## Abstract

We formulate Lagrangian perturbation theory to solve the non-linear dynamics of a self-gravitating fluid within the framework of the post-Newtonian approximation in general relativity, using the (3 + 1) formalism. Our formulation coincides with the Newtonian Lagrangian perturbation theory developed by Buchert for scales much smaller than the horizon scale, and with the gauge-invariant linearized theory in longitudinal gauge conditions for the linear regime. These conditions are achieved by using the gauge-invariant quantities at the initial time, when the linearized theory is valid. The post-Newtonian corrections in the solution of the trajectory field of fluid elements are calculated in their explicit forms. Thus our formulation allows us to investigate the evolution of large-scale fluctuations involving relativistic corrections from the early regime, such as the decoupling time of matter and radiation, until today. As a result, we are able to show that naive Newtonian cosmology for the structure formation will be a good approximation even for perturbations with scales not only inside but also beyond the present horizon scale in longitudinal coordinates. Although the post-Newtonian corrections are small, it is shown that they have a growing transverse mode, which is not present in Newtonian theory or in the gauge-invariant linearized theory. Such post-Newtonian-order effects might produce a characteristic appearance of large-scale structure formation, for example through the observation of anisotropies in the cosmic microwave background radiation (CMB). Furthermore, because our approach has a straight-forward Newtonian limit, it will also be convenient for numerical implementation based on the presently available Newtonian simulations. Our results easily allow us to perform a simple order estimation of each term in the solution, which indicates that post-Newtonian corrections cannot be neglected in the early evolution of density fluctuations, compared with Newtonian perturbation solutions.

Original language | English |
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Pages (from-to) | 64-88 |

Number of pages | 25 |

Journal | Monthly Notices of the Royal Astronomical Society |

Volume | 306 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1999 Jun 11 |

Externally published | Yes |

## Keywords

- Cosmology: theory
- Gravitation
- Instabilities
- Large-scale structure of Universe

## ASJC Scopus subject areas

- Astronomy and Astrophysics
- Space and Planetary Science