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LQC into irregular cosmo

  1. Nov 15, 2009 #1


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    Loop Quantum Cosmology has so far deal mainly with highly uniform models like what is normally studied in classical cosmology---homogeneous and isotropic universes.

    But in the past couple of years several papers have appeared that relax the uniformity assumptions---they get into studying universes that are NOT homogeneous and isotropic.

    The same has happened in LQG black hole research---people have started studying asymmetric collapse, and asymmetric bounce. The singularity is eliminated as usual but anisotropic models can give you different results. I don't want to get into a discussion of LQG black hole research. Just mention it in passing.

    Have to go now, when I get back I will get some links to LQC papers that treat inhomog and anistropic cases.


    Back now. By coincidence what I think is one of the most interesting relaxed-uniformity Loop Cosmo papers is co-authored by Francesca Vidotto who used to post here a lot and still comes by sporadically. I think her first real contact with LQG was when she attended the QG School at Zakopane in March 2007. The ski was still good so in the morning they had tutorials on covariant LQG, canonical LQG, Triangulations QG, Asymptotic Safety, and in the afternoon they went skiing. Zakopane is a resort in the Polish mountains. So in a sense this paper began there:

    Stepping out of Homogeneity in Loop Quantum Cosmology
    Carlo Rovelli, Francesca Vidotto
    Classical and Quantum Gravity 25:225024,2008
    16 pages
    (Submitted on 29 May 2008)
    "We explore the extension of quantum cosmology outside the homogeneous approximation, using the formalism of loop quantum gravity. We introduce a model where some of the inhomogeneous degrees of freedom are present, providing a tool for describing general fluctuations of quantum geometry near the initial singularity. We show that the dynamical structure of the model reduces to that of loop quantum cosmology in the Born-Oppenheimer approximation. This result corroborates the assumptions that ground loop cosmology, sheds light on the physical and mathematical relation between loop cosmology and full loop quantum gravity, and on the nature of the cosmological approximation. Finally, we show that the non-graph-changing Hamiltonian constraint considered in the context of algebraic quantum gravity provides a viable effective dynamics within this approximation."

    Today we saw on arxiv the follow-up to Rovelli-Vidotto

    Triangulated Loop Quantum Cosmology: Bianchi IX and inhomogenous perturbations
    Marco Valerio Battisti, Antonino Marciano, Carlo Rovelli
    21 pages
    (Submitted on 13 Nov 2009)
    "We develop the 'triangulated' version of loop quantum cosmology, recently introduced in the literature. We focus on the 'dipole' cosmology, where space is a three-sphere and the triangulation is formed by two tetrahedra. We show that the discrete fiducial connection has a simple and appealing geometrical interpretation and we correct the ansatz on the relation between the model variables and the Friedmann-Robertson-Walker scale factor. The modified ansatz leads to the convergence of the Hamiltonian constraint to the continuum one. We then ask which degrees of freedom are captured by this model. We show that the model is rich enough to describe the (anisotropic) Bianchi IX Universe, and give the explicit relation between the Bianchi IX variables and the variables of the model. We discuss the possibility of using this path in order to define the quantization of the Bianchi IX Universe. The model contains more degrees of freedom than Bianchi IX, and therefore captures some inhomogeneous degrees of freedom as well. Inhomogeneous degrees of freedom can be expanded in representations of the SU(2) Bianchi IX isometry group, and the dipole model captures the lowest integer representation of these, connected to hyper-spherical harmonic of angular momentum j=1."


    To provide a little background, there are classical lop-sided, irregular versions of the big crunch or big bang, in which (for example) things collapse/expand at different rates in different directions. These irregular classical solutions are described using terms like Bianchi I, Bianchi IX, Kasner, and Taub. The Bianchi series classifies partially restricted symmetries of space. Kasner and Taub are classical solutions of the GR equation with reduced symmetry that can lead to seemingly chaotic highly complicated singularities, one being the socalled "Mixmaster".
    Last edited: Nov 16, 2009
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  3. Nov 16, 2009 #2


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    More papers getting into the anisotropic case. The assumption of isotropy in Loop Cosmology does seem to be gradually breaking down, or easing off.

    Loop quantum cosmology of Bianchi I models
    Abhay Ashtekar, Edward Wilson-Ewing
    33 pages, 2 figures
    (Submitted on 19 Mar 2009)
    "The 'improved dynamics' of loop quantum cosmology is extended to include anisotropies of the Bianchi I model. As in the isotropic case, a massless scalar field serves as a relational time parameter. However, the extension is non-trivial because one has to face several conceptual subtleties as well as technical difficulties. These include: a better understanding of the relation between loop quantum gravity (LQG) and loop quantum cosmology (LQC); handling novel features associated with the non-local field strength operator in presence of anisotropies; and finding dynamical variables that make the action of the Hamiltonian constraint manageable. Our analysis provides a conceptually complete description that overcomes limitations of earlier works. We again find that the big bang singularity is resolved by quantum geometry effects but, because of the presence of Weyl curvature, Planck scale physics is now much richer than in the isotropic case. Since the Bianchi I models play a key role in the Belinskii, Khalatnikov, Lifgarbagez (BKL) conjecture on the nature of generic space-like singularities in general relativity, the quantum dynamics of Bianchi I cosmologies is likely to provide considerable intuition about the fate of generic space-like singularities in quantum gravity. Finally, we show that the quantum dynamics of Bianchi I cosmologies projects down exactly to that of the Friedmann model. This opens a new avenue to relate more complicated models to simpler ones, thereby providing a new tool to relate the quantum dynamics of LQG to that of LQC."

    Loop quantum cosmology of Bianchi type II models
    Abhay Ashtekar, Edward Wilson-Ewing
    26 pages
    (Submitted on 7 Oct 2009)
    "The improved dynamics of loop quantum cosmology is extended to include the Bianchi type II model. Because these space-times admit both anisotropies and non-zero spatial curvature, certain technical difficulties arise over and above those encountered in the analysis of the (anisotropic but spatially flat) Bianchi type I space-times, and of the (spatially curved but isotropic) k=+/-1 models. We address these and show that the big-bang singularity is resolved in the same precise sense as in the recent analysis of the Bianchi I model. Bianchi II space-times are of special interest to quantum cosmology because of the expected behavior of the gravitational field near generic space-like singularities in classical general relativity.

    Another related trend reinforces the connection between LQC and the full theory---covariant LQG (based primarily on spin foam models).

    Loop Quantum Cosmology and Spin Foams
    Abhay Ashtekar, Miguel Campiglia, Adam Henderson
    11 pages
    (Submitted on 23 Sep 2009)
    "Loop quantum cosmology (LQC) is used to provide concrete evidence in support of the general paradigm underlying spin foam models (SFMs). Specifically, it is shown that:

    i) the physical inner product in the timeless framework equals the transition amplitude in the deparameterized theory;

    ii) this quantity admits a convergent vertex expansion à la SFMs in which the M-th term refers just to M volume transitions, without any reference to the time at which the transition takes place;

    iii) the exact physical inner product is obtained by summing over just the discrete geometries; no 'continuum limit' is involved; and,

    iv) the vertex expansion can be interpreted as a perturbative expansion in the spirit of group field theory.

    This sum over histories reformulation of LQC also addresses certain other issues which are briefly summarized."
    Last edited: Nov 17, 2009
  4. Nov 16, 2009 #3


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    LQC into irregular cosmo, and connecting with the full LQG theory

    MTd2 spotted this paper today. It fits in with the topic here.

    On the spinfoam expansion in cosmology
    Carlo Rovelli, Francesca Vidotto
    6 pages
    (Submitted on 16 Nov 2009)
    "We consider the technique introduced in a recent work by Ashtekar, Campiglia and Henderson, which generate a spinfoam-like sum from a Hamiltonian theory. We study the possibility of using it for finding the generalized projector of a constraint on physical states, without first deparametrising the system. We illustrate this technique in the context of a very simple example. We discuss the infinities that appear in the calculation, and argue that they can be appropriately controlled. We apply these ideas to write a spinfoam expansion for the 'dipole cosmology'."

    An earlier paper in loop cosmology by Rovelli and Vidotto was mentioned in post #1 of this thread. Here the authors refer to work by Ashtekar et al that was mentioned in post #2.
    Last edited: Nov 16, 2009
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