Structure formation from unstable domain walls

NUCLEAR

PH VS I C S B

Nuclear Physics B406 (1993) 45 2—470 North-Holland

Structure formation from unstable domain walls S. Lola

*

and G.G. Ross

**

Department of Physics, University of Oxford, I Keble Road, Oxford OXI 3NP, UK Received 12 January 1993 Accepted for publication 29 April 1993

We discuss the field theory expectations for very light scalar states which may give rise to a late stage of spontaneous symmetry breaking, paying particular regard to the spontaneous breaking of the discrete symmetries which commonly occur in compactified string or Grand Unified theories. We consider the possibility that the large scale structure of the universe is generated due to domain walls associated with this late phase transition. Efficient dissipation of the wall energy in a dark matter component may occur leading to large local density fluctuations which generate a component of the large scale structure today. Provided the walls decay at an early time the magnitude of fluctuations needed to explain the present day structure is consistent with induced fluctuations in the cosmic microwave background radiation of order i0~. The model also predicts early quasar formation (at redshifts 10).

1. Introduction In recent years there has been a lot of interest in the possibility of explaining the observed large scale structure of the universe as a result of a phase transition associated with very light scalar fields. In these schemes either the topological defects of the theory [11 or the dynamical evolution of the fields themselves [2] induce large local density fluctuations in the matter component and these subsequently grow to reach the current values. It is of some interest to ask whether such fields are to be expected in reasonable particle physics models. Hill et al. [11 suggested they be identified with “schizon” fields, pseudo-Goldstone bosons associated with a symmetry of the tree level scalar potential which is broken by the Yukawa interactions of the theory [31. The scalar mass is generated radiatively in these models and is related to the Yukawa couplings. If these are small, as might be expected if they are related to neutrino masses [1], it is plausible the mass will be in a cosmologically interesting range. More generally, pseudo-Goldstone fields with masses in this range appear in a wide class of Grand Unified and String Theories. These are associated with the * * *

Supported by the Republic of Greece State Scholarships Foundation. SERC Senior Research Fellow.

0550-3213/93/$06.00 © 1993

—

Elsevier Science Publishers B.V. All rights reserved

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spontaneous breaking of continuous symmetries exact at the tree level but broken by (small) non-renormalisable terms in the effective potential. Such structure arises automatically in theories with discrete symmetries and the symmetry can naturally give a pseudo-Goldstone mass in a cosmologically interesting domain. Although such symmetries may arise in schizon theories, it is not necessary that the terms breaking the continuous symmetry be associated with Yukawa couplings. The advantage of theories with spontaneously broken discrete symmetries is that the symmetry also controls the radiative corrections to the pseudo-Goldstone mass so that their smallness may be maintained naturally to all orders. It is known that such symmetries commonly arise in compactified string models or in Grand Unified models; indeed in the supersymmetric versions some form of discrete symmetry is essential for a viable theory. In what follows we will discuss late phase transitions in models with spontaneously broken discrete symmetries. In sect. 2 we discuss the description of pseudo-Goldstone bosons in non-supersymmetric and supersymmetric models and the reason we expect them to occur in GUTs and superstring theories. In sect. 3 we discuss the multiple vacuum structure in a theory with light pseudo-Goldstone fields and the properties of the associated domain walls. We pay particular regard to the possibility these vacua be non-degenerate and the associated domain walls be unstable. Sect. 4 discusses the implications domain walls associated with a late phase transition have for structure formation. We show how unstable domain walls can play a significant role in producing structure and consider the spectrum of primordial fluctuations they can generate. We also point out that unstable domain walls lead naturally to early quasar formation. Sect. 5 presents some numerical estimates of structure formed in some of the field theory models introduced in sects. 2 and 3. Finally in sect. 6 we present a summary and our conclusions.

2. Pseudo-Goldstone bosons in unified models In realistic models of superstrings there are usually several discrete groups. We have found that these give rise to similar phenomena to the case of a single ZN discrete group [4] but with a large value for N (equivalent to the product of the component factors). For this reason we consider here only the simplest case with a single discrete group factor ZN, N> 4. The fields in the theory transform as ar r = 0, 1, 2, N 1, where a is the Nth root of unity, and the effective potential is constructed from ZN invariant combinations of the fields. The lagrangian of a complex scalar field 1, transforming as a then has the form ..

.,

—

~N

~

~j~*N

4

MN

+ MN4

+...

.

(1)

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In the above equation we have included the leading terms involving the modulus and the argument of ~. We have made the coupling A real by absorbing its phase in the field cP. The coupling A’ is of order unity. The non-renormalisable terms of dimension > 4 arise because we have an effective field theory generated by physics at some (high) scale, M. In Grand Unified theories M may be the Grand Unified scale (> 1016 GeV) at which a gauge symmetry is broken leaving a discrete gauge symmetry. Thus non-renormalisable terms are suppressed by inverse powers of this scale and so are naturally very small. In superstring theories M may be the compactification scale which is expected to be close to the Planck scale M~ 1018 GeV, again giving a strong suppression of non-renormalisable terms. In both cases, however, additional suppression occurs if the effective low energy theory is supersymmetric. Since supersymmetry offers the best hope for a solution to the mass hierarchy problem the need to explain why the electroweak breaking scale is so much less than the unification scale we will discuss this case shortly. If jx2 is positive, the effective potential for cP has a minimum for non-vanishing value of the modulus and leads to spontaneous symmetry breaking of the discrete ZN group. In this case it is convenient to reparameterise I~as —

—

=

(p

+

v) e1O/’(~,

(2)

where v eia is the VEV (vacuum expectation value) of I~,while p and 0 are real scalar fields. The potential of the field 0 is then 2A’v” =

ON

M~4c05(

+

Na).

(3)

This exhibits the main feature of a spontaneously broken discrete symmetry. In the absence of the non-renormalisable terms the lagrangian is U(1) invariant and 0 is the Goldstone boson associated with the spontaneous breaking of this symmetry. The non-renormalisable terms explicitly break this continuous symmetry and induce a mass for the pseudo-Goldstone boson, 0, given by

2 m

N2V =

0

,

V 0

2vN4 A’ MN_

(4)

For M much larger than v the mass is very small. Note that the potential of eq. (3) has an N-fold degeneracy corresponding to 0/v 0/v + 2ir/N. In the early universe we may expect field configurations corresponding to these different vacua to develop in causally disconnected domains. These domains will be separated by domain walls and the energy stored in these domain walls may have significant cosmological implications. —*

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We consider now the supersymmetric case. Keeping only leading terms, the potential in a supersymmetric theory has the form ~ V(cP)=~~, ,.~

2

(5)

where the sum is over the m chiral superfields, ~ in the theory, P(~P’)is the superpotential, and the chiral superfields are replaced by their scalar component fields cI when evaluating the RHS of eq. (5). If there is a ZN symmetry under which ~I’transforms as a, the superpotential has the form ~p2N

P=AMN3

+A’M2N3

+

...

(6)

.

However if supersymmetry exists it must be broken since the quarks and leptons do not have degenerate scalar partners. The supersymmetry breaking terms have to be small (on the unification scale) in order that the mass splitting between particles that belong to the same supermultiplets be ~ 1 TeV 0(10—15 M) otherwise the cancellation of loop diagrams involving fermionic and bosonic partners, needed to solve the hierarchy problem, is spoilt. The supersymmetry breaking is parameterised by soft terms (with couplings having dimensions of mass) of the form =

2

—

2

~0ft/H~I +AP~+h.c., (7) where ~xand A are of order m3/2, the gravitino mass characterising the breaking of supersymmetry (m3/2 1 TeV). The last term of eq. (7) is just the superpotential with the superfields, V’, replaced by their scalar components, c~7. The general form of the potential resulting from eqs. (5) and (7) is L =a

B~cp~* +,.~2I~j2

+(M~6I~I2N_1+

+F(111) AA)NN

+...,

(8)

where f3 is a constant determined by the couplings in the superpotential, F is some function of I and we have shown only the leading dependence on the argument of ~ coming from eq. (5). In this case, with the parameterisation of eq. (2), the potential for the pseudoGoldstone field becomes 2~’~ AA 2vN NO V(0) (M~N_l + ~)MN_4cos(_ +Na). (9) ~v =

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By comparison with eq. (3) we can see that since A is small (A

=

0(m5/2)

=

0(10- ‘°M)),the mass of the pseudo-Goldstone boson is now much smaller than in the non-supersymmetric case for a given value of v/M < 1 and N.

3. Vacuum and domain wall structure From eqs. (3) and (9) we see that there is an N-fold degeneracy in 0 vacua corresponding to the symmetry 0 0 + 2irv/N. Associated with these vacua will be domain walls lying between regions of different vacua. It is normally assumed that there is a period of inflation needed to generate the observed homogeneity of the universe. If this occurs after the domain wall production and if the reheat temperature is below that leading to ZN symmetry restoration (and the thermal energy is substantially less than the potential barrier between different vacua), then there will be a dilution of the domain walls and the observable universe may lie in a single vacuum state However for the late phase transitions of interest here this will not be the case and domain walls created after inflation may be expected. These are the walls associated with the appearance of domains of different vacua with size given by the horizon at the time of the late phase transition. Art immediate question concerns the stability of the different vacua and their associated domain walls. In the case discussed in the last section all vacua are degenerate. After the late phase transition causally disconnected regions will be based on a random sample of any of these vacuum states. As a result horizon size domain walls will continually be created as these causally disconnected regions come within the horizon. However, if the vacuum degeneracy can be lifted, at a redshift Za corresponding to the point at which the temperature drops below the difference in energy of the different vacua, the true vacua will grow in all domains to fill the whole universe. As a result there will be a maximum size for the domain walls corresponding to the horizon size at Za~ (We refer to this phenomenon rather loosely as “unstable” domain walls.) Since it is the largest domain wall that contributes most significantly to density fluctuations it is important to know whether there is a maximum size for a domain wall. It is possible to add soft terms to break the discrete symmetry but this is contrived and raises the question why there should be any discrete symmetry to start with. It is possible to generate small breaking terms if the discrete symmetry is derived from an anomalous gauge symmetry. However there are reasons why such a discrete symmetry coming from an anomalous symmetry may not be acceptable. It has been argued that only gauge symmetries (continuous or discrete) —~

~.

*

3t/(4s~-)will ultimately cause a population of more than During inflation the fluctuations <02) = H one minimum [5j. We do not consider this possibility here.

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are stable against large gravitational corrections. If this is so, only such symmetries are able to protect a (pseudo) Goldstone boson from acquiring a large mass. In this case are domain walls always stable? At first sight the answer appears to be yes. Even if one allows for arbitrary interactions amongst a number of fields transforming non-trivially under the discrete gauge symmetry, the existence of the underlying ZN symmetry means that if the vacuum state breaks this symmetry it will have an N-fold degeneracy and each degenerate vacuum state will be stable. However if, as is likely, the interactions between the fields make more than one field transforming non-trivially under the discrete symmetry group acquire a VEV, it is possible to generate a situation in which the vacuum degeneracy is apparently lifted. As we just discussed, if one of these fields acquires its VEV before or during inflation the observable universe will have a unique value for its VEV. After inflation the effective potential describing the remaining fields will have an approximate discrete symmetry and the vacua will not be exactly degenerate. 3.1. UNSTABLE DOMAIN WALLS

3.1.1. Non-supersymmetric models. To illustrate this consider adding to the above theory a second field 1’ transforming as a2 under the ZN group. Allowing for additional terms of the form ~j~N_2r~~r + h.c., r E ~ and dropping constants of order unity the effective potential describing the 0’ field becomes, instead of eq. (3), N/2 vN2rv ~r I rO’ V(0)= ~2MNr 4CosL~7+(N_2r)a+rI3), (10) where v’e~ is the ~D’VEV, for N even. For N odd, the potential has the same form, only now the sum goes from 0 to (N 1)/2. If the ‘i field acquires a large VEV, v (v > v’), before inflation ends the observable universe will have a unique value for a in eq. (2). Following from eq. (10) we see that for even N the leading term has r N/2 which has an N/2-fold degeneracy in 0’, while for N odd there exists an (N 1)/2 degeneracy. However, in both cases, the non-leading terms split this degeneracy. As a result if 4i’ has a small VEV so that after inflation thermal effects cause all the 0’ vacua to be populated, domains of different vacua will form but these vacua will be unstable withtothe by terms 2/(v’M) relative thesplitting barrier determined between vacua (we suppressed by powers of v assume v2/(v’M) is small, possible since v
=

—

22 —

N 2

2

MN/24’ v ,N/2—

1

(11)

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Structure formation from domain walls

for N even, and m2

)

2 2vv~(N_S)/ 2( N—i 2 M7~~2

=

v ,(N— 3)/2

6p=2vSM(N5)/2

(12)

for N odd. This example illustrates a general phenomenon that occurs when several fields with different ZN transformation properties acquire VEVs and suggests that unstable domain walls are quite likely to result from spontaneously broken discrete symmetries. We will see that the major problem in the models where the fields have the simplest possible transformation properties is that the terms that split the degeneracy are quite large, and comparable to the dominant terms. As a result the walls annihilate very early. This is not the case if the two fields transform as higher powers of a. If ~I transforms as am and çj’ as a’~under the symmetry group and assuming that n > m and N/n is integer in order to simplify the analysis, the potential is

=

r0’ ~

~

N/n

V(0)

r=O

2MN_(

m)r]/m-4

N +

nr

—

m

a

+

rP).

(13)

Let us now consider a specific example which will be useful as a model for structure formation. We assume a symmetry group Z 27 under which

(~) 2~-i

exp

~,

I 2’rri cP’—’exp~3-~—

(14)

~‘.

Then the terms giving rise to the potential for the pseudo-Goldstone boson are ~,9

~

M7 (15) M Provided that v3/M2 << v’ the first term will be the dominant one, while the second will split the degeneracy between the nine vacuum states of the theory. The scalar potential then will be v’9

V=—

5

‘90’\ cosl—I+ ~ v’

)

v3

v’M2

cos

80’\

—I v’

+...,

(16)

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where from now on we absorb factors of two in the definition of the VEVs of the fields. 3.1.2. Supersymmetric models. Similar results follow in the supersymmetric case but now there exists a much larger suppression of the pseudo-Goldstone mass. To illustrate this we consider the supersymmetric version of the example that we constructed in the non-supersymmetric case with a symmetry group Z~ and superfields transforming as 1!’ a, ~l”—s a3. The dominant terms in the scalar potential are —÷

v’9

V=A —cos M6

90’ v’

M’4

cos

M8

—

v’

I

)

+

150’

v~v’15

+144

80’\

v~v’8 +

cos

—

+...

(17)

.

The first term in the potential dominates for v3 —
M2

v~v’6

and

144

M9


which indicates that the lowest VEV, v’, has to be less than 4 case the mass of the pseudo-Goldstone boson is

=

(18) ><

103M. In this

81m 3/2~,

(19)

while the difference in potential energy between minima, öp, is 8

3p=m

v~v’ 8

or 144 M14 (20) 3/2 M depending on which is the next to leading term in the scalar potential. To summarise this section, we have shown how an effective theory with a discrete symmetry describing the light fields descending from an underlying theory with a large mass scale M can give rise to pseudo-Goldstone bosons with masses in a cosmologically interesting range. These masses are “naturally” small and stable against radiative corrections. The pseudo-Goldstone bosons may themselves acquire VEVs and generate a late phase transition. Taking account of a possible early stage of inflation we have shown that vacuum degeneracy can be split and the associated domain walls that are formed by the late stage of spontaneous symmetry breaking of the discrete symmetry may be unstable.

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Structure formation from domain walls

Finally we comment on the trigger for the discrete symmetry breaking and the associated magnitude of the scalar field VEVs. In eq. (1) ~2 triggers the spontaneous breaking of the symmetry and sets the scale for the VEV. The magnitude of ~ais not determined in GUTs so there is no general expectation. In string theories with supersymmetry breaking in the hidden sector all scalar fields get a positive (mass)2 at the compactification scale of order the gravitino mass (typically in the TeV range). Radiative corrections may drive the (mass)2 negative leading to the expectation from eq. (1) that the VEV is 0(TeV/i,I~).However the symmetries of such theories often have flat directions with with A 0 and in this case the VEV is given by the scale at which the radiative corrections drive the (mass)2 negative. Since these corrections depend only logarithmically on the scale, the resultant VEV may be far from the starting point, the compactification scale. For this reason it often follows that several fields will acquire VEVs of widely different values so that the case discussed above leading to approximate discrete symmetries and unstable domain walls is not unreasonable. =

4. Implications for structure formation We turn now to a discussion of the possible implications for structure formation that may result from a late phase transition. There are two possible mechanisms: domain wall formation [1]or structure formation during a relatively slow roll of the scalar field towards its minimum [2]. We consider the latter possibility first. In this case it is necessary for the scalar field to have (through thermal effects or otherwise) an initial value near a minimum so that it flows to it and does not create any domain walls. In the case of interest here, since the theory has a discrete symmetry, thermal effects will respect this symmetry and thus cannot set initial conditions near only one of the degenerate vacua. However, as we have discussed, it is possible that a period of inflation will cause the observable universe to have field values laying in the domain of attraction of a single minimum. During inflation we expect that there will be contributions to the pseudo-Goldstone potential which disappear after inflation. As a result the pseudo-Goldstone fields will have a unique VEV but this may not be at the minimum of the potential and so after inflation it may relax flowing to its true minimum. The idea of ref. [21is that the coherent motion of the scalar field becomes non-relativistic during the slow-roll period before it starts to oscillate about its minimum. If this happens the coherently moving domains will create density fluctuations of order one when the energy stored in the scalar field is converted into matter energy. In ref. [2] some 100 e-folds of expansion were found necessary to reduce the velocity of the scalar field and to lead to the desired level of fluctuations. For this to happen the initial value of 0 in eq. (3) must be close to the maximum 6 —av. Solving the equation of motion for the 0 field for the =

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potential of eq. (3) we find that 0 must be within 3 x 102_4 X iO~ rad of the maximum for one e-fold of expansion during the slow-roll phase and less than iO~ rad for 100 e-folds! We do not know of any mechanism that will arrange such a fine tuning in the initial conditions and for this reason we consider that this mechanism is unlikely to be realised in the models of interest here. We turn now to the second possibility, that of domain wall production. For the case of the potential of eq. (1) the wall between two adjacent minima can locally be considered as lying on the x—y plane and is described by the equation 0=2__arctan(sinh(~~)),

(21)

~1being the width of the wall,

The energy surface energy density density of of thethe solution wall is isNV(O), where 0 is given by eq. (21), thus(22) the NO o.=fV

0cos

dz=~V0~1.

(23)

Then the energy density stored inside the wall is, as expected, proportional to V0: (24) Since the latter is a constant of the theory, it is a non-redshifting quantity. The domain wall network is not a static system [61. In the case there are more than two minima, walls between adjacent minima are favoured in order to minimise the energy due to rapid spatial variation of the fields. Bubbles containing one vacuum in a sea of another vacuum tend to collapse under surface tension dissipating their energy on the scale of the bubble radius. Although the collapsing bubbles may bounce, simulations indicate that they lose a large fraction of their energy within several bounces [7]. Walls bounding the same vacua but in the opposite sense (e.g. 1,2 and 2,1 vacua) may annihilate dissipating their energy in pseudo-Goldstone fields on a scale ~l. Numerical simulations [81show that, as a result of these various processes, the walls will efficiently disappear so that only walls at the horizon size remain at any one time. However this conclusion must be slightly modified if there are multiple vacua, for walls between non-adjacent minima will first split giving rise to additional bubbles [61, and the minimisation of surface tension of walls between random vacua can only proceed once there are enough bubbles to allow on average at least one bubble of each vacuum type. This

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means the horizon should encompass at least N bubbles for an N-fold vacuum degeneracy. As a result there will be a range of wall sizes from the horizon size, R, to the size R/N’/3. The bubble collapse processes discussed above will be efficient in eliminating bubbles below this size. While a small effect, this has important phenomenological implications for it will allow bubbles associated with an N-fold degeneracy of vacua to generate a spectrum of fluctuations over this range of scales. The walls obviously correspond to density fluctuations and, since they stretch during expansion, their energy grows relative to energy in the form of radiation or matter. As a result the dominant contribution to density fluctuations comes from the largest walls namely those at the present horizon. The density fluctuations due to the walls generate an anisotropy in the cosmic microwave background (CMBR) given by (oT/T) GOR 11() [9], where RH is the horizon size today. Using the current bounds on (or measurements of) these fluctuations provides a stringent upper bound on the allowed magnitude of the density fluctuations coming from domain walls of the present horizon size. This bound implies an even stronger bound on the smaller walls relevant at the time structure formation takes place. As a result stable, non-interacting walls cannot account for galaxy formation [8]. One possible way out of this impasse is to postulate that the walls interact strongly [101.The effect of the interactions is to generate friction slowing down the network of walls, allowing a comoving scale to freeze in. In addition, efficient dissipation of the wall energy in matter is achieved. However in these schemes some fine tuning is required in order that the interactions do not lead to an unacceptable increase of the mass of the field. An alternative possibility, which does not involve such fine tuning, is to assume that the walls are non-interacting, but are unstable due to a non-degeneracy of the vacua causing them to decay through a combination of annihilation and collapse processes at a redshift Za~ Then the value of the CMBR anisotropy is ~T/T~GURH(Za).

(2.5)

This gives a weaker constraint on the domain wall energy and raises the possibility that unstable domain walls play a role in structure formation. Indeed if the wall decay occurs before recombination even the thermal fluctuations of eq. (25) will be erased by rescattering while density fluctuations in a cold dark matter background will still grow. In order to discuss this question more fully we need to determine the contribution of the domain walls to density fluctuations. 4.1. DOMAIN WALLS IN CDM MODELS

As discussed above, bubble collapse and domain wall annihilation processes convert wall energy to pseudo-Goldstone bosons. To have interesting cosmological

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effects these pseudo-Goldstone bosons must be very light and will not have decayed by the present time We will assume that dark matter is predominantly in the form of cold dark matter (CDM). This is the most interesting possibility for, since CDM does not couple to photons, fluctuations induced in CDM before photon decoupling through its gravitational coupling to the density fluctuations in the pseudo-Goldstone relics of the walls will not be communicated directly to the CMBR and so no stronger constraint on the domain wall density from the CMBR anisotropy is generated. After photon decoupling non-dark matter will gravitate to the overdense regions of the CDM that have been created. Because of this it is possible for there to be a period of structure growth before decoupling without affecting the microwave background and, thus increasing the effect walls can have on structure formation. ~.

4.2. LARGE SCALE STRUCTURE

Domain walls can complement the CDM model in explaining galaxy formation because their energy density is concentrated at the largest scales and a domain wall component can offer an explanation of the observed structure at large scales. Numerical simulations of density variations in a CDM dominated universe have big errors at the larger scales. Even so they indicate that at large distances more energy density than the one predicted by the CDM models needs to be added to that of the cold dark matter in order to obtain agreement with the data [11]. This overdensity is of magnitude op/p 0(0.2) for scales of approximately 40—50 Mpc and drops to Op/p 0(0.1) for scales of 100 Mpc (assuming a Hubble constant of 100 km s~ Mpc We will discuss whether unstable domain walls can supply this spectrum of fluctuations. Let R be the horizon size when the unstable walls disappear. We take it to correspond_today to R’, the largest scale at which overdensity is observed (R’ RH/ 1/i + Za 100 Mpc). As discussed above the expectation is that roughly N vacuum bubbles per horizon remain for an N-fold set of vacua giving domain walls separated by a distance L (being roughly R/N’/3). The density fluctuations averaged at a scale cL are determined by ‘~

‘).

=

Op ~(Za)

P

where

=

p

tn

~(Za)

~~(Za),

Pm

LPm

(26)

3 and p~ 0(l+ z) 0is the current value for the critical density. The dominant contribution to density fluctuations from walls comes from the walls existing at the latest time because the energy in a wall grows linearly with R. *

Pm

=

p~

The only kinematically allowed channels are into photons or

(~*massless)

neutrinos. These are

forbidden by the discrete symmetry and will only occur at a rate suppressed in amplitude by powers of the pseudo-Goldstone boson mass.

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In the case of interest here with unstable walls the latest time corresponds to the redshift Za at which the true vacuum spreads to fill the whole universe. As noted above the walls present at this time will have a range of sizes from R/N1/3 to R. These on their decay produce a spectrum of density fluctuations between these scales. The energy produced by a bubble is proportional to its surface area. The surface area of N bubbles of size R/N”3 is N’~”3greater than one bubble of size R and so the smaller scale bubbles. Thus the total contribution from the smaller bubbles to Op/p is larger than the largest bubbles giving a non-uniform distribution over a (small) range of scales from R/N1/3 to R in qualitative agreement with the discrepancy between the cold dark matter distribution and the observed spectrum [11]. In the case that Za> 1000, that is if the walls decay before recombination, as we have mentioned above the fluctuations due to the domain walls themselves are erased by rescattering. Then the major fluctuations to the CMBR arise due to (i) fluctuations in the gravitational potential on the latter scattering surface or (ii) intrinsic fluctuations of the gravitational field on the last scattering surface [121. The second effect provides the larger contribution to the anisotropy at small angular scales (0 << 1°)while the first effect (Sachs—Wolfe effect) is the dominant contribution to the anisotropy at large angular scales. However, the fluctuations at the large angular scales today were of super-horizon size at decoupling. In the models of domain walls that we discuss, the larger (in distance) fluctuations, which are produced by the overdensities that arise due to the domain walls, correspond to the horizon size at Za~ that is they are sub-horizon at the time of decoupling. It is possible to calculate the angular size of the induced fluctuations in the CMBR by comparing the scale that corresponds to the horizon size at Za with the formula for the proper distance D(0) along the geodesic that joins two sources as a function of 0. D(O) is given by [13]

D(0)

=

H 0(1+z) [i -(1 +z)1/2] sin(~)~

(27)

where H0 is the Hubble constant today. Thus we find that 0(rads) ~ 1/2 and hence an annihilation redshift Za> 1000 corresponds to fluctuations at angular scales less than 1°.Note that fluctuations of OT/T at these angular scales have to exist in order that the observed large scale structure is obtained, whether or not the source of the density fluctuations are domain walls. It is in principle possible to distinguish between fluctuations coming from interacting stable walls and the unstable walls considered here by measurements of the CMBR anisotropy at very small angular scales. It has been observed that an additional distortion of the CMBR arises due to collapsing wall bubbles, because the redshift of light entering the bubble is different from the blueshift when it exits [14]. In models with stable walls this effect, in contrast with the distortions caused

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by infinite walls, arises at very small angular scales (s~1°for the brighter spots) and can be calculated exactly, giving quantitatively the same result with the distortions that arise due to infinite walls. The resulting fluctuations are non-gaussian, but still consistent with the COBE results, which are obtained for angular scales > However, such fluctuations, which in the case of unstable walls would in general appear at even smaller angular scales, do not survive in our picture. This is because if the walls annihilate before recombination as in the specific models that we have discussed, all these fluctuations will be erased. This indicates that if non-gaussian fluctuations are found in future measurements at angular scales of the order of 10, models with interacting, stable walls will be favoured. 70~

4.3. EARLY QUASAR FORMATION

We have emphasised how an N-fold degeneracy of vacua gives a characteristic spectrum of bubble sizes which leads to observable effects in the density fluctuations. Here we point out another observable phenomena that is associated with unstable bubbles. There are two possible mechanisms for converting the energy in walls to energy in the form of the light pseudo-Goldstone fields. The bubbles may shrink under surface tension or the walls may collide and annihilate. When averaged over L the distinction between bubble collapse processes and annihilation processes is unimportant. However the latter deposits the wall energy in a region of the wall thickness of size ~1whereas the former does so over the bubble radius L. As a result the annihilation processes will lead to local density fluctuations which are much bigger. This is important because the density fluctuations from wall annihilation become non-linear at much higher redshifts and can explain why early quasar formation is possible; a desirable property because cold dark matter on its own can only account for quasar formation at the observed redshifts (Zq 5 [15]) with difficulty. To quantify this possibility we note that the scale, Lq, relevant for averaging fluctuations leading to quasars is such that the associated volume contains at least one quasar mass. Thus 3,

Mq

L

=La q

1+z. 1 +Zq

=

41TL~p~(1+Zq)

U ,

L.=

(1

.

+Za)2pc(1 +zq)

(28)

Here Mq is the quasar mass and the redshift Za enters because before wall decay the fluctuations are wall driven whereas after wall decay they grow linearly. Quasar production occurs at the redshift Zq when density fluctuations averaged over Lq become non-linear. To be significant the contribution of these fluctuations to the average density fluctuations averaged over L should be non-negligible (of order 0.1). Only the wall annihilation process can give rise to early quasar

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formation and the question is how important are they likely to be? For a sine-Gordon potential in 2-D (two-dimensional space—time), this effect does not occur at all. The solitons pass through each other and the final distribution is a “mirror-picture” of the initial one. This is due to a conformal invariance, which is not present in 4-D and we do not anticipate this soliton behaviour to persist. We expect that efficient energy dissipation through wall collision and annihilation will occur if the time scale, tL, that is needed for the field to roll down the potential is comparable to the time scale ~1(c 1) it takes for two non-interacting walls to pass through each other. This is indeed what happens, since the study of the dynamics of the pseudo-Goldstone field, as well as of the domain walls of the theory indicates that both the scales ~L and ~1are given by rn~. However in the case of stable walls the probability of wall collision is small and so for these walls bubble shrinkage will be the dominant mechanism. In the case of unstable walls the situation is different. At the latest times of wall survival (the time at which structure formation from walls is most important) the vacuum non-degeneracy causes the bubbles of true vacuum to expand rapidly. As a result bubbles of true vacuum will grow and collide and annihilation between walls will be a significant mechanism eliminating the energy stored in walls. The energy left in the wake of these collisions in the form of pseudo-Goldstone fields will act as centres of gravitational attraction for the CDM sea. The wall decay redshift, Za~ which up to now we have treated as a free parameter, is calculable in the discrete symmetry models discussed above. For non-degenerate vacua there is always a critical bubble radius above which it is energetically favourable for the bubble of true vacua to expand gaining more volume energy than is lost in surface energy. Once the horizon exceeds this critical radius bubbles of the true vacuum will expand everywhere at the speed of light to fill the whole universe and this occurs at the same time in all horizon volumes [16]. The redshift at which this occurs is given for the model of eq. (10) by =

OPRH

2/3

—1.

Za=

(29)

0~

We will discuss in the next section the possibility that this mechanism will produce quasars at a redshift of up to ten.

5. Model calculations We are now in a position to determine whether unstable domain walls of the type following from a spontaneously broken discrete symmetry can give a significant contribution to structure formation. We use eqs. (26)—(29) to determine the contribution to the overall density fluctuations coming from domain walls and a

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quasar production at zq 10. We take the horizon at the time that the walls annihilate to correspond today to a scale 100 Mpc; i.e. the walls have to disappear at a redshift 3600. We then demand the overdensity at 100 Mpc to be of order 0.1 and find, as a function of the model parameters, the predicted overdensity for a scale L 100/N 1/3 as well as the exact annihilation redshift and the amount of mass in the non-linear region at a redshift zq 10. We initially consider unstable walls in the first non-supersymmetric model considered in subsect. 3.1.1 with the parameters given by eqs. (11) and (12). In these models the main characteristic is that the terms that split the degeneracy are quite large comparable to the dominant terms, tending to lead to a very early annihilation redshift for the vast range of model parameters that we are interested in. We have found that even for very large values of N solutions exist only in the parameter range where v
=

=

v’=6X1O7GeV, v=8.6X i0~GeV, z. 1 3626, =

L1=lOOMpc,

Op —

=0.1,

P1

L2=48Mpc,

Op —

=0.2,

P2 2M®, (30) 10, Mg 10’ where the subindices 1, 2 refer to scales R’ and R’/N’/3 respectively, while M® is the solar mass. For the above model parameters we find that v3/M2 v’ is reversed. This indicates that models with even smaller vacuum splitting would be more natural. For example starting with a Z 36 symmetry and the transformation properties t1 exp(2iri/36)c1 and 1’ —s exp[4(2~ri/36)]~’with the potential 9 90 v4 80 v’ + v~M3C05(v,) + (31) Zq

=

—*

~ ~(~)

...

.

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Here the dominant term is as in the previous example but the subdominant term being much smaller. The condition for the above potential to be valid is now v4/M3
6.00 x iO~GeV, Za

v

2.82 x 10 GeV,

=

3612,

=

Op L 1=lOOMpc,

=0.1,

—

P1

L2

=

Op

48 Mpc,

—

=

P2 Zq

=

0.2,

10, (32)

2M®, Mg

10’

with the input parameters comfortably within the allowed region. Passing to the supersymmetric models with soft symmetry breaking terms of subsect. 3.1.2 and eq. (17), we find that there exist solutions for the following values of the input parameters: =

1.39 x i0~GeV,

v

=

1.98 x i0~GeV.

(33)

For a supersymmetric example with the Z 36 symmetry, built in analogy with the non-supersymmetric case above, we have =

1.38 x i0~GeV,

v

=

2.98 x lO~6eV.

(34)

In the above we have not detailed the values of the astrophysical observables, because they are approximately the same as those given in eqs. (30) and (32) respectively. The conditions under which we derived the potential are comfortably satisfied for these values of v’ and v. For a given symmetry group the suppression of the particle mass and energy density in the supersymmetric models is much larger than in the non-supersymmetric case and therefore the VEVs of the fields are higher. In all the examples that we have discussed here we have a theory with nine minima. This is because we find that this case gives a nice fit with the data over a wide range of scales. Models with a smaller N may also explain large scale structure but at a smaller range of scales e.g. for N 5 and L1 100 Mpc, L2 58. Larger N’s lead to an even wider range of scales than with N 9 and are also acceptable. However there is a limit as to how large N can be, since as N increases the quantitative agreement with the observed spectrum of fluctuations is less satisfactory. =

=

=

=

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A characteristic feature apparent from all these examples is that the domain wall contribution to density fluctuations is of just the order needed to complement the CDM predictions for a variety of scalar potentials. It is interesting that the favoured choice of quasar parameters leads to contributions to Op/p in the correct range and almost independent of the choice of scalar potential. Similarly the quasar mass is fixed once its production redshift is specified, M= q

3

4~tn

,

3p~(1+Zq)3(1 +Z

(35)

3 5)

and again a reasonable mass results for the favoured value of zq independent of the choice of scalar potential. In the discussion of unstable domain wall bubbles that has been presented, the basic point is that there is an upper cut-off of the domain wall size which results in density fluctuations peaked at a given wavelength with a relatively narrow distribution. As we discussed this may explain the possible discrepancy between the CDM predictions and measurements over a limited range. However discrepancy, including the recent COBE measurements, extends from 10 to 1000 Mpc and this range cannot be described by the domain walls related to a single phase transition. There are two possibilities which could explain such a wide range of discrepancy. The first is to assume that structure is generated by a series of phase transitions which lead to walls with different surface energy. The second is that domain walls associated with a phase transition occurring before an inflationary era may have a spectrum of sizes peaked at a scale which is sub-horizon size today [5]. We have examined the structure that is to be expected in ref. [17], were we have found that in principle it is possible to explain the observed large scale structure by a single phase transition.

6. Summary and conclusions In summary, our analysis indicates that the class of pseudo-Goldstone boson fields resulting from spontaneous breaking of discrete symmetries can have important implications for cosmology, domain walls produce density fluctuations which grow with the domain wall size. These can be reconciled with the CMBR limits if the walls are not stable. The largest walls produced just before decay may generate structure at large scales which may offer a resolution of the discrepancy between cold dark matter predictions and observation. In this case local density fluctuations are generated at a high level and trigger early quasar production. We are grateful to Z. Lalak, C. Lacey and S. Sarkar for useful discussions and to D. Schramm for stimulating our interest in late phase transitions.

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