Skip to main content
SpringerLink
Log in

Probability of inflation in loop quantum cosmology

  • Research Article
  • Published:
General Relativity and Gravitation Aims and scope Submit manuscript

Abstract

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 (Hollands and Wald in Gen Relativ Gravit, 34:2043, 2002; Kofman et al. in J High Energy Phys 10:057, 2002); (Gibbons and Turok in Phys Rev D 77:063516, 2008). In particular, for a quadratic potential, Kofman et al. (J High Energy Phys 10:057, 2002) have argued that the probability of inflation with at least 65 e-foldings is close to one, while Gibbons and Turok (Phys Rev D 77:063516, 2008) 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 Kofman et al. (J High Energy Phys 10:057, 2002), Gibbons and Turok (Phys Rev D 77:063516, 2008), Ashtekar and Sloan (Phys Lett B 694:108, 2010): 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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Komatsu, E., et al.: Seven-Year Wilkinson microwave anisotropy probe (WMAP) Observations: Cosmological interpretation, (2010), eprint [arXiv:hep-th/1001.4538]

  2. Gibbons G.W., Hawking S.W., Stewart J.: Nucl. Phys. B 281, 736 (1987)

    Article  MathSciNet  ADS  Google Scholar 

  3. Page D.N.: Phys. Rev. D 36, 1607 (1987)

    Article  MathSciNet  ADS  Google Scholar 

  4. Hawking S.W., Page D.N.: Nucl. Phys. B 298, 789 (1988)

    Article  MathSciNet  ADS  Google Scholar 

  5. Hollands, S., Wald, R.M.: Comment on Inflation and Alternative Cosmology, eprint [arXiv: hep-th/021000]

  6. Corichi A., Karami A.: On the measure problem in slow roll inflation and loop quantum cosmology. Phys. Rev. D 83, 104006 (2011)

    Article  ADS  Google Scholar 

  7. Kofman L.A., Linde A., Mukhanov V.F.: Inflationary theory and alternative cosmology. J. High Energy Phys. 10, 057 (2002)

    Article  MathSciNet  ADS  Google Scholar 

  8. Gibbons G.W., Turok N.: The measure problem in cosmology. Phys. Rev. D 77, 063516 (2008)

    Article  ADS  Google Scholar 

  9. Ashtekar A., Pawlowski T., Singh P.: Quantum nature of the big bang. Phys. Rev. Lett. 96, 141301 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  10. Ashtekar A., Pawlowski T., Singh P.: Quantum nature of the big bang: improved dynamics. Phys. Rev. D 74, 084003 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  11. Ashtekar A., Pawlowski T., Singh P., Vandersloot K.: Loop quantum cosmology of k = 1 FRW models. Phys. Rev. D 75, 024035 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  12. Bentivegna E., Pawlowski T.: Anti-deSitter universe dynamics in LQC. Phys. Rev. D 77, 124025 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  13. Ashtekar A., Corichi A., Singh P.: Robustness of key features of loop quantum cosmology. Phys. Rev. D 77, 024046 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  14. Ashtekar A., Sloan D.: Loop quantum cosmology and slow roll inflation. Phys. Lett. B 694, 108 (2010) eprint [arXiv:gr-qc/0912.4093]

    Article  MathSciNet  ADS  Google Scholar 

  15. Linde A.: Inflation and string cosmology. Prog. Theor. Phys. Suppl. 163, 295 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  16. Bojowald M.: Inflation from quantum geometry. Phys. Rev. Lett. 89, 261301 (2002)

    Article  MathSciNet  ADS  Google Scholar 

  17. Singh P.: Cosmological dynamics and dualities with Randall–Sundrum braneworlds. Phys. Rev. D 73, 063508 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  18. Agullo, I., Ashtekar, A., Nelson, W.: (2011, in preparation)

  19. Ashtekar A., Lewandowski J.: Background independent quantum gravity: a status report. Class. Quantum Gravity 21, R53 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  20. Rovelli C.: Quantum Gravity. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  21. Thiemann T.: Introduction to Modern Canonical Quantum General Relativity. Cambridge University Press, Cambridge (2007)

    Book  Google Scholar 

  22. Ashtekar A., Baez J., Krasnov K.: Quantum geometry of isolated horizons and black hole entropy. Adv. Theor. Math. Phys. 4, 1 (2000)

    MathSciNet  MATH  Google Scholar 

  23. Agullo I., Fernando Barbero J., Borja G.E.F., Diaz-Polo J., Villasen E.J.S.: Detailed black hole state counting in loop quantum gravity. Phys. Rev. D 82, 084029 (2010)

    Article  ADS  Google Scholar 

  24. Bojowald M.: Absence of singularity in loop quantum cosmology. Phys. Rev. Lett. 86, 5227 (2001)

    Article  MathSciNet  ADS  Google Scholar 

  25. Ashtekar A., Wilson-Ewing E.: Loop quantum cosmology of Bianchi I models. Phys. Rev. D 79, 083535 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  26. Ashtekar A., Wilson-Ewing E.: Loop quantum cosmology of Bianchi type II models. Phys. Rev. D 80, 123532 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  27. Wilson Ewing E.: Loop quantum cosmology of Bianchi type IX models. Phys. Rev. D 82, 043508 (2010)

    Article  ADS  Google Scholar 

  28. Mena Marugan G., Martin-Benito M.: Hybrid quantum cosmology: combining loop and fock quantizations. Int. J. Mod. Phys.A 24, 2820 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  29. Taveras V.: Corrections to the Friedmann equations from LQG for a universe with a free scalar field. Phys. Rev. D 78, 064072 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  30. Singh P.: Are loop quantum cosmos never singular?. Class. Quantum Gravity 26, 125005 (2009)

    Article  ADS  Google Scholar 

  31. Ashtekar, A.: Loop quantum cosmology: an overview. Gen. Relativ. Gravit. 41, 707 (2009); The big bang and the quantum. In: Alimi, J.-M., Füzfa, A. (eds.) AIP Conference Proceedings, vol. 1241, pp. 109–121 (2010), eprint [arXiv:hep-th/1005.5491]

  32. de Laplace, P.S.: Théorie analytique des probabilités (Courcier, Paris, 1812). A philosophical essay on probabilities (trans: Dale, A.I.). Springer, New York (1995)

  33. Belinsky V.A., Khalatnikov I.M., Grishchuk L.P., Zeldovich Y.B.: Inflationary stages in cosmological models with a scalar field. Phys. Lett. B 155, 232 (1985)

    Article  MathSciNet  ADS  Google Scholar 

  34. Liddle A.R., Parson P., Barrow J.D.: Formalising the slow roll approximation in inflation. Phys. Rev. D 50, 7222 (1994)

    Article  ADS  Google Scholar 

  35. Sloan, D.: (2011, in preparation)

  36. Sigh P., Vandersloot K., Vereshchagin G.V.: Non-Singular bouncing universes in loop quantum cosmology. Phys. Rev. D 74, 043510 (2006)

    Article  ADS  Google Scholar 

  37. Foster, S.: Scalar field cosmologies and the initial space-time singularity, (1998), eprint [arXiv:gr-qc/9806098]

  38. Taveras V., Yunes N.: The Barbero–Immirzi parameter as a scalar field: K-inflation from loop quantum gravity?. Phys. Rev. D 78, 064070 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  39. Hollands S., Wald R.M.: An alternative to inflation. Gen. Relativ. Gravit. 34, 2043 (2002)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  40. Germani C., Nelson W., Sakellariadou M.: On the onset of inflation in loop quantum cosmology. Phys. Rev. D 76, 043529 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  41. Grain J., Barrau A.: Cosmological footprints of loop quantum gravity. Phys. Rev. Lett. 102, 081301 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  42. Grain J., Cailleteau T., Barrau A., Gorecki A.: Fully loop-quantum-cosmology-corrected propagation of gravitational waves during slow-roll inflation. Phys. Rev. D 81, 024040 (2010)

    Article  ADS  Google Scholar 

  43. Grain J., Barrau A., Cailleteau T., Mielczarek J.: Observing the big bounce with tensor modes in the cosmic microwave background: phenomenology and fundamental LQC parameters. Phys. Rev. D 82, 123520 (2010)

    Article  ADS  Google Scholar 

  44. Mielczarek J., Cailleteau T., Grain J., Barrau A.: Inflation in loop quantum cosmology: dynamics and spectrum of gravitational waves. Phys. Rev. D 81, 104049 (2010)

    Article  ADS  Google Scholar 

  45. Barrau, A.: Inflation and Loop Quantum Cosmology, (2010), eprint [arXiv:gr-qc/1011.5516]

  46. Ashtekar, A., Kaminski, W., Lewandowski, J.: Quantum field theory on a cosmological, quantum space-time. Phys. Rev. D 79, 064030 (2009), eprint [arXiv:gr-qc/0901.0933]

  47. Cortez, J., Mena Marugan, G.A., Olmedo, J., Velhinho, J.M.: A unique Fock quantization for fields in non-stationary spacetimes. JCAP 1010, 030 (2010). Uniqueness of the Fock quantization of fields with unitary dynamics in nonstationary spacetimes. Phys. Rev. D 83, 025002 (2011)

Download references

Author information

Authors and Affiliations

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

    Abhay Ashtekar & David Sloan

  2. Institute for Theoretical Physics, Utrecht University, Utrecht, The Netherlands

    David Sloan

Authors
  1. Abhay Ashtekar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. David Sloan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Abhay Ashtekar.

Additional information

It is a pleasure to contribute this paper to the special volume honoring Josh Goldberg who, together with Peter Bergmann, led the first major research center in gravitational physics in the US for several decades.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ashtekar, A., Sloan, D. Probability of inflation in loop quantum cosmology. Gen Relativ Gravit 43, 3619–3655 (2011). https://doi.org/10.1007/s10714-011-1246-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10714-011-1246-y

Keywords

Search

Navigation