Testing a minimal one-parameter background extension with DESI DR1 BAO data and standard sirens

The Hubble tension becomes particularly acute when Baryon Acoustic Oscillation (BAO) data are forced to accommodate a local Hubble-constant prior. Rather than proposing a full alternative cosmology, we investigate whether a minimal one-parameter extension of the background expansion can absorb this mismatch. The modification introduces a single effective density term, $\Omega_{\mathrm{rheo}}$, to the Friedmann equation, interpreted phenomenologically as a delayed or non-instantaneous macroscopic background response. Under a strict local prior of $H_0 = 73.0\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$, this extension reduces the full-covariance $\chi^2$ against 12 recent DESI DR1 BAO measurements from $209.1$ (for a standard $\Lambda$CDM baseline) to $22.4$. While this represents a conditional consistency test rather than a global preference over $\Lambda$CDM, the framework distinguishes itself by yielding a falsifiable multi-messenger prediction: a redshift-dependent offset, $\Delta d_L/d_L \simeq \epsilon \ln(1+z)$, between electromagnetic and gravitational-wave luminosity distances. This effect remains negligible for local events such as GW170817, but reaches the sub-percent to percent level at cosmological redshifts, offering a direct observational discriminant for future standard-siren surveys.