## Abstract

We investigate the second-order gravitational scalar perturbations for a barotropic fluid. We derive the effective energy-momentum tensor described by the quadratic terms of the gravitational and the matter perturbations. We show that the second-order effective energy-momentum tensor is gauge dependent. We impose three gauge conditions (longitudinal, spatially flat, and comoving gauges) for radiation and dust. The resulting energy-momentum tensor is described only by a gauge-invariant variable, but the functional form depends on the gauge choice. In the matter-dominated epoch with dustlike fluid background, the second-order effective energy density and pressure of the perturbations evolve as 1/a2 in all three gauge choices, like the curvature density of the Universe, but they do not provide the correct equation of state. The value of this parameter depends also on the gauge choice. In the radiation-dominated epoch, the perturbations in the short-wave limit behave in the same way as the radiationlike fluid in the longitudinal and the spatially flat gauges. However, they behave in a different way in the comoving gauge. As a whole, we conclude that the second-order effective energy-momentum tensor of the scalar perturbation is strictly gauge dependent.

Original language | English |
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Article number | e043531 |

Journal | Physical Review D |

Volume | 102 |

Issue number | 4 |

DOIs | |

State | Published - 15 Aug 2020 |

### Bibliographical note

Funding Information:We are grateful to Gungwon Kang, Jai-chan Hwang, Viatcheslav Mukhanov, and Alexei Starobinsky for helpful discussions. This work was supported by the grant from the National Research Foundation funded by the Korean government, Grants No. NRF-2017R1A2B4010738, No. NRF-2020R1A2C1013266 (I. C.), No. NRF-2019R1A2C2085023 (J. G.), and No. NRF-2018-R1D1A1B0-7048945 (S. H. O.). J. G. also acknowledges the Korea-Japan Basic Scientific Cooperation Program supported by the National Research Foundation of Korea and the Japan Society for the Promotion of Science (Grant No. 2018K2A9A2A08000127) and the Asia Pacific Center for Theoretical Physics for hospitality while this work was completed and for Focus Research Program “The origin and evolution of the Universe” where parts of this work were presented and discussed.

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