Parametric Resonance of Higgsed Vector Dark Matter: Inflationary Initial Conditions and Sourced Displacements

Parametric resonance in a Higgsed Abelian sector provides an efficient mechanism for producing vector dark matter, but its viability depends crucially on the origin of the initial dark-Higgs displacement that seeds the resonance. In this work, we investigate this initial-condition problem in a weakly coupled Abelian-Higgs theory with potential $V={\lambda }_4({\phi }^2-v^2{)}^2\mathrm{/}4$, using the calibrated nonlinear broad-resonance relic map together with a stochastic inflationary analysis of the dark-Higgs condensate. We show that a minimal light-spectator realization fails under standard inflationary duration: while broad resonance and isocurvature constraints require ${\phi }_0\mathrm{/}{H}_I\gtrsim 3.3\times {10}^4,$ the stochastic equilibrium and finite-duration random walk produce only $\phi \mathrm{/}{H}_I=\mathcal{O}(1)\mathrm{.}$ This large displacement mismatch is robust against order-of-magnitude variations in the resonance efficiency and broadness threshold, establishing a model-independent obstruction to the stochastic branch. We then identify a distinct classically sourced branch, generated by a negative Hubble-induced mass, in which the condensate tracks a time-dependent minimum, ${\phi }_0=\kappa H_{\ast }\mathrm{/}\sqrt{{\lambda }_4},$ and the radial fluctuation remains heavy during inflation. In this case, the fixed-$e\mathrm{/}{\lambda }_D$ relic scaling shifts from $m_X\propto {\lambda }_4^{5\mathrm{/}8}{H}_I^{-3\mathrm{/}2}$ to $m_X\propto {\kappa }^{-3\mathrm{/}2}{\lambda }_4{H}_{\ast }^{-3\mathrm{/}2}\mathrm{.}$ We derive the simultaneous consistency conditions for this sourced branch, including broad resonance, adiabatic tracking, perturbativity, sub-Planckian displacement, thermal non-erasure, spectator backreaction, and control of inflationary vector fluctuations. Our results establish that Higgsed-vector resonance is not merely a dark-matter production mechanism, but a sensitive probe of the inflationary and reheating dynamics that determine its initial conditions.