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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
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
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
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 . ..
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 . ~, _
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
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.