A Cuscuton Representation of the Loop Quantum Cosmology Bounce

Abstract

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 $H\propto\sinθ$, while the Einstein constraint gives $ρ=ρ_c\sin^2(θ/2)$, 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 $Θ=\sinθ-θ$, 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.