We use numerical simulations of the time evolution of global textures to investigate the relationship between ordering dynamics and energy density in an expanding universe. Events in which individual textures become fully wound are rare. The energy density is dominated by the more numerous partially wound configurations, with median topological charge alpha ~ 0.44. This verifies the recent supposition (Borrill et al. 1994) that such partially wound configurations should dominate the cosmic microwave background

Kogut, A J

Phillips, N G

CERN Document Server

Field Ordering and Energy Density in Texture Cosmology
Nicholas G. Phillips
1
Department of Physics, University of Maryland, College Park, Maryland 20742
A. Kogut
2
Hughes STX Corporation, Laboratory for Astronomy and Solar Physics, Code 685,
NASA/GSFC, Greenbelt, MD 20771
To appear in Physical Review Letters
ABSTRACT
We use numerical simulations of the time evolution of global textures to investigate the
relationship between ordering dynamics and energy density in an expanding universe.
Events in which individual textures become fully wound are rare. The energy density
is dominated by the more numerous partially wound congurations, with median
topological charge 

' 0:44. This veries the recent supposition [11] that such
partially wound congurations should dominate the cosmic microwave background.
PACS numbers: 98.80.Cq, 98.65.Dx
1
Electronic Address: phillips@zwicky.gsfc.nasa.gov
2
Electronic Address: kogut@stars.gsfc.nasa.gov
I. Introduction
Recent observations [1, 2] of anisotropy in the cosmic microwave background (CMB)
provide strong evidence for primordial density perturbations. Several models predict
the statistical distribution of these perturbations. In inationary models, density
perturbations result from parametric amplication of quantum uctuations during
the inationary epoch [3], while in topological defect models the density perturbations
result from large-scale ordering eects following a phase transition with broken
symmetry [4, 5, 6, 7].
Observational tests of these models are complicated by the fact that
cosmological models predict the CMB anisotropy to be a single realization of a
stochastic process whose properties (power spectrum, phase correlations, etc.) are
predicted only for an ensemble average of equivalent realizations. Any single
realization may be expected to vary about the ensemble average. Although a large
body of work exists providing analytic derivation of ensemble average properties of
inationary models, the non-linear nature of defect models preclude exact analytic
results. Tests of defect models must use either time-consuming numerical simulations
or an approximate analytic model.
Global textures are one class of defect models for which an approximate
analytic model has been proposed [7]; however, the predictions of this approximate
model do not agree with the results of a complete simulation [8]. One of the main
assumptions of this model is that the CMB anisotropy is dominated by the most
energetic isolated defects. We use a numerical simulation of the complete theory to
investigate this assumption and nd that the dominant contribution comes instead
from the far more numerous but less energetic defects. This was rst suggested in [11]
for a non-expanding universe. By doing the simulation for an expanding universe, we
demonstrate that the less energetic defects must be considered in any model of the
texture generated CMB anisotropy.
Topological defects such as monopoles, domain walls, or cosmic strings involve
regions of space where the order eld remains in the higher-energy symmetric state.
The energy density in these models is dominated by the order eld potential. Textures
[6] are a class of topological defects with the property that the broken symmetric phase
vacuum manifold of the order eld is of the same dimension as the spatial geometry.
This allows the order eld always to stay on the vacuum manifold, regardless of initial
conditions. Perturbations in the energy density are driven solely by the order eld's
kinetic energy.
The topological charge of a region of space is dened as the fraction of the
vacuum manifold covered as the order eld is mapped into the manifold [9]. At the
time of the phase transition, the order eld in each causal domain will randomly
fall into any of the degenerate vacua. As the dierent domains come into causal
contact, the eld may develop a non-trivial topological charge, forming a wound
conguration or \knot". Derrick's theorem [10] states that the spatial extent of any
region of non-zero gradient energy must shrink, causing the order eld to become
2
increasingly tightly wound around the vacuum manifold. The kinetic energy in this
wound conguration will eventually become greater than the potential barrier to the
symmetric state, lifting the order eld from the vacuum manifold and allowing the
texture to \unwind" as it reduces the topological charge.
Since this unwinding process is so energetic, it is often viewed as the dominant
contribution to the energy density. Turok & Spergel [7] provide an analytic model of
the CMB anisotropy resulting from a distribution of unwinding spherical knots. The
results of this model are not reproduced by numerical simulations [8]. On the other
hand, Borrill et. al. have pointed out that unwinding events are rare: they found
only 40 such events in 1000 simulations [9]. This agrees with analytic predictions
[6] (see also [4, 15]). They also nd partially-wound texture congurations can still
generate anisotropies similar to those generated by events which do unwind [11].
We have simulated the order eld dynamics in the non-linear  model
approximation to investigate the distribution of gradient energy density. We identify
both the wound and partially-wound events in our simulation. The unwinding events
are indeed rare: we nd only 3 unwindings events out of over 400 texture events.
Although unwindings are the most energetic individual events, most of the energy
density is associated with the more numerous partially-wound texture congurations.
II. Analysis
Our calculations are done in a spatial at Robertson-Walker homogeneous universe,
with metric given by
ds
2
=  dt
2
+ a(t)
2
(dx
2
+ dy
2
+ dz
2
) = a( )
2
( d
2
+ dx
2
+ dy
2
+ dz
2
): (1)
We consider the evolution of a real global eld 
a
, a = 1; : : : ; 4, whose dynamics are
specied by the Lagrangian density
L =  
1
2
@

@

  V
0
(jj
2
  
2
(T ))
2
: (2)
The function 
2
(T ) is such that the eld  undergoes a phase transition at T = T
c
from a symmetric phase (T > T
c
) to a broken symmetry phase (T < T
c
). The
symmetric phase corresponds to 
2
(T ) < 0, giving the eld an O(4) symmetry. When

2
(T ) > 0, the eld falls into the broken symmetry phase with O(3) symmetry. The
vacuum manifold becomes S
3
= O(4)=O(3). For the range of dynamics we consider,
we assume T  T
c
and 
2
(T ) = 
2
0
, a constant.
Textures form as horizons meet, which has the length scale of Hubble's constant
H. The natural length scale for textures is the inverse of the radial excitation mass of
the eld. It is at this scale that we expect a collapsing \knotted" texture to unwind
[12]. Borrill [13] has pointed out that for a GUT scale phase transition, these scales
dier by 10
50
at the time of decoupling. Any attempt to numerically simulate this
model can not hope to cover this range of scales and an approximation must be made.
3
In this work, we follow [8] and use the non-linear  model (NLSM). This takes the
eld's mass to innity, forcing  to always be on the vacuum manifold.
The NLSM model is obtained when the Lagrangian density is rewritten as
L =  
1
2
@

@

   (jj
2
  1)
2
; (3)
where the constraint that the eld stay on the vacuum manifold is now imposed via
the Lagrange multiplier  (the eld rescaling 
a
! 
a
=
0
is assumed). Variation of
this Lagrangian leads to the equations of motion


a
+ 2
_a
a
_

a
 r
2

a
=
h
jrj
2
  j
_
j
2
i

a
: (4)
To investigate this theory via computer simulation, we simulate the order eld
evolution on a discrete cubic lattice. We map 
a
(x; t) ! 
a
i;n
, where i labels the
spatial cells and n is the temporal index. Without loss of generality, we take the grid
spacing x = y = y = 1 and set the scale via our choice of the time interval  .
Varying a discrete version of Eq. (4) [8], we get the equations of motion

a
i;n+1
= 
i

a
i;n
+ 
a
i;n
(5)
where

a
i;n
= 
a
  (a
2
n 
1
2
=a
2
n+
1
2
)
a
n 1
+ (a
2
n
=a
2
n+
1
2
)r
2

a
i;n

2
: (6)
(a
2
n
1
2
denotes a
2
(
n

1
2
 ).) 
i
is chosen to enforce the constraint j
i;n+1
j
2
= 1.
Squaring Eq. (5) and solving for  gives:

i
=  
i;n

i;n

q
1  j
i;n
j
2
+ j
i;n

i;n
j
2
(7)
Expect for the case mentioned below, we always take the positive square root.
The NLSM implementation provides no method by which a knotted
conguration unwinds. Since  is constrained to the vacuum manifold, the equations
of motion no longer allow for unwinding. In eect, we have assumed that the spatial
scale for unwindings is smaller than the inter-cell distance. If we can recognize a
wound conguration once it has collapsed to be comparable to the cell size, we may
add an unwinding mechanism\by hand". For each spatial cell, we dene an alignment

i
=
1
6
X
j

i

j
; (8)
where the sum on j runs over the i
th
cell's six nearest neighbors. A cell is counter-
aligned if 
i
< 0, when it has become completely wound on the scale of the cell
spacing. This can be thought of as a situation where the topological charge within a
333 cube centered around the counter-aligned cell becomes> 0:5. The unwinding
then proceeds by choosing the negative sign in Eq. (7): the cell is moved across the
4
vacuum manifold to a conguration that removes the topological charge [8]. Eq. (7)
is thus modied to become

i
=  
i;n

i;n
+ sign(
i
)
q
1  j
i;n
j
2
+ j
i;n

i;n
j
2
(9)
and the knotted textures are unwound once they shrink below the cell grid scale. The
energy density at each point is given by the  -component of the stress energy tensor:
 =
1
2
j
_
j
2
+
1
2
@
i
@
i
 (10)
and we use the obvious discretised version in our analysis.
III. Numerical Results
We use the above algorithm to investigate the distribution of the energy density during
the eld evolution from the time of decoupling onward. Since the decoupling time is
long after the time of the phase transition, we consider eld evolution starting from
an initially smooth eld conguration. This is achieved by initializing each point on
the simulation grid to be uniformly distributed on the vacuum manifold and allowing
the equations of motion to evolve the eld until it \settles down". Initially, the 
i
's
will be uniformly distributed in [ 1; 1]. If the unwinding mechanism (Eq. (9) instead
of Eq. (7)) were used, it would lead to an unstable eld evolution. Instead, we use Eq.
(7), choosing strictly the positive square root and thus keeping the gradients small.
We continue until the total number of counter-aligned cells stops decreasing. Then
the unwinding mechanism is enabled and Eq. (9) is used to evolve the eld. This
procedure allows the eld evolution algorithm to generate the required smooth initial
conditions. The late time behaviour of the eld is insensitive to the precise time that
the unwinding is enabled.
We have run a simulation of a 64
3
element spatial cube for 32 conformal time
units (c=1, x=1). Since periodic boundary conditions are used, this is the maximum
length of time the simulation can run for and still be viewed as physical. We took
 = 0:2. The evolution was insensitive to taking  ! =2. The evolution was also
insensitive to using a 32
3
elements spatial cube. The simulation became stable for
 > 10, at which time the unwinding was enabled.
The simulation shows the expected behaviour as regions of above average
energy density form, shrink in spatial extent while the energy density rises, and then
re-expand while the energy density falls back to the average background level. The
time scale for this rise and fall is of order   3. We call such a process an event.
Typically we nd the energy density at the center of an event to be at least 4 above
the average.
Fig. 1 shows the energy density vs cell alignment for the 20 most energetic
events. The alignment decreases and the energy increases as the texture becomes
increasingly wound. There were three unwinding events ( < 0) in this simulation:
5
the energy density increases by an order of magnitude as the knotted texture unwinds.
The rest of the events never become completely wound. The alignment decreases to a
minimum  > 0 and then starts to increase again. Shortly after the alignment stops
decreasing the energy density stops increasing and the event dissipates.
Fig. 2a shows the distribution of minimum alignments for all events. The
distribution's median is  ' 0:48. The most likely event is one where the texture
never becomes completely knotted. Although the energy density in these partially-
wound events is an order of magnitude below the fully knotted conguration, they
are so numerous as to dominate the total energy density of the simulation. Fig. 2b
shows the energy density integrated over time for all events in the simulation vs the
minimum alignment of each event. This distribution's median is 

' 0:44. The
dominant energy contribution in a texture cosmology comes from partially wound
congurations, with only minor contributions from unwinding knots.
IV. Discussion
The Sachs-Wolfe eect [14] relates CMB anisotropy to perturbations in the
gravitational potential, and is usually given in terms of an integral over the the time
rate of change of the gravitational potential. For texture models, the gravitational
potential results from the kinetic energy density of the order eld. There does not
appear to be a great variation in the spatial prole of the individual events; hence,
the contribution of each texture event to the CMB anisotropy should be proportional
to the integrated energy density in the event.
Numerical simulations of the time evolution of global textures demonstrate
that the integrated energy density is dominated by partially-wound congurations
and not by unwinding knots. The gradient energy density due to texture events is
centered about textures with alignment 

' 0:44. This result proves Borrill et. al.
[11] claim, but now in the more realistic scenario of an expanding universe.
Eorts to obtain observable predictions for specic texture models have
either relied on computationally expensive numerical simulations or on simplifying
assumptions. The rst method suers from logistical problems in that proper
assessment of cosmic variance requires hundreds to thousands of realizations for each
choice of model parameters. The second method typically suers from assumptions
(spherical symmetry, uncorrelated placement of subsequent textures, fully wound
textures) that may not be well founded. We have demonstrated that any simplied
texture model must consider the full distribution of partially-wound events along
with the rarer high-energy events. We are currently attempting to derive a
statistical description (distribution of shapes, spatial and temporal correlations)
of the gravitational potential in texture cosmologies. If a suciently compact
description can be obtained which retains the distinctive features of a full-blown
numerical simulation, signicant computational savings may be realized in Monte
Carlo simulation of CMB anisotropy for texture models. These techniques may
6
then be incorporated into other analytic models [15, 16] to compare specic texture
cosmologies with the observed CMB anisotropy on various angular scales.
Acknowledgements
We would like to thank Shoba Veeraraghavan for her time and helpful discussions.
One of us (NGP) would also like to thank Fred Lang of Catholic University of America
for making the funds and resources available to facilitate this work.
7
References
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8
Figure 1: Evolution of texture events viewed in the energy density-alignment plan.
Diamonds denote points in the evolution where the alignment was negative (an
unwinding event).
9
Figure 2: (a) Distribution of minimum alignments of event histories. (b) Energy
density (arbitrary units) integrated over time, sorted by the minimum alignment of
each event. The median for energy distribution is 

'0:44, from texture events that
are only partially wound.
10


Field ordering and energy density in texture cosmology

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