Pith. sign in

REVIEW 5 minor 45 references

Radiative gravitons cannot measurably decohere matter-wave interferometers; the exact visibility loss is the characteristic function of the initial graviton state at the branch-displacement difference.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · deepseek-v4-flash

2026-08-01 09:07 UTC pith:KPZFRSMG

load-bearing objection Exact displacement-operator treatment confirms radiative graviton decoherence is negligible; the Gaussian trajectory caveat doesn't change the bottom line.

arxiv 2607.20867 v1 pith:KPZFRSMG submitted 2026-07-23 gr-qc hep-thquant-ph

Graviton-induced which-path decoherence in matter-wave interferometry

Hiroki Matsui This is my paper
classification gr-qc hep-thquant-ph
keywords which-path decoherencegravitonmatter-wave interferometrylinearized quantum gravityquadrupole radiationsqueezed vacuumprimordial gravitational wavesdecoherence exponent
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper tries to prove that the which-path decoherence caused by gravitons, the quantum particles of the gravitational field, is completely negligible for matter-wave interferometry, now and in the foreseeable future. Because the graviton field is driven linearly by the mass distribution, the paper solves the branch-dependent evolution exactly: the loss of interference visibility is the overlap of two coherent graviton states, computed as the characteristic function of the initial graviton state at the difference of the branch-induced displacements. In the vacuum, that decoherence exponent equals half the mean number of gravitons radiated by the difference between the two branches; for a smooth Gaussian trajectory it reduces to (8/15)(m/m_P)^2(u/c)^4, ranging from about 10^-89 to 10^-61 across representative experiments. For the strongly squeezed graviton background left over from inflation, the exponent is at most about 10^-27. So radiative gravitons could not erase interference fringes in any planned experiment.

Core claim

The central claim: for any initial graviton state, the reduced coherence factor is D = e^{iΔΦ} Tr[ρ_g D[Δα]], where Δα is the difference of the coherent displacements induced by the two branches. In the vacuum, Γ = -ln|D| = ½N_Δ, half the mean number of gravitons the difference source would radiate. In the nonrelativistic quadrupole limit with a Gaussian closed trajectory, Γ_vac = (8/15) G m² d⁴/(ℏ c⁵ τ⁴) = (8/15)(m/m_P)^2(u/c)^4. For a general two-mode squeezed vacuum, the response is reweighted by factors between e^{-2r} and e^{2r}; for the inflationary squeezed background, the rapidly oscillating squeezing phase averages out, leaving Γ_inf = (π/20) Ω_inf (m d² H_0/ℏ)², which with Ω_inf =

What carries the argument

The central object is the displacement operator D[α] acting on each graviton mode, with amplitude α determined by the Fourier transform of the branch stress tensor. Because the matter–graviton coupling is linear, the branch evolution is exactly a coherent displacement and the Magnus expansion terminates; the visibility factor is the characteristic function of the initial graviton state at the difference displacement Δα. The workhorse identities are Γ = -ln|Tr[ρ_g D[Δα]]|, the vacuum relation Γ_vac = ½N_Δ, and the angular average ∫dΩ Λ_{ij,kl}A^{ij}A^{kl*} = (8π/5)A^{ij}A^*_{ij}, which turns the master formula into an ω^5-weighted quadrupole integral. The Gaussian trajectory model then suppli

Load-bearing premise

The numerical negligibility conclusion stands on the assumption that each branch follows a smooth Gaussian trajectory and that branch-dependent apparatus quadrupole contributions are negligible; if real interferometers have sharp accelerations whose high-frequency content is not suppressed, the ω^5-weighted quadrupole integral could make Γ much larger than the quoted values.

What would settle it

Take a realistic branch trajectory with sharp split/recombine pulses (for example, piecewise-constant acceleration with duration τ_acc) and evaluate the quadrupole integral (38) numerically. If the resulting Γ for an aggressive proposed setup (m ≈ 10^-14 kg, d ≈ 2.5×10^-4 m, τ ≈ 0.5 s) reaches 10^-3 or larger, the paper's negligibility conclusion for real experiments is falsified; otherwise the Gaussian estimate is confirmed as a lower bound. A direct experimental test would be measuring visibility loss of 0.1% or more in such a setup, which the paper predicts radiative gravitons cannot produc

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The vacuum decoherence exponent is exactly half the mean number of gravitons radiated by the difference source, so visibility loss is nothing but the distinguishability of the two branch-conditioned radiation states.
  • For the Gaussian closed trajectory, the exponent scales as (m/m_P)^2(u/c)^4, so both Planck-mass suppression and nonrelativistic suppression appear explicitly; the trajectory profile contributes only an O(1) factor.
  • Across representative matter-wave setups, the vacuum exponent sits between 10^-89 and 10^-61, and even an optimistic inflationary squeezed background raises it to at most ~10^-27, far below a 10^-3 visibility-loss benchmark.
  • Because the effect is radiative graviton decoherence, it is separate from the static Newtonian phase used in gravity-entanglement proposals; radiative gravitons would neither obstruct nor assist those experiments.

Where Pith is reading between the lines

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

  • Editorial: The exact formula suggests a sharper diagnostic: measure visibility loss versus branch separation d and duration τ and test the predicted (m/m_P)^2(u/c)^4 scaling; any visibility loss far above that scaling would point to non-gravitational noise rather than radiative gravitons.
  • Editorial: The Gaussian trajectory is the softest spot of the numerical estimates. The ω^5 factor in the quadrupole integral means sharp acceleration features, such as beam-splitter kicks or grating pulses, could contribute much more than the Gaussian tail; recomputing Γ for piecewise-constant-acceleration trajectories would bound this amplification and either rescue or overturn the 10^-89-to-10^-
  • Editorial: The phase-averaging step for inflation assumes the squeezing phase oscillates rapidly across the experimental frequency band; for narrowband or resonant matter-wave setups this cancellation may be incomplete. Evaluating the exact squeezed-state expression with a realistic source response and frequency-dependent phase would show whether the ~40-order enhancement is robust or an artifact
  • Editorial: By showing radiation gravitons are negligible, the paper implicitly sharpens the goal of tabletop quantum-gravity tests: their noise budget is not set by propagating gravitons, so any observed gravitational decoherence would have to come from the constraint (Newtonian) sector or from non-radiative field modes.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 5 minor

Summary. The paper derives the loss of coherence in a matter-wave interferometer due to tracing out propagating gravitons in linearized quantum gravity. For branch-dependent classical stress tensors, the interaction is a linear drive, so each branch imprints a coherent displacement on the graviton field; the reduced off-diagonal matter element is the characteristic function of the initial graviton state at the displacement difference (Eqs. (24)–(25)). In vacuum, the decoherence exponent equals half the mean number of gravitons radiated by the difference source (Eq. (32)), reducing in the nonrelativistic quadrupole limit to Eq. (38) and, for a Gaussian closed trajectory, to Γ_vac = (8/15)Gm²d⁴/(ℏc⁵τ⁴) (Eq. (49)). For a general two-mode squeezed vacuum the exact phase-dependent weight is derived (Eqs. (66)–(67)); for the inflationary background the phase-sensitive term is argued to average out, leaving Γ_inf = (π/20)Ω_inf(md²H_0/ℏ)² (Eq. (84)). Numerical estimates for five representative setups give vacuum exponents from 10⁻⁸⁹ to 10⁻⁶¹ and an inflationary contribution up to ~10⁻²⁷, leading to the conclusion that radiative graviton which-path decoherence is negligible in current and foreseeable matter-wave interferometers.

Significance. If the result holds, the paper provides a clean exact formula for radiative graviton decoherence and a transparent order-of-magnitude estimate that this effect will not obstruct matter-wave tests of quantum gravity. The strengths are the closed-form displacement-operator solution (Sec. III), the explicit relation Γ_vac = (1/2)N_Δ, and the complete derivations in Appendices A–D. I checked the angular average (8π/5), the Gaussian integral (8), and the coefficients 8/15 and π/20, and they are correct; the numerical entries in Table I also match Eqs. (49) and (84). No ad hoc free parameters are introduced beyond experimental proxy values and observational inputs. The central claim is not circular: the decoherence exponents are derived, not assumed. The stress-test concern about the Gaussian trajectory and the neglect of branch-dependent apparatus quadrupoles is a real modeling caveat, but it is not load-bearing for the qualitative negligibility conclusion, because the dimensional suppression (m/m_P)²(u/c)⁴ persists for any nonrelativistic closed trajectory with these mass and velocity scales.

minor comments (5)
  1. [Abstract and Sec. VIII] The claimed enhancement 'approximately 40 orders of magnitude' is inconsistent with the table. For the BMV/QGEM proxy, Eqs. (86)–(87) give Γ_inf/Γ_vac ≈ 3×10³³, i.e. about 33.5 orders, and the ratio ranges from ~10²⁶ (Na cluster) to ~10⁴³ (MAQRO) across Table I. Please correct or qualify this number.
  2. [Sec. V after Eq. (38)] The neglect of branch-dependent apparatus quadrupoles is acknowledged and is O(m/M) for recoil, but the paper does not explicitly state why the Gaussian trajectory is a safe proxy for the negligibility conclusion. A sentence noting that the dimensionless suppression (m/m_P)²(u/c)⁴ remains for any nonrelativistic closed trajectory, so sharp accelerations cannot change the order of magnitude, would address the reader's concern.
  3. [Sec. VIIB around Eq. (71)] The phase-averaging step ('largely cancels') is plausible because the squeezing phase has a huge frequency gradient ~2ω/H_0, but the residual oscillatory integral is not quantified. A short estimate or bound would make this step more rigorous; the final conclusion is unchanged even if the cosine term does not fully cancel.
  4. [Eq. (85)] The symbol G is used both for Newton's constant and for the radiation-era transfer factor (G ≃ 0.39). This is confusing; please use a different symbol, e.g. script G or g_*.
  5. [Abstract] The sentence beginning 'Hence, the which-path decoherence ... is therefore negligible' is redundant ('Hence' and 'therefore'). Also update the 'approximately 40 orders' phrase as noted above.

Circularity Check

0 steps flagged

No circularity: the decoherence exponents are derived from the linearized-gravity coupling and displacement-operator algebra, with external trajectory and cosmological parameters, not fitted to the predicted result.

full rationale

The derivation chain is self-contained: from the linearized coupling in Eq. (6), the branch evolution is solved exactly as a coherent displacement in Eq. (16); tracing out the graviton field gives the coherence factor as the characteristic function of the initial graviton state in Eqs. (24)-(25); the vacuum exponent follows as Eq. (27), the nonrelativistic quadrupole reduction as Eq. (38), and the Gaussian estimate Eq. (49) is obtained by substituting the explicitly assumed trajectory Eq. (42). At no point is the predicted exponent Γ used to define one of its own inputs: the trajectory parameters m, d, τ and the cosmological inputs H0 and Ω_inf are external quantities, not fitted to Γ. The manuscript also states its own limitations: Sec. V notes that the apparatus contribution is 'setup-dependent and is not considered here', the Table I caption says the values 'do not model the complete physical configurations of the individual experiments and should be regarded only as order-of-magnitude estimates', and the Discussion restricts the analysis to 'freely propagating gravitons in Minkowski spacetime'. These are scope caveats, not circular reductions. The only self-citation, Ref. [23] (Kanno, Matsui, Mukohyama), appears in a background survey of primordial-graviton studies and is not load-bearing; no uniqueness or ansatz is imported from it. The inflationary phase-averaging step relies on an external result [39], and Ω_inf = 10^-16 is taken from standard CMB bounds and transfer functions, not from the decoherence formula being tested. I therefore find no circular step.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

The derivation uses only standard linearized gravity and displacement-operator algebra. The only domain assumptions are the classical-source model, the quadrupole truncation, and the inflationary phase-averaging hypothesis; no new particles, forces, or fields are introduced, and no free parameters are fitted to the decoherence result. Ω_inf enters as an external observational input.

axioms (6)
  • domain assumption Linearized quantum gravity (Fierz–Pauli action, TT gauge, canonically normalized graviton field) is a valid low-energy description at the scales of tabletop experiments.
    Used throughout Secs. II–IV; standard EFT for gravitons coupled to matter, but its validity at laboratory scales is precisely what experiments aim to test.
  • domain assumption The matter branches act as classical conserved stress tensors T_b, and the initial state factorizes as ρ_in = ρ_m ⊗ ρ_g.
    Eq. (13)–(15). This is the standard coherent-source approximation; it omits quantum backreaction and initial matter–graviton correlations.
  • standard math The Magnus expansion terminates at second order because [H_int(t), H_int(t′)] is a c-number for linear coupling.
    Sec. IIIB; this is the mathematical basis of the exact displacement-operator solution.
  • domain assumption The nonrelativistic quadrupole approximation dominates, with higher multipoles suppressed by ωL/c and apparatus quadrupole contributions neglected.
    Sec. V, leading to Eq. (38); the paper notes the apparatus contribution is setup-dependent and not considered.
  • domain assumption Inflation produces a two-mode squeezed vacuum whose squeezing phase varies rapidly across the laboratory frequency band, so the phase-sensitive term averages to zero.
    Sec. VIIB, converting Eq. (67) into Eq. (71); no estimate of the residual is given.
  • domain assumption The Bunch–Davies vacuum matched through a de Sitter-to-radiation transition gives occupation numbers n_k ∝ k⁻⁴, connected to a nearly scale-invariant Ω_gw.
    Appendix C and Eq. (80)–(85); standard cosmological tensor-mode result cited to Refs. [21,38].

pith-pipeline@v1.3.0-alltime-deepseek · 202 in / 14500 out tokens · 192384 ms · 2026-08-01T09:07:35.359361+00:00 · methodology

0 comments
read the original abstract

We derive which-path decoherence for matter-wave interferometry that arises from tracing out the gravitons of linearized quantum gravity. Because the gravitons are driven linearly by the matter source, the branch-dependent evolution can be solved exactly, and the reduced coherence is given by the characteristic function of the initial graviton state, evaluated at the difference of the branch-induced field displacements. In vacuum, the decoherence exponent equals half the mean number of gravitons radiated by the difference source. For a smooth Gaussian trajectory, it reduces to $\Gamma_{\rm vac}=(8/15)\,Gm^2d^4/(\hbar c^5\tau^4)$, ranging from $10^{-89}$ to $10^{-61}$ across representative matter-wave platforms. For a general squeezed graviton vacuum, we obtain an exact expression in which squeezing either suppresses or enhances the vacuum response, depending on the squeezing phase. For the strongly squeezed state produced by inflation, this expression reduces to $\Gamma_{\rm inf}=(\pi/20)\,\Omega_{\rm inf}(md^2H_0/\hbar)^2$ and reaches at most $\sim10^{-27}$ for optimistic parameters. Hence, the which-path decoherence induced by radiative gravitons is therefore negligible in current matter-wave interferometers.

Figures

Figures reproduced from arXiv: 2607.20867 by Hiroki Matsui.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic of the which-path configuration. Branch 2 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Mass–kinematic-scale dependence of vacuum graviton decoherence in the Gaussian quadrupole model. The descending [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

45 extracted references · 30 linked inside Pith

  1. [1]

    Joos and H

    E. Joos and H. D. Zeh,The Emergence of classical properties through interaction with the environment,Z. Phys. B59(1985) 223

    1985

  2. [2]

    H. P. Breuer and F. Petruccione,The theory of open quantum systems. Oxford University Press, Oxford, 2002

    2002

  3. [3]

    W. H. Zurek,Decoherence, einselection, and the quantum origins of the classical,Rev. Mod. Phys.75(2003) 715 [quant-ph/0105127]

    Pith/arXiv arXiv 2003

  4. [4]

    C. J. Riedel,Evidence for gravitons from decoherence by bremsstrahlung,1310.6347

    Pith/arXiv arXiv

  5. [5]

    D. L. Danielson, G. Satishchandran and R. M. Wald, Black holes decohere quantum superpositions,Int. J. Mod. Phys. D31(2022) 2241003 [2205.06279]

    Pith/arXiv arXiv 2022

  6. [6]

    D. L. Danielson, G. Satishchandran and R. M. Wald, Killing horizons decohere quantum superpositions,Phys. Rev. D108(2023) 025007 [2301.00026]

    Pith/arXiv arXiv 2023

  7. [7]

    D. L. Danielson, G. Satishchandran and R. M. Wald, Local description of decoherence of quantum superpositions by black holes and other bodies,Phys. Rev. D111(2025) 025014 [2407.02567]

    Pith/arXiv arXiv 2025

  8. [8]

    Baym and T

    G. Baym and T. Ozawa,Two-slit diffraction with highly charged particles: Niels Bohr’s consistency argument that the electromagnetic field must be quantized,Proc. Nat. Acad. Sci.106(2009) 3035 [0902.2615]

    Pith/arXiv arXiv 2009

  9. [9]

    Carney, P

    D. Carney, P. C. E. Stamp and J. M. Taylor,Tabletop experiments for quantum gravity: a user’s manual,Class. Quant. Grav.36(2019) 034001 [1807.11494]

    Pith/arXiv arXiv 2019

  10. [10]

    R. P. Feynman and F. L. Vernon, Jr.,The Theory of a general quantum system interacting with a linear dissipative system,Annals Phys.24(1963) 118

    1963

  11. [11]

    M. P. Blencowe,Effective Field Theory Approach to Gravitationally Induced Decoherence,Phys. Rev. Lett. 111(2013) 021302 [1211.4751]

    Pith/arXiv arXiv 2013

  12. [12]

    Anastopoulos and B

    C. Anastopoulos and B. L. Hu,A Master Equation for Gravitational Decoherence: Probing the Textures of Spacetime,Class. Quant. Grav.30(2013) 165007 [1305.5231]

    Pith/arXiv arXiv 2013

  13. [13]

    Oniga and C

    T. Oniga and C. H.-T. Wang,Quantum gravitational decoherence of light and matter,Phys. Rev. D93(2016) 044027 [1511.06678]

    Pith/arXiv arXiv 2016

  14. [14]

    Parikh, F

    M. Parikh, F. Wilczek and G. Zahariade,The Noise of Gravitons,Int. J. Mod. Phys. D29(2020) 2042001 [2005.07211]

    Pith/arXiv arXiv 2020

  15. [15]

    Kanno, J

    S. Kanno, J. Soda and J. Tokuda,Noise and decoherence induced by gravitons,Phys. Rev. D103(2021) 044017 [2007.09838]

    Pith/arXiv arXiv 2021

  16. [16]

    Toroš, A

    M. Toroš, A. Mazumdar and S. Bose,Loss of coherence and coherence protection from a graviton bath,Phys. Rev. D109(2024) 084050 [2008.08609]

    Pith/arXiv arXiv 2024

  17. [17]

    Parikh, F

    M. Parikh, F. Wilczek and G. Zahariade,Quantum Mechanics of Gravitational Waves,Phys. Rev. Lett.127 (2021) 081602 [2010.08205]

    Pith/arXiv arXiv 2021

  18. [18]

    Parikh, F

    M. Parikh, F. Wilczek and G. Zahariade,Signatures of the quantization of gravity at gravitational wave detectors, Phys. Rev. D104(2021) 046021 [2010.08208]

    Pith/arXiv arXiv 2021

  19. [19]

    Kanno, J

    S. Kanno, J. Soda and J. Tokuda,Indirect detection of gravitons through quantum entanglement,Phys. Rev. D 104(2021) 083516 [2103.17053]

    Pith/arXiv arXiv 2021

  20. [20]

    L. H. Ford,Gravitational radiation by quantum systems, Annals Phys.144(1982) 238

    1982

  21. [21]

    L. P. Grishchuk and Y. V. Sidorov,Squeezed quantum states of relic gravitons and primordial density fluctuations,Phys. Rev. D42(1990) 3413

    1990

  22. [22]

    Kanno and J

    S. Kanno and J. Soda,Detecting nonclassical primordial gravitational waves with Hanbury-Brown–Twiss interferometry,Phys. Rev. D99(2019) 084010 [1810.07604]

    Pith/arXiv arXiv 2019

  23. [23]

    Kanno, H

    S. Kanno, H. Matsui and S. Mukohyama, Hanbury-Brown-Twiss interferometry and quantum nature of primordial gravitational waves in Hořava-Lifshitz gravity,Phys. Rev. D111(2025) 104077 [2412.19514]

    Pith/arXiv arXiv 2025

  24. [24]

    Kanno, J

    S. Kanno, J. Soda and A. Taniguchi,Coherent State Description of Gravitational Waves from Binary Black Holes,Phys. Rev. Lett.136(2026) 061404 [2508.17947]

    arXiv 2026

  25. [25]

    Dorlis, N

    P. Dorlis, N. E. Mavromatos, S. Sarkar and S.-N. Vlachos,Superradiant Axionic Black-Hole Clouds as Seeds for Graviton Squeezing,Phys. Rev. Lett.135 (2025) 151501 [2507.01689]

    arXiv 2025

  26. [26]

    Dorlis, N

    P. Dorlis, N. E. Mavromatos, S. Sarkar and S.-N. Vlachos,Squeezed gravitons from superradiant axion fields around rotating black holes,Phys. Rev. D113 (2026) 026023 [2507.23475]

    arXiv 2026

  27. [27]

    Kanno, J

    S. Kanno, J. Soda and A. Taniguchi,Binary gravitational waves as probes of quantum graviton states, Phys. Rev. D113(2026) 123542 [2510.23326]

    Pith/arXiv arXiv 2026

  28. [28]

    S. Bose, A. Mazumdar, G. W. Morley, H. Ulbricht, M. Toroš, M. Paternostro et al.,Spin Entanglement Witness for Quantum Gravity,Phys. Rev. Lett.119 (2017) 240401 [1707.06050]

    Pith/arXiv arXiv 2017

  29. [29]

    Marletto and V

    C. Marletto and V. Vedral,Gravitationally Induced Entanglement between Two Massive Particles Is Sufficient Evidence of Quantum Effects in Gravity,Phys. Rev. Lett.119(2017) 240402 [1707.06036]

    Pith/arXiv arXiv 2017

  30. [30]

    Matsumura and K

    A. Matsumura and K. Yamamoto,Gravity-induced entanglement in optomechanical systems,Phys. Rev. D 102(2020) 106021 [2010.05161]

    Pith/arXiv arXiv 2020

  31. [31]

    D. Miki, A. Matsumura and K. Yamamoto,Feasible generation of gravity-induced entanglement by using optomechanical systems,Phys. Rev. D110(2024) 024057 [2406.04361]

    arXiv 2024

  32. [32]

    Fujita, Y

    T. Fujita, Y. Kaku, A. Matsumura and Y. Michimura, Inverted oscillators for testing gravity-induced quantum entanglement,Class. Quant. Grav.42(2025) 165003 [2308.14552]. 12

    Pith/arXiv arXiv 2025

  33. [33]

    D. Miki, A. Matsumura and K. Yamamoto, Entanglement and decoherence of massive particles due to gravity,Phys. Rev. D103(2021) 026017 [2010.05159]

    Pith/arXiv arXiv 2021

  34. [34]

    Takeda, S

    H. Takeda, S. Tomizuka and T. Tanaka,Gravitationally Induced Quantum Decoherence of Macroscopic Objects, 2606.04099

    Pith/arXiv arXiv

  35. [35]

    Magnus,On the exponential solution of differential equations for a linear operator,Commun

    W. Magnus,On the exponential solution of differential equations for a linear operator,Commun. Pure Appl. Math.7(1954) 649

    1954

  36. [36]

    Blanes, F

    S. Blanes, F. Casas, J. A. Oteo and J. Ros,The Magnus expansion and some of its applications,Phys. Rept.470 (2009) 151

    2009

  37. [37]

    Weinberg,Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity

    S. Weinberg,Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. John Wiley and Sons, New York, 1972

    1972

  38. [38]

    Maggiore,Gravitational wave experiments and early universe cosmology,Phys

    M. Maggiore,Gravitational wave experiments and early universe cosmology,Phys. Rept.331(2000) 283 [gr-qc/9909001]

    Pith/arXiv arXiv 2000

  39. [39]

    Allen, E

    B. Allen, E. E. Flanagan and M. A. Papa,Is the squeezing of relic gravitational waves produced by inflation detectable?,Phys. Rev. D61(2000) 024024 [gr-qc/9906054]. [40]Planckcollaboration,Planck 2018 results. VI. Cosmological parameters,Astron. Astrophys.641(2020) A6 [1807.06209]

    Pith/arXiv arXiv 2000

  40. [41]

    Pedalino, B

    S. Pedalino, B. E. Ramírez-Galindo, R. Ferstl, K. Hornberger, M. Arndt and S. Gerlich,Probing quantum mechanics with nanoparticle matter-wave interferometry,Nature649(2026) 866

    2026

  41. [42]

    Kovachy, P

    T. Kovachy, P. Asenbaum, C. Overstreet, C. A. Donnelly, S. M. Dickerson, A. Sugarbaker et al.,Quantum superposition at the half-metre scale,Nature528(2015) 530

    2015

  42. [43]

    Kaltenbaek et al.,Research campaign: Macroscopic quantum resonators (MAQRO),Quantum Sci

    R. Kaltenbaek et al.,Research campaign: Macroscopic quantum resonators (MAQRO),Quantum Sci. Technol.8 (2023) 014006

    2023

  43. [44]

    H. Pino, J. Prat-Camps, K. Sinha, B. P. Venkatesh and O. Romero-Isart,On-chip quantum interference of a superconducting microsphere,Quantum Sci. Technol.3 (2018) 025001 [1603.01553]

    Pith/arXiv arXiv 2018

  44. [45]

    Watanabe and E

    Y. Watanabe and E. Komatsu,Improved Calculation of the Primordial Gravitational Wave Spectrum in the Standard Model,Phys. Rev. D73(2006) 123515 [astro-ph/0604176]. [46]BICEP, Keckcollaboration,Improved Constraints on Primordial Gravitational Waves using Planck, WMAP, and BICEP/Keck Observations through the 2018 Observing Season,Phys. Rev. Lett.127(2021) 1...

    Pith/arXiv arXiv 2006

  45. [47]

    P. J. Mohr, D. B. Newell, B. N. Taylor and E. Tiesinga, CODATA recommended values of the fundamental physical constants: 2022,Rev. Mod. Phys.97(2025) 025002 [2409.03787]

    Pith/arXiv arXiv 2022