# Imprints of quantum gravity in the cosmic microwave background

## Abstract

It has been shown that the spectrum of quantum gravity contains at least two new modes in addition to the massless graviton: a massive spin-0 and a massive spin-2. We calculate their power spectrum during inflation and we argue that they could leave an imprint in the cosmic microwave background should their masses be below the inflationary scale.

## 1 Introduction

A fully consistent theory of quantum gravity remains an elusive dream. While there are promising ideas on how to build a consistent quantum theory of gravity such as string theory or loop quantum gravity, there is no consensus yet on the right theoretical approach to the problem. The only strong constraint that one can impose is that classical general relativity must emerge as a low energy effective action of any viable quantum theory of gravity. If one is interested in physics at energies below the Planck scale or some \(10^{19}\ \hbox {GeV}\), it turns out that this constraint is enough to enable us to make predictions in quantum gravity independently of the ultraviolet theory. The framework that enables such predictions is that of effective field theory.

The application of effective field theory methods to quantum gravity [1, 2, 3, 4, 5, 6] is a powerful approach for making low-energy quantum predictions without needing detailed knowledge of the ultraviolet physics. It has the advantage of separating out the ultraviolet from the infrared physics, opening up the possibility for model-independent predictions in the latter while keeping the ultraviolet nature of the theory encoded in the Wilson coefficients. Thus, in principle, it gives us a way to probe quantum gravity experimentally up to the Planck scale. At this energy scale, contributions from an infinite number of curvature invariants would become important, signaling the breakdown of the effective theory approach.

A key prediction of this effective field theory approach is that there are more degrees of freedom than just the massless graviton in quantum gravity [7, 8, 9, 10, 11]. Indeed, there are at least two new massive fields with spin-0 and spin-2. In this paper we propose to search for the imprints of these new bosons in the cosmic microwave background as they could be excited during inflation if the scale of inflation larger than the masses of these fields. Indeed, any light gravitational field can be excited during inflation as these fields couple to the energy-momentum tensor like the metric does. This is true for spin-0 and spin-2 bosons. Our main result is a calculation of the power spectrum associated with these massive modes in quantum gravity. Experiments on earth, and the Eöt-Wash experiment in particular only set very weak bounds on the masses of these new modes. These masses must be larger than \(1 \times 10^{-12}\ \hbox {GeV}\) [12], otherwise a modification of Newton’s 1 / *r* law would have been observed by torsion pendulum experiments. Depending on their masses and the scale of inflation, these new modes could have been excited in full analogy to the gravitational wave power spectrum expected from the excitation of the massless graviton mode.

## 2 Linearisation around de Sitter

## 3 Power spectrum of massive gravitational waves

## 4 Inflationary observables

*r*is twice the standard value because of the contribution of the massive spin-2.

*r*without assuming the consistency condition. At \(k_*=0.01\,\text {Mpc}^{-1}\), which corresponds to the decorrelation scale of

*r*and \(n_t\), they found \(r_{0.01}<0.091\) and \(-0.34<n_t<2.63\) [17]. The upper bound translates into

*H*and only holds if the tensor tilt turns out to be red. The only possible way to solve for \(m_2\) without using any estimation for

*H*is through a direct measurement of the tensor power spectrum. This would naturally lead to separate bounds on

*H*and \(m_2\). Quantum gravity thus predict a blue tilt spectrum if the mass of the massive spin-2 field is bellow the scale of inflation with essentially no correction to the gravitational wave power spectrum.

*N*[15]:

## 5 Conclusion

In this short paper, we have shown that quantum gravity could leave an imprint in the cosmic microwave background in the form of massive gravitational waves. This is a model independent prediction which does not dependent on the ultra-violet completion of quantum gravity. Besides the power spectra of the inflaton and the usual gravitational waves (massless spin-2), quantum gravity predicts power spectra for the massive spin-2 and spin-0 modes present in the effective action. It is interesting that the massive spin-0 could actually be the inflaton itself, as it is the case in Starobinsky’s inflationary model. Interplays between the scalaron and the Higgs field could also be of interest and have been studied in [19].

If the masses of the massive spin-2 and spin-0 waves are below the scale of inflation, CMB observations have the potential to extend the search for quantum gravity massively. Currently, the bounds from the Eöt-Wash torsion pendulum experiment simply imply that their masses are larger than \(1 \times 10^{-12}\ \hbox {GeV}\). The discovery of the power spectrum for gravitational waves would fix the scale of inflation and thereby enable one to set a limit on the masses of these new modes or with a bit of luck prelude to a discovery of the new states predicted by quantum gravity. Future detectors could find the value of *r* to within \(\pm 0.001\) [20, 21], meaning that we would have strong evidence either for or against lighter than the scale of inflation quantum gravitational fields.

We conclude by noting that the massive gravitational modes discussed here could also represent a large component of dark matter as discussed in [12]. As explained in [12], these modes would then have to be very light and would essentially be massless during inflation. If these massive gravitational modes are truly the missing dark matter, then we expect that they will leave an imprint in the CMB as discussed here.

## Notes

### Acknowledgements

This work supported in part by the Science and Technology Facilities Council (grant number ST/P000819/1) and by the National Council for Scientific and Technological Development (CNPq - Brazil).

## References

- 1.S. Weinberg, in
*General Relativity. An Einstein Centenary Survey*, ed. by S.W. Hawking, W. Israel (Cambridge University Press, Cambridge, 1979) p. 790Google Scholar - 2.A.O. Barvinsky, G.A. Vilkovisky, The generalized Schwinger-de Witt technique and the unique effective action in quantum gravity. Phys. Lett.
**131B**, 313 (1983)ADSCrossRefGoogle Scholar - 3.A.O. Barvinsky, G.A. Vilkovisky, Phys. Rep.
**119**, 1 (1985). https://doi.org/10.1016/0370-1573(85)90148-6 ADSMathSciNetCrossRefGoogle Scholar - 4.I.L. Buchbinder, S.D. Odintsov, I.L. Shapiro,
*Effective Action in Quantum Gravity*(IOP, Bristol, 1992), p. 413Google Scholar - 5.J.F. Donoghue, Phys. Rev. D
**50**, 3874 (1994). https://doi.org/10.1103/PhysRevD.50.3874. [arXiv:gr-qc/9405057]ADSCrossRefGoogle Scholar - 6.X. Calmet, Int. J. Mod. Phys. D
**22**, 1342014 (2013). https://doi.org/10.1142/S0218271813420145. arXiv:1308.6155 [gr-qc]ADSMathSciNetCrossRefGoogle Scholar - 7.X. Calmet, Mod. Phys. Lett. A
**29**(38), 1450204 (2014). https://doi.org/10.1142/S0217732314502046. arXiv:1410.2807 [hep-th]ADSCrossRefGoogle Scholar - 8.X. Calmet, I. Kuntz, S. Mohapatra, Eur. Phys. J. C
**76**(8), 425 (2016). https://doi.org/10.1140/epjc/s10052-016-4265-8. arXiv:1607.02773 [hep-th]ADSCrossRefGoogle Scholar - 9.K.S. Stelle, Classical gravity with higher derivatives. Gen. Relativ. Grav.
**9**, 353 (1978). https://doi.org/10.1007/BF00760427 ADSMathSciNetCrossRefGoogle Scholar - 10.X. Calmet, R. Casadio, A.Y. Kamenshchik, O.V. Teryaev, Phys. Lett. B
**774**, 332 (2017). https://doi.org/10.1016/j.physletb.2017.09.080. arXiv:1708.01485 [hep-th]ADSCrossRefGoogle Scholar - 11.X. Calmet, B. Latosh, Eur. Phys. J. C
**78**(3), 205 (2018). https://doi.org/10.1140/epjc/s10052-018-5707-2. arXiv:1801.04698 [hep-th]ADSCrossRefGoogle Scholar - 12.X. Calmet, B. Latosh, Eur. Phys. J. C
**78**(6), 520 (2018). https://doi.org/10.1140/epjc/s10052-018-6005-8. arXiv:1805.08552 [hep-th]ADSCrossRefGoogle Scholar - 13.A. Hindawi, B.A. Ovrut, D. Waldram, Phys. Rev. D
**53**, 5583 (1996). https://doi.org/10.1103/PhysRevD.53.5583. [arXiv:hep-th/9509142]ADSMathSciNetCrossRefGoogle Scholar - 14.M.C. Guzzetti, N. Bartolo, M. Liguori, S. Matarrese, Riv. Nuovo Cim.
**39**(9), 399 (2016). https://doi.org/10.1393/ncr/i2016-10127-1. arXiv:1605.01615 [astro-ph.CO]ADSCrossRefGoogle Scholar - 15.A. De Felice, S. Tsujikawa, Living Rev. Rel.
**13**, 3 (2010). https://doi.org/10.12942/lrr-2010-3. arXiv:1002.4928 [gr-qc]CrossRefGoogle Scholar - 16.M. Fasiello, R.H. Ribeiro, JCAP
**1507**(07), 027 (2015). https://doi.org/10.1088/1475-7516/2015/07/027. arXiv:1505.00404 [astro-ph.CO]ADSCrossRefGoogle Scholar - 17.Y. Akrami et al. [Planck Collaboration], arXiv:1807.06211 [astro-ph.CO]
- 18.D. Baumann, https://doi.org/10.1142/9789814327183_0010 arXiv:0907.5424 [hep-th]
- 19.X. Calmet, I. Kuntz, Eur. Phys. J. C
**76**(5), 289 (2016). https://doi.org/10.1140/epjc/s10052-016-4136-3. arXiv:1605.02236 ADSCrossRefGoogle Scholar - 20.T. Matsumura et al., Proc. SPIE Int. Soc. Opt. Eng.
**9143**, 91431F (2014). https://doi.org/10.1117/12.2055794 CrossRefGoogle Scholar - 21.F. R. Bouchet et al., COrE: Cosmic Origins Explorer - A White Paper. (2015)Google Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Funded by SCOAP^{3}.