could the next generation of cosmology experiments exclude supergravity?

PHYSICAL REVIEW D 69, 105021 ~2004!

Could the next generation of cosmology experiments exclude supergravity?

A. Barrau*Laboratory for Subatomic Physics and Cosmology, CNRS-IN2P3/Joseph Fourier University, 53 av des Martyrs,

38026 Grenoble cedex, France

N. Ponthieu†

Institute for Space Astrophysics, CNRS-INSUE/Paris-Sud University 91405, Orsay, France~Received 5 August 2003; revised manuscript received 12 January 2004; published 28 May 2004!

Gravitinos are expected to be produced in any local supersymmetric model. Using their abundance predic-tion as a function of the reheating energy scale, we argue that the next generation of cosmic microwavebackground experiments could exclude supergravity or strongly favor ‘‘thermal-like’’ inflation models ifBmode polarized radiation were detected. Galactic cosmic-ray production by evaporating primordial black holesis also investigated as a way of constraining the Hubble mass at the end of inflation. Subsequent limits on thegravitino mass and on the related grand unification parameters are derived.

DOI: 10.1103/PhysRevD.69.105021 PACS number~s!: 04.65.1e, 04.70.Dy, 14.80.Ly, 98.70.Vc

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I. INTRODUCTION: GRAVITINOSIN THE EARLY UNIVERSE

Although not yet experimentally discovered, supersymetry ~SUSY! is still the best—if not the only—natural extension of the standard model of particle physics. It coprovide a general framework to understand the origin offundamental difference between fermions and bosonscould help to resolve the difficult problem of mass hierchies, namely, the instability of the electroweak scale wrespect to radiative corrections. In global supersymmetry,generator spinorsj are assumed to obey]mj50 @1#. If onewants to deal with local supersymmetry, or supergravity, tcondition must be relaxed andj becomes a function of thespace coordinatesx. New terms, proportional to]mj(x),must be canceled by introducing a spin 3/2 particle, cathe gravitino, as vector bosons are introduced in gauge thries. The gravitino is part of anN51 multiplet which con-tains the spin 2 graviton~see Ref.@2# for an introductivereview! and, in the broken phase of supergravity, supHiggs effects make it massive through the absorption ofNambu-Goldstone fermion associated with the SUSY breing sector.

It has long been known that if the gravitino is unstabsome severe constraints on its mass must be considereorder to avoid entropy overproduction@3#: m3/2*10 TeV.On the other hand, if the gravitino is stable, its mass shosatisfy m3/2&1 keV @4# to keep the gravitinos densitsmaller than the full Universe density (V3/2,V tot). In spiteof the huge dilution, those constraints are not fully evadedinflation as gravitinos should be reproduced by scatterprocesses off the thermal radiation after the Universereheated@5–11#. As the number density of such secondagravitinos is expected to be proportional to the reheattemperature, it is possible to relate the energy scale of in

*Electronic address: [email protected]; URL: httplpsc.in2p3.fr/ams/aurelien/aurel.html

†Electronic address: [email protected]

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tion with the requirement that they are not overproducedIn the first part of this paper, the next generation of co

mic microwave background~CMB! detection experiments isconsidered as a way of possibly excluding supergravity. Ishown that the energy scale of inflation required to prodan observable tensor mode in the background radiation iscompatible with local supersymmetry in the standard cosmlogical scenario. In the second part, a new way of constraing the gravitino mass, based on evaporating primordblack holes, is investigated. Taking into account thatblack hole masses cannot be much smaller than the Humass at the formation epoch, it is suggested that a detecof cosmic rays produced by the Hawking mechanism wolead to a lower bound on the reheating scale and, therefon the gravitino mass. Links with grand-unified models agiven, as an example, in the conclusion. Finally, the basicthe propagation model used to relate the source term tolocal spectrum are given in the Appendix.

II. TENSOR MODE IN THE COSMOLOGICALBACKGROUND

Observational cosmology has recently entered a newthanks to several experiments dedicated to the Cmeasurements,1 e.g., Maxima, BOOMERanG, ACBARDASI, CBI, VSA, ARCHEOPS, and WMAP. They givestrong evidence in favor of the inflationary scenario: a dsity extremely close to the critical value, a nearly scalevariant power spectrum, and a Gaussian structure of theturbations. Furthermore, in addition to the temperatanisotropies, the polarization of the CMB has also beencently observed@12,13#. For the time being, only the evenparity E mode has been detected and the odd-parityB modeis still to be discovered. The latter is of specific importanas it would probe the primordial gravitational waves throutensor perturbations. Their amplitude can be expressed

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A. BARRAU AND N. PONTHIEU PHYSICAL REVIEW D69, 105021 ~2004!

the Hubble parameter and the potential of the scalar fidriving inflation @14#:

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Figure 1 shows the 1s sensitivity of the Planck satelliteto polarization, as computed withCMBFAST.2 On the sameplot, theB mode polarization in a standardLCDM cosmol-ogy with an inflation energy scaleEinfl;1016 GeV ~dottedline! is also represented. Increasing~lowering! Einfl wouldresult in increasing~lowering! the amplitude of the primor-dial B mode thus making it easier~more difficult if not im-possible! to detect. On the contrary, the level of the expecB mode induced by weak lensing is fixed and rather ac

2http://www.cmbfast.org

FIG. 1. Sensitivity (1s) to polarization of the Planck satellit~solid line! versus the expectedB mode polarization in a standarLCDM cosmology with an inflation energy scale of;1016 GeV~dotted line!. Planck should provide significant detection of thtensor mode, especially at low multipole, where reionizationboosts the power spectrum. TheB mode induced by weak lensing ialso represented~dot-dashed line! and dominated the primordiaspectrum for,<200.

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rately predicted since it results from lensing effects onpolarizationE mode due to scalar perturbations that are nwell constrained. The challenge in the detection of the pmordial B mode and the estimation ofEinfl is then to have asensitive enough experiment and to avoid contaminationweak lensing. For the Planck experiment, the major hopthe detection at low, thanks to the high reionization opticadepth suggested by WMAP@16#. In the case of limited skycoverage experiments, the weak lensing contribution whave to be removed.

With the Planck sensitivity, theB mode should be detected (3s) if Einfl.1016 GeV @17,18#. This case would bein severe conflict with most supersymmetric models. Indein mSUGRA, the gravitino mass is, by construction, epected to lie around the electroweak scale, i.e., in theGeV–1 TeV range@19#. Considering that deuterium and3Heshould not be overproduced by photodissociation of4He be-low 700 GeV and that deuterium should not be destroybeyond the allowed observational values@20# above 700GeV @21#, the reheating temperature must remain lower th23109 GeV if the branching ratio of gravitinos into photonand photinos is assumed to be unity and lower th531011 GeV with a conservative branching ratio of 1/1The large difference between those limits and the enescale required to produce a measurable amountB mode po-larization makes the exact value of the branching ratiogravitinos into photons and photinos irrelevant. A detectof the polarizationB mode by the Planck satellite woultherefore disfavor mSUGRA instandardcosmology.

In gauge-mediated SUSY breaking alternative scenarmostly interesting in accounting for a natural suppressionthe rate of flavor-changing neutral-current due to the lenergy scale, the situation is even more constrained. Incase, gravitinos are the lightest supersymmetric particlesrequiring their density not exceed the total density impoan upper limit onTRH between 106 and 103 GeV for massesbetween 10 MeV and 100 keV@22#. Although some refinedmodels can relax those constraints@23#, local supergravitywould, in this case also, be in serious trouble if the reheatemperature were high enough to be probed by the Plaexperiment.

A possible way to get around these conclusions is tosume that a substantial amount of entropy was releasedthe gravitinos and moduli production, that would dilute theaccording to the entropy conservation (n/s.cte). Such ascenario can be realized while keeping the inflationary schigh, e.g., in thermal inflation@24,25#. Some studies@26#even show that a wide modulus mass region (mF

'10 eV–104 GeV) would be allowed but it requires in moscases a very small reheating temperature. Recently, thevaton scenario@27# has also attracted considerable interestit generates a huge amount of entropy through a scalarthat dominates the radiation at a given epoch. One canargue that a detection of tensor mode polarization wostrongly favor ‘‘thermal-like’’ inflation scenarios if supergravity is to remain as the preferred extension of the standmodel of particle physics. Interestingly, if evidence in favof local supersymmetry were obtained either by colliders

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COULD THE NEXT GENERATION OF COSMOLOGY . . . PHYSICAL REVIEW D69, 105021 ~2004!

by independent astroparticle experiments, this could evea very promising observational signature for thermal infltion.

Fortunately, the Planck satellite is not expected to beultimate experiment to study the CMB polarization and seral improvements can be expected in the future. Howeas pointed out in Refs.@28,29#, there remains a lower limit tothe removal of the polarizationB mode foreground inducedby gravitational lensing which sets at present time the lowlimit on the detectable inflation scale to a few tim1015 GeV. This scale remains, however, particularly intereing if the fundamental scalars driving the phenomenonrelated with grand unification since it lies around the GUenergy~between 1015 GeV and 331016 GeV depending onwhether supersymmetry is considered or not!. It thereforemakes sense to improve the polarization sensitivity to rethe capability to probe the typical GUT scale where inflaticould have occurred if the gravitino limit is ignored.

III. COSMIC RAYS FROM EVAPORATING BLACK HOLES

Another interesting way to experimentally probe theheating temperature would be to look for evaporating pmordial black holes~PBH’s!. Such black holes should havformed in the early Universe if the density contrast was henough on small scales. Many different possible scenahave been suggested to allow for an important PBH den~see Ref.@30# for a review!: a dustlike stage@31#, generalfirst order phase transitions@32#, a scale in the power spectrum @33,34#, to mention only the currently most discusspossibilities. Such PBH’s of massM should evaporate, fol-lowing a Planck-like spectrum with temperatureT5hc3/(16pkGM), which was derived by Hawking@35# us-ing the usual quantum mechanical wave equation for alapsing object with a postcollapse classical curved meinstead of a precollapse Minkowsky one. If those black hoare present in our galaxy~even with densities as low aVPBH;1029), the emitted quanta should contribute to tobserved cosmic rays. Among them, two kinds of particare especially interesting: antiprotons and gamma rays.tiprotons are useful because the astrophysical backgrocoming from spallation of cosmic rays on the interstelmedium ~so-called secondary particles! is very small ~theratio p/p is smaller than 1024 whatever the considered energy! and very well known@36#. A tiny excess due to evaporating black holes could therefore be easily probed in theenergy range@37# since the shape of the PBH spectrumdominated by fragmentation processes and is then softerthe secondary spectrum. Gamma rays, coming both fromrect emission and from the decay of neutral pions, takevantage of the very small optical depth of the Universe;100 MeV radiation @38#: the source emission can bprobed up to redshiftsz;700. Furthermore, the signal tnoise ratio is optimal at this energy as the PBH spectrbecomes softer (dN/dE}E21→dN/dE}E23) above 100MeV ~roughly corresponding to the QCD confinement sca!because of partons hadronization and integrated redshiffects @39#.

Using those cosmic rays, the experiments are curre

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sensitive to PBH’s with masses between 1012 and 1014 g.Those values can be intuitively understood as resulting frtwo opposite effects. On the one hand, the temperaturevors the light~i.e., hot! black holes but their number densitis very small: by integrating the Hawking flux over energy,is straightforward to show that the mass spectrummustbeproportional toM2 below M* 5531014 g ~the initial massof a black hole whose lifetime is equal to the age of tUniverse! whatever the details of the formation mechanis@40#. This is mostly due to the fact that the low-mass behior is fully governed by the evaporation process, as obtaiby writing dn/dM5(dn/dMi)3(dMi /dM) where Mstands for the current mass value andMi for the initialone. The evolution termdMi /dM is simply determined fromMi'(3at1M3)1/3, where a'$7.8ds51/213.1ds51%31024

g3 s21 accounts for the number of available degrees of frdom with ds51/2590 andds51527 in the standard mode@41#. On the other hand, the ‘‘number density’’ effect favothe heavy black holes but their low temperature makesemission rate very small, especially when heavy hadronsconsidered.

The important point for this study is that only black holformed after inflation would contribute to the observed phnomena as those formed before were exponentially diluFurthermore, whatever the considered formation mechaneither through the usual collapse of high density-contrastmordial Gaussian fluctuations or for near critical phenome@42#, the PBH mass at the formation epoch is close tohorizon mass at the same time. It cannot be larger asconsidered points would not be in causal contact and it cnot be much smaller as they would, in this case, more prably have formed before~as taken into account in the usuPress-Schechter formalism!. It means that if the evaporatioprocess were detected, the Hubble mass at the reheatingshould be small enough not to induce a cutoff in the PBmass spectrum which would make the light black hoabundance totally negligible. The best upper limit availaon the density of PBH’s aroundM* 5531014 g, taking intoaccount both the details of the source term evolution andbackground from galaxies and quasars, is currenVPBH(M* ),3.331029 @43#.

Fortunately, some hope for future detection is still posible thanks to antideuterons: those nuclei are expected tvery rarely formed by spallation processes below a few Gfor kinematical reasons. The threshold for an antideuteproduction isE517 mp ~total energy! in the laboratory, 2.4times higher than for antiproton production. The centermass is, therefore, moving fast and it is very unlikelyproduce an antideuteron nearly at rest~in the 100 MeV–1GeV range! in the laboratory. On the other hand, they coube emitted in this energy range by evaporating PBH’s acould be probed by the new generation of cosmic-ray detors: the AMS experiment@44# and the GAPS project@45#.To obtain this result, a coalescence model~see Ref.@46# fora review! was used, based mainly on phase space consations: the antideuteron density in momentum space isportional to the product of the proton density with the proability of finding a neutron within a small sphere of radiusp0

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Figure 2 shows the possible values of the reheating tperature as a function of the density of PBH’s at 531014 gfor different PBH-induced antideuteron flux at 100 Me~ranging from 231027 m22 s21 sr21 GeV21, the maximumvalue consistent with the gamma-ray upper limit, down2310210 m22 s21 sr21 GeV21). They were obtained withconservative values of all the free parameters enteringmodel, astrophysical quantities being totally bounded byexhaustive study of the heavy nuclei data@48#. As expected,there is a degeneracy between theD flux and VPBH: thesame amount of particles can be produced either by anormalization of the black hole spectrum and a cutoff inhigh mass range~i.e., a low reheating temperature value! orby a low normalization of the black hole spectrum andcutoff in the low mass range~i.e., a high reheating temperature value!. This means that, in the case of detection,should be possible to give a lower limit on the reheattemperature. Of course, the larger the antideuteron flux,better the constraint onTRH. As shown on this figure, for afixed value of the flux, whatever the value ofVPBH, thereheating temperature value cannot be arbitrarily low sithe mass spectrum cannot be cut much above masses rocorresponding to temperatures of the order of theD mass~i.e. TBH; a few GeV andTRH; a few 108 GeV). The other

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way round, whatever the value ofTRH, the density of blackholes cannot be arbitrarily low since even without any cutthe source term must remain high enough to account forconsidered flux. Naturally, this approach assumes thatmeasured antideuterons are indeed produced by evaporblack holes. The only other serious candidate as a sourclight antinuclei in the low energy range are annihilating spersymmetric particles. It has been demonstrated@49# thatonly neutralinos with masses around 100–200 GeV cocontribute to the observed antideuteron flux. As this mrange will be probed by the Large Hadron Collider, it shoube possible to distinguish between antideuterons inducedPBH’s and by SUSY particles~some reconstruction problemcould occur if the mass spectrum is strongly degeneraespecially between the lightest neutralinos and charginosthis would hide the lightest supersymmetric particles onlymasses in the TeV range!.

In the case where they are indeed coming from blaholes, Fig. 3 gives the reheating temperature value as a ftion of the measuredD flux. This result was obtained byvarying values of the 100 MeV antideuteron spectrum cobined with the upper limit coming from Refs.@43# and @37#(VPBH,331029) for the corresponding reheating sca~evaluated by the previously given method!. As expected, thelimit becomes more stringent when the measured fluxhigher and diverges when it goes to the maximum allowvalue ~otherwise it contradicts previously given limits!.When compared with the upper bound coming from bbang nucleosynthesis, this translates into a lower limit ongravitino massm3/2. This can be derived by solving thBoltzmann equation for the gravitino number densityn3/2@21#:

FIG. 2. Possible reheating temperaturesTRH as a function ofthe PBH density~normalized to the critical density! for differentantideuteron flux at 100 MeV: 231027, 231028, 231029,2310210 from right to left in m22 s21 sr21 GeV21.

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It is important to notice that a great amount of work halso been recently devoted to the nonthermal production

FIG. 3. Lower limit on the reheating temperatureTRH as a func-tion of the 100 MeV antideuteron flux.

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gravitinos and moduli fields~dilaton and modulus fields appearing in the framework of superstring theories whichquire mass through the nonperturbative effects of the susymmetry breaking!. Most papers claim that the upper limon the reheating temperature must be drastically decre~by up to 7 orders of magnitude@51#!. Those results beingstill controversial, they were not taken into account in thwork but they can only reinforce our conclusions and iprove our limits.

IV. PROSPECTS AND CONCLUSION

It must be pointed out that such possible constraintsthe gravitino mass can be translated into constraints on mfundamental parameters, making them very valuable insearch for the allowed parameter space in grand unified mels. As an example, in models leading naturally to mscales in the 102–103 GeV range through a specific dilatovacuum configuration in supergravity, the gravitino mass cbe related with the GUT parameters@52#

m3/25S 5p1/2l

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with MGUT;1016 GeV and a gauge couplingaGUT;1/26.The superpotential value in the dilaton direction definesmagnitude of the coupling constantl of the self-interacting24 multiplet. Figure 5 shows how the lower value onlevolves as a function of the reheating temperature whcould be probed by the previously given method, for thrdifferent branching ratios. Although not very constraininthis lower limit of the order 1.431023 over the full testedrange forB51 could be one of the first experimental costraints onl.

FIG. 4. Lower limit on the gravitino mass as a function of thmeasured antideuteron flux for three different branching ratios.

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The next generation of CMB experiments will face a nesituation. Important efforts are devoted to the search forpolarizationB mode @53# and the sensitivity should reacscales of inflation of order 1015–1016 GeV. This value isslightly higher than the GUT scale if supersymmetry isnored~i.e., if gravitinos production is expected not to haoccurred!, and slightly lower than the GUT scale if supesymmetry is taken into account~i.e., in the case gravitinosare expected to be produced by scattering processes!. Con-sidering that the grand unified scale is the highest natvalue for the reheating temperature, this means that, if anificant amount of entropy was not released after the moproduction, it should not be possible to detect those tenmodes in both scenarios.

On the other hand, cosmic-ray experiments could be ssitive enough to investigate the allowed reheating temptures if small black holes were formed at the end of inflatioIn this case, important limits could be derived on the graitino mass and on the related GUT parameters.

ACKNOWLEDGMENTS

The authors would like to thank P. Salati, D. Maurin,Taillet, and F. Donato who developed the propagation moused in this work. We also would like to thank A. Lucotte fvery valuable information on the neutralino detection.

APPENDIX: ANTIDEUTERON FLUX COMPUTATION

In this two-zone approach, the geometry of the MilWay is a cylindrical box whose radial extension isR520 kpcfrom the galactic center, with a matter~stars! disk whosethickness is 2h5200 pc and a diffusion halo whose extentthe major source of uncertainty~taken into account in the

FIG. 5. Lower limit on the coupling constantl as a function ofthe reheating temperature.

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analysis!. The five parameters used areK0 , d @describing thediffusion coefficientK(E)5K0bRd], the halo half heightL,the convective velocityVc , and the Alfven velocity Va .They are varied within a given range determined by anhaustive and systematic study of cosmic ray nuclei data@48#.The same parameters as employed to study the antiprflux @37# are used again in this analysis. The antideuterdensity produced by evaporating PBH’s per energy binc Dobeys the following diffusion equation:

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could the next generation of cosmology experiments exclude supergravity?

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