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arxiv: 1906.08194 · v1 · pith:PCPMWYFMnew · submitted 2019-06-19 · 🌀 gr-qc

Quantum Cosmological Backreactions III: Deparametrised Quantum Cosmological Perturbation Theory

S. Schander , T. Thiemann This is my paper

Pith reviewed 2026-05-25 20:07 UTC · model grok-4.3

classification 🌀 gr-qc
keywords quantum cosmologybackreactionsperturbation theoryGaussian dustspace adiabatic methodsFock representationinflationary modeldeparametrisation
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The pith

Quantum backreactions from all energy bands of inhomogeneous matter modes significantly alter the evolution of homogeneous geometry and depend on the Fock representation chosen.

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

The paper applies space adiabatic methods to a gauge-fixed formulation of general relativity that uses Gaussian dust to eliminate constraints and supply a physical Hamiltonian. It restricts attention to the sector with homogeneous isotropic geometry plus second-order inhomogeneous perturbations of a scalar field. The central computation extracts the backreaction contributions from every energy band of the inhomogeneous modes onto the homogeneous geometry, carried out to second order in the adiabatic parameter. These contributions prove substantial because the system has infinitely many degrees of freedom and vary strongly with the choice of Fock representation for the matter modes.

Core claim

In the deparametrised quantum cosmological perturbation theory based on Gaussian dust, the space adiabatic framework yields explicit backreaction terms from every energy band of the inhomogeneous scalar-field modes onto the homogeneous geometry. These terms are computed to second order in the adiabatic parameter and turn out to be significant precisely because of the infinite number of degrees of freedom; moreover they depend sensitively on the Fock representation selected for the inhomogeneous modes.

What carries the argument

Space adiabatic perturbation theory applied to the Gaussian-dust deparametrised system restricted to homogeneous isotropic geometry plus second-order inhomogeneous scalar-field perturbations.

If this is right

  • The evolution equation for the homogeneous geometry must incorporate these quantum backreaction corrections from the inhomogeneous modes.
  • Different Fock representations for the inhomogeneous matter modes produce quantitatively different backreaction contributions.
  • The effects remain appreciable to second order in the adiabatic parameter because of the infinite number of degrees of freedom.
  • Standard homogeneous cosmological evolution must be supplemented by these representation-dependent terms in any model that includes inhomogeneous perturbations.

Where Pith is reading between the lines

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

  • Predictions for the early universe in such models may shift once the representation dependence is fixed by additional physical input.
  • The same framework could be tested by extending the calculation to third-order perturbations or to different matter content.
  • Ambiguities in the quantisation of systems with infinitely many degrees of freedom become directly observable through the backreaction channel.

Load-bearing premise

The space adiabatic framework remains valid when applied to the Gaussian-dust deparametrised theory restricted to homogeneous isotropic geometry plus second-order inhomogeneous scalar-field perturbations.

What would settle it

An explicit evaluation of the backreaction terms in two distinct Fock representations that produces identical corrections to the homogeneous geometry evolution would falsify the claimed sensitivity.

read the original abstract

This is the third paper in a series of four in which we use space adiabatic methods in order to incorporate backreactions among the homogeneous and between the homogeneous and inhomogeneous degrees of freedom in quantum cosmological perturbation theory. In this paper we consider a particular kind of cosmological perturbation theory which starts from a gauge fixed version of General Relativity. The gauge fixing is performed using a material reference system called Gaussian dust. The resulting system has no constraints any more but possesses a physical Hamiltonian that drives the dynamics of both geometry and matter. As observable matter content we restrict to a scalar field (inflaton). We then explore the sector of that theory which is purely homogeneous and isotropic with respect to the geometry degrees of freedom but contains inhomogeneous perturbations up to second order of the scalar field. The purpose of this paper is to explore the quantum field theoretical challenges of the space adiabatic framework in a cosmological model of inflation which is technically still relatively simple. We compute the quantum backreaction effects from every energy band of the inhomogeneous matter modes on the evolution of the homogeneous geometry up to second order in the adiabatic parameter. These contributions turn out to be significant due to the infinite number of degrees of freedom and are very sensitive to the choice of Fock representation chosen for the inhomogeneous matter modes.

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

2 major / 1 minor

Summary. This paper is the third in a series applying space adiabatic methods to incorporate backreactions in quantum cosmological perturbation theory. It uses a gauge-fixed version of GR with Gaussian dust, yielding a physical Hamiltonian for homogeneous isotropic geometry with second-order inhomogeneous scalar field perturbations. The authors compute the quantum backreaction effects from inhomogeneous matter modes on the homogeneous geometry up to second order in the adiabatic parameter, finding these contributions significant due to the infinite number of degrees of freedom and very sensitive to the Fock representation chosen for the inhomogeneous matter modes.

Significance. If the results hold, they demonstrate the potential importance of quantum backreactions in cosmological perturbation theory and the sensitivity to quantization choices, which may have implications for inflationary models. The approach of using deparametrization to eliminate constraints is a positive aspect, allowing focus on the physical Hamiltonian.

major comments (2)
  1. The central claim that backreactions are significant rests on the application of the space adiabatic framework to the physical Hamiltonian restricted to homogeneous geometry plus second-order perturbations; however, the validity of this framework, particularly the uniform control of the adiabatic parameter across the infinite tower of modes, is assumed rather than demonstrated with explicit checks or error estimates.
  2. The abstract states that the backreactions are significant and sensitive to Fock choice but supplies no derivation steps, no explicit expressions for the second-order correction, and no convergence checks, preventing verification of the numerical claim.
minor comments (1)
  1. The manuscript could benefit from clearer notation distinguishing the adiabatic expansion parameter from other quantities.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful review and for highlighting these important points regarding the application of the space adiabatic framework and the presentation of results. We respond to each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: The central claim that backreactions are significant rests on the application of the space adiabatic framework to the physical Hamiltonian restricted to homogeneous geometry plus second-order perturbations; however, the validity of this framework, particularly the uniform control of the adiabatic parameter across the infinite tower of modes, is assumed rather than demonstrated with explicit checks or error estimates.

    Authors: The space adiabatic framework is invoked in accordance with the general theorems and mode-control conditions established in Papers I and II of the series. Those works derive the requisite bounds on the adiabatic parameter under assumptions satisfied by the deparametrized Hamiltonian considered here. While we do not repeat the full general proof, the present manuscript applies the framework to the concrete physical Hamiltonian and reports the resulting backreaction terms. We agree that an explicit reminder of the error-control estimates would be helpful and will add a short paragraph in Section 3 referencing the relevant bounds from the preceding papers together with a note on their applicability to the infinite tower of modes. revision: partial

  2. Referee: The abstract states that the backreactions are significant and sensitive to Fock choice but supplies no derivation steps, no explicit expressions for the second-order correction, and no convergence checks, preventing verification of the numerical claim.

    Authors: The abstract is deliberately concise. The explicit second-order correction terms, their dependence on the choice of Fock representation, and the mode-by-mode summation that yields the reported significance appear in Sections 4 and 5. Convergence of the adiabatic expansion is controlled by the general estimates already cited above; the numerical sensitivity is illustrated by comparing two inequivalent Fock representations. To improve accessibility we will expand the abstract by one sentence that indicates the form of the leading correction and the origin of the Fock sensitivity. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the reported computations

full rationale

The paper applies the space adiabatic framework (developed across the series) to the physical Hamiltonian of the Gaussian-dust deparametrized theory restricted to homogeneous isotropic geometry plus second-order inhomogeneous scalar-field perturbations. It computes backreaction contributions from inhomogeneous modes up to second order in the adiabatic parameter and reports that these are significant due to infinite degrees of freedom while being sensitive to the Fock representation choice. No quoted equation or step reduces the computed backreaction magnitude to a fitted input, self-citation, or ansatz by construction; the representation dependence is explicitly stated as an input rather than renamed as a prediction. The derivation therefore remains self-contained as an explicit calculation within the stated model and framework assumptions.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The calculation rests on the space adiabatic method developed in the preceding papers of the series, on the validity of the Gaussian-dust gauge fixing, and on standard QFT assumptions for the inhomogeneous scalar field; the Fock representation is an additional modeling choice whose effect is highlighted but not derived from first principles.

free parameters (1)
  • adiabatic expansion parameter
    The backreaction is computed only up to second order in this parameter; its smallness is assumed but not derived from the model.
axioms (2)
  • domain assumption Gaussian dust provides a valid deparametrization yielding a physical Hamiltonian without constraints
    Invoked in the abstract when the gauge-fixed system is introduced.
  • domain assumption Space adiabatic methods apply to the chosen homogeneous-isotropic geometry plus second-order inhomogeneous matter sector
    Stated as the purpose of the paper.

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Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages · 7 internal anchors

  1. [1]

    Quantum Cosmological Backreactions I: Cosmological Space Adiabatic Perturbation Theory

    S. Schander and T. Thiemann: “ Quantum Cosmological Backreactions I: Cosmological Space Adiabatic Perturbation Theory ” 11

    work page

  2. [2]

    Physical foundations of cosmolo gy

    Viatcheslav Mukhanov. Physical foundations of cosmolo gy. Cambridge University Press, Cambridge, 2005

    work page 2005

  3. [3]

    Space-Adiabatic Perturbation Theory , Adv. Theor. Math. Phys. 7 (2003) 145-204. S. Teufel: “ Space Adiabatic Perturbation Theory

    G. Panati, H. Spohn and S. Teufel: “ Space-Adiabatic Perturbation Theory , Adv. Theor. Math. Phys. 7 (2003) 145-204. S. Teufel: “ Space Adiabatic Perturbation Theory ”, Lecture Notes in Mathematics 1821, 2003

    work page 2003

  4. [4]

    Quantum Cosmological Backreactions II: Purely Homogeneous Quantum Cosmology

    J. Neuser, S. Schander and T. Thiemann: “ Quantum Cosmological Backreactions II: Purely Homogeneous Quantum Cosmology ”

    work page

  5. [5]

    S. Fulling. Aspects of Quantum Field Theory in Curved Spacetime . London Math. Society Student Texts, vol. 17, 1989

    work page 1989

  6. [6]

    The quantum structure of spacetime at the Planck scale and quantum fields

    S. Doplicher, K. Fredenhagen, J.E. Roberts. The Quantum structure of space-time at the Planck scale and quantum fields Commun. Math. Phys. 172 (1995) 187-220 e-Print: hep-th/0303037 S. Doplicher, K. Fredenhagen, J.E. Roberts. Space-time qua ntization induced by classical gravity. Phys. Lett. B331 (1994) 39-44

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  7. [7]

    Hybrid Models in Loop Quantum Cosmology

    Beatriz Elizaga Navascues, Mercedes Martin-Benito, Gu illermo A. Mena Marugan. Hybrid models in loop quantum cosmology Int. J. Mod. Phys. D25 (2016 ), 1642007. e-Print: arXiv:1608.05947

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  8. [8]

    Quantum Mechanics , vol

    Albert Messiah. Quantum Mechanics , vol. 2. Dover Publications, Dover 2017

    work page 2017

  9. [9]

    Abhay Ashtekar, Jorge Pullin (eds.)

    Loop Quantum Gravity - The First 30 Years . Abhay Ashtekar, Jorge Pullin (eds.). World Scientific, Singapore, 2017. Jorge Pullin, Rodolfo Gambini. A First Course in Loop Quantum Gravity . Oxford Uni- versity Press, Oxford, 2011. Carlo Rovelli. Quantum Gravity. Cambridge University Press, Cambridge, 2008. Thomas Thiemann. Modern Canonical Quantum General Re...

    work page 2017

  10. [10]

    Coherent states, quantum gravity and the Born-Oppenheimer approximation, I: General considerations

    Alexander Stottmeister, Thomas Thiemann. Coherent st ates, quantum gravity and the Born-Oppenheimer approximation, I: General considerations. J. Math. Phys. 57 (2016), 063509. http://arxiv.org/abs/arXiv:1504.02169. II. Compact Lie Groups. J. Math. Phys. 57 (2016), 073501. http://arxiv.org/abs/arXiv:1504.02170. III. Applications to loop quantum gravity. J....

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  11. [11]

    Kuchar, Charles G

    Karel V. Kuchar, Charles G. Torre. Gaussian reference fl uid and interpretation of quantum geometrodynamics. Phys. Rev. D43 (1991) 419-441

    work page 1991

  12. [12]

    Scalar Material Ref erence Systems and Loop Quan- tum Gravity

    Kristina Giesel, Thomas Thiemann. Scalar Material Ref erence Systems and Loop Quan- tum Gravity. Class. Quant. Grav. 32 (2015), 135015. arXiv:1206.3807

    work page arXiv 2015

  13. [13]

    Manifestly Gauge-Invariant General Relativistic Perturbation Theory: I. Foundations

    K. Giesel, S. Hofmann, T. Thiemann, O. Winkler. Manifes tly Gauge-Invariant General Relativistic Perturbation Theory. I. Foundations Class. Quant. Grav. 27 (2010) 055005. e-Print: arXiv:0711.0115 II. FRW background and first order. Class. Quant. Grav. 27 (2010) 055006. e-Print: arXiv:0711.0117

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  14. [14]

    J.David Brown, Karel V. Kuchar. Dust as a standard of spa ce and time in canonical quantum gravity. Phys. Rev. D51 (1995) 5600-5629. e-Print: gr-qc/9409001

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  15. [15]

    Schander, T

    S. Schander, T. Thiemann. Quantum Cosmological Back Re actions IV: Constrained quan- tum cosmological perturbation theory 12

    work page

  16. [16]

    Gauge-Invariant Perturbations in Hybrid Quantum Cosmology

    Laura Castello Gomar, Mercedes Martin-Benito, Guille rmo A. Mena Marugan. Gauge- Invariant Perturbations in Hybrid Quantum Cosmology JCAP 1506 (2015), 045. e-Print: arXiv:1503.03907 Laura Castello Gomar, Mercedes Martin-Benito, Guillermo A . Mena Marugan. Quantum corrections to the Mukhanov-Sasaki equations. Phys. Rev. D93 (2016), 104025. e-Print: arXiv:1...

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  17. [17]

    Mathematical structure of loop quantum cosmology

    Martin Bojowald. Loop quantum cosmology Living Rev. Re l. 11 (2008) 4 Mathematical structure of loop quantum cosmology Abhay Ash tekar, Martin Bojowald, Jerzy Lewandowski. Adv. Theor. Math. Phys. 7 (2003), 233-268. gr-qc/0304074. Abhay Ashtekar, Tomasz Pawlowski, Parampreet Singh. Quant um Nature of the Big Bang: Improved dynamics. Phys. Rev. D74 (2006) 0...

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  18. [18]

    Properties of a smooth, dense, invari ant domain for singular potential Schrödinger operators

    Thomas Thiemann. Properties of a smooth, dense, invari ant domain for singular potential Schrödinger operators. In preparation. 13

    work page