P1246990943ptyqL

1
cosmological evolution of cosmic string loops
astro-ph/0511646
in collaboration with christophe ringeval and
francois bouchet
mairi sakellariadou kings college london

coslab
2006 lorentz center


2
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
cosmological evolution of cosmic string loops

mairi sakellariadou
coslab 2006 -- lorentz
center

3
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
1
2
early numerical simulations revealed dynamical
processes at scales
3
interstring separation , curvature scale
, wiggliness
3-scale model
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
1
kibble 1985
2
bennett bouchet 1988, 1989, 1990
sakellariadou vilenkin 1990

allen shellard 1990
3
austin, copeland kibble 1993
cosmological evolution of cosmic string loops

mairi sakellariadou
coslab 2006 -- lorentz
center

4
1
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
results
1
1, 2
1
vincent, hindmarsh sakellariadou 1997
2
vincent, antunes hindmarsh 1998
cosmological evolution of cosmic string loops

mairi sakellariadou
coslab 2006 -- lorentz
center

5
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
2
faster relaxation
1
bennett bouchet 1990
2
vachaspati vilenkin 1984
cosmological evolution of cosmic string loops

mairi sakellariadou
coslab 2006 -- lorentz
center

6
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
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 2006 -- lorentz
center

7
scaling function
I is the loops length in units of the horizon
size
stationary, for all values of down
to
after a transient regime , it reaches a
self-similar evolution
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
cosmological evolution of cosmic string loops

mairi sakellariadou
coslab 2006 -- lorentz
center

8
the rescaled distibution
as a function of at equally spaced physical
times spread over the dynamic range of the
simulations
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
the scaling regime propagates from the large
scales towards the small ones
self-intersections give rise to more numerous
smaller loops so that a constant energy flow
cascades from long strings to smallest loops
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
cosmological evolution of cosmic string loops

mairi sakellariadou
coslab 2006 -- lorentz
center

9
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
rescaled distribution
typical distance between infinite strings
MDE
RDE
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
the correlations associated with remain
at constant physical lengths during the
subsequent evolution
cosmological evolution of cosmic string loops

mairi sakellariadou
coslab 2006
-- lorentz center

10
the discretisation effects concern the
smallest loops they should not influence the
string properties on larger scales
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
scaling
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
cosmological evolution of cosmic string loops

mairi sakellariadou
coslab 2006
-- lorentz center

11
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
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
cosmological evolution of cosmic string loops

mairi sakellariadou
coslab 2006
-- lorentz center

12

• 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 2006
-- lorentz center

13
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 2006
-- lorentz center

14
astro-ph/0511792 martins shellard (ref.1)

• dynamic range (in conformal time) of order 3
  (and up to 6)

in our simulation
dynamic range
dynamic range (in conformal time)
8 RDE 17 MDE dynamic range
(in physical time) 520
308

• resolution

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??
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)
cosmological evolution of cosmic string loops

mairi sakellariadou
coslab 2006
-- lorentz center

15
astro-ph/0511792 martins shellard
evolution of the position of the maximum
MDE
dynamical range (proportional to the conformal
time)
 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. 
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 2006
-- lorentz center

16
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 2006
-- lorentz center

17
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
cosmological evolution of cosmic string loops

mairi sakellariadou
coslab 2006
-- lorentz center