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

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 DH(z), DM(z), DV(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.