Hybrid Inflation, Dark Energy And Dark Matter

It has been suggested that the dark energy density \rho_v ~ 10^{-12} eV^4 in the universe is associated with a metastable (false) vacuum, while the true vacuum has a vanishing cosmological constant. By including supergravity corrections we show how this is naturally realized in realistic supersymmetric hybrid inflation models. With a fundamental supersymmetry breaking scale ~ TeV, the LSP is not a suitable candidate for cold dark matter. We consider axion physics to overcome this and simultaneously provide a resolution of the MSSM \mu problem.

Recent studies of the cosmic microwave background radiation [1], Supernovae 1a [2] and large scale structure [3], taken collectively, present a fairly compelling case for a dark (vacuum) energy density ρ v ∼ 10 −12 eV 4 . Indeed, ρ v is estimated to provide almost 70% of the critical energy density, with matter (including baryons and possibly neutrinos) making up the remaining 30% or so. Understanding the origin of ρ v poses one of the most fundamental theoretical challenges, namely how ρ v ∼ 10 −120 M 4 P happens to be so much smaller than M 4 P , where M P = 2.4 × 10 18 GeV denotes the reduced Planck mass. Another related problem is to understand how ρ v and the matter density ρ m which, in principle, can be expected to scale very differently with the universe expansion, are of comparable magnitudes today.
It is conceivable that ρ v is associated with a false vacuum energy, with the true vacuum possessing a zero cosmological constant [4,5,6]. In this admittedly modest approach to the problem, one tries to identify the origin of ρ v and also ensure that the false vacuum is sufficiently long lived. To this we wish to add in this paper an important new ingredient, namely inflation. This would help us explain how the universe got stuck in the false vacuum in the first place.
The model described below is organized within the framework of supersymmetric hybrid inflation [7] which is associated with the breaking of some gauge symmetry G to H 0 , where H 0 could be the MSSM gauge group or something larger. A remarkable feature of these models is that the symmetry breaking scale of G is estimated from the quadrupole anisotropy, δT T , to be of order 10 16 GeV, the supersymmetric GUT scale, M GU T . A nice, and perhaps the simplest, example of G is the MSSM gauge symmetry supplemented by a gauged U(1) B−L symmetry [8]. To realize (ρ v ) 1/4 ∼ 10 −3 eV we assume, following [6], that the fundamental supersymmetry breaking scale in nature is ∼ TeV [9], so that the gravitino mass m 3/2 ∼ TeV 2 /M P more or less coincides with (ρ v ) 1/4 . Furthermore, following [6], a new (acceleressence) sector containing a chiral superfield χ is introduced, which communicates with other sectors only via gravity. The χ sector will be arranged to yield a potential which has a false (metastable) minimum separated by ρ v from the true minimum with zero cosmological constant.
We will see that during inflation driven from the visible sector, taking supergravity corrections into account, the scalar component of χ acquires a mass of order the Hubble constant H, causing it to be trapped in the false minimum at the origin. If the barrier separating the two minima is sufficiently high, the field stays stuck in the false vacuum even after inflation ends. Because the gravitino is ultralight, the MSSM sector does not provide a suitable cold dark matter (CDM) candidate. Potential CDM candidates include stable relics from the supersymmetry breaking sector [6], or a suitable pseudogoldstone boson [10], and finally axions that we shall shortly discuss.
The model consists of three components namely, the visible sector, a strongly coupled supersymmetry breaking hidden sector, and the acceleressence sector which we will refer to as G, T and χ sectors respectively. The G sector, as we shall see, consists of the MSSM superfields and additional ones used to implement inflation and the axion mechanism. We do not need to specify the details of the supersymmetry breaking sector except to note that it contains a (possibly composite) chiral field T , whose auxiliary component has a vev F T ∼ TeV 2 . The T sector communicates via gauge interactions with the visible sector, so that the supersymmetric partners of the known (SM) particles can acquire masses in the range of M Z to TeV. The χ sector, following [6], allows us to relate the observed vacuum energy density to a false vacuum energy density. As stated before, this sector consists of a chiral superfield χ which communicates with the two sectors G and T only via gravity. With the superpotential W acc = σ 3 and including soft supersymmetry breaking terms, the χ potential takes the form where σ, A can be made real and positive by proper phase rotations of the fields. Here, both A and m are of order 10 −3 eV, and V 1 is adjusted to make the total energy density vanish at the absolute minimum which lies at χ = 3A+ also has a local (false) minimum at χ = 0 which is separated from the true minimum by ρ v .
It is possible to make the lifetime of this metastable state (much) greater than the age of the universe. The dark energy conundrum could be explained if the field χ is trapped at the origin rather than in the true minimum. We will show that supersymmetric hybrid inflation provides a natural mechanism to drive the χ field to the false minimum thereby realizing the acceleressence scenario.
The G sector contains the superpotential responsible for the simplest model of hybrid inflation [7,11] where φ,φ denote a conjugate pair of non-G singlet superfields, S is a gauge singlet superfield and a U(1) R symmetry is imposed under which S → e iα S, φφ → φφ, and W inf → e iα W inf .
The parameters κ and M can be made real and positive by field redefinitions. In the unbroken supersymmetric limit, vanishing of the F -and D-terms imply that the supersymmetric To realize inflation, S is displaced from its present day location to values that exceed M. The appearance of a vacuum energy density of order κ 2 M 4 induces radiative corrections to the tree level potential, with the result [7,11]. Thus, M is of order 10 16 GeV, the supersymmetric GUT scale [7]. The scalar spectral index in this class of models is estimated to be n s = 0.99 ± 0.01 [7,11].
Let us now include supergravity corrections that link the inflaton and the χ sector. The supergravity corrections coming from supersymmetry breaking in the strongly-coupled sector are small during inflation and would only play a significant role near the end of inflation, by which time the χ field is trapped in the false minimum. Assuming minimal supergravity, the scalar potential corresponding to a superpotential W and Kähler potential K is given by [12] where ij * is the inverse of the Kähler metric and the indices i, j run through all chiral fields.
We can parametrize, without explicit details of the supersymmetry breaking sector, the supergravity mediated supersymmetry breaking effects on the visible and χ sector by explicitly including a constant term W 0 in the superpotential. The presence of W 0 ensures the cancellation of the cosmological constant so that the vacuum energy at the global minimum is zero. The size of supersymmetry breaking in the T sector implies that to leading order in 1/M P (provided there are no Planckian vevs).
With the minimal Kähler potential K 1 = SS † + φφ † +φφ † from the inflationary sector and K 2 = χχ † from the acceleressence sector, the scalar potential is given by (we employ the same notation for superfields and their corresponding scalar components) where W = W inf + W acc + W M SSM + W 0 , and the ellipsis represent contributions from the MSSM fields. With |φ| = |φ| along the D-flat direction of the scalar potential, the tadpole term −2κM 2 m 3/2 S + h.c. induces a shift in the vevs [13]: The corresponding F -terms are The supergravity corrections play an important role during inflation. With φ =φ = 0 and |S| > M, the scalar potential is given by where only the dominant lower order terms are displayed, and the higher order terms in χ can be safely ignored for our discussion. Note that during inflation, the χ field acquires a positive mass squared larger than M P H, is suppressed compared to H, and therefore χ rapidly settles at the origin during inflation.
With the end of inflation, the effective potential for χ is given by Eq.
from which the term Aχ 3 in Eq.(2) can be generated, where A ∼ σ F T M P ∼ σ10 −3 eV. As for the quartic term, it just comes from the usual F -term squared, i.e. W i K −1 ij * W * j * . Thus after inflation, the χ sector scalar potential takes the form which is essentially equivalent to Eq.(2).
The next question we would like to address is that of dark matter. The superlight gravitino with mass ∼ 10 −3 eV is not a suitable dark matter candidate which forces us to look for alternative CDM candidates. One plausible candidate would be the lightest field in the supersymmetry breaking hidden sector as one would expect it to have quantum numbers not shared by fields in the other sectors and hence, be stable [6]. Another plausible candidate could be a pseudogoldstone boson such as the majoron, associated with a spontaneously broken global U(1) B−L symmetry [10]. We will focus here on axion CDM introducing a P Q symmetry U(1) P Q [14], since the associated physics can also be exploited to resolve the MSSM µ problem [15,16]. Implementation of this mechanism turns out to be not entirely straightforward.
The axion mechanism is easily implemented in models in which the gravitino mass, With m 3/2 ∼ 10 −3 eV in our present case, the above scenario cannot be realized in the simple way outlined above. Furthermore, superpotential terms such as N 2N 2 /M P give rise to F -term contributions ≫ TeV 2 , which can be disastrous for the χ sector, through nonminimal Kähler terms such as d 4 θN † Nχ † χ/M 2 P . We will attempt to implement the axion mechanism with a single G-singlet superfield N, by retaining only the superpotential term and letting m N , the coefficient of the mass term associated with the real component of N, also called the saxion, be a free parameter to be determined from the consistency requirements.
Namely, that the µ problem is resolved with a N vev of order 10 11 GeV in order to generate axion dark matter [15], and that there are no cosmological problems associated with the N field. How m N acquires the desired mass scale requires a more complete analysis of supersymmetry breaking which is beyond the scope of this paper. The cosmological evolution of the saxion field turns out to be somewhat non-trivial. The R and P Q charges of the various superfields are listed in Table I. The potential responsible for breaking the axion symmetry is taken to be where a negative mass squared term for the N field may, for instance, be induced via radiative corrections [16]. The second term follows from the superpotential in Eq.(11) after electroweak symmetry breaking. A constant term V 2 has been included to set V P Q to zero at the true minimum. Requiring f a = | N | ∼ 10 11 GeV [17] yields m N ∼ λ × 10 −5 GeV ∼ 10 −7 GeV, with λ ∼ 10 −2 so that the µ term ∼ 100 GeV. The saxion mass then is also of this magnitude. Since we have a very light and consequently a long lived (essentially stable) scalar we should ensure that no cosmological difficulties arise as a consequence. Note that in Eq.(12) we could introduce an additional quartic term γ|N| 4 , with γ ∼ 10 −38 . This latter coupling, whose origin like that of m N we will not discuss here, will be useful in cosmology.
The values for m N and γ proposed here suggest the presence of heavy fields that link the N superfields with the supersymmetry breaking sector.
In contrast to the χ field that remains trapped at the origin both during and after inflation, the saxion field must reach its minimum to implement proper breaking of the U(1) axion symmetry. In principle, it could stay at the origin during inflation. However, axion models are often plagued by the domain wall problem [18] and we prefer to circumvent this by letting N roll away from the origin during inflation. This can be accomplished by introducing suitable non-minimal Kähler potential terms. Consider, for instance, the Kähler potential so that, during inflation, the relevant potential involving the N field is given by where β = (κ 1 − 1) > 0. For β 10 −1 , the field N is rapidly driven to √ βM P . Note that the induced mass-squared term for S is suppressed relative to H 2 by a factor of β 3 , so that the inflationary scenario described earlier remains intact. As the Hubble induced mass drops below m N after reheat, which happens at a temperature of order 10 5 GeV, the N field moves, because of the quartic term γ|N| 4 , to a new minimum at around 10 13 GeV. A further drop in temperature to 10 2 GeV leads to the appearance of electroweak vevs, in which case the potential in Eq.(12) effectively takes over, and the N field reaches its true minimum value of around 10 11 GeV. This creates a cosmological problem since the energy stored in the N field (∼ λ 2 × 10 12 GeV 4 ) is comparable to the radiation energy density (∼ 10 8 GeV 4 ) and, with N having a lifetime that far exceeds the age of the universe, N would become the dominant component in the universe.
One mechanism for overcoming this is to invoke an epoch of thermal inflation [19]. We will not provide any details here since a similar problem was encountered in [20] where the decay of a heavy particle was employed to dilute sufficiently the saxion energy density. Of course, the release of entropy also dilutes any pre-existing baryon asymmetry and a mechanism should be found to resolve this problem [21]. Finally, let us note that in the presence of axions, the gravitino is replaced by the axino, with mass ∼ 10 −7 eV (for λ ∼ 10 −2 ), as the LSP. Its contribution to the energy density of the universe, like the gravitino, is negligible.
Cold dark matter comes from axions and possibly also the saxion.
Some remarks about the R-axion are in order here.
where the R-axion decay constant f R ∼ r i r j v i v * j K ij * , and r i and v i are the R charges and vevs of the fields respectively. With the large R-singlet vev of | φ | = | φ | ≃ M and hidden sector fields (generically labeled T ) with vevs T ∼ F T O(TeV), we expect that Substituting Eqs. (16) and (17) in Eq. (15), we obtain an R-axion mass of ∼ 10 GeV which is consistent with the astrophysical constraints.
In conclusion, we have explored a scenario in which supersymmetric hybrid inflation could play an essential role in understanding the origin of dark energy. Even though the true vacuum has a zero cosmological constant (how this comes about is beyond the scope of this paper), supergravity corrections during inflation can trap acceleressence field at the origin, which happens to be a local (false) minimum. The energy density scale separating the true vacuum from the false one is arranged to be of order