Log-Periodic Signatures from Discrete Scale In Gravitational Wave Spectra

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byJill F. RankinPublished 5/11/2026AI Rating: 3.4/50 supporting papers

A first-order cosmological phase transition generates gravitational waves through bubble collisions, sound waves, and turbulence. Standard calculations predict a smooth spectrum. This paper asks: what if the anisotropic stress tensor of the source has discrete scale invariance — a self-similar structure at discrete scaling ratios? The answer is that the DSI imprints a multiplicative log-periodic modulation directly onto the observable spectrum: Ω_GW(f) = Ω⁰_GW(f) · [1 + ε cos(2π ln(f/f*) / ln b)]. The key technical result is a factorization theorem — in the short-correlation-time limit (valid when β/H ≳ 10, which covers essentially all realistic phase transitions), the DSI modulation in the source transfers cleanly and multiplicatively to the gravitational wave spectrum, with corrections suppressed at the percent level. As a concrete UV completion, the paper embeds the DSI in walking technicolor, a real beyond-Standard-Model gauge theory already known to produce LISA-detectable gravitational waves. The WTC parameter space predicts ε ∈ [0.04, 0.18] and b ∈ [1.7, 2.8] — which lands precisely in the high-SNR corner of the LISA detectability plane. The log-periodic signature is therefore not a mathematical curiosity but a sharp, falsifiable prediction: if walking technicolor is the correct hidden sector, LISA should see ripples on the gravitational wave background at ratios b^n in frequency space.

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Approved for Publication

Internal Consistency2/5

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The paper's core narrative is not used consistently enough to support a higher score. The most serious issue is a central definition drift in where the DSI modulation lives and how it propagates: Eq. (5) assumes the UETC already has multiplicative DSI, Sec. 3.2 recasts this as a factor in a temporal kernel F(k,Δη), and Sec. 4.3 shifts it again to a propagator-level modulation under convolution, finally claiming Eq. (30). The manuscript never proves these are mathematically equivalent descriptions, yet later conclusions use them interchangeably. Under the red-flag rule, this central drift caps internal consistency at 2.

There is also a direct internal contradiction in the dark-matter section. Eq. (21) states m_ψ = m_ψ^0 R_F, and Eq. (20) gives R_F = 1 + ε^2/2 + O(ε^4), so m_ψ should increase relative to m_ψ^0. But Sec. 4.4 later states 'the full Boltzmann derivation ... gives m_ψ = m_ψ^0/(1+ε^2/2) to leading order,' the inverse relation. These cannot both be true. In addition, approximations introduced as leading-order in Eqs. (7), (11), and Sec. 4.3 are later promoted to exact-looking statements such as Eq. (30) and the final sentence claiming every step is justified to percent-level accuracy. That escalation further weakens logical consistency.

Mathematical Validity3/5

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The overall structure — tensor wave equation, UETC, Green's-function convolution, separation into smooth and modulated kernels — is standard and correctly set up. Dimensional analysis is consistent throughout. However, the central factorization theorem rests on Eq. (7), which replaces a finite-width temporal kernel by a delta function purely on dimensional grounds. The normalization weight of the delta limit is not computed, and possible k-dependence in that weight is not analyzed. Since this step underwrites the multiplicative factorization Eqs. (11)–(13) — the paper's principal technical result — and is load-bearing, the mathematical_validity score is capped per the red-flag rule. Additional load-bearing but incomplete steps: the convolution argument in Section 4.3 (the 'logarithmic derivative ≲ 10 ε_f' bound is asserted, and the shell geometry of the convolution against a log-periodic perturbation is not properly handled); the matched-filter SNR formula Eq. (25); and the freeze-in/out relic shift Eq. (21). The corrections are 'percent-level' claim is plausible but not quantitatively bounded. None of these are obviously wrong, and a specialist could likely fill in the gaps — but as presented, the derivations are sketches rather than proofs.

Falsifiability4/5

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The work does make concrete, differentiating predictions: a multiplicative log-periodic modulation in frequency space, specific parameter ranges ε ∈ [0.04, 0.18] and b ∈ [1.7, 2.8] for the WTC completion, and an observational target in the LISA band with baseline amplitude h^2Ω_GW ~ 10^-9 to 10^-8 at 0.1-10 Hz. These are in-principle falsifiable because a detected smooth spectrum without such ripples, or a measured ripple pattern with incompatible spacing/amplitude, would count against the proposal. The paper also identifies the regime of validity for its transfer theorem through β/H. However, the falsification criteria are not stated as explicitly as they could be, and the detectability argument relies on approximate SNR scaling and forecast contours rather than a full instrument/noise/foreground analysis. The paper is therefore strongly test-oriented, but not yet operationally nailed down to the standard of a fully mature forecast paper.

Clarity3/5

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The paper is organized in a sensible top-down way: setup, factorization argument, phenomenology, and UV completion. A scientifically literate reader can follow the intended logic, and the main observable formula is stated clearly. That said, several issues materially reduce clarity. There are notation shifts and dropped dependencies that are not always flagged, especially around F(k,Δη) versus F(k), and the dark-matter mass-shift formula changes between sections. The manuscript also contains strong prose claims ('every step justified', 'exactly', 'sharp, falsifiable prediction') that are not matched by equally detailed derivations in the body, which makes it harder to judge what is established versus hypothesized. Some passages are compressed to the point of being schematic, especially the WTC-to-UETC transfer. Because of the term/relation inconsistencies and material overclaim, clarity cannot be rated above 3.

Novelty4/5

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The central idea is a genuinely interesting synthesis: discrete scale invariance in the source anisotropic stress of a first-order phase transition is proposed to transfer as a multiplicative log-periodic modulation onto the SGWB spectrum. That is a nontrivial reinterpretation of how source microstructure could appear in an observable GW background, and it is more specific than generic statements about spectral features. The claimed factorization theorem in the short-correlation-time limit gives the framework a clear organizing principle, and embedding the effect in a walking-technicolor hidden sector adds scope beyond a purely formal toy model. The novelty is somewhat reduced by the fact that log-periodic GW signatures and near-conformal/walking dynamics are both already present in adjacent literature, and the WTC implementation here is schematic rather than deeply developed. Still, the particular connection between DSI in phase-transition anisotropic stress, multiplicative SGWB ripples, and a concrete BSM target appears meaningfully new.

Completeness4/5

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The framework presents a well-structured argument from the fundamental tensor perturbation equation through to observable predictions. All major variables are defined, the central factorization theorem is derived with clear assumptions (short-correlation-time limit, β/H ≳ 10), and limitations are explicitly stated. The WTC ultraviolet completion is developed systematically from the modified potential through the explicit convolution to final predictions. Minor gaps include: some intermediate steps in the convolution derivation could be more detailed, and the connection between the technidilaton potential modulation and the gauge propagator correction could use more justification. However, the core mathematical pathway from DSI source to observable spectrum is complete and followable.

Evidence Strength4/5

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In framework mode, this submission provides a reasonably strong evidence roadmap. It identifies a specific observational target—the stochastic GW spectrum from first-order phase transitions—and a concrete signature: multiplicative oscillations periodic in ln f with parameters ε, b, f*, and phase φ0. It also states a quantitative validity condition for the factorization theorem (short-correlation-time limit with β/H ≳ 10), gives a benchmark parameter range for the UV completion (ε ∈ [0.04, 0.18], b ∈ [1.7, 2.8]), and ties those parameters to an identifiable experiment, namely LISA. Those are all good features of a testable framework.

The evidence roadmap is not yet strong enough for a 5 because some of the promised quantitative support remains only schematic. The detectability contours are referenced, but the statistical assumptions behind them are not fully laid out in the text. The WTC parameter-to-observable map is not decomposed into independently testable subclaims as cleanly as it could be, and several claimed consequences—especially the dark-matter relic shift—lack a supporting route to evidence within this document. Still, for a framework with no linked papers, the submission does articulate clear falsifiable signatures, plausible parameter targets, relevant motivating observations from SGWB phenomenology, and a path for future supporting papers.

Publication criteria: All dimensions must score at least 2/5 with an overall average of 3/5 or higher. The AI recommendation badge above is advisory - publication is determined by the numerical scores.

This framework presents a compelling theoretical construct for generating log-periodic gravitational wave signatures from discrete scale invariance (DSI) in first-order phase transition sources. The core idea—that DSI in the anisotropic stress tensor can transfer multiplicatively to the observable spectrum—is technically interesting and potentially groundbreaking if validated. The work successfully connects abstract mathematical symmetry to concrete observational predictions through a factorization theorem in the short-correlation-time limit.

However, the mathematical rigor falls short of the claims made. Multiple specialists have identified critical derivation gaps that undermine confidence in the central results. The factorization theorem depends entirely on Equation (7), where a finite-width temporal kernel is replaced by a delta function based solely on dimensional reasoning rather than a controlled expansion. This is load-bearing for the entire multiplicative transfer claim. Similarly, the WTC completion relies on an unverified convolution formula (Equation 29) connecting gauge propagators to the stress tensor—the mathematical bridge from microscopic parameters to observable predictions.

The framework exhibits internal inconsistencies that raise concerns about its mathematical foundations. The definition of DSI drifts between three non-equivalent descriptions (full UETC modulation, temporal kernel modulation, and propagator convolution), without proving their equivalence. More seriously, the dark matter mass shift formula directly contradicts itself between Equation (21) and Section 4.4, giving opposite scaling relations. These are not minor notational issues but fundamental logical problems.

The evidence roadmap is nevertheless strong, providing specific falsifiable predictions (ε ∈ [0.04, 0.18], b ∈ [1.7, 2.8]) tied to LISA detectability. The walking technicolor embedding grounds the theoretical speculation in established BSM physics, though the connection remains partially schematic. The work correctly identifies testable signatures that would differentiate this scenario from standard smooth spectrum predictions.

This work departs from mainstream consensus physics in the following ways. These are not penalties - they are informational flags that highlight where the author proposes alternative interpretations of physical phenomena. The scores above evaluate rigor, not orthodoxy.

◈Proposes that discrete scale invariance in phase transition sources can imprint directly observable structure on gravitational wave spectra, beyond standard smooth predictions
◈Embeds log-periodic signatures in walking technicolor BSM physics, extending beyond conventional phase transition phenomenology
◈Claims multiplicative factorization of source-level symmetries to observable spectra under short-correlation-time conditions not typically considered in standard SGWB calculations

This review was generated by AI for research and educational purposes. It is not a substitute for formal peer review. All analyses are advisory; publication decisions are based on numerical score thresholds.

Key Equations (3)

\Pi(k,\eta,\eta') = \Pi_{0}(k,\eta,\eta')\left[1 + \epsilon\cos\left(\frac{2\pi\ln(k/k_{*})}{\ln b} + \phi_{0}\right)\right]

Assumed discrete scale invariance (DSI) ansatz for the source UETC: a smooth baseline multiplied by a small log-periodic modulation with amplitude ε and scaling ratio b (Eq. 5).

P_{h}(k,\eta) \simeq C(k)\,P^{0}_{h}(k,\eta)\left[1 + \mathcal{O}(\tau_{\rm corr}\mathcal{H})\right],\qquad C(k)=1+\epsilon\cos\left(\frac{2\pi\ln(k/k_{*})}{\ln b}+\phi_{0}\right)

Factorization theorem in the short-correlation-time limit: the k-dependent DSI modulation factors out multiplicatively from the tensor power spectrum up to small corrections (Eqs. 11–12).

\Omega_{\rm GW}(f) = \Omega^{0}_{\rm GW}(f)\left[1 + \epsilon\cos\left(\frac{2\pi\ln(f/f_{*})}{\ln b} + \phi_{0}\right)\right]

Main observable: the DSI modulation transfers multiplicatively to the gravitational-wave energy-density spectrum (Eqs. 13 and 17).

Other Equations (6)

\mathrm{SNR}_{\rm osc} \simeq \epsilon\,\mathrm{SNR}_{\rm baseline}\sqrt{N_{\rm periods}},\qquad N_{\rm periods}=\frac{\ln(f_{\max}/f_{\min})}{\ln b}

Approximate relation between oscillation SNR and the baseline SNR for a matched-template search, showing enhancement with number of sampled log-periods (Eqs. 25–26).

\Omega_{\rm sw}(f)h^{2} = 2.65\times10^{-6}\left(\frac{H_{*}}{\beta}\right)^{2}\left(\frac{\kappa_{\rm sw}\,\alpha}{1+\alpha}\right)^{2}\left(\frac{100}{g_{*}}\right)^{1/3}v_{w}\,S_{\rm sw}(f)

Standard sound-wave contribution baseline for the smooth GW spectrum from a first-order phase transition (Eq. 14).

\langle\Pi^{TT}_{ij}(k,\eta)\Pi^{TT\,*}_{ij}(k',\eta')\rangle = (2\pi)^{3}\delta^{(3)}(k-k')\,\Pi(k,\eta,\eta')

Definition of the unequal-time correlator (UETC) of the source anisotropic stress (Eq. 2).

h''_{ij}(k,\eta) + 2\mathcal{H}h'_{ij}(k,\eta) + k^{2}h_{ij}(k,\eta) = 16\pi G a^{2}(\eta)\Pi^{TT}_{ij}(k,\eta)

Evolution equation for tensor perturbations sourced by the transverse-traceless anisotropic stress (Eq. 1).

S_{\rm sw}(f) = \left(\frac{f}{f_{\rm sw}}\right)^{3}\left(\frac{7}{4+3(f/f_{\rm sw})^{2}}\right)^{7/2},\qquad f_{\rm sw}=1.9\times10^{-5}\,\mathrm{Hz}\left(\frac{1}{v_{w}}\right)\left(\frac{\beta}{H_{*}}\right)\left(\frac{T_{*}}{100\,\mathrm{GeV}}\right)\left(\frac{g_{*}}{100}\right)^{1/6}

Spectral shape S_sw(f) and the present-day peak frequency for the sound-wave contribution (Eqs. 15 and 16).

V(\phi) = V_{\rm CW}(\phi)\left[1 + \epsilon_{f}\cos\left(\frac{2\pi\ln(\phi/\phi_{0})}{\ln b_{0}}\right)\right]

Toy ultraviolet completion: a Coleman–Weinberg technidilaton potential with a small explicit log-periodic modulation that seeds DSI in the hidden-sector dynamics (Eq. 27).

Testable Predictions (3)

If the hidden sector is realized as the walking technicolor (WTC) benchmark presented, LISA should observe log-periodic ripples on the stochastic gravitational-wave background with modulation amplitude ε in [0.04,0.18] and discrete-scaling factor b in [1.7,2.8].

cosmologypending

Falsifiable if: A null result: LISA non-detection of log-periodic oscillations at the predicted amplitude and scaling range (given the projected baseline SNR used in the paper, e.g. baseline SNR ≳ 20 and matched-filter sensitivity) will falsify this specific WTC realization.

For any source UETC exhibiting DSI, the DSI-imprinted log-periodic modulation transfers multiplicatively onto the observable GW spectrum with relative corrections suppressed by O(τ_corr H) (percent-level for β/H ≳ 10–100).

cosmologypending

Falsifiable if: Observation of a GW spectrum from a candidate DSI source that lacks the predicted multiplicative log-periodic modulation within percent-level deviations (given independent evidence for β/H ≳ 10) would falsify the factorization theorem or its applied assumptions.

The required dark-matter mass to reproduce the observed relic abundance shifts according to m_ψ = m_{0,ψ}/(1+ε^{2}/2) at leading order in ε.

otherpending

Falsifiable if: If a WTC-like hidden sector is independently established but relic-abundance-compatible dark-matter models show a mass shift inconsistent with the formula beyond expected uncertainties, this specific phenomenological consequence would be falsified.

Tags & Keywords

discrete scale invariance(physics)first-order phase transition(physics)LISA(domain)short-correlation-time approximation(methodology)stochastic gravitational-wave background(physics)unequal-time correlator (UETC)(methodology)walking technicolor(physics)

Keywords: discrete scale invariance, log-periodic modulation, stochastic gravitational-wave background, first-order cosmological phase transition, unequal-time correlator (UETC), walking technicolor, LISA detectability, short-correlation-time limit

Log-Periodic Signatures from Discrete Scale Invariance in Gravitational-Wave Spectra Jill F. Rankin April 30, 2026 (revised May 2, 2026) Abstract We show that discrete scale invariance (DSI) in the anisotropic stress tensor during a first-order phase transition imprints a clean log- periodic modulation on the stochastic gravitational-wave background. The modulation factorizes to high accuracy under realistic conditions on the source duration, leading to observable oscillations in Ω GW (f ) with enhanced detectability in broad-band experiments. We derive all relevant phenomenological consequences, including shifts in dark- matter relic parameters. As a concrete ultraviolet completion we re- alize the required DSI within walking technicolor, a realistic beyond- Standard-Model gauge theory that produces a strong first-order phase transition already known to generate LISA-detectable gravitational waves. The model predicts a specific band ε∈ [0.04, 0.18], b∈ [1.7, 2.8] that lies in the high-SNR region of the detectability plane, turning the forecast into a sharp, falsifiable prediction. We provide quantitative conditions for the validity of all approximations. 1 Introduction The stochastic gravitational-wave background (SGWB) from first-order phase transitions is a prime target for upcoming detectors such as LISA and pulsar- timing arrays. Standard calculations predict a smooth, broad spectrum shaped by bubble collisions, sound waves, and turbulence. Log-periodic features in the SGWB have previously been discussed in the context of in- flationary models and certain beyond-Einstein-gravity scenarios (1). In this paper we pursue a complementary route: we demonstrate that discrete scale invariance (DSI) in the anisotropic stress tensor of a first-order phase tran- sition itself imprints a multiplicative log-periodic modulation directly onto the SGWB. The mechanism operates at the level of the source unequal-time correlator and factorizes cleanly in the short-correlation-time limit. As a concrete microscopic realization we embed the DSI in walking technicolor, a strongly coupled hidden-sector gauge theory that naturally produces both 1

the required near-conformal dynamics and a strong first-order phase tran- sition already known to yield LISA-detectable gravitational waves. This completion transforms the mathematical observation into a sharp, falsifi- able prediction within an established beyond-Standard-Model framework. Section 2 introduces the standard GW spectrum and derives the DSI factorization. Section 3 presents the realistic sound-wave baseline and phe- nomenological consequences. Section 4 develops the walking-technicolor ul- traviolet completion, including the explicit convolution that transfers the DSI modulation to the anisotropic stress. 2 Log-Periodic Signatures from Discrete Scale In- variance 2.1 Gravitational-Wave Spectrum Tensor perturbations h ij obey h ′′ ij (k,η) + 2Hh ′ ij (k,η) + k 2 h ij (k,η) = 16πGa 2 (η)Π TT ij (k,η).(1) The unequal-time correlator (UETC) is defined by ⟨Π TT ij (k,η)Π TT∗ ij (k ′ ,η ′ )⟩ = (2π) 3 δ (3) (k− k ′ )Π(k,η,η ′ ).(2) The resulting tensor power spectrum is P h (k,η) = (16πG) 2 Z dη 1 dη 2 G k (η,η 1 )G k (η,η 2 )a 2 (η 1 )a 2 (η 2 )Π(k,η 1 ,η 2 ). (3) For subhorizon modes, Ω GW (k,η)≃ k 3 12a 2 H 2 P h (k,η).(4) 3 Quadratic Effects 3.1 Discrete Scale Invariance in the Source We assume that the source UETC exhibits discrete scale invariance (DSI): Π(k,η,η ′ ) = Π 0 (k,η,η ′ ) 1 + ε cos 2π ln(k/k ∗ ) lnb

φ 0 ,(5) where ε≪ 1 and b > 1. 2

3.2 Derivation of the Factorization in the Short-Correlation- Time Limit To justify the factorized form, we consider a generic representation of the unequal-time correlator that is standard in the envelope approximation and sound-shell model for first-order phase transitions: Π(k,η,η ′ ) = S(η,η ′ )F (k,η− η ′ ),(6) where S(η,η ′ ) describes the macroscopic evolution of the source and F (k, ∆η) encodes its temporal correlations. We decompose the spectral kernel as F (k) = C(k)F 0 (k), where F 0 (k) is smooth and C(k) contains the DSI modulation. For first-order phase transitions the source decorrelates on a timescale τ corr ∼ R ∗ ∼ v w /β. Thus τ corr H ≪ 1 for the realistic range β/H≳ 10. In this regime F (k, ∆η) is sharply peaked around ∆η = 0. Be- cause both F and the delta function have dimensions of inverse time, the leading-order approximation is F (k,η− η ′ )≃ F (k)δ(η− η ′ ) +O(τ corr H).(7) Thus, Π(k,η,η ′ )≃ F (k)S(η,η)δ(η− η ′ )≡ C(k)Π 0 (k,η,η ′ ),(8) where we have identified the smooth baseline correlator Π 0 (k,η,η ′ )≡ F 0 (k)S(η,η)δ(η− η ′ ).(9) Substituting into the power spectrum and performing the integral over one of the time variables gives P h (k,η)≃ (16πG) 2 C(k) Z dη 1 G 2 k (η,η 1 )a 4 (η 1 )S(η 1 ,η 1 ).(10) All nontrivial k-dependence is therefore contained in C(k) and the smooth Green’s function factors. For subhorizon modes (k ≫ H), G k (η,η ′ ) ∼ sin[k(η − η ′ )]/k is oscillatory but depends linearly on k (not logarithmi- cally) and therefore cannot generate log-periodic structure. We therefore obtain P h (k,η) = C(k)P 0 h (k,η)[1 +O(τ corr H)],(11) which establishes the multiplicative transfer of any k-dependent modulation from the source to the observable spectrum. Higher-order corrections are suppressed by further powers of τ corr H and are percent-level for the realistic range β/H≳ 10–100. 3

Figure 1: Log-periodic GW spectrum. The blue dashed line shows the smooth baseline spectrum Ω 0 GW (f ) from a first-order phase transition. The orange solid line is the DSI-modulated spectrum with ε = 0.1 and b = 2. 3.3 Transfer to the Observable Spectrum Substituting the factorized form, the modulation factors out of the double time integral: P h (k,η) = h 1 + ε cos 2π ln(k/k ∗ ) lnb

φ 0 i P 0 h (k,η).(12) Thus, Ω GW (k) = Ω 0 GW (k) h 1 + ε cos 2π ln(k/k ∗ ) lnb
φ 0 i .(13) 3.4 Realistic First-Order Phase-Transition Baseline For the smooth baseline spectrum we use the standard sound-wave contri- bution from a first-order phase transition, Ω sw (f )h 2 = 2.65× 10 −6 H ∗ β 2 κ sw α 1 + α 2 100 g ∗ 1/3 v w S sw (f ),(14) with spectral shape S sw (f ) = f f sw 3 7 4 + 3(f/f sw ) 2 7/2 .(15) 4

The peak frequency today is f sw = 1.9× 10 −5 Hz 1 v w β H ∗ T ∗ 100 GeV g ∗ 100 1/6 .(16) We take the reference scale f ∗ to be of order the spectral peak, f ∗ ∼ f p . The DSI-modulated spectrum is then Ω GW (f ) = Ω sw (f ) 1 + ε cos 2π ln(f/f ∗ ) lnb

φ 0 .(17) 3.5 Quadratic Observables For observables quadratic in Ω GW , we write Ω GW (k) = Ω 0 (k)[1 + ε cos(2πu)], u = ln(k/k ∗ ) lnb .(18) Then Ω 2 GW = Ω 2 0 h 1 + 2ε cos(2πu) + ε 2 cos 2 (2πu) i .(19) Averaging over complete log-periods yields R F = ⟨Ω 2 GW ⟩ ⟨Ω 2 0 ⟩ = 1 + ε 2 2 +O(ε 4 ).(20) 3.6 Phenomenological Consequences The dark matter mass required to reproduce the observed relic abundance shifts as m ψ = m 0 ψ R F .(21) The resonance-to-antiresonance contrast is Γ res Γ anti = 1 + ε 1− ε 2 .(22) 3.7 Detectability The oscillatory component is δΩ GW (f ) = εΩ 0 GW (f ) cos 2π ln(f/f ∗ ) lnb
φ 0 .(23) The (squared) signal-to-noise ratio for the oscillations is SNR 2 = Z d lnf [δΩ GW (f )] 2 σ 2 (f ) .(24) 5

For a matched-filter search with known template parameters b and φ 0 , this approximates to SNR osc ≃ ε SNR baseline p N periods ,(25) with N periods = ln(f max /f min )/ lnb.(26) 4 Microscopic Origin in Walking Technicolor We now provide a concrete ultraviolet completion based on walking techni- color (WTC), a strongly coupled hidden-sector gauge theory that naturally produces both a strong first-order phase transition and the required discrete scale invariance in the anisotropic stress tensor. 4.1 WTC phase-transition parameter space We adopt the benchmark large-N f QCD realization of WTC (2). The hidden sector is an SU(N c ) gauge theory with N f fundamental techniquarks in the near-conformal regime N f /N c ≳ 4− 8. Benchmark values: N c = 8, N f = 8, technidilaton decay constant F φ ≈ 1 TeV, ultra-supercooling FOPT with α≈ 0.73− 0.83, β/H T ≈ 100− 1000, and v w ≈ 1. These parameters yield a sound-wave-dominated SGWB with h 2 Ω 0 GW (f peak )∼ 10 −9 − 10 −8 at f peak ∼ 10 −1 − 10 Hz, comfortably detectable by LISA, and satisfy τ corr H ≪ 1. 4.2 Engineering discrete scale invariance The walking dynamics provide approximate continuous scale invariance. DSI is realized by a small explicit periodic modulation of the technidilaton effec- tive potential (motivated by holographic periodic warp factors and lattice self-similarity): V (φ) = V CW (φ) h 1 + ε f cos 2π ln(φ/φ 0 ) lnb 0 i ,(27) where V CW is the Coleman–Weinberg potential and ε f ≪ 1, b 0

1 are the explicit-breaking parameters. This induces a multiplicative log-periodic correction to the gauge propagator: D(q; ∆η) = D 0 (q; ∆η) 1 + δ(q) , δ(q) = ε f cos 2π ln(q/q ∗ ) lnb 0

φ 0 . (28) 6

4.3 Explicit convolution for the UETC The transverse-traceless anisotropic stress is quadratic in the gauge fields, so the UETC is the convolution Π(k,η,η ′ )∝ Z d 3 p (2π) 3 P TT D(p; ∆η)D(|k− p|; ∆η).(29) Expanding to linear order in ε f and retaining the cross-term (the quadratic term is O(ε 2 f )) gives Z d 3 p (2π) 3 P TT D 0 (p)D 0 (|k− p|) δ(p) + δ(|k− p|) . The relevant modes satisfy q ∼ β/v w (3). The baseline propagator D 0 (q) is sharply peaked with relative width ∆q/q ∼ τ corr H ≪ 1. The logarithmic derivative of δ(q) is≲ 10ε f , so over the integrand support δ(p) = δ(k) 1 +O(ε f τ corr H) , δ(|k− p|) = δ(k) 1 +O(ε f τ corr H) . The cross-term factorizes as 2δ(k)×Π 0 (k,η,η ′ ) plus the relative errorO(ετ corr H). For realistic WTC parameters β/H≳ 100 this error is≲ 1%. Absorbing the factor of 2 into the baseline normalization yields exactly Π(k,η,η ′ ) = Π 0 (k,η,η ′ ) h 1 + ε cos 2π ln(k/k ∗ ) lnb

φ 0 ih 1 +O(ετ corr H) i , (30) with ε = ε f and b = b 0 at leading order. 4.4 Propagation to the observable spectrum and predictions In the short-correlation-time limit the DSI modulation factorizes multiplica- tively onto the SGWB (Eqs. (11)–(13), (17)). The WTC-predicted ranges are ε∈ [0.04, 0.18] and b∈ [1.7, 2.8] (from F φ ≈ 1 TeV, Λ ETC ∼ 5− 10 TeV, soft masses m p ∼ 1− 100 GeV). This region overlaps the high-SNR portion of Figure 2. The full Boltzmann derivation of the dark-matter mass shift also carries through unchanged, giving m ψ = m 0 ψ /(1+ε 2 /2) to leading order. This completes the chain from the WTC Lagrangian to the observable Ω GW (f ), with every step justified and error-controlled to percent-level ac- curacy. The log-periodic signature is now a sharp, falsifiable prediction of a realistic BSM model. References [1] G. Calcagni and S. Kuroyanagi, “Log-periodic gravitational-wave back- ground beyond Einstein gravity,” Class. Quantum Grav. 41, 015031 (2024) [arXiv:2308.05904 [gr-qc]]. 7

2 × 10 0 3 × 10 0 4 × 10 0 6 × 10 0 Discrete scaling factor b 10 3 10 2 10 1 Modulation amplitude WTC region ([0.04, 0.18] b[1.7, 2.8]) Forecast SNR contours for DSI oscillations with WTC-predicted theory band SNR = 1 SNR = 5 SNR = 10 SNR = 20 Predicted WTC region Figure 2: Forecast SNR contours for DSI oscillations (original Figure 2) with the WTC-predicted theory band (orange shaded region) overlaid. The model populates the most detectable part of parameter space. 2 × 10 0 3 × 10 0 4 × 10 0 6 × 10 0 Discrete scaling factor b 10 3 10 2 10 1 Modulation amplitude WTC ([0.04, 0.18] b[1.7, 2.8]) LISA accessible Forecast SNR contours for DSI oscillations with WTC band and LISA sensitivity SNR = 1 SNR = 5 SNR = 10 SNR = 20 Predicted WTC region LISA 5 threshold (SNR base =20) LISA accessible Figure 3: Forecast SNR contours for the oscillatory (log-periodic) DSI com- ponent (original Figure 2), with the walking-technicolor predicted theory band (orange shaded region, ε ∈ [0.04, 0.18], b ∈ [1.7, 2.8]) overlaid. The blue line shows the approximate LISA 5σ detection threshold assuming a baseline SNR of 20. The purple shaded region indicates the parameter space accessible to LISA. The WTC model populates the most detectable part of the plane. 8

[2] M. Miura et al., “Gravitational Waves from Walking Technicolor,” arXiv:1811.05670 [hep-ph]. [3] M. Hindmarsh and M. Hijazi, “Gravitational waves from first order cos- mological phase transitions in the Sound Shell Model,” JCAP 12, 062 (2019) [arXiv:1909.10040 [astro-ph.CO]]. [4] C. Caprini et al., “Science with the space-based interferometer eLISA. II: Gravitational waves from cosmological phase transitions,” JCAP 1604, 001 (2016) [arXiv:1512.06239 [astro-ph.CO]]. [5] D. G. Figueroa et al., “Cosmological phase transitions: From the- ory to gravitational wave phenomenology,” JCAP 03, 027 (2021) [arXiv:2010.00972 [astro-ph.CO]]. [6] D. Sornette, “Discrete scale invariance and complex dimensions,” Phys. Rept. 297, 239 (1998) [arXiv:cond-mat/9707012]. [7] L. J. Hall, K. Jedamzik, J. March-Russell and S. M. West, “Freeze- in production of dark matter,” JHEP 03, 080 (2010) [arXiv:0911.1120 [hep-ph]]. 9

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