Beautiful ILQGS talk by Hal and Aldo today--LambdaEPRL

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The cosmological constant Λ is a small constant space-time curvature. One can think of it as a residual or vacuum curvature. In cosmology it corresponds to a residual longterm percentage distance growth rate of H∞ = 1/173 percent per million years. After expansion has made the average density of matter essentially nil, and the Hubble growth rate has declined as much as it can, it will have leveled out at that asymptotic value.

Curvature is inverse area. The Planck curvature, for example, is one over the Planck area lP2. So what is the cosmological constant Λ in terms of the Planck unit of curvature? Λ = 2.90 x 10-122 lP-2

Riello and Haggard will show you how to work that curvature constant Λ into standard EPRL spin foam QG dynamics. It is interesting to see what kind of changes you need to make in EPRL so that it becomes what they call "ΛEPRL"
 
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They are reporting on work that is to appear by them in collaboration two others, so here's the title of the talk and list of researchers involved:
SL(2,C) Chern-Simons theory and spin foam gravity with a cosmological constant
Hal Haggard & Aldo Riello with Muxin Han & Wojciech Kamiski

===quote http://relativity.phys.lsu.edu/ilqgs/haggardriello112514.pdf slide 4==
Plan of the talk

1. Construction and definition of the model ΛEPRL
2. The equations of motions (EoM)
3. Focus on 3d curved geometries, and their reconstruction from the EoM
4. Towards a deformed phase space for curved quantum geometries
==endquote==

Have some things to do getting ready for Thanksgiving holiday. Have to get back to this later as time permits. I really like this presentation. The slides are exceptionally clear!
 
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Aldo and Hal's presentation starts by giving motivation
===quote http://relativity.phys.lsu.edu/ilqgs/haggardriello112514.pdf slide 1==
Motivation

The cosmological constant is non-zero: Λ = 2.90 x 10-122 lP-2

Why quantum groups in 4D?

Seek a (possibly more general) constructive route

We are led to couple Loop Quantum Gravity to Chern-Simons theory, the result has strong relations with previous Hamiltonian studies

A more geometric language that casts light on the asymptotics
==endquote==

Indeed the earlier way that Λ was introduced into Spinfoam was as a quantum group deformation parameter and this always seems in some way artificial and unmotivated. It sounds like they have found a more natural way to incorporate the cosmological constant in the theory!

It will be good to turn to the audio and listen to how they discuss this point when they elaborate on this slide.
 
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It's great that they recover the Regge action with cosmo constant! It is interesting that they find that the cosmological curvature constant must be quantized!
Here are their next two slides (text only without illustrative figures):
===quote http://relativity.phys.lsu.edu/ilqgs/haggardriello112514.pdf ==
What do we gain?

We enlarge the usual framework, taking tools from Chern-Simons theory into spinfoams

We develop a new description of curved simplices in 3 and 4d,
where holonomies also encode fluxes

We obtain both Λ > 0 and Λ < 0,
the sign being determined dynamically at the semiclassical level

We find that Λ must be quantized

Main results from the asymptotics
Disclaimer: for now the construction is at the single 4-simplex level only

The equations of motion define non-perturbatively curved 4-simplices, of positive and negative curvature.

The Regge action for curved 4-simplices augmented by the cosmological term is recovered exactly
[work in progress on an extra term that we seem to obtain]

SRegge = ∑triangles atΘt + ΛV4
==endquote==

This is just what it should be! The cosmological curvature constant is supposed to multiply the volume, in Regge action. This leaves me curious to know what the "extra term" that they mention getting might signify.
To have audio handy, for listening with the slides, I'll post the wav link again:
http://relativity.phys.lsu.edu/ilqgs/haggardriello112514.wav
 
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Interestingly enough, Riello and Haggard found that the Cosmological Curvature Λ is necessarily QUANTIZED. Let's try to get a handle on that and understand better what it means. They give the observed value of Lambda in terms of the planck unit of curvature. BTW how do you write the usual Planck curvature, lP-2, if your calculator doesn't remember the conventional Planck length?

I recall all the Planck units by remembering just two key ones. I remember that ħc is force area, and that c4/G is the Planck unit of force. So in this case I can quickly get reciprocal area if I simply multiply 1/ħc by the force. And that gives c3/Għ. So that is the Planck curvature. In words, cee-cubed-over-hbar-G, or cee-cubed-over G-hbar.

So let's see if google calculator will tell us the value of Λ = 2.90 x 10-122 c3/Għ, in ordinary units.
Paste in "2.9*10^(-122) c^3/(hbar*G) in m^-2"
The last bit ("in m^-2") is just to get it to give the curvature in standard metric units of reciprocal square meters.

To get a hands-on feel for what the cosmological curvature constant really means, it means the residual longterm distance growth rate is aiming to be 1/173 percent per million years and that rate squared is Lambda up to a factor of 3.
Paste in "3*(1/17300 per million years/c)^2 in m^-2" and you get essentially the same number as before.
Aldo and Hal's Lambda gives 1.11, and my asymptotic expansion rate H∞ of 1/173 percent gives 1.12, so maybe they should increase their 2.9 just a wee bit, but we basically agree.
 
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