Running of the Scalar Spectral Index From a Scale-Dependent Shear Fraction at the Bounce

If the conditions at a bounce vary with the scale crossing it, the spectral index runs. This paper computes the running produced when the shear fraction at crossing scales with wavenumber as a power law, f(k) = f* (k/k*)^gamma, and compares it with the Planck measurement. At the fiducial scaling exponent gamma = 0.12, the running at the pivot is alpha = -0.002109 and the running of the running is beta = -0.00012656. The Planck 2018 value is alpha = -0.0045 plus or minus 0.0067, so the prediction deviates by 0.36 standard deviations, comfortably inside the measurement. The same comparison bounds the scaling exponent: gamma above 1 would push the running outside the two-standard-deviation band. The observed near-absence of running therefore translates into a real constraint on how strongly the bounce conditions may vary across the decade of scales the microwave background probes. The study is computed from stated assumptions, not measured.