Geometric phases and cyclic isotropic cosmologies

# Geometric phases and cyclic isotropic cosmologies

Leonardo Banchi Department of Physics, University College London, Gower Street, London WC1E 6BT, UK    Francesco Caravelli Invenia Labs, 27 Parkside Place, Parkside, Cambridge CB1 1HQ, UK London Institute of Mathematical Sciences, 35a South Street, London W1K 2XF, UK
###### Abstract

In the present paper we study the evolution of the modes of a scalar field in a cyclic cosmology. In order to keep the discussion clear, we study the features of a scalar field in a toy model, a Friedman-Robertson-Walker universe with a periodic scale factor, in which the universe expands, contracts and bounces infinite times, in the approximation in which the dynamic features of this universe are driven by some external factor, without the backreaction of the scalar field under study. In particular, we show that particle production exhibits features of the cyclic cosmology. Also, by studying the Berry phase of the scalar field, we show that contrarily to what is commonly believed, the scalar field carries information from one bounce to another in the form of a global phase which occurs to be generically non-zero. The Berry phase is then evaluated numerically in the case of the effective Loop Quantum Cosmology closed Universe. We observe that Berry’s phase is nonzero, but that in the quantum regime the particle content is non-negligible.

Berry phase, cyclic cosmology, Bogoliubov coefficients

## I Introduction

Our current understanding of the Universe and of its history relies on several puzzling observational features. In fact, soon after the discovery of the Cosmological Microwave Background (CMB), theoretical physicists realized that the early universe had to follow a rapid expanding phase. Inflation was the first candidate for generating such expansion, and a more refined version of inflation is still at the basis of the current standard cosmological model. There are compelling evidences for a rapid expanding phase, which have culminated in the recent results of the BICEP and PLANCK collaborations Planck1 (); Planck2 (); Planck3 (). The current standard model of the universe involves an inflation field, but current experimental results, meanwhile ruling out many proposed models, seem not to exclude several other proposals alternative to inflationLehners (). Inflation has the advantage of the Occam’s razor, as it accounts for all the puzzles involved with our current understanding of the Universe, such as its flatness and acceleration, its entropy and the horizon problem in one shot.

However, an alternative and interesting proposal relies on the idea of cyclicality of the Universe, a concept almost as old as General Relativity. Cosmological bounce models have a long history, with its first proposal put forward by Tolman tolman34 (), and have been proposed in several different contexts osccuni (). In particular, it has been shown that oscillating cosmological universes can provide a solution to both flatness and horizon problems, as long as maximum of the expansion in each cycle is increased in the next cycle.

Quantum gravitational phenomena play a role only in the extremely early Universe. In this scenario, quantum gravity is the force that makes the bounce occur. In fact, a Big Bounce has been observed in several quantum gravitational cosmological theories, as for instance Loop Quantum Cosmology lqc1 (); lqc2 (); lqc3 (), Epkyrotic Stringsekp1 (); ekp2 (); ekp3 (), Asymptotically safe non-local gravity Calcagni1 () and in General Relativity with matter-torsion interaction bouncegr1 () just to name few.

In the present paper, we ask ourselves whether such bouncing phenomena has some physical, long-term effects at the quantum level. At the classical level, we find this unlikely to occur. However, one could argue that considering the Universe conformal factor as an external parameter, in a bouncing cosmology a Berry phase might emerge at the quantum level. This is the scenario we are interested in: does a scalar field maintain a quantum memory from previous bounces? We answer this question in the modest nonphysical scenario of a scalar field in a bouncing cosmology, in which the scalar field does not lead to any backreaction on the underlying metric; yet, this toy model features many of the stylized characteristics of bouncing cosmologies birreldavies (); mukwin (). The Berry phase has been already considered in the cosmological setting, by for the case of inflationary models bpm ().

In general, the Berry phase is not a measurable quantity if the scalar field is isolated. However, if the scalar field is entangled with other fields as considered for instance in Albrecht (), then a Berry phase is measurable. To see this, let us consider a scalar field , in the state at the beginning of the -th cycle and let be the propagator which maps . If the scalar field’s state does not change from one cycle to the other, i.e. , then the bounce produces no observable effects. However, even though the field is unchanged after one cycle, its state can pick a geometric phase which in principle can be measured as in interferometry. For instance, for the sake of argument, we consider the simplest case of two scalar fields and which are in the entangled state , although our argument can be extended to more complicated cases. After a cycle, because of the Berry’s phase, the state is a different one . We argue that this phenomenon is poorly described by analysing, for instance, the particle content coefficients only, in particular if these are calculated perturbatively and can vanish identically, and that a nonperturbative approach is more appropriate for calculating these phases. We will describe how to define a simple observable in Section IV, and we will provide an exact formula for the operator . We will provide both explicit formulae for specific scale factor functions and provide general results for smooth scale factors which are periodic.

The paper is organized as follows. In section II we introduce the scalar field in curved spacetime, the settings and the notation, assuming that the Universe is cyclic. In section III we study particle production in the toy model we introduce, both in a Kronig-Penney approximation and solving numerically the mode equation. In section IV we study the Berry Phase. Conclusions follow.

## Ii Scalar field in a cyclic toy universe

We are interested in the quantum features of a scalar field in a Friedman-Robertson-Walker (FRW) space time defined by the metric

 ds2=dt2−a2(t)(dx2+dy2+dz2), (1)

which describes an evolving isotropic flatspace with a dynamic scale factor . In order to simplify many of the calculations, we introduce the conformal time coordinate. In these coordinates, the metric reads:

 ds2=a(η)2[dη2−(dx2+dy2+dz2)], (2)

where is the scale factor in the new coordinate.

As a simplifying assumption, we consider a scale factor which oscillates according to a local time and that as considered in many nonsingular bouncing cosmology models ekp1 (); lqc1 (); Calcagni1 (), and that can be expanded in Fourier series. We assume that space is locally flat, i.e. that the curvature of space is zero. In addition to this, we assume that the expansion and contracting phases of the universe last an equal amount of conformal time, which is a scenario ruled out from the physical viewpoint.

In modelling the universe with these features, we were inspired by the two major current theories of bouncing cosmologies: the ekpyrotic scenario ekp1 (); ekp2 (); ekp3 () and loop quantum cosmology lqc1 (); lqc2 (); lqc3 (); asht1 (); merc1 (). Meanwhile the former has been affected with the recent results from the Planck collaboration Planck1 (), the latter has shown to be a long-living toy model. In general, the ekpyrotic scenario relies on an asymmetric expanding and contracting phases, differently from our case, meanwhile the loop quantum cosmology approach is symmetric in its expanding and contracting phases. It has been argued however that the ekpyrotic scenario suffers of singularities cail1 (). Although violating the Tolman principle tolman34 (), one could claim that a bounce occurring in a quantum gravity scenario might require an extension of the principles of thermodynamics, similarly to what happens for black holes. Thus, in this respect, we are more interested in understanding some of the features of a quantum field when the universe is cyclical. This simple approach has the advantage that one can carry on several calculations analytically.

The action for a scalar field on a curved background is

 S=∫dx4√−g[gμν∂μΦ∂νΦ∗−V(g)ΦΦ∗] , (3)

where we consider the curvature dependent potential , being for the metric (2) The equations of motion for a scalar field of this form are given by:

 1√−g∂μ(gμν√−g∂νΦ)+V(g)Φ=0 (4)

It is convenient to work with a field re-definition useful in the conformal time coordinates, , in which the equations of motion for the scalar field are given by: , with . If one works with conformally coupled fields, , then the dependence of the curvature on the modes disappears. Moreover, we assume space isotropy, so all the quantities only depend on .

The classical field can be quantized by introducing bosonic creation and annihilation operators () winitzky (): the expansion

 ^Φ(→x,η)=∫ d3→k√2(2π)3a(η)2× ×[ei→k⋅→xϕ∗k(η)^a→k+e−i→k⋅→xϕk(η)^a†→k] , (5)

defines the mode operators from the quantum field and the mode functions satisfy the equation

 [d2dη2+ω2k(η)]ϕk(η)=0 . (6)

Once the mode operators are determined, the vacuum state can be defined as the eigenstate of the annihilation operators with eigenvalue 0, i.e. . However, different solutions of (6) define different mode operators and thus different vacua. The physical vacuum at a certain conformal time corresponds to the (instantaneous) lowest energy state and it is given by the solution of (6) with initial conditions

 ϕk(η0) =1√ωk(η0), ϕ′k(η0) =iωk(η0)ϕk(η0) . (7)

From now on we call the solution of (6), (7). At another conformal time , since the vacuum satisfies different initial conditions, the theory has a different particle content described by new creation and annihilation operators . The mode operators at and are related by the Bogoliubov transformation

 ^b→k=αk^a→k+β∗k^a†→k , (8)

where and are independent on and defined by

 αk =W[ϕk(η,η1),ϕk(η,η0)∗]2i , (9) β∗k =W[ϕk(η,η1),ϕk(η,η0)]2i , (10)

being the Wronskian. The new vacuum is related to the old one via

 |Ω(η1)⟩=∏→k1|αk|1/2exp(−β∗k2αk^a†k^a†−k)|Ω(η0)⟩ . (11)

When and the above transformation defines the operator which maps the state of the th cycle to the next one. In the following sections we study the vacuum in a bouncing universe, discussing the possible creation of particles and the occurrence of a geometric phase after one or more periods.

## Iii Particle production

In a bouncing universe the scale factor of the universe oscillates. If one assumes a conformally coupled scalar field, the effect of gravity is included in the -dependent frequency through the mass of the scalar field, which is periodic with a certain conformal period . As it can be expected, in general the particle content at conformal times and is the not the same. In this section we show that the low-energy vacua at different periods are not the same in general, i.e. , even though the functions and satisfies the same differential equation (6).

We consider the universe in its ground state at conformal time . As discussed before, an observer at a different conformal time sees different particles which are related to the original ones by the Bogoliubov transformation (8). For instance let us call -particles the particles created by the operators at conformal time and -particles the ones created by at conformal time . The vacuum state at is the one annihilated by so, by definition, the number of -particles in is exactly zero. Nonetheless, contains many pairs of -particles and the average density of -particles in is winitzky ():

 nk =|βk|2 , βk =−ϕ′k−iωkϕk2i√ωk , (12)

where in the second equality we used (10). For instance, for sudden transitions between two instantaneous vacua, the Bogoliubov coefficients are given by

 αk =12(√ωk(η1)ωk(η0)+√ωk(η0)ωk(η1)), (13) βk =12(√ωk(η1)ωk(η0)−√ωk(η0)ωk(η1)). (14)

Therefore if the universe is cyclic and performs a sudden transition from to , then so the particle content is not changed. On the other hand, when the universe evolves smoothly the Bogoliubov coefficients are obtained by means of (10) from the solution of the differential equation (6).

The equation (6) is known in the literature as the Hill equation and has many interesting properties Hills (). In particular, because of Floquet theorem, all the solutions of (6) can be written as

 ϕk(η)=eiμkηc+kf−k(η)+e−iμkηc−kf−k(η), (15)

where the functions are periodic, , and are called the Floquet exponents. The coefficients describing the mode function are readily obtained by imposing (7), whereas depends on and, in general, can be real, complex or integer. When is complex the solutions are not stable, i.e. they diverge for . We argue that this is not a physical occurrence so we skip the discussion of this eventuality. More interestingly, when is an integer (or a fraction) the solution (15) is periodic of period (or a multiple of ). In this case, not only the geometry, but also the quantum fields display a cyclic dynamics during the evolution, so . From the mathematical point of view, can be designed to allow for periodic solutions HillTipo () for each value of , but typically, for physical conformal factors, the periodic solutions appear only for specific discrete values of the parameters, e.g. for some specific values of .

In general is a real number, making the solution Eq.(15), and thus the quantum fields, an aperiodic function of . When the vacuum state is not periodic the average density of particles after periods is evaluated from Eqs. (12) and (15):

 nk(Π) =sin(TΠμk)2ω2∣∣g+k+g−k∣∣2, (16) g±k =f±k(0)[μk∓ωk(0)]f±k(0)∓if±k′(0) . (17)

The Floquet exponent can be evaluated perturbatively using the techniques reported in the Appendix A while the quantities depends on the initial value. Therefore, is zero only when is an integer or when . We will discuss below these possibilities for some particular cases.

### iii.1 Kronig-Penney approximation

The equation for the modes of a bouncing universe are formally similar to those of Bloch electrons in a spatially periodical lattice, where the space dimension is the conformal time. Instead of for , we now have a relativistic energy given by , with the periodicity encoded within . One can thus use a similar approach to those of used for tunneling of electrons from one lattice site to the other, or for calculating the bands of a solid. In this section we will assume for simplicity a scale factor which oscillates as:

 a(t)=a0+b sin(rt). (18)

As is merely a redefinition of time, we consider hereon. We note that this toy model requires for a regular cosmology in which the scale factor never reaches zero. For this model, we find that the conformal time is

 η(t) = 2tan−1(a0tan(t2)+b√a20−b2)√a20−b2 (19) + 2π(θ(t−π)−∥∥b2−t2π∥∥)√a20−b2

where refers to the integer part, which is everywhere continuous and monotonic, allowing for to be inverted . From this we obtain a closed formula for given by:

 a(η) = b sin(2tan−1(Ξtan(12ηΞ)−ba0))+a0 (20)

with . It is easy to show that the cyclicity of this universe depends on the parameters and , , with . From now on, we will assume .

In what follows, we will assume also that our potential can be approximated by step functions. In particular, we would like to treat the effective potential as a a 2-valued function, as in the Konig-Penney approximation for electrons on a Bloch lattice. In the standard case of a scalar field in a potential with many local minima, as shown in Coleman1 (); Coleman2 () the field can move through tunnelings from one local minimum to another. In our case instead, the interpretation is different, as the potential is now driven by an external factor, i.e. the bounce of the Universe. This will allow to treat the modes close to the bouncing and contracting phases of this toy Universe as a constant, and thus the local solution being a plane waves. An example of this approximation is given in Fig. 1. We use a full width at half maximum (FWHM) approach, i.e., the energy is approximated by and in the low and high energy regimes, with being the local bounce, meanwhile being the local time of the maximum expansion, and the time of the switch at half maximum, and are defined by the two (locally in each cycle) solutions of the equation for :

 ωk(η)=12(ωk(ηbounce)+ωk(ηcontr)), (21)

in which the smaller solution is , and the larger given by , denoting the transition from the bounce to the maximum expansion and viceversa, respectively. We denote the -th recurrence of the as , as for a lattice.

For each mode , we solve the contact equation at each transition point, from one bounce to the other one, imposing the continuity and differentiability of the solution. If is the solution within at the -th bounce, and at the -th bounce, one can use a transfer matrix approach for calculating the transition coefficients. If is the wavefunction in the contracting phase, one can write the contact equations between the various solutions at :

 ϕnk(ηninf) = ~ϕnk(ηninf) ∂ηϕnk(ηninf) = ∂η~ϕnk(ηninf) ~ϕnk(ηnsup) = ϕn+1k(ηnsup) ∂η~ϕnk(ηnsup) = ∂ηϕn+1k(ηnsup) (22)

and if one expands the various terms in Fourier modes:

 ϕnk(η)=Aneiωl(k)η+Bne−iωl(k)η ~ϕnk(η)=Cneiωu(k)η+Dne−iωu(k)η (23)

as a result of eqns (22), one can write a matricial equation:

 (An+1Bn+1)=T(AnBn). (24)

In general, the transfer matrix is a rotation due to Bloch theorem, , and in particular each element is the wronskian between the two solutions:

 T=(T11T12T21T22)=(W[ϕn,ϕn+1]W[ϕn,ϕ†n+1]W[ϕ†n,ϕn+1]W[ϕ†n,ϕ†n+1]). (25)

Thus, the element represents times the Bogoliubov coefficient which represents , connecting the modes from one bounce to the next. Due to the symmetry of the problem, the transfer matrix depends only on the mode , once one has fixed the mass, and the shape of the function . From the physical point of view, we are calculating the tunneling of the modes between one bounce to the other, i.e. from one ground state to the other in this approximation.

The elements of the transfer matrix derived in the Kronig-Penney approximation of the cosmological bounce. We define and . The transfer matrix elements are given by , with . Thus we have:

 T11 = e−iωl(k)Δη2ωl(k)ωu(k)[2ωl(k)ωu(k)cos(ωl(k)Δη) + i(ωl(k)2+ωu(k)2)sin(ωu(k)Δη)], T12 = −T21 = im2a2(ηcontr)−a2(ηbounce)2ωu(k)ωl(k) ⋅ e−i(ωu(k)+ωl(k))¯¯η[e2iωu(k)ηinf−e2iωu(k)ηsup], = im2a2(ηcontr)−a2(ηbounce)2ωu(k)ωl(k) ⋅ e−i(ωu(k)+ωl(k))¯¯ηe2iωu(k)(ηinf+Δη2)sin(ωu(k)Δη) ∝ m2a2(ηcontr)−a2(ηbounce)2ωu(k)ωl(k)sin(ωu(k)Δη), T22 = 14ωl(k)ωu(k)e−i[(2ωu(k)+ωl(k))ηinf+(ωu(k)−ωl(k))ηsup] ⋅ [eiωl(k)ηinf(e2iωu(k)ηinf−e2iωu(k)ηsup) ⋅ (ωu(k)2+ωl(k)2) + 4ei[(ωl(k)+ωu(k))ηinf+ωu(k)ηsup] ⋅ ωu(k)ωl(k)cos(ωu(k)Δη)],

which one can verify to satisfy the condition of the transfer matrix: (i) , (ii) for or , , meanwhile . Since we are interested in the off-diagonal elements of this matrix, we note that for , the Bogoliubov coefficient goes to zero; in addition to this, it is proportional to the difference between the maximum and minimum conformal factors, . It is also clear, that the Bogoliubov coefficient is proportional to , and thus the faster the Universe expands and contracts, the less particles are produced.

In Fig. 2 we plot the Bogoliubov coefficient between one bounce and the next in the Kronig-Penney approximation for various values of and . As we will see, many of the phenomenological features of this approximation are retained in the exact solution of the differential equation with a smoother approximation of , as for instance interference patterns and particle production which goes to zero as . We note, however, that in this approximation the transfer matrix is independent from the bounce number which we used to diversify one bounce from the other. One can, in fact, interpret the Kronig-Penney approximation as a 2-way WKB approximation at the Bounce and at the point of maximum expansion in which one has . As a last comment to conclude this section, we note that due to the discontinuous approximation of the potential, several interference effects which are spurious occur. This will be solved in the next section, by means of a smoother approximation.

### iii.2 Mathieu fields

The Kronig-Penney approximation is a first, rough approach to calculating the Bogoliubov coefficients. In this section we work on the analytical solution of the differential equation which describes the evolution of the modes. Differently from the case discussed before, we consider a simple smooth approximation of the conformal factor given by the function:

 a2(η)=a0+bcos(2η) , (26)

whose period is . Within this approximation, the differential equation (6) is the well-known Mathieu equation, whose solutions are given by (15) with the following analytical expressions

 e±iμkηf±k= C(a0m2+k2,−12bm2,η)± (27) iS(a0m2+k2,−12bm2,η) ,

where is the even Mathieu function, is the odd Mathieu function and, in this case, the Mathieu exponent abramowitz (). Mathieu functions have a long history of applications in quantum mechanics (see e.g. Slater Slater ()) with important applications to the theory of optical lattices OpticalO (); Optical (). From (16) one finds that after full oscillations the average number of quasi-particles is

 nk =∣∣ ∣∣1−g2k2gksin(πμkT)∣∣ ∣∣2, (28) gk =C(a0m2+k2,−12bm2,0)S′(a0m2+k2,−12bm2,0)ωk(0) . (29)

For the present analysis we are interested only in the regions where is real but, for the sake of completeness, it is worth mentioning that there are some zones in the parameter space where the Mathieu exponent can become complex and the solutions of eqn. (6) may grow exponentially as Enghi (). From Fig. 3(a) one can see that, for small values of , the energy gap is sufficiently small to allow for the creation of some particles. Similar interference effects are found also using the Kronig-Penney approximation, although emphasized due to the non-smooth approximation in of . It is interesting to see that, for fixed small , there are some values of where no particles are created. This can be explained by noting that for some values of the Mathieu exponent can be an integer number, making in (29). The locus of points where is integer is shown in Fig.3(bottom).

On the other hand, when we expect that the system remains in its ground state during the evolution because of the large energy gap which prevents the creation of particles. In more mathematical terms, indeed we find that in this limit making .

An interesting question is whether the number of particles remains finite after many oscillations. Indeed, if the current lifetime of the universe is much larger than the oscillation time, then in our universe we can expect to observe extra particles due to the oscillating geometry. Calling the total number of particles we find

 N=π2∫∞0dk(1−g2kgk)2k2+O(T−1/2) , (30)

where the correction is due to the stationary phase approximation and can be neglected for . In our numerical simulations is finite for any reasonable value of and . Moreover, for large energy gaps, namely when we still find that , showing that the particle content of the universe is cyclic.

We have discussed some limits where not only the geometry but also the quantum fields show a cyclic or almost cyclic evolution. In the next section we ask whether this peculiarity shows some quantum features in the form of a non-zero geometric phase.

## Iv Berry’s phase

The Berry phase is a geometrical phase acquired by a quantum mechanical system during its evolution. It was introduced in BerryAdia () for adiabatic evolutions and then promptly extended to more general cases BerryNonadia (). This geometrical phase is gauge-invariant and can be regarded as a generalization of the Aharonov-Bohm phase, which on the other hand appears only in specific cases when the fields are coupled to electromagnetic radiation. Aside from its fundamental implications, the Berry phase has been used also to gain further understanding of complicated phenomena such as quantum phase transitions QPT1 (); ZhuQPT () and the dynamics of entanglement Dario (); Entanglement (), while its non-abelian generalization provides an alternative avenue for quantum computation Zanardi ().

In a cyclic universe one can interpret the occurrence of a Berry phase as a memory of the quantum system on the previous cycle. From this perspective, we are indeed interested in showing that for the case of cyclic cosmologies, such phases could occur to be non-zero. As a matter of fact, assuming that the Universe is topologically equivalent to a sphere in which the radius for the spectator scalar field represents an external forcing undergoing a loop in the space of parameters. Thus, the conditions for the emergence of a non trivial phase are present. Moreover, we have previously discussed that, if a Universe is cyclic, the vacuum states can display a periodic evolution or almost periodic evolution, depending on the geometry of the space-time. For cyclic evolutions, thus, a non-zero Berry phase can emerge in the ground state.

The Berry phase for a state which undergoes a cyclic evolution of period is given by BerryNonadia ()

 ϕB=i∫Π0⟨Ω(η)|∂∂ηΩ(η)⟩dη . (31)

We specify to the case in which is the ground state of the universe at conformal time , and we assume that the universe was in its ground state at time . Exploiting the relation (11) between ground states at different conformal times one can show that the Berry connection is

 ⟨Ω|Ω′⟩ =12∑→k(−|αk|′|αk|−α′kαk|βk|2+β∗k′βk) (32) =∑→k14(α′kαk−α∗k′α∗k)+β∗k′βk−α∗kα′k2 . (33)

The above equation can be written in a more convenient form by defining the quantity

 γk=vkv′k+iωvk , (34)

which satisfies the differential equation

 γ′k=1−2iωkγk−iω′kγ2k , (35)

with the initial condition . Indeed, by substituting (10) in (33) one finds

 ϕB =∫Π0∑→kω′k(η)2R[γk(η)]dη , (36) =2π∫Π0dη∫∞0dkk2ω′k(η)R[γk(η)] . (37)

Before evaluating the Berry phase for some relevant parameters, we study the dependence on of the integrand . When it is , and hence . Therefore, one expects that the integral over in eqn. (37) can be performed up to a cut-off value without introducing significant errors. In particular, here we evaluate the Berry’s phase numerically by solving the coupled differential equation and eqn. (35) for several values of and then using , where the latter integral is evaluated numerically. For instance, in Fig. 4, we show that for , while on other hand the integral is non-zero and increases for increasing .

From the form of the Berry connection one can see that, if there is no particle creation during the evolution of the universe, i.e. for each value of , then the Berry phase is exactly zero. In a non-trivial cyclic evolution on the other hand, some particles are created during the bounce, but then after one period the fields go back to their initial state. This observation, that will be clarified in the following, motivates our interpretation of the Berry phase as a memory of the dynamics of the universe. However, as in eqn. (37) the constraint about the periodicity of the initial state is not explicit, we start our analysis by introducing the -dependent Berry phase defined by

 ϕB(k) =∫Π0ω′k(η)2R[γk(η)]dη . (38)

Indeed, in section III.2 we have shown that, for a given cosmological factor, there are some pairs where there is no particle creation after a bounce. In other terms, particles are created during the evolution of the universe but after a cycle the ground state is exactly the initial ground state. In particular, for the simple cosmological factor (26), we have shown that the locus of points where define different curves (see Fig. 3) in the space where the Mathieu exponent takes integer values. We now evaluate the -dependent Berry phase along those curves and show that the Berry’s phase is non-zero even when the ground state is perfectly periodic, thus signaling a non-trivial quantum evolution of the universe. The results are shown in Fig. 5. Moreover, we observe that is larger for smaller values of where the corresponding mass in the curves Fig. 3 is smaller.

The results presented in Fig. 5 show a rigorous findings of a non-zero Berry phase acquired after one cycle by some discrete set of modes for which . However, in the universe there are many modes and, as discussed in the section III, in general the vacuum after one period is different from . The closeness of these two vacua can be evaluated with the fidelity

 F =|⟨Ω(η0)|Ω(η0+Π)⟩|2=∏→k|αk|−1 (39) =e−12∑→klog(1+|βk|2)≃e−N/2 ,

where is the mean density of particles and where, in the last equality, we assumed that . We claim that, even when is different from , and thus the evolution is not perfectly cyclic, if the two vacua are almost indistinguishable (). In that limit we can thus consider the field evolution approximately periodic and evaluate the Berry phase.

In Section III we have shown that, as far as the conformal factor (26) is concerned, the condition can be obtained when the gap is large, i.e. when . In the inset of Fig. 6 we show that this holds also for the more complicated geometry of eqn. (18). Given that in these conditions the universe is cyclic, in Fig. 6 we evaluate the resulting Berry phase of eqn. (37) and we find that it is non-zero. A non-trivial Berry phase seems thus a general occurrence in cyclic universes. To motivate this further, we consider more general scale factors with higher harmonics in the Fourier expansion (26), namely we set and we choose so that the dominant frequency is .

As shown in Fig. 7, when the number of particles increases with , although especially for . In all the cases discussed so far, the Berry phase is non-zero after a bounce so we argue that it is a generic feature of cyclic universes.

In order to show that the Berry phase heirs information of past evolution of the Universe in this toy model, we consider the accumulated phase in the limit . In this case an analytical result can be obtained (see Appendix A for a full derivation), showing that wavelengths at which Berry’s phase is zero can be enumerated and given by:

 k∗n+ = Π(2n−m2E4πn2) (40)

where and is the length of a cycle measured in conformal time. These wavelengths contain information on the life cycle of the Universe as eqn. (40) shows.

Measuring the Berry’s phase requires the introduction of extra degree’s of freedom coupled with the field. To show this point, we use a simplified argument and we consider a further term in the potential in (3) of the form , where is a spin-1/2 operator in the direction. Depending on the value of the spin along the direction, this extra coupling renormalizes the mass. Because, as we have shown, the Berry’s phase depends on the mass of the field, after one cycle the state of the coupled system formed by the field and the spin is

 1√2(eiϕ+B|Ω+⟩|Sz=+12⟩+eiϕ−B|Ω−⟩|Sz=−12⟩) , (41)

namely there is a different Berry’s phase depending on the state of the spin. The relative Berry’s phase can then be measured, e.g., with . Although simplified, our argument shows that the Berry’s phase can have measurable effects when the field is coupled to other external degrees of freedom.

## V The case of LQC

In the preceding sections we have tried to keep the discussion as general as possible. In fact, eqn. (40) applies to any scale factor which is periodic in conformal time. In this section, we will calculate the Berry phase using the effective loop quantum cosmology equations for a closed Universe. We start by introducing the standard Friedmann equations in a genericmatter-dominated Universe for in the dust approximation, in which and . The first Friedmann equation becomes dodelson ():

 H=(˙aa)2=8πG3ρ0a3−1a2, (42)

and if for simplicity we consider the case , then it is a known fact that the scale factor takes the form . Inevitably, such equation describes a Universe in which also a Big Bang is present, as dynamically the scale factor eventually becomes zero at values of conformal time .

These equations describe a Universe in which there is a maximum expansion (the Big Crunch), followed by a recollapse. It is a widely accepted idea that during the recollapse phase, when the linear size of the scale factor is of the order of the Planck length, Quantum Gravity phenomena take place asht1 (); merc1 (). Loop Quantum Cosmology (LQC) is one example in which a Bounce due to quantum corrections occurs. After the bounce, a new expansion phase is recovered, and thus it is possible to observe a cyclic dynamics of the scale factor. Thus, the machinery we have developed can be applied to the case of Loop Quantum Cosmology if considering purely semiclassical Friedmann equations.

Here we consider the effective corrected equations derived in a recent paper ed1 (), where eqn. (42) is corrected by a term which contains a quantum gravitational term

 (˙aa)2 = (8πG3ρ0a3−1a2)(1−ρρLQC) (43) = (8πG3ρ0a3−1a2)(1−ρ0a3ρLQC).

In general, equations of the type above strongly depend on the details of the quantization procedure, but the phenomenology described is well captured by eqn. (43), which differs in the exact values of . The effective model of eqn. 43 agrees with other quantization procedures cor1 () up to higher orders in the quantization procedure, but differs from well known results in the field asht1 (). Another important comment to be made here is that the derivatives with respect to “time”, are now with respect to an external scalar field which takes the role of a clock, which is a standard procedure in Quantum Gravity approaches asht1 (); merc1 (). Notwithstanding these important remarks and without going into the merit of the quantization procedure, the dynamical interpretation of equation (43) is clear: at large scale factor, matter forces the Universe to recollapse, meanwhile at scale factors comparable to the Planck length, the quantum regime kicks in, forcing the Universe to re-expand. This beautiful and appealing picture is the mechanism for which the singularity of the Big Bang is avoided, and which we considered in this paper insofar in a stylized manner. In order to proceed with our calculations, we need to recast the eqn. (43) into a form which is explicitly solvable as a function of the conformal time . It is easy to use the definition of conformal time, and turn eqn. (43) into eqn. (44) below:

 a′′(η) = a3(η)(−3A(1−2a(η)3ρLQC)2a(η)3 (44) + A(1−1a(η)3ρLQC)a(η)3+38a(η)5ρLQC)

for the case . One can in general obtain periodic solutions to the scale factor, as in Figs. 8,9. It is a general feature, usually discussed in the literature of Loop Quantum Cosmology, that the Universe remains surprisingly close to the classical trajectories predicted by eqn. (44) also in the quantum regime. In this case, we can in fact use the semiclassical method described in the body of this paper, and evaluate the Berry phase numerically after each cycle.

We stress that the Berry phase can be observed only if the dynamics of quantum wave function of the scalar field is cyclic. This implies that depending on the parameters and , we have to check whether the particle content of the Universe is negligible. In Fig. 8 we plot from the top to the bottom, the scale factor, the particle content and the Berry phase respectively obtained numerically for the case , i.e. when the density of the Universe is smaller but comparable to the effective density due to LQC corrections. We observe that the particle content due to the smooth Bounce. In this case, we observe a non-zero Berry phase which implies that at the quantum level the scalar field carries information from one Bounce to the other.

In Fig. 9 we show the same plots obtained for the case , and thus in the regime in which the Bounce is abrupt. In this case, we observe a high particle content, which although invalidating the calculation of the Berry phase, implies that the scalar field carries information through the Bounce in the form of particles. This picture might seem to differ however from the results obtained in (asht1 (); woy1 (); sin1 ()), where it has been shown that the dynamics has effectively no memory from the previous process. As a first comment, it is worth to note in barb1 () it has been discussed that although similar, the polymeric quantization differs from the standard harmonic oscillator quantization used here. Moreover, our analysis relies on the existence of a spectator scalar field which is quantized in a standard way, and is forced (parametrically) by the dynamics of the scale factor. Thus it is not the Universe per se carrying the phase, but the scalar field living on it.

Before concluding this section it is worth mentioning that, although in this paper we focused on standard semi-classical methods, the rigorous origin of the Berry’s phase might be further clarified with a full-quantum treatment, without any semiclassical approximation, given that the Hilbert space structure is exactly known lqcz1 (). Specifically, it may be possible to generalize recent techniques Dario (), where the transition between quantum to classical behavior is explicitly discussed, to the case of loop quantum gravity.

## Vi Conclusions

In the present paper we have studied the quantum features of scalar field both in cyclic and bouncing toy Universes, and in the case of Loop Quantum Cosmology effective Friedman equations. Among the many alternatives to the inflaton field, there are many approaches in which the Universe is considered cyclic. Among these, we mentioned Loop Quantum Cosmology and the Ekpyrotic String approach. Our motivation was to understand whether a scalar field does hold memory of the previous bounces in subsequent ones, thus changing the physics at each bounce. We studied the particle content after many bounces in a general manner, and observed numerically that even when the ground state is periodic, and hence there are no particles after one cycle, the universe keeps track of its history via a non-zero Berry phase. We argue that this is potentially a physically-relevant phenomenon if the scalar field is entangled, and have provided general arguments why such phases should be expected for more general, periodic evolutions of the conformal factor of the Universe. As a result of our analysis, we have observed within various approximations, as for instance the Kronig-Penney and the case of Mathieus approximation, the presence of particles, and noted that in cases where the particle content present at each new bounce is negligible, one can reliably observe the presence of a Berry phase. In fact, after having introduced a differential equation describing the Berry’s connection as function of time, we have obtained a numerical solution of the Berry’s phase, showing that this is non-zero for our toy model, but providing a general framework, based on the Fourier expansion of the energy of the scalar field, for calculating it perturbatively. We have observed that when the ground state is cyclic, for a particular mode, one can observe a non-zero Berry’s phase. In addition to this, we have observed that values for which the Berry phase are zero, contain information on the norm of the scale factor of the Universe over the cycle. This gauge-invariant phase is of geometrical origin and provides a unified framework to explain important observable phenomena like the Aharonov-Bohm effect. Being a measurable quantity, a non-zero Berry’s phase can provide a figure of merit to demonstrate that the universe is cyclic. This phase becomes physically relevant if the scalar field is entangled to another field which can be measured. In this case, after each bounce the physical state of the system changes, and thus the physics changes as well.

In the first part of this paper, we worked initially within the framework of a toy model and thus there are some further limitations to this approach. First of all, we have considered modes which cross the bouncing phase without any new physics to intervene, as for instance Quantum Gravity. The paper does not describe that phase, and treats the Universe as a smoothly oscillating ball. Yet, we believe that this phenomenon is interesting per se from the theoretical standpoint, as also for generic Fourier expansions of the scale factor a non-zero Berry’s phase was observed. In the second part of this paper, we applied these ideas to the case of Loop Quantum Cosmology (LQC), where a description of the quantum phase of these oscillations is provided. Also, an extra appeal of this approach for the present paper is that in LQC the quantum description is very close to the classical one, and thus we do not require the use of a full quantum gravitational approach. In the case of LQC, we find that for the case of a specific semiclassical Friedman equations for which there are oscillations of the scale factor, one can observe two specific regimes. In the first regime, in which the effective quantum density is comparable to the density of the Universe at , we observed numerically small particle content and non-zero Berry phase. In the other regime, in which the quantum effective density is much higher than the density of the Universe, one observes non-negligible particle content.

Our results are general enough to be considered in other cases in details. In fact, given the standard quantization procedure of the scalar field, we have considered the case in which the only requirement is that the scale factor has a Fourier expansion. We studied some specific cases, but used known relations for the Hill equation, which underlies the evolution of the modes of the scalar field. .In addition to the remarks above, nothing prevents this same analysis to be repeated in the case of particles with spin other than zero, which are far more common than scalar particles. As a final remark, we stress that there are other ways in which a Berry’s phase could be originated for non-cyclic cosmologies. This is the case, for instance, if the scalar field undergoes a quantum phase transition in the early Universe QFU (); QPT1 (); QPT2 () and as shown in bpm () for inflationary models as well.

## Acknowledgments

We would like to thank Lorenzo Sindoni and Alessandra Gnecchi for comments on an earlier draft of this paper, and to Mercedes Martin-Benito and Edward Wilson-Ewig for helping us and guiding through the results in Loop Quantum Cosmology in a revised version of this article.

L.B. is currently supported by the ERC grant PACOMANEDIA.

## Appendix A General considerations and Floquet’s theorem

The differential equation in eqn. (6),

 ϕ′′+ωk(η)ϕ=0 , (45)

is called Hill’s equation when