Lema\^{i}tre-Tolman-Bondi model: fractality, bang time, and Hubble law I. Initial conditions and compatibility of density and velocity laws

TL;DR

This paper systematically studies Lemaître-Tolman-Bondi models with various initial conditions, analyzing their compatibility with large-scale structure, fractal density, and the Hubble law, to understand cosmic evolution and the zero-velocity surface.

Contribution

It explores the relationships between initial conditions like bang time, fractal density, and velocity law in LTB models, providing insights into their compatibility with large-scale observations.

Findings

Fractal density and simultaneous bang time estimate the scale of negligible Hubble law deviation.

Non-simultaneous bang time arises when fractal density and linear Hubble law are assumed.

The zero-velocity surface is identified as the scale where deviations from linear Hubble law become small.

Abstract

We start a systematic study of the Lema\^{i}tre-Tolman-Bondi (LTB) model as applied to the large scale structure and its evolution. Here we study three possible initial conditions of the LTB models which are asymptotically FRW at large scales: bang time, fractal density (with fractal dimension D=2), and velocity law. Any two of these determine the third one. Fractal density and simultaneous bang time provide a quantitative estimate for the scale beyond which the deflection from the linear Hubble law is small. This border may be identified with the zero-velocity surface. For fractal density and linear Hubble law it is shown that the bang time is necessarily non-simultaneous.