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FERMILAB-Pub-89/128-A * NSF-ITP-89-87 

May 1989 

COSMIC EVOLUTION OF NON-TOPOLOGICAL SOLITONS I. 

Joshua A. F'rieman and Angela V. Olinto 

NASA/Fermilab Astroph ysics Center 

Fermi National Accelerator Labomtory 

Batavia, IL 60510 

Marcel0 Gleiser 

Institute for Theoretical Physics 

University of California 

Santa Barbara, CA 99106 

Charles Alcock 

Institute for Geophysics and Planetary Physics 

Lawrence Livermore National Laboratory 

Livermore, CA 94550 

ABSTRACT 

Nontopological solitons are stable field configurations which may be formed in a primordial phase transition. We study their cosmic evolution and examine the possibility that 

such objects could contribute significantly to the energy density of the Universe. As the 

Universe cools, initially all but the largest lumps evaporate into free particles; those which 

survive may subsequently enter a brief accretion phase before they 'keeze out' at a final 

size. Although the minimum critical charges which survive depend on particle masses and 

couplings, we develop an analysis which applies to a wide class of models. In most cases, 

solitons of moderate size survive the evaporation process only if there is a significant charge 

asymmetry or if they form at a temperature well below their binding energy per charge. 

Operated by Unlversities Research Association Inc. under contract with the United States Department of Energy 
I. Introduction 

Non-topological solitons (hereafter, NTS’s) are solutions of classical field theories 

which are stable by virtue of a conserved Noether charge carried by fields confined to 

a finite region of space. The prototype for such structures is the phenomenological bag 

model for hadrons, in which light quarks are confined in a false vacuum region by Yukawa 

coupling to a scalar field.’ In recent years, the bag model has genetically mutated into 

a rich variety of species, which include Q balls,2 abnormal n~clei,~ strange quark matter 

nuggets: cosmic neutrino as well as ~calar*9~9* and gauge field’ non-topological solitons. Despite the proliferation of models, they all share a number of family resemblances 

which allow them to be classified and discussed in general terms. The key feature of all 

models is that, for particle number Q greater than some minimum Qmin, the confined 

soliton state has lower energy than the free particle state. 

Recently, it has been suggested that non-topological solitons may be abundantly produced in a phase transition in the early uni~erse.*~~~~~~~ However, as pointed out by Alcock 

and Farhi” and Alcock and Olinto12 in the context of strange matter, at sufficiently high 

temperature, such structures are in general unstable to evaporation into free (unconfined) 

particles. Non-topological solitons are states of low energy but high order. At temperatures T > IQ, the NTS binding energy per particle, free particles have lower Helmholtz 

free energy, F = E-TS, than the NTS state, due to the entropy term. (This holds as long 

as the temperature is not very near the critical temperature for the phase transition.lO) 

At lower temperatures, the energy term dominates, and NTS’s are the preferred states. 

If we focus on the evolution of a single soliton of initial charge (or particle number) 

Q, as the universe cools the lump begins to evaporate. If it is large enough to survive 

down to temperatures T 5 IQ, it may then enter an accretion phase. Eventually, the 

2 
accretion or evaporation rate becomes negligible compared to the expansion rate, and 

the system ‘freezes out’ with a fixed charge. Thus, all solitons with initial charge less 

than a critical number Q, die before they reach the accretion or freeze-out phase. Those 

with charges Q > Q, are born frozen, ;.e., their evaporation and accretion rates are 

always negligible. However, evaporation is so efficient that Q, is often larger than the 

charge inside the particle horizon at the time of formation; in these cases, it is unlikely 

that any solitons formed in the phase transition survive to the present. At the other 

extreme, if NTS’s form at a temperature well below IQ, then we expect Q, - Qmin. In 

addition, it has recently been suggested that,13 if the particles are sufficientli strongly 

interacting to maintain chemical equilibrium at temperatures significantly below IQ, then 

a new population of solitons may be built up via accretion and fusion processes at low 

temperature. However, this latter possibility is only feasible if Qmin is very small, of 

order unity (and only if the mass scale is less than about lo4 GeV). Otherwise formation 

of new solitons by free particle collisions is statistically suppressed at low temperature, 

and the system cannot reach complete chemical equilibrium. Since, in almost all models 

considered in the literature, Qmin >> 1, we have relegated discussion of this possibility to 

an Appendix. 

In this paper, we follow the cosmological evolution of non-topological solitons kom 

birth, through the adolescent struggle for survival, to placid old age. In the next section, 

we outline the basic properties of NTS’s which we will need; a wide variety of models 

are encapsulated in a small number of parameters. For completeness, in Section 111, we 

review several of the NTS formation mechanisms which have been proposed. In Section 

IV, we set up the formalism for treating NTS evolution. We discuss the evolution of single 

NTS lumps in Section V; this treatment is relevant if Qmin is not close to one, in which 

3 
case the system generally does not reach chemical equilibrium. In Section VI, we discuss 

NTS destruction by ‘boiling’, ie., bubble nucleation, rather than surface evaporation. Our 

conclusions follow in Section VIII. 

Throughout most of the paper, we shall assume the universe carries a net charge 

asymmetry, and that the particles involved are long-lived. In the subsequent paper of 

this series (hereafter, paper 11, in preparation), we drop the assumption of asymmetry 

and discuss how NTS’s evolve as random charge fluctuations in a universe with zero net 

charge. There, we also consider models with unstable particles and discuss cosmological 

constraints on particle physics models with NTS’s. 

11. Properties of Non-topological Solitons 

In all NTS models, a partially ‘confined’ bose or fermi field t$, carrying a conserved 

additive quantum number, couples to a scalar field u; the vacuum expectation value uo of 

the ‘confining’ field generates part of the q5 mass, e.g., m+ = mo +m+(uo). (In the case of 

Q-balls,2 a single complex scalar field performs the roles of both t$ and u.) For appropriate 

couplings and sufficiently large charge, Q > Qmin, the lowest energy configuration consists 

not of free q5 particles roaming the vacuum, but of a spherical ‘false vacuum’ region in which 

u 1/ u’, surrounded by a domain wall where u rapidly approaches its ground state value. In 

the NTS interior, m+(u’) < m+(uo), and the effectively light t$ particles are trapped by the 

mass gap at the domain wall. Since, classically, a non-zero charge implies a time-dependent 

field, the confined particles carry kinetic energy, which varies as an inverse power of the 

volume. The potential (volume) and gradient (surface) energies vary with a positive power 

of the volume. As a consequence, for fixed charge Q, the total energy is minimized at 

a finite radius R, and the resulting configuration is in hydrostatic equilibrium. We will 

consider solutions which carry no currents, in which case the lowest energy configurations 

4 
are spherically symmetric. 

For large charges Q >> Qmin, the masses of all NTS models follow a simple scaling 

law: 

M(Q) = am#QP. (1) 

For NTS’s with non-aero volume energy (bag constant), the radius scales as 

On the other hand, models in which the interior potential energy density is degenerate 

with the vacuum, U(d) = U(Q), are confined by surface tension alone; 5~14 in this case, 

In this paper, we will focus almost exclusively on models with non-zero volume energy, 

ie., we will assume the radius obeys Eqn. (2). The dimensionless parameters p,a, and p 

depend on coupling constants in the theory. We shall only assume that they satisfy the 

criteria of NTS stability. The stability conditions are 

and 

(5) - dlM 

dQ2 

Eqn. (4) ensures stability aga-lst decay into free 4 partic-zs, while Eqn. (5) expresses 

stability against fission into smaller soliton fragments. Eqns. (l), (4), and (5) imply the 

upper bound p 5 1, and that the minimum stable charge is approximately Qmin E a1/(l-P) 

(for p < 1). The binding energy per unit charge is defined as the energy required to remove 

a 4 particle from a soliton, 

IQ = + M(Q) - M(Q + 1) . 

5 
Using Eqn. (l), we can express this in terms of the mass of the free 4 particles as 

f(Q) 1~/m+ = 1 - a[(Q + 1)’ - QP] N 1 - apQP-’. (7) 

Note that for p not very close to one, f(Q) II! 1 - p(Q/Q,;,)P-’, SO f(Q) + 1 as Q/Qmin 

grows large. 

Two notes are in order about the scaling laws, Eqns. (1) and (2). First, although 

they are strictly valid only in the limit Q >> Qmin, we will use them as approximate 

formulae for all Q larger than Qmin. Second, these scaling laws break down at very large 

charge, Qmaz - [(pla)(mpl/m+)2]31zP, when gravity becomes important. Since we are 

interested in only moderately sized lumps, Q << Qmaz, we can ignore this complication; 

for discussions of soliton stars, we refer the reader to ref. 15. 

Although the analysis we develop holds for general p, we will often focus on two 

particdar cases, p = 1 and 3/4, to demonstrate how the resdts vary with p. In the 

literature, many models have p essentially equal to one, e.g., Q-balls, fermion NTS’S, 

interacting scalar NTS’s, bag models, and strange matter. In this case, the requirement 

of stability is just a < 1, and the binding energy per charge is independent of Q, i.e., 

f(Q) = 1 -a. (Actually, in these models, due to the surface energy correction, p is slightly 

smaller than one; this ensures stability against fission.) For example, for strange matter, 

f(Q) N 0.05 and m+ = mn z 1 GeV. Configurations with p = 3/4 arise in a scalar 

model in which the confined field has little or no self-interaction.6~T~6 Weshall see that the 

qualitative evolution of NTS’s in the early universe is sensitive to both p and Qmin, as 

well as the mass scale m+. 

6 
111. Primordial Formation of NTSs 

One can envision several mechanisms by which non-topological solitons may be formed 

in the early universe. In principle, such objects arise in a symmetry-breaking phase transition if suitably large regions carrying a net charge Q > Qmin become trapped in the 

false vacuum state. In the simplest scenario, the NTS interior u = 0‘ corresponds to a 

metastable local minimum of the potential U(0) (dashed curve, Fig. l), and (i) may or (ii) 

may not be a phase of restored symmetry. In case (i), the universe at high temperature 

is initially in the ‘NTS vacuum’ and undergoes a first order phase transition to the true 

vacuum at a critical temperature T, - 60. Charge prefers to live in the false vacuum, 

where the 4 mass is smallest. Thus, as bubbles of the true vacuum phase nucleate, grow 

and eventually percolate, the charge may become highly concentrated in the shrinking 

‘NTS vacuum’ regions. Such a mechanism was proposed by Witten for the formation of 

strange matter nuggets in the QCD phase transition.* 

If the false vacuum is not a phase of restored symmetry, ie., case (ii) above, then NTS’s 

may form in a second order phase tran~ition.~ In this instance, during the transition, a 

fraction of space evolves to the metastable minimum 0’ instead of the true vacuum 60. 

The relative probability of ending up in the NTS phase is given roughly by the Boltzmann 

formula, Pfol.e/Ptrue = exp( -8(iAf/T~). Here, To is the Ginzburg temperature, the 

temperature at which thermal fluctuations between the true and false vacua freeze out, 

(G is the correlation length at the Ginzburg temperature, and Af is the difference in free 

energy density between the true and false vacua. We may interpret TG as the temperature 

at which NTS’s are first formed; typically Ta is just below T‘, the critical temperature 

for the phase transition. The size distribution of the proto-NTS false vacuum regions is 

calculable from percolation theory, while the scale size is set by the correlation length 

7 
(G. Generally, false vacuum regions with sizes R >> (G are exponentially suppressed; 

NTS’s with very large charges are correspondingly rare. This holds particularly if the 

probability pfalrc 5 1/3, when only finite NTS regions form. On the other hand, if 

Pfal,c 2 1/3, both the true vacuum and the NTS phase initially percolate, forming a 

web of infinite interconnecting segments. In this percolating-NTS case, the infinite false 

vacuum regions eventually pinch off into finite blobs when the pressure squeezing them 

from the true vacuum becomes dynamically important. If the vacuum asymmetry is small, 

Le., Af << Ti, pinch-off may occur well after the phase transition is completed; the finite 

NTS’s formed in this way are thus born at a temperature Tpinch << To, and have a greater 

chance of surviving. 

In a large class of models, the false vacuum NTS interior is not a metastable minimum 

of the potential, but is instead an unstable, symmetry preserving local maximum (solid 

curve, Fig. 1). In this case, solitons cannot form simply by charge piling up in a metastable 

phase. To see how formation occurs in this case,lo consider the structure of an NTS (of fixed 

charge) as a function of increasing temperature. Due to finite temperature quantum effects, 

when the temperature reaches the range near TG, the NTS radius expands enormously. 

If we now imagine starting with a dilute gas of solitons at T - 0 and similarly dialing 

up the temperature, then at T - To, the gas percolates. Reversing the film, as the 

temperature drops below TG, the Universe, initially in the ‘NTS vacuum’, shatters into 

soliton fragments, which subsequently shrink as they cool. (We note that this process may 

happen at a modest temperature compared to uo: a large charge asymmetry can delay the 

transition, since charge prefers the NTS vacuum.) The essential idea is that, very close to 

To, the kee energy of the NTS state becomes small compared to the state of a gas of free 

4’s. As a result, solitons can be formed at little cost. Thus, at temperatures T - TG, we 

8 
expect a population of NTS's in thermal equilibrium, with an exponential size distribution. 

(Although the initial NTS geometries are expected to be non-spherical, relaxation due to 

surface tension occurs rapidly.) 

From this discussion, several general features emerge. In the scenarios outlined above, 

charge concentration into solitons may arise either due to a net cosmological charge asymmetry (like the baryon asymmetry), or due to random fluctuations in a Universe containing 

zero net charge. We will treat the first case in this paper. In addition, if thermal equilibrium is approximately maintained down to a temperature of order Tc, as above, we expect 

a quasi-exponential size distribution of solitons produced in a phase transition. 

. 

Finally, independent of what may happen in a phase transition, NTS's may form at 

temperatures below TG by several mechanisms. First, as mentioned in the Introduction, in 

models with small Qmjn, solitons may form via accretion and fusion after the temperature 

falls below the binding energy IQ. Second, some field theory models give rise to both 

topological and non-topological solitons. For example, if 4 and Q are complex scalar fields 

in a theory with the symmetry breaking U(1) x V(1)' 4 U(l), the spectrum contains both 

cosmic strings and non-topological solitons. For appropriate couplings, the strings carry 

trapped charged aero modes which, when excited, act as a supercurrent." In this case, 

NTS's can be spawned in the decay of charged superconducting strings, when the charge 

condensate separates &om the underlying topological vortex.*J' As a final speculative 

possibility, in some models non-topological solitons might form in the gravitational collapse 

of density perturbations, due to unstable spatial fluctuations in the classical c field. 

Thus, there are a variety of ways in which non-topological solitons may be born in 

the early universe. However, in the temperature range Tc 2 T 3 IQ, they are not the 

lowest kee energy states of given charge, and are vulnerable to evaporation. To study 

9 
this process, we review the properties of NTS’s in equilibrium at finite temperature in 

the next section. We shd confine our attention to temperatures at least slightly below 

the Ginzburg temperature TG, so that finite temperature corrections to the structure of 

NTS’s’O may be neglected. 

IV. Non-topological Solitons in Equilibrium 

We are interested in the question of whether NTSs formed in a primordial phase 

transition could have survived to contribute significantly to the energy density of the 

Universe. First consider NTSs in thermal equilibrium, where a distribution of lumps of 

as1 charges Q coexist with free 4 particles. In kinetic equilibrium, the number density of 

NTSs of charge Q at temperature T is 

I where g9 is the internal partition function for a NTS of charge Q, p(Q) is the NTS 

chemical potential and M(Q), defined in Section 11, is the NTS mass. In Eqn. (8), we 

have made the reasonable assumption that NTSs are nonrelativistic, M(Q) > T, and 

nondegenerate, p(Q) << T. At temperatures T << m+, the distribution of free tp particles 

obeys an expression analogous to Eqn. (8). 

Chemical equilibrium between the relative number densities of 4’s and Q’s is established if the accretion and evaporation reactions, (Q+l) t) (Q)+tp, occur rapidly compared 

to the expansion rate. In this case, the chemical potentials obey 

Inverting Eqn. (8) and substituting in Eqn. (9), we obtain the fa.milia.r Saha equation 

10 
where IQ is the NTS binding energy per charge [Eqn. (S)]. Unless the minimum stable 

NTS charge is very small, Qmin 2 1, the reactions above are in fact not sufficient to 

establish chemical equilibrium. If, as an initial condition, the universe contains only q5 

particles, accretion done obviously cannot generate solitons. The missing ingredient is the 

fusion reaction, q5 + rp + ... + q5 + Qmin. 

Typically, one expects chemical equilibrium to be maintained until either the fusion 

or the accretion/evaporation rates become slower than the expansion rate. In principle, 

the proper way to study this freeze-out process is to integrate the (infinite set of) coupled 

rate equations for the number densities of all charges and free particles, ;.e., numerically 

solve the Boltzmann equations for the system. We do not follow this approach. Instead, 

in Appendix A, we estimate freeze-out number densities using the “poor cosmologist’s” 

approximation: the freeze-out densities are roughly the equilibrium densities at the epoch 

when the reaction rates drop below the expansion rate. 

Such an approach, however, assumes that chemical equilibrium is initially established 

and then lost. For most NTS models, this is unjustified. In particular, when Qmin is not 

very small (as is usually the case) and/or there is a potential energy barrier between the true 

and false vacua (the NTS vacuum is a metastable minimum), then, as we show in Appendix 

A, the fusion rate is strongly suppressed and can never approach equilibrium. (Essentially, 

this is analogous to what would happen in big bang nucleosynthesis if the lightest stable 

element had a large baryon number.) In the limit that fusion can be neglected, the total 

number of NTSs is a non-increasing function of time. Since new lumps are not being 

produced, it makes sense instead to follow the evolution of individual lumps of initial 

charge 8 separately. (We shall also assume that accretion of lumps by lumps is rare.) We 

develop this approach in the next Section. 

11 
V. Evolution of Single Lumps 

To follow the evolution of individual solitons, we need to calculate the emission and 

absorption rates of massive q5 particles by NTSs of charge Q. To do this, we apply detailed 

balance arguments in the usual way.ll First, we massage the Saha equation into a more 

convenient form. Since 4 is a charged field, we assume it has g+ = 2 degrees of freedom. 

In addition, up to corrections of order T/m+, we expect gq+l/gq cz 1, and we can further 

approximate M(Q + l)/M(Q) = 1 + O(p/Q). Substituting into Eqn. (lo), we find 

In chemical equilibrium, the rate at which NTSs emit 4's is equal to the absorption 

rate. Since the binding energy grows with an increase in charge, absorption is energetically 

favored. Thus, if the particles are sufficiently strongly interacting, we expect a geometric 

cross section for absorption, 

where the NTS radius R(Q) is given by Eqn. (2). Here, f+(< 1) is the absorption efficiency, 

which depends on coupling constants in the theory; we discuss it in more detail in Appendix 

B. The accretion rate per unit volume is simply 

where V+ = (T/2~m#)'/~ isthe mean velocity of free non-relativisitic 4 particles. The 

emission rate from a lump of charge Q + 1 is proportional to the number density of lumps 

times the evaporation rate, yetrap, 

12 
The two rates must be equal in chemical equilibrium; using Eqns. (11-14), the evaporation 

rate from a NTS is 

Although the evaporation rate was derived from the assumption of chemical equilibrium, Eqn. (15) applies more generally; in particular, it is valid whenever the q5 particles 

are kept in kinetic non-ddgenerate equilibrium [Eqn. (8)], e.g., by collisions with the ambient plasma. (The power of detailed balance is that it applies to situations outside of 

chemical equilibrium.) Thus, the accretion and evaporation rate from a lump of charge Q 

is 

Eqn. (16) describes the evolution of an individual soliton in a thermal bath of 4 particles. 

Before applying it, we must expose two additional hidden assumptions.l' i) As it evaporates 

or accretes, an NTS is cooled or heated with respect to the environment; we assume, 

however, that it is kept in good thermal contact with the surrounding medium. ii) We 

assume the exterior r$ gas is dilute, so that the flow of particles into or out of the lump is 

not inhibited. 

For the remainder of this paper, we suppose there is a primordial asymmetry of 4's 

over anti-r$'s, q+ [ns - n4]/n7, where n7 = 2.4T3/n2 is the photon number density 

at temperature 2'. We shall further assume that 4 and 6 are stable, but that they can 

annihilate into a pair of mosaleaa particles, say, neutrinos or photons. (These assumptions 

will be relaxed in paper 11.) The latter assumption guarantees that annihilation takes place 

inside as well as outside solitons. For stable r$'s, the Q asymmetry can be written as 

74 = 2.5 x 10-*f14h2(GeV/rn+) , 

13 
where h is the Bubble constant in units of 100 km/sec/Mpc, and n+ is the 4 energy 

density in units of the critical density. Requiring that 4 particles and charges do not 

overclose the universe, R4h2 5 1, yields an upper bound on 774 as a function of m4. We 

define j as the fraction of 4 particles in the NTS phase, so the number density of free 4 

particles is n#(T) = q+(1 - j)nr. The initial value of j depends on the efficiency with 

which NTSs are formed; e.g., in case (ii) of Section 111, we expect j; N pfalrc 5 0.5, while 

for case (i), Witten envisioned j; - 0.9. In general, however, j is a temperature dependent 

function which increases (decreases) during the accretion (evaporation) epoch. With these 

definitions, the expression for the evolution of a lump becomes 

where f(Q) is the binding energy per q5 mass [Eqn. (7)]. 

The evolution of a non-topological soliton is determined by three temperature scales. 

The evolution begins at the formation temperature Ti, which is of order the Ginzburg 

temperature TQ or smaller, depending on the formation mechanism. Once formed, the 

competition between the two terms in the brackets of Eqn. (18) determines the fate of 

the soliton. These terms are equal at the “turnaround” temperature Tt, which marks the 

transition between evaporation and accretion phases. Finally, the freeze-out temperature 

TF is defined as the temperature at which both the emission and absorption rates fall 

below the expansion rate, (1Q[/QH), = 1. Consider a model with Tj - m+. From 

Eqns. (17) and (18), at T - m+ evaporation dominates over accretion due to the small 

asymmetry term. Thus, only those charges which are large enough to survive down to Tt 

or TF are present today. There are two exceptions to this general picture. 1) If 4 is an 

unstable particle with a lifetime shorter than the age of the universe, the asymmetry is 

not constrained by Eqn. (17), and may be large. 2) If 4 is stable, but only annihilates into 

14 
particles more massive than f(Q)m4, then annihilation inside NTSs is suppressed at low 

temperature. In this case, NTSs can grow by accreting both 4’s and J’s, and the absorption 

rate is independent of 74. We discuss both of these scenarios in paper 11. 

To study lump evolution, it is convenient to introduce the dimensionless temperature 

variable, 

ZE~+/T . (19) 

During the radiation-dominated early universe, t = 0.3g,’12(mp~/m$)z2 (where g+ is 

the effective number of relativistic degrees of freedom), so the evolution equation can be 

rewrit ten, 

. 

From the discussion above, there are several possible fates that may befall a NTS with 

stable 4’s: a) death by evaporation; b) evaporation followed by freeze-out, 2j < ZF < 22; c) 

evaporation followed by accretion and then freeze out, zj < zt < ZF; d) accretion followed 

by freeze-out, tt < zj < ZF; and e) frozen at birth, zj > XF. Which of these fates obtain 

depends on the mass scale, the NTS charge, the asymmetry parameter, and the formation 

temperature. 

The turnaround temperature T’, at which accretion takes over from evaporation, is 

given implicitly by 

Eqn. (21) may be derived either by equating the accretion and evaporation terms in Eqn. 

(20) or by setting nq = n~+1 in Eqn. (11). In Figure 2 we show zt = m+/Tt as a function 

of q+(1- j), for several values of f(Q). Two trends are obvious. For fixed binding energy 

f(Q), turnaround occurs ‘later’, ie., at larger zt, as 776 decreases, since the supply of kee 

15 
4 particles is dropping. Thus, from Eqn. (17)) for a large mass scale m+, the evaporation 

phase is relatively long in units of the mass scale. Second, NTSs with lower binding energy 

per unit mass turn around later, as expected. For models with p < 1, f(Q) + 1 as Q + 00, 

so lumps with large charge spend less time in the evaporation phase than their smaller 

brethren. (The structure in Fig. 2 at very large asymmetry, 2 1, is only relevant if 4 is 

unstable, so we postpone discussion of it to paper 11.) 

Now consider the freeze-out temperature TF. During the radiation-dominated epoch, 

the expansion rate is H = 1.67gi/2T2/mpl = 1.67g:/a(m:/mp1)~-2. To obtain an analytic 

estimate of ZF, we consider case (b) separately from (c) and (d). For case (b), freeze-out 

occurs during the evaporation phase, when the absorption term can be neglected; using 

Eqn. (20), we find 

ZF = [In ( 2,62gt/2) f+P2 +43.9-In(%) GeV - (l-F)lnQ] , (ZF <zt) . (22) f (Q) 

Note that as the mass scale m+ or the charge Q increases, freeze-out occurs earlier, i.e., 

at smaller 2. Also, for large relative binding energy f (Q), NTSs freeze out at early times 

since more strongly bound objects evaporate less efficiently. 

For fixed mass scale, the freeze-out epoch XF grows as Q drops (unless f+ < 1, see 

Appendix B). Therefore, for small enough charge freeze-out will occur crfter turnaround. 

We denote the charge at the transition point, XF = 22, by Qeq; for Q < Qeq, we have 

ZF > zt, ;.e., cue (c) or (d) above. From Eqns. (17), (21) and (22), 

For p = 3/4 and 1, Qeq scales as mz4 and mg6 respectively. The dominant dependence 

in Eqn. (23) resides in the first two terms. Thus, approximately independent of p, for 

m+ > 0(104) GeV, we have Qeq < 1. Since Qeq is the largest charge which can have an 

accretion phase, this mass scale marks an important boundary. 

16 
Given that NTSs are generally born in the evaporation phase (except for case (d)), we 

would like to know how large the initial charge must be in order to survive to the accretion 

or freeze-out epoch. Approximately integrating Eqn. (20) during the evaporation phase, 

the smallest charge which survives is 

where we have assumed XF,Z~ >> z; and have used a standard approximation for the 

exponential integral. In arriving at Eqn. (24), we have also made use of the fact that, for 

Q >> 1, f(Q) is approximately independent of Q. For models with p = 3/4, the surviving 

charge scales as mG2, while for solitons with p = 1, Q, scales with mG3. 

F'rom Fig. 2, we see that typically zt - O(10-100). Thus, comparing Eqns. (23) and 

(241, we find Q,(zci) > Qeq if f(Q)t; < 17 + ln(m+/GeV). That is, unless NTSs form at 

a temperature well below f(Q)m#, cases (c) and (d) are eliminated: all those that would 

accrete before freeze-out in fact do not survive. If NTSs are created in a phase transition, 

we generally expect them to form at a temperature within an order of magnitude of their 

mass scale, .1 < zj < 10, for which we find Q, > Qeq. This can be circumvented, on 

the other hand, if NTSs form well after the phase transition is completed, for example by 

pinch-off in the asymmetric vacuum model, or by the decay of cosmic strings. In these 

cases, xi > 1, Q,(zi) can be substantially reduced, and an accretion phase is possible. 

For example, with p = 3/4 and f(Q) = 1 we obtain, for zi = 1 and zi = 10, respectively, 

Q,(l) - 1033(GeV/m+)2 and Q,(lO) - 102'(GeV/m+)2. 

For NTSs formed at relatively small values of z;, then, the only hope of survival is 

to start with sufficiently large charge, Q > Q,. However, this runs into two potential 

difficulties. First, for most of the formation mechanisms discussed in Section 111, large 

17 
charges are exponentially suppressed and thus may be rather rare. As a result, their 

contribution to the present energy density of the universe is likely to be small. Second, 

it is difficult to imagine forming lumps with a charge larger than that contained in the 

particle horizon at the time of formation. The total charge inside the horizon is 

4 48R+h2x3 GeV 

312 (F) ' 

Q&) = 1.2 x 10 

9* 

(25) 

For example, for p = 1, with f(Q) z zi N 1, we have Q,(l) > QH(~) for m,p > lo-' 

GeV; that is, solitons formed at early epochs (zi - 1) in these models are never large 

enough to survive (Q > Q,) if their formation is limited by causality. Only if they form 

at a sufficiently late epoch, since Q, depends exponentially on f(Q)zi, can this problem 

be overcome. For models with p = 3/4, on the other hand, at zi = 1, we find Q,(l) < 

QH(~) ifm,p < lo6 GeV. In this case, models with sufficiently small mass scale can 

have surviving NTSs which form early. Again, at later formation epochs, the situation 

can change substantially; e.g., for zi = 10 and p = 3/4, we find Q,(lO) < Q~(10) for 

m,p < 10l2 GeV. 

In Figure 3 we summarize our results for the case in which there is a primordial charge 

asymmetry. Here we have chosen f+P2 = 1, 9. = 100, &h2 = 1, f(Q) = 1 and the two 

cases p = 3/4 and 1. As a function of mass scale, we compare the charge inside the 

horizon at formation, QH(z~), with the smallest charge that survives evaporation, Q,(zi), 

for 2; = 1 and 10, obtained by numerically integrating the rate equation, Eqn. (20). For 

models with Q,(zi) > Q~(zi), even horizon-sized solitons disappear by evaporation. For 

f(Q)~i = 10, both models have a window of survival if m,p < 10l2 GeV. We also show the 

charge Qeq at which freeze-out and turnaround coincide, ZF = ~t. Recall that if Q > Qeq, 

the soliton will freeze out before turnaround; in the opposite case, there may be accretion 

before freeze-out. However, for all models and mass scales, Fig. 3 shows that Qeq << Q, 

18 
so that these charges die before they have a chance to grow. In Fig. 4, we show results of 

the numerical evolution of Q(z) for a sample model with p = 3/4 and m+ = lo4 GeV, in 

the neighborhood of Q,(l) N For a very narrow range of Qi, the value of the charge 

at freeze-out depends sensitively on the initial charge. Outside this range, the final charge 

is either zero (for Qi < Q,) or 31 Qi (for Qi > Q,). 

As noted above, for m+ 5 lo4 GeV, it is possible to have ZF > zt, so that an 

accretion phase is possible before freeze-out occurs. Thus, even if all solitons formed at 

zi - 1 evaporate away, a later period of accretion starting from individual charges can 

build up new solitons before freeze-out As noted in Section IV, this is only feasible 

for very small values of the minimum charge, Qmin - 1, so that free t$ particles can act as 

seeds for new lumps. We return to this possibility in Appendix A. 

VI. NTS Boiling 

Although NTS's may be formed at a phase transition at T,, they are not the lowest 

free energy phase until the universe cools below Tt. In the last section we studied the 

struggle for survival of NTSs against surface evaporation. In this section we discus8 the 

boiling of NTSs, i.e., bubble nucleation of the lowest Gee energy phase (massive particles 

in the true vacuum) throughout the volume of a NTS. 

Imagine heating up a NTS from zero temperature. Surface evaporation starts at 

temperatures just above Tt (or TF, if T' < TF). Bubble formation is negligible for temperatures just above Tt, due to the energy cost associated with the bubble surface. However, 

at higher temperatures (2' - Ta), bubble nucleation throughout the volume may dominate the evaporation process. The probability of bubble nucleation is very sensitive to the 

surface tension of the bubble wall, Ub, which is determined by the mass scale and coupling 

constants. NTSs can survive the boiling phase if ab is greater than a critical value derived 

19 
below. 

Following Ref. 12, we calculate the thermodynamic work expended to create a bubble 

of kee 4's inside a NTS: 

where T is the radius of the bubble, Pb is the pressure inside the bubble, and PNT~ is the 

pressure inside the NTS. W is maximized at the critical radius T, = 2cb/(Pb - PNTS). 

Bubbles with radius T < T, shrink away while those with T 2 T~ grow. The nucleation 

rate, I', is determined by the abundance of critical size bubbles, 

where 

The soliton is assumed to be in pressure equilibrium with its surroundings. Therefore, 

PNTS is just equal to the exterior thermal radiation pressure. The pressure inside the 

bubble, Pb, has the same thermal component, plus the pressure due to the massive 4 gas, 

P4. Hence, 

Pb - PNTs = P# = (2/7"~~'2T5''ezp(-~~/T) , (29) 

and 

At temperature T, the characteristic number density of bubbles, nb, is of order rt, 

where t is the expansion timescale, so nb N 0.39.-'l2mplT2erp(-w,/T) . If nb >> nNTs, 

the charge density inside the soliton, the NTS phase is so far out of thermodynamic 

equilibrium that the probability of remaining in such a phase is negligible. (The NTS would 

20 
likely not have been formed, let alone survived.) The condition nb < nNTS translates into 

a constraint on the surface tension ub > Umin, where 

Here we have assumed a spherical NTS lump of charge Q and radius RQ given by Eqn. 

On the other hand, if Ub is too large, boiling may never occur. Boiling will only 

dominate surface evaporation if the surface area of the NTS lump is smaller than the 

surface area of all the bubbles inside the lump. Boiling is more effective than surface 

. 

evaporation if ub < Umaz, where Umaz is given implicitly by: 

Both umin and umoz depend on the specific NTS model. In Fig. 5, we plot umin(z) 

(lower curve) and uma,(x) (upper curve) for different values of m+, where we set Q = 

Q~o~(mg,z = 1) [Eqn. (25)]; we have chosen f(Q) = 1, except for the case m+ = 1 GeV, 

where we set f(Q) = .OS, characteristic of strange matter. We have also set p = p = 1, 

since the dependence on p (- O(1)) is only logarithmic, and the difference between p = 1. 

and p = .75 is unnoticeable. For ub < umin, there is copious boiling and NTSs disintegrate. 

Between the two curves, amin < ub < umoz, boiling dominates over surface evaporation, 

and for ub > Umaz boiling can be neglected. 

Rom Fig. 5, we see that, for models with f(Q) N 1, boiling is important if xi - 1. 

In these models, the surface tension q, must be larger than about .3m$ to avoid boiling. 

Alternatively, for smaller Ub, NTSs can survive if they form later. More generally, the 

21 
efficiency of boiling is large for f(Q)zi 5 1. For example, for strange matter, f(Q) 11 0.05, 

2; 2! 10, and strange lumps can only survive iP2 bb 2 .006mi (much larger than the 

theoretically expected value4). 

VIII. Conclusion 

We have discussed the cosmic evolution of non-topological solitons, developing a general analysis which applies to virtually all classes of models, and thus separating general 

features from model-specific results. We have found that, for a wide range of parameters 

and models, primordially formed NTSs of moderate charge disappear quickly, either via 

surface evaporation or bubble nucleation, leaving behind a distribution of free massive particles. For a range of mass scales, depending on the scaling parameter p, there is a window 

of survival between the smallest charges which survive evaporation, Q,, and the largest 

that can be causally formed, Qw. This window, in turn, depends on the epoch of NTS 

formation; in scenarios with delayed NTS formation, the chances of survival are enhanced, 

and, if m,#, 5 lo' GeV, sufficiently small charges may then undergo an accretion phase. 

Our results pertain to models constrained to have small charge asymmetry by the existence 

of stable q5 particles. If this constraint is relaxed, the results can chaage considerably (d. 

paper 11). 

22 
Acknowledgements 

This work originated at the Aspen Center for Physics 1988 summer workshop, and 

much of it was completed while two of us (JF and AO) were visitors at the Institute for 

Theoretical Physics, Santa Barbara. We thank both institutions for their hospitality. This 

research was supported in part by NASA grant NAGW-1340 and the DOE at Fermilab, by 

NSF Grant PHY-82-17853, the DOE, and NASA at the University of California at Santa 

Barbara, and by DOE contract W-7405-ENG-48 at Livermore. 

Appendix A 

In this Appendix, we consider a system of interacting r$ particles and solitons which 

reaches chemical equilibrium for a time in the early universe. Recall that this might occur 

in two ways. 1) If Qmin - 1, chemical equilibrium can be established by the accretion 

and evaporation reactions, (Q + 1) c) (Q) + 4. (This possibility was recently studied 

by Griest and Kolb13 in the context of a particular toy NTS model with p = 3/4.) 2) 

In most NTS models, unless the coupling constants are specially chosen, Qmin > 1. In 

this case, the fusion and dissociation reactions 4 + 4 + ... + 4 t.) (Q) are also required to 

establish complete chemical equilibrium. However, at the temperatures at which fusion 

can be effective in generating new solitons, T < Tt, these reactions are likely to be strongly 

suppressed compared to the expansion rate, for reasons given at the end of this Section. 

In both cases, in chemical equilibrium the chemical potentials are related by 

where p(Q),p(r$) are given implicitly by Eqn. (8). We define the mass defect by 

BQ = em+ - M(Q) = Qm4(l- aQP-') Qm+f(Q) , 

23 
and note that f(Q) + f(Q) [Eqn. 71 as p -+ 1 or as Q -+ 00, independent of p. Then, from 

Eqns. (l), (8), (A.l) and (A.2), the number density of solitons of charge Q in chemical 

equilibrium can be written 

It is convenient to introduce the charge fraction of solitons of charge Q, YQ z nQQ/N, 

where the total charge density is given by 

Note that, in general, the charge fraction is different from the mass fraction. From Eqns. 

(A.3) and (A.4), we have 

and the constraint 

CyQ=j . Qmir 

The charge fraction is obtained by simultaneously solving Eqns. (A.5) and (A.6). 

The qualitative behavior of the NTS population is relatively simple to understand. 

At high temperature, the relative abundance of NTSs with large Q is suppressed by the 

small asymmetry factor - $-I. At lower temperature, the exponential and Q1+(3p/2) 

terms favor the formation of large Q objects. The trend from small to large Q shifts at the 

'turnaround' temperature defined in Section V: in the Saha equation, we find n~ = ~Q+I 

at T = Tt. Since f(Q) is approximately constant for large Q, zt is roughly Q-independent. 

Thus the turnaround is rapid: slightly above Tt, large charges are strongly suppressed, 

while slightly below it, they are strongly favored. As the temperature drops through Tt, 

24 
large charges quickly build up until the available 'fuel' of free particles is depleted, thereby 

shutting the system out of chemical equilibrium. 

At high temperature, z < zt, YQ falls rapidly to zero with increasing Q. Consequently, 

in this regime, only the first few terms contribute significantly to the sum in Eqn. (A.6), 

and the equilibrium charge fraction may be easily calculated. Thus, if XF < zt, ie., 

if freeze-out occurs before turnaround, the final abundance YQ[xF(Q)] can be estimated 

analytically. As discussed in Section V, this is always the case for m+ > 10' GeV. At 

1: < 1:t, we expect the total NTS charge fraction to be small, j a 1. To zeroth order of 

approximation, we can then estimate YQ by setting j = 0 in Eqn. (A.5). 

Alternatively, we can obtain a more transparent expression for YQ(ZF) without approximation. Setting g+ = 2 as in most cases of interest, the freeze-out abundance for 

XF < zt may be written 

where ZF(Q) is given by Eqn. (22), f(Q) by Eqn. (7) and f(Q) by Eqn. (A.2). If we 

approximate f(Q) = f(Q), valid at large Q, this may be rewritten using Eqn. (23), 

This gives the keeze-out abundance for Q >> Qcq and thus ZF < xt. The case ZF - xt 

must in general be handled numerically. 

We finish this section by discussing the fusion of free 4's into a soliton for models in 

which Qmin is not small. This rate must be faster than the expansion rate in order to 

maintain chemical equilibrium. We shall show that this is unlikely at temperatures below 

25 
m+. We estimate the rate by assuming that a soliton is formed when Q 2 Qmjn 4 particles 

get together in a fluctuation with a charge density comparable to the NTS’s charge density. 

This criterion can be made rigorous by considering the effective u potential U(Q,~) in a 

background of non-zero charge density q. At a charge density q - qNTS, the unconfined 

phase Q = no becomes degenerate with the NTS phase and at higher densities the NTS 

phase is preferred. If the vacuum potential U(u,O) has no barrier (see Fig. l), then a 

soliton forms whenever a sufficiently dense charge fluctuation occurs. However, if there 

is a barrier between the NTS state and the unconfined phase (dashed curve in Fig. l), a 

charge fluctuation is not sufficient to form an NTS: in addition, the u field must tunnel 

through the barrier, leading to an extra suppression of the fusion rate. We will ignore this 

factor, so the estimate below will yield an upper bound on the formation rate. 

For simplicity we focus on the case p = 1; the results may be easily generalized to 

other models. Consider a fluctuation region with fixed volume Vmin = (4~/3)P’m~~Q~i~, 

the size of a NTS with charge Qmjn, containing an average charge Q = BV’in, where the 

mean charge density Q N (2.4v+/x2)T3. The probability of finding a charge Q in a region 

with average charge Q is of order 

P(Q, Q) 21 (27rQ)-’/’ezp [ - (*it”] dQ . 

Here we have assumed Q is of order a few or more, so that the Gaussian approximation 

to the Poisson distribution is reasonably accurate, and have ignored antiparticles. We 

are interested in a fluctuation with charge Q > Qmjn. At the temperatures of interest, 

T C m+, we can safely assume v+ < (m4/pT)3, so that Qmin > Q. Thus, the probability 

of a fluctuation with charge large enough to make a soliton is 

(A.lO) 

26 
The number density of minimum charge NTSs is n,,l N P(Q > Qmin)/Vmin, and their 

abundance relative to free 4 particles is approximately 

(A.ll) 

Thus, if Qmin is large and/or the asymmetry T+ is small, the formation of solitons from 

thermal fluctuations is strongly suppressed at temperatures below m+ (z > 1). 

Appendix B 

The factor f+ measures the probability of absorption into a NTS of an’ incoming 

particle 4, relative to its geometric cross-section. Since the mass term for 4 generally 

drops abruptly (and monotonically) as the NTS surface is approached from the exterior, 

aa incoming 4 particle effectively sees a square well potential inside the NTS. Since there is 

no potential barrier, 4 particles have no difficulty entering the NTS, but they can as easily 

escape at the other side. The incoming 4’s (massive outside the NTS) have energies higher 

than the binding energy of the massless 4 gas inside the NTS. Therefore, f+ is determined 

by the probability of an incoming 4 being trapped by scattering down to lower energies. 

The probability of at least one collision inside a soliton of radius RQ is given by 

pc = 1 - ezp(-RQ/A+), where A+ is the mean kee path of 4’s inside the NTS. Thus, 

A+ = (n,u+)-l, where n, is the number density of scatterers and cr+ is the cross section €or 

&scattering. If RQ > A+, the incident 4 will have enough scatterings to be trapped, and 

f+ = pc N 1. On the other hand, if RQ < A+, then p, N RQ/A+ < 1. In this case, there 

will typically be at most one scattering, and f+ is given by the probability of scattering 

und losing enough energy in one collision to be trapped. 

If a 4 particle has initial kinetic energy E outside the NTS, it will have energy Ei = 

E + rn+ inside the NTS. (Here, we are assuming 4 is massless inside an NTS.) The particle 

27 
is captured if it scatters down to a final energy inside the NTS, Et < m+. In the case 

of multiple scattering, capture generally occurs, since the incident energy E - T 5 m+. 

In the single scattering regime, A+ 5 RQ, the capture probability is model dependent. 

If we consider isotropic cross sections, on average half of the incident energy is lost per 

collision. If the target particles responsible for capture are not 4 particles, the condition 

for trapping in one scattering is roughly E < m+. However, if the dominant scatterers are 

4's, the incident energy must be lower; otherwise, the initially bound 4 will be scattered 

out of the NTS, with no net gain in charge. In this case, the condition for single scatter 

capture is approximately E 5 IQ, and thus T 5 IQ. 

The mean free path A+ is also model dependent. In the simplest NTS models, the 

possible interactions are either 4-4 or 4-u scattering. In some models, although the crosssection for 4-a scattering is appreciable, the density of u particles inside the NTS is small 

(e.g., if u is very heavy). In these models, 4-4 scattering can occur either through a 4' 

term in the Lagrangian, or through the exchange of a u particle. 

For example, consider a model where 4-4 scattering through a g4' interaction dominates, with cross-section u, = g2/327rEt. If the 4's inside the NTS are non-degenerate, 

the density of scatterers is n, = 3Q/(47rR$). Then 

1 -2p/3 

f+=I-ezP[-(g) ] ' 

where 

In this case, f+ - 1 for Q >> Q. (multiple scattering), and f+ 3 (Q/Q.)'-2p/S for Q < 

Q.. At z = 1, for models with p = 3/4, we find Q. - 106(p/g)', while for p = 1, 

Q. - 10Q(p/g)6. Thus, the approximation f+ = 1 is valid for large charges, but for small 

28 
charges the scattering probability is suppressed. Consequently, small charges freeze out 

earlier; for Q < Q,, 

and 

In principle, the distribution of NTS's with charge Q << Q+ may be frozen out from the 

beginning (ZF 5 zi) and thus survive. For these objects, the mean free path is much 

larger than the NTS size, so the timescale to repopulate the evaporated tail of the interior 

q5 distribution is longer than the expansion timescale. In this case, the initial distribution 

of small NTSs is preserved, and may contribute significantly to the energy density of the 

universe. On the other hand, if 4 is kept in thermal equilibrium by scattering with other 

particles, the mean free path is much shorter and we expect f+ N 1 down to smaller values 

of Q. 

29 
References 

1. A. Chodos, R. J. Jaffe, K. Johnson, C. B. Thorn, and V. F. Weisskopf, Phys. Rev. 

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30 
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31 
Figure Captions 

1. Schematic cr (confining field) potentials for theories with NTS solutions, with a presumed NTS phase at cr = 0 and true vacuum cro = 2.5 (arbitrary units). For the 

dashed curve, the ‘NTS vacuum’ is metastable in (the absence of charge), while for 

the solid curve it is unstable. 

2. The turnaround epoch xt = m,p/T as a function of the asymmetry parameter q, for 

different values of the binding energy per unit mass, f(Q) = 0.05, .25, .36, .5, and 1. 

3. The minimum charge which survives evaporation, Q,(zi), is shown as a function of 

the mass scale nz+ for the two cases p = .75,1, and for xi = 1 and 10. Also shown are 

I the charge inside the horizon, QH(z~) and the charge below which an accretion phase 

~ 

is possible before freeze-out, Qcq. 

4. A sample numerical integration of Q(z), for the parameters f(Q) = 1, m+ = lo‘ GeV 

and p = 3/4. In this example, Q, N 

5. The surface tension crmin and cma, as a function of z, for mass scales m,p = 

lo2, lo5, lOI5 GeV, for f(Q) = 1. Also shown is the strange matter case, f(Q) = 0.05, 

I rn4 = mn N 1 GeV. Boiling dominates evaporation for ub < CmaS. 

32 
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