Why do we live in a 4D world: Can cosmology, black holes and branes give an answer?

TL;DR

This paper derives a universal scalar field potential compatible with black holes, p-branes, and inflation across multiple dimensions, providing a theoretical explanation for why our universe appears four-dimensional.

Contribution

It formulates a single equation governing scalar potentials in various dimensions and shows four as the maximum admissible dimension in the low-energy regime.

Findings

The scalar potential varies with dimension but follows a universal equation.

Four dimensions emerge as the maximum eigenvalue in the low-energy limit.

The approach links string theory, black holes, and cosmology through a common mathematical framework.

Abstract

We derive the general form of the cosmological scalar field potential which is compatible both with the existence of black holes and p-branes related to string/M theory and with multidimensional inflationary cosmology. It is shown that the scalar potential alters non-trivially from dimension to dimension yet always obeys one single equation where the number of spacetime dimensions is a free parameter. Using this equation we formulate an eigenvalue problem for the dimensionality parameter. It turns out that in the low-energy regime of sub-Planckian values of the inflaton field, i.e., when the Universe has cooled and expanded sufficiently enough, the value four arises as the largest admissible (eigen)value of this parameter.