Dark Matter from Starobinsky Supergravity
DDark Matter from Starobinsky Supergravity
Andrea Addazi Dipartimento di Fisica, Universit`a di L’Aquila, 67010 Coppito, AQ LNGS, LaboratoriNazionali del Gran Sasso, 67010 Assergi AQ, Italy
Maxim Yu. Khlopov
Centre for Cosmoparticle Physics Cosmion; National Research Nuclear UniversityMEPHI (Moscow Engineering Physics Institute), Kashirskoe Sh., 31, Moscow115409, Russia; APC laboratory 10, rue Alice Domon et L´eonie Duquet 75205 ParisCedex 13, France
Abstract
We review our recent results on dark matter from Starobinsky super-gravity. In this context, a natural candidate for Cold Dark Matter is thegravitino. On the other hand, assuming the supersymmetry broken at scalesmuch higher than the electroweak scale, gravitinos are super heavy parti-cles. In this case, they may be non-thermally produced during inflation, inturn originated by the scalaron field with Starobinsky’s potential. Assum-ing gravitinos as Lightest supersymmetric particles (LSSP), the non-thermalproduction naturally accounts for the right amount of cold dark matter.Metastability of the gravitino LSSP leads to observable effects of their de-cay, putting constraints on the corresponding Unstable or Decaying DarkMatters scenarios. In this model, the gravitino mass is controlled by theinflaton field and it runs with it. This implies that a continuous spectrum ofsuperheavy gravitinos is produced during the slow-roll epoch. Implicationsin phenomenology, model building in GUT scenarios, intersecting D-branesmodels and instantons in string theories are discussed.
Direct searches for Weakly Interacting Massive Particles supersymmetric Dark Mat-ter candidates do not give positive results as well as TeV-scale supersymmetry wasnot found at the LHC. It may move the supersymmetry scale to much higher ener-gies. On the other hand, the Starobinsky R + ζR model [1] shows a substantiallygood agreement with Recent Planck data [2]. In particular, Starobinsky’s model isconformally equivalent to a scalar-tensor theory and the scalar field is a slow-rollinginflaton. . This may motivate a supergravity reformulation of the old Starobinskymodel, assuming supersymmetry spontaneously broken at very high scales. E-mail: [email protected] a r X i v : . [ g r- q c ] F e b consistent embedding of the old Starobinsky model in supergravity is not so easyas naively expectable. For instance, it was realized the the first Starobinsky supergrav-ity model proposed in Refs.[3, 4] entails a tachyonic instability of the Goldstino field forlarge values of the inflaton field. Recently, these issues were revisited in Refs.[5, 6] andin Refs.[7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]: Starobinsky super-gravity was reformulated in frameworks of non-linear Volkov-Akulov supersymmetryand no-scale invariance. This new class of models allows to reformulate a consistent R + ζR supergravity without any pathologically unstable moduli fields. Recently, theconsistency of Starobinsky supergravity with Null and Weak energy conditions was alsodiscussed in Ref.[25]. In Ref.[25], we have also demonstrated that the Strong Energycondition is violated in a large region of parameter spaces, compatible with inflation.However, new Starobinsky supergravity models do not only consistently containold Starobinsky inflation but they may provide a new candidate of dark matter. Inparticular, these models must predict the presence of gravitinos, which, if in turnassumed as the lightest supersymmetric particle, may provide a natural candidate forcold dark matter. Of course, gravitino mass is highly dependent by supersimmetrybreaking scale. So that Recent LHC constraints on TeV-ish SUSY may motivate theanalysis of superheavy gravitinos. On the other hand, this opens new issues regardingthe production of gravitons: may they account for the right amount of cold darkmatter? In Ref.[24], we have provided the analysis regarding this issue.Here we will review our recent results obtained in Ref.[24]. We have studied R + ζR supergravity with local supersymmetry broken at scales higher than the inflatonreheating. We will show how super-heavy gravitinos are non-thermally produced duringStarobinsky’s inflation. In this mechanism, the right Cold Dark Matter abundance,without any WIMP-like thermal miracle. On the other hand, assuming the gravitinomass heavier than then the inflaton mass, the suppression of the gravitino thermalproduction was shown in Ref. [46]. The formulation of the Starobinsky supergravity is based on the following Lagrangian[14, 15, 16, 17, 20]: L = − [ − L V R + L Φ( z, ¯ z )] D + ζ [ W α ( V R ) W α ( V R )] (1) R = log L S ¯ S where the standard Einstein-Hillbert action is recovered by the first term, the higherderivative term R term is originated from the kinetic term of the (real) superfield V R , S is the so dub compensator field of minimal supergravity, Φ( z, ¯ z ) is the K¨ahlerpotential of the z, ¯ z fields and L is the linear multiplet.For a discussion of the gravitino mass from supergravity, it is convenient to considerthe off-shell formulation of the minimal Starobinsky lagrangian during inflation. ItsK¨ahler potential reads as K = − T + ¯ T − Φ( z, ¯ z )] , W I → . (2)The gravitino mass is directly controlled by the K¨ahler potential and the superpotentialas m ˜ G = e K / W M P l = e − √ φ W M P l (3)Let us remark that Eq.(3) implies a direct connection among the gravitino mass andthe inflaton field. In other words, gravitino mass is a functional of the inflaton fieldand it runs with it. This will turn out to imply important predictions in gravitino massspectrum. Let us also note that W I = 0 → m ˜ G = 0 . Naturally, the gravitino is massless only in the supersymmetry and R-symmetry pre-serving phase. As a consequence, the fact that the gravitino mass depends on theinflaton field is not relevant in this case.In our model, as mentioned above, we shall assume that supersymmetry is sponta-neously broken at scales higher than the inflation reheating . This means that duringinflation, the superpotential W is a constant W >
0. So that, the relation amongthe gravitino mass and the inflaton field is not more trivial. We will see in the nextsections that this will imply that a continuos spectrum of gravitinos will be producedduring the inflation .On the other hand, ss mentioned above, the condition W I → W I = 0 corresponds to a vacuumstate which is invariant under the R-symmetry and SUSY. So that, it seems that these ymmetries may act as a sort of projection from the full general case to the onesproviding a successful inflation. However, the condition W I = 0 cannot be compatiblewith our dark matter model, because implying massless gravitinos during inflation. Sothat, we suggest that U R (1) and SUSY are spontaneously broken before (or at leastduring) the slow-roll epoch. On the other hand, after the inflation epoch, the rapidrolling down of the superpotential is already assumed. Under these hypothesis, thesuperpotential may only be set to a constant non-zero value → W = const (cid:54) = 0.To show that this condition does not destabilize the model is straightforward. Forinstance, it will imply that the G -term will get an extra constant term:∆ G = log W + log ¯ W = const (4)which in turns provides a constant term for the V F,D -terms:∆ V F = − W ¯ W (5)2 ζ ∆ V D = − . So that, the spontaneous symmetry breaking of U R (1) and SUSY during inflationonly implies that the inflaton potential is shifted by a constant, which may be reab-sorbed in the cosmological term. But what is important is that this demonstratesthat the spontaneous symmetry breaking of U R (1) and SUSY cannot destabilize themoduli fields, i.e. it does not contribute with new extra dangerous interactions term in W I . For instance, the R-symmetry implemented in the action has fixed the structureof the potential under the condition on W I . One can see that the only effect of a W = const (cid:54) = 0 during the inflation is the shift of the Starobinsky’s potential of aconstant factor and the z I fields remain stabilized. In Ref. [24], we have calculated the the production rate of gravitinos during inflation.One can estimate the energy density of the gravitinos produced during inflation as ρ ˜ G ( η e ) = (cid:104) m ˜ G (cid:105) n ˜ G ( η e ) = (cid:104) m ˜ G (cid:105) H e (cid:18) a ( η e ) (cid:19) P (6)where η is the time-like cosmological time variable, n ˜ G is the number density of grav-itinos, H e is the Hubble rate at the and of the slow-roll epoch time t e ; where P is thepower of emission of gravitinos from the expanding background which can be calculated (cid:104) m ˜ G (cid:105) (in log scale in the y -axis).In particular, the oscillating epoch effectively starts at φ/M P (cid:39)
1. On the other hand, theslow-roll effectively starts at φ/M P (cid:39)
6. ∆ φ/M P ∼ ÷ from a Bogoliubov transformation of creation and destruction operators associated tothe gravitino field in the expanding FRW background; where (cid:104) m ˜ G (cid:105) is the average massof gravitinos produced during the slow-roll, which is (cid:104) m ˜ G (cid:105) (cid:39) (cid:104) e − √ φ (cid:105) ∆ N W M P l (cid:39) . W M P l (7)considering the inflationary plateau has a width of ∆ φ (cid:39) M P l , i.e ∆ N = log a f /a i (cid:39)
60 e-folds. In first approximation, one may set in Eq.(7) (cid:104) φ (cid:105) (cid:39) ∆ φ/ ρ ˜ G ( t ) ρ R ( t ) = ρ ˜ G ( t R e ) ρ R ( t R e ) (cid:18) T R T e (cid:19) (8)where ρ ˜ G ( t Re ) /ρ R ( t Re ) is the after-Reheating epoch ratio among gravitinos and raditionand where t is the present cosmological time. ρ ˜ G ( t Re ) /ρ R ( t Re ) during the reheating epoch -inflaton decays to SM particles- isestimated as ρ ˜ G ( t Re ) ρ R ( t Re ) (cid:39) π (cid:18) ρ ˜ G ( t e ) M P l H ( t e ) (cid:19) (9) et us remind that the inflaton mass sets the characteristic scale for the Hubble con-stant calculated in t e : H ( t e ) ∼ m φ and ρ ( t e ) ∼ m φ M P l . This impliesΩ ˜ G h ∼ (cid:18) T Rh GeV (cid:19) (cid:18) ρ ˜ G ( t e ) ρ c ( t e ) (cid:19) (10)where ρ c ( t e ) = 3 H ( t e ) M P l / π is the critical energy density during t e . Finally, Eq.(10)can be rewritten as Ω ˜ G h (cid:39) Ω R h (cid:18) T Rh T (cid:19) π (cid:18) (cid:104) m ˜ G (cid:105) M P l (cid:19) n ˜ G ( t e ) M P l H ( t e ) (11)Eq.(11) is very useful: it relates the gravitino abundance with the gravitino mass, theinflaton mass and the reheating termperature. The inflaton mass is of the order of m φ (cid:39) GeV or so. On the other hand, the reheating temperature is T Rh /T (cid:39) . × .These parameters are fixed for a successful inflation and reheating. So that, the correctabundance of dark matter is obtained for a gravitino mass of (cid:104) m ˜ G (cid:105) (cid:39) (10 − ÷ × m φ (cid:39) ÷ GeV, in turn constraining W in Eq.(7). This means that the SUSYsymmetry breaking scale is expected to be around the GUT scale 10 ÷ GeV. Thiscertainly leads to other indirect implications in particle physics beyond the standardmodel. In fact, if supersymmetry must be broken around the GUT scale, it does notbe helpful for couplings unification in GUT scenarios like SU (5) and SO (10). As aconsequence our model seems to motivate non-supersymmetric GUT scenarios in whichthe couplings unification is reobtained adding extra non-minimal multiplets (See forexample Ref.[50] for a revival of non-supersymmetric SO (10) models by introducinghigher multiplets and considering RG corrections beyond the tree-level relations.). Onthe other hand, generically, these multiplets must be added in order to obtain a realisticspectrum of SM Yukawa and neutrino mass matrix [49]. An alternative paradigm tothe unification is provided by intersecting D-branes models or quiver string theories.In this case, starting from N = 1 supersymmetry, it can be broken at the GUT scale(here only taken as conventional) without destabilizing the construction, i.e. tachyonsin D-brane worldsheets are avoided. This certainly seems to be a more promising classof models with respect to other attempts to construct intersecting D-brane modelswithout supersymmetry, which in general are expected to be plagued by tachyons. Letus also note that in intersecting D-brane models, supersymmetry may be dynamicallybroken by the Euclidean D-brane instantons [51]. Phenomenology
Certainly, Superheavy gravitino dark matter cannot be searched by direct detectionexperiments or in TeV-scale collider physics. However, we will comment how super-heavy gravitinos may be detected in very high energy indirect detection experiments,i.e. high energy cosmic rays observations.The spontaneously symmetry breaking of the gauged U R (1) parity may allow tonew effective operators destabilizing the gravitino and opening new decay channels toStandard Model particles. The new effective operators which may be generated aredependent by the details of the R-symmetry breaking. Of course, in realistic models,such operators must be very suppressed. Otherwise, the gravitino cannot be a good(meta)stable candidate for dark matter.For example, U (1) R may be spontaneously broken by the a scalar singlet field s contained in a supersymmetric chiral field S . Supposing that R ( S n ) = − R ( L ), being R the charge operator of U (1) R , one may introduce effective superpotentials like W sHL = 1 M n − S n HL (12)where M is an effective suppression scale generated by UV completion of the model.This generates an effective operator O sHL = 1 M n − φ nS ˜ hl L where ˜ h is the Higgsino field -it mixes the Higgsino field with neutrinos. On the otherhand, one can always introduce the operator L ˜ GV V = − i M P l ¯ ψ µ [ γ ν , γ ρ ] γ µ λF νρ (13)coupling the gravitino with W ± , Z, γ . Neutral gauginos mix with higgsinos, and theirmass eigenstates are neutralinos. So that, from (12) and (13), neutralinos mediate two-body decays ˜ G → γν, Zν, V R ν . In particular ˜ G → γν is the particularly interestingsince it may be constrained by very high energy gamma rays and neutrinos. A peaked2-body decay distribution is predicted. The associated decay rate isΓ (0)˜ G → γν = µ π cos θ W m ν m χ m G M P l (cid:32) − m ν m G (cid:33) (cid:32) m ν m G (cid:33) (14)where m ν is taken equal to heaviest neutrino, assumed to be m ν (cid:39) .
07 eV; µ ∼ ( (cid:104) φ S (cid:105) ) n − M n − . et us note that in the case of n = 1, the gravitino is rapidly destabilized and themodel should be easily ruled out In high scale supersymmetry breaking, assuming m χ (cid:39) GeV and m ˜ G (cid:39) GeV, the decay rate is of only Γ (cid:39) − eV correspondingto τ (cid:39) s. For a cosmologically stable gravitino the decay rate must be suppresseddown to 1 Gyr or so, i.e. of 10 − ÷− orders.It should be noted that if gravitino lifetime is smaller than the age of the Uni-verse, physics of the corresponding Unstable Dark Matter scenario should involvesome additional stable particles - candidates to the modern dark matter. Moreoverhigh energy neutrino and gamma background from gravitino decay lead to observableconsequences [53, 54] that may exclude this possibility. If the R-symmetry is sponta-neously broken before inflation, (cid:104) φ S (cid:105) (cid:39) GeV, Assuming M (cid:39) M P l , we must have( (cid:104) φ s (cid:105) /M P l ) ( n − (cid:39) − ÷ . This may be obtained for n = 4. Operators with n < n > G → γν with two photons and neutrino peaks of energy E CM (cid:39) m ˜ G / (cid:39) ÷ GeV. The observation of a so high energy neutrinos and photons could be astrong indirect evidence in favor of our scenario. In particular, these very high energyneutrinos can be observed by AUGER, Telescope Array, ANTARES and IceCube. andwhile eventually they could not be explained by any possible astrophysics sources.
The R + ζR supergravity could be UV completed by string theory. Often in literatureand in textbooks, one can find the following statement: in the limit of α (cid:48) = l s → superstrings reduce to supergravity models. However, this is not completely corrected.For instance, non-perturbative stringy corrections can generate new effective superpo-tential terms, even if not allowed by abelian symmetries at perturbative level. As aconsequence, non-perturbative stringy corrections may destabilize the gravitino. Thismay have dangerous or phenomenologically healthy implications (discussed above) forour model depending on the unknown global proprieties of the Calabi-Yau compactifi-cation.It is conceivable that the initial U (1) R gauge symmetry may be broken by EuclideanD-brane instantons of open superstring theories or worldsheet instantons in heterotic uperstring theory (See [47] for a review on this subject).For example, a µHL superpotential can be generated by E µHL is phenomenologically dangerous. In fact, we mustconsider its interplaying with gravitino couplings with gauge bosons W ± , Z, γ, V R andtheir related gauginos, as mentioned above. As discussed in the previous section thisshould imply a very fast gravitino decay. So that, non-perturbative stringy instantonsgenerating the µHL superpotential must be suppressed in non-perturbative regime. Ifspecific non-perturbative RR or NS-NS fluxes are wrapped by the instantonic EuclideanD-brane, such a suppression may be possible [48]. Calling N N.P. the non-pertubativesuppression factor, this can screen the the bare decay rate as Γ = N N.P. Γ . A suppres-sion factor N N.P. (cid:39) − in order to get a gravitino cosmological life-time of at least1 Gyr or so. In this paper, we have reviewed Superheavy gravitino dark matter in Starobinskysupergravity with supersymmetry broken at high scales. We have reviewed how grav-itinos may be non-thermally produced during inflationary slow-roll. As a consequenceparameters of the inflaton potential and of gravitino dark matter are interconnected.This model provides a new peculiar prediction: Super-Heavy Gravitinos are producedwith a continuos mass spectrum, following the inflaton field .We have commented about possible phenomenological implications of our scenario.In particular, our model suggests possible two-body decays ˜ G → γν producing veryhigh energy peaks of neutrinos and photons, of E CM (cid:39) ÷ TeV. The detectionof these very high energy neutrinos with a peak-like two-body decay distribution couldprovide a strong indirect hint in favor of our model.Finally, we have commented on possible open issues regarding the UV completionof Starobinsky supergravity model in superstring theories. In particular, U R (1) is notenough to protect the gravitino by non-perturbative stringy instantons. The gravitinowould be destabilized very fast if operators mixing the Higgsino with neutrino weregenerated by non-perturbative solutions. This seems to be another problem toward a The parameters space of gravitinos mass may change if a consistent amount of Primordial Black Holes[52, 55, 56] were produced during the early Universe. We did not consider this other possible contribution. ealistic UV embedding of our model in string theory in addition to the problem ofstring moduli stabilization during inflation. Acknowledgments
A would like to thank Fudan University of Shanghai and Hefei USTC - and inparticular Yifu Cai and Antonino Marcian`o - for hospitality during the preparationof this paper. The work by MK was performed within the framework of the Cen-ter FRPP supported by MEPhI Academic Excellence Project (contract 02.03.21.0005,27.08.2013).
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