CoChat
Sign Up
Cited by 25Cosmology and Gravitation TheoriesNoncommutative and Quantum Gravity TheoriesRelativity and Gravitational Theory

Quantized gravitational waves in the Milne universe

Open access
Tanaka, Takahiro, Sasaki, Misao·1997·Physical Review D
DOI 10.1103/physrevd.55.6061
Open this paper in CoChat — free accountOpen PDF

Abstract

The quantization of gravitational waves in the Milne universe is discussed. The relation between positive frequency functions of the gravitational waves in the Milne universe and those in the Minkowski universe is clarified. Implications to the one-bubble open inflation scenario are also discussed.

AI snapshot

Written from the paper’s full text.

TL;DR

The paper quantizes gravitational waves in the Milne universe and constructs positive-frequency mode functions corresponding to the Minkowski vacuum. It shows these Milne modes are unitarily and gauge-equivalent to Minkowski modes, but a naive Sachs-Wolfe calculation in Milne coordinates gives an infrared-divergent temperature anisotropy.

Key findings

  • Positive-frequency gravitational-wave functions in the Milne universe can be chosen by analyticity in the lower half complex \(t\)-plane, and their normalization is fixed by canonical quantization in the Rindler universe.
  • The Milne and Minkowski gravitational-wave mode functions are related by a unitary transformation followed by a gauge transformation, so the chosen Euclidean vacuum in Milne is the conventional Minkowski vacuum.
  • For the Sachs-Wolfe effect, the Milne-coordinate calculation of \(\langle a_\ell^2\rangle\) is infrared divergent: near \(p=0\), the integrand behaves as \(\sim 1/p^2\).
  • The corresponding Minkowski-coordinate calculation is infrared finite: the \(k\)-integration does not diverge around \(k=0\). The paper attributes the difference to the gauge-dependent choice of the last scattering surface.

Methods

  • Theoretical study of quantized gravitational waves in Minkowski and Milne coordinates, using traceless Lorentz gauge plus synchronous gauge conditions and tensor harmonics on flat Euclidean 3-space and hyperbolic 3-space.
  • Normalization is determined by analytically continuing to the Rindler universe, reducing the constrained gravitational-wave action to a physical variable, and imposing canonical commutation relations.
  • The Sachs-Wolfe effect is computed for gravitational-wave perturbations in both Milne and Minkowski coordinates, comparing multipole moments \(\langle a_\ell^2\rangle\) and analyzing the effect of gauge transformations on the last scattering surface.

Limitations

  • Stated: the last scattering surface is specified by hand and is not gauge invariant; the paper argues the infrared divergence arises from this unphysical setting and expects it to be removed in a realistic open inflationary model.
  • Stated: the model is an empty Minkowski/Milne toy model, with no physical scalar quantity determining the constant-temperature slice and no intrinsic temperature fluctuations.
  • Evident from the text: the analysis does not include a full open-inflation model, inflaton-gravity coupling, or a realistic background, so the results are not directly applied to observed CMB anisotropies.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1103/physrevd.55.6061

Cites

27 of 27 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

0 of 36 cited sources checked · 0 supported · 0 flagged

  • Findingreasoning
    § II. MASSLESS SCALAR FIELD IN THE MILNE UNIVERSE · p. 5

    On the ξ = constant hypersurface in the Milne universe, the quantity similar to the Klein-Gordon inner product gives a normalization different from the correct Klein-Gordon inner product by the factor e^{πp}/2 sinh πp.

    “Since the hypersurface is not a Cauchy surface, it gives a normalization different from the correct Klein-Gordon inner product by the factor e πp 2 sinh πp .”
    Quote in the paperNot a substantial claim
  • Findingreasoning
    § II. MASSLESS SCALAR FIELD IN THE MILNE UNIVERSE · p. 4

    When P_pl is analytically continued to the Rindler universe by Eq. (2.12) with the upper sign, it plays the role of a positive frequency function.

    “Note that when P pℓ is analytically continued to the Rindler universe by Eq. (2.12) with the upper sign, it plays the role of a positive frequency function.”
    Quote in the paperSubstantial claim
  • Findingcitation
    § II. MASSLESS SCALAR FIELD IN THE MILNE UNIVERSE · p. 4

    The equivalence of the two representations of positive frequency functions for the Minkowski vacuum (2.14) and those for the Euclidean vacuum in the Milne universe (2.20) can be directly proven by using the formula [21,22].

    “The equivalence of the two representations of positive frequency functions for the Minkowski vacuum (2.14) and those for the Euclidean vacuum in the Milne universe (2.20) can be directly proven by using the formula [21,22]”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [21] U. H. Gerlach, Phys.rev. D 28 , 761 (1983).

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [22] W. Magnus, F. Oberhettinger and R. S. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer, New York, 1966), pp. 92. source
  • Findingreasoning
    § V. SACHS-WOLFE EFFECT IN THE MILNE AND MINKOWSKI UNIVERSES · p. 16

    After the gauge transformation, the mode functions become the Milne synchronous ones.

    “After the gauge transformation, (4.14), the mode functions become the Milne synchronous ones, H ( e ) pℓm µν = ˜ H ( e ) pℓm µν .”
    Quote in the paperSubstantial claim
  • Findingcitation
    § II. MASSLESS SCALAR FIELD IN THE MILNE UNIVERSE · p. 5

    The formula gives a unitary transformation relation of the scalar modes.

    “the above formula gives a unitary transformation relation of the scalar modes,”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: There exists a small imaginary part in t as t - iε.
  • Classified as probably not a substantial claim. 8
  • May combine two claims. 8

  • Classified as probably not a substantial claim.

    Findingnone given
    § III. QUANTIZATION OF GRAVITATIONAL WAVES IN MINKOWSKI AND MILNE UNIVERSES · p. 5

    The Euclidean vacuum prescription gives the usual Minkowski vacuum for the Minkowski universe for gravitational waves.

    “This prescription gives the usual Minkowski vacuum for the Minkowski universe for gravitational waves as well.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § A. even parity · p. 12

    The newly defined mode functions satisfy the traceless Lorentz gauge condition but not the Milne synchronous gauge condition; therefore a gauge transformation is necessary to compare them with the Milne mode functions.

    “The newly defined mode functions H ( e ) pℓm µν also satisfy the traceless Lorentz gauge condition (3.3), but as mentioned before it does not fix the gauge completely. In fact, H ( e ) pℓm µν satisfy the Minkowski synchronous gauge condition (3.4) but not the Milne synchronous gauge condition (3.35). Thus to compare H ( e ) pℓm µν with H ( e ) pℓm µν , a gauge transformation is necessary.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § V. SACHS-WOLFE EFFECT IN THE MILNE AND MINKOWSKI UNIVERSES · p. 16

    A is finite at p→0, and hence H(e) is finite at p→0.

    “Observing Eq. (4.9), A is finite at p → 0. Hence so is H ( e ) pℓm µν .”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § V. SACHS-WOLFE EFFECT IN THE MILNE AND MINKOWSKI UNIVERSES · p. 15

    In the Milne universe, the rms temperature anisotropy is infrared divergent; the p-integration in Eq. (5.6) diverges near p=0.

    “Near p = 0, the integrand behaves as ∼ 1 /p 2 , and so the p -integration in (5.6) is infrared divergent.”
    • Also stated in: § VI. SUMMARY · p. 16 · plain
      “However, a naive application of the Sachs-Wolfe formula to the Milne universe results in infrared divergent temperature fluctuations although the state is set in the Minkowski vacuum one.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The last scattering surface is at ξ=ξ_ls in the Milne universe and at t=t_ls in the Minkowski universe, and there are no intrinsic temperature fluctuations when photons are emitted from this surface.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationreasoning
    § I. INTRODUCTION · p. 1

    In the one-bubble open inflation model, a second inflation is required for entropy production.

    “So in this model, the second inflation is required for entropy production.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation
    § I. INTRODUCTION · p. 1

    The one-bubble open inflation scenario realizes an open universe (Ω0 < 1) in inflationary cosmology.

    “Recently, a scenario which realizes an open universe (Ω 0 < 1) in the context of inflationary cosmology has been discussed by many authors [1-6].”
    Quote in the paperNot a substantial claim0 of 6 sources checked
    • cites [1]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [2]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [3]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [4]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [5]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [6]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • May combine two claims.

    Interpretationnone given
    § V. SACHS-WOLFE EFFECT IN THE MILNE AND MINKOWSKI UNIVERSES · p. 16

    In the present model, the last scattering surface is defined by hand, so it does not have gauge invariant meaning.

    “In the present model, the last scattering surface is defined by hand. So it does not have gauge invariant meaning.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The last scattering surface is at ξ=ξ_ls in the Milne universe and at t=t_ls in the Minkowski universe, and there are no intrinsic temperature fluctuations when photons are emitted from this surface.

  • Classified as probably not a substantial claim.

    Interpretationreasoning
    § V. SACHS-WOLFE EFFECT IN THE MILNE AND MINKOWSKI UNIVERSES · p. 14

    The gauge must be fixed to make the problem definite.

    “Thus to make the problem definite the gauge must be fixed.”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim.

    Premisenone given
    § II. MASSLESS SCALAR FIELD IN THE MILNE UNIVERSE · p. 3

    The natural extension of the Milne coordinates covers the (spherical) Rindler universe.

    “The region covered by a natural extension of the Milne coordinates is known as the (spherical) Rindler universe.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § V. SACHS-WOLFE EFFECT IN THE MILNE AND MINKOWSKI UNIVERSES · p. 16

    The divergence is not a crisis of theoretical framework but is solely due to the unphysical situation considered here.

    “We claim that it is not the case but the divergence is solely due to the unphysical situation we have considered here.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The last scattering surface is at ξ=ξ_ls in the Milne universe and at t=t_ls in the Minkowski universe, and there are no intrinsic temperature fluctuations when photons are emitted from this surface.

  • May combine two claims.

    Interpretationreasoning
    § V. SACHS-WOLFE EFFECT IN THE MILNE AND MINKOWSKI UNIVERSES · p. 14

    Only even parity modes contribute to the Sachs-Wolfe effect because ˇhχχ is the only component that causes it.

    “Since ˇ h χχ = ξ -2 h χχ is the only component that causes the Sachs-Wolfe effect, only the even parity modes contribute to it.”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim.

    Premisecitation
    § I. INTRODUCTION · p. 1

    A method to handle the gravitational wave perturbation in an open inflationary universe is still unclear because of the existence of gauge degrees of freedom.

    “a method to handle the gravitational wave perturbation is still unclear because of the existence of gauge degrees of freedom.”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites [11]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [12]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Probably not a substantial claim: may not pass S4 (based).

    Premisenone given
    § I. INTRODUCTION · p. 1

    The time coordinate of the Milne universe gives the hyperbolic time-slicing of Minkowski spacetime.

    “The time coordinate of the Milne universe gives the hyperbolic time-slicing of Minkowski spacetime.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning
    § V. SACHS-WOLFE EFFECT IN THE MILNE AND MINKOWSKI UNIVERSES · p. 16

    If a scalar quantity determines the time slice of constant temperature, it will be coupled with gravity and the background spacetime will be no longer Minkowskian (nor purely de Sitter).

    “Then such a quantity will be inevitably coupled with gravity and background spacetime will be no longer Minkowskian (nor purely de Sitter).”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § V. SACHS-WOLFE EFFECT IN THE MILNE AND MINKOWSKI UNIVERSES · p. 14

    The background spacetime is empty, so nothing physically determines the last scattering surface.

    “Since the background spacetime is empty, there is nothing that physically determines the last scattering surface.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Study design and conductnone given
    § 1. even parity · p. 21

    The reduced equations for even and odd parity are consistent with the mode functions obtained in Section 3.

    “Of course, Eqs. (A26) and (A28) are consistent with the mode functions obtained in Section 3.”
    • Also stated in: § 2. odd parity · p. 25 · plain
      “Of course, Eqs. (A59) and (A61) are consistent with the mode functions obtained in Section 3.”
    Quote in the paperNot a substantial claim
    • Rests on an assumption: The Milne universe does not contain a Cauchy surface, but its extension, the Rindler universe, does.
16 of 16

  • The authors state this as an assumption.

    There exists a small imaginary part in t as t - iε.

    “assuming the existence of a small imaginary part in t as t -iǫ”

  • The paper relies on this without support.

    The supercurvature mode may be sensitively affected by the effect of gravity.

    “But a preliminary analysis suggests that the supercurvature mode may be sensitively affected by the effect of gravity (although the result in [12] will not be changed).”

  • The authors state this as an assumption.

    The last scattering surface is at ξ=ξ_ls in the Milne universe and at t=t_ls in the Minkowski universe, and there are no intrinsic temperature fluctuations when photons are emitted from this surface.

    “Here we assume that the last scattering surface is at ξ = ξ ls in the case of the Milne universe and at t = t ls in the case of the Minkowski universe, and that there is no intrinsic temperature fluctuations when photons are emitted from this surface.”

  • The paper relies on this without support.

    The Milne universe does not contain a Cauchy surface, but its extension, the Rindler universe, does.

    “The Milne universe does not contain a Cauchy surface but its extension, i.e., the Rindler universe, does.”
4 of 4

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

Each contradiction between two passages listed here was confirmed by an independent check.

  • Odd-parity derivation refers to an even-parity mode

    Minor

    The appendix defines Q as an odd-parity variable, but its final comparison uses the even-parity H component. That component is inconsistent with the odd-parity derivation.

1 of 1

Cites

Flagged first, then by citations

Research works printed but not linked: 1

  1. M. Bucher, 1995, [Nucl. Phys. B (Proc. Suppl.)

Cited by

Sorted by citations

Related

Authors
Tanaka, T., & Sasaki, M.
Affiliations
The University of Osaka (JP)
ORCID
Venue
Physical Review D
Publisher
American Physical Society (APS)
Published
1997-05-15
Volume
55
Issue
10
Pages
6061-6080
ISSN
  • 0556-2821 (ISSN-L)
  • 1089-4918
DOI
10.1103/physrevd.55.6061
OpenAlex
W2007957836
Semantic Scholar
083016a786d87d994380777dd4946d8eb7c86923
Cited by
25
References
25
Citations by year
2021: 1 · 2020: 1 · 2015: 1 · 2013: 1
FWCI
3.47
Type
journal article
Language
en
Open access
Licence
link.aps.org/licenses/aps-default-license (vor)
Locations
  • Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
  • arXiv (Cornell University)
Topics
  • Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Cosmology and Gravitation Theories
  • Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Noncommutative and Quantum Gravity Theories
  • Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
Keywords
Milne universe, open inflation, de Sitter space, quantum field theory in curved spacetime

About these checks

Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.

Claims read by deepseek-v4.1-flash under definition version 3 (extract-v3) on 2026-11-05.

Classified by jev-1.13-20260917 (gold-guideline-v0.2) on 2026-11-05.

Statistics and arithmetic are recomputed, not judged.

Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.

What counts as a claim ›How we measure confidence ›How we check this ›

Are you an author, or do you see a problem with this page? Request a review