Primordial Gravitational Waves in Lava-Void Cosmology

Primordial Gravitational Waves in Lava-Void Cosmology

By C. Rich

To integrate primordial gravitational waves (PGWs) into Lava-Void Cosmology while maintaining the unified relativistic viscous fluid framework, these tensor perturbations are interpreted as gravitational wave modes amplified during the viscous quasi-de Sitter (inflationary) phase and sourced by anisotropic stresses arising from multifractal turbulent fluctuations in the fluid velocity field. At Planck-scale Reynolds numbers, intermittent vortex structures generate transient quadrupolar anisotropies, acting as a primordial source analogous to quantum vacuum fluctuations in scalar-field inflation but emergent from hydrodynamic turbulence.

This mechanism produces a stochastic gravitational wave background with a nearly scale-invariant or slightly tilted spectrum, consistent with current upper limits on the tensor-to-scalar ratio (r < 0.036 from Planck and BICEP/Keck constraints as of 2025). The helical nature of turbulence (from the chiral vortical effect extension) introduces parity violation, yielding circularly polarized GWs and potential TB/EB cross-correlations in CMB polarization—distinguishing signatures testable by future experiments (e.g., LiteBIRD, CMB-S4, LISA, or pulsar timing arrays).

Post-inflation, tensor modes propagate freely with minor viscous damping, contributing negligibly to late-universe dynamics but imprinting B-mode polarization on the CMB via the reionized and last-scattering surfaces.

Mathematical Formulation

Tensor perturbations are described in the transverse-traceless (TT) gauge, with the metric:
\[ ds^2 = -dt^2 + a^2(t)(\delta_{ij} + h_{ij}^{TT}(t,x)) dx^i dx^j \]

where \( h_{ij}^{TT} \) satisfies \( \partial_i h^{ij} = h_i^i = 0 \). In Fourier space, the mode functions \( h_\lambda(k,t) \) evolve via the sourced wave equation:
\[ \ddot{h}_\lambda + 3H \dot{h}_\lambda + \frac{k^2}{a^2} h_\lambda = 16\pi G \Pi_\lambda(k,t) \]

where \( \Pi_\lambda \) is the TT-projected anisotropic stress:
\[ \Pi_{ij} = -\frac{1}{a^3} [\rho(v_i v_j – \frac{1}{3} \delta_{ij} v^2) + \sigma_{ij}]^{TT} \]

During the quasi-de Sitter phase (\( H \approx \text{constant} \), \( \epsilon = -\dot{H}/H^2 \ll 1 \)):
\[ h_\lambda(k,\eta) \approx \frac{H(\eta_*)}{\sqrt{2k^3}} (1 + ik\eta) e^{-ik\eta} \]

Primordial power spectrum at horizon exit (\( k = aH \)):
\[ \Delta_t^2(k) = \frac{k^3}{2\pi^2} P_h(k) = \frac{2H^2}{\pi^2 M_{Pl}^2} \left( \frac{k}{aH} \right)^{n_t} \bigg|_{exit} \]

Tensor-to-scalar ratio:
\[ r = \frac{\Delta_t^2(k_*)}{\Delta_s^2(k_*)} \approx 16\epsilon_{visc} f_{turb} \]

For chirality:
\[ \Delta_t^R(k) – \Delta_t^L(k) \propto \langle \mathbf{v} \cdot \omega \rangle \frac{H^3}{M_{Pl}^4} \]
yielding circular polarization fraction \( P \sim 10^{-2} – 10^{-1} \) (potentially detectable).

Numerical integration (extending existing Friedmann solvers with tensor Boltzmann hierarchy, e.g., in modified CLASS) can propagate modes to recombination, predicting B-mode power:
\[ C_l^{BB} \approx r \cdot C_l^{BB,prim} + \text{lensing} \]
plus helical contributions to TB/EB. This extension resolves the treatment of tensor modes within the turbulent fluid paradigm, offering falsifiable predictions while aligning with null detections to date.

C. Rich