Time Crystals in Cosmology: Thesis, V ersion 2 Under development

A single action with R + R² gravity and three time crystals describes inflation and its end, the change in the electron mass at recombination and, with the third crystal (an axion crystal), dark matter . With Planck, ACT DR6, DESI DR2, P antheon+ and weak lensing, and without local measurements, it gives H₀ = 70.6 ± 0.8 km/s/Mpc (ΛCDM: 68.1 ± 0.2), with an electron 1.12 ± 0.47 % heavier at recombination. With the neutrino mass free it gives H₀ = 71.2 ± 1.1 and Σm _ ν = 0.12 ± 0.08 eV . With the minimum mass set by oscillations, it beats ΛCDM without using the local H₀ (ln B = +0.9; +3.1 if the hierarchy is inverted). Before inflation. The pre-universe is a de Sitter phase inherited from the previous aeon (conformal cyclic cosmology) with a dynamical vacuum ρ_ vac ∝ H⁴ . It is unstable: Ḣ̇ = −2H²(1 − H²/H*²), and the inherited gravitational waves make it decay . Its curvature leaves the scalaron on the plateau if H* ≥ 4.58 M = 1.36×10¹⁴ GeV (chapter 4, Appendix B). If the displacement of Ψ₂ arises from a ξRρ² coupling, ≳ 150 e-folds of tachyonic phase are needed (H* ≳ 7 M) and the imprint of the previous aeon is not visible (4.5). Ψ₁, the early crystal. It sustains Starobinsky inflation and ends it by melting; its latent heat reheats almost instantly . It predicts n _ s ≈ 0.97 4 and r = 0.0025 ± 0.0004, measurable by LiteBIRD and Simons Observatory . Ψ₂, the late crystal. It leaves inflation displaced, frozen by Hubble friction, and melts to its symmetric vacuum after recombination (mass 1.5×10⁻²⁹ eV). While it is displaced, the electron is heavier: recombination happens earlier , r _ d drops to 143.6 Mpc and H₀ rises. It satisfies the equivalence principle, atomic clocks and quasars. Its initial displacement (ρ_pre ≈ 0.28 M _ P) is taken as an initial condition, as in ΛCDM (10.16); the coupling to teleparallel torsion (2.5) remains an exploratory reading. Dark sector. Dark matter is a QCD axion of 95–450 μeV , the only one compatible with the model's inflation scale. Dark energy is a constant; a tachyonic field at the top of a cosine reproduces it, but the data fix it at w today < −0.944. The lensing S8 stays at 0.805 ± 0.011 with baryonic feedback. Statistics. MCMC with exact CLASS at every point. With Σm ν = 0.06 eV the evidence _ ties with ΛCDM without local H₀ (ln B = −0.4); with SH0ES or H0DN it is decisively f avoured (ln B = 8–17); with CCHP there is no preference. The tension with SH0ES drops from 4.6σ to 1.8σ (1.2σ with Σm _ ν free). Page 1 of 95Time Crystals in Cosmology: Thesis, Version 2 Declared risks. The age of the universe drops to 13.38 ± 0.13 Gyr (13.27 with Σm _ ν free); e± annihilation leaves the electron 2–5 % heavier when neutrons decouple, with helium at 1.0–2.0σ, and narrows the displacement of Ψ₂ to 0.14–0.21 M _ P; the mass of the radial mode of Ψ₂ is a fine-tuning that the thesis declares as an assumption (9.4; quantum correction ∼ 10²⁴ times m²); and the key parameters are fitted rather than derived. Where the thesis does not derive a value, it adopts the ΛCDM treatment (10.16). The model's own predictions for this decade: r ≈ 0.0025; a CMB lensing amplitude 2–3 % larger than in ΛCDM; a sound horizon r _ d = 143.6 Mpc, 2.8 % smaller , measurable with BAO and an independent H₀; slightly high primordial helium (Y _p ≈ 0.251); Σm ν ≈ 0.1 eV _ compatible with cosmology; and a 95–450 μeV axion, with a pulsed 23–109 GHz radio line in magnetars.