Baryon fraction from the BAO amplitude: A consistent approach to parametrizing perturbation growth

Galaxy clustering constrains the baryon fraction 1b=1m through the amplitude of baryon acoustic oscillations and the suppression of perturbations entering the horizon before recombination. This produces a different prerecombination distribution of baryons and dark matter. After recombination, the gravitational potential responds to both components in proportion to their mass, allowing robust measurement of the baryon fraction. This is independent of new-physics scenarios altering the recombination background (e.g. early dark energy). The accuracy of such measurements does, however, depend on how baryons and cold dark matter (CDM) are modeled in the power spectrum. Previous template-based splitting relied on approximate transfer functions that neglected part of information. We present a new method that embeds an extra parameter controlling the balance between baryons and dark matter in the growth terms of the perturbation equations in the CAMB Boltzmann solver. This approach captures the baryonic suppression of CDM prior to recombination, avoids inconsistencies, and yields a clean parametrization of the baryon fraction in the linear power spectrum, separating out the simple physics of growth due to the combined matter potential. We implement this framework in an analysis pipeline using effective field theory of largescale structure with halo occupation distribution-informed priors and validate it against noiseless Lambda CDM and early dark energy cosmologies with Dark Energy Spectroscopic Instrument (DESI)-like errors. The new scheme achieves comparable precision to previous splitting while reducing systematic biases, providing a more robust way to baryon-fraction measurements. In combination with big bang nucleosynthesis constraints on the baryon density and Alcock-Paczy & nacute;ski estimates of the matter density, these results strengthen the use of baryon fraction measurements to derive a Hubble constant from energy densities, with future DESI and Euclid data expected to deliver competitive constraints.