Cosmological perturbation theory and quantum gravity

  • Romeo Brunetti
  • Klaus Fredenhagen
  • Thomas-Paul Hack
  • Nicola Pinamonti
  • Katarzyna Rejzner
Open Access
Regular Article - Theoretical Physics


It is shown how cosmological perturbation theory arises from a fully quantized perturbative theory of quantum gravity. Central for the derivation is a non-perturbative concept of gauge-invariant local observables by means of which perturbative invariant expressions of arbitrary order are generated. In particular, in the linearised theory, first order gauge-invariant observables familiar from cosmological perturbation theory are recovered. Explicit expressions of second order quantities are presented as well.


Models of Quantum Gravity Classical Theories of Gravity 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


  1. [1]
    V. Acquaviva, N. Bartolo, S. Matarrese and A. Riotto, Second order cosmological perturbations from inflation, Nucl. Phys. B 667 (2003) 119 [astro-ph/0209156] [INSPIRE].
  2. [2]
    R. Brunetti, K. Fredenhagen and K. Rejzner, Quantum gravity from the point of view of locally covariant quantum field theory, Commun. Math. Phys. 345 (2016) 741 [arXiv:1306.1058] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    R. Brunetti, K. Fredenhagen, T.-P. Hack, N. Pinamonti and K. Rejzner, in preparation.Google Scholar
  4. [4]
    N. Bartolo, S. Matarrese and A. Riotto, CMB Anisotropies at Second-Order. 2. Analytical Approach, JCAP 01 (2007) 019 [astro-ph/0610110] [INSPIRE].
  5. [5]
    J.D. Brown and K.V. Kuchar, Dust as a standard of space and time in canonical quantum gravity, Phys. Rev. D 51 (1995) 5600 [gr-qc/9409001] [INSPIRE].
  6. [6]
    M. Bruni, S. Matarrese, S. Mollerach and S. Sonego, Perturbations of space-time: Gauge transformations and gauge invariance at second order and beyond, Class. Quant. Grav. 14 (1997) 2585 [gr-qc/9609040] [INSPIRE].
  7. [7]
    B. Dittrich, Partial and complete observables for canonical general relativity, Class. Quant. Grav. 23 (2006) 6155 [gr-qc/0507106] [INSPIRE].
  8. [8]
    B. Dittrich and J. Tambornino, Gauge invariant perturbations around symmetry reduced sectors of general relativity: Applications to cosmology, Class. Quant. Grav. 24 (2007) 4543 [gr-qc/0702093] [INSPIRE].
  9. [9]
    M. Duetsch and K. Fredenhagen, Perturbative algebraic field theory and deformation quantization, Submitted to: Fields Inst. Commun. (2000) 151 [hep-th/0101079] [INSPIRE].
  10. [10]
    B. Eltzner, Quantization of Perturbations in Inflation, arXiv:1302.5358 [INSPIRE].
  11. [11]
    K. Fredenhagen and K. Rejzner, Batalin-Vilkovisky formalism in the functional approach to classical field theory, Commun. Math. Phys. 314 (2012) 93 [arXiv:1101.5112] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    K. Fredenhagen and K. Rejzner, Batalin-Vilkovisky formalism in perturbative algebraic quantum field theory, Commun. Math. Phys. 317 (2013) 697 [arXiv:1110.5232] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    K. Fredenhagen, R. Rejzner Perturbative Construction of Models of Algebraic Quantum Field Theory, in Advances in algebraic quantum field theory, R. Brunetti, C. Dappiaggi, K. Fredenhagen and J. Yngvason eds., Springer (2015), arXiv:1503.07814.
  14. [14]
    T.-P. Hack, Quantization of the linearized Einstein-Klein-Gordon system on arbitrary backgrounds and the special case of perturbations in inflation, Class. Quant. Grav. 31 (2014) 215004 [arXiv:1403.3957] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    S. Hollands, Renormalized Quantum Yang-Mills Fields in Curved Spacetime, Rev. Math. Phys. 20 (2008) 1033 [arXiv:0705.3340] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    I. Khavkine, Local and gauge invariant observables in gravity, Class. Quant. Grav. 32 (2015) 185019 [arXiv:1503.03754] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    D. Langlois and F. Vernizzi, A geometrical approach to nonlinear perturbations in relativistic cosmology, Class. Quant. Grav. 27 (2010) 124007 [arXiv:1003.3270] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    J.M. Maldacena, Non-Gaussian features of primordial fluctuations in single field inflationary models, JHEP 05 (2003) 013 [astro-ph/0210603] [INSPIRE].
  19. [19]
    K.A. Malik and D. Wands, Cosmological perturbations, Phys. Rept. 475 (2009) 1 [arXiv:0809.4944] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    V.F. Mukhanov, CMB-slow, or how to estimate cosmological parameters by hand, Int. J. Theor. Phys. 43 (2004) 623 [astro-ph/0303072] [INSPIRE].
  21. [21]
    K. Nakamura, Second-order gauge invariant cosmological perturbation theory: Einstein equations in terms of gauge invariant variables, Prog. Theor. Phys. 117 (2007) 17 [gr-qc/0605108] [INSPIRE].
  22. [22]
    K. Nakamura, Recursive structure in the definitions of gauge-invariant variables for any order perturbations, Class. Quant. Grav. 31 (2014) 135013 [arXiv:1403.1004] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  23. [23]
    H. Noh and J.-c. Hwang, Second-order perturbations of the Friedmann world model, Phys. Rev. D 69 (2004) 104011 [INSPIRE].ADSGoogle Scholar
  24. [24]
    J.-c. Hwang and H. Noh, Fully nonlinear and exact perturbations of the Friedmann world model, Mon. Not. Roy. Astron. Soc. 433 (2013) 3472 [arXiv:1207.0264] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    T. Prokopec and J. Weenink, Uniqueness of the gauge invariant action for cosmological perturbations, JCAP 12 (2012) 031 [arXiv:1209.1701] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  26. [26]
    M. Reuter, Nonperturbative evolution equation for quantum gravity, Phys. Rev. D 57 (1998) 971 [hep-th/9605030] [INSPIRE].ADSMathSciNetGoogle Scholar
  27. [27]
    M. Reuter and F. Saueressig, Renormalization group flow of quantum gravity in the Einstein-Hilbert truncation, Phys. Rev. D 65 (2002) 065016 [hep-th/0110054] [INSPIRE].ADSMathSciNetGoogle Scholar
  28. [28]
    C. Rovelli, Partial observables, Phys. Rev. D 65 (2002) 124013 [gr-qc/0110035] [INSPIRE].
  29. [29]
    S. Sonego and M. Bruni, Gauge dependence in the theory of nonlinear space-time perturbations, Commun. Math. Phys. 193 (1998) 209 [gr-qc/9708068] [INSPIRE].
  30. [30]
    T. Thiemann, Reduced phase space quantization and Dirac observables, Class. Quant. Grav. 23 (2006) 1163 [gr-qc/0411031] [INSPIRE].
  31. [31]
    F. Vernizzi, On the conservation of second-order cosmological perturbations in a scalar field dominated Universe, Phys. Rev. D 71 (2005) 061301 [astro-ph/0411463] [INSPIRE].

Copyright information

© The Author(s) 2016

Authors and Affiliations

  • Romeo Brunetti
    • 1
  • Klaus Fredenhagen
    • 2
  • Thomas-Paul Hack
    • 3
  • Nicola Pinamonti
    • 4
    • 5
  • Katarzyna Rejzner
    • 6
  1. 1.Dipartimento di MatematicaUniversità di TrentoPovoItaly
  2. 2.II Institute für Theoretische PhysikUniversität HamburgHamburgGermany
  3. 3.Institute für Theoretische PhysikUniversität LeipzigLeipzigGermany
  4. 4.Dipartimento di MatematicaUniversità di GenovaGenovaItaly
  5. 5.INFN, Sezione di GenovaGenovaItaly
  6. 6.Department of MathematicsUniversity of YorkYorkUnited Kingdom

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