Primordial Graviton Entanglement in Quasicrystalline Inflation

Abstract

We study the entanglement structure of gravitons in quasicrystalline inflation. Both parity-even and parity-odd interactions generate graviton pairs with nonzero total momentum. For opposite-helicity pairs with identical momenta, the parity-even contribution can be nonzero, whereas the parity-odd contribution vanishes. The transition from inflation to radiation domination produces additional conventional two-mode squeezing. We derive the resulting normal and anomalous covariances. The initially nonzero reciprocal blocks of the full field-momentum covariance remain nonzero under homogeneous evolution, although individual components can vanish. We show that this homogeneous, helicity-preserving evolution acts unitarily within each helicity sector and therefore preserves the entanglement between the complete right- and left-helicity sectors. Nevertheless, it can eliminate entanglement in the reduced state of a co-propagating pair. We illustrate this distinction using a solvable four-mode model.

The paper asks how spatial quasiperiodicity changes primordial graviton pairing and whether the resulting entanglement survives homogeneous evolution into radiation domination. It specifies an icosahedral, rank-six reciprocal structure and two quadratic tensor interactions: a parity-even kinetic modulation and a parity-odd three-spatial-derivative operator. First-order active and passive Bogoliubov kernels are derived, with closed-form time integrals for finite-duration interaction pulses. Both interactions create pairs satisfying $\mathbf k+\mathbf p=\mathbf G_{\mathbf n}$, permitting co-propagating rather than exclusively opposite-momentum partners. For opposite helicities at identical momenta, the even amplitude can remain nonzero, while the chosen odd operator vanishes through a factor proportional to $kp(p-k)$. Subsequent homogeneous evolution redistributes correlations between normal and anomalous covariances. Invertibility of the field–momentum transfer matrices establishes that an initially nonzero complete reciprocal covariance block cannot disappear, although individual entries can vanish. Entanglement between the complete right- and left-helicity sectors is preserved because homogeneous evolution factorizes across that partition. Reduced-pair entanglement need not survive. An exactly solvable four-mode Gaussian example gives $\widetilde\nu_-\ge\tfrac12\cosh(2s)e^{-2r}$, so the selected co-propagating pair is separable whenever $\cosh(2s)e^{-2r}\ge1$, irrespective of phases; $r$ and $s$ parameterize initial and subsequent squeezing. The calculation assumes a fixed classical medium, weak controlled interactions, and negligible spectator backreaction. It does not establish medium formation or lifetime, quantize phasons, or supply a unique nonlinear completion. The four-mode example is not the full multimode EFT output; the complete entanglement network remains unanalyzed.