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Gravity of quantum vacuum and the cosmological constant problem Wang, Qingdi 2018

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Gravity of quantum vacuum and thecosmological constant problembyQingdi WangB.Sc., Wuhan University, 2008M.Sc., The University of British Columbia, 2011A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)July 2018c© Qingdi Wang 2018	The	following	individuals	certify	that	they	have	read,	and	recommend	to	the	Faculty	of	Graduate	and	Postdoctoral	Studies	for	acceptance,	the	dissertation	entitled:		Gravity	of	Quantum	Vacuum	and	the	Cosmological	Constant	Problem		submitted	by	 Qingdi	Wang	 	 in	partial	fulfillment	of	the	requirements	for	the	degree	of	 Doctor	of	Philosophy	in	 Physics		Examining	Committee:	William	G.	Unruh	Supervisor		Philip	C.E.	Stamp	Supervisory	Committee	Member		Mark	Van	Raamsdonk	Supervisory	Committee	Member	Joanna	Karczmarek	University	Examiner	Jingyi	Chen	University	Examiner			Additional	Supervisory	Committee	Members:	Gordon	Semenoff	Supervisory	Committee	Member	Ludovic	Van	Waerbeke	Supervisory	Committee	Member																																																																																																																																																																																															ii	AbstractWe investigate the gravitational property of the quantum vacuum by treat-ing its large energy density predicted by quantum field theory seriouslyand assuming that it does gravitate to obey the equivalence principle ofgeneral relativity. We find that the quantum vacuum would gravitate dif-ferently from what people previously thought. The consequence of thisdifference is an accelerating universe with a small Hubble expansion rateH ∝ Λe−β√GΛ → 0 instead of the previous prediction H = √8piGρvac/3 ∝√GΛ2 → ∞ which was unbounded, as the high energy cutoff Λ is takento infinity. In this sense, at least the “old” cosmological constant problemwould be resolved. Moreover, it gives the observed slow rate of the acceler-ating expansion as Λ is taken to be some large value of the order of Planckenergy or higher. This result suggests that there is no necessity to introducethe cosmological constant, which is required to be fine tuned to an accuracyof 10−120, or other forms of dark energy, which are required to have pecu-liar negative pressure, to explain the observed accelerating expansion of theUniverse.iiiLay SummaryBased on two fundamental principles of modern physics — the uncertaintyprinciple of quantum mechanics and the equivalence principle of generalrelativity, this study suggests that the space we live in is not as static asit appears. It is constantly moving. At each point, it oscillates betweenexpansion and contraction. As it swings back and forth, the two almostcancel each other but a very small net effect drives the universe to expandslowly at an accelerating rate. This process happens at very tiny scales,billions and billions times smaller even than an electron. This researchproposes an original idea to resolve one of the most important problems infundamental physics—the cosmological constant problem and provides anexplanation for the origin of “dark energy” which drives the acceleratingexpansion of the universe.ivPrefaceThis work is a further development with major corrections to the numericalresults from the following publication: [1] Qingdi Wang, Zhen Zhu, andWilliam G. Unruh, “How the huge energy of quantum vacuum gravitates todrive the slow accelerating expansion of the Universe”, Phys. Rev. D 95,103504 (2017).• Chapter 2 to 9 are basically from the published paper [1] with a lit-tle bit more details and revisions. In particular, chapter 5.6 and 5.7contains major corrections to the numerical work, chapter 7 containsmore discussions about different metrics, chapter 9 contains more dis-cussions about the singularity issue. We also include a different modelmodel (unsuccessful but still interesting) in chapter 10.• Sam Cree repeated the numerical simulation in our published work[1] and found that his numerical result does not match with ours. Hehelped identifying the problem, improving the numeric technique andcorrecting the old result.• The original work presented in this thesis was carried out by QingdiWang who also developed the conception and method of this researchwith various degrees of conception, methods, consultation and editingsupport from William G. Unruh. The numerical part of this research(Section 5.7 (also part of Section 10.3) and Appendix B) was mainlyconducted by Zhen Zhu who also contributed some method develop-ment of this research and helped editing part of the manuscript of[1].vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 The formulation of the cosmological constant problem . . 43 The fluctuating quantum vacuum energy density . . . . . . 74 Differences made by the inhomogeneous vacuum—a simplemodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.1 Beyond the FLRW metric . . . . . . . . . . . . . . . . . . . . 104.2 The fluctuating spacetime . . . . . . . . . . . . . . . . . . . . 114.3 Methods and assumptions in solving the system . . . . . . . 165 The solution for a(t,x) . . . . . . . . . . . . . . . . . . . . . . . 185.1 Parametric resonance . . . . . . . . . . . . . . . . . . . . . . 185.2 The solution for P (t,x) . . . . . . . . . . . . . . . . . . . . . 215.3 The global Hubble expansion rate H . . . . . . . . . . . . . . 225.4 A more intuitive explanation . . . . . . . . . . . . . . . . . . 255.5 Meaning of our results . . . . . . . . . . . . . . . . . . . . . . 275.6 The slow varying condition . . . . . . . . . . . . . . . . . . . 28viTable of Contents5.7 Numerical verification . . . . . . . . . . . . . . . . . . . . . . 306 The back reaction . . . . . . . . . . . . . . . . . . . . . . . . . 376.1 A simplified toy model . . . . . . . . . . . . . . . . . . . . . 396.2 General case . . . . . . . . . . . . . . . . . . . . . . . . . . . 477 The more general metrics . . . . . . . . . . . . . . . . . . . . 547.1 The full metric and Einstein equations . . . . . . . . . . . . 547.2 The Mixmaster-type metric . . . . . . . . . . . . . . . . . . . 557.3 An alternative derivation from geodesic deviation equation . 577.4 The physical picture . . . . . . . . . . . . . . . . . . . . . . . 607.5 The Raychaudhuri equation . . . . . . . . . . . . . . . . . . 608 Similarity of effects of vacuum energy in non-gravitationalsystem and gravitational system . . . . . . . . . . . . . . . . 638.1 Value of vacuum energy is relevant in Casimir effect . . . . . 648.2 Effect of vacuum energy on the motion of mirrors . . . . . . 668.3 Analogies between the motion of mirror and the motion ofa(t,x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679 The singularities at a(t,x) = 0 . . . . . . . . . . . . . . . . . . 699.1 Is singularity an end or a new beginning? . . . . . . . . . . . 699.2 Resolving singularity by multiplying a . . . . . . . . . . . . . 709.3 Singularities do not cause problems . . . . . . . . . . . . . . 7110 A different model with a large bare cosmological constant(unsuccesful) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7510.1 The formulation of the cosmological constant problem is de-stroyed by density fluctuations of quantum vacuum . . . . . 7510.2 New relation between λeff and λb . . . . . . . . . . . . . . . . 7710.3 Numerical simulation . . . . . . . . . . . . . . . . . . . . . . 8110.4 Meaning of the results . . . . . . . . . . . . . . . . . . . . . . 8610.5 Problem of this model . . . . . . . . . . . . . . . . . . . . . . 8711 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92viiTable of ContentsAppendicesA Real Massless Scalar Field . . . . . . . . . . . . . . . . . . . . 98B Wigner-Weyl Description of Quantum Mechanics and Nu-meric simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 104C Fourier transforms of the coefficients in (6.42) . . . . . . . . 109viiiList of Figures3.1 Plot of the expectation value of the square of the energy den-sity difference as a function of spacial separation Λ∆x. . . . . 94.1 Constant energy density requires negative pressure . . . . . . 155.1 Plot of the normalized covariance χ as a function of temporalseparation Λ∆t. . . . . . . . . . . . . . . . . . . . . . . . . . . 195.2 Plot of the power spectrum density of the varying part ofΩ2(t,0) (except for the constant Ω20 part). . . . . . . . . . . . 265.3 Numeric result for log |ao(t)| when one scalar field is present.The slope represents the Hubble expansion rate H. It showsthat as Λ increases, H also increases. The prediction (5.25)is not applicable in this case since the slow varying conditionis violated when Ω2 fluctuates to values smaller than ∼ Λ3. . 315.4 Numeric result for log |ao(t)| when one scalar field with a neg-ative bare cosmological constant −λb ∼ Λ3.5 are present. Theslope represents the Hubble expansion rate H. It shows thatas Λ increases, H decreases. The linear fit log(H/Λ) vs Λgives the parameter α ∼ e−4.1 ≈ 0.017, β ∼ 0.072. . . . . . . . 325.5 Top: numeric result for five scalar fields without a bare cos-mological constant and its linear fit log(H/Λ) vs Λ; Bottom:five scalar fields with a negative bare cosmological constant−λb ∼ Λ3.5 and its linear fit log(H/Λ) vs Λ. The sloperepresents the Hubble expansion rate H. In both cases asΛ increases, H decreases. The linear fit gives the param-eters α ∼ e−3.8 ≈ 0.022, β ∼ 0.14 for five scalar fieldswithout the bare cosmological constant and the parametersα ∼ e−4.2 ≈ 0.015, β ∼ 0.28 for five scalar fields with the barecosmological constant. . . . . . . . . . . . . . . . . . . . . . . 33ixList of Figures5.6 Top: numeric result for ten scalar fields without a bare cos-mological constant and its linear fit log(H/Λ) vs Λ; Bot-tom: ten scalar fields with a negative bare cosmological con-stant −λb ∼ Λ3.5 and its linear fit log(H/Λ) vs Λ. Theslope represents the Hubble expansion rate H. In both casesas Λ increases, H decreases. The linear fit gives the pa-rameters α ∼ e−3.7 ≈ 0.025, β ∼ 0.6 for ten scalar fieldswithout the bare cosmological constant and the parametersα ∼ e−3.5 ≈ 0.03, β ∼ 0.83 for ten scalar fields with the barecosmological constant. . . . . . . . . . . . . . . . . . . . . . 345.7 Top: numeric result for twenty scalar fields without a barecosmological constant and its linear fit log(H/Λ) vs Λ; Bot-tom: twenty scalar fields with a negative bare cosmologicalconstant −λb ∼ Λ3.5 and its linear fit log(H/Λ) vs Λ. Theslope represents the Hubble expansion rate H. In both casesas Λ increases, H decreases. The linear fit gives the param-eters α ∼ e−2.6 ≈ 0.074, β ∼ 1.9 for twenty scalar fieldswithout the bare cosmological constant and the parametersα ∼ e−2.1 ≈ 0.12, β ∼ 2.4 for twenty scalar fields with thebare cosmological constant. . . . . . . . . . . . . . . . . . . . 359.1 Plot of the correction function f(Θ) =+∞∑m=1xm sin 2mΘ aroundthe spacetime singularity at Θ = Ωt+ K ·k = pi/2, where thesum in the plot is from m = 1 to m = 20000. . . . . . . . . . 739.2 Plot of the derivative of the correction function df(Θ)dΘ =+∞∑m=12mxm cos 2mΘaround the spacetime singularity at Θ = Ωt + K · k = pi/2,where the sum in the plot is from m = 1 to m = 20000. . . . 7310.1 An illustration of the probability density distribution P (Ω2).The shape of the graph is obtained from the histogram ofa sample of Ω2(t) we used in the numerical simulation 10.3.This is for one boson and one fermion field. The tails fall ase−κ|Ω2+λb/3| (as shown in FIG. 10.2) for large argument whereκ is some constant. . . . . . . . . . . . . . . . . . . . . . . . . 7810.2 Plot of log(Ω2)which shows the tail of P (Ω2) falls as e−κ|Ω2+λb/3|for large argument where κ is some constant. . . . . . . . . . 79xList of Figures10.3 Numerical result for the dependence of log |a| on the barecosmological constant λb as |λb| is small. The cutoff Λ =1. 100 samples are averaged for each line. Planck units areused for convenience. The matter fields are one Boson fieldand one Fermion field. The magnitude of 〈ρ +∑3i=1 Pi〉 forboth fields are set equal but with opposite sign, i.e. we set〈ρ+∑3i=1 Pi〉 = 0 in the simulation. It shows that the Hubbleexpansion rate decreases as −λb increases. . . . . . . . . . . . 8210.4 Plot of logH over |λb| when |λb| is small. The fitting resultshows that β˜ ∼ 6. Planck units are used for convenience. . . . 8310.5 Numerical result for the dependence of log |a| on the barecosmological constant λb as |λb| is large. The cutoff Λ =1. 400 samples are averaged for each line. Planck units areused for convenience. The matter fields are one Boson fieldand one Fermion field. The magnitude of 〈ρ +∑3i=1 Pi〉 forboth fields are set equal but with opposite sign, i.e. we set〈ρ+∑3i=1 Pi〉 = 0 in the simulation. It shows that the Hubbleexpansion rate decreases as −λb increases. For larger −λb theslope of log |a| grows too slow that the numerical roundingerrors seem to dominant. . . . . . . . . . . . . . . . . . . . . . 8410.6 Plot of logH over√|λb| when |λb| is large. The fitting re-sult shows that α ∼ e18, β ∼ 14. Planck units are used forconvenience. The cutoff Λ = 1. . . . . . . . . . . . . . . . . . 85A.1 Plot of correlation coefficient ρx,x′ as a function of time sep-aration Λ∆t in the case ∆x = 0. . . . . . . . . . . . . . . . . 100A.2 Plot of correlation coefficient ρx,x′ as a function of spatialseparation Λ∆x in the case ∆t = 0. . . . . . . . . . . . . . . . 100xiAcknowledgementsI would like to express the deepest appreciation to my supervisor, ProfessorWilliam George Unruh, whose expertise and generous guidance made itpossible for me to work on a topic that was of great interest but also highlyrisky. It was lucky to had the opportunity to work with him. Without hisguidance this dissertation would not have been possible.I would also like to thank the many people with whom I discussedthe related work and have benefited greatly from such discussion: An-drei Barvinsky, Gordon W. Semenoff, Mark Van Raamsdonk, Philip C. E.Stamp, Daniel Carney, Yin-Zhe Ma, Daoyan Wang, Michael Desrochers,Giorgio Torrieri, Niayesh Afshordi, Roland de Putter, Je´roˆme Gleyzes andOlivier Dore´ for helpful discussions and criticisms. I especially thank MichaelDesrochers for helping revising the manuscript.I would also specially thank Sam Cree who helped to identify and correctthe mistakes we made in the numerical calculations in [1].Most importantly, none of this would have been possible without the loveand patience of my family. Especially I want to mention my grandfatherJiehua Wang, my parents Huan Wang and Shuangxiu Wang. My family hasbeen a constant source of love, concern, support and strength all over theseyears. I would like to express my heart-felt gratitude to my family.xiiDedicationTo my grandfather Jiehua WangxiiiChapter 1IntroductionThe two pillars that much of modern physics is based on are QuantumMechanics (QM) and General Relativity (GR). QM is the most successfulscientific theory in history, which has never been found to fail in repeti-tive experiments. GR is also a successful theory which has so far managedto survive every test [2]. In particular, the last major prediction of GR–the gravitational waves, has finally been directly detected on Sept 2015 [3].However, these two theories seem to be incompatible at a fundamental level(see e.g. [4]). The unification of both theories is a big challenge to moderntheoretical physicists.While the test of the combination of QM and GR is still difficult in lab,our Universe already provides one of the biggest confrontations betweenboth theories: the Cosmological Constant Problem [5]. Quantum field the-ory (QFT) predicts a huge vacuum energy density from various sources.Meanwhile, the equivalence principle of GR requires that every form of en-ergy gravitates in the same way. When combining these concepts together,it is widely supposed that the vacuum energy gravitates as a cosmologicalconstant. However, the observed effective cosmological constant λeff is sosmall compared with the QFT’s prediction that an unknown bare cosmolog-ical constant λb (2.7) has to cancel this huge contribution from the vacuumto better than at least 50 to 120 decimal places! It is an extremely difficultfine-tuning problem that gets even worse when the higher loop correctionsare included [6].In 1998, the discovery of the accelerating expansion of the Universe [7,8] has further strengthened the importance of this problem. Before this,one only needs to worry about the “old” cosmological constant problem ofexplaining why the effective cosmological constant is not large. Now, onealso has to face the challenge of the “new” cosmological constant problemof explaining why it has such a specific small value from the observation,which is the same order of magnitude as the present mass density of theUniverse (coincidence problem).This problem is widely regarded as one of the major obstacles to fur-ther progress in fundamental physics (for example, see Witten 2001 [9]). Its1Chapter 1. Introductionimportance has been emphasized by various authors from different aspects.For example, it has been described as a “veritable crisis” (Weinberg 1989, [5]p.1), an “unexplained puzzle” (Kolb and Turner 1993 , [10] p.198), “the moststriking problem in contemporary physics” (Dolgov 1997 [11] p.1) and even“the mother of all physics problems” , “the worst prediction ever”(Susskind2015 [12] chapter two). While it might be possible that people working ona particular problem tend to emphasize or even exaggerate its importance,those authors all agree that this is a problem that needs to be solved, al-though there is little agreement on what is the right direction to find thesolution [13].In this thesis, we make a proposal for addressing the cosmological con-stant problem. We treat the divergent vacuum energy density predicted byQFT seriously and assume that it does gravitate to obey the equivalenceprinciple of GR. We notice that the magnitude of the vacuum fluctuationitself also fluctuates, which leads to a constantly fluctuating and extremelyinhomogeneous vacuum energy density. As a result, the quantum vacuumgravitates differently from a cosmological constant. Instead, at each spatialpoint, the spacetime sourced by the vacuum oscillates alternatively betweenexpansion and contraction, and the phases of the oscillations at neighboringpoints are different. In this manner of vacuum gravitation, although thegravitational effect produced by the vacuum energy is still huge at suffi-ciently small scales (Planck scale), its effect at macroscopic scales is largelycanceled. Moreover, due to the weak parametric resonance of those oscilla-tions, the expansion outweighs contraction a little bit during each oscilla-tion. This effect accumulates at sufficiently large scales (cosmological scale),resulting in an observable effect—the slow accelerating expansion of the Uni-verse. Our proposal harkens back to Wheeler’s spacetime foam [14, 15] andsuggests that it is this foamy structure which leads to the cosmological con-stant we see today.This thesis is organized as follows: in chapter 2, we first review severalkey steps in formulating the cosmological constant problem; in chapter 3, wepoint out that the vacuum energy density is not a constant but is constantlyfluctuating and extremely inhomogeneous; in chapter 4, we investigate thedifferences made by the extreme inhomogeneity of the quantum vacuum byintroducing a simple model; in chapter 5, we give the solutions to this modelby solving the Einstein field equations and show how metric fluctuationsleads to the slow accelerating expansion of the Universe; in chapter 6, weinvestigate the back reaction effect of the resulting spacetime on the matterfields propagating on it; in chapter 7, we generalize our results to moregeneral metrics; in chapter 8 we discuss the role played by vacuum energy in2Chapter 1. Introductionnon-gravitational physics and gravitational physics; in chapter 9 we discussthe singularity issue; in chapter 10 we introduce another unsuccessful butinteresting model.The units and metric signature are set to be c = ~ = 1 and (−,+,+,+)throughout except otherwise specified.3Chapter 2The formulation of thecosmological constantproblemThe cosmological constant problem arises when trying to combine GR andQFT to investigate the gravitational property of the vacuum:Gµν + λbgµν = 8piGTvacµν , (2.1)where Gµν ≡ Rµν − 12Rgµν is the Einstein tensor and the parameter λb isthe bare cosmological constant.One crucial step in formulating the cosmological constant problem isassuming that the vacuum energy density is equivalent to a cosmologicalconstant. First, it is argued that the vacuum is Lorentz invariant and thusevery observer would see the same vacuum. In Minkowski spacetime, ηµν isthe only Lorentz invariant (0, 2) tensor up to a constant. Thus the vacuumstress-energy tensor must be proportional to ηµν (see, e.g. [16], [13])T vacµν (t,x) = −ρvacηµν . (2.2)The above vacuum equation of state (2.2) is then straightforwardly gen-eralized from inertial coordinates to arbitrary coordinates by replacing ηµνwith gµν1,T vacµν (t,x) = −ρvacgµν(t,x). (2.3)Then from the principle of general covariance, it is asserted that T vacµνhas also to be a constant times gµν when gµν describes a real gravitationalfield:T vacµν (t,x) = −ρvacgµν(t,x). (2.4)1Note that the gµν here is still describing flat spacetime. Do not be confused withthe gµν in (2.4), which is describing curved spacetime (with none-zero Riemann curvaturetensor components).4Chapter 2. The formulation of the cosmological constant problemIf T vacµν does take the above form (2.4), the vacuum energy density ρvac hasto be a constant, which is the requirement of the conservation of the stress-energy tensor∇µT vacµν = 0. (2.5)The effect of a stress-energy tensor of the form (2.4) is equivalent to thatof a cosmological constant, as can be seen by moving the term 8piGT vacµν in(2.1) to the left-hand sideGµν + λeffgµν = 0, (2.6)where,λeff = λb + 8piGρvac; (2.7)Or equivalently by moving the term λbgµν in (2.1) to the right-hand sideGµν = −8piGρvaceff gµν , (2.8)where,ρvaceff = ρvac +λb8piG. (2.9)So anything that contributes to the energy density of the vacuum acts likea cosmological constant and thus contributes to the effective cosmologicalconstant. Or equivalently we can say that the bare cosmological constantacts like a source of vacuum energy and thus contributes to the total effectivevacuum energy density. This equivalence is the origin of the identificationof the cosmological constant with the vacuum energy density.Following the above formulations, the effective cosmological constant λeffor the total effective vacuum energy density ρvaceff are the quantities that canbe constrained and measured through experiments. While solar system andgalactic observations have placed a small upper bound on λeff , large scalecosmological observations provide the most accurate measurement. It is in-terpreted as a form of “dark energy”, which drives the observed acceleratingexpansion of the Universe [7, 8].Based on the assumption of homogeneity and isotropy of the Universe,the metric has the cosmology’s standard FLRW form, which is, for thespatially flat case,ds2 = −dt2 + a2(t) (dx2 + dy2 + dz2) . (2.10)Then by applying the equations (2.6) or (2.8) for the above special metric(2.10), one obtains the contributions to the Hubble expansion rate H = a˙/a5Chapter 2. The formulation of the cosmological constant problemand the acceleration of the scale factor a¨ from λeff and/or ρvaceff are3H2 = λeff = 8piGρvaceff , (2.11)a¨ =λeff3a =8piGρvaceff3a. (2.12)The solution to the dynamic equation (2.12) isa(t) = a(0)eHt, (2.13)where H is determined by the initial value constraint equation (2.11).According to the Lambda-CDM model of the big bang cosmology, theeffective cosmological constant is responsible for the accelerating expansionof the Universe as shown in (2.12) and contributes about 69% to the currentHubble expansion rate [17]:λeff = 3ΩλH20 ≈ 4.32× 10−84(GeV)2, (2.14)orρvaceff = Ωλρcrit ≈ 2.57× 10−47(GeV)4, (2.15)where Ωλ = 0.69 is the dark energy density parameter, H0 is the currentobserved Hubble constant and ρcrit =3H208piG is the critical density.Unfortunately the predicted energy density of the vacuum from QFTis much larger than this. It receives contributions from various sources,including the zero point energies (∼ 1072(GeV)4) of all fundamental quan-tum fields due to vacuum fluctuations, the phase transitions due to thespontaneous symmetry breaking of electroweak theory (∼ 109(GeV)4) andany other known and unknown phase transitions in the early Universe (e.g.from chiral symmetry breaking in QCD (∼ 10−2(GeV)4), grand unification(∼ 1064(GeV)4) etc)[13, 18]. Each contribution is larger than the observedvalue (2.15) by 50 to 120 orders of magnitude. There is no mechanism in thestandard model which suggests any relations between the individual contri-butions, so it is customary to assume that the total vacuum energy densityis at least as large as any of the individual contributions [13]. One thus hasto fine tune the unknown bare cosmological constant λb to a precision of atleast 50 decimal places to cancel the excess vacuum energy density.6Chapter 3The fluctuating quantumvacuum energy densityThe vacuum energy density is treated as a constant in the usual formulationof the cosmological constant problem. While this is true for the expectationvalue, it is not true for the actual energy density.That’s because the vacuum is not an eigenstate of the local energy den-sity operator T00, although it is an eigenstate of the global Hamiltonianoperator H =∫d3xT00. This implies that the total vacuum energy all overthe space is constant but its density fluctuates at individual points.To see this more clearly, consider a quantized real massless scalar fieldφ in Minkowski spacetime as an example:φ(t,x) =∫d3k(2pi)3/21√2ω(ake−i(ωt−k·x) + a†ke+i(ωt−k·x)), (3.1)where the temporal frequency ω and the spatial frequency k in (3.1) arerelated to each other by ω = |k|.The vacuum state |0〉, which is defined asak|0〉 = 0, for all k, (3.2)is an eigenstate of the Hamiltonian operatorH =∫d3xT00 =12∫d3k ω(aka†k + a†kak), (3.3)where T00 is defined asT00 =12(φ˙2 + (∇φ)2). (3.4)7Chapter 3. The fluctuating quantum vacuum energy densityBut, |0〉 is not an eigenstate of the energy density operatorT00(t,x) =12∫d3kd3k′(2pi)312(√|k||k′|+ k · k′√|k||k′|)·(aka†k′e−i[(|k|−|k′|)t−(k−k′)·x] + a†kak′e+i[(|k|−|k′|)t−(k−k′)·x]−akak′e−i[(|k|+|k′|)t−(k+k′)·x] − a†ka†k′e+i[(|k|+|k′|)t−(k+k′)·x]), (3.5)because of the terms of the form akak′ and a†ka†k′ .Direct calculation shows the magnitude of the fluctuation of the vacuumenergy density diverges as the same order as the energy density itself,〈(T00 − 〈T00〉)2〉=23〈T00〉2, (3.6)where〈T00〉 = Λ416pi2, (3.7)where Λ is the effective QFT’s high energy cutoff. (For more details on thiscalculation, see equation (A.6) in Appendix A.) Thus the energy densityfluctuates as violently as its own magnitude. With such huge fluctuations,the vacuum energy density ρvac is not a constant in space or time.Furthermore, the energy density of the vacuum is not only not a constantin time at a fixed spatial point, it also varies from place to place. In otherwords, the energy density of vacuum is varying wildly at every spatial pointand the variation is not in phase for different spatial points. This resultsin an extremely inhomogeneous vacuum. The extreme inhomogeneity canbe illustrated by directly calculating the expectation value of the square ofdifference between energy density at different spatial points,∆ρ2 (∆x) =〈{(T00 (t,x)− T00 (t,x′))2}〉43 〈T00(t,x)〉2, (3.8)where ∆x = |x−x′| and we have normalized ∆ρ2 by dividing its asymptoticvalue 43〈T00〉2 (the curly bracket {} is the symmetrization operator which isdefined by (A.2)). The behavior of ∆ρ2 for the scalar field (3.1) in Minkowskivacuum is plotted in FIG. 3.1, which shows that the magnitude of the energydensity difference between two spacial points quickly goes up to the order of8Chapter 3. The fluctuating quantum vacuum energy density2 4 6 8 100.00.20.40.60.81.01.21.4LDxDΡ2HDxLFigure 3.1: Plot of the expectation value of the square of the energy densitydifference as a function of spacial separation Λ∆x.〈T00〉 itself as their distance increases by only the order of 1/Λ. (For moredetails on the calculations and how the energy density fluctuates all overthe spacetime, see Appendix A.)As the vacuum is clearly not homogeneous, equation (2.11) is not valid asit depends on a homogeneous and isotropic matter field and metric. There-fore a new method of relating vacuum energy density to the observed Hubbleexpansion rate is required.9Chapter 4Differences made by theinhomogeneous vacuum—asimple modelThe extreme inhomogeneity of the vacuum means its gravitational effectcannot be treated perturbatively, so another method is required. As solu-tions to the fully general Einstein equations are difficult to obtain, we willfirst look at a highly simplified model.4.1 Beyond the FLRW metricTo describe the gravitational property of the inhomogeneous quantum vac-uum, we must allow inhomogeneity in the metric. This is accomplished byallowing the scale factor a(t) in the FLRW metric (2.10) to have spatialdependence,ds2 = −dt2 + a2(t,x)(dx2 + dy2 + dz2). (4.1)The full Einstein field equations for the coordinate (4.1) areG00 = 3(a˙a)2+1a2(∇aa)2− 2a2(∇2aa)= 8piGT00, (4.2)Gii = −2aa¨− a˙2 −(∇aa)2+∇2aa+ 2(∂iaa)2− ∂2i aa= 8piGTii, (4.3)G0i = 2a˙a∂iaa− 2∂ia˙a= 8piGT0i, (4.4)Gij = 2∂iaa∂jaa− ∂i∂jaa= 8piGTij , i, j = 1, 2, 3, i 6= j, (4.5)where ∇ = (∂1, ∂2, ∂3) is the ordinary gradient operator with respect to thespatial coordinates x, y, z.104.2. The fluctuating spacetimeBy choosing the above simplest inhomogeneous metric (4.1), we are as-suming a mini-superspace type model, and will choose which of these equa-tions do apply later. This treatment might result in inconsistencies as gen-eral vacuum fluctuations of the matter fields posses rich structures that theymay not produce spacetime described by the metric (4.1). To fully describethe resulting inhomogeneous spacetime, one needs a more general metric.However, as a first approximation, using (4.1) is relatively easy to calculateand leads to interesting results. We are also going to do the calculations formore general metrics in chapter 7.4.2 The fluctuating spacetimeThe role played by the value of vacuum energy density in the above equations(4.2), (4.3), (4.4) and (4.5) is different from (2.11). The value of vacuumenergy density is no longer directly related to the Hubble rate H throughthe equation (2.11). This is evident from the 00 component of the Einsteinequation (4.2). The equation (2.11) is only the special case of (4.2) whenthe spatial derivatives ∇a and ∇2a are zero, which requires that the mat-ter distribution is strictly homogeneous and isotropic. However, as shownin the last section, the quantum vacuum is extremely inhomogeneous andnecessarily anisotropic, which requires ∇a and ∇2a be huge. This can beseen through the ij component of the Einstein equation (4.5). In fact, dueto symmetry properties of the quantum vacuum, we have the expectationvalue of shear stress Tij on the right side of (4.5)〈Tij〉 = 0, i, j = 1, 2, 3, i 6= j. (4.6)Meanwhile, Tij must fluctuate since the quantum vacuum is not its eigen-state either, and the magnitude of the fluctuation is on the same order ofthe vacuum energy density 〈T 2ij〉 ∼ 〈T00〉2 . (4.7)This means that the Tij is constantly fluctuating around zero with a hugemagnitude of the order of vacuum energy density. As a result, in (4.5),the spatial derivatives of a(t,x) must also constantly fluctuate with hugemagnitudes.More importantly, since the scale factor a(t,x) is spatially dependent, thephysical distance L between two spatial points with comoving coordinatesx1 and x2 is no longer related to their comoving distance ∆x = |x1 − x2|114.2. The fluctuating spacetimeby the simple equation L(t) = a(t)∆x and the observed global Hubble rateH is no longer equal to the local Hubble rate a˙/a. Instead, the physicaldistance and the global Hubble rate are defined asL(t) =∫ x2x1√a2(t,x)dl (4.8)andH(t) =L˙L=∫ x2x1a˙a(t,x)√a2(t,x)dl∫ x2x1√a2(t,x)dl, (4.9)where the line element dl =√dx2 + dy2 + dz2.Equation (4.9) shows the key difference between the gravitational behav-ior of quantum vacuum predicted by the homogeneous FLRW metric (2.10)and by the inhomogeneous metric (4.1).For the homogeneous metric (2.10), the scale factor a is spatially inde-pendent and (4.9) just reduces toH(t) =a˙a(t). (4.10)In this case, there are only two distinct choices for Hubble rates on a spatialslice t = Const under the initial value constraint equation (2.11)a˙a= ±√8piGρvac3, (4.11)which implies that all points in space have to be simultaneously expand-ing or contracting at the same constant rate (Here we do not include thecosmological constant λ).But for the inhomogeneous metric (4.1), the scale factor a is spatially de-pendent and there is much more freedom in choosing different local Hubblerates at different spatial points of the slice t = Const under the correspond-ing initial value constraint equation (4.2).In fact, the local Hubble rates must be constantly changing over spatialdirections within very small length scales. This can be seen from the initialvalue constraint equations (4.4), which can be rewritten as∇(a˙a)= −4piGJ, (4.12)124.2. The fluctuating spacetimewhere J = (T01, T02, T03) is vacuum energy flux2.The solution to (4.12) or (4.4) isa˙a(t,x) =a˙a(t,x0)− 4piG∫ xx0J(t,x′) · dl′, (4.13)where dl′ = (dx′, dy′, dz′) and x0 is an arbitrary spatial point. The abovesolution (4.13) shows that the difference in the local Hubble rates a˙/a be-tween x0 and x1 is determined by the spatial accumulations (integral) of thevacuum energy flux J. Similar to the shear stress, J has zero expectationvalue〈J〉 = 0 (4.14)but huge fluctuationsJ =√〈J2〉 ∼ 〈T00〉 ∼ Λ4 → +∞, (4.15)which implies that the local Hubble rates differ from point to point due tothe fluctuations. The average of the absolute value of a˙/a can be estimatedwith the constraint equation (4.2)√√√√〈( a˙a)2〉∼√G 〈T00〉 ∼√GΛ2. (4.16)By using (4.13), we find that the difference in local Hubble rates becomescomparable with itself for points separated by only a distance of the order∆x ∼ 1√GΛ2as Λ→ +∞:∆(a˙a)∼ 4piGJ∆x ∼√GΛ2 ∼√√√√〈( a˙a)2〉. (4.17)Up to this point, we have used the equations (4.2), (4.4) and (4.5). Theseequations are all initial value constraint equations which do not contain thescale factor’s second order time derivative a¨. To get the information about2One might notice that (4.12) requires ∇ × J = 0, which means that to produce themetric of the form (4.1), the energy flux of the matter field needs to be curl free. Asmentioned in the last paragraph of section 4.1, this is not true for general matter fields,but here as a first approximation we will use (4.12) to estimate the magnitude of changein a˙/a.134.2. The fluctuating spacetimethe time evolution of a(t,x), we also need to use (4.3). A linear combinationof equations (4.2) and (4.3) gives,G00 +1a2(G11 +G22 +G33) = −6a¨a, (4.18)where all the spatial derivatives of a cancel and only the second order timederivative left. Therefore we reach the following dynamic evolution equationfor a(t,x):a¨+ Ω2(t,x)a = 0, (4.19)whereΩ2 =4piG3(ρ+3∑i=1Pi), ρ = T00, Pi =1a2Tii. (4.20)(4.19) is just a generalization of the second Friedman equation. Its so-lution depends on the property of Ω2, especially its sign.If still treating the energy density ρ ≡ constant, then to satisfy theconservation equation (2.5), one must have P = P1 = P2 = P3 = −ρ that 3Ω2 =4piG3(ρ+ 3P ) = −8piGρ3< 0, if ρ > 0. (4.21)In this case, gravity becomes “repulsive” and the solution to (4.19) is justthe exponential expansion (2.13).However, when ρ is not a constant, fundamental difference happens—thesign of Ω2 may change. For example, consider a real massless scalar field φ,its stress energy tensor for a general spacetime metric gµν isTµν = ∇µφ∇νφ− 12gµν∇λφ∇λφ. (4.22)Direct calculation using the inhomogeneous metric (4.1) gives thatρ+3∑i=1Pi = 2φ˙2, (4.23)where all the spatial derivatives and explicit dependence on the metric a arecanceled. Thus we obtainΩ2 =8piGφ˙23> 0. (4.24)3This is easy to understand by considering the matter illustrated in Fig.4.2. The mattermust have negative pressure to be able to do negative work to the environment to maintainconstant energy density.144.2. The fluctuating spacetime 𝐹 𝐹𝜌=constant𝑃 = −𝜌Figure 4.1: Constant energy density requires negative pressureIn this case, gravity is still attractive 4 as usual and (4.19) describes aharmonic oscillator with time dependent frequency. The most basic behav-ior of a harmonic oscillator is that it oscillates back and forth around itsequilibrium point, which implies that the local Hubble rates a˙/a are period-ically changing signs over time. By using equation (4.17) you can find thata˙/a must also have this periodic sign change in a given spatial direction.Physically, these fluctuating features of a˙/a imply that, at any instantof time, if the space is expanding in a small region, it has to be contractingin neighboring regions; and at any spatial point, if the space is expandingnow, it has to be contracting later.These features result in huge cancellations when calculating the aver-aged H through (4.9). The observable overall net Hubble rate can be smallalthough the absolute value of the local Hubble rate |a˙/a| at each individualpoint has to be huge to satisfy the constraint equation (4.2). In other words,while the instantaneous rates of expansion or contraction at a fixed spatialpoint can be large, their effects can be canceled in a way that the physical4This is true if the matter fields satisfy normal energy conditions. We will assume thatΩ2 > 0 even after considering all the contributions from known and unknown fundamentalfields, i.e. gravity is always attractive as usual, no mysteries “dark energy” with peculiarnegative pressure.154.3. Methods and assumptions in solving the systemdistance (4.8) would not grow 10120 times larger than what is observed.This picture of fluctuating spacetime is not completely new. It is similarto the concept of spacetime foam devised by John Wheeler [14, 15] that in aquantum theory of gravity spacetime would have a foamy, jittery nature andwould consist of many small, ever-changing, regions in which spacetime arenot definite, but fluctuates. His reason for this “foamy” picture is the sameas ours—at sufficiently small scales the energy of vacuum fluctuations wouldbe large enough to cause significant departures from the smooth spacetimewe see at macroscopic scales.The solution for a(t,x) will be given by equations (5.4), (5.8) and (5.9)in the next chapter 5 to describe this foamy structure more precisely.4.3 Methods and assumptions in solving thesystemIn principle, we need a full quantum theory of gravity to solve the evolutiondetails of this quantum gravitational system. Unfortunately, no satisfactorytheory of quantum gravity exists yet.In this paper, we are not trying to quantize gravity. Instead, we arestill keeping the spacetime metric a(t,x) as classical, but quantizing thefields propagating on it. The key difference from the usual semiclassicalgravity is that we go one more step—instead of assuming the semiclassicalEinstein equation, where the curvature of the spacetime is sourced by theexpectation value of the quantum field stress energy tensor, we also take thehuge fluctuations of the stress energy tensor into account. In our method, thesources of gravity are stochastic classical fields whose stochastic propertiesare determined by their quantum fluctuations, i.e. our method is usingstochastic gravity framework [19]. 5The evolution details of the scale factor a(t,x) described by equation(4.19)depends on the property of the time dependent frequency Ω(t,x) given by(4.20). For both simplicity and clarity, in the following chapters we investi-gate the properties of Ω by considering the contribution from a real masslessscalar field φ, whose stress energy tensor is given by (4.22). (4.24) shows thatΩ2 is not explicitly dependent on the metric a(t,x). However, the resultingspacetime sourced by this massless scalar field φ does have back reactioneffect on φ itself. This is because φ obeys the equation of motion in curved5The difference from the usual stochastic gravity framework is that we do not try toregularize the divergent stress energy tensor.164.3. Methods and assumptions in solving the systemspacetime∇µ∇µφ = 1√−g∂µ(√−ggµν∂νφ) = 0, (4.25)which reduces to∂t(a3∂tφ)−∇ · (a∇φ) = 0 (4.26)for the special metric (4.1).Incorporating the back reaction effect by solving both the Einstein equa-tions (4.2), (4.3), (4.4), (4.5) for the metric a and the equation of motion(4.26) for the field φ at the same time is difficult. Fortunately, solving thesystem in this way is unnecessary. Physically, the quantum vacuum locallybehaves as a huge energy reservoir, so that the back reaction effect on itshould be small and can be neglected. In our method, we will first assumethat the quantized field φ is still taking the flat spacetime form of (3.1)for field modes below the effective QFT’s high frequency cutoff Λ. We use(3.1) to calculate the stochastic property of the time dependent frequencyΩ and then solve (4.19) to get the resulting curved spacetime described bythe metric a(t,x). This will be done in the next chapter 5.We then investigate the back reaction effect in chapter 6by quantizingthe field φ in the resulting curved spacetime. It turns out that, while theresulting spacetime is fluctuating, the fluctuation happens at scales whichare much smaller than the length scale 1/Λ. Therefore the corrections to thefield modes with frequencies below the cutoff Λ is quite small and thus theflat spacetime quantization (3.1) is valid to high precision. (See equations(6.38) (or (6.69)), (6.39) and (6.41) for quantitatively how high this precisionis.) In this way we justify neglecting the aforementioned back reaction.Empirically, this must be true since ordinary QFT has achieved greatsuccesses by assuming flat Minkowski background and using the expansion(3.1). So if our method is correct, (3.1) has to be still valid even the back-ground spacetime is no longer flat but wildly fluctuating at small scales. Inother words, the resulting spacetime should still looks like Minkowskian forlow frequency field modes. Long wavelength fields ride over the Wheeler’sfoam as if it is not there. This is similar to the behavior of very long wave-length water waves which do not notice the rapidly fluctuating atomic soupover which they ride.17Chapter 5The solution for a(t,x)In this chapter we give the solution for the local scale factor a(t,x).5.1 Parametric resonanceOne important feature of a harmonic oscillator with time dependent fre-quency is that it may exhibit parametric resonance behavior.If the Ω(t,x) is strictly periodic in time with a period T , the property ofthe solutions of (4.19) has been thoroughly studied by Floquet theory [20].Under certain conditions (for example, the condition (5.28)), the parametricresonance phenomenan occurs and the general solution of (4.19) is (see e.g.Eq(27.6) in Chapter V of [21])a(t,x) = c1eHxtP1(t,x) + c2e−HxtP2(t,x), (5.1)where Hx > 0, c1 and c2 are constants. The P1 and P2 are purely periodicfunctions of time with period T . They are in general functions oscillatingaround zero. The amplitude of the first term in (5.1) increases exponen-tially with time while the second term decreases exponentially. Thereforethe first term will become dominant and the solution will approach a pureexponential evolutiona(t,x) ' eHxtP (t,x), (5.2)where we have absorbed the constant c1 into P (t,x) by letting P (t,x) =c1P1(t,x).Physically, the exponential evolution of the amplitude of a(t,x) is easyto understand. If Ω is strictly periodic, the system will finally reach asteady pattern of evolution (when the second term in (5.1) has been highlysuppressed). In this pattern, after each period of evolution of the system, aincreases by a fixed ratio, i.e. a(t + T,x) = µxa(t,x), which results in theexponential increase since after n cycles, a(t+ nT,x) = µnxa(t,x). Here theµx is related to the Hx by Hx =lnµxT .185.1. Parametric resonance2 4 6 8 10-1.0-0.50.00.51.0LDtΧFigure 5.1: Plot of the normalized covariance χ as a function of temporalseparation Λ∆t.Due to the stochastic nature of quantum fluctuations, the Ω(t,x) in(4.19) is not strictly periodic. However, its behavior is still similar to a pe-riodic function. In fact, Ω exhibits quasiperiodic behavior in the sense thatit is always varying around its mean value back and forth on an approxi-mately fixed time scale. To see this, we calculate the following normalizedcovariance:χ (∆t) = Cov(Ω2(t1,x),Ω2(t2,x))(5.3)=〈{(Ω2(t1)−〈Ω2(t1)〉) (Ω2(t2)−〈Ω2(t2)〉)}〉〈(Ω2 − 〈Ω2〉)2〉 ,where ∆t = t1 − t2 and we have dropped the label x in the second line ofthe above definition (5.3) since the final result is independent with x.Explicit expression for χ as a function of ∆t is given by (A.12), whichis plotted in FIG. 5.1. It describes how Ω2 at different times change aroundtheir mean values together. We say that two Ω2 separated by time difference∆t are positively (negatively) correlated if χ(∆t) > 0(< 0), since it meansthat they are most likely to be at the same (opposite) side of their meanvalue 〈Ω2〉.195.1. Parametric resonanceFIG. 5.1 and (A.12) show that Ω2 at different times are strongly corre-lated at close range. Especially, the negative correlation is strongest when∆t ∼ 2/Λ, which implies that if at t = 0 the Ω2 is above its mean value〈Ω2〉, then at t ∼ 2/Λ, it is most likely below 〈Ω2〉. So basically, Ω2 variesaround its mean value quasiperiodically on the time scale T ∼ 1/Λ.This quasiperiodic behavior of Ω should also lead to parametric reso-nance behavior seen in (5.2), instead with a difference in that Hx wouldbecome time dependent, i.e. the solution would take the following forma(t,x) ' e∫ t0 Hx(t′)dt′P (t,x), (5.4)where P (t,x) here is no longer a strictly periodic function as in (5.2) buta quasiperiodic function with the same quasiperiod of the order 1/Λ as thetime dependent frequency Ω(t,x). (The solution (5.8) for P (t,x) in the nextsection 5.2 reveals this property.)The physical mechanism is similar. The system will also reach a fi-nal steady evolution pattern. In this pattern, after each quasiperiod ofevolution of the system, a will increase by an approximately fixed ratio.Suppose that during the ith cycle of quasiperiod Ti, a increases by a fac-tor µix, i.e. a(t + Ti,x) = µixa(t,x). Then after the n cycles, we havea(t +n∑i=1Ti,x) =(n∏i=1µix)a(t,x). Because the quasiperiods Ti and thefactors µix are generally different from each other, the exponent in (5.4)would need to take the form of integration.The detailed oscillating behavior of P (t,x) is not observable at macro-scopic scales. However, the factor of the exponential increase e∫ t0 Hx(t′)dt′can be observed. In fact, inserting (5.4) into (4.8), the observable physicaldistance would becomeL(t) = L(0)eHt, (5.5)whereL(0) =∫ x2x1√P 2(t,x)dl (5.6)and the global Hubble expansion rate H isH =1t∫ t0Hx(t′)dt′. (5.7)In the next two sections, we are going to give the solution for P (t,x)and the global Hubble expansion rate H.205.2. The solution for P (t,x)5.2 The solution for P (t,x)The magnitude of the time dependent frequency Ω is of the order∼√G 〈T00〉 ∼√GΛ2, while Ω itself varies roughly with a characteristic frequency Λ (thishas been shown by FIG. 5.1). Then according to (4.19), the scale factor awould oscillate with a period that roughly goes as T = 2pi/Ω ∼ 1/√GΛ2 1/Λ, as Λ → ∞, where 1/Λ is the time scale on which the Ω itself wouldchange significantly.So comparing to the oscillating period T of the scale factor a, the vari-ation of Ω itself is very slow, although the time 1/Λ is already very shortfor large Λ. Therefore, during one period of the oscillation of a, Ω is almostconstant since it has not have a chance to change significantly during sucha short time scale. In this sense the time dependent frequency Ω is slowlyvarying and the evolution of the scale factor a is an adiabatic process.The leading order solution of the equation (4.19) for a harmonic oscillatorwith the slowly varying frequency Ω can be obtained by a first order WKBapproximation. This adiabatic approximation neglects the small exponentialfactor in (5.4). It gives the solution P (t,x) which is describing the oscillatingbehavior of a(t,x). The result is,P (t,x) =A0√Ω(t,x)cos(∫ t0Ω(t′,x)dt′ + θx). (5.8)The P (t,x) above is a quasiperiodic function with the same quasiperiod ofthe order 1/Λ as the time dependent frequency Ω(t,x) just as expected. Thetwo constants of integration A0 and θx in (5.8) can be determined by theinitial values a(0,x) and a˙(0,x).The quantum vacuum is fluctuating everywhere, but its statistical prop-erty must be still the same everywhere. Correspondingly, the statisticalproperty of P (t,x) must also be the same everywhere, which requires thatthe constant A0 to be independent with respect to the spatial coordinatex. In addition, the constant A0 can be chosen as any nonzero value sincethe scale factor a multiplying by any nonzero constant describes physicallyequivalent spacetimes.The initial phase θx at different places must be dependent on x. Inapplying the initial value constraint equation (4.13), neglecting the smallexponential factor in (5.4) and neglecting the relatively small time derivativeterms of the slowly varying frequency Ω, we obtain the result,tan θx =Ω(0,x0)Ω(0,x)tan θx0 +4piGΩ(0,x)∫ xx0J(0,x′) · dl′, (5.9)215.3. The global Hubble expansion rate Hwhere θx0 is the initial phase of the scale factor a at an arbitrary spatialpoint x0.In solutions (5.8) and (5.9) we see the fluctuating nature of spacetimeat very small scales as described in the previous chapter 4.2. In particular,(5.9) shows that the phases of a(t,x) vary on a given initial Cauchy slice;some locations contract while others expand. In this new physical picturethe catastrophic vacuum energy density is confined to very small scales.5.3 The global Hubble expansion rate HAs the system is adiabatic, the parametric resonance effect is weak. Theadiabatic solution (5.8) in the last section does not include the parametricresonance and thus misses the small exponential factor expected in (5.4). Inthis section we go beyond the adiabatic approximation and investigate theexact strength of the weak parametric resonance.When considering the weak parametric resonance effect, the constantA0 in (5.8) would become time and space dependent and take the followingformA(t,x) = A0e∫ t0 Hx(t′)dt′ (5.10)in order to satisfy (5.4).To determine how the Hx(t) depends on the spacetime dependent fre-quency Ω(t,x), we consider the adiabatic invariant of a harmonic oscillatorwith time dependent frequency, which is defined asI(t,x) =EΩ, (5.11)whereE =12(a˙2 + Ω2a2). (5.12)Replace the constant A0 in (5.8) by A(t,x) and then plug it into theabove expression (5.11) we get thatI(t,x) =12A2(t,x), (5.13)where we have neglected the time derivatives of A and Ω in the above equa-tion (5.13), which are higher order infinitesimals. I is invariant in the firstorder adiabatic approximation. When going to higher orders, I will slowlychange with time. Through the relation (5.13) between I and A we can ob-tain how the A(t,x) changes by investigating how accurately the adiabaticinvariant is preserved and how it changes with time.225.3. The global Hubble expansion rate HIt has been proved by Robnik and Romanovski [22, 23] that, in fullgenerality (no restrictions on the function Ω(t,x)), the final value of theadiabatic invariant for the average energy I¯ = E¯/Ω is always greater orequal to the initial value I0 = E0/Ω0 (see the references [22, 23] for precisedefinition about the average energy). In other words, the average value ofthe adiabatic invariant I¯ = E¯/Ω for the mean value of the energy neverdecreases, which is a kind of irreversibility statement. It is conserved onlyfor infinitely slow process, i.e. an ideal adiabatic process.Therefore, in the case of our quasiperiodic frequency Ω(t,x) in (4.19), I¯will also always increase. Moreover, it will increase by a fixed factor aftereach quasiperiod of evolution, which results in an exponentially increasingI¯. This is in fact evident because of the weak parametric resonance effect.In the following we investigate this exponential behavior in detail.First we construct the evolution equation for the adiabatic invariant I.Do the canonical transformationa =√2I/Ω sinϕ, (5.14)a˙ =√2IΩ cosϕ. (5.15)Then the evolution equations for a and its conjugate momentum a˙ transferto the evolution equation for the new action variable I and the angle variableϕ,dIdt= −I Ω˙Ωcos 2ϕ, (5.16)dϕdt= Ω +Ω˙2Ωsin 2ϕ. (5.17)Integrating (5.16) yieldsI(t) = I(0) exp(2∫ t0Hx(t′)dt′), (5.18)whereHx(t′) = − Ω˙2Ωcos 2ϕ. (5.19)The Hx(t′) in the above equation (5.19) is just the same with the Hx(t′)defined in (5.4) and (5.10), which can be seen by applying equation (5.13).Thus equation (5.19) constructed the dependence of Hx(t′) on the timedependent frequency Ω(t′,x).235.3. The global Hubble expansion rate HThe observable global Hubble expansion rate H is the average of Hx(t′)over time, which was defined by equation (5.7). Plugging (5.19) into (5.7)gives,H = Re(−1t∫ t0Ω˙2Ωe2iϕdt′). (5.20)When the slow varying condition (5.35) holds, from equation (5.17) we knowthat dϕ/dt is positive, i.e. ϕ is a monotonic function in time. Thus we canchange the integral in (5.20) from the integration over t′ to integration overϕ′:H = Re(−1t∫ ϕϕ0Ω˙2Ωe2iϕdt′dϕ′dϕ′), (5.21)where ϕ0 = ϕ(0) and ϕ = ϕ(t).To evaluate H, we formally treat ϕ as a complex variable and close thecontour integral in the upper half plane. The integrand in (5.21) has nosingularities for real ϕ if the slow varying condition (5.35) holds. Equation(5.17) implies that ϕ ∼ Ωt ∼ √GΛ2t, so the length of the interval ϕ−ϕ0 ∼√GΛ2t goes to infinity as Λ→ +∞. Hence the principle contribution to theintegral in (5.21) comes from the residue values at singularities ϕ(k) insidethe contour:H =1tRe(2pii∑kRes(− Ω˙2Ωe2iϕdtdϕ, ϕ(k))). (5.22)Each term in (5.22) gives a contribution containing a factor exp(−2 Imϕ(k)).So the dominant contribution in (5.22) comes from the singularities near thereal axis, i.e. those with the smallest positive imaginary part. To keep thecalculation simple, we retain only those terms. Since Ω(t) varies quasiperi-odically with a characteristic time τ ∼ 1/Λ, the number of singularities nearthe real axis would roughly be on the order t/τ ∼ Λt. Therefore the H in(5.22) is roughlyH ∼ Λ exp (−2 Imϕ(k)) . (5.23)Let t(k) be the (complex) “instant” corresponding to the singularity ϕ(k):ϕ(k) = ϕ(t(k)) ∼ Ω t(k). In general, |t(k)| has the same order of magnitudeas the characteristic time τ ∼ 1/Λ of variation of the Ω. Remember thatΩ ∼ √GΛ2, thus the order of magnitude of the exponent in (5.23) isImϕ(k) ∼ Ωτ ∼√GΛ. (5.24)245.4. A more intuitive explanationTherefore, inserting (5.24) into (5.23) givesH = αΛe−β√GΛ, (5.25)where α and β are two dimensionless constants which depend on the varia-tion details of the time dependent frequency Ω(t,x). Therefore H becomesexponentially small in the limit of taking Λ to infinity. This is a manifes-tation of the well-established result that the error in adiabatic invariant isexponentially small for analytic Ω [22, 24]. In fact, the technique we usedin deriving (5.25) is very similar to the one used in deriving the error inadiabatic invariant in the pages “160− 161” of [24].5.4 A more intuitive explanationSo far we have obtained our key result (5.25) for the global Hubble expansionrate H. To understand the mechanism of weak parametric resonance better,we give a more intuitive explanation in this section.Consider the following simplest parametric oscillator:x¨+ ω2(t)x = 0, (5.26)whereω2(t) = ω20 (1 + h cos γt) . (5.27)The behavior of the above harmonic oscillator with time dependent fre-quency has been thoroughly studied (see e.g. eq(27.7) in Chapter V of [21]).The parametric resonance occurs when the frequency γ with which ω(t)varies is close to any value 2ω0/n, i.e.γ ∼ 2ω0n, (5.28)where n is an integer. The strength of the parametric resonance is strongestif γ is nearly twice ω0, i.e. if n = 1. As n increases to infinity, the strengthof the parametric resonance decreases to zero. This is easy to understandsince as n increases, the varying frequency γ of ω(t) becomes slower com-pared to the oscillator’s natural frequency ω0 and as n→∞, (5.26) reducesto an ordinary harmonic oscillator with constant frequency which has noparametric resonance behavior.Now let us go back to Eq.(4.19) for a(t,x). The time dependent fre-quency Ω(t,x) in (4.19) is more complicated than the ω(t) given in our255.4. A more intuitive explanation0.0 0.5 1.0 1.5 2.00.00.51.01.52.0Frequency HLLPowerPower SpectrumFigure 5.2: Plot of the power spectrum density of the varying part of Ω2(t,0)(except for the constant Ω20 part).example (5.27). However, it can be written in a similar form:Ω2(t,0) = Ω20(1 +∫ 2Λ0dγ (f (γ) cos γt+ g (γ) sin γt)), (5.29)whereΩ20 =〈Ω2〉=GΛ46pi, (5.30)and f(γ), g(γ) are operator coefficients, whose exact form are given by(A.15) and (A.16) in Appendix A. The behavior of Ω2(t,x) for an arbitraryx is the same with Ω2(t,0) except phase differences. The power spectrumdensity of the varying part of Ω2(t,0) (except for the constant Ω20 part) givenby (A.18) is plotted in FIG. 5.2.Unlike the case (5.27) where the ω(t) varies with a single frequency γ, theΩ(t,0) in (5.29) varies with frequencies continuously distributed in the range(0, 2Λ) with a peak around 1.7Λ (see FIG. 5.2). From (5.30) we have that,as taking the cutoff frequency Λ to infinity, Ω0 ∼√GΛ2  2Λ. Because ofthe continuity of the spectrum of Ω, we can always find integers n such that265.5. Meaning of our resultsifn ≥√G6piΛ, Λ→ +∞, (5.31)then2Ω0n∈ (0, 2Λ) . (5.32)So Ω(t,x) always contains frequencies 2Ω0/n that may excite resonances.From (5.31) we see that n → ∞ as taking the cutoff Λ to infinity. Whileas n increases, the relative magnitude of the resonance frequency 2Ω0/n de-creases comparing to the a(t,x)’s natural frequency Ω0. Then for reasonssimilar to the simplest parametric oscillator (5.26), the strength of the para-metric resonance of (4.19) would also decrease to zero. This weak parametricresonance effect leads to the global Hubble expansion rateH → 0, as Λ→ +∞. (5.33)5.5 Meaning of our resultsIt is interesting to notice that both (2.13) and (5.5) give the exponentialevolution and predict an accelerated expanding Universe. However, theunderlying mechanisms are completely different, which leads to oppositeresults on the predicted magnitude of the observable Hubble expansion rateH.The solution (2.13) is based on the assumption that quantum vacuumenergy density is constant all over the spacetime, which is a necessary re-quirement if one suppose that vacuum acts as a cosmological constant. Thisassumption leads to a huge Hubble expansion rateH =√8piGρvac3∝√GΛ2 → +∞ (5.34)as taking the high energy cutoff Λ to infinity.Our proposal (5.5) is based on the fact that quantum vacuum energydensity is constantly fluctuating and extremely inhomogeneous all over thewhole spacetime. This fact leads to a small Hubble expansion rate given by(5.25) which goes to zero as taking the high energy cutoff Λ to infinity.If we can literally take the cutoff Λ in (5.25) to infinity, then H = 0.In this sense, at least the “old” cosmological constant problem would beresolved.In principle, this effective theory is valid only up to a large but finitecutoff Λ, which leads to a tiny but nonzero H. Since H → 0 as Λ → +∞,275.6. The slow varying conditionthere always exists a very large cutoff value of Λ such that H =√ΩλH0 ≈1.2 × 10−42 GeV to match the observation, where H0 is current observedHubble constant.So our result suggests that there is no necessity to introduce the cosmo-logical constant, which is required to be fine tuned to an accuracy of 10−120,or other forms of dark energy, which are required to have peculiar negativepressure, to explain the observed accelerating expansion of the Universe.The exact value of Λ cannot be determined since we do not know thevalues of the two dimensionless parameters α and β in (5.25). In principle,we need the knowledge of all fundamental fields in the Universe to determineα and β, this deserves further investigations in the future and might providesome hint on elementary particle physics.We will use a couple of scalar fields to estimate the order of magnitudeof the parameters α and β in the numerical simulation presented in section5.7.5.6 The slow varying conditionThe key requirement for our derivation of (5.25) to work is that Ω2 is slowlyvarying that the whole process is adiabatic. The mathematical descriptionof the slow varying condition is (see equation (49.1) in Chapter VII of [24])∆Ω ∼ TdΩ/dt Ω, (5.35)where T ∼ 2pi/Ω is the period of the oscillation of a. Then the abovecondition can be rewritten asΩ˙Ω2 1. (5.36)If there is only one scalar field we have Ω2 = 8piG3 φ˙2 and(dΩdt)2= 8piG3 φ¨2.Using (3.1), we have the expectation values〈Ω2〉=8piG31(2pi)3∫d3k12ω=8piG314pi2∫ Λ0k3dk =16piGΛ4, (5.37)〈(dΩdt)2〉=8piG31(2pi)3∫d3k12ω3=8piG314pi2∫ Λ0k5dk =19piGΛ6. (5.38)285.6. The slow varying condition(5.37) just gives 〈Ω2〉 ∼ GΛ4 as expected, (5.38) gives 〈dΩ/dt〉 ∼ √GΛ3,therefore, we would have〈Ω˙〉〈Ω2〉 ∼√GΛ3GΛ4=1√GΛ 1, as Λ→ +∞, (5.39)i.e. the slow varying condition (5.35) is satisfied on average for one scalarfield.However, the quantum fluctuation of Ω2 is as big as its expectation value,so there is still possibility that Ω2 = 8piGφ˙2/3 fluctuates to values smallerthan√GΛ3 or even close to 0 where the slow varying condition (5.35) isnot satisfied. These extreme points have large contribution to the growth ofthe amplitude of a and destroy the key result H = αΛe−β√GΛ as Λ→ +∞(Eq.(5.25)).To resolve this problem, one has to make sure Ω always satisfy (5.35)(or at least the probability of violating (5.35) is low enough). This can bedone in two ways: i) increase the number of fields or ii) add a small negativebare cosmological constant in the Einstein equations. We will discuss thistwo solutions in the following.I). Adding more fieldsThe real Universe contains many different quantum fields. From centrallimit theorem, when more fields are added, the probability distribution forΩ2 would approach Gaussian. Moreover, the magnitude of the fluctuationof Ω2 would become relatively smaller compared to its expectation value.In fact, if we have n fields, the expectation value of Ω2 goes as nGΛ4 whilethe magnitude of the fluctuation (standard deviation of Ω2) goes as√nGΛ4.Therefore the probability for Ω2 to fluctuate to values smaller than√GΛ3goes to zero as the number of fields n go to infinity, i.e. the probability forΩ2 violating (5.35) becomes vanishingly small.In addition, when more fields are added, the expectation value of Ω2grows but the time scale of the variation of Ω2 stay the same, this makesthe variation of Ω2 becomes even slower for the same cutoff Λ. So the cutoffneeded to match the observed rate of accelerating expansion is reduced.II). Adding a small cosmological constantWhen a bare cosmological constant λb is included, Ω2 becomesΩ2 =4piG3(ρ+3∑i=1Pi)− λb3, ρ = T00, Pi =1a2Tii. (5.40)If −λb is greater than√GΛ3, then the slow varying condition (5.36) will295.7. Numerical verificationalways be satisfied. This can reduce the number of fields needed to achieveour key result (5.25).5.7 Numerical verificationIn this section, Planck units will be used, so all instances of Newton’s con-stant are set to unity, G = 1.The main idea is to rewrite the time dependent frequency Ω(t) in phasespace. (To see more details about this numeric method, please check Ap-pendix B. Here we only list the most crucial results.) For a real masslessscalar field, we haveΩ2({xk}, {pk}, t) = 8pi3∫d3kd3k′(2pi)3xkxk′ωω′ sinωt sinω′t+ pkpk′ cosωt cosω′t− 2xkpk′ω sinωt cosω′t.(5.41)This is the Weyl transformation of the operator Ωˆ2(t). Here {xk, pk} arephase space points of a particular field mode with momentum k. Approxi-mately, for a particular choice of {xk}, {pk}, we can get an classic equationfor a:a¨({xk}, {pk}, t) + Ω2({xk}, {pk}, t)a({xk}, {pk}, t) = 0 (5.42)The observed value ao(t) is the average of a({xk}, {pk}, t) over the Wignerpseudo distribution function W ({xk}, {pk}, t), which is based on the wavefunction of the quantum field:ao(t) =∫ (∏kdxkdpk)a({xk}, {pk}, t)W ({xk}, {pk}, t). (5.43)If the quantum field is in its ground state, we haveW ({xk}, {pk}, t) =∏k1pie−p2kω−x2kω (5.44)which means {xk}, {pk} are all Gaussian variables. Based on this obser-vation, our method to simulate this equation is as following: i) at first wegenerate a set of random Gaussian numbers for {xk}, {pk} ; ii) we solve theequation (5.42) for this particular set of numbers; iii) then we repeat the pro-cess for another set of random numbers until a certain amount of repetitions;iv) The result ao(t) is the average over all samples we have generated.305.7. Numerical verification0 50 100 150Time0200400600800log(|a|)Λ=10Λ=20Λ=30Λ=40Λ=50Figure 5.3: Numeric result for log |ao(t)| when one scalar field is present.The slope represents the Hubble expansion rate H. It shows that as Λincreases, H also increases. The prediction (5.25) is not applicable in thiscase since the slow varying condition is violated when Ω2 fluctuates to valuessmaller than ∼ Λ3.Fig. 5.3 is the result for only one scalar field contributing to Ω2. Theslope represents the Hubble expansion rate H. It shows that as Λ increases,H also increases. The prediction (5.25) is not applicable in this case sincethe probability for the slow varying condition (when Ω2 fluctuates to valuessmaller than ∼ Λ3) can not be neglected as explained in the last section.Fig. 5.4 is the result for one scalar field with a negative bare cosmologicalconstant −λb ∼ Λ3.5 contributing to Ω2. It shows that as Λ increases, Hdecreases. The linear fit log(H/Λ) vs Λ gives the parameter α ∼ e−4.1 ≈0.017, β ∼ 0.072. In this case, the prediction (5.25) is observed since λbmakes Ω2 always greater than Λ3 and has it be adiabatic.Fig. 5.5 is the result for five scalar fields with and without a bare cosmo-logical constant −λb ∼ Λ3.5 contributing to Ω2. In both cases as Λ increases,H decreases and the prediction (5.25) is observed. The linear fit gives theparameters α ∼ e−3.8 ≈ 0.022, β ∼ 0.14 for the case without the bare cos-mological constant and the parameters α ∼ e−4.2 ≈ 0.015, β ∼ 0.28 for thecase with the bare cosmological constant.Fig. 5.6 is the result for ten scalar fields with and without a bare cosmo-logical constant −λb ∼ Λ3.5 contributing to Ω2. In both cases as Λ increases,315.7. Numerical verification0 200 400 600 800 1000Time-20020406080log(|a|)Λ=10Λ=20Λ=30Λ=40Λ=500 10 20 30 40 50 60Λ-8-7-6-5-4log(H/Λ)log(H/Λ)=−0.072Λ−4.1Figure 5.4: Numeric result for log |ao(t)| when one scalar field with a negativebare cosmological constant −λb ∼ Λ3.5 are present. The slope represents theHubble expansion rate H. It shows that as Λ increases, H decreases. Thelinear fit log(H/Λ) vs Λ gives the parameter α ∼ e−4.1 ≈ 0.017, β ∼ 0.072.325.7. Numerical verification0 500 1000 1500 2000Time050100log(|a|)Λ=5Λ=10Λ=15Λ=20Λ=25Λ=30Λ=35Λ=400 10 20 30 40Λ-11-10-9-8-7-6-5-4log(H/Λ)log(H/Λ)=−0.14Λ−3.80 500 1000 1500 2000Time-20-10010203040log(|a|)Λ=5Λ=10Λ=150 5 10 15 20Λ-9-8-7-6-5log(H/Λ)log(H/Λ)=−0.28Λ−4.2Figure 5.5: Top: numeric result for five scalar fields without a bare cos-mological constant and its linear fit log(H/Λ) vs Λ; Bottom: five scalarfields with a negative bare cosmological constant −λb ∼ Λ3.5 and its linearfit log(H/Λ) vs Λ. The slope represents the Hubble expansion rate H. Inboth cases as Λ increases, H decreases. The linear fit gives the parametersα ∼ e−3.8 ≈ 0.022, β ∼ 0.14 for five scalar fields without the bare cosmolog-ical constant and the parameters α ∼ e−4.2 ≈ 0.015, β ∼ 0.28 for five scalarfields with the bare cosmological constant.335.7. Numerical verification0 2000 4000 6000 8000 10000Time050100log(|a|)Λ=3Λ=4Λ=5Λ=6Λ=7Λ=82 3 4 5 6 7 8 9Λ-9-8-7-6-5log(H/Λ)log(H/Λ)=−0.60Λ−3.70 2000 4000 6000 8000 10000Time-20020406080log(|a|)Λ=3Λ=4Λ=5Λ=62 4 6Λ-9-8-7-6-5log(H/Λ)log(H/Λ)=−0.83Λ−3.5Figure 5.6: Top: numeric result for ten scalar fields without a bare cosmo-logical constant and its linear fit log(H/Λ) vs Λ; Bottom: ten scalar fieldswith a negative bare cosmological constant −λb ∼ Λ3.5 and its linear fitlog(H/Λ) vs Λ. The slope represents the Hubble expansion rate H. Inboth cases as Λ increases, H decreases. The linear fit gives the parametersα ∼ e−3.7 ≈ 0.025, β ∼ 0.6 for ten scalar fields without the bare cosmolog-ical constant and the parameters α ∼ e−3.5 ≈ 0.03, β ∼ 0.83 for ten scalarfields with the bare cosmological constant.H decreases and the prediction (5.25) is observed. The linear fit gives theparameters α ∼ e−3.7 ≈ 0.025, β ∼ 0.6 for the case without the bare cosmo-logical constant and the parameters α ∼ e−3.5 ≈ 0.03, β ∼ 0.83 for the casewith the bare cosmological constant.Fig. 5.7 is the result for twenty scalar fields with and without a barecosmological constant −λb ∼ Λ3.5 contributing to Ω2. In both cases as Λincreases, H decreases and the prediction (5.25) is observed. The linear fitgives the parameters α ∼ e−2.6 ≈ 0.074, β ∼ 1.9 for the case without thebare cosmological constant and the parameters α ∼ e−2.1 ≈ 0.12, β ∼ 2.4for the case with the bare cosmological constant.We can see from Fig. 5.5, 5.6 and 5.7 that as more fields are added, the345.7. Numerical verification0 5000 10000 15000 20000Time-200204060log(|a|)Λ=2Λ=2.5Λ=3Λ=3.52 3 4Λ-10-9-8-7-6log(H/Λ)log(H/Λ)=−1.9Λ−2.60 5000 10000 15000 20000Time-20-1001020304050log(|a|)Λ=2Λ=2.5Λ=32 3Λ-10-9-8-7-6-5log(H/Λ)log(H/Λ)=−2.4Λ−2.1Figure 5.7: Top: numeric result for twenty scalar fields without a barecosmological constant and its linear fit log(H/Λ) vs Λ; Bottom: twentyscalar fields with a negative bare cosmological constant −λb ∼ Λ3.5 andits linear fit log(H/Λ) vs Λ. The slope represents the Hubble expansionrate H. In both cases as Λ increases, H decreases. The linear fit gives theparameters α ∼ e−2.6 ≈ 0.074, β ∼ 1.9 for twenty scalar fields without thebare cosmological constant and the parameters α ∼ e−2.1 ≈ 0.12, β ∼ 2.4for twenty scalar fields with the bare cosmological constant.355.7. Numerical verificationfitting parameter β increases, which implies that the parametric resonanceeffect becomes weaker and the cutoff needed to match the observation isreduced. The observed H is on the order of 10−61 ∼ e−140, so for five fieldsthe cutoff needed is about Λ ∼ 140/β ∼ 140/0.14 ∼ 1000, for ten fields thecutoff needed is about Λ ∼ 140/β ∼ 140/0.6 ∼ 230, for twenty fields thecutoff needed is about Λ ∼ 140/β ∼ 140/1.9 ∼ 74.If we also add a negative bare cosmological constant, the cutoff neededwould be further reduced. For a bare cosmological constant on the order of−λb ∼ Λ3.5, the cutoffs needed are Λ ∼ 140/β ∼ 140/0.28 ∼ 500 for fivefields, Λ ∼ 140/β ∼ 140/0.83 ∼ 168 for ten fields Λ ∼ 140/β ∼ 140/2.4 ∼ 58for twenty fields.36Chapter 6The back reactionIn this chapter, we investigate the back reaction effect by quantizing thefield φ in the resulting curved spacetime to justify our method of using thequantized field expansion (3.1) in Minkowski spacetime as an approximation.The standard way to quantize the scalar field φ in a generic curvedspacetime gµν is by first defining the following inner product on a spacelikehypersurface Σ with induced metric hij and unit normal vector nµ (see e.g.[16, 25]):(φ1, φ2) = −i∫Σ(φ1∂µφ∗2 − φ∗2∂µφ1)nµ√hd3x, (6.1)where h = dethij and φ1, φ2 are solutions to the equation (4.25). The aboveinner product is independent of the choice of Σ.One then choose a complete set of mode solutions uk of (4.25) which areorthonormal in the product (6.1):(uk, uk′) = δ(k− k′), (6.2)(u∗k, u∗k′) = −δ(k− k′), (6.3)(uk, u∗k′) = 0. (6.4)Then the field φ may be expanded asφ =∑k(akuk + a†ku∗k). (6.5)For the flat Minkowski spacetime, i.e. gµν = ηµν , (4.25) reduces to theusual wave equationφ¨−∇2φ = 0. (6.6)In this case, the mode solutions are usually chosen asuk(t,x) =1(2pi)3/21√2ωe−i(ωt−k·x), (6.7)where ω = |k|. Plugging (6.7) into (6.5) just gives the usual quantum fieldexpansion (3.1).37Chapter 6. The back reactionFor our specific metric (4.1), (4.25) reduces to (4.26). In this case, sincethe rate of accelerating expansion is extremely small, the back reaction effectdue to the macroscopic expansion of the Universe is only important on largecosmological time scales. For this reason, we only worry about the backreaction due to the wildly fluctuating spacetime at small scales. i.e. weneglect the small exponential factor in (5.4) and use the form of the a basedon the solution (5.8):a(t,x) =A0√Ω(t,x)cos (Θ(t,x)) , (6.8)whereΘ(t,x) =∫ t0Ω(t′,x)dt′ + θx. (6.9)Then (4.26) becomesA20Ωcos2 Θφ¨−∇2φ (6.10)−3A202(Ω˙Ω2cos2 Θ + sin 2Θ)φ˙+(∇Ω2Ω+ tan Θ∇Θ)· ∇φ = 0.In order to understand the effect from back reaction, we need to find outhow the mode solutions of the above equation (6.10) in the resulting curvedspacetime change from the mode solutions (6.7) of the equation (6.6) in theflat Minkowski spacetime.Physically, the correction to (6.7) should be small for wave modes withfrequencies lower than the cutoff frequency Λ. That is because the wavelength of those field modes is larger than 2pi/Λ, while our spacetime fluc-tuates on the length scale 2pi/Ω ∼ 1/(√GΛ2)  2pi/Λ. The relatively longwave length modes should not be sensitive to what is happening on smallscales. This is analogous to the situation of sound waves traveling in themedium such as air or water or solids. The medium is constantly fluctuatingat atomic scales, but this fluctuation does not affect the propagation of thesound wave whose wavelength is much larger than the atomic scale. Sim-ilarly, the propagation of the field modes in the “medium”–the spacetime,which is constantly fluctuating on scales much smaller than the wavelengthof the field modes, should also not be affected.Mathematical demonstration will be given in the following sections.386.1. A simplified toy model6.1 A simplified toy modelIt is complicated to obtain the mode solutions of (6.10) for a generic stochas-tic function Θ(t,x) whose stochastic property is determined by the quantumnature of the field φ. To illustrate the underlying physical mechanism moreclearly, we start with a simplified toy model by restricting the phase angleΘ(t,x) defined by (6.9) to take the following form:Θ(t,x) = Ωt+ K · x, (6.11)where both Ω and K are constants and they have the same order of magni-tude Ω ∼ |K| ∼ √GΛ2.Of course this toy model does not describe the real spacetime sourcedby the quantum vacuum since the Ω is by no means a constant but alwaysvarying, although the varying is slow compared to it own magnitude. How-ever, this toy model possesses the key property needed — the spacetime isconstantly fluctuating. It will be convenient for visualizing the back reactioneffect from a fluctuating spacetime.After setting the Ω ≡ Constant and the phase angle Θ(t,x) to be theform of (6.11), the equation of motion (6.10) for φ becomes(1 + cos 2 (Ωt+ K · x)) φ¨−∇2φ−3Ω sin 2 (Ωt+ K · x) φ˙+ tan (Ωt+ K · x) K · ∇φ = 0, (6.12)where we have set A0 =√2Ω such that the average of the coefficientA20Ω cos2 Θ before φ¨ is 1 for convenience.In the flat spacetime case (6.6), each mode solution uk in (6.7) containsonly one single frequency. However, for the above fluctuating spacetime case(6.12), high frequencies mixes with low frequencies and each mode solutionmust contain multiple frequencies. In fact, since (6.12) describes a strictlyperiodic system with time period pi/Ω and spatial period pi/|K|, each modesolution uk must change from (6.7) to the following form:uk(t,x) = e−i(ωt−k·x)c0 + +∞∑m=−∞m6=0cmei2m(Ωt+K·x) , (6.13)where cm are constants.396.1. A simplified toy modelInserting (6.13) into (6.12) and using the orthogonality of e2im(Ωt+K·x),we obtain the following infinite system of linear equations:mth equation:m−2∑n=−∞(−1)m+nK · (k + 2nK) cn+[12(ω − 2 (m− 1) Ω)2 − 32Ω (ω − 2 (m− 1) Ω)−K · (k + 2 (m− 1) K)]cm−1+[(ω − 2mΩ)2 − (k + 2mK)2]cm+[12(ω − 2 (m+ 1) Ω)2 + 32Ω (ω − 2 (m+ 1) Ω)+K · (k + 2 (m+ 1) K)]cm+1++∞∑n=m+2(−1)m+n+1K · (k + 2nK) cn= 0, m = 0,±1,±2,±3, . . . (6.14)In the above calculations, we have used the Fourier series expansiontanx = −2+∞∑n=1(−1)n sin 2nx (6.15)to expand the term tan(Ωt+ K · x) in (6.12).For the equations of m ≤ −1, we successively add the (m+1)th equationto the mth equation by the order from m = −∞ to m = −1; and for theequations of m ≥ 1, we successively add the (m − 1)th equation to themth equation by the order from m = +∞ to m = 1. Most terms can beeliminated by these elementary row operations and the above infinite system406.1. A simplified toy modelof linear equations (6.14) becomesif m ≤ −1,12(ω − 2 (m− 1) Ω) (ω − (2m+ 1) Ω) cm−1+[32(ω − 2mΩ) (ω − (2m+ 1) Ω)− (k + 2mK) · (k + (2m+ 1) K)]cm+[32(ω − 2(m+ 1)Ω) (ω − (2m+ 1) Ω)− (k + 2(m+ 1)K) · (k + (2m+ 1) K)]cm+1+12(ω − 2 (m+ 2) Ω) (ω − (2m+ 1) Ω) cm+2 = 0;if m = 0,−2∑n=−∞(−1)nK · (k + 2nK) cn+[12(ω + 2Ω) (ω − Ω)−K · (k− 2K)]c−1+(ω2 − k2) c0+[12(ω − 2Ω) (ω + Ω) + K · (k + 2K)]c1++∞∑n=2(−1)n+1K · (k + 2nK) cn = 0;if m ≥ 1,12(ω − 2 (m− 2) Ω) (ω − (2m− 1) Ω) cm−2+[32(ω − 2(m− 1)Ω) (ω − (2m− 1) Ω)− (k + 2(m− 1)K) · (k + (2m− 1) K)]cm−1+[32(ω − 2mΩ) (ω − (2m− 1) Ω)− (k + 2mK) · (k + (2m− 1) K)]cm+12(ω − 2 (m+ 1) Ω) (ω − (2m− 1) Ω) cm+1 = 0. (6.16)416.1. A simplified toy modelTo characterize the property of the solutions of this system more clearly,we define the following parameters for convenience: =ωΩ, υ =|k|Ω, δ =|K|Ω, cos γ =K · k|K||k| . (6.17)As mentioned before that our effective theory has a cutoff Λ such thatonly modes with ω, |k| ≤ Λ are relevant, which are much smaller than Ω ∼|K| ∼ √GΛ2 as Λ grows large. Therefore, we are only interested in thesolutions of (6.14) or (6.16) when ω, |k|  Ω, i.e. when , υ → 0.Dividing both sides of (6.16) by Ω2 and doing some necessary algebraic426.1. A simplified toy modelmanipulations, (6.16) can be rewritten asif m ≤ −1,[(m− 1)− 2]cm−1+[ (3− 2δ2)m− 32− 2mδ2+∞∑n=1(2m+ 1)n− υ2m+ 1((4m+ 1) δ cos γ + υ)+∞∑n=0(2m+ 1)n ]cm+[ (3− 2δ2) (m+ 1)− 32− 2(m+ 1)δ2+∞∑n=1(2m+ 1)n− υ2m+ 1((4m+ 3) δ cos γ + υ)+∞∑n=0(2m+ 1)n ]cm+1+[(m+ 2)− 2]cm+2 = 0;if m = 0,−2∑n=−∞(−1)n (2nδ2 + δυ cos γ) cn + [−1 + 2δ2 + 2− δυ cos γ + 22]c−1+(2 − υ2) c0 (6.18)+[−1 + 2δ2 − 2+ δυ cos γ +22]c1 ++∞∑n=2(−1)n+1 (2nδ2 + δυ cos γ) cn = 0;if m ≥ 1,[(m− 2)− 2]cm−2+[ (3− 2δ2) (m− 1)− 32− 2 (m− 1) δ2+∞∑n=1(2m− 1)n− υ2m− 1 ((4m− 3) δ cos γ + υ)+∞∑n=0(2m− 1)n ]cm−1+[ (3− 2δ2)m− 32− 2mδ2+∞∑n=1(2m− 1)n− υ2m− 1 ((4m− 1) δ cos γ + υ)+∞∑n=0(2m− 1)n ]cm+[(m+ 1)− 2]cm+1 = 0. 436.1. A simplified toy modelAs , υ → 0, the leading order asymptotic solution for {cn} of the abovesystem of linear equations (6.18) depends only on the leading order of the co-efficients before {cn}. By keeping only the leading term for each coefficient,(6.18) is asymptotic to the following infinite system of linear equations:BC = 0, (6.19)where the infinite matrix B isB =. . ...................... . ..· · · −3(3− 2δ2)−2(3− 2δ2) −1 0 0 0 0 · · ·· · · −3 −2(3− 2δ2)−(3− 2δ2) − 2 0 0 0 · · ·· · · 0 −2 −(3− 2δ2)−32 − δυ cos γ 1 0 0 · · ·· · · 6δ2 −4δ2 −1 + 2δ2 2 − υ2 −1 + 2δ2 −4δ2 6δ2 · · ·· · · 0 0 −1 −32 − δυ cos γ 3− 2δ2 2 0 · · ·· · · 0 0 0 − 2 3− 2δ2 2(3− 2δ2) 3 · · ·· · · 0 0 0 0 1 2(3− 2δ2) 3(3− 2δ2) · · ·. .. ...................... . .and C = (· · · , c−3, c−2, c−1, c0, c1, c2, c3, · · · )T .We will denote the matrix elements of B by bmn with −∞ < m,n < +∞.In order to have a nonzero solution, the determinant of B must be zero. Thisgives us the dispersion relation that  and υ must satisfy in the asymptoticregime , υ → 0.The determinant can be calculated by Laplace expansion:det(B) = b00M00 ++∞∑n=−∞n6=0(−1)nb0nM0n, (6.20)where M0n is the 0, n minor of B, i.e. the infinite determinant that resultsfrom deleting the 0th row and the nth column of B. Due to the symmetryproperty of B, we have that, for each n 6= 0,b0n = b0,−n, M0n = −M0,−n, (6.21)446.1. A simplified toy modelwhich implies that all the terms inside the summation symbol∑of (6.20)exactly cancel. Therefore, only the first term in (6.20) survive and thus wehave thatdet(B) = M00(δ2)(2 − υ2) = 0, (6.22)which leads to2 = υ2, (6.23)or equivalentlyω2 = k2. (6.24)This proves that the usual dispersion relation still holds for low frequencyfield modes.After setting 2 = υ2, we start solving the infinite system (6.19).First, we rewrite (6.19) as the following form:+∞∑n=−∞n6=0bmncn = −bm0c0, m = 0,±1,±2,±3, · · · (6.25)Notice that the matrix elements of B has the following symmetry properties:bmn = −b−m,−n, ifm,n 6= 0 (6.26)bm0 = b−m,0, b0n = b0,−n. (6.27)The above symmetry properties leads to the following relationcn = −c−n, n 6= 0, (6.28)which implies that we only need to solve cn for n > 0 to solve the wholesystem.For convenience, we define the following new variables xn bycn = c0xn, n 6= 0. (6.29)Then using the relation (6.28), the infinite system of linear equations (6.25)simplifies to the following infinite recurrence equations:(4− 2δ2)x1 + 2x2 = 32+ δ cos γ, (6.30)(3− 2δ2)x1 + (3− 2δ2) 2x2 + 3x3 = 12, (6.31)(m− 2)xm−2 +(3− 2δ2) (m− 1)xm−1+(3− 2δ2)mxm + (m+ 1)xm+1 = 0, if m ≥ 3, (6.32)456.1. A simplified toy modelwhere the dependence on  in the equation (6.19) or (6.25) has been elim-inated by introducing the new variables xn, n 6= 0 through (6.29) and thesolution for xn depends only on δ.In order to find for the sequence {xm}, we define the following newvariables:ym = (m− 1)xm−1 +mxm, m ≥ 3. (6.33)Then the recurrence equations (6.32) becomeym−1 + 2(1− δ2)ym + ym+1 = 0, m ≥ 3. (6.34)Sequences satisfying (6.34) must take the following form:ym = D cos (mϑ+ ψ) , m ≥ 3, (6.35)where D and ψ are two constants and ϑ is determined bycosϑ = −1 + δ2, sinϑ = δ√2− δ2. (6.36)Combining (6.35) and (6.33), the general formula for xm can be obtainedby iterationxm =1m(Dm∑n=3(−1)m−n cos(nϑ+ ψ) + (−1)m2x2)=(−1)mm(−D sec(ϑ2) sin((m− 2)ϑ2+mpi2)(6.37)· sin((m+ 3)ϑ2+ ψ +mpi2)+ 2x2), m ≥ 3.Replacing the cm in (6.13) by xm through (6.29) we obtain that, as→ 0, the mode solution uk(t,x) is asymptotic touk(t,x) = c0e−i(ωt−k·x)1 +  +∞∑m=−∞m 6=0xmei2m(Ωt+K·x) , (6.38)where xm is determined by (6.28), (6.29), (6.30), (6.31), (6.32) and (6.37).Using the orthogonality of ei2m(Ωt+K·x), the relative magnitude of thecorrection to uk from the usual plane wave mode e−i(ωt−k·x) in Minkowski466.2. General casespacetime can be characterized by applying Parseval’s identity:|∆uk(t,x)| =  +∞∑m=−∞m6=0x2m12. (6.39)From the solution (6.37) we know that as m→∞,x2m ∼1m2. (6.40)Thus the summation inside the bracket of (6.39) converges and the correc-tion|∆uk(t,x)| ∼ → 0, as → 0. (6.41)Thus we have demonstrated that the low frequency wave modes (ω ≤ Λ) arealmost not affected by the fluctuating spacetime with much higher frequency(Ω ∼ √GΛ2).6.2 General caseThe methods used and results obtained in the last section for the particularsimplified toy model (6.12) can be generalized to the generic case (6.10). Tostart, we rewrite (6.10) to the following form:(1 + f1) φ¨−∇2φ− Ω0f2φ˙+K0f3 · ∇φ = 0, (6.42)wheref1 =A20Ωcos2 Θ− 1, (6.43)f2 =3A202(Ω˙Ω2cos2 Θ + sin 2Θ)/Ω0, (6.44)f3 =(∇Ω2Ω+ tan Θ∇Θ)/K0, (6.45)Ω0 = 〈Ω〉 , K0 = 〈|∇Θ|〉 . (6.46)For convenience, we choose the constant A0 such that the average of f1〈f1(t,x)〉 = 0. (6.47)476.2. General caseUnlike the toy model (6.12) we used in the last section, (6.42) is notstrictly periodic. However, (6.42) is quasiperiodic and its quasiperiod isthe same as the period of (6.12). This property is reflected in the Fouriertransforms f1(ω,k), f2(ω,k) and f3(ω,k) of the functions f1(t,x), f2(t,x)and f3(t,x) respectively which are defined byf1(t,x) =∫dωd3k f1(ω,k)ei(ωt+k·x), (6.48)f2(t,x) =∫dωd3k f2(ω,k)ei(ωt+k·x), (6.49)f3(t,x) =∫dωd3k f3(ω,k)ei(ωt+k·x). (6.50)For the function f1(t,x) defined by (6.43), after setting the constant A0by (6.47) and considering the slow varying property of Ω(t,x) and Θ(t,x)in both temporal and spatial directions, its leading order goes asf1(t,x) ∼ cos 2Θ, (6.51)which implies that the Fourier transform f1(ω,k) would have two peakscentered atω = ±2Ω0, |k| = 2K0. (6.52)For the function f2(t,x) defined by (6.44), the second term which in-cludes the factor sin 2Θ is dominant since the first term which includes thefactor Ω˙/Ω2 goes as ∼ 1/Λ→ 0 due to the slow varying condition describedby (5.37) and (5.38). Thus, its leading order goes asf2(t,x) ∼ 3 sin 2Θ, (6.53)which implies that the Fourier transform f2(ω,k) would also have two peakscentered atω = ±2Ω0, |k| = 2K0. (6.54)Similarly, for the function f3(t,x) defined by (6.45), the second termwhich includes the factor tan Θ is dominant since the absolute value of thefirst term which includes the factor ∇Ω/(ΩK0) also goes as ∼ 1/Λ→ 0 dueto the slow varying property of Ω in spatial directions. Thus, its leadingorder goes asf3(t,x) ∼ tan Θ∇ΘK0. (6.55)486.2. General caseThen using the Fourier series expansion (6.15) for tan Θ, we know that theFourier transform f3(ω,k) would have infinitely many peaks centered atω = ±2nΩ0, |k| = 2nK0, n = 1, 2, 3, · · · . (6.56)(For a rough calculation of the above Fourier transforms, see Appendix C)In addition, we have the zero frequency component (see (C.6) in Ap-pendix C)fi(ω = 0,k = 0) ∼ 0, i = 1, 2, 3. (6.57)In summary, the system described by (6.42) is very similar to the systemdescribed by the simplified toy model (6.12). The only difference is that theFourier transforms of the coefficients f1, f2, and f3 in (6.42) spread aroundcenter points given by (6.52), (6.54) and (6.56) while the Fourier transformsof the corresponding coefficients in (6.12) are ideal delta functions exactlylocated at same points given by (6.52), (6.54) and (6.56).Therefore, the mode solution of (6.42) would take the form similar to(6.13):uk(t,x) = e−i(ωt−k·x)c0 + ∫ω′ 6=0k′ 6=0dω′d3k′ uk(ω′,k′)ei(ω′t+k′·x) , (6.58)where uk(ω′,k′) is non-negligible only when ω′,k′ are taking values aroundthe centers given by (6.52), (6.54) and (6.56).Inserting (6.58) into (6.42) and replacing the coefficients f1(t,x), f2(t,x)and f3(t,x) in (6.42) by the equations (6.48), (6.49) and (6.50) and thenusing the orthogonality of ei(ω′t+k′·x), we obtain the following uncountablyinfinite system of linear equations which are similar to (6.14):(ω′,k′)th equation :[(ω − ω′)2 − (k + k′)2]uk (ω′,k′)+∫dω′′d3k′′[ (ω − (ω′ − ω′′))2 f1 (ω′′,k′′)−iΩ0(ω − (ω′ − ω′′)) f2 (ω′′,k′′) (6.59)−iK0(k +(k′ − k′′)) · f3 (ω′′,k′′) ]uk (ω′ − ω′′,k′ − k′′) = 0,where we have defined the notation uk(0,0) = c0δ(0,0) for convenience.496.2. General caseTo characterize the property of the solutions of this system more clearly,we define the following parameters similar to (6.17) for convenience: =ωΩ0, υ =|k|Ω0, δ =K0Ω0, cos γ =k · k′|k||k′| ,cosµ =k · f3|k||f3| , cosµ′ =k′ · f3|k′||f3| , cosµ′′ =k′′ · f3|k′′||f3| .Dividing both sides of (6.59) by Ω20 gives(ω′,k′)th equation :[(− ω′Ω0)2−(υ2 +k′2Ω20+ 2υ|k′|Ω0cos γ)]uk(ω′,k′)+∫dω′′d3k′′[(−(ω′Ω0− ω′′Ω0))2f1(ω′′,k′′)−i(−(ω′Ω0− ω′′Ω0))f2(ω′′,k′′)(6.60)−iδ(υ cosµ+( |k′|Ω0cosµ′ − |k′′|Ω0cosµ′′))|f3(ω′′,k′′) |]·uk(ω′ − ω′′,k′ − k′′) = 0.Similar to the toy model case, as , υ → 0, the leading order solutionof (6.60) for uk(ω′,k′) satisfies the following uncountably infinite system of506.2. General caselinear equations:if (ω′,k′) = (0,0) :(2 − υ2) δ (0,0) c0+∫dω′′d3k′′[(ω′′Ω0)2f1(ω′′,k′′)− i(ω′′Ω0)f2(ω′′,k′′)+ iδ( |k′′|Ω0cosµ′′)|f3(ω′′,k′′) |]uk (−ω′′,−k′′) = 0,(6.61)if (ω′,k′) 6= (0,0) :(−if2(ω′,k′)− iδυ cosµ|f3(ω′,k′)|) c0+[(ω′Ω0)2−(k′Ω0)2]uk(ω′,k′)+∫ω′′ 6=ω′k′′ 6=k′dω′′d3k′′[(ω′Ω0− ω′′Ω0)2f1(ω′′,k′′)+ i(ω′Ω0− ω′′Ω0)f2(ω′′,k′′)− iδ( |k′|Ω0cosµ′ − |k′′|Ω0cosµ′′)|f3(ω′′,k′′) |]uk (ω′ − ω′′,k′ − k′′) = 0,where we have used the property (6.57) in obtaining (6.61) from (6.60).The above uncountably infinite system of linear equations (6.61) can alsobe written formally in matrix form similar to (6.19). We use similar nota-tions that denoting the matrix here by B and its elements by b(ω′,k′),(ω′′,k′′)for convenience.In order to have nonzero solutions, the determinant of the uncountablyinfinite matrix B has to be zero, which gives the dispersion relations that and υ must be satisfied in the asymptotic region , υ → 0.The “determinant” of B can be formally calculated through Laplaceexpansion similar to (6.20):detB = b(0,0),(0,0)M(0,0),(0,0) (6.62)+∫ω′′ 6=0k′′ 6=0dω′′d3k′′(−1)(ω′′,k′′)b(0,0),(ω′′,k′′)M(0,0),(ω′′,k′′),516.2. General casewhere M(0,0),(ω′′,k′′) is the (0,0), (ω′′,k′′) minor of B, i.e. the ‘determinant’resulting from deleting the (0,0)th row and (ω′′,k′′)th column of B.Notice that since f1(t,x), f2(t,x) and f3(t,x) are all real, their Fouriertransforms f1(ω,k), f2(ω,k) and f3(ω,k) defined by (6.48), (6.49) and (6.50)must satisfy the following relations:f1(ω,k) = f1(−ω,−k)∗,f2(ω,k) = f2(−ω,−k)∗,f3(ω,k) = f3(−ω,−k)∗, (6.63)where the ∗ means complex conjugate.The above symmetry property (6.63) leads tob(0,0),(ω′′,k′′) = b(0,0),(−ω′′,−k′′), (6.64)M(0,0),(ω′′,k′′) = −M(0,0),(−ω′′,−k′′), if (−ω′′,−k′′) 6= (0,0),which implies that all the terms inside the integral symbol∫of (6.62) exactlycancel. Therefore, only the first term in (6.62) survives and thus we havedetB = M(0,0),(0,0)(2 − υ2) = 0, (6.65)which gives again the usual dispersion relation2 = υ2 or ω2 = k2. (6.66)After setting the dispersion relation (6.66), we only need to solve the(ω′,k′) 6= (0,0)th equations in (6.61) since detB = 0 implies that the(ω′,k′) = (0,0)th equation is redundant.For convenience, we define new variables xk(ω′,k′) similar to the xndefined in (6.29):uk(ω′,k′) = c0xk(ω′,k′), (ω′,k′) 6= (0,0). (6.67)Then (6.61) can be rewritten asif (ω′,k′) 6= (0,0) :[(ω′Ω0)2−(k′Ω0)2]xk(ω′,k′)+∫ω′′ 6=ω′k′′ 6=k′dω′′d3k′′[(ω′Ω0− ω′′Ω0)2f1(ω′′,k′′)+ i(ω′Ω0− ω′′Ω0)f2(ω′′,k′′)−iδ( |k′|Ω0cosµ′ − |k′′|Ω0cosµ′′)|f3(ω′′,k′′) |]xk (ω′ − ω′′,k′ − k′′)= if2(ω′,k′) + iδ cosµ|f3(ω′,k′)|. (6.68)526.2. General caseReplacing the uk(ω′,k′) in (6.58) by xk(ω′,k′) through (6.67) we obtainthat, as → 0, the mode solution uk(t,x) is asymptotic touk(t,x) = c0e−i(ωt−k·x)1 + ∫ω′ 6=0k′ 6=0dω′d3k′ xk(ω′,k′)ei(ω′t+k′·x) , (6.69)where xk(ω′,k′) is determined by (6.68).Analogous to (6.37) in the simplified toy model, xk(ω′,k′) would also goasxk(ω′,k′) ∼ 1m, (6.70)when ω′,k′ taking values around the centersω′ ∼ ±2mΩ0, |k′| ∼ 2mK0, m = 1, 2, 3, · · · (6.71)(xk(ω′,k′) is negligible if ω′,k′ is far away from these centers).Due to Parseval’s theorem, (6.70) implies that the integral inside thebracket of (6.69) converges which is similar to (6.39) and thus the correctionto uk(t,x) also goes as .Therefore, when we quantize the scalar field φ in our wildly fluctuatingspacetime by expanding it in terms of the annihilation and creation operatorsaccording to (6.5), the leading order would still be the form of the Minkowskiquantum field expansion (3.1). The correction to the dispersion relationω2 = k2 and the plane wave mode e−i(ωt−k·x) are on the order ∼ . Inaddition, the extra wave modes which mixing in (6.38) or (6.69) are allmodes with frequencies higher than Ω0 ∼√GΛ2, which is much larger thanour effective QFT’s cutoff Λ. These extremely high frequency modes beyondthe cutoff are irrelevant to our low energy physics. This also explains whythe ordinary QFT works by assuming fixed Minkowski spacetime. The smallscale structure averages out in its effect on the long wavelength low energyfields.In summary, we have argued that although our spacetime sourced by thequantum vacuum is highly curved and wildly fluctuating, the back reactionof the resulting spacetime on the quantum field sitting on it is small. Thisjustifies our method of neglecting back reaction and using the quantum fieldexpansion (3.1) in Minkowski spacetime at the beginning.53Chapter 7The more general metricsIn previous chapters we assume the simplest inhomogeneous metric (4.1)to describe the spacetime resulting from the inhomogeneous vacuum. Inthis chapter, we try to generalize the result to more general inhomogeneousmetrics.7.1 The full metric and Einstein equationsWe can always choose a spacelike hypersurface and construct the followinggeneral synchronous coordinate (at least locally) (see pages 42-43 of [4]):ds2 = −dt2 + hab(t,x)dxadxb, a, b = 1, 2, 3. (7.1)For the above metric (7.1), we employ the initial value formulation ofgeneral relativity. In this formulation, the Einstein equation is equivalent tosix equations for the evolution of the second fundamental formk˙ab =−R(3)ab − (trk)kab + 2kackcb+4piGρhab + 8piG(Tab − 12habtrT),(7.2)plus the usual four constraint equations,R(3) + (trk)2 − kabkab = 16piGρ, (7.3)Dakab −Db(trk) = 8piGjb, (7.4)where kab =12 h˙ab, kab = hachbdkcd, trk = habkab, ρ = T00, jb = habT0a,trT = habTab, R(3) is the 3-dimensional spatial curvature and Da is thederivative operator associated with hab.Taking trace on both sides of (7.2) and then combining with (7.3) gives:habk˙ab − kabkab = −4piG (ρ+ trT ) . (7.5)547.2. The Mixmaster-type metricIt is interesting to notice that there are no spatial derivatives included onthe left hand of the above equation (7.5). The key evolution equation (4.19)for a(t,x) we used in previous chapters is just the special case of the aboveequation (7.5).Direct calculation using the expression (4.22) shows that, the contribu-tion from a real massless scalar field to the right-hand side of (7.5) isρ+ trT = 2φ˙2, (7.6)where all the spatial derivatives of φ and all the explicit dependence on themetric gµν in the definition of stress energy tensor (4.22) are canceled. Itis also interesting to notice that the above exact expression (7.6) is exactlythe same with the corresponding expression (4.23) for the simplest inhomo-geneous metric (4.1) case.7.2 The Mixmaster-type metricWe first consider the following special case:hab(t,x) =a2(t,x) 0 00 b2(t,x) 00 0 c2(t,x) , (7.7)which is similar to the metric of Mixmaster universe [26].The spacetime described by the above coordinate (7.7) possesses morefreedoms than (4.1) and thus would exhibit richer structures. In this case,the expansion rate at the same point becomes directionally dependent.Along the three principle axes xˆ, yˆ and zˆ, which are eigenvectors of thesymmetric matrix hab in (7.7), the expansion rates a˙/a, b˙/b and c˙/c can bedifferent. This means that, at one same point, the space can be expandingin one or two directions and contracting on the other two or one directions.Under the coordinate system (7.7), equation (7.5) becomesa¨a+b¨b+c¨c= −4piG(ρ+3∑i=1Pi), (7.8)where P1 = T11/a2, P2 = T22/b2, P3 = T33/c2. This equation is a general-ization of the key evolution equation (4.19) we used in previous chapters.Leta¨a= −Ω21(t,x),b¨b= −Ω22(t,x),c¨c= −Ω23(t,x), (7.9)557.2. The Mixmaster-type metricthen (7.8) immediately leads toΩ21(t,x) + Ω22(t,x) + Ω23(t,x) = 4piG(ρ+3∑i=1Pi). (7.10)Unlike equation (4.19), here the time dependent frequencies Ω2i (t,x0) donot necessarily go exactly the same as Ω2 = 4piG3(ρ+3∑i=1Pi). However, asthe functions a, b and c are alternately symmetric, Ω21, Ω22 and Ω23 must havethe same statistical properties. Especially, their expectation values must beequal 〈Ω2i (t,x0)〉=4piG3〈(ρ+3∑i=1Pi)〉, i = 1, 2, 3. (7.11)Moreover, since Ω2 is slowly varying, Ω2i should also be slowly varyingfunctions, since otherwise we would have three fast varying functions sumtogether and precisely cancel each other to give a slowly varying function,which is almost impossible in the system with such huge quantum fluctu-ations (This argument is from probability sense. To prove this, one needto also investigate other Einstein equations. This needs more study in thefuture.). Thus the evolution of a, b and c are also adiabatic processes thatthe solutions would be similar to (5.4):a ' e∫ t0 H1x0 (t′)dt′P˜1(t,x0), (7.12)b ' e∫ t0 H2x0 (t′)dt′P˜2(t,x0), (7.13)c ' e∫ t0 H3x0 (t′)dt′P˜3(t,x0), (7.14)where P˜i are quasiperiodic functions with the same quasiperiods as the timedependent frequencies Ωi.Also, on average, we haveHi = H = αΛe−β√GΛ, i = 1, 2, 3, (7.15)whereHi =1t∫ t0Hix0(t′)dt′. (7.16)Therefore, the determinant of hab goes ash(t,x0) = dethab(t,x0) = a2b2c2 ' exp(23∑i=1Hit)3∏i=1P˜ 2i , (7.17)567.3. An alternative derivation from geodesic deviation equationand the observable physical volume would be,V (t) =∫ √h(t,x)d3x = V (0)e3Ht, (7.18)which also gives the slow accelerating expansion of the Universe on cosmo-logical scale.7.3 An alternative derivation from geodesicdeviation equationThe dynamic equation (7.8) can also be derived from the geodesic deviationequation. It is easier to understand the physical meaning of (7.8) from thisalternative derivation.We first review the formalism of geodesic deviation equation. We willfollow the same notation in [4].Let γs(t) denote a smooth one-parameter family of geodesics, that is, foreach s ∈ R, γs is a geodesic parameterized by the affine parameter t andthe map (t, s) → γs(t) is smooth, one-to-one and has smooth inverse. Thecollection of these curves defines a smooth two-dimensional surface. We maychoose (t, s) as coordinates of this surface.There are two natural vector fields: the tangent vectors to the geodesicsT a =(∂∂t)aand the deviation vectors Xa =(∂∂s)awhich represent thedisplacements to infinitesimally nearby geodesics. The quantityva = T b∇bXa (7.19)gives the rate of change along a geodesic of the displacement to an infinites-imally nearby geodesic and may be interpreted as the relative velocity of aninfinitesimally nearby geodesics. The quantityaa = T c∇cva = T c∇c(T b∇bXa)(7.20)may be interpreted as the relative acceleration of an infinitesimally nearbygeodesic in the family.The curvature of spacetime tells how the geodesics move. It is easyto derive the following geodesic deviation equation from the definition ofRiemann curvature tensor (see derivation of Eq.(3.3.18) in page 47 of [4])aa = −RacbdXbT cT d, (7.21)577.3. An alternative derivation from geodesic deviation equationwhich shows that the relative acceleration between two neighboring geodesicsis proportional to the curvature.For synchronized coordinates, the curves {γx(t) : x = Constant} areall geodesics. This is a three-parameter family of geodesics (parametersx, y, z). To apply the geodesic deviation equation (7.21), we first pick thesub-family of geodesics: {γx(t) : −∞ < x < +∞, y = y0, z = z0}. For thisone-parameter family of geodesics, we apply (7.21) to the geodesic γx0(t)which goes through an arbitrary point x = x0 = (x0, y0, z0).The tangent vector and the deviation vector of γx0(t) expressed in thecoordinate (7.7) areT a =(∂∂t)a= (1, 0, 0, 0), (7.22)Xa =(∂∂x)a= (0, 1, 0, 0). (7.23)The relative velocity isva = T b∇bXa= T b∂bXa + T bΓabcXc= Γa01=(0,a˙a, 0, 0). (7.24)The relative acceleration isaa = T b∇bva= T b∂bΓa01 + TbΓabcΓc01= ∂0Γa01 + Γa01Γ101=(0,a¨a, 0, 0). (7.25)Also expressing (7.21) in the coordinate (7.7) givesaa = −Ra010. (7.26)Comparing (7.25) and (7.26) we obtain thata¨a= −R1010. (7.27)587.3. An alternative derivation from geodesic deviation equationPicking the other two sub-family of geodesics: {γy′(t′) : x′ = x0, y′ =Constant, z′ = z0} and {γz′(t′) : x′ = x0, y′ = y0, z′ = Constant}, andfollowing the same procedures we obtain thatb¨b= −R2020, (7.28)c¨c= −R3030. (7.29)Summing (7.27), (7.28) and (7.29) together we obtain thata¨a+b¨b+c¨c= −Rµ0µ0, (7.30)where the sum over µ can range over all four coordinates, not just thethree spatial ones, since the symmetries of the Riemann tensor requires thatR0000 = 0.The right-hand side of the above equation is just minus the time-timecomponent of the Ricci tensorR00 = Rµ0µ0. (7.31)Einstein equation has an equivalent form which directly relates the Riccitensor with matter field stress energy tensor (see e.g. Eq. (4.3.23) in page72 of [4]):Rµν = 8piG(Tµν − 12gµνT), (7.32)where T = gµνTµν is the trace of the stress energy tensor. Specially, thetime-time component of (7.32) expressed in the coordinate (7.7) is justR00 = 4piG(ρ+3∑i=1Pi). (7.33)Therefore, replacing the Rµ0µ0 in the right-hand side of (7.30) by (7.33) weobtain the same equation as (7.8):a¨a+b¨b+c¨c= −4piG(ρ+3∑i=1Pi). (7.34)An interesting fact that needs to be pointed out is that the key dy-namic equations (4.19), (7.5), (7.8) are all just the time-time component ofEq.(7.32) expressed in the corresponding coordinates.597.4. The physical picture7.4 The physical pictureIt is easy to understand the physical meaning of our key dynamic equation(7.8) from the last section’s derivation. (7.8) describes how the geodesicsγx(t) which are infinitesimally close to γx0(t) move in the wildly fluctuatingspacetime. It shows that the sum of the relative accelerations a¨/a, b¨/b, c¨/cis proportional to the energy density ρ plus the pressures P1, P2, P3 of thematter fields.Geometrically, the geodesics γx(t) with |x − x0| sufficiently small forman infinitesimally small ellipsoid of test particles around x0. The solutions(7.12), (7.13), (7.14) for a, b, c show that the ellipsoid is alternatively ex-panding and contracting in the three principal directions xˆ, yˆ, zˆ. Moreover,due to the weak parametric resonance effect, the expansion wins out a littlebit that the average volume of the ellipsoid would gradually increase. Thiseffect accumulates on the cosmological scale, which gives the slow acceler-ating expansion of the Universe.7.5 The Raychaudhuri equationIf hab take the most general formhab(t,x) =a2(t,x) d(t,x) e(t,x)d(t,x) b2(t,x) f(t,x)e(t,x) f(t,x) c2(t,x) , (7.35)then the dynamic equation (7.5) becomesa2h∗11ha¨a+b2h∗22hb¨b+c2h∗11hc¨c(7.36)+dh∗12hd¨d+eh∗13he¨e+fh∗23hf¨f+ F (hab, h˙ab) = −4piG(ρ+ trT ),where h = det(hab) is the determinant of the matrix (7.35), h∗ab is the ma-trix’s (a, b) cofactor and F is a nonlinear function of the metric componentshab and their first time derivatives h˙ab.(7.36) is very difficult to handle. However, it is in fact equivalent tothe well known Raychaudhuri equation. Its physical meaning is easier tounderstand in this way. In the following we will first review some basicsabout Raychaudhuri’s equation (see Section 9.2 of [4] for details).Consider a general spacetime gab and a smooth congruence of timelikegeodesics which are parameterized by proper time τ . So the vector field, ξa,607.5. The Raychaudhuri equationof tangents is normalized to unit length, ξaξa = −1. One then define thetensor field Bab, the expansion θ, shear σab, and twist ωab byBab = ∇bξa, (7.37)θ = Babhab, (7.38)σab = B(ab) −13θhab, (7.39)ωab = B[ab], (7.40)where the spatial metric hab is defined byhab = gab + ξaξb. (7.41)Along any geodesic in the congruence, θ measures the average rate ofexpansion of the infinitesimally nearby surrounding geodesics; ωab measurestheir rotation; and σab measures their shear, i.e. an initial sphere in thetangent space which is Lie transported along ξa will distort toward an el-lipsoid with principle axes by the eigenvectors of σab , with rate given by theeigenvalues of σab .Raychaudhuri’s equation is a differential equation for the expansion θ:ξc∇cθ + 13θ2 + σabσab − ωabωab = −Rabξaξb. (7.42)If the congruence is hypersurface orthogonal, we have ωab = 0 that the aboveequation (7.42) becomesξc∇cθ + 13θ2 + σabσab = −8pi(Tab − 12Tgab)ξaξb, (7.43)where we have used the equivalent version of Einstein equation (7.32) toreplace the Ricci tensor Rab.For synchronized coordinate (4.1) or (7.7) or (7.1), the curves {γx(t) :x = Constant} are all geodesics and they also form a congruence. Let ξabe the tangent vectors of them, i.e.ξa =(∂∂t)a, (7.44)then (7.43) can be expressed in the coordinate (4.1) or (7.7) or (7.1) asθ¨ +13θ2 + σabσab = −4piG (ρ+ trT ) . (7.45)617.5. The Raychaudhuri equationFor the simplest inhomogeneous metric (4.1), we haveσab = 0, (7.46)andθ = 3a˙a, (7.47)i.e. the average expansion θ is just 3 times the local Hubble expansion ratea˙/a; Then (7.45) becomesθ¨ +13θ2 = 3a¨a= −4piG(ρ+3∑i=1Pi), (7.48)which is just the key equation (4.19) for a time dependent harmonic oscillatorwe used in previous chapters.For the mixmaster-type metric (7.7), we haveθ =a˙a+b˙b+c˙c(7.49)andθ¨ +13θ2 + σabσab =a¨a+b¨b+c¨c. (7.50)Then (7.45) becomesa¨a+b¨b+c¨c= −4piG(ρ+3∑i=1Pi), (7.51)which is just equation (7.8).For the most general metric (7.1), (7.45) just becomes (7.36). This caseis difficult to handle. Further investigations are needed in the future.However, the results we obtained for the mixmaster-type metric (7.7)suggest that, for the most general case (7.35), the eigenvalues λ2i (t,x) of thematrix hab should also evolve adiabatically similar to a2, b2 and c2. In otherwords, we expect that the results (7.12), (7.13), (7.14) and (7.15) can begeneralized to λi in the most general case and the physical volume of spacewould expand asV (t) =∫ √h(t,x)d3x=∫ √λ21λ22λ23d3x= V (0)e3Ht, (7.52)where H is determined by (5.25).62Chapter 8Similarity of effects ofvacuum energy innon-gravitational system andgravitational systemVacuum fluctuations and their associated vacuum energies are direct con-sequences of the Heisenberg’s uncertainty principle of quantum mechanics.Although it is still controversial [27], various observable effects are oftenascribed to the existence of vacuum energies and have been experimentallyverified, which strongly suggests the reality of vacuum fluctuations. Thesevacuum fluctuation effects include the spontaneous emission [28], the Lambshift [29], the anomalous magnetic moment of the electron [30, 31] and theCasimir effect [32–35]. The reality of the vacuum energy associated to thespontaneous symmetry breaking of electroweak theory has also been con-firmed by the discovery of the Higgs boson at the LHC [36, 37].If we assume that the vacuum fluctuations do exist as evidenced by theabove listed observable effects, then according to the equivalence principle,the associated vacuum energies would gravitate as well as all other formsof energy. This has been experimentally demonstrated by, for example, thegravitational test of Lamb shift energy [38–40]. The gravitational propertyof Casimir energy has not been tested experimentally, but has been demon-strated theoretically with the conclusion that it does gravitate according toequivalence principle [41–44].However, in the literature, the value of vacuum energy density is usu-ally thought to play a different role in non-gravitational systems and ingravitational systems. The actual value of the vacuum energy density isgenerally regarded as irrelevant in non-gravitational contexts based on theargument that only energy differences from the vacuum are measurable;while when gravity is present, the actual value of the energy matters, notjust the differences, since the source for the gravitational field is the entire638.1. Value of vacuum energy is relevant in Casimir effectenergy momentum tensor that its large value may be potentially disastrous.We argue differently in this section with the following points: (i) thevalue of vacuum energy density can also be relevant in non-gravitationalcontexts; (ii) the huge value of vacuum energy density is not a direct ob-servable and that it is not disastrous in a theory of gravity. Moreover,there is essentially no difference between the roles played by vacuum energyin non-gravitational systems and in gravitational systems. In other words,although technically more complicated when gravity is included, the gravi-tational effect of the vacuum energy on spacetime metric is intrinsically thesame as its effect on material bodies when gravity is excluded.8.1 Value of vacuum energy is relevant inCasimir effectLet us first consider the Casimir effect. The Casimir force is usually derivedby calculating the change in vacuum energy due to the presence of theconducting plates, which acts as mirrors to reflect electromagnetic waves (Wewill call them mirrors in the following). This derivation is straightforward,but loses some important physical details about what is going on in thesystem [45, 46]. Due to quantum fluctuations, the zero point fields constantlyimpinge on both sides of the mirror and then reflect back, which transmitmomentum to the mirror and thus result in forces on both sides of the mirror.The Casimir stress (force per unit area) is just the difference between thepressure exerted by the electromagnetic field vacuum from inside and outsideS(t, x, y) = T insidezz − T outsidezz , (8.1)where we have set that the two parallel mirrors are normal to the z axis.Since the vacuum fluctuations between the two mirrors are different from thevacuum fluctuations outside, the expectation values of T insidezz and Toutsidezzwould be different and thus gives a net average force. Although both T insidezzand T outsidezz are divergent, this average force is finite since the quartic diver-gent Minkowski zero point fluctuations are canceled after the subtraction in(8.1) and one obtains the well known Casimir stress [46]〈S〉 = − pi2240d4. (8.2)Thus the effect of the value of zero point energy disappears in the calcu-lations. It is for this reason that although the Casimir effect is usually648.1. Value of vacuum energy is relevant in Casimir effectregarded as evidence of the reality of zero point energy, the actual value ofits energy density is thought to be irrelevant in this effect.However, the value of zero point energy density does have an effect. Notethat (8.2) only gives the expectation value of the Casimir stress S, but S isnever a constant, it fluctuates. That’s because the amount of momentumcarried by the zero point fields which impinge on both sides of the mirror isconstantly fluctuating due to the fact that the vacuum is not an eigenstateof the zz component of the stress energy tensor Tzz. The magnitude of thefluctuation of each Tzz is large and diverges as the same order of the vacuumenergy density 〈T00〉. For a perfect mirror, since the fields on the two sidesfluctuate independently of each other, the mean-squared stresses on the twosides simply add, resulting in the magnitude of the fluctuation of the netstress also diverges as〈∆S2〉=〈(S − 〈S〉)2〉 ∼ 〈T00〉2 →∞. (8.3)For more realistic imperfect mirrors which become transparent for frequen-cies higher than its plasma frequency Λ, the 〈T00〉 in (8.3) contains contri-butions only from field modes of frequencies lower than Λ and the meansquared value of the net stress S goes as〈∆S2〉 ∼ 〈T00〉2 ∼ Λ8. (8.4)The plasma frequency Λ in (8.4) acts as an effective cutoff which depends onthe microstructure of the mirror. It is similar but distinct from the effectiveQFT’s cutoff Λ in (5.25), which depends on the microstructure of spacetime.Therefore, the value of zero point energy density is still physically signif-icant even in non-gravitational system. Its value appears in (8.3) and (8.4)to characterize the strength of Casimir stress fluctuation, which implies thatthe net Casimir stress is constantly fluctuating with huge magnitudes aroundits small mean value (8.2). Due to this huge fluctuation, at almost any in-stant, the magnitude of the stress at each single point of the mirror is aslarge as the value of the zero point energy density.However, this effect is strong only at small scales. Its measurable effectbecomes small at larger scales. In practice, the measurements must be takenover some finite time interval T and some finite surface area of order l2. Moreprecisely, what the force detector measures is the time and surface averageS¯ =∫dtdxdyf(t, x, y)S(t, x, y), (8.5)where the averaging function f satisfies∫dtdxdyf(t, x, y) = 1. (8.6)658.2. Effect of vacuum energy on the motion of mirrorsThe exact shape of the averaging function depends on the measuring appa-ratus. On physical grounds one can choose f to be a single peak over a timeinterval T comparable to the experimental resolving time and over a spa-tial region of area l2 comparable to the resolution of the measuring device.Although the magnitude of the fluctuations of the net stress S is formallyinfinite as shown in (8.3), the magnitude of the measurable fluctuations ofits average S¯ is finite. This is because the effect of the vacuum fluctuationsat small scales is significantly weakened when averaging over larger scales.The calculations have been done by G Barton in [47] with the conclusionthat, for the realistic case where l cT , the mean squared deviation〈∆S¯2〉=〈(S¯ − 〈S¯〉)2〉 = constantT 8, (8.7)where the “constant” here is a pure number as could have been foreseenon dimensional grounds. The above equation (8.7) shows that〈∆S¯2〉in-creases as T decreases, which means that the better the measuring device,the stronger fluctuation due to the effect of the value of the zero point energydensity can be measured. And in principle, using a perfect instantaneousmeasuring device (T → 0), one can measure the infinite fluctuations of theCasimir stress on a perfect mirror due to the infinite value of zero pointenergy density. In practice, however,〈∆S¯2〉is too small to be measured fora real force detector whose resolving time T is too large [47].8.2 Effect of vacuum energy on the motion ofmirrorsThe value of zero point energy density also has effects on the dynamic motionof small material bodies. Imagine that we place a single mirror of very smallsize in the vacuum and then release it. The mirror would experience afluctuating force exerted by the quantum field vacuum and starts to move.The equation of motion of the mirror, which is called quantum Langevinequation, can be generally described byX¨ = F(t,X, X˙, φ, φ˙, . . .), (8.8)where X is the mirror’s position, φ represents the field interacting with themirror which is usually taken to be a scalar field for simplicity and we haveset the mirror’s mass M = 1 for convenience. The average force in this casewould be zero because of symmetry〈F 〉 = 0, (8.9)668.3. Analogies between the motion of mirror and the motion of a(t,x)and similar to the Casimir stress fluctuation (8.3), the force here also un-dergoes wild fluctuations with a magnitude〈F 2〉 ∝ 〈T00〉2 →∞. (8.10)The mathematically infinite fluctuating force F gives infinite instanta-neous accelerations of the mirror through (8.8). Similar to the case of infiniteCasimir stress fluctuation (8.3), this infinite fluctuating force and infinite in-stantaneous acceleration make sense since they are also only significant atvery small scales and will not result in infinite fluctuation of the mirror’sposition at observable larger scales. In fact, the mirror would oscillate backand forth with very high speeds, but its range of motion is still small [48–52].More precisely, suppose that the mirror is initially located at X(0) = 0with velocity X˙(0) = 0 and is then released at t = 0. The magnitudeof its acceleration X¨(t) and velocity X˙(t), which can be characterized bythe quantity〈X¨2(t)〉and〈X˙2(t)〉, is large. But, the magnitude of therange of the mirror’s fluctuating motion, which can be characterized by theobservable mean squared displacement〈X2(t)〉, is still small.In this sense, the value of vacuum energy density is still relevant evenin non-gravitational physics. This value appears in the equation (8.10) tocharacterize the strength of the force fluctuations acting on the mirror atsmall scales and it may have small observable effects at larger scales such asdiffusions predicted in [50–52].8.3 Analogies between the motion of mirror andthe motion of a(t,x)Although technically more complicated in gravity, the basic dynamic equa-tion of motion (4.19) satisfied by the scale factor a(t,x) is in fact very similarto the equation of motion (8.8) satisfied by the mirror’s position X(t). Con-sider only the contribution from the massless scalar field φ, (4.19) is just thefollowing same form as the equation (8.8)a¨ = F(a, φ˙), (8.11)whereF(a, φ˙)= −8piG3φ˙2a. (8.12)Also, the average of the fluctuating force F(a, φ˙)is zero due to symmetry〈F(a, φ˙)〉= 0, (8.13)678.3. Analogies between the motion of mirror and the motion of a(t,x)and its magnitude of fluctuation〈F 2(a, φ˙)〉∝ 〈T00〉2 →∞. (8.14)The above two statistical properties (8.13) and (8.14) satisfied by the “force”driving the “motion” of the scale factor a are the same with the statisticalproperties (8.9) and (8.10) satisfied by the force driving the motion of themirror. In this sense, the role played by the value of the vacuum energydensity in gravitational system is similar to its role in non-gravitationalsystem.Concretely speaking, the vacuum energy density results in large instan-taneous acceleration X¨ and velocity X˙ of the mirror, but the observableposition fluctuations of the mirror, which can be characterized by the quan-tity〈X2〉, is not large. Analogously, the vacuum energy density results inthe large instantaneous “acceleration” a¨ and “velocity” a˙ of the scale fac-tor, but the observable physical distance defined by (4.8), whose value isdetermined by the quantity〈a2〉, is also not large. These properties abouta(t,x) are evident from the solutions (5.4), (5.8) and (5.25), from which wecan see that the quantities〈a¨2〉and〈a˙2〉are as large as 〈T00〉2 and 〈T00〉respectively, while the magnitude of the quantity〈a2〉is on the order 1.In this sense, the role played by vacuum energy in gravitational systemis similar to its role in the non-gravitational mirror systems—it appearsboth at (8.10) and (8.14) to show the strongness of vacuum fluctuations atmicroscopic scales (for mirrors, microscopic means atomic scale; for gravity,microscopic means Planck scale) and their observable effects are both smallat macroscopic scales.By this same kind of mechanism, the violent gravitational effect producedby the vacuum energy density is confined to Planck scales, and its effectat macroscopic scales—the accelerating expansion of the Universe, due tothe weak parametric resonance is so small that, it is only observable afteraccumulations on the largest scale—the cosmological scale.68Chapter 9The singularities at a(t,x) = 0In our way of vacuum gravitating, the space is alternatively expanding andcontracting at each spatial point, and, during each such cycle, the expansionoutweighs the contraction a little bit due to the weak parametric resonanceeffect. This process gives a slowly increasing amplitude A(t,x) of the scalefactor a(t,x), whose observable effect is just the accelerating expansion ofour Universe.Probably one of the biggest concerns about this physical picture is theappearance of singularities at points a(t,x) = 0—according to the solution(5.8), the scale factor a(t,x) must go through zero whenever the space at xswitches from contraction phase to expansion phase. In this section, we aregoing to discuss this issue of singularities.9.1 Is singularity an end or a new beginning?Singularities are a generic feature of the solution of Einstein field equationsunder rather general energy conditions (e.g. strong, weak, dominant etc.),which is guaranteed by Penrose-Hawking singularity theorems [53–58]. Inthis paper, since we investigate the gravitational property of quantum vac-uum without modifying either QFT or GR, the appearance of singularitiesis inevitable—QFT predicts a huge vacuum energy, and according to GR,huge energy must collapse to form singularity.The Raychaudhuri equation (7.42) serves as a fundamental lemma forthe Penrose-Hawking singularity theorems. This lemma says that if thestrong energy condition is satisfied, the expansion θ, which is defined by(7.38), must satisfy the following inequality:1θ≥ 1θ0+τ3, (9.1)where θ0 is the initial value of θ. If θ0 is negative, then (9.1) implies that1/θ must pass through zero, i.e. θ → −∞, within a proper time τ ≤ 3/|θ0|.This usually signals an encounter with a curvature singularity (although notnecessarily) and in our case it is indeed a singularity.699.2. Resolving singularity by multiplying aIt is usually thought that the Einstein field equations break down at sin-gularities and thus the spacetime evolution will stop once the singularity isformed. However, it is not the case for our solution to the key dynamic evo-lution equation (4.19), which describes the oscillating motion of a harmonicoscillator. It is natural for a harmonic oscillator to pass its equilibriumpoint a(t,x) = 0 at maximum speed without stopping. From the calcula-tion (7.47) we know that θ is just 3a˙/a that as a → 0+, θ → −∞ and astime goes on, a quickly passes 0 and θ jumps discontinuously from −∞ to+∞ and the spacetime evolution start again. In this process, the metric ais still continuous, although the expansion θ is not.So in our solution, the singularity immediately disappears after it formsand the spacetime continues to evolve without stopping. Singularities arenot endings but new beginnings. They serve as the turning points at whichthe space switches from contraction phase to expansion phase.9.2 Resolving singularity by multiplying aIn order to understand better why in our solution the singularity is not theend of spacetime evolution, it is helpful to review one crucial step in deriving(4.19) from (4.18). Rigorously speaking, we can only obtain the followingequation from (4.18):− a¨a= Ω2(t,x), (9.2)which is not equivalent to (4.19). To get (4.19), we need one more step—multiply both sides of (9.2) by a.Mathematically, a is not allowed to be zero in (9.2) since it is in thedenominator. In fact, when writing down the Einstein field equations (4.2),(4.3), (4.4) and (4.5), it has been presumed that a 6= 0 since if a = 0, themetric would become degenerate (g = det(gµν) = −a6 = 0), the curvaturewould become infinite and the Einstein tensor are simply not defined there.But, after the inequivalent algebraic manipulation of multiplying bothsides of (9.2) by a, a is allowed to evolve to zero in the resulting equation(4.19) since there is nothing wrong for a harmonic oscillator to go throughits equilibrium point. In this sense, we have smoothly extended the solu-tion beyond the singularity by the mathematical operation of multiplyingboth sides of (9.2) by a (or more generally by some power of the metricdeterminant).The idea of resolving a singularity by mulptiplying Einstein equationswith some power of the determinant of the metric is not new. Einstein him-709.3. Singularities do not cause problemsself had proposed this idea with his collaborator Rosen in 1935 (for whichthey credited this idea to Mayer) [59]. Ashtekar used a similar trick in hismethod of “new variables” to develop an equivalent Hamiltonian formula-tion of GR [60]. It is also proposed by Stoica that the equations obtainedafter multiplying the usual Einstein equations by some power of the metricdeterminant are actually more fundamental than the usual Einstein equa-tions [61–69]. In this sense, we argue that our spacetime with singularitiesdue to the metric becoming degenerate (a = 0) is a legitimate solution ofGR.9.3 Singularities do not cause problemsWhile singularities are natural and inevitable in solutions to Einstein’s equa-tions, we must discuss the consequences they bring to this calculation.Will the singularities cause serious problems? At least in our case we donot feel they cause problems.To see this, we investigate how the singularities affect the propagationof the field modes in our toy model (6.12).In this toy model, the spacetime have singularities at the hypersurfacesΘ = Ωt+ K · x = (n+ 12)pi, n = 0,±1,±2,±3, · · · (9.3)Using the relation xm = −x−m, which is evident from (6.28) and (6.29), theasymptotic mode solution (6.38) becomesuk(t,x) = c0e−i(ωt−k·x)(1 + 2i+∞∑m=1xm sin 2mΘ). (9.4)At the singularities (9.3), the correction terms sin 2mΘ of (9.4) are all zeroand thus we haveuk(t,x) = c0e−i(ωt−k·x) (9.5)So the value of uk is normal at singularities.However, the stress energy tensor (4.22) for the field φ, which sources theEinstein field equations (4.2), (4.3), (4.4) and (4.5), mainly contains time719.3. Singularities do not cause problemsand spatial derivatives of φ:3(a˙a)2+1a2(∇aa)2− 2a2(∇2aa)= 4piG(φ˙2 +1a2(∇φ)2)(9.6)−2aa¨− a˙2 −(∇aa)2+∇2aa+ 2(∂iaa)2− ∂2i aa= 8piG((∂iφ)2 +12(a2φ˙2 − (∇φ)2)), (9.7)2a˙a∂iaa− 2∂ia˙a= 8piGφ˙∂iφ, (9.8)2∂iaa∂jaa− ∂i∂jaa= 8piG∂iφ∂jφ, i, j = 1, 2, 3, i 6= j. (9.9)Especially, the Ω2 in our key dynamic equation (4.19) is (see Eq.(4.24))Ω2 =8piG3φ˙2. (9.10)Therefore, one has to investigate how the derivatives of φ behave at thesingularities to assess the influence of the singularities on our calculations.To do this, we plot the correction functionf(Θ) =+∞∑m=1xm sin 2mΘ (9.11)to the mode solution uk defined by (9.4) and its derivative around the sin-gularity at Θ = pi/2. The sum in the plots are from m = 1 to m = 20000.The result are shown in FIG. 9.1 and FIG. 9.2.It can be seen from FIG. 9.1 and FIG. 9.2 that the derivative of f(Θ)is almost constant except suddenly goes to infinity when approaching thespacetime singularity at Θ = pi/2. Therefore, the property of Ω2 does notchange beyond the singularity. It suddenly goes to infinity as a→ 0.This sudden change at the singularity should not alter the dynamics ofour key equation (4.19) for the two following reasons: i) a passes through0 within an extremely short period ∆t since a harmonic oscillator reachesmaximum speed at the equilibrium point. So the change in the momentumof the oscillator due to the singularity ∆P = F∆t = −Ω2a∆t as a→ 0 willbe small (Ω2 → +∞ but a→ 0 so F does not necessarily blow up); ii) FromFIG. 9.1 we see that the correction to Ω2 around a = 0 is symmetric thatthe effect of F should be canceled that ∆P → 0.729.3. Singularities do not cause problems1.50 1.52 1.54 1.56 1.58 1.60 1.62 1.64-2-1012Figure 9.1: Plot of the correction function f(Θ) =+∞∑m=1xm sin 2mΘ aroundthe spacetime singularity at Θ = Ωt + K · k = pi/2, where the sum in theplot is from m = 1 to m = 20000.1.50 1.52 1.54 1.56 1.58 1.60 1.62 1.64-10 000010 00020 00030 00040 000Figure 9.2: Plot of the derivative of the correction function df(Θ)dΘ =+∞∑m=12mxm cos 2mΘ around the spacetime singularity at Θ = Ωt+K·k = pi/2,where the sum in the plot is from m = 1 to m = 20000.739.3. Singularities do not cause problemsSo we have argued that a goes through 0 is not a problem for the keyequation (4.19) we used. But is it a problem for other Einstein equations(9.6), (9.7), (9.8) and(9.9)? If we stick the solution (5.8) for a into theseequations, the left-hand side would blow up at a = 0, which requires thatthe time and spatial derivatives of φ on the right-hand side must also blowup (see eq. (9.8), (9.9)). As we can see from the plot of the correction to themode solution uk of φ in FIG. 9.1, φ˙ and ∇φ do blow up at a = 0, so oursolution (5.8) for a is also consistent with these other Einstein equations.The blow up of φ˙ at a = 0 seems invalidated our calculation at firstglance since we have implicitly assumed that Ω2 behaves normal all thetime to be able to vary slowly, which is the key property we used to solve(4.19). However, because of the two reasons we have argued, this should notalter the dynamics of (4.19).74Chapter 10A different model with alarge bare cosmologicalconstant (unsuccesful)In this chapter we introduce a different model with a large negative barecosmological constant. This model was motivated to cure the original modelpublished in [1] after the mistakes in the numerical calculation were found.In this model, instead of discarding the bare cosmological constant λb andtaking the high energy cutoff Λ to infinity, we keep the bare constant λband take it to negative infinity with Λ fixed. This model does have someadvantages over the original one, but unfortunately a fatal flaw was founded.We include this model here since even unsuccessful attempt could still bevaluable in scientific research.10.1 The formulation of the cosmologicalconstant problem is destroyed by densityfluctuations of quantum vacuumConsider a new model6 in which we keep the bare cosmological constant λb:ds2 = −dt2 + a2(t,x)(dx2 + dy2 + dz2). (10.1)6See chapter 4 for more details about this model.7510.1. The formulation of the cosmological constant problem is destroyed by density fluctuations of quantum vacuumG00 = 3(a˙a)2+1a2(∇aa)2− 2a2(∇2aa)= 8piGT00 + λb, (10.2)Gii = −2aa¨− a˙2 −(∇aa)2+∇2aa+ 2(∂iaa)2− ∂2i aa= 8piGTii − λba2, (10.3)G0i = 2a˙a∂iaa− 2∂ia˙a= 8piGT0i, (10.4)Gij = 2∂iaa∂jaa− ∂i∂jaa= 8piGTij , i, j = 1, 2, 3, i 6= j, (10.5)where ∇ = (∂1, ∂2, ∂3) is the ordinary gradient operator with respect to thespatial coordinates x, y, z.A linear combination of equations (10.2) and (10.3) gives,G00 +1a2(G11 +G22 +G33) = −6a¨a, (10.6)where all the spatial derivatives of a cancel and only the second order timederivative left. Therefore we reach the following dynamic evolution equationfor a(t,x):a¨+ Ω2(t,x)a = 0, (10.7)whereΩ2 =4piG3(ρ+3∑i=1Pi)− λb3, ρ = T00, Pi =1a2Tii. (10.8)(10.7) is just a generalization of the second Friedman equation (2.12). Inthe usual renormalization approach, λb is chosen very precisely to cancel theexpectation value of the first term ρ +∑3i=1 Pi in (10.8). Suppose that wehave successfully fine-tuned λb to an accuracy of 10−122 that the expectationvalue of the Ω2 in (10.8)〈Ω2〉 = −λeff/3 = −1.86× 10−122 (10.9)in Planck units. Because of the huge fluctuations in ρ+∑3i=1 Pi, Ω2 wouldhave large chances to fluctuate to both large negative and positive values.When Ω2 fluctuates to negative values, a roughly grows exponentiallya ∼ e|Ω|t ∼ e√GΛ2t ∼ eEP t  e10−61EP t, (10.10)where EP is Planck energy, the high energy cutoff Λ ∼ EP has been taken tobe Planck energy in the above estimation and 10−61 is the current order of7610.2. New relation between λeff and λbmagnitude of Hubble expansion rate in Planck units. When Ω2 fluctuates toa positive value, (10.7) becomes a harmonic oscillator with time dependentfrequency and parametric resonance 7 would happen that a would roughlygo asa ∼ eH˜tP, (10.11)where P is a quasiperiodic function oscillating around 0. In this case, Ω ∼√GΛ2 while the varying frequency of Ω itself is roughly Λ. As taking Λ ∼EP , the two frequency closes to each other that the parametric resonancewould be strong.Therefore, the density fluctuations would cause large deviations from(2.13) and the universe would still explode even one has successfully fine-tuned λb to an accuracy of 10−122. Moreover, the relation (2.7) which de-scribes the dependence of the observed effective cosmological constant λeffon the bare cosmological constant λb is destroyed by these fluctuations. Anew relation between λeff and λb is needed.10.2 New relation between λeff and λbSince the matter fields 8 in (10.8) go as4piG3(ρ+3∑i=1Pi)∼ (±)GΛ4, (10.12)the time dependent frequency Ω2 would average around〈Ω2〉 ∼ −λb3±GΛ4. (10.13)Something interesting happens if −λb is taken to the range −λb  Λ2 ≥GΛ4, assuming Λ ≤ EP (see FIG. 10.1). In this limit, i) the probabilitythat Ω2 fluctuates to negative values goes to zero; ii) the strength of theparametric resonance goes to zero. Thus the Hubble expansion rate H → 0and then the effective cosmological constant λeff = 3H2 → 0 as −λb → +∞.In the following we give an expression for the dependence of λeff on λb inthis regime.7See chapter 5.1 for more detailed explanation about the parametric resonance effect.8The sign of (10.12) is not known. In principle all known and unknown fundamentalfields in nature would contribute. Boson fields give positive signs to ρ but Fermion fieldsgive negative signs to ρ. Sine we do not have the knowledge about all of them, the signcan not be determined.7710.2. New relation between λeff and λb0Ω2P(Ω2 )-λb/3Λ20GΛ4|<>|Figure 10.1: An illustration of the probability density distribution P (Ω2).The shape of the graph is obtained from the histogram of a sample of Ω2(t)we used in the numerical simulation 10.3. This is for one boson and onefermion field. The tails fall as e−κ|Ω2+λb/3| (as shown in FIG. 10.2) for largeargument where κ is some constant.7810.2. New relation between λeff and λb0Ω2Log P(Ω2 )-λb/3Λ2Figure 10.2: Plot of log(Ω2)which shows the tail of P (Ω2) falls ase−κ|Ω2+λb/3| for large argument where κ is some constant.7910.2. New relation between λeff and λbDue to quantum fluctuation, we assume that Ω2 can possibly fluctuateto any values. Different values of Ω2 give different dynamics of the equa-tion(10.7). All possible values of Ω2 can be divided into two cases: i) Ω2 < 0or Ω ∼ Λ; ii) Ω Λ.Case I: Ω2 < 0 or Ω ∼ Λ. The probability that Ω2 fluctuates to thesevalues is exponentially suppressed, since ρ +3∑i=1Pi approximately obeyschi-square distributions. In fact, a free quantum field may be viewed as acollection of decoupled, time-independent, harmonic oscillators. At groundstate, each oscillator satisfies Gaussian distribution, and the sum of Gaussiandistributions are still Gaussian distributions. Since ρ +3∑i=1Pi is in generalcontains squares of the time and spatial derivatives of the fields, it wouldapproximately obey chi-square distributions. The probability density func-tion f(x) of a chi-square distribution roughly goes as ∼ e−κx as x large (asshown in FIG. 10.2), where κ is some constant. Therefore, the contributionto the growth of a from this case is proportional toH ∝ e−β˜(−λb3 − 4piG3 〈ρ+∑3i=1 Pi〉) ∝ eβ˜λb , as − λb  Λ2 ≥ GΛ4, (10.14)where we have absorbed the numerical factor 1/3 into the constant β˜. FIG.10.4 shows that β˜ = 6.09 for the matter fields used in the numerical simu-lation 10.3.Case II: Ω Λ. In this case, since Ω itself varies on the time scale 1/Λ,which is much longer than the time scale 1/Ω of the oscillation of a, thisprocess is basically adiabatic 9. It has been well-established that the errorin adiabatic invariant is exponentially small [22, 24]. Thus we would expectthat in this case the strength of the parametric resonance would also beexponentially suppressed. To obtain more detailed estimation, we only needto follow exactly the same steps of Chapter 5.3 with Ω ∼ √GΛ2 replaced byΩ ∼√4piG3 〈ρ+∑3i=1 Pi〉 − λb/3 ∼√±GΛ4 − λb/3. Then the contributionto the growth of a from this case is proportional toH ∝ Λe−β√±GΛ4−λb/3Λ ∼ Λe−β√−λbΛ , as − λb  Λ2 ≥ GΛ4, (10.15)where we have absorbed the numerical factor into β.9See chapter 5.2 for detailed analysis about this process.8010.3. Numerical simulationWhen |λb| is large, the contribution to H from (10.14) drops faster thanthe contribution from (10.15). So (10.15) is dominant:H = αΛe−β√−λbΛ , as − λb  Λ2 ≥ GΛ4, (10.16)where α, β > 0 are two dimensionless constants. This gives a differentrelation between the observed effective cosmological constant λeff and thebare cosmological constant λb:λeff = 3H2 = α2Λ2e−2β√−λbΛ , as − λb  Λ2 ≥ GΛ4. (10.17)10.3 Numerical simulationWe follow the same numerical method as described in chapter 5.7 and ap-pendix B. The difference is that the source of gravity is taken to be oneBoson field and one Fermion field. The contribution to 〈Ω2〉 from the Bosonfield is set to be GΛ4/6pi and from the Fermion field is set to be −GΛ4/6pi,i.e. we set 〈ρ+∑3i=1 Pi〉 = 0 in the simulation.The numerical simulation for the regime |λb| is small is shown in FIG.10.3 and FIG. 10.4. FIG. 10.3 shows that the Hubble expansion rate Hdecreases as −λb increases. FIG. 10.4 shows that this decrease is exponentialas predicted by (10.14) and the parameter β˜ there is around 6 in this setting.Note that the red line (λb = 0) in FIG 10.3 represents the case 〈ρvac〉 =0 (or λeff has been tuned to zero in the usual sense). It clearly showsthat the universe is exploding instead of stay still because of the densityfluctuation of quantum vacuum, i.e. even if one has successfully fine-tunedthe cosmological constant to the usually required accuracy of 10−122, theproblem of rapid exponential expansion of the universe driven by the largevacuum energy is still not resolved.The numerical simulation for the regime |λb| is large is shown in FIG.10.5 and FIG. 10.6. FIG. 10.5 shows that the Hubble expansion rate Hdecreases as −λb increases. FIG. 10.6 shows that this decrease is exponentialas predicted by (10.16) and the parameters α ∼ e18, β ∼ 14 in this setting.The error bar for the slope of the lines in FIG. 10.5 is big that the fittingparameters α and β obtained in FIG. 10.6 may not be accurate. Theseparameters would also depend on the matter fields we used. But FIG. 10.6at least gives an estimation that α is somewhere between e10 to e20 and βis somewhere between 10 to 20. These values are not accurate but it is notimportant for our model to work as explained in the next subsection.8110.3. Numerical simulation0 1000 2000 3000 4000 5000Time0100200300400500600log(|a|)λb=0.00λb=-0.03λb=-0.06λb=-0.09λb=-0.12λb=-0.15λb=-0.18Figure 10.3: Numerical result for the dependence of log |a| on the barecosmological constant λb as |λb| is small. The cutoff Λ = 1. 100 samplesare averaged for each line. Planck units are used for convenience. Thematter fields are one Boson field and one Fermion field. The magnitude of〈ρ+∑3i=1 Pi〉 for both fields are set equal but with opposite sign, i.e. we set〈ρ +∑3i=1 Pi〉 = 0 in the simulation. It shows that the Hubble expansionrate decreases as −λb increases.8210.3. Numerical simulation0 0.05 0.1 0.15 0.2|λb|-3.5-3-2.5-2log(H)log(H)=-6.09|λb|-2.21Figure 10.4: Plot of logH over |λb| when |λb| is small. The fitting resultshows that β˜ ∼ 6. Planck units are used for convenience.8310.3. Numerical simulation0 10000 20000 30000 40000 50000Time020406080log(|a|)λb=-3λb=-3.075λb=-3.15λb=-3.225λb=-3.3Figure 10.5: Numerical result for the dependence of log |a| on the barecosmological constant λb as |λb| is large. The cutoff Λ = 1. 400 samplesare averaged for each line. Planck units are used for convenience. Thematter fields are one Boson field and one Fermion field. The magnitude of〈ρ+∑3i=1 Pi〉 for both fields are set equal but with opposite sign, i.e. we set〈ρ +∑3i=1 Pi〉 = 0 in the simulation. It shows that the Hubble expansionrate decreases as −λb increases. For larger −λb the slope of log |a| grows tooslow that the numerical rounding errors seem to dominant.8410.3. Numerical simulation1.75 1.8|λb|1/2/Λ-8-7.5-7-6.5log(H)log(H)=-14.5(|λb|1/2/Λ)+18.5Figure 10.6: Plot of logH over√|λb| when |λb| is large. The fitting resultshows that α ∼ e18, β ∼ 14. Planck units are used for convenience. Thecutoff Λ = 1.8510.4. Meaning of the results10.4 Meaning of the resultsIn this section we use Planck units for convenience.The usual relation between λeff and λb which missed the important effectfrom the large density fluctuation of quantum vacuum is given byλeff = λb + 8piρvac, (10.18)Since ρvac (∼ 1 if take the cutoff Λ = 1) is larger than λeff = 5.6 × 10−122(Planck units is used here) by 50 to 122 orders of magnitude depending onthe cutoff energy scale Λ, λb has to be extremely fine-tuned to a precisionof at least 50 decimal places.However, once the effect of the density fluctuation is included, the quan-tum vacuum gravitates in a different way that the relation between λeff andλb becomes the equation (10.17). Rewrite (10.17) as− Λ2βlog(λeff) =√−λb − Λ2βlog(α2Λ). (10.19)The numerical simulation in 10.3 gives an estimation that α is somewherebetween e10 to e20 and β is somewhere between 10 to 20, for one Boson fieldand one Fermion field. If we take Λ = 1, i.e. the Planck energy, we wouldhave− Λ2βlog(λeff) ∼ 10, Λ2βlog(α2Λ) ∼ 1. (10.20)In this case, since the term − Λ2β log(λeff) is only different from the termΛ2β log(α2Λ) by 1 order of magnitude, the term√−λb only need to be tunedto an accuracy of 10−1 or λb only need to be tuned to an accuracy of 10−2to satisfy (10.19).In general, the difference in the order of magnitude between the twoterms − Λ2β log(λeff) and Λ2β log(α2Λ) in (10.19) is mainly determined by thevalue of α. α cannot be precisely determined because of the limitation ofnumerical simulation and the lack of the knowledge about the contribution toρ+∑3i=1 Pi from all fundamental fields in nature. But this does not matter.Even if α could take values in the range from 1 to e100, we would have log(α2)in the range from 0 to 200. This is different from the term − log(λeff) = 279by at most the order of 102. Then√−λb at most needs to be tuned toan accuracy of 10−2 and λb at most needs to be tuned to an accuracy of10−4. Therefore, the extreme fine-tuning of the bare cosmological constantto match the observation is no long needed.8610.5. Problem of this model10.5 Problem of this modelIn this model, we still presumed that the inhomogeneous “spatially flat”FLRW metric of the form (10.1). Unfortunately, it turns out that Einsteinequations with a large negative bare cosmological constant cannot be fittedin this metric.In fact, since we require that −λb  Λ2 ≥ GΛ4, i.e. the cosmologi-cal constant is dominant over the zero point fluctuations. Therefore, thesolution to the Einstein equations must be a small perturbation from thesolution of vacuum Einstein equations in which the only term in the stress-energy tensor is a negative cosmological constant term, which is well knownto be the anti-de Sitter space.Let us derive the solution in detail. We start with the Einstein fieldequations without matter fields but with a bare cosmological constant:Gµν + λbgµν = 0. (10.21)Then we have the homogeneous FLRW metricds2 = −dt2 + a2(t)dΣ2, (10.22)wheredΣ2 =11− kr2dr2 + r2(dθ2 + sin θ2dϕ2), (10.23)where k is a constant representing the curvature of the space.The Einstein equations areG00 = 3(a˙a)2+3ka2= λb, (10.24)G11 =11− kr2(−2aa¨− a˙2 − k) = − λba21− kr2 , (10.25)G22 = r2(−2aa¨− a˙2 − k) = −r2λba2, (10.26)G33 = r2 sin2 θ(−2aa¨− a˙2 − k) = −r2 sin2 θλba2, (10.27)G0i = Gij = 0, i, j = 1, 2, 3 = r, θ, ϕ, i 6= j. (10.28)Since λb is negative, k must also be negative according to (10.24).Equations (10.25), (10.26) and (10.27) are the same. A Linear combina-tion of (10.24), (10.25), (10.26) and (10.27) givesG00 +3∑i=1giiGii = −6a¨a= −2λb. (10.29)8710.5. Problem of this modelTherefore, we havea¨− λb3a = 0. (10.30)For a negative bare cosmological constant λb, the solution to (10.30) underthe constraint equation (10.24) area(t) =√3kλbcos(√−λb3t+ γ), (10.31)where λb, k < 0, γ is an arbitrary phase constant. This gives the solution tothe full Einstein equations (10.24), (10.25), (10.26), (10.27) and (10.28).Next let us add fluctuating matter fields to the Einstein equations:Gµν + λbgµν = 8piGTµν . (10.32)Assuming the metric takes the following form to account for the inhomo-geneities produced by quantum fluctuations:ds2 = −dt2 + a2(t, r, θ, ϕ)dΣ2, (10.33)Then the Einstein equations becomesG00 = 3(a˙a)2+3ka2+4kr∂raa3+(∇a)2a4− 2∇2aa3= λb + 8piGT00, (10.34)G11 =11− kr2(− 2aa¨− a˙2 − k − (∇a)2a2+∇2aa+ 2(1− kr2)(∂ra)2a2−(1− kr2)∂2raa− kr∂raa)= − λba21− kr2 + 8piGT11, (10.35)G22 = r2(−2aa¨− a˙2 − k − (∇a)2a2+∇2aa+2(∂θa)2r2a2− ∂2θar2a− (1r+ kr)∂raa)= −r2λba2 + 8piGT22, (10.36)G33 = r2 sin2 θ(− 2aa¨− a˙2 − k − (∇a)2a2+∇2aa+2(∂ϕa)2r2 sin2 θa2− ∂2ϕar2 sin2 θa− ∂θar2 tan θa− ∂rara− kr∂raa)= −r2 sin2 θλba2 + 8piGT33, (10.37)8810.5. Problem of this modelG0i = −2∂i(a˙a)= 8piGT0i, i = 1, 2, 3 = r, θ, ϕ, (10.38)G12 =∂θara+2∂θa∂raa2− ∂r∂θaa= 8piGT12, (10.39)G13 =∂ϕara+2∂ϕa∂raa2− ∂r∂ϕaa= 8piGT13, (10.40)G23 =∂ϕatan θa+2∂ϕa∂θaa2− ∂θ∂ϕaa= 8piGT23, (10.41)where(∇a)2 = g˜ij∂ia∂ja= (1− kr2)(∂ra)2 + 1r2(∂θa)2 +1r2 sin2 θ(∂ϕa)2, (10.42)∇2a = 1√|g˜|∂i(√|g˜|g˜ij∂ja)= (1− kr2)∂2ra+(2r− kr)∂ra+1r2∂2θa+1r2 tan θ∂θa+1r2 sin2 θ∂2ϕa, (10.43)and g˜ is the metric components of dΣ2 defined by (10.23). The linear com-bination like (10.29) also givesG00 +3∑i=1giiGii = −6a¨a(10.44)= −2λb + 8piG(ρ+3∑i=1Pi),where all the spatial derivatives of a cancel and ρ = T00, Pi = giiTii (nosummation for i here). Therefore we obtaina¨+ Ω2(t, r, θ, ϕ)a = 0, (10.45)whereΩ2 =4piG3(ρ+3∑i=1Pi)− λb3. (10.46)Since we have −λb  Λ2 ≥ GΛ4 ∼ Tµν (assuming Λ ≤ EP ), the solutionto the new inhomogeneous Einstein equations would just be a small pertur-bation around (10.31), which can be obtained by WKB approximation and8910.5. Problem of this modelthe weak parametric resonance effect described in section 5.1:a(t, r, θ, ϕ) = e∫ t0 H(t′,r,θ,ϕ)dt′√3kΩ(0, r, θ, ϕ)λbΩ(t, r, θ, ϕ)· cos(∫ t0Ω(t′, r, θ, ϕ)dt′ + γ(r, θ, ϕ)). (10.47)The phase γ(r, θ, ϕ) is different from point to point. In fact, plugging (10.47)into (10.38) givesΩ(0, r, θ, ϕ) tan γ(r, θ, ϕ) = Ω(0, r0, θ0, ϕ0) tan γ(r0, θ0, ϕ0)+ 4piG∫ r,θ,ϕr0,θ0,ϕ0J(0, r′, θ′, ϕ′) · dl′,(10.48)where J = (T01, T02, T03) is the energy flux and we have neglected the smallexponential factor and the relatively small time derivative terms of the slowlyvarying frequency Ω in the calculation.Solution (10.47) for a looks good but unfortunately the large negativeλb actually requires large negative k, which means that the spatial curva-tures have to become large everywhere. Then the spatial slice would havehyperboloid geometry with large curvature, which can not be average out toget flat space. This destroys the argument that the spatial averaging doneby long wavelength fields would produce a flat spatial spacetime which hasbeen shown in chapter 6. For this reason, this model does not work well.90Chapter 11ConclusionsStarting from two fundamental principles of modern physics — the uncer-tainty principle of quantum mechanics and the equivalence principle of gen-eral relativity, we have shown that quantum vacuum would gravitate in acompletely different way from what people previously thought. The gravi-tational effect produced by the huge vacuum stress energy is still huge, butconfined to Planck scales. At each Planck size region, the spacetime oscil-lates between expansion and contraction. As it swings back and forth, thetwo almost cancel each other but the expansion wins out a little bit.This physical picture might look crazy at first glance, but it is just theprediction of applying quantum mechanics and general relativity togetherto the quantum vacuum. 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CambridgeUniversity Press, 1995.97Appendix AReal Massless Scalar FieldIn this appendix we give the calculation details about how the quantumvacuum fluctuates all over the spacetime by using the massless scalar field(3.1) as an example.We first define the covariance of the energy density operator at twospacetime points x = (t,x) and x′ = (t′,x′)Cov(T00(x), T00(x′))= 〈{(T00(x)− 〈T00(x)〉 )(T00(x′)− 〈T00(x′)〉 )}〉, (A.1)where the curly bracket {} in (A.1) is the symmetrization operator which isdefined as, for any two operators A and B,{AB} = 12(AB +BA) . (A.2)Inserting (3.1) and (3.4) into (A.1) gives the following resultCov(T00(x), T00(x′))=12∫d3kd3k′(2pi)6(ωω′ + k · k′)22ω2ω′· cos((ω + ω′)∆t− (k + k′) ·∆x),(A.3)where ∆t = t − t′ and ∆x = x− x′ are time and space separation of thetwo spacetime points x and x′.If x and x′ are timelikely separated, we can find a reference frame to set∆x = |∆x| = 0. In this case, evaluation of the integral in (A.3) for a highfrequency cutoff |k| = Λ givesCov(T00(x), T00(x′))(A.4)=124pi4∆t8([−(Λ∆t)6 + 21(Λ∆t)4 − 72(Λ∆t)2 + 36] cos(2Λ∆t)+6[(Λ∆t)5 − 8(Λ∆t)3 + 12Λ∆t] sin(2Λ∆t)+12[(Λ∆t)3 − 6Λ∆t] sin(Λ∆t) + 36 [(Λ∆t)2 − 2] cos(Λ∆t) + 36).98Appendix A. Real Massless Scalar FieldIf x and x′ are spacelikely separated, we can find a reference frame to set∆t = 0. In this case, evaluation of the integral in (A.11) for a high frequencycutoff |k| = Λ givesCov(T00(x), T00(x′))=132pi4∆x8([2(Λ∆x)4 − 34(Λ∆x)2 + 33] cos(2Λ∆x)− [12(Λ∆x)3 − 50Λ∆x] sin(2Λ∆x)+16[(Λ∆x)2 − 6] cos(Λ∆x)− 64Λ∆x sin(Λ∆x) + 63) (A.5)As ∆t and ∆x goes to 0, both (A.4) and (A.5) reduces to the varianceof the energy density,〈(T00 − 〈T00〉)2〉=23(Λ416pi2)2=23〈T00〉2 (A.6)We then investigate the Pearson product-moment correlation coefficientρx,x′ =Cov(T00(x), T00(x′))σxσx′, (A.7)whereσx =√〈(T00(x)− 〈T00(x)〉)2〉. (A.8)The correlation coefficient ρx,x′ shows by its magnitude the strength ofcorrelation between two random variables. ρx,x′ is positive if the energy den-sity T00 at x and x′ are most possibly lying on the same side of the vacuumexpectation value 〈T00〉 = Λ4/(16pi2). Thus a positive correlation coeffi-cient ρx,x′ implies the energy density at x and x′ tend to be simultaneouslygreater than, or simultaneously less than the expectation value. Similarly,a negative ρx,x′ implies the energy density tend to lie on opposite sides ofthe expectation value. We will call the energy density T00 at x and x′ arepositively correlated if ρx,x′ > 0 or negatively correlated (anticorrelation) ifρx,x′ < 0.Because of transnational invariance, ρx,x′ is only dependent on the tem-poral and spatial separation ∆t = t− t′,∆x = x− x′. For the real masslessscalar field (3.1), the behavior of the correlation coefficient ρx,x′ as a func-tion of temporal separation Λ∆t for the case of ∆x = 0 and as a function ofspatial separation Λ∆x for the case of ∆t = 0 are plotted in FIG. A.1 andA.2 respectively.99Appendix A. Real Massless Scalar Field2 4 6 8 10-1.0-0.50.00.51.0LDtΡxx¢Figure A.1: Plot of correlation coefficient ρx,x′ as a function of time sepa-ration Λ∆t in the case ∆x = 0.2 4 6 8 10-0.20.00.20.40.60.81.0LDxΡxx¢Figure A.2: Plot of correlation coefficient ρx,x′ as a function of spatialseparation Λ∆x in the case ∆t = 0.100Appendix A. Real Massless Scalar FieldIn the temporal direction, i.e. the case of ∆x = 0 (Fig. A.1), thecorrelation coefficient goes quickly from 1 down to around −0.9 in a timescale around ∆t = 1.9/Λ and then goes up to 0.7 in a time scale around∆t = 3.8/Λ and then goes down and up alternatively from positive val-ues to negative values with decreasing amplitudes. It roughly oscillates as− cos(2Λ∆t)/(Λ∆t)2 with a period pi/Λ as ∆t is large. Thus at the ex-tremely small time scales ∆t ∼ 1.9/Λ, (Λ → +∞), the energy density arestrongly anticorrelated. In other words, if at some time the value of the en-ergy density is larger than its expectation value, for example, by an amountof 0.82 〈T00〉, after a short time ∆t = 1.9/Λ, its value is most likely to besmaller than the expectation value, for example, by an amount of 0.74 〈T00〉.The difference is 1.56 〈T00〉 only after such a short time.In the spatial direction, i.e. the case of ∆t = 0 (Fig. A.2), the correla-tion coefficient goes quickly from 1 down to around −0.14 in a length scalearound ∆x = 3.24/Λ and then goes up to 0.03 in a length scale around∆x = 5.4/Λ and then goes down and up alternatively from positive valuesto negative values with decreasing amplitudes. Compared to the temporaldirection, the decay in the oscillation amplitude of the correlation coefficientis faster in spatial direction. It roughly oscillates as 2 cos(2Λ∆x)/(Λ∆x)4with a period pi/Λ as ∆x is large. These properties show that the strengthof the correlation between energy densities at close range in spatial directionis not as strong as in the temporal direction. For larger spatial separations,ρx,x′ approaches zero and the vacuum energy density T00 at different x andx′ fluctuate independently. These properties result in extreme spatial in-homogeneities of the quantum vacuum which can be characterized by thequantity ∆ρ2 defined by (3.8) in chapter 3.The quantity ∆ρ2 is related to ρx,x′ by∆ρ2 = 1− ρx,x′ . (A.9)The behavior of ∆ρ2 has been plotted in FIG. 3.1.Next we calculate the χ(∆t) defined by (5.3) in section 5.1. Wick ex-pansion of (5.3) givesχ(∆t) =〈φ˙(t1,x)φ˙(t2,x)〉2+〈φ˙(t2,x)φ˙(t1,x)〉22〈φ˙2(t,x)〉2 , (A.10)where the correlation function can be calculated directly by inserting (3.1)〈φ˙(t1,x)φ˙(t2,x)〉=14pi2∫ Λ0k3e−ik∆tdk. (A.11)101Appendix A. Real Massless Scalar FieldPlugging (A.11) into (A.10) gives the following resultχ(∆t) =16Λ8∆t8(36(−2 + Λ2∆t2) cos(Λ∆t)+(36− 72Λ2∆t2 + 21Λ4∆t4 − Λ6∆t6) cos(2Λ∆t)+ 6(6 + 2Λ∆t(−6 + Λ2∆t2) sin(Λ∆t)+ Λ∆t(12− 8Λ2∆t2 + Λ4∆t4) sin(2Λ∆t))). (A.12)The behavior of χ(∆t) has been plotted in FIG. 5.1. It is closely relatedto the correlation coefficient ρx,x′ as a function of time difference ∆t in thecase ∆x = 0 (FIG. A.1).Next we derive the equation (5.29) in section 5.1. First, Ω2(t,0) can beexpanded asΩ2(t,0) =8piG3∫ω,ω′≤Λd3kd3k′(2pi)3√ωω′2(A.13)·[(aka†k′ + a†kak′)cos(ω − ω′) t+ i(−aka†k′ + a†kak′) sin (ω − ω′) t+(−akak′ − a†ka†k′)cos(ω + ω′)t+ i(akak′ − a†ka†k′)sin(ω + ω′)t].Specially, the vacuum state |0〉 is an eigenstate of the operator coefficients ofthe first two terms in the above expression (A.13). If k 6= k′, the eigenvaluesof the operator coefficients of the first two terms are zero. Thus in this case,the first two terms have to both take zero values. If k = k′, the second termis zero since in this case ω = ω′ and thus the factor sin(ω − ω′)t = 0. Soonly the first term survives and gives the expectation value of Ω2(t,0):Ω20 =〈Ω2〉=8piG3∫ω≤Λd3k(2pi)3ω2=GΛ46pi. (A.14)For the operator coefficients of the last two terms in the expression(A.13), the vacuum state |0〉 is not an eigenstate. So the last two termsare constantly fluctuating, and the time varying of Ω2 comes from these twoterms.After some algebraic manipulations, (A.13) can be rewritten as the formof (5.29) for the vacuum state |0〉, wheref(γ)dγ =16pi2Λ4∫γ≤ω+ω′≤γ+dγd3kd3k′(2pi)3√ωω′2(−akak′ − a†ka†k′), (A.15)102Appendix A. Real Massless Scalar Fieldg(γ)dγ =16pi2Λ4∫γ≤ω+ω′≤γ+dγd3kd3k′(2pi)3√ωω′2i(akak′ − a†ka†k′). (A.16)Evaluating the above integrals gives the expectation values〈f(γ)dγ〉 = 〈g(γ)dγ〉 = 0, (A.17)and their fluctuations〈(f(γ)dγ)2〉=〈(g(γ)dγ)2〉(A.18)={435( γΛ)7 dγ2Λ , if 0 ≤ γ ≥ Λ,− 435(40− 140 γΛ + 168( γΛ)2 − 70 ( γΛ)3 + ( γΛ)7) dγ2Λ , if Λ ≤ γ ≥ 2Λ.The above expression (A.18) gives the power spectrum density of the varyingpart of Ω2(t,0) (except the constant Ω20 part), which has been plotted in FIG.5.2.103Appendix BWigner-Weyl Description ofQuantum Mechanics andNumeric simulationsThis chapter explain the principle of the numeric calculations in the maintext. Same as the numeric part in the main text, we set G = 1 in this chap-ter. Wigner functions and Weyl transforms of operators offer a formulationof quantum mechanics that is equivalent to the standard approach givenby the Schro¨dinger equation. The Wigner distribution function is a quasidistribution function in the phase space. For a particular quantum wavefunction ψ(x), its Wigner function is defined asW (x, p) =∫dye−ipyψ(x+y2)ψ∗(x− y2) (B.1)The Weyl transform of an quantum operator Aˆ is defined asA(x, p) =∫dye−ipy〈x+ y2|Aˆ|x− y2〉 (B.2)Then the expectation value of the operator Aˆ under the state ψ(x) can bewritten as〈Aˆ〉 =∫ ∫dxdpW (x, p)A(x, p) (B.3)These two transformations give the Wigner-Weyl discription for quantummechanics. The expectation values of physical quantities are obtained byaveraging their Weyl transforms over phase space.For a harmonic oscillator with frequency ω and m = 1, the ground stateWigner function is a Gaussian distribution function for both x and pW0(x, p) =1pie−p2ω−x2ω (B.4)We can easily check that the Weyl transform of an operator H(xˆ) ( or H(pˆ))is simply replaced the operator xˆ by x (or pˆ by p). Other than that, another104Appendix B. Wigner-Weyl Description of Quantum Mechanics and Numeric simulationsparticular transform we are going to use in this write-up isxˆpˆ→ xp+ i2; pˆxˆ→ xp− i2(B.5)We can see that the transform of the product does not necessarily equal tothe product of transforms. In the following part we are going to get thegeneral expression for the transform of the product.Before that we notice that Weyl transform can be used to construct theoriginal operator , i.e.〈x|Aˆ|y〉 = 12pi∫dpA(x+ y2, p)eip(x−y) (B.6)Using this formula we can construct the transform of product of two states:∫dy〈x+ y2|AˆBˆ|x− y2〉e−ipy=∫dzdy〈x+ y2|Aˆ|z〉〈z|Bˆ|x− y2〉e−ipy=14pi2∫dzdydp1dp2eip1(x+y2−z)e−ip2(x−y2−z)e−ipy·A(x+ y/2 + z2, p1)B(x− y/2 + z2, p2)=14pi2∫dz1dz2dp1dp2eiz1(p2−p)eiz2(p−p1)·A(x+ z12, p1)B(x+z22, p2)(B.7)Here we definez1 =y2+ z − x (B.8)z2 = −y2+ z − x (B.9)We Taylor-expand A(x+ z12 , p1) and B(x+z22 , p2) around x and haveA(x+z12, p1) =∞∑n=01n!A(n)(x, p1)(z1/2)n (B.10)B(x+z22, p2) =∞∑n=01n!B(n)(x, p2)(z2/2)n (B.11)105Appendix B. Wigner-Weyl Description of Quantum Mechanics and Numeric simulationsand use the facts12pi∫dxxneixy = (−i)nδ(n)(y) (B.12)and ∫dyδ(n)(y)f(y) = (−1)nf (n)(0) (B.13)Therefore, we can write the Weyl transform of operator AˆBˆ as∑n,min(−i)m2n+mn!m!A(n,m)(x, p)B(m,n)(x, p) (B.14)The generalized FRW scale factor a satisfies the equationa¨+ Ω2(t)a = 0 (B.15)in whichΩ2(t) =8pi3φ˙2(t) (B.16)Now we replace all the quantities by operators, assuming that operators stillsatisfy the previous equation¨ˆa+ Ωˆ(t)2aˆ = 0 (B.17)withΩˆ2(t) =8pi3˙ˆφ2(t) (B.18)For a massless real scalar field, we can write it asφˆ =∫d3k(2pi)3/2(xˆk cos(ωkt) +1ωkpˆk sin(ωkt)) (B.19)in whichxˆk =√12ωk(b†k + bk) (B.20)pˆk = i√ωk2(b†k − bk) (B.21)are the generalized xˆ pˆ operators for each field modes.We can write the Weyl transformation of the Ωˆ(t)Ω({xk}, {pk}, t)2 = 8pi3∫∫d3kd3k′(2pi)3xkxk′ωkωk′ sinωkt sinωk′t+ pkpk′ cosωkt cosωk′t− 2xkpk′ωk sinωkt cosωk′t(B.22)106Appendix B. Wigner-Weyl Description of Quantum Mechanics and Numeric simulationsThis expression is quadratic in xk and pk, so if we apply it to (B.14), onlym + n ≤ 2 terms survive. Assuming a({xk}, {pk}, t) is the Weyl transformof operator aˆ, we have the equation for a asa¨+ Ω2a+i2∑k(∂Ω2∂xk∂a∂pk− ∂Ω2∂pk∂a∂xk)(B.23)−18∑k,k′(∂2Ω2∂xk∂xk′∂2a∂pk∂pk′+∂2Ω2∂pk∂pk′∂2a∂xk∂xk′− 2 ∂2Ω2∂xk∂pk′∂2a∂pk∂xk′)= 0The observed value a is the average over Wigner function W ({xk}, {pk}, t)ao(t) =∫ (∏kdxkdpk)a({xk}, {pk}, t)W ({xk}, {pk}, t) (B.24)If the quantum field is in the ground state, then by (B.6)W ({xk}, {pk}, t) =∏k1pie− p2kωk−x2kωk (B.25)Local approximation Generally the equation (B.23) depends on not onlythe value of Ω and a on a particular phase space point (x, p), but also on theneighboring values (i.e. derivatives). If our solution a is ”smooth” enoughin the phase space then we can neglect the last two derivative terms in the(B.23). It can be simplified toa¨+ Ω2a = 0. (B.26)Assuming the length of the Universe is L. We can replace the integral bysummations. For simplicity, we definet˜ → 2pitL(B.27)x˜n →√2ω 2pinLx 2pinL(B.28)p˜n →√2ω 2pinLp 2pinL(B.29)The equation can be written asa¨+23L2Ω(t˜)2a = 0 (B.30)107Appendix B. Wigner-Weyl Description of Quantum Mechanics and Numeric simulationswithΩ(t˜)2 =∑~n, ~n′√nn′(x˜~nx˜~n′ sinnt˜ sinn′t˜+ p˜~np˜~n′ cosnt˜ cosn′t˜− x˜~np˜~n′ sinnt˜ cosn′t˜))=[∑~n√n(x˜~n sinnt˜− p˜~n cosnt˜)]2 (B.31)Here ~n = (n1, n2, n3), n1,2,3 ∈ Z and n = |~n|. {x˜~n} {p˜~n} are random Gaus-sian variables with unit standard deviation. We can solve the equation fora randomly generated set of {x˜~n} and {p˜~n}, and repeat. The result ao(t) isthe average over all solutions as long as our sample size is big enough.108Appendix CFourier transforms of thecoefficients in (6.42)In this appendix, we demonstrate the property of the spectrum of the co-efficients in (6.42) given by (6.52), (6.54) and (6.56). Observing that thecos 2Θ, sin 2Θ and tan Θ in (6.51), (6.53) and (6.55) respectively can all bedecomposed as Fourier series sum of the form ei2nΘ, where n = ±1,±2, · · · ,we only need to analyze the spectrum of ei2nΘ.For simplicity, we only analyze the time component Fourier transformof ei2nΘ. The spatial part has similar property. The phase angle Θ isdetermined by Ω through (6.9) while Ω is determined by (5.29). The powerspectrum of Ω2 is given by (A.18) (illustrated in FIG. 5.2).Calculation of the Fourier transform of ei2nΘ exactly based on (5.29) iscomplicated. For simplicity, we assume that Ω taking the following simpleform which is similar to (5.27)Ω = Ω0(1 + h cos γt), (C.1)where γ take the peak value of the power spectrum (A.18) which is around∼ 1.7Λ (see FIG. 5.2) and h < 1 to make sure that Ω > 0.Then we haveΘ = Ω0t+hΩ0γsin γt. (C.2)Using the Jacobi-Anger expansion we haveei2nΘ =+∞∑m=−∞Jm(2nhΩ0γ)ei(2nΩ0+mγ)t, (C.3)where Jm is the mth Bessel function of the first kind.As |m| → ∞, we have|Jm(2nhΩ0γ)| ∼ 1m!(nhΩ0γ)|m|, (C.4)109Appendix C. Fourier transforms of the coefficients in (6.42)which drops faster than the exponential. Therefore, the Fourier transformof ei2nΘ is centered around 2nΩ0.To estimate the magnitude of the Fourier coefficients of ei2nΘ aroundzero frequency, we evaluate the Bessel function form ∼ −2nΩ0/γ ∼√GΛ→∞. (C.5)In this case, the zero component Fourier coefficient is asymptotic to (see[70])|Jm(−hm)| ∼ e−(ν−tanh ν)|m|√2pi|m| tanh ν → 0, (C.6)since ν is determined by h = sech ν < 1 that we always have ν− tanh ν > 0.When calculating the Fourier transform of ei2nΘ exactly based on (5.29),the spectrum becomes continuous instead of discrete. But the distributionof the spectrum should be similar.110

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