Relativity and Cosmology

   

Calculation of the Hubble Parameter from Geometry

Authors: Patrick L. Nash

It is shown that the field equations of Einstein gravity sourced by a real massless scalar inflaton field $\varphi$, with inflaton potential identically equal to zero, cast on an eight-dimensional pseudo-Riemannian manifold $\mathbb{X}_{4,4}$ (a spacetime of four space dimensions and four time dimensions) admit a solution that exhibits temporal exponential \textbf{deflation of three of the four time dimensions} and temporal exponential inflation of three of the four space dimensions. [The signature and dimension of $\mathbb{X}_{4,4}$ are chosen because its tangent spaces satisfy a triality principle \cite{Nash2010} (Minkowski vectors and spinors are equivalent).] Comoving coordinates for the two \textbf{unscaled} dimensions are chosen to be $(x^4 \leftrightarrow \textrm{ time}, x^8 \leftrightarrow \textrm{ space})$. The $x^4$ coordinate corresponds to our universe's observed physical time dimension. The $x^8$ coordinate corresponds to a compact spatial dimension with circumference $C_8$. $C_8$ determines the initial value of the Hubble parameter $H$. Most importantly, this model describes an initially inflating/deflating Universe created with inflaton potential identically equal to zero, which is an initial condition that is exponentially more probable than an initial condition that assumes an initial inflaton potential of order of the Planck mass. This model predicts that the Hubble parameter $H$ during inflation is $ H = \frac{\pi}{3 \, C_8} $.

Comments: 11 Pages.

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Submission history

[v1] 2013-08-27 00:24:23

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