|
Items in cart [0] |
Questions? Call 888-624-8373 |
| Copyright © 2009. National Academy of Sciences. All rights reserved. Terms of Use and Privacy Statement |
Below are the first 10 and last 10 pages of uncorrected machine-read text (when available) of this chapter, followed by the top 30 algorithmically extracted key phrases from the chapter as a whole.
Intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text on the opening pages of each chapter.
Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.
Do not use for reproduction, copying, pasting, or reading; exclusively for search engines.
OCR for page 322
The Formation and Evolution of Domain Walls
WILLIAM H. PRESS AND BARBARA S. RYDEN
Harvard-Smithsonian Center for Astrophysics
and
DAVID N. SPERGEL
Princeton University
INTRODUCTION
Symmetr, breakings in the early universe can produce stable topolog-
ical defects: monopoles, cosmic strings and domain walls. While cosmic
strings have attracted significant attention as possible seeds for galaxy for-
mation (see Press and Spergel 1989 for review). Domain walls-sheet-like
defects produced when the low energy vacuum has isolated degenerate
minima however, have been relegated to the list of cosmologically unde-
sirable objects.
Domain walls were banished to the cosmological dog house by Zel'do-
vich et al. (1975), who noted that the energy density in domain walls falls
much less fast, as the universe expands, than does the energy density in ra-
diation or even matter. Thus, stable domain walls would quickly dominate
the universe. Vilenkin (1985) reviews much of the early work on domain
walls, which confirmed the dangers of an early-time domain wall producing
phase transition. However, the failure of existing scenarios of galaxy for-
mation have motivated a domain wall revival. Hill et al. (1989) suggested
that a late-time (post-decoupling) phase transition could have produced
cosmologically interesting "light" domain walls. They discussed a model
proposed by Hill and Ross (1988a, b) in which a pseudo-Golds/one boson
associated with the neutrino family of fermions undergoes a spontaneous
symmetry breaking and acquires a mass on the order of mV2/MGUT- Hill
et at speculate that the light, thick domain walls produced by these late-
time phase transitions could account for much of the observed large-scale
structure. Earlier, Wasserman (1986) had suggested that late-time phase
transitions might explain the bubble-like topology seen in the CfA redshift
322
OCR for page 323
HIGH-ENERGY ASTROPHYSICS
323
survey. Independently, Dimopolous and Star~nan (1989) also proposed
a family of models in which a technicolor axion acquires a mass at low
temperature phase transition. These models also produce domain walls;
however, they remain severely constrained by stellar evolution bounds on
axion properties.
Unlike cosmic strings, whose dynamics have been studied in detail,
little work has been done on domain wall dynamics. Many of the basic
features of domain wall dynamics are not understood: How does the
energy density in walls scale as the universe expands? Is most of the energy
density in infinite walls or in closed domain wall "bags"? What is the
lifetime of such bags? What is the typical wall velocity? Do most wall
intersections lead to reconnection? How do wall oscillations damp? How
does the characteristic wall size grow as the universe expands? Bill Press,
Barbara Ryden and I have developed a computer simulation of domain
wall evolution. This talk will describe our results (Press et al. 1989; Ryden
et al. 1990) and discuss its implication for domain wall seeded large-scale
structure.
Rather than following the motion of infinitely thin domain walls (e.g.,
Kawano 1989), our computer code follows the evolution of a scalar field,
a, whose dynamics are determined by its Lagrangian density. The topology
of the scalar field determines the evolution of the domain walls. This
approach properly treats both wall dynamics and reconnection. We ran
10 separate 1024 x 1024 numerical simulations. A plot of the comoving
wall area A times conformal time per comoving volume V, as a function
of elapsed (conformal) time since the phase transition, is shown in Figure
1. One sees that, during the epoch when the conformal time (that is, light
travel distance) ~ is much greater than the wall thickness and much smaller
than the length of an edge of the box, the wall area is well fitted by a power
law
A/V or ~v.
with an exponent ~ not very different from 1, the value that implies a
scale-free evolution with, at all epochs, about one domain wall per horizon
volume. The average wall velocity was mildly relativistic, 0.4 c.
We also ran 10 200 x 200 x 200 numerical simulations. The dynamics
of walls in a three dimensional simulation is similar to that in a two
dimensional simulation.
Figure 2 shows the domain wall structure in the 2 dimensional universe.
Most of the walls are part of an infinite network that percolate across the
simulation. Only a small fraction of the energy density is in small bubbles.
This is a very different situation from that for cosmic strings, whose evo-
lution and reconnection can leave behind a significant spectrum of smaller
OCR for page 324
324
.8
.6
\
AMERICAN AND SOVIET PERSPECTIVES
.4
.2
1
~10242 ~
l' I I I I lll I , , I, llll I I 10 1~1,
10 100 1000
conformal time r'
FIGURE 1 A linear-log plot of the comov~ng wall area per unit oomov~ng volume of the
two~imensional simulations, multiplied by the conformal time q. Results are shown for
ten 1024 X 1024 wall simulations. The dashed line is the best fitting relation of the folm
RAIN = a + bin (~7/Wo), fit in the interval JO < ~ < 100, marked By the dotted lined
loops (Albrecht and lbrok 1985; Bennett and Bouchet 1988; see Press and
Spergel 1989, for additional references). The reason for the difference is:
(i) Wall bubbles are formed, we find, relatively rarely (per horizon volume),
and (ii) We find no instances of a wall bubble being formed in a configura-
tion that is able to persist without immediately collapsing, self-intersecting,
and radiating away its energy content as oscillatory excitations of Me ~ field
(i.e., schizons). This difference from string loops is not unexpected, gener-
ically, simply as a consequence of the different dimensionality of a loop
(one-dimensional) and a bubble (two-dimensional). Thin wall simulations
confirm the behavior seen in these thick wall calculations ~awano 1989~.
It is also supported by recent work of Andrew (1989a), who finds that
spherical bubbles can "bounce" only a few times at most, and out to dis-
tances several times We wall ~ic~ess, before dissipating. Prow (1989b)
also finds that there is a tendency for nonspherical bubbles to become
more spherical dunug the eartr stages of their collapse. This conclusion is
supported by what we see in our evolutions.
An immediate consequence of these findings is that, as a consequence
OCR for page 325
HIGH-ENERGY ASTROPHYSICS
32fS
an, ]~ -
:~
.~
FIGURE 2 Pictures of the domain wall network in a 1024 X 1024 simulation with a wall
thickness of wo = 5 Slab symmetry is here assumed in the third dimension (out of the
page). Snapshots are taken at conformal times (a) 7~ = 9.6, (by If = 32, and (c3 77 = 93.
(E.g., at ~ = 93 the horizon size is about JO times the wall thickness and 1110 times the
size of the picture.) Ibe gray scale map is chosen so that walls are gray on the side facing
domains where ~ is positive, black on the side facing domains where ~ is negative.
Of the observed lack of quadrupole microwave anisotropy, it is not possible
to hide any significant amount of matter in walls at the present epoch.
For the wall geometries that we see develop, one will always have OTT
~ I, so that What' (the fraction of critical density in walls at the
present epoch) must be ~ 10-4.
This implies that walls cannot seed the formation of structure on
smaller scales. Consider a comoving scale L" At a redshift of (Cto/L)2, when
this scale was on order the horizon scale, the domain walls generated a
. ..
OCR for page 326
326
AMERICAN AND SOVIET PERSPECTIVES
W::
FIGURE 2b
perturbation ~wal~0(ctO~)3. This wall-induced perturbation grew linearly
up to today and its current amplitude is w0Wall(ctolL) The wall-buluced
perwrba~ns with the highest amplitude are Pose near He size of the hon-
zon. The density perturbations on the horizon scale create a quadrupole
anisotropy in the microwave background through the Sachs-Wolfe effect,
with amplitude [TIC ~ [pip ~ await Thus, if walls move freezer in the
manner computed in this paper, and if they survive to the present, then
it is impossible for them both to generate large scale structure, and to be
consistent with quadrupole microwave background anisotropy limits.
We investigated not only potentials that produce single domain walls,
but also potentials that produce a network of walls and strings (Ryden et al.
1990~. These networks arise In anion models where the U(11 Pecce~-Quinn
symmetry is broken into ZN discrete symmetries. If N = 1, Me walls are
. ~, _
OCR for page 327
HIGH-ENERGY ASTROPHYSICS
327
~) I i,) ~
\~
/-\1
~/j / '_
at\ 0 ~ \~-~ ~ ~`
/ ~
FJ,GURE 2c
By/
Jim
1 ~-\ -
bounded by strings and the network quickly disappears. For N > 1, the
network of walls and strings behaved qualitatively just as the wall network
shown in both figures. This both confirms our rather pessimistic view that
domain walls can not play an important role in the formation of large
scale structure and implies that axion models with multiple minimum can
be cosmologically disastrous (see Everett and Vilenkin 1982; Vilenkin and
Everett 1982; and Vachaspati and Vilenkin 1984~.
ACKNOWLEDGMENTS
We thank Larry Widrow, Terry Walter, Id Lauer, David Nelson,
Glenn Starkman, Curt Call=, Dave Schram~n, Marcelo Gleiser, and Doug
Eardley for helpful discussions. This work was supported in part by the
OCR for page 328
328
AMERICAN AND SOVIET PERSPECTIVES
National Science Foundation: at Harvard University (PHY-86 04396), at
Princeton University (NSF Presidential Young Investigator Award), and at
the Institute for Theoretical Physics, Santa Barbara (PHY-82-17853), with
supplementary support from me National Aeronautics and Space Admin-
istration. W.H.P. and D.N.S. thank the Institute for Theoretical Physics
for hospitality during He early stages of this wore D.N.S. acknowledges
support from the Alffed P. Sloan Foundation.
REFERENCES
Albrecht, As, and P. Steinhardt. 1982. Phys. Rev. Lett. 48: 1220.
Albrecht, A, and N. Crow 1985. Phys. Rev. Lett. 54: 1868.
Bennett, D., and F. Bouchet. 1988. Phys. Rev. Lett. 60: 257.
Bertschinger, E., and P.N. Watts 1988. Ap. J., 316: 489.
Brandenberger, R. N. Kaiser, D.N. Schramm, and N. Ibrok. 1987. Phys. Rev. Lett 59:
2371.
Dimopoulos, S., and G. Starkman. 1990. In preparation.
Everett, AK., and A V~lenkin. 1982. Nucl. Phys., B207: 43.
Fneman, J.A, G.B. Gelmini, M. Gleiser, and E.W. Kolb. 1988. Phys. Rev. Lett, 60: 2101.
Guth, A. 1981. Phys. Rev. D 23: 347.
Hill, CT, and G.G. Ross. 1988a. Phys Lett. B205: 125.
Hill, CT, and G.G. Ross 1988b. Nuclear Phyla B311: 253.
Hill, COO., D.N. Schramm, and J.N. Fry. 1989. Comments on Nucl. Part. Phys. 19: 25.
Kawano, L 1989. The Evolution of Domain Walls in the Early Universe. FERMII^B
Pub~91~-a.
Linde, A 1982a. Phys Lett. lOBB: 389.
Iinde, A 1982b. Phys. Lett. 114B: 431.
Press, W.H., B.P. Flanne~y, Say Teukolsky, and W.l: Vetterling. 1986. Numencal Reapes:
The Art of Scientific Computing. Cambridge University Press, New York.
Press, W.H., and D. Spergel. 1989. Physics Today, 42~33: 29.
Press, W.H., B. Ryden, and D.N. Spergel. 1989. Astrophys. J. 347: 590.
Ryden, B.S., W.H. Press, and D.N. Spergel. 1990. Ap I. 357, 293.
Vachaspati, 1:, and A V~lenkin. 1984. Phys. Rev. D 30: 2036.
V~lenkin, A 1985. Phys Reports 121: 263.
moleskin, A, and NE. Everett. 1982. Phys Rev. Lett. 48: 1867.
Wasserman, I. 1986. Phys. Rev. Lett. 57: 2234.
W~drow, ~ 1989a. Phys. Rev. D 39, 5376.
Wldrow, ~ 1989b. Phys Revd 40, 1002.
Zel'dovich, Ya.B., I.Yu. Kobzarev, and LB. Okun. 1975. Sov. Phys. JETP 40: 1.
Representative terms from entire chapter:
energy density
