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## ALMOST ZERO MEASURED CURVATURE OF SPACE ACTUALLY MEANS S4(1/h) or S3(1/h) WITH h^2 COSMOLOGICAL CONSTANT WORKS

So the WMAP and Planck missions official data analysis suggests that the curvature of space is almost zero but actually positive with an error bar.  If I tell them that, look, the actual curvature is $h^2$ then they would call that zero as well because to their experimental accuracy it is zero.  But in this case there need be no dark energy since there is an ambient intrinsic curvature of the universe.  Again the amazing new feature of the $S^4(1/h)$ model is that it explains the redshift, has no expansion and it actually explains why energy is quantized in our universe.  So you don’t need Big Bang and you don’t need string theory and you have probably the most parsimonious explanation of the dark energy.  There IS no dark energy and there was no Big Bang.

## 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.