Equations & Glossary

The gravitational action with the Holst term. The second term (proportional to \(1/\gamma\)) is the parity-violating contribution from the Barbero-Immirzi parameter. This is the starting point of the entire framework -- standard Einstein gravity plus a topological-like correction that becomes physically relevant when torsion is present.

Torsion is algebraically determined by the spin density of matter -- it is not dynamical and does not propagate. In the minimal Einstein-Cartan-Holst framework, torsion is a constrained field that can be integrated out, leaving observable consequences in the matter sector.

After integrating out torsion, this parity-violating contact term emerges. \(J^{(A)}\) is the fermionic axial current \((\bar\psi\gamma_\mu\gamma_5\psi)\). The \(\gamma^2/(\gamma^2+1)\) factor encodes the Barbero-Immirzi parameter's physical effect. This interaction provides the repulsion that drives the cosmological bounce and is the seed of all parity-odd phenomena in the framework.

The key modification to standard cosmology. At low density, this reduces to standard GR. As \(\rho \to \rho_{\rm crit}\), the Hubble parameter \(H \to 0\) and the universe bounces rather than forming a singularity. The correction factor \([1 - \rho/\rho_{\rm crit}]\) is the single most important prediction of the framework.

No free parameters -- the critical density is determined entirely by the Barbero-Immirzi parameter \(\gamma = 0.274\). This is a sub-Planckian density, meaning the bounce occurs before quantum gravity effects dominate and the semiclassical description remains valid.

The 1+3 covariant decomposition of the Friedmann equation including shear \(\sigma\) and vorticity \(\omega\). Shear contributes positively (like extra energy density), while vorticity contributes negatively (like negative curvature). The vorticity term is the bridge between the bounce mechanism and late-time dark energy.

The full dark energy model. \(\Xi\) is the inflationary suppression factor and \(c_\omega\) controls the vorticity contribution. Important caveat: the equation of state \(w = -1\) at late times is assumed, not derived from first principles. The framework provides a mechanism for the magnitude of \(\Lambda\) but not its equation of state.

Converts a Planck-scale parity-odd effect into the observed dark energy scale. The exponential dilution factor \(\mathcal{D}_{\rm inf}\) arises from inflationary expansion. Reproducing the observed \(\rho_\Lambda \sim 10^{-123}\,M_{\rm Pl}^4\) requires \(N_{\rm tot} \approx 92\) e-folds -- a testable, quantitative requirement.

Standard formula relating the polarization rotation angle \(\beta\) to the EB cross-power spectrum. A non-zero \(C_\ell^{EB}\) is a smoking-gun signal of cosmological parity violation. This is a model-independent relation used by all CMB experiments searching for birefringence.

The surviving positive prediction of the framework. The birefringence angle is independent of \(f_a\) due to an exact cancellation in the ALP coupling. With \(\theta_i \sim \mathcal{O}(1)\) from natural misalignment and \(C_\gamma\) fixed by the photon-torsion coupling, the prediction is sharp and testable by LiteBIRD (launch early 2030s, JAXA JFY2032, \(\sigma \sim 0.03°\)).

Phenomenological fit for galaxy spin asymmetry as a function of redshift. \(A_0 \sim 0.003\) sets the amplitude. Status: contested. There is a 9--12 order-of-magnitude coupling gap between the theoretical torsion-induced mechanism and the observed signal amplitude. This observable is included for completeness but is not a robust prediction.

Order-of-magnitude estimate from a fermion loop diagram. The first term is the perturbative one-loop contribution; \(\delta_{\rm NY}\) is the Nieh-Yan topological correction. In practice, \(\alpha/M\) is treated as a phenomenological parameter because the full non-perturbative calculation is not available. Its value is constrained by requiring consistency with the observed \(\rho_\Lambda\).

A prediction with no free parameters in the cubic sector from matter-dominated contraction (Cai et al. 2009). Algebraically verified at three momentum configurations. The full-commutator polynomial (6,2,−18,10,−66,18) was derived from Cai's own intermediate vertex contributions via 2×(Eqs. 34+35+36) at ε = 3/2 using exact rational arithmetic—not a fit, but the physics-derived result. The published (3,1,−9,5,−66,9) are the single-ordering coefficients. No independent verification of −35/8 found in the 2020–2024 literature. Approximately 300× larger than standard inflation (\(f_{\rm NL} \sim 10^{-2}\)) and opposite in sign. Mechanism-independent (holds for ECH, LQC, or any nonsingular bounce). Testable by SPHEREx at ~5.5σ (template-corrected) and MegaMapper at ~7.5σ. Bayesian comparison (600K+ MC): bounce favored at ~8-17:1 over tuned multifield (prior-dependent).

Relates the measured non-Gaussianity amplitude (using a local-template estimator) to the true matter-bounce value. The template projection factor \(r \approx 0.85\text{--}0.90\) (CMB Fisher near 0.90, LSS/SDB nearer 0.85; validated by \(\ell\)-space Fisher at 0.878 and injection recovery at \(r_{\rm meas} = 0.90 \pm 0.01\)). With the physics-derived full-commutator polynomial (6,2,−18,10,−66,18), the mismatch is intrinsic. The 10–15% loss comes from folded and intermediate bispectrum configurations where the bounce and local shapes differ. This correction degrades the naive SPHEREx significance from 6.2σ to ~5.0–5.5σ.

The spectral consistency relation linking the non-Gaussianity amplitude to the scalar spectral index \(n_s\). The coefficient \(c\) is bounded: explicit cubic-action prefactors give \(c \approx -0.7\), while including mode-function amplitude changes near the singular point \(\epsilon = 3/2\) gives \(c \approx -10\). At the Planck best-fit \(n_s = 0.9649\), this gives \(f_{\rm NL} \in [-4.35,\;-4.02]\) — a 1–8% correction, well within SPHEREx measurement uncertainty \(\sigma \approx 0.7\).