arXiv:hep-ph/0212005v2 24 Jan 2003 e [7] see talrefil au.Ti tg nswe h Hub- the mass, curvaton when the ends of stage order the This H of frozen becomes value. effectively constant field ble remains large field equa- a the the at and in term motion, friction of a tions as acts epoch expansion post-inflationary Hubble the smaller In the much is constant. mass Hubble it its the if if than or value squared, expectation mass negative large a a has acquires field attract curvaton not the did it but [2–6]. recently [1], until ago interest perturbations years much nearly generating described be of first the must mechanism was beyond This constant inflation, Hubble of constant. curvature the nature of that the generation on requirement The depend not size. does observed perturba- the to the lead to of curvaton can tions this the considerable, of is contribution adiabatic density the into energy decay converted at are If field fluctu- ones. curvaton isocurvature sub-dominant the the of is inflation ations that After field, inflation. “curvaton” another during the in field, fluctuations isocurvature scalar instead but fluctu- (CMB) ations, background microwave respon- cosmic are the field for inflaton sible the in perturbations the not that models. inflationary there- possible and constrain inflation, severely during potential fore the on perturbations depend produced only The the re-entry. until horizon constant of moment remains which essentially perturbation become field classical and a exit, inflaton horizon rolling after soon slowly in” “freeze the the picture of usual the fluctuations In quantum formation. generate structure can for seeds inflation homo- the universe, and observable isotropy our the of explaining geneity to in- addition as In known expansion, flation. superluminal of period a through † o natraieipeetto ftecrao model curvaton the of implementation alternative an For h sa e pi h following the is up set usual The idea the in interest revived a been has there Recently went universe early the that believed widely now is It ∼ m φ twihpittecrao trsoscillating starts curvaton the point which at , h bu aa nentoa etrfrTertclPhysi Theoretical for Center International Salam Abdus The pcrm h uvtnms hudb ml uiginflation during small ob be To should candidates. mass curvaton curvaton many the therefore spectrum, and scalars, many n hra vprto.Tecrao opigt atrsh matter to coupling curvaton nucl ( The from constraints come evaporation. non model curvaton thermal and the and masses on soft constraints Other the tial. suppress which symmetries, invoking eaayetecrao cnroi h otx fsupersymm of context the in scenario curvaton the analyze We .INTRODUCTION I. e.g. h uvtnseai nsprymti theories supersymmetric in scenario curvaton The h ∼ < 10 − 8 † o yia otmasses soft typical for uiginflation During : aik Postma Marieke Nvme 2002) (November 1 m atr t nrydniyrdsit as shifts red cold density as energy behaves curvaton its The matter; potential. quadratic the in est nrdainrdsitn as energy shifting the red radiation the with in decay, density dominated, inflaton radiation After becomes universe. the universe of factor scale the tt fmxdmatter, mixed of state rcino h oa nrydniyi h nvre To temperature universe. reheat the the in overproduction density gravitino energy total avoid significant the a of be to fraction siz- needs density be energy to curvaton perturbations the the and able, For inflation), magnitude. during correct the constant of Hubble much the be decay), than should smaller inflaton mass curvaton after (the invariant decays scale curvaton nearly (which the adiabatic that be requires should perturbations produced The scales. all at high absent superpoten- the are some non-renormalization in tial absent to the operators up since that such assures absent helps, theorem work SUSY or to Here small model very order. curvaton be cur- the should the For in terms needed values scenario. field large vaton the for potential flat [11]. (pseudo) a level is boson. tree field Goldstone curvaton at the terms that pro- is mass that possibility soft Another symmetry from Heisenberg fields a scalar tects is in- generaliza- there and gauge thereof, models, by gravity tions inflation no-scale forbidden In D-term are [10]. In masses variance pro- soft masses. to induced soft invoked Hubble large be from can scalars contradiction in tect perturba- Symmetries tilt, a all spectral observations. to for large lead with constant a fields with Hubble heavy spectrum the Such tion of [8,9]. order fields the scalar of induces and masses supersymmetry, soft breaks density universe energy early the finite possible in the many However, therefore theories candidates. and curvaton SUSY fields, scalar (SUSY). many supersymmetry contain of context never the this if or decays. curvaton universe, the the until curva- dominate happens, the to until becomes grow ton perturbations into The transformed ones. are adiabatic perturbations isocurvature the ter, ∼ hr r eea osrit ntecrao model. curvaton the on constraints several are There the lift can potential the in terms Non-renormalizable in scenario curvaton the reanalyze we paper this In TeV). s taaCsir 1 40 ret,Italy Trieste, 34100 11, Costiera Strada cs, oytei,gaiiooverproduction, gravitino eosynthesis, t.Sprymti hoiscontain theories Supersymmetric ety. rnraial em ntepoten- the in terms -renormalizable udb eysalt aif these satisfy to small very be ould m anasaeivratperturbation invariant scale a tain ≪ H hscnb civdby achieved be can This . i.e. aito n oddr mat- dark cold and radiation , ρ I ∝ ρ a φ − ∝ 4 uigthis During . a − 3 with , a should be sufficiently low. Curvaton decay should not of inflation and the moment the potential approaches a destroy the nucleosynthesis predictions. Furthermore, quadratic form by the parameter q: thermal damping and evaporation should be taken into δφ δφ account. = q , (6) φ φ In the next section we discuss these various constraints osc ∗ on the curvaton model in more detail. In section III we where the subscript osc denotes the quantity at the onset consider the possible (soft) mass terms that can arise in of oscillations, when H m. For a quadratic or flat SUSY theories, and analyze the parameter space for the potential the evolution Eqs.∼ (2, 3) for φ and δφ are the various cases. We end with conclusions. same and q = 1. The oscillating curvaton field behaves as non- II. CONSTRAINTS relativistic matter. After inflaton decay, the energy den- sity becomes a mixture of radiation and matter, and isocurvature perturbation are transformed into adiabatic A. Density perturbations ones. This period ends when the curvaton becomes to dominate the energy density or, if that never happens, The density perturbations for the curvaton field have when it decays. been analyzed in [3]. We briefly review their results. The prediction of the curvaton model for the curvature Consider the curvaton with minimal kinetic term and perturbation can be written as (for small ηφ): scalar potential V (φ). We can expand the field in a clas- sical part plus quantum fluctuations 1/2 rdecq H∗ ζ = , (7) P 3π φ∗ φ(x)= φ + δφ(x). (1) where the subscript dec means the corresponding quan- The unperturbed and perturbed curvaton field satisfy re- tity evaluated at the time of φ-decay, i.e., when H Γ . ∼ φ spectively Furthermore, we have defined the parameter r as the ra- tio of energy density in the curvaton field to the total ¨ ˙ φ +3HI φ + Vφ =0 , (2) energy density in the universe r = ρφ/ρ. The COBE data requires
¨ ˙ 2 1/2 δφk +3HI δφk + (k/a) + Vφφ δφk =0 . (3) (COBE) = 4.8 10−5, Pζ × n(COBE) = 0.93 0.13. (8) Here δφk are the Fourier components of δφ. An over- ± dot denotes ∂/∂t, and a subscript φ denotes ∂/∂φ. We Moreover, the non-detection of tensor perturbations sets have made the first order approximation δ(V )= V δφ. φ φφ an upper limit on the energy density during inflation The fluctuations of a generic massive scalar field gen- 14 erated during de Sitter stage are then found to be on H∗ < 10 GeV. (9) superhorizon scales (where the gradient term in Eq. (3) ∼ is negligible) B. Initial conditions 3/2−νφ H∗ k δφk 3 , (4) | |≃ 2k aH∗ The initial field value of the curvaton at the begin- ning of inflation is a free parameter. The only constraint where the subscript denotes the time of horizon exit. is that the energy density in the curvaton field is sub- ∗ 2 2 Further, νφ = (9/4 m/H ). For ηφ = (m/3H ) 1 2 2 − ≪ dominant. The inflaton energy density is ρI HI MP, one has 3/2 νφ ηφ and the spectrum is nearly scale ∼ − ≃ with HI the Hubble constant during inflation. This invariant. To be more precise, the spectral tilt of the means that for masses m2 H2 a typical initial field perturbation is given by ≪ I value will be large φ MP. We consider scalar≫ potentials of the form d ln φ nφ P =2ηφ 2ǫH , (5) ≡ d ln k − 1 λ V (φ)= m2φ2 + φ4+n. (10) 2 M n and it is assumed that ǫ H/H˙ 1. H ≡ ≪ The field remains overdamped until H m when the The non-renormalizable operators, suppressed by the field starts oscillating in the potential.∼ The potential Planck scale or some other ultraviolet cutoff, will gen- 2 2 2 rapidly approaches a quadratic form, after which the frac- erate an effective mass δmHO = ∂ V/∂φ Vφφ >H for tional perturbation δφ/φ remains constant. We param- large initial field values. The curvaton is≡ underdamped, eterize the change in fractional density between the end and decreases exponentially fast.
2 The effective mass squared term during inflation can fractional energy density in the curvaton field increases be positive or negative. The finite energy density in the until a maximum value at curvaton decay: early universe breaks supersymmetry. This leads to soft 2 α mass terms of the form m2 = cH2. The sign of c is ρφ 1 φosc ΓI I rdec = , (13) determined by the K¨ahler potential. ρ ∼ h M 2 m dec P If the effective mass squared is negative and the initial curvature amplitude is large, the curvaton field will ap- with α = 0 for ΓI > m, and α = 1/2 for ΓI < m. proach its minimum exponentially fast. If on the other We have assumed instant reheating, with reheating tem- 1/4 −1 hand, the curvaton is initially at the origin, or at some perature T g∗ √MPΓI . Above T 10 GeV the ∼ ∼ effective degrees of freedom g∗ = (100), whereas round small field value, the field is highly damped. It will ap- O 2 MeV temperatures it drops to g∗ = (10). Further, we proach the minimum at a rate φ m t (for constant m). O 2 ∝ used Γ h2m, with h the coupling, yukawa or gauge, At t HI /m the motion goes non-linear and the classi- φ ∼ cal field∼ reaches the minimum exponentially fast. We will of the curvaton to matter. The curvaton field dominates assume that inflation last sufficiently long for the field to the energy density in the universe at decay for r > 1/2. be in its minimum.‡ As can be seen from Eq. (13), to avoid a period∼ of cur- If the mass squared is positive during inflation, and the vaton dominated inflation, the curvaton cannot dominate field has large initial values, it will decrease exponentially the energy density before the onset of oscillation. This fast until m < HI , and the field becomes overdamped. translates into The field value∼ typically freezes at φ HI . After that, the field decreases only linearly until the∼ classical motion φosc
3 For lower temperatures T < hφ the running of the at its minimum. Substituting this in the equation for yukawas is modified leading to∼ an effective thermal mass the fluctuations yields a positive effective mass squared 4 4 2 2 2 Mth = h T /φ . Thermal damping is unimportant when meff = [(n+4)(n+3) 1]cH . When the effective mass is of the order H damping− is efficient, and the fluctuations δM 2 th < 1 for H>m (18) will decrease exponentially fast during the last 60 e-folds 2 < H ∼ of inflation. This gives a bound m/HI 1/60. Therefore, for the curvaton scenario∼ to work within It is important to note that even before H ΓI there is 2 ∼ 1/4 the context of supersymmetry we need to invoke symme- a dilute plasma with temperature T (TI MPH) [15]. Another effect to be considered is∼ thermal evaporation tries to protect the scalar field from obtaining large soft by particles which are in equilibrium with the thermal masses. When inflation is driven by D-terms, gauge invari- bath, that is for which meff hφ < T [14]. For a cou- pling in the scalar potenital V∼ hφ2χ2 the cross section ance forbids the appearance of soft masses for the scalar 2 2∋ fields [10]. The scalar mass are set by the low energy for φχ-scattering is σ h /Ecm. The typical center of ∼ SUSY breaking, and are typically of the order m3/2. mass energy is Ecm √Tm, the mean of the typical χ- ∼ energy ( T ) and φ∼-energy ( m). The thermal decay At the end of inflation, the vacuum energy density stored ∼ 3 2 2 ∼ in D-terms is transferred to kinetic and F -terms, which rate is Γth σT h T /m. If the condensate evapo- ∼ ∼ > do induce mass terms of the δm2 H2. The sign of the rates thermally when H ΓI , the isocurvature curvaton ∼ are not converted into adiabatic∼ perturbations, and the mass term depends on the K¨ahler potential. curvaton scenario does not work. This happens when No-scale type gravity models possess an extra, so- called Heisenberg symmetry, which forbids soft mass
Γth > H & T > hφ. (19) terms at tree level [11]. Masses are induced radiatively, ∼ and are suppressed by loop factors All thermal constraints can be circumvented if the in- c flaton sector and curvaton sector decouple completely, δm2 h2H2, (20) ∼ 4π2 as is proposed in reference [6]. In their scenario the in- flaton is a hidden sector field which decays into hidden with c = (1), which can be positive or negative de- sector radiation, while the curvaton field is responsible pending onO whether gauge or yukawa coupling dominate, for reheating of the MSSM sector. Then, before curva- and on which field (hidden sector, matter field, dilaton) ton decay there is no thermal bath of MSSM particles, is the inflaton. For δm2 > m2, where the curvaton and thermal effects are negligibly small. mass is typically set by| low| energy SUSY breaking, the mass squared can be negative during inflation. Since the gravitino mass decouples in no-scale gravity, there is no III. VARIOUS MODELS problem related to gravitino overproduction. Pseudo-Goldstone bosons are protected from soft cor- The finite energy density in the early universe breaks rections; the mass is set by the breaking scale of the supersymmetry, leading to soft masses in the scalar po- global symmetry. The analysis for this case is, apart from tential which are generically of the order H [8]. Soft constraints concerning gravitino production, the same as mass terms are both induced by non-renormalizable for standard model curvaton fields. as by supergravity corrections. In global SUSY, non- renormalizeble terms in the K¨ahler potential of the form 4 −2 † † 2 A. Negative mass squared — no-scale inflation δK = d θMP φ φI I lead to mass terms δm (ρ/M 2) H2. Supergravity corrections to the La-∼ P R∼ grangian likewise induce mass terms of the order H2. In this section we consider the parameter space for The mass squared can be either positive or negative, de- curvaton fields with no-scale type masses of the form pending on the specific K¨ahler potential. δm2 10−2h2H2. (21) However, the curvaton scenario cannot work for δm = ∼− (H), since this would give large deviations from| scale| Odependence in the density fluctuations, in conflict with For the induced mass to be the dominant term, we also observations. The COBE requirement Eqs. (5, 8) trans- require that during inflation the effective mass is nega- tive, i.e., lates into m/HI < 10. But the constraint can be made stronger. For a positive mass squared a large coher- m2 < δm2 (22) ence length for the fluctuations is needed, which requires | | m/HI < 1/40. For a negative mass squared, there Then during inflation the φ-field is driven to its classical is another bound. After horizon exit the zero mode minimum and fluctuations evolve according to Eqs. (2, 3). Since the mass is negative m2 = cH2, the classical field is φ (H2M n/λ)1/(n+2). (23) − min ∼
4 In the post-inflationary evolution the field keeps track- 0.0001 ing its instantaneous minimum until δm2 m2, at | | ∼ which point the mass becomes positive, and the field 1e-06 freezes until H m. The higher order terms in the ∼ 1e-08 COBE potential Eq. (10) are of the same order as the mass Γ th term at the minimum, and therefore do not alter the 1e-10 m<0 conclusion that the fluctuation spectrum Eq. (5) is flat. h 1e-12 However, the higher order terms are important for the nucleosynthesis evolution of the fluctuations. With the classical field 1e-14 at its minimum, it follows from Eq. (3) that the fluc- 2 1e-16 tuations have a positive effective mass squared mfluc = −2 2 2 + [(n + 4)(n + 3) 1]10 h H . While the zero mode 1e-18 decreases for δm −>m, fluctuations are overdamped and 0.0001 0.01 1 100 10000 1e+06 1e+08 remain frozen.| This| leads to a q-factor Eq. (6) m (GeV) FIG. 1. Parameter space for the curvaton model in no-scale φ∗ 2 2 2 2 q = , (24) type gravities with a soft mass term δm 10− h H dur- φ ≈− ′ osc ing inflation, for the parameters M/√λ = MP, n = 2, and with φosc the classical minimum Eq. (23) at δm ΓI > m. The curvaton scenario works for masses and cou- 0.1hH m. The fraction of energy density stored| | in ≈ plings in the shaded area. the curvaton∼ field at decay is 2 α 1 φ ΓI In the same coupling range damping by thermal masses r = osc . (25) dec h M 2 m can become important. P The CMB constraint Eqs. (7, 8) then reads There are no constraints from gravitino production, as the gravitino mass can be arbitrary high in no-scale 1 α H m2M n n+2 Γ models. Further, φosc 5 the magnitude of the condensate. 1 During inflation a condensate is formed with magni- 0.01 tude given by Eq. (11). The curvature perturbation is given by Eq. (7) with q 1. If the curvaton comes to 0.0001 non−ren. terms ≈ dominate the universe, r 1 (see Eq. (13)), and the 1e-06 ≈ Γ COBE constraint becomes 1e-08 th h m2 Γ α 1e-10 h < 10−22 I non−dominance GeV2 m 1e-12 ∼ dominance m 10−4H . (30) 1e-14 ≈ I 1e-16 nucleosynthesis For larger values of the coupling, the curvaton energy 1e-18 density remains sum-dominant. Then r < 1 and the 0.0001 0.01 1 100 10000 1e+06 1e+08 1e+10 COBE constraint reads m(GeV) H3 Γ α FIG. 2. Parameter space for models with a positive mass h = 10−34 I I , m GeV2 m squared during inflation, for ΓI >m. The curvaton scenario works for masses and couplings in the shaded area. 4 14 10 m < HI < 10 GeV. (31) ∼ ∼ 0.0001 −9 For couplings in the range 10 m/ GeV 6 < 3 < 7 masses: m 10 GeV for n = 2, m 10 GeV for n = 4, 0.0001 and m < 10∼8 GeV for n = 6. For∼ larger masses the ef- ∼ 2 1e-06 fective mass meff Vφφ sets the initial conditions during ≡ 2 2 inflation. As long as m H the perturbation are Γ eff I 1e-08 th nearly scale invariant and the≪ curvaton scenario is possi- 1e-10 ble. We will analyze this possibility in this subsection. h The effective mass Vφφ sets the initial value of the cur- 1e-12 n=6 vaton, which from Eq. (11) is 1e-14 n=4 1 nucleosynthesis 4 n n+4 3 H M 1e-16 n=2 φI 2 . (34) ∼ 8π (n + 4)(n + 3)λ 1e-18 0.0001 0.01 1 100 10000 1e+06 1e+08 1e+10 (GeV) When H meff the curvaton field starts oscillating. Af- m ter a few oscillation∼ the potential approaches a quadratic FIG. 4. Parameter space when the effective mass is set by form. We approximate q 1. During the first initial 6 8 10 higher order terms φ (n = 2), φ (n = 4), or φ (n = 4). oscillation the curvaton energy≈ density changes as Here M = MP, λ =1, ΓI > m, and κ 1. The dashed line → 3 corresponds to the r = 1 line in the absence of higher order m(H) a0 ρ(H)= ρ(HI ), (35) terms (from Fig. 2), which is added for comparison. m(HI ) a where a is the scale factor of the universe. The field after shifting the plot by some power of κ (both to lower amplitude at the onset of quadratic oscillations, when 2 1/2+α 2 h and m values, see Eqs. (36,37)). Taking ΓI < m, or m(H) = m, is φosc = κ φI . Here we have n n M /λ < MP has a similar effects, the COBE bounds parametrized m = κ Vφφ. shifts to lower couplings and smaller masses. The COBE constraints Eq. (8) for domination r 1 p To summarize, the non-renormalizable terms can be translate into ∼ neglected only for sufficiently small masses. But in a 4 −16 m< 10 GeVCm, h< 10 Ch (n = 2) large part of parameter space, where the curvaton dom- 7 −8 m< 10 GeVCm, h< 10 Ch (n = 4) (36) inates the energy density, their contribution to the mass 8 −6 term during inflation is dominant, unless these opera- m< 10 GeVCm, h< 10 Ch (n = 6) tors are absent to a very high order. If the higher order −1/n 1/2+α α With Cm = κ(Mλ /MP) and Ch = κ (ΓI /m) . terms dominate, the curvaton scenario may still work, For the non-domination case r < 1, the COBE data re- but smaller couplings and masses are needed. quires α 1/n 3/4 10−20m5/4κ−3/4+α ΓI M/λ (n = 2) D. D-term inflation m MP α M/λ1/n h< 10−15mκ−1/2+α ΓI (n = 4) m MP 9/8 In D-term inflation there is no Hubble induced soft α M/λ1/n 10−13m7/8κ−3/8+ α ΓI (n = 6) m MP mass term during inflation. However at the end of infla- tion, the energy in D-terms is transferred to F -terms and (37) 4 14 2 2 AndHI is constraint 10 m/κ < HI < 10 GeV. kinetic energy and a mass term m = cH , with c = (1) ∼ ∼ n n is induced. O The constraints are shown in Fig. 4 for M /λ = MP , ΓI >m, and κ 1. Note that one can read off from the If c < 1 the field approaches its minimum Eq. (23). → 2 plot the parameter space where m > Vφφ during infla- The only difference with the no-scale model discussed in tion and higher order terms can be neglected, namely the section III A is the initial field value φ∗. If we ignore region to the left of the COBE constraints. In most of the the change in δφ/φ from turning on the Hubble induced curvaton dominated region in Fig. 2 the non-normalizable mass, then φ∗ drops out of the COBE constraint. There- terms cannot be neglected. Curvaton domination can fore the parameter space is given by Fig. 1, with the im- only occur if the non-renormalizable terms are absent to portant difference that the constraint meff < 0 of Eq. (22) some very high order, or if the effective mass during in- does not apply. flation is set by the higher order terms. This conclusion For c > 1 the field value decreases at the end of in- 3 does not change when Γ < m, since both the COBE flation, according to Eq. (35), and φ (m/H ) φ∗ I osc ∼ end constraints and the requirement r 1 have the same ΓI (for ΓI < m) with Hend the Hubble value at the end of dependence. ∼ inflation. Ignoring the change in δφ/φ from turning on For small masses, m = κ Vφφ with k 1, Fig. 4 the Hubble induced mass, the only difference with sec- indicates the couplings h needed for the curvaton≪ scenario tions III C and III B is that the coupling in the COBE p 7 constraints Eqs (30,31) and Eqs (36,37)) should be re- But that means that also their mass during inflation is placed by h h(m/H )3. This decreases parameter too large (δm2 H2), and they cannot be the curvaton. → end ∼ space, and moves the domination curve to the right. Right-handed sneutrinos can have small couplings to matter. In the sea saw model, neutrinos obtain masses −3 mν h Hu /MN < 10 eV, where Hu is the expec- IV. CONCLUSIONS ∼ h i h i tation value of the∼ Higgs field and MN the sneutrino mass. In [17] leptogenesis from an sneutrino dominated The finite energy density during inflation breaks su- universe is considered. In this model, sneutrino has mass 7 −12 persymmetry, and induces soft masses of the order of the MN 10 GeV and h < 10 . The sneutrino can be the Hubble constant for all scalars. The inflationary pertur- curvaton∼ if there are no∼ constraints from gravitino pro- bation spectrum for these fields is highly non-Gaussian, duction. Moreover, if the sneutrino mass is generated at and therefore they cannot play the role of the curvaton. some GUT scale, one needs to explain why MN MGUT To avoid this conclusion one has to invoke symmetries, and why non-renormalizable operators are absent≪ to very to keep the soft mass terms small during inflation. high order. 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