Quantum instability of an oscillating universe

TL;DR

This paper investigates the quantum stability of an oscillating universe model with a scalar field, showing that quantum instabilities prevent stable stationary states under certain conditions.

Contribution

It demonstrates that quantum instabilities, beyond previously known effects, hinder the existence of localized stationary states in an oscillating universe with a scalar field.

Findings

Quantum instability prevents stationary states with localized wave functions.

Previous sources of instability like particle production are assumed absent.

The analysis is based on solutions to the Wheeler-DeWitt equation.

Abstract

An oscillating, compact Friedmann universe with a massive conformally coupled scalar field is studied in the framework of quantum cosmology. The scalar field is treated as a perturbation and we look for solutions of the Wheeler-DeWitt equation describing stable stationary states of the model. We assume that the previous sources of quantum instability that have been discussed in the literature (particle production, and tunnelling to zero size) are absent. We then show, under rather general assumptions, that a further source of quantum instability prevents the existence of stationary states with localized wave function in the direction of the scalar-field modes.

Equations

$$\varphi (\mathbf{x},t)=\frac{1}{a(t)}\sum _n{\varphi }_n(t)Q_n(\mathbf{x}),$$

$$\mathcal{H}=-p_a^2-U(a)+\sum _n\left(p_n^2+{\omega }_n^2(a){\varphi }_n^2\right),$$

$$U(a)=a^2\left(k-a^2\rho (a)\right),$$

$${\omega }_n^2(a)=n^2+m^2{a}^2,$$

$$\mathcal{H}\Psi (a,\{{\varphi }_n\})=0,$$

$$\left[{\hslash }^2\frac{{\partial }^2}{\partial a^2}-U(a)+\sum _n\left(-{\hslash }^2\frac{{\partial }^2}{\partial {\varphi }_n^2}+{\omega }_n^2(a){\varphi }_n^2\right)\right]\Psi =0.$$

$$J=i({\Psi }^{\ast }\nabla \Psi -\Psi \nabla {\Psi }^{\ast }),$$

$$\mathcal{P}\propto |\Psi {|}^2.$$

$$\underset{a\to \infty }{\lim }\Psi (a,{\varphi }_n)=0,$$

$$\underset{{\varphi }_n\to \pm \infty }{\lim }\Psi (a,{\varphi }_n)=0.$$

$$\underset{a\to 0}{\lim }\Psi (a,{\varphi }_n)=0.$$

$$\Psi (a,\{{\varphi }_n\})=\exp \left[-\frac{S(a)}{\hslash }-\frac{1}{2\hslash }\sum _n{S}_n(a){\varphi }_n^2\right],$$

$$\mathrm{Re}\left[S_n(a)\right]>0,$$

$${S^{\prime }}^2-U(a)-\hslash S^{\prime \prime }+\hslash \sum _n{S}_n=0,$$

$$\left[S^{\prime }{S_n}^{\prime }-S_n^2+{\omega }_n^2(a)-\frac{\hslash }{2}{S_n}^{\prime \prime }\right]{\varphi }_n^2+\frac{1}{4}{{S_n}^{\prime }}^2{\varphi }_n^4=0.$$

$${S^{\prime }}^2-U(a)=0,$$

$$S^{\prime }{S_n}^{\prime }-S_n^2+{\omega }_n^2=0.$$

$$S^{\pm }(a)=\pm i{\int }_a^{a_2}\sqrt{-U(a^{\prime })}d{a}^{\prime }$$

$${\Psi }_0(a)=A_{+}{e}^{-S^{+}(a)}+A_{-}{e}^{-S^{-}(a)}=A_{+}{e}^{-i{\int }_a^{a_2}\sqrt{-U(a^{\prime })}d{a}^{\prime }}+A_{-}{e}^{+i{\int }_a^{a_2}\sqrt{-U(a^{\prime })}d{a}^{\prime }}(a_1<a<a_2),$$

$${\bar{S}}_2^{+}(a)\equiv +{\int }_{a_2}^a\sqrt{U(a^{\prime })}d{a}^{\prime }>0,(a>a_2),$$

$${\Psi }_0(a)=B_2{e}^{-{\bar{S}}_2^{+}(a)}(a>a_2),$$

$$B_2=e^{-i\pi /4}{A}_{+}=e^{i\pi /4}{A}_{-}.$$

$${\bar{S}}_1^{+}\equiv +{\int }_a^{a_1}\sqrt{U(a^{\prime })}d{a}^{\prime }>0,(0<a<a_1),$$

$${\Psi }_0(a)=B_1{e}^{-{\bar{S}}_1^{+}(a)}(0<a<a_1).$$

$$B_1=A_{+}{e}^{i\pi /4}{e}^{-iQ}=A_{-}{e}^{-i\pi /4}{e}^{iQ},$$

$$Q={\int }_{a_1}^{a_2}\sqrt{-U(a)}da.$$

$$Q=\left(n+\frac{1}{2}\right)\pi ,$$

$$B_1=e^{i\pi /4}{A}_{-}=B_2.$$

$$\left[-\frac{d^2}{d{a}^2}+U(a)\right]{\Psi }_0={ϵ}_r{\Psi }_0.$$

$$\Psi (a,\{{\varphi }_n\})=A_{+}\exp \left[-S^{+}(a)-\frac{1}{2}{S}_n^{+}(a){\varphi }_n^2\right]+A_{-}\exp \left[-S^{-}(a)-\frac{1}{2}{S}_n^{-}(a){\varphi }_n^2\right](a_1<a<a_2)$$

Full text

I Introduction

In recent years there has been much interest in the possible existence of absolutely stable models describing a universe that exists forever in a static or oscillating state. Apart from the intrinsic interest of such models, they can play the role of an ‘eternal seed’, providing a starting point for the emergent universe scenario (see, e.g, Mulryne:2005ef and references therein).

Construction of classical oscillating cosmologies is relatively straightforward. See Ref. Dabrowski1 for a general class of oscillating models. A particularly simple oscillating model of a closed Friedmann universe was discussed in Ref. Graham:2011nb : it includes a negative cosmological constant $\Lambda$ and a matter component having the effective equation of state $P=w\rho$ with $w=-2/3$ . A negative $\Lambda<0$ causes an expanding universe to turn around and recollapse, while the $w=-2/3$ matter component provides an energy density $\rho_{w}=\rho_{0}a^{-3(1+w)}=\rho_{0}a^{-1}$ which causes a collapsing universe to bounce. The evolution equation for this model has solutions where the scale factor changes sinusoidally with time, so the model has been dubbed “a simple harmonic universe” (SHU). We will give below another example of matter content leading to a simple sinusoidal variation of the square of the scale factor.

In such models the universe would classically oscillate forever between two scale factor values $a_{1}$ and $a_{2}$, but one needs to watch for potential quantum instabilities. A first source of instability is that the oscillation of the universe can induce particle production, which would be manifested by exponentially growing modes of quantum fields. The analysis in Refs. MV2 ; Graham2 has shown that unstable modes are indeed present for a significant part of the parameter space in a number of oscillating models, including SHU. However, there remains a significant, complementary range of the parameter space where the models do not exhibit any unstable modes. We shall assume here that we are in a parameter range where our oscillating model is stable under particle production. A second possible source of quantum instability is non perturbative: it was argued in Refs. MV1 (following the earlier work of Dabrowski ) that oscillating models are generically unstable with respect to collapse to zero volume via quantum tunneling. However, the presence or absence of this instability is linked to the choice of a boundary condition, at zero volume, for the wave function $\Psi$ of the universe, an issue which concerns the deep UV regime of quantum gravity. To fix ideas, we shall assume below (following an old suggestion of DeWitt DeWitt ) that $\Psi$ vanishes on singular universes, thereby avoiding the instability under tunneling to zero size111We note in passing that, in the quantum supergravity model of Ref. Damour , there exist exact quantum states describing seed universes where $\Psi$ vanishes (without fine-tuning) both for small and large volumes.. However, we shall see that this assumption is not important for reaching our main result.

In the present paper we shall study an oscillating universe in the framework of quantum cosmology, where both the scale factor $a$ and the quantum field modes $\phi_{n}$ are described by a wave function $\Psi(a,\phi_{n})$. This wave function is defined not only in the classical range $a_{1}<a<a_{2}$, but also extends beyond it into the classically forbidden regions. A stable oscillating universe would correspond to a stationary state described by a wave function which is localized around the classical solution (corresponding to a periodic function $a(t)$ oscillating between $a_{1}$ and $a_{2}$, and to the classical ground state $\phi_{n}=0$), and which exponentially decreases away from this solution, both in the $a$ and $\phi_{n}$ directions. Our main result is that such stationary states generally do not exist.

II Wheeler-DeWitt equation

We consider a compact ($k=+1$) Friedmann universe filled with homogeneous matter of energy density $\rho(a)$, and with a massive conformally coupled scalar field $\phi({\bf x},t)$, whose background value vanishes, and which is treated as a perturbation at the quantum level.222A minimally coupled field can be analyzed in a similar way with essentially the same results. The precise form of the function $\rho(a)$ will not be important for our analysis. We expand the field $\phi$ as

[TABLE]

where $Q_{nlr}({\bf x})$ are suitably normalized harmonics on the 3-sphere, $n=1,2,...$. We have suppressed the indices $l,r$ for brevity. After appropriate rescalings (see, e.g., Vilenkin:1987kf ), the Hamiltonian of this model can be written as

[TABLE]

where $p_{a}$ and $p_{n}$ are momenta conjugate to $a$ and $\phi_{n}$ respectively,

[TABLE]

[TABLE]

$k$ is the curvature parameter (here taken to be +1) and $m$ is the mass of the field $\phi$.

In quantum theory the momenta are replaced by differential operators, $p_{a}\to-i\hbar\partial/\partial a$, $p_{n}\to-i\hbar\partial/\partial\phi_{n}$, and the classical Hamiltonian constraint ${\cal H}=0$ is replaced by the Wheeler-DeWitt (WDW) equation for the wave function of the universe,

[TABLE]

or explicitly

[TABLE]

Here we write the Planck constant $\hbar\equiv 1$ only as a formal bookkeeping parameter, to clarify the use of the WKB approximation.

The probabilistic interpretation of the wave function $\Psi$ is a matter of debate. Some authors interpretation ; Halliwell have advocated that the probability should be expressed in terms of the conserved current,

[TABLE]

where the gradient $\nabla$ is with respect to the superspace variables $\{a,\phi_{n}\}$. This approach has a number of attractive properties, but it fails in the case of a stationary state, where $J=0$. An alternative approach HP1 is to use the ‘naive Schrödinger measure’,

[TABLE]

It has been argued in HP1 that the two approaches agree under certain assumptions. We will not attempt to resolve this issue here. In the present work, it will be enough to argue from the quadratic-in-$\Psi$ structure of both expressions, Eqs. (7) and (8), that $\Psi$ should be constrained to tend to zero far away from the support of the classical solution.

In particular, as our model is intended to describe the quantum version of a classical solution where a periodic scale factor $a^{\rm classical}(t)$ oscillates between $a_{1}$ and $a_{2}$, and where $\phi^{\rm classical}({\bf x},t)=0$, we should require that the wave function $\Psi(a,\phi_{n})$ satisfy

[TABLE]

and

[TABLE]

As already mentioned, we shall also tentatively require that $\Psi$ vanishes near zero-volume, singular universes:

[TABLE]

It is possible that a UV-complete theory of gravity might resolve the singularity at $a=0$, replacing it with a non-singular Planck-size nugget. The universe may then tunnel back and forth between the semiclassical oscillating regime and the nugget, resulting in a stationary quantum state. Eq. (11) would then be an approximate boundary condition, requiring that the probability of finding the universe at $a\lesssim 1$ is very small.

Using a standard technique Banks ; Lapchinsky ; HH , an approximate semiclassical solution to the WDW equation can be constructed as a linear combination of $\phi_{n}$-Gaussian terms of the form

[TABLE]

where $S(a)$ and $S_{n}(a)$ are complex functions of $a$, to be determined. For a stationary state the functions $S_{n}(a)$ must satisfy the stability condition Vachaspati ,

[TABLE]

which ensures that the modulus of the wave function decreases with increasing amplitude of the field fluctuations $\phi_{n}$.

Substituting the ansatz of Eq. (12) into the WDW equation (6) we obtain the conditions

[TABLE]

[TABLE]

Our approximation consists of disregarding the terms of order ${\cal O}(\hbar)$ and ${\cal O}(\phi_{n}^{4})$; a detailed discussion of its applicability can be found in Ref. Hong . We then have the following WKB equations for the functions $S(a)$ and $S_{n}(a)$,

[TABLE]

[TABLE]

From now on we set $\hbar=1$.

III Solving the WDW equation for the quantum dynamics of the scale factor: $\Psi_{0}(a)$

We first consider the part $\Psi_{0}(a)\equiv\exp\left[-\frac{1}{\hbar}S(a)\right]$ of the wave function that depends only on the scale factor. We will add later the $\phi_{n}$-dependent factor $\exp\left[-\frac{1}{2\hbar}\sum_{n}S_{n}(a)\phi_{n}^{2}\right]$.

In our notation, an oscillating universe corresponds to a WDW potential $U(a)$ that is negative, $U(a)<0$, in the oscillation range $a_{1}<a<a_{2}$, and positive, $U(a)>0$, in the classically forbidden regions on both sides of this range. The classical version of this model admits solutions where $a(t)$ oscillates periodically between the turning points $a_{1}$ and $a_{2}$.

The solutions of the zeroth order equation (16) in the classically allowed range are

[TABLE]

with upper and lower signs corresponding to contracting and expanding universe, respectively. For a stationary state the $a$-part of the wave function should be a superposition of such terms,333The standard WKB pre-exponential factors $\propto[-U(a)]^{-1/4}$ can be determined by solving the WDW equation to second order in $\hbar$. These factors do not affect our conclusions and we disregard them here.

[TABLE]

with $A_{\pm}={\rm const}$. In the range $a>a_{2}$ we have exponentially growing and decreasing solutions $\exp{\pm{\bar{S}}_{2}^{+}(a)}$ with

[TABLE]

where the bar over $S$ indicates that the solution is in a classically forbidden region, and the index $2$ that we are in the range beyond $a_{2}$. They can be matched to those in the allowed region using the WKB connection formulas LandauLifshitzQM . The boundary condition (9) selects the decreasing solution

[TABLE]

and the WKB connection formulas give the relations

[TABLE]

In the second classically forbidden region $0\leq a<a_{1}$ we also have exponentially growing and decreasing solutions, $\exp{\pm{\bar{S}}_{1}^{+}(a)}$, with

[TABLE]

and the boundary condition (11) can be approximately implemented by selecting

[TABLE]

The matching conditions at $a=a_{1}$ then require

[TABLE]

where

[TABLE]

It follows that $Q$ must be equal to

[TABLE]

where $n$ is an integer, and that

[TABLE]

Eq. (27) is just the Bohr-Sommerfeld quantization condition. It must be satisfied because of our double boundary condition (11), (9). One way to enforce it (if it is not automatically satisfied as in the quantum supergravity model of Ref. Damour ) is to fine-tune the matter content of the universe. For example, one can introduce an adjustable amount of radiation with energy density $\rho_{r}=\epsilon_{r}a^{-4}$, where $\epsilon_{r}={\rm const}$. The WDW potential is then replaced by $U(a)-\epsilon_{r}$ and the WDW equation for $\Psi_{0}(a)$ becomes

[TABLE]

This has the same form as the standard energy eigenvalue equation, and the condition (27) determines the spectrum of allowed values of $\epsilon_{r}$.444In our semiclassical treatment the quantity $\epsilon_{r}$ characterizing the amount of radiation is assumed to be a continuous parameter. It is not clear that such continuously tunable parameters will exist in the full quantum theory. One might expect that, on the contrary, $\epsilon_{r}$ could be quantized. Then there is no guarantee that its spectrum will allow us to enforce our boundary conditions.

IV Completing the solution with $\phi_{n}$-dependent factors

The full wave function, including scalar field perturbations, can be written as

[TABLE]

in the classically allowed region and

[TABLE]

in the two classically forbidden regions. The functions $S_{n}^{\pm}(a)$ and ${\bar{S}}_{n\{1,2\}}(a)$ satisfy Eq. (17) with $S(a)=S^{\pm}(a)$ and $S(a)={\bar{S}}_{\{1,2\}}^{+}(a)$, respectively. Continuity requires that these functions have to match at the turning points $a_{1}$ and $a_{2}$ Vachaspati :

[TABLE]

[TABLE]

The matching conditions (32), (33), will play a central role in our reasonings.

IV.1 Uniqueness of the solution for $S_{n}(a)$ in the outer classically forbidden region $a>a_{2}$

We start our analysis by considering the outer classically forbidden region, $a>a_{2}$. In this region $S(a)={\bar{S}}_{2}^{+}(a)\equiv+\int_{a_{2}}^{a}\sqrt{U(a^{\prime})}da^{\prime}>0$ is real (and growing with $a$). It is then convenient to define an auxiliary Euclidean time variable $\tau$ by

[TABLE]

This Euclidean time grows with $a(>a_{2})$, and can be set to be zero when $a=a_{2}$.

Then Eq. (17) becomes

[TABLE]

where a dot stands for a derivative with respect to $\tau$.

Let us first note that Eq. (35) is a first-order ordinary differential equation (ODE). Its general solution will therefore contain only one arbitrary constant. We must impose on the solution $S_{n}(\tau)$ to satisfy the stability inequality (13). We are going to show that it is possible to satisfy the inequality (13) in the entire region $a>a_{2}$, but that this condition uniquely determines a solution of (35). This surprisingly strong effect of imposing the apparently benign sign inequality (13) is due to the exponential behavior of the generic solution of the Euclidean-time equation (37).

Eq. (35) is a nonlinear ODE of the Riccati type. As is well-known such a Riccati equation can be transformed into a linear second-order ODE by representing $S_{n}$ in the form of a logarithmic derivative:

[TABLE]

Then $u_{n}(\tau)$ must satisfy the linear equation

[TABLE]

The general solution of the second-order ODE (37) contains (for each mode number $n$) two arbitrary constants, say $C_{n+}$ and $C_{n-}$. However, the logarithmic derivative $\dot{u}_{n}/u_{n}$ parametrizing $S_{n}(\tau)$ only depends on the ratio $C_{n+}/C_{n-}$ so that solving (37) leads to the same number (namely one) of arbitrary constants as solving the original equation, Eq. (35), for $S_{n}(\tau)$. We look for a solution of (37) such that $S_{n}=-{\dot{u}}_{n}/u_{n}$ satisfies the inequality (13).

Let us first discuss the asymptotic behavior of the mode functions at large $a$ by approximating the general solution of Eq. (37) by means of the WKB approximation,

[TABLE]

where $C_{n\pm}$ are two a priori complex constants. We shall later indicate that the results derived from the latter WKB approximation can be confirmed by using corresponding exact solutions of Eq. (37).

The integral in the exponents read

[TABLE]

Its asymptotic behavior for large values of $a$ can be expressed as

[TABLE]

For all of the known (and most of the hypothesized) forms of matter, the energy density $\rho(a)$ decreases with $a$ or at most remains constant. Then $U(a)$ does not increase faster than $a^{4}$ and the above integral diverges at $a\to\infty$. Hence, if $C_{n+}\neq 0$, the first term in (38) dominates at large $a$ and the representation (36) yields an asymptotically real $S_{n}(a)$ satisfying

[TABLE]

Here, the crucial minus sign in front of $\omega_{n}(a)$ (which implies that $S_{n}(a)$ is generically negative for large $a$) comes from the minus sign in front of ${\dot{u}}_{n}/u_{n}$ in Eq. (36), and the latter minus sign was uniquely determined by the various signs in the Riccati equation (35).

To avoid violation of the regularity condition (13), we should therefore set $C_{n+}=0$, so that

[TABLE]

Therefore, within the WKB approximation (38) for $u_{n}(\tau)$, the choice $C_{n+}=0$ (for all $n$’s) ensures that the regularity condition (13) is satisfied in the whole region $a>a_{2}$. Let us actually show that this result remains true without making use of the WKB approximation to solve the linear ODE (37). Indeed, let us first remark that Eq. (37) is a real equation, which therefore admits two independent real solutions. Actually, Eq. (37) has the general form of Schrödinger’s (fixed energy) one-dimensional equation in a classically forbidden interval. It is shown in Messiah’s Quantum Mechanics treatise Messiah (using Floquet-theory-type reasonings) that among the real solutions normalized to satisfy $u_{n}(\tau=0)=1$ there is a unique one, say $u_{n}^{-}(\tau)$, such that

[TABLE]

on the entire interval of variation of $\tau$, with $u_{n}^{-}(\tau)\to 0$ as $\tau$ tends to the upper limit of its interval of variation555$\tau$ varies between 0 and $\tau_{m}=\int_{a_{2}}^{\infty}da/\sqrt{U(a)}$. Depending on the oscillating-universe model, $\tau_{m}$ can be infinite or finite. However, what is crucial for our results is that the total “number of e-folds” $\int\omega_{n}d\tau=\int^{\infty}_{a_{2}}da\sqrt{n^{2}+m^{2}a^{2}}/\sqrt{U(a)}$ be infinite, which is the case in all models, as we already pointed out.. As second independent (real) solution one can take the solution $u_{n}^{+}(\tau)$ with initial data $u_{n}^{+}(\tau=0)=1$ and ${\dot{u}}_{n}^{+}(\tau=0)=0$. The latter solution is similar to the WKB solution $\cosh\left[\int^{\tau}_{0}\omega_{n}(\tau^{\prime})d\tau^{\prime}\right]$: it stays positive, grows monotonically (and exponentially), as $\tau\to\tau_{m}$, and is such that

[TABLE]

Redoing our reasoning above with the exact general solution

[TABLE]

leads to the same conclusions: if $C_{n+}\neq 0$, we have $S_{n}(a)=-\dot{u}_{n}/u_{n}\sim-\omega_{n}(a)\to-\infty$ as $a\to\infty$. The only stable solution is obtained for $C_{n+}=0$, and it satisfies

[TABLE]

Summarizing: we have shown that there is a unique solution of Eq. (35) in the outer classically forbidden range $a>a_{2}$ that is physically acceptable (namely satisfying the condition (13)). It is given by Eq. (46). Note that this unique solution is real, and therefore satisfies not only ${\rm Re}[S_{n}(a)]>0$, but actually the stronger condition $S_{n}(a)>0$.

In view of our matching condition (33), the above constructed unique stable solution yields, for each mode number $n$, a specific boundary condition for $S^{\pm}_{n}(a)$ at their outer boundary, say

[TABLE]

where we defined

[TABLE]

Here, we have introduced the notation $\mu_{n}$ for the uniquely defined sequence of (minus) the logarithmic derivatives of $u_{n}^{-}(\tau)$, evaluated at $\tau=0$. The real, positive numbers $\mu_{n}$ are characteristic numbers associated with the functions $\omega^{2}_{n}(\tau)$ (considered in the outer forbidden region). They will play a crucial role in the rest of our analysis.

IV.2 Non-existence of a solution for $S^{\pm}_{n}(a)$ in the classically allowed region matching the

outer solution ${\bar{S}}^{\rm outer}_{n}(a)$

In the classically allowed range, where ${S}^{\prime}(a)$ is pure imaginary, it will be convenient to introduce an auxiliary (Minkowskian) time variable $t$ which is related to $a$ by the classical equation of motion

[TABLE]

Then Eq. (17) becomes

[TABLE]

where a dot now stands for a derivative with respect to $t$.

Eq. (50) is again a nonlinear ODE of the Riccati type (which is now complex). We transform it into a linear second-order ODE by representing $S_{n}$ as the following (complex) logarithmic derivative:

[TABLE]

Then $u_{n}(t)$ must satisfy the linear equation

[TABLE]

with $a(t)$ satisfying Eq. (49). Note that this is the Minkowskian-time version ($t\to i\tau$) of the corresponding forbidden-range equation (37). As in the previous subsection, though the general solution of the second-order ODE (52) contains two arbitrary constants, say ${C}_{n}$ and ${\tilde{C}}_{n}$, the logarithmic derivative $\dot{u}_{n}/u_{n}$ parametrizing $S_{n}(t)$ only depends on the ratio ${\tilde{C}}_{n}/C_{n}$ so that solving (52) does not introduce more arbitrary constants than solving the original Riccati equation, Eq. (50), for $S_{n}(t)$.

The upper choice of sign in Eq. (49) corresponds to an expanding universe. The corresponding classical solution $a(t)$ varies from $a_{1}$ to $a_{2}$ with ${\dot{a}}=0$ at both ends, and its quantum-mechanical counterpart is the first term in Eq. (30). For the lower choice of sign, the universe contracts back from $a_{2}$ to $a_{1}$. Combining the two solutions we obtain a periodic function $a(t)$, of period $T$: $a(t+T)=a(t)$. Eq. (52) (which happens to coincide with the classical equation for scalar field modes in the oscillating universe background) is a Hill equation, and the stability of its solutions can be analyzed using Floquet theory Floquet .

We recall that Floquet theory consists in expressing the general solution of Eq. (52) as a linear combination of the two eigenmode solutions of Eq. (52) that reproduce themselves, modulo a multiplicative factor $\lambda_{n}^{\pm}$, after a period $T$. The eigenvalues $\lambda_{n}^{\pm}$ satisfy $\lambda_{n}^{+}\lambda_{n}^{-}=1$ and are either complex conjugated with unit modulus ($\lambda_{n}^{\pm}=e^{\pm i\alpha_{n}}$), or real ($\lambda_{n}^{\pm}=e^{\pm\beta_{n}}$). The first case ($\lambda_{n}^{\pm}=e^{\pm i\alpha_{n}}$) leads to stable solutions of the Hill equation of the general form

[TABLE]

Here $P_{n}(t)$ and ${\tilde{P}}_{n}(t)$ are complex-conjugated periodic functions with the same period $T$ as $a(t)$, $C_{n}$ and ${\tilde{C}}_{n}$ are arbitrary complex coefficients, and $\gamma_{n}\equiv\alpha_{n}/T$ is real (and, say, positive: $\gamma_{n}>0$). In the unstable case ($\lambda_{n}^{\pm}=e^{\pm\beta_{n}}$), the general solution of Eq. (52) can still be written in the form (53), but with the understanding that $\gamma_{n}\equiv-i\beta_{n}/T$ be purely imaginary. In that case, the two periodic functions $P_{n}(t)$ and ${\tilde{P}}_{n}(t)$ are both real.

We recall that the function $u_{n}(t)$, entering the representation (51) (and satisfying the Hill equation Eq. (52)) is just an auxiliary function that we introduced to solve the Riccati equation (50). We must find some solution $u_{n}(t)$ such that the corresponding function $S_{n}=-i\frac{\dot{u}_{n}}{u_{n}}$ satisfies two different requirements: (i) $S_{n}^{\pm}(a)$ must satisfy the matching conditions (32), (33), together with our strong outer matching condition (47); and (ii) they must satisfy (for all values of $a$) the stability inequality (13).

If we did not have the extra outer matching condition (47), we might have considered the possibility of constructing a sequence of functions $S_{nN}^{\pm}(a)$ entering various branches of a wave function of the form $\Psi(a,\phi_{n})=\sum_{N}\Psi_{N}(a,\phi_{n})$, with $\Psi_{N}(a,\phi_{n})$ containing an $N$-dependent factor $\exp\left[-\frac{1}{2}S_{nN}^{\pm}(a)\phi_{n}^{2}\right]$. Such a wave function might describe successive cycles of oscillations of a quantum universe that would only differ in the quantum dispersion of the $\phi_{n}^{\prime}s$. However, our outer matching condition (47), in conjunction with the matching conditions (32), (33), require $S_{n}(t)$ to be a periodic function of the Minkowskian time $t$, i.e. $S_{n(N+1)}^{+}(a)$ to be equal to $S_{nN}^{+}(a)$, and $S_{n(N+1)}^{-}(a)$ to be equal to $S_{nN}^{-}(a)$. Indeed, suppose we follow the wave function (30) for one oscillation period, as $a$ changes from its maximum value $a_{2}$ to $a_{1}$ and back. We must start from a given value $S_{n}^{-}(a_{2})=\mu_{n}$. In the first half-period $S_{n}^{-}(a_{2})=\mu_{n}$ evolves into some corresponding uniquely defined $S_{n}^{-}(a_{1})$, which must equal $S_{n}^{+}(a_{1})$ in view of the junction condition (32). Then, in the second half-period $S_{n}^{+}(a_{1})=S_{n}^{-}(a_{1})$ evolves into some corresponding $S_{n}^{+}(a_{2})$. The latter value must, in view of the other junction condition (33), be equal to $S_{n}^{-}(a_{2})$, which must (in view of the fixed outer matching condition (47)) be again equal to the same value $S_{n}^{-}(a_{2})=\mu_{n}$, with which we started the cycle. Q.E.D.

The mode functions $u_{n}(t)$ are not generally periodic, and, because of the multiplicative change of the eigenmodes after one period, their logarithmic derivative $S_{n}=-i\frac{\dot{u}_{n}}{u_{n}}$ is also not generally periodic. It is easily seen that the only way to construct a periodic $S_{n}(t)$ is to use just one of the two eigenmodes in Eq. (53). Using only the first mode, i.e. setting ${\tilde{C}}_{n}\to 0$ in Eq. (53), yields

[TABLE]

This yields a periodic $S_{n}(t)$ both in the unstable case ($\gamma_{n}\equiv-i\beta_{n}/T$, with real $P_{n}(t)$), and in the stable case ($\gamma_{n}\equiv\alpha_{n}/T>0$ with a complex periodic function $P_{n}(t)$). However, in the unstable case, the so-constructed periodic $S_{n}(t)$ is purely imaginary, and therefore does not satisfy the regularity condition (13), expressing that the $\phi_{n}$ wave function is localized near $\phi_{n}=0$. We thus reach the physically reasonable conclusion that stationary states of an oscillating universe do not exist in the presence of unstable scalar field modes, i.e. in presence of particle creation effects. As said in the Introduction, we assume here that all the $\phi_{n}$ modes are stable, therefore we continue our discussion by focussing on the stable case.

For a stable mode, $u_{n}(t)$, with $\gamma_{n}\equiv\alpha_{n}/T>0$, the construction (54) leads (for general values of $a$) to a complex-valued $S_{n}(t)$ (because $P_{n}(t)$ is a complex, periodic function) whose real part oscillates periodically around $\gamma_{n}>0$: $\langle{\rm Re}[S_{n}(t)]\rangle=\gamma_{n}$. However, we must ensure that the regularity condition $Re[S_{n}(t)]>0$, Eq. (13), is satisfied not only on average, but at any time $t$ during an entire oscillation period. It is, however, easy to show that this is automatically the case.

Indeed, let us decompose the complex number $u_{n}(t)$ in modulus and phase,

[TABLE]

with $\rho$ and $\theta$ real. Then

[TABLE]

so that

[TABLE]

Substituting (55) in Eq. (52) we obtain

[TABLE]

Taking the imaginary part of this equation yields

[TABLE]

From Eq. (59), we see (barring singular cases were $\rho$ might vanish or become infinite) that ${\rm Re}[S_{n}]={\dot{\theta}}$ will always keep the same sign during an entire oscillation period. As its average $\gamma_{n}$ has been chosen to be positive, we conclude that the stability condition (13) will remain satisfied all over the classically allowed range. Note in passing that Eq. (59) is actually just Kepler’s area law, applied to the Newtonian dynamical equation (52) for the motion of a particle moving in the complex plane under the influence of a time-dependent central force.

Summarizing so far: the matching conditions (32), (33),(47), impose the periodicity of $S_{n}$ as a function of $t$, and there is a unique solution of Eq. (50) in the classically allowed range $a_{1}<a<a_{2}$ that is periodic, and satisfies the condition (13)). It is given by Eq. (54) (with $\gamma_{n}>0$).

However, that’s where the good news end. Indeed, we must satisfy not only periodicity of $S_{n}(t)$, but also the precise boundary condition $S_{n}(a_{2})=\mu_{n}$, where the sequence of characteristic numbers $\mu_{n}>0$ is uniquely fixed by the structure of the function $\omega_{n}^{2}(a)$ (for $a>a_{2}$).

Let us first show that the values of $S_{n}(a_{1})$ and $S_{n}(a_{2})$, as obtained from Eq. (54), are some, uniquely defined, real numbers, say

[TABLE]

Indeed, the fact that the $t$-evolution of $a$ is defined by Eq. (49), i.e.

[TABLE]

implies that we have time-reversal symmetry under $t\to t^{\prime}=-t+{\rm const.}$. Actually, a moment of thought shows that the function $a(t)$ (and thereby any function constructed from $a(t)$) is even under time-reversal around both turning points. If we choose the origin of $t$ such that $a(0)=a_{1}$, we have $a(-t)=a(t)$. But, similarly, if we shift $t$ into $t^{\rm new}=t-\frac{1}{2}T$ so that the origin of $t^{\rm new}$ corresponds to $a(0^{\rm new})=a_{2}$, we also have $a(-t^{\rm new})=a(t^{\rm new})$. The potential in Hill’s equation (52) inherits these two symmetries. Therefore, the ($a_{1}$ and $a_{2}$) time reversals of the first Floquet solution, $e^{i\gamma_{n}t}P_{n}(t)$, must be equal to the second Floquet solution, $e^{-i\gamma_{n}t}{\tilde{P}}_{n}(t)$, so that (using a star to denote complex conjugation)

[TABLE]

Applying this result to both turning points, $t=0$ and $t^{\rm new}=0$, we have

[TABLE]

In words: $P_{n}(a_{j})$ is real, and $\dot{P}_{n}(a_{j})$ is pure imaginary at both turning points, $j=1,2$, so that

[TABLE]

are two real numbers, uniquely defined by the function $\omega_{n}^{2}(t)$.

The bad news is, however, that, as we have explicitly checked on some numerical examples, no miracle occurs, and the uniquely defined outer boundary condition ${\bar{S}}_{n}^{\rm outer}(a_{2})=\mu_{n}$ at $a_{2}$ generally differs from the (uniquely defined) necessary, periodic inner boundary condition:

[TABLE]

A simple case where we checked the non equality (65) is an oscillatory universe filled with (positive) radiation, together with a negative (fermioniclike Damour ) component $-C_{F}/a^{6}$, i.e. (with $C_{r}>0$ and $C_{F}>0$)

[TABLE]

so that (with $k=+1$)

[TABLE]

It is easily seen that Eq. (61) yields a harmonic oscillator equation for the squared scale factor, so that $a^{2}(t)=\alpha+\beta\cos(2\pi\frac{t}{T})$, say with $0<\beta<\alpha$. As a consequence, the potential $\omega_{n}^{2}(t)$ in Hill’s equation is varying sinusoidally for all modes numbers $n$:

[TABLE]

In other words, the Hill equations (52) reduce to Mathieu’s equation (and to their Euclidean avatars, with $\omega_{n}^{2}(\tau)=n^{2}+m^{2}\alpha\pm m^{2}\beta\cosh\left(2\pi\frac{\tau}{T}\right)$ in the forbidden regions $a>a_{2}$, or $a<a_{1}$, respectively). Using then both the analytical knowledge on Mathieu’s solutions, and a sample of numerical simulations, we have verified the generic absence of equality (65).

IV.3 Quasi-uniqueness of the solution for $S_{n}(a)$ in the inner classically forbidden region $a<a_{1}$, and its

failure to match the $S_{n}(a)$’s in the other intervals.

At this stage, we have already proven the impossibility to find stable quantum states of our system. It is, however, interesting to complete our analysis by considering the inner classically forbidden region $a<a_{1}$. This will show the existence of further impossibilities.

It is again convenient to define a Euclidean time variable $\tau$ by

[TABLE]

This Euclidean time grows when $a(<a_{1})$ varies between $a_{1}$ and 0, and can be set to be zero when $a=a_{1}$. The situation is then symmetric to our discussion of the $a>a_{2}$ region, with $\tau$ growing from 0 to some (finite or infinite) maximum value $\tau_{m}$. The equations to solve are exactly the same as those in the outer classically forbidden domain, namely the Riccati equation (35), and its linear second-order transform (37), obtained by representing $S_{n}(\tau)$ in the form (36).

The only difference with the discussion in the previous subsection concerns the total number of e-folds

[TABLE]

that will enter the approximate WKB solution (where $D_{n\pm}={\rm const}$)

[TABLE]

when considering its limit as $\tau\to\tau_{m}$, i.e. as $a\to 0$. [As in the previous subsection, one could work with the exact solutions corresponding to $\exp\pm\int^{\tau}_{0}\omega_{n}(\tau^{\prime})d\tau^{\prime}$ and reach the same conclusions, as long as $N_{1}$ is infinite or at least sufficiently large.] The explicit expression of $N_{1}$ reads

[TABLE]

Let us first assume that $a^{2}\rho(a)\to 0$ (or at least $<1$) as $a\to 0$ (this is notably the case for the SHU model Graham:2011nb ). Then $U(a)\approx a^{2}$ as $a\to 0$ so that both $N_{1}$ and $\tau_{m}=\int_{0}^{a_{1}}da/{\sqrt{U(a)}}$ are logarithmically infinite. In this case, we can rigorously reach the same conclusion as in the discussion of the $a>a_{2}$ region. Any nonzero value for the constant $D_{n+}$ leads to the asymptotic behavior

[TABLE]

which implies

[TABLE]

Therefore, there is a unique solution of the Riccati equation (35) for $S_{n}(a)$ in the domain $a<a_{1}$ that satisfies the stability requirement (13). In the WKB approximation it is given by setting $D_{n+}=0$ in Eq. (71), and is approximately given by

[TABLE]

Like in the $a>a_{2}$ case, the reasoning of Ref. Messiah shows the existence of an exact counterpart of this solution (with $S_{n}(a)$ not exactly equal to, but close to $+{\omega_{n}(a)}$), constructed as $S_{n}(a)=-\dot{u}_{n}^{-}/u_{n}^{-}$ where $u_{n}^{-}(\tau)$ is the unique (positive, monotonously decreasing, and asymptotically vanishing) solution of Eq. (37) (normalized to equal 1 at $\tau=0$).

Suppose now that, as $a\to 0$, $U(a)$ either decreases slower than $a^{2}$ or even increases. [This would be the case if, e.g., $a^{2}\rho(a)\to-\infty$ as $a\to 0$, as it happens for models including a term of the type $\rho_{F}(a)=-C_{F}/a^{6}$.] Then both $\tau_{m}$ and $N_{1}$ are finite. In that case, the inequality (13) is not powerful enough to select a mathematically unique solution. However, as

[TABLE]

the number $N_{1}$ of e-folds will grow linearly with the mode number $n$, and will therefore be very large for an infinite number of modes. As a consequence, there will be a quasi-uniqueness of the solutions satisfying (13), in the sense that for an infinite number of values of $n$ one must fine-tune the only integration constant $D_{n+}/D_{n-}$ entering $S_{n}(a)$ to be increasingly closer to zero, as $n$ increases.

And again, we have verified that the so-determined boundary condition for $S_{n}(a_{1})$, namely

[TABLE]

will not match any would-be periodic allowed-region solution (which must have ${\bar{S}}_{n}^{\rm inner}(a_{1})=\gamma_{n}+\delta_{n}^{(1)}$).

V Conclusions

Let us recap our logic and our main results. We investigated stationary quantum states of a compact universe filled with homogeneous matter and a massive scalar field conformally coupled to the curvature. The background value of the scalar field was taken to be vanishing, and the quantum scalar field was accordingly treated as a perturbation. The matter energy density $\rho(a)$ was assumed to be such that the unperturbed model has a classically allowed range for the scale factor, $0<a_{1}<a<a_{2}$, with the values of $a$ beyond this range being classically forbidden. Our main assumption is that the wave function vanishes at $a\to\infty$; we also tentatively required that the wave function also vanishes at $a=0$, but this is not necessary for reaching our main results. The unperturbed wave function describes a universe oscillating periodically between $a_{1}$ and $a_{2}$. We required that the usually considered source of quantum instability of an oscillating universe (particle creation due to parametric amplification of the modes of $\phi({\bf x},t)=\frac{1}{a(t)}\sum_{n}\phi_{n}(t)Q_{n}({\bf x})$) be absent.

If a stationary state exists, its full wave function should be given by a linear combination of Gaussian terms of the form

[TABLE]

with the functions $S_{n}(a)$ satisfying the stability condition ${\rm Re}[S_{n}(a)]>0$ in the three intervals $0<a<a_{1}$, $a_{1}<a<a_{2}$, and $a>a_{2}$, together with the matching conditions (32), (33). This condition ensures that the dependence on the amplitudes of the scalar field modes $\phi_{n}$ is Gaussian, rather than inverse Gaussian, so that the wave function does not grow with increasing $|\phi_{n}|$. We found, however, that the stability condition cannot be satisfied in the full range of variation of $a$, and is generally violated for an infinite number of modes.

With a suitable choice of boundary conditions, the stability condition ${\rm Re}[S_{n}(a)]>0$ can be satisfied in one of the three intervals, $0<a<a_{1}$, $a_{1}<a<a_{2}$, or $a>a_{2}$, but then it is violated in the two other intervals. Our conclusion is particularly sharp if we start imposing the stability condition in the outer classically forbidden range $a>a_{2}$. This determines, in a unique manner, the $S_{n}(a)$ function in the domain $a>a_{2}$. The matching conditions at the turning points $a_{2}$ and $a_{1}$ then determine $S_{n}(a)$ in the other intervals. However, this continuation to $a<a_{2}$ does not lead to a consistent, and stable solution. [If one tries to continue $S_{n}(a)$ in the classically allowed range starting from the unique $S_{n}(a_{2})=\mu_{n}$, one ends up, after one cycle of oscillation, with an inconsistent new (complex) value of $S_{n}(a_{2})$. Moreover, the value of $S_{n}(a_{1})$ obtained after the first half cycle would also be complex and inconsistent with stability in the $a<a_{1}$ domain.]

The conclusion is thus that stationary quantum states for an oscillating universe do not exist. We have verified that a similar analysis goes through, with the same conclusions, for a minimally coupled scalar field.

Our analysis was very general: we did not assume any specific form of the matter density function $\rho(a)$, requiring only that it should yield a classically allowed range of $a$ flanked by two classically forbidden ranges. One may be concerned that the same kind of analysis could be used to demonstrate non-existence of stationary states in ordinary quantum mechanics. There is, however, an important difference between the two cases. The WDW equation is hyperbolic, while the stationary Schrödinger equation is elliptic, with all derivative terms having the same sign. By applying our type of analysis to the case where the sign of the kinetic term for the scale factor is changed (with a corresponding change in the confining potential $U(a)$), i.e. for a Schrödinger equation of the type

[TABLE]

we have shown that our method meets no obstacle for constructing stable solutions, with ${\rm Re}[S_{n}(a)]>0$. The crucial difference is that we find that the sign of the term $S^{\prime}{S_{n}}^{\prime}$ in Eq. (17) is reversed, which implies that one must correspondingly reverse the signs of the logarithmic derivatives on the right-hand-sides of the representations (51) and (36) (the other equations remaining the same). This apparently minor sign change has a drastic effect on the selection of the stable solutions in the two forbidden domains. Indeed, when having $S_{n}=+\frac{{\dot{u}}_{n}}{u_{n}}$, the naturally dominating growing Euclidean solutions $u_{n}^{+}(\tau)\approx C_{+}\exp\left[\int^{\tau}_{0}\omega_{n}(\tau^{\prime})d\tau^{\prime}\right]$ yield the stable value $S_{n}(a)\approx+\omega_{n}(a)$, so that no fine tuning of the integration constants is needed to select a stable solution. One can then start (using Floquet theory) by constructing a stable, periodic function $S_{n}(a)$ in the domain $a_{1}<a<a_{2}$. This defines positive, real boundary values $S_{n}(a_{(1,2)})=\gamma_{n}+\delta_{n}^{(1,2)}$, which can then be continued into stable solutions in the two classically forbidden intervals $a<a_{1}$ and $a>a_{2}$.

Our result appears to point to some quantum instability of an oscillating universe. Mathematically, it is rooted in the fact that our WDW equation (6) is a Klein-Gordon equation, $({\widehat{P}}^{2}+M^{2})\Psi=(-\hbar^{2}\Box+M^{2})\Psi=0$ (with signature $-++++\cdots$), where $a$ is the time variable, and in which the squared mass is the following function on superspace:

[TABLE]

This squared mass is positive in a connected domain of the variables $(a,\phi_{n})$ which is centered along the line $(a_{1}<a<a_{2},\phi_{n}=0)$ and extends away from it in all the $\phi_{n}$ directions (and also in part of the regions where $a<a_{1}$ or $a>a_{2}$, if $|\phi_{n}|$ is large enough), but it is negative in two other (disconnected) domains: an infinite region centered along the infinite half-line $(a>a_{2},\phi_{n}=0)$ and extending around it in the $\phi_{n}$ directions, and a third region centered along the interval $(0<a<a_{1},\phi_{n}=0)$ and extending around it in the $\phi_{n}$ directions. In the latter two domains (centered around the classically forbidden lines, $(0<a<a_{1},\phi_{n}=0)$ and $(a>a_{2},\phi_{n}=0)$), the quantum particle-universe described by the WDW equation is a tachyonic particle, which gives rise to strong, exponential instabilities in its quantum propagation in superspace. These tachyonic exponential instabilities are the mathematical roots of the instabilities we found in our analysis.

The physical nature of these instabilities remains unclear. Rubakov et al Rubakov1 ; Rubakov2 found a similar instability in the under-barrier wave function of a tunneling universe. They argued that particles must be copiously (”catastrophically”) produced in the course of tunneling. An alternative explanation suggested in Refs. Hong1 ; Hong is ”critical branching”, meaning that the universe is a superposition of different quasi-classical branches, some of which contain a larger number of particles, leading to a smaller barrier that needs to be penetrated. The wave function after tunneling may then be dominated by states with high occupation numbers, even if such states are strongly suppressed in the initial wave function. Furthermore, it was shown in Hong that the initial state can always be fine-tuned to remove the multi-particle branches, so that the wave function exhibits no instability.

This situation is different from the case of an oscillating universe, where the instability is present for any choice of the quantum state. It is still possible that our result is a reflection of critical branching, which now cannot be removed by a choice of the quantum state because it is present in two under-barrier regions rather than one. On the other hand, even though the effect of critical branching is certainly present, it is not clear that it is accounted for in our formalism, which seems to disregard the effect of particle occupation numbers on the expansion of the universe. So it is conceivable that our result is a reflection of an instability due to some nonperturbative particle production mechanism. This issue remains an intriguing problem for future research.

Acknowledgments

T.D. thanks Gerald Dunne, Jens Hoppe, Sergiu Klainerman, Marcos Mariño, Vasily Pestun, Valery Rubakov and André Voros for informative discussions. A.V. is grateful to the Institut des Hautes Etudes Scientifiques for its hospitality during the conception of this work. A.V. acknowledges support by the National Science Foundation under grant PHY-1820872.