Early Universe: Hadronic Crystals Coherent Micro Gravitational Wave Emitters PHYSICS Part II

Hadronic Crystal Modeling: Skyrme-Type SU (2) Lattice Structures [18,25,27]

To model the hadronic phase of the early Universe immediately below the QCD confinement scale, we will consider Skyrmion lattice configurations motivated by recent experimental observations of optical Skyrme lattices accelerating in free space [43] and by the nuclear-theory analyses of dense Skyrmion matter (25-27). Quantified derivative aspects detailing Skyrmions, Solitons, SU (2), and the Baryon representative Coherent Lattice Modes, alongside explanations of the Skyrme Lagrangian,, which describes the dynamics of the pion fields (represented by the matrix U(x)∈SU(2)) in a model that accounts both the pion interactions and the stability of the solitons (Skyrmions), appear within these physics literature.

Hadronic degrees of freedom are described by the classical Skyrme Lagrangian for an SU (2) chiral field ρ ρ π τ π ∈ = . , 2 , i f U x t SU U e ( ) ( ) , representing the pion isotriplet. Lagrangian density is per (25-27):

( ) ( )

2 2 † † † 2 1 , 16 32

v f Tr U U Tr U U U U e µ µ   = ∂ ∂ + ∂ ∂   L (1)

µ π Objects and their meanings -fπ: the pion decay constant, sets kinetic scale for pion dynamics; e: dimensionless Skyrme parameter controlling the nonlinear stabilizing term. xµ µ ∂ = ∂∂ : spacetime derivative; Minkowski metric signature (+,-,-,-), with μ=0,1,2,3 while 0 → time & 1,2,3 → spatial coordinates, also similarly, , ensures Lorentz invariance; Tr: Trace of a matrix (sum of diagonal elements) over SU(2) matrices, and since U is a 2×2 matrix, the trace ensures the Lagrangian is a scalar; †U=I U or † -1 =U U Hermitian conjugate since ( ) ( ) ( ) ( ) 2 ; exp U SU U x i x fπ τ π ∈ = ⋅ ρρ , where: ( ) x π = ρ pion field; τ = ρ Pauli matrices.

Keynote 1: Structure and role of terms (i) Nonlinear sigma-

2 † 16 f Tr U U µ µ π ∂ ∂ describes pion propagation;

model term ( ) (ii) Skyrme term ( )

2 † † 2 1 , 32 v Tr U U U U e µ   ∂ ∂   prevents the collapse of soliton solutions (evades Derrick’s theorem) and stabilizes topological solitons (Skyrmions). The corresponding energy density is kin ε = − + Π L (2) where Πkin denotes canonical kinetic contributions.

Keynote 2: Physical interpretation (i) the baryons appear as the topological solitons whose baryon number equals the winding number of the map U: S3→SU (2); (ii) the commutator [A,B]=AB-BA, captures the non-Abelian structure of the SU (2) chiral fields.

Periodic arrangements of Skyrmions produce hadronic crystals, consistent with dense-matter Skyrmion crystal analyses (18,25-27). These lattices form the basis for our gravitational-wave emission modeling.

Equations of Motion, referring Equations (1) & (2)

Starting from the Skyrme Lagrangian:

( ) ( )

2 2 † † † 2 1 , 16 32 v f Tr U U Tr U U U U e µ π µ µ   = ∂ ∂ + ∂ ∂   L , we define the left-invariant current:

† L U U µ µ = ∂ . This allows the Lagrangian to be thus expressed compactly as: ( ) ( )

2 2

2 1 , 16 32 v f Tr L L Tr L L e µ π µ µ   = − +   L . Variation with respect to U yields the Euler–Lagrange equations of motion:

      ∂ + =         .This represents the typical field equation for the Skyrme model.

2 2 1 , , 0 v v L L L L e f µ µ µ π

Conserved Baryon (Topological) Current

The model possesses a conserved topological (baryon) current, Bμ {a 4-vector, μ=0,1,2,3}: having ( ) 2 1 , 0 24 v v B Tr L L L B µ µ αβ µ α β µ π = ∈ ∂ = , i.e., current is

0 0 B B t ∂ = + ∇⋅ = ∂

ρ that implies baryon number does not change with time: 3 0 B d xB = ∫ {B=1 → nucleon & B=2→ deuteron}, with B an integer equalling baryon number (topological charge). Spatial components Bi represent baryon flow; ϵ^μναβ: Levi–Civita tensor (totally antisymmetric symbol), having properties, like sign changes when two indices are swapped, ε^0123=+1 or zero if any indices repeat, ensures the expression is fully antisymmetric with a proper Lorentz scalar density. For the finite-energy boundary conditions conserved ( ) ( )1 U x →∞→ , the spatial slice compactifies to S3, and the mapping ( ) 3 3 : 2 ~ space U S SU S → is characterized by an integer winding number corresponding to the baryon number B ∈Z . Therefore, Skyrmions carry baryon number.

Explicit Field Parametrization Process Skyrme field U(x) can be expressed in terms of pion fields πa (x) using the exponential (with having nonlinear ( ) ( ) τ π   =      , where a τ are the Pauli matrices. Alternatively, in unit quaternion form: ( ) ( ) ˆ cos sin , U f i f π π π π τ π π π = + ⋅ = ρ a a i x U x exp fπ sigma model) parametrization:

. This form is particularly useful for the hedgehog ansatz of Skyrmions.

Hedgehog Ansatz Skyrmions The hedgehog ansatz is: ( ) ( ) ( ) ( ) ( ) ˆ U x cos F r i x sin F r τ = + ⋅ , for a single baryon (B=1) solution with the boundary conditions ( ) 0 0 ( ) , F F π ∞ = = . Solving the typical Skyrme equation for the F(r) yields the Skyrmion profile.

Micro–Gravitational-Wave Emissions: Linearized General Relativity [19,20,22,31]

The gravitational radiation from a time-varying quadrupole distribution is computed using the standard formulation of linearized general relativity. For an observer at distance r, the leading-order GW strain is

2

4 2 2 ij ij d Q G h t c r dt = (3) where Q_(ij )is the mass quadrupole moment ( )

2 3 1 , 3 ij i j ij Q x t x x x d x ρ δ   = −     ∫ & (4) having ρ(x,t): mass density; δ_ij: Kronecker delta.

( ) A perfectly spherical distribution has Qij=0; hence any coherent anisotropic oscillation in the Skyrmion lattice produces quadrupolar radiation typically.

In the early Universe, Skyrmion crystal oscillations in the GHz–THz regime generates rapidly varying quadrupoles ( ) ( ) 0 ~ sin xx Q t Q t ω (5) with typical microscopic scale

35 2 0 ~ 10 . Q kg pm −

(6) For r~Gpc, it yields

30 ~ 10 h − & (7) consistent with microscopic emitter expectations.

Simulation Algorithm: With Numerical Process

We implement lattice-based numerical evolution scheme to compute Skyrmion-lattice dynamics having corresponding GW signal, based on above literature and Sec. 2.4 references.

Algorithm Outline (1) Spatial Discretization: Partition the simulation volume into a cubic lattice of the spacing Δx; (2) Initial Conditions: Construct a Skyrmion crystal (hexagonal, BCC, or FCC symmetry) with initial density ρ_0; (3) Small Perturbations: Apply displacements δx to excite lattice modes; (4) Force Evaluation: Compute lattice forces from energy gradients: F=-∇E. Leapfrog Time Integration:

( ) ( ) ( ) ( ) 2 1 2 x t t x t v t t a t t + ∆ = + ∆+ ∆

(8) Quadrupole Calculation: Evaluate Qij (t)using Equation (4) Gravitational-Wave Calculation: Compute strain via Equation (3).

Overview of Integrated Analytical and Computational Framework [7,20,21,34,38]

Our Multiscale Analysis Couples Four Complementary Components: • Gravitational-wave Emission Theory: based on standard linearized GR [20, 38], extended to coherent ensembles and Kerr-like rotational enhancements. • Hadronic Crystal Modeling / Quantum Hadron Lattice (QHL): Using Skyrme-type fields and an effective scalar-field Lagrangian with spontaneous symmetry breaking [7, 21, 34]. • Curved-spacetime Scalar-Field Evolution: including one-loop quantum corrections as well as gravitational- wave–induced metric perturbations. • 1D and 2D Numerical Simulations: Using finite- difference discretization, FFT analysis, Runge–Kutta + leapfrog integrators, coherence diagnostics, and Lyapunov stability evaluations.

Algorithm 1 - Multiscale Simulation Workflow [2,8,14,29,36,37]

Input: Early-universe physical parameters {T,B,ρb,h(t)}. Output: GW strain spectrum h(f), scalar-field evolution ϕ(x,y,t), coherence measure C(t), and mode structure. Step 1: Physical Initialization (i) Read constants G,c,mp,rp,ℏ; (ii) Set QCD parameters - temperature T∼100 to 200 MeV, baryon density, curvature parameter α [14, 37]; as well as (iii) Place the typical baryonic delta-function sources at lattice sites xi. Step 2: Single-Particle Quadrupole Analysis - Compute intrinsic quadrupole 2 Qi p p m r ε = , rotational frequency

2 , ~ 2 i i p p i I m r I ω = η

2 single 4 4 h G Q c R ω = (9) This yields hsingle∼10-36-10-33. Step 3: With Collective Emission coherent N baryon ensemble, single h h total coh Kerr N = Γ Γ (10) , and strain τ τ Γ (11) Kerr-like enhancement [29, 36]:

With coherence amplification: coh ~ coh cycle Kerr 3 1 GM a c R Γ = + (12) Typical early-universe domains yield: 22 21 total~10 10 h − − − (13) within reach of projected GHz detectors [2, 8]. Step 4: Scalar-Field Crystal Formation - Solve static field equation of the form

2 3( ) i i dv g d φ δ χ χ φ ∇ = − − ∑ (14) This produces kink–antikink lattice structures. Step 5: 1D/2D Time Evolution - Evolve via curved-space Klein–Gordon equation of the form □g 0 eff V φ φ ∂ − = ∂ (15) with metric like 2 2 2 2 2 (1 ( ) ( )( ) ds h t dt a t dx dy = − + + + (16) Effective potential includes classical, quantum (Coleman– Weinberg), curvature, as well as gravitational phase-locking terms.

2 1 2 eff V ρ φ = ∇ +

Step 6: With Diagnostics (i) Energy density ( , , ) ( ) i x y t C t e θ = ;(iii) the Fourier ;(ii) having the Coherence

2 ( ) k φ ; as well as (iv) with the Lyapunov exponent 1 ( ) ln (0) r T T r δ λ δ   =    . Generate Output: (i) scalar-field maps ϕ(x,y), (ii) energy- density surfaces, (iii) FFT spectra, (iv) GW strain curves, & (v) Lyapunov stability maps.

spectrum

Quadrupole Moment Evolution in Skyrmion and Scalar-Field Crystals

Lattice Oscillations and Quadrupole Dynamics [15, 16, 20, 22, 25, 26, 27]: The numerically evolved Skyrmion- based hadronic crystal displays collective oscillatory deformation modes, generating a time-dependent mass quadrupole moment equation having: ( ) ( ) 0 sin xx Q t Q t ω ≈ , via Equation (5), with a natural consequence of high effective stiffness and small lattice spacing of the early-universe hadronic matter, having Skyrme-type lattices within dense QCD causing oscillation frequencies spanning GHz–THz domain [15, 16, 25, 26, 27].

The characteristic quadrupole amplitude from simulations: 35 2 0 10 . Q kg pm − ≈ ,is consistent with baryonic microstructures at the QCD epoch. The second derivative ij Q, entering the standard quadrupole radiation formula [20, 22], yields calculable typical metric perturbations:

( ) ( ) 4 2

ij ij G h t Q t c r = 

, with strain amplitudes: 30 ~ 10 h − at 1 r Gpc ≈ . These values would correspond to single-crystal or small-domain emission, prior to including with coherence or collective effects.

Simulation Graphics giving Figures with Descriptions and Captions: Hadronic cystal graphics show how: (i) Skyrmion Lattice Geometry evolves with Early-Universe Hadronic Phase (per Figure 1); (ii) Energy-Density Slice of the Skyrmion Crystal (per Figure 2); (iii) Quadrupole Moment produced by collective oscillations of the Skyrmion crystal with Gravitational-Wave Strain, demonstrating coherence of lattice vibrations (Figure 3); (iv) Power Spectral Density of the Gravitational-Wave Signal (per Figure 4); (v) Scalar-Field Profile of the Hadronic Kink–Antikink crystal structures between baryonic source points (per Figure 5); (vi) 2D Field Map and Curvature-Enhanced Compression, representing early-universe gravitational focusing near the center (Figure 6); (vii) Fourier Spectrum of Lattice Oscillations oscillations that enhance generation of gravitational waves (per Figure 7); (viii) Quantum Phase locking enables collective GW emission with dramatic enhancement of coherence factor (per Figure 8); (ix) Evolutions of coherence having Lattice Synchronization leading to enhancement of the gravitational wave signal (per Figure 9); (x) Lyapunov Exponent Heatmap of chaoticity showing how the early-universe hadronic crystals could sustain coherent oscillations over many cycles, even in the presence of thermal fluctuations (per Figure 10); (xi) Predicted Gravitational Wave Strain vs. Temperature of hadronic crystal domains under varying QCD-scale temperatures (according to the Figure 11); (xii) Multiscale Workflow from Skyrmion Fields to Gravitational Waves (according to the Figure 12).

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Keynote Aspects with: Representative field configuration of having modeled early-universe hadronic crystal; shown is a two-dimensional slice of the Skyrmion lattice obtained from the SU(2) Skyrme Lagrangian (Equation 1) using periodic boundary conditions. Color scale indicates that the baryon- density–related topological charge density having equation relating ( ) ( ) 0 2 1 , 24 ijk i j k B x y Tr L L L π = − ∈ , with arrows denoting the in-plane pion field direction. The hexagonal/ BCC-like tiling emerges from energy minimization of the Skyrme term, consistent with those crystalline Skyrmion phases reported in nuclear-matter studies [18, 25, 26, 27]. The Figure 1 illustrates periodic modulation of chiral orientation as well as presence of all those distinct Skyrmion cores which tend to form a coherent hadronic micro-lattice.

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Keynote aspects with: Mid-plane energy-density map ( ) ( ) ( ) 2 1 , 2

eff x y V ρ φ φ = ∇ + , showing localized having soliton nodes corresponding to Skyrmion centers. The central white contours represent constant-energy hypersurfaces, revealing the spatial periodicity and crystalline topology of the hadronic lattice. Enhanced compression near the origin reflects curvature-induced amplification (Algorithm 1), demonstrating the influence of early-universe spacetime gradients on localization of fields.

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Keynote aspects with: above Figure shows Quadrupole Dynamics Moment and Gravitational-Wave Strain plots. (A) Time evolution of quadrupole component Qxx(t) which is produced by having collective oscillations of the Skyrmion crystal. The simulation uses a leapfrog scheme with the perturbative displacements around equilibrium sites. Oscillation frequency lies well within the GHz–THz band, characteristic of microscopic hadronic modes. (B) Corresponding gravitational-wave strain hxx (t), obtained using typical linearized GR quadrupole formula ( ) ( ) 4 2 ij ij G h t Q t c r =  . Peak amplitudes reach h∼10-30 for a source distance of 1 Gpc. Then, such strain inherits the clean harmonic structure of the quadrupole moment, demonstrating coherence of lattice vibrations. Note that strain amplitudes suppressed by a very large asymmetric factor ϵ∼10(-32) relative to the maximal quadrupole estimate, reflecting the extreme near- sphericity and slow dynamical evolution of the configuration, conforms with relation: true max h h =∈× having ϵ of the order of ( ) 4 32 / 10 v c − ∼ . Such strong suppression can arise from near- perfect spherical symmetry, very slow quadrupole evolution, as well as additional velocity suppression factors, like (v/c)n. This explains the plot scale of about 10-62 versus hmax of about 10-30 here.

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Keynote aspects with: showing Fourier-domain power spectral density (PSD) of strain signal h(t). Dominant peak appears at the principal lattice oscillation mode in the GHz– THz range, with sub-harmonic as well as overtone features seen due to nonlinear interactions within the Skyrmion crystal. The narrowness of the spectral peak indicates high temporal coherence, having consistency with the collective- mode amplification mechanism described earlier as also in Sec. 3.3.2.

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Keynote Aspects with: One-dimensional solution of the static field equation (Equation 14) showing alternating kink and antikink structures between baryonic source points.

m x v x x φ     = −         with m=√(2λ) v, manifesting as well the baryon-induced perceptible discontinuity: ( ) ( ) ' ' i i x x g φ φ + − − = − is plotted Here, the analytical form ( ) ( ) 0 tanh 2 alongside numerical data, demonstrating excellent agreement. The alternating discontinuities in the 'x ϕ across lattice sites (Figure 5(ii)) illustrate how baryon sources generate a classical hadronic crystal, supporting DisContinuum Physics theory [46].

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Keynote aspects with: Two-dimensional simulation of the field ϕ(x,y) with curvature factor 2 ( ) 1 curve G r r α − = + . The result shows lattice contraction near the center, representing also early-universe gravitational focusing. Regions of high local curvatures amplify the Skyrmion-to-Skyrmion interaction strength, increasing coherence and raising further predicted GW emissivity of centrally located domains.

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2 k φ from a 2D field evolution run. Peaks at k=2π/d reveal the fundamental lattice spacing d, while secondary peaks correspond to the harmonics and the nonlinear couplings.

Keynote aspects with: the Fourier amplitude ( )

The high-k tail reflects thermal noise and curvature-induced steepening. These spectral signatures track the mode-locking and collective oscillations that enhance GW generations.

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Keynote aspects with: Response of the local lattice mode amplitude to an oscillatory gravitational background h(t), modeling phase-locking potential ( ) 2 lock 0 cos g V t µ ω θ φ = + . The plot(b) system displays synchronization plateaus: “Arnold tongues”, when the gravitational driving frequency

2 2 A µ

( ) γ ∆= ∆+ , having peak at exact resonance, the width controlled by damping γ, while increasing μ broadens the locking regions. Phase locking dramatically increases coherence factor Γcoh, enabling collective GW emissions.

satisfies g φ ω ω ≈ , and the plateau:

A(Δ): Amplitude of the local lattice mode response as a function of detuning that measures how strongly the lattice oscillation locks to the external gravitational driving. Δ: Detuning parameter, ; g g φ ω ω ω ∆= − = gravitational driving frequency, φ ω = intrinsic lattice mode frequency. When Δ=0, the system is exactly on resonance. μ: Coupling strength between the gravitational background and the lattice mode. It comes from the phase-locking potential: ( ) 2 0 cos lock g V t µ ω θ φ = + . Larger μ→ stronger forcing → wider synchronization plateau. γ: Damping or decoherence rate of the lattice mode. It

determines the width of the resonance peak, the stability of phase locking, and dissipation in the system. Smaller γ→ sharper the peak Larger γ→ the broader but lower resonance.

2 2 A µ

Why It Is Called a “Plateau”: The function ( )

γ ∆= ∆+ has a maximum at resonance: max A µ γ = (when Δ=0). Near Δ=0, the amplitude changes slowly with frequency - this flat region is the synchronization plateau (Arnold tongue center). Physical Interpretation: At resonance - the lattice mode phase locks to the gravitational drive, oscillations become then coherent, i.e., coherence factor Γcoh increases, enhancing collective GW emission. Off resonance, phase drifts, amplitude drops 1 ∆, and the synchronization breaks (Table 1).

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Keynote aspects with: Time evolution of the coherence order parameter ( ) i C t e θ = . When gravitational phase locking is active, thermally jittered, weakly correlated state

0.2 C < system transitions to a near-perfect synchronization ( 9) 0. C ≈ . This behavior underpins the enhancement of the GW signal: coherent domains contribute strain coh N ∝ Γ .

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Keynote aspects with: Heatmap of the Lyapunov exponent λ(x0, p0) across initial-conditions space, computed using ( ) ( ) 1 ln 0   =       r T δ λ δ Equation:

to evaluate dynamical stability, where ( ) r t δ represents the separation between neighboring geodesic trajectories. We obtain @ 0 λ ≈ : marginal coherence; blue regions (λ<0) correspond to stable, crystal- like evolution motion; yellow to red regions (λ>0) indicate chaotic, decohered states. The existence of broad stability basins implies that early-universe hadronic crystals could sustain coherent oscillations over many cycles, even in the presence of thermal fluctuations. Here, x0= initial position; p0= initial momentum and the Equation tell how sensitive that specific orbit is to small perturbations. The heatmap over ( ) 0 0 , x p shows stable vs chaotic regions of the system. T: Total evolution time over which separation is measured. Longer T gives a more reliable estimate of asymptotic stability. δr(0): Initial separation between two nearby trajectories:

T r

( ) ( ) ( ) 1 2 0 0 0 r r r δ = − . It must be very small to probe local stability. δr(T): Separation between those trajectories after time ( ) ( ) ( ) 1 2 : T T r T r T r δ = − . Growth of this separation determines chaotic behavior. r(t): Phase-space vector of the system: r(t)=(x(t),p(t)). In a gravitational/geodesic system, this may represent - spatial coordinates, conjugate momenta, or full geodesic state vector. Physical Interpretation: The formula measures exponential divergence: ( ) ( ) 0 T r r T eλ δ δ ≈ Case 1: λ>0→ Chaotic - nearby trajectories diverge exponentially, sensitive dependence to initial conditions, strong mixing, and possible enhanced GW emission due to irregular motions. Case 2: λ=0→ Marginal / periodic - Neutral stability, as well as regular oscillatory motions. Case 3: λ<0→ Stable / Attractor - Trajectories converge, dissipative stabilization, and phase locking possible. Why Heatmap Is Useful: Plotting λ(x0,p0) reveals - stability islands (dark/blue regions), chaotic seas (bright/red regions), resonance bands, “Arnold tongues”, separatrix boundaries. Thus, this allows mapping - mode-locking regions, geodesic instability zones, and transition to collective behavior (Table 2).

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Keynote aspects with: Simulated maximum strain amplitude hmax (T) for hadronic crystal domains under varying QCD-scale temperatures. Increasing temperature reduces coherence due to enhanced thermal noise, suppressing Γcoh and thus the total strain. At lower temperatures (70–120 MeV), the strain approaches h∼10-22 - 10-21, consistent with the collective- emission model of Secs. 3.3.2 & 3.7. These values fall within the sensitivity window of the proposed GHz–THz detectors.

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Keynote aspects with: Above flowchart illustrating integrated pipeline used in this work overall: (1) Skyrme- type hadronic crystal initialization; (2) static-field solution and energy-density mapping; (3) the nonlinear time evolution via curved-space Klein–Gordon equation typically; (4) extraction of quadrupole dynamics; (5) Fourier and coherence diagnostics; (6) computations of microscopic and collective GW strain. The diagram provides a structural overview connecting QCD-scale microphysics to observable high-frequency gravitational-wave signatures. Power Spectral Density and Gravitational-Wave Output: The gravitational-wave luminosity was computed using the relativistic quadrupole power expression [19, 31]:

5 ... ... 5 GW ij ij G P Q Q c =

Simulations yield a distinct spectral peak associated with the fundamental lattice oscillations mode, in full agreement with the expected modal structure of Skyrmion or pion- field crystals [18, 21]. The resulting power falls in the range of having 5 ~ 10 GW P − to 10-3 W per lattice site, suggesting that the dense domains comprised of having 12 10 > sites sites (typical of early-universe correlation lengths) would contribute significantly to the stochastic high-frequency GW backgrounds. In Figure 2 (Energy density slice) this has shown as the crystalline structure, while Figure 3A & 3B these are all shown to be time evolution of the Qxx(t) with having corresponding strain waveform. Stability and Parameter Dependence: The Skyrmion and the scalar-field lattices exhibit dynamic stability across evolution intervals tstable∼10-10 s, even under thermal perturbations of order T∼150 MeV and curvature gradients typical of the QCD epoch [14, 37]. The Key Scaling Behaviors: (i) strain amplitude increases with baryon density ρb; (ii) frequency increases with the lattice stiffness according to Skyrme parameter e and pion- field couplings fπ; (iii) curvature compression, introduced via Gcurve (r), enhances high-k modes and amplifies short- wavelength oscillations, feeding into the GW power spectrum.

These results confirm the robustness of hadronic crystalline configurations and their significance as micro- GW emitters in the early universe.

Multiscale Computational and Analytical Framework

Our Above Workflow Integrated Four Components: (1) Gravitational-wave emission modeling via linearized GR [20, 38]; (2) Hadronic lattice modeling using Skyrme-type effective with having overall field theories [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 33, 42, 43]; (3) Curved-spacetime scalar-field evolution through a Klein–Gordon system including gravitationally induced phase locking; (4) High-resolution numerical simulations employing finite-difference and leapfrog methods with Fourier-space diagnostics.

Algorithm 1 per Sec. 2.5 summarizes the full computational pipeline, from initialization of QCD-era physical parameters to the extraction of coherent strain spectra h(f).

Gravitational-Wave Emission: Single, Collective, and Coherent Sources

Single-Baryon Quadrupole Emissions: For a single baryonic emitter, using standard typically quadrupole radiation

2

4 4 single G Q h R c ω = , where

2 p p Q m r =∈ , ω=ℏ/ theory [20, 38]:

2 ~ p p I m r . For typical early-universe parameters: 36 33 ~ 10 10 single h − − − , consistent with microscopic quadrupole expectations as well as prior theoretical bounds. These appear within Sec. 2.5 Algorithm 1 as well.

(2I), and Coherent Collective Emission from Baryon Clusters: Having cluster of N baryons coherently locked within a hadronic crystal domain, according to Step 3 of Sec. 2.5 Algorithm 1: total single coh Kerr h Nh = Γ Γ , with coherence factor τ τ Γ ~ coh coh cycle and Kerr-like enhancement (frame-dragging analogue) 3 1 Kerr GM a c R Γ ≈+ . Our simulations, with results

6 8 ~ 10 10 coh Γ − , kerr 2 Γ < . For realistic early-universe cluster sizes:

Figure 11, show:

12 15 ~ 10 10 N − , coherent strain reaches with typical values 22 21 ~ 10 10 total h − − − , placing these signals squarely within the projected detection band for GHz–THz GW detectors [2, 8].

Scalar-Field Lattice Formation and Energy Density Structure

Scalar-field effective theory with potential

( ) ( ) 2 2 2 4 V v λ φ φ = − produces kink–antikink crystals [6, 7, 21], with lattice sites anchored by baryonic delta sources, per results shown in Figure 5 as well. One-Dimensional Crystals: Solutions of equation:

2 2 2 2 n d v g dx φ λ φ φ = − − ∑ ( ) x nd δ − yield analytic kink ( )

m x v x x φ   = −    , m=√(2λ) v, with baryon- induced derivative jumps. These kinks typically form periodic crystalline chains, consistent with Skyrmion or pion-field lattice analogues all shown in Figure 5(ii) and Sec. 2.5 Algorithm 1 Step 4.

profiles ( ) ( ) 0 tanh 2

Two-Dimensional Crystals, Curvature, and Quantum Corrections: The 2D simulations incorporate: Coleman– Weinberg loop corrections to the potential. Curvature amplification ( ) 2 1 curve G r r α − = + . Thermal noise ~ 150 T MeV . These ingredients produce honeycomb and square crystalline textures, with curvature-induced compression toward the origin, referring to Figures 1, 2 & 6.

Energy density ( ) ( ) 2 1 , 2 eff p x y V φ φ = ∇ + exhibits localized nodes at baryon positions, matching theoretical expectations for topological and nontopological soliton lattices, brought out per Figure 2 results.

Fourier Structure, Mode Locking, and Coherence

2 k φ show peaks at the fundamental lattice wavenumber k=2π/d, with harmonics indicating nonlinear mode coupling. These peaks correspond directly to the GW spectral peaks shown as per Figure 7. High-k tails reflect curvature and thermal effects.

Fourier spectra ( )

The coherence measure ( )

i C t e θ = , referring to Figure 9, reveals plateau regions characteristic of mode- locking, shown results per Figure 8 (Arnold-tongue) behavior especially when gravitational oscillations drive the lattice: ( ) ~ '' g V φ ω ω φ = . These locked phases contribute significantly to the enhancement factor Γcoh.

Lyapunov Stability and Onset of Chaos

We computed local dynamic stability by the Lyapunov

( ) ( ) 1 ln 0

  =      . Results of the various cases: (i) λ<0: stable crystalline phase; (ii) λ≈0: marginal, metastable coherence; (iii) λ>0: chaotic, decohered lattice dynamics. These are graphically demonstrated as Lyapunov heatmaps, per results of Figure 10 confirming that coherent GW- emitting phases correspond to low-λ regions, while high- temperature or curvature-dominated regimes lead to that decoherence and suppressed GW output.

r T

δ λ δ T r

exponent

Synthesis: Total Gravitational-Wave Output from Early-Universe Hadronic Crystals

With above results, spanning Figures 1 to 12 combining overall action encompassing effectively: (i) intrinsic quadrupole emission with radiation ( ) 0 ( ) xx Q t Q sin t ω ≈ , (ii) collective enhancement N, (ii) that coherence amplification Γcoh, (iii) the Kerr-like frame-dragging ΓKerr, (iv) curvature compression of lattice modes, as well as (v) phase-locking between gravitational and the overall scalar-field oscillations, we obtain peak strain amplitudes, shown per Figure 11:

22 21 ~ 10 10 total h − − − for hadronic crystal domains formed near the QCD epoch. These strains lie within forecasted detection thresholds for next-generation GHz-to-THz high-frequency gravitational-wave having ongoing detectors, including resonant cavities and plasma/microwave interferometers [2, 8].

The results strongly suggest that hadronic crystals in the early universe functioned as coherent, micro-scale gravitational-wave emitters, potentially contribute to possibly high-frequency stochastically gravitational waves backgrounds.

Implications for Early-Universe Physics

Hadronic Crystallization and Microstructure Formations: Emergence of kink–antikink and Skyrmion lattices in our simulations is consistent with prior analytical predictions having solitonic ordering in strongly interacting matter [21, 35]. During the QCD epoch (T∼150 - 200 MeV), with having the short-range interactions between baryons, the presence of the coherent pion fields, as well as the typical rapid cooling of the universe may have naturally favored the formation of transient yet long-lived microcrystalline domains. Our results suggest that such structures provide a physical mechanism for converting thermal and/or scalar field-driven oscillations into coherent, directed quadrupole radiations. High-Frequency Gravitational Waves and the Early Stochastic Backgrounds: The predicted strain levels,

22 21 ~ 10 10 total h − − − , fall within projected sensitivity of the ongoing proposed high-frequency detectors operating in the GHz–THz range [2, 8]. If such signals were produced at possible cosmological densities, they would contribute to a primordial stochastic background at frequencies orders of magnitude higher than those probed by LIGO/Virgo/KAGRA or pulsar timing arrays.

All these high-frequency backgrounds remain essentially unexplored but are predicted in several with beyond- standard cosmological scenarios, including axion-driven inflationary turbulence, alongside primordial magnetic-field instabilities [14, 37], the cosmic string cusp radiation [9], and phase-transition dynamics [5]. Our model adds also that QCD-specific mechanism capable of generating the narrow- band, coherent, high-frequency components would be quite possible – that are typically distinct from the broadband turbulence signatures.

Coherence Mechanisms and Enhancement Factors

A central main feature of our results is the emergence of significant collective amplification. We have shown algorithmically derived mathematical coherence factors have Γcoh ∼ 106 - 108, arising from gravitational–field locking (analogous to the driven nonlinear oscillators) and mode synchronizations within the scalar-field lattice. This behavior echoes general phenomena of the phase locking, resonance, and nonlinear mode coupling observed normally in condensed matter systems as well as nonlinear optics [42, 43]. The analogy typically between the hadronic crystals and optical Skyrme lattices, as systems supporting solitons with long- range interactions, suggests that coherence within nonlinear field theories may be a universal feature across disparate physical regimes. Quite notable also is Kerr-like enhancement ΓKerr which accounts for rotational-frame typical dragging effects in dense, rotating baryonic clusters. Although these are smaller in magnitude, this factor contributes to the total strain that well reflects the general relativistic coupling of angular momentum to GW emission [29, 36].

These effects overall imply that early-universe hadronic matter may have supported emergent collective behavior analogous to well synchronized oscillators, potentially generating coherent radiation at microphysical scales.

Robustness of Lattice Formation and Wave Emissions

Our simulations, with graphic results plotted Figures 1 to 12 demonstrate that the lattice structures remain stable across time intervals characteristic of the QCD epoch (∼10-10 s), even in the presence of: (i) thermal fluctuations at T∼150 MeV, (ii) curvature gradients from near-horizon cosmological expansion, (iii) localized perturbations in the baryon density, and (iv) the typically well entrenched nonlinear interactions between field modes.

Lyapunov exponent analysis reveals distinct dynamical regimes: (i) low-λ coherent phases that sustain GW emission, (ii) marginally coherent phases where locking intermittently breaks, as well as (iii) high-λ chaotic phases, where GW output drops sharply.

These findings highlight a potential link between the thermal/dynamical state of early-universe matter and the spectral structure of any resulting high-frequency GW backgrounds.

Observational Prospects and Experimental Outlook Although present-day GW observatories are not sensitive to the GHz–THz regime, multiple detectors under development aim to explore this range:

• Microwave-cavity resonators [8],
• Axion haloscope–inspired interferometers [4],
• Superconducting radiofrequency (SRF) detectors,
• Josephson-junction arrays,
• Plasma-mirror interferometric designs [2].

The predicted strains for coherent hadronic crystal emission fall within theoretical reach of all these next- generation instruments. A detection, or even a meaningful constraint would have profound implications with the early QCD physics, nature of strong interactions within that typically knowhow present cosmological context, and the existence of soliton microstructure evolving networks well in the first microseconds after the Big Bang.

Outlook

Synthesis of Skyrmion physics, early-universe cosmology, nonlinear scalar-field dynamics, as well as gravitational-wave theory offers a compelling framework for understanding microstructure-induced gravitational radiation. If hadronic crystals had potentially formed during the QCD epoch, they could have generated a high- frequency GW background emissions whose coherence as well as spectral character carry imprints of primordial strong-interaction physics.

This study opens the possibility that gravitational waves could serve as a probe of QCD-scale phenomena, providing a new observational window into the microphysics of the early universe. As experimental efforts mature, the prospect of detecting such high-frequency signals provides strong motivation with deeper theoretical, numerical, and phenomenological investigations onto hadronic solitons, Skyrmion crystals, and early-universe phase coherence measurements.

Summary of Key Findings

In this work, we investigated the hypothesis that hadronic crystals - Skyrmion lattices as well as scalar-field kink–antikink networks- formed transiently during the QCD epoch and acted as potential coherent micro–sources of high- frequency gravitational waves (GWs). By combining Skyrme- model effective field theory, curved-spacetime scalar- field dynamics, as well as quadrupole-based gravitational radiation theory, we produced a unified analytical with practical workable computational theoretical framework capable of quantifying the gravitational-wave emission from microstructure of the baryonic matter. Overall, the principal findings include: • Formation of Hadronic Crystalline Phases: Numerical simulations reveal that the typical nonlinear pion and scalar fields can form ordered crystal structures under the QCD-era temperatures and cosmological expansion conditions. These structures potentially and typically remain dynamically stable over characteristic timescales of the early universe. • Quadrupole Oscillations in Microstructured Matter: Lattice vibrations generate a time-dependent mass quadrupole moment with having natural frequencies in typically GHz–THz range, aligning with expectations for microphysical oscillations of strongly interacting fields. • Significant Gravitational-Wave Emission: Single- element quadrupole emission is weak, as expected, but collective, coherent effects dramatically enhance the output. Predicted strain amplitudes reach

22 21 ~ 10 10 total h − − − , for typically, physically reasonable coherence lengths and cluster sizes. • Role of Coherence and Curvature: Phase locking between the gravitational oscillations and expected typical scalar-fields modes leads to large coherence factors. We have mathematically computed, simulating curvature compression and Kerr-like rotational couplings that act as sub-leading but non-negligible amplification mechanisms. • Potential Cosmological Signatures: The resulting GW spectrum is seen to consist of the well also typically narrow-band and high-frequency, distinguishing it from describable turbulence-driven known backgrounds associated with phase transitions or cosmical strings. Consequently, hadronic crystals may contribute unique components to the primordial stochastic GW backgrounds. Conclusions: Our results thus reinforce the feasibility of QCD- scale microstructured matter to be a novel source of high- frequency gravitational radiation and highlight the broader significance of the existent potentially typical nonlinear field dynamics in early-universe environments. Though transient, the emergence of ordered baryonic and scalar-field lattices provides an efficient mechanism for effectively converting local oscillations into coherent GW emission.

This work strengthens the argument that gravitational waves may serve as a new observational window into strong-interaction physics during the earliest microseconds of cosmic history. As high-frequency gravitational-wave detection technologies continue to advance, the possibility to probe QCD-era microphysics via gravitational signals becomes increasingly compelling.

Several natural extensions follow from this study

• Fully 3D Simulations of Skyrmion and Scalar-Field Lattices: Moving beyond 1D and 2D domains is essential for capturing defects, dislocations, torsional modes, and the full geometrical complexity of realistic hadronic crystals. A 3D lattice may thus potentially yield stronger coherence and richer GW spectra. • Incorporating More Realistic QCD Potentials: We expect having the future models to integrate chiral Lagrangians, Polyakov-loop dynamics, color superconductivity, and quark–gluon interactions. These refinements may alter the stability and oscillation spectra of hadronic crystals and modify GW emissions. • Nonlinear General Relativistic Backreactions: Including full GR backreaction will clarify whether GW emission modifies the lattice evolutions, coherence persistence, or mode structure, particularly in dense or rapidly rotating regions. • Coupling to Magnetic Fields and Axionlike Interactions: Primordial magnetic fields and axion–pion couplings may interact strongly with hadronic crystal oscillations. This potentially could produce hybrid electromagnetic–gravitational signatures or resonant enhancement analogous to axion-photon conversion. • Predictions for Experimental Detections: With detector enhanced concepts in development - such as microwave-cavity resonators, superconducting radio- frequency systems, and plasma-mirror interferometers - there is opportunity to produce direct, testable predictions for high-frequency GW observatories. Mapping model parameters to measurable spectral features will be a priority. • Implications for Physics: Cosmological Phenomena: Hadronic-crystal-induced GWs may influence baryon inhomogeneities, seed primordial curvature perturbations, or contribute to pre-BBN microphysics. Investigations of all these possible key typical connections could deepen our understanding of QCD- era cosmology.

Outlook

The intersection of QCD soliton physics, nonlinear field theory, as well as typical gravitational-wave cosmology represents a rapidly emerging area of research. As theoretical tools sharpen to expand such experimental capabilities onto the GHz–THz range, the prospect of uncovering gravitational signatures of microphysical processes in the first microseconds of the universe will then also become increasingly reality.

This work provides a foundational step in that direction, offering a roadmap for future theoretical, computational, and experimental efforts aimed at unveiling the natural evolving physics with gravitational-wave fingerprints of the early strong-interaction era.