Gravitating sources with zero energy density in cosmology
A cosmological source can have identically zero comoving energy density and still gravitate through its pressure. This differs from an energy density passing through zero at an isolated time. Maintaining zero density with nonzero pressure requires energy exchange in a nonstatic FLRW universe. We construct proportional-pressure and constant-pressure backgrounds, including steady-state, bouncing, recollapsing, and $Λ$CDM expansion histories, and relate them to creation pressure, bulk viscosity, and time-dependent vacuum descriptions. Scalar fields can realize zero density along a trajectory; imposing it as an identity on a timelike $P(X,ϕ)$ domain selects $P=A(ϕ)\sqrt{X}$, locally a signed potential-free cuscuton for nonzero $A$. We formulate linear perturbations without dividing by the vanishing background density. For a nondegenerate $P(X,ϕ)$ scalar with a timelike field gradient, zero density and negative pressure cannot coexist with positive kinetic and gradient coefficients when the interaction leaves its principal kinetic structure unchanged. Two models couple the scalar to number-conserving dust through a field-dependent mass. The proportional-pressure trajectory is a homogeneous saddle and has no growing power-law dust mode on its accelerating branch in the matter-sourced quasistatic subhorizon approximation. A distinct constant-pressure model reproduces the $Λ$CDM expansion history, but in the same approximation the dust mode that initially grows during matter domination reaches a maximum at low redshift and then decays. These restrictions apply to the stated scalar class and models. Exact Bianchi I solutions demonstrate separately conserved zero-density sources supported by anisotropic stress. An inverse-pressure construction distinguishes the density of an interacting constituent from an inferred dark-energy density that may cross zero.