A crossing theorem for CMB-distance-transparent dark-energy deformations

Abstract

Cosmological distances constrain dark-energy evolution through weighted integrals of the expansion history. This paper isolates a sign-change consequence for deformations around a flat [Formula: see text]CDM background. Let [Formula: see text], with [Formula: see text] continuous, and hold fixed [Formula: see text], the present matter and radiation sectors, flatness, the present dark-energy normalization, and standard early-universe physics. If [Formula: see text] is nonzero and one-signed, the exact comoving distance to last scattering [Formula: see text] is a strictly monotonic function of [Formula: see text] and cannot return to its [Formula: see text]CDM value at any nonzero amplitude. Hence an exactly CMB-distance-transparent nontrivial deformation must cross [Formula: see text]. Linearization gives the positive-kernel moment condition [Formula: see text], with [Formula: see text], which localizes crossings in smooth templates. For [Formula: see text], [Formula: see text] of the kernel mass lies below [Formula: see text], with quartiles [Formula: see text]; the smooth one-crossing templates considered here give [Formula: see text]. We perform a background posterior analysis using the public 13-component DESI DR2 BAO vector and full covariance together with the Planck 2018 distance-prior likelihood. In the CPL family, conditional on a positive crossing, [Formula: see text] at 68% credibility, with a 95% interval [Formula: see text]; [Formula: see text] of all retained draws cross in [Formula: see text]. The first-order transparent direction and an exact nuisance-conditioned transparent manifold remain inside the corresponding two-sided 95% posterior ranges. These posterior diagnostics quantify the transparency geometry without excluding either transparent diagnostic at two-sided 95% credibility. These are background-level results, not a full CMB/PPF likelihood or a microphysical phantom model.