A phenomenological a⁴ temporal scaling in late-time dark-matter momentum-exchange models and the S8 tension

We explore a phenomenological variant of the Dark Scattering / interacting dark energy framework (Simpson 2010; Pourtsidou & Tram 2016; Baldi & Simpson 2015), in which cold dark matter experiences a late-time velocity drag. Whereas standard Dark Scattering models parametrize the drag as proportional to the dark energy density, we test an alternative temporal scaling of the form a⁴, applied to the CDM velocity divergence in the Newtonian gauge. We emphasize that this functional form is adopted empirically and is not derived from first principles. Implemented in a modified version of CLASS v3.3.4 and analyzed via MCMC against BAO (DESI DR1, BOSS DR12), the Planck acoustic-scale prior, a BBN prior, and the DES Year 3 weak-lensing S8 measurement, this variant shows a 2.1σ preference for non-zero drag (α₀ = 2.7 ± 1.3), reducing the S8 tension while leaving the CMB acoustic scale and BAO sound horizon unchanged — consistent with the known behavior of momentum-exchange dark-sector models. This deposit includes the manuscript, the patch against CLASS v3.3.4, the likelihood and MCMC driver scripts, and the full MCMC chains for reproducibility. The central mechanism is not novel; the contribution is limited to testing a specific temporal scaling within an established model family.