A Cuscuton Representation of the Loop Quantum Cosmology Bounce

Loop Quantum Cosmology (LQC) replaces the big bang singularity of the homogeneous universe by a bounce, usually described by the modified Friedmann equation H 2=ρ(1-ρ/ρ c)/(3M p 2). We show that this background dynamics follows from a local cuscuton effective theory, whose scalar equation is a constraint rather than a wave equation. One way to establish this is to write the cuscuton in terms of an angular coordinate θ, identified with the LQC polymerization angle 2λb. Its constraint gives H proportional to θ, while the Einstein constraint gives ρ=ρ c 2(θ/2), exactly reproducing the LQC bounce. To our knowledge, this is the first closed-form, local, generally covariant realization of the exact standard flat-FLRW LQC background dynamics for minimally coupled matter satisfying null energy condition, without introducing additional local dynamical degrees of freedom. A branchwise Legendre transformation establishes a canonical equivalence between the clock-gauge-fixed homogeneous mimetic system in the case of vanishing mimetic dust energy density and the cuscuton systems. It selects the constant-tension cuscuton coordinate Θ= θ-θ, while θ retains its interpretation as the LQC polymerization angle. This equivalence does not establish agreement of the inhomogeneous theories or their perturbations. The angular action also admits an explicit branched f(K) representation, where K is the mean extrinsic curvature of spatial hypersurfaces. The construction is therefore an effective covariant representation of the LQC holonomy correction, not a derivation from full loop quantum gravity.