# Classicalization of Quantum Fluctuations at the Planck Scale
in the $R_{\rm h}=ct$ Universe

Fulvio Melia<EMAIL_ADDRESS>Department of Physics, the Applied Math
Program, and Department of Astronomy,
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 time.
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 mechanics. 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
ground state with a time-independent frequency. They were therefore
effectively quantum harmonic oscillators with classical correlations in phase
space from the very beginning.

###### keywords:

cosmology , quantum fluctuations , inflation

###### PACS:

04.20.Ex , 95.36.+x , 98.80.-k , 98.80.Jk

††journal: Phys. Lett. B

##  1 Introduction

In standard inflationary $\Lambda$CDM cosmology, the early universe underwent
a phase of quasi-exponential expansion due to the action of a scalar
(inflaton) field with a near-flat potential, a process typically referred to
as ‘slow-roll’ inflation [1, 2, 3, 4, 5].

The concept of inflation was introduced to solve several global
inconsistencies with standard Big Bang cosmology, notably its horizon problem,
but an even more important consequence of inflation was realized with the
introduction of quantum effects [6, 7, 8, 9, 10]. Small inhomogeneous
fluctuations on top of an otherwise isotropic and homogeneous background are
now broadly believed to be the explanation for the anisotropies in the cosmic
microwave background (CMB) and the formation of large-scale structure [11].
Beginning as quantum seeds in the distant conformal past, well before the
Planck time $t_{\rm Pl}$, these inhomogeneities grew with the expansion of the
universe and produced a near scale-free power spectrum as they crossed the
Hubble horizon [12]. Indeed, the small departure from a perfectly scale-free
distribution is viewed (in this model) as a consequence of ‘Hubble friction,’
whose strength and duration are directly attributable to the detailed shape of
the inflaton potential. Considerable work has been carried out over the past
four decades to use this framework in order to constrain the physical
conditions in the early universe [13, 14, 15, 16, 17], at energies up to the
grand unified scale at $\sim 10^{15}$ GeV.

But the CMB anisotropies and galaxy clusters are classical, so we are faced
with the problem of understanding how quantum fluctuations in the inflaton
field transitioned into purely classical objects—an issue closely related to
the long-standing ‘measurement’ problem in quantum mechanics, i.e., how does a
deterministic outcome appear in a measurement process performed on a quantum
system prepared in a superposed state? Different authors use a range of
descriptions to characterize the distinctions between quantum and classical
states (see, e.g., refs. [18, 19, 20]) but, at a fundamental level, a
successful transition from the former to the latter requires the substantial
elimination of interference between the various degrees of freedom in a
superposed quantum state and the subsequent appearance of macroscopic
variables. The second condition is the emergence of classical correlations in
the phase space of such canonical variables. In other words, classical
trajectories (e.g., orbits) with well-defined values of these variables need
to be established.

The age-old (mostly heuristic) Copenhagen interpretation suggests that a
quantum system remains in its superposed state until an observer performs a
measurement on it, the interaction of which causes it to ‘collapse’ into one
of the eigenstates of the observable being measured [21]. The original version
of this concept has since been refined with the phenomenon of decoherence, in
which the system becomes entangled with its environment [22], consisting of a
very large number of degrees of freedom, bringing the original quantum state
into an ensemble of classical looking ones. The manifestation of this problem
specifically in the cosmological context has sometimes been framed in the
sense that the CMB anisotropies constitute a measurement of the field variable
[23, 24, 25].

The formalism based on decoherence, however, is subject to considerable
debate. It has been argued by some [26, 27, 28] that decoherence by itself
does not solve the problem of a single outcome. As we shall discuss later in
this paper, this mechanism is arguably even less likely to be relevant to the
early universe—a ‘closed’ system where the distinction between the quantum
state and an ‘environment’ is essentially nonexistent. Other types of
approaches, such as Bohmian mechanics [29, 30], do not by themselves predict
testable features that may be verified or refuted.

Our entry point into this discussion is motivated by several significant new
developments in the updated analysis of the latest Planck data release [31],
which build on many attempts made over the past two decades to find specific
features associated with the primordial power spectrum, ${\mathcal{P}}(k)$.
For example, several large-angle anomalies have been present in the CMB maps
since the 1990’s. The most recent studies [32, 33] of the Planck data have
extended our ability to examine whether these issues are more likely due to
instrumental or other systematic effects, or whether they truly represent real
characteristics in ${\mathcal{P}}(k)$. And in § 3.1, we shall briefly discuss
why these anomalies may now be viewed as possibly being due to a cutoff
$k_{\rm min}$ in the power spectrum. Its value, however, calls into question
whether slow-roll inflation remains viable [64], at least based on the
inflaton potentials proposed thus far. As we shall explore in § 3.2, a more
interesting interpretation for the appearance of $k_{\rm min}$ is that this
represents the first mode to exit from the Planck scale into the semi-
classical universe at about the Planck time. If correct, this interpretation
would clearly have a considerable impact on the quantum-to-classical
transition.

In this paper, we focus on how this interpretation could alter our view of the
long-standing classicalization problem in the early universe. The conventional
view has been that quantum fluctuations were seeded in the Bunch-Davies
vacuum, though without much guidance other than they emerged in the ground
state. But if they appeared at the Planck scale at about the Planck time, the
manner in which the quantum states were ‘prepared’ may have some bearing on
the classicalization problem. In § 2, we shall summarize the current status
with the quantum-to-classical transition of fluctuations in the inflaton
field, and then transition into a discussion of the analogous situation for a
non-inflationary scalar field with the zero active mass equation-of-state
(relevant to the above interpretation of $k_{\rm min}$) in § 3. We shall
describe the impact of these new developments on the classicalization question
in § 4 and end with our conclusions in § V.

##  2 Classicalization of Quantum Fluctuations in the Inflaton Field

###  2.1 Quantum Fluctuations in the Inflaton Field

The essential steps for deriving the perturbation growth equation are by now
well known, and one may find many accounts of this procedure in both the
primary and secondary literature. For the sake of brevity, we here show only
some key results—chiefly those that will also be relevant to our discussion of
quantum fluctuations at the Planck scale in §§ 3 and 4—and refer the reader to
several other influential publications for all the details.

The perturbed Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime for
linearized scalar fluctuations is given by the line element [34, 35, 36, 37]

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle(1+2A)\,dt^{2}-2a(t)(\partial_{i}B)\,dt\,dx^{i}-$ (1)
$\displaystyle\hskip-7.22743pta^{2}(t)\left[(1-2\psi)\delta_{ij}+2(\partial_{i}\partial_{j}E)+h_{ij}\right]\,dx^{i}\,dx^{j}\,,\quad$

where indices $i$ and $j$ denote spatial coordinates, $a(t)$ is the expansion
factor, and $A$, $B$, $\psi$ and $E$ describe the scalar metric perturbations,
while $h_{ij}$ are the tensor perturbations.

For small perturbations about the homogeneous scalar field $\phi_{0}(t)$,

$\phi(t,\vec{x})=\phi_{0}(t)+\delta\phi(t,\vec{x})\;,$ (2)

one can identify the curvature perturbation ${\mathcal{R}}$ on hypersurfaces
orthogonal to worldlines in the comoving frame written as a gauge invariant
combination of the scalar field perturbation $\delta\phi$ and the metric
perturbation $\psi$ [34]:

${\mathcal{R}}\equiv\psi+\left({H\over\dot{\phi}}\right)\,\delta\phi\;.$ (3)

A solution for ${\mathcal{R}}$ may then be obtained by (i) using an expansion
in Fourier modes,

${\mathcal{R}}(\tau,{\bf x})\equiv\int{d^{3}{\bf
k}\over(2\pi)^{3/2}}\;{\mathcal{R}}_{\bf k}(\tau)e^{i{\bf k}\cdot{\bf x}}\;,$
(4)

and (ii) inserting the linearized metric (Eq. 1) into Einstein’s equations,
which together yield the following perturbed equation of motion for each mode
k:

${\mathcal{R}}_{\bf k}^{\prime\prime}+2\left({z^{\prime}\over
z}\right){\mathcal{R}}_{\bf k}^{\prime}+k^{2}{\mathcal{R}}_{\bf k}=0\;.$ (5)

Overprime denotes a derivative with respect to conformal time $d\tau=dt/a(t)$,
and we have defined the variable

$z\equiv{a(t)(\rho_{\phi}+p_{\phi})^{1/2}\over H}\;.$ (6)

Using the canonically normalized Mukhanov-Sasaki variable [35, 36]

$u_{\bf k}\equiv z{\mathcal{R}}_{\bf k}\;,$ (7)

Equation (5) may be recast into the more familiar form of a parametric
oscillator with a time-dependent frequency, known (in this context) as the
Mukhanov-Sasaki equation,

$u_{\bf k}^{\prime\prime}+\omega_{\bf k}^{2}(\tau)\,u_{\bf k}=0\;,$ (8)

where the time-dependent frequency is

$\omega_{\bf k}(\tau)^{2}\equiv k^{2}-{z^{\prime\prime}\over z}\;.$ (9)

In Minkowski spacetime, the scale factor $a(\tau)$ is constant and
$\omega_{\bf k}=k$, which reduces Equation (8) to that of the more basic
harmonic oscillator with a constant frequency.

Since $\omega_{\bf k}(\tau)$ depends solely on $\tau$ and $k$ (not the
direction of k), the most general solution to the Mukhanov-Sasaki equation may
be written

$u_{\bf k}=a_{\bf k}^{-}u_{k}(\tau)+a_{\bf-k}^{+}u_{k}^{*}(\tau)\;,$ (10)

where $u_{k}(\tau)$ and its complex conjugate are linearly independent
solutions to Equation (8), and are the same for all Fourier modes with
$k=|{\bf k}|$. In general, however, the integration constants $a_{\bf k}^{-}$
and $a_{\bf-k}^{+}$ may depend on the direction of k. From Equations (4) and
(7), one may thus write the complete solution to the Mukhanov-Sasaki equation
as

$u(\tau,{\bf x})=\int{d^{3}{\bf k}\over(2\pi)^{3/2}}\left[a_{\bf
k}^{-}u_{k}(\tau)e^{i{\bf k}\cdot{\bf x}}+a_{\bf
k}^{+}u_{k}^{*}(\tau)e^{-i{\bf k}\cdot{\bf x}}\right],\quad$ (11)

which is manifestly real since $a_{\bf k}^{+}=(a_{\bf k}^{-})^{*}$.

Our derivation of Equation (11) has made no reference to the definite size of
the inhomogeneities. This result therefore merely shows us the dynamics of
such inhomogeneities if they came into existence. To address their presence
and magnitude, one must consider states at the quantum level, where the
fluctuation field becomes an operator. The next step is therefore the
quantization of the field, noting that the Hilbert space for the quantum field
will be a direct product of individual Hilbert spaces for the Fourier modes.

The field $u(\tau,{\bf x})$ is quantized like the harmonic oscillator, except
that the frequency (Eq. 9) is now time dependent, which is itself a
consequence of the curved spacetime within which these modes are evolving. To
properly normalize the quantum fluctuation, one must therefore choose the
vacuum carefully, since the spacetime is generally not Minkowskian.

The field $u(\tau,{\bf x})$ and its canonical conjugate ‘momentum’
$\pi(\tau,{\bf x})\equiv u^{\prime}(\tau,{\bf x})$ are promoted to quantum
operators $\hat{u}$ and $\hat{\pi}$, satisfying the standard equal-time
canonical quantization relations

$\displaystyle[\hat{u}(\tau,{\bf x}_{1}),\hat{\pi}(\tau,{\bf
x}_{2})]=i\delta^{3}({\bf x}_{1}-{\bf x}_{2})$
$\displaystyle[\hat{u}(\tau,{\bf x}_{1}),\hat{u}(\tau,{\bf x}_{2})]=0$
$\displaystyle[\hat{\pi}(\tau,{\bf x}_{1}),\hat{\pi}(\tau,{\bf x}_{2})]=0\;.$
(12)

The constants of integration $a_{\bf k}^{-}$ and $a_{\bf k}^{+}$ in the mode
expansion of $u(\tau,{\bf x})$ become operators $\hat{a}_{\bf k}^{-}$ and
$\hat{a}_{\bf k}^{+}$, so that

$\hat{u}(\tau,{\bf x})=\int{d^{3}{\bf k}\over(2\pi)^{3/2}}\left[\hat{a}_{\bf
k}^{-}u_{k}(\tau)e^{i{\bf k}\cdot{\bf x}}+\hat{a}_{\bf
k}^{+}u_{k}^{*}(\tau)e^{-i{\bf k}\cdot{\bf x}}\right].\quad$ (13)

We thus see from Equations (12) and (13) that

$\displaystyle[\hat{a}^{-}_{\bf k},\hat{a}^{+}_{{\bf
k}^{\prime}}]=\delta^{3}({\bf k}-{\bf k}^{\prime})$
$\displaystyle[\hat{a}^{-}_{\bf k},\hat{a}^{-}_{{\bf k}^{\prime}}]=0$
$\displaystyle[\hat{a}^{+}_{\bf k},\hat{a}^{+}_{{\bf k}^{\prime}}]=0\,,$ (14)

which allows us to interpret $\hat{a}_{\bf k}^{-}$ and $\hat{a}_{\bf k}^{+}$
as annihilation and creation operators, respectively, creating and
annihilating excitations, or particles, of the field $u(\tau,{\bf x})$. And in
the usual fashion, the quantum states in the Hilbert space are constructed by
a repeated application of the creation operator,

$|n_{{\bf k}_{1}},n_{{\bf k}_{2}},\cdots\rangle={1\over\sqrt{n_{{\bf
k}_{1}}!\,n_{{\bf k}_{2}}!\cdots}}\left\\{(\hat{a}_{{\bf k}_{1}}^{+})^{n_{{\bf
k}_{1}}}(\hat{a}_{{\bf k}_{2}}^{+})^{n_{{\bf
k}_{2}}}\cdots\right\\}|0\rangle,\;$ (15)

starting with the vacuum state $|0\rangle$, which is defined by

$\hat{a}_{\bf k}^{-}|0\rangle=0\;.$ (16)

In de Sitter space, where $a(\tau)=-(H\tau)^{-1}$ (in terms of the Hubble
constant $H$ during inflation), the exact solution to Equation (8) is

$u_{k}(\tau)=A_{k}{e^{-ik\tau}\over\sqrt{2k}}\left(1-{i\over
k\tau}\right)+B_{k}{e^{ik\tau}\over\sqrt{2k}}\left(1+{i\over k\tau}\right)\;,$
(17)

where $A_{k}$ and $B_{k}$ are integration constants to be fixed by the choice
of vacuum. In a time-independent spacetime, the normalization of the
wavefunction is obtained by imposing canonical quantization or, equivalently,
by minimizing the expectation value of the Hamiltonian in the vacuum state.
For example, in Minkowski space, one would find from Equation (8) with
$\omega_{k}=k$, and the Hamiltonian written in terms of $u_{k}$ and $\pi_{k}$,
that

$u_{k}(\tau)={e^{-ik\tau}\over\sqrt{2k}}\;\;{\rm(Minkowski)}\;.$ (18)

In de Sitter spacetime, such a solution would therefore correspond to
$|\tau|\rightarrow\infty$, and $A_{k}=1$, $B_{k}=0$. This is interpreted to
mean that in the remote conformal past, all modes of current interest were
much smaller than the Hubble radius, allowing us to ignore curvature effects
on the mode normalization, thereby defining a unique physical vacuum known as
the Bunch-Davies vacuum [38].

With the time evolution

$u_{k}(\tau)={e^{-ik\tau}\over\sqrt{2k}}\left(1-{i\over
k\tau}\right)\;\;{\rm(de\;Sitter)}\;,$ (19)

Equation (13) gives us the full form of the field operator $\hat{u}(\tau,{\bf
x})$, which may be used to find the quantum states of the inflaton scalar
field. As we shall see in § 3, much of this derivation remains intact for the
(non-inflationary) numen field as well, though with several crucial
differences we shall discuss shortly. (As we shall see in § 3.2, it will be
necessary to clearly distinguish between inflaton scalar fields and 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 state with a wavefunctional
that gives the probability amplitudes for different field configurations in
Fourier space. Though we have developed the expression for the field operator,
$\hat{u}(\tau,{\bf x})$, in the Heisenberg picture, finding the various
occupation numbers in a quantum state such as Equation (15) is more easily
accomplished using a functional approach in the Schrödinger picture (see,
e.g., refs. [39, 40]).

As long as the various $k$-modes are all independent, the Hilbert space for
the field operator $\hat{u}(\tau,{\bf x})$, containing eigenstates
$|u\rangle$, is simply a direct product of Hilbert spaces for the different
Fourier components. If a state is initially decomposed as $|\Psi\rangle$, the
probability amplitude of measuring the field configuration $u(\tau,{\bf x})$
is then the wave functional $\Psi[u(\tau,{\bf x})]\equiv\langle u|\Psi\rangle$
where, in the Schrödinger approach, one may factor it into independent mode
components according to:

$\Psi[u(\tau,{\bf x})]=\prod_{k}\Psi_{\bf k}[u_{k}(\tau)]\;,$ (20)

each of which is a solution to Schrödinger’s equation:

$i{\partial\Psi_{\bf k}\over\partial\tau}=\hat{H}_{\bf k}\Psi_{\bf k}\;.$ (21)

The Hamilton operator in Fourier space may be written

$\hat{H}={1\over 2}\int d^{3}{\bf k}\,\hat{H}_{\bf k}={1\over 2}\int d^{3}{\bf
k}\left[\hat{\pi}_{\bf k}\hat{\pi}_{\bf k}^{\dagger}+\omega_{\bf
k}^{2}\hat{u}_{\bf k}\hat{u}_{\bf k}^{\dagger}\right]\;.$ (22)

The solution to Equations (21) and (22) for a harmonic oscillator is well
known, and has the form of a Gaussian function:

$\Psi_{\bf k}[u_{\rm k}(\tau)]=N_{\bf k}(\tau)e^{-\Omega_{\bf k}(\tau)(u_{\bf
k})^{2}}\;,$ (23)

where

$|N_{\bf k}|\equiv\left({2{\rm Re}\\{\Omega_{\bf k}\\}\over\pi}\right)^{1/4}$
(24)

is the normalization factor, and

$\Omega_{\bf k}\equiv-{i\over 2}{f_{k}^{\prime}\over f_{k}}$ (25)

is written in terms of the function $f_{k}$, the solution to Equation (8),
i.e.,

$f_{k}={1\over\sqrt{2k}}\left(1-{i\over k\tau}\right)e^{-ik\tau}\;.$ (26)

The functional $\Psi[u(\tau,{\bf x})]$, representing a quantum fluctuation in
the inflaton field, is a superposition of many separate Fourier states
$\Psi_{\bf k}$ (Eqn. 20). Somehow, this quantum description, which includes
interference between the various modes and no a priori classical correlations
in phase space between the field amplitudes and their canonical partners, must
have transitioned into classical fluctuations, characterized in part by a
diagonalized density matrix and well defined dynamical trajectories. How this
might have happened is the issue we shall address next.

###  2.2 The Quantum to Classical Transition

The question concerning whether or not the state $\Psi[u(\tau,{\bf x})]$ can
complete the quantum to classical transition based on the factors we have
summarized above has received considerable attention over the past four
decades, and is still an open one [41, 42, 43, 44, 39, 45, 46, 47, 48, 49, 50,
51]. As noted in the introduction, the cosmological scalar perturbations must
satisfy at least two conditions to classicalize: (i) the system must undergo
decoherence, so that quantum interference becomes negligible and macroscopic
variables appear. We see a universe with well-defined, classical perturbation
field values, not superpositions of multiple components represented by
Equation (20). (ii) One must see the establishment of classical correlations
in phase space, meaning that the allowed dispersion in the canonical variables
$(u,\pi)$ becomes insignificantly smaller than their classical orbital values.

Different authors have proposed somewhat different schemes for this
transition, indicating that no consensus has yet been reached. In part, this
is due to the fact that the conventional rules one may rely on in ordinary
quantum mechanical applications are not necessarily all available in
cosmology. To begin with, the system being studied quantum mechanically is the
entire Universe. There is no possible separation into a subsystem of interest,
its environment and the observer. Moreover, there is just one universe,
obviating any possibility of adopting the statistical ensemble interpretation
of quantum mechanical measurements. Even more importantly, the observer is
here a consequence of the quantum to classical transition, and could not have
played any causal part in it. For these (and other) reasons, the system being
studied is thus unusual by typical quantum mechanical standards.

A common step taken by the various approaches to solving this problem is to
relegate most of the very large number of degrees of freedom in
$\Psi[u(\tau,{\bf x})]$ to an ‘environment’ and then to ignore them, allowing
one to evolve the reduced density matrix $\hat{\rho}$ for the subset of
remaining (presumably more interesting) observables. Under some circumstances,
also involving suitable time averaging, this procedure eventually diagonalizes
$\hat{\rho}$, representing a complete mitigation of any interference effects,
which is interpreted as the emergence of classical behavior.

This is not entirely satisfactory, however, because decoherence has not yet
successfully solved the long-standing ‘measurement’ problem in quantum
mechanics. Certainly, the diagonalization of $\hat{\rho}$ removes certain
quantum traits from the system, but it is not clear that a non-interfering set
of simultaneous co-existing possibilities is necessarily classical (see, e.g.,
ref. [52]). Note, for example, that changing the Hilbert space of the quantum
system to a different basis would destroy the diagonal nature of the density
matrix. The interpretation of $\hat{\rho}$ is therefore subject to various
observer-dependent choices. What is lacking is a clear understanding of how to
choose the basis and an interaction specific eigenstate for the preferred
observable. In our everyday experience, some progress may be made by allowing
the measurement device to ‘select’ the basis, and adopting the ensemble
interpretation for the density matrix. But these features are obviously
missing in the cosmological context [46, 53].

To solve the classicalization problem using decoherence in a cosmological
setting, one must therefore identify a physical mechanism and a preferential
basis it selects. It is also necessary to find a criterion for separating the
large number of degrees of freedom into the ‘interesting set’ and ‘the
environment’ dictated by the physical problem at hand. At least some of the
proposed treatments then appeal to ‘a specific realization’ of the stochastic
variables [39, 49], sounding very much like a conventional ‘collapse’ from the
statistical description of the universe to one of the members in the
statistical ensemble. But no insight is provided into how and when such a
transition occurred in the real universe.

To address the second requirement, we begin by noting that in the remote
conformal past, where $\Omega_{\bf k}\rightarrow k/2$, the wavefunctional
$\Psi_{\bf k}$ represented the ground state of a harmonic oscillator,
consistent with the previously described Bunch-Davies vacuum. For the largest
modes (those of relevance to the structure we observe today), the function
$\Omega_{\bf k}$ acquired a non-trivial time dependence and Equation (23)
evolved into a squeezed state—meaning that, for these modes, there exists a
direction in the $(u_{\bf k},\pi_{\bf k})$ plane where the dispersion is
exponentially small, while the dispersion in a perpendicular direction is very
large. The resultant linear combination of $u_{\bf k}$ and $\pi_{\bf k}$ for
which the dispersion is minimized is often viewed as the emerging classical
phase-space trajectory.

A convenient tool to study this process, and its possible relevance to the
classicalization question, is the Wigner function (see, e.g., the reviews in
refs. [54, 55]; for a more pedagogical account, see also ref. [56]), defined
by

$\displaystyle{\mathcal{W}}(u_{\bf k}^{R},u_{\bf k}^{I},\pi_{\bf
k}^{R},\pi_{\bf k}^{I})$ $\displaystyle=$ $\displaystyle{1\over(2\pi)^{2}}\int
dx\,dy$ (27) $\displaystyle\hskip-57.81621pt\Psi_{\bf k}\left(u_{\bf
k}^{R}-{x\over 2},u_{\bf k}^{I}-{y\over 2}\right)e^{-i\pi_{\bf
k}^{R}x-i\pi_{\bf k}^{I}y}\times\qquad$
$\displaystyle\hskip-28.90755pt\Psi_{\bf k}\left(u_{\bf k}^{R}+{x\over
2},u_{\bf k}^{I}+{y\over 2}\right)\;,$

in terms of the real and imaginary parts of $u_{\bf k}$ and $\pi_{\bf k}$.
Quantum mechanics inherently deals with probabilities, while classical physics
deals with well-defined trajectories in phase space. The Wigner function
provides a means of representing the density distributions in $u_{\bf k}$ and
$\pi_{\bf k}$ for comparison with the ensemble of trajectories one would get
using classical means.

The literature on Wigner functions is extensive (including the aforementiond
reviews in refs. [54, 55, 56]). Its properties suggest that it behaves like a
probability distribution in $(u,\pi)$ phase space, except that it can
sometimes take on negative values, so it is generally not a true probability
distribution. Moreover, points in the $(u,\pi)$ space to not represent actual
states of the system because the values of $u_{\bf k}$ and $\pi_{\bf k}$
cannot be determined precisely at the same time. Nevertheless, ${\mathcal{W}}$
can be used to visualize correlations between $u_{\bf k}$ and $\pi_{\bf k}$,
particularly for Gaussian states, such as we have in Equation (23), for which
the Wigner function is indeed always positive definite.

It is trivial to show in the case of Equation (23) that the explicit form of
${\mathcal{W}}$ is

$\displaystyle{\mathcal{W}}(u_{\bf k}^{R},u_{\bf k}^{I},\pi_{\bf
k}^{R},\pi_{\bf k}^{I})$ $\displaystyle=$
$\displaystyle{1\over\pi^{2}}e^{-{\rm Re}\\{\Omega_{\bf k}\\}\left[(u_{\bf
k}^{R})^{2}+(u_{\bf k}^{I})^{2}\right]}\times$ (28)
$\displaystyle\hskip-93.95122pte^{-(\pi_{\bf k}^{R}+{\rm Im}\\{\Omega_{\bf
k}\\}u_{\bf k}^{R})^{2}/{\rm Re}\\{\Omega_{\bf k}\\}}e^{-(\pi_{\bf k}^{I}+{\rm
Im}\\{\Omega_{\bf k}\\}u_{\bf k}^{I})^{2}/{\rm Re}\\{\Omega_{\bf
k}\\}},\qquad$

with which one may clearly see the effect of strong squeezing as $\Omega_{\bf
k}$ changes dramatically with time (Eqns. 25 and 26) during the cosmic
expansion.

As long as $\Omega_{\bf k}\rightarrow k/2$, ${\mathcal{W}}$ is peaked over a
small region of phase space, representing the Wigner function of a coherent
state, i.e., the ground state of a harmonic oscillator. We shall return to
this in § 4 below, where we consider the corresponding Wigner function for the
numen quantum fluctuations. We shall see that this limiting situation takes on
added significance in that case, given that $\Omega_{\bf k}$ actually remains
constant for a numen field. But $\Omega_{\bf k}$ is definitely not constant
here. As $\tau$ advances, ${\mathcal{W}}$ spreads and acquires a cigar shape
typical of squeezed states [57], with a drastically reduced dispersion around
$\pi_{\bf k}$ and a correspondingly enormous dispersion around $u_{\bf k}$.
But the overall dispersion may be minimized, as we alluded to earlier, by
choosing an appropriate linear combination of $u_{\bf k}$ and $\pi_{\bf k}$,
an outcome that some argue represents the emerging correlation in classical
phase space.

One may therefore think of a squeezed state as a state with the minimal
uncertainty, though not in terms of the original $(u,\pi)$ variables [39, 47].
In the cosmological context, the inflationary expansion created an uncertainty
on the value of the field and its conjugate momentum that was much larger than
the minimum uncertainty implied by the Heisenberg uncertainty principle. The
minimum uncertainty was instead associated with a new pair of ‘rotated’
canonical variables. But this outcome has also been challenged as not
necessarily representing a classical system [44, 51].

Take the following situation as an example. Consider an electron in a minimal
wavepacket localized at the origin, with an uncertainty $\Delta x$ in position
and $\Delta p=\hbar/(2\Delta x)$ in momentum. Then form a superposition of
this state with an identical one after a translation by a large distance $D$.
The overall uncertainty is now $\sim D\hbar/(2\Delta x)$, which can be
increased arbitrarily by simply choosing a sufficiently large distance $D$.
This situation is clearly analogous to one of our squeezed states, but the
superposition we have created is nonetheless not classical.

And then there is the issue of definite outcomes, also known as the
‘measurement problem’ [58], as we alluded to in the introduction. This aspect
of quantum mechanics has been with us for over a century. Even if decoherence
were successful in diagonalizing the density matrix, it cannot solve the
definite outcome problem, which is far worse in cosmology than it is in
typical laboratory situations. The usual approach of adopting the Copenhagen
interpretation of quantum mechanics, in which a measurement ‘collapses’ the
state vector of the system into an eigenstate corresponding to the measurement
result, does not work in the early Universe, where there were no measurement
devices or observers present.

While decoherence might have diagonalized the reduced density matrix of the
system to an ensemble of classically observable universes, it does not explain
how a certain universe was singled out to be observed [28]. Perhaps the
Copenhagen interpretation is just not well suited to cosmology. In addition,
if decoherence resulted from an environment comprised of the ‘non-interesting’
degrees of freedom in $\Psi[u(\tau,{\bf x})]$, how could this happen when
inflation would have driven all such fields towards their vacuum states?
Moreover, why do we have the privilege of deciding which degrees of freedom to
relegate to the background based solely on whether our current technology
allows us to observe them today?

We shall now divert our attention away from the traditional inflationary
scenario we have been describing, and seriously consider the implications of a
novel feature emerging from the latest release of the Planck data: the
observational measurement of a cutoff in the primordial spectrum bears
directly on the question of how and when quantum fluctuations were generated
in the early Universe. We shall study what new ideas and constraints this
brings to the classicalization process.

##  3 Quantum Fluctuations at the Planck Scale

###  3.1 Emergence of a Cutoff in the primordial power spectrum
${\mathcal{P}}(k)$

All three of the major satellite missions designed to study the CMB—COBE [59];
WMAP [60]; and Planck [31]— have uncovered several anomalies in its
fluctuation spectrum. The two most prominent among them are: (1) an
unexpectedly low level—perhaps even a complete absence—of correlation at large
angles (i.e., $\theta\gtrsim 60^{\circ}$), manifested via the angular
correlation function, $C(\theta)$; and (2) relatively weak power in the lowest
multipole moments of the angular power spectrum, $C_{\ell}$. Their origin,
however, is still subject to considerable debate, many arguing in favor of a
misinterpretation or the result of unknown systematics, such as an incorrect
foreground subtraction (see refs. [61, 62] for reviews). This uncertainty is
also fueled in large part by a persistent lack of clarity concerning how
quantum fluctuations were seeded in the early universe [63, 64].

The large-scale anomalies stand in sharp contrast to our overall success
interpreting the CMB anisotropies at angles smaller than $1^{\circ}$. Over the
past several decades, a concerted effort has therefore been made in attempting
to identify features in the primordial power spectrum, ${\mathcal{P}}(k)$,
responsible for their origin. For example, in their study of the CMB angular
power spectrum, Shafieloo & Souradeep [65] assumed an exponential cutoff at
low $k$-modes, and identified a turnover generally consistent with the most
recent measurement we shall discuss below (see Eq. 29). This early treatment
was based on WMAP observations [60], however, not the higher precision Planck
measurements [31] we have today, so the reality of a non-zero $k_{\rm min}$
remained somewhat controversial.

In followup work, Nicholson et al. [66] and Hazra et al. [67] inferred a ‘dip’
in ${\mathcal{P}}(k)$ on a scale $k\sim 0.002$ Mpc-1 for the WMAP data,
confirming the outcome of an alternative approach by Ichiki et al. [68] that
identified an oscillatory modulation around $k\sim 0.009$ Mpc-1. In closely
aligned work, Tocchini et al. [69] modeled both a dip at $k\sim 0.035$ Mpc-1
and a ‘bump’ at $k\sim 0.05$ Mpc-1. Like the others, though, these features
were based solely on WMAP observations and therefore appeared to be merely
suggestive rather than compelling. Tocchini et al. [70] improved on this
analysis considerably, and found evidence for three features in
${\mathcal{P}}(k)$, one of which was a cutoff at $\sim 0.0001-0.001$ Mpc-1 at
a confidence level of $\sim 2\sigma$. Complementary work by Hunt & Subir [71,
72] confirmed the likely existence of a cutoff $k<5\times 10^{-4}$ Mpc-1 but,
as before, also concluded that more accurate Planck data would eventually be
needed to confirm these results more robustly.

The subsequent Planck observations have not only largely confirmed these
earlier results, but have provided us with a greatly improved precision in the
‘measurement’ of $k_{\rm min}$. For example, the Planck Collaboration [73] fit
a cutoff to the CMB angular power spectrum and found a value $\sim 3-4\times
10^{-4}$ Mpc-1. Still, though these studies all pointed to the likely
existence of a cutoff in ${\mathcal{P}}(k)$, a non-zero $k_{\rm min}$ could
not be claimed with a confidence level exceeding $1-2\sigma$.

This situation improved considerably when, instead of looking solely at the
power spectrum, the impact of a cutoff was also considered on the angular
correlation function, independently of the angular power spectrum. Two
separate (though complementary) studies of the latest Planck data release have
provided more compelling evidence that the two large-angle features in the CMB
anisotropies may be real. The first of these [32] demonstrates that the most
likely explanation for the missing large-angle correlations is a cutoff,

$k_{\rm min}={4.34\pm 0.5\over r_{\rm cmb}}\;,$ (29)

in the primordial power spectrum, ${\mathcal{P}}(k)$, where $r_{\rm cmb}$ is
the comoving distance to the surface of last scattering. For the
Planck-$\Lambda$CDM parameters, $r_{\rm cmb}\approx 13,804$ Mpc [31], and we
therefore have $k_{\rm min}=(3.14\pm 0.36)\times 10^{-4}$ Mpc-1. A zero cutoff
(i.e., $k_{\rm min}=0$) is ruled out by the $C(\theta)$ data at a confidence
level exceeding $\sim 8\sigma$.

A subsequent study [33] focused on the CMB angular power spectrum itself
(i.e., $C_{\ell}$ versus $\ell$), and its results (i) confirmed that the
introduction of this cutoff in ${\mathcal{P}}(k)$ does not at all affect the
remarkable consistency between the standard inflationary model prediction and
the Planck measurements at $\ell>30$ [31], where the underlying theory is
widely believed to be correct; and (ii) showed that such a cutoff ($k_{\rm
min}$) also self-consistently explains the missing power at large angles,
i.e., the low multipole moments ($\ell=2-5$). The cutoff optimized by fitting
the angular power spectrum over the whole range of $\ell$’s is $k_{\rm
min}=(2.04^{+1.4}_{-0.79})\times 10^{-4}$ Mpc-1, while a fit to the restricted
range $\ell\leq 30$, where the Sachs-Wolfe effect is dominant [74], gives
$k_{\rm min}=(3.3^{+1.7}_{-1.3})\times 10^{-4}$ Mpc-1. The outcome based on
the CMB angular power spectrum therefore rules out a zero cutoff at a
confidence level $\gtrsim 2.6\sigma$. In either case, the inferred value of
$k_{\rm min}$ is fully consistent with the cutoff implied by missing
correlations in $C(\theta)$, and one concludes that both of these large-angle
anomalies are probably due to the same truncation, i.e., $k_{\rm min}\sim
3\times 10^{-4}$ Mpc-1, in ${\mathcal{P}}(k)$. The confidence with which one
may make such a claim depends on whether the cutoff is used to address the
missing power at low $\ell$’s, or the missing correlations at large angles.
Nevertheless, the fact that the same $k_{\rm min}$ apparently solves both
anomalies makes the assumption of a cutoff quite reasonable.

These results reinforce the perception that the small-angle anisotropies (for
$\ell\gtrsim 30$), which are mostly due to acoustic oscillations, are well
understood, while the fluctuations associated with angular correlations at
$\theta\gtrsim 60^{\circ}$, due to the Sachs-Wolfe effect, continue to be
problematic for the standard inflationary picture [59, 60, 31]. The evidence
for a non-zero $k_{\rm min}$ speaks directly to the cosmological expansion
itself. At $\ell\lesssim 30$, we are probing ever closer to the beginning of
inflation, culminating with the cutoff $k_{\rm min}$, signaling the very first
mode that would have crossed the horizon when the quasi-de Sitter phase
started [36, 64, 17].

But if one insists on inflation simultaneously fixing the horizon problem and
accounting for the observed primordial power spectrum, ${\mathcal{P}}(k)$, the
implied time at which the accelerated expansion began would have suppressed
the comoving size of the universe to a tenth of the required value [64, 17].
Moreover, neither a radiation-dominated, nor a kinetic-dominated, phase
preceding inflation could have alleviated this disparity [75, 76].

###  3.2 A Reinterpretation of $k_{\rm min}$

To be clear, the measurement of $k_{\rm min}$ does not completely rule out
inflation, nor even the idea that some slow-roll variant may eventually be
constructed to address the inconsistency described above. At a minimum,
however, the currently proposed inflaton potentials $V(\phi)$ require at least
some modification. Moreover, a cutoff in ${\mathcal{P}}(k)$ does not argue
against the influence of a scalar field, $\phi$, nor anisotropies arising from
its quantum fluctuations, but there is now some motivation to question whether
$V(\phi)$ was truly inflationary.

Over the past decade, some evidence has been accumulating that the cosmic
fluid may possess a zero active mass equation-of-state, $\rho+3p=0$ (in terms
of its total energy density $\rho$ and pressure $p$), supported by over 27
different kinds of observation at low and high redshifts (see Table 2 in ref.
[77] for a recent summary of these results). Such a universe lacks a horizon
problem [78, 79], so the lack of a fully self-consistent inflationary paradigm
may be telling us that the universe does not need it. This is the key
assumption we shall make to reinterpret $k_{\rm min}$ in this paper. In
addition to the growing body of empirical evidence favoring this approach,
there is also theoretical support for the zero active mass equation-of-state
from the ‘Local Flatness Theorem’ in general relativity [80].

As noted earlier, we shall clearly distinguish between the roles played by a
non-inflaton $\phi$ and a conventional inflaton field by informally refering
to the former as a ‘numen’ field, based on our inference that it may represent
the earliest form of substance in the universe. Its equation-of-state is
assumed to be $\rho_{\phi}+3p_{\phi}=0$, and we shall see shortly why this
property appears to provide a more satisfactory interpretation of $k_{\rm
min}$ than an inflaton field.

The background numen field is homogeneous, so its energy density $\rho_{\phi}$
and pressure $p_{\phi}$ are simply given as

$\rho_{\phi}={1\over 2}{\dot{\phi}}^{2}+V(\phi)\;,$ (30)

and

$p_{\phi}={1\over 2}{\dot{\phi}}^{2}-V(\phi)\;.$ (31)

The zero active mass equation-of-state therefore uniquely constrains the
potential to be

$V(\phi)={{\dot{\phi}}^{2}}\;,$ (32)

with the explicit solution

$V(\phi)=V_{0}\,\exp\left\\{-{2\sqrt{4\pi}\over m_{\rm Pl}}\,\phi\right\\}\;,$
(33)

in terms of the Planck mass

$m_{\rm Pl}\equiv{1\over\sqrt{G}}\;.$ (34)

Some may recognize this as a special member of the category of minimally
coupled fields explored in the 1980’s [81, 82, 83, 84], intended to produce
so-called power-law inflation. But unlike the other fields in this cohort, the
numen field’s zero active mass equation-of-state makes it the only member of
this group that does not inflate, since the Friedmann equations with this
density and pressure lead to an expansion factor $a(t)=t/t_{0}$, written in
terms of the age of the universe, $t_{0}$. This normalization of $a(t)$ is
appropriate for a spatially flat FLRW metric, which the observations appear to
be telling us.

With this expansion factor, the conformal time may be written

$\tau(t)=t_{0}\ln a(t)\;,$ (35)

such that the zero of $\tau$ coincides with $t=t_{0}$. The parameter $z$ in
Equation (6) thus becomes

$z={m_{\rm Pl}\over\sqrt{4\pi}}a(t)\;,$ (36)

so that $z^{\prime}/z=1/t_{0}$ and $z^{\prime\prime}/z=1/t_{0}^{2}$. The
resulting curvature perturbation equation analogous to Equation (8) may thus
be written

$u_{\rm k}^{\prime\prime}+\alpha_{\rm k}^{2}u_{\rm k}=0\;,$ (37)

where the frequency (analogous to Eqn. 9) is now given by the expression

$\alpha_{\rm k}\equiv{1\over t_{0}}\sqrt{\left(2\pi R_{\rm h}\over\lambda_{\rm
k}\right)^{2}-1}\;,$ (38)

in terms of the proper wavelength of mode $k$,

$\lambda_{\rm k}\equiv{2\pi\over k}a(t)\;.$ (39)

In this expression, the quantity $R_{\rm h}\equiv c/H=ct$ is the apparent (or
gravitational) radius [85], which defines the Hubble horizon in a spatially
flat universe. The most critical difference between Equations (8) and (9), and
Equations (37) and (38), is that here both $R_{\rm h}$ and $a(t)$ scale
linearly with $t$, and therefore the frequency $\alpha_{\rm k}$ of the numen
quantum fluctuations is always time-independent. This is because the ratio
$R_{\rm h}/\lambda_{k}\propto kR_{\rm h}/a(t)$ is constant for each $k$, and
therefore numen fluctuations do not criss-cross the horizon; once
$\lambda_{\rm k}$ is established upon the mode’s exit into the semi-classical
universe, it remains a fixed fraction of $R_{\rm h}$ as they expand with time.
This feature is crucial to understanding how and why numen quantum
fluctuations provide a more satisfactory explanation than inflation for the
origin of $k_{\rm min}$.

The solution to Equation (37) is that of the standard harmonic oscillator:

$u_{\rm k}(\tau)=\left\\{\begin{array}[]{ll}B(k)\,e^{\pm i\alpha_{\rm
k}\tau}&\mbox{($2\pi R_{\rm h}>\lambda_{\rm k}$)}\\\ B(k)\,e^{\pm|\alpha_{\rm
k}|\tau}&\mbox{($2\pi R_{\rm h}<\lambda_{\rm k}$)}\end{array}\right.\;.$ (40)

That is, all modes with $\lambda_{\rm k}<2\pi R_{\rm h}$ oscillate, while the
super-horizon ones do not, mirroring the behavior of the more conventional
inflaton field. Here, however, the mode with the longest wavelength relevant
to the formation of structure is the one for which $\lambda_{\rm k}(\tau)=2\pi
R_{\rm h}(\tau)$, i.e.,

$k^{\rm numen}|_{\rm min}=1/t_{0}\;.$ (41)

One’s intuition would immediately suggest that $k^{\rm numen}|_{\rm min}$
ought to be identified with the cutoff $k_{\rm min}$ measured in the CMB, and
it is not difficult to demonstrate why that has to be the case for a numen
scalar field, as we shall see shortly.

A slightly different (and simpler) definition of the Planck scale than that
appearing in Equation (34) is based on the length $\lambda_{\rm Pl}$ at which
the Compton wavelength $\lambda_{\rm C}\equiv 2\pi/m$ for mass $m$ equals its
Schwarzschild radius $R_{\rm h}\equiv 2Gm$. That is,

$\lambda_{\rm Pl}\equiv\sqrt{4\pi G}\;.$ (42)

The Compton wavelength $\lambda_{\rm C}$ grows as the gravitational radius
$R_{\rm h}$ shrinks, so the standard inflationary picture conflicts with
quantum mechanics in its interpretation of wavelengths shorter than
$2\pi\lambda_{\rm Pl}$ (the factor $2\pi$ arising from the definition of
$\lambda_{\rm Pl}$ in terms of $R_{\rm h}$). This is a serious problem for the
standard model because the fluctuation amplitude $A_{s}$ measured in the CMB
anisotropies requires quantum fluctuations in the inflaton field to have been
born in the Bunch-Davies vacuum, long before the Planck time $t_{\rm
Pl}\equiv\lambda_{\rm Pl}$, a conundrum commonly referred to as the “Trans-
Planckian Problem” [86].

The numen field can completely avoid this inconsistency if we argue that each
mode $k$ emerged into the semi-classical universe when

$\lambda_{\rm k}=2\pi\lambda_{\rm Pl}\;,$ (43)

and then evolved subject to the oscillatory solution in Equation (40). As we
shall see shortly, the fact that each succeeding $k$-mode emerges at later
times in this picture produces a near scale-free power spectrum
${\mathcal{P}}(k)$ with $n_{s}\lesssim 1$ [63]. Such an idea—that modes could
have been born at a particular spatial scale—has already received some
attention in the past, notably by Hollands & Wald [87]. In their case,
however, the fundamental scale was not related to $\lambda_{\rm Pl}$. Others
supporting this proposal include Brandenberger et al. [88] and Hassan et al.
[89].

It is not difficult to understand why the fundamental scale for the numen
field must be $\lambda_{\rm Pl}$. If we interpret $k^{\rm numen}|_{\rm min}$
to be $k_{\rm min}$, the latter defines the time $t_{\rm min}$ at which the
first quantum fluctuation emerged out of the Planck domain into the semi-
classical universe. From Equation (29) and the expression for $r_{\rm cmb}$ in
a universe with zero active mass,

$r^{\rm numen}|_{\rm cmb}={c\over H_{0}}\ln(1+z_{\rm cmb})\;,$ (44)

one therefore finds that

$t_{\rm min}={4.34\,t_{\rm Pl}\over\ln(1+z_{\rm cmb})}\;.$ (45)

Its dependence on $z_{\rm cmb}$ is so weak that $t_{\rm min}$ is approximately
equal to $t_{\rm Pl}$ regardless of where the last scattering surface was
located. For example, in Planck-$\Lambda$CDM, $z_{\rm cmb}\sim 1080$, for
which $t_{\rm min}\sim 0.63\,t_{\rm Pl}$. But even if we were to adopt a very
different value $z_{\rm cmb}=50$, the first quantum fluctuation would have
emerged at $t_{\rm min}\sim 1.1\,t_{\rm Pl}$.

If the spatially largest fluctuation we see in the CMB was due to a numen
fluctuation, one concludes from this analysis that it must have emerged out of
the Planck regime at roughly the Planck time. In other words, this fluctuation
would have physically exited into the semi-classical universe shortly after
the Big Bang—indeed, it would have appeared as soon as it could, given what we
currently understand about the Planck time $t_{\rm Pl}$. No other scalar field
introduced thus far, inflaton or otherwise, has this very interesting
property.

But if a Bunch-Davies vaccum in the remote conformal past is not used for
these fluctuations, how does one then determine the normalization constant
$B(k)$ of the modes in Equation (40)? As we saw in § 2.1, a principal
complication with the inflaton field is the significant spacetime curvature
encountered by its quantum fluctuations as they cross the Hubble radius. This
led to the introduction of a Bunch-Davies vacuum in the distant conformal
past, where the modes could have been seeded in Minkowski space. We made
reference to the fact that this situation actually creates an inconsistency
with quantum mechanics, often referred to as the trans-Planckian problem. This
issue is largely beyond the scope of the present paper, however, because the
numen field completely avoids this inconsistency.

Even though the numen quantum fluctuations emerged at the Planck scale—with a
wavelength comparable to the gravitational radius $R_{\rm h}$—the zero active
mass equation-of-state in the cosmic fluid (leading to Eqn. 32) ensures that
the frame into which they emerged from the Planck regime was geodesic. That
is, in spite of the Hubble expansion, the universe was always in free fall,
with zero internal acceleration. One can easily confirm this from the fact
that the frequency $\alpha_{\rm k}$ in Equation (38) is always time-
independent. One therefore does not need an ad hoc construction of a Bunch-
Davies vacuum, and we may simply set

$B(k)={1\over\sqrt{2\alpha_{\rm k}}}$ (46)

for the numen quantum fluctuations (in Eqn. 40), following the same
minimization of the Hamiltonian argument that led to Equation (18).

We complete this brief survey by considering the spectrum ${\mathcal{P}}(k)$
one should expect from the birth of quantum fluctuations at the Planck scale.
Right away, we can see from Equation (38) that the difference between
$\alpha_{\rm k}$ and $k$ is what distorts the primordial power spectrum away
from a pure, scale-free distribution, as we shall confirm below. This is most
easily recognized if we rewrite the mode frequency in the form $\alpha_{\rm
k}=\sqrt{k^{2}-{1/t_{0}^{2}}}$ (which is $<k$).

From the definition of $u_{\rm k}$ and the curvature perturbation, we see that

$|{\mathcal{R}}_{\rm k}|^{2}={2\pi\over m_{\rm Pl}^{2}}{1\over\alpha_{\rm
k}a^{2}}\;.$ (47)

The power spectrum ${\mathcal{P}}_{\mathcal{R}}(k)$ is defined as

${\mathcal{P}}_{\mathcal{R}}(k)\equiv{k^{3}\over 2\pi^{2}}|{\mathcal{R}}_{\rm
k}|^{2}\;,$ (48)

and therefore

${\mathcal{P}}_{\mathcal{R}}(k)={1\over(2\pi)^{2}}\left[{a(t_{k})\over
a(t)}\right]^{2}\left[1-\left({k_{\rm min}\over
k}\right)^{2}\right]^{-1/2}\;.$ (49)

What is not known yet is how these quantum fluctuations devolved into grand
unified theories particles, after which the perturbation amplitude remained
frozen. Nevertheless, it is likely that the dynamics of this decay/evolution
is associated with a particular length (or energy) scale, $L_{*}$ [90, 91],
not unlike the Planck scale $\lambda_{\rm Pl}$.

Mode $k$ reached this scale at $a(t_{k}^{*})=L_{*}k/2\pi$ or, equivalently, at
time $t_{k}^{*}=t_{0}L_{*}k$. And so Equation (49) may be re-written

${\mathcal{P}}_{\mathcal{R}}(k)={1\over(2\pi)^{2}}\left[{\lambda_{\rm Pl}\over
L_{*}}\right]^{2}\left[1-\left({k_{\rm min}\over
k}\right)^{2}\right]^{-1/2}\;,$ (50)

which one then needs to compare with the observed CMB power spectrum
${\mathcal{P}}(k)=A_{s}(k/k_{0})^{n_{s}-1}$. In the context of the standard
model, the Planck optimizations [31] give $A_{s}=(2.1\pm 0.04)\times 10^{-9}$
and $n_{s}=0.9649\pm 0.0042$. And it is not difficult to see that
$L_{*}$$\sim$$3.5\times 10^{3}\lambda_{\rm Pl}$. Therefore, with the Planck
scale set at $m_{\rm Pl}\approx 1.22\times 10^{19}$ GeV, one finds that
$L_{*}$ corresponds to an energy of roughly $3.5\times 10^{15}$ GeV,
remarkably consistent with the energy scale expected in grand unified
theories. Of course, much of this is mere speculation at the present time,
given that the physics of this process lies beyond the standard model.
Nevertheless, the quantum fluctuations in this picture would have oscillated
until $t\sim 3.5\times 10^{3}t_{\rm Pl}$, after which the numen field would
have devolved into grand unified theory particles, with a freezing of the
perturbation amplitude thereafter.

The primordial power spectrum (Eq. 50) is almost scale-free, but not exactly.
Using the conventional definition of the scalar index, we find that

$n_{s}=1+{d\,\ln{\mathcal{P}}_{\mathcal{R}}(k)\over d\,\ln
k}=1-{1\over(k/k_{\rm min})^{2}-1}\;.$ (51)

The index $n_{s}$ is therefore slightly less than $1$, and we confirm that the
deviation from a pure scale-free distribution is due to the aforementioned
difference between $\alpha_{\rm k}$ and $k$, which in the end arises from the
Hubble expansion (or ‘frictional’) term ${\mathcal{R}}^{\prime}_{k}$ in the
growth Equation (5).

At least qualitatively, this result agrees with the value of $n_{s}$ measured
by Planck. Of course, the numen optimization may produce a different outcome
than that seen with $\Lambda$CDM, but it would be difficult to see why the
‘red’ tilt ($n_{s}<1$) should be converted into ‘blue’ ($n_{s}>1$) with a
change in background cosmology. An average of $n_{s}$ over $k$ in Equation
(51) gives $\langle n_{s}\rangle\sim 0.96$ over the range $k_{\rm min}\lesssim
k\lesssim 50k_{\rm min}$. The spectral index would approach closer to one at
larger values of $k$.

The caveat here, of course, is that there are still several unknowns with this
process that make it impossible for us to know all the factors influencing the
value of $n_{s}$. For example, we have assumed that the length scale at which
the modes emerge from the Planck regime is always fixed at $\lambda_{\rm Pl}$.
But this constraint depends critically on the nature of quantum gravity and
its transition into general relativity. It may turn out that the actual length
scale varies as the Universe expands. If so, this evolution would produce an
additional deviation of $n_{s}$ from one. As a quick illustration, suppose we
were to represent such a variation with the scaling $\lambda_{\rm
PL}\rightarrow\lambda_{\rm Pl}k^{-\beta}$. In that case, Equation (51) would
become

$n_{s}=1-2\beta-{1\over(k/k_{\rm min})^{2}-1}\;,$ (52)

and the Planck data would then imply that $\beta\lesssim 0.02$.

##  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 inflaton field stems directly
from the nature of the wave functional $\Psi[u(\tau,{\bf x})]$ describing its
fluctuations (Eqn. 20). As discussed in § 2.2, there is no consensus yet on
how the interference 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—unlike their inflaton counterparts, each of the numen
fluctuations emerged with a single $k$-mode. Pure, single $k$-mode quantum
states were established from the very beginning as a result of the distinct
time at which they entered the semi-classical universe out of the Planck
domain.

A numen fluctuation born at time $t_{\rm b}$, had a unique comoving wavenumber
defined by the relation

$k_{\rm b}\equiv{2\pi a(\tau_{\rm b})\over\lambda_{\rm Pl}}$ (53)

or, more explicitly,

$k_{\rm b}={2\pi\over\lambda_{\rm Pl}\,t_{0}}t_{\rm b}\;.$ (54)

Thus, the oscillating modes in Equation (40) should more accurately be written
as follows:

$u_{\rm k}(\tau,\tau_{\rm b})={1\over\sqrt{2\alpha_{\rm k}}}e^{\pm
i\alpha_{\rm k}\tau}\delta(k-k_{\rm b})\;,$ (55)

where clearly $k_{\rm b}$ is uniquely related to $\tau_{\rm b}$ via Equations
(35) and (54). In other words, it is not sufficient to merely track the
temporal evolution of a numen quantum fluctuation. One must also specify its
time of birth.

For the numen quantum fluctuation operator, one thus has

$\displaystyle\hat{u}(\tau,\tau_{\rm b},{\bf x})$ $\displaystyle=$
$\displaystyle{1\over(2\pi)^{3/2}}\int{d^{3}{\bf k}\over\sqrt{2\alpha_{\rm
k}}}\delta(k-k_{\rm b})\times$ (56) $\displaystyle\left[\hat{a}_{\bf
k}^{-}e^{i({\bf k}\cdot{\bf x}-\alpha_{\rm k}\tau)}+\hat{a}_{\bf
k}^{+}e^{-i({\bf k}\cdot{\bf x}-\alpha_{\rm k}\tau)}\right].\qquad$

The integral is straightforward to evaluate, and one finds that the numen
field operator, analogous to Equation (13) for the inflaton field, is

$\hat{u}(\tau,\tau_{\rm b},{\bf x})={k_{\rm b}^{2}\over\sqrt{\pi\,\alpha_{{\rm
k}_{\rm b}}}}\,j_{0}(k_{\rm b}|{\bf x}|)\left[\hat{a}_{{\bf k}_{\rm
b}}^{-}e^{-i\alpha_{{\rm k}_{\rm b}}\tau}+\hat{a}_{{\bf k}_{\rm
b}}^{+}e^{i\alpha_{{\rm k}_{\rm b}}\tau}\right]\;,$ (57)

where $j_{0}(k_{\rm b}|{\bf x}|)$ is a spherical Bessel function of the first
kind. We emphasize again that, in this picture, the entire numen quantum
fluctuation born at $\tau_{\rm b}$ is characterized by a single wavenumber
$k_{\rm b}$. A schematic diagram showing the amplitude of this operator at any
given time, $\tau\geq\tau_{\rm b}$, is shown in figure 1.

Figure 1: Schematic diagram showing the amplitude (plotted along $\hat{z}$) of
a numen field operator projected onto the $xy$-plane. The spatial variation
(in comoving coordinates x) is proportional to the spherical Bessel function
of the first kind, $j_{0}(k_{\rm b}|{\bf x}|)$, where $k_{b}$ is the unique
wavenumber corresponding to the time $\tau_{\rm b}$ at which the fluctuation
emerged into the semi-classical universe across the Planck scale $\lambda_{\rm
Pl}$ (see Eqns. 54 and 57).

Several of the characteristics we see in the numen field were anticipated by
the model proposed by Hollands and Wald [87], though their basic concept and
scales were quite different. The essence underlying the numen field’s behavior
is its inferred potential given in Equation (33). As in our case, Hollands and
Wald also started by considering the dynamics of a scalar field in a
background cosmological setting, though the equation-of-state in their cosmic
fluid was inconsistent with the zero active mass condition. Nevertheless, in
both their and our cases we require that the fluctuations were born in their
ground state at a fixed length scale. The work described in this paper is
strongly informed by the recent Planck data release and its subsequent
analysis, which were not available at the time Hollands and Wald proposed
their model. As we have seen, the latest measurements strongly suggest that
the cutoff in the observed primordial spectrum is associated with the very
first quantum fluctuation that exited the Planck domain at about the Planck
time. There is no room, in this proposal, for the basic length scale at which
the numen fluctuations were born to be anything but the Planck wavelength.

But though the proposal by Hollands and Wald lacked the rigor imposed by the
latest Planck observations, already Perez et al. [51] noted a very important
trait of quantum fluctuations born in this fashion. They pointed out that such
a model clearly demonstrates the need for some process to be responsible for
their birth, playing a role analogous to that of quantum mechanical
measurement. The birth of the mode, they argued, is effectively the step in
which quantum mechanical uncertainty is removed. This is precisely the
situation we describe in Equation (55), whereby each quantum fluctuation
emerging into the semi-classical universe is characterized by a unique
wavenumber, constrained by the time at which the fluctuation was created.

Perez et al. [51] also wondered what process could be associated with the
particular time of such an occurrence. With the insights we have gained from
the Planck data, we can now suggest that the length scale at which the numen
modes were born is not at all random. It carries significant physical meaning.
As we have seen, the Planck length is the scale at which the Compton
wavelength equals the Schwarzschild radius, meaning that below this scale is
the realm of quantum gravity. If the picture we are describing in this paper
is correct, the birth of numen quantum fluctuations was associated with the
release of these modes into the semi-classical universe, where gravity is
adequately described by (the classical theory of) general relativity.
Moreover, as far as we know, the Planck scale never changes. Thus, as quantum
fluctuations stochastically exited in time, their wavenumber had to reflect
their time of ‘birth’ and it is this correlation that built the near scale-
free power spectrum seen in the cosmic microwave background [31, 63].

The manner in which numen fluctuations were born thus already removes a major
hurdle in the classicalization process. Rather than having to deal with
quantum mechanical interference between many degrees of freedom, here we have
distinct quantum fluctuations possessing unique wavenumbers. Two issues
remain, however, one having to do with how these modes acquired classical
correlations in phase space and the mechanism that converted a homogeneous,
isotropic universe into an inhomogeneous one. We shall consider the latter
next, and then revisit the Wigner function (Eqn. 27) to resolve the former.

###  4.2 Anisotropies

A missing ingredient from much of the past discussion concerning
classicalization in the early universe has been the process by which a
perfectly homogeneous and isotropic state transformed into an inhomogeneous
and anisotropic state described by the density fluctuations. A chief reason
for this handicap has been the absence of an obvious ‘external’ source of
asymmetry. Broadly speaking, such a transition can be effected as part of an
‘R’ process, e.g., measurement or collapse, but not a U process, i.e., a
unitary evolution via a Schrödinger type of equation. It has been recognized
that without a measurement-like process, the required transition could not
have happened. In hindsight, the problem has actually been a lack of
appreciation for the importance of the gravitational (or ‘apparent’) horizon
$R_{\rm h}(\tau)=c/H(\tau)$ [85].

Yes, the universe is homogeneous and isotropic, but only when described using
the ‘community’ coordinates of myriads of observers dispersed throughout the
cosmos [92]. But from the perspective of a single observer at a fixed
spacetime point, the universe does not appear to be homogeneous. His/her
description of the physical state of the system, using their coordinates
centered at their location, must take into account the effects of spacetime
curvature. Take $R_{\rm h}$, as a prime example. The gravitational radius is
an apparent horizon that separates null geodesics approaching the observer
from those that are receding. One could not argue that such a divided
congruence of null geodesics is consistent with homogeneity. But let us affirm
that there is no conflict between these two descriptions, because the universe
is indeed homogeneous when viewed relative to the comoving frame.

Given the thesis developed in this paper, it should be clear why the role of
$R_{\rm h}$ is central to the manner in which numen quantum fluctuations were
born and how they classicalized. It is this gravitational horizon that
delimited the size of the fluctuations, which were isotropic, but nevertheless
initially restricted in size to the Planck scale. Remember that $\lambda_{\rm
Pl}$ actually equals $R_{\rm h}$ at the time, $t_{\rm b}$, when mode $k$
emerged into the semi-classical universe.

Crucially, an apparent horizon in the cosmological context is not an event
horizon [85]. It may turn into one in our asymptotic future, but has not been
static up until today. It has been growing, and causally-connected regions of
spacetime continue to change as $R_{\rm h}$ expands. The ‘measurement-like’
process that created the numen fluctuations at the Planck scale therefore also
introduced an inhomogeneity relative to the observer who appeared later, at
time $t>t_{\rm b}$, outside the region bounded by $R_{\rm h}(t_{\rm b})$. In
the numen context, the largest CMB fluctuation we see today corresponds to
$R_{\rm h}$ at decoupling. Since $\lambda_{\rm max}$ (corresponding to $k_{\rm
min}$) grew at the same rate as $R_{\rm h}$, this happens to be the largest-
size mode created right after the Big Bang, at the Planck time $t_{\rm Pl}$.
This region has been exposed to us by the expanding universe, so that we can
now see the impact of that first quantum fluctuation on the CMB. The seeding
of numen fluctuations at the Planck scale not only created pure, single
$k$-mode fluctuations, but also explains why these quantum fluctuations have
always been finite in size, and why the universe we see is therefore a
patchwork of inhomogeneities. At least in this context, the breakdown of
spherical symmetry was not due to some unknown process in quantum mechanics
itself. It was the result of an inherent property of the FLRW spacetime when
we take into account the physical characteristics of the gravitational (or,
apparent) horizon.

###  4.3 Classical Phase Space Correlations

A distinguishing feature of quantum fluctuations in the inflaton field is that
their frequency changed with time in response to the spacetime curvature they
encountered as they grew into the expanding universe. This led to the very
strong time dependence in $\Omega_{\bf k}$ (Eqn. 25) which, according to the
Wigner function (Eqn. 28), then produced a highly squeezed state. For the
various reasons outlined in § 2.2, however, it is not clear that this
restructuring of the dispersions about $u_{\bf k}$ and $\pi_{\bf k}$ is
consistent with emerging classical correlations in phase space.

The situation with numen quantum fluctuations was completely different,
principally because their frequency $\alpha_{\rm k}$ (Eqn. 38) was constant in
time. As we alluded to earlier, this property results from the zero active
mass equation-of-state ($\rho_{\phi}+3p_{\phi}=0$), which produced a constant
expansion rate with $a(t)\propto t$. The numen modes therefore mimicked a
classical harmonic oscillator.

Figure 2: Schematic diagram showing the Wigner function of the coherent numen
state (see Eqn. 63). The state begins at the right, where $\alpha_{{\rm
k}_{\rm b}}\tau=\pi/4$, and moves clockwise about $(0,0)$. The second peak is
plotted at $\alpha_{{\rm k}_{\rm b}}\tau=3\pi/4$, and the third at
$\alpha_{{\rm k}_{\rm b}}\tau=5\pi/4$. In generating these plots, the
following values were chosen for the constants in Equation (63): $\alpha_{{\rm
k}_{\rm b}}=3$, and $\mu=3$.

Such a coherent state is considered to be the most classical of all states
because their Wigner function peaks over tiny regions in phase space, and
retains this configuration for all times. Given a value of $u_{\bf k}$ for
such a state, one may then obtain the corresponding canonical momentum
$\pi_{\bf k}$ very close to the value calculated using classical dynamics. The
Wigner function thus follows a classical trajectory, with minimal outward
spread in all phase-space directions.

To see this quantitatively, let us examine the Wigner function for a numen
fluctuation,

${\mathcal{W}}_{\Psi_{{\rm k}_{\rm b}}}(u_{{\rm k}_{\rm b}},\pi_{{\rm k}_{\rm
b}})\equiv\int{dy\over\pi}\,\Psi_{{\rm k}_{\rm b}}^{*}(u_{{\rm k}_{\rm
b}}+y)e^{2i\,\pi_{{\rm k}_{\rm b}}y}\Psi_{{\rm k}_{\rm b}}(u_{{\rm k}_{\rm
b}}-y)\;,$ (58)

which is reminiscent of the quantum harmonic oscillator. Its derivation is
very well known, so we won’t dwell on the details (see, e.g., ref. [56]). With
the ground-state solution to the Schrödinger Equation (21),

$\Psi_{{\rm k}_{\rm b}}[u_{{\rm k}_{\rm b}}(\tau)]=\left({\alpha_{{\rm k}_{\rm
b}}\over\pi}\right)^{1/4}e^{-\alpha_{{\rm k}_{\rm b}}{u^{2}_{{\rm k}_{\rm
b}}}/2}\;,$ (59)

we may easily evaluate the integral and find that

${\mathcal{W}}_{\Psi_{{\rm k}_{\rm b}}}(u_{{\rm k}_{\rm b}},\pi_{{\rm k}_{\rm
b}})={1\over\pi}e^{-\alpha_{{\rm k}_{\rm b}}{u^{2}_{{\rm k}_{\rm
b}}}-{\pi^{2}_{{\rm k}_{\rm b}}}/\alpha_{{\rm k}_{\rm b}}}\;.$ (60)

To see how ${\mathcal{W}}_{\Psi_{{\rm k}_{\rm b}}}$ varies in time, let us
briefly return to Equations (40) and (46), and solve for the time evolution of
$u_{{\rm k}_{\rm b}}(\tau)$ and $\pi_{{\rm k}_{\rm b}}(\tau)$ in terms of
their values at $\tau=0$: $u_{{\rm k}_{\rm b}}(0)$ and $\pi_{{\rm k}_{\rm
b}}(0)$. We find that

$\displaystyle u_{{\rm k}_{\rm b}}(0)$ $\displaystyle=$ $\displaystyle u_{{\rm
k}_{\rm b}}(\tau)\cos(\alpha_{{\rm k}_{\rm b}}\tau)-{\pi_{{\rm k}_{\rm
b}}\over\alpha_{{\rm k}_{\rm b}}}\sin(\alpha_{{\rm k}_{\rm b}}\tau)$
$\displaystyle\pi_{{\rm k}_{\rm b}}(0)$ $\displaystyle=$
$\displaystyle\pi_{{\rm k}_{\rm b}}(\tau)\cos(\alpha_{{\rm k}_{\rm
b}}\tau)+\alpha_{{\rm k}_{\rm b}}u_{{\rm k}_{\rm b}}\sin(\alpha_{{\rm k}_{\rm
b}}\tau)\;.\quad$ (61)

Now take the Wigner function at time $\tau=0$ to represent the lowest energy
state for the field amplitude $u_{{\rm k}_{\rm b}}(0)$ shifted by a constant
$\mu$. That is,

${\mathcal{W}}_{\Psi_{{\rm k}_{\rm b}}}(u_{{\rm k}_{\rm b}},\pi_{{\rm k}_{\rm
b}},0)={1\over\pi}e^{-\alpha_{{\rm k}_{\rm b}}(u_{{\rm k}_{\rm
b}}-\mu)^{2}-\pi^{2}_{{\rm k}_{\rm b}}/\alpha_{{\rm k}_{\rm b}}}\;.$ (62)

With the use of Equation (61), this becomes

$\displaystyle{\mathcal{W}}_{\Psi_{{\rm k}_{\rm b}}}(u_{{\rm k}_{\rm
b}},\pi_{{\rm k}_{\rm b}},\tau)$ $\displaystyle=$
$\displaystyle{1\over\pi}e^{-\alpha_{{\rm k}_{\rm b}}[u_{{\rm k}_{\rm
b}}(\tau)-\mu\cos(\alpha_{{\rm k}_{\rm b}}\tau)]^{2}}\times$ (63)
$\displaystyle\hskip-36.135pte^{-[\pi_{{\rm k}_{\rm b}}(\tau)+\alpha_{{\rm
k}_{\rm b}}\mu\sin(\alpha_{{\rm k}_{\rm b}}\tau)]^{2}/\alpha_{{\rm k}_{\rm
b}}}\;.\quad$

The Wigner function thus oscillates back and forth with an amplitude $\mu$ and
angular frequency $\alpha_{{\rm k}_{\rm b}}$—a coherent state with a very
small dispersion relative to the classicalized trajectory in the $(u_{{\rm
k}_{\rm b}},\pi_{{\rm k}_{\rm b}})$ plane (see fig. 2).

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 frequency $\alpha_{{\rm k}_{\rm b}}$, means that they
functioned exactly like the coherent states of a conventional quantum harmonic
oscillator. They therefore acquired classical correlations in the $(u_{{\rm
k}_{\rm b}},\pi_{{\rm k}_{\rm b}})$ phase space from the very beginning.

##  5 Conclusions

The Planck data suggest that the primordial power spectrum has a cutoff,
$k_{\rm min}$, probably associated with the first quantum fluctuation to
emerge out of the Planck domain into the semi-classical universe. Moreover,
this birth must have occurred at about the Planck time—the earliest moment
permitted by our current theories following the Big Bang. It is easy to
appreciate the physical significance of such an interpretation, because our
current (classical) theory of gravity is valid only down to the Planck scale,
not beyond. It is therefore tempting to view the manner in which these (numen)
quantum fluctuations were born as a physical consequence of the transition
from quantum gravity to general relativity.

Insofar as the question of classicalization is concerned, this picture
effectively removes the hurdles faced by inflation. Leaving aside the related
issue that the existence of $k_{\rm min}$ challenges the viability of slow-
roll inflation to simultaneously explain the fluctuation spectrum and mitigate
the horizon problem, which has been dealt with elsewhere [64], one cannot
ignore the challenge of explaining how fluctuations in the inflaton field
successfully transitioned from the quantum to classical domains.

With the interpretation we have examined in this paper, two factors stand out
clearly as the most influential. These are (i) the birth of quantum
fluctuations with a fixed length scale at the Planck time constitutes a
‘process’ that effectively replaces the need for a ‘measurement’ in quantum
mechanics; and (ii) the implied scalar field potential has an equation-of-
state consistent with the zero active mass condition in general relativity.
The spacetime associated with the expansion profile in this cosmology would
have allowed the fluctuations to emerge in their ground state, with a time-
independent frequency, $\alpha_{\rm k}$. This makes all the difference because
the numen quantum fluctuations were therefore essentially quantum harmonic
oscillators, with classical correlations of their canonical variables from the
very beginning.

Should the proposal we have made in this paper turn out to be correct, an
obvious question concerns the need for inflation in cosmology. For a paradigm
that has been with us for over four decades, it is somewhat troubling that no
complete theory yet exists. Some of the complications appear to be that the
data tend to shy away from its predictions as the measurement precision
continues to improve, rather than confirm the basic inflationary premise with
increasing confidence. The large-scale anomalies in the CMB anisotropies are a
good illustration of this point. Today, after three major satellite missions
have completed their work studying the CMB, one would reasonably have expected
all of the nuances associated with the primordial power spectrum
${\mathcal{P}}(k)$ to have fallen into place. But they don’t, and it is
difficult to discount the observational constraint that a zero $k_{\rm min}$
is now ruled out by the data at a relatively high level of confidence [32,
33].

On the flip side, inflation may not even be necessary to solve the horizon
problem, which actually does not exist in cosmologies that avoided an early
phase of deceleration [78, 93]. In that case, the fact that this paradigm
struggles explaining the angular correlation function in the CMB, and that its
quantum fluctuations have no obvious path to classicalization, may be
indicators that it simply never happened. The many observational tests already
completed to examine this question would not reject such a view (see, e.g.,
Table 2 in ref. [77]).

## Acknowledgments

I am grateful to the anonymous referee for an exceptional review of this
manuscript and for suggesting several improvements to its presentation.

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