L26: CMB, Inflation, and Baryon Asymmetry

What the CMB Tells Us

The cosmic microwave background (CMB) is the afterglow of the early universe — light emitted 380,000 years after the Big Bang when atoms first formed and photons could travel freely. Its temperature fluctuates by roughly one part in 100,000 across the sky, encoding the spectrum of initial density variations.

Two numbers characterize that spectrum. The spectral tilt n_s: a value of 1 means perfectly scale-invariant fluctuations (equal power at every wavelength). The observed value n_s = 0.9649 ± 0.0042 (Planck 2018) is slightly less than 1 — a red tilt, meaning slightly less power at short wavelengths. The tensor-to-scalar ratio r: how much primordial gravitational radiation was produced. Planck 2018 + BICEP/Keck give r < 0.036 (95% CL).

Standard inflation

Requires a hypothetical inflaton field with an arbitrary potential V(φ). Different potentials give different n_s and r. The agreement with Planck data constrains model parameters — it does not derive them. n_s is fitted, not predicted.

GTE: MDL initial state

No inflaton field. The MDL principle selects the unique minimum-description initial configuration. n_s is derived from binary entropy and geometry. r = 0 exactly — tensor modes would cost extra description bits and are therefore MDL-forbidden.

Deriving n_s in Four Steps

The GTE spectral index follows a four-step certified chain, each step CatAL.

1

Binary entropy H(Z₂) = ln 2

The CMCA center cell under Rule 110 implements a NAND gate: p(L,1,R) = 1 − L·R = NAND(L,R). The NAND gate is the universal binary operation. The entropy of a fair NAND gate is ln 2 — forced by the polynomial's structure, not chosen.

2

Z₇ superselection — only Z₂ drives primordial perturbations

The Φ_MDL field has seven Z₇ winding sectors. The DHR theorem forbids coherent perturbations across Z₇ sectors at the flat-metric (cosmological) scale, because winding number is a topological invariant. Only the Z₂ binary sublayer remains as a coherent quantum fluctuation driver. If the full Z₇ contributed, n_s would be 0.901 — 15σ off. The superselection is what makes the prediction work.

3

Weyl miracle: mode density = 1 at the Planck scale

The number of modes with wavenumber ≤ k on the unit 3-sphere S³ follows N(k) = k³/3 (Weyl law). The logarithmic derivative at k=1: dN/d(ln k)|ₖ₌₁ = Vol(S³)/(2π²) = 2π²/(2π²) = 1 exactly. The S³ volume and the Weyl denominator are both 2π² — they cancel. Mode density = 1.

4

β_G = ln 2 / (2π²) → n_s = 1 − β_G

The running rate is the binary entropy multiplied by the mode density: β_G = (ln 2) × 1 / (2π²) = ln 2 / (2π²) = 0.035124. The spectral index: n_s = 1 − β_G = 0.96488.

n_s = 1 − ln 2 / (2π²) = 0.96488

Planck 2018: 0.9649 ± 0.0042 | Deviation: +0.004σ | Zero free parameters | 14 Lean theorems, CatAL

Why ln 2 specifically? Because the NAND gate is binary — it operates on two states, giving entropy ln 2. If the CA were ternary, entropy would be ln 3 and n_s would be wrong. The binary structure of Rule 110 is not adjustable — it is forced by the GTE polynomial.

r = 0: No Primordial Gravitational Waves

The GTE MDL initial state predicts r = 0 exactly — no primordial gravitational waves. The argument is structural: primordial tensor modes require extra information to specify beyond the scalar density field. MDL-minimality therefore selects an initial state with zero tensor fluctuations.

Sharp falsification criterion: Any confirmed detection of r > 0.001 at 5σ significance by LiteBIRD (projected launch 2028, σ_r ≈ 0.001) or CMB-S4 would definitively falsify the GTE MDL initial state derivation. Most standard inflation models predict r > 0.001. Current constraint: r < 0.036 (95% CL).

The Baryon Asymmetry: Why Matter Exists

For every billion photons in the CMB, there is exactly one more baryon than antibaryon. This baryon asymmetry η_B ≈ 6.1 × 10⁻¹⁰ is the source of all atomic matter. Without it, every baryon would have annihilated with an antibaryon — no atoms, no stars, no us.

The Standard Model can explain baryogenesis in principle (Sakharov conditions: B-violation, CP violation, out-of-equilibrium processes), but the SM-predicted asymmetry is far too small. New physics is needed — and GTE provides it through the FKTT mechanism.

The GTE mechanism: the lightest right-handed neutrino N₁ (a kink excitation of Φ_MDL) decays with CP violation, generating a lepton asymmetry that electroweak sphalerons partially convert to a baryon asymmetry. The key is the kink-top coupling:

g_kink-top = ε_FN = exp(−π/N_c) = exp(−π/3) ≈ 0.3506

N_c = 3 colors (CatAL). BPS instanton action per CMCA tape = π/N_c. Not fitted — derived from the kink structure of the GTE field theory.

η_B = 6.109 × 10⁻¹⁰

Planck 2018 + BBN: (6.10 ± 0.06) × 10⁻¹⁰ | Deviation: +0.15σ | Zero free parameters | CatAL

Key Points

GTE predicts n_s = 1 − ln 2/(2π²) = 0.96488 from a four-step chain: NAND binary entropy → Z₇ superselection → Weyl mode density → spectral index. All 14 intermediate theorems are CatAL (zero sorry, zero axioms).
Planck 2018 measures n_s = 0.9649 ± 0.0042. The GTE prediction deviates by +0.004σ — the most precisely confirmed quantitative prediction in the GTE programme.
The MDL initial state predicts r = 0 (no primordial gravitational waves). LiteBIRD and CMB-S4 can falsify this at σ_r ≈ 0.001.
The baryon asymmetry η_B = 6.109 × 10⁻¹⁰ is derived from the FKTT mechanism: BPS kink-top coupling ε_FN = exp(−π/3), all CatAL. Matches Planck 2018 + BBN at +0.15σ.

CMB, Inflation, and Baryon Asymmetry

What the CMB Tells Us

Deriving ns in Four Steps

r = 0: No Primordial Gravitational Waves

The Baryon Asymmetry: Why Matter Exists

Key Points

See Also

Deriving n_s in Four Steps