Higher Dimensional Dust Cosmological Implications of a Decay Law for $\Lambda$ Term : Expressions for Some Observable Quantities

TL;DR

This paper explores higher-dimensional cosmological models with a decaying cosmological constant proportional to the acceleration of the universe, deriving observable quantities and comparing with known models, showing potential differences from four-dimensional cases.

Contribution

It generalizes the decay law of the cosmological constant to higher dimensions and derives observable cosmological quantities within this framework.

Findings

The cosmological constant decreases as inverse square of time.

Derived expressions for distance measures and look-back time in higher dimensions.

Model reduces to known four-dimensional cases for specific parameters.

Abstract

In this paper we have considered the multidimensional cosmological implications of a decay law for $\Lambda$ term that is proportional to $\beta \frac{\ddot {a}}{a}$, where $\beta$ is a constant and $a$ is the scale factor of RW-space time. We discuss the cosmological consequences of a model for the vanishing pressure for the case $k=0$. It has been observed that such models are compatible with the result of recent observations and cosmological term $\Lambda$ gradually reduces as the universe expands. In this model $\Lambda$ varies as the inverse square of time, which matches its natural units. The proper distance, the luminosity distance-redshift, the angular diameter distance-redshift, and look back time-redshift for the model are presented in the frame work of higher dimensional space time. The model of the Freese {\it et al.} ({\it Nucl. Phys. B} {\bf 287}, 797 (1987)) for $n=2$ is retrieved for the particular choice of $A_{0}$ and also Einstein-de Sitter model is obtained for $A_{0} = {2/3}$. This work has thus generalized to higher dimensions the well-know result in four dimensional space time. It is found that there may be significant difference in principle at least, from the analogous situation in four dimensional space time.