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REVIEW 3 major objections 5 minor 36 references

Covariant diffusion and drift of the stochastic GW background with LISA

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Detecting a stochastic gravitational-wave background with LISA could pin down the two parameters of Lorentz-invariant graviton diffusion, κ1 and κ2, to about 10^-56 kg m^2 s^-3 — over twelve orders of magnitude tighter than CMB blackbody…

desk verdict Solid application paper: the analytic diffusion solution and LISA forecast are new and worth engaging, but the 12-order CMB improvement claim is conditional on species-independent kappas, a caveat the authors concede only at the end. read the letter →

arxiv 2505.24620 v1 pith:FWVO3H6R submitted 2025-05-30 gr-qc astro-ph.COhep-th

Giorgio Mentasti , Arad Nasiri This is my paper
classification gr-qcastro-ph.COhep-th
keywords stochasticgravitationalwavebackgroundgravitondiffusiondriftquantumgravityphenomenologyLISALorentzinvarianceBoltzmannequationFisherforecast
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

This paper argues that the stochastic gravitational-wave background LISA is designed to detect carries a low-frequency imprint of a Lorentz-invariant quantum-gravity effect: gravitons undergo a tiny random walk in momentum (diffusion, $\kappa_1$) plus a systematic frequency drift ($\kappa_2$) while staying exactly on the light cone. Because the accumulated effect grows with the spacetime area swept by one wavelength, it is largest at millihertz frequencies, making LISA far more sensitive than the CMB photons used to bound the same two parameters. The authors solve the modified Boltzmann equation analytically, show how the background spectrum is distorted by the first and second frequency derivatives of the unperturbed spectrum, and run a Fisher forecast for two realistic source models. With a detected background from a first-order phase transition or from primordial black holes, LISA would constrain $\kappa_1$, $\kappa_2$ at the $10^{-56}\,\mathrm{kg}\,\mathrm{m}^2\,\mathrm{s}^{-3}$ level, improving CMB bounds by more than twelve orders of magnitude.

What carries the argument

The load-bearing object is the covariant stochastic geodesic equation $dp^\mu + \Gamma^\mu_{\alpha\beta} p^\alpha p^\beta d\lambda = \kappa_2 p^\mu d\lambda + p^\mu dW_\lambda$, with $\langle dW_\lambda^2\rangle = 2\kappa_1 d\lambda$, which preserves $p^2=0$. Its Fokker–Planck counterpart is the modified Boltzmann equation (3.6) for the graviton distribution $f$; in FRW coordinates with comoving energy $\epsilon = aE$ and time variable $dT = a^2 d\eta$, this becomes a diffusion equation whose heat kernels $K_\pm$ are given in terms of modified Bessel functions $I_{\pm(1-\kappa_2/\kappa_1)}$. The paper uses the small-diffusion expansion of this kernel, valid for $\kappa_1 \Delta T / \epsilon \ll 1$, to express the perturbed spectrum as $\Omega_{\mathrm{GW}} \simeq \Omega_0 + \kappa_1 \psi_1 + \kappa_2 \psi_2$ times the amplitude, and feeds this into the LISA likelihood (Fisher matrix) to forecast parameter uncertainties.

What would settle it

Take LISA data from a confidently detected first-order phase-transition or primordial-black-hole background, fit the broken power law jointly with the derivative-based distortion in equation (5.2), and read off the marginalized posterior on $\kappa_1$ and $\kappa_2$; if the posterior is centered at zero with a width well below the forecast $\sim 10^{-56}\,\mathrm{kg}\,\mathrm{m}^2\,\mathrm{s}^{-3}$, the claim that LISA can detect diffusive graviton transport at that level is falsified for that source.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that a Lorentz-invariant, parameter-light modification of graviton propagation can be tested with the shape of the stochastic gravitational-wave background rather than with dispersion-relation violations. The graviton distribution obeys the modified Boltzmann equation $p^\mu \partial_\mu f + \Gamma^\sigma_{\mu\nu} p^\sigma p^\mu \partial_{p^\nu} f = (3\kappa_1-\kappa_2) \frac{1}{E^2}\partial_E(E^2 f) + \kappa_1 \frac{1}{E^2}\partial_E(E^3 \partial_E f)$, which the authors solve analytically via heat kernels expressed with modified Bessel functions. In the observationally relevant small-$\kappa$ regime, the solution reduces to $f(T,\epsilon) = f_i(\epsilon) + \left[2(3-\kappa_2/\kappa_1) f_i/\epsilon + (6-\kappa_2/\kappa_1) f_i' + \epsilon f_i''\right] \kappa_1 \Delta T + O\left((\kappa_1 \Delta T/\epsilon)^2\right)$, so the distortion of $\Omega_{\mathrm{GW}}$ depends only on the first two frequency derivatives of the unperturbed spectrum. Fisher forecasts using LISA's five-year sensitivity then give marginal $1\sigma$ errors on $\kappa_1$ and $\kappa_2$ at the $10^{-56}\,\mathrm{kg}\,\mathrm{m}^2\,\mathrm{s}^{-3}$ scale for phase-transition backgrounds and down to $10^{-59}$–$10^{-60}$ in the primordial-black-hole scenario, over twelve orders of magnitude stronger than CMB blackbody constraints.

Load-bearing premise

The forecast rests on the assumption that the stochastic geodesic equation with a single Wiener process and constants $\kappa_1$, $\kappa_2$ is the correct effective transport law for gravitons, with no dependence on species, spin, or waveform; if quantum gravity's effect is not of this minimal form, the quoted LISA sensitivities may not correspond to any physical parameters.

Editorial extensions

If this is right

  • If LISA detects a stochastic background whose spectrum has broken-power-law features, the same data stream yields a direct measurement of $\kappa_1$, $\kappa_2$ at roughly $10^{-56}\,\mathrm{kg}\,\mathrm{m}^2\,\mathrm{s}^{-3}$ or better, instead of the indirect CMB constraints.
  • The diffusion and drift distortion is tied to frequency derivatives of the unperturbed spectrum, so any background with non-trivial spectral shape carries a distinct, difficult-to-mimic signature that breaks degeneracy with amplitude and peak frequency.
  • Because the effect accumulates with affine time and dominates at low frequencies, LISA's millihertz band is the natural place to look; ground-based interferometers lack both a detected cosmological background and the same low-frequency leverage.
  • If no background is bright enough or no spectral features are present, the same Fisher machinery provides upper limits on $\kappa_1$, $\kappa_2$ rather than measurements, scaling as $\sigma_\kappa \sim 10^{-56}(10^{-11}/\Omega_b)\,\mathrm{kg}\,\mathrm{m}^2\,\mathrm{s}^{-3}$.
  • The forecast bounds are model-dependent in amplitude but not in mechanism: any source whose $\Omega_{\mathrm{GW}}(\nu)$ is not a pure power law is distorted in a predictable way.

Reading between the lines

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

  • Editorial inference: if $\kappa_1$ and $\kappa_2$ are waveform-dependent, as the conclusion leaves open, the photon-based CMB bounds and the graviton-based LISA bounds may not be directly comparable; a field-level treatment of diffusion for spin-two waves could change the predicted spectral distortion and therefore the interpretation of a $10^{-56}$ bound.
  • Editorial inference: the forecast's $\sigma_\kappa \propto 1/\Omega_b$ scaling suggests that even a background with modest amplitude but a sharp spectral feature could outperform the fiducial cases, because the signal-to-noise enters through the mismatch between $\Omega_0$ and its derivatives.
  • Editorial inference: the paper's perturbative regime requires $\kappa_1\Delta T/\epsilon \ll 1$; near $\kappa_1 \sim 10^{-55}$ for mHz frequencies, the effect on the spectrum is already drastic, so a detection of the background with no distortion would push the bound substantially below $10^{-56}$, while a large measured distortion would require the full heat-kernel solution rather than the Taylor
  • Editorial inference: a negative drift $\kappa_2 \le 0$ is handled only perturbatively because the heat-kernel expressions are ill-behaved near zero frequency; if future data favored $\kappa_2 < 0$, one would need a separate treatment of the low-frequency tail before trusting the forecast.
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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 a Lorentz-invariant stochastic correction to the massless geodesic equation, parameterized by a diffusion coefficient κ1 and a drift coefficient κ2, and applies it to the propagation of a stochastic gravitational wave background. After relating the GW energy-density spectrum to the graviton phase-space distribution through the Wigner function (Section 2), the authors adopt the modified Boltzmann equation of [19,20] (Section 3), solve it analytically in FRW via heat kernels (Section 4), and derive a first-order perturbative solution. They then perform a Fisher forecast for LISA using broken power-law spectra motivated by first-order phase transitions and primordial black holes (Section 5), reporting projected 1σ sensitivity to κ1,κ2 and claiming an improvement over CMB blackbody bounds by more than 12 orders of magnitude.

Significance. If the effective transport equation is correct and the parameters κ1,κ2 are universal across massless species, this paper proposes a qualitative new, falsifiable probe of Lorentz-invariant quantum-gravity diffusion in the mHz band. The analytic heat-kernel solution (Section 4 and Appendix A) is a useful technical contribution, and the Fisher pipeline is described in enough detail to be reproduced. The authors are also transparent about the idealized LISA assumptions and about the open question of species and waveform dependence, which is an important strength of the presentation.

major comments (3)
  1. [Abstract; §5.1; §6] The headline claim that LISA can improve CMB bounds by over 12 orders of magnitude assumes that the κ1,κ2 constrained by gravitons are the same parameters as those constrained by CMB photons. Section 6 explicitly allows that 'the diffusion and drift parameters might as well differ between species of massless particles, or even depend on Lorentz-invariant properties of a given waveform [33].' Without an argument for universality, the comparison in the abstract and Section 5.1 is not between like and like. The paper should either provide such an argument or qualify the claim, e.g., 'under the assumption of species-universal κ1,κ2.'
  2. [§5.1, Fig. 3] The quoted constraints are internally inconsistent. For the first-order phase-transition cases, the numbers in the text (κ1 ≤ 1.8×10−56, κ2 ≤ 3.7×10−56 kg m2 s−3) are the upper ends of the 68% intervals of a posterior centered at the nonzero fiducial values (top-left panel of Fig. 3). For the PBH cases, the quoted values (κ1 ≤ 6.5×10−60, κ2 ≤ 2.6×10−59 kg m2 s−3) are instead the 1σ errors around a posterior centered at κ1 = κ2 = 10−56 (bottom-right panel), not upper bounds on the parameters. A true upper-limit forecast requires a null fiducial or reporting of the full posterior interval. As written, the abstract's statement that LISA can 'constrain the parameters down to a value of κ1, κ2 ≲ 10−56' does not follow from the PBH panels, which actually measure κ1 ≈ 10−56 with a few × 10−60 error.
  3. [§2–§3] The central forecast is conditional on equation (3.6) being the correct effective transport equation for the graviton distribution, but this equation is imported from [19,20] rather than derived here. The Wigner-function construction in Section 2 shows that the unperturbed f satisfies the Boltzmann equation and is related to ΩGW, but it does not establish that the modified equation (3.6), derived from a point-particle stochastic geodesic equation, applies to spin-2 fields. Since Section 1 notes that a Lorentz-invariant field-level diffusion for hμν is lacking, the paper should state this identification as an explicit assumption of the forecast, or provide additional support for its validity in the graviton sector.
minor comments (5)
  1. [Fig. 1 caption] The caption contains typos: 'rations' should be 'ratios' and 'time laps' should be 'elapsed time'.
  2. [Abstract; §5.1] The CMB bound from [19] is never stated numerically, so the 'over 12 orders of magnitude' claim cannot be checked. Please give the reference value and the explicit ratio.
  3. [§4, Eqs. (4.4)–(4.5)] The text establishes a unique physical heat kernel only for κ2 > 0, yet the Fisher forecast in Fig. 3 allows negative κ2 in the posterior. Either impose κ2 ≥ 0 as a prior or justify the use of the perturbative solution in the negative-κ2 region.
  4. [Eq. (4.1) and below] The notation I±(1−κ2/κ1) is confusing; it should be written as I_{±(1−κ2/κ1)} throughout Section 4.
  5. [Abstract] The abstract's single value κ1,κ2 ≲ 10−56 conflates the phase-transition and PBH scenarios; the PBH scenario gives considerably stronger projected constraints, and the abstract should make clear which scenario underlies the quoted number.

Circularity Check

1 steps flagged · score 2.0 of 10

Forecast is a self-contained forward calculation; only a minor, non-load-bearing self-citation supports the stochastic-geodesic form of the model.

  1. uniqueness imported from authors [Introduction (p. 2); Eqs. (3.2)-(3.4)]
    "This was further reinforced in [20], where it was shown that the Lorentz-invariant diffusion is equivalent to a stochastic correction to the massless geodesic equation. The uniqueness of the effect, however, is lost in curved spacetime where there can be curvature corrections to the covariance of the noise. Insisting on minimal coupling brings back the uniqueness and makes it the leading order correction to the geodesic equation."

    The two-parameter stochastic geodesic equation (3.3) is the model input for the whole forecast, and its uniqueness/equivalence is attributed to Ref. [20], a paper coauthored by A. Nasiri, one of the present authors. The text invokes that self-cited result as the justification for treating (3.3) as the forced leading-order correction. This is a mild self-citation circularity at the level of model justification, but it is not a reduction of the forecast: the paper then solves this imported equation analytically and builds a Fisher forecast against an external LISA noise curve, with no parameter fitted to the forecast target. Ref. [19], which is independent of the present authors, provides the underlying diffusion/Boltzmann equation and the CMB benchmark.

full rationale

The core quantitative result is a Fisher forecast, not a fit: the authors assume fiducial values κ̄1=κ̄2=10^-56, evolve the modified Boltzmann equation (3.6) with the analytic solution (4.7), map f to ΩGW via Eq. (2.7), and compute the covariance from Eq. (B.1) using the external LISA sensitivity curve of Ref. [28]. No parameter is fitted to a subset of data and then 'predicted'; the CMB bound is external and from Ref. [19], and the comparison with that bound is a forward calculation of how a detectable background would constrain the same two-parameter model. The only self-citation is Ref. [20], used to justify the stochastic-geodesic form and uniqueness of the Lorentz-invariant diffusion. That is a minor justification-level issue, not a construction-level circularity: the forecast would stand even if the uniqueness claim were weakened, since it only constrains the parameters of the adopted model. The paper explicitly flags the species-dependence caveat in Sec. 6, so the photon-vs-graviton comparison is presented as a possibility rather than disguised; this is a physical/validity limitation, not a circular step. No known empirical pattern is renamed, and no uniqueness theorem from the present authors is used to forbid alternatives in a way that determines the numerical constraints. I therefore do not find a significant circularity; the score reflects only the one non-load-bearing self-citation.

Assumptions & free parameters 4 free parameters · 5 assumptions · 1 invented entities

The central forecast rests on the imported diffusion model, the chosen spectral templates, and idealized LISA assumptions. The free parameters are the model's κ1,κ2 and the background amplitude/peak frequency; the axioms are the model validity, the photon-graviton parameter equality, the broken power law templates, and the data assumptions. No new entities are introduced beyond the stochastic noise inherited from [19,20].

free parameters (4)
  • κ1 = 1e-56 kg m^2 s^-3 (fiducial)
    Diffusion noise parameter from [19,20]; not fitted here, but the Fisher forecast is centered on this fiducial value.
  • κ2 = 1e-56 kg m^2 s^-3 (fiducial)
    Drift parameter; same treatment as κ1.
  • Ωb = 1e-11 (phase transition) or 1e-8 (PBH)
    Amplitude of the unperturbed GW background; assumed detectable in the forecast.
  • νb = 1 mHz or 5 mHz
    Peak frequency of the unperturbed background; assumed in the forecast scenarios.
assumptions (5)
  • domain assumption The stochastic geodesic equation (3.3) with a single Wiener process is the unique Lorentz-invariant correction to massless particle propagation
    Imported from [19,20]; not derived in this paper; underlies the modified Boltzmann equation (3.6).
  • domain assumption The diffusion and drift parameters κ1,2 are the same for gravitons as for the CMB photons constrained in [19]
    Used when comparing LISA bounds to CMB bounds; authors concede in Section 6 that parameters may differ between species.
  • domain assumption The unperturbed GW spectra are described by the broken power law (5.5) with exponents from [29,30,32]
    The Fisher forecast integrates these template shapes; the PBH exponents are chosen to mimic figure 2 of [32] (Section 5.1).
  • domain assumption LISA data analysis assumptions: perfect subtraction of other sources, stationary isotropic Gaussian background, Gaussian noise with known PSD
    Explicitly stated in Section 6 as a limitation of the forecast.
  • domain assumption The perturbative expansion κ1ΔT/ϵ << 1 is valid for the fiducial values and the LISA band
    The analytic result (4.7) is derived in this regime; the paper notes κ1ΔT/ϵ is order 1 for κ1~1e-55, so at fiducial 1e-56 the expansion is marginal but acceptable.
invented entities (1)
  • Lorentz-invariant stochastic noise dW_λ on the light cone
    purpose: Generates the covariant diffusion and drift of massless particles (gravitons) via equation (3.3)
    Postulated in [19,20]; no direct detection; existing CMB constraints only bound the parameters, and the LISA forecast is a proposed future probe.

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Cite this review

Pith. "Pith review of Covariant diffusion and drift of the stochastic GW background with LISA." pith.science (2026). https://pith.science/paper/FWVO3H6R

@misc{pith2026250524620,
  author       = {Pith},
  title        = {Pith review of: Covariant diffusion and drift of the stochastic GW background with LISA},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FWVO3H6R}},
  note         = {Machine review of arXiv:2505.24620}
}
abstract

We study the covariant diffusion and drift of massless particles on the light cone within the context of quantum gravity phenomenology. Unlike modified dispersion relations that violate Lorentz invariance and grow with frequency, this model introduces a stochastic correction to the massless geodesic equation while preserving Lorentz invariance, and is dominant at lower frequencies due to the larger spacetime support of long-wavelength modes. The effect is phenomenologically described by just two diffusion and drift parameters, $\kappa_1$ and $\kappa_2$, whose values are already constrained by measurements of the CMB blackbody spectrum. We show that a direct measurement and characterization of a gravitational wave (GW) background frequency spectrum can improve bounds on these diffusion and drift parameters by over 12 orders of magnitude compared to those from the CMB. In particular, we find that detecting a GW background sourced by realistic models of first-order phase transitions or primordial black holes (PBH) with LISA can constrain the parameters down to a value of $\kappa_1,\,\kappa_2\lesssim 10^{-56}\,\text{kg}\,\text{m}^2\text{s}^{-3}$.

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Works this paper leans on

36 extracted references · 13 canonical work pages

  1. [33]

    Dowker, Introduction to causal sets and their phenomenology , Gen

    F. Dowker, Introduction to causal sets and their phenomenology , Gen. Rel. Grav. 45 (2013), no. 9 1651–1667

    work page 2013

  2. [1]

    Kobakhidze, C

    A. Kobakhidze, C. Lagger, and A. Manning, Constraining noncommutative spacetime from GW150914 , Phys. Rev. D 94 (2016), no. 6 064033, [ arXiv:1607.03776]

    arXiv 2016

  3. [2]

    Maselli, P

    A. Maselli, P. Pani, V. Cardoso, T. Abdelsalhin, L. Gualtieri, and V. Ferrari, From micro to macro and back: probing near-horizon quantum structures with gravitational waves , Class. Quant. Grav. 36 (2019), no. 16 167001, [ arXiv:1811.03689]

    arXiv 2019

  4. [3]

    S. B. Giddings, S. Koren, and G. Trevi˜ no,Exploring strong-field deviations from general relativity via gravitational waves, Phys. Rev. D 100 (2019), no. 4 044005, [ arXiv:1904.04258]

    arXiv 2019

  5. [4]

    Agullo, V

    I. Agullo, V. Cardoso, A. D. Rio, M. Maggiore, and J. Pullin, Potential Gravitational Wave Signatures of Quantum Gravity , Phys. Rev. Lett. 126 (2021), no. 4 041302, [ arXiv:2007.13761]

    arXiv 2021

  6. [5]

    Garcia-Chung, J

    A. Garcia-Chung, J. B. Mertens, S. Rastgoo, Y. Tavakoli, and P. Vargas Moniz, Propagation of quantum gravity-modified gravitational waves on a classical FLRW spacetime , Phys. Rev. D 103 (2021), no. 8 084053, [arXiv:2012.09366]

    arXiv 2021

  7. [6]

    Calcagni, Lorentz violations in multifractal spacetimes , Eur

    G. Calcagni, Lorentz violations in multifractal spacetimes , Eur. Phys. J. C 77 (2017), no. 5 291, [arXiv:1603.03046]. – 17 –

    arXiv 2017

  8. [7]

    Wang, Exploring the CPT violation and birefringence of gravitational waves with ground- and space-based gravitational-wave interferometers, Eur

    S. Wang, Exploring the CPT violation and birefringence of gravitational waves with ground- and space-based gravitational-wave interferometers, Eur. Phys. J. C 80 (2020), no. 4 342, [arXiv:1712.06072]

    arXiv 2020

Show all 36 references
  1. [8]

    Yunes, K

    N. Yunes, K. Yagi, and F. Pretorius, Theoretical Physics Implications of the Binary Black-Hole Mergers GW150914 and GW151226 , Phys. Rev. D 94 (2016), no. 8 084002, [ arXiv:1603.08955]

    2016 arXiv

  2. [9]

    Calcagni, S

    G. Calcagni, S. Kuroyanagi, S. Marsat, M. Sakellariadou, N. Tamanini, and G. Tasinato, Gravitational-wave luminosity distance in quantum gravity , Phys. Lett. B 798 (2019) 135000, [arXiv:1904.00384]

    2019 arXiv

  3. [10]

    Calcagni, S

    G. Calcagni, S. Kuroyanagi, S. Marsat, M. Sakellariadou, N. Tamanini, and G. Tasinato, Quantum gravity and gravitational-wave astronomy , JCAP 10 (2019) 012, [ arXiv:1907.02489]

    2019 arXiv

  4. [11]

    Belgacem et al., Testing modified gravity at cosmological distances with LISA standard sirens , JCAP 07 (2019) 024, [ arXiv:1906.01593]

    LISA Cosmology W orking Group Collaboration, E. Belgacem et al., Testing modified gravity at cosmological distances with LISA standard sirens , JCAP 07 (2019) 024, [ arXiv:1906.01593]

    2019 arXiv

  5. [12]

    Ellis, N

    J. Ellis, N. E. Mavromatos, and D. V. Nanopoulos, Comments on Graviton Propagation in Light of GW150914, Mod. Phys. Lett. A 31 (2016), no. 26 1675001, [ arXiv:1602.04764]

    2016 arXiv

  6. [13]

    Vasileiou, A

    V. Vasileiou, A. Jacholkowska, F. Piron, J. Bolmont, C. Couturier, J. Granot, F. W. Stecker, J. Cohen-Tanugi, and F. Longo, Constraints on Lorentz Invariance Violation from Fermi-Large Area Telescope Observations of Gamma-Ray Bursts , Phys. Rev. D 87 (2013), no. 12 122001, [ar...

    2013 arXiv

  7. [14]

    Ellis, R

    J. Ellis, R. Konoplich, N. E. Mavromatos, L. Nguyen, A. S. Sakharov, and E. K. Sarkisyan-Grinbaum, Robust Constraint on Lorentz Violation Using Fermi-LAT Gamma-Ray Burst Data , Phys. Rev. D 99 (2019), no. 8 083009, [ arXiv:1807.00189]

    2019 arXiv

  8. [15]

    Cao et al., Stringent Tests of Lorentz Invariance Violation from LHAASO Observations of GRB 221009A , Phys

    LHAASO Collaboration, Z. Cao et al., Stringent Tests of Lorentz Invariance Violation from LHAASO Observations of GRB 221009A , Phys. Rev. Lett. 133 (2024), no. 7 071501, [ arXiv:2402.06009]

    2024

  9. [16]

    Piran and D

    T. Piran and D. D. Ofengeim, Lorentz invariance violation limits from GRB 221009A , Phys. Rev. D 109 (2024), no. 8 L081501, [ arXiv:2308.03031]

    2024 arXiv

  10. [17]

    Albert et al., Probing Quantum Gravity using Photons from a flare of the active galactic nucleus Markarian 501 Observed by the MAGIC telescope , Phys

    MAGIC, Other Contributors Collaboration, J. Albert et al., Probing Quantum Gravity using Photons from a flare of the active galactic nucleus Markarian 501 Observed by the MAGIC telescope , Phys. Lett. B 668 (2008) 253–257, [ arXiv:0708.2889]

    2008 arXiv

  11. [18]

    Collaboration, A

    H.E.S.S. Collaboration, A. Abramowski et al., Search for Lorentz Invariance breaking with a likelihood fit of the PKS 2155-304 Flare Data Taken on MJD 53944 , Astropart. Phys. 34 (2011) 738–747, [arXiv:1101.3650]

    2011 arXiv

  12. [19]

    Philpott, F

    L. Philpott, F. Dowker, and R. D. Sorkin, Energy-momentum diffusion from spacetime discreteness , Phys. Rev. D 79 (2009) 124047, [ arXiv:0810.5591]

    2009 arXiv

  13. [20]

    Albertini, A

    E. Albertini, A. Nasiri, and E. Panella, Stochastic dark matter: Covariant Brownian motion from Planckian discreteness, Phys. Rev. D 111 (2025), no. 2 023514, [ arXiv:2409.02188]

    2025 arXiv

  14. [21]

    C. R. Contaldi, F. Dowker, and L. Philpott, Polarization Diffusion from Spacetime Uncertainty , Class. Quant. Grav. 27 (2010) 172001, [ arXiv:1001.4545]

    2010 arXiv

  15. [22]

    Caprini, M

    C. Caprini, M. Hindmarsh, S. Huber, T. Konstandin, J. Kozaczuk, G. Nardini, D. Roper Pol, G. Servant, and V. Vennin, Science with the space-based interferometer elisa. ii: Gravitational waves from cosmological phase transitions, JCAP 04 (2016) 001, [ arXiv:1512.06239]

    2016 arXiv

  16. [23]

    Bartolo, C

    N. Bartolo, C. Caprini, V. Domcke, D. G. Figueroa, M. Gasperini, M. Hindmarsh, C. Hogan, M. Kamionkowski, E. D. Kovetz, M. Liguori, S. Matarrese, A. Riotto, and G. Tasinato, Science with the – 18 – space-based interferometer lisa. iv: Probing inflation with gravitational waves...

    2016 arXiv

  17. [24]

    Bartolo, C

    N. Bartolo, C. Caprini, S. Codis, D. G. Figueroa, X. Gao, S. Kuroyanagi, S. Matarrese, and P. Collaboration, Science with the space-based interferometer lisa. v: Probing dark matter and primordial black holes , JCAP 11 (2018) 055, [ arXiv:1712.01219]

    2018 arXiv

  18. [25]

    O. A. Fonarev, Wigner function and quantum kinetic theory in curved space–time and external fields , Journal of Mathematical Physics 35 (1994), no. 5 2105–2129

    1994

  19. [26]

    Garcia-Compean and F

    H. Garcia-Compean and F. J. Turrubiates, Ground-state Wigner functional of linearized gravitational field, Int. J. Mod. Phys. A 26 (2011) 5241–5259, [ arXiv:1109.1036]

    2011 arXiv

  20. [27]

    S. R. Groot, W. A. Leeuwen, C. G. van Weert, and C. G. Weert, Relativistic kinetic theory: principles and applications. North Holland, 1980

    1980

  21. [28]

    Bartolo et al., Probing anisotropies of the Stochastic Gravitational Wave Background with LISA , JCAP 11 (2022) 009, [ arXiv:2201.08782]

    LISA Cosmology W orking Group Collaboration, N. Bartolo et al., Probing anisotropies of the Stochastic Gravitational Wave Background with LISA , JCAP 11 (2022) 009, [ arXiv:2201.08782]

    2022 arXiv

  22. [29]

    Caprini, R

    LISA Cosmology W orking Group Collaboration, C. Caprini, R. Jinno, M. Lewicki, E. Madge, M. Merchand, G. Nardini, M. Pieroni, A. Roper Pol, and V. Vaskonen, Gravitational waves from first-order phase transitions in LISA: reconstruction pipeline and physics interpretation , JCA...

    2024 arXiv

  23. [30]

    Lewicki and V

    M. Lewicki and V. Vaskonen, Gravitational waves from bubble collisions and fluid motion in strongly supercooled phase transitions, Eur. Phys. J. C 83 (2023), no. 2 109, [ arXiv:2208.11697]

    2023 arXiv

  24. [31]

    J. R. Espinosa, D. Racco, and A. Riotto, A Cosmological Signature of the SM Higgs Instability: Gravitational Waves, JCAP 09 (2018) 012, [ arXiv:1804.07732]

    2018 arXiv

  25. [32]

    Bartolo, V

    N. Bartolo, V. De Luca, G. Franciolini, M. Peloso, D. Racco, and A. Riotto, Testing primordial black holes as dark matter with LISA , Phys. Rev. D 99 (2019), no. 10 103521, [ arXiv:1810.12224]

    2019 arXiv

  26. [34]

    S. A. Salcedo, T. Colas, and E. Pajer, An Open Effective Field Theory for light in a medium , JHEP 03 (2025) 138, [ arXiv:2412.12299]

    2025 arXiv

  27. [35]

    J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics. Quantum physics, quantum information and quantum computation. Cambridge University Press, 10, 2020

    2020

  28. [36]

    Dodelson and F

    S. Dodelson and F. Schmidt, Modern Cosmology. Elsevier Science, 2020. – 19 –

    2020

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.