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Relativity and Cosmology Blog

Relativity and Cosmology

This is a Blog on relativity and cosmology for engineers and the like. You are welcome to comment upon or question anything said on my website (relativity-4-engineers), in the eBook or in the snippets I post here.

Comments/questions of a general nature should preferably be posted to the FAQ section of this Blog (http://cr4.globalspec.com/blogentry/316/Relativity-Cosmology-FAQ).

A complete index to the Relativity and Cosmology Blog can be viewed here: http://cr4.globalspec.com/blog/browse/22/Relativity-and-Cosmology"

Regards, Jorrie

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How Aeons turned into Zeons

Posted May 12, 2015 12:00 PM by Jorrie

Quite a lot has been written on this CR4 Blog about the standard Lambda-Cold-Dark Matter (LCDM) cosmological model and its equations. Arguably the most important equation of the model is the evolution of the expansion rate over cosmological time. In other words, how the Hubble constant H has changed over time. If one knows this function, most of the other LCDM equations can be derived from it, because it fixes the expansion dynamics.

The changing H is most simply expressed in this variant of the Friedman equation, an exact solution of Einstein's field equations for a spatially flat and perfectly homogeneous universe.

(1) H2−Λc2/3 = 8/3 π G ρ

Here H is the fractional expansion rate at time t, Λ is Einstein's cosmological constant, G is Newton's gravitational constant and ρ is the mass density equivalent of the changing energy density of matter and radiation at time t. This energy density includes dark matter, but no 'dark energy', because Λ appears as a constant spacetime curvature on the left side of the equation. The c2 converts square curvature (length-2) to a squared fractional expansion rate (time-2), to be on par with H2.[1]

As you can check, G ρ gives SI units of 1/s2, which is the natural unit of the squared fractional expansion rate. Since Λ is a constant spacetime curvature, it is convenient to replace Λc2/3 with a constant 'Hubble rate' H2 , which represents the square of the Hubble constant of the 'infinite future', when cosmic expansion will effectively have reduced matter energy density to zero.

(2) H2−H2 = 8/3 π G ρ

Since we can measure the present value of H, labeled H0 (H-naught) and also how it has changed over time, it allows us to use Einstein's GR to determine the value of H. If we assume that radiation energy is negligible compared to other forms (as is supported by observational evidence), then we can express eq. (2) as:

(3) H2H2 = (H02 H2 )S3

The present observed fractional expansion rate tells us that all large scale distances are now growing by 1/144 % per million years. S is the 'stretch factor' by which wavelengths of all radiation from galaxies have increased since they were emitted.[2] Since S3 tells us by how much the volume has increased, it also tells us by what factor density and hence spacetime curvature has decreased since the observed emission.

If the present H0 would have stayed constant, all large scale distance would have doubled in the next 14.4 Gy. This 'doubling time' is however slowly increasing, to eventually double all large scale distances every 17.3 Gy. Or stated differently, all distances will eventually grow at H = 1/173 % per million years.

The 17.3 Gy 'doubling time' is a sort of natural time scale set by Einstein's cosmological constant. An informal study by a group of Physics-Forums contributors suggested that the 17.3 Gy time-span could be a natural timescale for the universe.[3] For lack of an 'official name' for it, the group called it a 'zeon', for no other good reason than the fact that it rhymes with aeon.

One light-zeon is 17.3 Gly in conventional terms and H0 causes a doubling in distances every 14.4/17.3 = 0.832 zeon. This makes H = 1 per zeon and H0 = 17.3/14.4 =1.201 per zeon.[3] Our present time is 13.8/17.3 ~ 0.8 zeon.

We can easily normalize equation (3) to the new (zeon) scale by dividing through by H2 (which then obviously equals 1).

(4) H2−1 = (1.2012−1) S3 = 0.443 S3

or

(5) H2 = 1 + 0.443 S3 ----> !NB!

This remarkably simple equation forms the basis of a surprisingly large number of modern cosmological calculations, as will be discussed in a follow-on Blog entry.

Here is a graph of the normalized H over 'zeon-time', which is obviously the x-axis (courtesy PhysicsForums).

The blue dot represents our present time, 0.8 zeon and a Hubble constant of 1.2 zeon-1. The long term value of H approaches 1.

Any questions before we proceed?

Regards, Jorrie

[1] The traditional unit of the Hubble constant as used by Edwin Hubble is kilometers per second per Megaparsec. From an educational p.o.v. it was an unfortunate choice, because it seems to indicate a recession speed, while it is really a fractional rate of increase of distance. It is a distance divided by a distance, all divided by time. So its natural unit is 1/time, or simply time-1.

[2] 'Stretch factor' S = 1/a, where a is the scale factor, as used in the LightCone calculator. S is also simply related to cosmological redshift z by S=z+1.

[3] Science Advisor 'Marcus' led a group of PhysicsForums members in fleshing out of this "universal scale", based on the cosmological constant.

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