Cosmological implications of supersymmetric axion models
aa r X i v : . [ h e p - ph ] F e b Cosmological implications of supersymmetric axionmodels
Masahiro Kawasaki, Kazunori Nakayama, Masato Senami ‡ Institute for Cosmic Ray Research, University of Tokyo, Kashiwa 277-8582, Japan
Abstract.
We derive general constraints on supersymmetric extension of axionmodels, in particular paying careful attention to the cosmological effects of saxion.It is found that for every mass range of the saxion from keV to TeV, severe constraintson the energy density of the saxion are imposed. Together with constraints from axinowe obtain stringent upper bounds on the reheating temperature. ‡ Now, at Department of Micro Engineering, Kyoto University. osmological implications of supersymmetric axion models
1. Introduction
One of the main problems of the standard model is the strong violation of CP invariancedue to non-perturbative effects of quantum chromodynamics (QCD). In general, QCDeffects generates the term in the lagrangian such as L θ = θg s π G aµν ˜ G µνa , (1)where g s is the QCD gauge coupling constant, G aµν is the field strength of the gluon,and ˜ G µνa = ǫ µνρσ G aρσ /
2. Experimentally θ must be smaller than about 10 − , but in thestandard model there seems to be no theoretical reasons that θ must be so small. This isthe well-known strong CP problem. As a solution to the strong CP problem, Peccei andQuinn [1] introduced anomalous global U(1) symmetry, which we denote U(1) PQ . Whenthis U(1) PQ is broken spontaneously, there appears a pseudo-Nambu Goldstone bosoncalled axion [2]. The axion dynamically cancels the θ -term effectively, and no strongCP violation is observed in true vacuum. In order not to contradict with terrestrialexperiments, astrophysical [3] and cosmological arguments [4], the breaking scale of PQsymmetry F a should lie in the range 10 GeV . F a . GeV. If F a is close to thisupper bound, the axion is an interesting candidate for the cold dark matter.On the other hand, another problem in the standard model is the quadraticdivergence of the radiative correction to the Higgs mass. In the standard model, themass of the Higgs is not protected by any symmetry, and hence naturally the Higgsboson is expected to obtain the mass of the cut-off scale. Thus to obtain hierarchicallysmall mass scale down to the weak scale requires unnatural fine-tuning. Supersymmetry(SUSY) [5] is the most motivated solution to this problem, since SUSY protects the massof scalar fields and the weak scale becomes stable against radiative correction. Fromcosmological points of view, SUSY also provides interesting candidates for the darkmatter. Due to the R -parity conservation, under which standard model particles havethe charge +1 and their superpartners have −
1, the lightest SUSY particle (LSP) isstable. So if the LSP is neutralino or gravitino, they are the dark matter candidates.Therefore, it seems reasonable to combine these two paradigms. In fact, axionmodels are easily extended to implement SUSY. As we will see, in SUSY extensionsof the axion models, many non-trivial cosmological consequences arise. In SUSY axionmodels, both the scalar partner of the axion, saxion , and the fermionic superpartner ofthe axion, axino , have significant effects on cosmology. But there have not been manystudies which treat both of them in spite of their importance [6, 7, 8, 9]. Moreover, theseearlier works were based on specific models and only the restricted parameter regionswere investigated. In this paper, we investigate all possible mass range of the saxion andcorresponding various cosmological bounds. Our results are easily applied to any axionmodels with slight modifications, and hence provide general cosmological constraints.Furthermore, since the saxion and axino densities depend on the reheating temperature T R after inflation, we can obtain constraints on T R . The important result is that theupper bound on T R becomes more stringent due to the late-decaying saxion and axino, osmological implications of supersymmetric axion models O (10) eV. We conclude inSec. 7.
2. Supersymmetric axion models
The axion is the pseudo-Nambu-Goldstone boson which appears due to the spontaneousbreaking of PQ symmetry. The axion obtains a mass from the effect of quantumanomaly, but this contribution is very small. It is estimated as m a ∼ × − eV(10 GeV /F a ).In SUSY extensions of axion models, the axion field forms supermultiplet, whichcontains a scalar partner (saxion) and fermionic superpartner (axino) of the axion [10].The saxion mass is expected to be of the order of the gravitino mass ( m / ). On theother hand the axino mass somewhat depends on models, but generically can be as largeas m / . Although the interactions of both particles with standard model particles aresuppressed by the PQ scale F a , they may cause significant effects on cosmology as willbe seen in Sec. 3. In SUSY models, superpotentials must satisfy the holomorphy. When the real U(1) PQ symmetry is combined with the holomorphic property of the superpotential, it isextended to complex U(1) symmetry, which inevitably includes scale transformation [11].The invariance under the scale transformation means the existence of a flat directionalong which the scalar field does not feel the scalar potential. The saxion corresponds tosuch a flat direction of the potential. But SUSY breaking effects lift the flat direction andthe saxion receives a mass of order m / . † The saxion field can develop to large fieldvalue during inflation, and begins to oscillate around its minimum when the Hubbleparameter H becomes comparable to the saxion mass m s ∼ m / . In general suchcoherent oscillation of the saxion field has large energy density and hence its late decaymay have significant effects on cosmology [6, 8, 9, 13].As an example, let us consider a model with the following superpotential: W = λX (Φ ¯Φ − F a ) , (2)where the superfields X, Φ and ¯Φ have the PQ charges 0 , +1 , − † In gauge-mediated SUSY breaking models, the saxion receives a logarithmic potential from gauge-mediation effects. If the PQ scalar is stabilized by the balance between the logarithmic potential andthe gravity-mediation effect, m s ∼ m / still holds [12] . osmological implications of supersymmetric axion models F a with X = 0. ‡ Including SUSY breaking massterms due to gravity-mediation effects, the scalar potential can be written as V = m / (cid:0) c X | X | + c | Φ | + c | ¯Φ | (cid:1) + | λ | (cid:8) | Φ ¯Φ − F a | + | X | (cid:0) | Φ | + | ¯Φ | (cid:1)(cid:9) , (3)where c X , c and c are O (1) constants which are assumed to be positive. The potentialminimum appears at | Φ | ∼ | ¯Φ | ∼ F a , but the initial amplitude of the saxion remainsundetermined. In general, if the saxion remains light during inflation, the field valuemay naturally take the value of the order of the reduced Planck scale M P . But the initialamplitude can be suppressed by introducing the Hubble-induced mass terms, which areinduced by the saxion coupling with inflaton field through supergravity effect, given by V H = H (cid:0) c ′ X | X | + c ′ | Φ | + c ′ | ¯Φ | (cid:1) (4)where c ′ X , c ′ and c ′ are O (1) constants. If either c ′ or c ′ are negative, the Φ or ¯Φ fieldroll away to ∼ M P during inflation. But if both coefficients are positive, the potentialminimum during inflation is also given by | Φ | ∼ | ¯Φ | ∼ F a , which almost coincides withthe low energy true minimum. Since there is a priori no reason that we expect that c /c is exactly equals to c ′ /c ′ , these two minima are separated by s ∼ "(cid:18) c c (cid:19) / − (cid:18) c ′ c ′ (cid:19) / F a , (5)where s denotes saxion field (here we have assumed c > c and c ′ > c ′ ). This simplemodel provides one realization of the scenario for the initial saxion amplitude s i tobe ∼ F a . For the case of s i ∼ M P , the cosmological constraints become much morestringent than the case of s i ∼ F a , and hence hereafter we consider only the latter case.For this scenario to work, the Hubble parameter during inflation H I should besmaller than ∼ F a , since otherwise the large Hubble mass term during inflation takes theall fields to the origin. Once they are trapped at the origin, the saxion has unsuppressedinteraction with particles in thermal bath and get a large thermal mass, which resultsin further trap of of the saxion at the origin. Thus, the oscillation epoch is significantlydelayed [9]. § Because delayed oscillation only makes the cosmological saxion problemsworse, we do not consider such a case. On the other hand, the axion field has isocurvatureperturbations with amplitude ∼ H I / ( πs i ) during inflation [16, 17], which leads to aconstraint on H I as H I . × GeV θ − i (cid:18) Ω m h . (cid:19) (cid:18) s i F a (cid:19) (cid:18) F a GeV (cid:19) − . , (6)where θ i denotes the initial misalignment angle of the axion, Ω m denotes the densityparameter of the nonrelativistic matter and h is the present Hubble parameter in units ‡ If the A -term contribution V A ∼ m / λXF a + h . c . is included, the X field can have the VEV | X | ∼ m / . But this does not modify the following arguments. § This is the case in the KSVZ (or hadronic axion) model [14]. In the DFSZ model [15] such a thermalmass may not arise because of the small coupling of the PQ scalar, but there arises another difficultyfrom domain wall formation. osmological implications of supersymmetric axion models . s i ∼ F a , the requirement H I . F a is in fact valid fromcosmological point of view. k Although we have presented a specific model above, the dynamics of the saxiondoes not depend on axion models much, once the initial amplitude and the mass of thesaxion are fixed. The saxion field starts oscillation at H ∼ m s with initial amplitude s i .As explained above, m s is likely of the order of m / and the natural expectation of theinitial amplitude is s i ∼ F a or s i ∼ M P .Now let us estimate the saxion abundance. First we consider the saxion abundancein the form of coherent oscillation. It is independent of the reheating temperature whenthe reheating temperature is high, i.e., Γ I > m s (Γ I : the decay rate of the inflaton ).The saxion-to-entropy ratio is fixed at the beginning of the saxion oscillation ( H ∼ m s ),and given by (cid:16) ρ s s (cid:17) (C) = 18 T osc (cid:18) s i M P (cid:19) (7) ≃ . × − GeV (cid:16) m s (cid:17) / (cid:18) F a GeV (cid:19) (cid:18) s i F a (cid:19) , (8)where T osc denotes the temperature at the beginning of the saxion oscillation. On theother hand, if Γ I < m s , the ratio is fixed at the decay of inflaton H ∼ Γ I , and thesaxion-entropy ratio is estimated as (cid:16) ρ s s (cid:17) (C) = 18 T R (cid:18) s i M P (cid:19) (9) ≃ . × − GeV (cid:18) T R GeV (cid:19) (cid:18) F a GeV (cid:19) (cid:18) s i F a (cid:19) . (10)The saxion is also produced by scatterings of particles in high-temperature plasma.For T R & T D ∼ GeV ( F a / GeV) , the saxions are thermalized through thesescattering processes and the abundance is determined as [10] (cid:16) ρ s s (cid:17) (TP) ∼ . × − GeV (cid:16) m s (cid:17) . (11)For T R . T D , this ratio is suppressed by the factor T R /T D . The result is (cid:16) ρ s s (cid:17) (TP) ∼ . × − GeV (cid:16) m s (cid:17) (cid:18) T R GeV (cid:19) (cid:18) GeV F a (cid:19) . (12)Here we have assumed that thermally produced saxions become nonrelativistic beforethey decay. This assumption is valid for the parameter regions we are interested in. Wecan see that the contribution from coherent oscillation is proportional to F a while thatfrom thermal production is proportional to F − a . Thus for small F a thermal productionmay be dominant. Note that this expression is valid for T R & m s . Otherwise the saxion k The saxion does not give rise to isocurvature fluctuation because it has large Hubble mass and itsquantum fluctuation is suppressed during inflation. osmological implications of supersymmetric axion models Figure 1.
Theoretical predictions for the saxion-to-entropy ratio. Thick blue linesrepresent contribution from the coherent oscillation ( ρ s /s ) (C) with s i ∼ F a and thinred ones represent thermal contribution ( ρ s /s ) (TP) . Solid, dashed and dotted linescorrespond to T R = 10 GeV,10 GeV and 1 GeV, respectively. cannot be produced thermally. As a result, the total saxion abundance is sum of thesetwo contributions, ρ s s = (cid:16) ρ s s (cid:17) (C) + (cid:16) ρ s s (cid:17) (TP) . (13)In Fig. 1 we show theoretical predictions for the saxion-to-entropy ratio with F a = 10 GeV, 10 GeV and 10 GeV. ¶ In the figures the thick blue lines representcontribution from the coherent oscillation ( ρ s /s ) (C) with s i ∼ F a and thin red onesrepresent thermal contribution ( ρ s /s ) (TP) . Solid, dashed and dotted lines correspondto T R = 10 GeV,10 GeV and 1 GeV, respectively. We can see that for F a . GeV ( F a . GeV), the contribution from thermal production dominates at T R & GeV ( T R & GeV).
Since the interaction of the saxion to other particles is suppressed by the PQ scale F a ,it has a long lifetime and decays at cosmological time scales, which leads to severalcosmological effects. First, let us consider the saxion decay into two axions, s → a . Ifwe parametrize PQ scalar fields Φ i asΦ i = v i exp (cid:20) q i σ √ F a (cid:21) , (14)where q i is the PQ charge of the i -th PQ field, and F a = pP i q i | v i | , the saxion andaxion are identified as s = Re[ σ ] and a = Im[ σ ]. The kinetic term is expanded as X i | ∂ µ Φ i | ∼ √ fF a s ! (cid:18) ∂ µ a∂ µ a + 12 ∂ µ s∂ µ s (cid:19) + . . . , (15) ¶ Axion overclosure bound ensures θ . i F a . GeV. Thus by tuning θ i , F a ∼ GeV is allowed.Late-time entropy production also makes such a large value of F a viable, although the low reheatingtemperature T R . osmological implications of supersymmetric axion models f = P i q i v i /F a . From this coupling, we can estimate the decay rate of the saxioninto axions as Γ( s → a ) ≃ f π m s F a . (16)If f ∼ τ s ≃ . × f − sec (cid:18) m s (cid:19) (cid:18) F a GeV (cid:19) (17)But for the model with superpotential Eq. (2), f can be zero at tree level due to thecancellation if c = c in Eq. (3). It is crucial for cosmological arguments whether thedominant decay mode is into axions or not, because axions produced in the decay donot interact with other particles and cosmological constraints can be relaxed if this isthe dominant decay mode. In this paper, we consider both possibilities f ∼ f ∼ m s & s → g ) ≃ α s π m s F a , (18)where α s denotes the SU(3) c gauge coupling constant. The emitted gluons producehadron jets which may affect the big bang necleosynthesis (BBN) as seen in Sec. 3On the other hand, the decay into two photons is always possible, which has thedecay rate, Γ( s → γ ) ≃ κ α π m s F a , (19)where α EM denotes the U(1) EM gauge coupling constant, and κ is a model dependentconstant of O (1). These photon produced in the decay also bring about cosmologicaldifficulty.In the DFSZ axion model, the PQ scalar has tree level coupling with the ordinaryquarks and leptons. For m s > m ui (2 m di ) where u i ( d i ) denotes the up-type (down-type)quark in the i -th generation ( i = 1 , ,
3) the saxion decays into a fermion pair with thedecay rate, Γ( s → u i ¯ u i ) = 38 π (cid:18) x − x + x − (cid:19) m s (cid:18) m ui F a (cid:19) (cid:18) − m ui m s (cid:19) / , (20)Γ( s → d i ¯ d i ) = 38 π (cid:18) xx + x − (cid:19) m s (cid:18) m di F a (cid:19) (cid:18) − m di m s (cid:19) / . (21)where x = tan β = h H u i / h H d i (through this paper, we set x = 5). Here it should benoticed that for m s . osmological implications of supersymmetric axion models m s & m s .
270 MeV). Therefore,for simplicity we neglect the effects of saxion decay into mesons for m s < s → l i ¯ l i ) = 18 π (cid:18) xx + x − (cid:19) m s (cid:18) m li F a (cid:19) (cid:18) − m li m s (cid:19) / . (22)Thus we can see that the saxion decay into heavier fermions is enhanced, as long as itis kinematically allowed. In fact, the decay into fermions may be the dominant modefor some mass region even if f = 1.In the KSVZ model, the decay of the saxion into quarks and leptons is suppressedbecause the saxion does not directly couple with them.Hereafter, we consider the following four typical cases labeled as model (a)-(d).Model (a) denotes the KSVZ model with f = 1 and model (b) denotes the KSVZ modelwith f = 0. Model (c) denotes the DFSZ model with f = 1 and model (d) denotes theDFSZ model with f = 0.
3. Cosmological constraints from saxion
Given the decay modes of the saxion, we can derive generic constraints on the saxiondensity depending on its lifetime and mass. As we will see, for almost all the mass range(1 keV . m s . Relativistic particles produced by decaying particles would contribute to the additionalradiation energy density, parametrized by the increase of the effective number ofneutrinos, ∆ N ν . The definition of N ν is given through the relation, ρ rad ( T ) = " N ν (cid:18) T ν T γ (cid:19) ρ γ ( T γ ) , (23)where ρ rad denotes the total relativistic energy density, T γ and T ν denote the temperatureof the photon and neutrino. In the standard model with three species of light neutrinos, N ν ≃ . N ν . Then the additional contribution ∆ N ν is given as ∆ N ν = 3( ρ rad − ρ γ − ρ ν ) /ρ ν . Theincrease of N ν speeds the Hubble expansion up and causes earlier freeze-out of the weakinteraction, which results in He overproduction. The recent analyses of primordial He osmological implications of supersymmetric axion models N ν ∼
3. Thus we conservatively adopt∆ N ν ≤ s → a [35].The increase of ∆ N ν changes the epoch of the matter-radiation equality and affectsthe structure formation of the universe. Thus, ∆ N ν is also constrained from cosmicmicrowave background (CMB), galaxy clustering, and Lyman- α forest. According tothe recent analyses [27, 28, 29, 30, 31], ∆ N ν ≫ + This constraint appliesto the saxion with lifetime τ s . sec.If the saxion decays mainly into axions, ∆ N ν is determined from the relation, ρ s s ∼ . g ∗ s ( T s ) − ∆ N ν T s , (24)where T s is the temperature at the decay of the saxion and g ∗ s counts the relativisticdegrees of freedom. Thus the requirement ∆ N ν ≤ ρ s s . . × − GeV (cid:18) g ∗ s ( T s ) (cid:19) (cid:18) T s (cid:19) . (25)Note that almost the same constraint is applied even if the saxion decay into axionsis suppressed, if its lifetime is longer than ∼ N ν . ∗ No constraint is imposedin this case if the branching ratio into axions is suppressed (see the case (b) and (d)of Figs. 2 and 3). The following analyses and the resulting constraints on the saxionabundance do not depend on whether the saxion dominates or not.
The saxion with its lifetime & − sec may affect BBN [34]. The saxion decays intoordinary particles either radiatively or hadronically. If the hadronic decay occurs atearly epoch ( τ s . sec), the main effect on BBN is p ↔ n conversion caused byinjected pions, which results in helium overproduction. At later epoch, photo- andhadro-dissociation processes of light elements take place efficiently. When s → a isthe dominant decay mode, the branching ratios into radiation or hadrons are small.Nevertheless, even a small fraction of the energy density of the saxion which goes intoradiation or hadrons may have impacts on BBN. In particular, if hadronic decay modesare open, the constraint is very stringent. + It is pointed out that including Lyman- α forest data raises the best-fit value of N ν [31], but ∆ N ν ≫ N ν is close to 1 and τ s > N ν at BBN and structure formation [32]. ∗ See Refs. [35, 36] for the case of thermal inflation driven by the saxion field trapped at the origin. osmological implications of supersymmetric axion models B r (cid:16) ρ s s (cid:17) . ( − – 10 − GeV for 10 sec . τ s . sec10 − GeV for 10 sec . τ s . sec , (26)for radiative decay, and B h (cid:16) ρ s s (cid:17) . ( − – 10 − GeV for 1 sec . τ s . sec10 − – 10 − GeV for 10 sec . τ s . sec , (27)for hadronic decay, where B r and B h denote the radiative and hadronic branchingratios, respectively (here B r includes the hadronic decay modes). Note that if theinjected photon energy (which is equal to the half of the saxion mass) is smaller thanthe threshold energy to destroy the light elements especially He, which is typically O (10)MeV, the photo- and hadro-dissociation constraints are much weakened. In particularBBN constraints are neglected for m s . . γ → n + p . The saxion with lifetime 10 sec . τ s . sec may affect the blackbody spectrum ofCMB. Since preserving the blackbody spectrum requires the photon number-violatingprocesses such as double-Compton scattering, to maintain thermal equilibrium betweenphotons and electrons, photons injected in the decay distort the CMB spectrum at t & sec when the double-Compton scattering becomes inefficient. The distortion ischaracterized by the chemical potential µ at t . sec when the energy transfer bythe Compton scattering is efficient, and Compton y -parameter at later epoch, whichcharacterizes the deviation of the CMB spectrum from thermal distribution due to theinverse Compton scattering by high energy electrons. They are constrained from COBEFIRAS measurement as | µ | . × − and y . . × − [37]. µ and y are related tothe injected photon energy δρ γ as [38, 39] δρ γ ρ γ ∼ . µ, (28)for 10 sec . τ s . sec, and δρ γ ρ γ ∼ y, (29)for 10 sec . τ s . sec. This in turn constrains the saxion energy density dependingon its branching ratio into radiation B r , as B r (cid:16) ρ s s (cid:17) . . × − GeV (cid:16) sec τ s (cid:17) / (10 sec . τ s . sec)6 . × − GeV (cid:16) sec τ s (cid:17) / (10 sec . τ s . sec) . (30) osmological implications of supersymmetric axion models γ )-ray background The two photon decay of the saxion with lifetime longer than ∼ sec may contributeto diffuse X( γ )-ray background. The mass of the saxion which has such a long lifetimeis typically smaller than 1 GeV. The photon with energy 1 keV . E γ . F γ ( E ) = E π Z t dt B γ n s ( z ) τ s (1 + z ) − dE ′ dE δ (cid:16) E ′ − m s (cid:17) , (31)where B γ denotes the branching ratio into two photons and n s ( z ) is the number densityof the saxion at the redshift z . E ′ is the energy of the photon at the instant ofproduction and E is the present redshifted energy, the relation between them is givenby E ′ = (1 + z ) E . Under the assumption of the flat universe (Ω Λ + Ω m = 1), thisexpression can be integrated yielding F γ ( E ) = B γ n s πτ s H g (cid:16) m s E (cid:17) × exp " H τ s √ Ω Λ ln (cid:0) √ Ω Λ g (cid:0) m s E (cid:1) − (cid:1) (cid:0) √ Ω Λ + 1 (cid:1)(cid:0) √ Ω Λ g (cid:0) m s E (cid:1) + 1 (cid:1) (cid:0) √ Ω Λ − (cid:1) (32)where n s denotes the present number density of the saxion, H denotes the presentHubble constant, and g ( x ) = (cid:2) Ω Λ + Ω m x (cid:3) − / . (33)On the other hand, the observed photon flux in the range 1 keV . E .
100 GeVis roughly given as F γ obs ( E ) ∼ (cid:0) E keV (cid:1) − . (0 . . E .
25 keV)57 × − (cid:0) E MeV (cid:1) − . (25 keV . E . × − (cid:0) E
100 MeV (cid:1) − . (4 MeV . E .
120 GeV) , (34)in the unit of cm − sec − sr − , from the observations of ASCA [42], HEAO1 [43],COMPTEL [44], and EGRET [45]. Thus, from the requirement F γ < F γ obs , the tightconstraint on the saxion density is derived. Note that even for the saxion lifetimelonger than the age of the universe, the small fraction of the decayed saxion at t < t contributes to the diffuse background and its abundance is limited.To estimate this constraint, let us consider the case with τ s > t . In this case, theX( γ )-ray spectrum of photons from the saxion decays has the maximum at E max = m s / F γ ( E max ) = B γ n s / (2 πτ s H ). Then from the condition F γ ( E max ) < F γ obs ( E max ), we obtain a constraint, B γ (cid:16) ρ s s (cid:17) . π m s τ s H s F γ obs (cid:16) m s (cid:17) ∼ . h × − GeV (cid:16) m s (cid:17) osmological implications of supersymmetric axion models × (cid:16) τ s sec (cid:17) (cid:18) F γ obs ( m s / − cm − sec − (cid:19) , (35)where s denotes the present entropy density. For the case with τ s < t , the photonenergy which gives the flux maximum deviates from E = m s / τ s ≪ t , it is given by E max = ( m s / H τ s √ Ω m / / . This leads tothe constraint B γ (cid:16) ρ s s (cid:17) . π m s s F γ obs ( E max ) ∼ . × − GeV (cid:16) m s (cid:17) (cid:18) F γ obs ( E max )10 − cm − sec − (cid:19) . (36) If the saxion decays after recombination era and the injected photon energy is relativelysmall ( m s . O (1) keV- O (1) MeV and 10 sec . τ s ), redshifted photons may leavethe transparency window until the present epoch [40]. Then, emitted photons interactwith and ionize the intergalactic medium (IGM), and they contribute as an additionalsource of the reionization. If this contribution is too large, the optical depth to thelast scattering surface is too large to be consistent with the WMAP data [33]. Herewe apply the results from Refs. [40, 46], simply assuming that if the decay-producedphoton leaves the transparency window, one-third of the photon energy is convertedto the ionization of the IGM (the remaining goes to the excitation and heating of theIGM). According to Refs. [40, 46], this is a good approximation when the decay occursbefore the reionization due to astrophysical objects takes place and most of hydrogenatoms exist in the form of neutral state. (The Gunn-Peterson test indicates that thereionization occurred at z ∼ B r (cid:16) ρ s s (cid:17) . (cid:16) ρ s s (cid:17) bound , (37)where ( ρ s /s ) bound can be read off from Fig. 2 of Ref. [46]. For example, for τ s & t , it isgiven by (cid:16) ρ s s (cid:17) bound ≃ . × − GeV (cid:16) τ s sec (cid:17) (cid:18) Ω b h . (cid:19) , (38)where Ω b denotes the density parameter of the baryonic matter. This constraint iscomplementary to the diffuse X( γ )-ray limit. For the saxion with its lifetime τ s > t , its energy density contributes to the dark matterof the universe, and hence the saxion density should be less than the observed matterdensity, Ω s h . Ω m h . In terms of the saxion-to-entropy ratio, this is written as ρ s s . . × − GeV (cid:18) Ω m h . (cid:19) . (39) osmological implications of supersymmetric axion models If the saxion mass is larger than about 1 TeV, the saxion can decay into SUSY particles.Here we suppose that the LSP is the lightest neutralino. The decay into SUSY particleswere investigated in detail in Ref. [48] and it was found that decay into gauginos hasroughly the same branching ratio as that into gauge bosons. Thus we should be carefulabout LSP overproduction from the saxion decay. The resultant abundance of theLSP depends on T s , and for T s & m LSP /
20 LSPs produced from the saxion decay arethermalized and have the same abundance as that expected in the standard thermal relicscenario of the LSP dark matter. In this case no upper bound on the saxion abundanceis imposed. On the other hand, if T s . m LSP /
20, the abundance of the LSP is given by ρ LSP s ≃ B s m LSP m s ρ s s + ρ thermalLSP s for n LSP ( T s ) h σv i < H ( T s ) , s π g ∗ ( T s ) m LSP h σv i T s M P for n LSP ( T s ) h σv i > H ( T s ) , (40)where B s denotes the branching ratio of the saxion into SUSY particles, h σv i denotesthe thermally averaged annihilation cross section of the LSP, and ρ thermalLSP denotes thecontribution from thermal relic LSPs taking account of the dilution from the saxiondecay. The LSP number density immediately after the saxion decay n LSP ( T s ) is definedas n LSP = 2 B s ρ s ( T s ) /m s . For deriving the constraint, we ignored the contribution tothe LSP production from thermal scattering processes. Moreover, the second line ofEq. (40) always results in overproduction of LSPs with the annihilation cross section forordinary neutralino dark matter. Thus for deriving the constraint, we consider only thefirst term of the first line of the right hand side of Eq. (40). The bound can be writtenin the form ρ s s . . × − GeV (cid:18) m s m LSP (cid:19) (cid:18) Ω m h . (cid:19) , (41)for m s & T s . m LSP /
20. This constraint can be relaxed if the annihilationcross section of the LSP is significantly large. We will revisit this issue in Sec. 5.Hereafter we set m LSP = 500 GeV as a reference value.Including all of these constraints, we can derive general upper bounds on the saxion-to-entropy ratio as a function of the saxion mass m s for models (a)-(d). In Fig. 2-4, weshow the results with F a = 10 , and 10 GeV, respectively. In each panel, theorange line represents the bound from ∆ N ν .
1, the thick-solid brown line representsthe bound from BBN, the thick-dotted purple line represents the bound from CMB,the thick-dot-dashed green line represents the bound from diffuse X( γ )-ray background,the thin-dot-dashed blue line represents the bound from reionization, the thin-dashedred line represents limit from the present matter density, and the thick-dashed gray linerepresents LSP overproduction limit from the saxion decay. We also show the theoreticalprediction for the saxion energy density in the figures for T R = 10 GeV and 1 GeV bythin-dotted black lines. osmological implications of supersymmetric axion models Figure 2.
Various cosmological constraints on the saxion abundance for F a =10 GeV. Thin dotted black lines represent theoretical prediction ρ s /s = ( ρ s /s ) (C) +( ρ s /s ) (TP) for T R = 10 GeV (upper) and T R = 1 GeV (lower) with s i = F a . Fourpanels correspond to different models. Model (a) : KSVZ with f = 1, model (b) :KSVZ with f = 0, model (c) : DFSZ with f = 1, model (d) : DFSZ with f = 0.
4. Constraints from axino
So far, we have ignored the cosmological effects of axino, the fermionic superpartner ofaxion. The mass of axino is model dependent, but it can be as heavy as the gravitinomass, m / [49, 22, 50]. For example, in the model of Eq. (2), the axino mass isestimated as m ˜ a ∼ | λX | . As noted earlier, taking into account the A -term potential like V A ∼ m / λXF a + h . c . , X can have the VEV of the order of m / . Thus in this modelthe axino mass is naturally expected to be m / . Hereafter for simplicity we assume m ˜ a ∼ m / .Axinos and gravitinos are produced through scatterings of particles in thermalbath. First, we assume either of them is the LSP, and hence their thermally producedabundance must not exceed the present abundance of the dark matter [51]. ♯ The ♯ As long as m s ∼ m / ∼ m ˜ a , which of them is lighter is not relevant. But if m ˜ a ≪ m / and m / & osmological implications of supersymmetric axion models Figure 3.
Same as Fig. 2, but for F a = 10 GeV. abundance of thermally produced axinos is calculated as [53] †† ρ ˜ a s ≃ . × − g s GeV (cid:16) m ˜ a (cid:17) (cid:18) GeV F a (cid:19) (cid:18) T R GeV (cid:19) , (42)where g s is the QCD gauge coupling constant. Thus for large m ˜ a ( ∼ m s ), the constrainton the reheating temperature becomes stringent. Note that axino thermal productionfor T R < T ˜ a similar to the case of the saxion decay. If T ˜ a & m LSP /
20, LSPs produced from theaxino decay are thermalized and the standard thermal relic scenario of the LSP darkmatter is maintained. If T ˜ a . m LSP /
20, the LSP abundance is determined by Eq. (40)after replacing T s and ρ s /s with T ˜ a and ρ ˜ a /s . †† Here we assume there is no entropy production after the reheating ends. If the saxion dominatesthe universe and decays before BBN, the axino and gravitino abundance can be reduced. osmological implications of supersymmetric axion models Figure 4.
Same as Fig. 2, but for F a = 10 GeV.
On the other hand, the thermally produced gravitinos have the abundance as [54] ρ TP3 / s ≃ . × − GeV (cid:18) m / (cid:19) (cid:16) m ˜ g (cid:17) (cid:18) T R GeV (cid:19) (43)for m / ≪ m ˜ g where m ˜ g denotes the mass of the gluino (here the logarithmicdependence on T R is omitted). Contrary to the axino, the constraint becomes severerwhen m / ( ∼ m s ) becomes smaller. Note that for m / . . m / . α forest data [55]. Thusthe gravitino mass in this region is strongly disfavored. (The case of ultra-light gravitino m / .
16 eV will be mentioned later.) In addition to the constraint from the presentmatter density, there may be another constraint coming from the late-decay of SUSYparticles into gravitinos or axinos, which may affect BBN. But the constraint is quitemodel dependent, and hence we do not consider it here.Including those constraints from the gravitino and axino, we derive the upperbounds on the reheating temperature for each saxion mass and show them in Figs. 5-7. osmological implications of supersymmetric axion models Figure 5.
Upper bounds on the reheating temperature T R for each model with F a = 10 GeV. The initial amplitude of the saxion is assumed to be s i ∼ F a . Thethin-short-dashed light blue line represents the bound from axino thermal production,and thin-dotted black line represents the bound from gravitino thermal production.The shaded region contradicts with the lowest possible reheating temperature [56].The other lines are the same as Fig. 2. Four panels correspond to different models.Model (a) : KSVZ with f = 1, model (b) : KSVZ with f = 0, model (c) : DFSZ with f = 1, model (d) : DFSZ with f = 0. In these figures, we have assumed that the initial amplitude of the saxion is given by s i ∼ F a . The thin-short-dashed light blue line represents the bound from axino thermalproduction, and thin-dotted black line represents the bound from gravitino thermalproduction. The other lines are the same as Fig. 2. Because the saxion-to-entropyratio ( ρ s /s ) (C) is proportional to ( T R s i ), the constraints on T R scale as s − i . However, itshould be noted that although the axino constraint is stringent for relatively large m s as can be seen from these figures, it may be significantly relaxed if the axino is muchlighter than the gravitino. osmological implications of supersymmetric axion models Figure 6.
Same as Fig. 5, but for F a = 10 GeV.
5. Dark matter candidates
We have seen that the very stringent bound on the reheating temperature is imposed forwide range of the saxion mass. It is typically stronger than the usual upper bound fromthe gravitino overproduction. It has some implications to the dark matter candidates.For example, it invalidates the gravitino dark matter for wide parameter regions. Inthis section we summarize the dark matter candidate in SUSY axion models. m s . TeV
First consider the case where the gravitino (axino) is the LSP. In our model, we assumethe axino has the mass comparable to the gravitino, and hence the axino (gravitino) isthe NLSP. Which of them is the lighter is not important because both have the similarproperties. The saxion also has the mass comparable to the gravitino. The stabilityof the saxion is not ensured by the R -parity, but since its decay rate is suppressed bythe PQ scale F a , the saxion lifetime can exceed the present age of the universe. If thisis the case, the saxion can be the dark matter. In addition, if θ . i F a ∼ GeV, theaxion can also play a roll of the dark matter. Therefore, we have four candidates for thedark matter, i.e. gravitino, axino, axion and saxion. However, the saxion dark matter osmological implications of supersymmetric axion models Figure 7.
Same as Fig. 5, but for F a = 10 GeV. is possible only when F a ∼ GeV and 1 keV . m s .
10 keV, as can be seen fromFig. 7, and hence is less attractive candidate. For 10 GeV . F a . GeV, theaxino dark matter is allowed for wide parameter regions, but the gravitino dark matteris excluded except for m s .
100 keV, as can be seen from Fig. 5 and Fig. 6. On theother hand, for F a & GeV, the axino or gravitino dark matter is almost impossibleand the axion is the most viable dark matter candidate, although other cosmologicalconstraints are severe (Fig. 7).To summarize, for m s . F a . GeV, and the axion may be dark matter candidate for F a & GeV. m s & TeV
For larger m s ( ≃ m / ), the saxion decay mode into SUSY particles opens. Then theLSP is assumed to be the lightest neutralino. As discussed in Sec. 3.7, the abundance isgiven by Eq. (40) if the decay occurs after the freeze-out of the LSP. For the neutralinowhich has small annihilation cross section, very low reheating temperature is needed toobtain the correct abundance of the dark matter as can be seen from Figs. 5-7. However,for the neutralino with larger annihilation cross section such as wino- or higgsino-like osmological implications of supersymmetric axion models T s is larger than the freeze-out temperature of the LSP T f ∼ m LSP / N ν ≫ m s & m s and F a . Of course, the axion isalso a good candidate for the dark matter for F a & GeV.
6. Ultra-light gravitino scenario
We have seen that for almost all the mass range of the gravitino, the reheatingtemperature is severely constrained. However, for an ultra-light gravitino with mass ofthe order of 1-10 eV, gravitinos are thermalized and their abundance is sufficiently lowerthan the dark matter abundance, and hence the reheating temperature is not constrainedfrom gravitino overproduction. Thus thermal leptogenesis using right-handed neutrino[57], which requires T R & GeV, may be possible.In SUSY axion model, the saxion oscillation also contributes to the present darkmatter density as given in Eqs. (8) and (10). We should ensure that this contributiondoes not exceeds the present matter density of the universe. From Eq. (8), ρ s /s isbounded as (cid:16) ρ s s (cid:17) (C) . . × − GeV (cid:16) m s
10 eV (cid:17) / (cid:18) F a GeV (cid:19) (cid:18) s i F a (cid:19) . (44)Thus for F a . GeV, the saxion abundance from coherent oscillation is smallerthan that of the dark matter. However, we should be aware that thermally producedsaxion and axino abundances are comparable to that of the gravitino for T R & GeV( F a / GeV) if m s = m ˜ a = m / , since both are thermalized in the earlyuniverse. Thus constraint on the gravitino mass may become more stringent by a factorof two, if thermal leptogenesis is assumed to work. In Fig. 8, we show allowed parameterregions on F a - T R plane for m s = 10 eV (indicated by (a)) and for m s = 1 keV (indicatedby (b)) with an assumption s i = F a . It can be seen that for m s .
10 eV, thermalleptogenesis is still possible as long as the PQ scale satisfies F a . × GeV.
7. Conclusions and discussion
We have investigated the cosmological constraints on supersymmetric axion modelswhich are motivated from particle physics point of view. It is found that the presenceof the saxion and the axino makes it rather difficult to construct a viable cosmological osmological implications of supersymmetric axion models Figure 8.
The shaded regions are exclude (a) for m s = 10 eV and (b) for m s = 1 keV. scenario which is free from any contradiction with observations. In particular, in almostall range of the gravitino mass (which is assumed to be the same order as the saxionmass), the strict upper bound on the reheating temperature is imposed, and it is morestringent than the bound from the usual gravitino problem [Figs. 5-7]. The constraintdepends on whether the main decay mode of the saxion is into axions or not. It shouldbe noted that although the axino constraint is stringent for relatively large m s as can beseen from our results, it may be significantly relaxed if the axino is much lighter thanthe gravitino. The axion is a good candidate for the cold dark matter, although theaxino or gravitino dark matter is also viable for some parameter regions.The obtained stringent bound on the reheating temperature has some implicationson the baryogenesis scenarios. As is well known, the standard thermal leptogenesisscenario using right-handed neutrino [57] is incompatible with the gravitino problemexcept for m / .
10 eV or m / &
10 TeV. Although the gravitino mass around m / ∼
10 GeV may also be compatible with thermal leptogenesis, in this region theNLSP decay into gravitino may cause another difficulty. The presence of saxion makesthis situation worse, and hence the standard thermal leptogenesis does not seem towork in supersymmetric axion models. The Affleck-Dine baryogenesis scenario [58] canwork well even for such a low-reheating temperature [59], except for the case of gauge-mediated SUSY breaking models for small m / , where the Affleck-Dine mechanism maysuffer from Q-ball formation [60]. On the other hand, for an ultra-light gravitino mass m / .
10 eV, thermal leptogenesis is still possible. Here it should be noticed that theseconstraints from the saxion strongly depend on the initial amplitude of the saxion s i andwe have assumed s i is roughly given by the PQ scale F a . A concrete example which givessuch initial amplitude is given in Sec. 2.1. Perhaps this is the smallest value expected osmological implications of supersymmetric axion models Acknowledgments
We thank Fuminobu Takahashi and Tsutomu Yanagida for helpful discussion andcomments. K.N. would like to thank the Japan Society for the Promotion of Science forfinancial support. This work was supported in part by the Grant-in-Aid for ScientificResearch from the Ministry of Education, Science, Sports, and Culture of Japan, No.18540254 and No 14102004 (M.K.). This work was also supported in part by JSPS-AFJapan-Finland Bilateral Core Program (M.K.).
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