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Classicalization of Quantum Fluctuations at the Planck Scale in the Rh=ct Universe

FULVIO MELIA (  [email protected] ) The University of Arizona https://orcid.org/0000-0002-8014-0593

Research Article

Keywords: cosmology, quantum uctuations, ination

Posted Date: May 17th, 2021

DOI: https://doi.org/10.21203/rs.3.rs-529452/v1

License:   This is licensed under a Creative Commons Attribution 4.0 International License. Read Full License Classicalization of Quantum Fluctuations at the Planck Scale in the Rh = ct Universe Fulvio Melia Department of Physics, the Applied Math Program, and Department of , The University of Arizona, Tucson AZ 85721

Abstract

The quantum to classical transition of fluctuations in the early universe is still not completely understood. Some headway has been made incorporating the effects of decoherence and the squeezing of states, though the methods and procedures continue to be challenged. But new developments in the analysis of the most recent Planck data suggest that the primordial power spectrum has a cutoff associated with the very first quantum fluctuation to have emerged into the semi-classical universe from the Planck domain at about the Planck . In this paper, we examine the implications of this result on the question of classicalization, and demonstrate that the birth of quantum fluctuations at the Planck scale would have been a ‘process’ supplanting the need for a ‘measurement’ in quantum . Emerging with a single wavenumber, these fluctuations would have avoided the interference between different degrees of freedom in a superposed state. Moreover, the implied scalar-field potential had an equation-of-state consistent with the zero active mass condition in general relativity, allowing the quantum fluctuations to emerge in their with a time-independent frequency. They were therefore effectively quantum harmonic oscillators with classical correlations in phase from the very beginning.

Key words: cosmology, quantum fluctuations, inflation PACS: 04.20.Ex, 95.36.+x, 98.80.-k, 98.80.Jk

1. Introduction horizon [12]. Indeed, the small departure from a perfectly scale-free distribution is viewed (in this model) as a con- In standard inflationary ΛCDM cosmology, the early uni- sequence of ‘Hubble friction,’ whose strength and duration verse underwent a phase of quasi-exponential expansion are directly attributable to the detailed shape of the infla- due to the of a scalar (inflaton) field with a near- ton potential. Considerable work has been carried out over flat potential, a process typically referred to as ‘slow-roll’ the past four decades to use this framework in order to con- inflation [1–5]. strain the physical conditions in the early universe [13–17], at up to the grand unified scale at 1015 GeV. The concept of inflation was introduced to solve several ∼ global inconsistencies with standard cosmology, But the CMB anisotropies and galaxy clusters are clas- notably its horizon problem, but an even more important sical, so we are faced with the problem of understanding consequence of inflation was realized with the introduction how quantum fluctuations in the inflaton field transitioned of quantum effects [6–10]. Small inhomogeneous fluctua- into purely classical objects—an issue closely related to tions on top of an otherwise isotropic and homogeneous the long-standing ‘measurement’ problem in quantum me- background are now broadly believed to be the explana- chanics, i.e., how does a deterministic outcome appear in tion for the anisotropies in the cosmic microwave back- a measurement process performed on a quantum system ground (CMB) and the formation of large-scale structure prepared in a superposed state? Different authors use a [11]. Beginning as quantum seeds in the distant conformal range of descriptions to characterize the distinctions be- past, well before the Planck time tPl, these inhomogeneities tween quantum and classical states (see, e.g., refs. [18–20]) grew with the expansion of the universe and produced a but, at a fundamental level, a successful transition from near scale-free power spectrum as they crossed the Hubble the former to the latter requires the substantial elimina- tion of interference between the various degrees of freedom Email address: [email protected] (Fulvio Melia). in a superposed and the subsequent appear-

Preprint submitted to Phys. Lett. B 10 May 2021 ance of macroscopic variables. The second condition is the Planck scale at about the Planck time, the manner in which emergence of classical correlations in the phase space of the quantum states were ‘prepared’ may have some bear- such canonical variables. In other words, classical trajecto- ing on the classicalization problem. In 2, we shall summa- ries (e.g., orbits) with well-defined values of these variables rize the current status with the quantum-to-classical§ tran- need to be established. sition of fluctuations in the inflaton field, and then transi- The age-old (mostly heuristic) Copenhagen interpreta- tion into a discussion of the analogous situation for a non- tion suggests that a quantum system remains in its super- inflationary scalar field with the zero active mass equation- posed state until an observer performs a measurement on of-state (relevant to the above interpretation of kmin) in 3. it, the interaction of which causes it to ‘collapse’ into one We shall describe the impact of these new developments§ on of the eigenstates of the observable being measured [21]. the classicalization question in 4 and end with our con- The original version of this concept has since been refined clusions in V. § with the phenomenon of decoherence, in which the system § becomes entangled with its environment [22], consisting of 2. Classicalization of Quantum Fluctuations in the a very large number of degrees of freedom, bringing the Inflaton original quantum state into an ensemble of classical look- ing ones. The manifestation of this problem specifically in 2.1. Quantum Fluctuations in the Inflaton Field the cosmological context has sometimes been framed in the sense that the CMB anisotropies constitute a measurement of the field variable [23–25]. The essential steps for deriving the perturbation growth The formalism based on decoherence, however, is subject equation are by now well known, and one may find many ac- to considerable debate. It has been argued by some [26– counts of this procedure in both the primary and secondary 28] that decoherence by itself does not solve the problem literature. For the sake of brevity, we here show only some of a single outcome. As we shall discuss later in this paper, key results—chiefly those that will also be relevant to our this mechanism is arguably even less likely to be relevant discussion of quantum fluctuations at the Planck scale in 3 and 4—and refer the reader to several other influential to the early universe—a ‘closed’ system where the distinc- §§ tion between the quantum state and an ‘environment’ is publications for all the details. essentially nonexistent. Other types of approaches, such as The perturbed Friedmann-Lemaˆıtre-Robertson-Walker Bohmian mechanics [29,30], do not by themselves predict (FLRW) spacetime for linearized scalar fluctuations is given testable features that may be verified or refuted. by the line element [34–37] Our entry point into this discussion is motivated by sev- ds2 = (1 + 2A) dt2 2a(t)(∂ B) dtdxi eral significant new developments in the updated analysis − i − a2(t) [(1 2ψ)δ + 2(∂ ∂ E)+ h ] dxi dxj , (1) of the latest Planck data release [31], which build on many − ij i j ij attempts made over the past two decades to find specific where indices i and j denote spatial coordinates, a(t) is the features associated with the primordial power spectrum, expansion factor, and A, B, ψ and E describe the scalar (k). For example, several large-angle anomalies have been metric perturbations, while h are the tensor perturba- Ppresent in the CMB maps since the 1990’s. The most recent ij tions. studies [32,33] of the Planck data have extended our abil- For small perturbations about the homogeneous scalar ity to examine whether these issues are more likely due to field φ (t), instrumental or other systematic effects, or whether they 0 φ(t, x)= φ (t)+ δφ(t, x) , (2) truly represent real characteristics in (k). And in 3.1, 0 we shall briefly discuss why these anomaliesP may now§ be one can identify the curvature perturbation on hyper- surfaces orthogonal to worldlines in the comovingR frame viewed as possibly being due to a cutoff kmin in the power spectrum. Its value, however, calls into question whether written as a gauge invariant combination of the scalar field slow-roll inflation remains viable [64], at least based on perturbation δφ and the metric perturbation ψ [34]: the inflaton potentials proposed thus far. As we shall ex- H ψ + δφ . (3) plore in 3.2, a more interesting interpretation for the ap- R≡ ˙ §  φ  pearance of kmin is that this represents the first mode to A solution for may then be obtained by (i) using an exit from the Planck scale into the semi-classical universe R at about the Planck time. If correct, this interpretation expansion in Fourier modes, 3 would clearly have a considerable impact on the quantum- d k ik·x (τ, x) k(τ)e , (4) to-classical transition. R ≡ (2π)3/2 R In this paper, we focus on how this interpretation could Z alter our view of the long-standing classicalization prob- and (ii) inserting the linearized metric (Eq. 1) into Ein- lem in the early universe. The conventional view has been stein’s equations, which together yield the following per- k that quantum fluctuations were seeded in the Bunch-Davies turbed equation of for each mode : ′ , though without much guidance other than they ′′ z ′ 2 k + 2 k + k k = 0 . (5) emerged in the ground state. But if they appeared at the R z R R   2 3 Overprime denotes a derivative with respect to conformal [ˆu(τ, x1), πˆ(τ, x2)] = iδ (x1 x2) time dτ = dt/a(t), and we have defined the variable − [ˆu(τ, x1), uˆ(τ, x2)] = 0 a(t)(ρ + p )1/2 [ˆπ(τ, x ), πˆ(τ, x )] = 0 . (12) z φ φ . (6) 1 2 ≡ H − + The constants of integration ak and ak in the mode ex- − + Using the canonically normalized Mukhanov-Sasaki vari- pansion of u(τ, x) become operatorsa ˆk anda ˆk , so that able [35,36] 3 d k − ik·x + ∗ −ik·x uk z k , (7) uˆ(τ, x)= aˆk uk(τ)e +a ˆk u (τ)e . ≡ R (2π)3/2 k Z Equation (5) may be recast into the more familiar form of   (13) a parametric oscillator with a time-dependent frequency, We thus see from Equations (12) and (13) that known (in this context) as the Mukhanov-Sasaki equation, − + 3 ′ [ˆak , aˆk′ ]= δ (k k ) ′′ 2 − uk + ωk(τ) uk = 0 , (8) − − [ˆak , aˆk′ ] = 0 + + where the time-dependent frequency is [ˆak , aˆk′ ] = 0 , (14) ′′ − + 2 2 z which allows us to interpreta ˆk anda ˆk as annihilation and ωk(τ) k . (9) ≡ − z creation operators, respectively, creating and annihilating In Minkowski spacetime, the scale factor a(τ) is constant excitations, or particles, of the field u(τ, x). And in the usual fashion, the quantum states in the Hilbert space are and ωk = k, which reduces Equation (8) to that of the more basic harmonic oscillator with a constant frequency. constructed by a repeated application of the creation oper- ator, Since ωk(τ) depends solely on τ and k (not the direction

k 1 + nk + nk of ), the most general solution to the Mukhanov-Sasaki k k 1 2 n 1 ,n 2 , = (ˆak1 ) (ˆak2 ) 0 , equation may be written | ···i nk1 ! nk2 ! ··· | i ···  − + ∗ (15) k p u = ak uk(τ)+ a−kuk(τ) , (10) starting with the vacuum state 0 , which is defined by | i − where uk(τ) and its complex conjugate are linearly inde- aˆk 0 = 0 . (16) pendent solutions to Equation (8), and are the same for all | i In de Sitter space, where a(τ) = (Hτ)−1 (in terms of Fourier modes with k = k . In general, however, the inte- the Hubble constant H during inflation),− the exact solution gration constants a− and|a+| may depend on the direction k −k to Equation (8) is of k. From Equations (4) and (7), one may thus write the complete solution to the Mukhanov-Sasaki equation as e−ikτ i eikτ i uk(τ)= Ak 1 + Bk 1+ , (17) 3 √2k − kτ √2k kτ d k − ik·x + ∗ −ik·x     u(τ, x)= ak uk(τ)e + ak uk(τ)e , where A and B are integration constants to be fixed by (2π)3/2 k k Z the choice of vacuum. In a time-independent spacetime, the   (11) + − ∗ normalization of the wavefunction is obtained by impos- which is manifestly real since ak =(ak ) . Our derivation of Equation (11) has made no reference to ing canonical quantization or, equivalently, by minimizing the definite size of the inhomogeneities. This result there- the expectation value of the Hamiltonian in the vacuum fore merely shows us the dynamics of such inhomogeneities state. For example, in Minkowski space, one would find if they came into existence. To address their presence and from Equation (8) with ωk = k, and the Hamiltonian writ- magnitude, one must consider states at the quantum level, ten in terms of uk and πk, that where the fluctuation field becomes an operator. The next e−ikτ uk(τ)= (Minkowski) . (18) step is therefore the quantization of the field, noting that √2k the Hilbert space for the quantum field will be a direct In de Sitter spacetime, such a solution would therefore cor- product of individual Hilbert for the Fourier modes. respond to τ , and Ak = 1, Bk = 0. This is inter- The field u(τ, x) is quantized like the harmonic oscilla- preted to mean| | →that ∞ in the remote conformal past, all modes tor, except that the frequency (Eq. 9) is now time depen- of current interest were much smaller than the Hubble ra- dent, which is itself a consequence of the curved spacetime dius, allowing us to ignore curvature effects on the mode within which these modes are evolving. To properly nor- normalization, thereby defining a unique physical vacuum malize the quantum fluctuation, one must therefore choose known as the Bunch-Davies vacuum [38]. the vacuum carefully, since the spacetime is generally not With the time evolution Minkowskian. e−ikτ i The field u(τ, x) and its canonical conjugate ‘momen- uk(τ)= 1 (de Sitter) , (19) ′ √ − kτ tum’ π(τ, x) u (τ, x) are promoted to quantum opera- 2k   torsu ˆ andπ ˆ,≡ satisfying the standard equal-time canonical Equation (13) gives us the full form of the field operator quantization relations uˆ(τ, x), which may be used to find the quantum states of

3 the inflaton scalar field. As we shall see in 3, much of this phase space between the field amplitudes and their canon- derivation remains intact for the (non-inflationary)§ numen ical partners, must have transitioned into classical fluc- field as well, though with several crucial differences we shall tuations, characterized in part by a diagonalized density discuss shortly. (As we shall see in 3.2, it will be neces- matrix and well defined dynamical trajectories. How this sary to clearly distinguish between inflaton§ scalar fields and might have happened is the issue we shall address next. those that do not produce an inflated expansion. We shall therefore informally refer to the latter as ‘numen’ fields.) We now turn our attention to describing the vacuum 2.2. The Quantum to Classical Transition state with a wavefunctional that gives the am- plitudes for different field configurations in Fourier space. The question concerning whether or not the state Though we have developed the expression for the field op- Ψ[u(τ, x)] can complete the quantum to classical transi- erator,u ˆ(τ, x), in the , finding the vari- tion based on the factors we have summarized above has ous occupation numbers in a quantum state such as Equa- received considerable attention over the past four decades, tion (15) is more easily accomplished using a functional ap- and is still an open one [41–44,39,45–51]. As noted in the proach in the Schr¨odinger picture (see, e.g., refs. [39,40]). introduction, the cosmological scalar perturbations must As long as the various k-modes are all independent, satisfy at least two conditions to classicalize: (i) the system the Hilbert space for the field operatoru ˆ(τ, x), contain- must undergo decoherence, so that quantum interference ing eigenstates u , is simply a direct product of Hilbert becomes negligible and macroscopic variables appear. We spaces for the different| i Fourier components. If a state is see a universe with well-defined, classical perturbation initially decomposed as Ψ , the probability amplitude of field values, not superpositions of multiple components measuring the field configuration| i u(τ, x) is then the wave represented by Equation (20). (ii) One must see the estab- functional Ψ[u(τ, x)] u Ψ where, in the Schr¨odinger lishment of classical correlations in phase space, meaning approach, one may factor≡ h it| intoi independent mode com- that the allowed dispersion in the canonical variables (u, π) ponents according to: becomes insignificantly smaller than their classical orbital values. Ψ[u(τ, x)] = Ψk[uk(τ)] , (20) Different authors have proposed somewhat different Yk schemes for this transition, indicating that no consen- each of which is a solution to Schr¨odinger’s equation: sus has yet been reached. In part, this is due to the fact ∂Ψk that the conventional rules one may rely on in ordinary i = HˆkΨk . (21) quantum mechanical applications are not necessarily all ∂τ available in cosmology. To begin with, the system be- The Hamilton operator in Fourier space may be written ing studied quantum mechanically is the entire Universe. 1 3 1 3 † 2 † There is no possible separation into a subsystem of inter- Hˆ = d k Hˆk = d k πˆkπˆ + ωkuˆkuˆ . (22) 2 2 k k est, its environment and the observer. Moreover, there is Z Z h i just one universe, obviating any possibility of adopting the The solution to Equations (21) and (22) for a harmonic statistical of quantum mechani- oscillator is well known, and has the form of a Gaussian cal measurements. Even more importantly, the observer function: is here a consequence of the quantum to classical transi- 2 −Ωk(τ)(uk) Ψk[uk(τ)] = Nk(τ)e , (23) tion, and could not have played any causal part in it. For these (and other) reasons, the system being studied is thus where 1/4 unusual by typical quantum mechanical standards. 2Re Ωk Nk { } (24) A common step taken by the various approaches to solv- | |≡ π   ing this problem is to relegate most of the very large num- is the normalization factor, and ber of degrees of freedom in Ψ[u(τ, x)] to an ‘environment’ ′ and then to ignore them, allowing one to evolve the reduced i fk Ωk (25) density matrixρ ˆ for the subset of remaining (presumably ≡−2 f k more interesting) observables. Under some circumstances, is written in terms of the function fk, the solution to Equa- also involving suitable time averaging, this procedure even- tion (8), i.e., tually diagonalizesρ ˆ, representing a complete mitigation of 1 i any interference effects, which is interpreted as the emer- f = 1 e−ikτ . (26) gence of classical behavior. k √ − kτ 2k   This is not entirely satisfactory, however, because deco- The functional Ψ[u(τ, x)], representing a quantum fluc- herence has not yet successfully solved the long-standing tuation in the inflaton field, is a superposition of many ‘measurement’ problem in . Certainly, separate Fourier states Ψk (Eqn. 20). Somehow, this quan- the diagonalization ofρ ˆ removes certain quantum traits tum description, which includes interference between the from the system, but it is not clear that a non-interfering various modes and no a priori classical correlations in set of simultaneous co-existing possibilities is necessarily

4 classical (see, e.g., ref. [52]). Note, for example, that chang- The literature on Wigner functions is extensive (includ- ing the Hilbert space of the quantum system to a different ing the aforementiond reviews in refs. [54–56]). Its proper- basis would destroy the diagonal nature of the density ma- ties suggest that it behaves like a probability distribution trix. The interpretation ofρ ˆ is therefore subject to various in (u, π) phase space, except that it can sometimes take on observer-dependent choices. What is lacking is a clear un- negative values, so it is generally not a true probability dis- derstanding of how to choose the basis and an interaction tribution. Moreover, points in the (u, π) space to not rep- specific eigenstate for the preferred observable. In our ev- resent actual states of the system because the values of uk eryday experience, some progress may be made by allowing and πk cannot be determined precisely at the same time. the measurement device to ‘select’ the basis, and adopt- Nevertheless, can be used to visualize correlations be- W ing the ensemble interpretation for the . But tween uk and πk, particularly for Gaussian states, such as these features are obviously missing in the cosmological we have in Equation (23), for which the Wigner function is context [46,53]. indeed always positive definite. To solve the classicalization problem using decoherence It is trivial to show in the case of Equation (23) that the in a cosmological setting, one must therefore identify a explicit form of is physical mechanism and a preferential basis it selects. It is W 1 R 2 I 2 also necessary to find a criterion for separating the large R I R I −Re{Ωk}[(uk ) +(uk) ] (uk ,uk, πk , πk)= e number of degrees of freedom into the ‘interesting set’ W π2 × R R 2 I I 2 and ‘the environment’ dictated by the physical problem e−(πk +Im{Ωk}uk ) /Re{Ωk}e−(πk+Im{Ωk}uk) /Re{Ωk}, (28) at hand. At least some of the proposed treatments then appeal to ‘a specific realization’ of the stochastic variables with which one may clearly see the effect of strong squeezing [39,49], sounding very much like a conventional ‘collapse’ as Ωk changes dramatically with time (Eqns. 25 and 26) from the statistical description of the universe to one of during the cosmic expansion. the members in the statistical ensemble. But no insight is As long as Ωk k/2, is peaked over a small region of provided into how and when such a transition occurred in phase space, representing→ W the Wigner function of a coher- the real universe. ent state, i.e., the ground state of a harmonic oscillator. We To address the second requirement, we begin by noting shall return to this in 4 below, where we consider the cor- § that in the remote conformal past, where Ωk k/2, the responding Wigner function for the numen quantum fluc- → wavefunctional Ψk represented the ground state of a har- tuations. We shall see that this limiting situation takes on monic oscillator, consistent with the previously described added significance in that case, given that Ωk actually re- Bunch-Davies vacuum. For the largest modes (those of mains constant for a numen field. But Ωk is definitely not relevance to the structure we observe today), the func- constant here. As τ advances, spreads and acquires a W tion Ωk acquired a non-trivial time dependence and Equa- cigar shape typical of squeezed states [57], with a drasti- tion (23) evolved into a squeezed state—meaning that, for cally reduced dispersion around πk and a correspondingly these modes, there exists a direction in the (uk, πk) plane enormous dispersion around uk. But the overall dispersion where the dispersion is exponentially small, while the dis- may be minimized, as we alluded to earlier, by choosing persion in a perpendicular direction is very large. The re- an appropriate linear combination of uk and πk, an out- sultant linear combination of uk and πk for which the dis- come that some argue represents the emerging correlation persion is minimized is often viewed as the emerging clas- in classical phase space. sical phase-space trajectory. One may therefore think of a squeezed state as a state A convenient tool to study this process, and its possi- with the minimal uncertainty, though not in terms of the ble relevance to the classicalization question, is the Wigner original (u, π) variables [39,47]. In the cosmological con- function (see, e.g., the reviews in refs. [54,55]; for a more text, the inflationary expansion created an uncertainty on pedagogical account, see also ref. [56]), defined by the value of the field and its conjugate momentum that was much larger than the minimum uncertainty implied by R I R I 1 the Heisenberg . The minimum uncer- (uk ,uk, πk , πk)= 2 dxdy W (2π) tainty was instead associated with a new pair of ‘rotated’ Z R I R x I y −iπk x−iπky Ψk uk ,uk e canonical variables. But this outcome has also been chal- − 2 − 2 × lenged as not necessarily representing a classical system  R x I y Ψk uk + ,uk + , (27) [44,51]. 2 2 Take the following situation as an example. Consider an   in terms of the real and imaginary parts of uk and πk. electron in a minimal wavepacket localized at the origin, Quantum mechanics inherently deals with , with an uncertainty ∆x in position and ∆p = ~/(2∆x) in while classical physics deals with well-defined trajectories momentum. Then form a superposition of this state with in phase space. The Wigner function provides a means of an identical one after a translation by a large distance D. representing the density distributions in uk and πk for com- The overall uncertainty is now D~/(2∆x), which can be parison with the ensemble of trajectories one would get us- increased arbitrarily by simply∼ choosing a sufficiently large ing classical means. distance D. This situation is clearly analogous to one of our

5 squeezed states, but the superposition we have created is persistent lack of clarity concerning how quantum fluctua- nonetheless not classical. tions were seeded in the early universe [63,64]. And then there is the issue of definite outcomes, also The large-scale anomalies stand in sharp contrast to our known as the ‘’ [58], as we alluded overall success interpreting the CMB anisotropies at angles to in the introduction. This aspect of quantum mechanics smaller than 1◦. Over the past several decades, a concerted has been with us for over a century. Even if decoherence effort has therefore been made in attempting to identify were successful in diagonalizing the density matrix, it can- features in the primordial power spectrum, (k), responsi- not solve the definite outcome problem, which is far worse ble for their origin. For example, in their studyP of the CMB in cosmology than it is in typical laboratory situations. The angular power spectrum, Shafieloo & Souradeep [65] as- usual approach of adopting the Copenhagen interpretation sumed an exponential cutoff at low k-modes, and identified of quantum mechanics, in which a measurement ‘collapses’ a turnover generally consistent with the most recent mea- the state vector of the system into an eigenstate corre- surement we shall discuss below (see Eq. 29). This early sponding to the measurement result, does not work in the treatment was based on WMAP observations [60], however, early Universe, where there were no measurement devices not the higher precision Planck measurements [31] we have or observers present. today, so the reality of a non-zero kmin remained somewhat While decoherence might have diagonalized the reduced controversial. density matrix of the system to an ensemble of classically In followup work, Nicholson et al. [66] and Hazra et al. observable universes, it does not explain how a certain [67] inferred a ‘dip’ in (k) on a scale k 0.002 Mpc−1 for universe was singled out to be observed [28]. Perhaps the the WMAP data, confirmingP the outcome∼ of an alternative Copenhagen interpretation is just not well suited to cosmol- approach by Ichiki et al. [68] that identified an oscillatory ogy. In addition, if decoherence resulted from an environ- modulation around k 0.009 Mpc−1. In closely aligned ment comprised of the ‘non-interesting’ degrees of freedom work, Tocchini et al. [69]∼ modeled both a dip at k 0.035 in Ψ[u(τ, x)], how could this happen when inflation would Mpc−1 and a ‘bump’ at k 0.05 Mpc−1. Like the∼ others, have driven all such fields towards their vacuum states? though, these features were∼ based solely on WMAP ob- Moreover, why do we have the privilege of deciding which servations and therefore appeared to be merely suggestive degrees of freedom to relegate to the background based rather than compelling. Tocchini et al. [70] improved on solely on whether our current technology allows us to ob- this analysis considerably, and found evidence for three fea- serve them today? tures in (k), one of which was a cutoff at 0.0001 0.001 We shall now divert our attention away from the tradi- Mpc−1 atP a confidence level of 2σ. Complementary∼ − work tional inflationary scenario we have been describing, and by Hunt & Subir [71,72] confirmed∼ the likely existence of a seriously consider the implications of a novel feature emerg- cutoff k < 5 10−4 Mpc−1 but, as before, also concluded ing from the latest release of the Planck data: the observa- that more accurate× Planck data would eventually be needed tional measurement of a cutoff in the primordial spectrum to confirm these results more robustly. bears directly on the question of how and when quantum The subsequent Planck observations have not only fluctuations were generated in the early Universe. We shall largely confirmed these earlier results, but have provided study what new ideas and constraints this brings to the us with a greatly improved precision in the ‘measurement’ classicalization process. of kmin. For example, the Planck Collaboration [73] fit a cutoff to the CMB angular power spectrum and found a value 3 4 10−4 Mpc−1. Still, though these studies 3. Quantum Fluctuations at the Planck Scale all pointed∼ − to× the likely existence of a cutoff in (k), a P non-zero kmin could not be claimed with a confidence level 3.1. Emergence of a Cutoff in the primordial power exceeding 1 2σ. spectrum (k) − P This situation improved considerably when, instead of looking solely at the power spectrum, the impact of a cut- All three of the major satellite missions designed to study off was also considered on the angular correlation function, the CMB—COBE [59]; WMAP [60]; and Planck [31]— have independently of the angular power spectrum. Two sepa- uncovered several anomalies in its fluctuation spectrum. rate (though complementary) studies of the latest Planck The two most prominent among them are: (1) an unex- data release have provided more compelling evidence that pectedly low level—perhaps even a complete absence—of the two large-angle features in the CMB anisotropies may correlation at large angles (i.e., θ & 60◦), manifested via be real. The first of these [32] demonstrates that the most the angular correlation function, C(θ); and (2) relatively likely explanation for the missing large-angle correlations weak power in the lowest multipole moments of the angular is a cutoff, power spectrum, Cℓ. Their origin, however, is still subject 4.34 0.5 kmin = ± , (29) to considerable debate, many arguing in favor of a misin- rcmb terpretation or the result of unknown systematics, such as in the primordial power spectrum, (k), where rcmb is the an incorrect foreground subtraction (see refs. [61,62] for re- comoving distance to the surface ofP last scattering. For the views). This uncertainty is also fueled in large part by a Planck-ΛCDM parameters, r 13, 804 Mpc [31], and cmb ≈ 6 we therefore have k = (3.14 0.36) 10−4 Mpc−1.A some modification. Moreover, a cutoff in (k) does not ar- min ± × P zero cutoff (i.e., kmin = 0) is ruled out by the C(θ) data at gue against the influence of a scalar field, φ, nor anisotropies a confidence level exceeding 8σ. arising from its quantum fluctuations, but there is now A subsequent study [33] focused∼ on the CMB angular some motivation to question whether V (φ) was truly infla- power spectrum itself (i.e., Cℓ versus ℓ), and its results tionary. (i) confirmed that the introduction of this cutoff in (k) Over the past decade, some evidence has been accumu- does not at all affect the remarkable consistency betweenP lating that the cosmic fluid may possess a zero active mass the standard inflationary model prediction and the Planck equation-of-state, ρ + 3p = 0 (in terms of its total measurements at ℓ > 30 [31], where the underlying the- density ρ and pressure p), supported by over 27 different ory is widely believed to be correct; and (ii) showed that kinds of observation at low and high redshifts (see Table 2 such a cutoff (kmin) also self-consistently explains the miss- in ref. [77] for a recent summary of these results). Such a ing power at large angles, i.e., the low multipole moments universe lacks a horizon problem [78,79], so the lack of a (ℓ = 2 5). The cutoff optimized by fitting the angu- fully self-consistent inflationary paradigm may be telling − lar power spectrum over the whole range of ℓ’s is kmin = us that the universe does not need it. This is the key as- +1.4 −4 −1 (2.04−0.79) 10 Mpc , while a fit to the restricted range sumption we shall make to reinterpret kmin in this paper. In ℓ 30, where× the Sachs-Wolfe effect is dominant [74], gives addition to the growing body of empirical evidence favor- ≤ +1.7 −4 −1 kmin = (3.3−1.3) 10 Mpc . The outcome based on the ing this approach, there is also theoretical support for the CMB angular power× spectrum therefore rules out a zero zero active mass equation-of-state from the ‘Local Flatness cutoff at a confidence level & 2.6σ. In either case, the in- Theorem’ in general relativity [80]. ferred value of kmin is fully consistent with the cutoff im- As noted earlier, we shall clearly distinguish between the plied by missing correlations in C(θ), and one concludes roles played by a non-inflaton φ and a conventional infla- that both of these large-angle anomalies are probably due to ton field by informally refering to the former as a ‘numen’ −4 −1 the same truncation, i.e., kmin 3 10 Mpc , in (k). field, based on our inference that it may represent the earli- The confidence with which one∼ may× make such a claimP de- est form of substance in the universe. Its equation-of-state pends on whether the cutoff is used to address the missing is assumed to be ρφ + 3pφ = 0, and we shall see shortly power at low ℓ’s, or the missing correlations at large angles. why this property appears to provide a more satisfactory Nevertheless, the fact that the same kmin apparently solves interpretation of kmin than an inflaton field. both anomalies makes the assumption of a cutoff quite rea- The background numen field is homogeneous, so its en- sonable. ergy density ρφ and pressure pφ are simply given as These results reinforce the perception that the small- 1 ˙2 angle anisotropies (for ℓ & 30), which are mostly due to ρφ = φ + V (φ) , (30) 2 acoustic oscillations, are well understood, while the fluctu- ◦ and ations associated with angular correlations at θ & 60 , due 1 p = φ˙2 V (φ) . (31) to the Sachs-Wolfe effect, continue to be problematic for φ 2 − the standard inflationary picture [59,60,31]. The evidence The zero active mass equation-of-state therefore uniquely for a non-zero kmin speaks directly to the cosmological ex- constrains the potential to be pansion itself. At ℓ . 30, we are probing ever closer to the V (φ)= φ˙2 , (32) beginning of inflation, culminating with the cutoff kmin, signaling the very first mode that would have crossed the with the explicit solution horizon when the quasi-de Sitter phase started [36,64,17]. But if one insists on inflation simultaneously fixing the 2√4π V (φ)= V0 exp φ , (33) horizon problem and accounting for the observed primor- (− mPl ) dial power spectrum, (k), the implied time at which the in terms of the Planck mass accelerated expansionP began would have suppressed the 1 comoving size of the universe to a tenth of the required mPl . (34) value [64,17]. Moreover, neither a radiation-dominated, nor ≡ √G a kinetic-dominated, phase preceding inflation could have Some may recognize this as a special member of the cat- alleviated this disparity [75,76]. egory of minimally coupled fields explored in the 1980’s [81–84], intended to produce so-called power-law inflation. But unlike the other fields in this cohort, the numen field’s 3.2. A Reinterpretation of kmin zero active mass equation-of-state makes it the only mem- ber of this group that does not inflate, since the Friedmann To be clear, the measurement of kmin does not completely equations with this density and pressure lead to an expan- rule out inflation, nor even the idea that some slow-roll sion factor a(t) = t/t0, written in terms of the age of the variant may eventually be constructed to address the in- universe, t0. This normalization of a(t) is appropriate for a consistency described above. At a minimum, however, the spatially flat FLRW metric, which the observations appear currently proposed inflaton potentials V (φ) require at least to be telling us.

7 With this expansion factor, the conformal time may be for mass m equals its Schwarzschild radius Rh 2Gm. written That is, ≡ τ(t)= t ln a(t) , (35) λPl √4πG . (42) 0 ≡ The Compton wavelength λ grows as the gravitational ra- such that the zero of τ coincides with t = t0. The parameter C z in Equation (6) thus becomes dius Rh shrinks, so the standard inflationary picture con- flicts with quantum mechanics in its interpretation of wave- mPl z = a(t) , (36) lengths shorter than 2πλPl (the factor 2π arising from the √4π definition of λPl in terms of Rh). This is a serious problem ′ ′′ 2 so that z /z = 1/t0 and z /z = 1/t0. The resulting curva- for the because the fluctuation amplitude ture perturbation equation analogous to Equation (8) may As measured in the CMB anisotropies requires quantum thus be written fluctuations in the inflaton field to have been born in the ′′ 2 Bunch-Davies vacuum, long before the Planck time t uk + αkuk = 0 , (37) Pl λ , a conundrum commonly referred to as the “Trans-≡ where the frequency (analogous to Eqn. 9) is now given by Pl Planckian Problem” [86]. the expression The numen field can completely avoid this inconsistency 2 if we argue that each mode k emerged into the semi-classical 1 2πRh αk 1 , (38) universe when ≡ t0 s λk −   λk = 2πλPl , (43) in terms of the proper wavelength of mode k, and then evolved subject to the oscillatory solution in Equa- 2π tion (40). As we shall see shortly, the fact that each suc- λk a(t) . (39) ceeding k-mode emerges at later in this picture pro- ≡ k duces a near scale-free power spectrum (k) with ns . 1 In this expression, the quantity R c/H = ct is the P h ≡ [63]. Such an idea—that modes could have been born at a apparent (or gravitational) radius [85], which defines the particular spatial scale—has already received some atten- Hubble horizon in a spatially flat universe. The most crit- tion in the past, notably by Hollands & Wald [87]. In their ical difference between Equations (8) and (9), and Equa- case, however, the fundamental scale was not related to λPl. tions (37) and (38), is that here both Rh and a(t) scale lin- Others supporting this proposal include Brandenberger et early with t, and therefore the frequency αk of the numen al. [88] and Hassan et al. [89]. quantum fluctuations is always time-independent. This is It is not difficult to understand why the fundamental because the ratio Rh/λk kRh/a(t) is constant for each ∝ scale for the numen field must be λPl. If we interpret k, and therefore numen fluctuations do not criss-cross the numen k min to be kmin, the latter defines the time tmin horizon; once λk is established upon the mode’s exit into at which| the first quantum fluctuation emerged out of the semi-classical universe, it remains a fixed fraction of Rh the Planck domain into the semi-classical universe. From as they expand with time. This feature is crucial to under- Equation (29) and the expression for rcmb in a universe standing how and why numen quantum fluctuations pro- with zero active mass, vide a more satisfactory explanation than inflation for the numen c origin of k . r cmb = ln(1 + zcmb) , (44) min | H The solution to Equation (37) is that of the standard 0 harmonic oscillator: one therefore finds that 4.34 tPl ±iαkτ tmin = . (45) B(k) e (2πRh >λk) ln(1 + zcmb) uk(τ)= . (40)  B(k) e±|αk|τ (2πR <λ ) Its dependence on zcmb is so weak that tmin is approxi-  h k mately equal to tPl regardless of where the last scatter- That is, all modes with λk < 2πRh oscillate, while the ing surface was located. For example, in Planck-ΛCDM, super-horizon ones do not, mirroring the behavior of the z 1080, for which t 0.63 t . But even if we were cmb ∼ min ∼ Pl more conventional inflaton field. Here, however, the mode to adopt a very different value zcmb = 50, the first quantum with the longest wavelength relevant to the formation of fluctuation would have emerged at t 1.1 t . min ∼ Pl structure is the one for which λk(τ) = 2πRh(τ), i.e., If the spatially largest fluctuation we see in the CMB was due to a numen fluctuation, one concludes from this anal- knumen = 1/t . (41) |min 0 ysis that it must have emerged out of the Planck regime One’s intuition would immediately suggest that knumen at roughly the Planck time. In other words, this fluctu- |min ought to be identified with the cutoff kmin measured in the ation would have physically exited into the semi-classical CMB, and it is not difficult to demonstrate why that has to universe shortly after the Big Bang—indeed, it would have be the case for a numen scalar field, as we shall see shortly. appeared as soon as it could, given what we currently un- A slightly different (and simpler) definition of the Planck derstand about the Planck time tPl. No other scalar field scale than that appearing in Equation (34) is based on the introduced thus far, inflaton or otherwise, has this very in- length λ at which the Compton wavelength λ 2π/m teresting property. Pl C ≡ 8 ∗ But if a Bunch-Davies vaccum in the remote conformal Mode k reached this scale at a(tk) = L∗k/2π or, equiv- ∗ past is not used for these fluctuations, how does one then alently, at time tk = t0L∗k. And so Equation (49) may be determine the normalization constant B(k) of the modes re-written in Equation (40)? As we saw in 2.1, a principal compli- −1/2 § 1 λ 2 k 2 cation with the inflaton field is the significant spacetime (k)= Pl 1 min , (50) PR (2π)2 L − k curvature encountered by its quantum fluctuations as they  ∗  "   # cross the Hubble radius. This led to the introduction of a which one then needs to compare with the observed CMB Bunch-Davies vacuum in the distant conformal past, where ns−1 power spectrum (k) = As(k/k0) . In the context of the modes could have been seeded in Minkowski space. the standard model,P the Planck optimizations [31] give We made reference to the fact that this situation actually −9 As = (2.1 0.04) 10 and ns = 0.9649 0.0042. And creates an inconsistency with quantum mechanics, often ± × 3 ± it is not difficult to see that L∗ 3.5 10 λPl. Therefore, referred to as the trans-Planckian problem. This issue is ∼ × 19 with the Planck scale set at mPl 1.22 10 GeV, one largely beyond the scope of the present paper, however, be- ≈ × 15 finds that L∗ corresponds to an energy of roughly 3.5 10 cause the numen field completely avoids this inconsistency. GeV, remarkably consistent with the energy scale expected× Even though the numen quantum fluctuations emerged in grand unified theories. Of course, much of this is mere at the Planck scale—with a wavelength comparable to the speculation at the present time, given that the physics of gravitational radius Rh—the zero active mass equation-of- this process lies beyond the standard model. Nevertheless, state in the cosmic fluid (leading to Eqn. 32) ensures that the quantum fluctuations in this picture would have oscil- the frame into which they emerged from the Planck regime 3 lated until t 3.5 10 tPl, after which the numen field was geodesic. That is, in spite of the Hubble expansion, the would have devolved∼ × into grand unified theory particles, universe was always in free fall, with zero internal acceler- with a freezing of the perturbation amplitude thereafter. ation. One can easily confirm this from the fact that the The primordial power spectrum (Eq. 50) is almost scale- frequency αk in Equation (38) is always time-independent. free, but not exactly. Using the conventional definition of One therefore does not need an ad hoc construction of a the scalar index, we find that Bunch-Davies vacuum, and we may simply set d ln R(k) 1 1 ns =1+ P = 1 2 . (51) B(k)= (46) d ln k − (k/kmin) 1 √2αk − The index ns is therefore slightly less than 1, and we confirm for the numen quantum fluctuations (in Eqn. 40), following that the deviation from a pure scale-free distribution is due the same minimization of the Hamiltonian argument that to the aforementioned difference between αk and k, which led to Equation (18). in the end arises from the Hubble expansion (or ‘frictional’) We complete this brief survey by considering the spec- ′ term k in the growth Equation (5). trum (k) one should expect from the birth of quantum AtR least qualitatively, this result agrees with the value fluctuationsP at the Planck scale. Right away, we can see of ns measured by Planck. Of course, the numen optimiza- from Equation (38) that the difference between αk and k tion may produce a different outcome than that seen with is what distorts the primordial power spectrum away from ΛCDM, but it would be difficult to see why the ‘red’ tilt a pure, scale-free distribution, as we shall confirm below. (ns < 1) should be converted into ‘blue’ (ns > 1) with a This is most easily recognized if we rewrite the mode fre- change in background cosmology. An average of ns over k 2 2 quency in the form αk = k 1/t (which is

9 4. Classicalization of Quantum Fluctuations at the Planck Scale

4.1. The Birth of Quantum Fluctuations at the Planck Scale

The acute classicalization problem plaguing the infla- ton field stems directly from the nature of the wave func- tional Ψ[u(τ, x)] describing its fluctuations (Eqn. 20). As discussed in 2.2, there is no consensus yet on how the in- terference between§ its mode components could have been completely removed. This problem does not exist for numen quantum fluctuations due to the way they were seeded— Fig. 1. Schematic diagram showing the amplitude (plotted alongz ˆ) unlike their inflaton counterparts, each of the numen fluctu- of a numen field operator projected onto the xy-plane. The spatial variation (in comoving coordinates x) is proportional to the spherical ations emerged with a single k-mode. Pure, single k-mode Bessel function of the first kind, j0(kb|x|), where kb is the unique quantum states were established from the very beginning wavenumber corresponding to the time τb at which the fluctuation as a result of the distinct time at which they entered the emerged into the semi-classical universe across the Planck scale λPl semi-classical universe out of the Planck domain. (see Eqns. 54 and 57). A numen fluctuation born at time t , had a unique co- b Wald [87], though their basic concept and scales were quite moving wavenumber defined by the relation different. The essence underlying the numen field’s behav- 2πa(τb) ior is its inferred potential given in Equation (33). As in kb (53) ≡ λPl our case, Hollands and Wald also started by considering or, more explicitly, the dynamics of a scalar field in a background cosmolog- 2π ical setting, though the equation-of-state in their cosmic kb = tb . (54) fluid was inconsistent with the zero active mass condition. λ t Pl 0 Nevertheless, in both their and our cases we require that Thus, the oscillating modes in Equation (40) should more the fluctuations were born in their ground state at a fixed accurately be written as follows: length scale. The work described in this paper is strongly

1 ±iαkτ informed by the recent Planck data release and its subse- uk(τ,τb)= e δ(k kb) , (55) √2αk − quent analysis, which were not available at the time Hol- lands and Wald proposed their model. As we have seen, the where clearly k is uniquely related to τ via Equations (35) b b latest measurements strongly suggest that the cutoff in the and (54). In other words, it is not sufficient to merely track observed primordial spectrum is associated with the very the temporal evolution of a numen quantum fluctuation. first quantum fluctuation that exited the Planck domain at One must also specify its time of birth. about the Planck time. There is no room, in this proposal, For the numen quantum fluctuation operator, one thus for the basic length scale at which the numen fluctuations has were born to be anything but the Planck wavelength. 1 d3k But though the proposal by Hollands and Wald lacked uˆ(τ,τ , x)= δ(k k ) b (2π)3/2 √2α − b × the rigor imposed by the latest Planck observations, already Z k − i(k·x−αkτ) + −i(k·x−αkτ) Perez et al. [51] noted a very important trait of quantum aˆk e +a ˆk e . (56) fluctuations born in this fashion. They pointed out that h i such a model clearly demonstrates the need for some pro- The integral is straightforward to evaluate, and one finds cess to be responsible for their birth, playing a role anal- that the numen field operator, analogous to Equation (13) ogous to that of quantum mechanical measurement. The for the inflaton field, is birth of the mode, they argued, is effectively the step in k2 b − −iαkb τ + iαkb τ which quantum mechanical uncertainty is removed. This uˆ(τ,τb, x)= j0(kb x ) aˆkb e +a ˆkb e , √παkb | | is precisely the situation we describe in Equation (55),  (57) whereby each quantum fluctuation emerging into the semi- where j0(kb x ) is a spherical Bessel function of the first classical universe is characterized by a unique wavenum- kind. We emphasize| | again that, in this picture, the entire ber, constrained by the time at which the fluctuation was numen quantum fluctuation born at τb is characterized by created. a single wavenumber kb. A schematic diagram showing the Perez et al. [51] also wondered what process could be amplitude of this operator at any given time, τ τb, is associated with the particular time of such an occurrence. shown in figure 1. ≥ With the insights we have gained from the Planck data, Several of the characteristics we see in the numen field we can now suggest that the length scale at which the nu- were anticipated by the model proposed by Hollands and men modes were born is not at all random. It carries sig-

10 nificant physical meaning. As we have seen, the Planck is no conflict between these two descriptions, because the length is the scale at which the Compton wavelength equals universe is indeed homogeneous when viewed relative to the Schwarzschild radius, meaning that below this scale is the comoving frame. the realm of quantum gravity. If the picture we are de- Given the thesis developed in this paper, it should be scribing in this paper is correct, the birth of numen quan- clear why the role of Rh is central to the manner in which tum fluctuations was associated with the release of these numen quantum fluctuations were born and how they clas- modes into the semi-classical universe, where gravity is ad- sicalized. It is this gravitational horizon that delimited the equately described by (the classical theory of) general rela- size of the fluctuations, which were isotropic, but neverthe- tivity. Moreover, as far as we know, the Planck scale never less initially restricted in size to the Planck scale. Remem- changes. Thus, as quantum fluctuations stochastically ex- ber that λPl actually equals Rh at the time, tb, when mode ited in time, their wavenumber had to reflect their time of k emerged into the semi-classical universe. ‘birth’ and it is this correlation that built the near scale-free Crucially, an apparent horizon in the cosmological con- power spectrum seen in the cosmic microwave background text is not an event horizon [85]. It may turn into one [31,63]. in our asymptotic future, but has not been static up un- The manner in which numen fluctuations were born thus til today. It has been growing, and causally-connected re- already removes a major hurdle in the classicalization pro- gions of spacetime continue to change as Rh expands. The cess. Rather than having to deal with quantum mechan- ‘measurement-like’ process that created the numen fluctu- ical interference between many degrees of freedom, here ations at the Planck scale therefore also introduced an in- we have distinct quantum fluctuations possessing unique homogeneity relative to the observer who appeared later, wavenumbers. Two issues remain, however, one having to at time t>tb, outside the region bounded by Rh(tb). In the do with how these modes acquired classical correlations in numen context, the largest CMB fluctuation we see today phase space and the mechanism that converted a homoge- corresponds to Rh at decoupling. Since λmax (correspond- neous, isotropic universe into an inhomogeneous one. We ing to kmin) grew at the same rate as Rh, this happens to shall consider the latter next, and then revisit the Wigner be the largest-size mode created right after the Big Bang, function (Eqn. 27) to resolve the former. at the Planck time tPl. This region has been exposed to us by the expanding universe, so that we can now see the im- pact of that first quantum fluctuation on the CMB. The 4.2. Anisotropies seeding of numen fluctuations at the Planck scale not only created pure, single k-mode fluctuations, but also explains A missing ingredient from much of the past discussion why these quantum fluctuations have always been finite in concerning classicalization in the early universe has been size, and why the universe we see is therefore a patchwork of the process by which a perfectly homogeneous and isotropic inhomogeneities. At least in this context, the breakdown of state transformed into an inhomogeneous and anisotropic spherical symmetry was not due to some unknown process state described by the density fluctuations. A chief reason in quantum mechanics itself. It was the result of an inher- for this handicap has been the absence of an obvious ‘exter- ent property of the FLRW spacetime when we take into ac- nal’ source of asymmetry. Broadly speaking, such a tran- count the physical characteristics of the gravitational (or, sition can be effected as part of an ‘R’ process, e.g., mea- apparent) horizon. surement or collapse, but not a U process, i.e., a unitary evolution via a Schr¨odinger type of equation. It has been recognized that without a measurement-like process, the 4.3. Classical Phase Space Correlations required transition could not have happened. In hindsight, the problem has actually been a lack of appreciation for A distinguishing feature of quantum fluctuations in the the importance of the gravitational (or ‘apparent’) horizon inflaton field is that their frequency changed with time in Rh(τ)= c/H(τ) [85]. response to the spacetime curvature they encountered as Yes, the universe is homogeneous and isotropic, but only they grew into the expanding universe. This led to the very when described using the ‘community’ coordinates of myri- strong time dependence in Ωk (Eqn. 25) which, according ads of observers dispersed throughout the cosmos [92]. But to the Wigner function (Eqn. 28), then produced a highly from the perspective of a single observer at a fixed space- squeezed state. For the various reasons outlined in 2.2, time point, the universe does not appear to be homoge- however, it is not clear that this restructuring of the disper-§ neous. His/her description of the physical state of the sys- sions about uk and πk is consistent with emerging classical tem, using their coordinates centered at their location, must correlations in phase space. take into account the effects of spacetime curvature. Take The situation with numen quantum fluctuations was Rh, as a prime example. The gravitational radius is an ap- completely different, principally because their frequency parent horizon that separates null geodesics approaching αk (Eqn. 38) was constant in time. As we alluded to earlier, the observer from those that are receding. One could not this property results from the zero active mass equation-of- argue that such a divided congruence of null geodesics is state (ρφ + 3pφ = 0), which produced a constant expansion consistent with homogeneity. But let us affirm that there rate with a(t) t. The numen modes therefore mimicked ∝ 11 W With the use of Equation (61), this becomes 5p 1.0 4 1 2 (u , π ,τ)= e−αkb [ukb (τ)−µ cos(αkb τ)] WΨkb kb kb π × 2 3p pk −[πkb (τ)+αkb µ sin(αkb τ)] /αkb 4 b e . (63) p 10 4 5 The Wigner function thus oscillates back and forth with an

amplitude µ and angular frequency αkb —a with a very small dispersion relative to the classicalized -5 u 5 kb trajectory in the (ukb , πkb ) plane (see fig. 2). -10 This result has a rather simple interpretation. The fact that numen fluctuations were born into the semi-classical universe in their ground state, with a time-independent fre- quency α , means that they functioned exactly like the Fig. 2. Schematic diagram showing the Wigner function of the coher- kb ent numen state (see Eqn. 63). The state begins at the right, where coherent states of a conventional quantum harmonic oscil-

αkb τ = π/4, and moves clockwise about (0, 0). The second peak is lator. They therefore acquired classical correlations in the k k plotted at α b τ =3π/4, and the third at α b τ =5π/4. In generat- (ukb , πkb ) phase space from the very beginning. ing these plots, the following values were chosen for the constants in

Equation (63): αkb = 3, and µ = 3. 5. Conclusions a classical harmonic oscillator. Such a coherent state is considered to be the most clas- sical of all states because their Wigner function peaks over The Planck data suggest that the primordial power spec- tiny regions in phase space, and retains this configuration trum has a cutoff, kmin, probably associated with the first for all times. Given a value of uk for such a state, one quantum fluctuation to emerge out of the Planck domain may then obtain the corresponding canonical momentum into the semi-classical universe. Moreover, this birth must πk very close to the value calculated using classical dynam- have occurred at about the Planck time—the earliest mo- ics. The Wigner function thus follows a classical trajectory, ment permitted by our current theories following the Big with minimal outward spread in all phase-space directions. Bang. It is easy to appreciate the physical significance of To see this quantitatively, let us examine the Wigner such an interpretation, because our current (classical) the- function for a numen fluctuation, ory of gravity is valid only down to the Planck scale, not be- dy yond. It is therefore tempting to view the manner in which ∗ 2i πkb y Ψk (ukb , πkb ) Ψ (ukb + y)e Ψkb (ukb y) , these (numen) quantum fluctuations were born as a physi- W b ≡ π kb − Z (58) cal consequence of the transition from quantum gravity to which is reminiscent of the quantum harmonic oscillator. general relativity. Its derivation is very well known, so we won’t dwell on the Insofar as the question of classicalization is concerned, details (see, e.g., ref. [56]). With the ground-state solution this picture effectively removes the hurdles faced by infla- to the Schr¨odinger Equation (21), tion. Leaving aside the related issue that the existence of kmin challenges the viability of slow-roll inflation to simul- 1/4 2 α b k k −α b ukb /2 Ψkb [ukb (τ)] = e , (59) taneously explain the fluctuation spectrum and mitigate π the horizon problem, which has been dealt with elsewhere   we may easily evaluate the integral and find that [64], one cannot ignore the challenge of explaining how fluc- 2 2 1 −αk uk −πk /αk tuations in the inflaton field successfully transitioned from (u , π )= e b b b b . (60) WΨkb kb kb π the quantum to classical domains.

To see how k varies in time, let us briefly return to With the interpretation we have examined in this paper, WΨ b Equations (40) and (46), and solve for the time evolution of two factors stand out clearly as the most influential. These ukb (τ) and πkb (τ) in terms of their values at τ = 0: ukb (0) are (i) the birth of quantum fluctuations with a fixed length and πkb (0). We find that scale at the Planck time constitutes a ‘process’ that effec- tively replaces the need for a ‘measurement’ in quantum πkb ukb (0) = ukb (τ) cos(αkb τ) sin(αkb τ) mechanics; and (ii) the implied scalar field potential has an − αkb equation-of-state consistent with the zero active mass con- π (0) = π (τ) cos(α τ)+ α u sin(α τ) . (61) kb kb kb kb kb kb dition in general relativity. The spacetime associated with Now take the Wigner function at time τ = 0 to represent the expansion profile in this cosmology would have allowed the lowest energy state for the field amplitude ukb (0) shifted the fluctuations to emerge in their ground state, with a by a constant µ. That is, time-independent frequency, αk. This makes all the differ- 2 2 ence because the numen quantum fluctuations were there- 1 −αk (uk −µ) −π /αk (u , π , 0) = e b b kb b . (62) WΨkb kb kb π fore essentially quantum harmonic oscillators, with classi- 12 cal correlations of their canonical variables from the very [19] M. Mijic, IJMP-D 6 (1997) 505 beginning. [20] W.-L. Lee & L.-Z. Fang, Europhys. Lett. 56 (2001) 904 24 Should the proposal we have made in this paper turn out [21] W. H. Zurek, PRD (1981) 1516 [22] E. Joos & H. D. Zeh, Z. Phys. B 59 (1985) 223 to be correct, an obvious question concerns the need for [23] C. Kiefer, D. Polarski & A. A. Starobinsky, Int. J. Mod. Phys. inflation in cosmology. 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14 Figures

Figure 1

Schematic diagram showing the amplitude (plotted along z) of a numen eld operator projected onto the xy-plane. Please see the Manuscript PDF le for the complete gure caption. Figure 2

Schematic diagram showing the Wigner function of the coherent numen state. Please see the Manuscript PDF le for the complete gure caption.