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CORE JHEP08(1999)009 9 brought to you by by you to brought 10 CERN Document Server Document CERN by provided . 8/18/99 RH T Accepted: , 7/28/99, Received: Gian.Giudice@cern.ch , Cosmological Phase Transitions, Neutrino and Gamma Astronomy. Many models of supersymmetry breaking, in the context of either su-

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HYPER VERSION CERN Theory Division, CH-1211E-mail: Geneva 23, Switzerland. [email protected] GeV. In this paper we showmay that be the much non-thermal more generation efficient of thanuli these the fields dangerous may thermal relics be production copiously after createdstate. inflation. by the Scalar Consequently, classical mod- gravitational the effects newto on the be, upper vacuum in bound some onof cases, the as gravitinos low reheating in as temperature the 100the is early GeV. thermal shown We Universe, production also which by studyinflationary can several the models. orders be non-thermal of production extremely magnitude, efficient in and realistic overcome supersymmetric Keywords: Abstract: pergravity or superstring theories,masses predict and the Planck-suppressed couplings. presence Typical ofthe examples particles are gravitino. the with scalar Excessive weak moduli production scale and the successful of predictions such of particles nucleosynthesis. inthese In relics particular, the the after early thermal inflation Universe production leads destroys of to a bound on the reheating temperature, Gian Francesco Giudice, Antonio Riotto and Igor Tkachev Non-thermal production of dangerousin relics the early universe View metadata, citation and similar papers at core.ac.uk at papers similar and citation metadata, View JHEP08(1999)009 . If produced X gravitational relics 1 = 1 supergravity models [1], supersymmetry is broken in some hidden sector N In In string models massless fields exist in all known ground states and parametrize One of the problemsenergy facing phenomena. Planck-scale physics Nonetheless, isinferred some indirectly the information through lack on its of high-energy effectsverse. predictivity physics on for the may If cosmological low- be we evolutionnot require of that significantly the early the modified Uni- successful andoverclose predictions that the of the Universe, big-bang severe energyclass nucleosynthesis constraints density of are are of supersymmetric imposed stable theories. on particles the does properties not of a large in the early Universe, suchat quanta very will late behave times, likelate eventually nonrelativistic dominating for matter the and nucleosynthesis energy decay oscillations to of with occur the large Universe (in amplitudesproblem until the it of [2]). case is the Typical too of zerothe examples scalar supersymmetric mode of partner fields, can gravitational of pose long-lived relicsthe the are even scalar coherent graviton – the a fields and spin-3/2 which moreand the gravitino parametrize serious seem moduli – almost supersymmetric fields ubiquitous flat – in the directions string quanta in theory. of modulithe space continuous vacuum degeneracies characteristicsible of examples supersymmetric of theories. gravitational Pos- expectation relics in value string parametrizes theory the are strength the of dilaton whose the vacuum gauge forces, and the massless 1. Introduction and gravitational-strength interactions communicate the breakingsector. down to In the these visible models thereorder often of exist scalar the and weak fermionicthe scale fields following, with and we masses gravitational-strength will of couplings the generically to refer ordinary to them matter. as In Contents 1. Introduction2. Gravitational production of scalar3. moduli Non-thermal production of gravitinos4. Conclusions 3 10 1 18 JHEP08(1999)009 X n (1.2) (1.3) (1.1) (1.4) TeV.) ∼ represents X 3 M T ∼ light n -particles is the essential X , . 2 light n |i . v | P . )GeV RH 9 12 tot M T and their lifetime becomes exceedingly − σ 10 2 W − 10 − 2 -abundance relative to the entropy density 'h M 8 10 . X mass, and here we have assumed X ' X (10 s X n Hn X . s n +3 RH T X dt dn can be produced in the early Universe because of thermal X GeV, gravitational relics are inconsistent with nucleosynthe- 16 is the total production cross section and ), moduli acquire masses of the order of the weak scale and, be- 10 2 P W ∼ M /M 1 P He; this requires that the 3 GUT M ∝ He and D nuclei by photodissociation, and thus successful nucleosynthesis M √ 4 ( tot ∼ σ O and its effect on the observable sector is screened, the cosmological problem of = Gravitational relics Comparing eqs. (1.1) and (1.3), one obtains an upper bound on the reheating RH T S S where long. On theM other hand, if supersymmetry breaking occurs at a scale larger than cause of their longbecomes lifetime, even pose more ascales, severe serious since in cosmological the models moduli problem are [3]. in lighter which than The supersymmetry problem is broken at lower Neglecting any initial number density,sity the of Boltzmann gravitational equation relics for during the the number thermalization den- stage after inflation is the number density of light particles in the thermal bath. The number density (The exact bound depends upon the at thermalization is obtained by solving eq. (1.2), scalar (and fermionic superpartner) gaugepactified singlets parametrizing dimensions. the size In of general,tion the these com- theory fields in are theof massless exact non-perturbative to effects supersymmetric all or limit ordersthe because in and usual of scenarios perturba- become supersymmetry-breaking in massive which contributions.M supersymmetry either In breaking because occurs at an intermediate scale If scatterings in the plasma.source of The cosmological slow problemsstroy decay because the rate the of decay products the of these relics will de- gravitational relics is relaxed.models In particular, with this anomaly-mediated is supersymmetry-breaking thegravitinos can case [4], of avoid in cosmological the difficulties recently-proposed which [5]. moduli and sis. Moreover, if the initial state after inflation is free from gravitational relics, the predictions [6, 7]. Thetion of most D stringent + bound comes from the resulting overproduc- temperature after inflation [7] at the time of reheating after inflation should satisfy [8] JHEP08(1999)009 effects. 3 non-thermal The goal of this paper is to point out that gravitational relics can be created One possibility is represented by the non-thermal effect of the classical gravita- Another possibility is represented by the non-thermal effects occuring right after The paper is organized as follows. In sect. 2 we show that the gravitational pro- 2. Gravitational production of scalar moduli Let us describescalar here particle the production. basic physics underlying the mechanism of gravitational reheating temperature in eq. (1.4)GUT is bosons too that low can to eventually allow produce for the the baryon creation asymmetrywith of [9]. dangerously superheavy large abundances by tional background on theparticle vacuum creation state mechanism during is similar theperturbations to that evolution the seed of inflationary the the generationtum formation of Universe. generation of gravitational large of The scale energyinflaton structures. density field fluctuations However, which the from dominated quan- inflationsub-dominant the scalar is mass density or associated of fermionic withbeen the field. the employed Universe, to generate Gravitational and non-thermal particle not populationsIn a creation of very particular, generic, has massive particles the recently [10,generated desired 11]. during abundance the of transition superheavydominated phase from dark as the matter the inflationary particleson result phase of may vacuum to the be quantum expansion a fluctuationseration of matter/radiation of the of background the density spacetime dark perturbations,background acting energy matter this density field. mechanism thatfamiliar contributes drives “particle Contrary to the production” to cosmic the effect expansion, the homogeneous of and relativistic gen- is field theory essentially ininflation the because external of fields. the rapid oscillationsis of the an inflaton interesting field(s). example Gravitino of production this kind. duction of scalar moduli whichefficient are than the minimally one coupled activated to bylead thermal gravity to scatterings may during extremely be reheating. stringent much constraints more OurIn findings on sect. the 3 reheating we temperatureshowing study after that the inflation. the non-thermal helicity-1/2 productionthe part of evolution of gravitinos of the in the gravitino thegravitinos Universe. can early in be This Universe, realistic efficiently leads supersymmetric excited tonon-thermal models during a production copious of of and gravitinos inflation.the dangerous have considered helicity-3/2 generation Previous components. only of studies the [12] However,equations subleading a of has effects thorough very of study recentlyequations of appeared presented the [13]. in complete Although our gravitino theagree paper derivation fully is of with different the the from gravitino analysis that of of ref. ref. [13]. [13], Finally, sect. our 4 final contains results our conclusions. JHEP08(1999)009 X (2.2) (2.5) (2.1) (2.3) (2.4) 6=0or X M , # 2 ξRX − , 2 ) . 0 (henceforth, the dots over i X + η x ~ . i = 0) is certainly of physical · ( 2 X x ~ k ~ · k , the action is i ξ ¨ a a k 2 ~ = i M − ) , x e 2 e 1) ∗ k ) d~ ) iωh k − ~ ) h η η t − k − ( 2 ( ( − ˙ k ) h ∗ k 2 ξ )=0 1 h 2 h a η X k k ~ † k − a ( ). The resulting mode equation is )= ( a a k 0 ~ − η ∇ ∗ k h +(6 η ( h ˙ 2 h (3) + ) ( ) 2 k δ k η 4 η dt a − ), we use the field mode expansion h ( ˙ ( h 2 2 2 = 2 a X k x = 2 k ω 3 i / , d~ M 2 3 2 2 d ) † k / − 1 + dt ds π dX )+ ,a 2 − k 2 η 1 (2 k ( ω k dη k " a ¨ h = h Z )( 3 2 η 2 a )= k ( 0 = ω 2 x η 3 a ( d X k h = Z 2 dt ds is 1/6 for conformal coupling and 0 for minimal coupling. The Z ξ = S is the Ricci scalar. After transforming to conformal time coordinate, where 6, conformal invariance is broken and particles are created. In the following R / The differential equation (2.3) can be solved once the boundary conditions are We start by canonically quantizing the action of a generic scalar massive field 6=1 Since the creation and annihilation operators obey the commutator relations functions denote derivatives with respect to interest. Just to givestring one theory, example, the in modulus thedimensions field toroidal is describing minimally compactification the coupled of volume to the gravity of type in the IIB the internal foursupplied. compactified dimensional effective Since action. theparticular annihilation basis, operator fixing is theWe just boundary choose a conditions the coefficient initial is of conditions equivalent as to an fixing expansion the in vacuum. a where the line element is , we obtain the normalization condition where The parameter equation of masslessspace-time conformally and, coupled in quanta this reduces case, to there the is equation no in particle flat production. When ξ we will be especiallycoupling. In interested particular, in the the case case of minimal of coupling scalar ( particles with non-conformal which, at the end, will bewhere identified the with a line modulus element field. is In given the by system of coordinates JHEP08(1999)009 I X )= )+ I η (2.6) (2.8) (2.7) ( ( .The ). We 0 2 P η k V h H k = H /M α | ρ C X | ( scalar particle. =3 = 0 is certainly )= V η 2 X H 2 ( P X 1 C η k M M into the one with the h 2 0 H of the Universe, that is η while the quantum state . mass 2 . To obtain the number ρ = 1 | = 0 k η η η X (1). To illustrate this effect, β V X, | † 2 O does not change in time in the = 0 . = IX η η 2 † dk k I H H 2 P C ∞ H 1 0 M C Z θ + ) 4 1 5 d 2 X η ( m 3 1 Z a 2 ' H π is identified with the inflaton field and 2 X 2 C I − M (any later time at which the particles are no longer )= = 1 1 η η ( δK = X ahler potential of the form -term is not dominating the energy density of the Universe n η F is the Hubble parameter and . During inflation, H i I † to be of the order of TeV, the present effective supersymmetry-breaking accounts for the mass term generated by any possible source of supersym- θI is the field which dominates the energy density 4 ) (the superscripts denote where the boundary condition is set), we obtain . The term (2.7) therefore provides an effective X X d η I 2 P ,where ( m 2 0 m R η M Let us now discuss the structure of the mass term of the ∗ k 2 H h 'h H k H One should note that the number operator is defined at conclude that moduli quantawith will a be mass gravitationally squared produced of in the the form early Universe Heisenberg representation. The flat directions inparticles, are the lifted supersymmetric by supersymmetry potential,superpotential. breaking corresponding An and important nonrenormalizable to observation effectssupersymmetry massless for in us the is is broken that (for inof instance the the inflaton). early because Universe In of global supergravitygravitational the interactions theories, and false supersymmetry the breaking vacuum supersymmetry-breaking is mass energyC squared transmitted density is by naturally a possibility. Indeed, in supergravity models which possess a Heisenberg symmetry, boundary condition at same mass term willafter appear inflation. during This example the canthe coherent be scalar oscillations easily moduli generalized, of fields and the the during inflaton resulting inflation potential field is of of the form Here density of the produced particles,vacuum we mode perform solution a Bogolyubov with transformation the from boundary the condition at where (approximated to be the vacuum state) defined at being created). Defining the Bogolyubov transformation as corresponding to a vanishing particle density at 3 the following number density of produced particles: consider a term in the K¨ scale. It is important to bear in mind – though – that the case β metry breaking whose during inflation, but persistspect in the zero-temperature limit. For this reason, we ex- ρ JHEP08(1999)009 , 3 X 1) a m X 1can GeV (2.9) N, n (2.10) ( 9 H C H SU , the ratio 1TeV(or C =10 ,itismore H (see ref. [11] H ∼ C C RH 2 X T / thereafter. This 3 m 3 m − a ∼ X m . . initial by the entropy density. In doing . On the other hand, we expect -particles will be independent of /s X X X . n I 5 GeV final /m s − m GeV. The results for scalar fields with X 13 10 ρ ≡ 13 10 γ × field starts oscillating on all scales and the 6 5 X =10 0 – the present-day energy density of produced ' I X m → s GeV RH n . In particular, notice that for small turns out to be extremely large for small H 9 T X C /s 10 m X n . Therefore, in the limit of relatively large 1 2 X m = 0) are summarized in fig. 1. This figure shows the limiting . . If the reader wishes to adopt different values for the reheating 1 X ξ as discussed above, which allows to extract the relevant physical − X 2 n X m /s ), the ratio M X 10 1 the number density of produced n − . In this limit – and & is better inferred from the ratio 10 X H X ' m n C I scales like From fig. 1 we infer that scalar moduli quanta are very efficiently produced by In our numerical results, we have normalized We have numerically integrated eq. (2.3), in presence of an inflaton with quadra- /m , since the field fluctuations will enter the oscillating regime at all scales at the An analytical study of the gravitational production of scalar moduli in the regime /s 1 -particles is an appropriate quantity, since it is independent of X X X convenient to use Furthermore, if one wants tolate include the time effect after of reheating,entropy significant increase entropy the release in data at the some in comoving volume, fig. 1 have to be divided by the amount of X so, we have assumed that the reheating temperature after inflation is for a complete discussiondensity of this point). Therefore, for small masses, the number which is constant atcalculation late is technically times difficult and inbetter the is regime strategy easy of is to very represented small compute by masses. adopting numerically. In observables this which However, regime are the a independent of n gravitational effects. In particular, for the realistic case andthemassoftheinflatonfieldis that at procedure determines a well-defined quantity, the total number of particles minimal coupling ( behaviour of quantities at arbitrarily small at small supersymmetry breaking makes nometric contribution flat to directions scalar unmodified masses, attype leaving tree-level. supersym- and No-scale supergravity the of untwisted the this sectors general from set orbifold of compactifications models. are special cases of temperature or the inflaton mass, the data in fig. 1 have to be multiplied by the ratio m epochinwhich m tic potential, up to the epoch when the field fluctuations are transformed into particles which redshift as be found in Ref. [2]. JHEP08(1999)009 is X RH X T m (2.11) (2.12) minimally X GeV. 13 in the Universe, c ρ =10 I . In the figure 2 m would be H . Therefore the bound in X H 1 , C − GeV, + . 9 X s 2 X Mpc n , does the number density of 1 m X X =10 − m m TeV = X for scalar moduli particles RH 2 X 1 T m TeV − H M C 10 7 11 ,and × I 10 4 m × . 4 2 h GeV. If one wishes to adopt different values of ' X 2 13 Ω h becomes as low as 100 GeV. Only when the parameter X as a function of Ω 12 =10 − /s I X 10 m n is shown in fig. 2. Again, this figure was calculated assuming . X /s X n . Were these particles stable, the parameter Ω c GeV and The ratio 9 /ρ is the Hubble rate in units of 100 km sec or to take into account some entropy release, the data in this figure have to X h I ρ =10 m Another way of presenting the bound in eq. (1.1) is to compute the energy approaches unity, regardless of the value of = X H RH Ω eq. (1.1) can be recast into the form T particles become acceptably small. density in scalar moduli particles normalized to the critical density expressed in units of the inflaton mass where coupled to gravity and with squared masses The parameter Ω and This result is seven orders ofnon-thermal magnitudes production above is the limit so inbecome efficient eq. completely that (1.1). irrelevant the In and thermal this theorder case, scatterings upper the to during bound get reheating on the reheating temperature in Figure 1: C JHEP08(1999)009 =0. (2.13) H C for ξ 6, where particle / =1 ξ . 2 as a function of 2 h X I GeV m 13 . Conventions are the same as in fig. 1. H 10 and Ω C /s 8 X n GeV RH 9 T . X 10 m 1 − γ is very close to the conformal value ξ Ratio of the energy density in scalar moduli particles minimally coupled to Figures 3 and 4 represent the ratio Our findings are also relevant for the idea of enhanced symmetries and the ground production shuts off for small From the data onedangerous can unless infer that scalar moduli particle production is extremely Figure 2: gravity to the critical density, as a function of be multiplied by the factor state of string theory [14].with maximally We know enhanced that symmetry, there i.e.under are points some often where points symmetry all in ofnot (which the the suffer may moduli moduli from space be are the charged oscillations. continuous cosmological moduli or For problem ordinary discrete).during generated moduli, inflation) by finite are These their likely temperature moduli to large and/orto leave coherent do the curvature the Universe effects in present a (say statestate minimum which until does of the not Hubble the correspond which parameter moduli the is potential. of system order oscillates.the of The Universe the The Universe when curvature moduli of remains theythe typically the vacuum decay, in dominate potential, is leading after the this a to energyUniverse point catastrophic to density of start consequences. of maximally out enhancedpotential in However, symmetry, for it this if the is state. moduli quite is For natural example, modified, for during it the inflation, can be even expected though the that the system remains in JHEP08(1999)009 001, ,for . ξ =0 I /m X m 1, respectively. . =0 I 9 /m X m 1, respectively. . 01 and . =0 of scalar moduli particles as a function of the parameter I =0 /s I /m = 0. The solid, long-dashed and short-dashed lines correspond to X X n H /m m C X m ,for ξ 001, 01 and . . Ratio of the energy density in scalar moduli particles to the critical density as The ratio =0 =0 I I = 0. The solid, long-dashed and short-dashed lines correspond to /m /m X X H m C m the symmetric state. Finitesymmetry temperature [14]. effects However, this alsobeing selection tend copiously rule to produced does prefer by not the states prevent gravitational of the effects, scalar higher unless quanta the from effective mass of the Figure 4: a function of Figure 3: JHEP08(1999)009 (3.5) (3.6) (3.1) (3.2) (3.3) (3.4) =0,which ψ . · ∂ )=0 a ψ , − =0 . ψ =0and , · ψ . · ] ψ =0 γ a b · b γ a b ,γ ψ γ γ γ a ( . mγ c ab γ [ γ m 4 1 mσ − + )=0 )+3 c = ψ ψ γ b +2 b · · ab ψ γ d ∂ ∂ a a 10 ψ γ c γ 2 components. Replacing eqs. (3.5)–(3.6) in − − ∂ / b / ∂ i ,σ 2 1 ψ ψ γ b · 5 · ± γ γ = γ γ b − / / ∂ =+1 ∂ γ , leaving the appropriate degrees of freedom for the abcd ( ( 5 a ψ i γ ψ · 2and m ≡ 0123 / γ abcd 3 a = =2 , satisfying the Rarita-Schwinger equation / ∂ a ± R R R a ψ · · γ γ ∂ ); we also use the convention i 2 − , = − a , R − , 6= 0, these constraints imply the conditions m Using the identity The free propagation of massive gravitinos in Minkowski space is described by =diag(+ For eliminate two spinors out of Here flat space indices areη denoted by Latin letters and are contracted by the metric The mass term in eq.itly (3.1) assuming explicitly that breaks supersymmetry supersymmetry. is However,choosing spontaneously we broken are a and implic- eq. supersymmetric (3.1) gauge followsfrom from such the that lagrangean. thegauge This Goldstino theories. choice field is is analogous fully to eliminated the unitary gauge in Yang-Mills eq. (3.1) becomes From eq. (3.4) we obtain the following two constraints 3. Non-thermal production of gravitinos In this section we compute thegravitational nonthermal backgrounds. production of Before gravitinos solving in thespace, timedependent relevant it equations of is motion useful in to curved recall the known resultsfour of Majorana gravitino spinors propagation in flat space. scalar excitations around theHubble point rate of during enhanced inflation. symmetrymoduli is fields Therefore, much has larger gravitational tomodel. than production be the of checked these case special by case, in order to assess the viability of a propagation of the spin JHEP08(1999)009 =0, (3.9) (3.7) (3.8) =0. (3.10) (3.11) (3.12) / ψ ∂ · ∂ ,andwecan a ∂ = . a ab ψ , σ , 2 spin states as physical δ ν / , µab abcd ψ 3 a ) ω ± σ d γ . 2 1 e µ bµa ρ c ¯ ψ e + C ν b 2 =0 P µ . e ) are comoving space coordinates, + 1 σ ∂ µ a 3 M ψ 2 ee ρ = ,x =0 abµ 2 µ − a D C ≡ ν ψ ,x a µ ˆ γ 1 + e ) 5 11 ν x = 0. Then, eq. (3.5) leads to γ m ∂ µνρσ µab ,D ψ µ − − C =( ˆ µνρσ γ · a ν / − x ~ i∂ e ( m γ ( µ i , ≡ 2 2 1 ∂ a µ γ + = = µ a R is the covariant derivative with respect to the spinorial µ e = 0 follows from the equation of motion in the massive µ µν D = 0 still allows gauge transformation subject to can be obtained (in first-order formalism) by solving the µab a ≡ ). Here A D 2 ω · = ψ µ C x · ˆ γ µab µ ∂ d~ ω γ D − 2 . 0 dη γ )( † µ η ψ ( = 0, eq. (3.1) possesses a gauge symmetry 2 a ≡ is the vierbein. m µ µ a = ¯ e ψ 2 For For our cosmological considerations, we are interested in the case of spatially Let us turn now to the case of curved space, in which the gravitino equation of ds where Greek letters denote space-timedices. indices and Gamma Latinfined matrices letters by and denote the tangent-space in- Levi-Civita symbol with curved indices are de- structure modified to also reproduce the mass term supergravity equation ofconnection motion are where treated as the independent gravitino, fields. the One finds vierbein, (see and e.g. ref. the [15]) spin The spin connection eq. (3.4), we obtain thatequation the propagation of physical states obeys the ordinary Dirac where choose a gauge fixing such that flat Friedmann-Robertson-Walker metrics, in whichas the line element can be written However, the choice and a further conditionan is required additional to fermionic fully component,particles. fix leaving the Notice gauge. only the complete the Thisthe analogy condition Lorentz with eliminates condition the case of spin-one particles, in which case and it isparticle, a the gauge gauge fixing choicedegree of requires in freedom. a the further massless condition, case. which eliminates Also,motion the for becomes scalar a massless spin-one JHEP08(1999)009 , p ). µν η ( and η 2 µ k ~ (3.15) (3.13) (3.14) a ψ m .Thus, x ~ · = k ~ =0gives i − µν R and pressure g ke dη/dt 3 ρ , d µ D· = a . + δ 1 R 1 − − ˙ a ψ m a ∼ a · i ) ˆ γ x = 0 , +2 µ µ a η,~ µ g 0 ( e ˆ γ µ γ , )+ − m ψ a µ direction. For simplicity, we . ψ i 2 2 ψ =0. · 3 2 P aδ · x m + ∂ . The condition ψ ˆ γ 0 3 µ = · ˙ 0 mM ˆ γ µ i ˆ γ ) γ − a µ ˆ γ 2 µ P e 0 − 2 4 γ +2 ˙ γ a/aψ a a M ψ 0 2 · , γ − + m +3ˆ +2˙ 0 γ 2 0 3 P /ˆ γ ¨ µ ∂ a 12 a +3 is along the ψ µ µ M ψ 2 µ ρ ˆ 2 γ γ )=0 µ k ( ~ γ µ − ∂ m a 3ˆ +ˆ 2 3 ψ is the scale factor, such that ˙ a a = − ψ − 2 4 a − · µ p µ 2 ψ ψ ˆ + γ ψ , µ · 0 µ m µ = i ∂ γ ∂ ∂ ˆ γ c 5ˆ ψ + µ i 1 ≡ − = ˆ γ γ 2 4 ˙ (ˆ a a a µ ˙ µ a ψ 2 3 ψ · =1 m i D i a X / ∂ ∂ ( 3 c + + i can be expressed in terms of the background energy density = = = c and 0 c µ ψ µ R ∂ . In this case, the covariant derivative in eq. (3.10) reduces to µ ˆ γ µν η and recovers the gauge fixing condition 2 ≡ − is the conformal time and 0 / ∂ a γ η Since spatial translations generate an exact symmetry of space-time, it is conve- Using the identity in eq. (3.3), we can rewrite eq. (3.8) in the Friedmann- − = The parameter ≡ are much smaller than the Planck mass. 2 = 0 µν the vierbein and the metric can be written as In the following we will considerand the choose equation the of motion coordinates for such a that single momentum mode using the Einstein equations the following algebraic constraint: g nient to expand the gravitino field in momentum modes, Here we havependent, also since considered in the generalbackground it possibility fields. is that In determined the the by flatc space-time time-varying gravitino limit supersymmetry-breaking mass with constant is gravitino mass, time one gets de- x Robertson-Walker metric as H where the dot denotesdropped the from derivative with eq. respect (3.12)is to the crucial conformal torsion for time. term, establishing the bilinearsiderations, Here consistency in it we of can the have supergravity be gravitino [16]. safelyis field. However, ignored sufficiently for This since our small. the term con- number During density the of produced cosmological gravitinos epochs we will consider, both where This expression agrees with the result found in ref. [13]. JHEP08(1999)009 (3.20) (3.22) (3.16) (3.18) (3.21) (3.17) (3.19) 2 helicity state ) indices are / correspond to 0 1 , 2 1 ± / µ ) (3.23) ± 3 2 P ψ 2 (spinorial indices) m / + and 2 . k µ 2 0 , , defines the / √ 1 P 2 2 , / ψ 2 / 0 1 / 1 m , 3 , 3 ψ . ± 0 , 2 ψ 1 + P / 2 ˆ γ 3 . k, 2 3 2 0 d 2 ψ , ( 2 ψ 3 1 ) and vectorial ( γ 2 γ 2 / , ˆ γ √ 2 1 / r 1 2 γ 1 2 m ˙ / 3 2 3 a / a 1 ψ iγ 1 ψ i − + +ˆ ± q and = 1 ψ 2 1 1 P ± 2 ˆ γ / µ 2 µ = ± 1 0 1 differ by a time-dependent function. ψ 1 / 1 i √ ˆ 1 γ 1 i P 2 , 3 ψ 13 ˆ ψ 2 γ 1 ψ ψ ˆ i γ / d i m µ 2 1 + 6 ˆ ± γ 1 ˆ γ 1 3 2 ∓ i P − ,P + 3 √ / ˆ γ = 2 =1 k 1 P , defined such that 3 i ˆ d / γ 2 4 P 0) 2 c 2 1 ˙ 1 3 a a ψ / / P 1 ± 3 − 1 2 3 2 3 ˆ γ 6 ,i, P r ± 2 ψ d 2 and 1 1 a P √ γ r r 2 , respectively. One can also verify that the normaliza- = = ∓ / 2 1 , / = = 2 2 = =ˆ = = and 3 ψ / / (0 3 3 1 2 3 0 1 d ψ 2 µ µ / 2 ± ± . Because of the antisymmetric properties of the Levi-Civita ψ = 0 does not contain time derivatives and therefore describes ψ ψ ψ ψ 1 1 µ k ~ P P √ 0 ψ ψ and R = 2 / 1 2 helicity states by explicitly constructing the helicity projectors. 1 µ / ± from ψ 3 P k ~ ± . It is convenient to describe the remaining independent fields with | k ~ onto µ ≡| 2and ψ / k 1 ± Notice that the linear combination 3 symbol, the equation It is now easy toproject verify that, in the flat limit and on mass-shell, eqs. (3.21) and (3.22) the and spin 1 (vectorial indices)the states, appropriate the Clebsch-Gordan helicity projectors coefficients can be decomposed using Since the gravitino field is built from the direct product of spin 1 We can show that, in the flat limit and on mass-shell, The helicity projectors acting on spinorial ( an algebraic constraint on the gravitino momentum modes, which is given by where two Majorana spinors drop the index tion chosen in eqs. (3.17)–(3.20)to insures canonical that kinetic the terms Rarita-Schwinger for lagrangean the leads fields adopted in ref. [13]. The states JHEP08(1999)009 , 2 / 3 , − a (3.24) (3.25) (3.26) (3.31) (3.30) (3.29) (3.27) (3.28) ,with W θ , ∼ ψ , ˙ 1 =sin mm . − 3 ˙ a a   a B 3 4 , , ∼ . − − φ and , a simple inspection 2 4

=0 =0 2 θ − m P 2 4 Φ 2 2 dt a d ˙ / / a a ) with constant gravitino 3 1 , ψ ψ . ∼ +3 2 coeﬃcient replacing the ˙ − /a # mM am are derived from eq. (3.14), / 2 2 # 2 4 µ 2

=cos

3 3 ˙ 2 a 2 a a ∂

  / γ m 2 Φ 3 A 2 dt kγ P Φ d ψ m dt 0 =2˙

d , 4 + M and

| a ˙ 2 m 0 ma − − iBγ a 2 m and P in terms of the superpotential ma 4 8 (Φ) + 2 ˙ + a a ∼ (Φ) + = M / − m 2 1 (Φ) W ˙ a µ V | A # 0 a + ψ ∂ V 2-helicity states satisfy the same equation " 2 2 γ | / 2 14 " P Φ 2 k P 2 a a P † 3 m 2 4 M 2 5˙ 2 P Φ W 2 ˙ 1 a a M + | i ± M 1 e 3 3 M +3 as + − = .Since 3 6 ma − 0 2 = = 4 ˙ V a a ∂ 2 − m m

0 − 2 4 2 4 a ˙ ˙ − a a a a 0 2 iγ P W γ ˙ 5 term. This coeﬃcient is determined by the ﬁeld scaling 3 a † − ¨ a a a a ¨ a a M Φ 5˙ 2 L∼ 3 2 2 ¨ a i a a/a + 2 2 2 + 0 2 2 W ∂ Φ 0 m m ). In general, however, the time dependendence of the gravitino ∂ 2

m iγ 1 + + " 2 for spin-0, spin-1/2, and spin-3/2 particles, respectively. 2 4 2 4 ma ˙ ˙ Φ 2 a a a a P 2 † ( / M Φ 2-helicity states, on the other hand, satisfy the more complicated evolu- 5 3 3 e a/ / − 1 a = 0), the coefficients in eq. (3.25) become = = = is related to the evolution of the scale factor. Let us consider the simple ± 2 in front of the ˙ ∼ / V m A B 2) = ˙ , and it can be simply obtained by recalling that the lagrangean density should µ m a ψ θ/ The The equations of motion for the fields tion described by eq. (3.25). In the de Sitter limit (¨ tan( mass ( ˙ and we can write the scalar potential of the kinetic terms in the corresponding lagrangeans shows that Using the expression for the gravitino mass mass case of a singleand chiral space superfield components Φ of with the Einstein minimal equation kinetic become terms. The diagonal time usual 3 with scale as an energy density, of motion as an ordinary Dirac particle, except for a 5 With the field redefinitiongrangean has in been eqs. diagonalized. (3.17)–(3.20), The the massive Rarita-Schwinger la- JHEP08(1999)009 2- : / 1 iB (3.38) (3.32) (3.37) (3.39) (3.33) (3.34) (3.35) (3.36) ± =0in + B A in terms of ≡ 2 , / G . i 1 k † ψ r − =1and a =0 ) with A . f k ~ 2 + # η, m u 2 2 ( 2 2 2 ˙ r / G G ˙ a a v 1 , 4 3 − , + ∓ G +6 − 0 . ma, r k ½ u a 4 6 ) = ) ½ + ˙ G ¨ a a G 0 k B 2 ~ =0 f − ω f η, ∓ + +2 ( . ≡ r = ˙ u iA Ω Ω 2 3 ( i ma h =1 = 1, see eq. (3.33). Defining k ˙ 2 m , 2 G G | + 3 i + 2 =1 B 15 G X r | , − ) satisfies the same equation of motion. The ,γ − ± x ~ + · · k ~ 2 4 ) k ~ 2 ˙ +Ω a a i η, ½ ole in the solution of the equation for the dη ω 2 − A 0 ( η e k ma − imau v R ( 2 ¨ a a i i / ∓ 0 ½ 2 3 k e ) + + 3 + = π ¨ d 2 2 4 f is used in eqs. (3.26) and (3.27), we obtain ≡ = ± a ¨ a a ). For simplicity, we now focus on a single polarization (2 ˙ 2 u 0 in the following. Choosing a gamma-matrix representation k ~ m G − γ m r − Z = 2 + / η, 2 5 ( 2 ˙ − ). The spinor k m rC a . To get a more transparent interpretation of this frequency Ω, it − " is analogous and it is obtained by just setting u ) satisfies the equation of motion (3.25), i.e. 2 k ~ + ,u / ma )= 3 + ¨ f η, x u )= ψ ( r k ~ η,~ u ( =( η, 2 ( / T r 1 v u ψ To proceed, we explicitly write the momentum expansion for We can now reduce the system (3.36) of first-order differential equations into a helicity states. eq. (3.37) can be rewritten as treatment of creation and annihilation operators When this expression for ˙ Here we have made use of the property An analogous equation hasthe been familiar derived in equation ref. foridentified [13]. a with Equation spin-1/2 (3.39) fermion is in identical a to time-varying background, if Ω is Replacing eqs. (3.28) and (3.29) in eq. (3.31), we obtain This condition plays an important rˆ state and omit the index such that where the previous equations. second-order differential equation for the function the spinor where JHEP08(1999)009 , 2 H − P 2or / M (3.41) (3.40) 2 Φ ∝ m is present. m P = M W and the frequency Ω , the Hubble rate 2 -term has the problem Φ W F 2 Φ ∂ . One can now verify that 2 | W/∂ 2 ∂ W is the value of the Hubble rate →− Φ I ∂ ω H , , . 2 | P −| 2 Φ gets contributions of order unity. In 2 | W | Φ 2 ˙ ∂ mM ∂ i ρ /V i /dt 2 00 ,where 2 Φ 3 16 I , as suggested in ref. [13]. Since V − − d P | 2 H P p and it is zero otherwise. The resulting number = M = M ∼ and fixed = ˙ 2 p G G max / = 3 G k | η n Φ . | and →∞ , supergravity corrections make inflation impossible to k 2 tends to the expression P n | Φ . The production of the helicity-1/2 gravitino is expected M G W Φ m ∼ 1 for ∂ | ' W + k 2 , to the equation describing the time evolution of the helicity-1/2 n | P M /dt , i.e. Φ k d | | Φ = | ρ To compute the abundance of helicity-1/2 gravitinos generated during and after Since the theory of production of helicity-1/2 gravitinos looks similar to the satisfies the following differential equation -term [19], or some particular properties based on string theory [20]. during inflation). However, ina the large evolution of amount the of UniverseFermi gravitinos distribution subsequent can to function be inflation, is produced.momentum rapidly During saturated the up inflaton to oscillations, some the maximum value of the in this limit the function of the oscillations corresponds towhich the is superpotential ‘eaten’ mass by parameter the gravitino ofdescribing when the supersymmetry Goldstino the is broken. production Therefore, oflimit eq. helicity-1/2 (3.39) of gravitinos in supergravity reduces, in the Goldstino in global supersymmetry and no suppression by powers of Therefore, in the limit of G that the flatness ofwords, the the potential slow-roll is parameter spoiled by supergravity corrections or, in other to be dominated by the fastest time-varying mass scalean in inflationary stage the in problem. thesupersymmetric early inflationary Universe, models one [17]. needsgeneric to supersymmetric A discriminate inflationary crucial among stage point various dominated to by keep an in mind is that a and the gravitino mass higher powers in Φ, The frequency of thethe oscillations problem, Ω namely depends the upon Goldstino all mass the parameter mass scales appearing in is useful to consider the limit simple one chiral field models based on superpotentials of the type start. To constructeither a invoke model accidental cancellations ofD [18], inflation or in a the period context of of inflation supergravity, dominated one by must a where case of helicity-1/2 fermionsresults with obtained a in the frequency theory Ω,During of inflation, one generation since of can Dirac the use fermions massconstant as during scales preheating in a present [21]. in guide time, the thenumber one frequency recent density Ω does can are be not approximately at expect most a significant production of gravitinos (the JHEP08(1999)009 , , 3 /κ 2 αµ µ 4 (3.43) (3.42) to the ∼ = 2 2 / i / 3 3 n ¯ n φφ h are oppositely ¯ φ and µ to the entropy den- and ∼ 2 φ . Asimilarresultmay / 3 being the scale factor – n RH max a is vanishing. T k – 2 3 − GeV and the ratio a is expected to be roughly of the , = 0 and the potential reduces to ) 15 W/∂S 2 . 2 ¯ max φ µ 10 ∂ k µ RH − = ∼ T φ α ¯ µ φφ 17 , κ ∼ ( . The superpotential (3.42) leads to hybrid κ ∼ | 2 . This is not a realistic situation – though – S √ / S | α s 3 = 2 n µ/ RH ∼ T √ W = .Thevalueof . Therefore, one expects ≡ /s 1 c 2 − / Φ -term inflation. 3 max GeV to reproduce the observed temperature anisotropy. 3 µ k n D 15 ∼ ∼ GeV and imposes a stringent upper bound on the reheating 2 9 / 1 TeV. Notice that the non-thermal production is about five 3 n 10 . , inflation ends because the false vacuum becomes unstable and the ∼ c RH rapidly oscillate around the minimum of the potential at RH T T ¯ φ (1), this violates the bound in eq. (1.1) by at least five orders of magni- summarizes the uncertainty in the estimate. If all the energy residing in ). The mass scales at the end of inflation change by an amount of order is a dimensionless coupling of order unity [22]. Here, plus supergravity and logarithmic corrections [18]. Therefore, in this regime 1 and O α − κ 4 φ µ within a time-scale µ = ( When Φ = Φ Our results on gravitino production are based on a simple one chiral superfield model, but we µ 4 = O α is fixed to be around 10 V orders of magnitude moreduring efficient the than reheating the stage, generation irrespectively through of thermal the scatterings value of while the field Φis rapidly oscillates around zero. The time-scale of the oscillations where the vacuum during inflation istemperature istantaneously would transferred to result radiation, to the reheating be be obtained in the case of entropy density would be sity is temperature of Notice that in this period, the Goldstino mass in the post-inflationary scenario,Universe. presumably characterized If by this a is matter-dominated the case, at reheating the final ratio density is therefore order of the inverse ofproblem the at time-scale hand. needed Let for us the give change a of realistic the example. mass scales Consider of the the superpotential If tude even if the Universe is trapped in theµ false vacuum and we have slow-roll inflation. The scale expect similar results to be valid in the case of a theory with more than one superfield. inflation. Indeed, for Φ fields where because such a highabout the reheating gravitino temperature generation through isIn thermal a collisions, already see more ruled the realistic bound outlate, scenario in by the in eq. which considerations number (1.4). reheating density and of thermalization gravitinos occur decreases sufficiently as charged under all symmetriesnormalized so inflaton that field their is product Φ is invariant. The canonically- JHEP08(1999)009 , 1 − I H ∼ 1. On the & H during inflation. C I H 4 / 1 V GeV is the height of the potential 15 10 ∼ 4 18 / 1 -term breaks supersymmetry, the Hubble rate V F 1, the generation can be so efficient that the standard for helicity-1/2 gravitinos in generic models of inflation ,where 4 . / /s 1 2 H / 6, i.e. they are conformally coupled, and if the effective mass 3 C /V / n RH T 6and / is comparable or larger than the Hubble rate, i.e. close to 1 . If produced with too large abundances, the late decays of these 2 6=1 P ξ H ξ M H C In this paper, we have also studied the non-thermal generation of gravitinos in The ultimate reason for such a copious generation of gravitinos is that the system GeV obtained from considerations about the production of scalar moduli through 9 This is a common feature of realistic models of supersymmetric inflation. since the frequency is set by the height of the potential and the gravitino mass. Again,the in non-thermal most production realistic of modelstheir of gravitinos thermal supersymmetric turns generation inflation, out during toexample, the that be reheat the stage much ratio more after inflation. efficient than We have found, for the early Universe. Theamounts, spin-3/2 as part the of theOn resulting gravitino the abundance is other is excitedhelicity-1/2 proportional only hand, Dirac in to the particle very the in spin-1/2 small adifferent small part background mass gravitino obeys whose scales mass. frequency the at issuperpartner equation a hand: of the combination of scalar the of field motion the superpotential whose of mass a parameter of normal the fermionic thermal collisions during the reheat stage. predictions of nucleosynthesis arelow safe as only 100 if GeV. This the10 upper final bound reheating is temperature considerably is smaller as than the bound of about during inflation. This genericallywe leads to should an stress overproduction that ofspecific the gravitinos. inflationary non-thermal However, model. gravitino production Therefore,search is it for quite will realistic become sensitive supersymmetric an to models important the ingredient of in inflation. the In conclusion, we have investigated theof non-thermal production particles in the with earlypowers mass Universe of in the TeV range and couplings to matter suppressed by 4. Conclusions gravitational relics may jeopardizenucleosynthesis. the We successful have shown predictionsin that of the scalar mass standard spectrum moduli of big-bang – supergravityin and which large string are theories amounts – generically as may present on populate a the the Universe result vacuum of quantumto the fluctuations entropy expansion density of of ratio the thegravity depends background and moduli upon space-time what fields. the acting isthat way their The these the effective resulting mass scalar generation during moduli number ofgravity and with are these after coupled moduli inflation. to fields We poses have no shown problem only if they couple to squared contrary, if relaxes to the minimum in a time-scale much shorter than the Hubble time is roughly given by JHEP08(1999)009 , , ]. 12 ]. (1993) 447 (1998) 4048 . (1999) 023501 (1999) 063504 81 hep-ph/9308292 B 318 (1983) 30; hep-ph/9403364 D59 ; D60 J. High Energy Phys. (1982) 59; (1984) 181. 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