CMB Test of the Higgs Origin of Dark-Photon Dark Matter

Existing laboratory, astrophysical, and direct-detection searches constrain the kinetic-mixing portal $\epsilon$ of dark-photon dark matter but do not determine the cosmological origin of the relic abundance. We show that cosmic microwave background (CMB) isocurvature provides an independent probe of Higgsed dark-photon production histories: two models with identical $(m_{\text{Ap}},\epsilon )$ and identical present-day abundance can produce distinct cold-dark-matter (CDM) isocurvature signatures if their hidden-scalar evolution differs. The relevant observable is the logarithmic response of the final dark-photon abundance to the inflationary dark-Higgs displacement. We develop a model-independent response formalism and demonstrate that any perturbative inheritance branch with conserved comoving yield necessarily satisfies $\text{qeff}\ge 2$. Consequently, a perturbative branch accounting for the full dark-matter abundance through $h\to \text{Ap}\text{Ap}$ requires an initial coherent displacement exceeding $3.5\times {10}^4{H}_{\ast }$, while ordinary stochastic fluctuations over $\mathcal{O}(60)$ inflationary e-folds are exponentially unlikely to generate the required field amplitude. Viable Higgsed dark-photon scenarios therefore require either subdominant abundance, coherent initial conditions, suppression of the inherited scalar response prior to freeze-out, or sufficiently cold momentum evolution of the produced vector population.