Appendix — The Geometry Behind Cosmic “Acceleration”

1. The Hypersphere as the Universe

In this model, the universe is not an infinite flat space. It is the 3‑sphere:

S3(R)

where:

  • R is the radius of a 4‑dimensional hypersphere

  • the boundary S3(R) is the universe

  • the interior is not physical space

Time is identified with the growth of R:

t≡R

So:

  • as time increases, R increases

  • as R increases, the universe expands

  • expansion is growth of the boundary, not motion in flat space

The intrinsic curvature of the 3‑sphere is:

K=1R2

Large R → small curvature → the universe appears locally flat.

 

2. The Flat FRW Assumption

Standard cosmology uses the flat FRW metric (curvature k=0):

ds2=−c2dt2+a2(t)(dr2+r2dΩ2)

where:

  • a(t) is the scale factor

  • r is a comoving radial coordinate

  • dΩ2 is the angular part

Redshift z is related to the scale factor by:

1+z=a(t0)a(tem)

Distances are then inferred assuming:

  • space is flat

  • geodesics are straight

  • curvature is negligible

When astronomers plot distance vs. redshift (the Hubble diagram), they see the curve bend upward at high z. In flat FRW, this bending is interpreted as acceleration, and a cosmological constant (dark energy) is added to make the equations fit.

 

3. The Hypersphere Metric and Hidden Curvature

For a hyperspherical universe, the spatial metric is:

ds2=R2(dχ2+sin⁡2χ (dθ2+sin⁡2θ dϕ2))

where:

  • R=R(t) is the hypersphere radius

  • χ is a radial angular coordinate on the 3‑sphere

Light travels along curved geodesics on this surface. As R grows:

  • geodesics stretch

  • distances grow faster than linearly

  • curvature decreases but never vanishes

When this 4‑D curved behaviour is forced into a 3‑D flat FRW framework, the true geometric relation between distance and redshift is distorted.

A smooth 4‑D curvature relation becomes a hyperbola in the 3‑D flat model.

This is the key:

The upward bend in the Hubble diagram is the projection of 4‑D curvature into a 3‑D flat description.

 

4. Expansion Redshift vs. Curvature Redshift

In flat FRW, redshift is purely expansion redshift:

1+zexp=a(t0)a(tem)

In the hypersphere model, there is an additional contribution:

z=zexp+zcurv

where:

  • zexp comes from the growth of R (expansion)

  • zcurv comes from light traveling through regions of changing curvature

As light moves across the hypersphere:

  • curvature changes along its path

  • geodesics bend

  • effective path length increases

  • wavelengths are stretched beyond pure expansion

This extra stretching is curvature redshift.

It grows with distance, so it becomes important precisely where dark energy is invoked in ΛCDM.

 

5. The Hyperbolic Projection and the Illusion of Acceleration

When we plot distance vs. redshift using a flat model, but the universe is actually hyperspherical:

  • the true 4‑D curvature relation is squeezed into 3‑D

  • the result is a hyperbolic curve

  • the hyperbola is misread as acceleration

Schematically:

  • true relation (in hypersphere coordinates):

Dtrue(z)∼curved, smooth function of R(t)

  • projected into flat FRW:

DFRW(z)∼hyperbola

The hyperbola is then interpreted as:

  • a cosmological constant

  • dark energy

  • a repulsive force

But in this model:

The hyperbola is not a new force. It is the shadow of 4‑D curvature on a 3‑D screen.

Once curvature redshift and hyperspherical geometry are included, the apparent acceleration disappears:

  • no cosmological constant is needed

  • no dark energy is required

  • the Hubble diagram becomes a natural consequence of geometry

 

6. Summary

This appendix has shown, in minimal mathematics, that:

  • the universe is a 3‑sphere S3(R) with radius R(t)

  • time is the growth of R

  • standard flat FRW cosmology misinterprets curvature as acceleration

  • 4‑D curvature, when projected into 3‑D, produces a hyperbolic distance–redshift relation

  • this hyperbola is mistaken for dark energy

  • redshift has two parts: expansion redshift and curvature redshift

  • including curvature redshift removes the need for a cosmological constant

In short:

What looks like cosmic acceleration is the projection of hyperspherical curvature into a flat model. Dark energy is a name we give to a geometric illusion.