Kasner and mixmaster behavior in universes with equation of statew>~1

Billyard et al. showed that the CLW scaling solution is stable to shear in expanding GR backgrounds, while in contracting GR backgrounds a canonical scalar is generically unstable to anisotropy (the shear instability), which motivates ekpyrotic phases with wφ≫1.

“Billyard et al. showed that the CLW scaling solution is stable to shear in expanding GR backgrounds [26], while in contracting GR backgrounds a canonical scalar is generically unstable to anisotropy (the shear instability), which motivates ekpyrotic phases with w φ ≫ 1 [27].”

During ekpyrotic contraction with wφ>1, the shear energy density ratio ρσ/ρφ ∝ a^{3(1+wφ)-6} → 0 as a→0, so the anisotropic contribution becomes negligible and the ekpyrotic phase acts as a strong attractor towards isotropy.

“Thus the anisotropic contribution becomes negligible during ekpyrotic contraction, and at the homogeneous level the ekpyrotic phase acts as a strong attractor towards isotropy, suppressing the growth of shear anisotropy.”

On the contracting branch, the field rolls down the steep negative potential into an ekpyrotic phase with wφ≫1; this super-stiff phase drives the homogeneous shear fraction Σ to zero exponentially, so the isotropic solution is a strong attractor.

“On the contracting branch (H < 0) the field rolls down the steep negative part of the potential (V < 0) into an ekpyrotic phase with w φ ≫ 1.As shown in Sec.IV A, this super-stiff phase drives the homogeneous shear fraction Σ to zero exponentially, so the isotropic solution is a strong attractor [27,34].”

In the ekpyrotic regime, the effective shear eigenvalue μeffΣ≃(3/2)(wφ-1) is positive for wφ>1, so along the contracting branch the shear mode decays and the ekpyrotic regime is a strong homogeneous attractor towards isotropy, unlike the marginally stable kinetic points with wφ=1.

“In the fixed-point language it is convenient to encode this as an effective shear eigenvalue µ eff Σ ≃ 3 2 w φ -1 ,(33) which is positive for w φ > 1.Along the contracting branch N → -∞, a mode with Σ ′ = µ eff Σ Σ decays as Σ(N ) ∝ e µ eff Σ N N →-∞ -----→ 0,(34) so the ekpyrotic regime is a strong homogeneous attractor towards isotropy [27], unlike the marginally stable kinetic points with w φ =”

A steep negative potential with wφ ≫ 1 makes the ekpyrotic regime a strong homogeneous attractor, with Σ ∝ e^{(q-2)N} → 0 along the contracting branch.

“By contrast, as shown in Sec.IV A, a steep negative potential with w φ ≫ 1 makes the ekpyrotic regime a strong homogeneous attractor [27,34]: Σ ∝ e (q-2)N → 0 along the contracting branch.”