Log-Periodic Signatures from Discrete Scale Invariance in the Stochastic Gravitational-Wave Background Walking Technicolor as a Candidate 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 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 de- tectability of the oscillatory component scales asSNR osc ≃(ε/ √ 2)SNR baseline p N periods , whereN periods =ln(f max /f min )/ lnbis the number of complete log-periods in the detector band andSNR baseline is the per-log-period baseline SNR, giving a useful enhancement over the naive ε suppression. As a candidate ultraviolet completion we explore the realization of 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 convolution calculation shows that, under the same short- correlation-time approximation, the DSI propagates from the technidilaton potential to the observable SGWB with errors≲1%. The WTC parameter space predicts ε ∈[0.04,0.18],b ∈[1.7,2.8], which occupies the high-SNR region of the LISA detectability plane, providing a sharp falsifiable target. All approximations are quantitatively bounded.

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 Signatures7 4.1 DSI-modulated energy-density spectrum . . . . . . . . . . . . . . . . . . .7 4.2 Matched-filter detectability . . . . . . . . . . . . . . . . . . . . . . . . . . .7 5 Ultraviolet Completion: Walking Technicolor9 5.1 Numerical validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9 5.2 Phase-transition parameter space . . . . . . . . . . . . . . . . . . . . . . . 11 5.3 Engineering discrete scale invariance . . . . . . . . . . . . . . . . . . . . . 11 5.4 Convolution for the UETC . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 5.5 WTC predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 6 Discussion15 7 Conclusions16 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 in condensed-matter physics [10] and in financial time-series analysis [11], 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 candidate ultraviolet (UV) completion we explore the realization of the required DSI within walking technicolor (WTC) [13], a strongly coupled hidden-sector gauge theory in the near-conformal regime. Two features motivate WTC as a host for DSI: (i) walking dynamics naturally provide approximate continuous scale invariance over a 3

wide range of energies, which can be broken to DSI by a small periodic modulation of the technidilaton effective potential — motivated (but not derived from first principles) 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. A companion paper [18] 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. [19]. 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 macroscopic source evolution (slowly varying on the Hubble timeH −1 ∗ ) andF(k,∆η) encodes temporal correlations (decaying onτ corr ≪ H −1 ∗ ). This separation holds when the source is stationary on timescalesτ corr ≪∆η ≪ H −1 ∗ : a good approximation for the envelope and sound-shell contributions [6,8], for whichSis approximately constant while F decays rapidly. 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 limitF(k,∆η) is sharply peaked at ∆η= 0. To bound the error, expandFabout ∆η= 0:F(k,∆η) =F(k,0)δ τ corr (∆η) +O(τ corr H ∗ ), whereδ τ corr is a nascent delta function of widthτ corr . Substituting into the double (η 1 ,η 2 ) integral of Eq. (4), theη 2 integral is dominated by the region|η 2 − η 1 |≲ τ corr . SinceG k (η,η 2 )a 2 (η 2 ) varies on timescales k −1 ≫ τ corr (for sub-horizon modes) andH −1 ∗ ≫ τ corr , it may be evaluated atη 2 =η 1 , giving F (k,η− η ′ )≃ F (k)δ(η− η ′ ) +O(τ corr H ∗ ),(9) whereF(k)≡ R F(k,∆η)d(∆η). The relative error isO(τ corr H ∗ ) =O(v w /(β/H ∗ )), bounded explicitly in Table 1. We decompose the spectral kernel asF(k) =C(k)F 0 (k), whereF 0 (k) is the smooth baseline kernel andC(k) = 1 +ε cos[2π ln(k/k ∗ )/ lnb+φ 0 ] 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 )F 0 (k)S(η 1 ,η 1 ).(10) C (k) depends only onk, not onη 1 , and therefore factors out of theη 1 integral exactly. The remaining integral, together with the prefactors, defines the smooth tensor power spectrum: P 0 h (k,η)≡ (16πG) 2 F 0 (k) Z dη 1 G 2 k (η,η 1 )a 4 (η 1 )S(η 1 ,η 1 ).(11) 5

We now show explicitly thatP 0 h acquires no log-periodic structure from the Green’s function. ExpandingG 2 k (η,η 1 ) =sin 2 [k(η − η 1 )]/k 2 = [1− cos(2k(η − η 1 ))]/(2k 2 ), the integral splits as Z dη 1 G 2 k a 4 S = I 0 (η) 2k 2 − e I(k,η) 2k 2 ,(12) whereI 0 (η) = R dη 1 a 4 (η 1 )S(η 1 ,η 1 ) is strictlyk-independent, and e I(k,η) = R dη 1 cos[2k(η− η 1 )]a 4 Sis an oscillatory Fourier transform of the slowly varying envelope. Integrating e I by parts once (boundary terms vanish since S = 0 outside the source epoch):   e I(k,η)   ≤ 1 2k Z   ∂ η 1 [a 4 S]   dη 1 ≲ H ∗ k I 0 (η),(13) sincea 4 Svaries on the Hubble timescale:|∂ η 1 [a 4 S]|≲ H ∗ a 4 S. For sub-horizon GW modes k ≫ H ∗ , the ratio | e I|/I 0 ≲ H ∗ /k. To boundH ∗ /kin terms of the factorization error, note that the characteristic GW wavenumber isk ∼ βa ∗ /(v w a 0 ), givingH ∗ /k ∼ H ∗ v w a 0 /(βa ∗ ) =v w /(β/H ∗ )≡ τ corr H ∗ . Thus | e I|/I 0 =O(τ corr H ∗ ) and Z dη 1 G 2 k a 4 S = I 0 (η) 2k 2  1 +O(τ corr H ∗ )  .(14) Thek-oscillations of e Ioccur on the linear scale ∆k ∼1/η ∗ (producing the sound-wave broken power-law spectral features [6]); in log-kspace this corresponds to ∆lnk ∼ 1/(kη ∗ )∼ τ corr H ∗ ≪ lnb. These sound-wave features are spectrally separated from the DSI modulation by the large factorlnb/(τ corr H ∗ )∼(β/H ∗ /v w )lnb≫1: there is no overlap in log-frequency space, andP 0 h acquires no log-periodic structure at periodlnb. Hence, combining Eq. (14) with the exact factorization of C(k): P h (k,η) = C(k)P 0 h (k,η)  1 +O(τ corr H ∗ )  ,(15) 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, with the Green’s-function contribution bounded explicitly by Eq. (13). Higher-order corrections are quantified in Table 1. Table 1: Relative error bound on the factorizationP h =C(k)P 0 h , forε= 0.1,v w = 1. Two independent corrections contribute. (i) Delta-function approximation, Eq. (9): relative error ≤ τ corr H ∗ ≡ v w /(β/H ∗ ), obtained by bounding the correction R dη 1 |G k (η,η 1 )|·O(τ corr )· |G k (η 2 ,η 1 )a 2 |relative to the leading term. (ii) Convolution factorization, Sec. 5.4: relative error≤(2π/ lnb)ε f τ corr H ∗ ≲13ε f τ corr H ∗ forb ≥1.5, using|dδ/d lnq| ≤ ε f ×2π/ lnb (the absolute derivative, uniformly bounded even at zeros ofδ). (iii) Green’s function oscillations, Eq. (13): relative error≤ H ∗ /k ≡ τ corr H ∗ from the Riemann–Lebesgue bound. The combined bound at ε = 0.1 is (1 + 2π/ lnb + 1)τ corr H ∗ ≈ 15τ corr H ∗ for b = 2. β/H ∗ τ corr H ∗ Combined bound Dominant term 100.10≲ 15%δ-fn 1000.01≲ 1.5%convolution 10000.001≲ 0.15%Green’s fn 6

Numerical validation. A numerical validation of the factorization theorem is presented in Sec. 5.1 (Fig. 4). 4 Observable Signatures 4.1 DSI-modulated energy-density spectrum Combining Eq. (6) with the factorization (15) 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  .(16) The fractional residual R(f )≡ Ω GW (f )− Ω 0 GW (f ) Ω 0 GW (f ) = ε cos  2π ln(f/f ∗ ) lnb
• φ 0  (17) is exactly a sinusoid inlnfwith period ∆lnf=lnb, amplitudeε, and phaseφ 0 . By construction, its Pearson correlation coefficient with a fixed-period cosine template equals r= 1.00 for any bandwidth spanning complete log-periods. This analyticr= 1 should be distinguished from the empiricalr= 0.81±0.04 reported in the companion FDTD paper [18]: in that setting, finite time-series length, Hann-window spectral leakage, and imperfect power-law envelope subtraction all reduce the observed correlation below unity. The FDTD value quantifies detection efficiency in a realistic finite-bandwidth experiment; r= 1 is the property of the underlying physics, recovered in the limit of infinite bandwidth and exact envelope knowledge. 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 ),(18) S sw (f ) =  f f sw  3  7 4 + 3(f/f sw ) 2  7/2 ,(19) with peak frequency f sw = 1.9× 10 −5 Hz 1 v w β H ∗ T ∗ 100 GeV  g ∗ 100  1/6 .(20) 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 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  .(21) 7

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 * Analytic: r = 1.00 (see text; cf. FDTD: r = 0.81 ± 0.04) ( 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. (16). 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. 8

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 = ε 2 Z [Ω 0 GW (f )] 2 σ 2 (f ) cos 2  2π ln(f/f ∗ ) lnb

• φ 0  d lnf.(22) Over N periods complete log-periods ⟨cos 2 ⟩ = 1/2, giving SNR 2 osc = ε 2 2 N periods SNR 2 bin ,(23) where SNR bin is the baseline SNR per log-period of width lnb: SNR 2 bin ≡ Z lnb [Ω 0 GW ] 2 σ 2 d lnf.(24) Hence SNR osc = ε √ 2 p N periods SNR bin ,(25) with N periods = ln(f max /f min ) lnb .(26) Throughout this paperSNR baseline ≡ SNR bin denotes the per-log-period baseline SNR; the total-band baseline SNR isSNR total = p N periods SNR bin . For LISA with effective band [f min ,f max ] = [10 −4 , 1]Hz(ln(f max /f min )≈9.21) the per-log-period baseline SNR at the WTC signal level isSNR bin ≈20–25; the factor p N periods / √ 2ranges from 2.6 atb= 2 to 1.7 atb= 5, providing meaningful amplification. We writeSNR osc ≈(ε/ √ 2)SNR bin p N periods in what follows; figures use SNR bin = 20 to set contours. 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. 5 Ultraviolet Completion: Walking Technicolor Before developing the WTC embedding it is worth stating explicitly which claims in this paper rest on what kind of argument. The phenomenological backbone (Sec. 3–4) follows from the UETC ansatz and the controlled-approximation bounds of Sec. 3.2; the UV-completion layer developed in this section is a candidate realization motivated by holography and near-conformal dynamics but not derived from a complete microscopic model. Table 2 makes this hierarchy explicit. We emphasize that the cited literature [10, 13, 16, 17] motivates the individual ingre- dients — DSI in near-conformal systems, walking dynamics, holographic warp factors, radion potentials — but the complete chain from a microscopic WTC Lagrangian to the periodic technidilaton potential (29) is not, to our knowledge, established in the literature. The construction below should be read as a plausibility argument for the existence of a UV completion within the 5.1 Numerical validation As an independent check on the factorization argument we evaluate Eq. (4) numerically for a separable UETC Π(k,η 1 ,η 2 ) =F 0 (k)C(k)S(η 1 ,η 2 )g(η 1 −η 2 ) with a tophat macroscopic 9

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. 10

Table 2: Scope and epistemic status of the principal claims of this paper. Rows 1–2 form the phenomenological backbone and are derived under explicit, quantified approximations. Rows 3–4 form the UV-completion layer and should be read as a candidate realization rather than a first-principles derivation. ClaimStatusSection DSI in UETC ⇒ log-periodic SGWB template derived3.2, 4.1 Factorization in short-correlation regimecontrolled approximation3.2, Table 1 WTC as DSI hostcandidate UV completion 5.3–5.5 Holographic origin of periodic warp factorconjectural motivation5.3 sourceSand a Gaussian temporal correlationg(∆η) of widthτ corr . Crucially the numerical computation uses the full Gaussian, not the δ-function limit invoked in Eq. (9). Figure 4 shows the resulting residualR(k) = (P h − P 0 h )/P 0 h alongside the analytic predictionε cos(2π ln(k/k ∗ )/ lnb+φ 0 ). Two features confirm the theorem: (i) atτ corr H ∗

0.05 the numerical residual matches the analytic template to within∼10 −16 , the floor of double-precision arithmetic; and (ii) repeating the calculation atτ corr H ∗ ∈{0.01,0.05,0.20} yields residuals that are pointwise identical to within numerical precision. Both observations follow from the fact that, whenC(k) depends only onkand is independent ofη 1 ,η 2 , it factors out of the double time integral algebraically — not merely up toO(τ corr H ∗ ). The error bounds in Table 1 arise instead from non-separable corrections (the convolution and Green’s-function terms): these are not probed by the present test and would require a more elaborate numerical setup. WTC framework, not as a derivation from first principles. 5.2 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.(27) 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,(28) comfortably within the LISA sensitivity band [1, 2], and satisfy τ corr H ∗ ≲ 0.01≪ 1. 5.3 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 11

10 0 10 1 wavenumber k (units with k * = 5) 0.15 0.10 0.05 0.00 0.05 0.10 0.15 R ( k ) ( P h P 0 h )/ P 0 h (a) = 0.10, b = 2.0, corr H * = 0.05, max dev. = 1.80e16 Factorization theorem: numerical residual vs.\ analytic template Analytic: cos(2ln(k/k * )/ln b + 0 ) Numerical: (P h P 0 h )/P 0 h 10 0 10 1 wavenumber k 0.15 0.10 0.05 0.00 0.05 0.10 0.15 R ( k ) (b) Independence of corr H * for separable UETCs: factorization is algebraic, not perturbative Analytic template Numerical ( corr H * = 0.01) Numerical ( corr H * = 0.05) Numerical ( corr H * = 0.2) Figure 4: Numerical validation of the factorization theorem (Sec. 3.2). The DSI-modulated tensor power spectrumP h (k) is computed by direct numerical integration of the double time integral (4) for a separable UETC Π =F 0 (k)C(k)S(η 1 ,η 2 )g(η 1 −η 2 ) with a Gaussian temporal correlationg(∆η) of widthτ corr (the full Gaussian; not theδ-function limit of Eq. (9)). (a) ResidualR(k) = (P h − P 0 h )/P 0 h (orange circles) plotted against the analytic predictionε cos(2π ln(k/k ∗ )/ lnb+φ 0 ) (gray line) atτ corr H ∗ = 0.05; maximum deviation ∼10 −16 (numerical floor). (b) The same residual evaluated atτ corr H ∗ ∈{0.01,0.05,0.20} collapses onto a single curve, confirming that the factorization is algebraic (independent ofτ corr H ∗ ) for separable UETCs. TheO(τ corr H ∗ ) corrections in Table 1 arise from non- separable structure (convolution and Green’s-function terms) that is beyond the scope of this clean test. 12

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,(29) 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]. Holographic origin ofε f andb 0 . Equation (29) arises naturally in the holographic dual of WTC without requiring fine-tuning of five-dimensional parameters. Modelling the technidilaton as the radion field in a Randall–Sundrum-type AdS 5 geometry [16], a small periodic modulation of the standard warp factor A(y) = ky of the form A(y) → ky + δA 0 sin(n p ky), δA 0 ≪ 1,(30) generates, at linear order inδA 0 , the multiplicative log-periodic correction to the 4D technidilaton potential [17] V 4 (φ)≈ V CW (φ)  1 + 4δA 0 cos  2π ln(φ/φ 0 ) lnb 0  +O(δA 2 0 )  ,(31) withε f = 4δA 0 andlnb 0 =kL/n p , whereLis the proper length of the extra dimension andn p is the number of warp-factor oscillation periods. For the ETC hierarchykL ≈ ln(Λ ETC /Λ TC )≈2–3 andn p = 3–4 (bothO(1) integers in AdS units), one obtains b 0 =e kL/n p ∈ [1.7,2.8]. The required amplitudeε f ∈[0.04,0.18] corresponds toδA 0 ∈ [0.01,0.045], a 1–5% warp-factor perturbation that is technically natural (protected by the approximate discreteφ→ φ+L/n p shift symmetry of the periodic modulation) and requires no independent fine-tuning. Both DSI parameters therefore emerge fromO(1) choices of the 5D geometry. The modulation in Eq. (29) induces a log-periodic correction to the gauge-field propa- gator at momentumq. We derive this at leading order inε f . In the near-conformal WTC regime, the technidilaton VEV⟨φ(q)⟩at renormalisation scaleqis related to its UV value by ⟨φ(q)⟩=⟨φ UV ⟩(q/q 0 ) −∆ φ , where ∆ φ is the technidilaton scaling dimension (∆ φ ≈1 near the quasi-fixed point). The gauge-boson mass is generated viam 2 V (φ) =y 2 ⟨φ⟩ 2 ; a modulation δV ∝ ε f cos(2π lnφ/ lnb 0 ) shifts the mass asδm 2 V /m 2 V =ε f cos(2π ln(q/q ∗ )/ lnb 0 ) +O(ε 2 f ) by the chain rule. Propagating to the full propagator at leading order in ε f : D(q; ∆η) = D 0 (q; ∆η) [1 + δ(q)], δ(q) = ε f cos  2π ln(q/q ∗ ) lnb 0

• φ 0  ,(32) withq ∗ ∼ q 0 . The modulation inherits the same log-periodb 0 as the potential, up to the conformal-dimension factor ∆ φ which isO(1) near the fixed point. Higher-order corrections enter at O(ε 2 f ). Note that |δ(q)|≤ ε f ≪ 1 so D is positive definite for all q. 13

5.4 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),(33) 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|)  .(34) We bound the two terms separately. Termδ(p). The baseline propagatorD 0 (p) is sharply peaked atp∼ q ∗ ∼ β/v w with relative half-width ∆p/p∼ τ corr H ∗ ≪1. The variation ofδacross this peak is bounded using the absolute derivative:     dδ d lnq     = ε f 2π lnb 0   sin

2π lnq/ lnb 0

• φ 0
    ≤ ε f 2π lnb 0 ,(35) uniformly bounded for allq, including near the zeros ofδ(where the logarithmic derivative d lnδ/d lnqwould diverge, but the divergence is integrable since the weightD 0 is smooth and |δ|→ 0). Integrating over the support | ln(p/k)|≲ τ corr H ∗ : |δ(p)− δ(k)|≲ 2πε f lnb 0 τ corr H ∗ ≲ 13ε f τ corr H ∗ (b 0 ≥ 1.5),(36) giving δ(p) = δ(k)[1 +O(ε f τ corr H ∗ )] over the support of D 0 (p). Termδ(|k−p|). Nearp≈k,|k−p|→0 andδ(|k−p|) oscillates rapidly. However, WTC gauge bosons have a mass gapm V ∼Λ TC , and for|k−p|≪ m V the propagator satisfiesD 0 (|k −p|)≲ D 0 (0)∼1/m 2 V . At the signal scalek ∼ q ∗ ≫ m V ,D 0 (q ∗ )∼ 1/(q 2 ∗ − m 2 V )≪1/m 2 V , soD 0 (0)/D 0 (q ∗ )∼ m 2 V /q 2 ∗ ≡ ρ ≪1. The contribution of the |k−p| →0 region is therefore suppressed byρrelative to the peak contribution from |k−p|∼ q ∗ , where the same slow-variation argument as forδ(p) applies. The combined bound is δ(p)≈ δ(|k−p|)≈ δ(k)  1 +O(ε f τ corr H ∗ ) +O(ρ)  ,(37) withρ≡ m 2 V /q 2 ∗ ≪ 1 in the WTC regime. The cross-term factorizes as 2δ(k)×Π 0 (k,η,η ′ ). The factor of 2 corresponds to the two linear cross-terms; absorbing it into a redefined baseline normalization setsε=ε f at leading order (the factor of 2 multiplies the baseline amplitude, not the modulation depth). Hence Π(k,η,η ′ ) = Π 0 (k,η,η ′ )  1 + ε cos  2π ln(k/k ∗ ) lnb
• φ 0   1 +O(ετ corr H ∗ ) +O(ρ)  , (38) withε=ε f andb=b 0 at leading order. For WTC benchmark parametersβ/H ∗ ≳100 and ρ∼ m 2 V v 2 w /β 2 ≲ 0.01, the combined relative correction is≲ 1.4% (Table 1). 14

5.5 WTC predictions Combining Eq. (38) with the short-correlation-time factorization theorem of Sec. 3.2, the DSI modulation propagates multiplicatively to the observable SGWB, recovering Eq. (16) up to controlledO(ετ corr H ∗ ) corrections (andO(ρ) mass-gap corrections from the WTC convolution, withρ ≡ m 2 V /q 2 ∗ ≲ 0.01). The WTC parameter space [13], spanned by F φ ≈ 1 TeV, Λ ETC ∼ 5–10 TeV, and soft masses m p ∼ 1–100 GeV, maps onto ε∈ [0.04, 0.18], b∈ [1.7, 2.8].(39) This band is shown in Figs. 2–3 and overlaps the high-SNR region of the LISA detectability forecast. The map from WTC parameters to (b,ε) follows from the holographic benchmark of Sec. 5.3:ε= 4δA 0 andb=exp(kL/n p ), withkL≈ ln(Λ ETC /Λ TC )≈ ln(5–10) = 1.6–2.3 andn p = 3–4 from the WTC benchmark [13];δA 0 ∈[0.01,0.045] from theO(1–5%) warp-factor perturbation range. For a baselineSNR bin = 20 the matched-filter SNR satisfies SNR osc ≳ 4 over most of the band (using Eq. (25) with the 1/ √ 2 factor). 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– 1000) 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|dδ/d lnq|·(∆q/q)≪1 — using the absolute derivative|dδ/d lnq| ≤ ε f ×2π/ lnb 0 , which is uniformly bounded for allq(the logarithmic derivatived lnδ/d lnqwould diverge at the zeros ofδ, but the absolute derivative does not; see Sec. 5.4) — is equally well controlled and introduces no additional tuning. FDTD analogy and cosmological causal structure. The companion FDTD pa- per [18] demonstrates log-periodic spectral imprinting in a controlled electromagnetic cavity; the correspondence to the cosmological FOPT warrants explicit comment. In the cavity, rigid static boundaries enforce global mode selection via discrete standing-wave conditions (Dirichlet or absorbing boundary conditions at the walls): the mode spectrum is shaped by the entire geometry simultaneously. In a FOPT no global boundary condition exists: bubbles nucleate independently within their past light cones, and causal horizons preclude global mode coherence. The structural role of the geometric boundary is instead played by the characteristic bubble spacingR ∗ ∼ v w /β, which acts as a local, dynamic filter. Modes withk ≫ R −1 ∗ are exponentially suppressed by the decay of the temporal correlation functionF(k,∆η) at large separations; modes withk ≪ R −1 ∗ see a nearly homo- geneous source and are coherently accessible. In the language of the companion framework paper [19],R ∗ plays the role of the boundary-conditioned mode density cutoff, with the plasma mean free path providing the dynamic spectral participation filter. Crucially, the factorization theorem of Sec. 3.2 relies only on the local conditionτ corr H ∗ ≪1 — set by β/H ∗ ≫1, independent of any global causal horizon structure. The FDTD result therefore 15

validates the mathematical mechanism of DSI imprinting (that a log-periodically structured boundary parameter transfers its signature multiplicatively to the power spectrum), while the factorization theorem independently establishes the validity of that transfer in the cosmological context via purely local causal arguments. Distinguishability from other spectral features. The log-periodic modulation (16) 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 (16) 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 (16) 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 [15] — 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, providing in principle an independent observational handle on the same (ε,b) parameters. 7 Conclusions We have demonstrated that discrete scale invariance in the anisotropic stress tensor of a first-order cosmological phase transition imprints a 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 ∗ )]. 16

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, characterized at leading order in τ corr H ∗ by three parameters (ε,b,φ 0 ). 3.Matched-filter detectability.SNR osc ≃(ε/ √ 2)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. 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 (16) 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]. 17

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