We play the following game. The input are (a) the empirical energy density of the universe (2.72 K) of the cosmic background radiation and (b) the Friedmann equations which are Einstein’s equations for metrics in the Friedmann-Robertson-Walker metrics

and (c) the empirically claimed radius of the observable universe.

The Friedmann equations are:

where is the MASS density and not the energy density and is the pressure. We had found the ENERGY density in the last post . We could not understand the possibility of a STATIC universe of radius 24 Gpc (which is the size of the OBSERVED universe. The static universe corresponds to which determines which is not a problem. The real problem is the first equation

So now let us set $\mu=4.14\times 10^{-14}$ as the energy density and let using the famous energy equivalence $\latex E=mc^2$. Then we find, for ,

This seems to produce too low of a curvature but the order of magnitude is not too great. In fact we can find that setting produces

This is encouraging because it tells us that if we have nothing in the universe except the mass density equivalent to the cosmic background radiation mean, approximately 2.7 K, then the Einstein/Friedmann equations for a STATIC universe would be satisfied for a radius of 623.644 Gpc. This is a universe where the mass density is being determined ONLY by the cosmic background radiation. If we increase in order to retain 1 on the right hand side we would have to decrease the radius. This is nice because it tells us that the radius of a static universe will be at most 623.644 Gpc and is then completely consistent with the observed radius of 24 Gpc.

The conclusion would be that if are careful to interpret as the MASS density rather than the energy density, we have a consistent model for a STATIC Einstein model.

Of course we should have some reasonable explanation for the Hubble redshift distance relation in a static universe. The idea that I had discovered some years ago applies here. The Hubble relation is explained as an ARTIFACT of spherical geometry having nothing to do with any OTHER physical phenomena. The spherical geometry enforces a deviation from the relationship between frequency and wavelength from to (essentially) . More precisely, on a sphere, frequency for light waves are quantized according to the eigenvalues of the Laplacian which occur as so there is a discrepancy between the frequency determined by the observed wavelength and the frequency at the source which grows in distance traveled by the light according to . We can implement this idea easily to match the Hubble graph qualitatively.

Here’s a quick and dirty model that produces Hubble ‘velocity’ as a function of distance based on the spherical geometry. The wavelength is . In an this corresponds to the -th eigenvalue of the Laplacian where . The following function (which includes a fudge factor to which I will return later) allows us to produce Hubble ‘velocities’ as a function of the radius of the universe.

> hubble_slope_for_universe_Gpc

function(gpc_radius,xMpc,fudge_factor){

Gpc<-3e25

c<-3e8

Halpha<-6.56281e-7

n<-gpc_radius*Gpc*fudge_factor/(Halpha)

2*(xMpc*Gpc)*1e-3/(sqrt(n^2+2)+n)

}

Although chi-by-eye is a bad habit, the issue here is to just ensure that we are not doing something completely absurd by checking that we get some reasonable numbers for ‘Hubble velocity’ by spherical geometry alone. So we can check that for a 30 Gpc universe at 400Mpc and a fudge factor of 1e-4 we obtain something between 25000 and 30000 as in the empirical graph.

> > hubble_slope_for_universe_Gpc(30,400,1e-4)*c [1] 26251.24

Of course this is not very tight. What this shows is that we can expect to constrain the radius of the universe between 24 Gpc and 624 Gpc which arise from considerations of empirical energy density of 2.7 K CMB temperature and still be able to explain the empirical redshifts without any expansion. Stability of the Einstein model dominated by radiation is known (with a speed of sound lower bound). Therefore we can have viable static models that does not violate Einstein’s equations and explains redshift, and can remove dark energy and dark matter. There are enormous benefits to such models since the cosmological constant problem is resolved in the model. Also Euclidean quantum field theory have no problems by work of Jaffee-Ritter. There is no singularity finite time in the past which I believe is the truth obviously. The problem thus is to fit the CMB anisotropy in a static Einstein universe. Recall that Einstein’s first reaction to Lemaitre was that the physics was atrocious and he was right about that. Quantization phenomena are probably GLOBAL geometric phenomena rather than phenomena that are purely microscopic. This is something that makes a great deal more sense in a static universe.

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