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REVIEW 2 major objections 4 minor 53 references

A global monopole network radiates Nambu-Goldstone bosons with a spectrum peaking at the Hubble scale, and if those bosons acquire a small mass they can account for the observed dark matter abundance down to masses around 10^-10 eV.

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2026-08-01 04:35 UTC pith:52VHXF5S

load-bearing objection Solid first measurement of the NG spectrum from monopole networks, undermined by an internal factor-130 error in the DM abundance formula. the 2 major comments →

arxiv 2607.22481 v1 pith:52VHXF5S submitted 2026-07-24 hep-ph astro-ph.CO

Nambu-Goldstone emissions from the cosmological evolution of global monopoles

Yukihiro Kanda , Naoya Kitajima This is my paper
classification hep-ph astro-ph.CO
keywords global monopolesNambu-Goldstone bosonspseudo-Nambu-Goldstone dark mattertopological defectslattice simulationsscaling networksoft spectrumsemilocal strings
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.

The paper asks whether the long-range dynamics of global monopoles—topological defects formed when a global O(3) symmetry breaks—can be a significant non-thermal source of Nambu-Goldstone (NG) bosons. Using large lattice simulations in an expanding spacetime, it shows the monopole network reaches a scaling regime and emits both NG modes with a soft spectrum peaking at the Hubble scale, while radial-mode emission is suppressed. The comoving number density of each NG mode grows linearly with conformal time, n_NG ≃ 0.7–0.9 v^2 H. If one or both NG modes later acquire a small mass, they become pseudo-NG dark matter; the paper's estimate gives Ω_NG h^2 ≃ 0.2 (m_NG/10^-13 eV)^{1/2} (v/10^14 GeV)^2, making masses down to ~10^-10 eV viable. A sympathetic reader would care because it provides a concrete, calculable dark matter production channel from monopoles and may explain NG emission from semilocal strings.

Core claim

The central discovery is that the scaling global monopole network is a copious and soft source of NG bosons: the spectrum of each of the two massless modes peaks at momenta of order the Hubble scale, the comoving number density increases linearly with conformal time (n_NG = C v^2 H with C ≈ 0.7–0.9), and the radial mode is subdominant. This linear growth is tied to the scaling of the network: monopole separation ξ ∝ τ, so the energy lost to radiation is roughly 2Hρ_M. When a soft mass is added, the abundance of the resulting pNG bosons is given by Eq. (15), and the paper maps the viable region in the (m_NG, v) plane, including the black-hole superradiance constraint. The same spectral shape

What carries the argument

The central object is the global O(3) monopole network, simulated on a lattice with both fat and physical monopole treatments. The key extraction method is writing the scalar triplet in spherical coordinates (ϕ_r, ϑ, φ) so that the two NG modes are read off from the kinetic terms (1/2)v^2 ϑ̇^2 and (1/2)v^2 sin^2ϑ φ̇^2. This lets the authors measure the NG number spectrum d(a^3 n)/d ln k and its linear growth, and relate it to the scaling radiation rate Γ ≃ 2Hρ_M.

Load-bearing premise

The dark matter estimate assumes the measured linear growth n_NG ∝ H and the monopole network's scaling behavior continue from the simulated window (vτ ≲ 10^3) all the way down to H ≃ m_NG for masses as low as 10^-10 eV, with the authors themselves flagging that confirming the suspected logarithmic scaling violation requires longer simulations.

What would settle it

Run a lattice simulation with enough dynamic range to reach H well below the lightest claimed pNG mass (10^-10 eV) while keeping the monopole core resolved; if the comoving NG number density stops growing linearly or the spectrum's IR peak moves away from the Hubble scale before that point, the abundance formula in Eq. (15) would need revision. Observationally, if dark matter is attributed entirely to this channel, the implied (m_NG, v) must lie on the predicted band, so an axion-like particle detection with parameters off that band would rule it out.

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

If this is right

  • NG bosons from global monopoles form a genuine non-thermal radiation background peaking at the horizon scale, with number density growing as n_NG ∝ H.
  • If one NG mode acquires a mass, the emitted pNG bosons can explain all of the observed dark matter for a wide band in the (m_NG, v) plane, including masses down to ~10^-10 eV.
  • The spectral shape closely resembles that from semilocal string networks, supporting the picture that semilocal string endpoints are global-monopole-like and dominate NG emission.
  • The emission is soft, so its observational signatures and constraints depend on infrared physics; gravitational-wave and primordial-black-hole constraints are not significant for v ≲ 10^16 GeV.
  • A logarithmic violation of scaling, if present, would enhance the predicted abundance through the factor F(m_r/m_NG) in Eq. (15).

Where Pith is reading between the lines

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

  • If the linear growth persists to H ≃ m_NG, pNG dark matter from monopoles would be produced with a very cold, non-thermal spectrum, potentially leaving distinct small-scale structure or isocurvature signatures compared with thermal or misalignment production.
  • The same soft, Hubble-peaked spectrum may be generic to scaling networks of global defects with multiple broken generators, suggesting a unified treatment of NG radiation from global strings, global monopoles, and semilocal strings.
  • The paper's announced analytical model of the emission mechanism could predict the coefficient C from monopole dynamics alone, which would let observers translate a measured dark matter abundance directly into constraints on v and the symmetry-breaking scale without relying on the simulation extrapolation.
  • A testable extension would be to include the soft mass term in the simulation itself, since the paper notes that this can create global strings or domain walls; the resulting multi-stage defect network would have its own distinctive gravitational-wave and particle-emission signatures.

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

2 major / 4 minor

Summary. This paper studies NG boson emission from the cosmological network of global O(3) monopoles using large lattice simulations (up to 4096^3, 5 realizations) in both the fat (PRS) and physical monopole treatments. It reports a scaling mean monopole separation, spectra of the two NG modes peaked near the Hubble scale with the radial mode suppressed, and a comoving number density growing as n_NG ≃ 0.7-0.9 v²H. The authors then estimate the relic abundance of pNG dark matter, Eq. (15), and draw an allowed region in the (m_NG, v) plane, concluding that pNG DM can be produced for masses down to ~10^-10 eV. The paper also argues that the spectral similarity to semilocal strings supports the endpoint-emission picture.

Significance. The numerical core is the paper's main strength: direct measurement of the emitted spectrum and number density, a 4096^3 ensemble with five realizations, and a comparison of fat and physical monopole treatments. The claim that the spectrum peaks near the Hubble scale with a soft power law is plausible and extends known global-string results to global monopoles. The connection to semilocal string endpoints is an interesting and falsifiable interpretation. However, the central DM abundance formula has an internal normalization problem and relies on a very long extrapolation in H, so the quantitative DM conclusion is not reliable as presented.

major comments (2)
  1. [Sec. IV, Eq. (15), Fig. 7] Eq. (15) is inconsistent with the measured n_NG reported in Sec. III C. With n_NG=C v²H, C≈0.85, and H_stop=m_NG=10^-13 eV, the standard conversion n_0/s_0=n_stop/s_stop gives T_stop≈1.5×10^7 eV (g_*s≈10.75), s_stop≈1.6×10^22 eV³, and n_stop≈8.5×10^32 eV³, so n_0/s_0≈5.4×10^10. With s_0≈2.3×10^-11 eV³ and ρ_crit,0 h^-2≈8.1×10^-11 eV⁴, this yields Ω_NG h²≈1.5×10^-3 at (m_NG=10^-13 eV, v=10^14 GeV), a factor ~130 below 0.2. Summing the two NG modes changes the factor to ~65. The F factor would need ζ/ζ_sc≈130^{3/2}≈1500, far outside the observed scaling range. The DM line in Fig. 7 is therefore shifted by more than an order of magnitude in v; the authors should recompute Eq. (15) and the figure.
  2. [Sec. IV/V, Eq. (17)] The DM prediction assumes n_NG∝H, or the logarithmic-growth parameterization of Eq. (17), continues from the last simulated time vτ≈10³, where H≈10^-5 v, down to H=m_NG for m_NG as low as 10^-10 eV; for v=10^14 GeV this is roughly 28 orders of magnitude in H. The form of Eq. (17) is taken from Ref. [32] and is not tested by the present simulations, and Sec. V states that verifying scaling violation requires longer simulations. The abundance estimates and Fig. 7 should therefore be presented as an extrapolation with the uncertainty quantified, not as a firm prediction.
minor comments (4)
  1. [Sec. III C, after Table II] The relation between the fitted coefficient A_NG in Table II and the quoted prefactor n_NG≈0.7-0.9 v²H is not explained; the text 'n_NG=C(vτ_i)²v²H' should state explicitly that C=A_NG and that (vτ_i)²=100 accounts for the conversion.
  2. [Eqs. (15)-(16)] The notation for F is inconsistent: Eq. (16) defines F(m_r/H), while Eq. (15) uses F(m_r/m_NG). The radial-mode mass m_r should be defined explicitly.
  3. [Sec. IV, Eq. (15)] Eq. (15) is cited to Ref. [35]; in view of the normalization issue above, the derivation of Eq. (15) should be shown explicitly in this paper rather than imported from a previous work.
  4. [General] No data or code availability statement is provided. Given the numerical nature of the central claim, the authors should state whether the lattice code or representative data sets will be released for reproducibility.

Circularity Check

0 steps flagged

No significant circularity: the NG spectrum and n_NG scaling are direct simulation measurements; the only self-citation (Eq. 15 from [35]) is a non-load-bearing normalization.

full rationale

The central derivation chain is not circular. The soft spectrum and linear growth n_NG ∝ H (Sec. III C, Table II) are extracted from independent lattice simulations of the O(3) global monopole network (Eqs. 4-10), not derived from the later abundance formula. The fit C_spec ≈ 0.03 in Eq. (14) is a calibration to those simulations, and converting it to Ω_NG h^2 in Eq. (15) is a standard cosmological projection. The only self-citation is the formula "estimated as follows [35]" for the relic abundance; [35] is the authors' own semilocal-string paper, but the monopole result (peak at Hubble scale, n_NG ≃ 0.7-0.9 v^2 H) stands on this paper's own numerics and does not reduce to [35]. Ref. [32] for logarithmic scaling violation is external. The skeptical concern that Eq. (15)'s coefficient 0.2 is inconsistent with Table II's measured n_NG is a possible arithmetic/derivation error, not a circularity; an incorrect prediction is not an input recycled as an output.

Axiom & Free-Parameter Ledger

4 free parameters · 6 axioms · 0 invented entities

The central spectrum measurement relies on standard topology and the PRS approximation; the pNG DM number additionally relies on fitted emission normalizations and on persistence of scaling far beyond the simulation window. No new entities are postulated.

free parameters (4)
  • λ0 = not stated
    Base quartic coupling in V = (λ/4)(φ^2 - v^2)^2; the paper uses time-dependent λ(t)=λ0 a^{2(s-1)} but never gives the numerical value, so the core size and UV cutoff are not fixed in the reported setup.
  • initial correlation length ℓ_i = 10 v^-1
    Initial Gaussian-field correlation length; chosen, not varied, so early-time transient effects are not checked.
  • emission normalization C_spec = ≈0.03
    Fitted in Eq. (14) to the simulated comoving number density; it sets the pNG DM abundance normalization in Eq. (15).
  • comoving number-density slope A_NG (C in n_NG=C v^2H) = 0.0085–0.0092 per vτ, i.e. C≈0.7–0.9
    Linear fit slopes from Table II (NG1/NG2, fat/physical) used to quote n_NG≈0.7–0.9 v^2H.
axioms (6)
  • standard math Global O(3) symmetry breaking produces stable global monopoles with long-range 1/r interactions (π2(S^2)=Z)
    Sec. II A; standard homotopy classification.
  • domain assumption The monopole network reaches a scaling regime ξ ∝ t in radiation domination
    Sec. III B; consistent with earlier work [25–31] and their fit, but the DM estimate assumes it persists far beyond the simulated time.
  • domain assumption The energy lost by the shrinking monopole network is transferred entirely to NG bosons
    Sec. III C before Eq. (13); if a significant fraction goes to radial modes or other channels, the abundance estimate is reduced (the authors argue radial emission is suppressed).
  • domain assumption The effective NG emission spectrum has IR cutoff at H and UV cutoff at the radial-mode mass, with q>1
    Used in Eq. (14), following Ref. [15]; needed to convert Γ(t) into n_NG.
  • ad hoc to paper The fat-monopole (PRS) approximation with λ(t)=λ0 a^{2(s-1)} preserves the long-wavelength emission
    Sec. III A; a numerical trick to keep the core resolved. The authors check the physical case (s=1) with a core-growth phase, but the fat case is the main long-time run.
  • domain assumption Introducing a soft mass m_NG at late times does not feed back into the monopole network until H ≃ m_NG
    Secs. IV and V; the authors note global strings/domain walls would form if mass terms are significant, which would change the picture.

pith-pipeline@v1.3.0-alltime-deepseek · 10103 in / 18819 out tokens · 191379 ms · 2026-08-01T04:35:09.909654+00:00 · methodology

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read the original abstract

We show the emission of the Nambu-Goldstone (NG) bosons from the cosmological evolution of global monopoles. The NG bosons are non-thermally produced from the dynamics of global monopoles such as the pair annihilation of monopole and anti-monopole. Our numerical lattice simulations demonstrate that the spectrum of NG bosons emitted from the scaling evolution of global monopoles has a peak around the horizon scale, as in the case with the global string and the semilocal string. We also estimate the abundance of the pseudo-Nambu-Goldstone (pNG) dark matter when one (or both) of the NG modes has a soft mass.

Figures

Figures reproduced from arXiv: 2607.22481 by Naoya Kitajima, Yukihiro Kanda.

Figure 1
Figure 1. Figure 1: FIG. 1. Snapshots of the evolution of the monopole network in the fat monopole regime from the simulation with [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The evolution of the mean monopole separation in the fat (left) and physical (right) monopole cases. The blue, green [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Spectrum of the comoving number density of the radial mode in the fat (left) and the physical (right) monopole [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Spectrum of the comoving number density of the first NG mode in the fat (left) and the physical (right) monopole [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Spectrum of the comoving number density of the second NG mode in the fat (left) and the physical (right) monopole [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The evolution of the mean comoving number density of the two NG modes (red and blue) and radial mode (green) [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Allowed parameter region in [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗

discussion (0)

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Reference graph

Works this paper leans on

53 extracted references · 37 linked inside Pith

  1. [1]

    T. W. B. Kibble, J. Phys. A9, 1387 (1976). [2] A. Vilenkin and E. P. S. Shellard,Cosmic Strings and Other Topological Defects(Cambridge University Press, 8 2000)

    1976

  2. [3]

    R. L. Davis, Phys. Lett. B180, 225 (1986)

    1986

  3. [4]

    Vilenkin and T

    A. Vilenkin and T. Vachaspati, Phys. Rev. D35, 1138 (1987)

    1987

  4. [5]

    Garfinkle and T

    D. Garfinkle and T. Vachaspati, Phys. Rev. D36, 2229 (1987)

    1987

  5. [6]

    Yamaguchi, M

    M. Yamaguchi, M. Kawasaki, and J. Yokoyama, Phys. Rev. Lett.82, 4578 (1999), arXiv:hep-ph/9811311

    Pith/arXiv arXiv 1999

  6. [7]

    Hagmann, S

    C. Hagmann, S. Chang, and P. Sikivie, Phys. Rev. D 63, 125018 (2001), arXiv:hep-ph/0012361

    Pith/arXiv arXiv 2001

  7. [8]

    Hiramatsu, M

    T. Hiramatsu, M. Kawasaki, T. Sekiguchi, M. Yam- aguchi, and J. Yokoyama, Phys. Rev. D83, 123531 (2011), arXiv:1012.5502 [hep-ph]

    Pith/arXiv arXiv 2011

  8. [9]

    Kitajima and M

    N. Kitajima and M. Uwabo-Niibo, Phys. Rev. D113, 103528 (2026), arXiv:2510.10708 [hep-ph]

    Pith/arXiv arXiv 2026

  9. [10]

    A. J. Long and L.-T. Wang, Phys. Rev. D99, 063529 (2019), arXiv:1901.03312 [hep-ph]

    Pith/arXiv arXiv 2019

  10. [11]

    Kitajima and K

    N. Kitajima and K. Nakayama, JHEP08, 068 (2023), arXiv:2212.13573 [hep-ph]

    Pith/arXiv arXiv 2023

  11. [12]

    Kitajima and K

    N. Kitajima and K. Nakayama, Phys. Lett. B846, 138213 (2023), arXiv:2306.17390 [hep-ph]

    Pith/arXiv arXiv 2023

  12. [13]

    Kawasaki, K

    M. Kawasaki, K. Saikawa, and T. Sekiguchi, Phys. Rev. D91, 065014 (2015), arXiv:1412.0789 [hep-ph]

    Pith/arXiv arXiv 2015

  13. [14]

    Gorghetto, E

    M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 [hep-ph]

    Pith/arXiv arXiv 2021

  14. [15]

    Gorghetto, E

    M. Gorghetto, E. Hardy, and G. Villadoro, JHEP07, 151 (2018), arXiv:1806.04677 [hep-ph]

    Pith/arXiv arXiv 2018

  15. [16]

    Kawasaki, T

    M. Kawasaki, T. Sekiguchi, M. Yamaguchi, and J. Yokoyama, PTEP2018, 091E01 (2018), arXiv:1806.05566 [hep-ph]

    Pith/arXiv arXiv 2018

  16. [17]

    Buschmann, J

    M. Buschmann, J. W. Foster, A. Hook, A. Peterson, D. E. Willcox, W. Zhang, and B. R. Safdi, Nature Com- mun.13, 1049 (2022), arXiv:2108.05368 [hep-ph]

    Pith/arXiv arXiv 2022

  17. [18]

    Saikawa, J

    K. Saikawa, J. Redondo, A. Vaquero, and M. Kaltschmidt, JCAP10, 043 (2024), arXiv:2401.17253 [hep-ph]

    Pith/arXiv arXiv 2024

  18. [19]

    J. N. Benabou, M. Buschmann, J. W. Foster, and B. R. Safdi, Phys. Rev. Lett.134, 241003 (2025), arXiv:2412.08699 [hep-ph]

    Pith/arXiv arXiv 2025

  19. [20]

    Hindmarsh, J

    M. Hindmarsh, J. Lizarraga, A. Lopez-Eiguren, and J. Urrestilla, Phys. Rev. Lett.124, 021301 (2020), arXiv:1908.03522 [astro-ph.CO]

    Pith/arXiv arXiv 2020

  20. [21]

    Hindmarsh, J

    M. Hindmarsh, J. Lizarraga, A. Lopez-Eiguren, and J. Urrestilla, Phys. Rev. D103, 103534 (2021), arXiv:2102.07723 [astro-ph.CO]

    Pith/arXiv arXiv 2021

  21. [22]

    Correia, M

    J. Correia, M. Hindmarsh, J. Lizarraga, A. Lopez- Eiguren, K. Rummukainen, and J. Urrestilla, Phys. Rev. D111, 063532 (2025), arXiv:2410.18064 [hep-ph]

    Pith/arXiv arXiv 2025

  22. [23]

    Barriola and A

    M. Barriola and A. Vilenkin, Phys. Rev. Lett.63, 341 (1989)

    1989

  23. [24]

    Achucarro and J

    A. Achucarro and J. Urrestilla, Phys. Rev. Lett.85, 3091 (2000), arXiv:hep-ph/0003145

    Pith/arXiv arXiv 2000

  24. [25]

    D. P. Bennett and S. H. Rhie, Phys. Rev. Lett.65, 1709 (1990)

    1990

  25. [26]

    Turok and D

    N. Turok and D. N. Spergel, Phys. Rev. Lett.66, 3093 (1991)

    1991

  26. [27]

    Yamaguchi, Phys

    M. Yamaguchi, Phys. Rev. D64, 081301 (2001), arXiv:hep-ph/0103130

    Pith/arXiv arXiv 2001

  27. [28]

    Yamaguchi, Phys

    M. Yamaguchi, Phys. Rev. D65, 063518 (2002), arXiv:hep-ph/0107230

    Pith/arXiv arXiv 2002

  28. [29]

    C. J. A. P. Martins and A. Achucarro, Physical Review D—Particles, Fields, Gravitation, and Cosmology78, 083541 (2008), arXiv:0806.2671 [hep-ph]

    Pith/arXiv arXiv 2008

  29. [30]

    Lopez-Eiguren, J

    A. Lopez-Eiguren, J. Urrestilla, and A. Ach´ ucarro, JCAP01, 020 (2017), [Erratum: JCAP 06, E01 (2017)], arXiv:1611.09628 [hep-ph]

    Pith/arXiv arXiv 2017

  30. [31]

    Sousa and P

    L. Sousa and P. P. Avelino, Phys. Rev. D96, 023521 (2017), arXiv:1703.09054 [astro-ph.CO]

    Pith/arXiv arXiv 2017

  31. [32]

    Nakano and W

    W. Nakano and W. Yin, (2026), arXiv:2607.05517 [hep- ph]

    Pith/arXiv arXiv 2026

  32. [33]

    Vachaspati and A

    T. Vachaspati and A. Achucarro, Phys. Rev. D44, 3067 (1991)

    1991

  33. [34]

    Hindmarsh, Phys

    M. Hindmarsh, Phys. Rev. Lett.68, 1263 (1992)

    1992

  34. [35]

    Kanda and N

    Y. Kanda and N. Kitajima, Phys. Lett. B873, 140230 (2026), arXiv:2510.07894 [hep-ph]

    arXiv 2026

  35. [36]

    G. W. Gibbons, M. E. Ortiz, F. Ruiz Ruiz, and T. M. Samols, Nucl. Phys. B385, 127 (1992), arXiv:hep- th/9203023

    arXiv 1992

  36. [37]

    Hindmarsh, Nucl

    M. Hindmarsh, Nucl. Phys. B392, 461 (1993), arXiv:hep-ph/9206229

    Pith/arXiv arXiv 1993

  37. [38]

    Achucarro, P

    A. Achucarro, P. Salmi, and J. Urrestilla, Phys. Rev. D 75, 121703 (2007), arXiv:astro-ph/0512487

    Pith/arXiv arXiv 2007

  38. [39]

    A. S. Nunes, A. Avgoustidis, C. J. A. P. Martins, and J. Urrestilla, Phys. Rev. D84, 063504 (2011), arXiv:1107.2008 [hep-ph]

    Pith/arXiv arXiv 2011

  39. [40]

    Ach´ ucarro, A

    A. Ach´ ucarro, A. Avgoustidis, A. M. M. Leite, A. Lopez- Eiguren, C. J. A. P. Martins, A. S. Nunes, and J. Ur- restilla, Phys. Rev. D89, 063503 (2014), arXiv:1312.2123 [hep-ph]

    Pith/arXiv arXiv 2014

  40. [41]

    Lopez-Eiguren, J

    A. Lopez-Eiguren, J. Urrestilla, A. Ach´ ucarro, A. Av- goustidis, and C. J. A. P. Martins, Phys. Rev. D96, 023526 (2017), arXiv:1704.00991 [hep-ph]

    Pith/arXiv arXiv 2017

  41. [42]

    Ach´ ucarro, A

    A. Ach´ ucarro, A. Avgoustidis, A. L´ opez-Eiguren, C. J. A. P. Martins, and J. Urrestilla, Phil. Trans. Roy. Soc. Lond. A377, 0004 (2019), arXiv:1912.12069 [astro- ph.CO]

    Pith/arXiv arXiv 2019

  42. [43]

    Hindmarsh, A

    M. Hindmarsh, A. Lopez-Eiguren, R. Sepp¨ a, and D. J. Weir, (2025), arXiv:2511.14204 [astro-ph.CO]

    arXiv 2025

  43. [44]

    J. R. C. C. C. Correia and C. J. A. P. Martins, Phys. Rev. D102, 043503 (2020), arXiv:2007.12008 [astro-ph.CO]

    Pith/arXiv arXiv 2020

  44. [45]

    D. G. Figueroa, A. Florio, F. Torrenti, and W. Valken- burg, JCAP04, 035 (2021), arXiv:2006.15122 [astro- ph.CO]

    Pith/arXiv arXiv 2021

  45. [46]

    Baeza-Ballesteros, D

    J. Baeza-Ballesteros, D. G. Figueroa, A. Florio, J. Lizarraga, N. Loayza, K. Marschall, T. Opferkuch, B. A. Stefanek, F. Torrent ´ ı, and A. Urio, JCAP06, 087 (2026), arXiv:2512.15627 [astro-ph.CO]

    arXiv 2026

  46. [47]

    W. H. Press, B. S. Ryden, and D. N. Spergel, Astrophys. J.347, 590 (1989)

    1989

  47. [48]

    N. D. Antunes, L. M. A. Bettencourt, and M. Kunz, Phys. Rev. E65, 066117 (2002), arXiv:cond- mat/0201149

    arXiv 2002

  48. [49]

    T. W. B. Kibble, Nucl. Phys. B252, 227 (1985), [Erra- tum: Nucl.Phys.B 261, 750 (1985)]

    1985

  49. [50]

    Cardoso, ´O

    V. Cardoso, ´O. J. C. Dias, G. S. Hartnett, M. Middle- ton, P. Pani, and J. E. Santos, JCAP03, 043 (2018), arXiv:1801.01420 [gr-qc]

    Pith/arXiv arXiv 2018

  50. [51]

    Aburatani, W

    D. Aburatani, W. Nakano, and W. Yin, (2026), arXiv:2606.31937 [hep-ph]

    Pith/arXiv arXiv 2026

  51. [52]

    Chang and Y

    C.-F. Chang and Y. Cui, Phys. Dark Univ.29, 100604 (2020), arXiv:1910.04781 [hep-ph]

    Pith/arXiv arXiv 2020

  52. [53]

    Gorghetto, E

    M. Gorghetto, E. Hardy, and H. Nicolaescu, JCAP06, 034 (2021), arXiv:2101.11007 [hep-ph]

    Pith/arXiv arXiv 2021

  53. [54]

    Chang and Y

    C.-F. Chang and Y. Cui, JHEP03, 114 (2022), arXiv:2106.09746 [hep-ph]

    Pith/arXiv arXiv 2022