A Modulated Two-Scalar Framework for Stable Non-Topological Solitons: Derivative Coupling, Spectral Stability, and Cosmological Bounds

Abstract

Motivated by dilaton-like conformal couplings in scalar-tensor gravity, we propose a covariant two-scalar field theory in which a dimensionless modulating scalar $\lambda(x)$ controls the kinetic, mass, and potential sectors of a complex scalar field $\Psi(x)$ through a $(1-\lambda)$ prefactor. The model possesses an exact, field-dependent $U(1)$ Noether current and a covariantly conserved stress-energy tensor. We demonstrate the existence of localized non-topological soliton (Q-ball) solutions and establish their classical orbital stability within the Grillakis--Shatah--Strauss (GSS) framework: formal nodal domain properties combined with high-resolution numerical diagonalization confirm the unconstrained radial Hessian operator possesses exactly one negative dilation mode ($n^- = 1$). Restricting perturbations to the constant-charge manifold removes this dilation instability, while the Vakhitov--Kolokolov condition $dQ/d\omega < 0$ ensures the constrained spectrum is strictly positive, establishing rigorous orbital stability. We verify this condition analytically via an explicit intermediate derivation in the thin-wall regime and numerically via adaptive shooting in the complementary thick-wall branch. Real-time hyperbolic integrations confirm non-linear stability and radiative relaxation under finite radial deformations. An explicit bare mass for $\lambda$ guarantees exponential Yukawa screening, shielding external long-range fifth forces. We formulate the linear perturbation scattering theory in the co-rotating frame, deriving the integrating factor that eliminates the advective gradient coupling and yields a self-adjoint effective radial scattering potential. In an expanding FLRW background, we delineate between the unfragmented homogeneous oscillating condensate behaving as cold dark matter ($\langle w \rangle \approx 0$) and the fragmented soliton phase generated via parametric resonance bands, deriving observational cluster collision bounds that constrain the macroscopic soliton parameter space.