Geometrical Origin of the Cosmological Constant

TL;DR

This paper derives a geometrical origin for the cosmological constant by modeling space-time as a submanifold in a hypercomplex manifold, leading to a space-time dependent, Lorentz invariant scalar that influences dark energy and cosmic acceleration.

Contribution

It introduces a novel geometric framework where the cosmological constant arises naturally from the embedding of space-time, linking it to a four-vector determined by Bianchi identities.

Findings

The cosmological constant is expressed as the norm of a four-vector U.

U can be determined from Bianchi identities and makes Lambda space-time dependent.

A specific solution shows Lambda decays as a power law with the scale factor.

Abstract

We show that the description of the space-time of general relativity as a diagonal four dimensional submanifold immersed in an eight dimensional hypercomplex manifold, in torsionless case, leads to a geometrical origin of the cosmological constant. The cosmological constant appears naturally in the new field equations and its expression is given as the norm of a four-vector $U$, i.e., ${\Lambda}=6g_{{\mu}{\nu}}U^{{\mu}}U^{{\nu}}$ and where U can be determined from the Bianchi identities. Consequently, the cosmological constant is space-time dependent, a Lorentz invariant scalar, and may be positive, negative or null. The resulting energy momentum tensor of the dark energy depends on the cosmological constant and its first derivative with respect to the metric. As an application, we obtain the spherical solution for the field equations. In cosmology, the modified Friedmann equations are proposed and a condition on ${\Lambda}$ for an accelerating universe is deduced. For a particular case of the vector $U$, we find a decaying cosmological constant ${\Lambda}\propto a(t)^{-6{\alpha}}$.