What exactly is the relation between LQG and the Friedmann equation?

[Flaneuse]

I'm just beginning to look into loop quantum cosmology, and I've recently run into some fundamental confusion concerning how exactly the Friedmann equation fits in. Is it just that the application of loop quantum gravity yields a modified/effective Friedmann equation, or is it the other way around, such that the Friedmann equation is essential to the actual derivation of loop quantum gravity/cosmology? (Or both?) I feel as if I've misunderstood something very basic and very important somewhere here, so any kind of clarification would be apprecaited.

 

[marcus]

main authority is Ashtekar, here are his papers
http://arxiv.org/find/grp_physics/1/au:+ashtekar/0/1/0/all/0/1
the most recent one is a review of LQC
http://arxiv.org/abs/1108.0893
Loop Quantum Cosmology: A Status Report
Abhay Ashtekar, Parampreet Singh
(Submitted on 3 Aug 2011 (v1), last revised 22 Aug 2011 (this version, v2))
The goal of this article is to provide an overview of the current state of the art in loop quantum cosmology for three sets of audiences: young researchers interested in entering this area; the quantum gravity community in general; and, cosmologists who wish to apply loop quantum cosmology to probe modifications in the standard paradigm of the early universe. An effort has been made to streamline the material so that, as described at the end of section I, each of these communities can read only the sections they are most interested in, without a loss of continuity.
Comments: 138 pages, 15 figures. Invited Topical Review, To appear in Classical and Quantum Gravity.

One thing that you may be looking for is on page 73. Equations (5.7 and 5.8) the quantum corrected Friedmann and Raychaudhuri equations
==quote==
... Squaring (5.4) and using (5.6) we obtain
H² =...=(8πG/3)ρ(1−ρ/_max) (5.7)

where H = a'/a denotes the Hubble rate and ρ_max is the maximum energy density given by ρ_max = 3/(8πGγ²λ²) ≈ 0.41ρ_Pl. This is the modified Friedmann equation we were seeking. Note that, in the expression of the effective constraint (5.6), it is the left hand side that is modified from b² to sin² λb/λ² due to the underlying quantum geometry. To arrive at the modified Friedmann equation, we have merely used the equation of motion for ν and trignometric identities to shift this modification to the right side.
Similarly, the modified Raychaudhuri equation can be obtained from Hamilton’s equation
for b:
a ̈/a =−(4πG/3)ρ(1−4 ρ/ρ_max )−4πGP(1−2 ρ/ρ_max ).

 

[marcus]

I like the name flaneuse. I think that the verb *flaner* means to lounge, loaf, or as we say to "hang out". It is a congenial activity. One needs to have an appreciation of idleness.

This 138 page paper of Ashtekar seem absurdly heavy to recommend to you. But at least he and his co-author DERIVE the modified Friedmann eqn. and the modified Friedmann acceleration eqn. (that they call the modified Raychaudhuri).

In other much shorter papers they seem to jump more quickly to this but perhaps they leave out some steps. I really don't know what to recommend. Maybe this will be some help to you even though it is a bit too long, complete, and detailed.

 

[member 11137]

main authority is Ashtekar, here are his papers
http://arxiv.org/find/grp_physics/1/au:+ashtekar/0/1/0/all/0/1
the most recent one is a review of LQC
http://arxiv.org/abs/1108.0893
Loop Quantum Cosmology: A Status Report
Abhay Ashtekar, Parampreet Singh
(Submitted on 3 Aug 2011 (v1), last revised 22 Aug 2011 (this version, v2))
The goal of this article is to provide an overview of the current state of the art in loop quantum cosmology for three sets of audiences: young researchers interested in entering this area; the quantum gravity community in general; and, cosmologists who wish to apply loop quantum cosmology to probe modifications in the standard paradigm of the early universe. An effort has been made to streamline the material so that, as described at the end of section I, each of these communities can read only the sections they are most interested in, without a loss of continuity.
Comments: 138 pages, 15 figures. Invited Topical Review, To appear in Classical and Quantum Gravity.

One thing that you may be looking for is on page 73. Equations (5.7 and 5.8) the quantum corrected Friedmann and Raychaudhuri equations
==quote==
... Squaring (5.4) and using (5.6) we obtain
H² =...=(8πG/3)ρ(1−ρ/_max) (5.7)

where H = a'/a denotes the Hubble rate and ρ_max is the maximum energy density given by ρ_max = 3/(8πGγ²λ²) ≈ 0.41ρ_Pl. This is the modified Friedmann equation we were seeking. Note that, in the expression of the effective constraint (5.6), it is the left hand side that is modified from b² to sin² λb/λ² due to the underlying quantum geometry. To arrive at the modified Friedmann equation, we have merely used the equation of motion for ν and trignometric identities to shift this modification to the right side.
Similarly, the modified Raychaudhuri equation can be obtained from Hamilton’s equation
for b:
a ̈/a =−(4πG/3)ρ(1−4 ρ/ρ_max )−4πGP(1−2 ρ/ρ_max ).

Where can I find a demonstration of relation (2.33) p.26? (gravitational part of the constraint?)
Thanks for help

 

[Flaneuse]

Agreed, on the subject of idleness, though you can probably tell that I took more time than I needed to to try to think of an appreciable name :).

As for the paper, yes, it is quite long, but I do appreciate the fact that they derive everything, and it is much less absurd a suggestion considering that it is nicely divided. Unfortunately though, I do not yet know enough physics to understand absolutely all of it.

One thing that you may be looking for is on page 73. Equations (5.7 and 5.8) the quantum corrected Friedmann and Raychaudhuri equations
==quote==
... Squaring (5.4) and using (5.6) we obtain
H² =...=(8πG/3)ρ(1−ρ/_max) (5.7)

where H = a'/a denotes the Hubble rate and ρ_max is the maximum energy density given by ρ_max = 3/(8πGγ²λ²) ≈ 0.41ρ_Pl. This is the modified Friedmann equation we were seeking. Note that, in the expression of the effective constraint (5.6), it is the left hand side that is modified from b² to sin² λb/λ² due to the underlying quantum geometry. To arrive at the modified Friedmann equation, we have merely used the equation of motion for ν and trignometric identities to shift this modification to the right side.
Similarly, the modified Raychaudhuri equation can be obtained from Hamilton’s equation
for b:
a ̈/a =−(4πG/3)ρ(1−4 ρ/ρ_max )−4πGP(1−2 ρ/ρ_max ).

What is meant by the ρ<sub>Pl in the maximum energy density, ρmax = 3/(8πGγ²λ²) ≈ 0.41ρPl? Apologies, I am not terribly familiar with the notation, so I am having a little trouble understanding what it is trying to suggest. What exactly is ρ the density of, and what exactly is Pl the pressure of?

Concerning the original question of the Friedmann equation, it seems here that it has been modified with considerations from loop quantum gravity in order to understand the implications of the theory. Is this correct, or am I just further confusing myself here?</sub>

 

[marcus]

Chère Flaneuse,
When you have Pl as a subscript it means "Planck". The "planck density" is a definite very high density. One Planck unit of energy per Planck unit of volume.

In the other equation there is a letter P by itself and they use this to stand for pressure.

They use the uppercase letter because the lowercase p is so easy to confuse with the rho, which they are using for energy density.

========================
BTW if you are familiar with ordinary cosmology you will remember that the usual Friedmann eqn only involves the density, but the other equation, the Friedmann acceleration or "Raychaud..." involves both the energy density and the pressure. It has a minus sign so that in order to get positive acceleration one must have negative pressure.
So that is what the uppercase P is doing in equation (5.8).
========================

As I recall from earlier Ashtekar papers, especially dealing with this simple case where things are homogeneous and isotropic, they have found that the quantum corrections that make gravity repellent begin to be important around 1% of Planck density, and the actual bounce, the turnaround, happens at about 41% of Planck density. But when isotropy is relaxed and more complicated cases considered the bounce would not necessarily happen at exactly the same density.

The Hubble parameter is negative during contraction and then becomes zero right at the moment of bounce (e.g. at 41% of Planck density) and then increases very rapidly during a period of faster-than-exponential inflation.

Can anyone help Blackforest please?

Where can I find a demonstration of relation (2.33) p.26? (gravitational part of the constraint?)
Thanks for help

 

[Flaneuse]

As I recall from earlier Ashtekar papers, especially dealing with this simple case where things are homogeneous and isotropic, they have found that the quantum corrections that make gravity repellent begin to be important around 1% of Planck density, and the actual bounce, the turnaround, happens at about 41% of Planck density. But when isotropy is relaxed and more complicated cases considered the bounce would not necessarily happen at exactly the same density.

The Hubble parameter is negative during contraction and then becomes zero right at the moment of bounce (e.g. at 41% of Planck density) and then increases very rapidly during a period of faster-than-exponential inflation.

Can anyone help Blackforest please?

marcus,
Thanks so much; your reply really cleared up a lot for me. Might you happen to know which paper deals specifically with the quantum bounce? I've been looking through the others, but there are enough of them that I'm having a bit of trouble finding it.

Blackforest,
Unfortunately, I am not nearly familiar enough with these things to be of much help, but I suggest maybe looking at some of the other Ashtekar papers in the first link from marcus. There are a few that look like they deal more specifically with the constraint (perhaps third, eighteenth, or twentieth in the list), though I have not looked at them, so I can't be entirely sure that that would be helpful.

 

[atyy]

http://arxiv.org/abs/1108.0893

Where can I find a demonstration of relation (2.33) p.26? (gravitational part of the constraint?)

I'm missing a couple of details, but that should be http://arxiv.org/abs/gr-qc/0702030, Eq 3.7.

A detailed derivation can be found by by reading http://arxiv.org/abs/1007.0402 from the start to Eq 2.28-2.31, followed by http://arxiv.org/abs/gr-qc/0612104, Eq 2.7-2.11.

I like the name flaneuse. I think that the verb *flaner* means to lounge, loaf, or as we say to "hang out". It is a congenial activity. One needs to have an appreciation of idleness.

Fascinating! Googling brings up http://www.danagoldstein.net/dana_goldstein/whats_a_flneuse.html which says George Sand was a flaneuse. I hear she's famous in her own right, but as a musician, I know her mainly for being Chopin's companion. From what I hear though, she was no idler.

 

[member 11137]

I'm missing a couple of details, but that should be http://arxiv.org/abs/gr-qc/0702030, Eq 3.7.

A detailed derivation can be found by by reading http://arxiv.org/abs/1007.0402 from the start to Eq 2.28-2.31, followed by http://arxiv.org/abs/gr-qc/0612104, Eq 2.7-2.11.

I am fascinated by the formalism of that constraint which seems to be very similar with the one obtained via another and totally different approach. Rules of the forum unfortunately forbids a discussion concerning that point. Anyway, thank you Marcus and atyy.

 

[marcus]

Might you happen to know which paper deals specifically with the quantum bounce?

I've been negligent. Other things got in the way.

The first LQC bounce papers were by Bojowald starting in 2001 but in 2006 Ashtekar got actively involved with what he and collaborators called "improved dynamics" LQC.
That was IIRC when we first saw the "modified Friedman equation" with its critical density rho_crit.

I will try to chase that down. If you just go back a little way you see things like page 7 eqn (3.1) of http://arxiv.org/pdf/0812.4703

Papers like that are apt to summarize and refer back to 2006. Let's try this:
http://arxiv.org/abs/gr-qc/0612104
At that point they were using (numerical) computer models of various cases but by some glitch they were getting a bounce at a different critical density. So that's not good. I'll look some more.

 

[Flaneuse]

Thanks so much :). And don't worry about it, feel free to take your time! No rush here, just a leisurely interest, for the moment.

 

[marcus]

I found something that may be right!
http://arxiv.org/abs/0710.3565

This is where they introduced a solvable equation model, as an alternative to numerical (computer) modeling, and got the same bounce result.

Robustness of key features of loop quantum cosmology
Abhay Ashtekar, Alejandro Corichi, Parampreet Singh
(Submitted on 18 Oct 2007)
Loop quantum cosmology of the k=0 FRW model (with a massless scalar field) is shown to be exactly soluble if the scalar field is used as the internal time already in the classical Hamiltonian theory. Analytical methods are then used
i) to show that the quantum bounce is generic;
ii) to establish that the matter density has an absolute upper bound which, furthermore, equals the critical density that first emerged in numerical simulations and effective equations;
iii) to bring out the precise sense in which the Wheeler DeWitt theory approximates loop quantum cosmology and the sense in which this approximation fails; and
iv) to show that discreteness underlying LQC is fundamental.

Finally, the model is compared to analogous discussions in the literature and it is pointed out that some of their expectations do not survive a more careful examination. An effort has been made to make the underlying structure transparent also to those who are not familiar with details of loop quantum gravity.
================
I think it was in this paper or right about this time they started using the abbreviation "sLQC" for
"solvable" Loop Quantum Cosmology. Because it employed a fully analytical approach.

It was also about this time that their estimate of the critical density, that maximum achieved right at the instant of the bounce, stabilized to about 41% of Planck density.
Before that a stray factor of two had somehow gotten in and they were saying 82%.

I am not sure but I think that the "modified Friedman equation" appeared about this time, which you have seen in the later papers like the 2008 one.
It has the factor (1 - ρ/ρ_crit)