Chaos Removal in the R qR2 gravity: the Mixmaster model · 2018-08-11 · Chaos Removal in the R +qR2 gravity: the Mixmaster model Riccardo Moriconi,1,2, Giovanni Montani,3,4, yand

Chaos Removal in the R + qR2 gravity: the Mixmaster model

Riccardo Moriconi,1, 2, ∗ Giovanni Montani,3, 4, † and Salvatore Capozziello1, 2, 5, ‡

1Dipartimento di Fisica, Universita di Napoli “Federico II”,Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy

2INFN Sezione di Napoli, Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy3Dipartimento di Fisica, Universita degli studi di Roma ”La Sapienza”, P.le A. Moro 5 (00185) Roma, Italy

4ENEA, Unita Tecnica Fusione, ENEA C. R. Frascati, via E. Fermi 45, 00044 Frascati (Roma), Italy5Gran Sasso Science Institute (INFN), Via F. Crispi 7, I-67100, L’ Aquila, Italy

(Dated: August 11, 2018)

We study the asymptotic dynamics of the Mixmaster Universe, near the cosmological singular-ity, considering f(R) gravity up to a quadratic corrections in the Ricci scalar R. The analysis isperformed in the scalar-tensor framework and adopting Misner-Chitre-like variables to describe theMixmaster Universe, whose dynamics resembles asymptotically a billiard-ball in a given domain ofthe half-Poincare space. The form of the potential well depends on the spatial curvature of the modeland on the particular form of the self-interacting scalar field potential. We demonstrate that thepotential walls determine an open domain in the configuration region, allowing the point-Universeto reach the absolute of the considered Lobachevsky space. In other words, we outline the existenceof a stable final Kasner regime in the Mixmaster evolution, implying the chaos removal near thecosmological singularity. The relevance of the present issue relies both on the general nature of theconsidered dynamics, allowing its direct extension to the BKL conjecture too, as well as the possibil-ity to regard the considered modified theory of gravity as the first correction to the Einstein-Hilbertaction as a Taylor expansion of a generic function f(R) (as soon as a cut-off on the space-timecurvature takes place).

PACS numbers: 04.50.Kd, 04.60.Kz

Introduction

The chaotic dynamics of the Mixmaster Universe[1],[2],[3] is a basic prototype of the local (sub-horizon)behaviour of the generic cosmological solution (the so-called BKL conjecture[4]). Investigating the stabilityof such a chaotic picture with respect to the pres-ence of matter [5],[6],[7] and space-time dimensionsnumber[8],[9],[10] has seen a great effort over the lastfour decades and the most significant issue was the proofof the chaos removal when a massless scalar field is in-volved in the dynamics [5]. Such a result is a consequenceof the capability manifested by the scalar field kineticenergy of affecting the second (quadratic) Kasner condi-tion, easily restated in the Hamiltonian picture, as shownin [11]. This property of the massless scalar field ac-quires intriguing perspectives when f(R) modified theoryof gravity are considered [12],[13],[14],[15],[17]. In fact,these alternative formulation of the gravitational field dy-namics can be represented by an equivalent scalar-tensorpicture: the scalar degree of freedom associated to theform of the function f is expressed via a self-interactingscalar field, coupled to the ordinary General Relativity[18],[19],[20],[21]. When implementing this scalar-tensorscheme to the Mixmaster Universe dynamics, a naturalquestion arise: the kinetic term of the scalar field re-moves the chaotic behaviour, but the presence of a po-tential term could restore it? Thus we can study, forspecific modified theories of gravity, if the Mixmasterchaos survives or not, simply characterizing the corre-

sponding scalar field potential. Here we analyse the mod-ified gravity theory corresponding to a quadratic correc-tion in the Ricci scalar to the ordinary Einstein-HilbertLagrangian, both because it is the simplest viable de-viation from General Relativity (apart from a cosmo-logical constant term), as well as the first correctionemerging from a Taylor expansion of a f(R) theory forvery small values of the space-time Ricci scalar, i.e. forvery law curvatures, like we observe today in the So-lar System[22]. The quadratic term in the Ricci scalarprovides an exponential-like potential term for the self-interacting scalar field, when a scalar-tensor reformula-tion of the model is considered. This case is particularlyappropriate to the analysis we pursue of the Mixmas-ter dynamics in terms of the Misner-Chitre-like variables[7],[25],[26],[30]. In fact, the kinetic term of the scalarfield is on the same footing of the anisotropy term con-tribution and, for the considered Lagrangian, also thepotential term is isomorphic to the spatial curvature ofthe model, i.e. the total potential term is constituted byequivalent exponential profile. In the asymptotic limittoward the initial singularity the total potential takesthe form of four potential walls, whose morphology de-termines if the configuration domain is closed or not. In-deed, we demonstrate how the whole domain, availablein principle, is a constant negative curvature space (half-Poincare space). We first analyse the case of the Mix-master Universe in the presence of a massless scalar field,demonstrating the open nature of its configuration spaceand the implied existence of a stable Kasner regime to the

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initial singularity. Then, we face a detailed study of thedynamics in the presence of the total potential and thestill open structure of the configuration domain. Thus,we demonstrate the non-chaotic nature of the MixmasterUniverse behaviour, as it is described by the scalar-tensorversion of the R2-gravity.

f(R) gravity

The f(R) theories of gravity are a direct generaliza-tion of the Einstein-Hilbert Lagrangian consisting in areplacement of the Ricci Scalar R by a general functionf(R)[13],[14],[15],[16]:

S =1

16π

∫d4x√−gf(R) (1)

where g is the determinant of the metric1. The intro-duction of the additional degree of freedom, related tothe presence of the f(R) term, can be translated into adynamics of a self-interacting scalar field coupled withthe Einstein-Hilbert Action, the so-called Scalar-Tensorframework. In this approach, a new auxiliary field χ isintroduced to get the following equivalent version of theaction (1):

S =1

16π

∫d4x√−g[f(χ)− f ′(χ)(R− χ)]. (2)

The variation of the action (2) with respect to χ providesf ′′(χ)(R − χ) = 0, implying χ = R if f ′′(χ) 6= 0. By aredefinition of the auxiliary field χ in the form ϕ = f ′(χ)the action becomes

S =1

16π

∫d4x√−g[ϕR− χ(ϕ)ϕ+ f(χ(ϕ))]. (3)

It is now possible to perform a conformal transformationon the metric gµν → ˜gµν = ϕgµν and a scalar field re-

definition ϕ ≡ f ′(R) → φ =√

316π ln f ′(R) in order to

obtain

S =

∫d4x√−g

[R

16π− 1

2∂αφ∂αφ− U(φ)

], (4)

where the potential term U(φ) has the form:

U(φ) =Rf ′(R)− f(R)

16π(f ′(R))2. (5)

For small values of the Ricci scalar, the first order cor-rection to the Einstein-Hilbert Lagrangian, is represented

1 We use the (−,+,+,+) signature of the metric and the geometricunit system (c = G = ~ = 1).

by a quadratic correction, i.e.

f(R) = R+ qR2. (6)

By this choice, the potential term (5) takes the form

U(φ) =1

64πq

(1− 2 exp−4

√π3 φ + exp−8

√π3 φ). (7)

This is the effective potential that emerges in the so calledStarobinsky-inflation model[22]. Such a model ensures a”slow-rolling” face and it is an inflationary model passingthe latest inflation constraint[23].

The Mixmaster model dynamics

Following the standard representation of the BianchiIX model[24] in the Misner variables[2],[3] the Einstein-Hilbert action takes the form:

Sg =

∫dt

[pαα+ p+β+ + p− ˙β− −

Ne−3α

24πHIX

], (8)

where the dynamics of the model implies the superHamil-tonian constraint

HIX ≡ −p2α + p2+ + p2− + 12π2e4αVIX(β±) = 0. (9)

Here α expresses the isotropic component of the Uni-verse(i.e. the volume of the universe) and the initial sin-gularity is reached for α→ −∞, while the traceless ma-trix βab = diag(β+ +

√3β−, β+−

√3β−,−2β+) accounts

for the anisotropy of this model. Furthermore, pα, p±are the conjugated momenta to α, β± respectively andVIX(β±) is the potential term depending only on β±,corresponding to the spatial curvature. If we executean ADM reduction of the dynamics[27], the Bianchi IXmodel resembles the behaviour of a two-dimensional par-ticle, evolving with respect to the time-like variable α inthe β+, β− plane. By other words, the system dynamicsis summarized by the time-dependent hamiltonian HIX :

−pα = HIX ≡√p2+ + p2− + 12π2e4αVIX(β±). (10)

Looking at the form of the potential term VIX(β±), itis possible, taking into account the three leading terms,to parametrize it as an infinitely steep potential well[3].This way, the point-Universe lives inside the triangu-lar region of the configuration space where the potentialterm is negligible; such a region it is individuate when

3

the following three conditions hold:

1

3+β+ +

√3β−

3α> 0,

1

3+β+ −

√3β−

3α> 0,

1

3− 2β+

3α> 0.

(11)

The presence of the “time” variable α in the relations (11)causes the outside motion of the potential walls and thecorresponding time-dependence of the domain allowed tothe point-Universe motion. Such a dependence can beremoved in the framework of the Misner-Cithre variablesτ, ζ, θ[24],[30] as standing in the Poincare Half-Plane:

α− α0 = −eτ 1 + u+ u2 + v2√3v

,

β+ = eτ−1 + 2u+ 2u2 + 2v2

2√

3v

β− = eτ−1− 2u

2v.

, (12)

where −∞ < τ < ∞, −∞ < u < +∞, 0 < v < +∞. Inthis scheme the role of the hamiltonian time is assigned toτ and the singularity is approach for τ →∞. The trans-formations (12) permit to rewrite the conditions (11) asindependent of the variable τ and thus the domain withinwhich the particle lives is fixed respect to the time vari-able. Making use of the transformations (12), the Hamil-tonian (10 in the free-potential case rewrites as

−pτ = HIX ≡√v2(p2u + p2v), (13)

and the point-Universe lives in the u, v plane inside theregion individuate when the following three conditionshold:

−u1 + u+ u2 + v2

> 0,

1 + u

1 + u+ u2 + v2> 0,

u(u+ 1) + v2

1 + u+ u2 + v2> 0.

(14)

As shown by [26], the asymptotic evolution towards thesingularity is covariantly chaotic because it is isomorphicto a billiard on the Lobachevsky plane. This demonstra-

tion is based on three points:i)the Jacobi metric in theu, v plane has a negative constant curvature; ii)the Lya-punov exponent, defined as in [28], are greater than zero;iii)the configuration space is (dinamically) compact. Theoccurrence of the these three properties ensures that thegeodesic trajectories cover the whole configuration space.

Mixmaster Universe in the R2-gravity

Now we analyse the case of the gravitational La-grangian (6) when the Bianchi IX model is considered.As starting point we consider the modified gravity model(4) in terms of the variables α, β+, β−, φ. Following thesame procedure of the previous section we get the gener-alized reduced hamiltonian −pα = H of the form

H ≡√p2+ + p2− + p2φ + 12π2e4αVIX + 4e6αU. (15)

In Eq.(15), we rescaled the zero point of α→ α− α0, sothat the spatial metric factor e3α → 1

(6π)e3α, and a redef-

inition of the scalar field amplitude φ→√

2(6π)φ is con-sidered too. A natural parametrization, in the Misner-Cithre - Poincare Half-Plane scheme, that reduces to therelations (12) if the scalar field is turned-off, reads as fol-lows

α− α0 = −eτ 1 + u+ u2 + v2√3v

,

β+ = eτ−1 + 2u+ 2u2 + 2v2

2√

3v,

β− = eτ−1− 2u

2vcos δ,

φ = eτ−1− 2u

2vsin δ,

(16)

where −∞ < τ <∞, −∞ < u < +∞ , 0 < v < +∞ and0 < δ < 2π. In this new system of variables the reducedHamiltonian takes the form:

−pτ = H ≡

√v2[p2u + p2v + 4

p2δ(1 + 2u)2

]+ e2τV. (17)

The introduction of the degree of freedom related to thescalar field implies that the point-Universe lives inside a3-dimensional domain in the configuration space u, v, δdetermined by the potential term:

4

e2τV = e2τ [12π2e−4eτξ(u,v)VIX(u, v, δ, τ) + 4e−6e

τξ(u,v)U(u, v, δ, τ)] =

= 12π2e2τ(e− 12eτ√

3v(u+u2+v2)

+ e− 6eτ√

3v(1+(1+2u) cos δ)

+ e− 6eτ√

3v(1−(1+2u) cos δ)

)+

+e2τ

8πq

(e− 12eτ√

3v(1+u+u2+v2) − 2e

− 6eτ√3v

(1+u+u2+v2−2√2π3(1+2u) sin δ)

+ e− 6eτ√

3v(1+u+u2+v2−4

√2π3(1+2u) sin δ)

), (18)

where ξ(u, v) = 1+u+u2+v2√3v

. Due to the exponential

forms of the terms in Eq.(18), when the singularity isapproached (τ → ∞) the point-Universe is confined tolive inside a 3-dimensional domain defined as the regionwhere all the exponents of the six terms are simultane-ously greater than zero. Looking the Eq.(18), the poten-tial term V behaves as an infinitely steep potential wellas in the Poincare variables (14). So for the evolution ofthe point-Universe it is possible to neglect the potentialeverywhere in a suitable domain. As first step we studythe case in absence of all the potential terms (V = 0), i.e.we deal with the Hamiltonian problem:

H = v

√p2u + p2v + 4

p2δ(1 + 2u)2

. (19)

The hamiltonian equations for this potential-free system(Bianchi I model with the massless scalar field) are

u =∂H

∂pu=v2

εpu , pu = −∂H

∂u=

8v2

ε

p2δ(1 + 2u)3

v =∂H

∂pv=v2

εpv , pv = −∂H

∂v= − ε

v

δ =∂H

∂pδ=

4v2

ε

pδ(1 + 2u)2

, pδ = −∂H∂δ

= 0.

(20)

It is possible to demonstrate, as we approach the singu-larity, that H is a constant of motion with respect to the“time” variable τ , following [26]. Thus, we perform thesubstitution H ' ε = const. inside Eq’s(20). It is nowpossible, by following the Jacobi procedure[29] and usingthe equations of motion (20), to write down the line ele-ment for the three-dimensional Jacobi metric in terms ofthe configuration variables, i.e.

ds2 =ε

v2

[du2 + dv2 +

(1 + 2u)2

4dδ2]. (21)

By a direct calculation we see that this metric has a neg-ative constant curvature (the associated Ricci scalar isR = − 6

ε ) and then the point-Universe moves over a neg-atively curved three-dimensional space. Furthermore, wecan find two singular values for the metric in correspon-dence to u = − 1

2 , v = 0. This feature allows us to restrict

the domain of the configuration space in which we willstudy the trajectories of the point-Universe to the funda-mental one identified by the inequalities − 1

2 < u < +∞,0 < v < +∞, 0 < δ < 2π. Indeed there is no wayfor the point-Universe trajectories to cross over the twoplanes u = − 1

2 , v = 0 (each choice of the Lobachevsky“half-space” is equivalent respect to the other one). Theintermediate step toward the general case of the potential(18), corresponding to the ordinary Mixmaster model inthe presence of a massless scalar field, takes place whenwe retain only the exponential terms due to the spa-tial curvature, namely V ' 12π2e−4e

τξ(u,v)VIX(u, v, δ, τ).Then, the point-Universe lives in the region where are si-multaneously satisfied the three following conditions

1 + (1 + 2u) cos δ > 0,

1− (1 + 2u) cos δ > 0,

u(u+ 1) + v2 > 0.

(22)

We now implement a numerical integration of the system

FIG. 1: The black lines represent the trajectories associatedto a points-Universe that bounce against the walls. Instead, thered lines describe the points-Universe witch directly approach thesingularity.

(20) in order to analyse the behaviour of the trajectoriesin the potential free region and then use this result forinterpreting the effect of the scalar curvature. As we

5

can see in the Fig.1 an opening of the domain emergesdue to the presence of the scalar field and it is possi-ble to individuate two families of trajectories: those onescorresponding to a point-Universe that bounces againstthe walls and turn back inside the domain (the blackones) and those corresponding to a particle that approachthe so called “absolute”[31] (the red ones), for valuesv → 0,∞, with no other bounces until the singular-ity. The presence of the trajectories of the second familyshows the removal of the oscillatory behavior of the Mix-master model coupled with a massless scalar field [4],[11].Let us see what happen if we consider the complete po-tential term(18). This time the restrictions on the dy-namics imply that the particle is confined inside a regionwhere all the six exponential terms in Eq.(18) are simul-taneously greater than zero. We can immediately removeone of the six conditions because the first exponent re-lated to the potential of the scalar field 1 + u + u2 + v2

is always greater than zero for any values of u, v takingin consideration. Thus, the five conditions that identifythe domain are

1 + (1 + 2u) cos δ > 0

1− (1 + 2u) cos δ > 0

u(u+ 1) + v2 > 0

1 + u+ u2 + v2 − 2√

2π3(1 + 2u) sin δ > 0

1 + u+ u2 + v2 − 4√

2π3(1 + 2u) sin δ > 0

(23)

We observe that the last of the conditions above natu-rally implies the validity of the fourth one too. Thus, weindeed deal with four potential walls only. As we can seein the Fig.2, taking into account also the potential termU(u, v, δ, τ) implies that the available configuration spacefor the point-Universe is clearly reduced with respect tothe case U = 0 (see Fig.1). However, trajectories yetexist(the red lines in the Fig.2) corresponding to a point-Universe that is able to reach the absolute for v → 0,∞.For this reason we can firmly conclude that a quadraticcorrection in the Ricci scalar to the Einstein-Hilbert ac-tion, that in the Scalar-Tensor theory is equivalent to thedynamics of a self-interacting scalar field (with potentialterms of the form (7)), is able to remove the never-endingbounces of the point-Universe against the walls. As aresult of the bounces against the infinite potential walls(which can be described by a reflection rule[25],[33]), soonor later the point-Universe reach a trajectory connectedwith the absolute. It is worth noting that the analysisabove is referred to the choice q > 0, in which case thesign of the scalar field potential is the same one of thescalar curvature. This choice is forced by the request thatthe additional scalar mode, associated to the quadraticmodification, be a real (non-tachyonic) massive one, ac-cordingly to the original Starobinsky approach in [22] anddemonstrated also in [36]. However, in the case q < 0, thescalar field potential would not contribute an infinite pos-

itive wall, but an infinite depression. Since in the regionof zero potential, the point-Universe has always positive“energy”, we can easily conclude that such a case overlapsthe non-chaotic potential-free one. We now observe that,in correspondence to the configuration region v = 0,∞and δ = 3π

2 , the scalar field acquires negative divergingvalues and its potential terms manifests a diverging be-haviour. Such a profile of the scalar field is typical of aBianchi I solution near the singularity [5] and the diverg-ing character of the potential term means that GeneralRelativity can not be asymptotically recovered. Rigor-ously speaking, the present result on the chaos structureapplies to a quadratic correction in the Ricci scalar only,because it is the first terms of the Taylor expansion of thefunction f(R) working nearby the singularity. Nonethe-less, our analysis has a general validity, as soon as, wetake into account a physical cut-off at the Planck time,where classical theory starts to fail and a quantum treat-ment is required. In fact, the Planckian cut-off wouldremove the φ and U(φ) divergences, allowing the Taylor

expansion for q . (ctcut)2

l2P, where tcut being the cut-off

time and lP the Planck length. Since tcut >lpc , we deal

with the (non-severe) restriction q . 1 for preservingthe general nature of our result. This estimation followsrequiring R > qR2 and remembering that for the caseof a Kasner solution, in the presence of a potential-freescalar field, the Ricci scalar behaves as R ∼ 1

t2s, where

ts is the synchronous time. We stress that qualitatively,a similar argument is at the ground of the non-chaoticnature of the Bianchi IX Loop Quantum dynamics in thesemi-classical limit [35]. However, the field φ(ts) admits,both for v → 0,∞ and δ = π/2, trajectories implying itspositive divergence. For such behaviours, correspondingto an open region in the initial condition, the potentialU(φ) approaches a constant value and φ is effectivelymassless. It is just the existence of these diverging pro-files at the ground of the chaos removal in the presentmodel. The massless nature of the potential along spe-cific trajectories is a good criterion for determining thechaotic properties of the Mixmaster Universe in a spe-cific non-expanded f(R) model. In fact, The behaviorof the free scalar field reads φf (ts) ∝ ln ts and the cor-responding kinetic energy density stands as 1/t2s. Then,for a given f(R) model, fixing the potential U(φ), thechaos removal is ensured by the validity of the conditionlimts→0 U(φf (ts))t

2s = 0. Clearly, the non-chaoticity is

ensured if such a limit holds for a non-zero measure setof trajectories.

Conclusions

The analysis above demonstrated how including aquadratic correction in the Ricci scalar to the Einstein-Hilbert Lagrangian of the gravitational field gives a deep

6

FIG. 2: The point-Universe lives inside the region marked by thewalls, where the conditions (23) are verified. We also sketch thetrajectories reaching the absolute.

insight on the nature of the Mixmaster singularity: theevolution of the scale factors is no longer chaotic and astable Kasner regime emerges as the final approach tothe singular point.

The relevance of this result is in its generality with re-spect to the behaviour of the cosmological gravitationalfield. In fact, on one hand, the result we derived in thehomogeneous cosmological setting, can be naturally ex-tended to a generic inhomogeneous Universe, simply fol-lowing the line of investigation discussed in [4],[5].

The basic statement, at the ground of the BKL con-jecture, is the space point decoupling in the asymptoticdynamics toward the cosmological singularity. Such a dy-namical property of a generic inhomogeneous cosmologi-cal model allows to reduce the behavior of a sub-horizonspatial region [4],[37] to the prototype offered by the ho-mogeneous Mixmaster Universe. We are actually stat-ing that the time derivative of the dynamical variablesasymptotically dominate their spatial gradients, limitingthe presence of the spatial coordinates in the Einsteinequation to a pure parametrical role. We are speaking ofa conjecture because the chaotic features of the point-likedynamics induce a corresponding stochastic behaviour ofthe spatial dependence and the statement above requiresa non-trivial treatment for its proof. Nonetheless a valu-able estimation of the spatial gradient behaviour, whenthe space-time takes the morphology of a foam, is pro-vided in [38]. When a scalar field is present the situa-tion is even more simple, because, after a certain num-

ber of iterations of the BKL map, in each space point,a stable Kasner regime takes place [39] and the validityof the solution is rigorously determined [40]. Thus, wecan extend our result to a generic inhomogeneous cos-mological model simply considering the dynamical vari-ables as space-time functions u = u(τ, xi), v = v(τ, xi)and δ = δ(τ, xi), which, in each space point, live in ahalf-Poincar space and are governed by an independentand morphologically equivalent dynamics. On the otherhand, the extension of General Relativity we consideredhere is the most simple and natural one, widely studiedin literatures in view of its implications on the primor-dial Universe features. Since the classical evolution is ex-pected to be predictive up to a finite value of the Universevolume, i.e. up to a given amplitude of the space-timecurvature, for sufficiently small coupling constant q val-ues, the present model can be considered as the quadraticTaylor expansion of a generic f(R) theory and we canthen guess that the non-chaotic feature is a very generaldynamical property, at least within the classical domainof validity of the f(R) theory. In this sense we traced avery general and reliable properties of the cosmologicalgravitational field in modified theories of gravity of sig-nificant impact on the so-called billiard representation ofthe generic primordial Universe[8],[25],[33].

∗ Electronic address: [email protected]† Electronic address: [email protected]‡ Electronic address: [email protected]

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