EFTA01141162.pdf

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Please reference the accession code and the first author of your article in your voicemail message. We will respond to you via email. EFTA01141163  PHYSICAL REVIEW D 89, 000000 (3OOOO Using cosmology to establish the quantization of gravity Lawrence M. Krauss School of Earth and Space Exploration and Department of Physics, Arizona State University•, Tempe Arizona 85287-1404 and Mount Stromlo Observatory, Research School of Astronomy and Astrophysics. Australian National Universiry, Weston, ACT 2611, Australia Frank Wilczek Center for Theoretical Physics. Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Received 4 October 2013) While many aspects of general relativity have been tested, and general principles of quantum dynamics demand its quantization, there is no direct evidence for that. It has been argued that development of detectors sensitive to individual gravitons is unlikely, and perhaps impossible. We argue here, however, that measurement of polarization of the cosmic microwave background due to a long wavelength stochastic background of gravitational waves from inflation in the early Universe would firmly establish the quantization of gravity. DOI: PACS numbers: O4.60.-m, 04.80.Cc El Direct detection of gravitational waves is an exciting magnitude, even within the inflationary scenario, depends frontier of experimental physics, with positive results on the rate of expansion during inflation. If the background anticipated soon (i.e., Ref. [I]). The anticipated signals is not observed, it could simply indicate a relatively small are classical disturbances, comprised of coherent super- rate of expansion. But detection is a plausible possibility, as positions of many individual quanta. The possibility of we describe, and major efforts are underway to achieve it. detecting individual gravitons is far more daunting. We should also emphasize that no essentially new pre- Indeed, recently Freeman, Dyson, and colleagues [2] have dictions or calculations are presented here; we are merely cogently estimated that it may in fact be infinitely more bringing to the foreground an implication of existing results daunting, namely, that it is likely to be impossible, to that seems particularly noteworthy. physically realize a detector sensitive to individual grav- The fact that quantization associated with gravity itons without having the detector collapse into a black hole appears to be an essential feature of a gravitational wave in the process. background generated by inflation is suggested by If that is the case, one might wonder whether we can ever existing calculations, including the following. A period directly validate any quantum effects associated with the of inflation in the early Universe results from a period of gravitational field. That would be ironic, not to say pathetic, quasi-de Sitter expansion associated with a scalar field since the apparent tension between quantum mechanics and in an almost flat potential. If one considers a quantized a full quantum treatment of general relativity has been one approximately massless scalar field in de Sitter space, of the driving forces in much of fundamental particle theory expanded into Fourier components with quantized over the past 30 years. mode functions, vs, then it is straightforward to calculate The purpose of this note is to point out that cosmology the zero-point quantum fluctuations of these mode provides a realistic observable that is directly tied to the functions, quantization of gravity. Specifically, observation of a cosmological gravitational wave background associated with an inflationary phase would provide, as a bonus, (vkvk') = Pr(k)6(k + (1) compelling evidence for the quantization of the gravita- tional field. It does so in a way which is at least heuristically where, on large scales the power spectrum P,,(k) equivalent to all laboratory experiments that probe quantum approaches phenomena—it couples quantum mechanical phenomena to a classical detector, effectively amplifying quantum 1 Pr = - (aH)2, (2) mechanical effects so that they are classically measurable. 2k3 The classical detector, in this case, is the expanding Universe. where a is the scale factor during the de Sitter expansion Let us emphasize at the start that such a cosmological and H is the Hubble expansion parameter associated with background has not yet been observed and that its predicted the de Sitter phase. 1 2014 American Physical Society EFTA01141164  BRIEF REPORTS PHYSICAL REVIEW D 89, 000000 (XXXX) Now consider the two helicity states of transverse background that was produced during the inflationary traceless metric perturbations, which we traditionally asso- epoch will require gravitational interactions and thus will ciate with classical gravitational waves. As first pointed out involve the gravitational constant C. We assume that the by Grishchuk in 1975 [3), the Fourier modes of these two background density can be usefully expanded as an analytic states,hr, are each governed by an action in de Sitter function of the coupling, as it would appear in any space that is identical to that of a massless scalar, with the perturbative approach to quantization. We also note that correspondence the dimensionless ratio GhH2/c5 is small for sub- Planckian inflation, i.e., inflation with curvature scale less 2 than the Planck length, while super-Planckian inflation is hk = uk. (3) theoretically dubious. The lowest-order effect, which (if aMPt nonzero) will dominate, therefore involves one power of C. Now if we want to form a dimensionless numerical Thus, if one treats these Fourier modes as quantum modes, measure of the strength of the gravitational background, then there will be zero-point fluctuations in each of the two we should take into account the following circumstance. modes that can be directly derived from Eq. (2), leading to a The energy density ps„ in gravitational radiation after power spectrum inflation ends gives a physical measure of the strength of the background, but it varies afterward with the length 4 H2 scale a of the expanding Universe as 1/a4. If we want to Pr k3 (4) extract a relic of the early Universe, we must compensate that factor. So we will look to combine C to the first Once these modes leave the horizon during the inflationary power, together with powers of H and the fundamental expansion, they freeze in, effectively amplifying the mode constants h, c, and L4, to produce a dimensionless invariant number while outside the horizon, and they return inside measure of the magnitude of the background. Thus, we the horizon as a coherent superposition of many quanta, require i.e., as a classical wave. These waves, originating as quantum fluctuations, then have a dimensionless power [E] ML3 [0[Hr [fir[cy [psmile] = L° = 72 (7) spectrum at the horizon, given by Ic3 28 2 This has a unique solution a = 2, /3 = 2, 7 = —4. Note that 112(k) = = =r ,,r (5) if factors of h and c are made explicit in Eq. (5), then our fri dimensional analysis is vindicated. In this calculation the initial mode number is small, thus Thus, the gravitational radiation background, measured implicating quantum gravity. invariantly, is proportional to h2. Since this is a positive While the fact that this calculation relies on mode power of h, we infer the essentially quantum-mechanical occupation originating in quantum fluctuations suggests nature of that phenomenon. Since no field other than that the calculated effect is essentially quantum mechanical, gravity is involved, we infer that quantization of the that conclusion is not logically forced. After all, many—in gravitational field is an essential ingredient. It is instruc- principle, all—classical effects can also be calculated tive to compare this result for graviton radiation in quantum mechanically, and sometimes that approach is cosmology with results for photon radiation in atomic even more direct or simpler. Our claim that a gravitational physics. h typically appears with a negative power in the wave background from inflation requires quantum effects decay rate of low-lying atomic levels. The point is that in gravity for its generation can, however, be based on more those levels themselves cannot be specified classically. general and perhaps firmer ground, without recourse to the Radiation from classical "Rydberg" orbits is classical and specific calculation outlined above, using simple dimen- contains no powers of h; however, there is no classical sional analysis. gravitational radiation from a classical de Sitter back- ig In the de Sitter limit, the inflationary epoch is charac- ground, and what radiation there is brings in positive powers of h. terized by a single parameter, the Hubble parameter H. Abstracting M, L, and T as dimensions of mass, length, and Inflation also in general predicts an almost flat spectrum time, we therefore have of Gaussian adiabatic primordial density fluctuations at the horizon, due to quantum fluctuations in the scalar field 1 driving inflation, which can generate all observed structure [8 ) =1.- (6) in the Universe and which appears to be in excellent quantitative agreement with observations of primordial (A bracketed quantity represents the dimensional content of temperature perturbations in the cosmic microwave back- that quantity.) A contemporary gravitational wave ground (CMB). If the inflation scale, H, is sufficiently EFTA01141165 BRIEF REPORTS PHYSICAL REVIEW D 89, 000000 (XXXX) large, horizon-sized gravitational waves will also produce These have sometimes been put forward as "quantum measurable CMB effects [4-7]. For inflation with a single gravitational" phenomena, but more properly they are scalar field, the ratio of the polarization power due to these manifestations of the ordinary quantum mechanics of gravitational wave perturbations to the power associated particles (i.e., neutrons) in classical gravitational fields. with temperature (i.e., scalar density) fluctuations, then Indeed, it is more natural to express the effect in terms of (i.e., Ref. [6]) the quantity, g, the gravitational acceleration near Earth's surface, which is the relevant aspect of the experimental H2int environment, and then C, which indicates intrinsically r = 0.01 (8) gravitational dynamics, does not appear at all. Similar (2.5 x 1013 GeV)1. remarks apply to scalar mode perturbations within infla- tionary models. Observations currently give an upper limit on this ratio to It is also possible, of course, that a fully realized theory be r < 0.11 [8], and it is possible that observations of quantum gravity would have other indirect consequences will be able to probe values of r that are far smaller that could be observed, e.g., the existence of unusual (i.e., Ref. [9]). Thus, a gravitational wave background due to inflation acwiated with the scale suggested by coupling interactions, or even that it would dictate the entirety of a "theory of everything:' Perhaps the most concrete ideas constant unification [4,10], which corresponds to II tr. 2.5 x 1013 GeV, could be observed in the near future. along these lines arise in gravity-mediated supersymmetry While the current observations of CMB temperature breaking, wherein quantum gravity effects make dominant fluctuations and the observed flatness of the Universe contributions to the masses of supersymmetric particles are strongly suggestive of an inflationary origin, the mere [15-17]. But those possibilities remain highly speculative. observation of polarization in the CMB compatible with a Through inflation, the Universe can act effectively as a gravitational wave background, as exciting as that may be, graviton detector built on an "impractical scale." It ampli- will not alone prove that it originates in quantum phenom- fies a quantum mechanical effect to where it can be detected ena associated with gravitation (i.e., Refs. [11,12]). as a classical, observable signal and may provide compel- Fortunately, there is a wide variety of consistency tests ling empirical support for the quantization of gravity. Thus, that can be performed to check for an inflationary origin we both illustrate and transcend, rather than contradict, the (i.e., see Ref. [131). These include a simple relationship arguments of Ref. [2]. between this ratio and the slope of the CMB temperature fluctuation power spectrum as a function of frequency. In ACKNOWLEDGMENTS addition, inflation predicts superhorizon size correlations in the gravitational wave spectrum that might be discernible We are grateful to Freeman Dyson for stimulating our (i.e., see Ref. [14]). interest in this question and to the organizers of the 90th If these consistency tests were satisfied quantitatively, birthday celebration for Dyson at the NTU in Singapore, we would thereby have reasonably unambiguous evidence where he lectured on this subject. We also thank Andrew that inflation did indeed occur and that linearized fluctua- Long, Subir Sabharwal, and Tanmay Vachaspati for useful tions in the gravitational field are quantized, with the power discussions and Freeman Dyson, Steve Weinberg, Edward spectrum originating in quantum zero-point fluctuations in Witten, and Xerxes Tata for comments on early drafts of the gravitational field. this manuscript. L. M. K. is supported by the U.S. We should contrast the joint appearance of G and h in Department of Energy at ASU and also by Australian Eqs. (7) and (8), which really does implicate quantization National University. F. W. is supported by the U.S. of the gravitational field, with other cases, including Department of Energy under Contract No. DE-FG02- specifically neutron interferometry, in which both appear. 05ER41360. [I] 1. Aasi a al. (LIGO Collaboration), Nat. Photonics 7, 613 (5] M. Kamionkowski and A. Kosowsky, Phys. Rev. D 57, 685 (2013). (1998). [2] T. Rothman and S. Boughn, Found. Phys. 36, 1801 (2006). (6] D. Baumann a at (CMBPoI Study Team), AIP Conf. Proc. [3] L. Grishchuk, Soy. Phys. JETP 40, 409 (1975). 1141, 10 (2009). 141 L. M. Krauss and M.I. White, Phys. Rev. Len. 69, 869 (7] L. Krauss, S. Dodelson, and S. Meyer, Science 328, 989 (1992). (2010). 3 EFTA01141166 BRIEF REPORTS PHYSICAL REVIEW D 89, 000000 OOOOO [8] E. Komatsu et al. (WMAP Collaboration), Astrophys. J. [13] A.R. Liddle and D. H. Lyth, Phys. Lett. B 291, 391 (1992). Suppl. Ser. 192, 18 (2011). [14] D. Baumann and M. Zaldarriaga, J. Cosmol. Astropart [9] L. Book, M. Kamionkowski, and F. Schmidt, Phys. Rev. Phys. 06 (2009) 013. Left. 108, 211301 (2012). [15] L. J. Hall, J. D. Lykken, and S. Weinberg, Phys. Rev. D 27, [10] S. Dimopoulos, S. Raby, and F. Wikzek, Phys. Rev. D 24, 2359 (1983). 1681 (1981). [16] S. K. Soni and H. A. Weldon, Phys. Lett. 126B, 215 Ill] L. M. Krauss, Phys. Lett B 284, 229 (1992). (1983). [12] K. Jones-Smith, L. M. Krauss, and H. Mathur, Phys. Rev. [17] Y. Kawamura, H. Murayama, and M. Yamaguchi, Phys. Left. 100, 131302 (2008). Rev. D 51, 1337 (1995). 4 EFTA01141167