Possible consistent extra time dimensions in the early universe
PPossible Consistent Extra Time Dimensions in the Early Universe
Patrick L. Nash ∗
475 Redwood Street Unit 801, San Diego, CA 92103-5864 (Dated: October 30, 2018)
Abstract
Gravity cannot be quantized unless the quantized theory is cast on a manifold whose concomi-tant number of physical space dimensions and number of physical time dimensions correspond tophysical reality, and not simply to the perception of reality. At present, the accepted number ofphysical time dimensions is dictated more by folklore than by science. True, there are theoremsthat restrict the number of possible time dimensions to one, but these only apply to rather specialscenarios. In this paper we discuss a model of the early universe in which the number of physicaltime dimensions is four, and formulate Theorem[10.1], which underlies an explanation of why theextra time dimensions do not source unphysical effects.In this paper we describe a new model of gravitational inflation that is driven by dark energyand “mediated” by a real massless scalar inflaton field ϕ whose potential is identically equal tozero. The coupled Einstein gravitational and inflaton field equations are formulated on an eight-dimensional spacetime manifold of four space dimensions and four time dimensions. We findexplicit solutions to these field equations that exhibit temporal exponential deflation of threeof the four time dimensions , and then study the dynamics of a massive complex scalar field ψ that propagates on the background ground state Einstein gravitational field to determine whetherits quantum fluctuations δψ are stable or unstable. We compute explicit approximate solutions tothe δψ field equations that are stable , meaning that the quantum fluctuations δψ of the field ψ donot grow exponentially with time. Instabilities driven by the momenta associated to the threeextra time dimensions do not appear in the physical solutions of the field equations of this model.
PACS numbers: 98.80.Cq, 04.62.+v, 04.50.KdKeywords: cosmology,extra time,inflation,deflation,dark energy ∗ Electronic address: [email protected] a r X i v : . [ phy s i c s . g e n - ph ] J un . INTRODUCTION Recent Planck 2013 data analysis [1] is in remarkable accord with a flat ΛCDM modelwith inflation, based upon a spatially flat, expanding universe whose dynamics are governedby General Relativity and sourced by cold dark matter, a cosmological constant Λ, and aslow-roll scalar inflaton field [2–4]. The main predictions of inflationary cosmology are alsoconsistent [5–8] with other recent observational data from important experiments such asWMAP [9] and the Sloan Digital Sky Survey [10–12], to name only two. Also, assuming thatthe large value of the tensor-to-scalar ratio in the cosmic microwave background radiationreported by the BICEP2 collaboration [13] is correct, then the energy scale of inflation isapproximately 2 × GeV, which is only roughly two orders of magnitude below the Planckscale of . × √ π GeV.However, parameterized ΛCDM inflationary cosmology may not represent fundamentalphysics. Here we discuss a radical new model of inflation/deflation that overcomes the wellknown issues of [1] the precise definition of the inflaton potential, [2] the precise definitionof the inflaton mass (many models of inflation omit the Higgs field that presumably gener-ates the inflaton mass, or do not explain why the Higgs field develops a non-zero vacuumexpectation value at such a high energy) and [3] re-heating. This model is based on the ideathat our universe has as many time dimensions as space dimensions.In this paper we describe a new model of gravitational inflation that is driven by darkenergy and “mediated” by a real massless scalar inflaton field ϕ whose potential is identicallyequal to zero. The coupled Einstein gravitational and inflaton field equations are formu-lated on an eight-dimensional spacetime manifold of four space dimensions and four time dimensions. For the case of a diagonal metric, two periodic solution classes (“ground” and“excited” classes) for the coupled Einstein field equations are obtained that exhibit temporalexponential deflation of three of the four time dimensions and temporal exponentialinflation of three of the four space dimensions. Moreover, for the ground state solution class the universe does not cool during inflation . Reheating [14] is a non-issue. Moreover,we show that the extra time dimensions do not generally induce exponentially rapid growthof fluctuations in quantum fields that propagate on the ground state solution class.Comoving coordinates for the two non-inflating/deflating dimensions are chosen to be( x , x ). The x coordinate corresponds to our universe’s observed physical time dimension,2hile the x coordinate corresponds to a new spatial dimension that is assumed to becompact with constant unit radius R ≡
1, i.e., the radius equals the Planck length. Thevalue of R can be adjusted to fit experimental data if need be. Dark energy is realized interms of a cosmological constant Λ, whose values are determined to be quantized in terms ofthe constant unit radius of the compact spatial x dimension. It is also interesting to notethat although the inflaton potential is zero, an effective inflaton potential for the groundstate is computed to be V eff = Λ cot ( x √ .In this model, after “inflation” the observable physical macroscopic world appears to aclassical observer to be a homogeneous, isotropic universe with three space dimensions andone time dimension. Also, under the assumption that an “arrow of time” exists for eachtimelike dimension, unphysical closed timelike curves are not observed in this model. Wealso prove that a well known so-called “single time” theorem does not apply to our model.
2. ARGUMENTS AGAINST PLURAL TIME DIMENSIONS
Everyone is familiar with arguments against the existence of extra time dimensions. Whilethere are many aspects to this issue, we address two arguments against the existence of extratime dimensions which stand out most clearly[15]:1. A spacetime with extra time dimensions may not carry a spin structure ⇔ spinorscannot be defined on the spacetime;2. Momenta corresponding to the extra time dimensions induce exponentially rapidgrowth of quantum fluctuations of the field; the universe is unstable. This instabilityis associated with the very largest momenta (shortest wavelengths).Issue [1] is not applicable here; it is known that our model spacetime carries a spin structure[16]. The issues raised in [2] may be investigated by studying the field equation for thepropagation of a massive complex scalar field ψ on the background gravitational field:0 = 1 (cid:112) det ( g α β ) ∂ µ (cid:20) (cid:113) det ( g α β ) g µ ν ∂ ν ψ (cid:21) − (cid:0) Λ m + ζ R (cid:1) ψ, (2.1)which is derivable from the Lagrangian L ψ = (cid:113) det ( g α β ) (cid:2) − g µ ν ∂ µ ψ ∗ ∂ ν ψ − ψ ∗ ψ (cid:0) Λ m + ζ R (cid:1) (cid:3) . (2.2)3ere R is the Ricci scalar, and m and ζ are real input parameters; the factors of Λ areincluded for convenience. The quantum fluctuations δψ of ψ satisfy a field equation similarto Eq.[2.1], but generally with a different rest mass parameter.It will be seen that Eq.[2.1] is not separable. Because Eq.[2.1] is not separable, there willnot be a simple dispersion relation, per se , relating a physical-frequency ( ω ↔ − ˙ ı ∂∂ x ln ψ )to the momentum wave vectors ( (cid:126)k, (cid:126)w ) that are defined two paragraphs below. In fact wefind decaying quasi-normal mode approximate solutions to Eq.[2.1]. Modes of this classare well known in black hole physics [17–19]: the authors of Ref.[18] discuss the theory ofquasi-normal modes of compact objects, including perturbations of (Schwarzschild, Reissner-Nordstr¨om, Kerr and Kerr-Newman) black holes; Ref.[17] provides a rigorous definition ofquasi-normal modes for a rotating black hole; the author proves that the local energy oflinear waves in certain black hole backgrounds decay exponentially once orthogonality tothe zero resonance is imposed. In the present early universe model, decaying quasi-normaltype modes appear as approximate solutions to Eq.[2.1]; however numerical experimentsindicate that the contribution to the complex frequency that is responsible for decay getssmaller as the order of the approximation increases (at least for a finite percentage of themodes). The consequence of this for the unitarity of the quantum field theory is not presentlyknown.We shall discuss two types of solution to Eq.[2.1], the first with nonzero coupling to theRicci scalar and the second with ζ = 0. It should be emphasized that the expected Yukawainteraction coupling of the massive complex scalar field ψ with the massless real scalarinflaton field ϕ is not included here. We do not wish to confuse the production/annhilationof ψ “particles” due to its interaction with the inflaton field with instabilities in the ψ fieldthat are sourced by the momenta associated to the extra time dimensions. Also, it is beyondthe scope of the present paper to consider the one-loop effective potentials and concomitantrenormalizations of both ϕ and ψ . These are important questions, but outside the scope ofthis work.In Sections 5 and 11 we seek “plane wave” solutions to Eq.[2.1] in terms of comovingwave vectors and coordinates of the form 4 = Ψ( x , x ) e ˙ ı ( (cid:126)k · (cid:126)r − (cid:126)w · (cid:126)R ) , where (cid:126)r = (cid:0) x , x , x (cid:1) T and (cid:126)k = ( k , k , k ) T ; (cid:126)R = (cid:0) x , x , x (cid:1) T and (cid:126)w = ( k , k , k ) T . (2.3)When discussing these solutions the following shorthand is employed:Λ k = (cid:126)k · (cid:126)k = k + k + k ;Λ w = (cid:126)w · (cid:126)w = k + k + k . (2.4)To summarize, in this paper we find explicit solutions of the coupled Einstein-inflatonfield equations, which are then substituted into Eq.[2.1]; solutions to the latter equation arethen determined for which the effects of multiple time dimensions are stable, meaning thatquantum fluctuations δψ of the ψ field do not grow exponentially with time. Exponentiallyrapid growth of quantum fluctuations δψ of the ψ field cause perturbation theory to breakdown when the norm of the term(s) involving δψ become of order one. The following “Single-Time” Theorem [20] has been asserted, incorrectly, to argue thatextra time dimensions cannot exist in viable physical models of the universe. To demonstratethat this theorem does not apply to the model in this paper, we need only quote the statementof the STT and highlight two points in its proof that clearly are not satisfied in our model.We quote the author of the STT:“Theorem 2 (Single-Time Theorem) [20]. Any BLACK HOLE solution with k > t such that g tt = 0 at the horizon.”“The starting point is ... the model action for D-dimensional gravity with several scalardilatonic fields φ a and antisymmetric n s -forms F s ... in a (pseudo-)Riemannian mani-fold M = R u × M × . . . × M n with the metric ds = g MN dz M dz N = we α ( u ) du + (cid:80) ni =0 e β i ( u ) ds i , w = ±
1, where u is a selected coordinate ranging in R u ⊆ R ; g i = ds i are metrics on d i -dimensional factor spaces M i of arbitrary signatures ε i = sign g i ; | g | = | detg MN | and similarly for subspaces; ... M i , i > M isallowed to be a space of constant curvature K = 0 , ± homogeneous andisotropic. There are no black holes and no black hole potentials, so the STT clearly doesnot apply.
3. NOTATION AND CONVENTIONS
This model is cast on X , , which is an eight-dimensional pseudo-Riemannian man-ifold that is a spacetime of four space dimensions, with local comoving spatial coordi-nates ( x , x , x , x ), and four time dimensions, with local comoving temporal coordinates( x , x , x , x ) (employing the usual component notation in local charts). Greek indicesrun from 1 to 8. Each of the x α has units (dimensions) of length. It is assumed that −∞ < x α < ∞ for α = 1 , , . . . , X , may be decomposed as X , = X , × S , where S is a one dimensional compact space that is homeomorphic to the unitcircle S . We need at least two charts to cover S , which are chosen to be x : S → R , − π R < x < π R and x (cid:48) : S → R , 0 < x (cid:48) < π R , where R ≡ R is introduced to give x the correct dimension. R is not associated to a dilatonicdegree of freedom since it is constant, R ≡ U be a subset of X , , p ∈ U and φ : U → R be a local chart. ∂∂ x α denotes thenatural basis of T p ( X , ) associated with the coordinates x α = [ φ ( p )] α , where [ . ] α denotesthe canonical projection of the α th component of x ∈ R . In the context of this paper aphrase such as “ x is a timelike dimension” means that the timelike vector ∂∂ x is tangentto the lines of constant spatial coordinates and constant ( x , x , x ), i.e. the curves of X , defined by ( x = K , x = K , x = K , x = K , x = K , x = K , x = K ), where K , K , K , K , K , K , K are seven constants.In this paper we obtain solutions to the coupled Einstein-inflaton field equations thatexhibit temporal exponential deflation of three of the four time dimensions. More precisely,6e record solutions that possess exponentially decreasing scale factors for three of the fourtimelike dimensions, where decreasing means decreasing with respect to the remaining fourthtime coordinate, which is chosen to be x . In effect we arbitrarily label the temporal direc-tion/dimension that does not deflate (or inflate, for that matter) the x - axis. The threeremaining time dimensions are labeled with ( x , x , x ) and they undergo deflation. Weassume that unknown quantum gravity effects sort things out shortly after the Universe iscreated so that the three mutually orthogonal deflating timelike directions and the remainingfourth mutually orthogonal non-inflating/non-deflating timelike direction (i.e., the temporaldimension whose scale factor is a constant) are the same for all values of ( x , x , x , x ).This is an unsatisfying assumption, but a necessary hypothesis given our current knowledgeof quantum gravity.Let g denote the pseudo-Riemannian metric tensor on X , . The signature of the metric g is (4 , ↔ (+ + + − − − − +). The covariant derivative with respect to the symmetricconnection associated to the metric g is denoted by a vertical double-bar. We employ theLandau-Lifshitz spacelike sign conventions [21]. g ↔ g αβ = g αβ ( x µ ) is assumed to carry theNewton-Einstein gravitational degrees of freedom. It is moreover assumed that the ordinaryEinstein field equations (on X , ) G αβ + g αβ Λ = 8 π G T αβ (3.1)are satisfied. Here G αβ denotes the Einstein tensor, Λ is the cosmological constant, G denotesthe Newtonian gravitational constant, and the reduced Planck mass is M P l = [ 8 π G ] − / .Natural units c = 1 = (cid:126) = 8 π G are used throughout. Lastly, if f = f ( x , x ) then f (1 , = ∂∂ x f ( x , x ) , f (0 , = ∂∂ x f ( x , x ) , f (1 , = ∂ ∂ x ∂ x f ( x , x ) ,f (2 , = ∂ ∂ x f ( x , x ) , f (0 , = ∂ ∂ x f ( x , x ) , etc. (3.2)
4. EINSTEIN FIELD EQUATIONS4.1. Preview
We seek solutions to the Einstein field equations Eq.[3.1] for which the observable phys-ical macroscopic early universe appears to a classical observer, after inflation ends, to bea homogeneous, isotropic universe with three space dimensions and one time dimension.7ccordingly the three space dimensions ( x , x , x ) are assumed to possess equal scale fac-tors, denoted a = a ( x , x ), and the three extra time dimensions ( x , x , x ) are assumed topossess equal scale factors, denoted b = b ( x , x ).We find two periodic (in x ) solution classes ( E , E ) to the Einstein field equationsEq.[3.1] on X , that exhibit inflation/deflation and describe a universe that is spatially flatthroughout the inflation era. Moreover it is found that Λ is quantized in terms of R . For( E , E ), during “inflation”, the scale factor a = a ( x , x ) for the three space dimensions( x , x , x ) exponentially inflates as a function of x , and the scale factor b = b ( x , x ) for thethree extra time dimensions ( x , x , x ) exponentially deflates as a function of x ; moreover,the scale factors for the x and x dimensions are constants equal to one.We define the cosmological relative expansion rate (cid:96) as (cid:96) = 12 ∂∂x ln (cid:18) ba (cid:19) . (4.1)In Section 5 it is shown that (cid:96) is independent of time and equal to (cid:96) = − (cid:113) Λ for solutionsin the first class E ; this solution corresponds to pure exponential time inflation of the scalefactor a ( x , x ) and pure exponential time deflation of the scale factor b ( x , x ). For thiscase | (cid:96) | coincides with the Hubble parameter H .For the class E solution we find that Λ is “quantized” in terms of the compact spatial x unit radius R ≡
1, and given by Λ = 12 1 R , (4.2)and therefore is one half the square of the Planck mass.Therefore (cid:96) verifies R | (cid:96) | = 16 , (4.3)The two relationships are reminiscent of semiclassical quantization relationships.For solutions in the second class E , which describes a transition from pure exponential“inflation,” (cid:96) = (cid:96) is generally time dependent. In this case the “quantized” value of Λ isΛ = 65 Λ . (4.4)We refer to Λ as the ground state value of the cosmological constant and Λ as its firstexcited state. 8athematical universes that possess positive relative expansion rate (cid:96) > x > The line element for inflation/deflation is assumed to be given by { ds } = (cid:8) a ( x , x ) (cid:9) (cid:104)(cid:0) dx (cid:1) + (cid:0) dx (cid:1) + (cid:0) dx (cid:1) (cid:105) − (cid:0) dx (cid:1) − (cid:8) b ( x , x ) (cid:9) (cid:104)(cid:0) dx (cid:1) + (cid:0) dx (cid:1) + (cid:0) dx (cid:1) (cid:105) + (cid:0) dx (cid:1) = a (cid:104)(cid:0) dx (cid:1) + (cid:0) dx (cid:1) + (cid:0) dx (cid:1) (cid:105) − b (cid:104)(cid:0) dx (cid:1) + (cid:0) dx (cid:1) + (cid:0) dx (cid:1) (cid:105) − (cid:0) dx (cid:1) + (cid:0) dx (cid:1) ; (4.5)where a = a ( x , x ) and b = b ( x , x ) carry the metric degrees of freedom in this model. Thereal massless scalar inflaton field is ϕ = ϕ ( x , x ). The action for the metric and inflatondegrees of freedom is assumed to be given by S = (cid:90) (cid:20) π G ( R − − g µ ν ∂ µ ϕ ∂ ν ϕ − ξ R ϕ + L SM (cid:21) (cid:113) det ( g α β ) d x. (4.6)Here Λ is the cosmological constant. The inflaton potential is zero; its action is purelykinematic, although ϕ (0 , ( x , x ) may be regarded as contributing to an effective inflatonpotential. For the remainder of the paper we assume minimal coupling of the inflaton withgravity and set ξ = 0. We also neglect the standard model degrees of freedom and set L SM = 0. The canonical stress-energy tensor for the real massless scalar inflaton field is T µ ν = − √ det( g αβ ) ∂∂ g µ ν L ϕ , L ϕ = (cid:112) det ( g α β ) (cid:2) − g µ ν ∂ µ ϕ ∂ ν ϕ (cid:3) . The distinct components are T = 12 a ( x , x ) (cid:2) ϕ (1 , ( x , x ) − ϕ (0 , ( x , x ) (cid:3) , (4.7)9 = 12 (cid:2) ϕ (1 , ( x , x ) + ϕ (0 , ( x , x ) (cid:3) , (4.8)(which clearly satisfies the weak energy condition), T = 12 b ( x , x ) (cid:0) ϕ (0 , ( x , x ) − ϕ (1 , ( x , x ) (cid:1) , (4.9) T = 12 ϕ (0 , ( x , x ) + 12 ϕ (1 , ( x , x ) , (4.10)and T = T = ϕ (0 , ( x , x ) ϕ (1 , ( x , x ) , (4.11)which is non-zero, in general. Due to the ( x , x ) dependence of the metric the (4 ,
8) and(8 ,
4) components of the Einstein tensor G = G are also, in general, non-zero.Let ρ = ρ ( x , x ) denote the effective energy density of the (classical) inflaton field and p = p ( x , x ) denote the effective pressure. ρ and p may be associated to T µ ν according to T µ ν = (cid:34) j =7 (cid:88) j =4 u µ ( j ) u ν ( j ) (cid:35) ( ρ + p ) + p g µ ν + [ δ µ δ ν + δ µ δ ν ] T (4.12)where u µ ( j ) = δ µj j = 4 , , , T = ρ = 12 (cid:2) ϕ (1 , ( x , x ) + ϕ (0 , ( x , x ) (cid:3) T = g p = 12 a ( x , x ) (cid:2) ϕ (1 , ( x , x ) − ϕ (0 , ( x , x ) (cid:3) . (4.13)Therefore w = pρ = T g T = ϕ (1 , ( x , x ) − ϕ (0 , ( x , x ) ϕ (1 , ( x , x ) + ϕ (0 , ( x , x ) . (4.14)When ϕ (1 , ( x , x ) = 0, as is the case for the Class E solution of Subsection[5], then w = − . (4.15)In this case the effective inflaton potential may be defined as V eff = 12 ϕ (0 , ( x , x ) . (4.16)Therefore for the Class E solution of Section[5], V eff = 54! Λ cot ( 12 x √ . (4.17)10 .4. Field Equations G µ ν + Λ g µ ν = 8 π G T µ ν (4.18)The distinct field equation components may be written as G = G = − a (1 , a − b (1 , b = 8 π G ϕ (0 , ϕ (1 , (4.19) G = 1 b (cid:2) ab (cid:0) a (0 , b (0 , − a (1 , b (1 , + a (cid:0) b (0 , − b (2 , (cid:1)(cid:1) + (cid:0) a (0 , − a (1 , + 2 a (cid:0) a (0 , − a (2 , (cid:1)(cid:1) b + 3 a (cid:0) b (0 , − b (1 , (cid:1)(cid:3) = 4 π G a (cid:0) − ϕ (0 , + ϕ (1 , (cid:1) (4.20) G = − a b (cid:2) b (cid:0) a (0 , − a (1 , + aa (0 , (cid:1) + ab (cid:0) a (0 , b (0 , − a (1 , b (1 , + ab (0 , (cid:1) + a (cid:0) b (0 , − b (1 , (cid:1)(cid:3) = 4 π G (cid:0) ϕ (0 , + ϕ (1 , (cid:1) (4.21) G = 1 a (cid:2) ab (cid:0) − a (0 , b (0 , + 3 a (1 , b (1 , + a (cid:0) b (2 , − b (0 , (cid:1)(cid:1) + 3 (cid:0) − a (0 , + a (1 , + a (cid:0) a (2 , − a (0 , (cid:1)(cid:1) b + a (cid:0) b (1 , − b (0 , (cid:1)(cid:3) = 4 π G b (cid:0) ϕ (0 , − ϕ (1 , (cid:1) (4.22) G = − a b (cid:2) a (cid:0) b (1 , − b (0 , (cid:1) + (cid:0) − a (0 , + a (1 , + aa (2 , (cid:1) b + ab (cid:0) − a (0 , b (0 , + 3 a (1 , b (1 , + ab (2 , (cid:1)(cid:3) = 4 π G (cid:0) ϕ (0 , + ϕ (1 , (cid:1) (4.23)11he components of T µ || µ α that are not identically zero must satisfy a b T µ || µ = 0 = 3 a (cid:0) − b (1 , (cid:0) ϕ (1 , (cid:1) + b (0 , ϕ (0 , ϕ (1 , + b (0 , (cid:1) + b (cid:0) − a (1 , (cid:0) ϕ (1 , (cid:1) + 3 a (0 , ϕ (0 , ϕ (1 , + a (cid:0) ϕ (1 , (cid:0) ϕ (0 , − ϕ (2 , (cid:1)(cid:1)(cid:1) a b T µ || µ = 0 = 3 a (cid:0) b (0 , (cid:0) ϕ (0 , (cid:1) − b (1 , ϕ (0 , ϕ (1 , (cid:1) − b (cid:0) − a (0 , ϕ (0 , + 3 a (1 , ϕ (1 , ϕ (0 , − a (cid:0) ϕ (0 , (cid:0) ϕ (0 , − ϕ (2 , (cid:1)(cid:1)(cid:1) . (4.24)Let L = a (1 , ( x ,x ) a ( x ,x ) , L = a (0 , ( x ,x ) a ( x ,x ) , L = b (1 , ( x ,x ) b ( x ,x ) and L = b (0 , ( x ,x ) b ( x ,x ) . The Euler-Lagrange equation for the inflaton field yields ϕ (2 , ( x , x ) − ϕ (0 , ( x , x ) + 3 ϕ (1 , ( x , x ) ( L + L ) − ϕ (0 , ( x , x ) ( L + L )= 0 , (4.25)or, equivalently, ϕ (2 , ( x , x ) + 3 ϕ (1 , ( x , x ) ( L + L ) + µ ϕ ( x , x )= ϕ (0 , ( x , x ) + 3 ϕ (0 , ( x , x ) ( L + L ) + µ ϕ ( x , x ) , (4.26)where µ is arbitrary.Lastly, the field equations demand that the constraint equation L + 3 L L + L − π G ϕ (1 , ( x , x ) − (cid:2) L + 3 L L + L − π G ϕ (0 , ( x , x ) (cid:21) = 29 Λ , (4.27)be satisfied.To solve the field equations we use the fact that P = ln ( a × b ), Q = ln (cid:0) ba (cid:1) and ϕ satisfy,respectively, uncoupled, linear and linear field equations of the form f (0 , ( x , x ) − f (2 , ( x , x ) = S ++ 3 P (1 , ( x , x ) f (1 , ( x , x ) − P (0 , ( x , x ) f (0 , ( x , x ) , (4.28)where f = P , Q or ϕ and S = 0 unless f = P in which case S = − Λ. General solutions tothese equations are substituted into the constraint Eq[4.27] and the remaining field equa-tions, which are then solved; this procedure yields the solutions in the classes E and E that are described below. 12 . CLASS E SOLUTION: EXACT TEMPORAL EXPONENTIAL INFLA-TION/DEFLATION
Using the technique described above we find that this model admits the following solu-tions, which are exponential in x , periodic in x and that are parameterized by Λ: Thescale factors are a = a ( x , x ) = a e ± √ Λ2 x (cid:114) sin (cid:16) x √ (cid:17) b = b ( x , x ) = b e ∓ √ Λ2 x (cid:114) sin (cid:16) x √ (cid:17) , (5.1)where a and b are constants. The relative expansion rate and the inflaton field are givenby (cid:96) = ∓ (cid:114) Λ2 ϕ = ± (cid:114)
56 ln (cid:20) tan (cid:18) √ x (cid:19)(cid:21) . (5.2)For this case the scale factors ( a, b ) have a spatial period equal to C , while the threefunctions ( a, b, ϕ ) possess a common spatial periodicity in the x coordinate whose nonzerominimum value is equal to the x -dimension spatial period C = 2 π R = π (cid:114)
2Λ = π | (cid:96) | , (5.3)Λ = Λ = 12 1 R (5.4)For this case | (cid:96) | coincides with the Hubble parameter H .Since a universe with many macroscopic times is not observed, one may identify thephysical solution for inflation by choosing the solution with the − sign in the equation for b . This solution then predicts the exponential deflation, with respect to time, of the scalefactor b associated with the three extra time dimensions. This coincides with the exponentialinflation, with respect to time, of the scale factor a associated with the three observed spatialdimensions. 13 . TEMPERATURE HISTORY OF INFLATION IN THE CLASS E BACK-GROUND
The reduced volume element d Ω on the hypersurface x = x = constant is d Ω = d Ω( x α ) = (cid:104)(cid:112) det ( g ) (cid:105) x = x d τ (6.1)where d τ = | dx ∧ dx ∧ dx ∧ dx ∧ dx ∧ dx ∧ dx | . (6.2)For the class E solution d Ω = d τ (cid:12)(cid:12)(cid:12) sin (cid:16) x √ (cid:17)(cid:12)(cid:12)(cid:12) , (6.3)which clearly does not inflate. The fact that d Ω vanishes on a set of measure zero is discussedin the
Conclusion .To compute the temperature history T = T ( x ) during inflation we assume that beforestandard model particles are created the chemical potential of the universe is zero. Wealso assume that the entropy density, energy density, and pressure describe only Λ and theinflaton ϕ field, and that their averages over x are functions s ( T ) , ρ ( T ) , and p ( T ) of thetemperature T alone.In analogy with the conventional definition of thermal equilibrium [see, for example,reference [22], p.150, Eq. (3.1.1)], we assume that the x -average entropy in a co-movingvolume element is fixed: s ( T ) (cid:28) d Ω d τ (cid:29) x = (cid:90) π − π s ( T, x ) (cid:12)(cid:12)(cid:12) sin (cid:16) x √ (cid:17)(cid:12)(cid:12)(cid:12) dx = constant , (6.4)where d τ = | dx ∧ dx ∧ dx ∧ dx ∧ dx ∧ dx | . Differentiating Eq.[6.4] with respect to x yields ddx (cid:2) s ( T ( x )) d Ω d τ (cid:3) = dT ( x ) dx s (cid:48) ( T ) d Ω d τ + s ( T ) ddx (cid:2) d Ω d τ (cid:3) = ˙ T ( x ) s (cid:48) ( T ) d Ω d τ + 0 = 0.Assuming that s (cid:48) ( T ) (cid:54) = 0 gives ddx T ( x ) = 0 . (6.5)For this case the universe does not cool during inflation. Reheating [14] is a non-issue.
7. THE PROBLEM OF THE STABILITY OF A MASSIVE COMPLEX SCALARFIELD PROPAGATING ON THE CLASS E BACKGROUND
If a massive complex scalar field ψ satisfies a linear field equation, as does the ψ fieldunder discussion, then a quantum fluctuation δψ of ψ satisfies a linear field equation that14s of the same form. Therefore the question of the possible existence of instabilities ofthe massive complex scalar ψ field that are sourced by the momenta (cid:126)w associated to theextra time dimensions, Issue [2] raised in Section[1], may be investigated by studying thepropagation of the complex scalar field ψ on the background gravitational field, but whichis not coupled to the inflaton field ϕ . In this paper the field equation for this scalar fieldis given in Eq.[2.1]. Substituting the background gravitational field of Eq.[5.1] into Eq.[2.1]and using Eq.[2.3] yields0 = − Ψ (2 , ( x , x ) + Ψ (0 , ( x , x ) + √ (cid:16) √ x (cid:17) Ψ (0 , ( x , x )+ Λ (cid:16) w e √ x − k e − √ x (cid:17) (cid:114) csc (cid:16) √ x (cid:17) Ψ( x , x ) − Λ (cid:20) m + 13 ζ (cid:16) (cid:16) √ x (cid:17)(cid:17)(cid:21) Ψ( x , x ) , (7.1)which is a non-separable linear wave equation. Here (cid:126)k = ( k , k , k ) T , (cid:126)w = ( k , k , k ) T ,Λ k = (cid:126)k · (cid:126)k = k + k + k and Λ w = (cid:126)w · (cid:126)w = k + k + k ; we have also usedthe fact that the Ricci scalar is Λ (cid:16) (cid:16) √ x (cid:17) + 8 (cid:17) for the class E backgroundgravitational field. In Eq.[7.1] we put Ψ( x , x ) = ψ (cid:16) √ x , √ x (cid:17) , and then fornotational simplicity set x = √ x (where − π ≤ x ≤ π ), and t = √ x . Since Λ = for this case, x = x . This yields a wave equation for the massive complex scalar ψ ( t , x )field that is given by0 = − ψ (2 , ( t , x ) + ψ (0 , ( t , x ) + cot ( x ) ψ (0 , ( t , x )+ 12 (cid:16) w e t − k e − t (cid:17) (cid:112) csc ( x ) ψ ( t , x ) − (cid:20) m + 13 ζ (cid:0) ( x ) (cid:1)(cid:21) ψ ( t , x ) . (7.2)Note that12 (cid:0) w e t/ − k e − t/ (cid:1) = k w (cid:18) wk e t/ − kw e − t/ (cid:19) = k w sinh (cid:20) t (cid:16) wk (cid:17)(cid:21) . (7.3)Eq.[7.2] is non-separable. Obviously one cannot write ψ ( t , x ) = T ( t ) X ( x ) then sep-arate functions of x from functions of t . On the other hand, for a separable linear partialdifferential equation one may separate variables, then introduce a separation constant and inprinciple solve the ODE for eigenmodes, subject to given boundary conditions, and deducea dispersion relation that directly relates a physical frequency ω to, say, the momentum15ave vectors ( (cid:126)k, (cid:126)w ). Eq.[7.2] does not admit a simple dispersion relation that relates aphysical-frequency ω to the momentum wave vectors ( (cid:126)k, (cid:126)w ).We study two special cases of Eq.[7.2], the first with ζ = 3 /
10 and the secondwith ζ = 0. The solution for ζ = 3 /
10 is expanded into a series of Fourier modesΣ ∞ n = −∞ f n ( t ) exp( i n x ). The second solution with ζ = 0 is expanded into a series of(two) Legendre polynomial modes, each of the form (cid:80) ∞ n =0 h n ( t ) (cid:104)(cid:113)(cid:0) n + (cid:1) P n (cos x ) (cid:105) .There is an expansion for each of the intervals − π ≤ x ≤ ≤ x ≤ π . The twoLegendre expansions must be matched at x = 0 and x = ± π . ζ = 3 / We take advantage of the coupling of the massive scalar field to the Ricci scalar to simplifythe wave equation. In Eq.[7.2] we first put ψ ( t, x ) = sin − ( x ) F ( t, x ). This yields0 = F (2 , ( t, x ) − F (0 , ( t, x )+ 12 (cid:0) k e − t/ − w e t/ (cid:1) (cid:112) csc ( x ) F ( t, x )+ 112 (cid:0) ζ + 6 m − (cid:1) F ( t, x ) + 112 (10 ζ −
3) csc ( x ) F ( t, x ) . (7.4)In order to eliminate the (10 ζ −
3) csc ( x ) F ( t, x ) term from this equation we henceforthassume in this section that ζ = 3 / F ( t, x ) = Σ ∞ n = −∞ f n ( t ) exp( i n x ) into Eq.[7.4], with ζ = 3 /
10, and then multiplying by π exp( − i m x ) dx and integrating from ( − π, π ) yields0 = d dt f m ( t ) + m eff2 f m ( t ) − (cid:0) w e t/ − k e − t/ (cid:1) ∞ (cid:88) n = −∞ { c m n f n ( t ) } = d dt f m ( t ) + m eff2 f m ( t ) − k w sinh (cid:20) t (cid:16) wk (cid:17)(cid:21) ∞ (cid:88) n = −∞ { c m n f n ( t ) } , (7.5)where the c m ,n are defined in Eq.[7.7] and m eff2 = m m (7.6)As expected, the square of the effective mass m eff of the scalar field receives contributionsfrom the spatial x Fourier modes. The square of the effective mass m eff also receives acontribution ζ = 3 /
10 from the coupling with the Ricci scalar.16he matrix elements c m ,n of (cid:112) csc ( x ) in the Fourier basis are c m ,n = 12 π (cid:90) π − π (cid:112) csc ( x ) exp[ i ( n − m ) x ] dx = √ (cid:0) (cid:1) Γ (cid:0) (cid:1) ( − N Γ ( ) Γ ( − N ) Γ ( + N ) if n − m = 2 N is even0 if n − m is odd . (7.7)We note that the ( − N Γ ( ) Γ ( − N ) Γ ( + N ) are rational numbers, γ N = ( − N Γ (cid:0) (cid:1) Γ (cid:0) − N (cid:1) Γ (cid:0) + N (cid:1) = (cid:26) , , , , , , , , , , , , , , , . . . (cid:27) (7.8)for N = 0 , , , . . . , , . . . . The c m ,n are the matrix elements of a Toeplitz matrix andverify c m ,n = c m − n , = c ,n − m . Accordingly, Eq.[7.5] may be written as0 = d d t f m ( t ) + (cid:26) m eff2 − k w sinh (cid:20) t (cid:16) wk (cid:17)(cid:21) c , (cid:27) f m ( t ) − k w sinh (cid:20) t (cid:16) wk (cid:17)(cid:21) ∞ (cid:88) n = 1 c n , [ f n + m ( t ) + f − n + m ( t ) ] (7.9)or 17 = d d t f m ( t ) + (cid:110) m eff2 − √ Y τ ) (cid:111) f m ( t ) − √ Y τ ) ∞ (cid:88) n = 1 ( − n Γ (cid:0) (cid:1) Γ (cid:0) − n (cid:1) Γ (cid:0) + n (cid:1) [ f n + m ( t ) + f − n + m ( t ) ] , (7.10)where τ = τ ( t ) = 13 t + log (cid:16) wk (cid:17) , (7.11) √ Y = 12 k w √ (cid:0) (cid:1) Γ (cid:0) (cid:1) (7.12)and m eff2 = m m . (7.13)Because Γ ( ) Γ ( − N ) Γ ( + N ) ∝ (cid:16) | N | (cid:17) / as N → ±∞ , the series expansions in Eq.[7.5] andEq.[7.10] exhibit a very long range coupling between modes. In general the infinite sums inEq.[7.5] and Eq.[7.10] will converge if | f n ( t ) | < (cid:16) | n | (cid:17) / as | n | → ∞ .In order to estimate the error of an approximate numerical solution to Eq.[7.10] an explicitapproximate form for (cid:112) csc ( x ) in terms of its Fourier components c m n is needed. We findthat (cid:112) csc ( x ) = ∞ (cid:88) n = −∞ (cid:8) c m n e ˙ ı n x (cid:9) = √ (cid:0) (cid:1) Γ (cid:0) (cid:1) ∞ (cid:88) m = −∞ e ı m x (cid:32) ( − m Γ (cid:0) (cid:1) Γ (cid:0) − m (cid:1) Γ (cid:0) + m (cid:1) (cid:33) = 2 (cid:112) π Γ (cid:0) (cid:1) Γ (cid:0) (cid:1) (cid:18) F (cid:18) ,
1; 56 ; e ix (cid:19) + F (cid:18) ,
1; 56 ; e − ix (cid:19) − (cid:19) (7.14)almost everywhere, except x = 0 , ± π . A plot of the difference (cid:112) csc ( x ) − √ π Γ ( ) Γ ( ) (cid:2) F (cid:0) , ; e ix (cid:1) + F (cid:0) , ; e − ix (cid:1) − (cid:3) is given in Figure[1]. Let n ∈ N , the natural numbers (excluding 0). Given an approximatenumerical solution that contains only a finite number of Fourier modes − n ≤ n ≤ n onemay make the substitution based on Eq.[7.14] (cid:112) csc ( x ) → √ (cid:0) (cid:1) Γ (cid:0) (cid:1) [ n ] (cid:88) m − [ n ] e i (2 m x (cid:16) ( − m Γ (cid:0) (cid:1) (cid:17) Γ (cid:0) − m (cid:1) Γ (cid:0) m (cid:1) (7.15)18hen computing the numerical error. Here, (cid:2) n (cid:3) gives the greatest integer less than or equalto n .We call the first line of Eq.[7.10] D [ f ]( k, w ; m ; t ) = d d t f m ( t ) + (cid:20) m eff2 − (cid:0) w e t/ − k e − t/ (cid:1) c , (cid:21) f m ( t ) (7.16)the diagonal contribution to Eq.[7.10]. Here c , = √ Γ ( ) Γ ( ) . D [ f ]( k, w ; m ; t ) = 0 D [ f ]( k, w ; m ; t ) = 0 possesses simple mathematical solutions for the two special cases k = 0 and w = 0, which provide much insight into this problem, but which are not phys-ical because the long range coupling with the other modes is completely neglected. Thesesolutions are discussed in the next section.Let J n ( z ) denote an ordinary Bessel function of the first type, Y n ( z ) denote an ordinaryBessel function of the second type (Neumann function), I λ ( z ) denote the modified Besselfunction of the first type and order λ , and K λ ( z ) denote a modified Bessel function of thesecond type (also known as the modified Bessel function of the third kind).If k = 0 then the general solution to D [ f ]( k = 0 , w ; m ; t ) = 0 is f m ( t ) = c K ı m eff (cid:0) w e t/ α (cid:1) + c I ı m eff (cid:0) w e t/ α (cid:1) , (7.17)where α = (cid:112) c , = 3 × / (cid:113) Γ (cid:0) (cid:1) Γ (cid:0) (cid:1) (7.18)and the ( c , c ) are arbitrary constants. Asymptotically, for large argument | z | , the modifiedBessel functions behave as K λ ( z ) ∼ e − z (cid:112) π z + · · · and I λ ( z ) ∼ e z (cid:113) πz + · · · [23]. Therefore f m ( t ) is a stable solution as w → ∞ if c = 0. Moreover, K ˙ ı ν ( z ) possesses a well knownintegral representation K ˙ ı ν ( z ) = (cid:90) ∞ cos ( ν ξ ) e − z cosh ( ξ ) d ξ, (7.19)valid for arg z < π , which enables a simple demonstration that f m ( t ) is stable as w → ∞ .19 stable solution to D [ f ]( k = 0 , w ; m ; t ) = 0 may be obtained by choosing an initialcondition for f (cid:48) m ( t ) f m ( t ) so that c vanishes. The required initial condition is f (cid:48) m (0) f m (0) = (cid:40) ∂∂ t K ı m eff (cid:0) w e t/ α (cid:1) K ı m eff ( w e t/ α ) (cid:41) t =0 = (cid:26) W ∂ log( K ı m eff ( W )) ∂W (cid:27) W = w α , (7.20)where α is defined in Eq.[7.18]. To make this solution also well-behaved as w → c = w w C .If one attempts to numerically integrate D [ f ]( k = 0 , w ; m ; t ) = 0 using a numericalquadrature algorithm then at each time step t n of the process one must maintain tightcontrol of the error (cid:26) f (cid:48) m ( t n ) f m ( t n ) (cid:27) numerical − (cid:40) ∂∂ t K ı m eff (cid:0) w e t/ α (cid:1) K ı m eff ( w e t/ α ) (cid:41) t = t n , (7.21)because a non-zero value of this difference will source a contribution from I ı m eff (cid:0) w e t/ α (cid:1) in the numerical solution, which may eventually exponentially overwhelm the true analyticalsolution. This problem is stiff.We note in passing that the behavior of the modified Bessel function of the second kindwith pure imaginary order, K iν ( z ), has been investigated by Balogh [24] and others. Baloghhas proved that K iν ( νp ) is a positive monotone decreasing convex function of p for 0 < p < p >
1, having a countably infinite number of zeros. Baloghgives asymptotic expansions of the zeros of K iν ( z ) and its derivative that are uniform withrespect to the enumeration of the zeros.If w = 0 then the stable solution to D [ f ]( k, w = 0; m ; t ) = 0 is f m ( t ) = c J ı m eff (cid:0) k e − t/ α (cid:1) + c Y ı m eff (cid:0) k e − t/ α (cid:1) , (7.22)where the ( c , c ) are arbitrary constants. 20 .1. Wave equation with ζ = 0 We put ζ = 0 in Eq.[7.2] to obtain the wave equation for a minimally coupled massivecomplex scalar field propagating through the background class E gravitational field:0 = − ψ (2 , ( t , x ) + ψ (0 , ( t , x ) + cot ( x ) ψ (0 , ( t , x ) − m ψ ( t , x ) + 12 (cid:16) w e t − k e − t (cid:17) (cid:112) csc ( x ) ψ ( t , x )= − ψ (2 , ( t , x ) + 1sin ( x ) ∂∂ x (cid:20) sin ( x ) ∂∂ x ψ ( t , x ) (cid:21) − m ψ ( t , x ) + 12 (cid:16) w e t − k e − t (cid:17) (cid:112) csc ( x ) ψ ( t , x ) . (7.23)Recall that − π ≤ x ≤ π and not ≤ x ≤ π . In order to obtain a solution to Eq.[7.23]we solve for ψ ( t , x ) on each of the two intervals I − = − π ≤ x < I + = 0 ≤ x ≤ π and then enforce appropriate continuity and boundary conditions: We assume that ψ ( t , x )and ∂∂ x ψ ( t , x ) are continuous, and that ψ ( t , − π ) = ψ ( t , π ), since x may be regardedas a canonical angular coordinate θ for the boundary of the unit circle. Note that, with X = cos x = cos θ, x = θ = ± arccos X , Eq.[7.23] maps to − ψ (2 , ( t, X ) + (cid:0) − X (cid:1) ψ (0 , ( t, X ) − Xψ (0 , ( t, X ) − m ψ ( t, X )+ 12 (cid:0) w e t/ − k e − t/ (cid:1) (cid:32) (cid:114) − X (cid:33) ψ ( t, X ) = 0 . (7.24)Let U ( x ) denote the unit step function U ( x ) = x < for x = 01 for x > ψ ( t , x ) may be expressed as ψ ( t , x ) = ψ ( t , x ) (cid:99) I − U ( − x ) + ψ ( t , x ) (cid:99) I + U ( x )= ψ − ( t , x ) U ( − x ) + ψ + ( t , x ) U ( x ) , (7.26)and ∂∂ x ψ ( t , x ) is given by ψ (0 , ( t , x ) ≡ ∂∂ x ψ ( t , x )= U ( − x ) ∂∂ x ψ − ( t , x ) + U ( x ) ∂∂ x ψ + ( t , x )+ δ ( x ) [ ψ + ( t , − ψ − ( t , . (7.27)21s stated above, we assume that ψ ( t , x ) and ∂∂ x ψ ( t , x ) are continuous, and that ψ ( t , − π ) = ψ ( t , π ). These conditions imply that ψ ( t , ψ − ( t , ψ + ( t , lim x → + ψ − ( t , x )lim x → − ψ + ( t , x ) , (7.28)whence ψ − ( t , ψ + ( t , , (7.29)and, using Eq.[7.29] in Eq.[7.27], ψ (0 , ( t , (cid:104) ψ (0 , − ( t , ψ (0 , ( t , (cid:105) = lim x → + ψ (0 , − ( t , x )lim x → − ψ (0 , ( t , x ) . (7.30)This implies that ψ (0 , − ( t , ψ (0 , ( t , ψ (0 , ( t , ψ (0 , − ( t , ψ (0 , ( t , x ∈ I − , − π ≤ x < ψ − ( t , x ) = ψ + ( t , − x ) , x ∈ I − . (7.33)Next, let us expand ψ ( t , x ) on each of the two intervals I − = − π ≤ x < I + = 0 ≤ x ≤ π in terms of a series of Legendre polynomials of the form ψ ± ( t , x ) ≡ ψ ( t , x ) (cid:99) I ± = ∞ (cid:88) n =0 h ( ± ) n ( t ) (cid:34)(cid:115)(cid:18) n + 12 (cid:19) P n (cos x ) (cid:35) . (7.34)In virtue of Eq.[7.33], h ( − ) n ( t ) = h (+) n ( t ). For brevity we put h (+) n ( t ) = h n ( t ). Beforesubstituting Eq.[7.34] into Eq.[7.23] and simplifying we record the following definitions.22et N denote the non-negative integers 0 , , , , . . . . Define, ∀ m , n ∈ N , d m n = d m , n = (cid:115)(cid:18) m + 12 (cid:19)(cid:115)(cid:18) n + 12 (cid:19) (cid:90) π P m (cos x ) P n (cos x ) sin ( x ) dx = (cid:115)(cid:18) m + 12 (cid:19)(cid:115)(cid:18) n + 12 (cid:19) (cid:90) − P m ( X ) P n ( X ) 1 √ − X dX = π Γ (cid:18) (cid:19) (cid:115)(cid:18) m + 12 (cid:19) (cid:18) n + 12 (cid:19) m + n (cid:88) j = | n − m | , step 2 (cid:40) (2 j + 1)( − j + m + n Γ (cid:0) − j (cid:1) Γ (cid:0) − j (cid:1) Γ (cid:0) j + 1 (cid:1) Γ (cid:0) j + (cid:1) Γ( − j + m + n + 1)Γ( j − m + n + 1)Γ( j + m − n + 1)Γ( j + m + n + 2) (cid:34) Γ (cid:0) ( j + m + n + 2) (cid:1) Γ (cid:0) ( − j + m + n + 2) (cid:1) Γ (cid:0) ( j − m + n + 2) (cid:1) Γ (cid:0) ( j + m − n + 2) (cid:1) (cid:35) (7.35)(of course, sin ( x ) = (cid:112) csc ( x ) sin x on 0 ≤ x ≤ π ). One observes that d m +1 , n = 0 = d m , n +1 ∀ m , n ∈ N . (7.36)Substituting Eq.[7.34] into Eq.[7.23], multiplying by (cid:104)(cid:113)(cid:0) m + (cid:1) P m (cos x ) (cid:105) sin x dx , m ∈ N , and integrating over the interval 0 ≤ x ≤ π yields¨ h m ( t ) + (cid:20) m + m ( m + 1) (cid:21) h m ( t ) + 12 (cid:16) k e − t − w e t (cid:17) ∞ (cid:88) n =0 d m n h n ( t ) = 0 . (7.37)For this case the effective mass m eff is given by m eff2 = 12 m + m ( m + 1) (7.38)We find that d m n ∝ (cid:16) n (cid:17) / for n >> m as n → ±∞ . the series expansion inEq.[7.37] also exhibits a very long range coupling between modes. The infinite sum inEq.[7.37] will converge if | h n ( t ) | < (cid:16) n (cid:17) / as n → ±∞ .The diagonal contribution to Eq.[7.37] is D [ h ]( k, w ; m ; t ) =¨ h m ( t ) + (cid:2) m eff2 − (cid:0) w e t/ − k e − t/ (cid:1) d m m (cid:21) h m ( t ) , (7.39)23hich has the same form as the diagonal contribution to Eq.[7.10].Setting to zero the diagonal contribution to either Eq.[7.10] or Eq.[7.37] yields an equationof the form − ¨ Y m ( t ) + B (cid:0) w e t/ − k e − t/ (cid:1) Y m ( t ) = m eff2 Y m ( t ) , (7.40)where m eff2 = + m + m , B = c , , for ζ = ,m eff2 = m + m ( m + 1) , B = B ( m ) = d m m , for ζ = 0 (7.41)
8. EXISTENCE OF STABLE SOLUTIONS
We recall the content of Issue [2],
Momenta corresponding to the extra time dimensions in-duce exponentially rapid growth of quantum fluctuations of the field; the universe is unstable.This instability is associated with the very largest momenta (shortest wavelengths) .In this paper we are concerned with exponentially rapid growth of quantum fluctuationsof the field . This causes perturbation theory to break down since | δψ | becomes of order onefaster than some positive power of t . Exponentially rapid growth of the field ψ itself is not aninstability. Instability of the field ψ means the norm | ψ | of the field goes to infinity in finitetime. This is not the issue under discussion. “Stable” means that solutions f m = δψ m ofEq.[7.10] and h m = δψ m of Eq.[7.37] exist that possess a norm that is non-exponentiallyincreasing as w increases. Then if | δψ | is initially small then | δψ | does not become of orderone exponentially with time.A formal argument can be made that demonstrates that stable solutions f m = δψ m toEq.[7.10] and h m = δψ m to Eq.[7.37] always exist. Since both Eq.[7.10] and Eq.[7.37] havesimilar forms, our discussion will focus on Eq.[7.10]; our conclusions will be valid for bothequations.The phrase stable solution means, most importantly, that the norm of the solution isfinite, for fixed time t and k , as w → ∞ [recall that (cid:126)k = ( k , k , k ) T , (cid:126)w = ( k , k , k ) T ,Λ k = (cid:126)k · (cid:126)k = k + k + k and Λ w = (cid:126)w · (cid:126)w = k + k + k ].The argument begins as follows. We affinely transform the physical time t to a new time24oordinate τ that is defined by setting t = 3 τ + 3 log (cid:18) kw (cid:19) (8.1)As we have seen above, under this transformation12 (cid:0) w e t/ − k e − t/ (cid:1) = k w (cid:18) wk e t/ − kw e − t/ (cid:19) = k w sinh (cid:20) t (cid:16) wk (cid:17)(cid:21) = k w sinh ( τ ) . (8.2)In terms of the new time coordinate τ , ψ = ψ ( τ, x ) satisfies the wave equation Eq.[8.10],below. Since the quantum fluctuations of ψ satisfy a field equation similar to Eq.[8.10], welimit our discussion of stability to solutions of Eq.[8.10] and its relatives.The inverse transformation to the map of the time coordinate t defined in Eq.[8.1] is τ = 13 t + log (cid:16) wk (cid:17) . (8.3)For fixed t and k , as w → ∞ then τ → ∞ . For fixed t and k a solution δψ ( τ, x ) iscalled stable if the norm (cid:107) δψ (cid:107) of the solution to the wave equation Eq.[8.10], below, satisfies0 ≤ (cid:107) δψ (cid:107) < ∞ as τ → ∞ . Note that an effective frequency-like parameter (cid:36) , with (cid:36) = 2 √ Y ∝ k w appears in Eq.[8.4] and Eq.[8.10], below. For fixed w , as k → ∞ then (cid:36) → ∞ and for fixed k , as w → ∞ then (cid:36) → ∞ . The dependence of δψ ( τ, x ) on (cid:36) as (cid:36) → ∞ does not enter into our definition of stability. δψ ( τ, x ) oscillates at higherand higher frequencies as (cid:36) → ∞ , but does not become unstable. Both non-resonant andresonant oscillations do not drive | δψ | to order one faster than some positive power t γ of t .Under the change of time coordinate defined in Eq.[8.1], Eq.[7.10] maps to0 = 19 d dτ h n ( τ ) + (cid:110) m eff2 − √ Y τ ) (cid:111) h n ( τ ) − √ Y τ ) ∞ (cid:88) m = 1 ( − m Γ (cid:0) (cid:1) Γ (cid:0) − m (cid:1) Γ (cid:0) + m (cid:1) [ h m + n ( τ ) + h − m + n ( τ ) ] , (8.4)where h n ( τ ) = f n ( t ) = δψ n ( t ), m eff2 = m + + n (see Eq.[7.13]) and √ Y = k w √ Γ ( ) Γ ( ) (see Eq.[7.12]); the independent ‘modes’ are labelled by n . From the formof this equation one sees that only the product k w , and not k and w independently, govern25he behavior of the solutions. Note that Eq.[8.4] has the property that the even modes andodd modes evolve independently of each other (because (cid:112) csc ( x ) is an even function of x ).According to the theory of systems of linear differential equations a solution associatedto Eq.[8.4] exists that is expressible in the form h n ( τ ) = (cid:80) ∞ j = −∞ h n ; j exp (cid:0)(cid:0) −√ a + j (cid:1) τ (cid:1) for n even (cid:80) ∞ j = −∞ h n ; j exp (cid:0)(cid:0) −√ a + j (cid:1) τ (cid:1) for n odd , (8.5)where the h n ; j , a , a are constants to be determined (see, for example, Ref.[25] Equation[7],page 177). We substitute Eq.[8.5] into Eq.[8.4] and then demand that the coefficient of each e j τ , j = −∞ . . . ∞ , vanish, where a = a if n is odd and a = a if n is even. This yieldsthe system of linear equations for the h n ; j given by0 = − h n ; j (cid:20)
320 + m n + 19 (cid:0) j − √ a (cid:1) (cid:21) + √ Y ( h n ; j − − h n ; j +1 )+ √ Y ∞ (cid:88) m =1 ( − m Γ (cid:0) (cid:1) Γ (cid:0) − m (cid:1) Γ (cid:0) + m (cid:1) ( h n − m ; j − − h n − m ; j +1 + h n +2 m ; j − − h n +2 m ; j +1 ) , (8.6)where a = a if n is odd and a = a if n is even.Note that if the h n ; j satisfy the three term recurrence relation √ Y ( h n ; j − − h n ; j +1 ) = β n (cid:20)
320 + m n + 19 (cid:0) j − √ a (cid:1) (cid:21) h n ; j , (8.7)where the β n are also to be determined, then0 = ∞ (cid:88) m = −∞ ( − m Γ (cid:0) (cid:1) Γ (cid:0) − m (cid:1) Γ (cid:0) + m (cid:1) (cid:20)
320 + m n − m ) + 19 (cid:0) j − √ a (cid:1) (cid:21) × ( β n − m − δ m , ) h n − m ; j , (8.8)which is block diagonal in the temporal index j .Except at the irregular singular points x = 0 ± π , both the massive complex scalar wavefunction ψ and its quantum fluctuation δψ are expressible in the form given byΨ( τ, x ) = C Ψ odd ( τ, x ) + C Ψ even ( τ, x )Ψ even ( τ, x ) = ∞ (cid:88) n = −∞ , n even ∞ (cid:88) j = −∞ h n ; j exp (( −√ a + j ) τ + ˙ ı n x )Ψ odd ( τ, x ) = ∞ (cid:88) n = −∞ , n odd ∞ (cid:88) j = −∞ h n ; j exp (( −√ a + j ) τ + ˙ ı n x ) , (8.9)26here the h n ; j satisfy Eq.[8.6]. Of course, the constants C and C are redundant andemployed only for emphasis. The scalar wave function Ψ( τ, x ) satisfies0 = − Ψ (2 , ( τ, x ) + 19 Ψ (0 , ( τ, x )+ (cid:18) m + 320 − √ Y sinh ( τ ) (cid:112) csc ( x ) (cid:19) Ψ( τ, x ) . (8.10)The mass parameter m is in general different for Ψ = ψ and Ψ = δψ .A sequence of approximate solutions to Eq.[8.6] may be defined as follows: Let n ∈ N ,the natural numbers (excluding 0). n will be called the index of the approximation. Foreach positive integer n ∈ N = 1 , , . . . we define an approximate solution to Eq.[8.6] interms of (2 n + 1) mode coefficients ( n ) h n ; j , − n ≤ n , j ≤ n that, in the limit n → ∞ ,will satisfy Eq.[8.6] exactly. Eq.[8.6] defines a system of linear equations for all of the h n ; j mode coefficients. In order to define a finite approximate solution set, we set all of themode coefficients h n ; j = 0 for −∞ < n , j < − n and n < n , j < ∞ . Thisresults in two homogeneous systems of linear equations for the (2 n + 1) mode coefficients ( n ) h n ; j , − n ≤ n , j ≤ n , one system for the even n modes and one system for the odd n modes (since the even n modes and odd n modes evolve independently of each other).To obtain non-trivial solutions to these two systems we solve for the complex values of a and a that make the determinants of their respective coefficient matrices vanish. Twodispersion relations thereby arise: one that relates n and both m and k w , the product ofthe magnitudes of the momentum wave vectors ( (cid:126)k, (cid:126)w ), to the complex parameter a for theodd n modes, and one that relates n and ( m, k w ) to the complex parameter a for theeven n modes.Recall that in this paper “stable” means that solutions Ψ = δψ of Eq.[8.10] possess a normthat is non-exponentially increasing as τ increases. Stable modes exist when − Real( √ a )+ n and − Real( √ a ) + n are both negative, or zero. To see this first define the correspondingapproximate solution ( n ) Ψ( τ, x ) to Eq.[8.10] as ( n ) Ψ odd ( τ, x ) = n (cid:88) n = − n + evn ( n ) , step 2 n (cid:88) j = − n ( n ) h n ; j e [( −√ a + j ) τ + ˙ ı n x ]( n ) Ψ even ( τ, x ) = n (cid:88) n = − n + odd ( n ) , step 2 n (cid:88) j = − n ( n ) h n ; j e [( −√ a + j ) τ + ˙ ı n x ] . (8.11)27ere we have defined two simple functions on N , which for n ∈ N , are given by evn ( n ) = n is even0 if n is odd , odd ( n ) = n is even1 if n is odd . (8.12)One sees that, of all of the (2 n + 1) wave modes, each of the form ( n ) h n ; j e [ ( −√ a + j ) τ + ˙ ı n x ], the j = n waves increase at the greatest rate for given a . If − Real( √ a ) + n and − Real( √ a ) + n are both negative, or zero, then the j = n waves havenon-increasing norms as time τ increases (either because w increases, t increases or k de-creases), so that the norm of ( n ) Ψ odd ( τ, x ) and ( n ) Ψ even ( τ, x ) also have non-increasing normsas time τ increases (here Ψ = δψ ). These stable modes are associated to so-called “stablezeros” ( a , a ) of the determinants of the coefficient matrices for the odd n labeled modesand the even n labeled modes, respectively. The “unstable zeros” ( a , a ) of the determi-nants of the coefficient matrices produce instabilities driven by the momenta associated tothe three extra time dimensions, and do not yield physical solutions of the field equations.In passing we remark that the error of an approximation may be provisionally defined tobe error[ ( n ) Ψ( τ, x )] = − ( n ) Ψ (2 , ( τ, x ) + 19 ( n ) Ψ (0 , ( τ, x )+ (cid:18) m + 320 − √ Y sinh( τ ) (cid:112) csc ( x ) (cid:19) ( n ) Ψ( τ, x ) . (8.13)However since ( n ) Ψ( τ, x ) contains only a finite number of Fourier modes it is reasonable tomake a substitution based on Eq.[7.15], (cid:112) csc ( x ) → √ ( ) Γ ( ) (cid:80) [ n ] m − [ n ] e i (2 m x (cid:16) ( − m Γ ( ) (cid:17) Γ ( − m ) Γ ( m ) ,in Eq.[8.13] when computing the numerical error ( (cid:2) n (cid:3) gives the greatest integer less than orequal to n ). Since the error is expected to be large for small n we defer error analysis.
9. COEFFICIENT MATRICES FOR THE ( n ) h n ; j MODE COEFFICIENTS INEQ.[8.6]
Let M = 12 m + 320 , (9.1)and recall that √ Y = k w √ Γ ( ) Γ ( ) . Fix n ∈ N ; we group the (2 n + 1) = n (2 n + 1)+( n + 1) (2 n + 1) mode coefficients ( n ) h n ; j , − n ≤ n , j ≤ n , into two vectors −→ h even and28 → h odd according to whether n is even or odd. When n is an odd positive integer then −→ h odd has dimension ( n + 1) (2 n + 1) = ( n + odd ( n ) ) (2 n + 1) and −→ h even has dimension n (2 n + 1) = ( n + evn ( n ) ) (2 n + 1). When n is an even positive integer then −→ h odd hasdimension n (2 n + 1) = ( n + odd ( n ) ) (2 n + 1) and −→ h even has dimension ( n + 1) (2 n + 1) =( n + evn ( n ) ) (2 n + 1). Note that −→ h odd always has an even number of components and that −→ h even always has an odd number of components.Explicitly, the i th component of the vector −→ h associated to ( n ) h n ; j has index i = 12 (2 n + 1) ( n + n − evn ( n ) odd ( n ) − odd ( n ) evn ( n ) ) + j + n + 1 (9.2)so that h even (cid:18)
12 (2 n + 1) ( n + n − odd ( n ) ) + j + n + 1 (cid:19) = ( n ) h n ; j , where n is even (9.3)and h odd (cid:18)
12 (2 n + 1) ( n + n − evn ( n ) ) + j + n + 1 (cid:19) = ( n ) h n ; j , where n is odd . (9.4)Here h ( i ) = (cid:110) −→ h (cid:111) i = i th component of −→ h .The inverse map ( n ; i, parity ) (cid:55)→ ( n , j ) is defined as follows: Let N (cid:51) parity = −→ h odd −→ h even .The inverse map ( n ; i, parity) (cid:55)→ ( n , j ) is given by j = ( i −
1) mod (2 n + 1) − n and n = 2 (cid:2) i − n +1 (cid:3) − n + odd ( parity ) evn ( n ) + evn ( parity ) odd ( n ), where (cid:2) i − n +1 (cid:3) gives the greatestinteger less than or equal to i − n +1 .Let µ denote the minimum value of n for either −→ h even or −→ h odd (thus, µ is either equalto − n or − n + 1). The components of these two vectors are ordered according to (( n = µ, j = − n ) , ( n = µ, j = − n + 1) , . . . , ( n = µ, j = n ) , ( n = µ + 2 , j = − n ) , ( n = µ + 2 , j = − n + 1) , . . . , ( n = µ + 2 , j = n ) , . . . , ( n = − µ, j = − n ) , ( n = − µ, j = − n + 1) , . . . , ( n = − µ, j = n )).With respect the obvious canonical bases each coefficient matrix is a square matrix andis equal to the sum D + A of a diagonal matrix D , which carries the a and M dependence,and an antisymmetric matrix √ Y A , where (cid:101) A = − A (the tilde denotes the transpose). Fora given n ∈ N , the coefficient matrix ( n ) D even + ( n ) A even corresponding to the even n modecomponents has dimensions equal to [( n + even ( n )) (2 n + 1)] × [( n + even ( n )) (2 n + 1)],29nd thus always has an odd number of rows and columns. The coefficient matrix ( n ) D odd + ( n ) A odd corresponding to the odd n mode components has dimensions equal to[( n + odd ( n )) (2 n + 1)] × [( n + odd ( n )) (2 n + 1)], and thus always has an even number ofrows and columns.The diagonal matrix ( n ) D has non-zero matrix elements − (cid:16) M + n + ( j − √ a) (cid:17) ,arranged (as described above) along the main diagonal in blocks, which are labelled by n ,of the 2 n + 1 possible j values.Because ( n ) A even always has an odd number of rows and columns it possesses a vanishingdeterminant, det (cid:0) ( n ) A even (cid:1) ≡
0. The antisymmetric ( n ) A odd corresponding to the odd n mode components always has an even number of rows and columns, and hence a possiblynon-vanishing determinant. However for this case the rows ( n ) A odd ( i ) , i = 1 , , . . . , [ n + odd ( n )] × (2 n + 1) of ( n ) A odd are not linearly independent andsatisfy ( n ) A odd (1) + ( n ) A odd (3) + · · · + ( n ) A odd (2 n + 1) = ( n + odd ( n )) (2 n +1) (cid:122) (cid:125)(cid:124) (cid:123) (0 , , . . . , . (9.5)Hence for this case A odd also has a vanishing determinant, det (cid:0) ( n ) A odd (cid:1) = 0.The antisymmetric matrix ( n ) A is related to an antisymmetric Toeplitz matrix ( n ) T = − (cid:94) ( n ) T whose non-zero upper triangular matrix elements are given by ( n ) T i ,i + (cid:96) (2 n +1)+1 = − γ (cid:96) , (cid:96) = 0 , , . . . , and ( n ) T i ,i + (cid:96) (2 n +1) − = γ (cid:96) , (cid:96) = 1 , . . . , where the γ (cid:96) are defined in Eq.[7.8] and i + (cid:96) (2 n + 1) + 1 ≤ [( n + odd ( n )) (2 n + 1)] for ( n ) T corresponding to the ( n ) A odd associated to the odd n modecomponents and i + (cid:96) (2 n + 1) + 1 ≤ [( n + evn ( n )) (2 n + 1)] for ( n ) T corresponding to the ( n ) A even associatedto the even n mode components. The remaining matrix elements of ( n ) T are either dictatedby antisymmetry, or are zero.The matrix elements of ( n ) A equal the matrix elements of ( n ) T except that they arepunctuated by additional zeros along the super diagonals of ( n ) T . ( n ) A even , n ≥ , has N additional zeros = N AZ = 2 (cid:0) [ n + evn ( n )] − (cid:1) replacements of non-zero matrix elements of ( n ) T by zero along its super diagonals. ( n ) A odd , n ≥ , has N AZ = 2 (cid:0) [ n + odd ( n )] − (cid:1) zeroreplacements of non-zero matrix elements of ( n ) T by zero along its super diagonals.The specification of the row and column indices ( r, c ) of a particular matrix element ( n ) A r c of ( n ) A that is zero, when ( n ) T r c is not zero, is somewhat involved, but may be30escribed in stages. Let the dimensions of ( n ) A be L × L . Define a mapping of the two-dimensional array ( n ) A r c , r, c = 1 , . . . , L to a one-dimensional array ( n ) B i , i = 1 , . . . , L by setting ( n ) B i = ( n ) B ( r − L + c = ( n ) A r c . Formally, i = ( r − L + c . Theinverse map ( n ; i, parity) (cid:55)→ ( r, c ) is given by r = ( i −
1) mod ( L ) + 1 and c = (cid:2) i − L (cid:3) + 1,where L = (2 n + 1) ( n + evn ( parity ) evn ( n ) + odd ( parity ) odd ( n )).Assume that the row and column indices { ( r, c ) j } ] N AZ j =1 of the N AZ matrix elements whosenon-zero ( n ) T r j c j values are to be replaced by zero are known, and that their corresponding { i j } ] N AZ j =1 values have been computed; sort the { i j } values in ascending order. Define therelative displacement coordinates (cid:8) I RDC j (cid:9)(cid:3) N AZ j =1 by I RDC1 = i , I RDC2 = i − i , . . . , I RDC j = i j − i j − , j = 2 , . . . , N AZ . The inverse map is simply i j = (cid:80) jh =1 I RDC h .We find that the (cid:8) I RDC j (cid:9)(cid:3) N AZ j =1 are given by two sequences, one for the odd n modecomponents and one for the even n mode components. Let N j = 2 n + 1 denote thenumber of j -values in the n th -order approximation. Let { N j } [ h ] denote the finite sequence { N j } [ h ] = h (cid:122) (cid:125)(cid:124) (cid:123) { N j , N j , . . . , N j } with h elements, and let { S } [ h ] denote a finite sequence repeated h times.For the case of the odd n mode components let N = n − evn ( n ) and α = (2 n (2 n (2 n − ( − n + 1) −
1) + ( − n + 1) ; the sequence (cid:8) I RDC j (cid:9)(cid:3) N AZ j =1 for the odd n mode components is (cid:8) I RDCodd j (cid:9)(cid:3) N AZ j =1 = {{ N j } [ N ] , α + N j , {{ N j } [ N ] , n, { N j } [ N ] , α } [ N − , { N j } [ N ] , α + N j , { N j } [ N ] } . (9.6)For the case of the even n mode components let N = n − odd ( n ) and β = (2 n (2 n (2 n + ( − n + 1) − − ( − n + 1) ; the sequence (cid:8) I RDC j (cid:9)(cid:3) N AZ j =1 for the even n mode components is (cid:8) I RDCeven j (cid:9)(cid:3) N AZ j =1 = {{ N j } [ N ] , β + N j , {{ N j } [ N ] , n, { N j } [ N ] , β } [ N − , { N j } [ N ] , β + N j , { N j } [ N ] } . (9.7)Only the first N AZ sequence elements are to be employed when evaluating the the relativedisplacement coordinates from the previous two sequence formulas.31
0. THEOREM: EXISTENCE OF STABLE SOLUTIONS
We record the
Theorem 10.1.
Let n ∈ N be the index of an approximation. Define an approximatesolution as described above to Eq.[8.6] in terms of (2 n + 1) mode coefficients ( n ) h n ; j , − n ≤ n , j ≤ n .Let ( n ) D n odd ( a , Y, M ) denote the determinant of the coefficient matrix of the ( n ) h n ; j forthe odd n labeled modes and ( n ) D n even ( a , Y, M ) denote the determinant of the coefficientmatrix of the ( n ) h n ; j for the even n labeled modes.Let n a denote the number of stable zeros of ( n ) D n odd ( a , Y, M ) and n a denote thenumber of stable zeros of ( n ) D n even ( a , Y, M ) .Then1. ( n ) D n odd ( a , Y, M ) is a trivariate polynomial in ( a , Y, M ) with rational coefficients;2. ( n ) D n even ( a , Y, M ) is a trivariate polynomial in ( a , Y, M ) with rational coefficients;3. For each positive integer n ∈ N = 1 , , . . . , stable zeros ( a , a ) of the the determinantsof the coefficient matrices ( n ) D n odd ( a , Y, M ) and ( n ) D n even ( a , Y, M ) exist that sat-isfy − Real ( √ a ) + n ≤ and − Real ( √ a ) + n ≤ ;4. if n is an odd positive integer then(a) ( n ) D n odd ( a , Y, M ) is of degree (n+1)(2 n+1) in a and M , and of degree n(n+1)in Y(b) ( n ) D n even ( a , Y, M ) is of degree n(2 n+1) in a and M , and of degree n in Y • If Y = 1 theni. n a = 2 n+2ii. n a = 2 n • If Y (cid:54) = 1 theni. n a ≥ n + 2 ii. n a ≥ n
5. if n is an even positive integer then a) ( n ) D n odd ( a , Y, M ) is of degree n(2 n+1) in a and M , and of degree n in Y(b) ( n ) D n even ( a , Y, M ) is of degree (n+1)(2 n+1) in a and M , and of degreen(n+1) in Y • If Y = 1 theni. n a = 2 nii. n a = 2 n+2 • If Y (cid:54) = 1 theni. n a ≥ n ii. n a ≥ n + 2 This relationship holds in the limit n → ∞ yielding exact stable solutions to Eq.[8.6], andtherefore to Eq.[8.10]. Partial Proof of Theorem[10.1]: For n = 1 , . . . , | D + B | of the sum of two n × n matrices D and B for the special case where one of the matrices is diagonal. We quote Theorem 13.7.3.[26]: Let B represent an n × n matrix, and let D represent an n × n diagonal matrix whosediagonal elements are d , . . . , d n . Then, | D + B | = (cid:88) { i ,i ,...,i r } d i · d i · · · · d i r | B { i ,i ,...,i r } | . (10.1)where { i , i , . . . , i r } is a subset of the first n positive integers 1 , . . . , n (and the summationis over all 2 n such subsets) and where B { i ,i ,...,i r } is the ( n − r ) × ( n − r ) principal submatrixof B obtained by striking out the i , i , . . . , i r th rows and columns. [The term in the sum[10.1] corresponding to the empty set is to be interpreted as | B | , and the term correspondingto the set { , , . . . , n } is to be interpreted as | D | = d · d · · · · d n .]In our case the matrix elements of D and B are defined in Section [9]. B = √ Y A isantisymmetric. Note that all principal submatrices B { i ,i ,...,i r } of the antisymmetric matrix B = √ Y A are also antisymmetric.The general proof of this theorem is a work in progress. Thus, for all other cases thanthose explicitly noted, the Theorem devolves to a Conjecture.33ote that according to Theorem[10.1], for given n , the degrees of the { a , a } equal thenumbers of independent symmetric and anti-symmetric matrix elements of a (2 n + 1) × (2 n + 1) real matrix. The sum of the degrees of the ( a , a ) equals (2 n + 1) .Table [10.2] summarizes the degrees of the polynomials ( n ) D n odd ( a , Y, M ) and ( n ) D n even ( a , Y, M ) found by direct calculation for n = 1 , . . . ,
9. The degree of M = m + is found to always equal the degree of a for ( n ) D n odd ( a , Y, M ) and always equalthe degree of a for ( n ) D n even ( a , Y, M ), and therefore is not displayed in this table. ( n ) D n odd ( a , Y, M ) ( n ) D n even ( a , Y, M ) a α Y β a γ Y δ for Y=1 (cid:72)(cid:72)(cid:72)(cid:72)(cid:72)(cid:72)(cid:72)(cid:72)(cid:72) n Degree α β γ δ n a n a n n (n+1) (2 n+1) n (n+1) 2 n 2 n+2 (10.2)The significance of this theorem is that each pair of “stable” zeros ( a , a ) defines a pairof stable propagation modes for a quantum fluctuation δψ . The numbers of these modesincreases with n as n → ∞ yielding a possibly complete set of decaying quasi-normal modesthat one may employ in cosmological perturbation calculations. Quantitative calculations ofthe quantum fluctuations that occur during inflation are deferred until we have an analyticalapproximation for a complete set of decaying quasi-normal mode functions for the massivecomplex scalar field. 34 Y = 1 , m = 0 The results of an example for n = 6, m = 0 and √ Y = k w √ Γ ( ) Γ ( ) = 1 are displayedin Figures [2] and [3], which present pairs of plots ( τ dependence, for fixed x ) and ( x dependence, for fixed τ ) of the stable scalar wave functions (6) ψ ( τ, x ) of Eq.[8.11]. For n = 6there are 169 = (2 n +1) mode coefficients, 78 with odd n and 91 with even n . Roughly 352digits of precision in the solved-for values of a and a are required to make the determinantsof the coefficient matrices vanish to double precision accuracy. The relevant values (roundedto 9 digits) of a and a for each plot are listed in the yellow box associated to the plot;space limitations do not permit their display in a table (if one is reading the pdf versionof the manuscript then zooming-in may be required, unless you have remarkable eyesight).Red curves correspond to the real part of the solutions and blue curves and correspond tothe imaginary part.The results of an example for n = 7, m = 0 and √ Y = k w √ Γ ( ) Γ ( ) = 1 are displayedin Figures [4] and [5], which also present pairs of plots ( τ dependence, for fixed x ) and ( x dependence, for fixed τ ) of the stable scalar wave functions (7) ψ ( τ, x ) of Eq.[8.11]. For n = 7there are 225 = (2 n + 1) mode coefficients, 120 with odd n and 105 with even n . Inthis case, about 400 digits of precision in the solved-for values of a and a are required tomake the determinants of the coefficient matrices vanish to double precision accuracy. Therelevant values of a and a for a plot are again listed in the yellow box (zooming-in mayalso be required, unless you have remarkable eyesight). Red curves correspond to the realpart of the solutions and blue curves and correspond to the imaginary part.The results of an example for n = 8, m = 0 and √ Y = k w √ Γ ( ) Γ ( ) = 1 are displayedin Figures [6] and [7], which present pairs of plots ( τ dependence, for fixed x ) and ( x dependence, for fixed τ ) of the stable scalar wave functions (8) ψ ( τ, x ) of Eq.[8.11]. For n = 8there are 361 = (2 n + 1) mode coefficients, 190 with odd n and 171 with even n . Inthis case, about 400 digits of precision in the solved-for values of a and a are required tomake the determinants of the coefficient matrices vanish to double precision accuracy. Therelevant values of a and a for a plot are again listed in the yellow box (zooming-in mayalso be required, unless you have remarkable eyesight). Red curves correspond to the realpart of the solutions and blue curves and correspond to the imaginary part.Qualitatively, corresponding n = 6, n = 7 and n = 8 plots agree, although there are35uantitative differences due to the low orders of approximation. The overall number ofmodes increases by four for each increment in n . In the limit n → ∞ it is reasonable tohope that this procedure generates a “complete” set of stable modes, in the sense that any stable wavefunction can be expressed as a linear combination of these modes.We arrive at a picture of stable quasi-normal modes that decay during inflation. Accord-ing to Theorem[10.1] this model possesses an infinite number of stable quasi-normal modes,such that the momenta associated with the extra time dimensions do not create instability.
11. CLASS E SOLUTIONS OF THE EINSTEIN EQUATIONS
The field equations admit a second class of solutions that is periodic in x and param-eterized by ξ , − √ < ξ < √ , and which has a time dependent (cid:96) . In this case the scalefactors are a = a (cid:34) cosh (cid:32) x (cid:114) Λ3 (cid:33)(cid:35) (1+5 ξ ) × (cid:34) tan (cid:32) (cid:114) √ Λ x (cid:33)(cid:35) ± √ √ − ξ (cid:34) sin (cid:32)(cid:114) √ Λ x (cid:33)(cid:35) (1 − ξ ) b = b (cid:34) cosh (cid:32) x (cid:114) Λ3 (cid:33)(cid:35) (1 − ξ ) × (cid:34) tan (cid:32) (cid:114) √ Λ x (cid:33)(cid:35) ∓ √ √ − ξ (cid:34) sin (cid:32)(cid:114) √ Λ x (cid:33)(cid:35) (1+ ξ ) , (11.1)where a and b are constants. Periodicity in x requires that Λ be quantized,Λ = Λ = 35 = 65 Λ (11.2)We also find that ϕ = (cid:112) − ξ ln (cid:104) cosh (cid:16) √ Λ x √ (cid:17) csc (cid:16)(cid:113) √ Λ x (cid:17)(cid:105) √ ∓ ξ ln (cid:34) tan (cid:32) (cid:114) √ Λ x (cid:33)(cid:35) (cid:96) = ∓ √ Λ ξ tanh (cid:16) √ Λ x √ (cid:17) √ ± ξ (cid:114) (cid:96) tanh (cid:32) x (cid:114) Λ3 (cid:33) . (11.3)36o exclude collapsing universe solutions, which correspond to ∂∂x ln ( a ) <
0, and solutionsfor universes with multiple macroscopic times, which correspond to ∂∂x ln ( b ) ≥
0, from theset of physical solutions we restrict ξ to the interval < ξ < √ .For this case the volume element d Ω = dτ cosh (cid:16) x (cid:113) Λ3 (cid:17) (cid:12)(cid:12)(cid:12) sin (cid:16) x (cid:113) Λ (cid:17)(cid:12)(cid:12)(cid:12) , is independentof both ξ and the choice of ± signs. The temperature history during inflation is non-trivialin this case. This and the possible stability of quantum fluctuations are of interest, but notinvestigated further in this paper.
12. CONCLUSION
We have presented a model of the very early universe that possesses equal numbers ofspace and time dimensions. In this model, after “inflation” the observable physical macro-scopic world appears to a classical observer to be a homogeneous, isotropic universe withthree space dimensions and one time dimension. Notwithstanding the often-expressed con-cern that the momenta associated to extra time dimensions source destabilizing quantumfluctuations, we have shown that the physical solutions δψ to Eq.[8.10] that propagate on theclass E solution to the coupled Einstein-inflaton equations possess decaying quasi-normalmodes, and have no instabilities that are sourced by the momenta (cid:126)w associated to the extratime dimensions.This model and the class E solution to the coupled Einstein-inflaton field equationsprovide simple answers to at least three of the important questions about the early universe,namely, how to the define the inflaton potential [27], how to the define the inflaton massand how to resolve the issue of reheating [14]. At the classical level the answers to thefirst two questions are zero inflaton potential and zero inflaton mass. Computable quantumfluctuations may shift the classical V inflaton = 0 value and the classical m inflaton = 0 value.How these values behave under renormalization are important questions, but beyond thescope of the present paper. The answer to the last question is that there is no obvious needfor reheating, since inflation/deflation is isothermal in this model for the class E solution(the ground state solution to the Einstein-inflaton field equations).We have also proved that a well known so-called “single time” theorem [20] does notapply to our model.Do closed timelike curves exist in this model? If we assume that an “arrow of time” exists37or each timelike dimension then this model does not admit unphysical closed timelike curves.The reason is simply that the tangent vector to any such curve is not everywhere future-directed, if it is timelike. Let x α = x α ( ξ ) , ≤ ξ ≤
1, with x α (0) = x α (1), parameterizea timelike closed curve in X , ; then ∂∂ ξ x α ( ξ ) for 0 ≤ ξ ≤ x , x , x , x ) 4-manifold that traverses the x dimension is not physical classically becausesuch a curve must move both forward and backward in physical time x .If we do not assume that an “arrow of time” exists for each extra timelike dimensionthen, if deflation for the scale factor associated to the extra time dimensions proceeded forsufficiently long ∆ x interval(s), then observationally the effects of closed timelike curvesmight be described as a difficulty in assigning classical spacetime coordinates to certainevents. A discussion of this is beyond the scope of this paper.In this model as presently formulated, inflation also goes on forever. This “feature”obviously must be eliminated from the model. This model must be generalized in a waythat leads to inflaton decay and also terminates inflation. Possible generalizations include:[1] modifing the inflaton Lagrangian to include a self-coupling; and [2] coupling the inflatonto a Higgs doublet (before the Higgs develops a non-zero vacuum expectation value, in orderto keep the inflaton massless) through a Yukawa interaction, while also generalizing the totalLagrangian in the model to incorporate Standard Model quarks, leptons and gauge bosons. Typically and approximately, inflation scenarios inflate a scale of the size of one billionththe present radius of a proton to the size of the present radius of a marble or a grapefruitin about 10 − seconds. In virtue of the Heisenberg Uncertainty Principle, and becausethe comoving ( x , x , x ) dimensions have undergone inflation while the x dimension hasnot, present epoch quantum fields that are functions of ( x , x , x , x , x ) are expected toalmost uniformly sample the region of the x dimension that they occupy. The spatial x average of functions of x are expected to appear in effective four dimensional spacetimetheories. The fact that the scale factors vanish on a set of measure zero may be handled ina straightforward manner by employing spatial x averages in physical calculations.38 ppendix A: Solution of the diagonal contribution to the mode equations (neglect-ing mode coupling) For both the ζ = 3 /
10 case and the second case with ζ = 0, one may solve the diagonalmode equations obtained from Eq.[7.10] and Eq.[7.37] using Mathieu functions. Howevercomplex manipulations involving such mode functions is limited by the state of Mathieufunction science. Instead, for both cases, to solve the uncoupled mode function equationswe make use of the following identity, which is easily verified using the recursion relationsfor the Bessel functions. First, let us define several quantities.Let m eff2 > , B > , ν, α and β be given constants, and n ∈ Z ;let J n = J n (cid:16) k √ B exp (cid:0) − t (cid:1)(cid:17) , K µ = K µ (cid:16) w √ B exp (cid:0) + t (cid:1)(cid:17) and I µ = I µ (cid:16) w √ B exp (cid:0) + t (cid:1)(cid:17) . Then using the recursion relations for the Bessel functions [23] itis straightforward to show that0 = ∂ ∂t [( αK n +˙ ı ν + βI n +˙ ı ν ) J n ] + (cid:20) m eff2 − B (cid:0) w e t/ − k e − t/ (cid:1)(cid:21) [( αK n +˙ ı ν + βI n +˙ ı ν ) J n ] − (cid:34) m eff2 + 19 (cid:18) n + ˙ ı ν (cid:19) (cid:35) [( αK n +˙ ı ν + βI n +˙ ı ν ) J n ]+ 12 k w B ( − αK n − ı ν + βI n − ı ν ) J n − + 12 k w B ( αK n +1+˙ ı ν − βI n +1+˙ ı ν ) J n +1 . (A.1)Employing this identity to solve the uncoupled mode function equations has its roots ina closely related technique due to Dougall [25], Section[15], pages 191-193, with appropriatemodifications that account for the asymmetric “potential” (cid:0) w e t/ − k e − t/ (cid:1) in thisproblem. Dougall gives the solution of Mathieu’s modified differential equation as a seriesof products of Bessel functions.The general solution to Eq.[7.40],0 = ¨ Y m ( t ) + (cid:20) m eff2 − B (cid:0) w e t/ − k e − t/ (cid:1) (cid:21) Y m ( t ) , may be obtained by introducing coefficient sets { c n , d n } , n = −∞ , . . . , ∞ and then setting α = ( − ˙ ı ) n c n and β = (˙ ı ) n d n in Eq.[A.1]. Next we sum over n from n = −∞ , . . . , ∞ and39hen re-label indices so that we may identify coefficients of (cid:80) [( αK n +˙ ı ν + βI n +˙ ı ν ) J n ] plusa remainder, which we require to vanish. This generates a three term recursion relation. Wefind that a general solution to Eq.[7.40] is given by Y m ( t ) = ∞ (cid:88) n = −∞ (cid:110) ˙ ı n J n (cid:16) e − t/ k √ B (cid:17) (cid:104) d n I n +˙ ıν (cid:16) e t/ w √ B (cid:17) + ( − n c n K n +˙ ıν (cid:16) e t/ w √ B (cid:17)(cid:105)(cid:111) . (A.2)The coefficient sets { c n , d n } , n = −∞ , . . . , ∞ are solutions of the same recurrencerelation 12 ˙ ı k w B ( C n − + C n +1 ) − (cid:34) m eff2 + 19 (cid:18) n + ˙ ı ν (cid:19) (cid:35) C n = 0 , (A.3)but with possibly distinct initial values, since a general solution to the three-term recurrencerelation Eq.[A.3] possesses two arbitrary constants.Eq.[A.2] may find application in calculating initial values in a numerical simulation, or inthe computation of the approximate cross section for the creation of particle/anti-particlepairs of ψ particles through the annihilation of ϕ quanta, if one generalizes this model toinclude an interaction of ψ with the inflaton ϕ of the form λ ψ ∗ ψ ϕ .40
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Rev. Mod. Phys. , 69:373–410, Apr 1997. (cid:45) (cid:45) (cid:45) (cid:180) (cid:45) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) csc (cid:72) x (cid:76) (cid:45) Π (cid:74) F (cid:74) , 1; ; (cid:227) (cid:45) (cid:228) x (cid:78) (cid:43) F (cid:74) , 1; ; (cid:227) (cid:228) x (cid:78) (cid:45) (cid:78) (cid:71) (cid:74) (cid:78) (cid:71) (cid:74) (cid:78) FIG. 1: Difference of (cid:112) csc ( x ) and its realization in terms of a Fourier series that may be evaluatedin closed form using hypergeometric functions. [Color online] T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) T (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) T (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) T (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61) (cid:45) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61) (cid:45) (cid:61) (cid:45) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61) (cid:43) (cid:61) (cid:43) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61) (cid:45) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61) (cid:45) (cid:61) (cid:45) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61) (cid:43) (cid:61) (cid:43)
FIG. 2: Stable approximate modes FOR ODD n ; pairs of plots ( τ dependence, for fixed x and x dependence, for fixed τ ). Total number of modes = 169 = 91 EVEN + 78 ODD n. n = 6 , m =0 , kw √ ( ) Γ ( ) = 2 √ Y = 2. The rows and columns are delimited by distinct values of a . [ a corresponds to odd n ; a corresponds to even n ]. [Color online] T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61) (cid:45) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61) (cid:45) (cid:61) (cid:45) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61) (cid:43) (cid:61) (cid:43) T (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61) (cid:45) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61) (cid:45) (cid:61) (cid:45) T (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61) (cid:43) (cid:61) (cid:43) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61) (cid:45) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61) (cid:45) (cid:61) (cid:45) T (cid:45) (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45)
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FIG. 3: Stable approximate modes FOR EVEN n ; pairs of plots ( τ dependence, for fixed x and x dependence, for fixed τ ). Total number of modes = 169 = 91 EVEN + 78 ODD n. n = 6 , m =0 , kw √ ( ) Γ ( ) = 2 √ Y = 2. The rows and columns are delimited by distinct values of a . [ a corresponds to odd n ; a corresponds to even n ]. [Color online] T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) T (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) T (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) T (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61) (cid:45) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61) (cid:45) (cid:61) (cid:45) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61) (cid:43) (cid:61) (cid:43) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61) (cid:45) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61) (cid:45) (cid:61) (cid:45) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45)
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FIG. 4: Stable approximate modes FOR ODD n ; pairs of plots ( τ dependence, for fixed x and x dependence, for fixed τ ). Total number of modes = 225 = 105 EVEN + 120 ODD n. n =7 , m = 0 , kw √ ( ) Γ ( ) = 2 √ Y = 2. The rows and columns are delimited by distinct values of a . [ a corresponds to odd n ; a corresponds to even n ]. [Color online] T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) x (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45) (cid:45) x (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61) (cid:45) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61) (cid:45) (cid:61) (cid:45) T (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61) (cid:43) (cid:61) (cid:43) T (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61) (cid:45) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61) (cid:45) (cid:61) (cid:45) T (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61) (cid:43) (cid:61) (cid:43) T (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61) (cid:45) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61) (cid:45) (cid:61) (cid:45) T (cid:45) (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45)
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FIG. 5: Stable approximate modes FOR EVEN n ; pairs of plots ( τ dependence, for fixed x and x dependence, for fixed τ ). Total number of modes = 225 = 105 EVEN + 120 ODD n. n = 7 , m = 0 , kw √ ( ) Γ ( ) = 2 √ Y = 2. The rows and columns are delimited by distinct values of a . [ a corresponds to odd n ; a corresponds to even n ]. [Color online] T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) T (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61) (cid:45) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61) (cid:45) (cid:61) (cid:45) T (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61) (cid:43) (cid:61) (cid:43) T (cid:45) (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61) (cid:45) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61) (cid:45) (cid:61) (cid:45) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45)
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FIG. 6: Stable approximate modes FOR ODD n ; pairs of plots ( τ dependence, for fixed x and x dependence, for fixed τ ). Total number of modes = 289 = 153 EVEN + 136 ODD n. n =8 , m = 0 , kw √ ( ) Γ ( ) = 2 √ Y = 2. The rows and columns are delimited by distinct values of a . [ a corresponds to odd n ; a corresponds to even n ]. [Color online] T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) x (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45) (cid:45) x (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) x (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45) (cid:45) x (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) x (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45) (cid:45) x (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:45) (cid:61) (cid:45) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61)(cid:45) (cid:43) (cid:61) (cid:43) T (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61) (cid:45) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) x (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61) (cid:45) (cid:61) (cid:45) T (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45) (cid:45) x (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61) (cid:43) (cid:61) (cid:43) T (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61) (cid:45) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61) (cid:45) (cid:61) (cid:45) T (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61) (cid:43) (cid:61) (cid:43) T (cid:45) (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61) (cid:45) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61) (cid:45) (cid:61) (cid:45) T (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;x (cid:61) (cid:61) (cid:61) (cid:43) (cid:61) (cid:43) (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) (cid:45) (cid:45) Ψ n (cid:61) Ψ MODES;T (cid:61) (cid:61) (cid:61) (cid:43) (cid:61) (cid:43)
FIG. 7: Stable approximate modes FOR EVEN n ; pairs of plots ( τ dependence, for fixed x and x dependence, for fixed τ ). Total number of modes = 289 = 153 EVEN + 136 ODD n. n = 8 , m = 0 , kw √ ( ) Γ ( ) = 2 √ Y = 2. The rows and columns are delimited by distinct values of a . [ a corresponds to odd n ; a corresponds to even n ]. [Color online]]. [Color online]