aa r X i v : . [ h e p - ph ] J un Brane Cosmology and KK Gravitinos
C. Bambi, F.R. Urban
University of Ferrara, Department of PhysicsINFN, sezione di Ferraravia Saragat 1, 44100, Ferrara, Italy
Abstract.
The cosmology of KK gravitinos in models with extra dimensions isconsidered. The main result is that the production of such KK modes is not compatiblewith an epoch of non–standard expansion after inflation. This is so because the BBNconstraint on the zero mode forces the reduced five dimensional Planck mass M downto values much smaller than the usual four dimensional one, but this in turn impliesmany KK states available for a given temperature. Once these states are taken intoaccount one finds that there is no M for which the produced KK gravitinos satisfyBBN and overclosure constraints. This conclusion holds for both flat and warpedmodels in which only gravity propagates in the full spacetime.
1. Introduction
The problem of overproduction of gravitinos, the supersymmetric partner of thegraviton, is a long–standing one in cosmology [1]. The gravitino interacts very weaklywith ordinary matter, its coupling being gravitationally suppressed, and this makes it along living particle which is never in thermal equilibrium after inflation.An unstable gravitino lighter than about 10 TeV decays after the BigBang Nucleosynthesis (BBN) and the entropy injected into the plasma can causephotodissociation of the light elements, altering their abundances [2]. Hence therequirement of a successful BBN severely constrains the produced amount of gravitinos[2]. If gravitinos are much heavier, their decay products are not dangerous for primordialnuclei since they harmlessly decay before BBN; however, if R –parity is a good symmetry(as it is assumed throughout the whole paper) their decay will produce a non–thermalabundance of light SUSY particles, either the lightest stable one (LSP), or some otherparticle which will later decay into it. The present day energy density stored in LSP asdark matter is constrained by cosmological observations [3]. A similar scenario holds fora light gravitino, lighter than about 100 GeV, which, if it is the LSP as it is likely is thecase, must (at least) not overclose the universe. These considerations lead to an upperlimit on the temperature at which thermal equilibrium had been established (usuallyreferred to as the reheating temperature T R ), this limit being around 10 ÷ TeV,depending on the model [2]. This is discussed in the forthcoming section. rane Cosmology and KK Gravitinos
2. The gravitino problem
Gravitinos are produced in several different ways, thermally and non–thermally.Thermal production [9] involves either inelastic scattering processes of thermalisedparticles, or decays of supersymmetric particles. For the reason explained in section 6.1,this last mechanism is uninteresting in braneworld cosmology. Non–thermal mechanisms[10] include perturbative and non–perturbative production by means of inflaton decay,or some other scalar fields (moduli, dilaton, radion), which is strongly model dependent,and which is not treated here, see section 6.2. The last option is gravitational particleproduction, which is discussed in 6.3. Hence, in what follows mainly thermal productionof gravitinos via inelastic scattering is considered.The zero mode gravitino abundance is usually expressed in terms of the gravitinonumber density to the entropy density ratio as Y / ( T ) = n / ( T ) s ( T ) . (1)Here s ( T ) = (2 π / g ∗ S T and g ∗ S is the number of “entropic” degrees of freedom, and T is the temperature of the system.The Boltzmann equation for the process under examination leads to the abundanceof the produced particles ddT Y / = − s h σv i Y rad HT , (2) rane Cosmology and KK Gravitinos Y rad is the equilibrium number density to entropy density ratio for relativisticparticles, and h σv i parametrises the thermally averaged cross section for the processunder scrutiny. H is the Hubble parameter which in standard cosmology is H =( ρ/ M ) / , where ρ is the energy density of the universe and M = 2 . · TeVis the reduced four dimensional Planck mass. In the radiation dominated epoch of theearly universe ρ ∝ T .Integrating this equation one finds the well known [2] expression for the abundanceat the BBN (given that the zero mode is not too heavy and thus survives at least till T ≃ Y / = 1 . · − m m / ! (cid:18) T R TeV (cid:19) , (3)where ˜ m is the gluino mass. Since the primordial gravitino abundance (3) is proportional to the reheatingtemperature, cosmological constraints on Y / translate into upper bounds on T R andhence on the inflationary model. If the gravitino is stable, its energy density today mustnot overclose the universe. In particular, it must not exceed the dark matter energydensity. This puts a bound on T R only for m / & B h ≈ m / ∼ B h is non–vanishing: the photon can be virtual and candecay into q ¯ q pair, with branching ratio B h ∼ α em / π ∼ . m / = 100 GeV ÷
30 TeV are summarised in the table below for B h = 1[ B h = 10 − [. More details are given in [2]. m / Y / m LSP constraint1 keV ÷
100 GeV 6 · − (cid:16) m / (cid:17) m / direct overclosure100 GeV 3 · − [3 · − ] any but m / BBN300 GeV 4 · − [4 · − ] any but m / BBN1 TeV 4 · − [3 · − ] any but m / BBN3 TeV 1 · − [3 · − ] any but m / BBN10 TeV 5 · − [2 · − ] any but m / BBN30 TeV 8 · − [2 · − ] any but m / BBN30 TeV ÷ TeV 6 · − (cid:16)
100 GeV m LSP (cid:17) any but m / daughters overclosureGravitino decay can also affect the CMBR spectrum if gravitino lifetime is in therange 10 − s, causing Bose–Einstein and/or Compton distortion, but the constraints rane Cosmology and KK Gravitinos m / &
30 TeV, the gravitino lifetime is τ / . m / ∼
100 GeV ÷ T R . − GeV (4)and several inflationary models have to be rejected or strongly fine–tuned.
So far only the standard cosmological expansion law H = ρ/ M has been considered,and it has been shown that the limits on the reheating temperature are very restrictive,especially in connection with inflation model building (it is non–trivial to lower thereheating temperature down to such values and still have successful inflation ‡ ).However, if the SM lived on a four dimensional Friedman–Robertson–Walkerhypersurface (the brane), embedded in an extra dimensional spacetime, the earlyuniverse would admit an epoch of non–standard expansion [7]. Several such modelshave been built in the last few years, the ones which are dealt with here being the ADD[13] and RS [14] models, which involve flat and warped extra dimensions respectively.These models show a peculiar feature when their cosmology is investigated. For instancethe model named “RSII”, where only one four dimensional three–brane is present tocatalyse gravity and warp the bulk five dimensional AdS spacetime, has a Friedmanequation of this kind § [7]: H = ρ M (cid:16) ρ λ (cid:17) , (5)where λ is the tension of the brane, which is related to the five dimensional Planck massas λ = 6 M /M . This equation says that at high energy densities the expansion of theuniverse was much faster than at later times, and went as T instead of T , togetherwith the unknown parameter M : the smaller M the faster the expansion.This five dimensional mass scale is constrained, in the “RSII” model, to be biggerthan about 10 TeV from measurements of the gravitational inverse square law in sub–mm range [14]. However such a bound might be inapplicable if the RSII model were ‡ At the moment good candidates in this direction are low scale gravity models, which may “naturally”provide reasonably low temperatures [12]. § This form for the Friedman equation holds more generally for five dimensional brane models, the maindifferences being extra terms such as cosmological constants, dark radiation, etc., which are neglectedhere. rane Cosmology and KK Gravitinos M &
10 TeV [16].At this point it is convenient to define a “transition” temperature T ∗ from standardcosmology to brane one, which can be extracted from ρ = 2 λ [8] T ∗ = (cid:18) π g ∗ (cid:19) / M M , (6)where g ∗ = g ∗ ( T ) counts the relativistic degrees of freedom at a given temperature T .If the dominant component of the universe is not radiation then this “temperature”approximately means the fourth root of the energy density, and parametrises the epochat which the transition occurs. In terms of this new quantity the Hubble parameter canbe cast as H = H d " (cid:18) TT ∗ (cid:19) . (7)Here H d stands for the standard four dimensional Hubble parameter.This new expansion law needs to be taken into account when the amount of gravitinoproduced in the early universe is calculated, that is, this expression has to be pluggedinto (2). Under the assumptions that T R ≫ T ∗ and T ∗ ≫ T , and that the extradimension does not change the coupling of the gravitino zero mode to the matter residingon the brane, instead of (3) the abundance at the BBN is approximately given by [8] Y / = 3 . · − m m / ! (cid:18) T ∗ TeV (cid:19) . (8)The main point here is that the former constraints on T R need now to be imposed on ∼ T ∗ , and thus on the unknown five dimensional mass scale. This conclusion involvedonly the zeroth gravitino: sections 5 and 4 are intended to extend the analysis to thefull spectrum of KK modes.
3. SUSY and extra dimensions
Supersymmetry and supergravity in the context of extra dimensions has beeninvestigated by several authors, primarily in connection with supersymmetry breakingby means of extra dimensional mechanisms [4, 5, 6]. The main reason of interest onthese models revolves around superstring theory, for it requires both supersymmetryand extra dimensions, although the path from such low–energy models and the fullunderlying string theory is far from being crystal clear. The cosmology of these modelshas not been studied yet, and, while it is expected that the well known main featuresof brane cosmology still hold, even relevant modifications could arise, primarily dueto extra field in the bulk (the gravitino) and model–dependent orbifolding boundaryconditions. This possibility is not explored further here, as the analysis presented in theforthcoming sections is readily extended to other cosmologies.In order for this work to maintain its validity in a broad class of models, this analysiswill be based mainly on two toy models, which reflect general features of supersymmetric rane Cosmology and KK Gravitinos
In this model the bulk spacetime is flat and contains only gravitons and gravitinos. Inconsidering the non–standard expansion epoch only the model with one extra dimensionis analysed, since in this case the modified Friedman equation (5) holds, whereas littleis known for the general case.The mass for each state can be expressed in two ways, depending upon thediagonalisability of the KK mass matrix k [4] m n = m + nR (9) m n = r m + (cid:16) nR (cid:17) , where the first case holds if the KK mass matrix is diagonalised, while the other one doeswhen it is not. Here R is the size of the extra dimension, while m is the zeroth mass,which can be either fixed by the extra dimensional parameters (this is the case if SUGRAis broken thanks to a mechanism which relies on the extra dimensions themselves), ornot [4]. Since there is no agreement on the way supergravity is broken, the zero modemass will be taken as a free parameter, while for simplicity the mass matrix is assumedto be diagonalisable. That specified, the mass gap between two nearby states is givenby ∆ m = 1 R = 2 πM M = (cid:18) π g ∗ (cid:19) / T ∗ M . (10)This expression can be straightforwardly generalised to N extra dimensions, except forthe last equality.Coming to the coupling constants, the situation is tricky and highly modeldependent. Several distinct possibilities arise, as these couplings could be set by theextra dimension parameters, or be completely unrelated to them. This especially truefor the ± / ± / /M )way to brane–stuck MSSM matter. However, the goldstino states will reveal themselvesto be not relevant in this study. Thus, the standard parametrisation for the cross sectionextracted from (3) is still valid, where of course the n –th KK gravitino mass m n has tobe taken into account. The second model to be dealt with is the warped one. Now a five dimensionalcosmological constant resides in the bulk, which makes it an
AdS spacetime. Onceagain, the Friedman equation receives a high–energy correction as in (5). k Henceforth the n –th KK gravitino mass will be just m n . rane Cosmology and KK Gravitinos m n = m + kx n e − πkR , (11)where x n is a solution of J ( x n ) = 0 ( J is the BesselJ function of the first kind), k isthe AdS curvature k = M M (cid:0) − e − πkR (cid:1) , (12)and R parametrises the size of the extra dimension. The same hypothesis done for theflat model concerning the zero mode holds here as well. The mass gap reads∆ m = ke − πkR ( x n − x n − ) ≃ ke − πkR (13)= (cid:18) π g ∗ (cid:19) / − e − πkR e πkR T ∗ M ≡ (cid:18) π g ∗ (cid:19) / F ( kR ) T ∗ M , where F ( kR ) is defined by the last equality.The coupling constants in this case may be different for different modes. The reasonfor this is that the effective coupling on the brane is given by two factors, the actualcoupling and the localisation of the wave function in the fifth dimension. Thus, a KKstate peaked on the brane under inspection will interact strongly, while a state located inthe other brane will be weakly interacting. The situation can be even more complicatedif there is more than one tower of gravitinos, as it is likely the case since five dimensionsrequire N=2 SUGRA at least.A considerable simplification is made here by using again the standard cross section,that is, the KK gravitinos tower is taken to be localised in the far away brane. If insteadthe tower resided on “our” brane and the couplings reflected the ones for gravitons(Planck suppressed the zero mode, highly enhanced the KK states), KK gravitinos wouldthermalise, while the zero mode would not, and the resulting picture would reflect theone reviewed in section 2.2. However, in section 4.3 the simultaneous presence of twoKK towers is investigated in more detail, in connection with the “twisted” model [6].
4. KK gravitinos and standard cosmology
Gravitinos are initially produced during a high temperature era. The total abundancefor a given KK mode is computed by integrating (2), where the n –th mode mass has to betaken into account. The upper limit for the integral is the highest temperature reachedin the early universe for which the relativistic plasma was in thermal equilibrium; thelower limit is the temperature at which thermal production stops, which, for each mode,is approximately equal to its mass.The abundance generated so far remains constant, except for some small jumps inthe total entropy density, until it is time for these gravitinos to decay. The numberdensity to entropy density ratio for the n –th gravitino mode at the BBN is Y n / = 1 . · − (cid:18) m m n (cid:19) (cid:18) − m n T R (cid:19) (cid:18) T R TeV (cid:19) . (14) rane Cosmology and KK Gravitinos n = 0 to n = ( T R − m ) / ∆ m ≃ T R / ∆ m , ora summatory over them. The result of the two operations (integral and summatoryrespectively) is, Y tot3 / ≃ − (cid:18) T R TeV (cid:19) { T R ∆ m + 2 ˜ m m ∆ mm (cid:18) m T R ln m m + T R (cid:19) }≃ − (cid:18) T R TeV (cid:19) { T R ∆ m + 2 ˜ m m ∆ mm } (15) Y tot3 / ≃ − (cid:18) T R TeV (cid:19) { T R ∆ m + 1 + 2 ˜ m m (cid:26) Y [ q ] − Y [ q + x ]+ m T R (cid:20) Y [ q ] − Y [ q + x ] + ∆ mm ( Y [ q ] − Y [ q + x ]) (cid:21)(cid:27) }≃ − (cid:18) T R TeV (cid:19) { T R ∆ m + 2 ˜ m m Y [ q ] } , (16)where the function Y a is the a –th logarithmic derivative the Gamma function, q = m / ∆ m , and x = ( T R + ∆ m ) / ∆ m . In order to obtain eqs. (15) and (16), the condition T R & m has been repeatedly used. As it can be easily seen these expressions arepractically equivalent once the mass gap ∆ m is smaller than the zero mode mass.Indeed, were this not the case then the integral would return a wrong answer: only(16) would be reliable.The lifetime of a heavy ( O (1) TeV) gravitino of mass m / is given, roughlyspeaking, by M /m / , while a light one is likely to be the LSP, since R –parityconservation is assumed. Moreover, high KK–number gravitinos could decay into lighterones, through processes such as KK n → KK m + X , where m < n and X is a StandardModel (SM) particle. This is connected with KK–number violation by the localisationof the process on the brane. Their amplitudes however can be either comparableor negligible with respect to the processes mentioned above (since also a transition KK → LSP + X violates KK–number). Below the reasons why the inclusion of theseprocesses should not significantly modify the results are outlined.Thus, there are two possible scenarios, depending on the zero mode mass. If thezero mode is heavy it will decay producing a non–thermal abundance of LSP particles,as its KK tower will as well. However, there is a subtlety here. In the standard case,a gravitino which is heavy enough to decay when the temperature of the plasma ishigher than the freeze–out temperature of the LSP pair annihilation process, it will notcontribute to the non–thermal abundance for the LSP. This places an upper limit onthe mass of the dangerous gravitinos (see the summary table in section 2.1).In standard cosmology the typical LSP freeze–out temperature is about 10 GeV,which corresponds to a time t ∼ (1 MeV /T ) ∼ − s. Since the gravitino lifetime is τ / ∼ (100 GeV /m / ) s, gravitinos with masses larger than m MAX ≃ TeV donot play any rˆole. On the other hand, if KK gravitinos decay quickly into lighter ones, rane Cosmology and KK Gravitinos ¶ m n > m NLSP , for the heavy gravitinos decay into light SUSYparticles which further decay into the LSP (which is the zero mode gravitino itself).In addition to the abundance therewith produced, one should thus take into accountmodes which survive, that is, those for which m n < m NLSP . Once more, if in additionwe have transitions between KK modes, only these gravitinos whose lifetime is longerthan the age of the universe t can survive, while heavier modes would have decayedinto them.Summarising, with the aforementioned simplifications, the KK gravitino towercould be split into four “bands”, keeping in mind that the lightest band may not existif the zero mode is heavy enough. The first band consists of the modes for which m n < m NLSP : once produced they will remain as non–thermal relics. If there are directtransitions between KK modes only those for which τ / > t contribute to the darkmatter today, but their abundance is fed by the decays of heavier modes. It will beseen that these light modes are not going to be very relevant, though. The secondband is that for which m ( N ) LSP < m n < m MAX : these gravitinos end up as out–of–equilibrium LSP relics, whoever the LSP is. In the third band superheavy gravitinoslive: they either decay into thermalised particles, and contribute nothing to non–thermalrelics abundances today, or decay into lighter KK modes, increasing their abundancesand tightening the constraints following from non–thermal gravitinos. The fourth band,which overlaps the second, and possibly also the first one, is the band for which gravitinodecays affect BBN: this band may admit less freedom for the parameters of the models,and goes approximately from 100 GeV to 30 TeV.
A first rough constraint can be obtained by using the previously computed total amountof gravitinos. Moreover, since the relevant mass gaps are going to be around ∆ m & + is then given by Y tot3 / ≃ · − T R M ∆ m . (17)A TeV zero mode gravitino forces the reheating temperature to be lower than order10 ÷ TeV, see (4) (the upper limit here reflects the fact that the gravitino couldbe much heavier that about 1 TeV). If, for example, T (0) R . TeV is taken, where ¶ The next–to–LSP (NLSP) mentioned here is not the first KK gravitino, but the lightest non–gravitinoMSSM particle. + Notice that in this case eqs. (15) and (16) are equivalent. rane Cosmology and KK Gravitinos M & · TeV. Were M smaller,other KK states would have become available at T (0) R , and the limits on the reheatingtemperature would need to be reconsidered.As an example consider M = 5 · TeV, which in turn means a mass gap ofabout 100 GeV. If for every KK gravitino produced back then there is now a 100 GeVLSP particle, one can ask that this amount be smaller than 10 − , see table in section2.1. Plugging the mass gap into (17), the new limit on the reheating temperature reads T R . TeV.For B h = 1, the BBN constraint however is found to be stronger: since each of the40 KK gravitinos in the mass range 1 ÷ T R . TeV [2], then T R is going to be 40 times smaller, that is T R .
25 TeV. Gravitinos outsidethis mass span allow for much higher reheating temperatures, so that one can neglectthem in this simple estimate. On the other hand, if B h = 10 − , the bound is weakerand comparable to the overclosure one. Indeed, looking at primordial abundances of Li and D, each one of the 40 KK gravitinos requires T R . · TeV, which becomes T R . · TeV once all the gravitinos are considered together.Finally, it is noteworthy that all these reheating temperatures are below T ∗ , asrequired for consistency, for M > TeV.Concluding, since the number of KK states available below a certain temperaturegrows very rapidly with the lowering of the five dimensional gravity scale, thecorresponding allowed reheating temperature drops noticeably. If the mass gap is small,for instance around 1 keV, the reheating temperature would be tightly constrainedaround T R ≃ m . This is no news since gravitons put similar upper bounds [18].Furthermore, as for gravitons, stronger bounds would be deduced if there is more thanone extra dimension and M is not too big, because the number of KK states grows as( T R / ∆ m ) N , where N is the number of extra dimensions. The results of the previous section apply here as well, the only relevant difference beingthe expression for the mass gap, which is given by (13). Thus, for instance, the standardchoice k ≃ M and kR ≃
12, gives a mass gap around 0.3 TeV. If this is the case, thenthe reheating temperature needs to be smaller than around 10 TeV.As a matter of fact, since these models involve SUSY and extra dimensions togetherthere is no need for the extra dimension to solve the hierarchy problem. This impliesmore freedom in the choice of the values for the parameters, which may be significantlydifferent from the example given above: (17) would still return the uppermost safe valuefor T R .It should be stressed here that these considerations are valid only if KK gravitino rane Cosmology and KK Gravitinos /M to the brane, while KK gravitons interact with 1 /M strength, sincetheir wave functions are peaked on the SM brane. This means that KK gravitons arenot able to give such restrictive constraints on the reheating temperature, while weaklycoupled gravitinos are. This subsection is devoted to some comments on the (seemingly realistic) possibilitythat if SUGRA is realised in an extra dimensional model then there will be more thanone tower of KK gravitinos, and these towers may be not all localised on the same brane.In view of [6] one can build a model in which there are two towers of KK gravitinos,living on opposite branes. One tower will have strongly enhanced interaction strengthsas KK gravitons have, and will most likely thermalise in the early universe, while theother tower will come with 1 /M couplings. Hence, the scenario is almost the same asin the previous section, but the number of degrees of freedom for a given temperature isnow different. In particular, if KK degrees of freedom are more numerous than MSSMones, a different law for Y / will be found.In order to illustrate this fact, one needs to specify the behaviour of the relativisticdegrees of freedom with the temperature. In such a picture this quantity can beapproximately expressed, for temperatures higher than the mass of the zero mode, as g ∗ ( T ) ≃ g MSSM ( T ) + (cid:0) g / + g (cid:1) T ∆ m , (18)where g MSSM ( T ) counts MSSM degrees of freedom and it is weakly dependent on thetemperature, while the second factor accounts for the relativistic KK gravitons andgravitinos in equilibrium ( g / = 4, g = 5). If the second term dominates, the totalamount of 1 /M interacting KK gravitinos will be given by Y tot3 / ≃ − T R M (cid:18) T R ∆ m (cid:19) / . (19)The effect of equilibrium KK states is to produce more efficiently dangerousgravitinos. This can be quantified by choosing the values for the parameters as inthe previous section, that is, k ≃ M and kR ≃
12. The corresponding reheatingtemperature is T R .
600 TeV, whereas 2 · TeV was found in sec. 4.2. Note thatthe assumptions made in deriving (19) hold here, since there are more than 10 KKstates available, while the MSSM degrees of freedom are much less, and since with thischoice of parameters KK gravitinos interact strongly. Furthermore, KK gravitons arepresent in the model of section 4.2 as well, and the estimates given there would need tobe improved taking these states into account if necessary. rane Cosmology and KK Gravitinos
5. KK gravitinos and brane cosmology
The above discussion can be generalised allowing for an epoch of non–standardexpansion, which would have taken place after T R but before BBN. The Friedmanequation is given by (7). The n –th mode abundance is thus given by (again the gluinoterm in the cross section is neglected): Y n / ≃ − (cid:18) T R TeV (cid:19) F [ 14 ,
12 ; 54 ; − (cid:18) T R T ∗ (cid:19) ] ≃ · − (cid:18) T ∗ TeV (cid:19) , (20)where F is the Gauss’ Hypergeometric function, and the last step implies T R ≫ T ∗ . Thisequation basically means that gravitinos are mainly produced around T ∗ , regardless of T R as long as it is much bigger than T ∗ itself. This is the result obtained for the zeromode in [8].Since all the gravitinos with masses lighter than T ∗ are produced in the amountpredicted by (20), while the production of heavier ones is strongly suppressed, the totalgravitino abundance will be given by Y tot3 / ≃ − T ∗ M ∆ m ≃ − , (21)where the last equality follows from (10). It is straightforward to conclude that KKgravitinos and the non–standard expansion epoch are not compatible with each other,the only possible way out being that all of the KK masses lie outside the range forwhich gravitinos are constrained by overclosure or BBN, but this seems unrealistic sinceit would require fine tuning of the zeroth mass together with T ∗ . This can be seen inanother way: the available number of KK states, inversely proportional to T ∗ , growsfaster than the amount which can be cut away by lowering T ∗ itself. Thus, once a small T ∗ is taken, as demanded by the zeroth gravitino bound, many KK states would becomeavailable below that temperature, which would require a further step downwards for T ∗ ,which in turn implies even more KK states available, and so on. There is no value for M for which a safe enough amount of gravitinos is produced. This is entirely due tothe relation between T ∗ and ∆ m .For instance, had the transition temperature been chosen around 10 TeV, asimposed by the zeroth gravitino constraint, the mass gap would have been around3 · − TeV, which means an enormous number (about 10 ) of KK states available atthat temperature. In this case one could hope that since the mass gap depends on two unknown parametersthere will be some parameters space for which the conclusion of the previous sectioncould be evaded. However, despite this fact, unless the gravitino zeroth mass and thetemperature scales on the scene are finely tuned, there is still no way one can get rid ofthe too many KK gravitinos. rane Cosmology and KK Gravitinos Y tot3 / ≃ − T ∗ M ∆ m ≃ − F ( kR ) . (22)One can easily see that this case is not better than the previous one, sincethe function 1 /F ( kR ) is always bigger than approximately 2.5. This means that Y tot3 / > · − , which is of course too much. If the zeroth gravitino constraint isimposed one would find ∆ m/ TeV ≃ − F ( kR ) . · − : tons of KK states areavailable in this scenario as well.One more comment is in order here. As in section 4.2, KK gravitons, if in thermalequilibrium, could play an important rˆole, namely they could enhance the abundancegiven by (22). If that were the case then (19) would approximately give the right amountof KK gravitinos at the BBN, where T R needs to be replaced by 2 T ∗ . However, afteradding these extra states the result (22) does not change much (recall it is an order ofmagnitude estimate).To conclude, it appears very difficult to reconcile an epoch of non standardexpansion and the presence of a KK tower of gravitinos, at least in these simplifiedtoy models, unless one demands a fine tuning between the parameters of the model. Itshould be stressed once more here that if the coupling constants are drastically differentthese conclusions do not hold; in particular, were KK gravitinos strongly interacting (asKK gravitons in RS–like models) they would be part of the thermalised plasma, whichcould not provide any constraint on the extra dimension free parameters.
6. More Gravitinos
In this section other gravitino production mechanisms are very briefly discussed:supersymmetric particle decays and non–thermal generation by scalar fields or by timevariable background metric of an expanding universe.
In addition to inelastic scattering, gravitinos can be produced in a hot plasma by decayof supersymmetric particles [9]. The partial decay width is roughlyΓ X ∼ π m X m / M , (23)where m X is the mass of the initial state. In standard cosmology, such a productionmechanism is relevant only for light gravitinos, i.e. for masses smaller than about 100keV, but gives rise to very strong bounds: if m / is in the range 1 ÷
100 keV, T R cannotexceed the mass of SUSY particles (0 . ÷ rane Cosmology and KK Gravitinos T R : the decay width is always given by (23) and,if the universe temperature is high enough that supersymmetric particles are relativisticand in thermal equilibrium (that is, their number density is not Boltzmann suppressed),the mechanism is too efficient for m / ≃ ÷
100 keV. The possibility that after inflationthe universe expansion is non–standard, i.e. T ∗ . m . − keV (see eqs. (10) and (13)), which further opens way to ahuge number of KK states. So, KK gravitinos and non–standard expansion in the earlyuniverse continue to be not compatible with each other, unless the extra dimensionalSUSY–breaking mechanism provides significantly different couplings. Perturbative and non–perturbative gravitino production by scalar fields in the earlyuniverse was considered for the first time in [10]. Depending on the particular framework,the mechanism may be completely negligible or the dominant gravitino source. In anycase, it does not influence the other production processes and hence cannot reconcilethe existence of gravitationally interacting KK gravitinos with non–standard expansionafter inflation.
Gravitational particle creation in braneworld cosmology has been very recently discussedin [19]. The mechanism is quite interesting, because it allows for the generation of manyvery weakly interacting or sterile particles, which today may account partially, or evencompletely, for the cosmological dark matter. The final abundance depends only on theparticle mass and, in order to be non–negligible, the universe would have had to undergoa period of braneworld regime. However, since this mechanism represents an additionalsource of gravitinos during the non–standard expansion epoch, it does not help. On thecontrary, even more dangerous particles would be produced.
7. Some considerations about KK gravitons
Up to now the focus has been mainly on gravitinos. It is quite natural to wonderwhether similar conclusions could be deduced by considering gravitons alone, whosebetter known properties furnish more reliable grounds for discussing BBN constraints.In ADD–like models, KK graviton interactions are 1 /M suppressed, so, thecorresponding lifetime is basically the same of KK gravitinos of equal mass, as long as weare not dealing with gravitino–goldstino states. However, since the graviton zero modeis massless and gravitons are not supersymmetric particles (and thus they have not acorresponding graviton R–parity), only those whose masses lie within 100 GeV ÷
30 TeVare useful. Indeed, heavier gravitons provide essentially no bounds, because their decaycan not affect BBN or produce stable and dangerous relic particles. Concerning lighter rane Cosmology and KK Gravitinos ÷
30 TeV shouldsuffice, that is, KK gravitons and non–standard expansion in ADD–like models are notcompatible as well.The situation is completely different in RS–like models, as here KK gravitonwavefunctions are peaked on “our” brane, so they interact much more strongly. Inthis case KK gravitons could thermalise, and no relevant bounds would be obtained. Ofcourse if some modes do not reach thermal equilibrium, they would be able to constrainthe reheating (or transition) temperature, even though these limits are expected to bemuch more shallow than what has been obtained previously.
8. Conclusion
We have considered the phenomenology of toy models where supergravity is realised andsubsequently broken in an extra dimensional setup, and we studied the cosmology of KKgravitino states which arise in that case. The most relevant conclusion is that, unless aconsiderable fine tuning between masses and parameters of the extra dimensional modelis required, it is not possible to allow for an epoch of non–standard expansion and, atthe same time, avoid KK gravitino overproduction. This is true for both flat and warpedextra dimensional models, as long as there is at least one weakly interacting tower ofKK gravitinos.We obtained constraints on the reheating temperature of the universe when thelatter is smaller than the transition temperature to standard expansion, i.e. when afterinflation the hot universe started out following the standard Friedman law, and nomodified expansion epoch has ever taken place thereafter. As expected, the availabilityof many KK states for a given temperature puts bounds on the reheating temperaturewhich are stronger than in the standard case, where only the zero mode is present. Wecomputed such upper limits in several illustrative scenarios, including a warped modelin which a tower of strongly interacting gravitinos in thermal equilibrium coexists witha gravitationally interacting one, as in the model proposed in [6].As far as high ( O (100) GeV or more) temperatures are concerned, our results arerelatively general, as they do not rely on ± / rane Cosmology and KK Gravitinos Acknowledgments
We wish to thank A.D. Dolgov for useful comments and encouragement. F.U. thanks H.Murayama for kind hospitality at UC Berkeley where the discussion which originatedthis work took place. F.U. is supported by INFN under grant n.10793/05.
References [1] M. Y. Khlopov and A. D. Linde, Phys. Lett. B , 265 (1984);J. R. Ellis, J. E. Kim and D. V. Nanopoulos, Phys. Lett. B , 181 (1984).[2] M. Bolz, A. Brandenburg and W. Buchmuller, Nucl. Phys. B , 518 (2001)[arXiv:hep-ph/0012052];M. Kawasaki, K. Kohri and T. Moroi, Phys. Rev. D , 083502 (2005) [arXiv:astro-ph/0408426];J. Pradler and F. D. Steffen, Phys. Rev. D , 023509 (2007) arXiv:hep-ph/0608344];J. Pradler and F. D. Steffen, Phys. Lett. B , 224 (2007) [arXiv:hep-ph/0612291];V. S. Rychkov and A. Strumia, Phys. Rev. D et al. [WMAP Collaboration], arXiv:astro-ph/0603449.[4] J. Bagger, F. Feruglio and F. Zwirner, JHEP , 010 (2002) [arXiv:hep-th/0108010];J. L. Hewett and D. Sadri, Phys. Rev. D , 015001 (2004) [arXiv:hep-ph/0204063].[5] J. Bagger and D. V. Belyaev, Phys. Rev. D , 025004 (2003) [arXiv:hep-th/0206024].[6] T. Gherghetta and A. Pomarol, Nucl. Phys. B , 3 (2001) [arXiv:hep-ph/0012378];J. Bagger and D. V. Belyaev, JHEP , 013 (2003) [arXiv:hep-th/0306063].[7] J. M. Cline, C. Grojean and G. Servant, Phys. Rev. Lett. , 4245 (1999) [arXiv:hep-ph/9906523];P. Binetruy, C. Deffayet, U. Ellwanger and D. Langlois, Phys. Lett. B , 285 (2000)[arXiv:hep-th/9910219].[8] N. Okada and O. Seto, Phys. Rev. D , 289 (1993);A. de Gouvea, T. Moroi and H. Murayama, Phys. Rev. D , 1281 (1997)[arXiv:hep-ph/9701244].[10] R. Kallosh, L. Kofman, A. D. Linde and A. Van Proeyen, Phys. Rev. D , 103503 (2000)[arXiv:hep-th/9907124];G. F. Giudice, A. Riotto and I. Tkachev, JHEP , 036 (1999) [arXiv:hep-ph/9911302].[11] W. Hu and J. Silk, Phys. Rev. Lett. , 2661 (1993);E. Holtmann, M. Kawasaki, K. Kohri and T. Moroi, Phys. Rev. D , 023506 (1999)[arXiv:hep-ph/9805405].[12] R. Maartens, D. Wands, B. A. Bassett and I. Heard, Phys. Rev. D , 041301 (2000)[arXiv:hep-ph/9912464].[13] N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, Phys. Lett. B , 263 (1998)[arXiv:hep-ph/9803315]; rane Cosmology and KK Gravitinos I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, Phys. Lett. B , 257 (1998)[arXiv:hep-ph/9804398].[14] L. Randall and R. Sundrum, Phys. Rev. Lett. , 3370 (1999) [arXiv:hep-ph/9905221];L. Randall and R. Sundrum, Phys. Rev. Lett. , 4690 (1999) [arXiv:hep-th/9906064].[15] S. Mizuno and K. i. Maeda, Phys. Rev. D , 123521 (2001) [arXiv:hep-ph/0108012].[16] J. D. Bratt, A. C. Gault, R. J. Scherrer and T. P. Walker, Phys. Lett. B , 19 (2002)[arXiv:astro-ph/0208133].[17] H. Davoudiasl, J. L. Hewett and T. G. Rizzo, Phys. Rev. Lett. , 2080 (2000)[arXiv:hep-ph/9909255].[18] J. L. Feng, A. Rajaraman and F. Takayama, Phys. Rev. D68