Understanding acoustic scale observations: the one-sided fight against $\Lambda$

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 $\theta_*$ 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 $\Lambda$CDM 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 $F_{\rm AP}(z)$ relative to $\Lambda$CDM predictions. These inequalities demonstrate that certain deviations from $\Lambda$CDM 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 $\Lambda$CDM, this tension arises primarily in directions that are inconsistent with the null-energy condition, so $\Lambda$CDM is favoured by current acoustic scale measurements unless the null-energy condition is violated.