UTMF v5.2
Gravitational Waves
Cosmology
Falsifiable Physics

UTMF v5.2: Gravitational Waves from Unified Topology

A falsifiable GW module tied to renormalization, thresholds, and cosmology

Dustin Beachy•September 2025
Abstract

The Unified Tuple–Matrix Framework (UTMF) unifies a topological BF-like gravity sector, an SO(10) gauge completion, and a tuple-curvature matter sector. Versions v5.0–v5.1 established one-loop RG flows with a quasi-invariant GUT boundary, two-loop SM running with thresholds and a linearized unification fit, benchmark dark-matter (DM) sectors (tuple-ALP and tuple-soliton), inflation (flattened quartic), and a reheating–leptogenesis link.

This v5.2 release adds a complete gravitational-wave (GW) module, providing a common redshift backbone and derivation-grade source physics with exact spectral shapes and closed-form amplitudes parameterized directly by UTMF inputs. We cover: (i) primordial inflationary tensors; (ii) first-order phase transitions (FOPT) in the tuple sector, with α and β/H★ from the finite-temperature potential V_eff; (iii) local (gauge) cosmic strings from tuple symmetry breaking; (iv) global ALP strings/domain walls; (v) preheating GWs from inhomogeneous inflaton decay; and (vi) rare bursts from tuple-soliton dynamics.

We include the exact sound-wave shape with the 8/7 constant, a scaling Nambu–Goto loop distribution n(l,t)=ζ/(l²t²) with ζ≃2.2, lattice-motivated preheating exponents (γ,δ)=(1,1), and worked FOPT benchmarks that land in the LISA band with

\Omega_{\rm sw}h^2\sim10^{-15}
. The module integrates with v5.1 falsification packs and specifies detector-band targets without introducing new free parameters.

Introduction and Scope

UTMF recap

The framework encodes curvature and flavor in braided tuples over a discrete/topological substrate, coupled to a BF-type gravitational sector and unified SO(10) gauge fields. Earlier releases demonstrated a UV-complete candidate, flavor-from-tuples, topological singularity avoidance, realistic RG flows and SM matching, DM and inflation benchmarks, and a falsification roadmap.

Goal of v5.2

Elevate falsifiability by adding a complete GW module whose observables are algebraic functions of UTMF parameters fixed or scanned in v5.0–v5.1:

  • FOPT parameters (α,β/H★,v_w) from the tuple-sector V_eff(τ,T)
  • Local strings from gauge-tuple breaking scale η (Gμ∝η²)
  • Global ALP strings/walls from (f_a,m_a)
  • Preheating from the v5.1 inflaton sector (m_φ,T_reh)
  • Primordial tensors from r in B3 (flattened quartic inflation)
  • Soliton bursts tied to tuple-defect dynamics

Cosmological Redshift Backbone

Let

\bar M_P
be the reduced Planck mass. For a radiation-era source at temperature T★ with entropy/gauge DOF (g*S,★,g★):

H_\star = \sqrt{\frac{8\pi^3}{90}}\ \frac{\sqrt{g_\star}\,T_\star^2}{\bar M_P}, \quad \frac{a_\star}{a_0} = \left(\frac{g_{*S,0}}{g_{*S,\star}}\right)^{\!1/3}\frac{T_0}{T_\star}

Observed peak frequency and present energy density:

f_0 \simeq 1.65\times10^{-5}\ \mathrm{Hz}\ \left(\frac{T_\star}{100\ \mathrm{GeV}}\right)\left(\frac{g_\star}{100}\right)^{\!1/6}
\Omega_{\rm GW,0}(f)=\Omega_r\ \mathcal T^2(f)\ \frac{1}{\rho_r}\left.\frac{d\rho_{\rm GW}}{d\ln f}\right|_\star

with

\Omega_r h^2\simeq 4.2\times10^{-5}
.

Primordial Inflationary Tensors (UTMF B3)

With tensor power

P_t=rA_s(k/k_\star)^{n_t}
and
n_t\simeq -r/8
:

\Omega_{\rm prim}h^2 \simeq 3.7\times 10^{-15}\left(\frac{r}{0.01}\right)

UTMF B3 points with r≃0.008 imply

\Omega_{\rm prim}h^2\sim 3\times 10^{-15}
(quasi-flat over mHz–Hz).

Tuple-Sector First-Order Phase Transition (FOPT)

High-T potential and critical data

Adopt:

V_{\rm eff}(\tau,T)= D\,(T^2-T_0^2)\,\tau^2 - E\,T\,\tau^3 + \frac{\lambda_T}{4}\tau^4

Coefficients from UTMF dof and couplings:

D=\frac{1}{24}\sum_b n_b a_b+\frac{1}{48}\sum_f n_f a_f, \quad E=\frac{1}{12\pi}\sum_b n_b a_b^{3/2}\quad(\text{bosons only})

with mappings a_g=2C_2 g² (gauge), a_s=y²/16 (scalars), a_f=y²/16 (fermions). Degenerate minima at T_c give:

\frac{v_c}{T_c}=\frac{2E}{\lambda_T}\equiv R, \quad T_c^2=\frac{T_0^2}{1-\frac{E^2}{D\lambda_T}}

Latent heat L and strength α=L/ρ_r:

L=\frac{8T_c^4}{\lambda_T^3}\Big(DE^2\lambda_T-E^4\Big)
\alpha=\frac{240}{\pi^2 g_\star}\,\frac{R^2}{4}\Big(D-\frac{R^2\lambda_T}{4}\Big)

Consistency requires E²/(Dλ_T)<1 and D-R²λ_T/4>0.

Nucleation and β/H★

In the thin-wall regime:

\frac{S_3}{T}\simeq \frac{16\pi}{3T}\,\frac{\sigma^3}{(\Delta V)^2}, \quad \sigma \simeq \frac{2\sqrt{2}}{3}\frac{E^3T_c^3}{\lambda_T^{5/2}}

and:

\frac{\beta}{H_\star}= T\frac{d}{dT}\Big(\frac{S_3}{T}\Big)\Big|_{T_\star}

For mild supercooling Δ=(T_c-T★)/T_c≪1:

\frac{\beta}{H_\star}\ \approx\ 0.0137\ \frac{R^5\,\lambda_T^{3/2}}{\big(D-\frac{R^2\lambda_T}{4}\big)^{2}}\ \Delta^{-3}

GW from sound waves (exact shape)

Peak frequency and amplitude (Caprini consensus):

f_{\rm sw,0}\simeq 1.9\times10^{-5}\ \mathrm{Hz}\ \frac{\beta/H_\star}{v_w}\left(\frac{T_\star}{100\,\mathrm{GeV}}\right)\!\left(\frac{g_\star}{100}\right)^{1/6}
\Omega_{\rm sw}h^2 \simeq 2.7\times 10^{-6}\ \Big(\frac{H_\star}{\beta}\Big)\Big(\frac{\kappa_{\rm sw}\,\alpha}{1+\alpha}\Big)^{2}\Big(\frac{100}{g_\star}\Big)^{1/3} v_w\ \Upsilon\ \mathcal S_{\rm sw}(f)

with finite-lifetime factor Υ=min(1,H★τ_sw). The exact unit-area shape used:

\mathcal S_{\rm sw}(f)=\frac{(f/f_{\rm sw})^{3}}{\Big[1+\frac{8}{7}(f/f_{\rm sw})^{2}\Big]^{7/2}}

A practical efficiency fit is κ_sw≃0.5α/(0.5+α). Optimization knob: maximizing α(R)∝R²(D-R²λ_T/4) gives:

R_{\max}=\sqrt{\frac{2D}{\lambda_T}}, \quad \alpha_{\max}\ \propto\ \frac{D^2}{\lambda_T}

Local (Gauge) Cosmic Strings from Tuple Breaking

For a gauge-tuple breaking scale η:

G\mu = 2\pi\left(\frac{\eta}{M_P}\right)^{2}

In the scaling regime, Nambu–Goto loop distribution (simulations):

n(l,t)=\frac{\zeta}{l^{2}\,t^{2}}, \quad \zeta\simeq 2.2

The stochastic background from loop emission (power Γ≃50) produces a radiation-era plateau ∝ζΓ(Gμ)² with a turnover near f_eq∼nHz. In UTMF, η traces the tuple-Higgs scale ∼√λ_c v_tuple.

Global ALP Strings/Domain Walls (N_DW=1)

For the tuple-ALP (f_a,m_a):

\mu_{\rm glob}\simeq \pi f_a^2 \ln\!\frac{L}{\delta}\quad(\delta\sim m_a^{-1})
\sigma \simeq 8\,m_a f_a^2

Annihilation at T_ann redshifts to:

f_{\rm wall,0}\simeq 1.65\times10^{-5}\ \mathrm{Hz}\ \left(\frac{T_{\rm ann}}{100\,\mathrm{GeV}}\right)\left(\frac{g_\star}{100}\right)^{1/6}
\Omega_{\rm wall}h^2 \sim \epsilon_{\rm GW}\,\Omega_r\,\Big(\frac{G\,\sigma}{H_{\rm ann}}\Big)^{2}, \quad \epsilon_{\rm GW}\ll 1

For global networks the GW channel is subdominant due to Goldstone radiation; nevertheless it provides a consistency constraint linking the ALP benchmark region to any reported SGWB.

Preheating GWs (Inhomogeneous Inflaton Decay)

Let m_φ and T★∼T_reh come from v5.1 inflation/reheating:

f_{\rm preh,0}\simeq 2.6\times10^7\ \mathrm{Hz}\ \left(\frac{m_\phi}{10^{13}\,\mathrm{GeV}}\right)\left(\frac{T_\star}{10^{10}\,\mathrm{GeV}}\right)^{-1}\left(\frac{g_\star}{100}\right)^{-1/6}
\Omega_{\rm preh}h^2 \sim \epsilon_{\rm preh}\,\Omega_r\,\Big(\frac{H_\star}{m_\phi}\Big)^{2}
\mathcal S_{\rm preh}(f)=\frac{(f/f_p)^3}{1+(f/f_p)}

(lattice-motivated exponents γ=δ=1). For non-resonant or modest resonance cases, ε_preh≲10⁻¹.

Tuple-Soliton Bursts

Localized reconnections/mergers of tuple solitons radiate bursts with characteristic strain:

h_c(f)\simeq \sqrt{\frac{2G}{\pi^2}}\ \frac{1}{D\,f}\,\sqrt{\frac{dE_{\rm GW}}{df}}

and, for narrowband events, h_c∼GE_GW/(π²Df²).

UTMF Plug-Ins and Worked Benchmarks

Plug-ins (all algebraic)

  • FOPT: (g,y)→(a_b,a_s,a_f)→(D,E,λ_T)→(R,α,β/H★,f_sw,Ω_sw)
  • Local strings: η→Gμ→Ω_cs(f)
  • Global networks: (f_a,m_a)→(μ,σ)→(T_ann,f_wall,Ω_wall)
  • Preheating: (m_φ,T_reh)→(f_preh,Ω_preh)
  • Primordial: r→Ω_prim

B-FOPT-1 (LISA-band example, T★∼100 GeV)

{N_s,a_s; N_g,a_g; N_f,a_f} = {3,0.8; 3,2.25; 96,0.08}, λ_T=0.40, g★=106.75:

E=0.3255, D=0.5413, R=1.6285>1; α=0.0418, β/H★=4.16×10³. With v_w=0.8:

f_{\rm sw,0}\approx 0.13\ \mathrm{Hz}, \quad \Omega_{\rm sw}^{\rm peak}h^2\approx 1.8\times 10^{-15}

B-FOPT-2 (optimized R at fixed D,λ_T)

Take λ_T=0.30 with D≃0.5413:

R_{\max}=\sqrt{2D/\lambda_T}=1.90, \quad E=R\lambda_T/2=0.285

α≃0.0556, β/H★≃6.05×10³, f_sw,0≈0.14 Hz,

\Omega_{\rm sw}^{\rm peak}h^2\approx 2.5\times10^{-15}
.

Falsification Playbook (GW)

LISA-band (mHz–Hz)

Use closed (f_sw,Ω_sw) to predict points directly from (D,E,λ_T,v_w,Δ). Non-detection prunes (R,λ_T); detection fixes (α,β/H★).

PTA/nHz

A measured plateau ∝(Gμ)² constrains η in the tuple breaking chain; inconsistent η disfavors the local-string branch.

kHz–MHz

Null results upper-bound ε_preh and/or disfavor extreme m_φ.

Broadband sum

Ω_prim+Ω_sw+Ω_cs+Ω_wall+Ω_preh yields a multi-component spectrum with relative scalings fixed by UTMF inputs—over-/under-shoots are crisp falsifiers.

Conclusions

We delivered a complete, derivation-grade GW module for UTMF that is algebraically closed in terms of existing v5.0–v5.1 parameters, exact in its key spectral shapes and loop statistics, and benchmarked with realistic degrees of freedom to produce LISA-band predictions at

\Omega h^2\sim 10^{-15}
. The formulas specify what to compute and how it scales, enabling direct, parameter-free cross-checks once the v5.1 scans are run.

This completes the GW leg of UTMF's falsification program and ties collider, neutrino, DM, CMB, and GW observables into a single calculational pipeline.

Appendices

A. High-T Potential: Derivation of α

Starting from:

V_{\rm eff}(\tau,T)= D\,(T^2-T_0^2)\,\tau^2 - E\,T\,\tau^3 + \frac{\lambda_T}{4}\tau^4

degenerate minima at T_c imply v_c/T_c = 2E/λ_T≡R and

T_c^2 =T_0^2/(1-\frac{E^2}{D\lambda_T})
. The latent heat:

L = -T\left.\frac{d\,\Delta V}{dT}\right|_{T_c}=\frac{8T_c^4}{\lambda_T^3}(DE^2\lambda_T - E^4)

With

\rho_r=(\pi^2/30)g_\star T_c^4
:

\alpha=\frac{L}{\rho_r} =\frac{240}{\pi^2 g_\star}\left(\frac{DE^2}{\lambda_T^2}-\frac{E^4}{\lambda_T^3}\right) =\frac{240}{\pi^2 g_\star}\frac{R^2}{4}\left(D-\frac{R^2\lambda_T}{4}\right)

Design window: E²/(Dλ_T)<1 and R>1 (strong FOPT) together imply E∈(λ_T/2,√(Dλ_T)) and D>λ_T/4.

B. Nucleation: β/H★ under mild supercooling

Using S_3/T≃(16π/3T) σ³/(ΔV)² with σ above and ΔV ≈ LΔ/T_c near T★:

\frac{\beta}{H_\star}=T\frac{d}{dT}\Big(\frac{S_3}{T}\Big)\Big|_{T_\star} \approx 0.0137\ \frac{R^5\lambda_T^{3/2}}{\big(D-\frac{R^2\lambda_T}{4}\big)^2}\ \Delta^{-3}

The prefactor is conventional; the strong Δ⁻³ scaling is robust.

C. Sound-Wave Spectrum: Exact Shape

We use the unit-area consensus form:

\mathcal S_{\rm sw}(f)=\frac{(f/f_{\rm sw})^{3}}{\Big[1+\frac{8}{7}(f/f_{\rm sw})^{2}\Big]^{7/2}}

with f_sw,0 and Ω_sw h² given in the main text, including the finite-lifetime factor Υ.

D. Local Strings: Loop Distribution and Plateau

Simulations give a scaling loop number density:

n(l,t)=\frac{\zeta}{l^2 t^2}, \quad \zeta\simeq 2.2

with gravitational power ΓGμ² per loop (Γ≃50). Integrating over redshift yields a radiation-era plateau Ω_cs∝ζΓ(Gμ)² and a knee near matter–radiation equality, enabling a direct constraint Gμ↔η.

E. Global ALP Networks: Wall Tension and Burst

For N_DW=1, σ=8 m_a f_a² and annihilation at H∼H_ann (set by friction/explicit breaking) produce:

\Omega_{\rm wall}h^2 \sim \epsilon_{\rm GW}\,\Omega_r\,\Big(\frac{G\,\sigma}{H_{\rm ann}}\Big)^{2}, \quad \epsilon_{\rm GW}\ll 1

with

f_{\rm wall,0}\simeq 1.65\times10^{-5}\,\mathrm{Hz}\,(T_{\rm ann}/100\,\mathrm{GeV})(g_\star/100)^{1/6}
.

F. Preheating: Peak and Shape

The peak redshifts from f_p∼m_φ/2π to:

f_{\rm preh,0}\simeq 2.6\times10^7\ \mathrm{Hz}\ \left(\frac{m_\phi}{10^{13}\,\mathrm{GeV}}\right)\left(\frac{T_\star}{10^{10}\,\mathrm{GeV}}\right)^{-1}\left(\frac{g_\star}{100}\right)^{-1/6}

with amplitude Ω_preh∼ε_preh Ω_r(H★/m_φ)², and a simple broken-power shape:

\mathcal S_{\rm preh}(f)=\frac{(f/f_p)^3}{1+(f/f_p)}\quad (\gamma=\delta=1)

G. Worked FOPT Benchmarks (step-by-step)

B-FOPT-1

Choose dof and coefficients {N_s,a_s; N_g,a_g; N_f,a_f} = {3,0.8; 3,2.25; 96,0.08}, λ_T=0.40, g★=106.75.

E = \frac{1}{12\pi}\Big[3(0.8)^{3/2}+3(2.25)^{3/2}\Big]=0.3255
D = \frac{1}{24}(3\cdot 0.8+3\cdot 2.25)+\frac{1}{48}(96\cdot 0.08)=0.5413
R = \frac{2E}{\lambda_T}=1.6285>1, \quad \alpha=\frac{240}{\pi^2 g_\star}\,\frac{R^2}{4}\Big(D-\frac{R^2\lambda_T}{4}\Big)=0.0418
\frac{\beta}{H_\star} \approx 0.0137\,\frac{R^5\lambda_T^{3/2}}{\big(D-\frac{R^2\lambda_T}{4}\big)^2}\,\Delta^{-3} \ \Rightarrow\ 4.16\times10^{3}\ \text{for}\ \Delta=0.05

For v_w=0.8: f_sw,0≈0.13 Hz,

\Omega_{\rm sw}^{\rm peak}h^2\approx 1.8\times10^{-15}
.

B-FOPT-2 (optimized R)

Fix D≃0.5413, choose λ_T=0.30.

R_{\max}=\sqrt{2D/\lambda_T}=1.90\Rightarrow E=R\lambda_T/2=0.285
\alpha\simeq 0.0556, \quad \frac{\beta}{H_\star}\simeq 6.05\times10^{3}

Then f_sw,0≈0.14 Hz,

\Omega_{\rm sw}^{\rm peak}h^2\approx 2.5\times10^{-15}
.

References

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[2] J. J. Blanco-Pillado, K. D. Olum, and B. Shlaer, The number of cosmic string loops, Phys. Rev. D.

[3] T. Hiramatsu, M. Kawasaki, K. Saikawa, and T. Sekiguchi, Gravitational waves from domain walls, Phys. Rev. D.

[4] D. G. Figueroa and F. Torrenti, Gravitational wave production from preheating, Phys. Rev. D.

[5] M. Maggiore, Gravitational Waves, Vol. 1 (Oxford University Press).