Gaussian Scalar Modulation of the Local Jeans Scale in a Quartic Superfluid Dark Matter Core

We consider a quartically self-interacting dark matter condensate coupled to a slowly varying real scalar background. The scalar field is described by a Gaussian profile that interpolates between a central value and an approximately constant value in the outer part of the superfluid core. We treat this profile as an external, non-fluctuating background and derive the local Gross–Pitaevskii equation, equation of state, and sound speed. We then study the local Jeans instability, retaining both interaction pressure and quantum pressure. In the interaction-pressure limit, the Jeans wavelength is independent of the local density because the quartic condensate obeys [Formula: see text]. At fixed local mass density, a positive scalar coupling increases the local effective mass and decreases both the sound speed and the Jeans wavelength. Within the frozen-background approximation, we derive a closed-form expression for the local Jeans modulation across the crossover between interaction and quantum pressure. We show that the specific hydrodynamic scaling [Formula: see text], where [Formula: see text], is recovered only when interaction pressure dominates. The calculation is a local diagnostic: the scalar profile, condensate density, and gravitational potential are not solved self-consistently, and scalar perturbations are neglected.