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–

1. 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

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