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Noncommutative geometry naturally includes inflation epoch

  1. Mar 9, 2009 #1


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    In ordinary GR geometry, the Higgs field cannot play the rôle of the inflaton, so to imagine inflation one has to introduce some exotic field that is not part of the Standard Model. Something completely made up must be introduced to make it work.

    Not so in the context of non-commutative geometry (NGC) say William Nelson and Mairi Sakellariadou.

    ...within the noncommutative geometry approach to unifying gravity and the standard model, it is possible to have an epoch of inflation sourced by the dynamics of the Higgs field.

    In addition, this type of noncommutative inflation could have specific consequences that would discriminate it from alternative models. In particular, since the theory contains all of the standard model fields, along with their couplings to the Higgs field, which in this scenario plays the rôle of the inflaton, a quantitative investigation of reheating should be possible. More significantly, the cosmological evolution equations for inhomogeneous perturbations differs from those of the standard Friedmann-Lemaître-Robertson-Walker cosmology [7]. This raises the possibility that signatures of this noncommutative inflation could be contained within the cosmic microwave background power spectrum.

    Natural inflation mechanism in asymptotic noncommutative geometry
    William Nelson, Mairi Sakellariadou
    3 pages
    (Submitted on 9 Mar 2009)
    "The possibility of having an inflationary epoch within a noncommutative geometry approach to unifying gravity and the standard model is demonstrated. This inflationary phase occurs without the need to introduce 'ad hoc' additional fields or potentials, rather it is a consequence of a nonminimal coupling between the geometry and the Higgs field."


    Nelson and Sakellariadou explain why the Higgs field does not work

    Unfortunately, it has proved difficult to naturally embed inflation within an underlying fundamental theory. Inflation most naturally occurs when the dynamics of the universe are dominated by the evolution of a scalar field, the inflaton, slowly rolling in its potential; the form of the potential defines the type of the inflationary model.

    There is only one scalar field within the standard model of particle physics, the Higgs field, and it is naturally hoped that this could play the rôle of the inflaton. However, it has been shown[4] that inorder for the Higgs field to produce the correct amplitude of density perturbations, its mass would have to be some 11 orders of magnitude higher than the one required by particle physics. This conclusion was however reached using general relativistic cosmology and here we re-examine the calculation in the context of cosmological noncommutative geometry[5, 6, 7].

    We already have an earlier paper by the same authors about NCG cosmology.
    Last edited: Mar 9, 2009
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  3. Mar 9, 2009 #2


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    Here's the earlier Nelson Sakellariadou paper, that appeared three months ago.

    Cosmology and the Noncommutative approach to the Standard Model
    William Nelson, Mairi Sakellariadou
    4 pages
    (Submitted on 9 Dec 2008)
    "We study cosmological consequences of the noncommutative approach to the standard model. Neglecting the nonminimal coupling of the Higgs field to the curvature, noncommutative corrections to Einstein's equations are present only for inhomogeneous and anisotropic space-times. Considering the nominimal coupling however, we obtain corrections even for background cosmologies. A link with dilatonic gravity as well as chameleon cosmology are briefly discussed, and potential experimental consequences are mentioned."

    A central feature of what WN and MS are doing is that they include a coupling between the Higgs field and the geometry. This is how they get inflation to work in the paper that came out today.

    Their work draws on this earlier paper by three Russians, the reference [4] mentioned earlier
    Standard Model Higgs boson mass from inflation
    Fedor L. Bezrukov, Amaury Magnin, Mikhail Shaposhnikov
    5 pages, 3 figures
    (Submitted on 29 Dec 2008)
    This is where it was found that the Higgs mass would have to be many orders of magnitude greater. (But they also seem to get around this by the same means, introducing a nonminimal coupling of the Higgs to the geometry.)

    I haven't figured out how important NCG is to this result, since Bezrukov et al do not use it.
    Last edited: Mar 9, 2009
  4. Mar 10, 2009 #3
    I would be curious to see more in-depth work about this, it seems like at the end they conclude their paradigm can produce much more specific calculations and predictions than they produce in just this introductory paper.

    Is the nature of what they're doing such that they will be able to produce more specific results once the exact Higgs mass is known?
  5. Mar 10, 2009 #4
    Why is that?
    Brian Greene in FABRIC OF THE COSMOS, Chapter 19 Deconstructing the Bang, spends several pages drawing what appear to be direct linkages between Einstein's cosmologcal constant and Guths' inflationary field:

  6. Mar 10, 2009 #5
    I checked Roger Penrose THE ROAD TO REALITY, Chapter 28.4 he says

    Again this seems to imply a linkage....Greene discusses two differences in inflationary vs cosmological fields, I could not figure out what Penrose had in mind...
    Last edited: Mar 11, 2009
  7. Mar 10, 2009 #6
    Natural Inflation Mechanism in asymptotic...

    Isn't that just the kind of thing you love to see!!!!!

    Reading the above paper, not following the math, it seems like the issue is whether a Higgs like (standard model) field coupled to geometry or whether the inflationary Higgs field was considerably different than todays is the better model....Seems like

    from Roger Penrose could sure have been different than todays...

    I have forgotten just how Guth came across his inflationary expansion but it has always seemed like a lucky find, a good overlay, rather than the culmination of a series of natural events...
  8. Mar 10, 2009 #7
    Naty1: Do you suppose that when Penrose says "a Higgs field", he is using this as a synonym for "a scalar field"?
  9. Mar 11, 2009 #8

    If you mean does Penrose know Higgs is scalar, I'm sure he does. It's the only scalar field in the standard model so I dout he'd miss it!

    If you mean is he using Higgs as a generic term for scalar, my interpretation of his descriptions leads me to say No....

    Both Greene and Penrose refer to the phase transition, rolling potential/mexican hat style associated with spontaneous symmetry breaking of the inflationary cycle....so the concepts are the same but maybe their formulations of the field transition are different...

    What I should have also mentioned: I like the idea in Nelsons paper even if Higgs could be included in GR...but right now I don't understand those Higgs in/out implications either way...
    Last edited: Mar 11, 2009
  10. Mar 15, 2009 #9


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    Yes, I think this is the case. One should not rely on popular science, since it makes lots of simplifications: in this case, presumably Penrose doesn't want to have to define a scalar field, so calls them "Higgs-type" fields, but which are not the "ordinary" Higgs.
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