Late-time entropy production due to the decay of domain walls

0.13 MB·English

by  Masahiro Kawasaki

Late-time entropy production due to the decay of domain walls
Masahiro Kawasaki and Fuminobu Takahashi
Institute for Cosmic Ray Research, University of Tokyo, Kashiwa 277-8582, Japan
(Dated: February 2, 2008)
It is shown that late-time decay of domain walls can dilute unwanted relics such as moduli, if
the universe was dominated by frustrated domain walls with tension σ = (1 ∼ 100TeV)3. Since
energy density of the frustrated domain walls decreases as slow as the inverse of the scale factor,
an overclosure limit on the axion decay constant f is also considerably relaxed. In fact f can be
a a
as large as the Planck scale, which may enable us to naturally implement the QCD axion in the
string scheme. Furthermore, in contrast to thermal inflation models, the Affleck-Dinebaryogenesis
can generate enough asymmetry to explain the present baryon abundance, even in the presence of
late-time entropy production.
5
0
PACSnumbers: 98.80.Cq,11.27.+d
0
2

n Particle physics beyond the standard model predicts inflation proposedby Lyth and Stewart [8, 9]. The ther-
a a number of new particles, some of which have long life- mal inflation is mini-inflation with the number of e-folds
J
timesanddecayatcosmologicaltimescales. Ifsuchlong- 10 which takes place at the weak scale. A comprehen-

7 ∼
lived particles are substantially produced in the early sive study of the thermal inflation was done in Ref. [10]
2
universe, they may give crucial effects on cosmology and and it was shown that for modulus mass between 10 eV

2 spoil success of the standard big-bang model. and10TeVthe thermalinflationsolvesthe cosmological
v moduli problem. However, the thermal inflation also di-
One well-known example of such dangerous relic par-
8 lutethebaryonnumberoftheuniverseandhenceweneed
ticles is gravitino in supergravity theories. Gravitinos
5
an efficient baryogenesis mechanism. In Refs. [10, 11] it
1 are produced during reheating after inflation and the
was pointed out that the Affleck-Dine mechanism does
0 abundance of the produced gravitino is proportional to
1 the reheating temperature. The gravitino with mass that job. But later it was shown that Q-ball formation
4 obstructs the baryogenesis [12].
0.1 10 TeV decays during or after the big-bang nu-
0 ∼ −
cleosynthesis (BBN) and may destroy the light elements
/ In this letter we propose an alternative model to di-
h synthesized in BBN. Thus, in order to keep success of
lute the dangerous cosmologicalrelics. In our model do-
p
BBNthereheatingtemperatureshouldbesufficientlylow
- mainwalls,whichare usually consideredas cosmological
p (for recent works, see [1, 2, 3, 4]).
disaster, play an important role. The domain walls are
e
Another class of dangerous relics are light scalar producedwhensomediscretesymmetryisspontaneously
h
: fields called moduli which appear in superstring theo- broken. The domain wall network in the universe be-
v
ries [5, 6, 7]. The moduli fields are expected to acquire comesverycomplicatediftherearemanydiscretevacua.
i
X masses of the order of the gravitino mass from nonper- The density of such domain wall network decreases very
r turbativeeffectsoftheSUSYbreakingandhavelonglife- slowly and quickly dominates the universe. In extreme
a
timessincetheyhaveonlythegravitationallysuppressed case, the domain walls become frustrated and their den-
interactions. During inflation the field value of moduli sity ρ decreases as ρ a−1 (a: the scale factor)
DW DW
generally deviates from the vacuum value due to extra which is slower than the scal∝ing evolution ρ a−3/2
DW
∝
SUSY breaking by the cosmic density. The deviation is inmatter-dominateduniverse. Furthermore,ifthescalar
of the order of the Planck scale M (= 2.4 1018 GeV) potentialhas atiny termwhichexplicitly breaksthe dis-
G
×
since no other mass scale exists. Then, when the Hub- crete symmetry, the vacua separated by domain walls
ble becomes less than the modulus mass, the modulus haveslightlydifferentenergiesandhencethedomainwall
field starts to osciilate with amplitude M . Because network is no longer stable and decays producing large
G
∼
the modulus denity is comparable to the cosmic density entropy. Itwillbeshownthatthe entropyproductionby
at onset of the oscillation, it soon dominates the den- the decay of domain wall is enough to dilute the moduli
sity of the universe and causes the serious cosmological density. Moreover, because the density of domain wall
problem (moduli problem). Since the abundance of the decreasesmuchmoreslowlythanthematter density,the
moduli is independent of the reheating temperature, the cosmic axion density is also diluted. As a result, the ax-
moduli problem is much more serious than the gravitino ion decay constant f as large as M is allowed, which
a G
problem. To solve the moduli problem (as well as the might enable us to implement the axion into the string
gravitino problem) it is necessary to dilute the oscillat- framework. In addition, the Affleck-Dine baryogenesisis
ing moduli by huge entropy production. So far the most compatible with our model as we will see below. There-
successfulmodeltoproducesuchlargeentropyisthermal fore, the domain walls provide more efficient and consis-
2
tent dilution mechanism than the thermal inflation. decay, at least several domain walls must be present in-
Firstwepresentourscenariothatthedecayofdomain sideonehorizon,otherwisetheoldinflationoccurssome-
walls generates large entropy to dilute unwanted relics where leading to unacceptably inhomogeneous universe.
such as moduli and gravitinos,and derive constraints on This requires that the decay temperature T satisfy the
d
tensionofthedomainwalls. Incontrasttotheothercan- following inequality: Td4 > σHd, where Hd ∼ Td2/MG
didates for late-time entropy production, domain walls is the Hubble parameter when the domain walls decay.
have an advantagethat the energy density decreasesrel- Thus the tension is bounded above:
atively slowly, which ensures huge entropy production.
2
T
To be specific, an equation-of-state w equals to 2/3 if σ <T2M (100TeV)3 d . (3)
the domain walls are completely frustrated, that−is, the d G ∼ 10MeV
(cid:18) (cid:19)
structureofthedomainwallnetworkremainsunchanged
On the other hand, the tension should be large enough
as the universe expands. Throughout this letter, we as-
to dominate the universe before the decay: H > H .
eq d
sumethisisthecase. Laterwegiveamodelwhichwould
That is,
lead to such frustrated domain walls.
Let us consider the evolution of the energy density of 1 8 −1
T8M2 3 T 3 H 3
domain walls, which eventually dominate the universe. σ > d G (100GeV)3 d i .
H ∼ 10MeV 1TeV
The energy density of the domainwalls at the formation (cid:18) i (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
(4)
is estimated as
Now let us estimate the the abundance of the mod-
ulus field after decay of the domain walls. For domain
ρ σH , (1)
DW,i ∼ i wallswiththetensionsatisfying(3)and(4),Heq ismuch
smaller than the modulus mass. Since the modulus field
whereσ is thetensionofthe wall,H the Hubble param-
i
dominates the universe as soon as it starts oscillating,
eteratthephasetransition. Wehaveassumedthatthere
it is reasonable to assume that the universe was domi-
are only a few domain walls per one horizon when they
nated by the modulus field when H = H . Then the
areformed. HereandinwhatfollowsweneglectO(1)nu- eq
modulus-to-entropy ratio is given by
merical factors. If the domain walls are fully frustrated,
the energydensityfalls offasthe inverseofthe scalefac- ρ H 4 T9M2
tor: ρ a−1. Fordefinite discussion,weassumethat mod T d d G, (5)
DW ∝ s ∼ d H ∼ σ3H
the universe is dominated by either the inflaton or mod- (cid:18) eq(cid:19) i
ulus fields when the domain walls are formed. Noting where we have used the fact that the Hubble parameter
that the energy density of an oscillating massive scalar falls off as a−1/2 while the domain walls dominate the
field decreases as a−3, the domain walls dominate the universe, an∝d Eq. (2) is substituted in the last equation.
∝
universe when the Hubble parameter becomes equal to This must satisfy the observational bound:
Heq ∼σ34Hi41MG−23. (2) ρmsod <κρsc =3.6×10−9κh2GeV, (6)
If the reheating occurs earlier than H = Heq, the whereρc isthecriticaldensity,κvariesfrom10−10 to0.2
radiation-dominated epoch might last for a while, which dependingonthemodulusmass[10],andhistheHubble
makes it easy for the domain walls to dominate the un- constant in units of 100 km/sec/Mpc. Thus the tension
verse. Thus, for conservative discussion, we assume that is further constrained as
theuniverseisdominatedbynonrelativisticparticlesun-
til the domain walls dominate the universe. σ >κ−31(500GeV)3 Td 3 Hi −13 . (7)
The decay of domain walls can proceed if there is a 10MeV 1TeV
(cid:18) (cid:19) (cid:18) (cid:19)
tinybiasbetweenvacua,δρ. Whenthisenergydifference
becomes comparable to the energy of the domain walls, To sum up, the domain walls with the tension σ = (1
∼
the decay occurs and the universe is reheated. Since the 100TeV)3, which decay just before the BBN starts, can
universe was dominated by the domain walls in our sce- dilute the moduli to the observationally allowed level.
nario, δρ is simply related to the decay temperature as Next we present a model that would lead to forma-
δρ T4. In order to have both large enough entropy tion of the frustrated domain walls. Let us consider the
pro∼ductdion and the successful BBN, we take the decay following superpotential:
temperature as low as 10MeV. Note that this value is
N N
justanexemplifiedvalueandthatthe successfuldilution W =√λ Z Φ Φ + 2ǫ Z Φ2, (8)
isactuallyrealizedforawide rangeofthe decaytemper- ij i j √λ ii i
i j i
ature, say, T = 10MeV 10GeV (see Eqs. (3), (4) and XX X
d
∼
(7)). It is the smallness of δρ that enables domain walls where Z and Φ are gauge-singlet superfields, and real
ij i
tobelong-lived. Letusnotethat,whenthedomainwalls couplingconstantsλ>0andǫareassumedtosatisfythe
3
inequality, λ ǫ [17]. These interactions respect S law. The minima are separated by the potential barrier
N
≫ | |
symmetry if Φ and Z are taken to be the fundamental represented by the third term in Eq. (9), if ǫ is larger
i ij
and bi-fundamental representations of S permutation than m2 /v2 λNm2 /m2. Then the tension of the
N 3/2 2 ∼ 3/2 0
group. The R-chargesare assignedas R =0 and R = walls is given by
Φ Z
2. We assume that Z always sits at the origin: Z =
ij ij
i0n,dwuhciecdhmcaanssbeterremalidzuedrinifgZriejleavcaqnutireepsoachp.ositiveHhubbile- σ1 ∼m3/2v22 ∼(1TeV)3(λN)−1 1mT3e/V2 1mTe0V 2.
(cid:16) (cid:17)(cid:16) (cid:17)(13)
ThenweobtainthefollowingeffectivepotentialforΦ ,
i On the other hand, if ǫ is smaller than m2 /v2, the po-
3/2 2
V(Φ) V m2 Φ 2 m2 (Φ2+Φ∗2) tential barrier is the last term in Eq. (9). The tension
≃ 0− 0Xi | i| − 3/2Xi i i is σ2 ∼ |ǫ|21v23 ≤ σ1, where the last equality holds when
+λ( Φi 2)2+4ǫ Φi 4, (9) ǫ ∼ m23/2/v22. Thus the tension of the domain walls in
| | | | this model can satisfy the bound derived in the previous
Xi Xi section,althoughλN<1mightbenecessaryinthelatter
where m3/2 is the gravitino mass, V0 =O(m40/λ) is cho- case. ∼
seninsuchawaythatthecosmologicalconstantvanishes
We assume that all Φ get positive Hubble-induced
i
inthetrueminimum,andwehaveusedλ ǫ. Herewe
massesthereforesitattheoriginuntiltheHubbleparam-
≫| |
haveassumedthe negativemassofthe orderofthe weak
eterbecomescomparabletom . Thenthosescalarfields
0
scaleattheorigin,m0 ΛEW 1TeV,whichisinduced startrollingdownandgetsettledsomwhereonSN−1 de-
∼ ∼
bytheSUSYbreakingeffects. Thethirdtermcomesfrom fined by Φ 2 = 2Nv2. When the Hubble parameter
the gravity-mediatedSUSYbreakingeffects. Due tothis i| i| 2
becomes equal to the gravitino mass, all Φ move to the
i
term,theglobalminimaofthepotentialliealongthereal P
real axes due to the third term in Eq. (9) and fall in
axesofΦ . Inordertostudythevacuumstructureofthis
i one of the vacua (12). Then the domain walls with the
potential, let us concentrate on the the real components
tension σ or σ are formed.
1 2
of Φ by setting ImΦ = 0. Rewriting the potential in
i i Domain walls must decay before the relevant BBN
terms of the real components φ √2ReΦ , we obtain
i i epoch. To this end, we introduce a R-symmetry vio-
≡
lating interaction that lifts the degeneracy of the vacua:
V(φ) V m2 φ2/2+λ( φ2)2/4+ǫ φ4, (10)
≃ 0− 0 i i i W =c′ W Φ3/M3 with W =m M2, leading to
Xi Xi Xi V =cmh 2 i i Φi3/MG+h.c.,hwhiereca3n/2dc′Garecoupling
where the third term in Eq. (9) is neglected since we are coAnstants3./F2PorsiimipliciGtywetakecbothrealandnegative
interestedinthe caseofm3/2<m0. The scalarpotential sothattruemPinimumisgivenbyφtruemin =(v2,...,v2).
given by Eq. (10) actually ag∼rees with the O(N) model The decay temperature is then expressed by
studied in Refs. [13, 14]. The approximate O(N) sym-
metry originates from the hierarchical couplings, λ and Td −3 c 14 m3/2 12 m0 34
(λN) 8 (1.4)
ǫ. Note that this potential is obtained by disregarding 100MeV ∼ 0.1 1TeV 1TeV
ImΦ , so it does not necessarily give a right answer, for (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
i
When the bias becomes comparable to the energy of the
instance, when one estimates the energy density inside
domainwalls,thetopologicaldefectsarenolongerstable
domain walls, althoughit is still useful to study the vac-
and decay into Φ-particles in the true vacuum. Not to
uum structure.
spoilthesuccessoftheBBN,however,Φ-particlesshould
The positions ofthe globalminima of V(φ) depend on
decay into standard model degrees of freedom. We as-
the sign of ǫ. For ǫ<0, there are 2N minima:
sume that Φ weakly couples to some heavy vector-like
i
φ±(i) =(0,...,0, v ,0,...,0), for i=1 N (11) quarks,whichenableΦ-particlestoradiativelydecayinto
min ± 1 ∼
the standard model gauge bosons as soon as the domain
with v1 m0/√λ. On the other hand, if ǫ > 0, the walls decay. Since the R-parity of Φ is even, the de-
≡
potential minima are given by cay processes into the lightest supersymmetric particles
(LSPs),whichmayoverclosetheuniverse,canbeavoided
φ =( v ,..., v ) (12)
min 2 2
± ± ifthe massofΦ is smallerthantwotimes the LSP mass.
with v m /√λN, where arbitrarycombination of the Thusourmodelofdomainwallscannaturallysatisfythe
2 0
≡
signs are allowed. In the following, we consider the case constraints necessary to induce successful dilution.
of ǫ > 0. Then all Z fields acquire masses of the or- As we saw in the above discussions, the domain wall
ij
der of v Λ in these minima. Also, the number of can be a viable candidate for late-time entropy produc-
2 EW
∼
the minima is 2N. Thus, for large N, the vacuum struc- tion. What differs from the other candidates is that the
ture becomes morecomplicated, comparedto the former energy density falls off very slowly: ρ a−1. Such a
DW
∝
case. Since there are many types of vacua, pair annihi- distinctive feature canleadto anotherimportant cosmo-
lation processes of walls are expected to be highly sup- logical consequence: the overclosure limit on the axion
pressed, which enables walls to deviate from the scaling decayconstantcanbe considerablyrelaxed. Ifwedonot
4
assume entropy production after axionbegins the coher- that the frustrated domain walls quickly dominate the
ent oscillation, the axion decay constant is constrained density ofthe universe andtheir decayproducesentropy
as F <1012GeV not to overclose the universe. This up- large enough to dilute the dangerous moduli. We have
a
perlim∼it is relaxed to 1015GeV, if the late-time entropy also given a concrete model which leads to frustrated
production due to the decay of nonrelativistic particles network of domain walls. Moreover, we have found that
occurs [15]. In the following, we show that the axion de- the late-time decay of the domain walls greatly relaxes
cay constant is further relaxed and can be as large the the constraint on the axion model and the axion decay
Planck scale if the domain-wallinduced entropy produc- constant f as large as the Planck scale is allowed.
a
tion occurs well below the QCD scale.
Up to here, we have assumed fully frustrated domain
TheaxionstartstooscillatewhentheHubble parame-
walls, however, this assumption can be weaken to some
ter becomes comparable to the mass: 3H m (T ).
osc a osc extent. Similarargumentsshowthatdomainwallswhose
≃
TheaxionmassmadependsonthetemperatureT as[16] energy evolves as ρ a−1−γ with γ<0.2(0.1) for
DW
0.1m (Λ /T)3.7 for T >Λ /π, κ = 0.2(10−10) works as∝well. Finally we∼make a com-
a QCD QCD
m (T) mentonbaryonnumbergenerationinthepresentmodel.
a ≃( ma for T∼<ΛQCDπ, The domain-wall decay also dilute pre-existing baryon
∼ (15) number. Therefore, there should be large baryon asym-
whereΛQCD ≃0.2GeV,andma ≃Λ2QCD/Fa istheaxion metry before the entropy production or baryon num-
mass at the zero temperature. Since the number density ber should be generated after the decay of the domain
of the axion decreases as a−3, the energy density of wall. Since the decay temperature is expected to be low
∝
the axion when the domain walls decay is ( 10MeV),itisunlikelytoproducethebaryonnumber
∼
after the decay. Then the most promising candidate for
1 H6
ρ = m (m (T )F2θ2) d , (16) baryogenesis is the Affleck-Dine mechanism. In order to
a|T=Td 2 a a osc a H6
osc produce sufficient baryon asymmetry the gravitino mass
where we have used H a−1/2 when the domain walls shouldbe relativelysmall, m3/2<10 MeV [10], as in the
dominate the universe. θ∝ O(1) denotes the initial am- gauge-mediated SUSY breaking∼scenarios [18]. In the
plitude of the axion field∼in the unit of F . The axion- caseofthethermalinflation,theAffleck-Dinemechanism
a
to-entropy ratio is then given by does not work due to Q-ball formation [12]. The crucial
point is that the thermal inflationrequires largemessen-
ρ F6θ2T9
a =1.3 102 a d , (17) ger scale, which leads to large Q-balls and small baryon
s × ξ(Tosc)5Λ8QCDMG6 numberinthebackgrounduniverse. However,ourmodel
allows relatively small messenger scale, since v is much
whereξ(T) m (T)/m 1. Thismustbesmallerthan 2
≡ a a ≤ smallerthanthevevoftheflaton. ThustheAffleck-Dine
thepresentvalueoftheratioofthecriticaldensitytothe
baryogenesis does work in the present scenario.
entropy. Thatis,theaxiondecayconstantshouldsatisfy
Acknowledgment: F.T. is grateful to K. Ichikawa
Fa < 3.6 1018GeVξ(Tosc)56θ−13 and M. Kakizaki for useful discussions. F.T. would like
∼ ×Ω h2 61 Λ 34 T −32 to thank the Japan Society for Promotion of Science for
a QCD d
.(18) financial support.
× 0.14 0.2GeV 10MeV
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
Therefore the domain-wall induced entropy production
opensuptheaxionwindowtothePlanckscale,ifξ(T )
osc
iscloseto1. Inotherwords,theaxionisagoodcandidate
for dark matter if F M . We need to check that T [1] M. Kawasaki, K.Kohri and T. Moroi, Phys.Rev. D 63,
a G osc
∼ 103502 (2001).
is smaller than 0.1GeV so that ξ(T ) is 1. To this
osc ∼ [2] K. Jedamzik, Phys.Rev.Lett. 84, 3248 (2000).
end, the evolution of the cosmic temperature before the
[3] R. H. Cyburt, J. R. Ellis, B. D. Fields and K. A. Olive,
decay of domain walls must be specified. If the decay is Phys. Rev.D 67, 103521 (2003).
approximatedtobeanexponentialdecaywithaconstant [4] M. Kawasaki, K. Kohri and T. Moroi,
decay rate, we obtain arXiv:astro-ph/0408426.
[5] G. D. Coughlan, W. Fischler, E. W. Kolb, S. Raby and
T 0.05GeV Td 0.26 Fa −0.13<0.1GeV. G. G. Ross, Phys. Lett. B 131, 59 (1983).
osc ≃ 10MeV M [6] T. Banks,D.B.Kaplan andA.E.Nelson,Phys.Rev.D
(cid:18) (cid:19) (cid:18) G(cid:19) ∼ (19) 49, 779 (1994) [arXiv:hep-ph/9308292].
[7] B. de Carlos, J. A. Casas, F. Quevedo and E. Roulet,
Therefore, ξ(T ) is close to 1 when T 10MeV and
osc d ∼ Phys. Lett. B 318, 447 (1993) [arXiv:hep-ph/9308325].
F M .
a ∼ G [8] D. H. Lyth and E. D. Stewart, Phys. Rev. Lett. 75, 201
In this letter we have investigated the dilution of un-
(1995) [arXiv:hep-ph/9502417].
wantedcosmologicalrelicssuchasmodulibyentropypro- [9] D. H. Lyth and E. D. Stewart, Phys. Rev. D 53, 1784
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5
[10] T. Asaka and M. Kawasaki, Phys. Rev. D 60, 123509 383, 313 (1996) [arXiv:hep-ph/9510461].
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D 56, 1281 (1997) [arXiv:hep-ph/9701244]. [17] Note that large hierarchy between λ and ǫ is not neces-
[12] S.Kasuya,M.KawasakiandF.Takahashi,Phys.Rev.D sary,sincethetensionofthedomainwallsisindependent
65, 063509 (2002) [arXiv:hep-ph/0108171]. of ǫ as long as ǫ is not too small. In fact, ǫ ∼ 0.1λ does
[13] H.Kubotani, Prog. Theor. Phys. 87, 387 (1992). work.
[14] H. Ishihara, H. Kubotani and Y. Nambu, KUNS-1070, [18] For such small m3/2, λ (m0) should be so small (large)
June1991. that thetension is larger than (1TeV)3. SeeEq. (13).
[15] M. Kawasaki, T. Moroi and T. Yanagida, Phys. Lett. B