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## THERE IS NO EXPANSION AND NO DARK ENERGY IN THE ACTUAL UNIVERSE

Here is a simple geometric explanation of the redshift in a static Einstein (1917) model which is a scaled $S^3$.  The issue turns on the fact that the obvious frequency-wavelength relation $f=c/\lambda$ that holds for wave propagation on the plane does not hold for wave propagation on a sphere.  Now Einstein’s static universe model is a scaled $S^3$.  The wave equation on an $n$-sphere,

$(1/c^2 \partial_t^2 - \Delta_{S^n})\psi = 0$

can be solved explicitly by very standard methods in physics, separation of variables and using spherical harmonics $Y_{\ell,m}(x)$ (see for example orthoganalpolnomialsinddimesnions for a treatment of spherical harmonics).  The frequency of the waves on $S^3(R)$ are

$\omega_{\ell} = (c/R)\sqrt{\ell(\ell+2)}$

This is in sharp contrast to the frequency of a wave on a circle of radius $R$ which is $\omega = c\ell/R$.  It is this discrepancy that explains the observed redshift.

An intrinsic curvature for a closed universe has a positive cosmological constant without any need for a dark energy, and of course the universe is static and will not have any expansion since redshift is a geometric phenomenon.  De Sitter relativity is one approach that can apply, but more interesting is the fact that quantization of energy can be explained easily also as a geometric feature.