The Active Universe Alexnder Glück, Helmuth Hüffel, Sš Ilijić, Gerld Kelnhofer Fculty of Physics, University of Vienn helmuth.hueffel@univie.c.t Deprtment of Physics, FER, University of Zgreb ss.ilijic@fer.hr Abstrct Active motion is concept in complex systems theory nd ws successfully pplied to vrious problems in nonliner dynmics. Explicit studies for grvittionl potentils were missing so fr. We interpret the Friedmnn equtions with cosmologicl constnt s dynmicl system, which cn be mde ctive in strightforwrd wy. These ctive Friedmnn equtions led to cyclic universe, which is shown numericlly. 1 Active Motion To ccount for self-driven motion s observed in biologicl systems, n dditionl degree of freedom, clled the "internl energy" e, ws introduced [Schweitzer et l. (1998), Schweitzer (003)] in the context of complex systems theory. First interprettions of this formlism were given with respect to niml movement nd complex motion of prticles in generl. It describes prticles who cn convert their internl energy e into mechnicl energy nd thus exhibit complex movement. Vrious pplictions of this model were studied extensively, for instnce in swrm theory [Schweitzer et l. (001), Glück et l. (010)]. Also for Quntum Field Theory it could be shown tht ctive dynmics is importnt for the behvior of systems fr from equilibrium, nmely such which exhibit spontneous symmetry breking [Glück et l. (008)]. Obviously this introduction of n internl vrible e enbles us to describe open dissiptive systems in generl wy. The vrible e functions s dynmicl quntity which models the energy flux between n open system nd its surrounding. We now wnt to discuss the implictions of ctive dynmics lso for grvittionl systems. The ctive formlism is pplied to the cosmologicl equtions of motion strightforwrdly in order to study the behvior of universe fr from equilibrium. Dynmics of prticle with position q nd momentum p undergoing ctive motion s discussed in [Glück et l. (009)] is determined by q i = p i, p i = U (1 e), (1) q i 1
where the evolution of the internl energy e is given by the eqution ė = c 1 c e c 3 eu(q). () The c i remin constnts to be fixed. The first integrl of the ctive dynmicl system reds q i + U(q i) ˆ t 0 dt e(t ) U(q i (t )) = C = const, (3) which cn be viewed s generliztion of the clssicl energy eqution. In the context of ctive motion the prticle is not simply driven to the minimum of the potentil U(q), its internl energy llows it to exhibit self-driven movement. Depending on the choice of the constnts c i nd on the structure of the potentil, the systems show vrious complex motion ptterns. These cn be studied nlyticlly by bifurction theory nd numericlly by simulting the evolution of q, p nd e in time. So fr, explicit investigtions were done only for hrmonic potentils (see [Glück et l. (009)] for detiled study of the nonliner dynmics). One cn nturlly sk for ctive motion of prticles driven by grvittionl interctions. We will now give formultion of this problem within the frmework of Friedmnn equtions. Active Friedmnn Equtions The Friedmnn equtions for the dimensionless normlized cosmologicl scle prmeter (t), describing the sptil evolution of flt, homogeneous nd isotropic universe with cosmologicl constnt Λ red ( ä = H0 Ωr,0 3 + Ω ) Ω Λ, (4) ȧ ( = H 0 Ωr,0 + Ω ) + Ω Λ, (5) where Ω nd Ω r,0 re the density prmeters of mtter nd rdition t present time t = 0 nd Ω Λ = is the density prmeter of the cosmologicl Λ 3H 0 constnt. (0) = 1 per definition nd H 0 = ȧ(0) denotes the current Hubble prmeter. If we introduce the following potentil U() = H 0 + Ω ) + Ω Λ, (6) then the nlogy with clssicl dynmicl system of single prticle with coordinte (t) nd energy C = 0 is evident. According to generl reltivity the constnt C is relted to the sptil curvture of the universe which, however, is tken to be zero due to recent observtions. The density prmeter
of Λ is fixed by the Friedmnn eqution (5) which t present time t = 0 gives the reltion Ω r,0 + Ω + Ω Λ = 1. (7) We now wnt to propose new phenomenologicl model for the development of the universe which is bsed on the ctive generliztion of the conventionl Friedmnn equtions. The ctive Friedmnn equtions follow from equtions (1) - (3) using (6) nd re given by ä = H 0 3 + Ω ė = c 1 c e + c 3 e H 0 ȧ = H 0 H 0 ˆ t 0 + Ω dt e(t )ȧ(t ) ) Ω Λ (1 e), (8) + Ω ) + Ω Λ, (9) + Ω Λ ) + 3 (t ) + Ω ) (t ) Ω Λ(t ) + H 0 C, (10) where c 1, c, c 3 re rbitrry but fixed rel constnts. For nottionl convenience we hve chosen H0 C/ s the corresponding constnt for the first order integrl in (10). Differing from the conventionl cse we llow for nonvnishing C in the ctive context, which hs to be determined yet. We first hve to ssign specific vlues to the density prmeters of mtter/rdition nd the cosmologicl constnt. Notice tht we cn t set Ω Λ = 0.7, since this results from eqution (7), which looks different in the ctive formultion, nmely Ω Λ = 1 Ω r,0 Ω C, (11) which is derived from eqution (10) by setting t = 0. Hence, the constnt C determines the density prmeter Ω Λ of the cosmologicl constnt. It cn be fixed by demnding tht the current ccelertion ä(0) of the universe in the ctive scheme should gree with the ccelertion vlue given by the conventionl Friedmnn equtions (thus ccounting for the experimentl result ä(0) > 0). For ny fixed e 0 := e(0) one finds C = e ( 0 3 1 e 0 Ω + Ω r,0 1). (1) Accordingly, Ω Λ = 1 ( 1 Ω (1 + e ) 0 1 e 0 ) Ω r,0(1 + e 0 ), (13) so tht Ω Λ is now completely determined by the density prmeters of mtter/rdition nd the initil condition for e, which is freely chosen. Depending on this initil condition Ω Λ my lso be brought to vnish. 3
Figure 1: Numericl result for the evolution of (t), with c 1 = 5, c = 1, c 3 = 1, e(0) = 100, H 0 = 1, Ω = 0.9 nd Ω r,0 = 0.01. (These re rough representtive vlues for the density prmeters, consult [Komtsu et l. (009)] for recent overwiev of cosmologicl prmeters.) This choice of prmeters leds to Ω Λ = 0.15. 3 Discussion Figure 1 shows simultion exmple of the ctive dynmics of the normlized cosmologicl scle prmeter (t). We observe oscilltory behvior with significntly smller mplitudes nd shorter periods in the pst. The scle prmeter (t) is oscillting round certin equilibrium point, which cn be clculted nlyticlly. The ctive Friedmnn equtions cn be rewritten s follows: ȧ = p ṗ = du() (1 e) (14) d ė = c 1 c e c 3 eu(). Equilibrium points re found by serching vlues (ã, p, ẽ), for which the right hnd side of (14) vnishes. Equilibrium points with ẽ = 1 re unstble, the cse ẽ 1 leds to the conditions p = 0, 0 = Ω Λ ã 4 Ω ã Ω r,0, (15) c 1 ẽ = c + c 3 U(ã). For the specific choice of prmeters nd initil conditions mde for the simultion shown in Figure 1, the numericl vlues of (ã, p, ẽ) cn be clculted 4
directly. The qurtic eqution for ã hs only one positive, rel-vlued solution, nmely ã = 1.01, resulting in (ã, p, ẽ) = (1.01, 0, 4.08). Figure 1 shows tht this vlue of the scle prmeter mrks the center point of the oscilltions. The stbility nlysis of the bove equilibrium point (ã, p, ẽ) cn be mde by linerizing the system (14) nd clculting the eigenvlues of the Jcobin mtrix. Our clcultion shows the presence of purely imginry pir of eigenvlues λ 1, = ±iω nd we re ner Hopf bifurction point (for bifurction theory, see [Kuznetsov (1995)]). In our cse ω = 1.19, so tht the period of oscilltions is given by T = π ω = 5.8 in units of the Hubble time H0 1, which grees well for lrge times with our numericl simultion. Summrizing we cn sy tht interpreting the Friedmnn equtions s nonliner dynmicl system nd hndling it within the frmework of ctive motion, cn give rise to oscilltory solutions for the scle prmeter. Hence we succeeded in constructing model for cyclic universe giving rise to sequence of expnsions nd contrctions without ny singulrity, yet ccounting for the observed sptil fltness nd the current ccelerted expnsion. Besides this prticulr exmple, the ctive Friedmnn equtions llow for lot of other possible scenrios which deserve further nlysis. A future tsk will be to find n interprettion of the vrible e within cosmologicl context. In prticulr, cosmologicl models which ssume the existence of extr dimensions seem to be promising cndidtes for providing n dequte nswer in this respect. Acknowledgments: We thnk Helmut Rumpf for vluble discussions. We re grteful for finncil support within the Agreement on Coopertion between the Universities of Vienn nd Zgreb. References [Schweitzer et l. (1998)] Schweitzer, F. & Ebeling W. & Tilch, B. [1998] Complex Motion of Brownin Prticles with Energy Depots, Phys. Rev. Lett. 80, 5044. [Schweitzer (003)] Schweitzer, F. [003] Brownin Agents nd Active Prticles, Springer, Berlin. [Schweitzer et l. (001)] Schweitzer, F. & Ebeling, W. & Tilch, B. [001] Sttisticl Mechnics of Cnonicl-Dissiptive Systems nd Applictions to Swrm Dynmics, Phys. Rev. E 64, 01110. [Glück et l. (008)] Glück, A. & Hüffel H. [008] Nonliner Brownin Motion nd Higgs Mechnism, Phys. Lett. B 659, 447. 5
[Glück et l. (009)] Glück, A. & Hüffel H. & Ilijić, S. [009] Cnonicl ctive Brownin motion, Phys. Rev. E 79, 0110. [Glück et l. (010)] Glück, A. & Hüffel H. & Ilijić, S. [010] Swrms with cnonicl ctive Brownin motion, rxiv:1005.3646. [Komtsu et l. (009)] Komtsu, E. et l. [009] Five-Yer Wilkinson Microwve Anisotropy Probe Observtions: Cosmologicl Interprettion, Astrophys. J. Suppl. 180, 330. [Kuznetsov (1995)] Kuznetsov, Y.A. [1995] Elements of pplied Bifurction Theory, (Springer, New York). 6