Probability of Inflation in Loop Quantum Cosmology (new Ashtekar Sloan paper)

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http://arxiv.org/abs/1103.2475
Probability of Inflation in Loop Quantum Cosmology
Abhay Ashtekar, David Sloan
34 pages, 3 figures
(Submitted on 12 Mar 2011)
"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."
 

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  • #2
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I must admit LQC has surprised my every time with something new, just when I think okay that's about all LQC has to offer, I should stick with string theory, LQC comes up with something amazing...
 
  • #3
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From what I saw in the paper, LQC simply assumes some ad hoc scalar field that drives inflation with some ad hoc scalar potential whose coefficients are fine-tuned to make the potential sufficiently flat, see eq. (4.27), for example. There is nothing generic in making such ad hoc choices for the inflationary potential. Inflation is not at all generic unless you give a good reason for why the inflaton potential has the right property to drive inflation. Likewise, LQC says absolutely nothing about the couplings of the inflaton to the Standard Model which would be important for reheating. To contrast this, in stringy models of inflation people first identify what the inflaton actually is then try to derive the shape of the inflaton potential, including the coefficients. Once you know this, you can make meaningful predictions such as the spectral tilt, tensor to scalar ratio, possible non Gaussianities, etc. That's where the interesting game really starts and that's the main reason why string theorists have focused on inflation
 
  • #4
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Well, smoit, you need to read a little deeper: Around Eq 4.27 is a discussion of how the potential is NOT fine tuned - it just needs to be "well approximated by a quadratic potential around its minimum". As in, a potential that admits a good Taylor series - that's actually incredibly generic. In the very paragraph following the equation you quote are the words "it is not unreasonable to expect that... <results hold> for a wide class of physically interesting potentials".

Ashtekar also discusses why this potential fits with WMAP data - and how this general scenario fits well with observations. I must admit I got a little lost with the derivations there, but it looks like what is being said is that on the whole you give LQC a scalar field with a fairly generic potential and what you get back is that almost all of phase space (using some clever way of measuring it) leads to exactly the observable inflationary results.
 
  • #5
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I believe that their result holds for the quadratic potential, which is just the old chaotic inflation where such a potential is simply assumed. However, in order to get slow roll inflation for a more generic case it is not enough to say that the potential at the minimum is well approximated by [tex]V\approx m^2\phi^2[/tex] because the [tex]\eta[/tex] slow roll parameter is directly affected by the higher order terms even when [tex] \phi<< m_{pl} [/tex] is sub-Planckian and the series expansion makes sense. By the way, it will most certainly not be a Taylor series but instead some asymptotic series.
In quantum field theory a scalar potential is generically expected to receive quantum corrections suppressed by powers of the cutoff scale with order one coefficients, in this case the natural scale would be the Planck scale. So, in the next order (quartic) the corrected scalar potential can be written as [tex] \tilde V\approx V(1+{\cal O}(1)\frac {\phi^2}{m_{pl}^2})[/tex], which results in a correction to the slow roll parameter [tex]\delta\eta={\cal O}(1)[/tex] which ruins slow roll inflation. Note that in this case the correction to the quadratic scalar potential is small and the expansion is fine since the vev [tex]\phi<< m_{pl}[/tex]. So to get a slow roll inflation one has to explain why the COEFFICIENTS of such higher order terms are really tiny, like [tex] \lambda \approx 10^{-13}[/tex] in the case of eq. (4.27). Just saying the the potential is well approximated my the quadratic contribution near the minimum, which is correct as a statement, is red herring because it is the coefficients of the higher order terms that directly affect the slow roll parameters and unless you can compute them from a more fundamental theory, statements that inflation is generic in LQC are just plain BS because LQC says nothing about the nature of the inflaton and its potential! There are no "generic" potentials that give you slow roll inflation in any theory! One has to work very hard to come up with a model where the coefficients of the higher order terms are naturally suppressed to maintain slow roll to get enough e-foldings.
 
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Actually, no. A coefficient of phi^4 of that order is exactly the kind discussed by Linde in his work on inflation. And you certainly do NOT expect order 1 planck scale corrections to a potential whose first term is of order 10^-6. In the units you're using, you'd actually expect a the coefficient in front of the fourth power to be of the order of the square of the coefficient in front of the second power, which exactly fits with the model they have.

Anyway, you completely misread if you think that genericity is to do with potentials - it's about initial data in phase space, as discussed in the measure section. The main point made is that 99% (more even) of solutions to LQC with this potential fit with WMAP 7. Contrast this to the Gibbons+Turok claim that the probability of the same for GR is 10^{-85} and you have a result, though it is pointed out WHY there is such a big discrepancy (originally I believe by Corichi).
 
  • #7
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Sorry, in the previous post I should have stated that the coefficient of the quartic term should be much less than [tex]\frac{m^2}{m_{pl}^2} [/tex], i.e. [tex]\lambda << 10^{-13}\sim \frac{m^2}{m_{pl}^2} [/tex].
This is easy to see if you try to compute the slow roll parameter [tex]\eta=m_{pl}^2\frac{V^{\prime\prime}}{V}[/tex]. In this case you get [tex]\eta=2\frac{m_{pl}^2} {\phi^2}+4\lambda\frac{m_{pl}^2}{m^2}[/tex], which is too large to get slow roll inflation unless [tex]\lambda << \frac{m^2}{m_{pl}^2} [/tex] and [tex] \phi>m_{pl}[/tex], but when [tex] \phi>m_{pl}[/tex], one must take into account all the higher powers and not cut off at the quartic order. Hence, unless there is a good reason for suppressing all the corresponding coefficients, you don't get enough e-foldings with a "generic" potential, i.e. with no fine tuning of all the coefficients in the expansion. I understand that the goal of the paper was to address the initial conditions but my point is that this solves nothing until you come up with a theory that explains the inflaton potential itself. You can study a bunch of ad hoc potentials and build hundreds of models, which is what people have done for decades but saying that since Linde did it in the past we might as well do the same does not address the actual problem, which is the much bigger one than the initial conditions.
 
  • #8
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And you certainly do NOT expect order 1 planck scale corrections to a potential whose first term is of order 10^-6.
Why not? Do you have some kind of symmetry that protects the scalar potential? In fact, to keep the mass squared term at order [tex]m^2\sim 10^{-12}m_{pl}^2 [/tex] you already have to do some extreme fine-tuning because SUSY is badly broken during inflation, but it's not just the leading quadratic term but all the higher order terms that can receive order one quantum corrections. For a generic scalar there is actually no particularly good reason why the quartic self-coupling should be so small or even perturbative! Recall that in the Standard Model the quartic self-coupling of the Higgs is perturbative but not that small, i.e.
[tex] \lambda =\frac{m_{H}^2}{2v^2}>>\frac{m_{H}^2}{m_{pl}^2}[/tex].
If you want any LQC, or string-inspired, inflationary model to be taken seriously you must present a damn-good reason for why the inflaton potential is so fine-tuned.
 
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  • #9
marcus
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Well, smoit, you need to read a little deeper: Around Eq 4.27 is a discussion of how the potential is NOT fine tuned - it just needs to be "well approximated by a quadratic potential around its minimum". As in, a potential that admits a good Taylor series - that's actually incredibly generic. In the very paragraph following the equation you quote are the words "it is not unreasonable to expect that... <results hold> for a wide class of physically interesting potentials".

Ashtekar also discusses why this potential fits with WMAP data - and how this general scenario fits well with observations. I must admit I got a little lost with the derivations there, but it looks like what is being said is that on the whole you give LQC a scalar field with a fairly generic potential and what you get back is that almost all of phase space (using some clever way of measuring it) leads to exactly the observable inflationary results.
I think that is a fair reading. The point is not to explain why there could be a scalar field with quadratic potential present at the bounce. They just assume a generic scalar field of the type that infation scenarists commonly employ, and get interesting results.

The results are robust (in the sense of not depending on details of the scalar field or some sort of "fine tuning".)
And they are impressive (in the sense of recovering measured features of the universe with high probability.)

Actually, no. A coefficient of phi^4 of that order is exactly the kind discussed by Linde in his work on inflation. And you certainly do NOT expect order 1 planck scale corrections to a potential whose first term is of order 10^-6. In the units you're using, you'd actually expect a the coefficient in front of the fourth power to be of the order of the square of the coefficient in front of the second power, which exactly fits with the model they have.

Anyway, you completely misread if you think that genericity is to do with potentials - it's about initial data in phase space, as discussed in the measure section. The main point made is that 99% (more even) of solutions to LQC with this potential fit with WMAP 7. Contrast this to the Gibbons+Turok claim that the probability of the same for GR is 10^{-85} and you have a result, though it is pointed out WHY there is such a big discrepancy (originally I believe by Corichi).
That seems to be right. Their reference [10] is to the paper by Corichi and Karami. Corichi pointed out the difficulty of defining a prior measure where there is a singuarity at the start of expansion. You have no natural place to specify initial conditions! So there is ambiguity.
The way it looks to me is that this March paper of Ashtekar is in part a response to the January paper of Corichi Karami. It resolves the ambiguity they pointed out.

And in doing so it really goes a lot farther. I really like the direction they are heading!
The key thing, for me, is this bounce hypersurface where H goes from neg to pos and where initial conditions can be defined.

Intuitively because gravity turns repellent (equation 2.4) near the critical density one would expect inhomogeneity to be smoothed out. So the quantum Friedmann model is applicable and bounce occurs coherently---at a definable moment in universe-time.

I also find the thermodynamics implications interesting: with attractive gravity the equilibrium state is clumpy and with repellent it is uniform. There seems to be a subtle evasion of a second law paradox.

Thanks for your comments!
 

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