Why r \ll 0.01: The Tensor-to-Scalar Ratio as a Falsification Test

The tensor-to-scalar ratio r is the most discriminating observable in early-universe cosmology. Standard slow-roll inflation generically predicts r \sim 0.01–0.1. The canvas model predicts r \ll 0.01. The difference is not a matter of parameter tuning—it follows from a fundamental structural difference in the inflationary mechanism.

What this paper provides:

· A derivation of the canvas model's prediction for r from first principles. In the canvas model, inflation occurs before any spacetime voxels exist. There is no metric, and therefore no propagating tensor modes are produced during the de Sitter phase. Tensor perturbations can only be generated during reheating, when the voxel lattice forms. These secondary tensor modes have an amplitude suppressed by the ratio of the lattice formation timescale to the Hubble time: r \sim (t_{\text{form}}/t_H)^2 \sim (\ell_P / H^{-1})^2 \sim 10^{-64}, effectively zero for all practical purposes.
· A comparison with standard inflation models. The paper surveys predictions for r across major inflationary scenarios: chaotic inflation (r \sim 0.13, already disfavored), natural inflation (r \sim 0.07, disfavored), Starobinsky R^2 inflation (r \sim 0.003), Higgs inflation (r \sim 0.003), and \alpha-attractors (r ranging from 10^{-4} to 10^{-2} depending on \alpha). The canvas model's prediction r \ll 0.01 is distinct and falsifiable.
· A clear falsification criterion. A detection of r \gtrsim 0.01 by upcoming CMB experiments (CMB-S4, LiteBIRD, Simons Observatory) would rule out the canvas model. A null result at r \sim 10^{-3} would rule out large classes of standard inflation models (chaotic, natural, Starobinsky, Higgs) while leaving the canvas model intact.
· A discussion of the consistency relation. In standard slow-roll inflation, r and n_s are linked by the consistency relation r = 8(1 - n_s), which with n_s \approx 0.965 gives r \approx 0.28, already ruled out. In the canvas model, there is no consistency relation because r and n_s arise from different mechanisms. The absence of a consistency relation is a prediction: n_s and r are independent observables, not linked by inflationary dynamics.
· A discussion of other CMB predictions: running of the spectral index \alpha_s = -2/N^2 \approx -6.6 \times 10^{-4}, non-Gaussianity f_{\text{NL}}^{\text{local}} \sim 1/N \sim 0.02, and zero isocurvature modes—all consistent with current data and with standard single-field inflation.

Why this matters:

The tensor-to-scalar ratio is the most powerful discriminator between the canvas model and standard inflation. The canvas model does not merely accommodate a small r—it demands it. This is the hallmark of a falsifiable prediction. Upcoming CMB experiments will provide a definitive test. A detection of r \gtrsim 0.01 falsifies the canvas model. A null result at r \sim 10^{-3} leaves the canvas model as one of the few frameworks consistent with the data.

Keywords: tensor-to-scalar ratio, r, canvas model, inflation, CMB, primordial gravitational waves, B-modes, falsifiability, CMB-S4, LiteBIRD, Simons Observatory, Starobinsky inflation, Higgs inflation, \alpha-attractors