(Im)probability of Inflation (Ashtekar Sloan 1003.2475)

[marcus]

The likelihood or actually the UNlikelihood of a satisfactory inflation episode---assuming various beyond-standard-model pictures of cosmology---has recently become a major issue.

The discussion revolves around the difficulty of putting a MEASURE on the range of possible initial conditions at the start of expansion. And in models involving a BOUNCE there is the issue of ENTROPY. How can the entropy of the gravitational field (the geometry of the universe) be defined? Assuming a satisfactory definition of entropy, what role might be played by the Second Law of Thermodynamics? Do observers before and after the bounce has different perspectives on the states of the universe and apply fundamentally different coarse-graining, and so on?

A conference on Challenges for Early Universe Cosmology was recently held at Perimeter and the talks wrestled again and again with the topics of Measure, Bounce, Geometric Entropy, Probability of Inflation.

At the end of the first talk of the conference (by Turok 12 July) a very interesting point was raised by someone in the audience (at time 1:07:40) who pointed out that Loop cosmology addresses these issue in a comaratively simple way. They referred to this paper of Ashtekar and Sloan:

http://arxiv.org/abs/1103.2475
Probability of Inflation in Loop Quantum Cosmology
Abhay Ashtekar, David Sloan
34 pages, 3 figures
Inflationary models of the early universe provide a natural mechanism for the formation of large scale structure. This success brings to forefront the question of naturalness: Does a sufficiently long slow roll inflation occur generically or does it require a careful fine tuning of initial parameters? In recent years there has been considerable controversy on this issue. In particular, for a quadratic potential, Kofman, Linde and Mukhanov have argued that the probability of inflation with at least 65 e-foldings is close to one, while Gibbons and Turok have argued that this probability is suppressed by a factor of ~ 10^-85. We first clarify that such dramatically different predictions can arise because the required measure on the space of solutions is intrinsically ambiguous in general relativity. We then show that this ambiguity can be naturally resolved in loop quantum cosmology (LQC) because the big bang is replaced by a big bounce and the bounce surface can be used to introduce the structure necessary to specify a satisfactory measure.
The second goal of the paper is to present a detailed analysis of the inflationary dynamics of LQC using analytical and numerical methods. By combining this information with the measure on the space of solutions, we address a sharper question than those investigated in the literature: What is the probability of a sufficiently long slow roll inflation WHICH IS COMPATIBLE WITH THE SEVEN YEAR WMAP DATA? We show that the probability is very close to 1.
The material is so organized that cosmologists who may be more interested in the inflationary dynamics in LQC than in the subtleties associated with measures can skip that material without loss of continuity.
=========

One of the main points of this paper is that the Loop bounce is simple enough that the universe forms a spacelike hypersurface at the moment of the bounce---making it straightforward to define a probability measure on the range of initial conditions.

Models suffering from a singularity at the start of expansion, or where something more elaborate happens (which may involve more complicated assumptions) seem to have a harder time establishing a plausible measure on the initial conditions. We saw a lot of that in the talks at the "Challenges" conference. With some models one had to put a measure not on initial conditions at bounce, but on hypothetical limiting states in the far distant future.

Here are videos of all the conference talks:
http://pirsa.org/C11008
The opening one, by Turok, at the top of this iist is the one that had the interesting comment towards the end (at time 1:07:40) referring to the Ashtekar et al result in arxiv 1103.2475.

 

[marcus]

The essential thing to note here is that IF you assume gravity/geometry is quantized in the style of LQG, this does away with the unlikelihood of adequate inflation---under Ashtekar's assumptions, problem solved. On the other hand if you assume some other early universe model it seems to be much more difficult to match the available data with high probability.

Speakers coming from various other models mentioned this problem of unlikely or inadequate inflation time and again.

Sir Roger Penrose gave a talk in which he needed to "transcend" the Second Law of Thermodynamics, to the things to work out. And in which assumed that over 100s of billions of years electrons would become massless----along with all the other particles. Mass itself would decay. His talk was very charming, but illustrates how people were going through contortions to address the issues that I mentioned---the challenges facing early U cosmo.

 

[marcus]

Perhaps the best talk to watch on these topics is one given by Abhay Ashtekar in May 2011 at the Madrid Loops conference. At time 27:00, or slightly before that, he starts a new topic
WMAP AND THE PROBABILITY OF INFLATION.

If you are interested in this topic you can save time by pausing the talk, dragging the time-button along the timebar until the clock says 26:00 or so, and then unpausing.

I'll get the link for downloading the Madrid video files. They play on "VLC", a free piece of software that is easy to get off the web.
Here's the listing of all the Madrid talks:
http://loops11.iem.csic.es/loops11/index.php?option=com_content&view=article&id=75&Itemid=73
Plenary talks have links to both slides PDF and video. Having the slides handy in a separate screen can sometimes help you follow the video.
Scroll down to Wednesday and click on Ashtekar's talk (Recent Advances in Loop Quantum Cosmology) or try this direct link:

http://loops11.iem.csic.es/loops11/...-Cosmology&catid=35:plenary-lectures&Itemid=1

 

[marcus]

The second link in the preceding post was wrong, sorry. I gave the link to Rovelli's talk instead, by mistake. Too late to edit. Here's the link to Ashtekar's talk (I hope this is right now):
http://loops11.iem.csic.es/loops11/index.php?option=com_content&view=article&id=181
Yes! That goes directly the abstract page for Ashtekar's talk, which has further links to the slides PDF and to the video.

And I gave the time as 27:00, which was off by a minute. The section starts around 28:00. Drag the time button to 28:00 and the slide will say "WMAP and the Probability of Inflation".

 

[bcrowell]

Interesting. I didn't know the relevant background for the Ashtekar-Sloan paper, so I found this helpful: Bojowald, Inflation from Quantum Geometry, http://arxiv.org/abs/gr-qc/0206054

"Quantum geometry predicts that a universe evolves through an inflationary phase at small volume before exiting gracefully into a standard Friedmann phase. This does not require the introduction of additional matter fields with ad hoc potentials; rather, it occurs because of a quantum gravity modification of the kinetic part of ordinary matter Hamiltonians. An application of the same mechanism can explain why the present-day cosmological acceleration is so tiny."

Marcus, maybe you could help me to figure out whether I'm understanding correctly.

I thought LQG had problems incorporating matter fields. Is this now a solved problem?

Bojowald describes a scalar field (like the traditional inflaton), but one that doesn't have to have a self-interaction potential with any specially cooked up properties. Is he just using a scalar field because it's easy to calculate with, or is LQG's inflation similar to traditional inflation in that you need a matter field that's a scalar? In other words, does LQG still produce inflation if you instead put in some other field such as a photon?

The Ashtekar-Sloan paper refers to "super-inflation." What's that?

 

[atyy]

I thought LQG had problems incorporating matter fields. Is this now a solved problem?

As far as I understand, it is better to treat LQC separately from LQG. LQC does have matter and gravity. It seems like a framework (But maybe one should read only papers after 2006 first to figure out what that is). LQG has not yet been shown to exist, nor to contain gravity or matter.

 

[bcrowell]

As far as I understand, it is better to treat LQC separately from LQG. LQC does have matter and gravity. It seems like a framework (But maybe one should read only papers after 2006 first to figure out what that is). LQG has not yet been shown to exist, nor to contain gravity or matter.

Hmm...how can LQG not contain gravity if it's a theory of quantum gravity?

 

[atyy]

Hmm...how can LQG not contain gravity if it's a theory of quantum gravity?

It may or may not, it's still work in progress. (Or as we know from string theory and AdS/CFT - theories without gravity can contain gravity;)

LQC, on the other hand does seem to have matter and gravity. However, it is background dependent (requires symmetry of the spacetime).

It would be nice to have a link between LQC and LQG, but I think that's also work in progress.

 

[bcrowell]

(Or as we know from string theory and AdS/CFT - theories without gravity can contain gravity;)

I'm sure you know what you're talking about, but I can't make sense of this statement.

 

[MTd2]

What is the justification for a quadratic potential and not other kind of potential?

 

[marcus]

Hi Ben,
at present there seem no serious obstacles to including matter in LQG AFAICS. Many of the current papers do include matter. In particular there was that landmark paper around December 2010 called "Spinfoam Fermions".
I suppose how one thinks about this depends on one's sense of the momentum or progress of research in the field. No responsible expert has identified any reason why it should be impossible to include matter in a fully satisfactory way. Meanwhile plenty remains to be done, and plenty is getting done.
It's a rapidly developing field, so to gauge the current status and rate of progress one has to watch the latest overview presentation.
http://loops11.iem.csic.es/loops11/index.php?option=com_content&view=article&id=76

The Ashtekar-Sloan paper refers to "super-inflation." What's that?

Inflation is understood to be (near) exponential growth of the scale factor. Super-inflation is a phase of super-exponential growth. The Hubble parameter, instead of being large and nearly constant (as in usual inflation) is increasing rapidly.

Before the bounce the H is negative, and at the bounce H = 0.
During the extremely brief period of superinflation, H increases from zero to nearly the Planck frequency.

Then you get inflation with H staying nearly constant and gradually declining.

The Hubble time 1/H we think of as a long time, like 14 billion years. But by the end of super-inflation H has increased to 93% of Planck frequency so that the Hubble time is around 10^-43 second. The bounce is what causes this rapid increase in H.

If anyone is new to the discussion and wants background, this is OK:
http://en.wikipedia.org/wiki/Inflation_(cosmology )
you will see that during inflation the scale factor ~ e^Ht with H approx constant.
In the paper Ben referred to, superinflation is described in section B-1 starting around page 16 or 17.
You will see that the bounce causes H to increase abruptly from 0 to near Planck scale.
So in that brief episode the growth is much much faster than exponential! This brief episode is what drives the (hypothetical) inflaton up its potential---gets it all charged up and prepared for a longer satisfactory period of ordinary inflation. Super-inflation itself may be too brief to afford much increase in the scale factor, so you need it to be followed by usual inflation.

 

[atyy]

I'm sure you know what you're talking about, but I can't make sense of this statement.

A regular QFT without gravity in a lower dimension holographically specifies a theory of gravity in a higher dimension. It's one of the most amazing things to have emerged in quantum gravity, not yet proven, but lots of evidence in its favour.

http://arxiv.org/abs/0909.0518
http://arxiv.org/abs/gr-qc/0602037

 

[bcrowell]

Thanks, atyy and marcus, for the explanations!

 

[MTd2]

What is the justification for a quadratic potential and not other kind of potential?

 

[marcus]

What is the justification for a quadratic potential and not other kind of potential?

Quadratic is a common type of inflaton potential which has traditionally been used in constructing inflation scenarios.
I think this probably goes way back---perhaps to the 1980s---to do the analysis, just for the sake of definiteness inflation scenarists would have needed to specify some type of potential.
So maybe you should ask that question of Alan Guth or Andrei Linde.

The point is that IF you assume inflaton with (for the sake of argument) that common type of potential THEN with other approaches YOU STILL HAVE PROBLEMS. As speakers at the Perimeter conference were explaining, adequate inflation is still improbable. But with LQC, making the same assumption, you do not have a fine-tuning problem.

You might find it interesting to listen to what Ashtekar has to say at around time 28:00 concerning the motivation for taking the possibility of inflation seriously. He emphasizes the observed large-scale structure (clusters of galaxies, superclusters, filaments, voids...). Numerical models of structure formation based the initial density fluctuation spectrum that would have arisen from inflation predict structure resembling what is observed.

It is an impressive match of theoretical model with observation. I don't think the "quadratic vs non-quadratic" issue is important here. It is just one common type of potential which one traditionally assumes. You have to introduce some potential and that is one which theorists have used for inflation for a long time. Think of it as "generic". It is the choice you don't have to explain. If you pick anything else, it would seem fancy, like a deliberate complication, and then you would need to give a reason.

The point is not "quadratic vs non-quadratic". It is that inflation (with a wide range of choice for the potential) gives a good match with observed largescale structure. I remain skeptical---I would like to see other possible explanations that do NOT involve inflation. But if you want to have inflation, and you assume the usual generic inflaton field, then getting the right amount of inflation is STILL improbable (unless, as Ashtekar points out, you base the analysis on Lqc.)

 

[MTd2]

It could be a discussion of quadratic vs. non-quadratic. What if the quadratic function is the peak value of an ensemble of eigenfunctions of LQG?

 

[marcus]

MTd2,
Re your question, I suggest you read the last paragraph on page 28. The paragraph which ends "...will continue to hold for a wide class of physically interesting potentials."

 

[bapowell]

One important distinction is between potentials that are local attractors and those that aren't. Quadratic inflation is one such potential, one that generically leads to eternal inflation. The other popular potential -- inverted, Mexican hat-type -- are not generically attractors, and without presupposing thermal equilibrium in the pre-inflationary universe, it is difficult to understand how the inflaton would find itself in the false vacuum.

 

[genneth]

As a dilettante:

1. The entropy issue seems like an old (and solved?) problem of time in general relativity (quantum doesn't necessarily add any new problems). It seems possible that current LQC is simply using a coordinate time and not in any sense a measurable time. This might be the perfect testing ground for Rovelli/Connes thermal time hypothesis, and show that time itself becomes poorly defined around the bounce, but away from it definitely flows *away*.

2. Aren't all potentials quadratic at the bottom? I seem to remember thinking this when the paper first came out and this issue was raised --- was the suggestion was then (as now?) that their analysis needed the lack of higher order terms to remain true because one does get pushed very far away from the bottom of the well?

 

[marcus]

MTd2,
Re your question, I suggest you read the last paragraph on page 28. The paragraph which ends "...will continue to hold for a wide class of physically interesting potentials."

2. Aren't all potentials quadratic at the bottom? ...

Hi Genneth! Yes, this was one of the points made in that paragraph I referred to on page 28 of the paper. I don't think there is any serious issue here---the results hold qualitatively for a wide class of potentials.

In his Madrid talk Ashtekar explains how to view the results.

 

[bapowell]

2. Aren't all potentials quadratic at the bottom? I seem to remember thinking this when the paper first came out and this issue was raised --- was the suggestion was then (as now?) that their analysis needed the lack of higher order terms to remain true because one does get pushed very far away from the bottom of the well?

Not ones without mass terms.

 

[MTd2]

I don't think there is any serious issue here---the results hold qualitatively for a wide class of potentials.

Not a wide class, for example, for example all odd powers (x,x^3,x^5..), as well as all polynomials whose even terms are big enough to have local minima.

 

[marcus]

MTd2,
Re your question, I suggest you read the last paragraph on page 28. The paragraph which ends "...will continue to hold for a wide class of physically interesting potentials."

Ashtekar calls it a wide class of physically interesting potentials.
If you have different ideas of what is "wide" from Ashtekar, or if you have different ideas of what is "physically interesting", that is fine.
these are subjective qualitative terms. You can just speak your own language and have your own meanings of words different from his. Perfectly fine. Would be stupid for me to argue about that.

 

[MTd2]

It would not be stupid to think what I proposed in the beginning, that is, LQG selects the interesting potentials, not the researcher.

 

[marcus]

It would not be stupid to think what I proposed in the beginning, that is, LQG selects the interesting potentials, not the researcher.

That is a nice idea, but it is asking a lot. Asking too much of the theory, that it should tell you at this point something about the potential of an inflaton field that we cannot even say for sure that it existed.

You are smart and it would do you a lot of good to actually listen to the talk. The analysis A. is presenting is of a natural PRIOR measure---the Liouville measure on initial conditions (at the bounce, defined on the bounce (hyper)surface.)

From a prior measure, as he explains several times, you can get guidance only if the probability of the region of phase space in question is either very close to zero or very close to one.

Then, performing analysis is not likely to change the qualitative conclusion. In this case a small region of phase space turned out to have a probability (a priori) of something like 0.999997.

In other words, adequate inflation, and a good fit with the WMAP observations, had probability very near one. Refining the probability measure (as with Bayesian analysis) is not likely to change that qualitative conclusion.

This does not show that inflation occurred, only that it COULD have, nor does it pick out any particular inflaton potential. It assumes an inflaton field and the Liouville measure prior is arrived at based on very broad assumptions. So it could have, in that context, if LQC is right.

But what it does do, I think, is call in question the basis for the select little Challenges for Cosmology conference organized by Turok, Steinhardt and a few others at Perimeter. http://pirsa.org/C11008
A central theme of that conference was the IMPROBABILITY of Inflation. Same Liouville measure method. It was premised on the experience with several non-Loop cosmo models that even if you assume an inflaton with some usual potential adequate inflation, giving the results we observe, was very very unlikely and required far-fetched extra assumptions or some fancy fine-tuning. Only certain alternatives were allowed to be presented involved---in Science when you have a conference on some problem like that you are expected to allow various ways to address the problem to be heard.

Turok gave the opening talk and underscored how improbable inflation was and how this presented a challenge to develop better cosmo models. Other speakers echoed this perception that inflation was unsatisfactory because so improbable in any scheme they knew leading up to the onset of expansion. How can we do without inflation? Since it is so unlikely.

The Challenges conference was conceived and organized in a somewhat peculiar way: the thematic Big Challenge was the Improbability of Inflation (what can we do about it?) yet they did not get Ashtekar to give the obvious presentation responsive to that theme! By contrast a central message in Ashtekar's talk was the Probability of Inflation. (The slide headline at around minute 26 or 28.)

There is an amusing moment right at the end of this talk.
http://pirsa.org/11070044/
Have a listen.
*Waves hands* "there must be some error in Ashtekar's analysis, only I haven't been able to find it" (or words to that effect.)