CPHT-RR076.092021, October 2021

Probability distribution for the quantum universe

Alex<EMAIL_ADDRESS>Hervé
<EMAIL_ADDRESS>and Nicolaos
<EMAIL_ADDRESS>
1 Department of Physics, School of Natural Sciences, NTUA
15780 Athens, Greece

2 CPHT, CNRS, Ecole polytechnique, IP Paris,
F-91128 Palaiseau, France

3 Department of Physics, University of Cyprus,
Nicosia 1678, Cyprus

Abstract

> We determine the inner product on the Hilbert space of wavefunctions of the
> universe by imposing the Hermiticity of the quantum Hamiltonian in the
> context of the minisuperspace model. The corresponding quantum probability
> density reproduces successfully the classical probability distribution in
> the $\hbar\to 0$ limit, for closed universes filled with a perfect fluid of
> index $w$. When $-1/3<w\leq 1$, the wavefunction is normalizable and the
> quantum probability density becomes vanishingly small at the big bang/big
> crunch singularities, at least at the semiclassical level. Quantum
> expectation values of physical geometrical quantities, which diverge
> classically at the singularities, are shown to be finite.

##  1 Introduction

Quantum effects are essential in understanding the early-universe dynamics. In
particular, it is important to determine the wavefunction of the universe,
which is expected to specify the initial conditions for the cosmological
evolution. A first attempt towards this direction has been undertaken by
Wheeler and DeWitt who defined a Schrödinger-like equation, known as the
Wheeler-DeWitt (WDW) equation [1], satisfied by such a wavefunction [2, 3, 4,
5, 6, 7]. Later, Hartle and Hawking [8] introduced a wavefunction for a closed
universe as a path integral satisfying the “no-boundary proposal” [9, 10, 11,
12, 13]. However, some issues arise from both viewpoints, which can be
resolved [14, 15]. This can be illustrated in the simpler framework of the
minisuperspace model, where the closed universe is assumed to be homogeneous
and isotropic with a positive cosmological constant, so that the scale factor
is the only dynamical degree of freedom:

$\bullet$ Field redefinitions $a=A(q)$ of the scale factor $a(t)$ leave the
classical action invariant. However, in the path integral approach, the
measures ${\cal D}a$ and ${\cal D}q$ are not equal, being related by a
Jacobian. Hence, there exist infinitely many inequivalent definitions for the
wavefunction in terms of path integrals of the form

$\int{{\cal D}N\over{\rm Vol(Diff)}}\int{\cal D}q\,e^{iS[N,q]}\,,$ (1.1)

where $N$ is the lapse function, $\rm Vol(Diff)$ is the volume of the time-
reparametrization group and $S$ is the gravitational action. This leads to
different results, as can already be seen at the semiclassical level [14].111
To compute the path integrals, one replaces $\int{\cal D}N/{\rm Vol(Diff)}$ by
$\int_{0}^{+\infty}{\rm d}\ell\times 1$, where $\ell$ is the invariant length
of the time interval, and the factor $1$ stands for a Fadeev-Popov
determinant, which turns out to be trivial. As explained in Sect. 5.3 of Ref.
[14], previous work often employs a gauge for the lapse function that depends
on the scale factor, $N(t)=\ell/a(t)$ [9, 11, 13]. In that case, the limits of
the time integral in the action $S$ depend on the trajectory $q(t)$.
Therefore, it is not valid to only vary the Lagrangian with respect to $q$ to
find the saddle points of the action. The authors proceed to treat the limits
of integration in the action as constants, but this changes the classical
problem to be quantized. The motivation for the gauge $N(t)=\ell/a(t)$ is that
the field redefinition $a=A(q)=\sqrt{q}$ renders the Lagrangian quadratic in
$q$. This then would seem to turn the path integral into a Gaussian one.
However one should keep track of the change in the limits of integration.
Indeed, the computation at the semiclassical level can be done in the gauge
$N(t)=\ell$, for which the limits of integration are independent of the
trajectory [14], yielding different results from [9, 11, 13].

$\bullet$ Each choice of field $q(t)$ yields a different classical Hamiltonian
${\cal H}$, which vanishes. At the quantum level, the constraints ${\cal H}=0$
translate into different WDW equations accompanied with additional ambiguities
in their forms, due to the non-commutativity of the canonical variable $q$ and
its conjugate momentum $\pi_{q}$ [1]. The ambiguities can be partially
implemented by replacing

$\pi_{q}^{2}={1\over\rho(q)}\,\pi_{q}\,\rho(q)\,\pi_{q}\leavevmode\nobreak\
\leavevmode\nobreak\ \longrightarrow\leavevmode\nobreak\ \leavevmode\nobreak\
-{i\over\rho(q)}{\partial\over\partial
q}\Big{(}\\!-i\rho(q){\partial\over\partial q}\Big{)}\,.$ (1.2)

$\bullet$ By imposing that the path-integral wavefunctions are solutions of
the WDW equations, one obtains that [14]222This resolution of the ambiguity in
form of the WDW equation differs from that derived in Ref. [9], which presents
the issue raised in Footnote 1.

$\rho(q)=A(q)^{-{3\over 4}}|A^{\prime}(q)|^{-{3\over 2}}\,.$ (1.3)

$\bullet$ For each prescription, one defines an inner product on the Hilbert
space,

$\langle\Psi_{1}|\Psi_{2}\rangle=\int{\rm
d}q\,\mu(q)\,\Psi_{1}(q)^{*}\Psi_{2}(q)\,,$ (1.4)

such that the corresponding Hamiltonian ${\cal H}$ is Hermitian. This
requirement determines the measure $\mu(q)$ on the Hilbert space to be [14]

$\mu(q)=A(q)\,A^{\prime}(q)^{2}\,\rho(q)\,.$ (1.5)

$\bullet$ The key point is that at the semiclassical level, one finds

$\Psi(q)\propto{\sqrt{|A^{\prime}(q)|\over\mu(q)}}\,,$ (1.6)

which implies that the inner product is independent of the prescription, at
this level of approximation. This strongly suggests that the various choices
of fields $q$ do not yield different quantum gravity theories with the same
classical limit. Hence, we may choose $q\equiv a$, which leads to

$\mu(a)=a\,\rho(a)\,.$ (1.7)

$\bullet$ Notice that in the literature, a widely used inner product on the
Hilbert space is based on the measure $\mu_{\rm HH}(a)=\rho(a)$ employed by
Hartle and Hawking in Ref. [8]. Their choice follows by imposing Hermiticity
of the Wheeler-DeWitt operator $a{\cal H}$ rather than ${\cal H}$.

The relation $\mu(a)=a\rho(a)$ arising from the Hermiticity of the quantum
Hamiltonian is valid in the more general context where the closed, homogeneous
and isotropic universe is filled with a perfect fluid of state equation
$p_{m}=w\rho_{m}$. In this work, we test the consistency of this measure by
showing that the quantum probability density reproduces in the $\hbar\to 0$
limit the classical probability distribution. In the context of quantum
cosmology see e.g. Refs. [16, 17, 18, 19]. To find the classical probability
distribution, we compute the ratio of the time the scale factor lies in the
infinitesimal range $[a,a+{\rm d}a]$ to the total duration of the cosmological
evolution. The total duration is finite when the index $w$ of the perfect
fluid satisfies $w>-1/3$. If $-1\leq w\leq-1/3$, the duration is infinite but
relative probabilities can be defined, leading again to a consistency between
the classical and quantum pictures.

Notice that when $w>-1/3$, an initial singularity exists at the classical
level. However, the fact that the wavefunction is normalizable implies that
the classical singularity is innocuous in the quantum case. This is similar to
the problem of the hydrogen atom, where the classical singularity at the
center of the atom is resolved quantum mechanically.333For $w=-1/3$, the
wavefunction is non-normalizable but $\int_{0}^{\alpha}{\rm d}a\mu|\Psi|^{2}$
is finite for any $\alpha>0$, implying the classical initial singularity to be
again innocuous.

Indeed, to obtain the measure $\mu$ for more general Hamiltonians in higher
$d$, one could generalize the derivation in the Appendix of Ref. [14], see Eq.
(A.21) - Eq. (A.25). This derivation is partially based on Section 9 of
DeWitt’s paper [1], where he obtains Eqs. (9.3) and (9.4) 444We mention that
$i$ factors in the r.h.s. of these equations are missing., valid for more
general Hamiltonians in arbitrary $d$. In fact, one has to include in all
steps of DeWitt’s proof a measure $\mu$ and $\rho$-like functions (arising
from operator ordering ambiguities). One could then generalize Eq. (A.25) of
Ref. [14] for a more general Hamiltonian in arbitrary $d$. To determine $\mu$
from this result, one must follow the procedure below Eq. (A.25) of Ref. [14]:
Since the functions $\Psi_{1}$ and $\Psi_{2}$ are arbitrary, the form of the
measure $\mu$ is extremely constrained. So one obtains the analogous result
given below Eq. (A.27) of Ref. [14].

The paper is organized as follows. In Sect. 2, we compute the classical
probability distribution in the case of a closed universe filled with a
perfect fluid with arbitrary index $w\in[-1,1]$. In Sect. 3, we derive the WDW
equation from a microscopic description of the fluid based on a scalar field.
In Sect. 4, we obtain the quantum probability density at the semiclassical
level and find perfect agreement with the classical probability distribution
in the $\hbar\to 0$ limit. A summary of our results can be found in the
conclusion, Sect. 5. The Hermiticity of the Hamiltonian is also important in
other approaches. In Ref. [20], the authors discuss the problem of time and
non-unitarity emerging in an adiabatic, WKB approach concerning quantum
gravity corrections to quantum field theory.

##  2 Classical probability distribution

In this section, we consider classical cosmological evolutions in order to
derive the associated probability distributions.

###  2.1 Classical cosmological evolution

Our starting point is gravity coupled to matter with action

${\cal S}=\int{\rm d}^{4}x\sqrt{-g}\left(\frac{R}{2}+{\cal L}_{m}\right),$
(2.8)

where ${\cal L}_{m}$ is the matter Lagrangian density. We work in units $8\pi
G=M_{\rm P}^{2}=1$. The equations of motion are

$R^{\mu\nu}-\frac{1}{2}g^{\mu\nu}R=T^{\mu\nu}\,,$ (2.9)

where $T^{\mu\nu}$ is the matter energy-momentum tensor. We will examine the
case where $T^{\mu\nu}$ describes a perfect fluid with equation of state
$p_{m}=w\rho_{m}$, where $\rho_{m}$ and $p_{m}$ are the energy density and
pressure, while the index $w$ is in the range $-1\leq w\leq 1$. For FRW metric

${\rm d}s^{2}=-{\rm d}t^{2}+a^{2}(t){\rm d}\Omega_{3}^{2}\,,$ (2.10)

where ${\rm d}\Omega_{3}^{2}$ is the line element of the unit 3-sphere, the
Friedmann equations are

$\displaystyle 3H^{2}$ $\displaystyle=\rho_{m}-\frac{3}{a^{2}}\,,$ (2.11)
$\displaystyle 3\,\frac{\ddot{a}}{a}$ $\displaystyle=-{1\over
2}\,(\rho_{m}+3P_{m})=-{1\over 2}\,(1+3w)\rho_{m}\,,$ (2.12)

with $H=\dot{a}/a$ denoting the Hubble parameter. Conservation of energy-
momentum specifies the energy density to be of form

$\rho_{m}=\left\\{\begin{array}[]{lll}\displaystyle{3\,a_{0}^{3w+1}\over
a^{3(w+1)}}\,,&a_{0}>0\,,&\mbox{for}\quad\displaystyle w\neq-{1\over 3}\,,\\\
\displaystyle{3\,V_{0}\over a^{2}}\,,&V_{0}>1\,,&\mbox{for}\quad\displaystyle
w=-{1\over 3}\,.\phantom{\\!\\!\overset{\displaystyle|}{|}}\end{array}\right.$
(2.13)

With this notation and for $w\neq-1/3$, the first Friedmann equation can be
written as

$\dot{a}^{2}=\left({a_{0}\over a}\right)^{3w+1}-1\,,$ (2.14)

which shows that there is always a turning point $(\dot{a}=0$) when $a=a_{0}$.
Two cases must be distinguished:

$\bullet$ $w>-1/3$:
In this case, the scale factor satisfies $a\leq a_{0}$. The universe is
decelerating, as follows from Eq. (2.12), and $\dot{a}\to\pm\infty$ when $a\to
0$. Therefore, the evolution starts/ends with a big bang/big crunch, and the
scale factor is maximal when $a=a_{0}$.

$\bullet$ $w<-1/3$:
In this case, the scale factor satisfies $a\geq a_{0}$. The trajectory is
accelerating and $\dot{a}\to\mp\infty$ when $a\to+\infty$. Thus, the evolution
describes a bouncing universe, with minimal size when $a=a_{0}$. There is a
contracting/expanding phase before/after the bounce, from/to $a=+\infty$.

In order to obtain the explicit solutions for $w\neq-1/3$, we use Eq. (2.14)
in the form

${a^{3w+1\over 2}\,{\rm d}a\over\sqrt{a_{0}^{3w+1}-a^{3w+1}}}=\pm{\rm d}t\,.$
(2.15)

The l.h.s. is always integrable at $a=a_{0}$. Upon integration, one sees that
the lifetime of the universe is finite for $w>-1/3$ and infinite for $-1\leq
w<-1/3$.555This will be related to the fact that the quantum wavefunction will
be normalizable for $w>-1/3$ and non-normalizable for $-1\leq w<-1/3$. Time as
a function of the scale factor is given by

$\displaystyle\pm t$ $\displaystyle=a_{0}\int_{1}^{a\over a_{0}}{x^{3w+1\over
2}\,{\rm d}x\over\sqrt{1-x^{3w+1}}}$ (2.16) $\displaystyle={2a_{0}\over
3(w+1)}\,\Big{(}{a\over a_{0}}\Big{)}^{{3\over
2}(w+1)}{}_{2}F_{1}\bigg{(}{1\over 2},{1\over 2}+{1\over 3w+1},{3\over
2}+{1\over 3w+1};\Big{(}{a\over
a_{0}}\Big{)}^{3w+1}\bigg{)}-a_{0}\,\sqrt{\pi}\,{\Gamma\Big{(}{1\over
2}+{1\over 3w+1}\Big{)}\over\Gamma\Big{(}{1\over 3w+1}\Big{)}}\,,$

where we have chosen the time origin such that the cosmological evolution is
symmetric under time reversal $t\to-t$. Notice that in the last line, one may
think that there is a pathology when the Gamma function in the numerator
diverges, i.e. when $1/2+1/(3w+1)=-n$ where $n\in\mathbb{N}$; that is when
$w=-1/3-2/(6n+3)$, which lies in the range $[-1,-1/3)$. However, this is only
an artefact of this way of writing things, since the first line is perfectly
well behaved for any $w\neq-1/3$.

Finally, for the remaining case $w=-1/3$, one finds two possible cosmological
evolutions described by the “$+$” or “$-$” branches of the following equation

$a=\pm\sqrt{V_{0}-1}\;t\,.$ (2.17)

###  2.2 Classical probability distributions for $w>-1/3$

The finite lifetime of the universe, as follows from Eq. (2.16), is

$\displaystyle\Delta t$ $\displaystyle=-2a_{0}\int_{1}^{0}{x^{3w+1\over
2}\,{\rm d}x\over\sqrt{1-x^{3w+1}}}\,$
$\displaystyle=a_{0}\,2\sqrt{\pi}\,{\Gamma\big{(}{1\over 2}+{1\over
3w+1}\big{)}\over\left|\Gamma\big{(}{1\over
3w+1}\big{)}\right|}\,.\phantom{\\!\\!\overset{\displaystyle|}{|}}$ (2.18)

It satisfies $\Delta t\to+\infty$ when $w\to(-1/3)_{+}$.

In order to compute the classical probability distribution, we must find the
time the scale factor spends in the interval $[a,a+{\rm d}a]$. This time turns
out to be

$\delta t=2|{\rm d}t|={2a^{3w+1\over 2}\,{\rm
d}a\over\sqrt{a_{0}^{3w+1}-a^{3w+1}}}\,,$ (2.19)

where the factor of $2$ takes into account the contracting and expanding
phases. Then, the classical probability distribution as a function of $a$
times ${\rm d}a$ is

$\displaystyle P_{cl}(a)\,{\rm d}a$ $\displaystyle={\delta t\over\Delta t}$
$\displaystyle={1\over\sqrt{\pi}}\,{\left|\Gamma\big{(}{1\over
3w+1}\big{)}\right|\over\Gamma\big{(}{1\over 2}+{1\over
3w+1}\big{)}}\,{a^{3w+1\over 2}\,{\rm d}a\over
a_{0}\sqrt{a_{0}^{3w+1}-a^{3w+1}}}\,,$ (2.20)

where $a\leq a_{0}$. By construction, the total probability is normalized to
unity,

$\int_{0}^{a_{0}}P_{cl}(a)\,{\rm d}a=1.$ (2.21)

Notice that the classical probability distribution is infinite at the turning
point $a=a_{0}$, where the rate of expansion is zero, and vanishes at $a=0$
where the rate of expansion/contraction is infinite.

###  2.3 Classical probability distributions for $-1\leq w\leq-1/3$

In this case, the lifetime of the universe is infinite but nevertheless we can
consider “relative probabilities.” To this end, let us define the quantity

$P_{cl}(a)\,{\rm d}a=\delta t=\left\\{\begin{array}[]{ll}2|{\rm
d}t|=\displaystyle{2a^{3w+1\over 2}\,{\rm
d}a\over\sqrt{a_{0}^{3w+1}-a^{3w+1}}}\,,&\mbox{for}\quad-1\leq\displaystyle
w<-{1\over 3}\,,\\\ \displaystyle|{\rm d}t|={{\rm
d}a\over\sqrt{V_{0}-1}}\,,\phantom{\\!\\!\overset{\displaystyle|}{|}}&\mbox{for}\quad\displaystyle
w=-{1\over 3}\,,\end{array}\right.$ (2.22)

which cannot be interpreted as an absolute probability distribution since its
integral over $a\in[a_{0},+\infty)$ or $a\in[0,+\infty)$ is infinite. However,
ratios of such quantities (at different values of the scale factor) can be
understood as relative probabilities.

##  3 Microscopic derivation of WDW equation for any state equation

In this section, our aim is to derive the WDW equation in minisuperspace, when
the closed universe is filled with a perfect fluid of index $w\neq-1,0$.666The
case $w=0$ can be studied as a limiting case, taking $w$ to be a very small
positive constant. The case $w=-1$ can be easily derived by adding a positive
cosmological constant and no matter field, see e.g. [14].

Let us consider the microscopic model involving a derivatively coupled scalar
field, with action [21, 22]

${\cal
S}=\int_{M}d^{4}x\sqrt{-g}\left\\{\frac{R}{2}+\mbox{sign}(w)\left(-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi\right)^{\frac{w+1}{2w}}\right\\}+{\cal{S}}_{\rm
boundary}\,,$ (3.23)

where $M$ is spacetime and

${\cal{S}}_{\rm boundary}=-\int_{\partial M}d^{3}y\sqrt{h}\,K$ (3.24)

is the Gibbons-Hawking-York boundary term. In the above expression, $\partial
M$ is a spacelike boundary of extrinsinc curvature $K$ and induced metric
$h_{ij}$. The energy-momentum tensor for the scalar field is given by

$T^{\mu\nu}=g^{\mu\nu}\,\mbox{sign}(w)\left(-\frac{1}{2}\partial_{\kappa}\phi\partial^{\kappa}\phi\right)^{\frac{w+1}{2w}}+\frac{w+1}{2|w|}\left(-\frac{1}{2}\partial_{\kappa}\phi\partial^{\kappa}\phi\right)^{\frac{1-w}{2w}}\partial^{\mu}\phi\partial^{\nu}\phi\,.$
(3.25)

It turns out to be of the form of the energy-momentum tensor of an
irrotational fluid,

$T^{\mu\nu}=(\rho_{m}+p_{m})\,u^{\mu}u^{\nu}+p_{m}\,g^{\mu\nu}\,,$ (3.26)

where

$p_{m}=\mbox{sign}(w)\left(-\frac{1}{2}\partial_{\kappa}\phi\partial^{\kappa}\phi\right)^{\frac{w+1}{2w}}\,,\qquad
u^{\mu}=\frac{\partial^{\mu}\phi}{\sqrt{-\partial_{\kappa}\phi\partial^{\kappa}\phi}}\,,$
(3.27)

and equation of state

$p_{m}=w\rho_{m}\,.$ (3.28)

In minisuperspace, the $4$-metric and the scalar field are of the form

${\rm d}s^{2}=-N(x^{0})^{2}\big{(}{\rm d}x^{0}\big{)}^{2}+a^{2}(x^{0}){\rm
d}\Omega_{3}^{2}\,,\qquad\phi=\phi(x^{0})\,,$ (3.29)

and the action becomes

${\cal S}=v_{3}\int
dx^{0}\left(-3\,\frac{a\dot{a}^{2}}{N}+3Na+\frac{\mbox{sign}(w)}{2^{\frac{w+1}{2w}}}\,\frac{a^{3}}{N^{\frac{1}{w}}}\,|\dot{\phi}|^{\frac{w+1}{w}}\right),$
(3.30)

where $v_{3}=2\pi^{2}$ is the volume of the unit $3$-sphere. Denoting $L$ the
Lagrangian, the generalized momenta are given by

$\displaystyle\pi_{a}=\frac{\partial
L}{\partial\dot{a}}=-6v_{3}\,\frac{a}{N}\,\dot{a}\,,\qquad\pi_{N}={\partial
L\over\partial\dot{N}}=0\,,$ $\displaystyle\pi_{\phi}=\frac{\partial
L}{\partial\dot{\phi}}=\frac{v_{3}}{2^{\frac{w+1}{2w}}}\,\frac{a^{3}}{N^{\frac{1}{w}}}\,\frac{w+1}{|w|}\,|\dot{\phi}|^{\frac{1}{w}}\,\,\mbox{sign}(\dot{\phi})\,.$
(3.31)

The classical Hamiltonian can then be expressed as

$\displaystyle\frac{{\cal H}}{N}$
$\displaystyle=-\frac{1}{12v_{3}}\,\frac{\pi_{a}^{2}}{a}-3v_{3}a+\frac{|w|^{w}}{(w+1)^{w+1}}\,\frac{2^{\frac{w+1}{2}}}{v_{3}^{w}}\,\big{(}\mbox{sign}(\pi_{\phi})\big{)}^{w+1}\,\frac{\pi_{\phi}^{w+1}}{a^{3w}}$
$\displaystyle=-\frac{1}{12v_{3}}\,\frac{\pi_{a}^{2}}{a}-3v_{3}a+C\,\frac{|\pi_{\phi}|^{w+1}}{a^{3w}}\,,$
(3.32)

where the constant $C$ is

$C=\frac{|w|^{w}}{(w+1)^{w+1}}\,\frac{2^{\frac{w+1}{2}}}{v_{3}^{w}}\,.$ (3.33)

It satisfies

$\frac{{\cal H}}{N}=-\frac{\partial L}{\partial N}=0\,,$ (3.34)

by the equation of motion of the lapse $N$, which takes the form

${3\over N^{2}}\,H^{2}=\rho_{m}-\frac{3}{a^{2}}\,,$ (3.35)

where

$\rho_{m}={C\over v_{3}}\,\frac{|\pi_{\phi}|^{w+1}}{a^{3(w+1)}}\geq 0\,.$
(3.36)

Note that $\pi_{\phi}={\rm const.}$, as follows from the Hamilton equation
$\dot{\pi}_{\phi}=-\partial{\cal H}/\partial\phi=0$. As a result, the scaling
of the energy density $\rho_{m}$ is the expected one for a perfect fluid with
equation of state (3.28). Comparing Eqs. (2.13) and (3.36), we find that

${C\over
v_{3}}\,|\pi_{\phi}|^{w+1}=\left\\{\begin{array}[]{ll}3a_{0}^{3w+1}\,,&\mbox{for}\quad\displaystyle
w\neq-\frac{1}{3}\,,\\\ 3V_{0}\,,&\displaystyle\mbox{for}\quad
w=-\frac{1}{3}\,.\phantom{\\!\\!\overset{\displaystyle|}{|}}\end{array}\right.$
(3.37)

To canonically quantize the theory, we replace

$\pi_{a}\leavevmode\nobreak\ \longrightarrow\leavevmode\nobreak\
-i\hbar\,\frac{\partial}{\partial a}\,,\qquad\pi_{\phi}\leavevmode\nobreak\
\longrightarrow\leavevmode\nobreak\ -i\hbar\,\frac{\partial}{\partial\phi}\,.$
(3.38)

However, because classically

${\pi_{a}^{2}\over a}={1\over
a\,\rho_{1}\,\rho_{2}}\,\pi_{a}\,\rho_{1}\,\pi_{a}\,\rho_{2}\,,$ (3.39)

where $\rho_{1}(a)$, $\rho_{2}(a)$ are arbitrary functions of the scale
factor, the process of quantization yields an ambiguity in the form of the WDW
equation of the wavefunction $\Phi(a,\phi)$,

${{\cal H}\over N}\,\Phi(a,\phi)\equiv{\hbar^{2}\over 12v_{3}}\,{1\over
a\rho_{1}\rho_{2}}\,{\partial\over\partial
a}\Big{[}\rho_{1}\,{\partial\over\partial
a}\big{(}\rho_{2}\Phi\big{)}\Big{]}-3v_{3}a\,\Phi+\frac{C}{a^{3w}}\left|-i\hbar\,\frac{\partial\Phi}{\partial\phi}\right|^{w+1}=0\,.$
(3.40)

Redefining

$\rho=\rho_{1}\rho_{2}^{2}\,,\qquad\omega={(\rho_{1}\rho_{2}^{\prime})^{\prime}\over\rho_{1}\rho_{2}}\,,$
(3.41)

one obtains the more convenient form

${{\cal H}\over
N}\,\Phi(a,\phi)\equiv\left\\{\frac{\hbar^{2}}{12v_{3}a}\left(\frac{1}{\rho}\,\frac{\partial}{\partial
a}\Big{(}\rho\,\frac{\partial}{\partial
a}\Big{)}+\omega\right)-3v_{3}a+\frac{C}{a^{3w}}\left|-i\hbar\,\frac{\partial}{\partial\phi}\right|^{w+1}\right\\}\Phi(a,\phi)=0\,,$
(3.42)

where $\rho$ and $\omega$ depend on $a$. We will see in the next section that
none of the functions $\rho$ and $\omega$ need to be known (at least) at the
semiclassical level [14], and to recover the classical probability
distribution from the quantum picture. Moreover, notice that there is no issue
of non-commutativity for the term $\pi_{\phi}^{w+1}$ since the canonical
variable $\phi$ does not appear in the Hamiltonian. Since Eq. (3.42) does not
depend on $\phi$, the wave function $\Phi(a,\phi)$ can be expressed as a
Fourier mode

$\Phi(a,\phi)=\Psi(a)\,e^{i\frac{\lambda}{\hbar}\phi}\,,$ (3.43)

where $\lambda$ is an arbitrary real constant. In that case, one has
$\pi_{\phi}=\lambda$ and $\Psi(a)$ satisfies

$\Psi^{\prime\prime}+\frac{\rho^{\prime}}{\rho}\,\Psi^{\prime}+\omega\,\Psi=\frac{\mathscr{P}(a)}{\hbar^{2}}\,\Psi\,,$
(3.44)

where

$\displaystyle\mathscr{P}(a)$
$\displaystyle=36v_{3}^{2}\,a^{2}\left(1-\frac{C|\lambda|^{w+1}}{3v_{3}a^{3w+1}}\right)$
$\displaystyle=\left\\{\begin{array}[]{ll}36v_{3}^{2}\,a^{2}\displaystyle\left(1-\Big{(}{a_{0}\over
a}\Big{)}^{3w+1}\right),&\mbox{for}\quad
w\neq-1/3\,,\phantom{\\!\\!\overset{\displaystyle|}{|}}\\\
36v_{3}^{2}\,a^{2}\left(1-V_{0}\right),&\mbox{for}\quad
w=-1/3\,.\phantom{\\!\\!\overset{\displaystyle|}{|}}\end{array}\right.$ (3.47)

Finally, as shown in Ref. [14], the quantum Hamiltonian for any fixed
$\lambda$ is Hermitian with respect to the inner product

$\langle\Psi_{1}|\Psi_{2}\rangle=\int_{0}^{+\infty}{\rm
d}a\,\mu(a)\,\Psi_{1}(a)^{*}\Psi_{2}(a)\,,$ (3.48)

where the measure $\mu(a)$ of the Hilbert space is given in Eq. (1.7).

##  4 Quantum probability density

Our goal is to derive the quantum probability distributions and show their
consistencies with the classical results.

Since we are interested in the classical limit, it is enough to solve the WDW
equation (3.44) at the semiclassical level using the WKB method. To this end,
we define

$\Psi=\exp\\!\Big{[}{i\over\hbar}\Big{(}\mathscr{F}_{0}+{\hbar\over
i}\mathscr{F}_{1}+{\cal O}(\hbar^{2})\Big{)}\Big{]}\,,$ (4.49)

which can be used in Eq. (3.44) to get

$\displaystyle\left\\{\\!\\!\begin{array}[]{rl}\mathscr{F}_{0}^{\prime
2}\\!\\!\\!\\!\\!&=-\mathscr{P}\,,\\\
\mathscr{F}_{1}^{\prime}\\!\\!\\!\\!\\!&=\displaystyle-{1\over
2}\Big{(}{\mathscr{F}_{0}^{\prime\prime}\over\mathscr{F}_{0}^{\prime}}+{\rho^{\prime}\over\rho}\Big{)}\end{array}\right.\Longrightarrow\quad\mathscr{F}_{1}=\ln\\!\Big{(}|\mathscr{P}|^{-{1\over
4}}\rho^{-{1\over 2}}\Big{)}+{{\rm const.}}\,.$ (4.52)

As anticipated before, one sees that the function $\omega$ plays no role at
the semiclassical level. Moreover, the expression of $\mathscr{F}_{1}$ implies
$\Psi\propto 1/\sqrt{\rho}$, so that the probability density $\mu|\Psi|^{2}$,
where $\mu=a\rho$, is independent of the function $\rho$, at least at the
semiclassical level.

In the quantum case, the scale factor $a$ can take any value from $0$ to
$+\infty$. When $w\neq-1/3$, the expression of the probability density then
depends on whether $a$ is greater or smaller than $a_{0}$:

$\bullet$ $a<a_{0}$ when $w>-1/3$, or $a>a_{0}$ when $w<-1/3$:
In this case, $\mathscr{P}(a)<0$ so that

$\mathscr{F}_{0}^{\prime}=\epsilon\,\sqrt{-\mathscr{P}}\quad\Longrightarrow\quad\mathscr{F}_{0}=\epsilon\,{4v_{3}\over
1-w}\,a^{2}\Big{(}{a_{0}\over a}\Big{)}^{3w+1\over
2}\Big{[}1+O\Big{(}\Big{(}{a\over a_{0}}\Big{)}^{3w+1}\Big{)}\Big{]}+{{\rm
const.}}\,,$ (4.53)

where $\epsilon=\pm 1$ and the $O$-term is independent of $\epsilon$. As a
result, the wavefunction takes the form

$\Psi(a)=\sum_{\epsilon}N_{\epsilon}(a_{0})\,{\displaystyle\exp\\!\Big{[}\epsilon\,{i\over\hbar}\,{4v_{3}\over
1-w}\,a^{2}\Big{(}{a_{0}\over a}\Big{)}^{3w+1\over
2}\big{[}1+O\big{(}({a/a_{0}})^{3w+1}\big{)}\big{]}\Big{]}\over\sqrt{\rho}\;a^{1-3w\over
4}\big{(}a_{0}^{3w+1}-a^{3w+1}\big{)}^{1\over
4}}\,\big{(}1+O(\hbar)\big{)}\,,$ (4.54)

where $N_{+}(a_{0})$, $N_{-}(a_{0})$ are integration constants that depend on
$a_{0}$ but not on $\hbar$. Note that for any scale factor in the range
$0<a<a_{0}$, the WKB approximation can be trusted. At $a=0$ and $a=a_{0}$, the
adiabaticity condition is not satisfied and one has to solve the full WDW
equation around these points and match the solutions to the WKB one in the
usual way, see e.g. Ref. [23].

The quantum probability density, using the measure in Eq. (1.7), is thus

$\displaystyle P(a)$ $\displaystyle=a\rho|\Psi|^{2}$
$\displaystyle={a^{3w+1\over
2}\over\sqrt{a_{0}^{3w+1}-a^{3w+1}}}\,\big{(}1+O(\hbar))\bigg{\\{}|N_{+}|^{2}+|N_{-}|^{2}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+N_{+}\bar{N}_{-}\exp\\!\Big{[}{i\over\hbar}\,{8v_{3}\over
1-w}\,a^{2}\Big{(}{a_{0}\over a}\Big{)}^{3w+1\over
2}\big{[}1+O\big{(}({a/a_{0}})^{3w+1}\big{)}\big{]}\Big{]}+\mbox{c.c.}\bigg{\\}}\,.$
(4.55)

However, in the $\hbar\to 0$ limit, the frequency of the oscillatory terms
blows up and the latter can be omitted. As a result, we obtain

$P(a)\simeq\Big{(}|N_{+}(a_{0})|^{2}+|N_{-}(a_{0})|^{2}\Big{)}{a^{3w+1\over
2}\over\sqrt{a_{0}^{3w+1}-a^{3w+1}}}\,\big{(}1+O(\hbar)\big{)}\,.$ (4.56)

Therefore, $P$ and $P_{cl}$ (see Eqs. (2.20) and (2.22)) become proportional
in the $\hbar\to 0$ limit.

$\bullet$ $a>a_{0}$ when $w>-1/3$, or $a<a_{0}$ when $w<-1/3$:
This is the classically forbidden region. In this case, we have
$\mathscr{P}(a)>0$ so that

$\mathscr{F}_{0}^{\prime}=\epsilon\,i\sqrt{\mathscr{P}}\quad\Longrightarrow\quad\mathscr{F}_{0}=\epsilon\,i\,3v_{3}\,a^{2}\Big{[}1+O\Big{(}\Big{(}{a_{0}\over
a}\Big{)}^{3w+1}\Big{)}\Big{]}+{\rm const.}\,,$ (4.57)

where $\epsilon=\pm 1$ and again the $O$-term are independent of $\epsilon$.
The wavefunction now takes the form

$\Psi(a)=\sum_{\epsilon}M_{\epsilon}(a_{0})\,{\displaystyle\exp\\!\Big{[}\\!-\\!{\epsilon\over\hbar}\,3v_{3}\,a^{2}\big{[}1+O\big{(}({a_{0}/a})^{3w+1}\big{)}\big{]}\Big{]}\over\sqrt{\rho}\,a^{1-3w\over
4}\big{(}a^{3w+1}-a_{0}^{3w+1}\big{)}^{1\over
4}}\,\big{(}1+O(\hbar)\big{)}\,,$ (4.58)

where $M_{+}(a_{0})$, $M_{-}(a_{0})$ are integration constants independent of
$\hbar$.

The quantum probability density is

$\displaystyle P(a)$ $\displaystyle=a\rho|\Psi|^{2}$
$\displaystyle={a^{3w+1\over
2}\over\sqrt{a^{3w+1}-a_{0}^{3w+1}}}\,\big{(}1+O(\hbar))\bigg{\\{}|M_{-}|^{2}\exp\\!\Big{[}{6\over\hbar}\,v_{3}\,a^{2}\big{[}1+O\big{(}({a_{0}/a})^{3w+1}\big{)}\big{]}\Big{]}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+M_{+}\bar{M}_{-}+\bar{M}_{+}M_{-}+|M_{+}|^{2}\exp\\!\Big{[}\\!-\\!{6\over\hbar}\,v_{3}\,a^{2}\big{[}1+O\big{(}({a_{0}/a})^{3w+1}\big{)}\big{]}\Big{]}\bigg{\\}}\,.$
(4.59)

Considering for $w>-1/3$ a normalizable state, we have

$1=\int_{0}^{a_{0}}a\rho|\Psi|^{2}{\rm
d}a+\int_{a_{0}}^{+\infty}a\rho|\Psi|^{2}{\rm d}a\,,$ (4.60)

where the first term in the r.h.s. is always finite whereas the second term is
finite only when $M_{-}=0$. More generally, for $w\neq-1/3$, imposing that a
finite $\hbar\to 0$ limit exists in Eq. (4.59), we conclude that $M_{-}=0$. In
this case, we obtain

$P(a)=|M_{+}(a_{0})|^{2}\,{a^{3w+1\over
2}\over\sqrt{a^{3w+1}-a_{0}^{3w+1}}}\,\exp\\!\Big{[}\\!-\\!{6\over\hbar}\,v_{3}\,a^{2}\big{[}1+O\big{(}({a_{0}/a})^{3w+1}\big{)}\big{]}\Big{]}\big{(}1+O(\hbar)\big{)}\,.$
(4.61)

As a result, in the classical limit, the quantum probability density in the
classically excluded region is exponentially suppressed. As expected, we have
$P(a)\to 0$ when $\hbar\to 0$.

$\bullet$ $w=-1/3$:
In this particular case, we have $\mathscr{P}(a)<0$ for all $a\geq 0$ and we
find

$\Psi(a)=\sum_{\epsilon}N_{\epsilon}(V_{0})\,{\displaystyle\exp\\!\Big{[}\epsilon\,{i\over\hbar}\,3v_{3}\sqrt{V_{0}-1}\;a^{2}\Big{]}\over\sqrt{a\rho}}\,\big{(}1+O(\hbar)\big{)}\,,$
(4.62)

where $\epsilon=\pm 1$. Therefore, the quantum probability is

$\displaystyle P(a)$ $\displaystyle=a\rho|\Psi|^{2}$
$\displaystyle=\,\big{(}1+O(\hbar))\bigg{\\{}|N_{+}|^{2}+|N_{-}|^{2}+N_{+}\bar{N}_{-}\exp\\!\Big{[}{i\over\hbar}\,6v_{3}\sqrt{V_{0}-1}\;a^{2}\Big{]}+\mbox{c.c.}\bigg{\\}}\,.$
(4.63)

The oscillatory terms can be omitted in the $\hbar\to 0$ limit, so that

$P(a)\simeq\big{(}|N_{+}|^{2}+|N_{-}|^{2}\big{)}\,\big{(}1+O(\hbar))\,,$
(4.64)

which becomes proportional to $P_{cl}$ (see Eq. (2.22)).

Summarizing, for any $w\in[-1,1]$ and any $a\geq 0$, the quantum probability
density $P$ in the $\hbar\to 0$ limit is proportional to the classical one
$P_{cl}$.

For $w>-1/3$, these densities are normalized to 1 and thus, comparing Eqs.
(4.56) and (2.20), we can identify777One could relate $N_{+}$ and $N_{-}$ to
$M_{+}$ by using the gluing relations between the two phases $a>a_{0}$ and
$a<a_{0}$, using Airy functions which solve the WDW equation when $a\simeq
a_{0}$ [23]. In that case we would identify $N_{+},N_{-},M_{+}$.

$|N_{+}(a_{0})|^{2}+|N_{-}(a_{0})|^{2}={1\over\sqrt{\pi}}\,{\left|\Gamma\big{(}{1\over
3w+1}\big{)}\right|\over\Gamma\big{(}{1\over 2}+{1\over 3w+1}\big{)}}\,{1\over
a_{0}}\,.$ (4.65)

Moreover, it is important to stress that although there exists an initial
singularity at the classical level, the fact that the wavefunction (at finite
$\hbar$) is normalizable suggests that the classical singularity at $a\to 0$
is harmless. In fact, the semiclassical probability density satisfies

$P(a)\underset{a\to 0}{\sim}{1\over\sqrt{\pi}}\,{\left|\Gamma\big{(}{1\over
3w+1}\big{)}\right|\over\Gamma\big{(}{1\over 2}+{1\over 3w+1}\big{)}}\,{1\over
a_{0}}\Big{(}{a\over a_{0}}\Big{)}^{3w+1\over 2}\,\big{(}1+O(\hbar)\big{)}\,,$
(4.66)

which vanishes at $a=0$ exactly as in the case of the probability density for
the position of the electron at the center of the hydrogen atom.

We may proceed to calculate at the semiclassical level the expectation value
of the extrinsic curvature $K$ associated with the spatial 3-sphere. The
extrinsic curvature is equal to $6/a^{2}$, yielding

$\langle K\rangle=\int_{0}^{+\infty}6\,{P(a)\over a^{2}}\,{\rm d}a\,.$ (4.67)

At the semiclassical level, in the neighborhood of $a=0$, the integrand is
proportional to $a^{3(w-1)/2}$, showing that the contribution to the
expectation value from the small $a$ region is finite, provided that

$w>{1\over 3}\,.$ (4.68)

Classically, we expect the very early universe to be dominated by the kinetic
energy of moduli fields, i.e. a perfect fluid of index $w=1$. Thus, although
this implies a curvature singularity at the classical level, this is resolved
at the quantum level since the expectation value of the extrinsic curvature is
finite. Notice that this would not be the case had we used the Hartle-Hawking
measure $\mu_{\rm HH}$ for the inner product of the Hilbert space.

For $-1\leq w\leq-1/3$, we have chosen an arbitrary normalization of $P_{cl}$
in Eq. (2.22) (since only ratios yielding relative probabilities make sense).
Hence, we conclude that in the $\hbar\to 0$ limit, the quantum system based on
non-normalizable wavefunctions reproduces consistently the classical relative
probabilities.

Notice that for $w\neq-1/3$, $P(a)$ becomes large when the scale factor is of
the order of $a_{0}$; that is, when the classical closed universe is at its
turning point and the energy density is extremal. The favored size of the
universe “created from nothing” is thus of the order of $a_{0}$, for which the
energy density is extremal.

Our approach can be readily generalized to higher dimensional cases when the
Wheeler-DeWitt equation is a separable equation. For example, we may add, as
in Ref. [8], a conformally coupled scalar field $\phi$. Upon suitable
rescalings ($\phi\to\chi/a$), the Hamiltonian (over the lapse function $N$)
splits into a sum of two factors,
${\cal{H}}_{1}(a,\rho(a))+{\cal{H}}_{2}(\chi)/a$, where
${\cal{H}}_{1}(a,\rho(a))$ is the Hamiltonian for the one-dimensional model we
consider in the present work, and ${\cal{H}}_{2}(\chi)$ is a harmonic
oscillator potential. Imposing the Hermiticity of the full Hamiltonian yields
a measure $\mu(a,\phi)=\mu(a,\rho)\times 1$, where the factor $1$ stands for
the usual “trivial” quantum mechanical measure on the Hilbert space spanned by
the eigenfunctions of ${\cal{H}}_{2}(\chi)$, while $\mu(a,\rho)$ is the
measure in Eq. (1.7). For product wavefunctions, the probability density
$P(a,\chi)$ is independent of $\rho$ (at least at the semiclassical level) and
yields the expected classical behavior in the $\hbar\to 0$ limit.

For more general $d$-dimensional minisuperspace models, one could employ the
Klein-Gordon norm, which is defined in terms of an integral over a
$(d-1)$-dimensional surface of the conserved Klein-Gordon current [1].
Interpreting the coordinate minisuperspace field in the direction
perpendicular to this surface as “time,” the Klein-Gordon norm and the
associated probabilities are conserved. However, the Klein-Gordon norm is not
positive definite and vanishes for real wavefunctions – see e.g. [16, 17, 18]
for discussions. In some simpler models, the problem of negative probabilities
can be avoided by suitably choosing the $(d-1)$-dimensional integration
surface (as shown by DeWitt in Ref. [1]), but it is not clear if this approach
can be extended in the more general cases.

Rather, we could seek to define a positive definite inner product based on the
modulus of the wavefunction squared, $|\Psi|^{2}$, and an appropriate positive
measure $\mu(a,\phi^{i})$. For a certain operator ordering choice, the
Hamiltonian becomes proportional to the laplacian $\nabla^{2}$ on the
minisuperspace manifold, with respect to an appropriate metric of Lorentzian
signature. Thus, the natural choice of measure $\mu(a,\phi^{i})$ that ensures
the Hermiticity of the Hamiltonian in this case is the volume element [16].
Imposing the Hermiticity of the Hamiltonian for generic operator orderings, we
expect the measure to be modified to $\mu(a,\phi^{i},\rho)$,888In fact, this
measure could be obtained by generalizing the derivation given in Appendix A.5
of Ref. [14]. where the function $\rho(a,\phi^{i})$ parametrizes the ordering
ambiguities in the higher $d$ cases at the semiclassical level.999Depending on
the model, more than one function maybe needed to parametrize the possible
ordering ambiguities. It would be interesting to investigate whether the
resulting probability densities are independent of $\rho$, as in the one-
dimensional cases. In addition, we would like to see if the expected classical
probability distribution will be recovered in the $\hbar\to 0$ limit. To this
extend, it would be interesting to compare with the seminal work [18], where
probability distributions are constructed based on the decoherent histories
approach to quantum mechanics.

Notice that as in some one-dimensional cases, it may turn out that the
wavefunction is non-normalizable with respect to this inner-product in
phenomenologically interesting higher $d$ cosmological models. So at best, we
can use the wavefunction in these cases to define relative probabilities. For
instance, it would be interesting to extend our approach to inflationary
models such as the Starobinsky model, involving the scale factor and a scalar
field with a potential suitable to realize inflation. Finding the measure on
the Hilbert space would allow us to define relative probabilities between
configurations where the inflaton slowly rolls along the plateau of the
potential and a configuration where the inflaton sits at the minimum of the
potential. In principle, the structure of probability density would allow us
to conclude whether initial conditions suitable for inflation are favored.

##  5 Conclusion

Field redefinitions of the scale factor provide us with different
prescriptions for implementing the no-boundary proposal and computing the
wavefunction as a path integral. These wavefunctions satisfy different WDW
equations that also admit ordering ambiguities occurring when quantizing the
Hamiltonians. At least at the semiclassical level, the inner products on the
corresponding Hilbert spaces turn out to be equivalent, leading to universal
predictions [14].

In this work we have shown that the quantum probability density reproduces the
classical probability distribution in the $\hbar\to 0$ limit, for a
homogeneous and isotropic closed universe filled with a perfect fluid with
equation of state $p_{m}=w\rho_{m}$, $w\in[-1,\,1]$. When the index $w$ is in
the range $(-1/3,1]$, the cosmology has a finite lifetime, between a big bang
and a big crunch, and the classical probability distribution can be normalized
to unity. Likewise the corresponding quantum wavefunction is normalizable. The
quantum probability density is vanishingly small at the big bang and big
crunch singularities, indicating that a singular universe could be excluded at
the quantum level. The quantum expectation value of the extrinsic curvature is
finite provided that $w>1/3$.

On the other hand, based on semiclassical considerations, we show that the
quantum wavefunction is not normalizable when the prefect fluid index
satisfies $-1\leq w\leq-1/3$, corresponding to accelerating cosmologies (for
$w<-1/3$) with infinite duration. One however may use the wavefunction to
define relative probabilities, with the classical behavior reproduced in the
$\hbar\to 0$ limit.

## Acknowledgements

The authors would like to thank Balthazar de Vaulchier for discussions. A.K
and N.T. would like to thank the Ecole Polytechnique for hospitality. The work
of A.K. is partially supported by the NTUA basic research grant no 65228100.

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