Skip to main content
SpringerLink
Log in

Nonminimal Couplings in the Early Universe: Multifield Models of Inflation and the Latest Observations

  • Chapter
  • First Online:
At the Frontier of Spacetime

Part of the book series: Fundamental Theories of Physics ((FTPH,volume 183))

Abstract

Models of cosmic inflation suggest that our universe underwent an early phase of accelerated expansion, driven by the dynamics of one or more scalar fields. Inflationary models make specific, quantitative predictions for several observable quantities, including particular patterns of temperature anistropies in the cosmic microwave background radiation. Realistic models of high-energy physics include many scalar fields at high energies. Moreover, we may expect these fields to have nonminimal couplings to the spacetime curvature. Such couplings are quite generic, arising as renormalization counterterms when quantizing scalar fields in curved spacetime. In this chapter I review recent research on a general class of multifield inflationary models with nonminimal couplings. Models in this class exhibit a strong attractor behavior: across a wide range of couplings and initial conditions, the fields evolve along a single-field trajectory for most of inflation. Across large regions of phase space and parameter space, therefore, models in this general class yield robust predictions for observable quantities that fall squarely within the “sweet spot” of recent observations.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Preprint MIT-CTP-4740.

  2. 2.

    On Einstein’s changing considerations of Mach’s principle, see [9] and references therein.

  3. 3.

    We have bracketed, for now, the important and rather subtle question of whether there remains any significant “frame dependence” for predictions from such multifield models. It seems clear that one may map predictions for observables from one frame to another in the case of single-field models [11, 12]. But making that mapping between frames in the presence of entropy (or isocurvature) perturbations—which can only arise in multifield models—seems to raise new subtleties [37].

  4. 4.

    In the case of Brans-Dicke-like couplings, with \(M_0 = 0\), one may rescale the fields \(\phi ^I\) to bring \(\mathcal{G}_{IJ} \rightarrow \delta _{IJ}\), and hence restore canonical kinetic terms, only for \(N \le 2\). For \(N > 2\), even with \(M_0 = 0\), one again finds that \(\mathcal{G}_{IJ}\) is not conformal to flat [40].

  5. 5.

    Because the field-space manifold is curved, one must work with a representation of the field fluctuations that is covariant with respect to reparameterizations of the field-space coordinates, as discussed in [32] and references therein. That form reduces to Eq. (2.17) to linear order in the field fluctuations, which will suffice for our purposes here.

  6. 6.

    To calculate \(f_{NL}\) properly, one must go beyond linear order in the fluctuations and calculate the genuine bispectrum, \(\langle \mathcal{R}_c (\mathbf{k}_1) \mathcal{R}_c (\mathbf{k}_2) \mathcal{R}_c (\mathbf{k}_3) \rangle \) [32, 42, 43]; upon performing the full calculation, we find a strong correlation between nonzero \(T_{RS}\) and sizable \(f_{NL}\) [32].

  7. 7.

    We have set \(M_0 = M_\mathrm{pl}\) in \(f (\phi ^I)\), since for \(\tilde{V} (\phi ^I)\) in Eq. (2.34), the global minimum of the potential occurs at \(\phi = \chi = 0\) rather than at any nonzero vacuum expectation value. Hence at the end of inflation, once \(\phi \) and \(\chi \) settle into the global minimum of the potential, \(f (\phi ^I) \rightarrow M_\mathrm{pl}^2 / 2\), recovering the usual gravitational coupling for general relativity.

  8. 8.

    Beyond his work on scalar-tensor gravitation, I also learned of Carl’s interest in quantum entanglement and Bell’s theorem [50] during my early visits with him—work that has also inspired some of my own recent research [51].

References

  1. C. H. Brans, Mach’s principle and a varying gravitational constant. Ph.D. dissertation, Princeton University, 1961

    Google Scholar 

  2. C.H. Brans, R.H. Dicke, Mach’s principle and a relativistic theory of gravitation. Phys. Rev. 124, 925 (1961)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. C.H. Brans, Mach’s principle and a relativistic theory of gravitation, II. Phys. Rev. 125, 2194 (1962); C.H. Brans, Mach’s principle and the locally measured gravitational constant in general relativity. Phys. Rev. 125, 388 (1962)

    Google Scholar 

  4. C. Will, Was Einstein Right? Putting General Relativity to the Test, 2nd ed. (Basic Books, New York, 1993 [1986])

    Google Scholar 

  5. J.D. Norton, Einstein, Nordström, and the early demise of Lorentz-covariant, scalar theories of gravitation. Arch. Hist. Exact Sci. 45, 17 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. D.I. Kaiser, When fields collide. Sci. Am. 296, 62 (2007)

    Article  Google Scholar 

  7. C.H. Brans, Varying Newton’s constant: a personal history of scalar-tensor theories. Einstein Online 04, 1002 (2010)

    Google Scholar 

  8. H. Goenner, Some remarks on the genesis of scalar-tensor theories. Gen. Rel. Grav. 44, 2077 (2012). arXiv:1204.3455 [gr-qc]

    Google Scholar 

  9. M. Janssen, Of pots and holes: Einstein’s bumpy road to general relativity. Ann. Phys. (Leipzig) 14(Supplement), 58 (2005)

    Google Scholar 

  10. Y. Fujii, K.-I. Maeda, The Scalar-Tensor Theory of Gravitation (Cambridge University Press, New York, 2003)

    MATH  Google Scholar 

  11. V. Faraoni, Cosmology in Scalar-Tensor Gravity (Springer, New York, 2004)

    Book  MATH  Google Scholar 

  12. S. Capozziello, V. Faraoni, Beyond Einstein Gravity: A Survey of Gravitational Theories for Cosmology and Astrophysics (Springer, New York, 2011)

    MATH  Google Scholar 

  13. S. Nojiri, S.D. Odintsov, Unified cosmic history in modified gravity: from \(F(R)\) theory to Lorentz non-invariant models. Phys. Rep. 505, 59 (2011). arXiv:1011.0544 [gr-qc]

    Article  ADS  MathSciNet  Google Scholar 

  14. C.G. Callan Jr., S.R. Coleman, R. Jackiw, A new improved energy-momentum tensor. Annals Phys. 59, 42 (1970)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. T.S. Bunch, P. Panangaden, L. Parker, On renormalization of \(\lambda \phi ^4\) field theory in curved spacetime, I. J. Phys. A 13, 901 (1980); T.S. Bunch, P. Panangaden, On renormalization of \(\lambda \phi ^4\) field theory in curved spacetime, II. J. Phys. A 13, 919 (1980)

    Google Scholar 

  16. N.D. Birrell, P.C.W. Davies, Quantum Fields in Curved Space (Cambridge University Press, New York, 1982)

    Book  MATH  Google Scholar 

  17. S.D. Odintsov, Renormalization group, effective action and Grand Unification Theories in curved spacetime. Fortsh. Phys. 39, 621 (1991)

    Article  MathSciNet  Google Scholar 

  18. I.L. Buchbinder, S.D. Odintsov, I.L. Shapiro, Effective Action in Quantum Gravity (Taylor and Francis, New York, 1992)

    Google Scholar 

  19. L.E. Parker, D.J. Toms, Quantum Field Theory in Curved Spacetime (Cambridge University Press, New York, 2009)

    Book  MATH  Google Scholar 

  20. T. Markkanen, A. Tranberg, A simple method for one-loop renormalization in curved spacetime. J. Cosmol. Astropart. Phys. 08, 045 (2013). arXiv:1303.0180 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  21. A.H. Guth, The inflationary universe: A possible solution to the horizon and flatness problems. Phys. Rev. D 23, 347 (1981); A.D. Linde, A new inflationary universe scenario: a possible solution of the horizon, flatness, homogeneity, isotropy, and primordial monopole problems. Phys. Lett. B 108, 389 (1982); A. Albrecht, P.J. Steinhardt, Cosmology for Grand Unified Theories with radiatively induced symmetry breaking. Phys. Rev. Lett. 48, 1220 (1982)

    Google Scholar 

  22. B.A. Bassett, S. Tsujikawa, D. Wands, Inflation dynamics and reheating. Rev. Mod. Phys. 78, 537 (2006). arXiv:astro-ph/0507632

    Article  ADS  Google Scholar 

  23. A.H. Guth, D.I. Kaiser, Inflationary cosmology: exploring the universe from the smallest to the largest scales. Science 307, 884 (2005). arXiv:astro-ph/0502328; D.H. Lyth, A.R. Liddle, The Primordial Density Perturbation: Cosmology, Inflation, and the Origin of Structure (Cambridge University Press, New York, 2009); D. Baumann, TASI Lectures on Inflation. arXiv:0907.5424 [hep-th]; J. Martin, C. Ringeval, V. Vennin, Encyclopedia inflationaris. arXiv:1303.3787 [astro-ph.CO]; A.H. Guth, D.I. Kaiser, Y. Nomura, Inflationary paradigm after Planck 2013. Phys. Lett. B 733, 112 (2014). arXiv:1312.7619 [astro-ph.CO]; A.D. Linde, Inflationary cosmology after Planck 2013. arXiv:1402.0526 [hep-th]

  24. B.L. Spokoiny, Inflation and generation of perturbations in broken-symmetric theory of gravity. Phys. Lett. B 147, 39 (1984); F.S. Accetta, D.J. Zoller, M.S. Turner, Induced-gravity inflation. Phys. Rev. D 31, 3046 (1985); F. Lucchin, S. Matarrese, M.D. Pollock, Inflation with a nonminimally coupled scalar field. Phys. Lett. B 167, 163 (1986); R. Fakir, W.G. Unruh, Induced-gravity inflation. Phys. Rev. D 41, 1792 (1990); D.I. Kaiser, Constraints in the context of induced-gravity inflation. Phys. Rev. D 49, 6347 (1994). arXiv:astro-ph/9308043; D.I. Kaiser, Induced-gravity inflation and the density perturbation spectrum. Phys. Lett. B 340, 23 (1994). arXiv:astro-ph/9405029; J.L. Cervantes-Code, H. Dehnen, Induced gravity inflation in the Standard Model of particle physics. Nucl. Phys. B 442, 391 (1995). arXiv:astro-ph/9505069

  25. L. Smolin, Towards a theory of spacetime structure at very short distances. Nucl. Phys. B 160, 253 (1979)

    Article  ADS  MathSciNet  Google Scholar 

  26. A. Zee, Broken-symmetric theory of gravity. Phys. Rev. Lett. 42, 417 (1979)

    Article  ADS  Google Scholar 

  27. D. La, P.J. Steinhardt, Extended inflationary cosmology. Phys. Rev. Lett. 62, 376 (1989); P.J. Steinhardt, F.S. Accetta, Hyperextended inflation. Phys. Rev. Lett. 64, 2740 (1990); R. Holman, E. W. Kolb, Y. Wang, Gravitational couplings of the inflaton in extended inflation. Phys. Rev. Lett. 65, 17 (1990); R. Holman, E.W. Kolb, S.L. Vadas, Y. Wang, Extended inflation from higher-dimensional theories. Phys. Rev. D 42, 995 (1991)

    Google Scholar 

  28. T. Futamase, K. Maeda, Chaotic inflationary scenario of the universe with a nonminimally coupled ‘inflaton’ field. Phys. Rev. D 39, 399 (1989); D.S. Salopek, J.R. Bond, J.M. Bardeen, Designing density fluctuation spectra in inflation. Phys. Rev. D 40, 1753 (1989); R. Fakir, S. Habib, W.G. Unruh, Cosmological density perturbations with modified gravity. Astrophys. J. 394, 396 (1992); R. Fakir, W.G. Unruh, Improvement on cosmological chaotic inflation through nonminimal coupling. Phys. Rev. D 41, 1783 (1990); N. Makino, M. Sasaki, The density perturbation in the chaotic inflation with nonminimal coupling. Prog. Theor. Phys. 86, 103 (1991); D.I. Kaiser, Primordial spectral indices from generalized Einstein theories. Phys. Rev. D 52, 4295 (1995). arXiv:astro-ph/9408044; S. Mukaigawa, T. Muta, S.D. Odintsov, Finite Grand Unified Theories and inflation. Int. J. Mod. Phys. A 13, 2839 (1998). arXiv:hep-ph/9709299; E. Komatsu, T. Futamase, Complete constraints on a nonminimally coupled chaotic inflationary scenario from the cosmic microwave background. Phys. Rev. D 59, 064029 (1999). arXiv:astro-ph/9901127; A. Linde, M. Noorbala, A. Westphal, Observational consequences of chaotic inflation with nonminimal coupling to gravity. J. Cosmol. Astropart. Phys. 1103, 013 (2011). arXiv:1101.2652 [hep-th]

  29. F.L. Bezrukov, M.E. Shaposhnikov, The Standard Model Higgs boson as the inflaton. Phys. Lett. B 659, 703 (2008). arXiv:0710.3755 [hep-th]

    Article  ADS  Google Scholar 

  30. D.H. Lyth, A. Riotto, Particle physics models of inflation and the cosmological density perturbation. Phys. Rept. 314, 1 (1999). arXiv:hep-ph/9807278; A. Mazumdar, J. Rocher, Particle physics models of inflation and the curvaton scenarios. Phys. Rep. 497, 85 (2011). arXiv:1001.0993 [hep-ph]; V. Vennin, K. Koyama, D. Wands, Encyclopedia curvatonis. arXiv:1507.07575 [astro-ph.CO]

    Google Scholar 

  31. D.I. Kaiser, A.T. Todhunter, Primordial perturbations from multifield inflation with nonminimal couplings. Phys. Rev. D 81, 124037 (2010). arXiv:1004.3805 [astro-ph.CO]

    Article  ADS  Google Scholar 

  32. D.I. Kaiser, E.A. Mazenc, E.I. Sfakianakis, Primordial bispectrum from multifield inflation with nonminimal couplings. Phys. Rev. D 87, 064004 (2013). arXiv:1210.7487 [astro-ph.CO]

    Article  ADS  Google Scholar 

  33. R.N. Greenwood, D.I. Kaiser, E.I. Sfakianakis, Multifield dynamics of Higgs inflation. Phys. Rev. D 87, 044038 (2013). arXiv:1210.8190 [hep-ph]

    Article  Google Scholar 

  34. D.I. Kaiser, E.I. Sfakianakis, Multifield inflation after Planck: the case for nonminimal couplings. Phys. Rev. Lett. 112, 011302 (2014). arXiv:1304.0363 [astro-ph.CO]

    Article  ADS  Google Scholar 

  35. K. Schutz, E.I. Sfakianakis, D.I. Kaiser, Multifield inflation after Planck: Isocurvature modes from nonminimal couplings. Phys. Rev. D 89, 064044 (2014). arXiv:1310.8285 [astro-ph.CO]

    Article  ADS  Google Scholar 

  36. M.P. DeCross, D.I. Kaiser, A. Prabhu, C. Prescod-Weinstein, E.I. Sfakianakis, Preheating after multifield inflation with nonminimal couplings, I: covariant formalism and attractor behavior. arXiv:1510.08553 [hep-ph]

  37. J. White, M. Minamitsuji, M. Sasaki, Curvature perturbation in multifield inflation with nonminimal coupling. J. Cosmol. Astropart. Phys. 07, 039 (2012). arXiv:1205.0656 [astro-ph.CO]; J. White, M. Minamitsuji, M. Sasaki, Nonlinear curvature perturbation in multifield inflation models with nonminimal coupling. J. Cosmol. Astropart. Phys. 09, 015 (2013). arXiv:1406.6186 [astro-ph.CO]; A. Yu. Kamenshchik, C.F. Steinwachs, Question of quantum equivalence between Jordan frame and Einstein frame. Phys. Rev. D 91, 084033 (2015). arXiv:1408.5769 [gr-qc]

  38. H. Kodama, M. Sasaki, Cosmological perturbation theory. Prog. Theor. Phys. Suppl. 78, 1 (1984); V.F. Mukhanov, H.A. Feldman, R.H. Brandenberger, Theory of cosmological perturbations. Phys. Rep. 215, 203 (1992); K.A. Malik, D. Wands, Cosmological perturbations. Phys. Rep. 475, 1 (2009). arXiv:0809.4944 [astro-ph]

    Google Scholar 

  39. R.H. Dicke, Mach’s principle and invariance under transformation of units. Phys. Rev. 125, 2163 (1962)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  40. D.I. Kaiser, Conformal transformations with multiple scalar fields. Phys. Rev. D 81, 084044 (2010). arXiv:1003.1159 [gr-qc]

    Article  ADS  Google Scholar 

  41. S.V. Ketov, Quantum Nonlinear Sigma Models (Springer, New York, 2000)

    Book  MATH  Google Scholar 

  42. J.M. Maldacena, Non-Gaussian features of primordial fluctuations in single field inflationary models. J. High Energy Phys. 0305, 013 (2003). arXiv:astro-ph/0210603

    Article  ADS  MathSciNet  Google Scholar 

  43. N. Bartolo, E. Komatsu, S. Matarrese, A. Riotto, Non-Gaussianity from inflation: theory and observations. Phys. Rep. 402, 103 (2004). arXiv:astro-ph/0406398; X. Chen, Primordial non-Gaussianities from inflation models. Adv. Astron., 638979 (2010). arXiv:1002.1416 [astro-ph]

    Google Scholar 

  44. C. Gordon, D. Wands, B.A. Bassett, R. Maartens, Adiabatic and entropy perturbations from inflation. Phys. Rev. D 63, 023506 (2001). arXiv:astro-ph/0009131; D. Wands, N. Bartolo, S. Matarrese, A. Riotto, An observational test of two-field inflation. Class. Quant. Grav. 19, 613 (2002). arXiv:hep-ph/0205253

  45. P.A.R. Ade et al. (Planck collaboration), Planck 2015 results, XIII: cosmological parameters. arXiv:1502.01589 [astro-ph.CO]

  46. M. Sasaki, E.D. Stewart, A general analytic formula for the spectral index of the density perturbations produced during inflation. Prog. Theor. Phys. 95, 71 (1996). arXiv:astro-ph/9507001; D. Wands, Multiple field inflation. Lect. Notes Phys. 738, 275 (2008). arXiv:astro-ph/0702187; D. Langlois, S. Renaux-Petel, Perturbations in generalized multifield inflation. J. Cosmol. Astropart. Phys. 0804 (2008), 017. arXiv:0801.1085 [hep-th]; C.M. Peterson, M. Tegmark, Testing multifield inflation: a geometric approach. arXiv:1111.0927 [astro-ph.CO]; J.-O. Gong, T. Tanaka, A covariant approach to general field space metric in multifield inflation. J. Cosmol. Astropart. Phys. 1103, 015 (2011). arXiv:1101.4809 [astrod-ph.CO]

  47. R. Kallosh, A. Linde, Nonminimal inflationary attractors. J. Cosmol. Astropart. Phys. 1310, 033 (2013). arXiv:1307.7938 [hep-th]; J.J.M. Carrasco, R. Kallosh, A. Linde, Cosmological attractors and initial conditions for inflation. arXiv:1506.00936 [hep-th], and references therein

    Google Scholar 

  48. M.A. Amin, M.P. Hertzberg, D.I. Kaiser, J. Karouby, Nonperturbative dynamics of reheating after inflation: a review. Int. J. Mod. Phys. D 24, 1530003 (2015). arXiv:1410.3808 [hep-ph]

    Article  ADS  MATH  Google Scholar 

  49. N. Barnaby, J. Braden, L. Kofman, Reheating the universe after multifield inflation. J. Cosmol. Astropart. Phys. 1007, 016 (2010). arXiv:1005.2196 [hep-th]; T. Battefeld, A. Eggemeier, J.T. Giblin, Jr., Enhanced preheating after multifield inflation: on the importance of being special. J. Cosmol. Astropart. Phys. 11, 062 (2012). arXiv:1209.3301 [astro-ph.CO], and references therein

  50. C.H. Brans, Bell’s theorem does not eliminate fully causal hidden variables. Int. J. Theor. Phys. 27, 219 (1988)

    Article  Google Scholar 

  51. J. Gallicchio, A.S. Friedman, D.I. Kaiser, Testing Bell’s inequality with cosmic photons: closing the setting-independence loophole. Phys. Rev. Lett. 112, 110405 (2014). arXiv:1310.3288 [quant-ph]

    Article  ADS  Google Scholar 

Download references

Acknowledgments

It is a great pleasure to thank Carl Brans for his kind encouragement over the years. I would also like to thank Torsten Asselmeyer-Maluga for editing this Festschrift in honor of Carl’s 80th birthday. I am indebted to Evangelos Sfakianakis for pursuing the research reviewed here with me, along with our collaborators Matthew DeCross, Ross Greenwood, Edward Mazenc, Audrey (Todhunter) Mithani, Anirudh Prabhu, Chanda Prescod-Weinstein, and Katelin Schutz. This research has also benefited from discussions with Bruce Bassett and Alan Guth over the years. This work was conducted in MIT’s Center for Theoretical Physics and supported in part by the U.S. Department of Energy under grant Contract Number DE-SC0012567.

Author information

Authors and Affiliations

  1. Department of Physics, Center for Theoretical Physics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA, 02139, USA

    David I. Kaiser

Authors
  1. David I. Kaiser
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to David I. Kaiser .

Editor information

Editors and Affiliations

  1. PT-SW, DLR Projektträger, Berlin, Germany

    Torsten Asselmeyer-Maluga

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Kaiser, D.I. (2016). Nonminimal Couplings in the Early Universe: Multifield Models of Inflation and the Latest Observations. In: Asselmeyer-Maluga, T. (eds) At the Frontier of Spacetime. Fundamental Theories of Physics, vol 183. Springer, Cham. https://doi.org/10.1007/978-3-319-31299-6_2

Download citation

Publish with us

Policies and ethics

Search

Navigation