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Conserved cosmological perturbation in Galileon models

We prove the existence of a fully nonlinear conserved curvature perturbation on large scales in Galileon-type scalar field models in two approaches. The first approach is based on the conservation of energy-momentum tensor of the Galileon field, which is also the familiar approach in understanding the conservation in $k$-essence or perfect fluid models. We show that the fluid corresponding to the Galileon field becomes perfect and barotropic on large scales, which is responsible to the conservation. The difference from $k$-essence model is that, besides the energy-momentum conservation, the Einstein equation must be employed in order to complete the proof of barotropy. In the second approach, we derive the fully non-perturbative action for the curvature perturbation $ζ$ in Galileon model on large scales, and argue that $ζ=const$ is indeed an exact solution on large scales. This conservation of curvature perturbation is important since it relates the later and the primordial universe.