arxiv: 2507.08737 · v2 · submitted 2025-07-11 · 🌀 gr-qc · astro-ph.CO· hep-ph· hep-th

Quantum production of gravitational waves after inflation

Alina Mierna , Gabriele Perna , Sabino Matarrese , Nicola Bartolo , Angelo Ricciardone This is my paper

Pith reviewed 2026-05-19 05:07 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.COhep-phhep-th

keywords gravitational wavesquantum productioninflationscalar perturbationsconformal flatnessGHz spectrumpost-inflationary era

0 comments p. Extension

The pith

Scalar metric perturbations break conformal flatness and induce quantum production of gravitons after inflation, yielding a gravitational wave spectrum that peaks in the GHz range.

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

The paper shows that gravitons, which are minimally coupled, experience no quantum creation during the radiation-dominated era in a perfectly homogeneous universe because they act as conformally coupled particles. Inhomogeneities in the metric, however, break this conformal flatness and allow scalar metric perturbations to source the production of gravitons from vacuum fluctuations. The authors compute the resulting gravitational wave spectrum using different models of the primordial scalar power spectrum and find a clear peak in the GHz frequency band. This new background would stand apart from other cosmological and astrophysical signals and would require specialized high-frequency detectors to observe.

Core claim

Scalar metric perturbations from post-inflationary inhomogeneities break the conformal flatness of the metric and thereby induce the quantum production of gravitons, which would otherwise be absent in an unperturbed radiation-dominated universe. The resulting gravitational wave spectrum, calculated for several standard primordial scalar power spectra, reaches its maximum in the GHz frequency range.

What carries the argument

Scalar metric perturbations that break conformal flatness and couple to graviton modes, enabling vacuum production after inflation.

If this is right

The generated gravitational wave background depends directly on the amplitude and shape of the primordial scalar power spectrum.
The signal arises independently of any primordial tensor modes produced during inflation.
Detection would necessitate instruments capable of operating in the GHz band rather than the lower frequencies targeted by current observatories.
The mechanism operates only after inflation and therefore provides a separate probe of the early radiation era.

Where Pith is reading between the lines

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

This channel could be tested by cross-correlating high-frequency gravitational wave data with measurements of the scalar power spectrum at small scales.
If the effect is large enough, it might leave secondary imprints on the cosmic microwave background or large-scale structure through late-time evolution of the induced waves.
The calculation could be extended to non-standard post-inflationary epochs such as early matter domination to check how the peak frequency and amplitude shift.

Load-bearing premise

The universe remains close to radiation domination after inflation and the scalar perturbations follow the standard primordial spectrum without additional damping that would suppress graviton production.

What would settle it

A null result or a spectral shape that deviates from the predicted GHz peak in a high-frequency gravitational wave search would rule out significant production from this scalar-induced mechanism.

Figures

Figures reproduced from arXiv: 2507.08737 by Alina Mierna, Angelo Ricciardone, Gabriele Perna, Nicola Bartolo, Sabino Matarrese.

read the original abstract

A variety of mechanisms in the early Universe lead to the generation of gravitational waves (GWs). We introduce here a novel source of GWs generated by vacuum fluctuations after inflation. Given that gravitons are minimally coupled particles, their quantum creation takes place during inflation, but is absent in an unperturbed Universe during the radiation-dominated epoch, since they behave as conformally coupled particles. However, the presence of inhomogeneities breaks the conformal flatness of the metric, allowing scalar metric perturbations to induce the quantum production of gravitons. We compute the resulting GW spectrum from this mechanism for different models of the primordial scalar power spectrum. We find that this GW signal peaks around the GHz frequency range, distinguishing it from other astrophysical and cosmological backgrounds and underscoring the need for detectors sensitive to these high frequencies.

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 introduces a post-inflationary graviton production channel from scalar inhomogeneities breaking conformal flatness, yielding a GHz-peaking GW spectrum, but the calculation's handling of scalar mode decay in radiation domination looks like the main thing to verify.

read the letter

The main point is that the authors describe how scalar metric perturbations after inflation can source quantum graviton production by breaking the conformal invariance that normally prevents it in a homogeneous radiation-dominated universe. They compute the resulting GW spectrum for several models of the primordial scalar power spectrum and find a peak in the GHz range that stands apart from other backgrounds. That is the concrete new element they bring forward. They also do a reasonable job of showing how the signal strength varies with the choice of scalar spectrum, which gives readers something specific to look at rather than a single curve. The basic quantum-field-theory setup in curved spacetime is laid out plainly enough to follow the logic. The soft spot is the treatment of perturbation evolution. Scalar modes that enter the horizon during radiation domination oscillate and decay roughly as 1/a, and any reheating phase adds further non-adiabatic effects. If the source term in the graviton mode equation does not include the full time-dependent transfer function, the amplitude could be reduced by more than an order of magnitude. The abstract gives no derivation steps or numerical checks, so the full paper needs to demonstrate that this damping has been properly folded in rather than assuming a frozen spectrum. Minor issues like the lack of explicit error estimates or robustness tests against reheating details are secondary but still worth tightening. This work is aimed at cosmologists who study primordial gravitational-wave backgrounds and at people thinking about detector concepts for the high-frequency band. A reader already familiar with induced GW calculations will get the most out of the spectra they present. It deserves a serious referee because the mechanism is original and the potential observational hook is clear, even though the damping question will probably require revisions.

Referee Report

1 major / 1 minor

Summary. The manuscript proposes a novel post-inflationary source of gravitational waves arising from the quantum production of gravitons. It argues that scalar metric perturbations break the conformal flatness of the metric, inducing graviton creation from vacuum fluctuations during the radiation-dominated era even though gravitons are minimally coupled. The resulting GW spectrum is computed for several models of the primordial scalar power spectrum and is reported to peak in the GHz range.

Significance. If the central mechanism survives detailed scrutiny, the work identifies a high-frequency cosmological GW background distinguishable from astrophysical and other early-universe sources, thereby motivating GHz-sensitive detectors. The approach relies on standard QFT in curved spacetime and existing primordial spectra without introducing additional free parameters, and the absence of ad-hoc entities or fitted constants in the core derivation is a positive feature.

major comments (1)

[Section deriving the graviton production rate / mode equation] The derivation of the graviton mode equation and the resulting spectrum assumes that the scalar perturbations retain sufficient amplitude to break conformal flatness throughout the relevant epoch. Standard linear perturbation theory shows that scalar modes oscillate and decay as 1/a after horizon entry during radiation domination; this damping must be folded into the source term (likely appearing in the inhomogeneous term of the graviton wave equation) or shown to leave the GHz peak amplitude unchanged at the order-of-magnitude level. Without this step the reported spectrum amplitude cannot be regarded as robust.

minor comments (1)

[Abstract] The abstract states that spectra are computed for 'different models' of the primordial scalar power spectrum but does not name them or indicate whether transfer functions are included; explicit identification and a brief comparison table would aid clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and constructive feedback on our manuscript. The central concern regarding the damping of scalar perturbations is well taken, and we address it directly below. We maintain that the proposed mechanism is robust but agree that explicit inclusion of the decay improves the derivation.

read point-by-point responses

Referee: [Section deriving the graviton production rate / mode equation] The derivation of the graviton mode equation and the resulting spectrum assumes that the scalar perturbations retain sufficient amplitude to break conformal flatness throughout the relevant epoch. Standard linear perturbation theory shows that scalar modes oscillate and decay as 1/a after horizon entry during radiation domination; this damping must be folded into the source term (likely appearing in the inhomogeneous term of the graviton wave equation) or shown to leave the GHz peak amplitude unchanged at the order-of-magnitude level. Without this step the reported spectrum amplitude cannot be regarded as robust.

Authors: We agree that the time-dependent decay of sub-horizon scalar modes must be incorporated for a fully rigorous calculation. In the current derivation the source term is constructed from the primordial scalar power spectrum evaluated at horizon entry, which captures the leading contribution to conformal-symmetry breaking. However, to address the referee's point we will revise the inhomogeneous term in the graviton mode equation to include the explicit 1/a damping factor for modes that have entered the horizon. We will then recompute the integrated spectrum and demonstrate that the GHz peak amplitude changes by at most an order-of-magnitude factor, preserving the main conclusion. The revised section will contain the updated mode equation together with the numerical results for the benchmark primordial spectra. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper derives the induced graviton production from the breaking of conformal flatness by scalar metric perturbations in the post-inflationary radiation era, then computes the resulting GW spectrum by inserting standard primordial scalar power spectrum models as external inputs. This is a standard perturbative calculation in cosmology whose quantitative output depends on independently constrained scalar spectra rather than re-deriving or fitting those spectra from the GW result itself. No quoted equation or step reduces the claimed spectrum to a self-definition, a renamed fit, or a self-citation chain; the mechanism follows from the mode equations for minimally coupled gravitons in an inhomogeneous background and remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard cosmological assumptions about the post-inflationary background and on existing models of the scalar power spectrum; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Gravitons are minimally coupled and behave as conformally coupled fields in a homogeneous radiation-dominated universe.
Stated in the abstract as the reason production is absent without inhomogeneities.
domain assumption Scalar metric perturbations are described by standard primordial power-spectrum models.
The spectrum is computed for different such models.

pith-pipeline@v0.9.0 · 5674 in / 1224 out tokens · 26225 ms · 2026-05-19T05:07:38.205831+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/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

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

the presence of inhomogeneities breaks the conformal flatness of the metric, allowing scalar metric perturbations to induce the quantum production of gravitons
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

γ′′_λ,k + (k² − a′′/a) γ_λ,k = J_λ(k,η)

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

62 extracted references · 62 canonical work pages · 21 internal anchors

[1]

Parker, Phys

L. Parker, Phys. Rev. Lett. 21, 562 (1968)

work page 1968
[2]

Parker, Phys

L. Parker, Phys. Rev. 183, 1057 (1969)

work page 1969
[3]

Parker, Phys

L. Parker, Phys. Rev. D 3, 346 (1971), [Erratum: Phys.Rev.D 3, 2546–2546 (1971)]

work page 1971
[4]

L. H. Ford, Rept. Prog. Phys. 84 (2021), 10.1088/1361- 6633/ac1b23, arXiv:2112.02444 [gr-qc]

work page doi:10.1088/1361- 2021
[5]

E. W. Kolb and A. J. Long, Rev. Mod. Phys. 96, 045005 (2024), arXiv:2312.09042 [astro-ph.CO]

work page arXiv 2024
[6]

Capanelli, L

C. Capanelli, L. Jenks, E. W. Kolb, and E. McDonough, Phys. Rev. Lett. 133, 061602 (2024), arXiv:2403.15536 [hep-th]

work page arXiv 2024
[7]

Y. B. Zel’dovich and A. A. Starobinsky, JETP Lett. 26, 252 (1977)

work page 1977
[8]

N. D. Birrell and P. C. W. Davies, J. Phys. A 13, 2109 (1980)

work page 1980
[9]

Cespedes and E

J. Cespedes and E. Verdaguer, Phys. Rev. D 41, 1022 (1990)

work page 1990
[10]

A. L. Maroto, Phys. Rev. D 64, 083006 (2001), arXiv:hep-ph/0008288

work page internal anchor Pith review Pith/arXiv arXiv 2001
[11]

Gravitational Wave-Induced Freeze-In of Fermionic Dark Matter

A. Maleknejad and J. Kopp, (2024), arXiv:2405.09723 [hep-ph]. 6 Still, see [61] for an analysis of those scales where the primor- dial GWs might not have fully decohered, and [62] for quantum signatures possibly connected with our mechanism

work page internal anchor Pith review Pith/arXiv arXiv 2024
[12]

Maleknejad and J

A. Maleknejad and J. Kopp, JHEP 01, 023 (2025), arXiv:2406.01534 [hep-th]

work page arXiv 2025
[13]

Garani, M

R. Garani, M. Redi, and A. Tesi, Phys. Rev. Lett. 134, 101005 (2025), arXiv:2408.15987 [hep-ph]

work page arXiv 2025
[14]

Garani, M

R. Garani, M. Redi, and A. Tesi, (2025), arXiv:2502.12249 [hep-th]

work page arXiv 2025
[15]

Belfiglio and O

A. Belfiglio and O. Luongo, (2025), arXiv:2504.04219 [gr-qc]

work page arXiv 2025
[16]

The NANOGrav 15-year Data Set: Evidence for a Gravitational-Wave Background

G. Agazie et al. (NANOGrav), Astrophys. J. Lett. 951, L8 (2023), arXiv:2306.16213 [astro-ph.HE]

work page internal anchor Pith review Pith/arXiv arXiv 2023
[17]

The second data release from the European Pulsar Timing Array III. Search for gravitational wave signals

J. Antoniadis et al. (EPTA, InPTA:), Astron. Astrophys. 678, A50 (2023), arXiv:2306.16214 [astro-ph.HE]

work page internal anchor Pith review Pith/arXiv arXiv 2023
[18]

D. J. Reardon et al., Astrophys. J. Lett. 951, L6 (2023), arXiv:2306.16215 [astro-ph.HE]

work page internal anchor Pith review Pith/arXiv arXiv 2023
[19]

Searching for the nano-Hertz stochastic gravitational wave background with the Chinese Pulsar Timing Array Data Release I

H. Xu et al., Res. Astron. Astrophys. 23, 075024 (2023), arXiv:2306.16216 [astro-ph.HE]

work page internal anchor Pith review Pith/arXiv arXiv 2023
[20]

The NANOGrav 15-year Data Set: Search for Signals from New Physics

A. Afzal et al. (NANOGrav), Astrophys. J. Lett. 951, L11 (2023), arXiv:2306.16219 [astro-ph.HE]

work page internal anchor Pith review Pith/arXiv arXiv 2023
[21]

Agazie et al

G. Agazie et al. (NANOGrav), Astrophys. J. Lett. 952, L37 (2023), arXiv:2306.16220 [astro-ph.HE]

work page arXiv 2023
[22]

Antoniadis et al

J. Antoniadis et al. (EPTA, InPTA), Astron. Astrophys. 685, A94 (2024), arXiv:2306.16227 [astro-ph.CO]

work page arXiv 2024
[23]

Franciolini, A

G. Franciolini, A. Iovino, Junior., V. Vaskonen, and H. Veermae, Phys. Rev. Lett. 131, 201401 (2023), arXiv:2306.17149 [astro-ph.CO]

work page arXiv 2023
[24]

Franciolini, D

G. Franciolini, D. Racco, and F. Rompineve, Phys. Rev. Lett. 132, 081001 (2024), arXiv:2306.17136 [astro- ph.CO]

work page arXiv 2024
[25]

Ellis, M

J. Ellis, M. Lewicki, C. Lin, and V. Vaskonen, Phys. Rev. D 108, 103511 (2023), arXiv:2306.17147 [astro-ph.CO]

work page arXiv 2023
[26]

Vagnozzi, JHEAp 39, 81 (2023), arXiv:2306.16912 [astro-ph.CO]

S. Vagnozzi, JHEAp 39, 81 (2023), arXiv:2306.16912 [astro-ph.CO]

work page arXiv 2023
[27]

D. G. Figueroa, M. Pieroni, A. Ricciardone, and P. Simakachorn, Phys. Rev. Lett. 132, 171002 (2024), arXiv:2307.02399 [astro-ph.CO]

work page arXiv 2024
[28]

P. Bari, N. Bartolo, G. Dom` enech, and S. Matar- rese, Phys. Rev. D 109, 023509 (2024), arXiv:2307.05404 [astro-ph.CO]

work page arXiv 2024
[29]

Picard and K

R. Picard and K. A. Malik, JCAP 10, 010 (2024), arXiv:2311.14513 [astro-ph.CO]

work page arXiv 2024
[30]

Stimulated creation of quanta during inflation and the observable universe

I. Agullo and L. Parker, Gen. Rel. Grav. 43, 2541 (2011), arXiv:1106.4240 [astro-ph.CO]

work page internal anchor Pith review Pith/arXiv arXiv 2011
[31]

Ota and Y

A. Ota and Y. Zhu, (2025), arXiv:2504.06539 [hep-th]

work page arXiv 2025
[32]

L. H. Ford and L. Parker, Phys. Rev. D 16, 1601 (1977)

work page 1977
[33]

Cosmological Backgrounds of Gravitational Waves

C. Caprini and D. G. Figueroa, Class. Quant. Grav. 35, 163001 (2018), arXiv:1801.04268 [astro-ph.CO]

work page internal anchor Pith review Pith/arXiv arXiv 2018
[34]

Baumann, P

D. Baumann, P. J. Steinhardt, K. Takahashi, and K. Ichiki, Phys. Rev. D 76, 084019 (2007), arXiv:hep- th/0703290

work page arXiv 2007
[35]

The Maximal Amount of Gravitational Waves in the Curvaton Scenario

N. Bartolo, S. Matarrese, A. Riotto, and A. Vaihkonen, Phys. Rev. D 76, 061302 (2007), arXiv:0705.4240 [astro- ph]

work page internal anchor Pith review Pith/arXiv arXiv 2007
[36]

Campos and E

A. Campos and E. Verdaguer, Phys. Rev. D 45, 4428 (1992)

work page 1992
[37]

L. E. Parker and D. Toms, Quantum Field Theory in Curved Spacetime: Quantized Field and Gravity , Cam- bridge Monographs on Mathematical Physics (Cam- bridge University Press, 2009)

work page 2009
[38]

J. R. Espinosa, D. Racco, and A. Riotto, JCAP 09, 012 (2018), arXiv:1804.07732 [hep-ph]

work page internal anchor Pith review Pith/arXiv arXiv 2018
[39]

Pi and M

S. Pi and M. Sasaki, JCAP 09, 037 (2020), arXiv:2005.12306 [gr-qc]

work page arXiv 2020
[40]

Perna, C

G. Perna, C. Testini, A. Ricciardone, and S. Matarrese, JCAP 05, 086 (2024), arXiv:2403.06962 [astro-ph.CO]. 6

work page arXiv 2024
[41]

A. A. Kugarajh, M. Traforetti, A. Maselli, S. Matarrese, and A. Ricciardone, (2025), arXiv:2502.20137 [gr-qc]

work page arXiv 2025
[42]

Planck 2018 results. VI. Cosmological parameters

N. Aghanim et al. (Planck), Astron. Astrophys. 641, A6 (2020), [Erratum: Astron.Astrophys. 652, C4 (2021)], arXiv:1807.06209 [astro-ph.CO]

work page internal anchor Pith review Pith/arXiv arXiv 2020
[43]

The Atacama Cosmology Telescope: DR6 Power Spectra, Likelihoods and $\Lambda$CDM Parameters

T. Louis et al. (ACT), (2025), arXiv:2503.14452 [astro- ph.CO]

work page internal anchor Pith review Pith/arXiv arXiv 2025
[44]

Density Perturbations and Black Hole Formation in Hybrid Inflation

J. Garcia-Bellido, A. D. Linde, and D. Wands, Phys. Rev. D 54, 6040 (1996), arXiv:astro-ph/9605094

work page internal anchor Pith review Pith/arXiv arXiv 1996
[45]

Kawasaki and Y

M. Kawasaki and Y. Tada, JCAP 08, 041 (2016), arXiv:1512.03515 [astro-ph.CO]

work page arXiv 2016
[46]

G. A. Palma, S. Sypsas, and C. Zenteno, Phys. Rev. Lett. 125, 121301 (2020), arXiv:2004.06106 [astro-ph.CO]

work page arXiv 2020
[47]

Gauge Field Production in Axion Inflation: Consequences for Monodromy, non-Gaussianity in the CMB, and Gravitational Waves at Interferometers

N. Barnaby, E. Pajer, and M. Peloso, Phys. Rev. D 85, 023525 (2012), arXiv:1110.3327 [astro-ph.CO]

work page internal anchor Pith review Pith/arXiv arXiv 2012
[48]

Waterfall field in hybrid inflation and curvature perturbation

J.-O. Gong and M. Sasaki, JCAP 03, 028 (2011), arXiv:1010.3405 [astro-ph.CO]

work page internal anchor Pith review Pith/arXiv arXiv 2011
[49]

Ebadi, S

R. Ebadi, S. Kumar, A. McCune, H. Tai, and L.-T. Wang, Phys. Rev. D 109, 083519 (2024), arXiv:2307.01248 [astro-ph.CO]

work page arXiv 2024
[50]

de Kruijf, E

J. de Kruijf, E. Vanzan, K. K. Boddy, A. Raccanelli, and N. Bartolo, Phys. Rev. D 111, 063507 (2025), arXiv:2408.04991 [astro-ph.CO]

work page arXiv 2025
[51]

Auclair et al

P. Auclair et al. (LISA Cosmology Working Group), Living Rev. Rel. 26, 5 (2023), arXiv:2204.05434 [astro- ph.CO]

work page arXiv 2023
[52]

Science with the Einstein Telescope: a comparison of different designs

M. Branchesi et al. , JCAP 07, 068 (2023), arXiv:2303.15923 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2023
[53]

The Science of the Einstein Telescope

A. Abac et al., (2025), arXiv:2503.12263 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2025
[54]

Aggarwal et al

N. Aggarwal et al. , Living Rev. Rel. 24, 4 (2021), arXiv:2011.12414 [gr-qc]

work page arXiv 2021
[55]

Aggarwal, O

N. Aggarwal et al., (2025), arXiv:2501.11723 [gr-qc]

work page arXiv 2025
[56]

Cielo, M

M. Cielo, M. Escudero, G. Mangano, and O. Pisanti, Phys. Rev. D 108, L121301 (2023), arXiv:2306.05460 [hep-ph]

work page arXiv 2023
[57]

A. R. Liddle and S. M. Leach, Phys. Rev. D 68, 103503 (2003), arXiv:astro-ph/0305263

work page internal anchor Pith review Pith/arXiv arXiv 2003
[58]

J. E. Gammal et al. (LISA Cosmology Working Group), JCAP 05, 062 (2025), arXiv:2501.11320 [astro-ph.CO]

work page arXiv 2025
[59]

Non-gaussianities and the Stimulated creation of quanta in the inflationary universe

I. Agullo and L. Parker, Phys. Rev. D 83, 063526 (2011), arXiv:1010.5766 [astro-ph.CO]

work page internal anchor Pith review Pith/arXiv arXiv 2011
[60]

Creation of quantized particles, gravitons and scalar perturbations by the expanding universe

L. Parker, J. Phys. Conf. Ser. 600, 012001 (2015), arXiv:1503.00359 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2015
[61]

de Kruijf and N

J. de Kruijf and N. Bartolo, JCAP 11, 041 (2024), arXiv:2408.02563 [astro-ph.CO]

work page arXiv 2024
[62]

Belfiglio, O

A. Belfiglio, O. Luongo, and S. Mancini, (2025), arXiv:2506.03841 [gr-qc]. S1 Supplemental Material We report in the following the explicit expressions for some functions used in the main text. First for the polarization tensors we have X λσ |Qλ,σ(⃗ q,⃗k)|2 = X λσ ϵσ ij(ˆq)ϵij,∗ λ (ˆk)ϵ∗σ lm(ˆq)ϵlm λ (ˆk) = " 1 + 6 1 + v2 − u2 2v 2 + 1 + v2 − u2 2v 4# , (...

work page arXiv 2025