The Scalar Spectral Index Through a Torsion-Regulated Bounce: The Shear Fraction Required by the Planck Tilt

This paper computes the primordial scalar spectral index produced when curvature perturbations cross the horizon through a torsion-regulated bounce, and inverts the calculation to find the bounce shear fraction that the observed tilt requires. In the model studied, the tilt obeys ns = 1 - 2 sqrt(f), with f the ratio of shear energy density to total density at crossing. A confined-matter value f = 0.000309 gives ns = 0.9648, a deviation of -0.014 standard deviations from the Planck 2018 value ns = 0.9649 plus or minus 0.0042. Inverting the relation, the Planck central value requires f = 0.000308, and the two-standard-deviation interval is f between 0.000178 and 0.000473. The required shear fraction is therefore small but far from zero: a bounce that is too isotropic produces too blue a spectrum, and one with f above 0.000473 is excluded by the tilt alone. The result converts a qualitative expectation, that bounce anisotropy reddens the spectrum, into a bounded interval that any complete bounce model must land inside. The study is computed from stated assumptions, not measured.