Echoes in multi-ALP scenarios

Shihabul Haque; Sourov Roy

REVIEW 4 major objections 3 minor 70 references

Echoes in multi-ALP scenarios

T0 review · 4 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper claims that N coherent axion-like particles amplify the axion echo power by a factor of N, with small mass splittings adding extra gain, while random phases suppress the signal below the single-ALP level.

desk verdict The coherent N-scaling result is solid and worth knowing; the mass-splitting boost for N=2 does not survive the paper's own approximations. read the letter →

arxiv 2507.16555 v4 pith:UEXNQ3L7 submitted 2025-07-22 hep-ph astro-ph.HE

Shihabul Haque , Sourov Roy This is my paper

classification hep-phastro-ph.HE

keywords axionechoaxion-likeparticlesmulti-ALPdarkmatterphoton-ALPcouplingstimulateddecaycoherentandincoherentphaseslarge-Napproximation

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

The paper develops the theory of axion echoes — returning photons produced when an outgoing beam stimulates the decay of axion-like-particle (ALP) dark matter — for the case where the dark matter consists of $N$ distinct ALPs rather than one. Its central claim is that if the $N$ fields oscillate coherently (same phase), the echo power is multiplied by $N$, as if there were a single ALP with a coupling $\sqrt{N}$ times larger, and that a narrow spread of ALP masses adds a further amplification factor, noticeable even at $N=2$. In the opposite, incoherent case of random phases, the paper claims the echo power is suppressed to at most the single-ALP value and typically below it. This matters because multiple ALPs arise naturally in many beyond-standard-model constructions, so echo-search projections and the interpretation of null results depend on which scenario is realized.

What carries the argument

The load-bearing object is the forced-oscillator equation for the first-order photon perturbation, $(\partial_t^2 + p^2)\mathbf{A}_p^1 = -i\sum_n \mathbf{B}_{kp}^n [e^{i(m_n^a-p)t} + e^{-i(m_n^a+p)t}]$, together with the resonance condition $p = m_n^a/2$, at which the amplitude grows linearly in time and the echo wave propagates backward. The multi-ALP argument converts the sum over $N$ fields into an integral over smooth mass and coupling distributions (Eq. 3.11), justified in the large-$N$ limit by the Law of Large Numbers, and then expands the mass integral in the small splitting $\epsilon = (m_M^a - m_L^a)/2$, producing the dimensionless coefficients $a_1$, $a_2$ and the amplification factor $Z(\epsilon,t)$. In the incoherent treatment the same sum is replaced by its root-mean-square amplitude, which removes the $N$ enhancement. These replacements — discrete sum to continuum integral, coherent summation versus RMS — are what carry the paper's results.

What would settle it

Numerically integrate the exact $N=2$ forced-oscillator equation (Eq. 3.3) with masses $m_L^a$ and $m_L^a + 2\epsilon$, equal couplings, $\epsilon = m_L^a$, and sample at $m_L^a t \sim 1$; if the echo power does not match $P_N^\epsilon$ from Eq. (3.35) (about $1.1\times 2P_{N=1}$), the continuum approximation fails at $N=2$ and the claimed enhancement is not established. A null echo search at the predicted boosted power would test the coherent multi-ALP claim directly.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is a scaling law for echo power: for $N$ coherent ALPs with equal masses and couplings, the first-order echo field amplitude is $N$ times the single-ALP amplitude, giving $P_N = N P_{N=1} = N g_{a\gamma\gamma}^2 (t/16) \rho \, dP_0/d\nu\big|_{k=m_a/2}$ (Eq. 3.23), equivalent to a single ALP with coupling $\sqrt{N}\,g_{a\gamma\gamma}$. When the masses are drawn from a narrow distribution with splitting $\epsilon$, a large-$N$ continuum replacement plus Taylor expansion in $\epsilon$ yields an additional factor $Z(\epsilon,t)>1$ in the power, bounded near $1.1N P_{N=1}$ at maximal $\epsilon=1$ with $m_a^L t\sim 1$; the paper therefore claims stronger projected bounds even for $N=2$ (Fig. 5). For incoherent phases, replacing the random drive by its root-mean-square value removes the $N$ enhancement entirely: $P_N=f_g P_{N=1}$ with $f_g\le 1$, and $f_g=1/3$ for a uniform coupling distribution, so the observable signal is at best that of a single ALP and often weaker.

Load-bearing premise

The paper's extra mass-splitting amplification rests on treating the $N$ ALPs as a smooth continuum distribution, an approximation its own appendix shows needs more than four ALPs for ten-percent accuracy, even though the $N=2$ amplification is computed from that large-$N$ formula.

Editorial extensions

If this is right

If coherent multi-ALP dark matter is realized, echo experiments effectively search for a coupling $\sqrt{N}$ larger than in the single-ALP case, so projected bounds on $g_{a\gamma\gamma}$ tighten by a factor $\sim\sqrt{N}$ for fixed signal power.
If the ALP masses are spread over a narrow band, the extra factor $Z(\epsilon,t)$ adds up to roughly 10% amplification at maximal $\epsilon=1$, enough for even an $N=2$ model to give stronger projected constraints than the single-ALP baseline (Fig. 5).
For random ALP phases, the echo power is at most equal to the single-ALP power (and one-third of it for uniform couplings), so a null search interpreted under the coherent assumption would claim sensitivity to multi-ALP dark matter that the incoherent scenario does not actually provide.
For ALPs with well-separated masses, such as Kaluza-Klein towers, only one resonance is active at a time and the signal matches the single-ALP case, so the $N$ amplification requires near-degenerate masses.
The mass range $m_L^a \in [2.5\times10^{-7}, 2.5\times10^{-3}]$ eV lies in the atmospheric transparency window, so multi-ALP echo searches would yield bounds complementary to haloscope-type dark-matter axion searches.

Reading between the lines

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

Beyond the paper: measuring echo power at two closely spaced outgoing frequencies should reveal the mass-distribution-dependent factor $Z(\epsilon,t)$ in the coherent case but no such structure in the incoherent case, giving an observational way to distinguish the two scenarios.
Beyond the paper: a direct numerical solution of the exact $N=2$ and $N=3$ forced-oscillator systems (without the continuum replacement) at $\epsilon=1$ would test whether the 'even $N=2$' amplification survives finite-$N$ fluctuations, which Appendix B's accuracy bound leaves open.
Beyond the paper: because the mass-splitting amplification depends on $p(m_L^a)$, $p'(m_L^a)$ and $p''(m_L^a)$, a sufficiently precise measurement of echo power versus frequency could in principle reconstruct moments of the ALP mass distribution, turning echo searches into a spectroscopy tool.
Beyond the paper: the incoherent suppression implies that ALP models with random phases, such as fields produced at different epochs, would be systematically harder to detect by the echo method than single-ALP dark matter of the same total density; null results should therefore carry a model-dependent suppression factor.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Bottom line: the coherent N-scaling result (PN = N P1) is correct and cleanly derived. The secondary claim that small mass splittings add an extra amplification, even for N=2, does not survive the paper's own approximations. The large-N continuum replacement used to derive it is not valid for a discrete, resonant system.

What the paper does well: the single-ALP review is compact, the forced-oscillator formulation makes the resonance condition p = ma/2 transparent, and the equal-mass coherent extension is genuinely new—the source term picks up a factor N and the power scales as N. The incoherent scenario, where random phases suppress the signal back to single-ALP levels, is a useful contrast and likely qualitatively correct even if the effective-drive treatment is heuristic. Appendix B's discussion of the Law of Large Numbers validity is honest and helpful.

The soft spot is Section 3.4. The mass-splitting enhancement is derived by replacing the discrete sum over N ALPs with a smooth integral over the mass distribution, then solving the averaged equation. For a driven oscillator at resonance, that interchange is not safe. At fixed photon momentum p, only the ALP with mn = 2p is on resonance; the off-resonant ALPs contribute bounded, oscillating terms. The continuum replacement smears the resonance and produces secular t^2 and t^3 terms that do not correspond to any physical drive. The stress-test note makes this concrete: for N=2 with masses mL and 3mL, at p = mL/2 the second ALP is off resonance by 2mL and cannot contribute to the same echo line, so the power cannot be 2.2 P1. The paper almost acknowledges this at Eq. (3.6), where it says that for distinct masses only one resonance dominates. Appendix B's N>4 criterion concerns fluctuations of the source sum, not this resonance-pole failure, so it does not rescue the N=2 claim.

The projected bounds in Fig. 5 also use a t-expansion beyond its perturbative validity: the paper gives the condition ϵ mL t = 1, but then replaces t with R/v⊥, which is orders of magnitude larger. The mass-splitting amplification and the associated sensitivity projections are therefore not reliable; they should be rederived treating the discrete spectrum exactly, or dropped.

Overall: the central N-scaling result is solid, the incoherent suppression is a plausible qualitative contrast, and the mass-splitting claims are the weak part—unfortunately the part the abstract leads with. A referee would catch this. The paper deserves peer review because the core result is useful and the fix is local. For the reader: the axion-echo community gets a useful N-scaling result; the multi-ALP modeling community gets a caution about continuum approximations. I'd bring it to a reading group as an instructive case of a large-N replacement failing in a resonant system.

Referee Report

4 major / 3 minor

Summary. This paper studies the back-scattered 'echo' radiation produced when a photon beam passes through a background of N axion-like particles (ALPs) coupled to the photon. After reviewing the single-ALP derivation, the authors generalize to multiple ALPs in two phase configurations: coherent (all ALP fields in phase) and incoherent (random phases). For the coherent equal-mass case they derive P_N = N P_{N=1} (Eq. 3.23), which improves coupling reach by sqrt(N). They further claim that small mass splittings produce an additional amplification, up to about 1.1N for epsilon=1 (Eq. 3.37), even for N=2, and present projected bounds in Fig. 5. For the incoherent case they find the N-dependence disappears and the signal is at most equal to the single-ALP one. The paper includes detailed appendices for the forced-oscillator solution, the large-N approximation, and the power calculation.

Significance. If the coherent N-scaling result holds, the paper provides a clean and potentially important effect: echo searches in multi-ALP models gain a factor sqrt(N) in coupling reach without any fitting to data. The derivation of P_N = N P_{N=1} is transparent and correct. The treatment of the incoherent case as a distinct scenario is also a useful warning that the N-enhancement is not generic. However, the additional mass-splitting amplification, which appears in the abstract and in Fig. 5, rests on a large-N continuum replacement applied to finite N. This is the paper's main advertised new quantitative claim, so its status determines the paper's overall impact. The appendices include a careful statement of the LLN validity condition (N>4 for uniform distributions at 10% accuracy), which is in tension with the N=2 application.

major comments (4)

[Sec. 3.4, Eq. (3.37), Fig. 5] The 'additional amplification' claim for N=2 is derived from the continuum LLN replacement rather than from the finite-N discrete equations. The paper's own Eq. (3.6) states that for distinct masses only one resonance dominates; the exact finite-N solution for p=m_L/2 has only the m_L ALP growing linearly, while the off-resonant ALPs give bounded oscillatory terms. The continuum approximation replaces the discrete sum by an integral (Eq. 3.11) and produces secular t^2 and t^3 terms (Eq. C.31) that are artifacts of the averaging. Therefore the N=2 enhancement in Fig. 5 is unsupported, and the abstract's 'even for a N=2 case' claim should be withdrawn or supported by a direct N=2 calculation.
[Sec. 3.1 and Appendix B] The LLN condition (B.15) requires N > 4 for uniform distributions at the 10% precision level, and this condition concerns the fluctuation of the source sum, not the resonance-pole structure. Since the mass-splitting formula (3.18) is used for N=2 and N=10 in Fig. 5, the finite-N validity is not established. The paper needs to either restrict the mass-splitting amplification to N satisfying the LLN bound and to cases where the continuum resonance condition is physically justified, or compute the finite-N sum directly.
[Sec. 3.2, Eq. (3.25)] The limiting procedure used to recover the equal-mass result from the distribution formalism, lim_{epsilon->0} epsilon p(ma) = 1/2, is an ad hoc normalization condition rather than a well-defined distributional limit. While this does not affect the direct derivation of Eq. (3.23), it obscures the status of the general-distribution formula (3.18) and should be clarified or replaced by a direct delta-function treatment.
[Sec. 3.4, Eq. (3.36)] The paper's own perturbative validity condition is epsilon m_L t = 1, giving t ~ 10^-11 sec for m_L ~ 10^-4 eV. The authors then replace t by R/v_perp, which is orders of magnitude larger, and conclude the echo should be detectable. However, the secular t^2 and t^3 terms used in Eq. (3.34) are derived under the assumption epsilon t << 1 (or at most epsilon m_L t ~ 1), so evaluating the power at R/v_perp is outside the domain of validity of the perturbative expansion. The projected bounds in Fig. 5 therefore require a non-perturbative treatment or a finite-N calculation for the relevant timescale.

minor comments (3)

[Sec. 5] There is a typo: 'strenghtening' should be 'strengthening'.
[Fig. 5 caption] The caption says 'mass ratio 3 corresponding to epsilon = 1'; since epsilon is defined as (m_M - m_L)/2, it would be clearer to state that epsilon/m_L = 1 for masses m_L and 3m_L.
[References] Several references, e.g., [1], [5], and [13], lack complete page or article-number information; the bibliography should be standardized to the journal's style.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the N-scaling and mass-splitting amplification follow from solving the forced-oscillator equations under stated distribution assumptions; no fitted parameter is renamed as a prediction.

full rationale

The derivation chain is self-contained rather than circular. The single-ALP echo power (Eq. 2.27) is obtained by solving the first-order forced-oscillator equation (2.16)-(2.25), and the multi-ALP coherent equal-mass result (Eqs. 3.20-3.23) starts from the N-term source and the fixed total density rho = (N/2) m^2 A0^2, so P_N = N P_1 follows from the coherent superposition of identical drives, with the N -> 1 limit reproduced by the normalisation condition (3.25). The variable-mass amplification (Eqs. 3.15-3.19, 3.34-3.37) is obtained by an explicit Taylor expansion of the continuum source Q(t) under the stated Law-of-Large-Numbers and narrow-distribution assumptions; its factors a1, a2, f(epsilon), and Z(epsilon,t) are derived quantities, not parameters fitted to the echo power. The incoherent RMS replacement (Eqs. 4.4-4.10) is likewise an explicitly stated modeling choice, and the suppression factor f_g (Eq. 4.15) is the mean-square coupling ratio rather than an output tuned to a target signal. The self-citations in the reference list are contextual and are not load-bearing for any uniqueness claim or ansatz. The finite-N validity caveat in Appendix B (N > 4 for 10% accuracy) and the use of epsilon = 1 in Figure 5 are accuracy and validity concerns about the large-N approximation, not evidence that a prediction reduces to an input by construction.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The core N-scaling result introduces no free parameters fitted to data; it depends on the modeling assumptions listed above. The paper's projected bounds add benchmark choices (isothermal profile, detector radius, outgoing power, maximum coupling) that are external inputs from prior literature.

free parameters (3)

N (number of ALPs) = 2 to 30 (scanned)
Model parameter varied to illustrate the N-scaling; not fitted to data. The claimed amplification for N=2 is highlighted in the abstract and Fig. 5.
epsilon (fractional mass splitting) = 0 to 1 (mass ratio up to 3)
Model parameter controlling the width of the mass distribution; chosen to show the mass-splitting effect.
g^M_aγγ (maximum ALP-photon coupling) = 10^-12 GeV^-1 (projection benchmark)
Chosen for the sensitivity estimate in Sec. 3.3 and Fig. 4; not fitted to data.

assumptions (6)

domain assumption ALP dark matter is described by a classical non-relativistic field an(t) = A0 sin(mn t + θn) with negligible spatial gradients.
Basis of the Maxwell equation reduction in Sec. 2 and the multi-ALP extension in Sec. 3.
domain assumption The total local dark matter density is fixed at ρ and shared equally among the N ALPs: ρ = (N/2) m^2 A0^2 (Eq. 3.22).
Used to convert field amplitude into density; the isothermal value 0.3 GeV/cm^3 is adopted for projections.
domain assumption In the coherent scenario all ALP phases are equal and set to zero; in the incoherent scenario phases are uniformly random in [0,2π].
Defines the two scenarios; the results differ drastically between them.
ad hoc to paper The discrete sum over N ALP contributions is replaced by N times the expectation value under smooth mass and coupling distributions (Law of Large Numbers).
Central approximation in Sec. 3.1; Appendix B shows it requires N > 4 for uniform distributions, which is inconsistent with the paper's N=2 application.
ad hoc to paper In the incoherent case the sum of N random-phase cosines is replaced by a single cosine of amplitude sqrt(sum(g_n m_n)^2) at the mean mass m* (Eq. 4.7).
Heuristic RMS treatment; only valid when the mass spread is small and the detector bandwidth covers the spread.
standard math First-order perturbation theory in the small coupling g = g_aγγ m A0 is valid, and the photon field is expanded as A = A0 + A1.
Standard perturbative treatment used in the echo literature; the paper estimates g t << 1 for the parameter ranges considered.

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Cite this review

Pith. "Pith review of Echoes in multi-ALP scenarios." pith.science (2026). https://pith.science/paper/UEXNQ3L7

@misc{pith2026250716555,
  author       = {Pith},
  title        = {Pith review of: Echoes in multi-ALP scenarios},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UEXNQ3L7}},
  note         = {Machine review of arXiv:2507.16555}
}

abstract

We present a theoretical study of axion echoes in the context of multiple ALP models. We begin by reviewing the single ALP case, deriving the conditions for resonance and echo formation. Starting from a set of $N$ ALPs coupled to the photon, we then derive the relevant echo equations for both coherent and incoherent configurations. In the former case, we show that the echo power scales with $N$ leading to sharper amplification and potentially improving sensitivity estimates discussed earlier in literature. Small mass splittings between the ALPs further increase this amplification, even for a $N=2$ case. In the incoherent scenario, we show that the random phases lead to a suppression of the echo power, eventually resulting in observable signals akin to or even weaker than the single ALP case. We also outline the potential experimental implications of our results and discuss prospects for detecting these echoes in a wide range of ALP masses.

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Works this paper leans on

70 extracted references · 54 canonical work pages

[1]

Peccei and H.R

R.D. Peccei and H.R. Quinn, CP conservation in the presence of pseudoparticles , Phys. Rev. Lett. 38 (1977)

work page 1977
[2]

Weinberg, A new light boson? , Phys

S. Weinberg, A new light boson? , Phys. Rev. Lett. 40 (1978) 223

1978
[3]

Wilczek, Problem of strong p and t invariance in the presence of instantons , Phys

F. Wilczek, Problem of strong p and t invariance in the presence of instantons , Phys. Rev. Lett. 40 (1978) 279

1978
[4]

Zhitnitskii, Possible suppression of axion-hadron interactions , Sov

A.P. Zhitnitskii, Possible suppression of axion-hadron interactions , Sov. J. Nucl. Phys. 31 (1980)

work page 1980
[5]

M. Dine, W. Fischler and M. Srednicki, A simple solution to the strong cp problem with a harmless axion , Phys. Lett. B 104 (1981) 199

work page 1981
[6]

Kim, Weak-interaction singlet and strong CP invariance, Phys

J.E. Kim, Weak-interaction singlet and strong CP invariance, Phys. Rev. Lett. 43 (1979) 103

1979
[7]

Shifman, A

M.A. Shifman, A. Vainshtein and V. Zakharov, Can confinement ensure natural cp invariance of strong interactions? , Nucl. Phys. B 166 (1980) 493

work page 1980
[8]

Berezhiani and M

Z. Berezhiani and M. Khlopov, Cosmology of spontaneously broken gauge family symmetry with axion solution of strong cp-problem , Z. Phys. C - Particles and Fields 49 (1991) 73

work page 1991

Show all 70 references

[9]

Berezhiani, A

Z. Berezhiani, A. Sakharov and M. Khlopov, Primordial background of cosmological axions , Sov. J. Nucl. Phys. 55 (1992) 1063

1992
[10]

Abbott and P

L.F. Abbott and P. Sikivie, A cosmological bound on the invisible axion , Phys. Lett. B 120 (1983) 133

1983
[11]

Preskill, M.B

J. Preskill, M.B. Wise and F. Wilczek, Cosmology of the invisible axion , Phys. Lett. B 120 (1983) 127

1983
[12]

Dine and W

M. Dine and W. Fischler, The not so harmless axion , Phys. Lett. B 120 (1983) 137

1983
[13]

Alonso- ´Alvarez, J.M

G. Alonso- ´Alvarez, J.M. Cline and T. Xiao, The flavor of QCD axion dark matter , J. High Energ. Phys. 07 (2023) 187

2023
[14]

Bauer, M

M. Bauer, M. Neubert and A. Thamm, Collider probes of axion-like particles , J. High Energ. Phys. 44 (2017)

2017
[15]

Mimasu and V

K. Mimasu and V. Sanz, Alps at colliders , J. High Energ. Phys. 2015 (2015)

2015
[16]

Caputo and G

A. Caputo and G. Raffelt, Astrophysical axion bounds: The 2024 edition , PoS COSMICWISPers (2024) 041

2024
[17]

O’Hare, Cosmology of axion dark matter , PoS COSMICWISPers (2024) 040

C.A.J. O’Hare, Cosmology of axion dark matter , PoS COSMICWISPers (2024) 040

2024
[18]

Marsh, Axion Cosmology, Phys

D.J.E. Marsh, Axion Cosmology, Phys. Rept. 643 (2016) 1 [ 1510.07633]

2016 arXiv
[19]

Sikivie, Axion Cosmology, Lect

P. Sikivie, Axion Cosmology, Lect. Notes Phys. 741 (2008) 19 [ astro-ph/0610440]

2008 arXiv
[20]

A. Kar, T. Kumar, S. Roy and J. Zupan, Searching for relativistic axions in the sky , J. Cosmol. Astropart. Phys. 08 (2023) 056. – 32 –

2023
[21]

A. Kar, S. Roy and P. Sarkar, Constraining eV-scale axion-like particle dark matter: insights from the M87 Galaxy , J. Cosmol. Astropart. Phys. 05 (2025) 100

2025
[22]

S. Roy, P. Sarkar, S. Sau and S. SenGupta, Exploring axions through the photon ring of a spherically symmetric black hole , J. Cosmol. Astropart. Phys. 11 (2023) 099

2023
[23]

Cand´ on, S

F.R. Cand´ on, S. Ganguly, M. Giannotti, T. Kumar, A. Lella and F. Mescia, Fresh look at the diffuse ALP background from supernovae , Phys. Rev. D 112 (2025) 015006

2025
[24]

Raffelt and L

G. Raffelt and L. Stodolsky, Mixing of the photon with low-mass particles , Phys. Rev. D 37 (1988) 1237

1988
[25]

Arza, Photon enhancement in a homogeneous axion dark matter background , Eur

A. Arza, Photon enhancement in a homogeneous axion dark matter background , Eur. Phys. J. C 79 (2019) 250

2019
[26]

Arza and P

A. Arza and P. Sikivie, Production and detection of an axion dark matter echo , Phys. Rev. Lett. 123 (2019) 131804

2019
[27]

Arza and E

A. Arza and E. Todarello, Axion dark matter echo: A detailed analysis , Phys. Rev. D 105 (2022) 023023

2022
[28]

A. Arza, A. Kryemadhi and K. Zioutas, Searching for axion streams with the echo method , Phys. Rev. D 108 (2023) 083001

2023
[29]

A. Arza, Q. Guo, L. Wu, Q. Yang, X. Yang, Q. Yuan et al., Listening for echo from the stimulated axion decay with the 21 centimeter array , Sci. Bull. 69 (2024) 2971

2024
[30]

Caputo, M

A. Caputo, M. Regis, M. Taoso and S.J. Witte, Detecting the Stimulated Decay of Axions at Radio Frequencies, J. Cosmo. Astropart. Phys. 03 (2019) 027

2019
[31]

Buen-Abad, J

M.A. Buen-Abad, J. Fan and C. Sun, Axion echoes from the supernova graveyard , Phys. Rev. D 105 (2022) 075006

2022
[32]

Y. Sun, K. Schutz, A. Nambrath, C. Leung and K. Masui, Axion dark matter-induced echo of supernova remnants, Phys. Rev. D 105 (2022) 063007

2022
[33]

Ghosh, J

O. Ghosh, J. Salvado and J. Miralda-Escud´ e, Axion gegenschein: Probing back-scattering of astrophysical radio sources induced by dark matter , arXiv:2008.02729 [astro-ph.CO] (2020)

2020 arXiv
[34]

Y. Sun, K. Schutz, H. Sewalls, C. Leung and K.W. Masui, Looking in the axion mirror: An all-sky analysis of stimulated decay , Phys. Rev. D 109 (2024) 043042

2024
[35]

Caputo, C.P.n

A. Caputo, C.P.n. Garay and S.J. Witte, Looking for Axion Dark Matter in Dwarf Spheroidals, Phys. Rev. D 98 (2018) 083024

2018
[36]

Todarello, F

E. Todarello, F. Calore and M. Regis, Anatomy of astrophysical echoes from axion dark matter, J. Cosmo. Astropart. Phys. 05 (2024) 040

2024
[37]

W. Yang, Y. Sun, Y. Wang, K. Schutz, Y. Li, C. Leung et al., Searching for Axion Dark Matter Gegenschein of the Vela Supernova Remnant with F AST , Astrophys. J. 988 (2025) 104

2025
[38]

Arvanitaki, S

A. Arvanitaki, S. Dimopoulos, S. Dubovsky, N. Kaloper and J. March-Russell, String axiverse, Phys. Rev. D 81 (2010) 123530

2010
[39]

Svrcek and E

P. Svrcek and E. Witten, Axions in string theory , J. High Energ. Phys. 06 (2006) 051

2006
[40]

Broeckel, M

I. Broeckel, M. Cicoli, A. Maharana, K. Singh and K. Sinha, Moduli stabilisation and the statistics of axion physics in the landscape , J. High Energ. Phys. 08 (2021) 059. – 33 –

2021
[41]

Kaplan and R

D.E. Kaplan and R. Rattazzi, Large field excursions and approximate discrete symmetries from a clockwork axion , Phys. Rev. D 93 (2016) 085007

2016
[42]

Choi and S.H

K. Choi and S.H. Im, Realizing the relaxion from multiple axions and its uv completion with high scale supersymmetry , J. High Energ. Phys. 01 (2016) 149

2016
[43]

Giudice and M

G.F. Giudice and M. McCullough, A clockwork theory , J. High Energ. Phys. 02 (2017) 036

2017
[44]

Bhattacharya, D

S. Bhattacharya, D. Choudhury, S. Maharana and T. Srivastava, Axion icebergs: Clockwork alps at hadron colliders , arXiv:2409.05983 [hep-ph] (2024)

2024 arXiv
[45]

Chadha-Day, J

F. Chadha-Day, J. Maxwell and J. Turner, Alp anarchy, J. Cosmo. Astropart. Phys. 2024 (2024) 056

2024
[46]

13 (2017) 584

CAST collaboration, New CAST Limit on the Axion-Photon Interaction , Nature Phys. 13 (2017) 584

2017
[47]

Kondo and H

D. Kondo and H. Murayama, Multiple Axions Save High-Scale Inflation , arXiv:2507.07973 [hep-ph] (2025)

2025
[48]

Dunsky, C.A

D.I. Dunsky, C.A. Manzari, P. Qu ´ ılez, M. Ramos and P. Sørensen,Resonant Landau-Zener Conversion In Multi-Axion Systems , arXiv:2507.06287 [hep-ph] (2025)

2025 arXiv
[49]

Turner, Cosmic and Local Mass Density of Invisible Axions , Phys

M.S. Turner, Cosmic and Local Mass Density of Invisible Axions , Phys. Rev. D 33 (1986) 889

1986
[50]

Duffy and P

L.D. Duffy and P. Sikivie, The Caustic Ring Model of the Milky Way Halo , Phys. Rev. D 78 (2008) 063508 [ 0805.4556]

2008 arXiv
[51]

Chakrabarty, Y

S.S. Chakrabarty, Y. Han, A.H. Gonzalez and P. Sikivie, Implications of triangular features in the Gaia skymap for the Caustic Ring Model of the Milky Way halo , Phys. Dark Univ. 33 (2021) 100838 [ 2007.10509]

2021 arXiv
[52]

Bastero-Gil, C

M. Bastero-Gil, C. Beaufort and D. Santos, Solar axions in large extra dimensions , J. Cosmo. Astropart. Phys. 10 (2021) 048

2021
[53]

ADMX collaboration, SQUID-Based Microwave Cavity Search for Dark-Matter Axions , Phys. Rev. Lett. 104 (2010) 041301

2010
[54]

ADMX collaboration, A Search for Invisible Axion Dark Matter with the Axion Dark Matter Experiment, Phys. Rev. Lett. 120 (2018) 151301

2018
[55]

ADMX collaboration, Extended Search for the Invisible Axion with the Axion Dark Matter Experiment, Phys. Rev. Lett. 124 (2020) 101303

2020
[56]

ADMX collaboration, Search for Invisible Axion Dark Matter in the 3.3–4.2 µeV Mass Range, Phys. Rev. Lett. 127 (2021) 261803

2021
[57]

ADMX collaboration, Piezoelectrically Tuned Multimode Cavity Search for Axion Dark Matter, Phys. Rev. Lett. 121 (2018) 261302

2018
[58]

ADMX collaboration, Dark matter axion search using a Josephson Traveling wave parametric amplifier, Rev. Sci. Instrum. 94 (2023) 044703

2023
[59]

Crisosto, P

N. Crisosto, P. Sikivie, N.S. Sullivan, D.B. Tanner, J. Yang and G. Rybka, ADMX SLIC: Results from a Superconducting LC Circuit Investigating Cold Axions , Phys. Rev. Lett. 124 (2020) 241101

2020
[60]

Jeong, S

J. Jeong, S. Youn, S. Bae, J. Kim, T. Seong, J.E. Kim et al., Search for Invisible Axion Dark Matter with a Multiple-Cell Haloscope , Phys. Rev. Lett. 125 (2020) 221302. – 34 –

2020
[61]

Y. Lee, B. Yang, H. Yoon, M. Ahn, H. Park, B. Min et al., Searching for Invisible Axion Dark Matter with an 18 T Magnet Haloscope , Phys. Rev. Lett. 128 (2022) 241805

2022
[62]

CAPP collaboration, First Results from an Axion Haloscope at CAPP around 10.7 µeV, Phys. Rev. Lett. 126 (2021) 191802

2021
[63]

H. Yoon, M. Ahn, B. Yang, Y. Lee, D. Kim, H. Park et al., Axion haloscope using an 18 T high temperature superconducting magnet, Phys. Rev. D 106 (2022) 092007

2022
[64]

Burns et al., Nasa probe study report: Farside array for radio science investigations of the dark ages and exoplanets (farside) , arXiv:1911.08649 [astro-ph.IM] (2019)

J.O. Burns et al., Nasa probe study report: Farside array for radio science investigations of the dark ages and exoplanets (farside) , arXiv:1911.08649 [astro-ph.IM] (2019)

2019 arXiv
[65]

Burns et al., A lunar farside low radio frequency array for dark ages 21-cm cosmology , arXiv:2103.08623 [astro=ph.IM] (2021)

J.O. Burns et al., A lunar farside low radio frequency array for dark ages 21-cm cosmology , arXiv:2103.08623 [astro=ph.IM] (2021)

2021 arXiv
[66]

X. Chen, J. Yan, L. Deng, F. Wu, L. Wu, Y. Xu et al., Discovering the sky at the longest wavelengths with a lunar orbit array , Phil. Trans. R. Soc. A 379 (2021)

2021
[67]

Taruya, A

A. Taruya, A. Nishizawa and Y. Himemoto, Hunting axion dark matter signatures in low-frequency terrestrial magnetic fields , arXiv:2504.06653 [hep-ph] (2025)

2025
[68]

Nishizawa, A

A. Nishizawa, A. Taruya and Y. Himemoto, Axion dark matter search from terrestrial magnetic fields at extremely low frequencies , arXiv:2504.07559 [hep-ph] (2025)

2025 arXiv
[69]

Benabou, C

J.N. Benabou, C. Dessert, K.C. Patra, T.G. Brink, W. Zheng, A.V. Filippenko et al., Search for axions in magnetic white dwarf polarization at lick and keck observatories , arXiv:2504.12377 [hep-ph] (2025)

2025 arXiv
[70]

Ross, Introduction to Probability Models, Elsevier, 12th ed

S.M. Ross, Introduction to Probability Models, Elsevier, 12th ed. (2014), 10.1016/C2012-0-03564-8. – 35 –

2014 doi

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Peccei and H.R

R.D. Peccei and H.R. Quinn, CP conservation in the presence of pseudoparticles , Phys. Rev. Lett. 38 (1977)

work page 1977

[2] [2]

Weinberg, A new light boson? , Phys

S. Weinberg, A new light boson? , Phys. Rev. Lett. 40 (1978) 223

1978

[3] [3]

Wilczek, Problem of strong p and t invariance in the presence of instantons , Phys

F. Wilczek, Problem of strong p and t invariance in the presence of instantons , Phys. Rev. Lett. 40 (1978) 279

1978

[4] [4]

Zhitnitskii, Possible suppression of axion-hadron interactions , Sov

A.P. Zhitnitskii, Possible suppression of axion-hadron interactions , Sov. J. Nucl. Phys. 31 (1980)

work page 1980

[5] [5]

M. Dine, W. Fischler and M. Srednicki, A simple solution to the strong cp problem with a harmless axion , Phys. Lett. B 104 (1981) 199

work page 1981

[6] [6]

Kim, Weak-interaction singlet and strong CP invariance, Phys

J.E. Kim, Weak-interaction singlet and strong CP invariance, Phys. Rev. Lett. 43 (1979) 103

1979

[7] [7]

Shifman, A

M.A. Shifman, A. Vainshtein and V. Zakharov, Can confinement ensure natural cp invariance of strong interactions? , Nucl. Phys. B 166 (1980) 493

work page 1980

[8] [8]

Berezhiani and M

Z. Berezhiani and M. Khlopov, Cosmology of spontaneously broken gauge family symmetry with axion solution of strong cp-problem , Z. Phys. C - Particles and Fields 49 (1991) 73

work page 1991