The ΛEPRL spin foam model presented 25 November at ILQGS by Haggard and Riello achieves an interesting quantization of the cosmological constant. Basically this is done on slide #10 around minute 15 of the audio.(adsbygoogle = window.adsbygoogle || []).push({});

http://relativity.phys.lsu.edu/ilqgs/haggardriello112514.pdf

http://relativity.phys.lsu.edu/ilqgs/haggardriello112514.wav

I want to talk about the basic philosophy of their generalization of Regge using simplices with homegeneous curvature, that they call "Λ Regge" and how the quantization comes about.

But before giving motivation and overview, I will just very briefly indicate what is shown on slide #10 and said in the audio for a few seconds starting around minute 15:30. You can drag the time button and listen selectively to that section of the talk, if you wish.

In their notation R_{Λ}is what in Jorrie's Lightcone calculator is called R_{∞}, the longterm asymptotic value of the Hubble radius. The Hubble radius, c/H, is currently about 14.4 Gly and is expected to grow and approach the limit of 17.3 Gly. The practical effect of the positive cosmological curvature constant Λ (a kind of residual vacuum curvature) is that the percentage distance expansion rate H (now about 1/144 % per million years) is not expected to decline to zero, but to decline ever more slowly and level off at 1/173% per million years. This places a bound on the growth of its reciprocal, c/H, the Hubble radius.

So in their equation on slide 10 they refer to the eventualarea of the Hubble sphere4π R_{Λ}^{2}. And they give a LQG theoretical argument, using their ΛEPRL spin foam model, that this area must bean integer multiple of the Planck area, scaled by the Immirzi parameter, namely γl_{P}^{2}.

So the picture is there is a big sphere with radius 17.3 billion lightyears, which will eventually be our cosmological event horizon, with distances larger than 17.3 Gly growing faster than light and objects near or on the horizon being redshifted without limit. so that time for them appears to stop.

And the theoretical result is that this large spherical area comprises an integer number of little Planck sized areas. In LQG it is not uncommon for the Planck area l_{P}^{2}to be multiplied by the Immirzi gamma number γ and that happens here.

So you can see the equation on slide 10 saying 4π R_{Λ}^{2}∈ γl_{P}^{2}ℕ

In other words the eventual cosmological horizon area is some natural number times gamma by the planck area.

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# ΛEPRL quantiz'n of cosmological horizon area in Planck units

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