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Inflation with the Starobinsky potential in loop quantum cosmology

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Abstract

A self-consistent pre-inflationary extension of the inflationary scenario with the Starobinsky potential, favored by Planck data, is studied using techniques from loop quantum cosmology (LQC). The results are compared with the quadratic potential previously studied. Planck scale completion of the inflationary paradigm and observable signatures of LQC are found to be robust under the change of the inflation potential. The entire evolution, from the quantum bounce all the way to the end of inflation, is compatible with observations. Occurrence of desired slow-roll phase is almost inevitable and natural initial conditions exist for both the background and perturbations for which the resulting power spectrum agrees with recent observations. There exist initial data for which the quantum gravitational corrections to the power spectrum are potentially observable.

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Notes

  1. For other approaches to bouncing models see, e.g. [5–8]. Also see [9] (and references therein) for a recent review.

  2. Here, in order to compute the mass parameter, we have assumed that the LQC corrections to the power spectrum are extremely small at the pivot scale \(k_*\). This assumption does indeed hold in the numerical results discussed here. However, in principle, this is inconsistent and a proper way to address this would require significant numerical work along the lines of [29]. There the authors find that, while this assumption is conceptually important, it leaves the main results practically unchanged for the quadratic potential; we expect the same to be true for the Starobinsky potential.

  3. This issue could be resolved for potentials, which are related to Starobinsky potential via \(\alpha \)-attractors, that have a plateau for a finite range in \(\phi \). For such potentials the space of initial conditions at the bounce will be compact and a regular measure suffices to talk about probabilities. For instance, the Higgs potential satisfies this criterion as it has a plateau region in the center and exponential walls on both sides.

  4. Here, we only show existence of at least one state that leads to power suppression. It should be noted that there also exist states that show power enhancement for \(\ell <30\). As of now, these states are at the same footing as the one chosen here that shows power suppression. The physical criteria to select states resulting in power suppression and the issue of their uniqueness are currently being investigated [39].

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Acknowledgments

We are grateful to Abhay Ashtekar for his constant guidance, extensive discussions and feedback at various stages of the preparation of this manuscript, and to Ivan Agullo for extensive discussions. We would also like to thank Aurelien Barrau and Parampreet Singh for fruitful discussions as well as the anonymous referees for their insightful comments which led to improvement in the manuscript. This work was supported by NSF Grant PHY-1505411, the Eberly research funds of Penn State and a Frymoyer Fellowship to BB.

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

    Béatrice Bonga & Brajesh Gupt

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  1. Béatrice Bonga
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Bonga, B., Gupt, B. Inflation with the Starobinsky potential in loop quantum cosmology. Gen Relativ Gravit 48, 71 (2016). https://doi.org/10.1007/s10714-016-2071-0

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