Chapter 17: Cosmological Probes

17.1 CMB Log-Periodic Oscillations — The Distinction Fingerprint

Theorem 17.1 (The smoking gun of nested distinctions)

$P(k) = P_0(k)[1 + A \cos(2\pi \log k / \log q + \phi)]$ with $A \sim 0.01-0.05$. Discrete tree branching — the hierarchical distinction structure of spacetime — imprints a periodic modulation in $\log k$ on the primordial power spectrum.

Test: Bayesian search in Planck 2018 data and future CMB-S4. Current Planck data show marginal $2-3\sigma$ features consistent with this prediction.

17.2 Dark Matter

DM = tree boundary modes — distinction states trapped at the spacetime boundary with suppressed couplings to bulk matter:

$m_\text{DM} \sim M_\text{Pl} \cdot q^{-d_\text{DM}} \sim 100$ GeV for $d_\text{DM} \sim 30$
$\sigma_\text{SI} \sim 10^{-46}-10^{-48}$ cm$^2$ — below current LZ/XENONnT limits, within DARWIN reach

17.3 Inflationary Predictions

Parameter	Tree Prediction	Planck 2018
$n_s$	$0.960-0.970$	$0.9649 \pm 0.0042$
$r$	$< 0.01$	$< 0.036$ (95%)
Log-periodic $A$	$0.01-0.05$	Unconstrained (search ongoing)

17.4 Lorentz Violation

$\delta c/c \sim q^{-d(E)}$ where $d(E) \approx \log(M_\text{Pl}/E)/\log q$. Current GRB/UHECR bounds: $\delta c/c < 10^{-15}$ at GeV-TeV, $< 10^{-22}$ at EeV. Tree predictions are safely below current bounds but potentially detectable with next-generation observatories.

17.5 Baryon Asymmetry

$\eta \approx \varepsilon_\text{CP} \cdot q^{-d_\text{CP}} \cdot T_\text{reheat}/M_\text{Pl} \sim 6 \times 10^{-10}$ for $d_\text{CP} \sim 40$, matching observation. The matter-antimatter asymmetry is encoded in the distinction parity structure at specific tree depths.

Next: Chapter 18: Tabletop Experiments →