:figures/full_fig_p036_18.png] view at source ↗
read the original abstract

In this paper, we study cosmological perturbations in a modified theory of loop quantum cosmologies, the so-called mLQC-I model. Our purposes are two-fold: First, using a method developed by Birrell and Davies, we identify an initial state in the remote contracting phase, which turns out to be stable, minimize particle creations and diagonalize the Hamiltonian, despite the fact that at this time some modes may be still outside of the Hubble horizon and not in their adiabatic states. Second, using the uniform asymptotic approximation method, we obtain the first-order approximate solutions of the mode function in terms of either the Airy functions, or the first or second kind of cylindrical functions, depending on the values of the wavenumber. In each case, the mode function contains two integration constants, which are uniquely determined by the initial state.

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.

Referee Report

1 major / 2 minor

Summary. The manuscript examines cosmological perturbations in the mLQC-I model of modified loop quantum cosmology. Using the Birrell-Davies method, it identifies an initial state in the remote contracting phase claimed to be stable, to minimize particle creation, and to diagonalize the Hamiltonian even for some super-Hubble modes. It then applies the uniform asymptotic approximation to derive first-order analytic solutions for the mode functions in terms of Airy functions or cylindrical functions of the first or second kind, with the two integration constants in each case fixed by the chosen initial state.

Significance. If the initial-state construction is shown to be consistent with the modified dynamics, the work supplies concrete analytic expressions and a principled way to set initial conditions in mLQC models. Such results can reduce dependence on purely numerical evolution and facilitate later comparison with CMB observables. The explicit use of uniform asymptotic methods is a methodological strength.

major comments (1)
  1. [Section on initial-state construction (Birrell-Davies application)] The central claim that the Birrell-Davies state diagonalizes the Hamiltonian and minimizes particle creation is load-bearing for the first purpose of the paper. The mode equation in mLQC-I contains a quantum-corrected effective potential arising from the modified Hubble parameter. The manuscript applies the standard Birrell-Davies conditions directly; an explicit verification that the off-diagonal matrix elements of the Hamiltonian vanish (or remain negligible) under the actual mLQC-I equation of motion, especially for the super-Hubble modes highlighted in the abstract, is required.
minor comments (2)
  1. [Abstract and § on uniform asymptotic approximation] The abstract refers to 'the first or second kind of cylindrical functions' without specifying the choice criterion; the main text should state the precise range of wavenumbers for which each form is adopted.
  2. [Perturbation equation section] Notation for the modified effective potential should be introduced once and used consistently when writing the mode equation, so that readers can see exactly where the mLQC-I corrections enter.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thorough review and valuable feedback on our manuscript concerning cosmological perturbations in mLQC-I. We address the major comment regarding the initial-state construction in detail below. We are confident that the clarifications and additions will strengthen the paper.

read point-by-point responses

  1. Referee: [Section on initial-state construction (Birrell-Davies application)] The central claim that the Birrell-Davies state diagonalizes the Hamiltonian and minimizes particle creation is load-bearing for the first purpose of the paper. The mode equation in mLQC-I contains a quantum-corrected effective potential arising from the modified Hubble parameter. The manuscript applies the standard Birrell-Davies conditions directly; an explicit verification that the off-diagonal matrix elements of the Hamiltonian vanish (or remain negligible) under the actual mLQC-I equation of motion, especially for the super-Hubble modes highlighted in the abstract, is required.

    Authors: We appreciate the referee's emphasis on this key aspect. The Birrell-Davies method selects the vacuum state by requiring that the mode functions satisfy conditions which ensure the Hamiltonian is diagonalized at the initial time, thereby minimizing particle creation. Although the effective potential in mLQC-I is modified by the quantum corrections to the Hubble parameter, the general formalism remains applicable as the mode equation is still of the form of a time-dependent harmonic oscillator. In the remote contracting phase, the background evolution allows us to choose the initial time sufficiently early such that the state is well-defined. To provide the requested explicit verification, particularly for super-Hubble modes, we will include in the revised version a direct computation of the off-diagonal elements of the Hamiltonian using the mLQC-I equation of motion and demonstrate that they vanish at the initial time for the chosen constants. This will be added as a new paragraph or appendix subsection. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation applies external Birrell-Davies method to modified background without self-referential reduction

full rationale

The paper's central step selects an initial state in the remote contracting phase via the Birrell-Davies method, then claims this state is stable, minimizes particle creation, and diagonalizes the Hamiltonian for the mLQC-I perturbation equation. This method is a standard external reference (Birrell & Davies, 1982) rather than a self-defined quantity or prior result by the same authors. The subsequent uniform asymptotic approximation for mode functions (Airy or cylindrical) is likewise a standard mathematical technique whose integration constants are fixed by the chosen state. No load-bearing step reduces by construction to a fitted parameter, self-citation chain, or ansatz imported from the authors' own prior work. The derivation therefore remains self-contained against external benchmarks and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review prevents exhaustive enumeration; the ledger is therefore minimal and reflects only explicitly named external inputs.

axioms (1)
  • domain assumption The mLQC-I background evolution is governed by the modified Friedmann equation of the model.
    Invoked implicitly when applying the Birrell-Davies method to the contracting phase.

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Works this paper leans on

104 extracted references · 104 canonical work pages · 52 internal anchors

  1. [1]

    Junction Conditions Across Regions I - II To match the mode functionν II k given by Eq.(124) to the oneν I k given by Eq.(79), we first note the relation betweenyandξfory≪1 (orζ≪ −ζ 0) given by Eq.(122), from which we find that y−3/2 ≃x be 1 4 x2 ,b≡ − 1 2 ξ2 0,x=− √ 2ξ≫1. (129) Then, from Eq.(128) we find that νII k (y)≃ 2 9 1/4  ˜ak (2π)1/2 Γ 1 2 +b +...

    work page

  2. [2]

    Then, using the asymptotical behavior ofU(b,x)and ¯U(b,x)forx≫1 givne by Eq.(128), we find that νII k (y)≃ 1 (2g)1/4  ˜ake−F1(y) + ˜bk r 2 π Γ 1 2 −b eF1(y)  ,(y≫1)

    Junction Conditions Across Regions II - III Fory>y 1 from Eq.(123) we find that F1(y)≡ Z y y1 q g(y)dy≃ 1 4 x2 +bln √ 2x , (132) but now withx≡ √ 2ξ>0, while the parameterbis still defined as that in Eq.(129). Then, using the asymptotical behavior ofU(b,x)and ¯U(b,x)forx≫1 givne by Eq.(128), we find that νII k (y)≃ 1 (2g)1/4  ˜ake−F1(y) + ˜bk r 2 π Γ 1 ...

    work page

  3. [3]

    let us first notice that fory≫y 2, we have Z y y2 q −g(y)dy≃y−y 2 = 2 3 (−ζ)3/2

    Junction Conditions Across Regions III - IV Finally, let us consider the junction conditions across Regions III and IV . let us first notice that fory≫y 2, we have Z y y2 q −g(y)dy≃y−y 2 = 2 3 (−ζ)3/2 . (139) Then, from the asymptotical behavior of Ai(−x)and Bi(−x)forx≫1 given by Eq.(82), we find that νIII k (y)≃ 1 π1/2    h ˜αk cos ˆy2 + ˜βk sin ˆy2 i...

    work page

  4. [4]

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

    work page 1981

  5. [5]

    Planck 2018 results. X. Constraints on inflation

    Planck, Y. Akramiet al., Planck 2018 results. X. Constraints on inflation, Astron. Astrophys. 641, A10 (2020), arXiv:1807.06211

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  6. [6]

    The Atacama Cosmology Telescope: DR6 Power Spectra, Likelihoods and $\Lambda$CDM Parameters

    ACT, T. Louiset al., The Atacama Cosmology Telescope: DR6 Power Spectra, Likelihoods andΛCDM Parameters, (2025), arXiv:2503.14452

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  7. [7]

    R. H. Brandenberger and J. Martin, Trans-Planckian Issues for Inflationary Cosmology, Class. Quant. Grav.30, 113001 (2013), arXiv:1211.6753

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  8. [8]

    E. Silverstein, TASI lectures on cosmological observables and string theory, inTheoretical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings, pp. 545–606, 2017, arXiv:1606.03640

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  9. [9]

    Inflation and String Theory

    D. Baumann and L. McAllister,Inflation and String TheoryCambridge Monographs on Math- ematical Physics (Cambridge University Press, 2015), arXiv:1404.2601

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  10. [10]

    Eternal inflation and the initial singularity

    A. Borde and A. Vilenkin, Eternal inflation and the initial singularity, Phys. Rev. Lett.72, 3305 (1994), arXiv:gr-qc/9312022

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  11. [11]

    Inflationary spacetimes are not past-complete

    A. Borde, A. H. Guth, and A. Vilenkin, Inflationary space-times are incompletein past di- rections, Phys. Rev. Lett.90, 151301 (2003), arXiv:gr-qc/0110012

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  12. [12]

    T. S. Bunch and P . C. W. Davies, Quantum Field Theory in de Sitter Space: Renormalization by Point Splitting, Proc. Roy. Soc. Lond. A360, 117 (1978)

    work page 1978

  13. [13]

    Loop Quantum Cosmology: A Status Report

    A. Ashtekar and P . Singh, Loop Quantum Cosmology: A Status Report, Class. Quant. Grav. 28, 213001 (2011), arXiv:1108.0893

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  14. [14]

    Becker, M

    K. Becker, M. Becker, and J. H. Schwarz,String theory and M-theory: A modern introduction (Cambridge University Press, 2006)

    work page 2006

  15. [15]

    Thiemann,Modern Canonical Quantum General RelativityCambridge Monographs on Mathematical Physics (Cambridge University Press, 2007)

    T. Thiemann,Modern Canonical Quantum General RelativityCambridge Monographs on Mathematical Physics (Cambridge University Press, 2007)

    work page 2007

  16. [16]

    Rovelli and F

    C. Rovelli and F. Vidotto,Covariant Loop Quantum Gravity: An Elementary Introduction to Quantum Gravity and Spinfoam Theory(Cambridge University Press, 2014)

    work page 2014

  17. [17]

    B. S. DeWitt, Quantum theory of gravity. i. the canonical theory, Phys. Rev.160, 1113 (1967)

    work page 1967

  18. [18]

    Ashtekar, New variables for classical and quantum gravity, Phys

    A. Ashtekar, New variables for classical and quantum gravity, Phys. Rev. Lett.57, 2244 (1986)

    work page 1986

  19. [19]

    Bojowald, Absence of a singularity in loop quantum cosmology, Phys

    M. Bojowald, Absence of a singularity in loop quantum cosmology, Phys. Rev. Lett.86, 5227 (2001)

    work page 2001

  20. [20]

    Li and P

    B.-F. Li and P . Singh,Loop Quantum Cosmology: Physics of Singularity Resolution and Its Impli- cations(Springer Nature Singapore, 2023), p. 1–55

    work page 2023

  21. [21]

    Agull´ o, A

    I. Agull ´o, A. Wang, and E. Wilson-Ewing, Loop quantum cosmology: relation between theory and observations, (2023), arXiv:2301.10215

    work page arXiv 2023

  22. [22]

    B.-F. Li, M. Motaharfar, and P . Singh, Constraining regularization ambiguities in loop quan- tum cosmology via CMB, Phys. Rev. D110, 066005 (2024), arXiv:2405.12296. 43

    work page arXiv 2024

  23. [23]

    Ashtekar, B

    A. Ashtekar, B. Gupt, D. Jeong, and V . Sreenath, Alleviating the Tension in the Cosmic Microwave Background using Planck-Scale Physics, Phys. Rev. Lett.125, 051302 (2020), arXiv:2001.11689

    work page arXiv 2020

  24. [24]

    Ashtekar, B

    A. Ashtekar, B. Gupt, and V . Sreenath,Cosmic Tango Between the Very Small and the Very Large: Addressing CMB Anomalies Through Loop Quantum Cosmology, Front. Astron. Space Sci.8, 76 (2021), arXiv:2103.14568

    work page arXiv 2021

  25. [25]

    Agullo, D

    I. Agullo, D. Kranas, and V . Sreenath, Anomalies in the CMB from a cosmic bounce, Gen. Rel. Grav.53, 17 (2021), arXiv:2005.01796

    work page arXiv 2021

  26. [26]

    Agullo, D

    I. Agullo, D. Kranas, and V . Sreenath, Large scale anomalies in the CMB and non-Gaussianity in bouncing cosmologies, Class. Quant. Grav.38, 065010 (2021), arXiv:2006.09605

    work page arXiv 2021

  27. [27]

    Mart ´ın-Benito, R

    M. Mart ´ın-Benito, R. B. Neves, and J. Olmedo, Alleviation of anomalies from the nonoscilla- tory vacuum in loop quantum cosmology, Phys. Rev. D108, 103508 (2023), arXiv:2305.09599

    work page arXiv 2023

  28. [28]

    Beetle, J

    C. Beetle, J. S. Engle, M. E. Hogan, and P . Mendonc ¸a, Quantum isotropy and the reduction of dynamics in Bianchi I, Class. Quant. Grav.38, 245001 (2021), arXiv:2102.03901

    work page arXiv 2021

  29. [29]

    Cortez, G

    J. Cortez, G. A. M. Marug ´an, and J. M. Velhinho, A Brief Overview of Results about Unique- ness of the Quantization in Cosmology, Universe7, 299 (2021), arXiv:2108.07489

    work page arXiv 2021

  30. [30]

    Bojowald, Space–Time Physics in Background-Independent Theories of Quantum Grav- ity, Universe7, 251 (2021), arXiv:2108.11936

    M. Bojowald, Space–Time Physics in Background-Independent Theories of Quantum Grav- ity, Universe7, 251 (2021), arXiv:2108.11936

    work page arXiv 2021

  31. [31]

    B.-F. Li, P . Singh, and A. Wang, Phenomenological implications of modified loop cosmolo- gies: an overview, Front. Astron. Space Sci.8, 701417 (2021), arXiv:2105.14067

    work page arXiv 2021

  32. [32]

    J. Yang, Y. Ding, and Y. Ma,Alternative quantization of the Hamiltonian in loop quantum cos- mology II: Including the Lorentz term, Phys. Lett. B682, 1 (2009), arXiv:0904.4379

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  33. [33]

    Emergent de Sitter epoch of the quantum Cosmos

    M. Assanioussi, A. Dapor, K. Liegener, and T. Pawłowski, Emergent de Sitter Epoch of the Quantum Cosmos from Loop Quantum Cosmology, Phys. Rev. Lett.121, 081303 (2018), arXiv:1801.00768

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  34. [34]

    Quantum Nature of the Big Bang: Improved dynamics

    A. Ashtekar, T. Pawlowski, and P . Singh,Quantum Nature of the Big Bang: Improved dynamics, Phys. Rev. D74, 084003 (2006), arXiv:gr-qc/0607039

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  35. [35]

    Dapor and K

    A. Dapor and K. Liegener, Cosmological coherent state expectation values in loop quantum gravity I. Isotropic kinematics, Class. Quant. Grav.35, 135011 (2018), arXiv:1710.04015

    work page arXiv 2018

  36. [36]

    Dapor, K

    A. Dapor, K. Liegener, and T. Pawłowski, Challenges in Recovering a Consistent Cosmol- ogy from the Effective Dynamics of Loop Quantum Gravity, Phys. Rev. D100, 106016 (2019), arXiv:1910.04710

    work page arXiv 2019

  37. [37]

    Han and H

    M. Han and H. Liu, Loop quantum gravity on dynamical lattice and improved cosmological effective dynamics with inflaton, Phys. Rev. D104, 024011 (2021), arXiv:2101.07659

    work page arXiv 2021

  38. [38]

    B.-F. Li, P . Singh, and A. Wang, Towards Cosmological Dynamics from Loop Quantum Gravity, Phys. Rev. D97, 084029 (2018), arXiv:1801.07313

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  39. [39]

    B.-F. Li, P . Singh, and A. Wang, Qualitative dynamics and inflationary attractors in loop cosmology, Phys. Rev. D98, 066016 (2018), arXiv:1807.05236

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  40. [40]

    Generic absence of strong singularities and geodesic completeness in modified loop quantum cosmologies

    S. Saini and P . Singh, Generic absence of strong singularities and geodesic complete- ness in modified loop quantum cosmologies, Class. Quant. Grav.36, 105014 (2019), arXiv:1812.08937. 44

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  41. [41]

    B.-F. Li, P . Singh, and A. Wang,Genericness of pre-inflationary dynamics and probability of the desired slow-roll inflation in modified loop quantum cosmologies, Phys. Rev. D100, 063513 (2019), arXiv:1906.01001

    work page arXiv 2019

  42. [42]

    Saeed, R

    J. Saeed, R. Pan, C. Brown, G. Clevear, and A. Wang, Universal properties of the evolution of the Universe in modified loop quantum cosmology, (2024), arXiv:2406.06745

    work page arXiv 2024

  43. [43]

    Primordial power spectrum from the Dapor-Liegener model of loop quantum cosmology

    I. Agullo, Primordial power spectrum from the Dapor-Liegener model of loop quantum cosmology, Gen. Rel. Grav.50, 91 (2018), arXiv:1805.11356

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  44. [44]

    B.-F. Li, P . Singh, and A. Wang,Primordial power spectrum from the dressed metric approach in loop cosmologies, Phys. Rev. D101, 086004 (2020), arXiv:1912.08225

    work page arXiv 2020

  45. [45]

    B.-F. Li, J. Olmedo, P . Singh, and A. Wang,Primordial scalar power spectrum from the hybrid approach in loop cosmologies, Phys. Rev. D102, 126025 (2020), arXiv:2008.09135

    work page arXiv 2020

  46. [46]

    N. D. Birrell and P . C. W. Davies,Quantum Fields in Curved SpaceCambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, UK, 1982)

    work page 1982

  47. [47]

    Penna-Lima, N

    M. Penna-Lima, N. Pinto-Neto, and S. D. P . Vitenti, New formalism to define vacuum states for scalar fields in curved spacetimes, Phys. Rev. D107, 065019 (2023), arXiv:2207.08270

    work page arXiv 2023

  48. [48]

    Elizaga Navascu ´es, G

    B. Elizaga Navascu ´es, G. A. M. Marug ´an, and T. Thiemann, Hamiltonian diagonalization in hybrid quantum cosmology, Class. Quant. Grav.36, 185010 (2019), arXiv:1903.05695

    work page arXiv 2019

  49. [49]

    F. W. J. Olver,Asymptotics and Special Functions(A K Peters, Ltd., 1997), p. 1–572

    work page 1997

  50. [50]

    S. E. Joras and G. Marozzi, Trans-Planckian Physics from a Non-Linear Dispersion Relation, Phys. Rev. D79, 023514 (2009), arXiv:0808.1262

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  51. [51]

    Effects of Nonlinear Dispersion Relations on Non-Gaussianities

    A. Ashoorioon, D. Chialva, and U. Danielsson, Effects of Nonlinear Dispersion Relations on Non-Gaussianities, JCAP06, 034 (2011), arXiv:1104.2338

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  52. [52]

    A Quantum Gravity Extension of the Inflationary Scenario

    I. Agullo, A. Ashtekar, and W. Nelson,A Quantum Gravity Extension of the Inflationary Sce- nario, Phys. Rev. Lett.109, 251301 (2012), arXiv:1209.1609

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  53. [53]

    An Extension of the Quantum Theory of Cosmological Perturbations to the Planck Era

    I. Agullo, A. Ashtekar, and W. Nelson,Extension of the quantum theory of cosmological pertur- bations to the Planck era, Phys. Rev. D87, 043507 (2013), arXiv:1211.1354

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  54. [54]

    The pre-inflationary dynamics of loop quantum cosmology: Confronting quantum gravity with observations

    I. Agullo, A. Ashtekar, and W. Nelson,The pre-inflationary dynamics of loop quantum cosmol- ogy: Confronting quantum gravity with observations, Class. Quant. Grav.30, 085014 (2013), arXiv:1302.0254

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  55. [55]

    K. N. Abazajianet al., Inflation Physics from the Cosmic Microwave Background and Large Scale Structure, Astropart. Phys.63, 55 (2015), arXiv:1309.5381

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  56. [56]

    Detailed analysis of the predictions of loop quantum cosmology for the primordial power spectra

    I. Agullo and N. A. Morris, Detailed analysis of the predictions of loop quantum cosmology for the primordial power spectra, Phys. Rev. D92, 124040 (2015), arXiv:1509.05693

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  57. [57]

    Anomaly freedom in perturbative loop quantum gravity

    M. Bojowald, G. M. Hossain, M. Kagan, and S. Shankaranarayanan,Anomaly freedom in perturbative loop quantum gravity, Phys. Rev. D78, 063547 (2008), arXiv:0806.3929

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  58. [58]

    Anomaly-free perturbations with inverse-volume and holonomy corrections in Loop Quantum Cosmology

    T. Cailleteau, L. Linsefors, and A. Barrau, Anomaly-free perturbations with inverse-volume and holonomy corrections in Loop Quantum Cosmology, Class. Quant. Grav.31, 125011 (2014), arXiv:1307.5238

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  59. [59]

    Observational Exclusion of a Consistent Quantum Cosmology Scenario

    B. Bolliet, A. Barrau, J. Grain, and S. Schander,Observational exclusion of a consistent loop quantum cosmology scenario, Phys. Rev. D93, 124011 (2016), arXiv:1510.08766

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  60. [60]

    The perturbed universe in the deformed algebra approach of Loop Quantum Cosmology

    J. Grain,The perturbed universe in the deformed algebra approach of Loop Quantum Cosmology, Int. J. Mod. Phys. D25, 1642003 (2016), arXiv:1606.03271. 45

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  61. [61]

    Preinflationary perturbations from the closed algebra approach in loop quantum cosmology

    B.-F. Liet al.,Preinflationary perturbations from the closed algebra approach in loop quantum cosmology, Phys. Rev. D99, 103536 (2019), arXiv:1812.11191

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  62. [62]

    T. Zhu, A. Wang, K. Kirsten, G. Cleaver, and Q. Sheng, Universal features of quantum bounce in loop quantum cosmology, Phys. Lett. B773, 196 (2017), arXiv:1607.06329

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  63. [63]

    T. Zhu, A. Wang, G. Cleaver, K. Kirsten, and Q. Sheng,Pre-inflationary universe in loop quantum cosmology, Phys. Rev. D96, 083520 (2017), arXiv:1705.07544

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  64. [64]

    Q. Wu, T. Zhu, and A. Wang,Nonadiabatic evolution of primordial perturbations and non- Gaussianity in hybrid approach of loop quantum cosmology, Phys. Rev. D98, 103528 (2018), arXiv:1809.03172

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  65. [65]

    G. A. Mena Marug ´an, A. Vicente-Becerril, and J. Y´ebana Carrilero, Comparing Analytic and Numerical Studies of Tensor Perturbations in Loop Quantum Cosmology, Universe10, 365 (2024), arXiv:2409.18302

    work page arXiv 2024

  66. [66]

    G. A. Mena Marug ´an, A. Vicente-Becerril, and J. Y´ebana Carrilero, Analytic and numerical study of scalar perturbations in loop quantum cosmology, Phys. Rev. D110, 043508 (2024), arXiv:2404.04595

    work page arXiv 2024

  67. [67]

    Alonso-Serrano, G

    A. Alonso-Serrano, G. A. Mena Marugan, and A. Vicente-Becerril, Analytic primordial power spectrum in the dressed metric approach to loop quantum cosmology and thermo- dynamics of spacetime, Int. J. Mod. Phys. D34, 2450062 (2025), arXiv:2307.06813

    work page arXiv 2025

  68. [68]

    Vector and tensor perturbations in Horava-Lifshitz cosmology

    A. Wang, Vector and tensor perturbations in Horava-Lifshitz cosmology, Phys. Rev. D82, 124063 (2010), arXiv:1008.3637

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  69. [69]

    T. Zhu, A. Wang, G. Cleaver, K. Kirsten, and Q. Sheng, Constructing analytical solutions of linear perturbations of inflation with modified dispersion relations, Int. J. Mod. Phys. A29, 1450142 (2014), arXiv:1308.1104

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  70. [70]

    T. Zhu, A. Wang, G. Cleaver, K. Kirsten, and Q. Sheng, Inflationary cosmology with non- linear dispersion relations, Phys. Rev. D89, 043507 (2014), arXiv:1308.5708

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  71. [71]

    T. Zhu, A. Wang, G. Cleaver, K. Kirsten, and Q. Sheng, Gravitational quantum effects on power spectra and spectral indices with higher-order corrections, Phys. Rev. D90, 063503 (2014), arXiv:1405.5301

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  72. [72]

    Gravitational quantum effects in the light of BICEP2 results

    T. Zhu and A. Wang, Gravitational quantum effects in light of BICEP2 results, Phys. Rev. D 90, 027304 (2014), arXiv:1403.7696

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  73. [73]

    T. Zhu, A. Wang, K. Kirsten, G. Cleaver, and Q. Sheng, High-order Primordial Perturbations with Quantum Gravitational Effects, Phys. Rev. D93, 123525 (2016), arXiv:1604.05739

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  74. [74]

    Qiao, G.-H

    J. Qiao, G.-H. Ding, Q. Wu, T. Zhu, and A. Wang, Inflationary perturbation spectrum in extended effective field theory of inflation, JCAP09, 064 (2019), arXiv:1811.03216

    work page arXiv 2019

  75. [75]

    T. Zhu, Q. Wu, and A. Wang, An analytical approach to the field amplification and particle production by parametric resonance during inflation and reheating, Phys. Dark Univ.26, 100373 (2019), arXiv:1811.12612

    work page arXiv 2019

  76. [76]

    G.-H. Ding, J. Qiao, Q. Wu, T. Zhu, and A. Wang, Inflationary perturbation spectra at next- to-leading slow-roll order in effective field theory of inflation, Eur. Phys. J. C79, 976 (2019), arXiv:1907.13108

    work page arXiv 2019

  77. [77]

    T. Zhu, A. Wang, G. Cleaver, K. Kirsten, and Q. Sheng, Power spectra and spectral indices ofk-inflation: high-order corrections, Phys. Rev. D90, 103517 (2014), arXiv:1407.8011. 46

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  78. [78]

    Q. Wu, T. Zhu, and A. Wang, Primordial Spectra of slow-roll inflation at second-order with the Gauss-Bonnet correction, Phys. Rev. D97, 103502 (2018), arXiv:1707.08020

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  79. [79]

    J. Qiao, T. Zhu, W. Zhao, and A. Wang, Polarized primordial gravitational waves in the ghost-free parity-violating gravity, Phys. Rev. D101, 043528 (2020), arXiv:1911.01580

    work page arXiv 2020

  80. [80]

    Detecting quantum gravitational effects of loop quantum cosmology in the early universe?

    T. Zhuet al., Detecting quantum gravitational effects of loop quantum cosmology in the early universe?, Astrophys. J. Lett.807, L17 (2015), arXiv:1503.06761

    work page internal anchor Pith review Pith/arXiv arXiv 2015

Showing first 80 references.