Catastrophic Production of Slow Gravitinos
CCatastrophic Production of Slow Gravitinos
Edward W. Kolb , Andrew J. Long , and Evan McDonough Kavli Institute for Cosmological Physics and Enrico Fermi InstituteThe University of Chicago, Chicago, IL 60637 Department of Physics and Astronomy, Rice University, Houston, TX 77005February 23, 2021
Abstract
We study gravitational particle production of the massive spin- / Rarita-Schwinger field, and its closerelative, the gravitino, in FRW cosmological spacetimes. For masses lighter than the value of the Hubbleexpansion rate after inflation, m / (cid:46) H , we find catastrophic gravitational particle production, whereinthe number of gravitationally produced particles is divergent, caused by a transient vanishing of the helicity-1/2 gravitino sound speed. In contrast with the conventional gravitino problem, the spectrum of producedparticles is dominated by those with momentum at the UV cutoff. This suggests a breakdown of effective fieldtheory, which might be cured by new degrees of freedom that emerge in the UV. We study the UV completionof the Rarita-Schwinger field, namely N = 1 , d = 4 , supergravity. We reproduce known results for modelswith a single superfield and models with an arbitrary number of chiral superfields, find a simple geometricexpression for the sound speed in the latter case, and extend this to include nilpotent constrained superfieldsand orthogonal constrained superfields. We find supergravity models where the catastrophe is cured andmodels where it persists. Insofar as quantizing the gravitino is tantamount to quantizing gravity, as is the casein any UV completion of supergravity, the models exhibiting catastrophic production are prime examples of 4-dimensional effective field theories that become inconsistent when gravity is quantized, suggesting a possiblelink to the Swampland program. We propose the Gravitino Swampland Conjecture , which is consistent withand indeed follows from the KKLT and Large Volume scenarios for moduli stabilization in string theory. a r X i v : . [ h e p - t h ] F e b ontents Introduction
Gravitational particle production (GPP) is ubiquitous in quantum field theories in curved spacetime. In a cosmo-logical context GPP was first noted by Erwin Schrodinger in 1939 [1]. In a modern context, gravitational particleproduction has risen to prominence as a mechanism to produced the observed dark matter density, beginning withearly work on heavy scalar fields [2–8], and later extended to spin-1/2 [9–11], spin-1 [9–12], spin-3/2 [13–17],spin-2 [18], and spin- s ( s > ) fields [19]. A gravitational origin of dark matter is particularly compelling giventhe dearth of evidence for the interactions between dark matter and the Standard Model of particle physics uponwhich conventional WIMP models are premised.More generally, early universe inflationary cosmology has risen to the fore of particle physics, in par-ticular, under the banner of Cosmological Collider Physics [20–22], and the Cosmological Bootstrap Pro-gram [23–25], with the hope that one may use the cosmic microwave background non-Gaussianity as a par-ticle detector [21, 26, 27, 27, 28]. Complementary to this, reheating as Cosmological Heavy Ion Collider [29],namely, reheating as a playground in which to study thermalization in quantum field theory, has seen a recentresurgence [29–33], as has reheating as a testing ground for Higgs physics [34]. With the rise of cosmology as aparticle physics laboratory, it is natural to continue the study of gravitational particle production and of inflationas a particle factory; the B-factory to the Cosmological Collider’s LHC. Many very recent works have proceededalong these lines.Parallel to the developments in particle cosmology have been many developments in string theory andsupergravity. Gravitational production of spin-3/2 particles was examined in detail two decades ago by severalgroups [13–17]. Since then, relatively little attention has been paid to this question (with exception of [35–38]).Meanwhile, string theory saw the advent of stabilized flux compactifications [39–41] and the string landscape[42, 43]. These string theory setups have very recently made contact with models of supergravity, with the anti-D3 brane of KKLT encapsulated by a nilpotent constrained superfield [44–48]. With all these developments, itis well past due to re-examine the gravitational production of gravitinos in the early universe.In this work we study in detail the dynamics of massive spin-3/2 fields in cosmological spacetimes. Themassive spin-3/2 carries within it a helicity-1/2 state and a helicity-3/2 state, which enjoy decoupled equationsof motion. The former has a sound speed v s which in general differs from , and can vanish if m / (cid:46) H (here m / is the mass of the spin-3/2 field). We compute the gravitational particle production of these fields in abackground cosmology generated by an oscillating scalar field, and find that for m / (cid:46) H there is a productionof particles with arbitrarily high- k . That the spectrum is dominated by modes in the deep UV (including at thecutoff of the effective field theory) is in stark contrast with the conventional gravitino problem [49, 50]. We thusrefer to this phenomenon as “catastrophic particle production.”One might hope that supergravity cures this instability. We demonstrate this is not always the case:there are a plethora of supergravity models that exhibit this instability and a plethora that do not. Indeed, thecatastrophic production we study in detail here was first observed in the context of a supergravity model byHasegawa et al. [37]. We demonstrate that a generic result in supergravity is a sound speed v s that differsfrom ; this was shown long ago [14, 17], and has been referred to in the literature as a “slow gravitino” [51].Whether the sound speed ever vanishes, v s = 0 , is dependent on details of the model, such as the precise3orm of the superpotential and K¨ahler potential. We derive a simple expression for the gravitino sound speed insupergravity models with an arbitrary number of chiral superfields, and find a simple geometric interpretationof slow gravitinos in terms of an angle in field space. We find for the sound speed of the helicity-1/2 state ofgravitinos v s = 1 − | (cid:126) ˙Φ | | (cid:126)F | ( | (cid:126) ˙Φ | + | (cid:126)F | ) (cid:0) − cos ( θ ) (cid:1) , (1.1)where θ is the angle between the field-space vector of the cosmological evolution, (cid:126) ˙Φ , and the F -term vector (cid:126)F . This matches Bastero-Gil and Mazumdar [17], and provides a simple geometric interpretation: The soundspeed vanishes whenever | (cid:126) ˙Φ | = | (cid:126)F | and cos( θ ) = 0 . We show this applies also in models of constrainednilpotent superfields, and with a small modification, to models of orthogonal constrained superfields. In allthese cases, we find models and parameter choices which exhibit catastrophic particle production, and modelswhich do not. We give an example of a toy model which interpolates between these two regimes. Note that theadditional fermionic degrees of freedom in supergravity, which allow the helicity-1/2 gravitino to decay, and theidentification of the gravitino as a linear combination of spin-1/2 fermions that may change with time, does notameliorate the breakdown of effective field theory that defines catastrophic production. This is in contrast withthe conventional gravitino problem, where the EFT is valid at all times (unlike the catastrophe), and where thechanging identity of the gravitino can significantly amelioriate the problem [52, 53] (see also [54] in the contextof orthogonal constrained superfield supergravity models).In summary, we find that quantization of the gravitino leads to an instability in certain quantum fieldtheories, such as the Rarita-Schwinger model, which, at the classical level, appear to be perfectly well behaved.In the context of supergravity, at energy scales above the supersymmetry breaking scale there is no distinctionbetween a graviton and a gravitino, the two being related by a linear supersymmetry transformation. Thus, theinstabilities found here are no less severe than any that emerge from quantizing gravity. The study of apparentlyconsistent field theories which become inconsistent when embedded in quantum gravity, and hence cannot becompleted into quantum gravity in the ultraviolet, is known as the Swampland Program [55] (for a review,see [56, 57]). With this in mind, we propose the Gravitino Swampland Conjecture . The conjecture is given inSec. 6 and elaborated upon in a separate article [58].The structure of this paper is as follows: In Sec. 2 we review the Rarita-Schwinger theory of a massivespin-3/2 field in flat space. In Sec. 3 we generalize this to an FRW space, and derive the equations of motion for amassive spin-3/2 field with a general time-dependent mass. In Sec. 4 we numerically compute the gravitationalparticle production of a constant-mass spin-3/2 field, and find catastrophic particle production of helicity-1/2particles in the case that m / (cid:46) H e , where H e is the Hubble parameter at the end of inflation. In Sec. 5 weconsider a variety of supergravity models, and observe cases where there are, and cases where there are not,catastrophic particle production. Motivated by this, in Sec. 6 we propose the Gravitino Swampland Conjecture.We conclude in Sec. 7 with a discussion of directions for future work. There are three appendices. Appendix Ais a brief discussion of frame fields used to promote the Minkowski-space field theory to curved space. AppendixB discusses how to extract the helicity components from the vector-spinor ψ µ . A final Appendix C reviews thesalient features of supergravity we employ to study GPP. Conventions.
We adopt the Landau-Lifshitz timelike conventions [59] for the signature of the metric4 sign[ η ] = +1 where η µν is the Minkowski metric), the Riemann curvature tensor ( R ρσµν = + ∂ µ Γ ρνσ · · · ),and the sign of the Einstein tensor G µν = +8 πG N T µν . To translate these conventions to other conventions, seethe introductory material in Misner, Thorne, and Wheeler [60]. Our sign conventions correspond to ( − , + , +) in their table. Two useful references for this section are Giudice, Riotto, and Tkachev [15] and Kallosh, Kofman, Linde, andVan Proeyen [13]. In the discussion we hew closer to the Giudice et al. treatment.The relativistic quantum theory of spin-3/2 particles was constructed in 1941 by Rarita and Schwinger[61]. The Rarita-Schwinger field ψ µ ( x ) is a “vector-spinor” resulting from the direct product of the ( , ) vectorrepresentation of the Lorentz group and the ( , ⊕ (0 , ) representation of the Lorentz group for a Dirac spinor,resulting in the reducible representation ( , ) ⊗ (cid:2) ( , ⊕ (0 , ) (cid:3) = ( , ⊕ ( , ⊕ (1 , ) ⊕ (0 , ) . (2.1)The ( , ⊕ (1 , ) reducible representation, corresponding to a spin-3/2 Rarita-Schwinger field, can be furtherdecomposed into irreducible representations corresponding to the ± and ± helicity states. One may addition-ally impose a reality, i.e., Majorana, condition on ψ µ ( x ) . This choice finds a natural home in supergravity, sincethe number of degrees of freedom of a massless Majorana ψ µ then matches that of the metric. In what followswe treat ψ µ ( x ) as Dirac and impose a Majorana condition only when explicitly stated.The properties of the Rarita-Schwinger field of mass m in Minkowski space are derived from the action S [ ψ µ , ¯ ψ µ ] = (cid:90) d x (cid:20) i ψ µ γ µρν ( ∂ ρ ψ ν ) − i (cid:0) ∂ ρ ¯ ψ µ (cid:1) γ µρν ψ ν + 2 m ¯ ψ µ Σ µν ψ ν (cid:21) (2.2)where Σ µν ≡ γ µν = γ [ µ γ ν ] = ( γ µ γ ν − γ ν γ µ ) = [ γ µ , γ ν ] and γ µρν = γ [ µ γ ρ γ ν ] = ( γ µ γ ρ γ ν − γ ν γ ρ γ µ ) .There are several equivalent forms for the action. The first two are S [ ψ µ , ¯ ψ µ ] = (cid:90) d x (cid:2) i ¯ ψ µ γ µρν ( ∂ ρ ψ ν ) + 2 m ¯ ψ µ Σ µν ψ ν (cid:3) = (cid:90) d x (cid:2) − i ( ∂ ρ ¯ ψ µ ) γ µρν ψ ν + 2 m ¯ ψ µ Σ µν ψ ν (cid:3) , (2.3)where in both expressions we have discarded a surface term resulting from integration by parts which will notcontribute to the equation of motion. In a manner reminiscent of the case for vectors, the µ = 0 component of ψ µ is not dynamical., i.e., ∂ ψ does not appear in the action. The third form of the action makes use of theidentity γ µρν = − i(cid:15) µρνσ γ γ σ , (2.4)where (cid:15) µρνσ is the Levi-Civita symbol with convention (cid:15) = +1 . Using this, the action can be written in themodern form S [ ψ µ , ¯ ψ µ ] = (cid:90) d x (cid:2) (cid:15) µσρν ¯ ψ µ γ γ σ ( ∂ ρ ψ ν ) + 2 m ¯ ψ µ Σ µν ψ ν (cid:3) (2.5a) = (cid:90) d x (cid:2) − (cid:15) µσρν ( ∂ ρ ¯ ψ µ ) γ γ σ ψ ν + 2 m ¯ ψ µ Σ µν ψ ν (cid:3) . (2.5b)5riting the action in this form makes transparent that ψ is not a dynamical degree of freedom since the termproportional to ∂ ψ vanishes by action of the Levi-Civita symbol.Variation of the action (2.3) with respect to ψ µ and ¯ ψ µ yield the field equations iγ µρν ( ∂ ρ ψ ν ) + 2 m Σ µν ψ ν = (cid:15) µσρν γ γ σ ( ∂ ρ ψ ν ) + 2 m Σ µν ψ ν = 0 (2.6a) − i ( ∂ ρ ¯ ψ µ ) γ µρν + 2 m ¯ ψ µ Σ µν = − (cid:15) µσρν ( ∂ ρ ¯ ψ µ ) γ γ σ + 2 m ¯ ψ µ Σ µν = 0 . (2.6b)The next step is to find the two constraint equations. They are found from operations on Eq. (2.6a): γ µ · [ iγ µρν ( ∂ ρ ψ ν ) + 2 m Σ µν ψ ν ] = 2 i ( γ µ γ ν ∂ µ ψ ν − η µν ∂ µ ψ ν ) + 3 mγ µ ψ µ = 0 (2.7a) ∂ µ · [ iγ µρν ( ∂ ρ ψ ν ) + 2 m Σ µν ψ ν ] = m ( γ µ γ ν ∂ µ ψ ν − η µν ∂ µ ψ ν ) = 0 . (2.7b)For m (cid:54) = 0 the constrain equations can be expressed as γ µ ψ µ = 0 and η µν ∂ µ ψ ν = 0 . (2.8)Enforcing these two constraints in Eq. (2.6a), after tedious manipulation of γ -matrices, leads to iγ µ ∂ µ ψ ν − mψ ν = 0 . (2.9)Thus, in Minkowski space the field equation for the spin- / field ψ ν is simply four copies of the Dirac equationlabeled by the spacetime index ν . The difference is that the original four components of ψ ν are reduced to twophysical components because of the constraint equations (2.8).We will remove the nondynamical degrees of freedom after an expansion of the field into Fourier modes: ψ µ ( t, x ) = (cid:90) d k (2 π ) ψ µ, k ( t ) e i k · x . (2.10)Consider a single Fourier mode for which we choose k along the third axis; k = (0 , , k z ) , i.e., k z ≡ k . FromEqs. (2.9) and (2.10), the equation of motion for that Fourier mode is iγ ∂ ψ µ, k − γ k z ψ µ, k − mψ µ, k = 0 . (2.11)The first constraint from Eq. (2.8), γ µ ψ µ = 0 , can be used to express ψ , k = − γ γ i ψ i, k . Using ∂ ψ , k from the -th component of the equation of motion, we can solve the second constraint equation for ψ , k . The constraintequations imply for the Fourier modes γ ψ , k = − γ i ψ i, k ψ , k = (cid:18) k z m + γ (cid:19) (cid:0) γ ψ , k + γ ψ , k (cid:1) . (2.12)The four spinor fields ψ . . . ψ separately obey the Dirac equation with canonically normalized kineticterms. However, they are not independent as there are two constraint equations. In Appendix B we discuss theconstruction of two orthogonal combinations of ψ . . . ψ that are helicity eigenstates, have correctly normalized6inetic terms, and satisfy the Dirac equation. The two states are (B.5)–(B.6): ψ / , k = 1 √ (cid:0) ψ , k + γ γ ψ , k (cid:1) ψ / , k = √ (cid:0) ψ , k − γ γ ψ , k (cid:1) . (2.13)The fields ψ / , k and ψ / , k are constructed from helicity projections, have canonical kinetic terms, are orthogo-nal, and obey the Dirac equation (see Appendix B).Using Eq. (2.13) and the constraint equations we can also express the four fields ψ µ, k in terms of the twoindependent fields ψ / , k and ψ / , k : γ ψ , k = − √ k z m γ γ ψ / , k γ ψ , k = 1 √ γ ψ / , k + 1 √ γ ψ / , k γ ψ , k = 1 √ γ ψ / , k − √ γ ψ / , k γ ψ , k = 2 √ k z m γ γ ψ / , k − √ γ ψ / , k . (2.14)The equations of motion for ψ / , k and ψ / , k can be found from the equations of motion for ψ µ, k : (cid:0) iγ ∂ − k z γ − m (cid:1) ψ / , k = 0 (cid:0) iγ ∂ − k z γ − m (cid:1) ψ / , k = 0 . (2.15)So the ψ / , k and ψ / , k are, as advertised, independent states which satisfy the same equation of motion as anordinary Dirac field, and the quantization can proceed as in the Dirac fermion case. In deriving Eq. (2.15) wehave assumed k = k z ˆ z . The obvious rotationally-invariant mode equation would be to replace k z γ in Eq. (2.15)by k · γ . The quantum field theory in curved spacetime is discussed in many excellent reviews and textbooks; see forinstance Freedman and Van Proeyen [62]. As a specific application, the phenomenon of gravitational particleproduction applied to a massive gravitino during inflation is discussed in early work [13, 15, 16], and see alsomore recent articles [35–37].
In a curved spacetime the Rarita-Schwinger action from Eq. (2.2) generalizes to S [Ψ µ ( x ) , ¯Ψ µ ( x ) , e βν ( x )] = (cid:90) d x √− g (cid:20) i µ ¯ γ µρν ( ∇ ρ Ψ ν ) − i ∇ ρ ¯Ψ µ )¯ γ µρν Ψ ν + 2 m ¯Ψ µ ¯Σ µν Ψ ν (cid:21) . (3.1) The derivation in Appendix B departs from Giudice et al. [15], but the final results are the same. ¯Ψ µ and Ψ µ yields the field equations i ¯ γ µρσ ∇ ρ Ψ σ + 2 m ¯Σ µσ Ψ σ = 0 and i ∇ ρ ¯Ψ µ ¯ γ µρσ − m ¯Ψ µ ¯Σ µσ = 0 . (3.2)Here we work in the frame-field formalism; see Appendix A for a brief review. The frame field is denotedby e αµ ( x ) where the first index is raised/lowered by the Minkowski metric η αβ and the second index by thespacetime metric g µν . Gamma matrices become local objects, ¯ γ µ ( x ) = e µα ( x ) γ α , while γ α obeys the Cliffordalgebra in Minkowski space. The Dirac conjugate of Ψ µ is given by ¯Ψ µ = Ψ † µ γ . The covariant derivativesare given by ∇ ρ Ψ ν = ∂ ρ Ψ ν + Γ ρ Ψ ν and ∇ ρ ¯Ψ µ = ∂ ρ ¯Ψ µ − ¯Ψ µ Γ ρ where Γ µ ( x ) is the spin connection, and seeAppendix A for an explicit expression. We assume that Ψ µ is minimally coupled to gravity, but we suppose thatit may also interact with other fields, and in this background the mass parameter may develop a time-dependence m = m ( η ) . For example, in Sec. 5 we will show that this can happen when the spin-3/2 field is embedded intotheories of supergravity as the gravitino.Specifying to the FRW background, the frame-field components and spin connection are given by e = a ( η ) , e ij = a ( η ) δ ij , e i = e i = 0 , Γ µ = a ( η ) H ( η ) η µν Σ ν , (3.3)where H ≡ ∂ η a/a is the Hubble expansion rate, which satisfies the Friedmann equation H = (cid:112) (8 πG/ ρ forenergy density ρ . The action (3.1) becomes S [ ψ µ ( η, x ) , ¯ ψ µ ( η, x )] = (cid:90) ∞−∞ dη (cid:90) d x (cid:20) i ¯ ψ µ γ µρν ∂ ρ ψ ν + 2 am ¯ ψ µ Σ µν ψ ν − iaH ¯ ψ µ (cid:0) γ µ η ν − γ ν η µ (cid:1) ψ ν (cid:21) (3.4)where all indices are raised/lowered with the Minkowski metric η µν . To ensure that that field’s kinetic term iscanonically normalized we have introduced ψ µ ( η, x ) according to ψ µ ( η, x ) = a / ( η )Ψ µ ( η, x ) . (3.5)This transformation partially absorbs the factor of √− g = a ; in the kinetic terms the remaining a is canceledby the inverse frame field e µα ∝ a − , and in the mass term the factor of a remains. Of course if one sets H = 0 and a = 1 , the Minkowski result (2.3) is recovered. In the FRW background the field equations (3.2) become iγ µρν ∂ ρ ψ ν − iaH (cid:0) γ µ η ν − γ ν η µ (cid:1) ψ ν + 2 am Σ µν ψ ν = 0 (3.6a) i∂ ρ ¯ ψ ν γ µρν + iaH ¯ ψ ν (cid:0) γ µ η ν − γ ν η µ (cid:1) − am ¯ ψ ν Σ µν = 0 . (3.6b)There are two ways in which this equation differs from the Rarita-Schwinger equation in flat space: the mass isreplaced by m → a ( η ) m and there is an additional Hubble-dependent term. This is unlike the case of a Diracspinor field for which the mass was the only source of conformal symmetry breaking. The field ψ µ can be viewed as a set of four Dirac or Majorana spinors labeled by µ = 0 , , , . From a naiveaccounting, it looks like this field describes degrees of freedom (or if the Majorana condition is imposed).However, the field equations (3.6) also contain constraints, which cut the degrees of freedom down by half.8t’s useful to first write the field equations (3.6) as R µ = 0 where R µ = R µν ψ ν (3.7a)and where R µν = iγ µρν ∂ ρ − iaH (cid:0) γ µ η ν − γ ν η µ (cid:1) + 2 am Σ µν (3.7b)is a linear differential operator. Written out explicitly, R = 2 iγ Σ ij ∂ i ψ j + iaHγ i ψ i + am γ γ i ψ i (3.7c) R i = iγ iρν ∂ ρ ψ ν − (cid:0) iaH + amγ (cid:1) γ i ψ + 2 am Σ ij ψ j , (3.7d)one can see that R µ = 0 is equivalent to Eq. (3.6). The first constrain equation is γ µ R µ = 0 = 2 i (cid:2) ( γ ρ ∂ ρ )( γ σ ψ σ ) − ( ∂ µ ψ µ ) (cid:3) − iaH ψ + (cid:0) iaHγ + 3 am (cid:1) ( γ σ ψ σ ) . (3.8)Note that γ µ R µ does not contain a term like ∂ ψ , which cancels between the two terms in square brackets. Toextract the second constraint, consider the differential operator D µ = ∂ µ + Γ µ + i mγ µ = ∂ µ + aH η µν Σ ν + i amη µν γ ν . (3.9)Applying this operator to the field equation gives D µ R µ = 0 = R − i a (cid:20) R + H − m (cid:21) ( γ i ψ i ) + i a (cid:2) m + H ) (cid:3) ( γ ψ ) + a ( ∂ η m ) γ ( γ i ψ i ) . (3.10)In this expression, R = − (6 ∂ η a ) /a is the Ricci scalar curvature, and the Friedmann equation enforces R = − πG ( ρ − p ) where p is the cosmological pressure. The last term in Eq. (3.10) arises in models that allowfor a time-dependent mass, and in Sec. 5 we will encounter nonzero ∂ η m in the context of supergravity where aMajorana ψ µ is the gravitino. Solving Eq. (3.10) gives γ ψ ( η, x ) = C c ( η ) γ i ψ i ( η, x ) (3.11)where the time-dependent coefficient, C c ( η ) , is given by (note appearance of ∂ η m ) C c = R + H − m H + m ) + 2 iγ a ∂ η m a ( H + m ) . (3.12)In the Minkowski limit with static mass we have C c = − . Since the field equation (3.6) is left invariant under spatial translations, x (cid:48) = x + (cid:15) , we anticipate that itssolutions can be labeled by a 3-vector k and will be proportional e i k · x . So we decompose the field Ψ µ ( x , x ) Ψ µ, k ( x ) labeled by -vectors k , which are called comoving wavevectors. We can writethe Fourier transform of the field Ψ µ and the comoving field ψ µ as Ψ µ ( x , x ) = (cid:90) d k (2 π ) Ψ µ, k ( x ) e i k · x and ψ µ ( η, x ) = (cid:90) d k (2 π ) ψ µ, k ( η ) e i k · x (3.13)where η is the conformal time.After using Eq. (3.13), the field equation (3.7) becomes a set of mode equations R k = 0 and R i k = 0 (3.14a)where R k = − γ ( γ · k )( γ · ψ k ) − γ ( k · ψ k ) + (cid:0) iaH + amγ (cid:1) ( γ · ψ k ) (3.14b) R i k = − iγ (cid:0) γ i γ j − η ij (cid:1) ∂ ψ j, k + (cid:20) (cid:16) γ ijl − γ lji (cid:17) k j + am (cid:16) γ i γ l − η il (cid:17)(cid:21) ψ l, k (3.14c) − (cid:2) k i + γ i (cid:0) k · γ + iaHγ − am (cid:1)(cid:3) γ ψ , k . The constraints in Eqs. (3.8) and (3.11) have Fourier representations of ( k · ψ k ) = (cid:2) − ( γ · k ) + iaHγ + am (cid:3) ( γ · ψ k ) (3.15) γ ψ , k = C c ( γ · ψ k ) (3.16)where we’ve introduced ψ k through ψ µ, k = { ψ , k , ψ k } . Recall that the time-dependent coefficient C c ( η ) isgiven by Eq. (3.12).The two constraints in Eqs. (3.15) and (3.16) reduce the degrees of freedom by half. Now we’d like todecompose the four spinor fields, ψ µ, k ( η ) into their constituent degrees of freedom. We will denote these by ψ / , k ( η ) and ψ / , k ( η ) since they correspond to the helicity-1/2 and helicity-3/2 states. This decomposition iseasier to identify by work in the frame where k = (0 , , k z ) . In this frame, the constraint from Eq. (3.15) gives ψ , k = (cid:0) C d + γ (cid:1) (cid:0) γ ψ , k + γ ψ , k (cid:1) , (3.17)where we have defined C d = k z a ( H + m ) (cid:0) iaHγ + am (cid:1) . (3.18)In the Minkowski limit C d = k z /m . We decompose the fields ψ µ, k onto ψ / , k and ψ / , k by writing [cf. For the sake of clarity and completeness, we note the following. The four-vectors k µ and x µ k to be a space-like 4-vector, so k µ = ( ω, k ) , and x µ = ( x , x ) . This defines k and x . The 4-vector product k · x = η µν k µ x ν = η k x + η ij k i x j = ωx − k · x . We will take k in the -direction ( z -direction), so k · x = k z x . The Dirac matrices γ µ are also contravariant vectors: γ µ = ( γ , γ ) , so k · γ = η µν k µ γ ν = η k γ + η ij k i γ j = η k γ − k · γ . When we take k inthe z -direction, k · γ = ωγ − k z γ . Now ψ µ = ( ψ , ψ ) is naturally a covariant 4-vector. So γ · ψ = γ ψ + γ i ψ i = γ ψ + γ · ψ .We also note a subtle difference: k · x = k z x and k · γ = k z γ , while γ · ψ = γ i ψ i . C d = k k /m, C c = − ) γ ψ , k = 2 √ C c γ C d γ ψ / , k (3.19a) γ ψ , k = 1 √ γ ψ / , k + 1 √ γ ψ / , k (3.19b) γ ψ , k = 1 √ γ ψ / , k − √ γ ψ / , k (3.19c) γ ψ , k = 2 √ C d γ γ ψ / , k − √ γ ψ / , k . (3.19d)Equation (3.19) implies [cf. Eq. (2.13)] ψ / , k = (cid:114) (cid:0) ψ , k − γ γ ψ , k (cid:1) ψ / , k = (cid:114) (cid:0) ψ , k + γ γ ψ , k (cid:1) . (3.20)After this decomposition, the mode equations (3.14) become (cid:2) iγ ∂ η − k z γ − am (cid:3) ψ / , k = 0 (3.21a) (cid:2) iγ ∂ η − k z (cid:0) C A + iC B γ (cid:1) γ − am (cid:3) ψ / , k = 0 (3.21b)where we have defined C A = 13( H + m ) (cid:20) ( m − H ) (cid:18) − R − H + 3 m (cid:19) − Hm ∂ η ma (cid:21) (3.22a) C B = 2 m H + m ) (cid:20) H (cid:18) − R − H + 3 m (cid:19) + ( m − H ) ∂ η mma (cid:21) . (3.22b)Here we note for future use that v s ≡ C A + C B = 19( H + m ) (cid:34)(cid:18) − R − H + 3 m (cid:19) + 4 ( ∂ η m ) a (cid:35) . (3.23)Although we have taken k = (0 , , k z ) to derive these relations, we know that the system is rotationally invariant,and so we can extend to the Lorentz-covariant mode equations: (cid:2) iγ ∂ η − k · γ − am (cid:3) ψ / , k = 0 (3.24a) (cid:2) iγ ∂ η − (cid:0) C A + iC B γ (cid:1) k · γ − am (cid:3) ψ / , k = 0 . (3.24b)In Minkowski space C A = 1 and C B = 0 and the mode equations reduce to Eq. (2.15).It is useful to compare Eq. (3.24) with the mode equation for a Dirac spinor field. We see that the helicity-3/2 mode function, ψ / , k , satisfies precisely the same equation of motion as the Dirac spinor mode function,while the equation of motion for a helicity-1/2 mode function, ψ / , k , differs from the corresponding equationfor a Dirac spinor if C A (cid:54) = 1 and C B (cid:54) = 0 . 11 .4 Helicity eigenspinors For each k , the mode equations (3.24) will admit be two solutions, which we can distinguish by writing ψ s / , k ( η ) and ψ s / , k ( η ) where the new label s takes values s = ± / or s = ± / as appropriate. In the Dirac represen-tation of the gamma matrices it is convenient to parametrize the spinor wavefunctions as ψ s / , k ( η ) = (cid:32) χ A, / , k ( η )(2 s/ χ B, / , k ( η ) (cid:33) ⊗ h s/ k and ψ s / , k ( η ) = (cid:32) χ A, / , k ( η )(2 s ) χ B, / , k ( η ) (cid:33) ⊗ h s ˆ k (3.25)where χ A, / , k ( η ) , χ B, / , k ( η ) , χ A, / , k ( η ) , and χ B, / , k ( η ) are complex-valued mode functions, and h λ ˆ k is a2-component complex column vector, called the helicity 2-spinor, which only depends upon ˆ k = k / | k | . Inparticular, h λ ˆ k is defined to be an eigenfunction of the helicity operator with eigenvalue λ : ˆ k · σ h λ ˆ k = λ h λ ˆ k for λ = ± . (3.26)Putting the Ans¨atze (3.25) into the mode equation (3.24) yields i∂ η (cid:32) χ A, / ,k ( η ) χ B, / ,k ( η ) (cid:33) = A / (cid:32) χ A, / ,k ( η ) χ B, / ,k ( η ) (cid:33) (3.27a) i∂ η (cid:32) χ A, / ,k ( η ) χ B, / ,k ( η ) (cid:33) = A / (cid:32) χ A, / ,k ( η ) χ B, / ,k ( η ) (cid:33) (3.27b)where the 2-by-2 complex matrices A / and A / are defined by A / ( η ) = (cid:32) am kk − am (cid:33) and A / ( η ) = (cid:32) am ( C A + iC B ) k ( C A − iC B ) k − am (cid:33) , (3.28)and k ≡ | k | . We will see below that the eigenvalues of A / and A / are λ / ± = ± ω / ,k where ω / ,k ≡ (cid:112) k + a m (3.29a) λ / ± = ± ω / ,k where ω / ,k ≡ (cid:112) v s k + a m (3.29b)where v s = C A + C B is given by Eq. (3.23). These relations let us interpret v s as the time-dependent sound speedof the helicity-1/2 modes, while the helicity-3/2 modes have unit sound speed. The coupled first-order modeequations (3.27) can be combined to obtain second-order mode equations for the helicity-3/2 mode functions ∂ η χ A, / ,k = − (cid:0) ω / ,k + ia Hm (cid:1) χ A, / ,k ∂ η χ B, / ,k = − (cid:0) ω / ,k − ia Hm (cid:1) χ B, / ,k (3.30a)and the helicity-1/2 mode functions ∂ η χ A, / ,k = − (cid:0) ω / ,k + ia Hm (cid:1) χ A, / ,k − (cid:0) ik∂ η C A (cid:1) χ B, / ,k ∂ η χ B, / ,k = − (cid:0) ω / ,k − ia Hm (cid:1) χ B, / ,k − (cid:0) ik∂ η C A (cid:1) χ A, / ,k . (3.30b)Notice that the helicity-3/2 mode functions decouple, whereas the helicity-1/2 mode functions remain coupledthrough the ∂ η C A term. We will see below that the mode functions must satisfy a constraint (3.38), which arisesfrom quantizing the field, and one can easily verify that the mode equations (3.27) respect this constraint.12 .5 Helicity-3/2 modes The helicity-3/2 mode equation (3.27a) can be brought into an approximately-diagonal form by performing atime-dependent transformation (cid:32) χ + , / ,k χ − , / ,k (cid:33) = U / (cid:32) χ A, / ,k χ B, / ,k (cid:33) where U / = (cid:32) cos ϕ / sin ϕ / − sin ϕ / cos ϕ / (cid:33) . (3.31)The matrix A / is diagonalized, U / A / U T / = diag( ω / ,k , − ω / ,k ) if the rotation angle ϕ / satisfies theequalities cos ϕ / = (cid:112) ω / ,k + am/ (cid:112) ω / ,k and sin ϕ / = (cid:112) ω / ,k − am/ (cid:112) ω / ,k . In the quasi-diagonalbasis, the mode equation (3.27a) becomes (cid:32) ∂ η χ + , / ,k ∂ η χ − , / ,k (cid:33) = (cid:32) − iω / ,k ∂ η ϕ / − ∂ η ϕ / iω / ,k (cid:33) (cid:32) χ + , / ,k χ − , / ,k (cid:33) , (3.32)where ω / ,k is given by Eq. (3.29a) and where ∂ η ϕ / = − ka Hmω / ,k . (3.33)Notice how the mode functions are diagonal for ∂ η ϕ / = 0 , and the solutions are χ ± , / ,k = e ∓ iω / ,k η . Theoff-diagonal terms vanish for m = 0 , which reflects the fact that m is the order parameter for the breakingof Weyl invariance in the helicity-3/2 modes. Additionally ∂ η ϕ / → as a → at the early times andas H → at late times. In other words χ + , / ,k would describe positive-frequency solutions and χ − , / ,k would describe negative-frequency solutions if the evolution is adiabatic ( ∂ η ϕ / → ). The function ∂ η ϕ / drives mixing between positive- and negative-frequency solutions. Thus an initial condition consisting of onlypositive-frequency modes can evolve into an admixture of negative frequency modes, which is a signal of particleproduction. Now we turn to the helicity-1/2 modes that obey Eq. (3.27b). It is convenient to rewrite A / as A / = (cid:32) am v s k e iζ v s k e − iζ − am (cid:33) , (3.34)where the time-dependent phase ζ ( η ) obeys cos ζ = C A /v s and sin ζ = C B /v s . Since A / is Hermitian, it canbe diagonalized by a unitary transformation (cid:32) χ + , / ,k χ − , / ,k (cid:33) = U / (cid:32) χ A, / ,k χ B, / ,k (cid:33) where U / = (cid:32) cos ϕ / e − iζ/ sin ϕ / e iζ/ − sin ϕ / e − iζ/ cos ϕ / e iζ/ (cid:33) , (3.35)and where the time-dependent angle ϕ / ( η ) should be chosen to satisfy cos ϕ / = (cid:112) ω / ,k + am/ (cid:112) ω / ,k and sin ϕ / = (cid:112) ω / ,k − am/ (cid:112) ω / ,k , which gives U / A / U † / = diag( ω / ,k , − ω / ,k ) . This transformation The transformation between basis is nothing more than a Bogoliubov transformation. (cid:32) ∂ η χ + , / ,k ∂ η χ − , / ,k (cid:33) = (cid:32) − iω / ,k ∂ η ϕ / − ∂ η ϕ / iω / ,k (cid:33) (cid:32) χ + , / ,k χ − , / ,k (cid:33) + i ∂ η ζ ω / ,k (cid:32) − a m v s k − v s k a m (cid:33) (cid:32) χ + , / ,k χ − , / ,k (cid:33) , (3.36)where the derivatives are given by ∂ η ϕ / = − kamv s ω / ,k (cid:104) aHv s − (cid:0) C A ∂ η C A + C B ∂ η C B (cid:1)(cid:105) ∂ η ζ = v − s (cid:0) C A ∂ η C B − C B ∂ η C A (cid:1) . (3.37)Recall that C A and C B were defined in Eq. (3.22). At late times, the spacetime becomes asymptoticallyMinkowski meaning a → , H → , and R → , and if m → const . as well, then C A → and C B → and both of the derivatives above vanish. At early times, in the quasi-de Sitter era, both of the derivatives arealso small, since H , R , and m are slowly varying. Similar to the helicity-3/2 case, in the asymptotic early andlate times χ ± , / ,k describe positive and negative frequency modes, and for the study of gravitational particleproduction, we will be interested in the value of χ − , / ,k at late times. In the quantized theory, the mode functions discussed previously, along with a set of ladder operators, areused to construct the field operator. The field operator and its conjugate momentum are required to obey ananticommutation relation, which imposes a normalization condition on the mode functions. In terms of thenon-diagonal mode equations (3.27), the normalization conditions are | χ A, / ,k | + | χ B, / ,k | = 1 and | χ A, / ,k | + | χ B, / ,k | = 1 , (3.38a)and in terms of the quasi-diagonal mode equations from Eqs. (3.32) and (3.36) these conditions are | χ + , / ,k | + | χ − , / ,k | = 1 and | χ + , / ,k | + | χ − , / ,k | = 1 . (3.38b)The relative plus sign between each of the pairs of terms is a consequence of Fermi-Dirac statistics, and it implies | χ | ≤ for each mode function. We now turn our attention to the gravitational production of spin-3/2 particles at the end of inflation. In orderto calculate the spectrum and total number of gravitationally-produced particles, we solve the mode equationssubject to the Bunch-Davies initial condition, and we read off the late-time behavior of the mode functions. Tobegin this section, we first describe how the initial conditions were chosen and how the spectrum was extractedfrom the late-time solutions, and then we present our numerical results.14 .1 Initial conditions & particle number spectrum
For the helicity-3/2 modes, the mode functions are required to obey the Bunch-Davies initial condition lim η →−∞ χ + , / ,k ( η ) = 1 and lim η →−∞ χ − , / ,k ( η ) = 0 , (4.1)such that only positive-frequency modes are present at early times. In terms of the mode functions in the non-diagonal basis, we have lim η →−∞ χ A, / ,k ( η ) = 1 / √ and lim η →−∞ χ B, / ,k ( η ) = 1 / √ , (4.2)which is obtained by inverting the transformation in Eq. (3.31), using the initial condition in Eq. (4.1), andnoting that a → and ω / ,k → k as η → −∞ . Evolution under the mode equation will populate the negative-frequency modes. We are interested in the solutions at late times, meaning that any given mode k has becomenonrelativistic k (cid:28) am and passed inside the horizon k (cid:29) aH . Having solved the mode equations, the spectrumof helicity-3/2 particles that results from gravitational particle production is then calculated from the late-timeamplitude of the negative-frequency modes as [9] n / ,k = k π | β / ,k | for | β / ,k | ≡ lim η →∞ | χ − , / ,k ( η ) | . (4.3)Here n / ,k is the comoving number density of helicity-3/2 particles per logarithmically-spaced wavenumberinterval. As a matter of numerical stability, we find it easier to solve the non-diagonal mode equations (3.27a).Then the late-time solution is χ − , / ,k ≈ χ B, / ,k , since the dispersion relation (3.29a) is approximately ω / ,k ≈ am , and ϕ / ≈ in the transformation from Eq. (3.31).We also implement the Bunch-Davies initial condition for the helicity-1/2 modes, which is now written as lim η →−∞ χ + , / ,k ( η ) = 1 and lim η →−∞ χ − , / ,k ( η ) = 0 . (4.4)After the mode equations are solved, the spectrum of helicity-1/2 particles is extracted by calculating n / ,k = k π | β / ,k | for | β / ,k | ≡ lim η →∞ | χ − , / ,k ( η ) | . (4.5)To express the initial condition in terms of the non-diagonal basis mode functions, we invert the transformationin Eq. (3.35) and use Eq. (4.4), which gives lim η →−∞ χ A, / ,k = lim η →−∞ (cid:32)(cid:114) C A v s + i (cid:114) − C A v s (cid:33) lim η →−∞ χ B, / ,k = lim η →−∞ (cid:32)(cid:114) C A v s − i (cid:114) − C A v s (cid:33) . (4.6)Although ϕ / → at early times, the phase ζ depends nontrivially on C A and v s , which leads to the expressionabove. Once again, for numerical stability we solve the non-diagonal mode equations (3.27b) and extract thelate-time solution as χ − , / ,k ≈ χ B, / ,k .To close this section we remark upon the impact of Fermi-Dirac statistics and Pauli blocking. Recall thatthe Fermi-Dirac statistics of the Rarita-Schwinger field leads to normalization conditions on the mode functions153.38), implying | χ − , / ,k | ≤ and | χ − , / ,k | ≤ . In other words, the occupation number cannot exceed foreach Fourier mode and each helicity, which is the phenomenon of Pauli blocking. As a result the spectra fromEqs. (4.3) and (4.5) are capped by n / ,k = n / ,k ≤ k / π .In the next subsection we will observe that a “catastrophic” particle production causes these spectra tosaturate at their upper limit for arbitrarily large k up to the UV cutoff. Using the mode equations and initial conditions described above, we solve for the evolution of the mode func-tions and extract the spectrum n k of gravitationally produced particles. The total number of particles in a regionof space with physical radius R is given by, N tot ( R, t ) = 43 a ( t ) πR (cid:90) /R d kk n k . (4.7)This is divergent if n k grows with increasing k , or equivalently, if | β k | falls more slowly than k − . Theparticle number at late times (when the particles are non-relativistic) in a bounded region of space is a physicalobservable, of the kind advocated for by [63]. Thus a UV divergence of N is physical, and its implicationsmust be understood. We refer to this divergent particle number as “catastrophic” particle production.We compute the particle production for both the helicity-1/2 and the helicity-3/2 modes, separately. Forsimplicity, both aesthetic and computational, our numerical study focuses on the canonical massive Rarita-Schwinger model, namely a spin- / field with constant mass. For the background spacetime, we assume aquadratic inflaton potential at the end of inflation.Numerical examples are shown in Fig. 1. The spectrum of helicity-3/2 modes are represented by the redcurves in the two panels. At low- k the spectra rise like n k ∝ k , corresponding to to | β k | ≈ , which isconsistent with these modes being maximally occupied and subject to Pauli blocking. At high- k the spectra dropoff, which is consistent with the evolution of these modes remaining approximately adiabatic. Since the modeequation for the helicity-3/2 modes of the Rarita-Schwinger field (3.24) is identical to the mode equation for aspin-1/2 Dirac field, we can compare our numerical results against previous studies in the literature [9], and bydoing so we find good agreement for the shape and amplitude of the spectrum.The spectrum of helicity-1/2 modes are given by the blue curves in each panel. In the right panel wherethe Rarita-Schwinger field’s mass is m = H e , we observe that the spectrum peaks at k ∼ few × a e H e , and dropsoff for larger k . However, in the left panel where the Rarita-Schwinger field is lighter, here only m = 0 . H e ,we observe that the spectrum continues to rise well past k = 2 a e H e . Numerical limitations prevent us fromevaluating the spectrum for larger k , but we expect that the spectrum will continue to rise as n k ∝ k indefinitely.We justify this claim in the following subsections, where we argue that the time-dependent sound speed of the From [63], p.5., “This procedure corresponds to counting particles present in the region after reheating and using the enclosednumber of particles as a measure of the physical size of the region... In the late universe we could use the number of dark matter particlesor equivalently the mass in dark matter particles enclosed in the volume. Not only is this measure of size a well defined physical choice,but the amplitude of perturbations at a given mass scale is directly related to the abundance of objects of that mass that will form in thelate Universe.” k spectrum, which saturates the Pauli blockinglimit. By contrast, the helicity-3/2 modes have a static sound speed and a time-dependent effective mass, whichis responsible for their particle production.Let us reiterate the main message of Fig. 1. The spectrum of helicity-1/2 modes in a spin-3/2 Rarita-Schwinger field with constant mass m = 0 . H e appears to grow without bound as n k ∝ k . We refer tothis UV-dominated spectrum as “catastrophic” gravitational particle production. The total number of particlesproduced is calculated by integrating this spectrum over all k , Eq. (4.7), and this quantity acquire a power-lawsensitivity to the UV cutoff. Similarly the net energy of these particles is UV dominated. One might argue thatthe calculation is not self-consistent in this regime, since we are solving for the evolution of a spectator field inthe external background induced by the inflaton, and neglecting the backreaction of the produced particles on theclassical background, which can be expected to be significant once their energy densities become comparable.Further, any theory containing gravity is an effective field theory with a cutoff at (or below) the Planck scale,so the divergent particle number should also be cutoff at some scale. We will instead argue that the calculationdisplaying catastrophic GPP is inconsistent because effective field theory itself has broken down, precisely dueto the on-shell particles with momenta equal to the cutoff. This breakdown of the EFT that manifests itselfonly when the gravitino is quantized is reminiscent of swampland conjectures that deal with EFT’s that becomeinconsistent when gravity is quantized. We discuss this point further in Sec. 7.On the other hand, for the helicity-3/2 modes or for the helicity-1/2 modes with m/H e (cid:38) we do notobserve catastrophic GPP. Here we claim that the numerical calculation is robust. If the Rarita-Schwinger fieldis stable, then its spin-3/2 particle provides a dark matter candidate, and the GPP calculation here furnishes aprediction for the dark matter relic abundance. We will explore this dark matter model further in future work. An important feature of the mode equations described above is that the sound speed v s = C A + C B will oscillateat the end of inflation, and that these oscillations can allow v s to vanish multiple times. We will see shortly thatthis behavior is a crucial determinant in whether catastrophic GPP takes place. We begin here by deriving ananalytic expression for the squared sound speed and determining the conditions under which it vanishes at theend of inflation.We assume that at the end of inflation the energy density is dominated by an inflaton field φ with potential V ( φ ) . Using the Friedmann equations, R = − M − ( ρ − p ) and H = M − ρ/ , Eq. (3.23) becomes v s = ( p − m M ) ( ρ + 3 m M ) + M (2 a − ∂ η m ) ( ρ + 3 m M ) . (4.8)Recall that v s enters the dispersion relation for the helicity-1/2 mode (3.29b), which is written as ω / ,k = v s k + a m . At late times ρ → , p → , and if ∂ η m → as well, then v s → , giving the usual dispersionrelation. At early times, during the quasi-de Sitter phase of inflation, p ≈ − ρ and if also ∂ η m ≈ then v s ≈ .However, near the end of inflation, the sound speed is expected to deviate from . The potential at the end of inflation may well differ from the potential that determines curvature fluctuations on scales probed byobservations. − − k/a e H e − − − − − − − n k / a e H e m/H e = 10 − helicity 1 / / − k/a e H e − − − − n k / a e H e helicity 1 / / m/H e = 1 Figure 1: Values of n k /a e H e = k | β k | / π as a function of k for two values of m/H e for helicity-3/2 and helicity-1/2. (The helicity-3/2 results are identical to the results for a Dirac fermion.) If m/H e = 0 . then v s vanishes duringthe evolution (see Fig. 3) and catastrophic particle production results. If m/H e = 1 . the v s does not vanish in theevolution (but is less than unity). There is no catastrophic GPP, but particle production is enhanced over helicity-3/2 whichhas v s = 1 . The oscillations for k (cid:38) can be compared to the oscillations seen in Figure 3 of Giudice, Riotto, andTkachev [16]. To better understand how the sound speed varies with time, we need to specify a model for the inflaton sothat ρ ( η ) and p ( η ) can be calculated, and we must also specify how the spin-3/2 field’s mass varies with time, sothat ∂ η m can be calculated. For simplicity we consider a quadratic potential, and the inflaton’s energy densityand pressure are given by ρ = ˙ φ + m φ φ and ρ = ˙ φ − m φ φ . (4.9)Here and below “dot” denotes d/dt where t is cosmic time. The inflaton mass is related to H e by m φ =2 H e M Pl /φ e . Again, we emphasize that this is the potential that describes the oscillation of the inflaton aboutits minimum after inflation, and it need not describe the inflaton potential when the scale factor was about 50 e -folds from the end of inflation when scales important for observable curvature fluctuations crossed the Hubbleradius. For the spin-3/2 field’s mass, we consider the simplest scenario first and assume that it is a constant sothat ∂ η m = 0 . Then it is possible for the squared sound speed (4.8) to vanish when p = 3 m M .It is easy to see that the sound speed cannot vanish during inflation: since ρ + 3 p < and ρ > , it followsthat p < and v s > . We study the evolution of v s ( η ) after inflation by numerically solving the inflaton’s fieldequation to calculate the pressure p . In Fig. 2 we show the pressure p as as a function of a/a e , and we comparewith m M for several different values of the spin-3/2 field’s mass m . If the spin-3/2 field’s mass is static, ∂ η m = 0 , then the sound speed v s vanishes when the pressure p equals m M [see Eq. (4.8)]. The blue curveshows the pressure p (in units of H e M ) and the gray-dashed lines show values of m M (same units) for18 a/a e − − p & m M P l ( un i t s o f H e M P l ) m/H e . . p m M Figure 2: The evolution of the pressure at the end of inflation in the chaotic inflation model. For a given choice of thespin-3/2 field’s mass m , the sound speed v s vanishes when p = 3 m M corresponding to intersections of the blue curvewith the corresponding gray-dashed curve. different choices of the spin-3/2 field’s mass m/H e . For a given mass choice, the sound speed vanishes v s = 0 whenever the blue and gray-dashed curves intersect. In this example, for m/H e (cid:38) . the sound speed nevervanishes, for m/H e (cid:39) . it vanishes just once, and for m/H e (cid:28) . it vanishes many times as the pressureoscillates at the end of inflation.The evolution of the sound speed is illustrated directly in Fig. 3, which presents v s for the same threevalues of m/H e that appear in Fig. 2. Every time v s touches zero the sound speed vanishes. Note that the upperlimit to the sound speed in unity, so there is no superluminal propagation. As the pressure damps away withtime, it eventually decreases below the threshold | p | < m M and subsequently the sound speed has no morezero-crossings and it remains positive. Here we seek to clarify how the vanishing sound speed leads to efficient production of the helicity-1/2 modes.In general, particle production results from non-adiabatic mode evolution. If the mode equation takes the formof ∂ η χ k + ω k ( η ) χ k = 0 with time-dependent dispersion relation ω k ( η ) , then A k ( η ) ≡ ∂ η ω k ω k (4.10)provides a dimensionless measure of the nonadiabaticity. When the dispersion relation changes rapidly and | A k | (cid:29) , efficient particle production results. (See, e.g., [64, 65] in the context of preheating.)The mode equation for the helicity-1/2 modes (3.30) has the dispersion relation ω k, / = v s k + a m from Eq. (3.29b). For simplicity, we consider the case of a constant mass, as studied in the numerics. The19 . . . . . . v s m/H e = 0 . . . . . . . v s m/H e = 0 . − − − a/a e . . . . . . . . v s m/H e = 1 . Figure 3: The evolution of v s as a function of a/a e for three values of m/H e . As illustrated in the upper panel, for m/H e (cid:28) after inflation, v s = 0 many times in the oscillation of the inflaton field. If m/H e (cid:39) . , while v s stilloscillates it vanishes only once (see the middle panel). As seen in the lower panel, if m/H e (cid:38) . the oscillationsbecome less important and the sound speed never vanishes. (Note the different lower limit in the bottom panel.) A k = a Hm + v s ( ∂ η v s ) k ( a m + v s k ) / (4.11)for ∂ η m = 0 and ∂ η a = a H . As illustrated in Fig. 3, if m (cid:46) . H e , the sound speed v s will vanish, and maydo so many times. At the moments when the sound speed vanishes, the adiabaticity parameter is given by, A k | v s =0 = Hm , (4.12)which is manifestly larger than unity since m/H < is a requirement for v s to ever vanish.Strikingly, Eq. (4.12) is independent of k , which implies that particles of arbitrarily high momentum canbe produced. This is precisely the “catastrophrophic particle production.” This is very different from the standardsituation of particle production, e.g., of spin-0 fields with v s = 1 and oscillating nonzero mass m ( η ) > , wherelarge values of k act to shut-off the violation of adiabaticity by making the denominator in Eq. (4.11) large.In fact, the violation of adiabaticity is even more severe than suggested by Eq. (4.12). The maximumvalue of A k , and hence maximum violation of adiabaticity, occurs shortly before and shortly after v s = 0 , when v s is small but finite. This occurs when the second term in the numerator of Eq. (4.11) is dominant, while thefirst term is dominant in the denominator of Eq. (4.11). In this case, one finds A k is given by, A k ≈ v s ( ∂ η v s ) k a m . (4.13)This approximation is valid while k lies in the range v s (cid:28) a m /k (cid:28) v s | ∂ η v s | /aH , which occurs, for arbi-trarily large k , for a transient period before v s vanishes, and again for a period after. Combining this constraintwith Eq. (4.13), we find, Hm (cid:28) | A k | (cid:28) | ∂ t ln v s | H Hm , (4.14)which brackets the nonadiabaticity away from v s = 0 for a mode of arbitrary k , including arbitrarily large k ,for example, k equal to the UV cutoff of the theory. This occurs for a very short period of time, reminiscent ofpreheating, wherein adiabaticity is violated as the inflaton crosses zero [64]. In particular, non-adiabatic particleproduction from a rapidly-varying mass parameter has been studied in Refs. [66–68], where it is observed thatan arbitrarily rapid change in the mass parameter leads to particle production in arbitrarily high- k modes.In the spirit of effective field theory, one might hope that is cured by UV completion. Fortunately, the UVcompletion is in hand: the UV completion of the massive Rarita-Schwinger field is supergravity, and the UVcompletion of supergravity is string theory. In the spirit of effective field theory, one might hope that the catastrophic particle production exhibited by theRarita-Schwinger model with m (cid:46) H e would be cured by UV completion. Fortunately, the UV completion isin hand: the UV completion of the massive Rarita-Schwinger field is supergravity, and the UV completion ofsupergravity is string theory. 21ndeed, the construction of a consistent quantum field theory for spin-3/2 particles in a general spacetimewas once thought to be problematic [69, 70] but the issues are resolved if the spin 3/2 field is coupled to asupercurrent as in supergravity (SUGRA). As we have alluded to many times, the physical spin-3/2 field inSUGRA is known as the gravitino. A salient feature of SUGRA models is that the gravitino mass is related to theenergy density and pressure and can be time dependent. The expression for the gravitino sound speed, Eq. (4.8),applies independent of SUGRA considerations, and indeed matches that given throughout the SUGRA literature[13, 15, 16]. On the other hand, whether v s vanishes or not (hence, catastrophic GPP or not) is dependent on thesupergravity model.Some discussion of SUGRA model building can be found in Appendix C for those unfamiliar withSUGRA. In SUGRA models there are several masses to keep track of; so when discussing SUGRA modelswe will denote the mass of the spin-3/2 gravitino by m / . Also, in SUGRA m / need not be real, so wewill often encounter | m / | . Here, we first consider a model with a single chiral superfield, then consider morerealistic models with more than one superfield. As a starting point, let us consider a single chiral superfield oscillating in its potential. This is the model studiedby Giudice, Riotto, and Tkachev [15, 16] and Kallosh, Linde, with Van Proeyen [13] when discussing gravitinoGPP (and discussed again by Kallosh, Linde, Van Proeyen, and Kofman [14]). Denoting as Φ the complex scalarcomponent of a chiral superfield by Φ , and ¯Φ the scalar field’s complex conjugate, we consider as a model theK¨ahler potential K and superpotential W given by, K (Φ , ¯Φ) = Φ ¯Φ , W (Φ) = m φ Φ . (5.1)We may express the complex scalar Φ in terms of real fields as Φ = √ φ e iσ/M Pl . With this choice of the K¨ahlerpotential and the superpotential, and after setting σ = 0 , the gravitino mass is given by [see Eq. (C.9)], m / = e K (Φ , ¯Φ) / M W (Φ) M = e φ / M m φ M φ , (5.2)where the middle expression is general for single-superfield models and the final expression is particular to theabove choices of K and W . For example, φ may be the inflaton field, and its evolution during inflation leads toa time-dependence for the gravitino mass m / = m / ( t ) . The scalar-field Lagrangian is , L = 12 ( ∂φ ) + 12 M φ ( ∂σ ) − V ( φ ) , (5.3)with potential given by Eq. (C.8) V ( φ ) = e K/M (cid:32)(cid:12)(cid:12)(cid:12)(cid:12) ∂ Φ W + WM ∂ Φ K (cid:12)(cid:12)(cid:12)(cid:12) − | W | M (cid:33) = e φ / M m φ φ (cid:18) φ M + φ M (cid:19) , (5.4)where again the middle expression is general for single-superfield models and the final expression obtains for K and W of Eq. (5.1). In the limit φ (cid:28) M Pl the potential is simply V = m φ φ .As a momentary digression, we note that this model is not complete. In particular, the potential is in-dependent of the angular field σ , and thus σ is massless. In general, one must introduce additional superfields22o stabilize σ , and additional terms in the K¨ahler potential to stabilize the fields added to stabilize σ (see,e.g., [71, 72]). Furthermore, in order to describe our universe the model must at least be able to accommodatethe Standard Model of particle physics, which includes the Higgs field, and any supersymmetric embedding ofthe Standard Model generally introduces many scalar fields.The shortcomings aside, the model has a seemingly miraculous feature: the sound speed v s = 1 at alltimes! We can see this by noting that the energy density ρ and pressure p contributed by φ are ρ = ˙ φ + V ( φ ) and p = ˙ φ − V ( φ ) , and that M (cid:0) a − ∂ η m / (cid:1) = 4 M ˙ m / is given by M ˙ m / = e φ / M m φ φ (cid:18) φ M + φ M (cid:19) ˙ φ = 2 V ˙ φ + 6 m / M ˙ φ . (5.5)Using (5.5) and the expressions for ρ and p in Eq. (4.8) results in v s = 1 ! To recap: v s = 1 is a result of thecancellation of the time dependence of ρ and p with the time dependence of ˙ m / .One might ask if the conspiracy that leads to v s = 1 is general in SUGRA, whether it is a feature ofall models with a single chiral superfield, or is it particular to this model of a single superfield with the K¨ahlerpotential and superpotential of Eq. (5.1). We now demonstrate that v s = 1 for all models with a single chiralsuperfield (as shown long ago in [13, 14]), and in general is not true in models with multiple superfields. We now consider the sound speed Eq. (4.8) in a model of N number of chiral superfields, Φ I , with field index I = 1 ...N . For a thorough discussion of the gravitino equations of motion in models of arbitrary superfields,we refer the reader to [14]. It was strongly emphasized in [14] that to compute the particle production in modelswith more than one chiral superfield, one must carefully track all of the spin- / fields in the theory (in contrastto what was done in [17]). Here we content ourselves to consider only the sound speed, with a vanishing soundspeed considered to be a diagnostic for catastrophic particle production.The first step is to find a general expression for ˙ m / . The time derivative can be written as a directionalcovariant derivative in field space. We write the time-derivative as a field-space derivative as, dd t = (cid:88) I (cid:104) ˙Φ I ∇ I + ˙¯Φ I ∇ ¯ I (cid:105) , (5.6)where ∇ represents a directional covariant derivative in field space on a complex manifold with metric G I ¯ J defined in Eq. (C.1). When operating on a scalar function f (Φ , ¯Φ) , the directional derivative is ∇ I f (Φ , ¯Φ) = ∂f (Φ , ¯Φ) /∂ Φ I = f, I and ∇ ¯ I f (Φ , ¯Φ) = ∂f (Φ , ¯Φ) /∂ ¯Φ I = f, ¯ I . The bar over an index denotes an operationwith respect to ¯Φ .For the gravitino mass given by M m / = e K (Φ , ¯Φ) / M W (Φ) [see Eq. (C.9)], we express M ˙ m / as M ˙ m / = e K (Φ , ¯Φ) / M n (cid:88) I =1 (cid:18) ˙Φ I ∂ I W (Φ) + 12 M ˙Φ I K, I W (Φ) + 12 M ˙¯Φ I K, ¯ I W (Φ) (cid:19) . (5.7)For the models we consider, K is symmetric under the interchange Φ I ←→ ¯Φ I . We further assume that theimaginary parts of Φ I are constant and we set them equal to zero, i.e., for each complex scalar ¯Φ I , we consider23nly Re Φ or | Φ | to be dynamical, as is the case in common supergravity cosmological models. These propertiesyield K ,I = K , ¯ I and ˙¯Φ I = ˙Φ I , and results in M (cid:12)(cid:12) ˙ m / (cid:12)(cid:12) = e K (Φ , ¯Φ) /M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) I ˙Φ I (cid:2) ∂ I + M − K ,I (cid:3) W (Φ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = e K (Φ , ¯Φ) /M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) I ˙Φ I D I W (Φ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (5.8)where D I ≡ ∂ I + M − K ,I .The next step is to express the energy density and pressure as ρ = (cid:80) I | ˙Φ I | + V and p = (cid:80) I | ˙Φ I | − V where V = e K/M (cid:80) I | D I W | − M | m / | . Using these expressions, along with the expression for M ˙ m / in Eq. (5.7), leads to the result v s = 1 − e K/M ( ρ + 3 M m / ) (cid:32)(cid:88) I | ˙Φ I | (cid:33) (cid:32)(cid:88) J | D J W | (cid:33) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) I ˙Φ I D I W (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (5.9)which is not equal to unity except for special cases. In particular, the sum-of-squares and the square-of-a-sumonly cancel in the trivial case where the “sum” is over 1 element, but not in general when there is more than onefield. This means that in general the v s = 1 result is peculiar to the case of only one chiral superfield [13, 14].The above expression matches the corresponding expression given in [17].We can simplify this further by defining the field space vectors (cid:126) ˙Φ and (cid:126)F , which have components ˙Φ I and e K/ D I W respectively. In this notation, one finds, ρ + 3 M pl m / = (cid:126) ˙Φ · (cid:126) ˙Φ + (cid:126)F · (cid:126)F , (5.10)where the dot product is with respect to the metric G I ¯ J , and similarly (cid:88) I ˙Φ I D I W = e − K/ M (cid:126) ˙Φ · (cid:126)F , (cid:88) J | D J W | = e − K/M (cid:126)F · (cid:126)F , (cid:88) I | ˙Φ I | = (cid:126) ˙Φ · (cid:126) ˙Φ . (5.11)From this we can express v s as v s = 1 − (cid:126) ˙Φ · (cid:126) ˙Φ)( (cid:126)F · (cid:126)F ) − ( (cid:126) ˙Φ · (cid:126)F ) ( (cid:126) ˙Φ · (cid:126) ˙Φ + (cid:126)F · (cid:126)F ) . (5.12)This has a simple geometric interpretation as an angle in field space. To appreciate this, consider the simple caseof a flat field-space metric G I ¯ J = δ I ¯ J . In this case, the dot product has the usual property (cid:126) ˙Φ · (cid:126)F = | (cid:126) ˙Φ || (cid:126)F | cos( θ ) ,where θ is the angle between the two vectors. The above can be written as, v s = 1 − | (cid:126) ˙Φ | | (cid:126)F | ( | (cid:126) ˙Φ | + | (cid:126)F | ) (cid:0) − cos ( θ ) (cid:1) . (5.13)Thus the gravitino is “slow”, i.e., v s < , in any case when cos( θ ) (cid:54) = 1 . As an interesting side note, theabove is proof of subluminal propagation of gravitinos in supergravity: since cos ( θ ) − ≤ , and f ( x, y ) =4 xy/ ( x + y ) ≤ , we see that v s ≤ in general supergravity models: gravitino propagation is sub-luminal, asexpected.One may also appreciate from Eq. (5.13) that a vanishing sound speed v s = 0 corresponds to a relativelysimple limit. Consider a case wherein the cosmological evolution, namely the field space trajectory as measured24y (cid:126) ˙Φ , is orthogonal to the breaking of supersymmetry, namely the vector (cid:126)F . In this cos( θ ) = 0 . The soundspeed vanishes when | ˙Φ | = | (cid:126)F | . In summary, (cid:126) ˙Φ · (cid:126)F = 0 and | (cid:126) ˙Φ | = | (cid:126)F | implies v s = 0 . (5.14)Importantly, in contrast with what one might expect given that the single superfield model has v s = 1 at alltimes, the above suggests that supergravity does not a priori protect against catastrophic particle production. Tounderstand this in more detail, we now consider particular supergravity cosmology constructions. An extensively studied class of supergravity models is that of nilpotent superfields [73]. These models contain asuperfield S ( x, θ ) that obeys a superspace constraint equation, S ( x, θ ) = 0 . (5.15)These models are reviewed in Appendix C. They provide the effective field theory of the KKLT constructionfor de Sitter in string theory [40], where S encodes the spontaneous breaking of supersymmetry by an anti-D3brane [44–48]. The constraint on S in turn imposes that the scalar component S is a fermion bilinear, and thebosonic sector of theory corresponds to setting S equal to . The cosmology of these models was developed in,e.g., [73–75].In simple models of this type one always has the property that the field evolution is orthogonal to the SUSYbreaking, since the field predominantly responsible for SUSY breaking has no dynamical scalar component. Thesound speed is given by Eq. (5.9), with the simplification that scalar component S of the nilpotent superfield, S ,is vanishing. A simple model realization is the following [74, 75], W = M S + W M Pl , K = M S ¯ Sf (Φ , ¯Φ) + 12 (Φ − ¯Φ) . (5.16)where M , f , and W have mass dimension . The S -dependence of W breaks supersymmetry, while the Φ dependence of the K¨ahler potential generates a scalar potential for Re Φ and Im Φ .We consider a phase of inflation that proceeds along the Re Φ ≡ φ/ √ direction and we set Im Φ = 0 ; thelatter has super-Hubble mass independent of the values of W and M or the precise form of f (Φ , ¯Φ) [74, 75],and thus can be self-consistently set to . In this background, the K¨ahler potential vanishes, K = 0 . The F -termvector (cid:126)F is given by (cid:126)F ≡ ( F Φ , F S ) = (0 , M M Pl ) , the gravitino mass is given by m / = W /M Pl , and thepotential is given by V (Φ , ¯Φ) = − W + M f (Φ , ¯Φ) .As a simple example, we consider f (Φ , ¯Φ) = M + m | Φ | M − , such that the potential becomes V = Λ + 12 m φ , (5.17)with cosmological constant Λ given by Λ ≡ M − W . Interestingly, the gravitino mass and SUSY breakinghave effectively been sequestered into the cosmological constant, and thus are (naively) decoupled from theHubble parameter during inflation. 25owever, at the end of inflation, we find ourselves in an identical situation to the constant-mass massiveRarita-Schwinger. The oscillations of ˙ φ have amplitude of approximately M Pl H e , while the SUSY breakinghas amplitude | (cid:126)F | = M M Pl , and the field space evolution and SUSY breaking are orthogonal at all times, (cid:126) ˙Φ · (cid:126)F = 0 . If M (cid:46) H e , then the condition for vanishing v s , Eq. (5.14), is satisfied once every oscillation, andcatastrophic particle production occurs. Meanwhile, the observed near-vanishing of the cosmological constant, Λ (cid:39) , dictates that M (cid:39) √ m / . From this one finds that for m / (cid:46) H e there is catastrophic particleproduction in the nilpotent superfield model Eq. (5.16).Somewhat surprisingly, the bound m / (cid:38) H to avoid the catastrophe is identical to that found in Sec. 4without any of the machinery of supergravity. This demonstrates that supergravity in itself, despite introducingnew degrees of freedom, does not always resolve the catastrophic particle production. A final class of models we consider is that of orthogonal constrained superfields [76–79]. These models supple-ment the nilpotent superfield model with an additional superfield Φ satisfying the orthogonality constraint, S · ( Φ − ¯Φ ) = 0 . (5.18)For a model comprised solely of Φ and S , in the unitary gauge the dynamical fields are only a single real scalar,and the gravitino, making this an ideal playground for cosmological model building [77–79]. Catastrophicgravitino production in this class of models has been noted previously in Ref. [37]. The sound speed in this typeof models has also been studied in e.g., Ref. [51].For simplicity we restrict ourselves to two fields. Consider the model [76–79], W = f (Φ) S + g (Φ) M Pl , K = 12 (Φ − ¯Φ) + S ¯ S, (5.19)with nilpotency constraint S = 0 and orthogonal constraint S · ( Φ − ¯Φ ) = 0 . The superpotential W = f (Φ) S + g (Φ) M Pl is the most general possible superpotential for these two fields given the superfield constraints. Thesecond constraint removes the D Φ W contribution to V , and correspondingly, removes the D Φ W contributionto the first term in the square brackets Eq. (5.9). Furthermore, the orthogonality constraint, in the unitary gauge,removes Im Φ from the spectrum [77], which implies K = 0 . More explicitly: ( v s ) orthogonal = 1 − ρ + 3 m / ) (cid:32)(cid:88) I | ˙Φ I | (cid:33) (cid:88) J (cid:54) =Φ | D J W | − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) I ˙Φ I D I W (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (5.20)Again we consider Φ = ¯Φ = φ/ √ . The scalar potential is V = | D S W | − M − | W | = (cid:0) | f | − | g | (cid:1) . (5.21)Note that D Φ W does not contribute to V due to the additional constraint equation of the orthogonal models.The gravitino mass is m / (Φ) = g (Φ) /M Pl , which has time derivative ˙ m / = M − ˙Φ ∂g (Φ) ∂ Φ . (5.22)26e find the sound speed is v s (Φ , ¯Φ) = 1 − ( ˙Φ + f ) (cid:34) f − M (cid:18) ∂g (Φ) ∂ Φ (cid:19) (cid:35) . (5.23)This applies to general f (Φ) and g (Φ) .As a simple example, consider a constant mass gravitino. That is, consider g (Φ) = m / M Pl . We canexpress the potential V so as to absorb the gravitino mass into the cosmological constant, as V = Λ + ˆ f ( φ ) ,where Λ ≡ f (0) − m / and ˆ f ( φ ) = f ( φ ) − f (0) . The sound speed can be computed from the above ordirectly from Eq. (4.8). It is given by v s = (cid:18) p φ ρ φ (cid:19) , (5.24)with p φ = ˙Φ − | ˆ f (Φ) | and ρ φ = ˙Φ + | ˆ f (Φ) | . The above is simply the equation of state of the field Re Φ ≡ φ/ √ . If φ is oscillating, the above crosses zero any time the pressure crosses zero, leading to catastrophicparticle production. Moreover, this applies for any choice of the gravitino mass, provided we set Λ = 0 upto small corrections. Notably, this model with a constant gravitino mass can not be saved by the restriction to m / > H , since the gravitino mass is absorbed into Λ and we take Λ (cid:39) up to very small corrections.Thus we see that orthogonal constrained superfield models with a constant-mass gravitino have the catas-trophic particle production detailed in Sec. 4 for any mass of the gravitino. However, this is not the fate of allorthogonal constrained superfield models.To understand the diverse possibilities, we consider a toy model that interpolates between the differingcases. We consider a constrained orthogonal superfield, and take a superpotential similar to those proposed inRef. [77]. We consider the model, W = f (Φ) S + g (Φ) M Pl , K = 12 (Φ − ¯Φ) + S ¯ S, (5.25)satisfying the superspace constraints, S = 0 and S · ( Φ − ¯Φ ) = 0 , and with f (Φ) = (cid:112) m Φ + a and g (Φ) = (cid:115) M α m Φ + b (5.26)The motivation for this seemingly ad hoc model is to parametrize in a simple way the cosmological constant,the Hubble constant, the time-dependence of the gravitino mass, and the size of the gravitino mass in vacuum.These are parametrized by a , m , α , and b respectively.We take Φ = ¯Φ = φ/ √ . The scalar potential is given by V = ( a − b ) + 12 m φ − M α m φ . (5.27)The first term is the cosmological constant. Demanding that it be near-zero sets a (cid:39) b , which we take to becase in what follows. This leaves as model parameters { α, b, m } .The gravitino mass is M Pl m / = (cid:115) M α m φ + b . (5.28)27n the vacuum φ = 0 , we have m / = b / /M Pl . The time-dependence of m / is controlled by the size of α .The gravitino sound speed in general is given by, Eq. (5.23) v s = 1 − φ (cid:18) b − α m φ b + α m φ + m φ (cid:19)(cid:20) (cid:18) b + m φ (cid:19) + ˙ φ (cid:21) . (5.29)The minimum of the potential is locally a m φ type . We consider m (cid:29) H such that the oscillations aremuch faster than Hubble, to a good approximation we can neglect the expansion and consider the backgroundsolution φ ( t ) = φ cos( mt ) .First consider a constant gravitino mass, α = 0 . We find the sound speed is v s ( t ) | α =0 = (cid:2) b + m φ cos(2 mt ) (cid:3) (cid:0) b + m φ (cid:1) . (5.30)This has zeros at t ∗ , defined by cos(2 mt ∗ ) = − bm φ . (5.31)Since | cos( x ) | ≤ , we deduce that v s ( t ) has zeros if b < m φ / . Incidentally, the Friedmann equation reads H (cid:39) (3 M P l ) − ( m φ /
2) = m φ / M , and m / = g = √ b , and hence we find zeros of v s whenever m / < H . This can be seen in the top panel of Fig. 4.Now let us consider a time-varying gravitino mass. For simplicity, take a vanishing vacuum gravitinomass, b = 0 , and maintain a = 3 b . We find v s ( t ) | b =0 = 12 (cid:2) (1 + 4 α ) + (1 − α ) cos(4 mt ) (cid:3) . (5.32)Note that we demand α ≤ / to ensure casuality. When α = 0 , this exhibits zeros at t = nπ/ m . However,for α (cid:54) = 0 , we see min v s ( t ) | b =0 = 4 α . (5.33)Thus the vanishing of the sound speed is cured by turning on a field-dependence of the gravitino mass. This canbe seen in the bottom panel of Fig. 4.Finally, we note that all of these examples have D S W (cid:54) = 0 and thus are within the regime of validity ofthe dS supergravity theory [80–85], and thus ostensibly in the regime of effective field theory. As a concreteexample, we fix α = b = 0 and a = Λ . Then we find for the sound speed v s = (cid:0)
2Λ + m φ cos(2 mt ) (cid:1) (cid:0)
2Λ + m φ (cid:1) = p ρ . (5.34) One may introduce additional corrections to f (Φ) to enforce the potential be positive definite at large φ , but these terms are notrelevant provided φ is restricted to small oscillations. . . . . . . v s ( t ) [ b = ] α = 0 α = 1 / α = 1 / α = 1 / mt . . . . . . v s ( t ) [ α = ] b = 0 b = 1 / b = 1 / b = 1 / b = 1 b = 100 Figure 4: Evolution of the gravitino sound speed in a toy background with φ = φ cos( mt ) . Here we fix m = φ = 1 .[Top Panel] For b = 0 but α (cid:54) = 0 , from Eq. (5.32) we find the zeros are lifted to v s = α . [Bottom Panel:] For α = 0 ,from Eq. (5.30) the sound speed has zeros whenever m / < H . All other regions of parameter space interpolate betweenthese examples. We see v s has zeros if m φ / (2Λ) > . Meanwhile, for SUSY breaking along the S -direction | D S W | = f = Λ + 12 m φ = Λ + 12 m φ cos( mt ) , (5.35)is given at the zeros of v s by | D S W | | v s =0 = Λ2 + 14 m φ , (5.36)which is manifestly positive, even in the limit a → . Thus D S W is well behaved.29 .5 Corrections to the K¨ahler Potential In the previous section we observed that a non-zero time dependence of the gravitino mass can lift the zeros ofthe sound speed v s . In the simple model specified by Eq. (5.26), the parameter α controls the time-dependence,and as can be appreciated from Fig. 4, raises the minimum value of v s from to α .This phenomenon can arise in a simple fashion due to small corrections to the K¨ahler potential. We notethat while the superpotential is protected from perturbative corrections by non-renormalization theorems [86,87],the K¨ahler potential is not. In the context of string theory, the K¨ahler potential receives corrections from boththe α (cid:48) and string loop expansions. The leading corrections in string compactifications have been computed ine.g. [88, 89].Corrections to the K¨ahler potential play a similar role to the parameter α of the previous section (seeFig. 4). As an illustrative example, consider the following K¨ahler potential, K (Φ , ¯Φ , S, ¯ S ) = K (Φ , ¯Φ , S, ¯ S ) + δK (Φ , ¯Φ) (5.37)with K = 12 (Φ − ¯Φ) + S ¯ S , δK = 12 c (Φ + ¯Φ) (5.38)Here K has a shift symmetry for Re Φ , that is broken by δK , with the breaking parametrized by the parameter c (cid:28) . In this case, the kinetic terms are no longer canonical, but can be made canonical by a simple rescaling.The potential gets a small contribution that is subdominant as long as c (cid:28) . The gravitino mass is now givenby, m / = e c φ / M W (Φ) M , (5.39)which has inherited a field-dependence from the breaking of the shift-symmetry of Re Φ .Consider an orthogonal constrained superfield model with g (Φ) = g M , with g a dimensionless con-stant. The time-dependence of the gravitino mass is given by, ˙ m / = c φ ˙ φM Pl g (5.40)The sound speed for c = 0 is given by Eq. (5.24). When c (cid:54) = 0 , we instead find, v s = p φ ρ φ + c M φ ˙ φ ρ φ e c φ /M . (5.41)For a potential with a minimum that is locally quadratic, such that the one can approximate φ (cid:39) φ cos( mt ) ,one finds the would-be zeros of v s , which occur when p φ = 0 (at mt = π/ ), are lifted to, v s | p φ =0 = c m e c φ / M . (5.42)Thus we see that corrections to the K¨ahler potential, much like the parameter α in Fig. 4, lifts the zeros of thesound speed, making it positive definite. Incidentally, corrections to the K¨ahler potential which mix S and Φ play a crucial role in the inflation scenarios of [74, 75]. As mentioned above, the leading corrections to theK¨ahler potential in compactifications of string theory are known and have been computed, see e.g. [88, 89].30 The Gravitino Swampland Conjecture (GSC)
From the exercises of the previous section we deduce that supergravity does not a priori cure the vanishing ofthe sound speed seen in the Rarita-Schwinger model. We take v s = 0 as a diagnostic for the catastrophe. Hencewe deduce that SUGRA does not necessarily cure the catastrophic particle production: There are a plethora ofsupergravity models which do exhibit the catastrophe, and a plethora of models which do not. This breakdownof theory when the gravitino is quantized (which we will expand on shortly) suggests a possible link to theswampland [55] (for a review, see [56, 57]).The swampland is the set of four-dimensional effective field theories, which are self-consistent as aneffective field theory coupled to classical gravity, but become inconsistent when gravity is quantized. From thisinconsistency one infers that the effective theory does not have a UV completion in quantum gravity. The leadingcandidate of quantum gravity is superstring theory, and in this context, quantizing gravity means quantizing notonly the graviton but also the gravitino(s). In this work we have seen examples of field theories that falter whenthe gravitino is quantized. Motivated by this, we propose the Gravitino Swampland Conjecture: The sound speed of gravitino(s) must be positive-definite, v s > , at all points in moduli spaceand for all initial conditions, in all 4d effective field theories that are low-energy limits of quantumgravity. The GSC conjecture forbids the constant-mass massive Rarita-Schwinger field with m / (cid:46) H during inflation.This also forbids the supergravity models shown to exhibit the vanishing sound speed, such as simple nilpotentsuperfield models with m / (cid:46) H and orthogonal constrained superfield models with constant m / .This conjecture is substantiated by the prominent proposals for moduli stabilization in string theory,namely the KKLT [40] and Large Volume [41] scenarios. In both cases, m / > H is required for the radialfield to be stabilized during inflation [90–92]. In the opposite limit, the inflationary vacuum energy destabilizesthe compactification, and the radial field will experience a runaway. In string theory setups which do allow m / < H , such as the KL model [90], consistency with the conjecture requires only the inclusion of smallcorrections to the K¨ahler potential, which are ubiquitious in string theory.A detailed discussion of the Gravitino Swampland Conjecture, the cosmological implications, and evi-dence from string theory, can be found in [58]. In this work we have considered the dynamics of massive spin-3/2 fields in curved spacetime. Considering sim-ple cosmological backgrounds, we have computed the gravitational particle production, following the proceduredeveloped in the context of gravitational production of dark matter [2–5, 7–11, 11–13, 15, 16]. We have foundthe surprising and striking result that the Rarita-Schwinger model, namely a constant mass spin-3/2 particle, hasdivergent (“catastrophic”) particle production if the spin-3/2 particle (the “gravitino”) is cosmologically light,namely lighter than the Hubble parameter of the cosmological spacetime, m / (cid:46) H . The physical effect driv-ing the production is a vanishing sound speed v s = 0 for the helicity-1/2 gravitino, which can be used as a31iagnostic for the catastrophic particle production. The particle number at late times (when the particles arenon-relativistic) in a bounded region of space is a physical observable: all observers should agree on how manybilliard balls there are on the pool table. Thus we are forced to take this catastrophe seriously.The production of particles of arbitrarily-high momentum implies a breakdown of effective field theory,since an infinite tower of irrelevant operators, for example a set of operators labelled by n , O n = c n m / ∂ n Λ n (cid:0) ¯ ψ µ ψ µ (cid:1) ∼ c n m / (cid:90) d k (cid:18) k Λ (cid:19) n ¯ ψ µ k ψ µ k , (7.1)make a non-negligible contribution to the equations of motion if particles with momenta near cutoff, k ∼ Λ ,are produced. The relevance of an infinite tower of operators is indicative of a total breakdown of effective fieldtheory. This feature, in addition to the physical menchanism underlying it, distinguishes the catastrophic pro-duction from the conventional gravitino problem [49, 50]. It bears strong resemblance to the Planck-suppressedoperators in large field inflation, which lead to the η -problem [93], and which is the domain of the SwamplandDistance Conjecture [94, 95]. The SDC, in effect, states that no single EFT can describe regions of moduli spaceseparated by Planckian distances, and thus that any EFT which purports to do so (i.e., a theory without any newdegrees of freedom entering the theory as a scalar field traverses a Planckian distance in field space) is in the“swampland” [55].Faced with this, one may impose an ad hoc UV cutoff on the Rarita-Schwinger model, to parametrize ourignorance of UV physics, and postulate that the particle production computation can only be computed up tothe cutoff, and moreover that the calculation can be trusted all the way up to the cutoff, despite the argumentabove. However, for the Rarita-Schwinger model we are fortunate to have the UV completion already in hand:supergravity (SUGRA), and string theory. In this work we have studied the gravitino sound speed in single fieldSUGRA models, multifield SUGRA models, nilpotent constrained SUGRA models, and orthogonal constrainedSUGRA models. We have shown SUGRA does not provide a single answer: it is model dependent whether v s = 0 , and hence whether there is catastrophic particle production. In simple cases, with the exception oforthogonal models, the restriction to gravitino masses greater than Hubble, m / > H , is a sufficient conditionto avoid catastrophic production. In the orthogonal models, one may lift the zeros of v s by additional terms inthe superpotential or K¨ahler potential which induce a time-dependence of the gravitino mass.From this we conclude that the new degrees of freedom of SUGRA do not a priori save the Rarita-Schwinger model with m / (cid:46) H . In supersymmetric theories of quantum gravity, such as superstring theory,quantizing the gravitino is part and parcel with quantizing the graviton. This motivates an extension of the stringswampland program [55] to effective field theories that become inconsistent when the gravitino is quantized,and along these lines, we have proposed the Gravitino Swampland Conjecture. Since the conjecture specificallypertains to the gravitino sound speed, this conjecture applies to theories with arbitrary field content and scalarpotential. This thus applies a plethora of example models beyond those studied here.This work illustrates the power of gravitational particle production to uncover new physics. It comple- We further note that the additional fermionic degrees of freedom in supergravity, which allow the helicity-1/2 gravitino to decay,and the identification of the gravitino as a linear combination of spin-1/2 fermions that may change with time, does not ameliorate thebreakdown of effective field theory that defines catastrophic production. This is in contrast with the conventional gravitino problem,where the EFT is valid at all times (unlike the catastrophe), and where the changing identity of the gravitino can significantly amelioriatethe problem [52, 53] (see also [54] in the context of orthogonal constrained superfield supergravity models). s + 1 / with s > , has been studied in detail in [27], and itwill be interesting to understand in detail how the spin- / case differs. We leave this, and other possibilities, tofuture work. Acknowledgements
The authors thank Mustafa Amin, Sylvester James Gates, Jr., Gian Giudice, Wayne Hu, Andrei Linde, AntonioRiotto, Marco Scalisi, Leonardo Senatore, Igor Tkachev, and Lian-Tao Wang for helpful comments. The workof E.W.K. and E.M. was supported in part by the US Department of Energy contract DE-FG02-13ER41958.
Appendix A Frame Fields
We will be concerned with quantum field theory in the expanding universe. To promote a familiar flat-spacequantum field theory to one in curved space we start with a relativistic field theory in Minkowski space. Forfields that transform under Lorentz transformations as scalars, vectors, or tensors, the procedure is to replacein the action or field equations the Minkowski metric η µν by g µν , replace all tensors by objects that behave astensors under general coordinate transformations, and replace all derivatives ∂ µ with covariant derivatives ∇ µ ;i.e., for the derivative of a scalar field the covariant derivative is just ∂ µ , and for a vector field V µ the covariantderivative is ∇ µ V ν = ∂ µ V ν − Γ αµν V α and ∇ µ V ν = ∂ µ V ν + Γ νµα V α , and similarly for tensors.This prescription fails for fields with half-integer spin, such as the Dirac electron field, which transformas spinors under (infinitesimal) Lorentz transformations. Another approach is required to promote a Minkowskifield theory with spinor fields to curved space. The frame field formalism is an elegant and general formalismof general relativity which will allow a field theory with spinors to be extended to curved spacetime.To start, erect at every spacetime point X a set of local inertial coordinates y αX . At that spacetime pointin terms of the local inertial coordinates the metric is simply the Minkowski metric η αβ . This implies that themetric in a general noninertial coordinate system can be expressed it terms of the Minkowski metric η αβ as g µν ( x ) = e αµ ( x ) e βν ( x ) η αβ , with e αµ ( x ) ≡ (cid:18) ∂y αX ∂x µ (cid:19) x = X , (A.1) The frame field was originally introduced by Cartan who called them rep´eres mobiles (moving frames); in German they are referredto as vierbeins (four-leg), and in the English literature they are usually called tetrads (Greek for “set of four”). We will follow thenotation of Freedman and Van Proeyen [62], and in the spirit of Cartan refer to them as “frame fields.” The early-letter Greek alphabet will refer to the local inertial frame with coordinates y αX , α = 0 , · · · , , and early-letter Latinalphabet for the spatial part y aX , a = 1 , , . Mid-alphabet Greek letters will refer to coordinates in the noninertial system x µ , µ =0 , · · · , , and mid-alphabet Latin letters for the spatial part x i , i = 1 , , . e αµ ( x ) is the frame field. Early-alphabet indices are raised/lowered by η αβ while mid-alphabet indicesare raised/lowered by g µν . The frame field transforms as a covariant vector, so it should be regarded as fourorthonormal covariant vectors (one timelike and three spacelike) rather than as a single tensor. By using framefields, general objects (like spinors) can be converted into proper local, Lorentz-transforming tensors with theadditional spacetime dependence absorbed by the frame fields.To promote a Minkowski-space field theory to curved spacetime, contract vectors and tensors into framefields (e.g., V α → e µα V µ ) and replace derivatives ∂ α by ∇ α , where ∇ α = e µα ∂ µ + e µα Γ µ , (A.2)with the spin connection Γ µ ( x ) defined as Γ µ ( x ) = Σ αβ e να g σν (cid:16) ∂ µ e σβ + Γ σµρ e ρβ (cid:17) . (A.3)In the above expression for Γ µ , the quantity Σ αβ is the generator of the Lorentz group associated with therepresentation under which the field transforms.The stress-energy tensor can also be found by varying the geometry via the frame field T µα = 1 | e | δS M δe αµ , (A.4)where | e | is the determinant of the frame field. Then, T µν = − η αβ e µα T νβ .The frame-field formalism can be used for any field (scalar, vector, etc.) but we will only employ itfor half-integer spins. The interested reader should consult a fuller discussion of frame-field formalism (e.g.,[96–98]). Appendix B Projection Operators
The four spinor fields ψ , k . . . ψ , k separately obey the Dirac equation with canonically normalized kineticterms. However, they are not independent as there are two constraint equations. Here we find two orthogonalcombinations of ψ , k . . . ψ , k that are helicity eigenstates, have correctly normalized kinetic terms, and satisfythe Dirac equation. Since ψ µ, k is a vector-spinor, helicity projection operators are constructed from projectionoperators for vectors and spinors. To simplify the analysis we will employ the freedom to choose the momentumin the z direction.First, the projection operators for the helicity ± / states of a spinor (with momentum in the z direction)are S ± = 12 (cid:0) ± iγ γ (cid:1) . (B.1)We note S + and S − are real and satisfy S + S + = S + , S + S − = 0 and S − S − = S − . The vector projectorsfor helicity s are constructed from polarization vectors ( (cid:15) s ) µ , with s = ± , . For z -directed momentum, thepolarization vectors are ( (cid:15) ± ) µ = 1 √ , ± , i,
0) and ( (cid:15) ) µ = 1 m (cid:16) k, , , − (cid:112) k + m (cid:17) . (B.2)34ote the normalization of the vectors: ( (cid:15) ∗ s ) µ ( (cid:15) r ) µ = − δ sr . The spin-1, helicity- s projectors ( M s ) µν are formedfrom the polarization vectors: ( M s ) µν ≡ − ( (cid:15) ∗ s ) µ ( (cid:15) s ) ν ( s = ± , . (B.3)This is a well-behaved helicity projector: ( M s ) µρ ( M r ) ρν = δ rs ( M s ) µν . Putting together the spinor projectors(B.1) and vector projectors (B.3) we find the projectors for helicities +3 / , − / , +1 / , − / : ( P ± / ) µν = S ± ( M ± ) µν ( P ± / ) µν = S ± ( M ) µν + S ∓ ( M ± ) µν . (B.4)One can easily demonstrate ( P s ) µν ( P r ) µν = δ rs ( P s ) µν .Now we define ψ / , k as ψ / , k = − ( (cid:15) ∗ + ) ν S + ( M + ) µν ψ µ, k + ( (cid:15) ∗− ) ν S − ( M − ) µν ψ µ, k ψ / , k = 1 √ (cid:0) ψ , k + γ γ ψ , k (cid:1) , (B.5)and we define ψ / , k as ψ / , k = −√ (cid:15) ∗ + ) ν S − ( M + ) µν ψ µ, k + √ (cid:15) ∗− ) ν S + ( M − ) µν ψ µ, k ψ / , k = √ (cid:0) ψ , k − γ γ ψ , k (cid:1) . (B.6)So defined, ψ / , k and ψ / , k are constructed from helicity projections, have canonical kinetic terms, are orthog-onal, and obey the Dirac equation. Appendix C Supergravity
We consider N = 1 supergravity in d = 4 dimensions with N chiral superfields. We denote chiral superfieldsby boldface, e.g., Φ . The field content of Φ consists of Φ ( x, θ ) = Φ( x ) + √ θχ Φ ( x ) + θθF Φ ( x ) where x represents the spacetime coordinates which form the bosonic coordinates on superspace, and θ is the Grassman-valued fermionic superspace coordinates. Here Φ( x ) is a complex scalar, χ is a chiral fermion, and F is anon-dynamical complex scalar; i.e., an auxiliary field. When dealing with a set of fields, we use a capital Romanalphabet superscript, as in Φ I ( x, θ ) with scalar component Φ I ( x ) . We denote the corresponding anti-chiralsuperfields as ¯ Φ ¯ I .After integrating out auxiliary fields, the Lagrangian for the scalar components Φ( x ) is given by L = L kinetic − V (Φ , ¯Φ) . The kinetic term is given by L kinetic = G I ¯ J g µν ∂ µ Φ I ∂ ν ¯Φ ¯ J . This defines a non-linear sigmamodel with a target space metric G I ¯ J . The target space metric is specified by derivatives of a real scalar, G I ¯ J ≡ ∂∂ Φ I ∂∂ ¯Φ ¯ J K (Φ , ¯Φ) . (C.1)The above property ensures that the target space manifold is a K¨ahler manifold. For a review of the differentialgeometry aspects we refer the reader to Green, Schwartz, and Witten [99]. We refer to the scalar K (Φ , ¯Φ) as theK¨ahler potential. 35or ease of notation, we henceforth denote partial derivative by a comma, X ,I ≡ ∂∂ Φ I , X , ¯ I ≡ ∂∂ ¯Φ ¯ I . (C.2)In this notation the metric on field space is given by G I ¯ J = K ,I ¯ J . Canonical kinetic terms correspond to a flatfield space manifold, e.g., in Cartesian coordinates, G I ¯ J = δ I ¯ J .There are conventional coordinate systems (i.e., bases for the fields Φ I ) that lead to canonical kineticterms for the real scalars. The first is the shift-symmetric K¨ahler potential K = 12 (cid:0) Φ + ¯Φ (cid:1) or K = 12 (cid:0) Φ − ¯Φ (cid:1) , (C.3)where the + choice endows K with a shift symmetry in Im Φ , and the − choice with a shift symmetry in Re Φ .Expanding the complex scalar in terms of real fields φ and a as Φ = φ + ia √ (C.4)one finds canonical kinetic terms for φ and a . Note the factor of / √ is necessary for the correct normalizationof the kinetic terms. An alternative coordinate system leading to canonical kinetic terms is, K = Φ ¯Φ . (C.5)Decomposing the complex scalar as Φ = 1 √ φe ia , (C.6)again one finds canonical kinetic terms, properly normalized by the factor of / √ .In string theory the former choice appears in the Large Volume Scenario (LVS) [41] as the approximateK¨ahler potential for the K¨ahler moduli parametrizing the small cycles, and the latter choice is the approximateK¨ahler potential for D -brane moduli [100, 101]. In string scenarios the K¨ahler potential is given by the logof the inverse volume, K = 2 log V − , where V is the volume of the compactification, and depends on allK¨ahler moduli. In KKLT [40], which has a single K¨ahler modulus, the K¨ahler potential is given by K = − (cid:0) T + ¯ T (cid:1) , where Re T is the volume modulus of the compactification. This choice of K¨ahler potentialgenerates a negative Ricci scalar on the field space, and non-canonical kinetic terms. Generalizing the prefactorfrom to α , with α a positive real parameter, leads to the α -attractor models of inflation [102–104].We now turn to the potential energy of the bosonic components of the chiral superfields Φ I . After inte-grating out the auxiliary fields F I , one finds V (Φ , ¯Φ) = e K (Φ , ¯Φ) /M (cid:16) G I ¯ J D I W D ¯ J ¯ W ( ¯Φ) − M − W ¯ W (cid:17) , (C.7)where W (Φ) is a holomorphic function referred to as the superpotential, and where D I denotes a K¨ahler covari-ant derivative: D I ≡ ∂ I + M − K ,I and D ¯ I ≡ ∂ ¯ I + M − K , ¯ I , where we again note that the comma subscriptdenotes a partial derivative. This can be compactly written as, V = e K/M (cid:0) | DW | − M − | W | (cid:1) , (C.8)36here the norm of D I W has been taken with respect to the metric G I ¯ J . This may alternately be written as V = F I F I − M | m / | , where F I = D I W is the VEV of the auxiliary fields, and m / is the gravitinomass: m / (Φ) ≡ e K/ M Pl M − W (Φ) . (C.9)This corresponds to the massive Rarita-Schwinger field with a real mass via the identification of the mass with | m / | . The F I comprise the components of a field space vector (cid:126)F , and the direction of (cid:126)F is referred to as the“direction of supersymmetry breaking.”We note that the recent SUGRA literature has focused on models of constrained superfields. The mostwell studied example is that of a nilpotent superfield S ( x, θ ) , satisfying the constraint equation S ( x, θ ) = 0 .These models are the effective field theory describing KKLT in the case that all bosonic degrees of freedomhave been frozen out, e.g., by placing the anti-D3 brane directly on an O3 orientifold plane. The cosmology ofthese models was developed in e.g., [73–75]. Solving the above order-by-order in θ imposes a relation betweenthe scalar component S and fermionic component χ S : S = χ S χ S / F S . This implies that the bosonic sectorof the theory corresponds to imposing S = 0 . Note that this solution only exists for F (cid:54) = 0 . Note also that (cid:104) F (cid:105) = D S W . Thus, the nilpotent superfield framework only applies when D S W (cid:54) = 0 . We will discuss thisfurther below.The recent gravitino literature [37] has focused on supergravity models wherein an additional constraint ison imposed at the level of the full supergravity action, i.e., before integrating out the auxiliary fields. These so-called “orthogonal constrained superfield” models were were developed in [77–79], and in [46] were shown to bethe effective field theory of an anti-D3 brane in a KKLT compactification where the scalar fields, parametrizingthe position of the brane, are explicitly kept in the spectrum. These models add an additional constraint, in termsof a chiral superfield Φ : S ( x, θ ) · (cid:0) Φ − ¯ Φ (cid:1) = 0 . This constraint removes the D Φ W contribution to the scalarpotential [77–79]. For example, in a model with superpotential W = f (Φ) S + g (Φ) , satisfying constraints S ( x, θ ) = 0 and S ( x, θ ) · (cid:0) Φ − ¯ Φ (cid:1) = 0 , the scalar potential is given by V (Φ , ¯Φ) = e K/M (cid:0) | D S W (Φ) | − M − | g (Φ) | (cid:1) = e K/M ( | f | − | g | ) . (C.10)where we note the would be D Φ W = g (cid:48) is absent from the scalar potential.Before we proceed further, we note a limitation of the constrained superfield models. The discussion ofconstrained superfield models above, and in the cosmology literature, has been generally limited to the bosonictruncation of de Sitter supergravity [80–85]. As emphasized in [105], the construction and self-consistency ofthe supergravity theory demands that D S W (cid:54) = 0 [80–84]. For example, for W ( T, S ) = W ( T ) + Sf ( T ) onehas D S W ( T, S ) (cid:12)(cid:12) S =0 = f ( T ) (cid:54) = 0 . This requirement can be appreciated from two perspectives. The first is thatthe full Lagrangian contains fermionic interactions that scale with /f , i.e., there are fermionic interactions ofthe form L dS − SUGRA ⊃ L int ∼ f ( T ) ¯ χχ ¯ χχ , (C.11)where χ can be the fermionic component of S or other fields. For the explicit interactions see [80–85]. Thiswould suggest f → corresponds to a strong-coupling limit of the theory. However, this too is misleading as to37he severity of the problem. Recall that that the nilpotency condition S = 0 translates into 3 equations: S ( x ) = 0 , S ( x ) χ S = 0 , S ( x ) F − χ S χ S = 0 . (C.12)If F = D S W (cid:54) = 0 one may find the non-trivial solution S = χ S χ S / (2 F ) . Inserting this back in the action leadsto dS supergravity. On the other hand, if F = D S W = 0 there is only the trivial solution: S ( x ) = 0 , χ S ( x ) =0 , and F = 0 . Substituting this back in the full supergravity action one finds ‘textbook’ supergravity with notrace of the S superfield, i.e., one does not arrive at dS SUGRA. 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