Coslab 2006 lorentz center cosmological evolution of cosmic string loops mairi sakellariadou king’s college london astro-ph/0511646 in collaboration with

coslab 2006 lorentz center

cosmological evolution of cosmic string loops

mairi sakellariadou

king’s college london

astro-ph/0511646

in collaboration with christophe ringeval and francois bouchet

cosmological evolution of cosmic string loops mairi sakellariadou coslab 2006 -- 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

t 1=t2 1=a2

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

ü ø

3-scale model : interstring separation , curvature scale , wiggliness

kibble 1985

bennett & bouchet 1988, 1989, 1990; sakellariadou & vilenkin 1990; allen & shellard 1990austin, copeland & kibble 1993

1

3

3

2

2

1


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

ø

ø øöð


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 1997

1

1

1, 2

1

2vincent, antunes & hindmarsh 1998

results:


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 19902

2

vachaspati & vilenkin 1984

lc

faster relaxation


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): 520 308

(100lc)3

dh0 = 0:063 (RDE) ; 0:057 (MDE)

comoving volume in the matter era; the observable universe occupies one eight of the box

(100lc)3

initial physical correlation length:

associated with the vv initial conditions

initial resolution physical length:

also associated with initial conditions

lc = 1=100

memory of the initial conditions lr = (Nppcl

3 )lc


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

lphys = ëdh

ñ=lc

ú1 / 1=d2h

dú0 / 1=d2h

transient energy excess, which signs the relaxation of the initial string network

the transient regime is longer for the smaller loops

dú0(ë)d2h

stationary, for all values ofdown to ù 5â 10à 3

dëdú0 = S(ë)d2

h

U

dëdn = ëd3

h

S(ë)

ú1 U

d2h = 28:4æ0:9

ú1 U

d2h = 37:8æ1:7

scaling function

dh = 3t (MDE) ; 2t (RDE)

I is the loops length in units of the horizon sizedh

after a transient regime , it reaches a self-similar evolution

ë

ë


the rescaled distibution as a function ofat equally spaced physical times spread over the dynamic range of the simulations

ëd3h(dn=dë)

t = 1:1 t = 0:8

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

ë ëlphys = ëdh

the scaling regime propagates from the large scales towards the small ones

ë

ë


power law squares fit of in where loops scale

ëd3h(dn=dë) ës < ë < ë1

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 stringsë1 = (U=ú1 )1=2=dh

S(ë) = C0ëà p

rescaled distribution : C ñ ëp+1d3h(dn=dë)

lc = 1=100

lphys = ëdh

peak around a constant value close to the initial physical correlation length associated with initial conditionsthe relaxation bump around the initial correlation length is progressively damped

lcthe overall maximum of distribution appears as a knee, lose to initial resolution length associated with inititial conditions

ëd3h(dn=dë)

lr

lr = (Nppcl3 )lc

MDE RDE

the correlations associated with remain at constant physical lengths during the subsequent evolution

lr; lc


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

lr

(40lc)3

ë = lphys=dh 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

lc

scaling

lc = 1=40 ; lr = (3=ppcl)lc

lc

the discretisation effects concern the smallest loops; they should not influence the string properties on larger scales

lr


ú : total string energy density

: total pressureP

M = a3ú : total network mass

W = a3P : total pressure work

Q = M 0+ 3HW : energy dissipation ratefor the RDE run (20 ppcl)(40lc)3

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

Q = 0

10à 3

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


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

là 5=2

ë > 2:10â 10à 3

ë < 2:10â 10à 3


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


astro-ph/0511792 martins & shellard (ref.1)

dynamic range:dynamic range (in conformal time): 8 RDE 17 MDEdynamic range (in physical time): 520 308

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??

lclr = 20ppcl

3


astro-ph/0511792 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

l=t = 0:075664 ñà 3:203

MDE

t ø ñ3MDE:« 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

lr


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 – »

no !

ø

øø 2â 10à 1

scaling up to 2â 10à 3

) ø=100


astro-ph/0511792 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

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Coslab 2006 lorentz center cosmological evolution of cosmic string loops mairi sakellariadou king’s college london astro-ph/0511646 in collaboration with