Appendix — The Geometry Behind the Hubble Tension

1. One universe, two Hubble measurements

The Hubble constant H0 is defined as:

H0=a˙(t)a(t)∣t=t0

In ΛCDM:

• Local measurements (supernovae, Cepheids) give H0≈73.

• Early‑universe measurements (CMB) give H0≈67.

The tension arises because both are interpreted inside a flat FRW framework that the universe does not obey.

 

2. Flat FRW vs hypersphere geometry

Flat FRW metric:

ds2=−c2dt2+a2(t)(dr2+r2dΩ2)

Hypersphere universe:

Space=S3(R),R=R(t)

Spatial metric on the 3‑sphere:

ds2=R2(t)(dχ2+sin⁡2χ (dθ2+sin⁡2θ dϕ2))

Key differences:

• FRW assumes flat space and treats curvature as a small correction.

• Hypersphere model makes curvature fundamental: K=1/R2.

 

3. Distance–redshift in flat FRW

In flat FRW, comoving distance to redshift z:

DFRW(z)=c∫0zdz′H(z′)

with H(z) computed assuming:

• flat geometry,

• dark energy (cosmological constant),

• no explicit hyperspherical curvature.

The observed upward bend in the Hubble diagram at high z is then interpreted as:

• acceleration of expansion,

• need for dark energy,

• lower inferred H0 from CMB.

 

4. Distance–redshift on a hypersphere

On S3(R), light travels along curved geodesics. A simple schematic form for the comoving distance is:

Dhyp(z)∼R0 χ(z)

with χ(z) obtained from integrating along the curved path:

χ(z)=∫0zdz′Hgeom(z′)

Here Hgeom(z) encodes:

• growth of R(t),

• intrinsic curvature K=1/R2,

• non‑flat geodesic structure.

When this curved relation is forced into a flat FRW description, the resulting D(z) appears hyperbolic:

DFRW(z)≈hyperbolic fit to Dhyp(z)

This hyperbola is misread as acceleration and leads to a biased H0 from early‑universe data.

 

5. Expansion redshift vs curvature redshift

Total redshift in the hypersphere model:

z=zexp+zcurv

• Expansion redshift (growth of R):

1+zexp=R0Rem

• Curvature redshift (light through changing curvature):

zcurv∝∫pathdℓR2(t)

In the early universe:

• R is small → curvature large → zcurv significant.

• Ignoring zcurv underestimates distances and mis‑infers a lower H0.

In the late universe:

• R is large → curvature small → zcurv negligible.

• Local H0 measurements are closer to the true geometric value.

 

6. Resolving the tension

When:

• hyperspherical curvature is included,

• the hyperbolic projection is recognised,

• curvature redshift is added to expansion redshift,

then:

H0CMB↑andH0local≈H0geom

The two values converge to a single geometric Hubble constant:

H0true≈one consistent value (e.g. ∼72 km/s/Mpc)

The “Hubble tension” is not a physical contradiction; it is the result of interpreting a curved hypersphere with a flat FRW ruler.