# New work in loop cosmology

1. Nov 4, 2003

### marcus

Some new names (new to me anyway)
Shinji Tsujikawa (has published with Robert Brandenberger, background in string research now interested in loop gravity)
Roy Maartens (also string background, has published with Ellis)
Parampreet Singh (one of the new LQG people in India)

"Loop quantum gravity effects on inflation and the CMB"
Tsujikawa, Singh, Maartens
http://arxiv.org/astro-ph/0311015 [Broken]

a short paper that explores how LQG can have caused all the inflation ( > 65 e-foldings) people ordinarily postulate AND in doing so have left a distinctive SIGNATURE on the CMB
that is exciting if other people confirm this because then one can
look at the power spectrum of the fluctuations of the microwave background and see if it checks out

or maybe some ekpyrotic colliding braneworld model fits the data better???

great if observation can help distinguish models

bravo to T,S,and M----hope more comes of this

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2. Nov 4, 2003

### marcus

LQG predicts matter-antimatter asymmetry?

Another of Parmpreet Singh's papers, this time co-authored with
Gaetano Lambiase of Uni-Salerno

"Matter-Antimatter Asymmetry Generated by Loop Quantum Gravity"

http://arxiv.org/gr-qc/0304051 [Broken]

abstract: "We show that LQG provides a new mechanism though which observed matter-antimatter asymmetry in the Universe can naturally arise at temperatures less than GUT scale..."

In a very hot universe (as at big bang time) light itself can give rise to matter and antimatter-----but in equal or very nearly equal amounts

it is assumed that in the very early universe matter and antimatter particles were present in equal or very nearly equal numbers

as the bath of energy cooled it was possible for most of the matter to annihilate with most of the antimatter leaving almost zero

the puzzle is, why not exactly zero?

why was there just a tiny overbalance so that a residue of ordinary matter remained and was predominate over the antimatter?

this asymmetry has been explained in various ways

so Lambiase and Singh have offered a new explanation
maybe it is interesting and maybe not---just something to keep tabs on. I only came across it because of Singh's new paper
that was posted to day, about the LQG basis for the inflation scenario

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3. Nov 5, 2003

### marcus

At last week's Strings meets Loops conference (AEI-Potsdamm) Marten Bojowald described the TSM results and showed a slide of
WMAP data----a graph of the "running" of the spectral index ns(k) with wavenumber k

the empirical coefficient from the WMAP data was in the same ballpark with that predicted by LQG

see Bojowald's slide talk at the AEI website, he gives the reference
map.gsfc.nasa.gov
for the WMAP data he uses

at around the wavenumber k = 1/1000 per Mpc, the empirical "running" coef a = - 0.077 (plus 0.05 minus 0.052)
and the loop cosmology prediction is a = - 0.04

notice that the empirical confidence interval is [- 0.12, - 0.03] or so, pretty wide and includes the prediction

this seems to be preliminary but verging on testability---which I think was the point Bojowald was making

possibilities are beginning to appear for testing and either disproving LQG or at least of eliminating some versions (if and where there are different ones) and so narrowing things down. it is very nice to have happening I think!

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4. Nov 5, 2003

### Ambitwistor

That's pretty interesting. I worked out the spectrum produced by Bojowald's inflation and found that the spectrum was inconsistent with observation. I concluded that a second phase of ordinary inflation would be needed to agree with observations, which is what this paper found, but I suspected that this second phase would erase all evidence of the first, quantum gravity-driven phase, and didn't have time to pursue it further. The authors appear to have found a way to make it work (though I can't tell for sure, because for some reason my copy of the PDF chops off after page 3).

5. Nov 5, 2003

### marcus

Re: Re: new work in loop cosmology

You mean your PDF copy of
http://arxiv.org/astro-ph/0311015 [Broken]
does not include the last page?

Let's see what we can do. If I understand you, the paper is
"Loop quantum gravity effects on inflation and the CMB"
by Tsujikawa, Singh, Maartens

[edit: rest of post proved unnecessary...]

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6. Nov 5, 2003

### jeff

Re: Re: new work in loop cosmology

I wouldn't mind seeing that.

7. Nov 5, 2003

### Ambitwistor

Re: Re: Re: new work in loop cosmology

I don't think I still have my notes ... as I recall, the upshot was that the spectral index was really blue-tilted, like n=10 or something (as opposed to the scale-invariant n=1). I had to make some approximations that I wasn't sure were really justified; I'm not an expert in inflation, and most of the formulas you find in inflationary papers are derived under assumptions that don't quite hold in Bojowald's case, with a modified Friedmann equation. However, I'm fairly sure that the predicted spectral index wouldn't change too dramatically if it was worked out carefully.

The basic idea is to start with Bojowald's expression for the scale factor a(t), calculate the Hubble parameter H = a'(t)/a(t), from that get the spectrum of the primordial curvature perturbation (I forget the formula), and its scaling will give you the spectral index. It's a pretty simple calculation, actually.

This was all within the context of the trivial free scalar model that Bojowald initially proposed; the obvious way to fix it is to put in an inflaton potential to give you a second phase of inflation to fix things up, as the paper in question did. It gets rid of the nice idea that you can have purely quantum gravity-driven inflation with no inflaton at all, but it still has the possibility of leaving a quantum gravity imprint.

Last edited: Nov 5, 2003
8. Nov 5, 2003

### Ambitwistor

Re: Re: Re: Re: new work in loop cosmology

Hmm. I changed PDF viewers and managed to get the last page of the paper; they say that the quantum inflationary era would produce n &asymp; 1.73. I don't know if my value was really n=10, but I do remember it was at least several times larger than 1. So maybe my calculation was wrong, although I haven't fully studied their paper to determine whether the calculations are really analogous. If they do disagree, I'd take their calculation over mine, since they appear to be quite knowledgeable about the ins and outs of inflationary calculations, whereas I tried to just taught myself the subject from scratch to do mine.

9. Nov 9, 2003

### marcus

cosmological constant in Lorentzian spin foam model

Another paper. I just came across it though it is
almost a year old:

http://arxiv.org/gr-qc/0211109 [Broken]

It puts a positive cosmological constant into the Barrett-Crane
Lorentzian spin-foam model.

It is by Philippe Roche and K. Noui
"Cosmological Deformation of Lorentzian Spin Foam Models"

There is a paper by Freidel, Livine, and Rovelli which cites
this one, which is how I happened to find it.

[edit: another Philippe Roche paper that seems interesting is
http://arxiv.org/q-alg/9710022 [Broken]
Buffenoir/Roche "Harmonic Analysis on the Quantum Lorentz Group".
They find the representations of the Quantum Lorentz Group q-SL(2,C)
and prove a couple of Plancherel formulas (pp. 47, 48) and a
Plancherel Theorem in the q-deformed case]

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10. Nov 12, 2003

### marcus

Re: cosmological constant in Lorentzian spin foam model

This puts a positive cosmological constant into the Barrett-Crane
Lorentzian spin-foam model and gets a "q-BC" or quantum-deformed Lorentzian BC model. The paper had an interesting follow-up by Girelli and Livine "Quantizing speeds with the cosmological constant", discussed in another thread.

Livine's thesis did not deal with q-deformed groups, so-called "quantum groups", and did not consider positive cosmological constant. Essentially the thesis could be redone putting in positive Lambda and the Girelli/Livine paper (http://arxiv.org/gr-qc/0311032 [Broken]) makes a start.

The thesis establishes a bridge between the Lorentzian BC foam model and the covariant version of loop gravity. Assuming the bridge works with the q-deformed Lorentz group and the q-BC model connects to some "q-covariant" loop gravity with positive cosmological constant, what does this look like?

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11. Nov 12, 2003

### marcus

Re: Re: cosmological constant in Lorentzian spin foam model

Here's a series of papers relevant to that:

Smolin "Quantum gravity with a positive cosmological constant"
http://arxiv.org/hep-th/0209079 [Broken]

Witten "A note on the Chern-Simons and Kodama wavefunctions"http://arxiv.org/abs/gr-qc/0306083

Witten's "Kodama is unphysical" paper was partly in response to Smolin's "Quantum Gravity With a Positive Cosmological Constant". Undeterred, Smolin continued the thread with two more papers which (among other things) explicitly address points made by Witten

Stephon Alexander, Justin Malecki, Lee Smolin
"Quantum Gravity and Inflation"
http://arxiv.org/abs/hep-th/0309045

Freidel, Smolin "The linearization of the Kodama state"
http://arxiv.org/abs/hep-th/0310224

Although Girelli/Livine does not use the Kodama state, it mentions the Kodama state in its first paragraph in connection with positive cosmological constant.
There seems to be some interest in the Kodama state or finding something like it that does not have the same shortcomings. Having something corresponding to it over in the spin foam department may make this easier.

By way of stabilizing notation, Girelli/Livine's way of writing the q-deformed Lorentz group is "Uq(SL(2,C))".

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12. Nov 12, 2003

### marcus

Re: Re: Re: cosmological constant in Lorentzian spin foam model

Yesterday a new paper on the Kodama state (as applied to quantum gravity) came out.

Alejandro Corichi and Jeronimo Cortez "Note on Self-duality and the Kodama State"
http://arxiv.org/hep-th/0311089. [Broken]

Perhaps someone could explain why this state, or ones like it, comes up in the context of a positive cosmological constant. Also Corichi/Cortez think that Witten's analysis was in a "pure Yang-Mills" context and does not necessarily apply in full to the corresponding use of the state in quantum gravity. I cannot tell if Corichi/Cortez arguments have merit. I will re-list this sequence of articles related to the Kodama state and/or positive Lambda

Smolin "Quantum gravity with a positive cosmological constant"
http://arxiv.org/hep-th/0209079 [Broken]

Witten "A note on the Chern-Simons and Kodama wavefunctions"http://arxiv.org/abs/gr-qc/0306083

Alexander, Malecki, Smolin
"Quantum Gravity and Inflation"
http://arxiv.org/abs/hep-th/0309045

Freidel, Smolin "The linearization of the Kodama state"
http://arxiv.org/abs/hep-th/0310224

Corichi, Cortez "Note on Self-duality and the Kodama State"
http://arxiv.org/hep-th/0311089 [Broken]

Ichiro Oda "A Relation Between Topological Quantum Field Theory and the Kodama State"
http://arxiv.org/hep-th/0311149 [Broken]

[edit: another recent posting, 17 November, referring to Witten's paper and the open questions around the Kodama state]

from Oda's abstract

"We study a relation between topological quantum field theory and the Kodama (Chern-Simons) state. It is shown that the Kodama (Chern-Simons) state describes a topological state with unbroken diffeomorphism invariance in Yang-Mills theory and Einstein’s general relativity in four dimensions. We give a clear explanation of ”why” such a topological state exists."

from the conclusions paragraph

"...the Kodama state automatically satisfies the quantum Ashtekar constraints.

Even if we have understood a relation between the Kodama (Chern-Simons) state topological quantum field theory, we have no idea whether such a topological state is relevant to the real world or not. Of course, one of the big problems in the future is to clarify whether or not the Lorentzian Kodama state is normalizable under an appropriate inner product or not."

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13. Nov 18, 2003

### nonunitary

I think one has to separate what those papers are saying.

As I see it, the most interesting question regarding the Kodama states refers to quantum gravity. I have seen the Corichi/Cortez paper you refer to and I think they don´t have much to say about gravity. As far as I can tell, their construction is relevant for the Maxwell field, where they find a neat relation between self-duality and the chern-simons state. It seems also that the argument that Witten results are mainly for Yang-Mills is due to Smolin and Freidel and not to Corichi/Cortez.
(By the way, the new paper by Oda is not related to gravity either.)

Going back to gravity, the main issue is whether the kodama state is physical or not. I see several problems with it. First, in order to be useful one has to go to the complex $SL(2,C)$ connections, which even after the work by Levine, Freidel etc. is not as rigurous as the real SU(2) case. Even when it is true that a proper inner product on the physical Hilbert space might save the day and make the state normalizable, there is still some work to be done to show that. Second, I think there is still debate on whether the kodama state truly represents a vacuum centered on classical de Sitter space. As far as I can tell, some leadrs like Ashtekar himself has never agreed that this is the case.

14. Nov 19, 2003

### marcus

Hello nonunitary, thanks for responding! I'd appreciate any help sorting these papers out. Please explain why the new Oda paper is not related to gravity. I got the impression it was from his several mentions of General Relativity.

For instance in the short "discussion" section at the end he mentions GR three times and also refers to the "quantum Ashtekar constraints". He seems to think his paper is relevant to gravity.

15. Nov 19, 2003

### marcus

Indeed their actual construction uses an Abelian gauge group. They observe, however, that the same construction can be carried out in the non-Abelian case and they discuss its relevance to gravity. Here's what they say in their conclusions paragraph.

"Perhaps the most important question is whether these results on Abelian gauge theories can teach us something about the non-Abelian case, and in particular about gravity.

Clearly we can say nothing about the issue of how a physical inner product on the space of solutions to the constraints might affect the normalizability of the state. However, we can still learn something from this simple model.

What we have seen is that we do not need to introduce self-dual variables and complicated reality conditions in order to get a “self dual” representation and vacuum state [13]; our variables were always real. The self-duality in the resulting quantum theory came from the quantum representation, in this case due to the choice of complex structure on phase space. One might hope that this new perspective on the problem may shed some light in the goal of constructing physically relevant vacuum states and representations for gravity, without the need to introduce self-dual complex connections.

Note also that even when we have considered a pure Abelian gauge field, this construction can also be done for linearizations of Yang-Mills [5] and gravity theories [18]."

16. Nov 19, 2003

### nonunitary

"For instance in the short "discussion" section at the end he mentions GR three times and also refers to the "quantum Ashtekar constraints". He seems to think his paper is relevant to gravity"

Marcus,

I think that the paper is not relevant for gravity, because he does not do anything new with gravity. He shows that the kodma state is a solution to the full constraints: We already new that for a long time. He writes an action for gravity, but then makes some ansaetze that yield a topological theory at the end. His only observation (I still don't know if trivial or non-trivial) is that formally, the kodama state is a solution of the theory anytime the theory is topological (in the action) of the Chern type. What I still need to figure out is "why" he thinks he has shown "why" the kodama state exists in Yang-Mills and gravity.

On the other hand the paper about abelian gauge fields has more concrete results even when more modest in its claims. The authors (Corichi/Cortez) do not claim to explain anything about Yang-Mills nor gravity but to point out to the fact that the kodama state is the natural (rigurous) vacuum state arising from the choice of a self-dual "complex structure". A complex structure is a tensor of type (1,1) (that is, a map from vectors to vectors) such that when squared it is minus the identity $(J^2=-I)$. This quantity is important (essential) in the construction of the Fock representation in field theory. Given a complex structure, one can decompose the (complexified) phase space into two parts: the eigenspace with eigenvalue $i$ and the eigespace with eigenvalue $-i$. Depending on the nature of the complex structure $J$ these eigespaces may have different names. In the ordinary field theory case (in Minkowski) the decomposition is called positive-negative frequency decomposition. This choice leads (as shown in Wald's book about QFT) to the ordinary Fock space. The choice that the paper makes for the abelian gauge field is the one that comes naturally from the hodge-dual operator (that for a Lorentzian manifold in 4D defines a complex structure). The decomposition is then the self-dual, antiself-dual decomposition. The interesting part is that they get the kodama state in a antural way adn then make contact with the linking number. Some of this connection were well known, namely the chern-simons- knot invariants connection is known since the work by Witten, and the relation between self-duality and linking number was also known for the maxwell field. This paper then brings these three elements together.

17. Nov 19, 2003

### nonunitary

Marcus, I didn't see your last post until after I submitted my reponse. The phase that you highlight:

"Note also that even when we have considered a pure Abelian gauge field, this construction can also be done for linearizations of Yang-Mills [5] and gravity theories [18]."

means that the linearizations of Yang Mills as considered by Freidel and Smolin and of gravity (as analyzed in the loopy context by Varadarajan), are abelian gauge fields (three copies in the case of gravity) and therefore, the construction they consider can be done "verbatim". However, that does not mean that they have said anything non-trivial about the main issue I was refering to previously: is the Kodama state a physicaly interesting state for gravity? Even before that, is it even well defined?

18. Nov 19, 2003

### marcus

nonunitary,

Thanks for the clarification on both points!