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Cosmological time crystals are created when a scalar field moves periodically through phase space in a spatially flat Friedmann-Robertson-Walker spacetime due to the presence of a limit cycle.
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For a detailed derivation of the full theory, see [ 21 ] . We follow the conventional notation of [ 22 ]
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Here we only consider the scalar perturbations. For a complete treatment of both scalar and tensor perturbations, see e.g. [ 6 ]
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Throughout we take M p 2 = 1 M_{p}^{2}=1 and adopt the mostly plus metric signature, opposite to [ 4 ] . Note, X X is sometimes defined in the literature without the factor of 1 / 2 1/2
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We can guarantee that c s 2 c_{s}^{2} is an even, perfectly periodic function if we choose one of the two initial conditions lying on the limit cycle where \mathaccentV d o t 05 F φ 0 ≈ 0 \mathaccentV{dot}05F{\varphi}_{0}\approx 0 so that c s 2 = 1 c_{s}^{2}=1 initially, given by ( 17
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