Thermal and Non-Thermal Production of Gravitinos in the Early Universe

Home , Gian Francesco Giudice

JHEP11(1999)036 GeV. 9 10 . RH T November 26, 1999 Accepted: November 12, 1999, riotto@nxth04.cern.ch , urich, Switzerland Received: Cosmology of Theories beyond the SM, Physics of the Early Universe. The excessive production of gravitinos in the early universe destroys ur Theoretische Physik [email protected] [email protected] onggerberg, CH-8093, Z¨ HYPER VERSION E-mail: ETH-H¨ Institut f¨ CERN Theory Division CH-1211 Geneva 23, Switzerland E-mail: However, it has been recentlyin realized the that early universe the can non-thermalby be generation several extremely of orders efficient gravitinos of and overcome magnitude,temperature. the leading thermal to production In much tighter this constraintsduction on paper, of the gravitinos, we reheating taking first intoand account investigate inflation that some is in fact likely aspects reheating tothe of is be inflaton not the followed field. instantaneous by thermal We aation pro- then prolonged of proceed stage gravitinos, by of providing further coherenttime-dependent the investigating oscillations background necessary the of with tools non-thermal to any gener- first study numerical number this results of regarding process superfields.ular the in supersymmetric non-thermal a We models. generation generic also of gravitinos present in the partic- Keywords: the successful predictions ofafter nucleosynthesis. inflation leads The thermal to generation the of bound gravitinos on the reheating temperature, Abstract: Igor Tkachev Gian Francesco Giudice and Antonio Riotto Thermal and non-thermal productiongravitinos of in the early universe JHEP11(1999)036 after inflation is RH T should be smaller than s 1 to the entropy density 2 / 3 n ) GeV [3]. We will come back to this point and present He and D nuclei by photodissociation, thus jeopardizing 9 4 10 − 8 (10 ∼ [4] for gravitinos with mass of the order of 100 GeV. 12 reheatinging reheatingand in a time-dependent background 11 5 7 − 2.1 Thermal production of dangerous relics2.2 in the case A of more appropriate instantaneous approach to thermal generation of gravitinos dur- equivalence3.1 The equation of motion3.2 of the Goldstino Non-thermal production3.3 in of gravitinos global in supersymmetry the Comments case on3.4 of the one gravitino chiral superfield decay Numerical results 16 for the case of one chiral superfieldperfields 19 18 10 24 However, it has been recently realized that the non-thermal effects occuring right Gravitinos can be produced in the early universe because of thermal scatterings in the successful nucleosynthesis predictionsnumber [2, density 3]. of As gravitinos a consequence, the ratio of the not larger than about 10 1. Introduction The overproduction of gravitinos representslogical a models major obstacle based in onare constructing supergravity cosmo- copiously [1]. produced during Gravitinosdecay the decay products evolution very destroy of late the the and early — universe if — they their energetic 3. Non-thermal production of gravitinos and the gravitino-Goldstino 4. Non-thermal production of gravitinos in the case of two chiral su- Contents 1. Introduction2. Aspects of thermal production of gravitinos during reheating 3 1 a detailed analysis of the thermal generation ofafter gravitinos inflation during because reheating. of the rapid oscillations of the inflaton field(s) provide an extra the plasma during the stageof of gravitinos reheating one after has inflation. to To require avoid that the the overproduction reheating temperature JHEP11(1999)036 2 / 1 ± 2 graviti- / 1 -term breaks ± F GeV is the height of bosons in the standard 15 2partofthegravitinois 10 / W 2 gravitinos in supergrav- 3 / 2 part obeys the equation of ∼ 1 ± / 4 ± 1 / 1 ± V ,where 2 gravitinos are asymptotically equivalent 4 / / 2 1 2 gravitino has been studied in refs. [5, 6] 1 / ± 1 /V ± 2 gravitinos in generic supersymmetric models of GeV) GeV [6]. RH / 1 T in the final number density of helicity 15 2 ± 2 Dirac particle in a background whose frequency is a P − 10 / / M 1 4 / ± 1 2 gravitinos turns out to be much more efficient than their V / ( 1 5 ± 10 for helicity [7]. On the contrary, the helicity 2 . /s / 3 2 / 3 RH m n T 2 gravitinos through the equation of motion of the corresponding Gold- / 1 ± The goal of this paper is twofold. In the first part of this work we will still This simple observation about the gravitino-Goldstino equivalence becomes cru- The equation describing the production of helicity The production of the helicity stino in global supersymmetryin is which expected the to scalarsmaller provide the fields than correct the after result planckian inflationrealistic in scale. supersymmetric oscillate the models with Luckily, case of this amplitudes inflation situation and [10]. is frequencies realizedconcern in ourselves most with of somethe aspects the reheating of the stage thermal afterprocess, production taking inflation. of into gravitinos account We during the will fact perform that reheating a is detailed far analysis from being of instantaneous. such a ity reduces, in the limitthe in Planck which the scale, amplitude to ofGoldstino the the in equation oscillating global describing field supersymmetry. is the Thispression smaller time identification by than evolution explains inverse of powers why of there the is helicity no sup- to amplitudes with corresponding externalthe gravitino Goldstinos mass for [9]. energies This much is analogous larger to than the longitudinal electroweak model behaving as Nambu-Goldstone bosons in the highcial energy when limit. the problemthermal of effects computing involves more the than abundance onehelicity of chiral superfield. gravitinos generated Describing the by production non- of nos and is a simple manifestationshell of amplitudes the with gravitino-Goldstino equivalence external theorem: helicity on- thermal generation during thethat reheat the ratio stage after inflation [5, 6] and it was claimed the potential during inflation.temperature, This leads to a very tight upper bound on the reheat inflation is roughly given by 10 starting from theenergy supergravity density lagrangean and andscalar in the field the pressure Φ simplest belonging ofapplication to case the in a the in universe single context which are chiral ofpresented supersymmetric superfield the in dominated new with ref. by inflation minimal [8]. models an kinetic has term. oscillating been recently An motion of a normal helicity combination of the different massmass scales parameter at of the hand: fermionic the superpartner rapidly of the varying scalar superpotential field whose supersymmetry, the Hubbleproduction rate of and helicity the gravitino mass [5, 6]. The non-thermal excited only in tiny amounts, asgravitino the mass resulting abundance is always proportional to the and very efficient source of gravitinos [5, 6]. The helicity JHEP11(1999)036 rapidly increases to a the scale factor of the a T 2 gravitinos and present the / 1 2 gravitinos during the stage of ,being 8 ± / / 1 3 − ± a 3 2 gravitino found in refs. [5, 6] starting from the / . 1 /s ± 2 / 3 . During this complicated dynamics, both gravitinos and n RH T 2 gravitino equation found in supergravity for one single chiral superfield, 2 gravitino production in a generic time-dependent background and with a / / 1 1 The paper is organized as follows. In section 2 we comment about the thermal In the second part of this work we willOur be aim dealing is with the to non-thermal provide produc- the reader with all the tools necessary to study the helic- reheating ± ± numerical results regarding the numbermetric density models of gravitinos containing in a particularthe single supersym- non-thermal chiral production superfield. of gravitinosis for Finally, relevant the in for case section realistic of 4 two supersymmetric chiral we models superfields, discuss of which inflation. we comment upon the decay rate of the helicity preheating for one single chiralkeeping superfield. all This the numerical supergravity analysis structure will be of performed the theory. production of gravitinos. In sectionof 3, the we show Goldstino how in toity derive a the generic equation time-dependent of motion background, we reproduce the helic- generic number of superfields.tion To of achieve this motion goal, ofsuperfields we the will and Goldstino derive show in the that, globalof master in supersymmetry the equa- the oscillating with case fields a of smallertion one generic than single of the number chiral Planck of motion scale, superfield of itsupergravity and the exactly lagrangean. amplitudes reproduces helicity the As equa- achiral special superfields, case, which is wemodels will particularly of concentrate relevant hybrid when on inflation. dealing theputation with case of We supersymmetric of will the two also number density present the of the first helicity complete numerical com- entropy are continously generated andto one has compute to the solve final a ratio set of Boltzmann equations 2. Aspects of thermal production of gravitinos during At the end of inflation theof energy density kinetic of the energy universe isinflaton and locked up energy in potential density a in combination energy the of zero-momentum mode the of inflaton the field, field. Thus, with the the universe bulk of the Inflation is followedfield. by a In prolonged thisbeen stage regime, entirely of the converted inflaton coherent intomaximum is oscillations radiation. value decaying, of and but The then the the slowly temperature inflaton inflaton decreases energy as has not yet universe. Only when the decayrate, rate of the the universe inflaton enters becomes the ofthe the radiation-dominated reheat order phase temperature of and the one Hubble can properly define tion of gravitinos during the preheating stage after inflation. ity JHEP11(1999)036 , I φ a and I ρ of the energy density of the inflaton field. 4 all . In the oversimplified treatment, the reheat tempera- RH T into radiation when the decay width of the inflaton energy, Γ is the Robertson–Walker scale factor. If we denote as φ a , the expansion rate of the universe. H ,where 3 executes coherent oscillations about the minimum of the potential. Averaged − a φ The reheat temperature is calculated quite easily [11]. After inflation the inflaton The process by which the inflaton energy density is converted to radiation is The simplest way to envision this process is if the comoving energy density in the There are two reasons to suspect that the inflaton decay width might be small. As we already mentioned, of particular interest is a quantity known as the reheat ∝ φ field is equal to over several oscillations, the coherentρ oscillation energy density redshifts as matter: ture is calculated by assumingthe an inflaton instantaneous field conversion of the energy density in the total inflaton energy density and the scale factor at the initiation of coherent at the end ofvery inflation much is unlike in theinflaton-dominated present a universe hot, cold, must high-entropy low-entropy somehow universe. state beradiation-dominated with defrosted After universe. and few inflation become degrees the a of frozen high-entropy freedom, known as “reheating” [11]. Theof reader reheating should is anticipated rememeber by that ais — stage very even of sensitive if preheating to the [12] the process —is model the inefficient; and efficiency in the of some model preheating models parameters.efficient, it In it is some not is models operative unlikely the at that all. process it Even removes if preheating is relatively In particular, already duringprocesses the of resonant rescattering decay [13] ofwith always the non-zero create inflaton momentum a field, sizeable [14]is back-reaction population which therefore of do likely inflaton not that quanta necessary partecipate a to stage in extract the during the remainingare which resonant inflaton going the decay. field to inflaton energy. analyze It This field in is is this exactly slowly section. the decaying stagezero we is mode of the inflatonduring (or preheating) the decays soft quanta into generatedto normal in form particles, the a which process then thermal ofprocess rescattering scatter background. is and the It thermalize same is as usually the assumed decay width that ofThe the a requisite decay free width flatness inflaton of of field. flaton this the field inflaton to potential otherpling suggests fields to a since weak other theric coupling fields. potential of theories is the where However, renormalized in- fields the this by nonrenormalization and restriction the their theorem inflaton maypling ensures superpartners. cou- be is a evaded that cancelation A inheating. in between second supersymmet- local Unless and supersymmetric reheating basiclarge is theories reason undesired delayed, gravitinos to gravitinos entropy are will suspect productionsis. produced be when weak overproduced, during they cou- leading re- decay to after a big-bang nucleosynthe- temperature, denoted as JHEP11(1999)036 , is 4 RH P (2.4) (2.5) (2.2) (2.1) (2.6) (2.3) T ∗ g πM in (2.3) 8 X 30) √ / 2 = Hn π . Pl -particles is the =( Pl M X R M ρ φ is ( Γ a p 4 / 1 , . Now if we assume that all . ∗ /a 2 light converted into radiation at this 3 g I 200 n a . |i I a v a 2 | . P . . RH tot 12 M T P 4 2 σ − Pl I =0 2 T ρ M − M 10 5 'h Pl π ∝ 10 3 . 8 X M X φ ' represents the number density of light particles X s n He and D nuclei by photodissociation, and thus instantaneously Γ 3 n 4 Hn X )= s T He, which would require that the relic abundance p a n 3 ( 4 is readily solved. The Boltzmann equation reads +3 2 ∼ / 1 H X X dt ∗ light dn g n 3 90 π relative to the entropy density at the time of reheating after leads to an expression for , equal to the radiation energy density, 8 3 φ ) 12 is the total cross section determining the rate of production of − /a = , one finds 2 I P 10 P a ( RH ∼ I /M )andΓ T ρ /M 1 , we can define the reheat temperature by setting the coherent energy 2 a ( = T ∝ /a H I is the effective number of relativistic degrees of freedom at temperature φ ∼ a ρ tot ∗ 1 g σ − t reheating . The result is Since thermalization is by hypothesis very fast, the friction term 3 ∼ RH The number density at thermalization in units of entropy density reads As mentioned in the introduction, the slow decay rate of the Under the approximationdangerous of gravitational instantaneous relic reheating, the number density of any 2.1 Thermal production of dangerous relics in the case of instantaneous available coherent energy density is value of Equating can be neglected andH using the fact that the universe is radiation-dominated, i.e. where is smaller than the gravitational relic and where inflation [4] density, successful nucleosynthesis predictions.resulting The overproduction of most D stringent + bound comes from the essential source ofgravitational the relics will cosmological destroy problems the because the decay products of the the Planck mass) oscillations, then the Hubble expansion rate as a function of in the thermal bath. T JHEP11(1999)036 be (2.7) should not (or the inflaton RH T RH , the inflaton energy T 1 − φ Γ ∼ t . , which implies that the entropy 8 / 3 has the following behaviour. When )GeV − [16, 11]. During this long stage, the 9 a 8 T / . This implies that 10 15 − a 6 RH 8 T ∝ (10 S . , but the inflaton energy has not yet been entirely is the number of relativistic degrees of freedom. In t ∗ φ RH Γ g T − e will violate the gravitino bound. Therefore, the universe ∝ is created: φ ρ ,where S RH 1 T − T 3 , dangerous relics such as gravitinos would be abundant during / 1 − ∗ through eq. (2.2)). The reheat temperature is calculated by assuming g GUT and then it decreases scaling as φ M , otherwise ∼ MAX H T . Indeed, the reheat temperature is best regarded as the temperature below When studying the production of dangerous relics during reheating, it is neces- In this early-time and prolonged regime of inflaton oscillations, the inflaton is However, the reheating process is not instantaneous. Right after inflation the In the discussion above, the crucial quantity which determines the abundance Comparing eqs. (2.6) and (2.5), one may obtain an upper bound on the reheating RH T φ RH If per comoving volume density gets converted entirely intodominated radiation phase. and Only the at this universeT point enters one the can radiation- properly define the reheat temperature converted into radiation. The temperature this regard it has a limited meaning [16, 11]. sary to take intomaximum account temperature the fact is that greater reheating than is not instantaneous and that the undergoes a very long periodis of dominated matter-domination during by whichpotential. the the energy oscillations These density of oscillationslifetime the last of till the inflaton inflaton the field field. cosmic around time the becomes minimumnevertheless of of decaying, the its order of the universe is not yet radiation-dominated. Finally, when decay rate Γ used as the maximum temperature obtained in the universe during reheating. The an instantaneous conversion of thewhen energy the density decay in width the ofuniverse. inflaton the field inflaton into energy radiation is equal to the the expansiondecay rate of width the of theΓ inflaton is expected to be much smaller than the Hubble rate, of dangerous relics after reheating is the reheat temperature nucleosynthesis and destroyHowever, the if good the agreement reheatingto of temperature create the satisfies superheavy theory the GUTasymmetry with [15]. gravitino bosons observations. bound, that it eventually decay is and too produce low the baryon which the universe expands asdecreasing a as radiation-dominated universe, with the scale factor temperature after inflation [3] the inflaton oscillations startbeen and a transferred small to portion radiation,value of the the temperature inflaton rapidly energy density grows has to reach a maximum JHEP11(1999)036 , . is φ φ = ρ φ (2.8) (2.9) RH M T . We will 2 / 3 .For n Pl , is determined 2 M / , defined in terms φ EQ 3 X n , , , M α p 2 =0 =0 =0 / and it is inconsistent to 1 φ and φ φ times the reheat temperature α . 2 φ 1 ρ ρ 3 2 φ φ α 2 . / ' 2 Γ − 3 RH 10 / 13 EQ 3 T must be small. From eq. (2.2), − m +Γ × − n is the total thermal average of RH X φ φ R T α |i − ,Γ v is Hρ Hρ | = φ 2 / φ 2 3 tot |i M +3 +4 σ n v M | h and that the reheat temperature φ R ˙ σ ρ ˙ 7 3 ρ h |i − v and | a φ tot α ∝ σ φ h 3 M T + . φ 2 α T / ∼ 3 = , and the number density of the gravitino, φ R Hn light ρ Γ n +3 2 / 3 ˙ n as |i must be approximately smaller than 10 v | φ σ α h and may be produced in interesting abundance to serve as dark-matter candidates [17]. GeV, ing reheating With the above assumptions, the Boltzmann equations describing the redshift It is useful to introduce two dimensionless constants, φ As an application of this, particles of mass as large as 2 1 13 RH where dot denotes time derivative. Here For a reheat temperature much smaller than assume that the decay rate of the inflaton field energy density into radiation is Γ solve the Boltzmann equationthe for the period gravitational of relics reheating assuming that throughout 10 the cross section times theand we Møller have flux neglected factor the givingenergy back-reaction density. of rise The the to equilibrium gravitino the energy abundance gravitino density on for production the the radiation gravitinos, radiation energy density, We will also assumeequilibrium. that This the is light by degreesshows no means of that, guaranteed, freedom even but are if thereheating in thermalization analysis is performed local does not in much thermodynamic not ref. different. occur, [17] production of gravitinosand interchange during in the energy density among the different components is the reheat temperature in terms of of Γ by the radiation temperature, Let us consider a model universe with three components: inflaton field energy, the largest temperature ofsubsection the is thermal to system provide afterdangerous a inflation. relics more The generated appropriate goal duringwe computation of will the of the focus process the next on of numberdangerous the density reheating. gravitational gravitino of relics. case, For but sake our of results may simplicity, be easily2.2 extended A to more other appropriate approach to thermal generation of gravitinos dur- maximum temperature is, in fact, much larger than T JHEP11(1999)036 . 2 / I 1 Φ 2 (2.14) (2.10) (2.15) (2.11) (2.13) (2.12) (2.16) / 3 . x ) 1 φ c as Γ . = T φ 0 2 / M R 2 3 H Pl ( . m 3 M .Itisconvenientto a I X 2 / α 3 , , i.e. before a significant . n π φ 4 3 becomes / 8 Γ 1 0 ≡ . )=Φ # ,whichleadsto r I 2 R 4 3 EQ I . x X − x X = 4 . φ / H 3 1 3 I I x − T ) x M x x R 2 ; φ , 2 2 φ I X 4 3 , M a Φ H − d/dx M x ,c 4 R 2 Φ . With this choice the system of equa- 2 / 2 / R R φ ρ 2 / Pl φ 1 φ 3 X 3 8 R g α , the effective number of degrees of freedom 2 α + + M − ≡ M ∗ )=0andΦ( 2 − x 2 aM φ + x x I g X π 2 x π (3) 3 R π I x 8 ∗ Φ Φ 30 x x ζ x M ( M x = g is simple to obtain [17] Φ 1 √ √ 4 3 = X and , given in terms of the temperature c x T √ 1 3 I " = ; c c X R = = 0, so the equation for 8 1 Φ 3 / − c − 2 1 )= a T EQ I ' 1 are given by = = = x − φ X I 3 I 3 ( 0 0 0 c X x Φ M R Φ R X φ ,c ' ρ of 4 φ / α 1 1 R I ≡ ,and c . Here, by early time, we mean 2 x φ Pl 4 c Φ ) in terms of the expansion rate at R / I M , 1 M = 1 x c π ,and x 3 ∗ 8 I = g 2 Φ 12 x r π ( φ ' = ρ 1 ' c is the equilibrium value of It is straightforward to solve the system of equations in eq. (2.11) with initial It is also convenient to work with rescaled quantities that can absorb the effect φ T EQ M conditions at fraction of the comoving coherenttimes energy Φ density is converted to radiation. At early tions (2.8) can be written as (prime denotes in the radiation: It is also convenientvariable, to so use we the define scale a factor, variable rather than time, for the independent Thus, the early time solution for Before numerically solving the systemtime of solution equations, for it is useful to consider the early- express The constants The temperature depends upon X of expansion of the universe. This may be accomplished with the definitions JHEP11(1999)036 ∝ in ap- φ with T α (2.18) (2.17) φ final ) GeV. M ), impos- 9 /s 10 GeV seems 2 ∼ / 3 10 12 I =10 n − H 9 RH is the entropy den- T , , which implies that s , but remains always 4 . This result can be 8 / . / for 1 H 3 H 4 / 8 − ! 1 / a I 1 while the total number of ) 8 . This means that most of 3 H ∗ / φ 2 1 1 ) φ Pl ,where 2 − φ Pl P /g M temperature of 10 2 Γ M RH I 3 I M I final T x ) /M Φ GeV. ∼

H 2 (200 / 4 3 /s scales as 12 t 3 / 2 4 1 / 8 / 10 10 m 3 T / 1 1 is smaller than 1 c n ∗ 6( 48. It is also possible to express ∼ ( ' 4 . . φ g / / maximum 3 ∗ 1 9 ) 9 16 g π RH 3 9 2 /s =1 ∗ .Wehavealsocheckedthat( π 2 MAX ' /T g / 5 230. The inflaton parameters have been chosen 5 T 2 3 / 12 4 X RH 2 n π / ' 1 φ T α , and that during the early-time evolution, 3) ∗ MAX α 77 / g . T RH 77 77 . . . This dependence is not present in the case of instan- T φ =0 =(8 M =0 =0 I RH MAX φ T x/x T MAX M T GeV, which gives 9 1, in the early-time regime > and obtain =10 I GeV, which leads to RH RH T has a maximum value of 13 x/x T T 10 . For We observe that the quantity ( For an illustration, in the simplest model of chaotic inflation We conclude that — even though the 8 The equation of motion of the number density of the gravitino is easily solved ' / 3 φ − sity, gradually increases with time when Γ to be in contradiction with the usually quoted upper bound of (10 relativistic degrees of freedom is the gravitinos are producedand at it the makes last sense stage to of talk reheating about when the inflaton decays terms of proximates well the usualthe estimate temperature one of gets the neglecting radiation the during non-trivial the evolution of period Γ M entropy is created induring the reheating early-time it regimetemperature is [16]. necessary is So greater to if than take one into is account producing the gravitinos fact that the maximum smaller than unity until the inflaton decays at Thus, ing the constraint (2.6) gives the usual upper bound on the reheating temperature to have taneous reheating, where thereheating number temperature density and not of on gravitinos the depends frequency only of the upon inflaton the oscillations. explained recalling that —comoving during the volume coherent is oscillation increasingis epoch and — continuously diluted the the entropy by abundance per number the of density entropy of the release. gravitinos just-produced has Weof a gravitinos have the dependence, also inflaton even oscillations checked though that weak, the on final the frequency which is obtained at numerically. The resultsgravitino are production plotted is such in that figure 1. The total cross section for the a JHEP11(1999)036 GeV. 9 2partofthe =10 / 1 ± RH T 10 is nonstandard and the physics leading to 1 − φ 2 gravitino has been found in refs. [5, 6] in the Γ / 1 ∼ ± t GeV is much more involved than is usually thought. This 9 10 . RH T The time dependence of the temperature, the radiation energy density, the GeV. One should keep in mind, however, that the thermal evolution of 9 10 . The equation of the helicity Goldstino equivalence RH case in which theby an energy oscillating density scalar and field the Φ pressure belonging to of a the single universe chiral are superfield dominated with minimal the bound observation might be relevant whenfor dealing the gravitino, with e.g. either ifrelics. a the different gravitino parameter is space very light, or with other kinds of dangerous the universe before the epoch gravitino can be efficiently excited duringThe the evolution non-thermal of the generation Universe after canproduction inflation. by be several extremely ordersmodels. efficient of and magnitude, overcome in realistic the supersymmetric thermal inflationary As shown in refs. [5,rapid 6], oscillations non-thermal of effects the occuringAs inflaton right we field(s) after noted inflation may in due lead the to to introduction, the copious this gravitino occurs production. because the helicity 3. Non-thermal production of gravitinos and the gravitino- Figure 1: inflaton energy density and the gravitino number density for the case T JHEP11(1999)036 , P (Φ) ,to M .In P 2 W / 3 M | 2 helicity m / Φ | 3 | ± (where Φ | W 2 Φ ∂ is present. P 2 gravitinos in the case in which and the gravitino mass / M . The frequency Ω corresponds to 1 H 2 particle, posseses only ± W / 2 Φ ∂ 11 2 gravitino with a single chiral superfield and / 1 ± 2 gravitino by finding the equation of motion of the corresponding Gold- / 1 where the longitudinal component of the gravitino effectively behaves as a ± 2 Goldstino [9]. 2 / / 3 in a time-dependent background Therefore, it appears of advantage to compute the equation of motion of the The equation of the helicity This does not come as a surprise and is in agreement with the gravitino-Goldstino 2 states. The equivalence theorem is valid in the limit of large energies compared / m 1 Let us now findof the the equation background of isimposed motion by dominated of their by equation the of aare Goldstino motion. set displaced when We from of will the the therefore scalar energy suppose minimaminima. fields that of density This the their following is scalar potential the fields what and happens trajectories are right free after to inflation oscillate and about during such the preheating stage. the scalar fields after inflationthe oscillate with planckian amplitudes scale. and frequencies Thissymmetric smaller models is than of exactly inflation what [10]. is realized in most3.1 of The the equation realistic of super- motion of the Goldstino in global supersymmetry and stino in globalproblem supersymmetry. involves more This than one procedurerect chiral is result superfield and for particularly is the welcome expected number when to density provide the of the cor- helicity denotes the superpotential), the Hubble rate minimal kinetic term is identicaltime-varying to background the with familiar frequency equation Ω,appearing for which in a depends the spin-1/2 problem, upon fermion namely all in the the a Goldstino mass mass scales parameter the superpotential mass parameter ofwhen the supersymmetry Goldstino which is is broken. ‘eaten’of by Therefore, helicity-1/2 the the gravitino gravitinos equation in describing the supergravity production reduces, in the limit of helicity spin 1 the frequency of the oscillations tends to the limit in which the amplitude of the oscillating scalar field is small, the equation describingsupersymmetry the and time no evolution suppression of by powers the of helicity-1/2 Goldstino in global kinetic term. In thisgravitinos section in we a would genericnumber like time-dependent of to chiral gravitational study superfields. background the and non-thermal production for a of generic states. The Goldstino,± a Majorana fermion, providesto for its missing (longitudinal) equivalence theorem. Ingravitino spontaneously acquires a broken mass supergravity,the through the Goldstino the initially which superhiggs massless disappears mechanismsive, from [18, the the 19], gravitino, physical which by is spectrum. absorbing a Majorana Before spin becoming 3 mas- JHEP11(1999)036 is i (3.4) (3.5) (3.2) (3.3) (3.1) , +h.c. j χ L 2 is the left-handed / P i ) 5 ¯ χ γ the number of multiplets ij − W N is nonvanishing 2 1 i =(1 − . L ) i i , ,being P i ) denote a set the scalar and fermionic † ,z χ N i i . L ε, . z W i χ ( ( and εP i V =0 2¯ 2Θ − =0 i6 and i 12 − i √ √ i6 i i i χ /z z ε χ Θ = = i∂ δ h µ i i h ∂W/∂z ∂ z χ µ ε = ε 2 gravitinos generated during the preheating stage, δ γ = δ i i runs from 1 to / i ¯ 1 Θ i χ i 2 W ± , . † + ) i i i ) z † z µ ∂ W i ( =( z i i -term is nonvanishing. Our findings can be easily generalized µ z W ∂ F = ) is the superpotential, i )= z i L ( ,z i W z ( = is the spinor parametrizing the infinitesimal supersymmetric transformation V ε W Given a generic background, supersymmetry is broken if The identification of the Goldstino requires a generalization of the standard The lagrangean is invariant under the following set of supersymmetric transfor- Consider a global supersymmetric theory with lagrangean -term. This happens if the expectation value of the matrix Θ where and we have defined the matrix Here fields respectively, projection operator. The index this is good approximation since theto non-thermal production overcome of the gravitinos is thermalfrequency expected generation of in the those oscillationsexpansion supersymmetric of of the the models universe scalar and in most fieldsoscillations. of which the is Neglecting the gravitinos the much are expansion generated larger of withinof than the the universe the first the will few also Goldstino rate make of more thetheory identification the charged transparent. under Finally, somethe we gauge system group, will some but not suppose concern that ourselves during with the a evolution of to include the possibilityD that supersymmetry is broken by some (time-dependent) procedure used in theminima of static their case, potential thatwe and will is neglect the when the cosmological expansion the constantthe of number the scalar vanishes. density universe. of fields In For helicity are the the practical following sitting purpose of at computing the and we use thecontracted. standard convention that the sum is intendedmations when the index JHEP11(1999)036 (3.8) (3.6) (3.7) (3.9) which (3.11) (3.10) (3.12) ρ -term is nonvan- F ,the i z ρ. = # , 2 j ) i Θ η, W † i Θ , . † † i Θ . j j . . Θ † Θ Θ +( i χ Θ Θ i fields may be expressed in terms of the 2 χ ij i − † i =0 Θ ) =0 dt + † dz ⊥ i ⊥ i Θ 13 i6 ij ⊥ i 0 i χ Θ ⊥ i δ P dt γ χ =Θ dz i χ W . The two operators project respectively onto = = h η Θ k = =( " ij ij Θ i ) ) ⊥ i † k k i χ ⊥ χ X P P ( Θ gives the total energy density of the system ( † Θ= Θ=Θ † can be rewritten as † are not linearly independent since they satisfy the following i Θ ⊥ i χ . This is not surprising since supersymmetry is broken in the i χ 1) ones. 2field / − N Making use of the definition (3.3), we find that, for a background of real fields, The spin 1 The Goldstone theorem tells us that the Goldstino is easily identified from the We introduce now the two projection operators the subspace orthogonal toitself. the Goldstone fermion and onto the Goldstone fermion — if the expansion of the universe is neglected — remains constant in time. sishing On the other hand,supersymmetry in comes the also case from of the a time-dependent background, the breakdown of piece in the matrix Θ wherewehavedefinedΘ Notice that the fields remaining ( relation In particular, for a constantcondition (time-independent) that background, supersymmetry one is recovers broken the if, usual for some field Therefore, the combination Θ supersymmetric transformation (3.2) early universe anytime somewhat form happens of during inflation nonvanishing andwhen energy the scalar density subsequent fields appears. stage oscillate of around preheating This the and is reheating minima of their potential. This condition tells us that one of the JHEP11(1999)036 i † i p (3.15) (3.16) (3.18) (3.17) (3.13) (3.20) (3.19) (3.14) and Θ i and we have k ~ · γ and the pressure ~ ρ = term; in the ultraviolet ˆ k . , η 2 ⊥ j k =0 χ ij ⊥ i χ W , † i † i j . ˙ . Θ χ 0 j , ˙ ij W j Θ . , 0 ˆ kγ +2Θ W ij i i ⊥ i W iγ 2 Θ ρ Θ χ ij + W ˆ † kη i 2 gravitinos during the preheating stage † i i = 0. The equation of motion of the scalar − / − Θ W Θ ˆ +2 χ, + i 14 1 η z ˆ ˆ k − kχ ˙ † p = η ± ˙ =Θ = 0 G i for the space-dependent part. The matrices Θ + − = = γ ˆ ˙ k ˆ χ Θ G † x ~ 0 i i η · 0 = † ˙ ¨ G k ~ φ χ 2, Im i i iγ 0 G iγ e G √ ˆ − k + iγ / ∼ i η = = i φ 2 ˙ χ ˙ ˆ k η χ = 0 0 i + iγ z iγ ¨ η can be expressed in terms of the energy density G and smaller than the Planck scale. We notice the non-trivial result that 2 / 3 m We now choose the nonvanishing vacuum expectation values of the scalar fields after inflation, when thefrequencies typical of the energy oscillations of ofmass the the scalar system fields) and are large field compared amplitudes to the (or gravitino the and fermionic fields read Inserting now the decomposition (3.11) into eq. (3.14), multiplying by Θ The matrix of the oscillating scalar fields respectively and makingmotion use of the eq. (3.15), we get the following equations of satisfy the following equation any time-dependent function has disappeared from the regime, eq. (3.20) iswould solved expect by from plane-waves general and arguments. particle production shuts off as one where we have defined the following combinations in the real direction, Re This equation is valid forgravitino-Goldstino a generic equivalence theorem number of — chiral isto superfields expected and to describe — provide the because the of necessary production the tool of helicity Differentiating eq. (3.16) with respectequation to time of and motion using of eq. the (3.17) we Goldstino find the master where the dots standused the for plane-wave ansatz derivative with respect to time, JHEP11(1999)036 G (3.23) (3.27) (3.22) (3.24) (3.25) (3.21) (3.28) (3.26) =0and i ˙ Θ = † ˙ G , † i =Θ i . ½ = 1. Notice that the matrix , , i = † 1 1 . =0 2 =0 Θ Θ W . Φ 2 η , 2 , the equation of motion of the Gold- η W ∂ , Φ ∂ ϕ × η 2 gravitino found in refs. [5, 6] starting 2 eff 0 ) 1 † 1 ∂ / ∂ =0 1 † 1 † m iγ ϕ 1 = 0 Θ Θ 2 G 0 Θ Θ χ . We now wish to show that the eq. (3.26) =0,wehaveΘ e ± 15 ˆ k − ϕ iγ i iγ = eff =ˆ = ˙ = φ − =˙ ˆ kη G m ⊥ in the following form 0 G =2 G † ∂ χ eff exp( − ˙ G 0 G G G satisfies the following differential equation ˙ η m iγ 0 → = G 2 iγ η |G| = 1, the only physical degree of freedom is the Goldstino N is given in eq. (3.3) for the case 1 Let us now consider the special case of one single chiral superfield Φ with minimal In a static background for which = 0 by virtue of eq. (3.12). The Goldstino equation is solved by plane-waves and ˆ kinetic term. We have χ — as expected — no particle production takes place. and the matrix Θ from a local supersymmetric theory, i.e. supergravity. found for one singlereproduces chiral — superfield for in amplitudes the of— the limit the scalar of field equation global Φ of supersymmetry much motion smaller correctly of than the the helicity Planck scale has manifestly absolute value equal to unity and we have used the fact that where Therefore, it is possible to rewrite By making a field redifinition and Eq. (3.16) simplifies to Eq. (3.26) is the equationground of given motion by of the a oscillating spin-1/2 mass fermion in a time-dependent back- stino becomes where JHEP11(1999)036 (3.31) (3.32) (3.33) (3.30) (3.29) (3.34) =0. —which µ ψ χ † i Θ µ ˆ γ i . 2 gravitino in the case of one / . 1 . , =0 ± i . µ −G σ ψ ψ i ψ † 1 = . ρ ˆ γ =0 , reduces to µ Θ , which coincides with the constraint D i ˙ P µ ψ µ 3 ν W =1 ψ ˆ † 1 γ i 0 ˆ X γ ψ i M 1 5 † 1 16 ˆ γ Θ c iγ γ Θ ρ µ Θ ¯ η ˆ the mixing term becomes γ = µ =1 3 i | 1 ¯ ˆ χ γ 0 +2 χ µνρσ 1 Φ Θ 1 | P ψ 1 p † 1 0 Θ ˆ γ Θ ≡ = −G =Θ µ c η R = 0 is present. Under these circumnstances, the equation of ψ 0 χ γ , in the limit of c and = 0 gives the following algebraic constraint µ ψ R D· is the covariant derivative and greek letters denote space-time indices. The ρ 2 D superfield We note that the constraint (3.30) may be recovered in the following alternative If we start with the supergravity lagrangean, the single chiral fermion Because of the antisymmetric properties of the Levi-Civita symbol, the equation The constarint (3.34) is easily generalized to the case of many superfields Θ = 0 does not contain time derivatives and provides another algebraic constraint 2 0 motion of the gravitino becomes Two degrees of freedom may be eliminated using eq.way. (3.30). The mixing term inGoldstino the is supergravity of lagrangean the between the form gravitino and the By using the definition (3.8) condition Choosing the gauge in which such a term vanishes is equivalent to require that single chiral superfield and minimal kinetic term. is the superpartner ofthe the role scalar of component the Goldstino inthe and the can gravitino chiral be supermultiplet gauged Φ away to — zero, plays so that no mixing between Let us first remindregarding the the reader equation of of motion some of of the the helicity basic results obtained in refs. [5, 6] 3.2 Non-thermal production of gravitinos in the case of one chiral (3.30). R where the matrix Here This condition gives ˆ JHEP11(1999)036 2 / , 1 ψ W (3.36) (3.41) (3.35) (3.39) (3.40) (3.38) (3.42) (3.37) ). 2 and x . d~ 2 / . 3 − ψ   4 2 2 3 ˙ a a dτ − . − , )( 2 , 2

2 τ / 2 / ( 2 2 3 / P 2 3 Φ / 2 dt 1 d 3 a m m 2 M ψ / 2 2 2 m , 3 ˙ / = a a 2 , are time-dependent func- 3 / 2 . a 2 3 / 2 # ˙ 2 3 / m ˆ am kG 2 ˙ # / 2 helicity states respectively, B 3 m +6

ds ψ 2 3 / ˙

3 + ˙ 4 6   m Φ a 1 a m ˙ dt a a Φ − d 2 a a , P 4 dt ±

2 2 d 4 8 and | + / / ˙

a a M 3 3 − 2 +2 2 in terms of the superpotential 2 A P ˙ / a − m 2 m a (Φ) + 2 3 2 / 2 M / 4 3 2 / 2and 3 − + m (Φ) + W / 2 3 / | 2 4 2 3 m and (Φ) 0 0 m 3 V ˙ ,where a a = m γ γ 2 2 V m P " Φ ± / B a a a a † 3 # 17 " M +3 1 − 0 2 Φ P 5˙ 5˙ 2 2 2 2 +3 2 | i i L 2 2 4 P e − 2 1 iγ P / ˙ M 1 a 6 a 3 2 3 2 4 ˙ + + 3 M + W 2 a a = as M ˙ | + / a a m 0 0 2 2 3 3 2 / ∂ ∂ − = = / A V 3 0 0 3 m ˙ − 5 3 a 2 4 2 4 ¨ a a L 4 ¨ a ˙ ˙ iγ iγ m a = ¨ a a 2 a a a a

= 2 2 − + 2 2 G − 2 P 2 4 / / W L 3 1 2 2 3 ¨ a a † ¨ ¯ ¯ a M ψ ψ a Φ − 2 2 2 / / = = 2 3 2 3 + = 2 2 m m 2 2 / / / / 3 1 W 3 2 3 + + Φ 1 L L ˙ 4 4 m m ∂

2 " /a /a 2 2 Φ 2 ˙ ˙ P a a † M Φ e 3 3 is the scale factor, = a = = V A The diagonal time and space components of the Einstein equation become B by explicitly constructing the helicitymay projectors be in diagonalized the as flat [6] limit [6]. The lagrangean Replacing eqs. (3.38) and (3.39) in eq. (3.41), one obtains [6] which may be shown to correspond to the on the gravitino momentumdegrees modes. of freedom Such and to a define constraint two allows physical Majorana to fermion remove states two extra where Using the expression forthe gravitino mass we can write the scalar potential They may be expressedfield in Φ. terms Here the time pressure is conformal and and the the energy line density element of is the scalar tions [6] JHEP11(1999)036 , 2 f 2 / / 2 3 1 and m ± 2 -term (3.44) (3.43) (3.45) / 3 ,where F ∼ 1 m − ) P 2 f M m ∼ − 2 gravitinos are e / 2 f 2 gravitinos from 3 /F 2 gravitino has to / m λ / 1 ± 1 m ± ± 2 gravitino, a fermion 2and / / 1 .Since( 1 . ± 2componentsofthegravitino † ± / . /F . 1 ) 2 f −G ± =0 m and is therefore small. This can be = =1 2 2 -suppressed cross sections. 2 / − ˙ 1 − P P e W 2 gravitinos do have a decay rate which 2 f B has the following limit [5, 6] ψ 0 / M M m 2 gravitino therefore reduces to 1 + iγ G † / ρ 2 2 ± 1 18 G . Similarly, the coupling of the helicity A ˆ k ± − 1 − P = p −

0 P ∂ /M G M 0 † 2 2 gravitinos at nucleosynthesis is the same one which / is used in eqs. (3.36) and (3.37), we obtain the ra- | 3 G iγ −→ /

2 gravitinos promptly decay (or rapidly thermalize), Φ 1 2 | has absolute value equal to unity, by making use of m / / . The matrix 3 ± 1 G P G ± m M is proportional to ( e f | Φ , the coupling between the helicity | P M 2 / 3 is the gaugino mass. This is the reason why, when dealing with thermal m 2 / 3 ∼ m F does not appear in the equation of motions, but then tacitly assumed that the However, this is not the case; helicity We are now in the position to show the gravitino-Goldstino equivalence explicitly. One can also use the gravitino-Goldstino equivalence to explain the remarkable ∼ P λ is suppressed by the gravitational coupling thus not leadingbig-bang to nucleosynthesis. a large undesired entropy production when they decay after field correspond to theHence, Goldstino, which the is total derivatively amplitude coupledbe for to proportional the the supercurrent. to decay rate thethe of mass present the splitting vacuum, helicity within where the we supermultiplets. suppose supersymmetry For is example, broken in by some with and its superpartner non-thermal effects andM how this phenomenon is strictly related to the fact that m may be generated duringmight preheating. think This that issue helicity deserves a closer look because one gravitino, with a gauge boson and a gaugino is proportional to treated on the same ground and have both consider the limit the coupling is suppressed by The equation of motion of the helicity easily understood in the following way. The helicity eq. (3.24). 3.3 Comments on the gravitinoWe decay bragged about achieving a large number density of helicity markable property [5, 6] When this expression for ˙ To do so, we neglect the expansion of the universe and the gravitino mass comoving number of helicity production of gravitinos during reheating, the helicity This equation is exactly reproducedtion in of the motion global of supersymmetric the limit Goldstino by (3.22). the equa- property that the matrix JHEP11(1999)036 . , - 2 2 / F H m 1 /ρ (3.46) 2 .Asthe Ω m 2 2 P ) 2 gravitino P 2 gravitinos / /M / 2 1 1 / /M 3 3 ± ± (Ω m 2 gravitino comes / ∼ ∼ 1 ± 2 gravitinos (which can 2 gravitinos may decay / / 1 1 -term like in the present H, ± ± F -term has the problem that the H is sufficiently small. F /s P 2 / Ω 3 M n . The helicity 2 19 P / 2 gravitinos during the stage of preheating . If the decay is kinematically allowed, the 1 H / 2 M ρ 1 H ± ∼ ∼ Θ= † stored in the oscillating scalar fields and what mea- 2 Ω 2 gravitino changes with time — the decay rate will P / 4 Θ P ρ 2gravitinosintofermionsandsfermionsisatmost 1 ρ / √ M ± 1 ρ/M ± ∼ Γ . At this moment, gravitinos decay and their decay products 2 / 3 3 /m 2 P M is the mass-splitting in the given light supermultiplet. As supersymme- 2 m These estimates are valid as long as the oscillating scalar fields dominate the Right after inflation and during the preheating stage, supersymmetry is badly 2 gravitino is given by the fermionic superpartners of the scalar fields whose / 1 smoothly interpolate between (3.46) and the more familiar rate Γ terms break supersymmetry incomposition the of present the vacuum. helicity This means that — when the does not drop during this epoch.± At later stages, the main contribution to the helicity from the fermionic superpartnersdecay of rate the (3.46) applies. coherentlyof oscillating This the scalar decay expansion fields of rate the and is universe; the tiny the and number always density smaller of than the the helicity rate be identified with thecase typical of energy decay into of gauge the bosons produced plus gravitinos). gauginos, the Similarly, decay in rate the is decay rate remains smaller thanthe the Hubble amount rate of tillduring after gravitinos the the per nucleosynthesis preheating epoch, comoving stagethe volume remains order generated frozen of till by the non-thermal age effects of the universe becomes of through the Goldstino mixture (3.8)of changes coherent with oscillations, time. the During main the contribution prolonged to stage the helicity energy density of the universe. The “composition” of the helicity where ∆ decay rate of the helicity destroy the light element abundances unless after inflation in theby case a in single which the oscillatingsupersymmetric energy scalar inflationary stage density field. dominated of by the anflatness It of universe is the is important potential dominated is to spoiled keep by in supergravity corrections mind or, that in other a words, generic the 3.4 Numerical results for theIn case this of subsection one we chiralthe wish superfield number to density provide of the the first helicity complete numerical computation of into lighter fermions and sfermions through a coupling proportional to ∆ is at most of the order of where Ω is the typical time-dependentresponsible frequency for the of non-thermal the production oscillations of of the the helicity scalar fields try breaking is transmitted by the gravitational force, at the preheating stage ∆ vacuum, but the parameter sures the breaking of supersymmetry is not a simple broken by the energy density JHEP11(1999)036 , 3 I = has H W (3.47) (3.48) (3.49) (3.50) ∼ G 2 and it is / 3 2or n / 2 max Φ k .Thevalueof φ , . M ) 3 max k k k = -term [22], or some − ( ∼ D † r W b k 1 for ) n ' k, η . k ( r 2 n , v / 1 ψ simplifies to ] )+ ˙ a ϕ , i.e. a 2 k is given in eqs. (3.36). One can use / ( k 1 eff , r a L ϕ B m + ) 0 0 2 iγ − / 2 iγ 3 e k, η / 20 ( i∂ + m r [ = u 2 2 gravitino in the supergravity approach with A / / 1 G − 1 1 ¯ gets contributions of order unity [10]. In simple = ψ ± ± = = X r = G /V eff 2 x ~ 00 · / k ~ 1 m i , the lagrangean V L 2 − 2 P / e )and 1 M 2 ψ are approximately constant in time, one does not expect a / ) 3 = k ˙ ϕ ) 3 G/G 0 eff π d ( η 0 iγ m can be as usual expanded in terms of Fourier modes of the form (2 γ − 2 2) / Z 1 i/ ψ ( = exp( 2 2 − / / 5 is the value of the Hubble rate during inflation). However, in the evolution 1 = − is a phase depending upon the conformal time. By making a field redifinition ψ I a 3, supergravity corrections make inflation impossible to start. To construct a ϕ H ϕ / is expected to be roughly of the order of the inverse of the time-scale needed 3 → The field The equation for the helicity Φ 2 / λ 1 max absolute value equal to unity, it is possible to rewrite it in the following form zero otherwise. The resulting number density is therefore √ where for the change of the mass scales of the problem at hand. model of inflation incancellations the [21], context of or supergravity, a one must period either of invoke accidental inflation dominated by a one single chiral superfieldferental equation has for been a spin-1/2 reducedWe fermion wish to in to a a time-varying more present background familiar in here refs. second-order a [5, dif- 6]. slightly different derivation. Since the matrix where significant production of gravitinos (the number density can be at most one chiral field models based on superpotentials of the type This is theeffective lagrangean mass for a spin-1/2 fermion in a time-varying background with k of the Universe subsequent to inflation,During a large the amount of inflaton gravitinos oscillations, may be theup produced. Fermi to distribution some function maximum is value rapidly of saturated the momentum slow-roll parameter where as a guide theduring recent results and obtained after in inflationthe the effective [20]. theory mass of During generation of inflation, Dirac since fermions the mass scales present in particular properties based on stringhere theory in [23]. Nevertheless, the weDuring inflationary are this not stage, interested period, but itquadratic rather or might on cubic be expression the that along subsequent the the superpotential oscillating stage scalar is of field. well-approximated preheating. by a ψ JHEP11(1999)036 , r b T )] and = k ( + (3.53) (3.55) (3.52) (3.51) (3.54) (3.56) r r a u ψ ) η ( , −

)and ) ,v k =1,thechoice k ) , related to the − 2 † k − | ( . ( b ˆ ( k r T r in terms of † r i β ¯ b u ) ψ | k ) ) k C F k η ( + + and , ( ( r 2 † r , b ) ˆ ≡ | + ) a . ) a k k ) ) k ) k and 2 rv α ( , E η k | ( [ | k 2 † r ( ( , † r ) k | + b r ∗ b ˆ + k k − k ) . u E ≡ v through the (time-dependent) ( F F k † − ω − | r ( † b =0 ( ( r u v )+ b −| ) † b ω b + k ± η )+ 1 ) 2 : ( ( u u k r − ∗ η k − ( ( a eff and · ) and = α r u k eff )ˆ . ) k a 2 k β v a a T ( ) | ( r am k k † r )] am )+ eff a and β ˆ a k k ) | − )+ ( 21 h m ( ( and k r + k )+ ( r ) a ( ), eq. (3.50) can be written as two uncoupled )+ i ( ) b u ψ † r 2 − u η ½ − a ) ) ( η a u ) ± u ( η η k − , η ∗ ( ( k 2 k ∗ + , ) ω − ( k − β , along the third axis and use the representation in which u η k ω ½ ω k F ( 2 + r α − ∗ k k β k u + X ,ru ( + + F E Re( k ) k 2 k 3 k ± k E )= )= d ( ¨ and u k k r r = = =diag( ( ( k = ψ X † Z . In order to calculate the number density, we must first ˆ a k k 0 ) α b 2 ˆ k k k γ η F E 3 a ( β α d )= 2 eff + η u Z ( m [ H + ≡ 2 )= r k η u . ( ) are the two-component eigenvectors of the helicity operators, and using = 0 ½ H k 2 k ( r ½ ω 0 − ψ In order to calculate particle production one wants to write the hamiltonian Defining Here we choose the momentum 3 : 3 = − 3 γ u where in terms of creationdefines and a annihilation new operators set that of are creation diagonal. and annihilation To do operators, this ˆ one Bogolyubov coefficients diagonalize the hamiltonian. In the basis of eq. (3.50) the hamiltonian is original creation and annihilation operators are imposed by theanticommutation fact relations that imposed themay upon gravitino be the is used to a creation normalize Majorana and the particle. annihilation spinors operators The canonical where the summation is over spin and the conditions results in a diagonal hamiltonian, second-order differential equations for where the equations of motion can be used to express where, a representation where The Bogolyubov coefficients willthe fact be that chosen the to canonical commutation diagonalize relations the imply hamiltonian. Using JHEP11(1999)036 , 1 2 P eff / − 1 m M ± (3.58) (3.57) =10 P | Φ | /M 0 . φ (0) = 0. The initial ± i u 0 . | eff 2 b | k = β ) such that the particle | iam k i 2 ( 0 and for a quadratic superpo- r ∓ | a 3 a )ˆ dk k − k (0) as a function of time and for a ( ∓ ∞ † r φ 0 a . Z =10 1 The evolution of the parameter M The power spectrum of helicity iku − ) =ˆ P η ( 3 such that 1 /M N =10 a 0 P 2 i (0) = φ 0 π ± 22 | u /M 0 = tential. quadratic superpotential. Theφ initial condition is Figure 2: in units of Figure 3: and gravitinos for the initial conditions i 0 | ;˙ φ . = eff φ V N a M | ,so m M 0 eff 0 h .Itis m = W ω eff φ/φ ∓ )= W m η ω 2 Φ .Noticein ( = 1 2 ∂ / r − it reduces to 1 = 1. We define X n 0 is (including the two degrees of freedom from the spin) P ,aswellasthe X n =10 τ M (0) = φ P ± tends to M u . This is expected since one can verify that, in the limit of 2, but we have retained φ /M / 0 2 ˙ M = ϕ/a | φ φ 2 φ 2. The supergravity po- Φ | τ / 2 M '− Let us now consider a The (quasi) particle number operator By solving the Einstein equa- We define the initial vacuum state Φ as the value of the scalar field at = φ eff 0 conditions corresponding to the no-particle state are dimensionless field particular that at large times, tential (3.41) isfor easily such computed superpotential.limit In the m plotted in figure 2 in units of V φ its complete supergravitythe form numerical in analysis. Itto is write useful the equation ofterms motions in of dimensionlessWe variables. introducetime the ˜ dimensionless and for tends to that the scalarby field the is condition normalized quadratic superpotential number density the moment whenstart. the oscillations tions (3.38)equation and of (3.39) motionfield, and for the the we scalar dependent have evolution of found the time- M JHEP11(1999)036 2 as / 1 eff ± [5, 6]. m 0 λφ √ 2 √ = | W as a function of time for 2 Φ ∂ | /s g n and for a quadratic superpotential. 1 − The power spectrum of helicity The evolution of the parameter The ratio =10 P 23 /M 0 a function of time and for a cubic superpotential. φ Figure 5: Figure 6: Figure 4: gravitinos for a cubic superpotential. φ M and 1 ∼ − 3. In the / is given in 3 , one expects Φ 4. A special =10 s . 1 / λ φ 4 − φ P √ GeV as required does not decrease M M λφ 13 = the potential (3.41) /M eff ∼ ∼ 0 = P 10 m φ k W . This expectation is 2 gravitinos is summa- M ' V φ / is expected to oscillate with maximum amplitude 2 gravitinos gives rise to 1 φ M / eff ± 1 | M m Φ ± ∼ | changes by an amount . If the mass of the inflaton Finally the ratio of the number Let us now consider a cubic su- The result of our numerical in- eff RH max rized in figurevalues 3 of for initial twom conditions. different Since This behaviourfigure 5. is well-confirmed by the numerical results given in limit confirmed by our numericalwhich results indicates aspectrum for cut-off in the reduces to k feature of thisproblem theory of gravitino is production thatan in expanding the universe can bepletely com- reduced tolem a in similar Minkowski prob- space-timesimple conformal by redefinition a of the scalar field. This explainseffective mass why the in units of theT reheat temperature perpotential figure 4 for the entropy density in a time scale tegration for the power spectrum of helicity density of gravitinos in units of with time, seemore, figure 5. Further- field is by the normalization of density per- turbations, we seethermal that particle production the oflicity non- he- a number density wellbound beneath (2.6) the [5, 6]. JHEP11(1999)036 will plus (4.1) 4 4 / , while is fixed 3 µ κ µ /ρ √ = 2 / 3 µ/ the Universe V n c Φ =2 i as a function of time φ h 4 / 3 /ρ g n .Theresult φ ahler potential for the super- , 2 The ratio µ − 2 2 φ 24 κ for a cubic superpotential. Figure 7: S = 0 and the potential reduces to = eff φ m . The superpotential (4.1) leads to hybrid inflation. in a , W | S 0 | 2 /κµ λφ 2 √ √ p ≡ . 0 GeV to reproduce the observed temperature anisotropy. = is given in fig- and one expects 15 λφ c 4 1 / , inflation ends because the false vacuum becomes unstable. The Φ √ 3 10 − c ) ρ indicate the energy 0 ∼ × . This expectation is ρ k λφ 0 2 gravitinos is summa- / √ λφ 1 is a dimensionless coupling of order unity [24, 1]. The canonically-norma- √ ± is minimal, the supergravity corections to the mass term of the inflaton field rapidly oscillates around the minimum of the potential at κ ∼ S φ The result of our numerical in- Finally, the ratio of the number When Φ = Φ chiral superfields max to be around 5 supergravity and logarithmic corrections [21].field If the K¨ cancel and they do not spoil the flatness of the potential. For Φ where lized inflaton field is Φ contradict the boundergy (2.6) density by in about therelativistic one scalar particles. order field of is magnitude transferred [5, to the 6] energy when density the of en- a hot gas of again confirmed by our numerical results which indicated a cut-offspectrum in the for field is trapped in the false vacuum and we have slow-roll inflation. The scale density stored in the massless oscillating scalar field ure 7. Here time scale ( As we already mentioned, constructinggravity requires a some model effort. of Realisticmass inflation supersymmetric of in models the the of inflation context inflaton requireachieve of field the in super- to the be context of much supergravityof smaller since the than supergravity inflaton corrections the potential spoil Hubble [10]. theinvolve rate. flatness more However, than some This exceptions one is are scalar hard known field. to and Consider they the usually superpotential 4. Non-thermal production of gravitinos in the case of two tegration for the powerhelicity spectrum of changes by an amount rized in figure 6. In this case Indeed, for Φ k density of gravitinos inentropy units density of the JHEP11(1999)036 ). 1 − (4.6) (4.4) (4.5) (4.3) (4.8) (4.7) (4.2) µ ( within O µ [6]. After 3 µ 2 − 10 ∼ . 2 / , † 1 3 n Θ , 2 =0 ⊥ 1 Θ η χ and . 2 − † ∆ . † 2 µ Θ 2 2 G 0 Θ / † ˙ . , 1 ) ∼ Θ iγ † α † 1 Θ α 2 † Θ µ + 2 gravitino with the Goldstino is well + RH − α max α / 0 T † ( − k ˙ 2 1 the corresponding fermionic degrees of 2 γ t G 0 + = 1 R i − 2 ± 2 ˙ α ⊥ 2 to the entropy density is [6] Θ − χ ˆ 25 k − 10 † 2 e χ 2 0 † 2 / Θ − 3 = ∼ iγ n 1 2 = and ˙ 0 z / + +Θ s 2 ∆ 3 γ 1 ˙ ˙ η i ∆ ∆ ⊥ 1 n χ W α χ = † 1 ∆=∆(0) − + can be expressed, making use of eq. (3.12), as α 2 η ˙ z χ 2 =Θ 1 k ˆ χ W and by + 2 ¨ η ∆= . Therefore, one expects 1 − and Φ µ 2 gravitinos in theories with more than one chiral superfield from 1 / 1 ∼ ± GeV and imposes aGeV stringent [6]. upper bound on the reheating temperature 9 5 10 10 ∼ . We generically denote the two chiral superfields involved in the generic problem The estimate (4.2) obtained in ref. [6] was based on the assumption that the RH RH justified, since the amplitudes of thehybrid oscillating inflation fields in are the far models below of supersymmetric the Planck scale. at hand by Φ Notice that ∆ satisfy the following differential equation Using eq. (3.17), the master eq. (3.20) becomes where atime-scale wherewehavedefined∆as T a supergravity point oftheorem. view, The identification we of make the helicity use of the gravitino-Goldstino equivalence Therefore during the time evolution of the system, ∆ will never vanish. which is solved by reheating takes place, the final ratio The mass scales at the end of inflation change by an amount of order of the field Φ rapidly oscillates around zero. The time-scale of the oscillations is This violates the boundT in eq. (2.6) by at least four orders of magnitude even if freedom. The combination ˆ results on the gravitino productionin for a a theory single with onethis more chiral assumption than superfield one is model superfield. areof justified. valid helicity In Instead the following, of we attacking wish the to problem show that of the production JHEP11(1999)036 , ) ∝ k ) ( t r (4.9) ( (4.10) (4.15) (4.11) (4.13) (4.14) (4.12) ψ in r ) t u ( ∗ − ,sincethe 1 2 gravitinos. )= ru − 2 gravitinos) I t / / ( − H 1 1 [ , . 0 ± out r ± ≡ ∼ dt u T r ) v 0 =0 t . ( during inflation. The η k I dt , η dt . ∆ ) H ∆ ) )] t ∆(0) 0 )] and t ( ∆(0) t · ( k , · ( ) , one has in r out r r t − k dt u ( µ u t ψ † V 0 dt ) ) ( ) in r † t α t t ∼ ( α v ( R ( ∞ 0 k k R e 4 . − V rr † / k e V 2 1 ) † ) | β G t 0 t to be the solution of the equation at G ( V ( ,ru † rr k † x ~ ) 0 · 0 ˆ k in β r ) sin [Ω in )+ k r k ~ | ˆ 0 0 k i t v ( becomes v t 0 ( r + ( ) as a perturbative potential, then we can 0 0 0 iγ t t r k in r ∞ 26 ψ iγ ( ∞ rr k ∞ X ) −∞ u V − k −∞ t 0 Ω β Z , eq. (4.5) can be recast in the form + i is given by ( V Z = rr k η − t 0 + η α e 2 −∞ u rr k kr 4 [ 2 Z ∞ α β ∞ i ∝ 2 N 1 ≡ i 4 )= ∞ − + α T r x ~ 2Ω · t 2Ω u k ( ~ ˙ 2 i α dt α/ + e ) out r ˙ R − 2 α t − )+Ω u ( t − = . If we treat ( in k = 0 )= η in k 0 t − ∞ rr η k ( is the spin index, in the late time region, eq. (4.10) possesses rr k η k ∼ + β exp 2 β V r ) k )= → )intermsof t → t + ( t x, t ~ ( η k ( ¨ k η η η in χ )], where , i.e. a plane-wave, eq. (4.9) can be written as an integral equation k ( and the Bogolyubov coefficient r t ψ ) Redefining ∞ t Ω ( i →−∞ ∗ + − frequency is set by the height of the potential t where in a given spin state is therefore the solution If we define Finding an exact solutiongoes to beyond eq. the (4.9), scopeapproximation or method of even to this estimate studying paper; the the we number problem will density limit numerically, of ourselves helicity to outline a standard Even though this approximationabout is the expected resonant to behaviourof offer of magnitude the only for system, part theinflation, we of number like believe density. the the it information In one providesthe typical described the minimum supersymmetric in right by hybrid a order the models time-scale superpotential of much (4.1), shorter the than the system Hubble relaxes time to e Decomposing andwehavelet solve eq. (4.10) by iteration. To the lowest order in u The corresponding number of Goldstinos (or, equivalently, helicity where the Bogolyubov coefficient JHEP11(1999)036 µ in ∼ µ Nucl. Phys. , , ∼ max ]. k Astrophys. Phys. Lett. Phys. Lett. ). We are (1992) 399. (1995) 879 t , , , . ( , confirming Astrophysical k 3 k 93 V µ ∼ B 373 1 for 3 ∝ max Limits on late decaying k k hep-ph/9907510 e V ∼ , µ particles from vacuum fluctu- ]; Nucl. Phys. 2 , Gravitino production after infla- / 3 ) changes by an amount k (1989) 699; t ( k Prog. Theor. Phys. 2 will be (1999) 009 [ After primordial inflation V / , 1 08 ± B 311 Inflation can save the gravitino Non-thermal production of dangerous relics The cosmology of decaying gravitinos Cosmological gravitino regeneration and de- , the Fourier transform of k .Since 27 hep-ph/9609441 e V k ' Production of spin Gravitino production in the inflationary universe ∞ Nucl. Phys. , Supersymmetry, supergravity and particle physics Cosmological consequences of particle creation during in- (1998) 7120 [ ; (1984) 181. J. High Energy Phys. . depends upon , D57 kr rapidly dies out for momenta ]. N k B 145 e V , (1985) 175; 1 Cosmological constraints on the lifetime of massive particles − µ (1984) 1. ∼ Phys. Rev. (1982) 59; (1983) 30; (1985) 1; hep-ph/9904206 , B 259 , hep-th/9907124 110 Phys. Lett. , , 294 hep-ph/9403364 J. J. Ellis, D.V. Nanopoulos and S. Sarkar, Rep. Phys. particles from nucleosynthesis S. Dimopoulos, R. Esmailzadeh, L.J. Hall and G.D. Starkman, J. Ellis, G.B.constraints Gelmini, on massive J.L. unstable Lopez, neutral relic D.V. particles Nanopoulos and S. Sarkar, and the effects on Big-Bang nucleosynthesis tion in the early Universe B 118 D.V. Nanopoulos, K.A. Olive and M.B Srednicki, 127 J. Ellis, J.E. Kimcay and D.V. Nanopoulos, [ flation A.L. Maroto and A. Mazumdar, ations [2] D. Lindley, [1] For a review, see H.P. Nilles, [3] J. Ellis, A.D. Linde and D.V. Nanopoulos, [6] G.F. Giudice, I. Tkachev and A. Riotto, [4] M. Kawasaki and T. Moroi, [5] R. Kallosh, L. Kofman, A. Linde and A.V. Proeyen, [7] D.H. Lyth and D. Roberts, and zero otherwise. The number of helicity the original estimate made in ref. [6]. References Acknowledgments We would like to thank F. Feruglio, R. Kolb and A. Linde for discussions. number of particles interested in large momenta and Ω a timescale JHEP11(1999)036 , , , 77 38 ]. Phys. ,hep- , Phys. Rev. ; , (1999) 103522 (1989) 2281. Phys. Rev. Lett. , High-energy equiv- Phys. Rev. Lett. annihilation experi- Phys. Rev. Lett. D39 D60 , hep-ph/9807278 − , e + ]. e (1999) 1 [ Phys. Rev. Recent progress in baryogenesis Phys. Rev. (1973) 312; , , Universe reheating after inflation 314 18 , Addison-Wesley, Menlo Park 1990. ]. Recent progress in baryogenesis Production of massive particles during Particle physics models of inflation and Reheating after inflation hep-ph/9809453 Particle production during out of equilibrium 28 ]. Phys. Rep. , Classical decay of inflaton Resonant decay of bose condensates (1990) 2491; , Lectures given at the ICTP Summer School in ]; Decaying particles do not ‘heat up’ the universe Sov. Phys. JETP hep-ph/9407247 The early universe D42 (1979) 105. ]. (1999) 063504 [ (1986) 471; Annual Review of Nuclear and Particle Science Broken supersymmetry and supergravity B 147 hep-ph/9610477 D60 B 175 Phys. Rev. (1995) 5438 [ ]. . , A. Riotto and A. Trodden, . , hep-th/9405187 Gravitational production of gravitinos The gravitino abundance in supersymmetric ‘new’ inflation models hep-ph/9603378 D51 (1985) 681. Theories of baryogenesis Phys. Rev. Nucl. Phys. Lower limit on the mass of a light gravitino from , (1997) 1607 [ , Phys. Lett. , 79 D31 (1994) 3195 [ hep-ph/9908333 ments alence theorem in spontaneously broken supergravity R. Casalbuoni, S.D. Curtis, D. Dominici, F. Feruglio and R. Gatto, the cosmological density perturbation M. Lemoine, [ hep-ph/9911257 phase transitions L. Kofman, A. Linde and A.A.73 Starobinsky, Phys. Rev. Y. Shtanov, J. Traschen and R. Brandenberger, (1996) 219 [ hep-ph/9901362 Lett. Rev. reheating ph/9901362, to appear in A. Riotto, High-Energy Physics and Cosmology, Miramare,hep-ph/9807454 Trieste, Italy, 29 Jun – 17 Jul 1998, S. Deser and B. Zumino, (1977) 1433. Spontaneous symmetry breaking and Higgsconstant effect in supergravity without cosmological [8] D.H. Lyth, [9] P. Fayet, [10] See, for example, D.H. Lyth and A. Riotto, [11] E.W. Kolb and M. S. Turner, [12] J.H. Traschen and R.H. Brandenberger, [13] S.Y. Khlebnikov and I.I. Tkachev, [14] S.Y. Khlebnikov and I.I. Tkachev, [15] For a recent review, see A. Riotto and M. Trodden, [17] D.J. H. Chung, E.W. Kolb and A. Riotto, [16] R.J. Scherrer and M.S. Turner, [18] D.V. Volkov and V.A. Soroka, [19] E. Cremmer, B. Julia, J. Scherk, S. Ferrara, L. Girardello, P. van Nieuwenhuizen, JHEP11(1999)036 , , ]; (1997) B 442 B 417 (1999) 6 (1996) 43 (1996) 241 (1997) 1841 (1999) 123006 B 412 (1998) 125013 ]. B 448 B 387 D56 hep-ph/9905242 Phys. Lett. Phys. Lett. , D59 , B 388 D58 Phys. Lett. , Phys. Rev. Phys. Lett. Phys. Lett. (1999) 014 [ , ]; , Phys. Rev. , Probing planckian physics: res- hep-ph/9406319 Production of massive fermions , 08 Phys. Rev. Phys. Lett. D-term inflation in superstring the- , , ]; ]. F-term inflation in superstring theories D-term inflation in superstring theories Nonequilibrium dynamics of fermions in (1994) 1886 [ 29 hep-ph/9804214 73 Large scale structure and supersymmetric inflation D-term inflation hep-ph/9804214 Preheating of fermions J. High Energy Phys. hep-ph/9903492 , Matter creation via vacuum fluctuations in the early uni- ]; ]. . Comments on D-term inflation (1998) 461 [ Dilaton stabilization in D-term inflation Minimalism in inflation model building Hybrid inflation in supergravity ]; Phys. Rev. Lett. (1998) 461 [ (1999) 91 [ , B 531 ]; ]; ]; ]. ]; ]; B 531 B 459 hep-ph/9706408 hep-ph/9806281 Hybrid inflation from supergravity D-terms hep-ph/9910437 , Nucl. Phys. , hep-ph/9707273 hep-ph/9806205 hep-ph/9809547 hep-ph/9807339 hep-ph/9703209 hep-ph/9606342 hep-ph/9606423 P.B. Greene and L. Kofman, [ a spatially homogeneous scalar background field G.F. Giudice, M. Peloso, A. Riotto, and I. Tkachev, [ at preheating and leptogenesis verse and observed ultrahigh-energy cosmic ray events V. Kuzmin and I. Tkachev, [ D.J.H. Chung, E.W. Kolb, A.onant Riotto production and of I. Tkachev, particlesspectrum during inflation and features in the primordial power [ E. Halyo, D.H. Lyth and A. Riotto, [ [ G. Dvali and A. Riotto, 28 [ (1998) 20 [ J.R. Espinosa, A. Riotto and G.G. Ross, Nucl. Phys. ories S.F. King and A. Riotto, (1998) 68 [ Phys. Lett. without fine tuning [20] J. Baacke, K. Heitmann and C. Patzold, [21] A. Linde and A. Riotto, [22] P. Binetruy and G. Dvali, [23] J.A. Casas, G.B. Gelmini and A. Riotto, [24] G. Dvali, Q. Shafi and R. Schaefer,