In recent papers it has been suggested that human locomotion may be modeled
as an inverse optimal control problem. In this paradigm, the trajectories are
assumed to be solutions of an optimal control problem that has to be
determined. We discuss the modeling of both the dynamical system and the cost
to be minimized, and we analyze the corresponding optimal synthesis. The main
results describe the asymptotic behavior of the optimal trajectories as the
target point goes to infinity

A Abramovici

A Albrecht

A Guth

A Linde

AA Starobinski

AA Tseytlin

AR Liddle

B Campbell

C Angelantonj

CE Reece

CM Caves

E Martinec

EJ Copeland

F Pegaro

G Veneziano

J Levin

JM Bardeen

K Behrndt

K Kounnas

KS Thorne

M Cvetic

M Gasperini

MK Gaillard

R Brustein

SW Hawking

V Mukhanov

English

arXiv

I present an outline for cosmological evolution in the framework of string theory with emphasis on a phase of dilaton-driven kinetic inflation. It is shown that a typical background of stochastic gravitational radiation is generated, with strength that may allow its detection in future gravity wave experiments.I present an outline for cosmological evolution in the framework of string theory with emphasis on a phase of dilaton-driven kinetic inflation. It is shown that a typical background of stochastic gravitational radiation is generated, with strength that may allow its detection in future gravity wave experiments

Brustein, Ram

CERN Document Server

arXiv:hep-th/9506045v1  7 Jun 1995BGU PH-95/05CERN-TH/95-147DILATON-DRIVEN INFLATIONIN STRING COSMOLOGYRam Brustein ∗,∗∗)Theory Division, CERN CH-1211 Geneva 23, SwitzerlandABSTRACTI present an outline for cosmological evolution in the framework of stringtheory with emphasis on a phase of dilaton-driven kinetic inflation. It isshown that a typical background of stochastic gravitational radiation is gen-erated, with strength that may allow its detection in future gravity waveexperiments.∗)Present address: Department of Physics, Ben-Gurion University,Beer-Sheva 84105, Israel.∗∗)Contribution to the proceedings of the International conferenceUnified symmetry in the small and in the large,February 2-5, 1995, Coral Gables , Florida.CERN-TH/95-147June 1994INTRODUCTIONI present an outline of cosmological evolution in the framework of stringtheory. The main emphasis is on a phase of dilaton-driven kinetic infla-tion and its possible observable consequences, in particular, a background ofstochastic gravitational radiation. The results concerning the produced spec-trum of gravitational radiation were obtained in [1, 2]. More details on vari-ous aspects of the suggested outline and additional references may be found in[3-11].POTENTIAL-DRIVEN INFLATIONInflationary evolution of the universe requires a source of energy to drivethe expansion. The conventional expectation is that the energy source isdominated by potential energy of scalar fields, called inflatons [12]. The in-flatons are expected to posses non-vanishing potential energy during somephase in their evolution in which inflationary expansion takes place. Eventu-ally, the inflatons settle down to the true minimum of their potential wherethe potential energy vanishes, thus depriving the universe of the necessarysource to drive its accelerated expansion. The inflationary phase ends andthe universe continues to expand sub-luminally until today. If one tries toimplement similar ideas in the framework of string theory, an apparent prob-lem is immediately encountered [13, 14]. String theory does indeed con-tain many scalar fields, called moduli, which seem particularly suitable forthe job of inflatons [15, 16]. Among the moduli the dilaton φ is an im-portant and universal field whose expectation value determines the stringcoupling parameter g2s ∼ 〈exp(φ)〉. It couples to all other fields with gravi-tational strength. If some scalar field, for example, one of the moduli fieldsacquires a non-vanishing potential so does the dilaton. The type of gener-ated dilaton potential depends on the details of the model. Two types aredistinguished, perturbative V (φ) ∼ exp(−αφ/MP l), and non perturbativeV (φ) ∼ exp(−exp(−βφ/MP l)), with particular numerical parameters α, β.The equations of motion for the resulting string dilaton-gravity, assumingisotropic and homogeneous universeds2 = −dt2 + a2(t)dxidxiφ = φ(t), (1)1are the followingH2 =8π3M2P l(12φ˙2 + V (φ))φ¨+ 3Hφ˙ = −dVdφ(2)The Hubble parameter, H , is related to the scale factor, a in the usual way,H ≡ a˙a and V is the potential. Consider, for example, the (unrealistic) caseof exponential potential V = V0 exp(−αφ/MP l) for which can solve eqs. (2)explicitlya(t) = a0 t16π/α2. (3)If the potential is steeper than the critical steepness α = 4√π, the dilatonkinetic energy becomes dominant over potential energy and the expansionis subluminal. The generic situation in string theory is that the potentialsin several models are steeper than critical and therefore potential-driven in-flation requires special situations and is generally speaking hard to obtain.Recently, some progress has been made towards characterizing requirementsfrom models in which potential-driven inflation could be supported [16, 17].DILATON-DRIVEN KINETIC INFLATIONThe outline for cosmological evolution that I present here relies heavily onthe fact that the kinetic energy of the dilaton tends to dominate the energydensity. Instead of trying to fight this tendency, one accepts it and turns thisfeature into a virtue, using it to drive kinetic energy dominated inflationaryevolution. Kinetic inflation was also discussed in [18]. The evolution startswhen the dilaton is deep in the weak-coupling region (φ≪ −1) and Hubbleparameter, H , is small. The evolution in this epoch is shown below to be ac-celerated expansion dominated by the dilaton kinetic energy and determinedby the vacuum solution of the string dilaton-gravity equations of motion [3].To describe the first phase in more detail, look for solutions of the effectivestring equations of motion in which the metric is of the isotropic, FRW typewith vanishing spatial curvature and the dilaton depends only on time. One2finds three independent first order equations for the dilaton and HH˙ = ±H√3H2 + U + eφρ− 12U ′ +12eφp (4a)φ˙ = 3H ±√3H2 + U + eφρ (4b)ρ˙+ 3H(ρ+ p) = 0 (4c)where U = eφV . Some sources in the form of an ideal fluid were included [3]as well. The (±) signifies that either (+) or (−) is chosen for both equationssimultaneously. The solutions of equations (4a-4c) belong to two branches,according to which sign is chosen. In the absence of any potential or sourcesthe (+) branch solution for {H, φ} is given byH(+) = ± 1√31t− t0φ(+) = φ0 + (±√3− 1) ln(t0 − t) , t < t0 (5)This solution describes either accelerated contraction and evolution towardsweak coupling or accelerated inflationary expansion and evolution from acold, flat and weakly coupled universe towards a hot, curved and stronglycoupled one. I assume that the initial conditions are such that the latteris chosen. In general, the effects of a potential and sources on this branchare quite mild. After a period of time, of length determined by the initialconditions, a “Branch Change” event from the dilaton-driven acceleratedexpansion era into what will eventually become a phase of decelerated ex-pansion has to occur. It occurs either when curvatures and kinetic energiesreach the string curvature or when quantum effects become strong enough.The correct dynamical description of this phase should, therefore, be stringyin nature. If the value of the dilaton is small throughout this stage of evo-lution, dynamics can be described by classical string theory in terms of atwo-dimensional conformal field theory. This stage is not yet well under-stood. At the moment, the only existing examples are not quite realistic[19, 20]. More ideas about this stage may be found in [21, 22]. The valuewhich the dilaton takes at the end of this epoch φend is an important param-eter. After the “Branch Change” event, the universe cools down and maybe described accurately, again, by means of string dilaton-gravity effective3theory. Now, however, radiation and matter are important factors. Thedilaton remains approximately at the value φend. The universe evolves asa regular Friedman-Robertson-Walker (FRW) radiation-dominated universe.TENSOR PERTURBATIONS AND RELIC GRAVITATIONALWAVESThe phase of accelerated evolution, described in the previous section,produces a typical and unique spectrum of gravitationl radiation. The basicmechanism is by now well known [23] (see [24, 25] for recent reviews). Quan-tum mechanical perturbations exist as tiny wrinkles on top of the classicallyhomogeneous and isotropic background. These wrinkles are then magnifiedby the accelerated evolution and become classical stochastic inhomogeneities.Below I sketch the derivation of the spectrum of tensor perturbations. Manytechnical elements are omitted here and can be found in gory details in [1].The classical solution (5) in conformal time η, where dt ≡ adη, is given bygµν = diag(a2(η),−a2(η)δij) i, j = 1, 2, 3 (6)wherea2(η) ∼ |η|1/2, φ(η) ∼ −√3 ln |η|+ φ0 (7)for η → 0−. One expands the metric around the classical solution g =gcs+δg where gcs is given in the previous equation and δgij = −a2(η)hij(η, ~x).The resulting equation of motion for each of the two independent tensorperturbation components is given in Fourier space byh′′k + 2a′ah′k + k2hk = 0 (8)and has the general solution hk = Ak + Bk ln |kη|. Initial conditions corre-sponding to quantum fluctuations at short scales hk ∼ 1/(a√k) exp[i(~k · ~x− kη)],determine hk|hk| ≃ ln |kη|√kaHC≃ ln |kη|. (9)The amplitude of stochastic tensor perturbations in x space is characterizedby |δhk| ∼ k3/2|hk|. From eq.(9) we obtain|δhk|2(η) ∼(HmaxMP l)2|kηmax|3 (ln |kη|)2 . (10)4The end of the dilaton-driven epoch is assumed [5] to take place when thecurvature scale H reaches the string scale Ms. In the Einstein frame, inwhich MP l is constant, the string scale depends upon the dilaton as Ms =exp(φ/2)MP l. Thus we assume the dilaton-driven era to end at conformaltime |η| = η1 where H1 ≃ (η1a(η1))−1 = Ms(η1) = exp(φ(η1)/2)MP l. At theend of the dilaton-driven era we thus have|δhk(η1)| ∼ H1MP l(kη1)3/2 ln(kη1) (11)This is the final result for the primordial spectrum of tensor perturbations.From the primordial spectrum one wishes to compute the observable spec-trum today. A nice feature of gravitational waves is that gravitons are af-fected practically only by the evolution of the background curvature sinceright after the “Branch Change” era. Thus the spectrum that should be seentoday should mainly reflect what happened in the very early universe pro-cessed through presumed known background evolution. While frequenciesshift according to the evolution of the background scale factor throughoutthe evolution, amplitudes of tensor perturbations freeze while outside thehorizon and evolve only when inside the horizon. If the dilaton-driven erais followed by a stringy phase characterized by an almost constant value ofH , we expect scales which went out of the horizon during the dilaton-drivenera to keep moving further outside and to reenter only much later, duringthe radiation, or possibly even matter dominated era. If we assume this tobe the case for all (comoving) scales larger than η−11 , we must also assumethat hk remains frozen, for all these scales, at the value given in eq.(11) untilreentry.The result is [2] that the part of the processed spectrum which lies belowa certain maximal frequency ωmax, the highest frequency amplified duringthe dilaton-driven era, is presently given, in the string frame, by|δhω| =√H0/Msz−1/4eq z1/2out exp (12φend) (ωωmax)12 ln(ωωmax) (12)where zout(k) = are(k)/aex(k) is the red-shift while the scale k−1 was outsidethe horizon, zeq is the red-shift from the matter-radiation equality epoch untiltoday, Ms is the present value of string scale (usually estimated to be about2−5 ·1017GeV ), H0 ∼ 10−18Hz is the present value of the Hubble parameter,5φend = φ(η1), andωmax =√H0Msz−1/4eq z−1/2out . (13)The fraction of energy in gravitational waves in units of the critical densityis given bydΩd lnω= z−1eq exp(φend) (ωωmax)3 ln2(ωωmax). (14)Equations (12-14) were derived assuming reentry during the radiation dom-inated era and should be taken as good estimates and not as numericallyaccurate expressions. The processed spectrum of gravitational radiation ispresented graphically in Figure 1,δ hωFrequency in Hertz δ hω maxω1/2ω1/2ω1/2Ω GW = 10−10Ω GW = 10−4102 106 10101 10−3010−2610−22Ω GW =10−161/2zoute φ endFigure 1. The characteristic spectral amplitude of gravitational waves |δhω |. Thesolid lines show several individual spectra for different values of zout and φend = 0.The thick dashed line shows the maximum amplitude |δhmaxω | as a function of zoutfor φend = 0. The dashed lines are lines of fixed φend and therefore lines of constantenergy density. ΩGW is the maximal amount of gravitational energy density at agiven φend. Also shown in the figure is a triangular shape marking the sensitivitygoals for detection of stochastic background h3/yr, of the “Advanced LIGO”.6Two possible devices may be able to detect the predicted stochastic grav-itational wave background, in the lower frequency region 1 − 104 Hz, largeinterferometers, such as the planned LIGO[26] and VIRGO[27] and in thehigher range of frequencies 106 − 109 Hz, room-size microwave cavities. Fora given set of parameters the amplitude grows as |δhω| ∼ ω1/2 and thereforeit may seem that the best sensitivity for detection is at the high end of thespectrum ω = ωmax. However, the noise in a given interferometer grows ashn ∼ ω5/4 [28]. Therefore for a given interferometer the best sensitivity ac-tually is in the lowest frequency range available. Microwave cavities may beoperated as gravity wave detectors [29] for the high frequency range 106−109Hz. For the MHz range specific suggestions [30, 31] have been implemented[32], but not operated as gravitational radiation detector. As can be seenfrom Figure 1, the required sensitivity for detection at the MHz region ishc ∼ 10−26 corresponding to h3/yr of the same order and therefore to a noiselevel of hn ∼ 10−23 [28], assuming a bandwidth of MHz. With attainable Qfactors of the order of 1011, this sensitivity goal does not seem out of reach.For the GHz region the required sensitivity is hc ∼ 10−28 corresponding tohn ∼ 10−24.ACKNOWLEDGMENTResearch supported in part by an Alon Grant. I would like to thank M.Gasperini, M. Giovannini, V. Mukhanov and G. Veneziano for enjoyable andfruitful collaboration and S. Finn, P. Michelson P. Saulson and N. Robertsonfor discussions about gravity wave detectors.REFERENCES[1] R. Brustein, M. Gasperini, M. Giovannini, V. Mukhanov and G.Veneziano, “Metric perturbations in dilaton-driven inflation”, CERN-TH. 7544/94, Phys. Rev. D (1995), in Press.[2] R. Brustein, M. Gasperini, M. Giovannini and G. Veneziano, CERNpreprint.[3] G. Veneziano, Phys. Lett. B265 (1991) 287.7[4] M. Gasperini and G. Veneziano, Astropart. Phys. 1 (1993) 317; M.Gasperini and G. Veneziano, Mod. Phys. Lett. A8 (1993) 3701;M. Gasperini and G. Veneziano, Phys. Rev. D50 (1994) 2519.[5] R. Brustein and G. Veneziano, Phys. Lett. B329 (1994) 429.[6] M. Gasperini, “Phenomenological aspects of the pre-big-bang scenarioin string cosmology”, in Proc. of the 2nd Journee Cosmologie, Paris,June 1994 (World Scientific P.C., Singapore), Torino University PreprintDFTT-24/94;M. Gasperini, “The inflationary role of the dilaton in string cosmology”,in Proc. of the First Int. Workshop on Birth of the Universe and Fun-damental Physics, Rome, May 1994 (Springer-Verlag, Berlin), TorinoUniversity Preprint DFTT-29/94.[7] G. Veneziano, “Strings, Cosmology, ...and a Particle” talk given at PAS-COS ’94, preprint CERN-TH.7502/94 (November 1994).[8] R. Brustein, “The role of the superstring dilaton in cosmology and par-ticle physics”, in Proc. of the XXIX Rencontres de Moriond, Meribel,March 1994;R. Brustein, “Cosmology and models of supersymmetry breaking instring theory”, Proc. of the SUSY94 Workshop, Ann Arbor, May 1994.[9] M. Gasperini and M. Giovannini, Phys. Rev. D47 (1993) 1529.[10] M. Gasperini and M. Giovannini and G. Veneziano, “Primordial mag-netic fileds from string cosmology”, preprint hep-th/9504083.[11] K. Behrndt and S. Forste, Nucl. Phys. B430 (1994) 441;E. J. Copeland, A. Lahiri and D. Wands, Phys. Rev. D50 (1994) 4880;C. Angelantonj, L. Amendola, M. Litterio and F. Occhionero, Phys.Rev. D51 (1995) 1607.[12] A. Guth, Phys. Rev. D23 (1981) 347;A. Linde, Phys. Lett 108B (1982) 389;A. Albrecht and P.J. Steinhardt, Phys. Rev. Lett. 48 (1982) 122;A. Linde, Phys. Lett 129B (1983) 177.[13] R. Brustein and P. J. Steinhardt, Phys. Lett. B302 (1993) 196.8[14] B. Campbell, A. Linde and K. Olive, Nucl. Phys. B335 (1991) 146.[15] P. Binetruy and M.K. Gaillard, Phys. Rev. D34 ( 1986) 3069;M. Cvetic and R. L. Davis, Phys. Lett. B296 (1992) 316;[16] T. Banks, M. Berkooz, S.H. Shenker, G. Moore and P.J. Steinhardt,“Modular cosmology”, preprint hep-th/9503114.[17] T. Damour and A. Vilenkin, “String theory and inflation”, preprint hep-th/9503149.[18] J. Levin, Phys. Rev. D51 (1995) 1536.[19] E. Kiritsis and K. Kounnas, Phys. Lett. B331 (1994) 51;E. Kiritsis and K. Kounnas, in Proc. of the 2nd Journee Cosmologie,Paris, June 1994 (World Scientific P.C., Singapore).[20] A. A. Tseytlin, Phys. Lett. B334 (1994) 315.[21] E. Martinec, Class.Quant.Grav.12 (1995) 941.[22] N. Deruelle and V.F. Mukhanov, “On matching conditions for cosmo-logical perturbations”, preprint gr-qc/9503050.[23] V. Mukhanov and G. V. Chibisov, JETP Lett. 33, (1981) 532;A. Guth and S. Y. Pi. Phys. Rev. Lett. 49 (1982) 1110;A.A. Starobinski, Phys. Lett. B117 (1982) 175;S.W. Hawking, Phys. Lett. B115 (1982) 295;J.M. Bardeen, P.S. Steinhardt and M.S. Turner, Phys. Rev. D28 (1983)679.[24] V. Mukhanov, H.A. Feldman and R. Brandenberger, Phys. Rep. 215(1992) 203.[25] A. R. Liddle and D. H. Lyth, Phys. Rep. 231, (1993) 1.[26] A. Abramovici et al., Science 256 (1992) 325.[27] B. Caron et. al, “Status of the VIRGO experiment”, preprint Lapp-Exp-94-15.9[28] K. S. Thorne, in 300 Years of Gravitation, S. W. Hawking and W. Israel,Eds. (Cambridge Univ. Press, Cambridge, 1987.[29] L. P. Grishchuk, Proc. 9th Int. Conf. on General Relativity and Gravi-tation, ed. E. Schmutzer, Cambridge Univ. Press, 1983.[30] F. Pegaro, E. Picasso, L. Radicati, J. Phys. A11 (1978) 1949.[31] C. M. Caves, Phys. Lett B80 (1979) 323.[32] C. E. Reece et al., Phys. Lett A104 (1984) 341.10

Optimal control models of the goal-oriented human locomotion

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