TY - JOUR

T1 - A post-Newtonian Lagrangian perturbation approach to large-scale structure formation

AU - Takada, Masahiro

AU - Futamase, Toshifumi

PY - 1999/6/11

Y1 - 1999/6/11

N2 - 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.

AB - 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.

KW - Cosmology: theory

KW - Gravitation

KW - Instabilities

KW - Large-scale structure of Universe

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U2 - 10.1046/j.1365-8711.1999.02441.x

DO - 10.1046/j.1365-8711.1999.02441.x

M3 - Article

AN - SCOPUS:0042916599

SN - 0035-8711

VL - 306

SP - 64

EP - 88

JO - Monthly Notices of the Royal Astronomical Society

JF - Monthly Notices of the Royal Astronomical Society

IS - 1

ER -