Understanding acoustic scale observations: the one-sided fight against Λ

Abstract The cosmic microwave background (CMB) and baryon acoustic oscillations (BAO) provide precise measurements of the cosmic expansion history through the comoving acoustic scale. The CMB angular scale measurement θ * is particularly robust, constraining the ratio of the sound horizon to the angular diameter distance to last scattering independently of the late-time cosmological model. For models with standard early-universe physics, this measurement strongly constrains possible deviations from ΛCDM at late times. We show that the null energy condition imposes strict inequalities on the BAO observables D H (z) , D M (z) , D V (z) and F AP (z) relative to ΛCDM predictions. These inequalities demonstrate that certain deviations from ΛCDM are impossible for any physical non-interacting dark energy model that respects the null energy condition within the context of FRW cosmological models. We also identify the regions of parameter space in the CPL parameterization w ( a ) = w 0 + w a (1 - a ) that can give predictions consistent with both the null energy condition and the observed CMB scale. While current DESI DR2 BAO measurements exhibit some joint-constraint parameter tensions with ΛCDM, this tension arises primarily in directions that are inconsistent with the null-energy condition, so ΛCDM is favoured by current acoustic scale measurements unless the null-energy condition is violated.