Gravitino and Polonyi production in supergravity
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Abstract
We study production of gravitino and Polonyi particles in the minimal StarobinskyPolonyi \(\mathcal {N}=1\) supergravity with inflaton belonging to a massive vector supermultiplet. Our model has only one free parameter given by the scale of spontaneous SUSY breaking triggered by Polonyi chiral superfield. The vector supermultiplet generically enters the action nonminimally, via an arbitrary real function. This function is chosen to generate the inflaton scalar potential of the Starobinsky model. Our supergravity model can be reformulated as an abelian supersymmetric gauge theory with the vector gauge superfield coupled to two (Higgs and Polonyi) chiral superfields interacting with supergravity, where the U(1) gauge symmetry is spontaneously broken. We find that Polonyi and gravitino particles are efficiently produced during inflation, and estimate their masses and the reheating temperature. After inflation, perturbative decay of inflaton also produces Polonyi particles that rapidly decay into gravitinos. As a result, a coherent picture of inflation and dark matter emerges, where the abundance of produced gravitinos after inflation fits the CMB constraints as a Super Heavy Dark Matter (SHDM) candidate. Our scenario avoids the notorous gravitino and Polonyi problems with the Big Bang Nucleosynthesis (BBN) and DM overproduction.
1 Introduction
The Planck data [1, 2, 3] of the Cosmic Microwave Background (CMB) radiation favors slowroll singlelargefield chaotic inflation with an approximately flat plateau of the scalar potential, driven by single inflaton (scalar) field. The simplest geometrical realization of this description is provided by the Starobinsky model [4]. It strongly motivates us to connect this class of inflationary models to particle physics theory beyond the Standard Model (SM) of elementary particles. A reasonable way of theoretical realization of this program is via embedding of the inflationary models into \(\mathcal {N}=1\) supergravity. It is also the first natural step towards unification of inflation with the Supersymmetric Grand Unified Theories (SGUTs) and string theory. Inflaton is expected to be mixed with other scalars, but this mixing has to be small. The inflationary model building in the supergravity literature is usually based on an assumption that inflaton belongs to a chiral (scalar) supermultiplet, see e.g., the reviews [5, 6] for details. However, inflaton can also belong to a massive \(\mathcal {N}=1\) vector multiplet instead of a chiral one. Since there is only one real scalar in a massive \(\mathcal {N}=1\) vector multiplet, there is no need of stabilization of its (scalar) superpartners, and the \(\eta \)problem does not exist because the scalar potential of a vector multiplet is given by the Dterm instead of the Fterm. The minimal supergravity models with inflaton belonging to a massive vector multiplet were proposed in Refs. [7, 8] by using the nonminimal selfcoupling of a vector multiplet, paramaterized by an arbitrary real function [9]. These models can accommodate any desired values of the CMB observables (the scalar tilt \(n_s\) and the tensortoscalar ratio r), because the corresponding singlefield (inflaton) scalar potential is given by the derivative squared of that (arbitrary) real function. However, all models of Refs. [7, 8] have the vanishing vacuum energy after inflation, i.e. the vanishing cosmological constant, and the vanishing vacuum expectation value (VEV) of the auxiliary fields, so that supersymmetry (SUSY) is restored after inflation and only a Minkowski vacuum is allowed. A simple extension of the models [7, 8] was proposed in Refs. [10, 11], where a Polonyi (chiral) superfield with a linear superpotential [12] was added to the action, leading to a spontaneous SUSY breaking and a deSitter vacuum after inflation.
A successful theoretical embedding of inflation into supergravity models is, clearly, a necessary but is not a sufficient condition. Even when these models are well compatible with the Planck constraints on the (\(r,n_{s}\)), they still may (and often, do) lead to incompatibility with the (Hot and Cold) Dark Matter (DM) abundance and the Big Bang Nucleosynthesis (BBN). A typical issue is known as the gravitino problem: in order to not ruin the BBN, gravitinos must not decay in the early thermal bath injecting hadrons or radiation during the BBN epoch [13, 14, 15, 16]. As is well known, the BBN is very sensitive to initial conditions, while each extra hadron or radiation can radically jeopardize the BBN picture, leading to disastrous incompatibility with cosmological and astrophysical data. In addition, the socalled Polonyi problem was also pointed out in the literature: Polonyi particles decay can also jeopardize the success of the BBN [17, 18, 19, 20, 21, 22]. Indeed, a generic supergravity model predicts a disastrous overproduction of gravitinos and/or Polonyi particles or neutralinos. Any specific predictions are modeldependent, because they are very sensitive to the mass spectrum and the parameter range under consideration. The mass pattern selects the leading production mechanism or channel: either thermal (WIMPlike) production or/and nonthermal production sourced by inflation and decays of other heavier particles. The last channel includes a possible (different) production mechanism due to evaporating Primordial Black Holes (PBH’s) that may be formed in the early Universe [23, 24, 25]. Our minimal estimation of the probability of the miniPBH’s formation at the long dustlike preheating stage after inflation gives such a low value that the successive evaporation of the miniPBH’s doesn’t lead to a significant contribution to gravitino production (Sect. 3). However, if inflation ends by a first order phase transition, the situation drastically changes, and copious production of miniPBHs in bubble collisions [26, 27] can lead to a huge gravitino overproduction, thus excluding the first order phase transition exit from inflation.
We consider the very specific, minimalistic and, hence, a bit oversimplified \(\mathcal {N}=1\) supergravity model of inflation, with inflaton belonging to a massive vector multiplet. We demonstrate that our model avoids the overproduction and BBN problems, while it naturally accounts for the right amount of cold DM. We assume both Polonyi field, triggering a spontaneous SUSY breaking at high scales, and the massive gravitino, produced during inflation, to be superheavy, and call it the SuperHeavy Gravitino Dark Matter (SHGDM) scenario. A production of superheavy scalars during inflation was first studied in Refs. [28, 29], whereas the gravitino production sourced by inflation was considered in Refs. [30, 31, 32], though without specifying a particular model. In this paper we apply the methods of Refs. [30, 31, 32] to the specific StarobinskyPolonyi supergravity model proposed in Refs. [10, 11]. The supersymmetric partners of known particles (beyond the ones present in the model) are assumed to be heavier than Polonyi and gravitino particles (in the context of Highscale SUSY), also in order to overcome several technicalities in our calculations. Some of the physical predictions of our model are (i) the Polonyi mass is a bit higher than two times of the gravitino mass, and (ii) the inflaton mass is slightly higher than two Polonyi masses. This implies that inflaton can decay into Polonyi particles that, in their turn, decay into a couple of gravitinos. We show that superheavy gravitinos produced from inflation and Polonyi decays can fit the cold DM abundance. The parameter spaces of inflation and cold DM are thus linked to each other in a coherent unifying picture.
Our paper is organized as follows. In Sect. 2 we review our model. In Sect. 3 we consider the gravitino and Polonyi particle production mechanisms. Section 4 is our conclusion and outlook.
2 The model
In this Section we briefly review the inflationary model of Refs. [10, 11]. We use the natural units with the reduced Planck mass \(M_{Pl}=1\).^{1}
When the function J is linear with respect to its argument (i.e. in the case of the minimal coupling of the vector multiplet to supergravity), our results agree with the textbook [33]. In the absence of chiral matter, \(\Phi =0\), our results also agree with Refs. [8, 9].^{4}

there is no need to “stabilize” the singlefield inflationary trajectory against scalar superpartners of inflaton, because our inflaton is the only real scalar in a massive vector multiplet,

any values of CMB observables \(n_s\) and r are possible by choosing the Jfunction,

a spontaneous SUSY breaking after inflation takes place at arbitrary scale \(\mu \),

there are only a few parameters relevant for inflation and SUSY breaking: the coupling constant g defining the inflaton mass, \(g\sim m_\mathrm{inf.}\), the coupling constant \(\mu \) defining the scale of SUSY breaking, \(\mu \sim m_{3/2}\), and the parameter \(\beta \) in the constant term of the superpotential. Actually, the inflaton mass is constrained by CMB observations as \(m_{ inf.}\sim \mathcal{O}(10^{6})\), while \(\beta \) is fixed by the vacuum solution, so that we have only one free parameter \(\mu \) defining the scale of SUSY breaking in our model (before studying reheating and phenomenology).
3 Gravitino and Polonyi production
In this Section, we consider the gravitino (\(\psi _{\mu }\)) and Polonyi (A) particles production during inflation and after it. We assume that all other (heavy) SUSY particles (not present in our model) have masses larger than those of Polonyi and gravitino (Highscale SUSY), with gravitino as the LSP (the lightest superpartner of known particles) and as the cold Dark Matter (SHGDM).
There are several competitive sources of particle production in our model. First, gravitino and Polonyi particles can be produced via Schwinger’s effect (out of vacuum) sourced by inflation. Since the mass of a Polonyi particle is higher than two gravitino masses (Sect. 2), the former is unstable, and decays into two gravitinos, \(A\rightarrow \psi _{3/2}\psi _{3/2}\). Second, both gravitino and Polonyi can be produced by inflaton decays, during oscillations of the inflaton field around its minimum after inflation. A competition between the gravitino/Polonyi creation during inflation and their production by inflaton decays is known to be very sensitive to the mass hierarchy. It is, therefore, very instructive to study them in our model (Sect. 2) that is minimalistic and highly constrained.
(i) Technical details about the power spectrum and our estimate of the normalized value of \(\mathcal {P}_{A}\) to be of the order \(\exp \left[ O(1)M_{A}/H_{inf}\right] \) are given in Appendix.
(iii) According to Eq. (43), relating Hubble scale, Polonyi mass and the desiderata Polonyi energydensity, there is about 8thordersofmagnitude suppression of the energydensity. According to (i), the normalized power spectrum \(\mathcal {P}_{A}\) cannot provide such suppression with our values for \(M_A\) and \(H_{inf}\). However, it comes from the dilution factor \((\tilde{a})^{3}=(a_{f}/a_{i})^{3}\) in Eq. (43).
The \(a(t_{f})\) refers to a cosmological time \(t_{f}\) close to reheating. The cosmological time \(t_{i}\) when particle production effectively started (and checked by our numerical simulations) is not far from the \(t_{i}\) because the inflaton field has an excursion of merely \(\Delta \Phi =\Phi (t_i)\Phi (t_f)\simeq 20\) that is proportional to \(\Delta t=t_ft_i\). But the relation between a(t) and the cosmological time t is exponential. This is the origin of the very large exponential suppression (by the 8th orders) between \(a_f\) and \(a_i\) despite of the fact that the effective time of particle production is very short. From the physical point of view, particles produced during \(t_{f}\) are diluted with the factor exponentially larger then \(a(t_i)\). Our results are in agreement with the standard expectations [60] that particle production is most efficient towards the end of inflation.
(iv) To get the masses \(M_A\) and \(m_{3/2}\equiv m_{\psi }\) in a different way, we have to add a few more assumptions about details of reheating. Since our SHGDM scenario is based on Starobinsky inflation, all cosmological parameters can be fixed modulo the efoldings number \(N_{e}\) that is between 50 and 60 for compatibility with CMB observations. This also allows us to estimate the error margin for the masses in question at about 20%.
In Fig. 1 we show a numerical simulation of the produced gravitino mass density as a function of the Polonyi mass.^{5}
The spectrum is composed of two contributions: (a) the Polonyi energydensity spectrum produced during inflation, converted into gravitinos after reheating; and (b) the energydensity spectrum of gravitinos produced during inflation. The first contribution largely dominates over the second one. Intriguingly, the Polonyi and gravitino masses inferred from inflation, reheating and leptogenesis bounds are well compatible with the correct amount of CDM.
The situation drastically changes when inflation ends by a first order phase transition when bubble collisions lead to copious production of black holes at \(\beta \sim 0.1\) with the mass (65) [26, 27]. Evaporation of these PBHs leads to a fraction of the total density by the end of preheating of the order \(10^{4}\) in the form of gravitino, and it results in the gravitino dominated stage at \(T \sim 10^{4} T_\mathrm{reh} \sim 10^5\, \mathrm{GeV}\). This huge gravitino overproduction excludes a possibility of the first order phase transition by the end of inflation. This is impossible in the singlefield inflation, also by considering an extra axionlike field and extra moduli decoupled during the slowroll epoch. The inflaton in our model has the characteristic Starobinsky potential that, as is well known, does not lead to any violent phase transition after inflation. The Polonyi field does not alter this situation. However, in a more general case, in which other scalar and pseudoscalar fields enter the slowroll dynamics, these extra fields can have scalar potentials ending in false minima during the reheating. In such case, the tunnelling from the false minima to the true minima can induce bubbles in the early Universe, catalising an efficient formation of PBHs. These issues deserve a more detailed investigation beyond the scope of our paper.
4 Conclusion and outlook
In this paper we studied the gravitino production in the context of StarobinskyPolonyi \(\mathcal {N}=1\) supergravity with the inflaton field belonging to a massive vector multiplet, and the mass hierarchy \(m_\mathrm{inf}>2M_A>4m_{3/2}\) close to the bounds (by the order of magnitude). On the one hand, we found the regions in the parameter space where the gravitino and Polonyi problems are avoided, and super heavy gravitinos can account for the correct amount of Cold Dark Matter (CDM). The dominating channel of gravitino production is due to decays of Polonyi particles, in turn, produced during inflation. On the other hand, we found that direct production of gravitinos during inflation is a subleading process that does not significantly change our estimates.
Intriguingly, our results imply that the parameter spaces of cold DM and inflation can be linked to each other, into a natural unifying picture. This emerging DM picture suggests a phenomenology in ultra high energy cosmic rays. For example, super heavy Polonyi particles can decay into the SM particles as secondaries in topbottom decay processes. Cosmological high energy neutrinos from primary and secondary channels may be detected in IceCube and ANTARES experiments. A numerical investigation of these channels deserves further investigations, beyond the purposes of this paper.
Our scenario offers a link to the realistic Supersymmetric Grand Unified Theories (SGUTs) coupled to supergravity. The superHiggs effect considered in Sect. 2 is associated with the U(1) gaugeinvariant supersymmetric field theory. This U(1) can be naturally embedded into a SGUT with a nonsimple gauge group. The Starobinsky inflationary scale, defined by either \(m_\mathrm{inf}\) or \(H_\mathrm{inf}\) is by three or two orders of magnitude lower, respectively, than the SGUT scale of \(10^{16}\) GeV. The SGUTs with the simple gauge group SU(5), SO(10) or \(E_6\) are well motivated beyond the Standard Model. However, the SGUTs originating from the heterotic string compactifications on CalabiYau spaces usually come with one or more extra U(1) gauge factors as e.g., in the following gauge symmetry breaking patterns: \(E_6\rightarrow SO(10)\times U(1)\), \(SO(10)\rightarrow SU(5)\times U(1)\), the “flipped” \(SU(5)\times U_X(1)\) and so on (see e.g., Ref. [50] for more). Alternatively, SGUTs can be obtained in the low energy limit of intersecting Dbranes in type IIA or IIB closed string theories. Also in this context, extra U(1) factors in the gauge group are unavoidable. For instance, the “flipped” \(SU(5)\times U_X(1)\) from the intersecting Dbrane models was studied in Refs. [51, 52, 53, 54].
We propose to identify one of the extra U(1) gauge (vector) multiplets with the inflaton vector multiplet considered here. This picture would be very appealing because it unifies SGUT, inflation and DM. Moreover, the extra U(1) gauge factor in the SGUT gauge group may also stabilize proton and get rid of monopoles, domain walls and other topological defects [55].
Our scenario also allows us to accommodate a positive cosmological constant, i.e. to include dark energy (see the end of Sect. 2) too. Further physical applications of our supergravity model for inflation and DM to SGUTs and reheating are very sensitive to interactions between the supergravity sector and the SGUT fields. Demanding consistency of the full picture including SGUT, DM and inflation may lead to further constraints. For instance, a matter field must be weakly coupled to inflaton – less then \(10^{3}\) – in order to preserve the almost flat plateau of the inflaton scalar potential. Among the other relevant issues, the Yukawa coupling of inflaton to a RightHanded (RH) neutrino is very much connected to the leptogenesis issue. Inflaton can also decay into RHneutrinos, in turn, decaying into SM visible particles. Of course, these remarks are very generic and have low predictive power, being highly modeldependent. But they motivate us for a possible derivation of our supergravity model from superstrings – see e.g., Ref. [50] for the previous attempts along these lines.
Another opportunity can be based on Refs. [56, 57], by adapting the PatiSalam model to become predictive in the neutrino mass sector and be accountable for leptogenesis in supergravity, as was suggested in Ref. [30]. Then one can generate a highly degenerate mass spectrum of RH neutrinos, close to \(10^{9}\, \mathrm{GeV}\), i.e. four orders smaller than the inflaton mass. In this approach an extra U(1) is necessary for consistency, while it can be related to the Higgs sector in supergravity.
It is worth mentioning that our hidden sector includes only inflaton and Polonyi, and it may have to be extended. The scale invariance of the (singlefield) Starobinsky inflation is already broken by the mixing of inflaton with Polonyi scalar, while its breaking is necessary for a formation of miniPBHs during inflation. The physical consequences of the inflatonPolonyi mixing demand a more detailed investigation, beyond the scope of this paper.
Our scenario can be reconsidered in the general framework of SplitSUSY and Highscale SUSY, by questioning its compatibility with the SM and the known Higgs mass value of 125.5 GeV in particular, e.g., along the lines of the comprehensive study in Ref. [58]. ^{7} Then the “unification help” from SUSY to GUT scenarios could be implemented in our model. In this case, several new decay channels are opened and new parameters enter. In particular, there is a scenario in which the Higgsino at \(1100\, \mathrm{TeV}\) scale is envisaged, with intriguing implications for future colliders. In such case, produced gravitinos can decay into Higgsinos that (in the form of neutralinos) could provide another candidate for Dark Matter. However, there also exist contributions from thermal production that may affect the Higgsino production. Actually, the upper bound on the scale of SplitSUSY, according to Fig. 3 of Ref. [58], is given by \(10^8\) GeV that already excludes compatibility of SplitSUSY with our scenario that requires a higher SUSY. In the case of Highscale SUSY, the upper bound in Fig. 3 of Ref. [58] is given by \(10^{12}\) GeV for a considerable part of the parameter space, so that this bound is again too low for our model. However, it is still possible to go beyond that bound in the case of Highscale SUSY, as is shown in Fig. 5 of Ref. [58], so that our SHGDM scenario is still allowed. A more detailed study of the compatibility of our scenario with the SM deserves further investigation, beyond the scope of this paper.
Footnotes
 1.
 2.
The primes and capital latin subscripts denote the derivatives with respect to the corresponding fields.
 3.
The vertical bars denote the leading field components of the superfields at \(\theta =\bar{\theta }=0\).
 4.
 5.
We chose the lower (on the left) intersection point in Fig. 1 because the higher (on the right) intersection point leads to the heavy Polonyi particle becoming a spectator during inflation and reheating that is inconsistent with our approach.
 6.
Similarly, we can ignore the massless fermion present in the spectrum of our model because the expansion of (conformally flat) FLRW universe does not lead to perturbative production of massless particles.
 7.
Our scenario is apparently incompatible with Low or Intermediatescale SUSY that imply a significantly lower gravitino mass and a substantial inflaton decay rate into gravitino pairs, see e.g., Ref. [59] for details.
Notes
Acknowledgements
S.V.K. was supported by a GrantinAid of the Japanese Society for Promotion of Science (JSPS) under No. 26400252, the Competitiveness Enhancement Program of Tomsk Polytechnic University in Russia, and the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan. S.V.K. would like to thank a FAPESP Grant 2016/013437 for supporting his visit in August 2017 to ICTPSAIFR in Sao Paulo, Brazil, where part of this work was done. The work by M.K. was supported by Russian Science Foundation and fulfilled in the framework of MEPhI Academic Excellence Project (contract 02.a03.21.0005, 27.08.2013). The authors are grateful to Y. Aldabergenov, A. Marciano, K. Hamaguchi, A. Kehagias, K. Kohri, A. A. Starobinsky and T. Terada for discussions and correspondence.
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