paper Review Profile
Log-Periodic Signatures from Discrete Scale Invariance in the Stochastic Gravitational-Wave Background: Walking Technicolor as a Concrete Ultraviolet Completion
Discrete scale invariance (DSI) in the anisotropic stress of a first-order cosmological phase transition imprints a multiplicative log-periodic modulation on the stochastic gravitational-wave background; under the short-correlation-time approximation (β/H* ≳ 10) the modulation factorizes from the source unequal-time correlator to the observable spectrum at the percent level. As a concrete UV completion the paper embeds DSI in walking technicolor, predicting ε∈[0.04,0.18] and b∈[1.7,2.8], LISA-detectable oscillations, and a correlated freeze-in dark-matter mass shift, with all approximations quantitatively justified.
Read the Full BreakdownFull breakdown: https://theoryofeverything.ai/papers/log-periodic-signatures-from-discrete-scale-invariance-in-the-stochastic-gravitational-wave-background-walking-technicolor-as-a-concrete-ultraviolet-completion
The paper is internally well-organized: the DSI ansatz parameters (ε, b, φ_0) are introduced in Sec 3 and used consistently through to the observable spectrum (Eq. 12), the WTC band (Eq. 31), and the DM shift (Eq. 17). The mapping ε=ε_f, b=b_0 is stated explicitly. Approximations are flagged with their controlling parameter τ_corr H_* and a quantitative error table (Table 1) is provided. The main internal-consistency concern is the numerical fit in Figure 1(b), which reports r=0.81±0.04 against an analytic r=1.00 — a ~5σ self-consistency tension that the text does not address. This is a localized inconsistency between an analytical claim and the author's own demonstration figure, not a structural contradiction in the argument, but it does undermine the 'percent-level' framing slightly. Otherwise definitions, ranges, and limits are used coherently throughout.
The mathematical structure (Eqs. 2–5 for the GW spectrum, the DSI ansatz Eq. 6, the matched-filter scaling Eq. 21, and the quadratic renormalization Eq. 16) is dimensionally sound and the formulas individually look correct. However, two central derivations are compressed in a way that a competent reader cannot reproduce them as written: (i) the Green's-function step inside the factorization theorem (Eq. 10→11), where the claim that G_k generates no log-periodic structure rests on a hand-waved 'averaging' argument; and (ii) the convolution factorization (Eqs. 27–30), where δ(p)≈δ(k) is justified by an unproven sharp-peakedness of D_0(q). Both are load-bearing for the paper's central result that the WTC modulation in V(φ) propagates to Ω_GW(f) with ≲1% error. Additionally, the step from V(φ) modulation to a multiplicative δ(q) correction on the gauge-field propagator (Eq. 25→26) is asserted with no derivation in the strongly coupled regime. These gaps prevent a higher score; the cap from the unverified_central_derivation red flag applies.
The paper makes several concrete, differentiating predictions: a multiplicative log-periodic modulation in the SGWB spectrum, a restricted parameter band ε∈[0.04,0.18] and b∈[1.7,2.8] for the walking-technicolor realization, a count of observable oscillation periods across the LISA band, and a scaling estimate for oscillatory matched-filter SNR. These are tied to identifiable observables and could be falsified by SGWB data that favor a smooth baseline over the proposed modulated template, or that exclude the predicted amplitude-period region. The predictions are quantitative and operationally meaningful because the main target is in an observational band explicitly discussed as accessible to LISA-like experiments. The dark-matter mass-shift prediction is much weaker experimentally, but it is secondary. The main limitation preventing a 5 is that the paper does not give a fully instrument-specific forecast with realistic noise/background systematics, nor does it state formal falsification criteria in a dedicated section; instead it provides a promising but simplified detectability estimate.
The paper is generally well organized and readable at a high level: the section flow is sensible, the main claimed result is easy to identify, and the phenomenological message is clear. A graduate-level reader can follow the broad argument without much difficulty. However, several clarity issues materially limit confidence. The manuscript overclaims in the abstract relative to the body, especially regarding a 'full convolution calculation' and the strength of quantitative justification. There is also notation/slippage: the treatment of Π as sometimes a full unequal-time correlator and sometimes effectively a k-only spectrum is not always flagged, and Eq. (16)–(17) introduces an apparent inconsistency in how the renormalization factor R_F feeds into the dark-matter mass relation. Some steps that are central to the argument—such as why the Green's-function/time integrals cannot distort the log-periodic structure beyond percent level, and how the WTC parameters concretely map to ε and b—are presented more as plausibility arguments than as fully transparent derivations. Because there is both material overclaim and term/symbol consistency slippage, clarity cannot exceed 3.
The core idea—propagating discrete scale invariance from a phase-transition source UETC into a log-periodic SGWB modulation and then embedding that in a walking-technicolor UV completion—is a genuinely interesting synthesis. The work is not merely restating standard first-order phase-transition GW phenomenology: its novelty lies in linking DSI, source correlators, observable spectral oscillations, and a concrete hidden-sector realization. The paper also attempts to connect the GW feature to an auxiliary dark-matter consequence. That said, the manuscript itself cites prior work on log-periodic GW backgrounds in nonstandard inflation/beyond-Einstein settings, and the DSI concept is of course established. The new contribution is therefore more a novel application/synthesis with claimed predictive consequences than a wholly new mathematical structure or mechanism. Because some steps in the UV completion remain heuristic rather than deeply derived, a 4 is more appropriate than a 5.
The paper is structured coherently and the intended logic is followable: it introduces a DSI ansatz for the source UETC, argues that short source correlation time allows multiplicative transfer of the modulation to the SGWB spectrum, and then proposes a walking-technicolor realization. However, the central argument is not fully developed at the level required for completeness. The core factorization result is presented as a theorem, but the derivation is only heuristic: the delta-function approximation in Eq. (9), the transition from Eq. (10) to Eq. (11), and the claim that Green-function oscillations cannot induce additional log-periodic structure are asserted rather than demonstrated under explicit conditions. Because this is the paper's main result, that gap is structural. Beyond that, several key quantities are insufficiently defined for reproduction, and important boundary/validity conditions are handled only partially. The paper does note the regime β/H* ≳ 10 and comments on failure for slower transitions, which is good, but it does not treat edge cases such as incomplete log-period coverage, finite detector response, non-delta UETC kernels, or sensitivity to deviations from the assumed separable form Π=S·F. The WTC completion is also incomplete: the modulation added to the potential is specified, but the chain from that potential to the quoted numerical prediction band for ε and b is not worked out. Since the paper explicitly claims that this chain is complete and quantitatively justified, that unmet goal further lowers completeness.
This submission presents a novel theoretical framework connecting discrete scale invariance in first-order phase transitions to log-periodic modulations in the stochastic gravitational-wave background, with a concrete walking technicolor UV completion. The work makes several sharp, falsifiable predictions that are directly testable with LISA observations. However, there are significant mathematical gaps in the central derivations that prevent the claimed quantitative rigor from being fully established. The mathematical specialists identified multiple high-risk derivation steps, particularly the factorization theorem (Eqs. 10-11) where the Green's function averaging argument is asserted rather than demonstrated, and the WTC convolution calculation (Eqs. 27-30) where the propagator factorization and amplitude mapping contain unjustified approximations. The internal consistency is also compromised by tensions such as the numerical fit in Figure 1(b) showing r=0.81±0.04 versus the analytic prediction r=1.00, representing a ~5σ discrepancy that undermines the percent-level accuracy claims. Despite these mathematical deficiencies, the work succeeds in proposing a distinctive observational template with clear parameter ranges (ε∈[0.04,0.18], b∈[1.7,2.8]) that occupy the high-SNR region of LISA's detectability plane, making it scientifically valuable and highly falsifiable.
Strengths
- +Sharp, falsifiable predictions with explicit parameter ranges and LISA detectability forecasts
- +Novel synthesis connecting discrete scale invariance, phase transition dynamics, and gravitational wave observations
- +Clear multiplicative template structure that preserves the log-periodic modulation through linear response theory
- +Concrete walking technicolor UV completion that naturally provides the required scale invariance breaking
- +Well-organized presentation with explicit approximation control and quantified error estimates
Areas for Improvement
- -Derive the factorization theorem rigorously rather than relying on qualitative Green's function averaging arguments
- -Provide complete derivation of the WTC convolution calculation, especially the propagator amplitude mapping in Eqs. 27-30
- -Resolve the internal tension between analytical predictions and numerical demonstrations (Figure 1b shows r=0.81 vs predicted r=1.00)
- -Strengthen the connection between technidilaton potential modulation and gauge field propagator response with explicit perturbation theory
- -Derive the freeze-in dark matter mass shift formula from first principles rather than asserting the quadratic dependence
Log-Periodic Signatures from Discrete Scale Invariance in the Stochastic Gravitational-Wave Background Walking Technicolor as a Concrete Ultraviolet Completion Jill F. Rankin Independent Researcher jill.rankin@g.austincc.edu May 2026(preprint) Abstract We show that discrete scale invariance (DSI) in the anisotropic stress tensor during a first-order cosmological phase transition imprints a clean, multiplicative log-periodic modulation on the stochastic gravitational-wave background (SGWB). Under the physically motivated short-correlation-time approximation (τ corr H ∗ ≪1, satisfied forβ/H ∗ ≳10), the DSI modulation factorizes from the source unequal-time correlator to the observable energy-density spectrum at the percent level, yielding Ω GW (f ) = Ω 0 GW (f ) 1 + ε cos 2π ln(f/f ∗ ) lnb
- φ 0 , with modulation amplitudeε≪1 and discrete scaling ratiob >1. Matched-filter detectability of the oscillatory component scales asSNR osc ≃ ε SNR baseline p N periods , whereN periods =ln(f max /f min )/ lnbis the number of complete log-periods in the detector band, giving a useful enhancement over the naive ε suppression. As a concrete ultraviolet completion we embed the required DSI within walking technicolor (WTC), a strongly coupled hidden-sector gauge theory that (i) naturally provides approximate continuous scale invariance broken to DSI by a small periodic modulation of the technidilaton potential, and (ii) produces a strong first-order phase transition already known to generate LISA-detectable gravitational waves. A full convolution calculation shows that the DSI propagates from the technidilaton potential to the observable SGWB with errors≲1%. The WTC parameter space predictsε ∈[0.04,0.18],b ∈[1.7,2.8], which occupies the high-SNR region of the LISA detectability plane, turning the forecast into a sharp, falsifiable prediction. We also derive an accompanying fractional shiftm ψ /m 0 ψ = (1 +ε 2 /2) −1 in the dark-matter freeze-in relic mass. All approximations are quantitatively justified.
Contents 1 Introduction3 2 Gravitational-Wave Tensor Power Spectrum4 3 Discrete Scale Invariance in the Source UETC5 3.1 DSI ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5 3.2 Factorization theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5 4 Observable Signatures6 4.1 DSI-modulated energy-density spectrum . . . . . . . . . . . . . . . . . . .6 4.2 Quadratic observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6 4.3 Matched-filter detectability . . . . . . . . . . . . . . . . . . . . . . . . . . .8 5 Ultraviolet Completion: Walking Technicolor10 5.1 Phase-transition parameter space . . . . . . . . . . . . . . . . . . . . . . . 10 5.2 Engineering discrete scale invariance . . . . . . . . . . . . . . . . . . . . . 10 5.3 Convolution for the UETC . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 5.4 WTC predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 6 Discussion11 7 Conclusions12 2
1 Introduction The stochastic gravitational-wave background (SGWB) from first-order cosmological phase transitions is among the most promising observational targets for current and next-generation gravitational-wave detectors. The Laser Interferometer Space Antenna (LISA) [1] will be sensitive to phase transitions occurring at temperaturesT ∗ ∼10–10 4 GeV , covering a broad class of beyond-Standard-Model (BSM) scenarios. Pulsar-timing arrays (PTAs) have now reported evidence for a gravitational-wave background at nano-hertz frequencies [3–5], with spectra consistent with — though not yet uniquely identified as — a cosmological phase-transition origin. In this environment, spectral features that go beyond the smooth envelope predicted by conventional calculations take on special importance: they carry direct information about the microphysics of the transition and the nature of any BSM sector responsible for it. Standard calculations of the SGWB from a first-order phase transition predict a broad-band spectrum shaped by three source contributions — bubble collisions [8], sound waves [6,7], and magneto-hydrodynamic turbulence [9] — each with a characteristic broken power-law profile. A variety of beyond-standard effects can modify this picture: strong supercooling can sharpen the bubble-collision peak [8]; non-runaway walls alter the sound-wave contribution [6]; and non-equilibrium dynamics can generate additional log contributions [7]. However, none of these mechanisms generically produces a coherent log-periodic oscillation superimposed on the spectrum. Discrete scale invariance (DSI) is the symmetry that does. A system is said to possess DSI with ratiob >1 if it is invariant only under the discrete rescalingx→ b n xfor integer n, rather than under all continuous dilations [10]. DSI arises in hierarchical lattice models, fractal structures, iterated-function-system attractors, and — crucially for our purposes — near-conformal gauge theories with explicit periodic modulations. Its universal observable consequence is a log-periodic correction to any power-law observable, F (x) = x D 0 1 + A cos 2π lnx lnb
- φ ,(1) arising from complex scaling dimensionsD n =D 0 ±2πin/ lnbin the spectrum of the dilatation operator [10]. DSI and its signatures have been studied extensively in condensed- matter physics [10], financial mathematics [11], seismology, and fractal geometry, but its imprint on the SGWB has received comparatively little attention. Log-periodic features in the SGWB have been discussed in the context of non-standard inflationary scenarios and beyond-Einstein-gravity models [12]. In this paper we pursue a more direct route: we show that DSI in the anisotropic stress tensor of a first-order phase transition itself imprints a multiplicative log-periodic modulation on the observable SGWB. The mechanism operates at the level of the source unequal-time correlator (UETC) and is not specific to any particular BSM sector. The key technical result is a factorization theorem: under the physically well-motivated short- correlation-time approximation, valid for all realistic first-order phase transitions with β/H ∗ ≳10, the DSI modulation passes through the double time-integral of the tensor power spectrum unchanged, at the percent level. As a concrete ultraviolet (UV) completion we realize the required DSI within walking technicolor (WTC) [13], a strongly coupled hidden-sector gauge theory in the near- conformal regime. Two features make WTC an ideal host for DSI: (i) walking dynamics naturally provide approximate continuous scale invariance over a wide range of energies, 3
which can be broken to DSI by a small periodic modulation of the technidilaton effective potential — motivated by holographic models with periodic warp factors and by RG-group limit-cycle structure near the quasi-fixed point; and (ii) the WTC phase transition is already known to generate LISA-detectable gravitational waves [13], placing the DSI- modulated prediction squarely in the observable band without requiring any new tuning. We perform an explicit convolution calculation that traces the DSI modulation from the technidilaton potential through the UETC to the observable Ω GW (f), with every approximation quantified. The resulting prediction is sharp: the WTC parameter space maps onto a specific bandε ∈[0.04,0.18],b ∈[1.7,2.8] in the DSI amplitude–ratio plane, which overlaps the high-SNR region of the LISA detectability forecast. A matched-filter search for the log-periodic template provides an optimal discriminant; the associated dark-matter mass shift and resonance contrast offer independent corroborating observables. A companion paper [17] demonstrates the log-periodic spectral imprinting mechanism in a controlled one-dimensional electromagnetic cavity using finite-difference time-domain (FDTD) simulations, providing a numerical proof of concept independent of gravitational- wave physics. The theoretical framework of dynamic mode-accessibility engineering that unifies both papers is developed in Ref. [18]. The paper is structured as follows. Section 2 reviews the tensor power spectrum and sets up notation. Section 3 states the DSI ansatz and derives the factorization theorem. Section 4 works out the observable signature, quadratic relic corrections, and matched-filter detectability. Section 5 develops the WTC UV completion. Section 6 discusses robustness, distinguishability, and extensions. Section 7 summarizes the main results. Throughout we use natural units c =ℏ = k B = 1 and metric signature (−, +, +, +). 2 Gravitational-Wave Tensor Power Spectrum Tensor metric perturbationsh ij in a flat Friedmann–Lemaˆıtre–Robertson–Walker (FLRW) background satisfy h ′′ ij (k,η) + 2Hh ′ ij (k,η) + k 2 h ij (k,η) = 16πGa 2 (η) Π TT ij (k,η),(2) where primes denote derivatives with respect to conformal timeη,H=a ′ /a,a(η) is the scale factor, and Π TT ij is the transverse-traceless projected anisotropic stress sourced by the phase transition. The two-point function of the source defines the unequal-time correlator,
Π TT ij (k,η) Π TT∗ ij (k ′ ,η ′ ) = (2π) 3 δ (3) (k−k ′ ) Π(k,η,η ′ ),(3) where statistical isotropy has been used to write Π as a function ofk=|k|. Solving Eq. (2) with the retarded Green’s function G k (η,η ′ ) gives the tensor power spectrum, 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 ).(4) The fractional GW energy density per logarithmic frequency interval, referred to the critical density today, is [7] Ω GW (k,η)≃ k 3 12a 2 H 2 P h (k,η),(5) valid for sub-horizon modesk ≫ H. In what follows we work in terms of the observed frequency f = k/(2πa 0 ). 4
3 Discrete Scale Invariance in the Source UETC 3.1 DSI ansatz We assume that the source UETC carries a discrete scale invariance with ratiob >1 and amplitude ε≪ 1: Π(k,η,η ′ ) = Π 0 (k,η,η ′ ) 1 + ε cos 2π ln(k/k ∗ ) lnb
- φ 0 ,(6) where Π 0 is the smooth DSI-free UETC,k ∗ is a reference scale, andφ 0 is an overall phase. Equation (6) is the leading-order expression consistent with invariance underk → b n kfor integern; the log-periodic modulation is the real part of the complex power-law correction associated with complex scaling dimensions [10]. 3.2 Factorization theorem For a first-order phase transition the UETC naturally separates into macroscopic (slow) and microscopic (fast) parts, Π(k,η,η ′ ) = S(η,η ′ )F (k,η− η ′ ),(7) whereS(η,η ′ ) describes the bulk source evolution andF(k,∆η) encodes temporal cor- relations. This form is standard in the envelope approximation and the sound-shell model [6, 8]. The phase-transition source decorrelates on the bubble radius/wall-speed timescale τ corr ∼ R ∗ ∼ v w /β, giving τ corr H ∗ ∼ v w β/H ∗ ≪ 1for β/H ∗ ≳ 10.(8) In this limit F (k, ∆η) is sharply peaked at ∆η = 0, and to leading order F (k,η− η ′ )≃ F (k)δ(η− η ′ ) +O(τ corr H ∗ ).(9) We decompose the spectral kernel asF(k) =C(k)F 0 (k), whereF 0 (k) is the smooth baseline kernel andC(k) carries the DSI modulation. Substituting into Eq. (4) and performing the η 2 integral using the delta function, 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) For sub-horizon modes the Green’s function satisfiesG k (η,η ′ )∼ sin[k(η− η ′ )]/k, which depends onkonly through an overallk −1 factor and oscillatory terms that average to a k-independent contribution on the relevant timescales. It therefore cannot generate log- periodic structure ink. Any such structure present inC(k) passes through the remaining time integral unchanged: P h (k,η) = C(k)P 0 h (k,η) 1 +O(τ corr H ∗ ) ,(11) whereP 0 h is the tensor power spectrum evaluated with the smooth UETC Π 0 . This is the factorization theorem: the DSI modulation transfers multiplicatively from the source to the tensor power spectrum. Higher-order corrections are suppressed by powers ofτ corr H ∗ and are at the per-cent level for the realistic range β/H ∗ ≳ 10–100 (see Table 1). 5
Table 1: Relative error of the factorization approximation as a function ofβ/H ∗ , forε= 0.1, v w = 1. Both the short-correlation-time correction and the convolution-factorization correction (Sec. 5.3) are shown. β/H ∗ τ corr H ∗ Relative error 100.10≲ 13% 1000.01≲ 1.3% 10000.001≲ 0.13% 4 Observable Signatures 4.1 DSI-modulated energy-density spectrum Combining Eq. (6) with the factorization (11) and using Eq. (5), the observable GW energy-density spectrum is Ω GW (f ) = Ω 0 GW (f ) 1 + ε cos 2π ln(f/f ∗ ) lnb
- φ 0 .(12) The fractional residual (Ω GW − Ω 0 GW )/Ω 0 GW is a pure sinusoid inlnfwith period ∆lnf= lnb, amplitude ε, and phase φ 0 . For the smooth baseline Ω 0 GW we adopt the standard sound-wave contribution [6, 7], 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 ),(13) S sw (f ) = f f sw 3 7 4 + 3(f/f sw ) 2 7/2 ,(14) with peak frequency f sw = 1.9× 10 −5 Hz 1 v w β H ∗ T ∗ 100 GeV g ∗ 100 1/6 .(15) Hereαis the transition strength,κ sw is the fraction of the released latent heat converted to fluid bulk motion,g ∗ is the number of relativistic degrees of freedom atT ∗ , andv w is the wall velocity. We set the DSI reference scalef ∗ ∼ f sw . Figure 1 shows the spectrum, residual, and log-period spacing for representative parameter values. 4.2 Quadratic observables Writing Ω GW (k) = Ω 0 (k)[1 +ε cos(2πu)] withu=ln(k/k ∗ )/ lnb, squaring, and averaging over complete log-periods gives the renormalization factor R F ≡ ⟨Ω 2 GW ⟩ ⟨Ω 2 0 ⟩ = 1 + ε 2 2 +O(ε 4 ).(16) This factor enters every observable that is quadratic in Ω GW . In particular, the freeze-in dark-matter mass [15] required to reproduce the observed relic abundance shifts as m ψ = m 0 ψ R F = m 0 ψ 1 + ε 2 /2 ,(17) 6
10 3 10 2 10 1 10 0 10 1 Frequency f [Hz] 10 13 10 12 10 11 10 10 10 9 10 8 10 7 h 2 GW ( f ) (a) = 0.10,b = 2.0, 0 = 0 Smooth baseline 0 GW DSI-modulated GW 10 3 10 2 10 1 10 0 10 1 Frequency f [Hz] 0 + R ( f ) / 0 (b) f * r = 0.81 ± 0.04 (analytic: r = 1.00) ( GW 0 )/ 0 Fixed-period fit: cos(2ln(f/f * )/ln b) 10 3 10 2 10 1 10 0 10 1 Frequency f [Hz] ln f = ln b f * (c) log-period spacing Figure 1: Log-periodic modulation of the SGWB. (a) Power spectrumh 2 Ω GW (f) (orange, solid) and smooth baselineh 2 Ω 0 GW (f) (blue, dashed) versus frequency, forε= 0.1,b= 2, φ 0 = 0. (b) Fractional residualR(f)≡[Ω GW (f)−Ω 0 GW (f)]/Ω 0 GW (f), showing the clean sinusoidal oscillation inlnfpredicted by Eq. (12). The orange curve is the fixed-period cosine fit. (c) Log-period spacing: vertical ticks mark frequencies where the modulation peaks (cos = +1), equally spaced by ∆ lnf = lnb. The reference scale f ∗ is indicated. 7
a downward fractional shift ofε 2 /2≃0.08%–1.6% across the WTC band. The resonance- to-antiresonance contrast — the ratio of maximum to minimum of Ω GW over one log-period — is Γ res Γ anti
1 + ε 1− ε 2 ,(18) ranging from 1.17 to 1.96 across ε∈ [0.04, 0.18]. 4.3 Matched-filter detectability The oscillatory component of the signal is δΩ GW (f ) = ε Ω 0 GW (f ) cos 2π ln(f/f ∗ ) lnb
- φ 0 .(19) The squared matched-filter signal-to-noise ratio for a search with fixed template parameters (b,φ 0 ) is SNR 2 osc = Z [δΩ GW (f )] 2 σ 2 (f ) d lnf,(20) whereσ(f) is the noise level of the experiment. Using⟨cos 2 ⟩= 1/2 over complete log-periods and the definition of the baseline SNR this reduces to SNR osc ≃ ε SNR baseline p N periods ,(21) with N periods = ln(f max /f min ) lnb .(22) For LISA with effective band [f min ,f max ] = [10 −4 , 1]Hz(ln(f max /f min )≈9.21) and baseline SNR baseline = 20, the factor p N periods ranges from 3.6 atb= 2 to 2.4 atb= 5, providing meaningful amplification of the intrinsically small ε signal. The detectability plane (bvs.ε) is shown in Figs. 2 and 3, with SNR contours at {1, 5, 10, 20} and the WTC prediction band overlaid. 8
23456 Discrete scaling factor b 0.01 0.05 0.10 0.50 Modulation amplitude SNR base = 20; LISA band [10 4 , 1] Hz Forecast SNR contours for DSI oscillations in the SGWB WTC [0.04, 0.18] b[1.7, 2.8] SNR=1 SNR=5 SNR=10 SNR=20 Figure 2: Forecast matched-filter SNR contours for the DSI oscillatory component in the (b,ε) plane, assumingSNR baseline = 20 and a LISA frequency band [10 −4 ,1]Hz. Contours are shown atSNR osc = 1,5,10,20. The orange shaded region is the WTC prediction band ε∈ [0.04, 0.18], b∈ [1.7, 2.8]. The model populates the high-SNR portion of the plane. 23456 Discrete scaling factor b 0.01 0.05 0.10 0.50 Modulation amplitude SNR base = 20; LISA band [10 4 , 1] Hz Forecast SNR contours with LISA 5 sensitivity and WTC prediction WTC [0.04, 0.18] b[1.7, 2.8] LISA 5 threshold (SNR osc = 5) LISA accessible (SNR osc 5) SNR=1 SNR=5 SNR=10 SNR=20 Figure 3: Same as Fig. 2, with the approximate LISA 5σdetection threshold (blue line, SNR osc = 5 forSNR baseline = 20) and LISA-accessible region (purple shading) overlaid. The WTC prediction band lies entirely within the LISA-accessible region. 9
5 Ultraviolet Completion: Walking Technicolor 5.1 Phase-transition parameter space We adopt the benchmark large-N f QCD realization of walking technicolor [13]. The hidden sector is anSU(N c ) gauge theory withN f fundamental techniquarks in the near-conformal windowN f /N c ≳4–8. Near this window the gauge coupling walks — evolves slowly over many decades of energy scale — providing approximate scale invariance; the theory is attracted toward a quasi-fixed point (the Banks–Zaks fixed point) before condensing at Λ TC . Benchmark values areN c = 8,N f = 8, technidilaton decay constantF φ ≈1TeV, with an ultra-supercooled first-order phase transition (FOPT) characterized by [13] α≈ 0.73–0.83, β/H ∗ ≈ 100–1000, v w ≈ 1.(23) These give a sound-wave-dominated SGWB with h 2 Ω 0 GW (f peak )∼ 10 −9 –10 −8 at f peak ∼ 0.1–10 Hz,(24) comfortably within the LISA sensitivity band [1, 2], and satisfy τ corr H ∗ ≲ 0.01≪ 1. 5.2 Engineering discrete scale invariance Walking dynamics provide approximate continuous scale invariance: the technidilatonφis the pseudo-Nambu–Goldstone boson of the approximate scale symmetry, and its effective potential is of Coleman–Weinberg form [14]. DSI arises when this symmetry is broken from continuous to discrete. We realize this by adding a small explicit periodic modulation, V (φ) = V CW (φ) 1 + ε f cos 2π ln(φ/φ 0 ) lnb 0 , ε f ≪ 1, b 0
1,(25) whereV CW is the Coleman–Weinberg potential [14]. Such modulations are motivated by two independent sources. First, in AdS/CFT dual descriptions of near-conformal dynamics, periodic warp factors in the extra dimension generate exactly this type of potential modulation in the 4D effective theory. Second, near the quasi-fixed point the RGβ-function has no zero; instead the integrated RG flow over one cycle inφ-space is zero, corresponding to a limit cycle rather than a fixed point — the RG-flow realization of DSI [10]. The modulation in Eq. (25) induces a multiplicative log-periodic correction to the gauge-field propagator at momentum q: D(q; ∆η) = D 0 (q; ∆η) [1 + δ(q)], δ(q) = ε f cos 2π ln(q/q ∗ ) lnb 0
- φ 0 .(26) 5.3 Convolution for the UETC The transverse-traceless anisotropic stress is bilinear 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),(27) 10
whereP TT projects onto the transverse-traceless sector. Expanding to linear order inε f and retaining only the cross-term (the self-term is O(ε 2 f )), Π(k)⊃ Z d 3 p (2π) 3 P TT D 0 (p)D 0 (|k−p|) δ(p) + δ(|k−p|) .(28) The relevant momenta satisfyq ∼ β/v w . The baseline propagatorD 0 (q) is sharply peaked with relative width ∆q/q ∼ τ corr H ∗ ≪1. The logarithmic derivative ofδ(q) satisfies |d lnδ/d lnq|= 2π/ lnb 0 ≲13 forb 0 ≥1.5, so over the support ofD 0 the modulation varies by at most 13ε f (∆q/q)∼ 13ε f τ corr H ∗ . Hence to high accuracy δ(p) = δ(k) [1 +O(ε f τ corr H ∗ )],(29) and the cross-term factorizes as 2δ(k)×Π 0 (k,η,η ′ ). Absorbing the factor of 2 into the baseline normalization yields Π(k,η,η ′ ) = Π 0 (k,η,η ′ ) 1 + ε cos 2π ln(k/k ∗ ) lnb
- φ 0 1 +O(ετ corr H ∗ ) ,(30) withε=ε f andb=b 0 at leading order. For WTC benchmark parametersβ/H ∗ ≳100 the relative correction is≲ 1% (Table 1). 5.4 WTC predictions Combining Eq. (30) with the short-correlation-time factorization theorem of Sec. 3.2, the DSI modulation propagates multiplicatively to the observable SGWB, exactly recovering Eq. (12). The WTC parameter space [13], spanned byF φ ≈1TeV, Λ ETC ∼ 5–10TeV, and soft masses m p ∼ 1–100 GeV, maps onto ε∈ [0.04, 0.18], b∈ [1.7, 2.8].(31) This band is shown in Figs. 2–3 and overlaps the high-SNR region of the LISA detectabil- ity forecast; for a baselineSNR baseline = 20 the matched-filter SNR for the oscillatory component satisfies SNR osc ≳ 5 over most of the band. The dark-matter mass shift (17) evaluates tom ψ /m 0 ψ = 1/(1 +ε 2 /2)∈[0.984,0.999] across the WTC band, a downward fractional shift of 0.1%–1.6%. While small, this is in principle measurable through precision relic-density determinations combined with independent constraints on the phase transition parameters. The chain from the WTC Lagrangian to the observable Ω GW (f) is now complete: every step has been individually justified and the cumulative relative error is below 2% for β/H ∗ ≳ 100. 6 Discussion Robustness of the factorization. The key approximation is the short-correlation-time limitτ corr H ∗ ≪1. Its validity requiresβ/H ∗ ≫1, i.e. a transition that completes rapidly compared to the Hubble time. This is satisfied for the WTC benchmark (β/H ∗ ∼100–
- and is a generic property of strong first-order transitions. Slow transitions with β/H ∗ ≲10 would require higher-order corrections, which can be computed systematically as an expansion inτ corr H ∗ . The separate factorization condition|d lnδ/d lnq|·(∆q/q)≪1 is equally well controlled and introduces no additional tuning. 11
Distinguishability from other spectral features. The log-periodic modulation (12) produces a coherent, phase-stable sinusoid inlnf, persisting overN periods ∼6–13 full oscillations across the LISA band forb∈[1.7,2.8]. This is qualitatively distinct from other known spectral features: (i) The kink at the crossover from sound-wave to turbulence domination is a single discontinuity in the spectral slope, not a periodic oscillation. (ii) A sharp bubble-collision peak is a feature of limited frequency extent, not a multi-period sinusoid. (iii) Stochastic backgrounds from astrophysical sources produce spectra that are smooth inlnfto high accuracy. A likelihood-ratio test between the smooth template Ω 0 GW and the DSI-modulated template (12) provides the optimal discriminant. The three-parameter family (b,ε,φ 0 ) can be mapped from the data by standard matched-filter techniques [2]. Parameter degeneracies. The phaseφ 0 merely shifts the oscillation inlnfand does not affect detectability;εandbcan be independently constrained from the oscillation depth and period respectively. The frequency resolution needed to resolve individual oscillations is ∆f/f ∼ lnb/(2π); forb= 2 this is ∆f/f ≈0.11, well within LISA’s capabilities over its four-year nominal mission. Alternative UV completions. The factorization result and the observable template (12) are model-independent consequences of DSI in the UETC, requiring onlyτ corr H ∗ ≪1. Walking technicolor is one concrete realization; other BSM models with approximate conformal symmetry and explicit periodic modulations — extended Higgs sectors with Coleman–Weinberg potentials modified by threshold corrections, Randall–Sundrum–type models with periodic radion potentials, or clockwork models [16] — are equally valid candidates and will produce the same spectral template with different (b,ε) values. A detection of log-periodic oscillations in the SGWB would uniquely fixbandε, allowing discrimination among UV completions. Multi-messenger signatures. Beyond gravitational waves, the DSI in the WTC potential generates log-periodic modulations in the technidilaton production rate and hence in the energy density of any dark-radiation component coupled to the hidden sector. Furthermore, the dark-matter mass shift (17) can be tested by combining future precision cosmological measurements of the matter power spectrum (which constrainsm ψ ) with direct phase-transition reconstructions from the GW signal (which constrain ε). 7 Conclusions We have demonstrated that discrete scale invariance in the anisotropic stress tensor of a first-order cosmological phase transition imprints a clean, multiplicative log-periodic modulation on the stochastic gravitational-wave background. The main results are: 1. Factorization theorem. In the physically motivated short-correlation-time limit (τ corr H ∗ ≪1, satisfied forβ/H ∗ ≳10), the DSI modulation passes from the source UETC to the observable Ω GW (f) at the per-cent level:P h =C(k)P 0 h [1 +O(τ corr H ∗ )]. 2.Universal spectral template. The observable signature is Ω GW = Ω 0 GW [1 + ε cos(2π ln(f/f ∗ )/ lnb+φ 0 )] — a sinusoid inlnfsuperimposed on the smooth baseline, fully characterized by three parameters (ε,b,φ 0 ). 12
3.Matched-filter detectability.SNR osc ≃ ε SNR baseline p N periods , withN periods = 6– 13 oscillations in the LISA band. 4.Walking technicolor UV completion. An explicit convolution calculation con- firms that the WTC potential modulation propagates to the SGWB with≲1% error. The WTC prediction bandε ∈[0.04,0.18],b ∈[1.7,2.8] sits in the high-SNR osc region of the LISA detectability plane. 5.Dark-matter mass shift.m ψ /m 0 ψ = (1 +ε 2 /2) −1 , a 0.1%–1.6% downward shift across the WTC band, providing an independent phenomenological handle. A non-detection by LISA would place sharp upper limits onεas a function ofb, directly constraining the allowed parameter space for near-conformal BSM phase transitions. A detection would simultaneously reveal the discrete scaling ratio, the DSI amplitude, and the phase of the modulation, providing a unique window into the self-similar structure of the hidden-sector dynamics. The log-periodic template (12) is simple, well-defined, and implementable in any LISA data-analysis pipeline via standard matched-filter methods. Acknowledgments The author thanks the gravitational-wave and beyond-Standard-Model communities for stimulating discussions. No external funding was received for this work. References [1]P. Amaro-Seoane et al. (LISA Collaboration), “Laser Interferometer Space Antenna,” (2017) [arXiv:1702.00786]. [2]C. Caprini et al., “Science with the space-based interferometer eLISA. II: Grav- itational waves from cosmological phase transitions,” JCAP 1604, 001 (2016) [arXiv:1512.06239]. [3]G. Agazie et al. (NANOGrav Collaboration), “The NANOGrav 15 yr Data Set: Evidence for a Gravitational-Wave Background,” Astrophys. J. Lett. 951, L8 (2023) [arXiv:2306.16213]. [4] D. J. Reardon et al. (PPTA Collaboration), “Search for an Isotropic Gravitational- Wave Background with the Parkes Pulsar Timing Array,” Astrophys. J. Lett. 951, L6 (2023) [arXiv:2306.16215]. [5] J. Antoniadis et al. (EPTA Collaboration), “The second data release from the European Pulsar Timing Array: V. Implications for massive black holes, dark matter and the early Universe,” Astron. Astrophys. 678, A50 (2023) [arXiv:2306.16227]. [6]M. Hindmarsh and M. Hijazi, “Gravitational waves from first-order cosmological phase transitions in the Sound Shell Model,” JCAP 12, 062 (2019) [arXiv:1909.10040]. [7] D. G. Figueroa, A. Florio, F. Guedes, and F. Torrenti, “Cosmological phase tran- sitions: From theory to gravitational wave phenomenology,” JCAP 03, 027 (2021) [arXiv:2010.00972]. 13
[8]J. R. Espinosa, T. Konstandin, J. M. No, and G. Servant, “Energy Budget of Cosmological First-Order Phase Transitions,” JCAP 06, 028 (2010) [arXiv:1004.0691]. [9]C. Caprini and R. Durrer, “Gravitational waves from stochastic relativistic sources: Primordial turbulence and magnetic fields,” Phys. Rev. D 74, 063521 (2006) [arXiv:astro-ph/0603476]. [10]D. Sornette, “Discrete scale invariance and complex dimensions,” Phys. Rep. 297, 239 (1998) [arXiv:cond-mat/9707012]. [11]D. Sornette, Critical Phenomena in Natural Sciences: Chaos, Fractals, Self- Organization and Disorder: Concepts and Tools, 2nd ed. (Springer, 2006). [12]G. Calcagni and S. Kuroyanagi, “Log-periodic gravitational-wave background beyond Einstein gravity,” Class. Quantum Grav. 41, 015031 (2024) [arXiv:2308.05904]. [13]M. Miura, K. Ohnishi, T. Sawanaka, and K. Yamawaki, “Gravitational waves from walking technicolor,” (2019) [arXiv:1811.05670]. [14] S. Coleman and E. Weinberg, “Radiative corrections as the origin of spontaneous symmetry breaking,” Phys. Rev. D 7, 1888 (1973). [15]L. J. Hall, K. Jedamzik, G. March-Russell, and S. M. West, “Freeze-in production of FIMP dark matter,” JHEP 03, 080 (2010) [arXiv:0911.1120]. [16]G. F. Giudice, Y. Kats, M. McCullough, R. Torre, and A. Urbano, “Clockwork/linear dilaton: structure and phenomenology,” JHEP 06, 098 (2018) [arXiv:1711.08437]. [17] J. F. Rankin, “Log-periodic spectral hierarchies in a boundary-driven electromagnetic cavity: evidence from FDTD simulations,” preprint (2026). [18]J. F. Rankin, “A phenomenological framework for mode-accessibility engineering in structured field environments,” preprint (2026). 14
⚑Derivation Flags (18)
- highEq. (10) → Eq. (11) — Argument that sub-horizon Green-function oscillations 'average to a k-independent contribution' is qualitative; no explicit averaging procedure, window, or estimate is provided to bound induced k-dependence relative to the DSI modulation.
If wrong: Even if C(k) factorizes, additional k-dependent phase/amplitude effects from the time integral could mimic or distort log-periodic structure, undermining the claim that only C(k) carries the oscillation and that transfer is multiplicative at percent level.
- highEq. (11) / Sec. 3.2 factorization theorem — The claim that the Green-function oscillatory terms average to a k-independent contribution and cannot affect log-periodic structure is asserted rather than derived. No bound is given for the residual k-dependent convolution terms.
If wrong: The central factorization theorem P_h(k)=C(k)P_h^0(k)[1+O(τ_corr H_*)] would not be established, and the observable template Eq. (12) would not follow with the claimed percent-level accuracy.
- highEq. (17) — The freeze-in dark-matter mass shift m_ψ=m_ψ^0/(1+ε^2/2) is asserted from the quadratic factor R_F without deriving why the freeze-in relic abundance depends quadratically on Ω_GW or why the inverse scaling with mass follows.
If wrong: The claimed correlated dark-matter mass shift, one of the listed main results, is unsupported.
- highEq. (21) — The matched-filter scaling SNR_osc ≃ ε SNR_baseline sqrt(N_periods) is not derived from Eq. (20) under the standard total-baseline-SNR definition. Averaging cos^2 over the same frequency band would instead give a factor of approximately ε/√2 times the total baseline SNR, not an additional sqrt(N_periods) enhancement.
If wrong: The LISA detectability contours and the claim that the WTC band lies in a high-SNR region are quantitatively unreliable.
- highEq. (26) — Claim that a periodic modulation in the technidilaton potential induces D(q;Δη)=D0(q;Δη)[1+δ(q)] with δ(q) a log-periodic cosine is asserted without an explicit derivation (e.g., via perturbation theory in εf or RG arguments translating V(φ) modulation into momentum-space propagator modulation).
If wrong: The UV completion link from WTC potential modulation to a DSI-modulated UETC (and thus to ε,b predictions) is unsupported; the main model-specific quantitative band Eq. (31) would not be mathematically established.
- highEq. (27)-(30) — Convolution factorization step δ(p)=δ(k)[1+O(εf τcorr H*)] relies on D0(q) being sharply peaked and on identifying Δq/q with τcorr H*. The mapping between momentum width of D0 and temporal correlation time is asserted, and the estimate uses |d ln δ/d ln q| but treats δ(q) itself (which changes sign) as if its logarithmic derivative were uniformly bounded.
If wrong: The claim that Π(k) inherits a clean multiplicative cosine with ≤1% error (Eq. (30)) may fail; ε could be renormalized, b shifted, or harmonics generated, impacting both universality and WTC parameter forecasts.
- highEq. (31) — Mapping of WTC parameter space onto ε∈[0.04,0.18], b∈[1.7,2.8] is presented as a result without showing the functional dependence of ε,b on (Fφ,ΛETC,mp) or providing an explicit computation/scan method.
If wrong: The paper's sharp falsifiable quantitative prediction band and its claimed occupancy of the high-SNR region would be unsupported.
- highEq. (9) — Replacement of the fast correlation kernel by F(k)δ(η−η′)+O(τcorr H*) is asserted without showing normalization and how the error term scales in the double integral of Eq. (4) given the Green functions and a(η) factors.
If wrong: Eq. (10) and therefore the claimed multiplicative factorization Eq. (11) may be invalid; the DSI modulation could be smeared or altered by finite correlation time, affecting the main observable template Eq. (12).
- highEq. 10–11, Sec 3.2 (Green's-function step) — The claim that G_k(η,η') ~ sin[k(η−η')]/k contributes only an overall k^{-1} factor plus oscillatory terms that 'average to a k-independent contribution' is asserted without explicit calculation. The actual time integral ∫ dη_1 G_k^2(η,η_1) a^4(η_1) S(η_1,η_1) can in principle carry k-dependence beyond k^{-2}, and whether it preserves the multiplicative log-periodic structure of C(k) is the heart of the factorization claim.
If wrong: If the Green's-function convolution generates additional k-dependent structure that mixes with C(k), the factorization theorem (Eq. 11) fails and the observable template Ω_GW = Ω_0_GW [1 + ε cos(...)] (Eq. 12) — the paper's central prediction — would acquire distortions in amplitude and phase that are not bounded by τ_corr H_*.
- highEq. 25–26 (potential modulation → propagator modulation) — The step from a periodic modulation of the technidilaton potential V(φ) to a multiplicative log-periodic correction δ(q) on the gauge-field propagator is asserted but not derived. The functional form D(q;∆η) = D_0(q;∆η)[1+δ(q)] with δ(q) inheriting precisely the same ε_f, b_0 as V(φ) requires a nontrivial calculation in the strongly coupled WTC theory.
If wrong: If the propagator does not inherit the same log-periodic structure (or inherits it with different ε, b), the identification ε=ε_f, b=b_0 fails and the WTC prediction band (Eq. 31) is not derived from the WTC Lagrangian.
- highEq. 28–29, Sec 5.3 (convolution factorization) — Going from Π(k) ⊃ ∫ d^3p D_0(p) D_0(|k−p|)[δ(p)+δ(|k−p|)] to δ(p) ≈ δ(k) requires that D_0(p) be sharply peaked at p ≈ k. The argument that ∆q/q ∼ τ_corr H_* is stated without derivation; in a relativistic plasma the gauge-field propagator's momentum support is typically not narrow in this sense. Additionally, evaluating δ(|k−p|) at p≈k gives δ(0)·argument issues that are not addressed.
If wrong: If D_0 is not sharply peaked or if δ(|k−p|) does not reduce to δ(k), then ε ≠ ε_f and b ≠ b_0 in general — the WTC parameter map onto (ε,b) ∈ [0.04,0.18]×[1.7,2.8] (Eq. 31) loses its derivation, and the central LISA-falsifiability claim is undermined.
- highEqs. (27)-(30) — The convolution factorization is compressed and contains nontrivial unproved replacements: δ(p) and δ(|k-p|) are replaced by δ(k), although the integration variables need not be localized at the external momentum k; the derivative estimate uses |d ln δ/d ln q|, which diverges near zeros of δ; and the factor of 2 from the two linear cross-terms is said to be absorbed into the baseline while still setting ε=ε_f.
If wrong: The claimed UETC ansatz Eq. (30), the mapping ε=ε_f and b=b_0, and the WTC prediction band ε ∈ [0.04,0.18], b ∈ [1.7,2.8] are not mathematically established.
- mediumEq. (16) — The averaging result R_F=1+ε^2/2 assumes averaging over complete log-periods with no residual weighting from Ω_0(k), detector response, or integration limits. The text does not derive conditions under which the linear term averages to zero for the relevant observable.
If wrong: Quadratic-observable corrections may include nonzero O(ε) terms or different O(ε^2) coefficients, invalidating the universal renormalization factor used later.
- mediumEq. (7) — UETC separability Π(k,η,η′)=S(η,η′)F(k,η−η′) is stated as 'standard' without specifying conditions (stationarity, time-translation invariance, approximations such as envelope/sound-shell) under which it holds for the anisotropic stress in an expanding background.
If wrong: Without this separability, the subsequent delta-function reduction and clean extraction of C(k) in Eq. (10)-(11) may fail; modulation could mix with time dependence and not factorize.
- mediumEq. 16–17 (quadratic observables and DM mass shift) — The renormalization R_F = ⟨Ω_GW^2⟩/⟨Ω_0^2⟩ = 1+ε^2/2 is correct as a log-average identity, but its application to the freeze-in DM mass via m_ψ = m_0_ψ/R_F is asserted without showing that the relic-abundance integral is indeed quadratic in Ω_GW with a linear inversion to m_ψ. The freeze-in mechanism does not obviously have this structure.
If wrong: The DM mass-shift prediction (Eq. 17), advertised as a corroborating multi-messenger observable, would not actually follow from the framework.
- mediumEq. 20–21 (matched-filter SNR) — The reduction from the matched-filter integral (Eq. 20) to SNR_osc ≃ ε SNR_baseline √N_periods assumes σ(f)/Ω_0_GW(f) is approximately constant across the band so that the ⟨cos^2⟩=1/2 averaging factorizes cleanly. This is plausible near the peak but not derived; LISA noise rises rapidly off-peak.
If wrong: The √N_periods enhancement factor could be substantially smaller, weakening the detectability claim and shifting the WTC band's position relative to the LISA SNR=5 contour.
- mediumFigure 1(b) — fit residual r = 0.81 ± 0.04 vs analytic r = 1.00 — The figure caption reports a fitted amplitude ratio r=0.81±0.04 against the analytic prediction r=1.00, a ~20% discrepancy with quoted uncertainty 0.04 — a ~5σ tension between the analytic template and its own numerical demonstration. This is not discussed in the text.
If wrong: Either the analytic template's amplitude normalization is off by ~20%, or the numerical implementation has a systematic. Either way the percent-level error claim of the factorization theorem is in tension with this self-consistency check.
- mediumTable 1 — Percent-level 'relative error' entries are given, but the method producing 13%, 1.3%, 0.13% is not shown; it appears to assume linear scaling with τcorr H* without demonstrating the proportionality constant or definition of 'relative error'.
If wrong: Quantitative claims about 'percent-level' accuracy for β/H*≳10–100 become unsupported; downstream detectability forecasts and 'sharp falsifiable' assertions would have weaker mathematical backing.
Mathematically, the submission presents a coherent perturbative framework in which a log-periodic multiplicative modulation in the source UETC is intended to pass through to the observed SGWB spectrum as an equally clean multiplicative modulation. The algebraic manipulations once that multiplicative structure is assumed are consistent (e.g., the log-period definition, residual properties, and small-ε expansions). The work is also logically organized so that the main claims are traceable to specific equations. However, the key load-bearing derivations are only outlined. The factorization theorem (Eq. (11)) depends on separability (Eq. (7)), a delta-function correlator approximation (Eq. (9)), and a qualitative averaging argument about the Green function’s k-dependence; without explicit bounds in the actual double-integral (Eq. (4)), the claimed universality and percent-level accuracy are not yet mathematically established. The UV completion adds further unproven steps (notably Eq. (26) and the convolution reduction to Eq. (30)), and the quantitative prediction band (Eq. (31)) is stated without derivation. As written, the mathematical structure is plausible but insufficiently demonstrated for the central quantitative conclusions.
⚑Derivation Flags (18)
- highEq. (10) → Eq. (11) — Argument that sub-horizon Green-function oscillations 'average to a k-independent contribution' is qualitative; no explicit averaging procedure, window, or estimate is provided to bound induced k-dependence relative to the DSI modulation.
If wrong: Even if C(k) factorizes, additional k-dependent phase/amplitude effects from the time integral could mimic or distort log-periodic structure, undermining the claim that only C(k) carries the oscillation and that transfer is multiplicative at percent level.
- highEq. (11) / Sec. 3.2 factorization theorem — The claim that the Green-function oscillatory terms average to a k-independent contribution and cannot affect log-periodic structure is asserted rather than derived. No bound is given for the residual k-dependent convolution terms.
If wrong: The central factorization theorem P_h(k)=C(k)P_h^0(k)[1+O(τ_corr H_*)] would not be established, and the observable template Eq. (12) would not follow with the claimed percent-level accuracy.
- highEq. (17) — The freeze-in dark-matter mass shift m_ψ=m_ψ^0/(1+ε^2/2) is asserted from the quadratic factor R_F without deriving why the freeze-in relic abundance depends quadratically on Ω_GW or why the inverse scaling with mass follows.
If wrong: The claimed correlated dark-matter mass shift, one of the listed main results, is unsupported.
- highEq. (21) — The matched-filter scaling SNR_osc ≃ ε SNR_baseline sqrt(N_periods) is not derived from Eq. (20) under the standard total-baseline-SNR definition. Averaging cos^2 over the same frequency band would instead give a factor of approximately ε/√2 times the total baseline SNR, not an additional sqrt(N_periods) enhancement.
If wrong: The LISA detectability contours and the claim that the WTC band lies in a high-SNR region are quantitatively unreliable.
- highEq. (26) — Claim that a periodic modulation in the technidilaton potential induces D(q;Δη)=D0(q;Δη)[1+δ(q)] with δ(q) a log-periodic cosine is asserted without an explicit derivation (e.g., via perturbation theory in εf or RG arguments translating V(φ) modulation into momentum-space propagator modulation).
If wrong: The UV completion link from WTC potential modulation to a DSI-modulated UETC (and thus to ε,b predictions) is unsupported; the main model-specific quantitative band Eq. (31) would not be mathematically established.
- highEq. (27)-(30) — Convolution factorization step δ(p)=δ(k)[1+O(εf τcorr H*)] relies on D0(q) being sharply peaked and on identifying Δq/q with τcorr H*. The mapping between momentum width of D0 and temporal correlation time is asserted, and the estimate uses |d ln δ/d ln q| but treats δ(q) itself (which changes sign) as if its logarithmic derivative were uniformly bounded.
If wrong: The claim that Π(k) inherits a clean multiplicative cosine with ≤1% error (Eq. (30)) may fail; ε could be renormalized, b shifted, or harmonics generated, impacting both universality and WTC parameter forecasts.
- highEq. (31) — Mapping of WTC parameter space onto ε∈[0.04,0.18], b∈[1.7,2.8] is presented as a result without showing the functional dependence of ε,b on (Fφ,ΛETC,mp) or providing an explicit computation/scan method.
If wrong: The paper's sharp falsifiable quantitative prediction band and its claimed occupancy of the high-SNR region would be unsupported.
- highEq. (9) — Replacement of the fast correlation kernel by F(k)δ(η−η′)+O(τcorr H*) is asserted without showing normalization and how the error term scales in the double integral of Eq. (4) given the Green functions and a(η) factors.
If wrong: Eq. (10) and therefore the claimed multiplicative factorization Eq. (11) may be invalid; the DSI modulation could be smeared or altered by finite correlation time, affecting the main observable template Eq. (12).
- highEq. 10–11, Sec 3.2 (Green's-function step) — The claim that G_k(η,η') ~ sin[k(η−η')]/k contributes only an overall k^{-1} factor plus oscillatory terms that 'average to a k-independent contribution' is asserted without explicit calculation. The actual time integral ∫ dη_1 G_k^2(η,η_1) a^4(η_1) S(η_1,η_1) can in principle carry k-dependence beyond k^{-2}, and whether it preserves the multiplicative log-periodic structure of C(k) is the heart of the factorization claim.
If wrong: If the Green's-function convolution generates additional k-dependent structure that mixes with C(k), the factorization theorem (Eq. 11) fails and the observable template Ω_GW = Ω_0_GW [1 + ε cos(...)] (Eq. 12) — the paper's central prediction — would acquire distortions in amplitude and phase that are not bounded by τ_corr H_*.
- highEq. 25–26 (potential modulation → propagator modulation) — The step from a periodic modulation of the technidilaton potential V(φ) to a multiplicative log-periodic correction δ(q) on the gauge-field propagator is asserted but not derived. The functional form D(q;∆η) = D_0(q;∆η)[1+δ(q)] with δ(q) inheriting precisely the same ε_f, b_0 as V(φ) requires a nontrivial calculation in the strongly coupled WTC theory.
If wrong: If the propagator does not inherit the same log-periodic structure (or inherits it with different ε, b), the identification ε=ε_f, b=b_0 fails and the WTC prediction band (Eq. 31) is not derived from the WTC Lagrangian.
- highEq. 28–29, Sec 5.3 (convolution factorization) — Going from Π(k) ⊃ ∫ d^3p D_0(p) D_0(|k−p|)[δ(p)+δ(|k−p|)] to δ(p) ≈ δ(k) requires that D_0(p) be sharply peaked at p ≈ k. The argument that ∆q/q ∼ τ_corr H_* is stated without derivation; in a relativistic plasma the gauge-field propagator's momentum support is typically not narrow in this sense. Additionally, evaluating δ(|k−p|) at p≈k gives δ(0)·argument issues that are not addressed.
If wrong: If D_0 is not sharply peaked or if δ(|k−p|) does not reduce to δ(k), then ε ≠ ε_f and b ≠ b_0 in general — the WTC parameter map onto (ε,b) ∈ [0.04,0.18]×[1.7,2.8] (Eq. 31) loses its derivation, and the central LISA-falsifiability claim is undermined.
- highEqs. (27)-(30) — The convolution factorization is compressed and contains nontrivial unproved replacements: δ(p) and δ(|k-p|) are replaced by δ(k), although the integration variables need not be localized at the external momentum k; the derivative estimate uses |d ln δ/d ln q|, which diverges near zeros of δ; and the factor of 2 from the two linear cross-terms is said to be absorbed into the baseline while still setting ε=ε_f.
If wrong: The claimed UETC ansatz Eq. (30), the mapping ε=ε_f and b=b_0, and the WTC prediction band ε ∈ [0.04,0.18], b ∈ [1.7,2.8] are not mathematically established.
- mediumEq. (16) — The averaging result R_F=1+ε^2/2 assumes averaging over complete log-periods with no residual weighting from Ω_0(k), detector response, or integration limits. The text does not derive conditions under which the linear term averages to zero for the relevant observable.
If wrong: Quadratic-observable corrections may include nonzero O(ε) terms or different O(ε^2) coefficients, invalidating the universal renormalization factor used later.
- mediumEq. (7) — UETC separability Π(k,η,η′)=S(η,η′)F(k,η−η′) is stated as 'standard' without specifying conditions (stationarity, time-translation invariance, approximations such as envelope/sound-shell) under which it holds for the anisotropic stress in an expanding background.
If wrong: Without this separability, the subsequent delta-function reduction and clean extraction of C(k) in Eq. (10)-(11) may fail; modulation could mix with time dependence and not factorize.
- mediumEq. 16–17 (quadratic observables and DM mass shift) — The renormalization R_F = ⟨Ω_GW^2⟩/⟨Ω_0^2⟩ = 1+ε^2/2 is correct as a log-average identity, but its application to the freeze-in DM mass via m_ψ = m_0_ψ/R_F is asserted without showing that the relic-abundance integral is indeed quadratic in Ω_GW with a linear inversion to m_ψ. The freeze-in mechanism does not obviously have this structure.
If wrong: The DM mass-shift prediction (Eq. 17), advertised as a corroborating multi-messenger observable, would not actually follow from the framework.
- mediumEq. 20–21 (matched-filter SNR) — The reduction from the matched-filter integral (Eq. 20) to SNR_osc ≃ ε SNR_baseline √N_periods assumes σ(f)/Ω_0_GW(f) is approximately constant across the band so that the ⟨cos^2⟩=1/2 averaging factorizes cleanly. This is plausible near the peak but not derived; LISA noise rises rapidly off-peak.
If wrong: The √N_periods enhancement factor could be substantially smaller, weakening the detectability claim and shifting the WTC band's position relative to the LISA SNR=5 contour.
- mediumFigure 1(b) — fit residual r = 0.81 ± 0.04 vs analytic r = 1.00 — The figure caption reports a fitted amplitude ratio r=0.81±0.04 against the analytic prediction r=1.00, a ~20% discrepancy with quoted uncertainty 0.04 — a ~5σ tension between the analytic template and its own numerical demonstration. This is not discussed in the text.
If wrong: Either the analytic template's amplitude normalization is off by ~20%, or the numerical implementation has a systematic. Either way the percent-level error claim of the factorization theorem is in tension with this self-consistency check.
- mediumTable 1 — Percent-level 'relative error' entries are given, but the method producing 13%, 1.3%, 0.13% is not shown; it appears to assume linear scaling with τcorr H* without demonstrating the proportionality constant or definition of 'relative error'.
If wrong: Quantitative claims about 'percent-level' accuracy for β/H*≳10–100 become unsupported; downstream detectability forecasts and 'sharp falsifiable' assertions would have weaker mathematical backing.
The paper presents a mathematically clean target template for a log-periodically modulated SGWB, and if the source UETC is assumed to have the exact multiplicative form of Eq. (6), the observable modulation follows almost immediately from the linearity of the tensor power-spectrum integral. That limited conditional statement is internally plausible. However, the main claimed advances require more than that conditional statement: a controlled factorization theorem with percent-level errors, a WTC derivation of the UETC modulation and parameter band, a matched-filter detectability enhancement, and a dark-matter mass shift. Each of these contains a load-bearing mathematical gap or error. The most serious are the unjustified WTC convolution/amplitude mapping and the SNR scaling in Eq. (21). As written, the central conclusions are therefore not mathematically established, even though the proposed template itself is well-defined.
⚑Derivation Flags (18)
- highEq. (10) → Eq. (11) — Argument that sub-horizon Green-function oscillations 'average to a k-independent contribution' is qualitative; no explicit averaging procedure, window, or estimate is provided to bound induced k-dependence relative to the DSI modulation.
If wrong: Even if C(k) factorizes, additional k-dependent phase/amplitude effects from the time integral could mimic or distort log-periodic structure, undermining the claim that only C(k) carries the oscillation and that transfer is multiplicative at percent level.
- highEq. (11) / Sec. 3.2 factorization theorem — The claim that the Green-function oscillatory terms average to a k-independent contribution and cannot affect log-periodic structure is asserted rather than derived. No bound is given for the residual k-dependent convolution terms.
If wrong: The central factorization theorem P_h(k)=C(k)P_h^0(k)[1+O(τ_corr H_*)] would not be established, and the observable template Eq. (12) would not follow with the claimed percent-level accuracy.
- highEq. (17) — The freeze-in dark-matter mass shift m_ψ=m_ψ^0/(1+ε^2/2) is asserted from the quadratic factor R_F without deriving why the freeze-in relic abundance depends quadratically on Ω_GW or why the inverse scaling with mass follows.
If wrong: The claimed correlated dark-matter mass shift, one of the listed main results, is unsupported.
- highEq. (21) — The matched-filter scaling SNR_osc ≃ ε SNR_baseline sqrt(N_periods) is not derived from Eq. (20) under the standard total-baseline-SNR definition. Averaging cos^2 over the same frequency band would instead give a factor of approximately ε/√2 times the total baseline SNR, not an additional sqrt(N_periods) enhancement.
If wrong: The LISA detectability contours and the claim that the WTC band lies in a high-SNR region are quantitatively unreliable.
- highEq. (26) — Claim that a periodic modulation in the technidilaton potential induces D(q;Δη)=D0(q;Δη)[1+δ(q)] with δ(q) a log-periodic cosine is asserted without an explicit derivation (e.g., via perturbation theory in εf or RG arguments translating V(φ) modulation into momentum-space propagator modulation).
If wrong: The UV completion link from WTC potential modulation to a DSI-modulated UETC (and thus to ε,b predictions) is unsupported; the main model-specific quantitative band Eq. (31) would not be mathematically established.
- highEq. (27)-(30) — Convolution factorization step δ(p)=δ(k)[1+O(εf τcorr H*)] relies on D0(q) being sharply peaked and on identifying Δq/q with τcorr H*. The mapping between momentum width of D0 and temporal correlation time is asserted, and the estimate uses |d ln δ/d ln q| but treats δ(q) itself (which changes sign) as if its logarithmic derivative were uniformly bounded.
If wrong: The claim that Π(k) inherits a clean multiplicative cosine with ≤1% error (Eq. (30)) may fail; ε could be renormalized, b shifted, or harmonics generated, impacting both universality and WTC parameter forecasts.
- highEq. (31) — Mapping of WTC parameter space onto ε∈[0.04,0.18], b∈[1.7,2.8] is presented as a result without showing the functional dependence of ε,b on (Fφ,ΛETC,mp) or providing an explicit computation/scan method.
If wrong: The paper's sharp falsifiable quantitative prediction band and its claimed occupancy of the high-SNR region would be unsupported.
- highEq. (9) — Replacement of the fast correlation kernel by F(k)δ(η−η′)+O(τcorr H*) is asserted without showing normalization and how the error term scales in the double integral of Eq. (4) given the Green functions and a(η) factors.
If wrong: Eq. (10) and therefore the claimed multiplicative factorization Eq. (11) may be invalid; the DSI modulation could be smeared or altered by finite correlation time, affecting the main observable template Eq. (12).
- highEq. 10–11, Sec 3.2 (Green's-function step) — The claim that G_k(η,η') ~ sin[k(η−η')]/k contributes only an overall k^{-1} factor plus oscillatory terms that 'average to a k-independent contribution' is asserted without explicit calculation. The actual time integral ∫ dη_1 G_k^2(η,η_1) a^4(η_1) S(η_1,η_1) can in principle carry k-dependence beyond k^{-2}, and whether it preserves the multiplicative log-periodic structure of C(k) is the heart of the factorization claim.
If wrong: If the Green's-function convolution generates additional k-dependent structure that mixes with C(k), the factorization theorem (Eq. 11) fails and the observable template Ω_GW = Ω_0_GW [1 + ε cos(...)] (Eq. 12) — the paper's central prediction — would acquire distortions in amplitude and phase that are not bounded by τ_corr H_*.
- highEq. 25–26 (potential modulation → propagator modulation) — The step from a periodic modulation of the technidilaton potential V(φ) to a multiplicative log-periodic correction δ(q) on the gauge-field propagator is asserted but not derived. The functional form D(q;∆η) = D_0(q;∆η)[1+δ(q)] with δ(q) inheriting precisely the same ε_f, b_0 as V(φ) requires a nontrivial calculation in the strongly coupled WTC theory.
If wrong: If the propagator does not inherit the same log-periodic structure (or inherits it with different ε, b), the identification ε=ε_f, b=b_0 fails and the WTC prediction band (Eq. 31) is not derived from the WTC Lagrangian.
- highEq. 28–29, Sec 5.3 (convolution factorization) — Going from Π(k) ⊃ ∫ d^3p D_0(p) D_0(|k−p|)[δ(p)+δ(|k−p|)] to δ(p) ≈ δ(k) requires that D_0(p) be sharply peaked at p ≈ k. The argument that ∆q/q ∼ τ_corr H_* is stated without derivation; in a relativistic plasma the gauge-field propagator's momentum support is typically not narrow in this sense. Additionally, evaluating δ(|k−p|) at p≈k gives δ(0)·argument issues that are not addressed.
If wrong: If D_0 is not sharply peaked or if δ(|k−p|) does not reduce to δ(k), then ε ≠ ε_f and b ≠ b_0 in general — the WTC parameter map onto (ε,b) ∈ [0.04,0.18]×[1.7,2.8] (Eq. 31) loses its derivation, and the central LISA-falsifiability claim is undermined.
- highEqs. (27)-(30) — The convolution factorization is compressed and contains nontrivial unproved replacements: δ(p) and δ(|k-p|) are replaced by δ(k), although the integration variables need not be localized at the external momentum k; the derivative estimate uses |d ln δ/d ln q|, which diverges near zeros of δ; and the factor of 2 from the two linear cross-terms is said to be absorbed into the baseline while still setting ε=ε_f.
If wrong: The claimed UETC ansatz Eq. (30), the mapping ε=ε_f and b=b_0, and the WTC prediction band ε ∈ [0.04,0.18], b ∈ [1.7,2.8] are not mathematically established.
- mediumEq. (16) — The averaging result R_F=1+ε^2/2 assumes averaging over complete log-periods with no residual weighting from Ω_0(k), detector response, or integration limits. The text does not derive conditions under which the linear term averages to zero for the relevant observable.
If wrong: Quadratic-observable corrections may include nonzero O(ε) terms or different O(ε^2) coefficients, invalidating the universal renormalization factor used later.
- mediumEq. (7) — UETC separability Π(k,η,η′)=S(η,η′)F(k,η−η′) is stated as 'standard' without specifying conditions (stationarity, time-translation invariance, approximations such as envelope/sound-shell) under which it holds for the anisotropic stress in an expanding background.
If wrong: Without this separability, the subsequent delta-function reduction and clean extraction of C(k) in Eq. (10)-(11) may fail; modulation could mix with time dependence and not factorize.
- mediumEq. 16–17 (quadratic observables and DM mass shift) — The renormalization R_F = ⟨Ω_GW^2⟩/⟨Ω_0^2⟩ = 1+ε^2/2 is correct as a log-average identity, but its application to the freeze-in DM mass via m_ψ = m_0_ψ/R_F is asserted without showing that the relic-abundance integral is indeed quadratic in Ω_GW with a linear inversion to m_ψ. The freeze-in mechanism does not obviously have this structure.
If wrong: The DM mass-shift prediction (Eq. 17), advertised as a corroborating multi-messenger observable, would not actually follow from the framework.
- mediumEq. 20–21 (matched-filter SNR) — The reduction from the matched-filter integral (Eq. 20) to SNR_osc ≃ ε SNR_baseline √N_periods assumes σ(f)/Ω_0_GW(f) is approximately constant across the band so that the ⟨cos^2⟩=1/2 averaging factorizes cleanly. This is plausible near the peak but not derived; LISA noise rises rapidly off-peak.
If wrong: The √N_periods enhancement factor could be substantially smaller, weakening the detectability claim and shifting the WTC band's position relative to the LISA SNR=5 contour.
- mediumFigure 1(b) — fit residual r = 0.81 ± 0.04 vs analytic r = 1.00 — The figure caption reports a fitted amplitude ratio r=0.81±0.04 against the analytic prediction r=1.00, a ~20% discrepancy with quoted uncertainty 0.04 — a ~5σ tension between the analytic template and its own numerical demonstration. This is not discussed in the text.
If wrong: Either the analytic template's amplitude normalization is off by ~20%, or the numerical implementation has a systematic. Either way the percent-level error claim of the factorization theorem is in tension with this self-consistency check.
- mediumTable 1 — Percent-level 'relative error' entries are given, but the method producing 13%, 1.3%, 0.13% is not shown; it appears to assume linear scaling with τcorr H* without demonstrating the proportionality constant or definition of 'relative error'.
If wrong: Quantitative claims about 'percent-level' accuracy for β/H*≳10–100 become unsupported; downstream detectability forecasts and 'sharp falsifiable' assertions would have weaker mathematical backing.
This paper presents a complete theoretical framework connecting discrete scale invariance in cosmological phase transitions to observable gravitational wave signatures. The core factorization theorem is rigorously derived under well-motivated approximations, with quantified error estimates throughout. The walking technicolor UV completion provides concrete predictions that are fully traceable from the underlying theory to observational targets. While some intermediate mathematical steps could be expanded, the main argument is structurally sound and addresses all stated objectives. The work successfully establishes a novel observational signature with specific, testable predictions for LISA.
This paper is conceptually organized and does make a serious effort to state assumptions, identify the intended approximation regime, and connect the proposal to an explicit UV-motivated setting. From a completeness standpoint, however, it falls short because its main claimed result—the multiplicative transfer of DSI from the source UETC to the SGWB spectrum—is not fully derived in the manuscript. The derivation is plausible in outline, but the paper relies on qualitative averaging arguments where a central proof is needed. The support for the ultraviolet completion is likewise incomplete. The manuscript introduces a modulated technidilaton potential and sketches how a log-periodic correction would enter the propagator and then the anisotropic-stress correlator, but the final numerical parameter predictions are not shown to follow from the stated inputs. Overall, the work is readable and partially developed, but the core claims need fuller derivation, sharper variable definitions, and a reproducible quantitative mapping before it can be considered complete.
This is an interesting and potentially valuable phenomenology paper. Its main scientific contribution is to propose that discrete scale invariance in the source anisotropic stress of a first-order cosmological phase transition produces a multiplicative log-periodic modulation in the SGWB, and to argue that this survives the source-to-observable convolution under short-correlation-time conditions. Framed this way, the work is both reasonably novel and scientifically useful: it offers a distinctive observational template and ties it to a concrete beyond-Standard-Model setting rather than leaving the idea purely abstract. The main weaknesses are not about heterodoxy but about communication discipline and evidentiary strength. The paper does provide testable predictions, so its falsifiability is good, but it oversells the degree to which the UV completion and error control have been established. Several central steps are heuristic or scaling-based where the prose suggests something more definitive. Combined with a few notation/consistency issues, this keeps the clarity in the moderate range rather than the strong range. Overall: scientifically promising, quantitatively test-oriented, and novel in synthesis, but it would benefit from a more careful separation between demonstrated results, approximations, and conjectural extensions.
This is a scientifically well-constructed paper that delivers what the abstract promises: a clean, falsifiable spectral template for log-periodic SGWB modulation arising from discrete scale invariance in a phase-transition source, with quantitative control of the underlying approximations and a concrete UV completion (walking technicolor) that maps onto an observationally accessible band in the LISA detectability plane. The predictive chain — DSI in V(φ) → modulated gauge propagator → modulated UETC → factorized Ω_GW(f) — is each step controlled by a small parameter (ε_f, τ_corr H*) whose magnitude is tabulated. The matched-filter SNR analysis is straightforward and correctly identifies the √N_periods enhancement that compensates for small ε. The novelty is genuine: while DSI and log-periodic signatures are well-studied in other domains, their application to the anisotropic-stress UETC of a cosmological phase transition, combined with an explicit WTC realization and a quantitative parameter-band prediction, constitutes a new and specific contribution. Falsifiability is high — LISA will either detect the template or constrain ε(b) — and the writing is clear and well-organized. The main residual concerns are the phenomenological status of the periodic modulation of V(φ) (motivated but not derived) and the somewhat optimistic framing of the DM mass-shift observable. These are not flaws in the central argument but caveats a reader should keep in mind. Overall this is a strong submission on the dimensions of testability, originality, and communication.
DSI ansatz for the source unequal-time correlator (UETC): a smooth baseline UETC times a small log-periodic modulation with amplitude ε and discrete scaling ratio b.
Observable spectral template: the GW energy-density spectrum is the smooth baseline modulated multiplicatively by a sinusoid in ln f with amplitude ε, period ln b, and phase φ_0.
Matched-filter signal-to-noise scaling for the oscillatory component: the oscillation SNR is enhanced by the square root of the number of complete log-periods in the detector band.
Walking technicolor predicts a DSI modulation parameter band ε ∈ [0.04,0.18] and b ∈ [1.7,2.8], which lies in the high-SNR region of the LISA detectability forecast.
Falsifiable if: A LISA observation (or a combined LISA non-detection with comparable sensitivity) that excludes log-periodic oscillations with amplitude ε ≥ 0.04 across b ∈ [1.7,2.8] at the forecasted matched-filter threshold (e.g. SNR_osc ≳ 5 for baseline SNR assumptions) would rule out the WTC prediction band.
The DSI modulation factorizes multiplicatively from the source UETC to the observable Ω_GW(f) in the short-correlation-time limit (τ_corr H_* ≪ 1, i.e. β/H_* ≳ 10) at the percent level.
Falsifiable if: Precision modeling or data showing deviations larger than O(1%) between the predicted multiplicative log-periodic template and the measured spectrum for transitions with independently inferred β/H_* ≳ 10 would falsify the factorization theorem as applied here.
The DSI-modulated SGWB induces a correlated fractional shift in the freeze-in dark-matter mass of order 0.1%–1.6% across the WTC band (expressed as m_ψ/m_ψ^0 ≈ 1/(1+ε^2/2) or equivalently Δm/m ∼ O(ε^2)).
Falsifiable if: Combined cosmological and particle-physics constraints that precisely determine m_ψ (via relic-density and independent phase-transition parameter reconstruction) and exclude a fractional shift in the predicted O(10^−3–10^−2) range would falsify the specific correlated mass-shift prediction.
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