coslab 2006 lorentz center cosmological evolution of cosmic string loops mairi sakellariadou king’s college london astro-ph/ in collaboration with christophe ringeval and francois bouchet

cosmological evolution of cosmic string loops mairi sakellariadou coslab lorentz center long, or, infinite strings : super-horizon sized strings loops : sub-horizon sized loops a cosmic string network is cosmologically acceptable due to the scaling regime of long strings intersections between super-horizon sized strings produce sub-horizon sized loops, so that the total energy density of long strings scales with the cosmic time as, instead of the catastrophic the universe is not overclosed only if the energy density in the form of loops is radiated away

early analytical studies predicted the scaling property of long strigs : the string network is dominated by only one length scale, the interstring distance which grows with the horizon early numerical simulations revealed dynamical processes at scales 3 -scale model : interstring separation, curvature scale, wiggliness kibble 1985 bennett & bouchet 1988, 1989, 1990; sakellariadou & vilenkin 1990; allen & shellard 1990 austin, copeland & kibble cosmological evolution of cosmic string loops mairi sakellariadou coslab lorentz center feature of the model: the small length scale reaches a scaling regime only if gravitational back reaction effect is considered, otherwise the kinky structure keeps growing w.r.t. horizon size

cosmological evolution of cosmic string loops mairi sakellariadou coslab lorentz center the main features of the 3 -scale model have been numerically confirmed in minkowski spacetime nearly all loops are produced at the lattice spacing size, which makes the evolution & scaling properties of the small scale structure strongly dependent on the cutoff if this feature persists whatever the lattice spacing, then the typical size of physical loops might be the string width particle production rather than gravitational radiation would be the dominant mode of energy dissipation from a string network vincent, hindmarsh & sakellariadou , vincent, antunes & hindmarsh 1998 results:

cosmological evolution of cosmic string loops mairi sakellariadou coslab lorentz center model: improved version of the bennett & bouchet nambu-goto string code II in a FLRW universe vachaspati & vilenkin initial conditions: the long strings path is a random walk of correlation length with a random tarnsverse velocity component of root mean squared amplitude 0.1 simulations are performed in a fixed unity comoving volume with periodic b.c. the initial scale factor is normalised to unity; the initial horizon size is a free parameter which controls the starting string energy within a horizon volume the evolution is stopped before the comoving horizon size fills the whole unit volume 1 1 bennett & bouchet vachaspati & vilenkin 1984 faster relaxation

cosmological evolution of cosmic string loops mairi sakellariadou coslab lorentz center two high resolution runs in MDE/RDE, performed in a comoving box and with an initial string sampling of 20 points per correlation length ( ppcl ) initial size of horizon : dynamic range (in conformal time): 8 17 dynamic range (in physical time): comoving volume in the matter era; the observable universe occupies one eight of the box initial physical correlation length: associated with the vv initial conditions initial resolution physical length: also associated with initial conditions memory of the initial conditions

cosmological evolution of cosmic string loops mairi sakellariadou coslab lorentz center evolution of energy density of long strings and of loops of physical size the time variable is the rescaled conformal time U : string mass per unit length transient energy excess, which signs the relaxation of the initial string network the transient regime is longer for the smaller loops stationary, for all values of down to scaling function I is the loops length in units of the horizon size after a transient regime, it reaches a self-similar evolution

cosmological evolution of cosmic string loops mairi sakellariadou coslab lorentz center the rescaled distibution as a function of at equally spaced physical times spread over the dynamic range of the simulations transient overproduction of loops preceding the scaling and the overall maximum of the loop distribution evolve in time; during the runs they peak at decreasing sizes wrt the horizon size best power law fit & systematic errors the distribution functions start to superimpose at the largest length scales during earlier times the non-scaling parts of the distribution function shift towards smaller self-intersections give rise to more numerous smaller loops so that a constant energy flow cascades from long strings to smallest loops the scaling regime propagates from the large scales towards the small ones

cosmological evolution of cosmic string loops mairi sakellariadou coslab lorentz center power law squares fit of in where loops scale lowest size of loops, in units of horizon size, for which the energy density remains stationary during the last 5% of simulation conformal time range typical distance between infinite strings rescaled distribution : peak around a constant value close to the initial physical correlation length associated with initial conditions the relaxation bump around the initial correlation length is progressively damped the overall maximum of distribution appears as a knee, lose to initial resolution length associated with inititial conditions MDERDE the correlations associated with remain at constant physical lengths during the subsequent evolution

cosmological evolution of cosmic string loops mairi sakellariadou coslab lorentz center influence of the initial resolution length on the rescaled loop distributions at the end of 3 small RDE runs having an initial sampling of 10, 20, 40 ppcl and a dynamic range of 45 in physical time is the loop length in units of the horizon size the finite resolution effects remain confined to length scales smaller than the initial correlation length of the string network and do not affect the loop scaling regime we have also checked the insensitivity of the loop distribution wrt initial random velocity scaling the discretisation effects concern the smallest loops; they should not influence the string properties on larger scales

cosmological evolution of cosmic string loops mairi sakellariadou coslab lorentz center : total string energy density : total pressure : total network mass : total pressure work : energy dissipation rate for the RDE run (20 ppcl) conservation of nambu-goto stress tensor: sharp negative peak at very beginning shows a strong energy loss rate in the form of numerically unresolved loops, during a brief period that the universe expands less than a factor of at any time loops with stricktly less than 3 points cannot be formed, so all triangle shaped loops are removed from the subsequent evolution (such a removal is not equivalent to a fixed physical size cutoff) study dissipation effects by testing the total stress energy conservation during the evolution

cosmological evolution of cosmic string loops mairi sakellariadou coslab lorentz center  the loop distribution, once it reaches the scaling regime, depends upon the physical loop length as roughly (MDE)  only loops with roughly have this power law distribution the finite numerical resolution allows us to probe only an expansion factor ~60  during the run, we observe the scaling to propagate towards small length scales for an even bigger simulation, the power law behavior would have reached much smaller loops  for loops such as there is some memory of the initial conditions, i.e. remaining correlation effects from the vachaspati-vilenkin network  the propagation of the scaling towards small scales shows that these initial correlations are progressively washed out during the cosmological evolution and seem to be transient effects note: the power law breaks down under a cutoff which is not known: it depends on the assumptions about the microscopic string model or gravitational back reaction

cosmological evolution of cosmic string loops mairi sakellariadou coslab lorentz center conclusions for the first time, evidence of a scaling evolution for string loops in both radiation and matter eras down to a few thousandths of the horizon size the loops scaling evolution is similar to the long strings one and does not rely on any gravitational back reaction effect it only appears after a relaxation period which is driven by a transient overproduction of loops, wrt the scaling value, whose length is close to the initial correlation length of the string network there is an axplosive-like formation of very small sized and numerically unresolved loops during the first stage of the simulations, suggetsing that particle production may briefly dominate the physical evoluton of a string network soon after its formation

cosmological evolution of cosmic string loops mairi sakellariadou coslab lorentz center astro-ph/ martins & shellard (ref.1) dynamic range: dynamic range (in conformal time): 8 RDE 17 MDE dynamic range (in physical time):  dynamic range (in conformal time): of order 3 (and up to 6) in our simulation  resolution : precision of the numerical calculation (code II of bennett & bouchet, 4 times better precision that martins & shellard) 20 ppcp for us equivalent to 75 ppcl in (ref. 1) apart the initial correlation length there is an additional correlation length coming from the # ppcl (we have segments) we have less N ppcl, so bigger segments in ref.1, cutoff is 14 ??

cosmological evolution of cosmic string loops mairi sakellariadou coslab lorentz center astro-ph/ martins & shellard « The dominant loop production scale starts out being about the size of the correlation length, but becomes progressively smaller as small-scale structure builds up on the strings. The evolution of the peak of the loop distribution, however, is clearly beginning to slow down at late times indicating that it is rising above the minimum simulation resolution and will approach scaling. » dynamical range (proportional to the conformal time) evolution of the position of the maximum MDE MDE : « scaling »: evolution of correlation length due to the cutoff the constant physical length during simulations is the correlation length in small scales due to the discretisation in initial coditions

cosmological evolution of cosmic string loops mairi sakellariadou coslab lorentz center note: « This paper (RSB) presents some evidence for the scaling of the overall loop distribution on intermediate length scales below the correlation length but still near it, roughly loop lengths x/10 – x » no !

cosmological evolution of cosmic string loops mairi sakellariadou coslab lorentz center astro-ph/ vachurin, olum & vilenkin (ref.2)  minkowski background  to get bigger dynamical range the authors glue simulations this can create artificial correlations in small length scales  loops smaller than ¼ of the horizon are artificially removed from the network (we do not remove any loops) we find that the distribution of loops grows steeply towards small scales, with a power index different than in ref.2 this is clear since we are in an expanding universe; the power law we found is indeed different between RDE and MDE