Cosmology
MatureThe S³ Friedmann equation is derived [E]; the precise cosmological expansion history matching CMB+BAO data is [M].S³ Cosmology
The Friedmann equation on a compact S³ universe. Finite volume, positive curvature, no Big Bang singularity — the universe nucleated as a topological bubble. Mach closure determines the curvature radius.
The standard cosmological model assumes a spatially flat, infinite universe originating from a Big Bang singularity. In contrast, the CPT framework models the universe as a compact S³ hypersphere. The universe did not begin from a point singularity but rather nucleated as a topological bubble with a finite initial volume.
The S³ Metric and Mach Closure
The S³ geometry is described by the metric:
The metric of the S³ universe
The curvature radius is not an arbitrary parameter. It is determined by the Mach closure condition , deeply linking the global geometry to the total mass content of the universe.
The Modified Friedmann Equation
Applying this metric to the Einstein field equations yields a modified Friedmann equation for an S³ universe:
The S³ Friedmann Equation
Remarkably, the curvature term acts in a manner similar to a cosmological constant , providing a potential geometric alternative to dark energy. At leading order, it substitutes for , though predictions for the Cosmic Microwave Background (CMB) differ at large angular scales.
Resolving the Hubble Tension
Combining Mach closure with the double-copy gravity relation yields a precise expression for the curvature radius:
Curvature Radius Prediction
The volume of this universe is .
The current "Hubble tension" — the discrepancy between local measurements of and those inferred from the CMB — may simply be an artifact of assuming a flat-universe formula (CDM) in a genuinely curved S³ universe.
Integrity Status [M]: Maturing
The fundamental S³ Friedmann equation is fully derived [E]. However, the precise cosmological expansion history and its exact matching to CMB and BAO (Baryon Acoustic Oscillation) data remains in the maturing stage [M].
Source Documents
The primary research documents underlying this page. All files are freely viewable and downloadable. They represent the working research as written — including integrity tags, equations, and finding IDs.
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