arxiv: 2605.08428 · v1 · submitted 2026-05-08 · 🌀 gr-qc

Recognition: 3 theorem links

· Lean Theorem

Bipartite temporal Bell inequality for squeezed coherent state of inflationary perturbations

Aurindam Mondal

Authors on Pith no claims yet

Pith reviewed 2026-05-12 01:13 UTC · model grok-4.3

classification 🌀 gr-qc

keywords temporal Bell inequalityinflationary perturbationssqueezed coherent stateprimordial perturbationsBunch-Davies vacuumquantum cosmologyBell operator expectation value

0 comments

The pith

Coherent initial states for inflationary perturbations produce no temporal Bell inequality violation.

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

The paper examines a temporal analogue of Bell's inequality applied to primordial perturbations when the standard Bunch-Davies vacuum is replaced by a coherent state. It derives an exact analytical form for the expectation value of the bipartite temporal Bell operator built from a single pseudo-spin component measured at two different times. No violation of the inequality appears for the coherent state. Adding squeezing yields values that differ only slightly from the pure squeezed-vacuum case at large squeezing parameters. This leads to the conclusion that distinguishing among possible initial states of inflation does not depend on seeing a temporal Bell violation. The temporal version also shows a dependence on the imaginary phase factor of the wave function that is absent from spatial Bell inequalities.

Core claim

Assuming a coherent state as the initial condition, the expectation value of the bipartite temporal Bell operator is obtained in closed form and remains below the classical bound for all parameter values. For the squeezed coherent state the same expectation value deviates only modestly from the corresponding squeezed-vacuum result when the squeezing parameter becomes large. Consequently, the violation (or non-violation) of the temporal Bell inequality cannot by itself discriminate between different candidate initial states of primordial perturbations.

What carries the argument

Bipartite temporal Bell operator formed from time-separated measurements of one component of the pseudo-spin operator.

If this is right

Temporal Bell tests remain usable in cosmology because they require only one observable component.
Absence of violation for coherent states means other observables must be used to detect quantum imprints of such initial conditions.
Small differences between squeezed coherent and squeezed vacuum results at large squeezing allow limited discrimination without a violation.
The dependence on the imaginary phase factor is a distinctive property of the temporal inequality.

Where Pith is reading between the lines

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

Precision measurements of the Bell-operator expectation value could still constrain squeezing even when no violation occurs.
The phase sensitivity unique to the temporal case offers a potential additional handle on the wave-function content of primordial fluctuations.
The same construction may be applied to other non-vacuum states without requiring full two-observable spatial Bell tests.

Load-bearing premise

A coherent state is a physically plausible initial condition for inflationary perturbations and single-component measurements at separated times suffice to build a meaningful temporal Bell test under cosmological constraints.

What would settle it

An explicit numerical evaluation of the derived expectation value for a chosen squeezing parameter and phase that exactly reproduces the squeezed-vacuum result instead of the reported slight difference.

Figures

Figures reproduced from arXiv: 2605.08428 by Aurindam Mondal.

**Figure 2.** Figure 2: Bipartite temporal Bell operator as a function of the real part of [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗

read the original abstract

We investigate the role of the bipartite temporal Bell inequality, an analogue of the spatial Bell inequality, in probing the quantum imprints of primordial perturbations when the initially chosen Bunch-Davies vacuum is replaced by a coherent state. Although it is based on the same principles of locality and realism, its primary advantage lies in the fact that it does not require two distinct set of observable for its construction. Instead, measurements performed on a single component of the pseudo-spin operator at different times are sufficient. Consequently, it is particularly well suited for cosmological scenarios, where observational constraints typically allow access to only one component of the pseudo-spin operator. Assuming a coherent state as the initial condition, we derive an analytical expression for the expectation value of the bipartite temporal Bell operator and demonstrate the absence of temporal Bell violation in such a scenario. Interestingly, the results for squeezed coherent state is found to differ - albeit slightly - from those of squeezed vacuum state for large values of the squeezing parameter. This suggests that the ability to distinguish among different initial states of primordial perturbations does not rely on the violation of temporal Bell inequality. Furthermore, the dependence of the temporal Bell inequality on a purely imaginary phase factor of the wave function appears to be an unique feature, which is entirely absent in the context of spatial Bell inequalities.

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.

Desk Editor's Note private letter to a colleague

The paper calculates no temporal Bell violation for coherent-state initial conditions in inflation, with only a minor difference from squeezed vacuum at large squeezing and a noted phase dependence.

read the letter

The key result is an analytical expression for the expectation value of the bipartite temporal Bell operator under a coherent initial state, showing no violation. The squeezed coherent case differs slightly from squeezed vacuum only at large squeezing parameters, and the inequality picks up a dependence on the imaginary phase of the wave function that spatial versions lack. This is framed as evidence that Bell violation is not needed to tell initial states apart in this setup. The work does a clean job of adapting the temporal test to cosmology, where only one pseudo-spin component is typically measurable at different times, avoiding the need for two separate observables. The calculation is presented as direct and parameter-dependent only through the squeezing value. That said, the coherent state is taken as given without strong observational anchoring, so the no-violation outcome tracks the input assumption rather than an independent test. The difference at high squeezing is small enough that it may not shift practical distinctions much. The derivation itself is not reproduced in the abstract, but the stress-test logic holds: the operator construction and time-evolved correlators show no internal contradiction. This is for people already working on quantum signatures of inflation and Bell-type tests in curved spacetime. A reader wanting broad new frameworks or observational predictions will find little, but those needing explicit results for coherent versus vacuum states will get a usable calculation. The paper is coherent on its own terms and deserves peer review to check the algebra and assess whether the phase dependence survives closer scrutiny.

Referee Report

0 major / 4 minor

Summary. The manuscript derives an analytical expression for the expectation value of the bipartite temporal Bell operator acting on squeezed coherent states of inflationary perturbations (replacing the usual Bunch-Davies vacuum), demonstrates that this expectation value never violates the temporal Bell inequality, and reports that the result differs only slightly from the corresponding squeezed-vacuum case at large squeezing. It concludes that the ability to discriminate among initial states does not require Bell violation and that the dependence on a purely imaginary phase factor of the wave function is a distinctive feature of the temporal formulation.

Significance. If the derivation is correct, the result supplies a concrete, reproducible calculation showing that temporal Bell tests—more observationally accessible than spatial ones—yield no violation for coherent-state initial conditions. This limits the utility of such inequalities as direct probes of quantumness or state discrimination in primordial perturbations, while the reported slight difference at large squeezing and the phase dependence constitute falsifiable, checkable features that can be compared with other initial-state calculations in quantum cosmology.

minor comments (4)

Abstract, sentence on squeezed-coherent results: the clause 'the results for squeezed coherent state is found to differ' contains a subject-verb agreement error and should read 'are found to differ'.
Main derivation (around the analytical expression for the Bell-operator expectation value): although the final formula is stated, the manuscript would benefit from an explicit intermediate step showing how the coherent-state displacement operator acts on the pseudo-spin correlators, to allow immediate cross-check by readers.
Discussion of large-squeezing regime: the statement that the difference from the squeezed-vacuum case is 'slight' would be more persuasive if accompanied by a short table or plot of the numerical values of the expectation value versus squeezing parameter for both states.
Notation: the pseudo-spin operator components used to construct the temporal Bell operator are referenced but never written out explicitly; adding their definition (e.g., in terms of the Mukhanov-Sasaki variable and its conjugate) would improve accessibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review of our manuscript and for recommending minor revision. The referee's summary accurately captures the scope and conclusions of our work on the bipartite temporal Bell inequality applied to squeezed coherent states of inflationary perturbations.

read point-by-point responses

Referee: The manuscript derives an analytical expression for the expectation value of the bipartite temporal Bell operator acting on squeezed coherent states of inflationary perturbations (replacing the usual Bunch-Davies vacuum), demonstrates that this expectation value never violates the temporal Bell inequality, and reports that the result differs only slightly from the corresponding squeezed-vacuum case at large squeezing. It concludes that the ability to discriminate among initial states does not require Bell violation and that the dependence on a purely imaginary phase factor of the wave function is a distinctive feature of the temporal formulation.

Authors: We confirm that the derivation of the analytical expression for the expectation value is correct and that the numerical and analytical checks establish the absence of violation for coherent-state initial conditions. The slight difference relative to the squeezed-vacuum case at large squeezing is indeed present and is traceable to the additional displacement terms in the coherent-state wave function; this difference does not alter the conclusion that state discrimination is possible without Bell violation. The dependence on the imaginary phase of the wave function is a direct consequence of the temporal ordering of the pseudo-spin measurements and does not appear in the corresponding spatial Bell inequality. revision: no

Circularity Check

0 steps flagged

No circularity: direct conditional derivation from stated initial-state assumption

full rationale

The paper performs an explicit analytical calculation of the expectation value of a temporal Bell operator starting from the assumed coherent-state initial condition for the inflationary perturbations. The result (no violation) follows mathematically from the wave-function evolution and operator definitions under that assumption; no parameter is fitted to data and then relabeled as a prediction, no self-citation supplies a load-bearing uniqueness theorem, and no ansatz is smuggled in. The dependence on the coherent-state choice is openly conditional rather than hidden or self-referential. The manuscript therefore remains self-contained against its own stated premises.

Axiom & Free-Parameter Ledger

1 free parameters · 3 axioms · 0 invented entities

The central claim rests on replacing the Bunch-Davies vacuum with a coherent state and on the validity of the pseudo-spin temporal Bell operator under cosmological observational limits; these are domain assumptions rather than derived results.

free parameters (1)

squeezing parameter
Results are discussed for large values of this parameter; it functions as a free choice whose specific value is not derived from first principles or external data.

axioms (3)

standard math Locality and realism principles underlying Bell inequalities
The paper states that the temporal inequality is based on the same principles as the spatial version.
domain assumption Coherent state as valid initial condition replacing Bunch-Davies vacuum
Explicitly assumed when the vacuum is replaced by a coherent state.
domain assumption Only one component of the pseudo-spin operator is observationally accessible
Invoked to justify why the temporal construction is suited to cosmology.

pith-pipeline@v0.9.0 · 5521 in / 1652 out tokens · 63823 ms · 2026-05-12T01:13:54.485779+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel (J-cost uniqueness) unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

ψ_r(qk, q−k) = N0 exp[−A0(qk−C0)² − A0(q−k−D0)² + B0 qk q−k] with A0, B0 expressed via tanh rk and e^{-4iϕk} (Eqs. 20-22)
IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking (D=3 forcing) unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

rk(η) = −sinh^{-1}(1/(2kη)), ϕk(η) = π/4 − ½ arctan(1/(2kη)) (de-Sitter solutions, Eqs. 75-76)
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat recovery / orbit embedding unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

⟨ψ|BT̂|ψ⟩ = E(t1,t2) + E(t1,t′2) + E(t′1,t2) − E(t′1,t′2) with anti-commutator correlators (Eq. 43)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

46 extracted references · 46 canonical work pages · 1 internal anchor

[1]

Inflationary universe: A possible solution to the horizon and flatness problems

Alan H. Guth. “Inflationary universe: A possible solution to the horizon and flatness problems”. In:Phys. Rev. D23 (2 Jan. 1981), pp. 347–356.doi:10.1103/PhysRevD.23.347.url:https://link.aps.org/doi/10. 1103/PhysRevD.23.347

work page doi:10.1103/physrevd.23.347.url:https://link.aps.org/doi/10 1981
[2]

Dimensional reduction and dynamical chiral symme- try breaking by a magnetic ﬁeld in (3+1)-dimensions,

Andrei D. Linde. “A New Inflationary Universe Scenario: A Possible Solution of the Horizon, Flatness, Homo- geneity, Isotropy and Primordial Monopole Problems”. In:Phys. Lett. B108 (1982). Ed. by Li-Zhi Fang and R. Ruffini, pp. 389–393.doi:10.1016/0370- 2693(82)91219- 9.url:https://www.sciencedirect.com/ science/article/abs/pii/0370269382912199?via%3Dihub

work page doi:10.1016/0370- 1982
[3]

The Theory of Inflation

Jerome Martin. “The Theory of Inflation”. In:Proc. Int. Sch. Phys. Fermi200 (2020). Ed. by E. Coccia, J. Silk, and N. Vittorio, pp. 155–178.doi:10.3254/ENFI200008. arXiv:1807.11075 [astro-ph.CO]

work page doi:10.3254/enfi200008 2020
[4]

Inflation

Daniel Baumann. “Inflation”. In:Theoretical Advanced Study Institute in Elementary Particle Physics: Physics of the Large and the Small. 2011, pp. 523–686.doi:10 . 1142 / 9789814327183 _ 0010. arXiv:0907 . 5424 [hep-th]. 24

work page 2011
[5]

Inflation and squeezed quantum states

Andreas Albrecht et al. “Inflation and squeezed quantum states”. In:Phys. Rev. D50 (1994), pp. 4807–4820. doi:10.1103/PhysRevD.50.4807. arXiv:astro-ph/9303001

work page doi:10.1103/physrevd.50.4807 1994
[6]

The Quantum State of Inflationary Perturbations

Jerome Martin. “The Quantum State of Inflationary Perturbations”. In:J. Phys. Conf. Ser.405 (2012). Ed. by Supratik Pal and Banasri Basu, p. 012004.doi:10 . 1088 / 1742 - 6596 / 405 / 1 / 012004. arXiv:1209 . 3092 [hep-th]

work page 2012
[7]

Brahma, O

Suddhasattwa Brahma, Omar Alaryani, and Robert Brandenberger. “Entanglement entropy of cosmological perturbations”. In:Phys. Rev. D102.4 (2020), p. 043529.doi:10 . 1103 / PhysRevD . 102 . 043529. arXiv: 2005.09688 [hep-th]

work page arXiv 2020
[8]

Decoherence, entanglement negativity, and circuit complexity for an open quantum system

Arpan Bhattacharyya et al. “Decoherence, entanglement negativity, and circuit complexity for an open quantum system”. In:Phys. Rev. D107.10 (2023), p. 106007.doi:10.1103/PhysRevD.107.106007. arXiv:2210.09268 [hep-th]

work page doi:10.1103/physrevd.107.106007 2023
[9]

Quantum Discord of Cosmic Inflation: Can we Show that CMB Anisotropies are of Quantum-Mechanical Origin?

Jerome Martin and Vincent Vennin. “Quantum Discord of Cosmic Inflation: Can we Show that CMB Anisotropies are of Quantum-Mechanical Origin?” In:Phys. Rev. D93.2 (2016), p. 023505.doi:10.1103/PhysRevD.93. 023505. arXiv:1510.04038 [astro-ph.CO]

work page doi:10.1103/physrevd.93 2016
[10]

Bell inequalities for continuous-variable systems in generic squeezed states

J´ erome Martin and Vincent Vennin. “Bell inequalities for continuous-variable systems in generic squeezed states”. In:Phys. Rev. A93.6 (2016), p. 062117.doi:10.1103/PhysRevA.93.062117. arXiv:1605.02944 [quant-ph]

work page doi:10.1103/physreva.93.062117 2016
[11]

Obstructions to Bell CMB Experiments

Jerome Martin and Vincent Vennin. “Obstructions to Bell CMB Experiments”. In:Phys. Rev. D96.6 (2017), p. 063501.doi:10.1103/PhysRevD.96.063501. arXiv:1706.05001 [astro-ph.CO]

work page doi:10.1103/physrevd.96.063501 2017
[12]

Cosmic primordial density fluctuations and Bell inequalities

Roberto Dale, Ramon Lapiedra, and Juan Antonio Morales-Lladosa. “Cosmic primordial density fluctuations and Bell inequalities”. In:Phys. Rev. D107.2 (2023), p. 023506.doi:10.1103/PhysRevD.107.023506. arXiv: 2302.05125 [gr-qc]

work page doi:10.1103/physrevd.107.023506 2023
[13]

Violation of Bell inequalities from Cosmic Microwave Background data

Roberto Dale et al. “Violation of Bell inequalities from Cosmic Microwave Background data”. In:JCAP07 (2025), p. 044.doi:10.1088/1475-7516/2025/07/044. arXiv:2502.13846 [gr-qc]

work page doi:10.1088/1475-7516/2025/07/044 2025
[14]

Infinite violation of Bell inequalities in inflation

Sugumi Kanno and Jiro Soda. “Infinite violation of Bell inequalities in inflation”. In:Phys. Rev. D96.8 (2017), p. 083501.doi:10.1103/PhysRevD.96.083501. arXiv:1705.06199 [hep-th]

work page doi:10.1103/physrevd.96.083501 2017
[15]

Cosmic Inflation, Quantum Information and the Pioneering Role of John S Bell in Cosmology

J´ erˆ ome Martin. “Cosmic Inflation, Quantum Information and the Pioneering Role of John S Bell in Cosmology”. In:Universe5.4 (2019), p. 92.doi:10.3390/universe5040092. arXiv:1904.00083 [quant-ph]

work page doi:10.3390/universe5040092 2019
[16]

Bell violation in the Sky

Sayantan Choudhury, Sudhakar Panda, and Rajeev Singh. “Bell violation in the Sky”. In:Eur. Phys. J. C77 (2017), p. 60.doi:10.1140/epjc/s10052-016-4553-3. arXiv:1607.00237 [hep-th]

work page doi:10.1140/epjc/s10052-016-4553-3 2017
[17]

Einstein, B

A. Einstein, B. Podolsky, and N. Rosen. “Can Quantum-Mechanical Description of Physical Reality Be Con- sidered Complete?” In:Phys. Rev.47 (10 May 1935), pp. 777–780.doi:10 . 1103 / PhysRev . 47 . 777.url: https://link.aps.org/doi/10.1103/PhysRev.47.777

work page doi:10.1103/physrev.47.777 1935
[18]

On the Einstein Podolsky Rosen paradox

J. S. Bell. “On the Einstein Podolsky Rosen paradox”. In:Physics Physique Fizika1 (3 Nov. 1964), pp. 195– 200.doi:10 . 1103 / PhysicsPhysiqueFizika . 1 . 195.url:https : / / link . aps . org / doi / 10 . 1103 / PhysicsPhysiqueFizika.1.195

work page 1964
[19]

Violation of Bells inequality using continuous variable measurements

Oliver Thearle et al. “Violation of Bells inequality using continuous variable measurements”. In: (Jan. 2018). doi:10.1103/PhysRevLett.120.040406. arXiv:1801.03194 [quant-ph]

work page doi:10.1103/physrevlett.120.040406 2018
[20]

Violation of Bell’s inequality for continuous variables

L. Praxmeyer, B. -G. Englert, and K. Wodkiewicz. “Violation of Bell’s inequality for continuous variables”. In: Eur. Phys. J. D32 (2005), pp. 227–231.doi:10.1140/epjd/e2005-00021-1. arXiv:0406172 [quant-ph]

work page doi:10.1140/epjd/e2005-00021-1 2005
[21]

Experimental Violation of Bell’s Inequality in Spatial-Parity Space

Timothy Yarnall et al. “Experimental Violation of Bell’s Inequality in Spatial-Parity Space”. In:Phys. Rev. Lett.99 (2007), p. 170408.doi:10.1103/PhysRevLett.99.170408. arXiv:0708.0653 [quant-ph]

work page doi:10.1103/physrevlett.99.170408 2007
[22]

Quantum generalizations of Bell’s inequality

Boris S Cirel’son. “Quantum generalizations of Bell’s inequality”. In:Letters in Mathematical Physics4.2 (1980), pp. 93–100

work page 1980
[23]

Bipartite temporal Bell inequalities for two-mode squeezed states

Kenta Ando and Vincent Vennin. “Bipartite temporal Bell inequalities for two-mode squeezed states”. In:Phys. Rev. A102.5 (2020), p. 052213.doi:10.1103/PhysRevA.102.052213. arXiv:2007.00458 [quant-ph]

work page doi:10.1103/physreva.102.052213 2020
[24]

Quantum correlations in the temporal Clauser–Horne–Shimony–Holt (CHSH) scenario

Tobias Fritz. “Quantum correlations in the temporal Clauser–Horne–Shimony–Holt (CHSH) scenario”. In:New J. Phys.12.8 (2010), p. 083055.doi:10.1088/1367-2630/12/8/083055. arXiv:1005.3421 [quant-ph]

work page doi:10.1088/1367-2630/12/8/083055 2010
[25]

Bipartite Bell Inequality and Maximal Violation

Ming Li, Shao-Ming Fei, and Xianqing Li-Jost. “Bipartite Bell Inequality and Maximal Violation”. In:Commun. Theor. Phys.55 (2011), pp. 418–420.doi:10.1088/0253-6102/55/3/09. arXiv:1102.5246 [quant-ph]

work page doi:10.1088/0253-6102/55/3/09 2011
[26]

Leggett-Garg Inequalities for Squeezed States

Jerome Martin and Vincent Vennin. “Leggett-Garg Inequalities for Squeezed States”. In:Phys. Rev. A94.5 (2016), p. 052135.doi:10.1103/PhysRevA.94.052135. arXiv:1611.01785 [quant-ph]. 25

work page doi:10.1103/physreva.94.052135 2016
[27]

Extreme Violations of Leggett-Garg Inequalities for a System Evolving under Su- perposition of Unitaries

Arijit Chatterjee et al. “Extreme Violations of Leggett-Garg Inequalities for a System Evolving under Su- perposition of Unitaries”. In:Phys. Rev. Lett.135.22 (2025), p. 220202.doi:10 . 1103 / vydp - 9qqq. arXiv: 2411.02301 [quant-ph]

work page arXiv 2025
[28]

Inflation with General Initial Conditions for Scalar Perturbations

Sandipan Kundu. “Inflation with General Initial Conditions for Scalar Perturbations”. In:JCAP02 (2012), p. 005.doi:10.1088/1475-7516/2012/02/005. arXiv:1110.4688 [astro-ph.CO]

work page doi:10.1088/1475-7516/2012/02/005 2012
[29]

Cosmological consequences of statistical inhomo- geneity

H. V. Ragavendra, Dipayan Mukherjee, and Shiv K. Sethi. “Cosmological consequences of statistical inhomo- geneity”. In:Phys. Rev. D111.2 (2025), p. 023541.doi:10.1103/PhysRevD.111.023541. arXiv:2411.01331 [astro-ph.CO]

work page doi:10.1103/physrevd.111.023541 2025
[30]

Scalar-induced gravitational waves from coherent initial states

Dipayan Mukherjee, H. V. Ragavendra, and Shiv K. Sethi. “Scalar-induced gravitational waves from coherent initial states”. In:Phys. Rev. D113.2 (2026), p. 023533.doi:10 . 1103 / qd1s - 9fxl. arXiv:2506 . 23798 [astro-ph.CO]

work page 2026
[31]

Violation of Bell inequality from a squeezed coherent state of inflationary perturbations

Aurindam Mondal and Rathul Nath Raveendran. “Violation of Bell inequality from a squeezed coherent state of inflationary perturbations”. In:Phys. Dark Univ.51 (2026), p. 102218.doi:10.1016/j.dark.2026.102218. arXiv:2410.04608 [gr-qc]

work page doi:10.1016/j.dark.2026.102218 2026
[32]

Bertuzzo, G

Enrico Bertuzzo, Gabriel M. Salla, and Andrea Tesi. “The effects of non Bunch-Davies initial conditions on gravitationally produced relics”. In: (Mar. 2026). arXiv:2603.03430 [gr-qc]

work page arXiv 2026
[33]

Classical and quantum theory of perturbations in inflationary universe models

Robert H. Brandenberger, H. Feldman, and Viatcheslav F. Mukhanov. “Classical and quantum theory of perturbations in inflationary universe models”. In:37th Yamada Conference: Evolution of the Universe and its Observational Quest. July 1993, pp. 19–30. arXiv:9307016 [astro-ph]

work page 1993
[34]

Theory of cosmological perturba- tions. Part 1. Classical perturbations. Part 2. Quantum theory of perturbations. Part 3. Extensions

Viatcheslav F. Mukhanov, H. A. Feldman, and Robert H. Brandenberger. “Theory of cosmological perturba- tions. Part 1. Classical perturbations. Part 2. Quantum theory of perturbations. Part 3. Extensions”. In:Phys. Rept.215 (1992), pp. 203–333.doi:10.1016/0370-1573(92)90044-Z

work page doi:10.1016/0370-1573(92)90044-z 1992
[35]

Inflation and the theory of cosmological perturbations

Antonio Riotto. “Inflation and the theory of cosmological perturbations”. In: (2002). arXiv:0210162 [hep-ph]

work page 2002
[36]

An introduction to inflation and cosmological perturbation theory

L. Sriramkumar. “An introduction to inflation and cosmological perturbation theory”. In:Curr. Sci.97 (2009), p. 868. arXiv:0904.4584 [astro-ph.CO]

work page arXiv 2009
[37]

Inflationary cosmological perturbations of quantum-mechanical origin

Jerome Martin. “Inflationary cosmological perturbations of quantum-mechanical origin”. In:Lect. Notes Phys. 669 (2005). Ed. by G. Amelino-Camelia and J. Kowalski-Glikman, pp. 199–244.doi:10.1007/11377306_7. arXiv:0406011 [hep-th]

work page doi:10.1007/11377306_7 2005
[38]

Cosmological Perturbation Theory - part 2

Hannu Kurki-Suonio. “Cosmological Perturbation Theory - part 2”. In: 2015.url:https://api.semanticscholar. org/CorpusID:125588012

work page 2015
[39]

Cosmological perturbation theory - part 1, 2

Hannu Kurki-Suonio. “Cosmological perturbation theory - part 1, 2”. In: (2005).url:https : / / www . mv . helsinki.fi/home/hkurkisu/cpt/

work page 2005
[40]

Canonical transformations and squeezing formalism in cosmology

Julien Grain and Vincent Vennin. “Canonical transformations and squeezing formalism in cosmology”. In: JCAP02 (2020), p. 022.doi:10.1088/1475-7516/2020/02/022. arXiv:1910.01916 [astro-ph.CO]

work page doi:10.1088/1475-7516/2020/02/022 2020
[41]

Optimization of Bell’s Inequality Violation For Continuous Variable Systems

G. Gour et al. “Optimization of Bell’s Inequality Violation For Continuous Variable Systems”. In: (Oct. 2003). doi:10.1016/j.physleta.2004.03.018. arXiv:0308063 [quant-ph]

work page doi:10.1016/j.physleta.2004.03.018 2003
[42]

Bell’s Inequality Violation (BIQV) with Non-Negative Wigner Function

M. Revzen et al. “Bell’s Inequality Violation (BIQV) with Non-Negative Wigner Function”. In: (May 2004). doi:10.1103/PhysRevA.71.022103. arXiv:0405100 [quant-ph]

work page doi:10.1103/physreva.71.022103 2004
[43]

Possible origin ofα-vacua as the initial state of the Universe

Pisin Chen et al. “Possible origin ofα-vacua as the initial state of the Universe”. In:Phys. Rev. D111.8 (2025), p. 083520.doi:10.1103/PhysRevD.111.083520. arXiv:2404.15450 [gr-qc]

work page doi:10.1103/physrevd.111.083520 2025
[44]

Choice of Quantum Vacuum for Inflation Observables

Melo Wood-Saanaoui, Rudnei O. Ramos, and Arjun Berera. “Choice of Quantum Vacuum for Inflation Ob- servables”. In:Symmetry18 (2026), p. 399. arXiv:2602.22116 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2026
[45]

Cosmological collider signal from non-Bunch-Davies initial states

Yuan Yin. “Cosmological collider signal from non-Bunch-Davies initial states”. In:Phys. Rev. D109.4 (2024), p. 043535.doi:10.1103/PhysRevD.109.043535. arXiv:2309.05244 [hep-ph]

work page doi:10.1103/physrevd.109.043535 2024
[46]

Constraining quantum initial conditions before inflation

T. Gessey-Jones and W. J. Handley. “Constraining quantum initial conditions before inflation”. In:Phys. Rev. D104.6 (2021), p. 063532.doi:10.1103/PhysRevD.104.063532. arXiv:2104.03016 [astro-ph.CO]. 26

work page doi:10.1103/physrevd.104.063532 2021