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Loop Quantum Gravity and the Planck Regime of Cosmology

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General Relativity, Cosmology and Astrophysics

Part of the book series: Fundamental Theories of Physics ((FTPH,volume 177))

Abstract

The very early universe provides the best arena we currently have to test quantum gravity theories. The success of the inflationary paradigm in accounting for the observed inhomogeneities in the cosmic microwave background already illustrates this point to a certain extent because the paradigm is based on quantum field theory on the curved cosmological space-times. However, this analysis excludes the Planck era because the background space-time satisfies Einstein’s equations all the way back to the big bang singularity. Using techniques from loop quantum gravity, the paradigm has now been extended to a self-consistent theory from the Planck regime to the onset of inflation, covering some 11 orders of magnitude in curvature. In addition, for a narrow window of initial conditions, there are departures from the standard paradigm, with novel effects, such as a modification of the consistency relation involving the scalar and tensor power spectra and a new source for non-Gaussianities. The genesis of the large scale structure of the universe can be traced back to quantum gravity fluctuations in the Planck regime. This report provides a bird’s eye view of these developments for the general relativity community.

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Notes

  1. 1.

    Strictly speaking, the BD vacuum refers to deSitter space; it is the unique ‘regular’ state which is invariant under the full deSitter isometry group. During slow roll, the background FLRW geometry is only approximately deSitter whence there is some ambiguity in what one means by the BD vacuum. One typically assumes that all the relevant modes are in the BD state (tailored to) a few e-foldings before the mode \(k_o\) leaves the Hubble horizon. Throughout this report, by BD vacuum I mean this state.

  2. 2.

    The curvature perturbations \(\fancyscript{R}_{\varvec{k}}\) fail to be well-defined at the ‘turning point’ where \(\dot{\phi }=0\), which occurs during pre-inflationary dynamics. However, they are much more convenient for relating the spectrum of perturbations at the end of inflation with the CMB temperature fluctuations. Therefore, we first calculate the power spectrum \(\fancyscript{P}_{\fancyscript{Q}}\) for Mukhanov–Sasaki variable \(\fancyscript{Q}_{\varvec{k}}\) and then convert it to \(\fancyscript{P}_{\fancyscript{R}}\), reported in Fig. 3.

  3. 3.

    During this phase, the scalar field is monotonic in time in the effective trajectory. Therefore we can use the scalar field as an ‘internal’ or ‘relational’ time variable with respect to which the background scale factor (and curvature) as well as perturbations evolve. This interpretation is not essential but very helpful in practice because of the form of the Hamiltonian constraint \(\hat{\mathbb {C}}_o\, \varPsi _o =0\) (for details, see e.g. [14]).

  4. 4.

    Properties of the eigenvalues of length operators [53–55] have not been analyzed in comparable detail. But since their definitions involve volume operators, it is expected that there would be no ‘length gap’.

  5. 5.

    Of course, this would not imply that the inflationary scenario does not admit an extension to the Planck regime. But to obtain it one would then have to await the completion of a full quantum gravity theory.

  6. 6.

    For scalar modes, the classical equation of motion involves also ‘an external potential’ \(\mathfrak {A}\). This has also to be replaced by a dressed effective potential \(\tilde{\mathfrak {A}}\). For details, see [12].

  7. 7.

    While this difference is conceptually important, because the states \(\varPsi _o\) of interest are so sharply peaked, in practice the deviations from effective trajectories are small even in the Planck regime. Of course the deviations from classical solutions are enormous in the Planck regime because \(\tilde{g}_{ab}\) is non-singular.

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Acknowledgments

This report is based on joint work with Ivan Agullo, Alejandro Corichi, Wojciech Kaminski, Jerzy Lewandowski, William Nelson, Tomasz Pawlowski, Parampreet Singh and David Sloan over the past six years. I am most grateful for this collaboration. I am also indebted to a very large number of colleagues especially in the LQG community for discussions, comments, questions and criticisms. This work was supported by the NSF grant PHY-1205388 and the Eberly research funds of Penn state.

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  1. Institute for Gravitation and the Cosmos and Physics Department, Penn State, University Park, PA, 16802, USA

    Abhay Ashtekar

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Correspondence to Abhay Ashtekar .

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  1. Faculty of Mathematics and Physics, Charles University, Praha 8, Czech Republic

    Jiří Bičák

  2. Faculty of Mathematics and Physics, Charles University, Praha 8, Czech Republic

    Tomáš Ledvinka

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Ashtekar, A. (2014). Loop Quantum Gravity and the Planck Regime of Cosmology. In: Bičák, J., Ledvinka, T. (eds) General Relativity, Cosmology and Astrophysics. Fundamental Theories of Physics, vol 177. Springer, Cham. https://doi.org/10.1007/978-3-319-06349-2_16

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