Pith. sign in

REVIEW 3 major objections 5 minor 52 references

Cosmological perturbations meet Wheeler DeWitt

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper establishes a semiclassical bridge between the Wheeler-DeWitt wavefunction of the universe and the wavefunction of standard cosmological perturbation theory, through an explicit phase and normalization identity verified in two…

desk verdict Careful WKB-to-perturbation-theory dictionary for mini-superspace WdW, with the one load-bearing mass-term check in Sec. 6.2 asserted rather than shown; worth refereeing after that gap is filled. read the letter →

arxiv 2412.19782 v2 pith:J5AQR6KX submitted 2024-12-27 hep-th astro-ph.COgr-qc

Federico Piazza , Siméon Vareilles This is my paper · ORCID
classification hep-thastro-ph.COgr-qc
keywords Wheeler-DeWittequationcosmologicalperturbationtheoryminisuperspacesemiclassicalapproximationwavefunctionoftheuniversegaugefixinginflationquantumcosmology
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

The paper asks how the non-perturbative Wheeler-DeWitt equation, which governs the wavefunction of the universe without reference to a background, relates to the Schrödinger wavefunction $\psi_P$ used in cosmological perturbation theory. It proposes a specific identity, Eq. (3.13): $\psi_P = e^{if(q)/\alpha}\,(\sqrt{-g}\,B_0 S)^{1/2}\,\psi + O(\alpha)$, where $\alpha$ is a dimensionless gravitational coupling, roughly $H_*^2/M_P^2$, that plays the role of $\hbar$ in a WKB expansion. The identity is verified in two mini-superspace models, an exponential potential and a slow-roll potential, each in both unitary and spatially flat gauges. If correct, the Wheeler-DeWitt equation reproduces perturbative quantum cosmology at leading order in $\alpha$, and the apparent absence of the perturbation mass term in the Wheeler-DeWitt equation is explained: the mass term is generated by a canonical phase redefinition rather than being present in the equation from the start.

What carries the argument

The load-bearing object is the semiclassical WKB decomposition $\Psi = e^{iS/\alpha}\psi$, with the dimensionless coupling $\alpha \sim H_*^2/M_P^2$ taking the place of $\hbar$. The function $S$ satisfies the Hamilton-Jacobi equation and defines a congruence of classical trajectories; $\psi$ then obeys a Schrödinger-like equation whose time derivative is provided by the vector field $B_\mu S\,B^\mu$. The bridge identity (3.13) ties $\psi$ to the perturbation-theory wavefunction $\psi_P$ through the normalization $(\sqrt{-g}\,B_0 S)^{1/2}$ and a field-dependent phase $e^{if(q)/\alpha}$. This phase is the mechanism that resolves the missing-mass puzzle: a canonical transformation that moves a total derivative from the Lagrangian into the wavefunction turns a term of the form $i\,\phi\,\partial_\phi\psi$ in the Wheeler-DeWitt equation into the $\phi^2$ potential term of perturbation theory.

What would settle it

Solve the Wheeler-DeWitt equation (3.3) numerically for the slow-roll potential (4.24) at small but nonzero $\alpha$, construct the conditional probability density (3.12), and compare it with the perturbation-theory density $|\psi_P|^2$ built from Eq. (4.25) through the bridge relation (3.13): the two should agree to order $\alpha$. A disagreement that grows with $\alpha$ would show the semiclassical identity is not the limit of the exact equation.

Watch

Extended reading notes

Core claim

The central claim is that the gap between the Wheeler-DeWitt equation $H\Psi=0$ and the Schrödinger equation $i\partial_t\psi_P = H\psi_P$ is closed by writing $\Psi = e^{iS/\alpha}\psi$, with $S$ a real solution of the Hamilton-Jacobi equation and $\alpha$ the small semiclassical parameter. The modulation $\psi$ obeys a Schrödinger-like equation in which the vector field $B_\mu S\,B^\mu$ acts as a time derivative. The paper then proposes the bridge relation $\psi_P = e^{if(q)/\alpha}(\sqrt{-g}\,B_0 S)^{1/2}\psi + O(\alpha)$, with the prefactor fixed by matching the conditional probability densities of the two approaches: perturbation theory conditions on a chosen clock variable, while the Wheeler-DeWitt wavefunction requires choosing a foliation of field space before a probability can be extracted. The phase $f(q)$ is fixed by requiring that the order-$\alpha^0$ spatial derivative terms in the semiclassical equation be converted into the familiar potential term of perturbation theory, a canonical transformation illustrated with a free-particle example. The construction is carried out explicitly for the two models, in which unitary gauge corresponds to using the scalar field as time and spatially flat gauge to using the scale factor as time.

Load-bearing premise

The comparison rests on a single premise: a super-Hubble patch of the universe can be modeled as one homogeneous system with a single scale factor and a single scalar field, so that the zero-momentum variables $\zeta$ and $\varphi$ in the paper are the physical perturbations; if spatial gradients, tensor modes, or nonzero-momentum effects matter, the relation (3.13) may not survive.

Editorial extensions

If this is right

  • At leading order in $\alpha$, the Wheeler-DeWitt equation reproduces the conditional probabilities of cosmological perturbation theory: the two wavefunctions represent the same state up to the phase and normalization of Eq. (3.13).
  • The perturbation-theory mass term is not an input to the Wheeler-DeWitt equation; it emerges from a canonical phase redefinition, so the absence of an explicit potential in the semiclassical equation is no longer an obstruction to matching perturbation theory.
  • In comoving, classically conserved variables, the semiclassical wavefunction has no diffusion at $\alpha=0$, so a state initially peaked on the classical trajectory remains peaked; deviations from the classical background appear only at order $\alpha$.
  • The higher time-derivative terms that distinguish the Wheeler-DeWitt equation from the perturbation-theory Schrödinger equation are suppressed by powers of $\alpha$, in the same way that the non-relativistic limit of the Klein-Gordon equation suppresses higher time derivatives.
  • Non-Gaussian terms already present in the Wheeler-DeWitt equation are enhanced by a factor $1/\epsilon$ and become sizable in the eternal-inflation regime $\alpha/\epsilon \gtrsim 1$, where they generate deterministic, computable deviations from the classical trajectory.

Reading between the lines

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

  • If the identity extends to nonzero momentum modes, the phase $e^{if/\alpha}$ should become a momentum-dependent phase in the wavefunction; equal-time correlators built from moduli would be unchanged, but the squeezing angle of each Fourier mode would differ between the two formalisms, offering a sharp place to look for Wheeler-DeWitt corrections.
  • The deterministic deviations computed here suggest a complementary picture to stochastic inflation: in the eternal-inflation regime the Wheeler-DeWitt equation predicts specific non-Gaussian tails, whereas stochastic approaches model the same regime as noise; a test could compare the probability tails from the WdW equation with the stochastic distribution for the same potential.
  • The phase-redefinition mechanism points toward an effective field theory of Wheeler-DeWitt corrections: working order by order in $\alpha$ should yield higher-derivative operators added to the perturbation-theory Hamiltonian, a route the paper notes as future work.
  • A numerical solution of the Wheeler-DeWitt equation at finite $\alpha$ for either model would turn the order-$\alpha$ mismatch identified in the slow-roll gauge into a quantitative prediction for the shift of the wavefunction's peak, which perturbation theory alone cannot provide.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies the relation between solutions of the Wheeler-DeWitt (WdW) equation and the wavefunction of standard cosmological perturbation theory, in a mini-superspace truncation. The authors introduce a dimensionless gravitational coupling α and use the WKB ansatz Ψ = e^{iS/α} ψ, with S satisfying the Hamilton-Jacobi equation and ψ obeying a Schrödinger-like equation. By comparing the conditional probability densities of the two frameworks, they propose Eq. (3.13): ψ_P = e^{if(q)/α} (√(-g) B_0 S)^{1/2} ψ + O(α). This relation is tested in two scalar-field models — a purely exponential potential (Model 1) and a slow-roll potential with nonzero η (Model 2) — each in the mini-superspace analogues of unitary and spatially flat gauges. The paper also discusses conditional probabilities, the apparent absence of a mass term in the WdW equation and its recovery through a phase redefinition, higher-time-derivative terms, and possible deviations from classical backgrounds. The central claim is that, at leading order in α, the WdW equation reproduces perturbative quantum cosmology, with the dictionary given by (3.13).

Significance. If fully established, the dictionary (3.13) provides a concrete and useful bridge between the WdW wavefunction and the standard cosmological perturbation-theory wavefunction. It would resolve an apparent puzzle — the WdW equation for ψ contains no potential term while perturbation theory has mass terms — by showing that the potential is generated by a canonical phase redefinition. The paper has notable strengths: the relation (3.13) is parameter-free, no numbers are fitted, two analytic models are worked out in two gauges, and the discussion of conditional probabilities as gauge fixing is conceptually clear. The authors also honestly flag the mini-superspace limitation. However, the most nontrivial verification, namely the nonzero-mass case in Sec. 6.2, is asserted rather than demonstrated, and the unitary-gauge check in Sec. 6.1 partly assumes the target relation. The central claim is therefore not yet fully established, although it is plausible and the explicit formulas that are shown are internally consistent.

major comments (3)
  1. [6.2] The phase-redefinition mechanism that is the heart of the paper's resolution of the missing-potential puzzle is not verified. After Eq. (6.18), the paper states that with f(φ,ρ) of Eq. (6.20), the left-hand side no longer contains a φ-derivative and the equation acquires a mass term matching (4.25). But the transformed equation for \tildeψ is never written. Since f is O(η/ϵ) and contains e^{-(-3+ϵ)ρ+ϵφ} with polynomial factors, its first and second derivatives generate many terms from ∂_φ^2, ∂_ρ^2, and the cross terms in (6.18); the cancellation is not visible. Please display the substitution and the resulting equation for \tildeψ, or provide a supplementary computation, and state the precise order in the slow-roll expansion at which the matching holds.
  2. [6.1] The check in unitary gauge is partly circular. Eq. (6.14) inserts the target relation (3.13) into the perturbative Schrödinger equation (4.12) to derive Eq. (6.15); the subsequent comparison with the WdW result (6.13) therefore tests consistency of the two derivations rather than independently verifying (3.13). The agreement at α=0 is reassuring, but the logical status of each step should be made explicit, or the derivation should be restructured so that (3.13) is the output rather than an input.
  3. [2.1 and Sec. 7] The paper's scope is the mini-superspace (zero-momentum) truncation, and the authors acknowledge this in the introduction and Sec. 2.1. This limitation is load-bearing for the claim that (3.13) is 'the' relation between WdW and cosmological perturbation theory: the k=0 variables ζ and φ are not the same as the physical Fourier modes ζ_k, and Sec. 4.1 explicitly notes that the imaginary parts of their wavefunctions behave differently (ζ does not become as squeezed as ζ_k). The manuscript should either frame the central claim as specifically mini-superspace, or provide an argument that the dictionary extends to nonzero k, before making the broader statement in the abstract and introduction.
minor comments (5)
  1. [Introduction] There is a typo 'Minkoswki' in the Introduction; it should be 'Minkowski'. Also, in Sec. 4, 'FLR W-gravity' appears to be a typo for 'FLRW'.
  2. [5.1] In Eq. (5.7), the distinction between 'black' and 'red' terms is invisible in monochrome print or for color-blind readers; please use explicit labels or markers instead of color alone.
  3. [3.4] The suppression of the B-term in Eq. (3.24) is demonstrated only for a simplified 1+1 model with a Gaussian ansatz. This is acceptable for a discussion section, but the later statement in Sec. 6.1 that higher χ-derivatives are 'effectively α-suppressed' should cite this heuristic derivation rather than presenting it as established for the full system.
  4. [2.2] The symbol ψ_P is used in Eq. (2.10) before it is defined as the perturbation-theory wavefunction; please define it at first use.
  5. [3.2] In Eq. (3.12) and (3.13), the expression √(-g B_0 S) presumes a sign convention; the text notes that the sign can be flipped when B_0 S is negative, but this should be stated explicitly at the point of the definition.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the WdW-to-perturbation-theory dictionary is an ansatz tested against independently derived equations; the self-citations are not load-bearing.

full rationale

The central relation (3.13) is proposed by comparing the probability densities (2.11) and (3.12), not by fitting a parameter or by assuming the perturbation-theory result; the phase f is left free and is later fixed by a canonical-transformation argument. The model checks are genuine consistency tests: in Sec. 5.1 and Sec. 6.1 the paper takes the independently derived WdW equation for psi, eqs. (5.7) and (6.13), and the independently derived perturbation-theory Schrodinger equation, eqs. (4.12) and (4.25), and uses the proposed relation (3.13) as a translation rule to compare them. The comparison is not vacuous: in Sec. 6.1 the two resulting equations disagree at order alpha, as shown by the red terms in (6.15) versus (6.13), which is exactly the content of the O(alpha) statement in (3.13). In Sec. 6.2 the phase f(phi,rho) in (6.20) is chosen 'by inspection' to eliminate the phi-derivative term, and the subsequent claim that the induced mass 'matches precisely' the perturbation-theory mass in (4.25) is asserted rather than displayed; this is an omitted verification step, not circularity, since f is not selected to force that mass. The self-citations, Refs. [6], [37], and [51], are contextual, interpretative, or future-work and are not load-bearing for (3.13); no uniqueness theorem is imported from the authors' earlier papers and no fitted input is renamed as a prediction. The main limitations, such as the mini-superspace patch assumption and the k=0 truncation, are correctness and scope concerns, not circularity. Therefore no circular step is established.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

No invented entities (no new particles, forces, or dimensions). The central claim depends on the WKB quantization of the WdW equation, the special choice of a real HJ function S from a congruence of solutions, and the mini-superspace patch truncation; all inputs are model scales, not numbers fitted to data.

free parameters (2)
  • alpha = 1/(L H_*) ≈ H_*^2/M_P^2 = ~10^-9 epsilon during observable inflation (Sec. 2.1)
    Dimensionless coupling that plays the role of hbar in the WKB expansion, defined in Eq. (2.5). It is a physical scale of the system, not fitted to observations in this paper, but it is the expansion parameter on which the central comparison (3.13) rests.
  • Slow-roll parameters epsilon, eta at phi=0 = epsilon, eta (model inputs, not fitted here)
    Enter the potential ansatz (4.22)-(4.24) and the background solution (6.3); they are part of the model definition, not free parameters adjusted to data.
assumptions (5)
  • domain assumption The WdW equation (3.3) with the covariant d'Alembertian is the correct quantum constraint for the system
    Sec. 3 states this is 'the only choice for p^2 which seems to make sense', citing [9]; the paper's central comparison assumes this factor ordering.
  • domain assumption The HJ solution S is generated by a surface-orthogonal congruence of classical solutions via B_mu S = qbar_dot_mu / H_* (3.8)
    Sec. 3.1 constructs S this way; in Model 2 surface orthogonality holds only barring O(epsilon^2) terms (Sec. 6).
  • ad hoc to paper The WKB ansatz Psi = e^{iS/alpha} psi with real S and slowly-varying psi represents the closest quantum analog of a chosen classical background
    Sec. 3 explicitly takes S real and ignores initial conditions; App. A.1 shows S is highly non-unique, so the choice is an input to the comparison.
  • domain assumption A comoving super-Hubble patch is described by the mini-superspace action (2.2)
    Sec. 2.1 separate-universe reasoning; this is the paper's own stated limitation.
  • domain assumption The conditional probability is given by the conserved current j^0 of Eq. (3.12), positive in the semiclassical regime
    Sec. 3.2; the paper notes j^0 is not positive-definite in general and that the issue should be resolved by third quantization.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Cosmological perturbations meet Wheeler DeWitt." pith.science (2026). https://pith.science/paper/J5AQR6KX

@misc{pith2026241219782,
  author       = {Pith},
  title        = {Pith review of: Cosmological perturbations meet Wheeler DeWitt},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J5AQR6KX}},
  note         = {Machine review of arXiv:2412.19782}
}
abstract

We study approximate solutions of the Wheeler DeWitt (WdW) equation and compare them with the standard results of cosmological perturbation theory. In mini-superspace, we introduce a dimensionless gravitational coupling $\alpha$ that is typically very small and functions like $\hbar$ in a WKB expansion. We seek solutions of the form $\Psi = e^{iS/\alpha} \psi$ that are the closest quantum analog of a given classical background spacetime. The function $S$ satisfies the Hamilton-Jacobi equation, while $\psi$ obeys a Schr\"odinger-like equation and can be given a probabilistic interpretation. The semiclassical limit suggests a specific relation between $\psi$ and the standard perturbation-theory wavefunction $\psi_P$. We verify this relation in two main examples: a scalar field with a purely exponential potential, of which simple scaling solutions are known and a slow-roll scenario expanded in the vicinity of the origin in field space. Each example is worked out in two different gauges, that are the minisuperspace equivalent of unitary gauge and spatially flat gauge. We discuss possible deviations from the classical background trajectory as well as the higher ``time" derivative terms that are present in the WdW equation but not in the perturbative approach. We clarify the \emph{conditional probability} content of the wavefunctions and how this is related with the standard gauge fixing procedure in perturbation theory.

Figures

Figures reproduced from arXiv: 2412.19782 by the authors.

Figure 1
Figure 1. The WdW wavefunction Ψpϕ, ρq (in blue) is approximately centered around the classical trajectory ¯ρpϕq. In order to obtain a probability from it one has to choose how to section the field space, i.e. which variable should be used as “time”. If we use ϕ, we are going to calculate the conditional probability dPpζ|ϕq (in green) of getting some value of ρ, or of ζ “ ρ ´ ρ¯pϕq, given that the scalar field evaluates ϕ. Th… view at source ↗
Figure 2
Figure 2. Two examples of solutions of the HJ equation associated with a given classical trajectory (think, green). Dotted in blue, some neighboring trajectories for each case. In the notation introduced in Sec. 3.1, such curves are examples of ζ i “ const. curves because they correspond to potential solutions of the classical equations of motion. A.2 Use of Fermi coordinates The above example shows that S is all but unique. … view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

52 extracted references · 21 canonical work pages

  1. [1]

    B. S. DeWitt, Quantum Theory of Gravity. 1. The Canonical Theory, Phys. Rev. 160, 1113–1148, 1967

    work page 1967

  2. [2]

    Kuchar, Ground state functional of the linearized gravitational field, J

    K. Kuchar, Ground state functional of the linearized gravitational field, J. Math. Phys. 11, 3322–3334, 1970

    work page 1970

  3. [3]

    Chakraborty, J

    T. Chakraborty, J. Chakravarty, V. Godet, P. Paul and S. Raju, The Hilbert space of de Sitter quantum gravity, JHEP 01, 132, 2024, [arXiv:2303.16315 [hep-th]]

    arXiv 2024

  4. [4]

    Creminelli, M

    P. Creminelli, M. A. Luty, A. Nicolis and L. Senatore, Starting the Universe: Stable Violation of the Null Energy Condition and Non-standard Cosmologies, JHEP 12, 080, 2006, [arXiv:hep-th/0606090]

    arXiv 2006

  5. [5]

    Cheung, P

    C. Cheung, P. Creminelli, A. L. Fitzpatrick, J. Kaplan and L. Senatore, The Effective Field Theory of Inflation, JHEP 03, 014, 2008, [arXiv:0709.0293 [hep-th]]. 30

    arXiv 2008

  6. [6]

    Piazza and F

    F. Piazza and F. Vernizzi, Effective Field Theory of Cosmological Perturbations, Class. Quant. Grav. 30, 214007, 2013, [arXiv:1307.4350 [hep-th]]

    arXiv 2013

  7. [7]

    S. A. Salcedo, M. H. G. Lee, S. Melville and E. Pajer, The Analytic Wavefunction, JHEP 06, 020, 2023, [arXiv:2212.08009 [hep-th]]

    arXiv 2023

  8. [8]

    Benincasa, Amplitudes meet Cosmology: A (Scalar) Primer, 2022, [arXiv:2203.15330 [hep-th]]

    P. Benincasa, Amplitudes meet Cosmology: A (Scalar) Primer, 2022, [arXiv:2203.15330 [hep-th]]

    arXiv 2022

Show all 52 references
  1. [9]

    Nicolis, F

    A. Nicolis, F. Piazza and K. Zeghari, Rotating cosmologies: classical and quantum, JCAP 10, 059, 2022, [arXiv:2204.04110 [hep-th]]

    2022 arXiv

  2. [10]

    J. J. Halliwell, Derivation of the Wheeler-De Witt Equation from a Path Integral for Minisuperspace Models, Phys. Rev. D 38, 2468, 1988

    1988

  3. [11]

    Vilenkin, The Interpretation of the Wave Function of the Universe, Phys

    A. Vilenkin, The Interpretation of the Wave Function of the Universe, Phys. Rev. D 39, 1116, 1989

    1989

  4. [12]

    J. J. Halliwell, INTRODUCTORY LECTURES ON QUANTUM COSMOLOGY, in 7th Jerusalem Winter School for Theoretical Physics: Quantum Cosmology and Baby Universes , pp. 159–243, 1989. [arXiv:0909.2566 [gr-qc]]

    1989 arXiv

  5. [13]

    Janssen, Slow-roll approximation in quantum cosmology, Class

    O. Janssen, Slow-roll approximation in quantum cosmology, Class. Quant. Grav. 38, 095003, 2021, [arXiv:2009.06282 [gr-qc]]

    2021 arXiv

  6. [14]

    J. B. Hartle, Space-time quantum mechanics and the quantum mechanics of space-time, in Les Houches Summer School on Gravitation and Quantizations, Session 57 , pp. 0285–480, 1992. [arXiv:gr-qc/9304006]

    1992 arXiv

  7. [15]

    Maldacena, The role of the wdw equation?, YouTube, 2015

    J. Maldacena, The role of the wdw equation?, YouTube, 2015

    2015

  8. [16]

    Lehners, Review of the no-boundary wave function, Phys

    J.-L. Lehners, Review of the no-boundary wave function, Phys. Rept. 1022, 1–82, 2023, [arXiv:2303.08802 [hep-th]]

    2023 arXiv

  9. [17]

    Maldacena, Comments on the no boundary wavefunction and slow roll inflation, 2024, [arXiv:2403.10510 [hep-th]]

    J. Maldacena, Comments on the no boundary wavefunction and slow roll inflation, 2024, [arXiv:2403.10510 [hep-th]]

    2024 arXiv

  10. [18]

    S. W. Hawking, The Boundary Conditions of the Universe, Pontif. Acad. Sci. Scr. Varia 48, 563–574, 1982

    1982

  11. [19]

    J. B. Hartle and S. W. Hawking, Wave Function of the Universe, Phys. Rev. D 28, 2960–2975, 1983

    1983

  12. [20]

    Brizuela, C

    D. Brizuela, C. Kiefer and M. Kr¨ amer, Quantum-gravitational effects on gauge-invariant scalar and tensor perturbations during inflation: The slow-roll approximation, Phys. Rev. D 94, 123527, 2016, [arXiv:1611.02932 [gr-qc]]

    2016 arXiv

  13. [21]

    Di Gioia, G

    F. Di Gioia, G. Maniccia, G. Montani and J. Niedda, Nonunitarity problem in quantum gravity corrections to quantum field theory with Born-Oppenheimer approximation, Phys. Rev. D 103, 103511, 2021, [arXiv:1912.09945 [gr-qc]]

    2021 arXiv

  14. [22]

    Chataignier, Gauge Fixing and the Semiclassical Interpretation of Quantum Cosmology, Z

    L. Chataignier, Gauge Fixing and the Semiclassical Interpretation of Quantum Cosmology, Z. Naturforsch. A 74, 1069–1098, 2019, [arXiv:1901.07104 [gr-qc]]

    2019 arXiv

  15. [23]

    A. Y. Kamenshchik, A. Tronconi and G. Venturi, The Born-Oppenheimer approach to Quantum Cosmology, 2020, [arXiv:2010.15628 [gr-qc]]. 31

    2020 arXiv

  16. [24]

    Chataignier and M

    L. Chataignier and M. Kr¨ amer, Unitarity of quantum-gravitational corrections to primordial fluctuations in the Born-Oppenheimer approach, Phys. Rev. D 103, 066005, 2021, [arXiv:2011.06426 [gr-qc]]

    2021 arXiv

  17. [25]

    Chataignier, Beyond semiclassical time, Z

    L. Chataignier, Beyond semiclassical time, Z. Naturforsch. A 77, 805–812, 2022, [arXiv:2205.09147 [gr-qc]]

    2022 arXiv

  18. [26]

    Maniccia, G

    G. Maniccia, G. Montani and S. Antonini, QFT in curved spacetime from quantum gravity: Proper WKB decomposition of the gravitational component, Phys. Rev. D 107, L061901, 2023, [arXiv:2302.10832 [gr-qc]]

    2023 arXiv

  19. [27]

    Chataignier, C

    L. Chataignier, C. Kiefer and P. Moniz, Observations in quantum cosmology, Class. Quant. Grav. 40, 223001, 2023, [arXiv:2306.14948 [gr-qc]]

    2023 arXiv

  20. [28]

    D. S. Salopek and J. R. Bond, Nonlinear evolution of long wavelength metric fluctuations in inflationary models, Phys. Rev. D 42, 3936–3962, 1990

    1990

  21. [29]

    Sasaki and E

    M. Sasaki and E. D. Stewart, A General analytic formula for the spectral index of the density perturbations produced during inflation, Prog. Theor. Phys. 95, 71–78, 1996, [arXiv:astro-ph/9507001]

    1996 arXiv

  22. [30]

    Wands, K

    D. Wands, K. A. Malik, D. H. Lyth and A. R. Liddle, A New approach to the evolution of cosmological perturbations on large scales, Phys. Rev. D 62, 043527, 2000, [arXiv:astro-ph/0003278]

    2000 arXiv

  23. [31]

    Pattison, V

    C. Pattison, V. Vennin, H. Assadullahi and D. Wands, Stochastic inflation beyond slow roll, JCAP 07, 031, 2019, [arXiv:1905.06300 [astro-ph.CO]]

    2019 arXiv

  24. [32]

    Artigas, J

    D. Artigas, J. Grain and V. Vennin, Hamiltonian formalism for cosmological perturbations: the separate-universe approach, JCAP 02, 001, 2022, [arXiv:2110.11720 [astro-ph.CO]]

    2022 arXiv

  25. [33]

    Creminelli, S

    P. Creminelli, S. Dubovsky, A. Nicolis, L. Senatore and M. Zaldarriaga, The Phase Transition to Slow-roll Eternal Inflation, JHEP 09, 036, 2008, [arXiv:0802.1067 [hep-th]]

    2008 arXiv

  26. [34]

    Goeller, P

    C. Goeller, P. A. Hoehn and J. Kirklin, Diffeomorphism-invariant observables and dynamical frames in gravity: reconciling bulk locality with general covariance, 2022, [arXiv:2206.01193 [hep-th]]

    2022 arXiv

  27. [35]

    Witten, Algebras, Regions, and Observers, 2023, [arXiv:2303.02837 [hep-th]]

    E. Witten, Algebras, Regions, and Observers, 2023, [arXiv:2303.02837 [hep-th]]

    2023 arXiv

  28. [36]

    Geng, Quantum Rods and Clock in a Gravitational Universe, 2024, [arXiv:2412.03636 [hep-th]]

    H. Geng, Quantum Rods and Clock in a Gravitational Universe, 2024, [arXiv:2412.03636 [hep-th]]

    2024 arXiv

  29. [37]

    Piazza, Glimmers of a post-geometric perspective, Class

    F. Piazza, Glimmers of a post-geometric perspective, Class. Quant. Grav. 40, 165014, 2023, [arXiv:2108.12362 [hep-th]]

    2023 arXiv

  30. [38]

    Khan, The Semiclassical Approximation: Its Application to Holography and the Information Paradox, 2023, [arXiv:2309.08116 [gr-qc]]

    R. Khan, The Semiclassical Approximation: Its Application to Holography and the Information Paradox, 2023, [arXiv:2309.08116 [gr-qc]]

    2023 arXiv

  31. [39]

    H. M. Van Horn and E. E. Salpeter, WKB Approximation in Three Dimensions, Phys. Rev. 157, 751–758, 1967

    1967

  32. [40]

    Chataignier, Relational observables, reference frames, and conditional probabilities, Phys

    L. Chataignier, Relational observables, reference frames, and conditional probabilities, Phys. Rev. D 103, 026013, 2021, [arXiv:2006.05526 [gr-qc]]

    2021 arXiv

  33. [41]

    Bozza, M

    V. Bozza, M. Giovannini and G. Veneziano, Cosmological perturbations from a new physics hypersurface, JCAP 05, 001, 2003, [arXiv:hep-th/0302184]. 32

    2003 arXiv

  34. [42]

    Celoria, P

    M. Celoria, P. Creminelli, G. Tambalo and V. Yingcharoenrat, Beyond perturbation theory in inflation, JCAP 06, 051, 2021, [arXiv:2103.09244 [hep-th]]

    2021 arXiv

  35. [43]

    Creminelli, S

    P. Creminelli, S. Renaux-Petel, G. Tambalo and V. Yingcharoenrat, Non-perturbative wavefunction of the universe in inflation with (resonant) features, JHEP 03, 010, 2024, [arXiv:2401.10212 [hep-th]]

    2024 arXiv

  36. [44]

    Lehners and J

    J.-L. Lehners and J. Quintin, Delicate curvature bounces in the no-boundary wave function and in the late universe, 2024, [arXiv:2403.15205 [gr-qc]]

    2024 arXiv

  37. [45]

    A. D. Linde, Eternally Existing Selfreproducing Chaotic Inflationary Universe, Phys. Lett. B 175, 395–400, 1986

    1986

  38. [46]

    A. D. Linde, ETERNAL CHAOTIC INFLATION, Mod. Phys. Lett. A 1, 81, 1986

    1986

  39. [47]

    A. H. Guth, Eternal inflation and its implications, J. Phys. A 40, 6811–6826, 2007, [arXiv:hep-th/0702178]

    2007 arXiv

  40. [48]

    Garriga and V

    J. Garriga and V. F. Mukhanov, Perturbations in k-inflation, Phys. Lett. B 458, 219–225, 1999, [arXiv:hep-th/9904176]

    1999 arXiv

  41. [49]

    J. M. Maldacena, Non-Gaussian features of primordial fluctuations in single field inflationary models, JHEP 05, 013, 2003, [arXiv:astro-ph/0210603]

    2003 arXiv

  42. [50]

    L. Hui, A. Nicolis and A. Podo To appear

  43. [51]

    Nitti, F

    F. Nitti, F. Piazza and A. Taskov, Relativity of the event: examples in JT gravity and linearized GR, JHEP 10, 092, 2024, [arXiv:2402.01847 [hep-th]]

    2024 arXiv

  44. [52]

    A. I. Nesterov, Riemann normal coordinates, Fermi reference system and the geodesic deviation equation, Class. Quant. Grav. 16, 465–477, 1999, [arXiv:gr-qc/0010096]. 33

    1999 arXiv

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.