KUL-TF-99/12

SU-ITP-99/19

CITA-99-16

hep-th/9907124

Gravitino Production After Inflation

Renata Kallosh, Lev Kofman, Andrei Linde, and Antoine Van Proeyen

Department of Physics, Stanford University, Stanford, CA 94305, USA

CITA, University of Toronto, 60 St George Str, Toronto, ON M5S 3H8, Canada

Instituut voor Theoretische Fysica, Katholieke Universiteit Leuven,

Celestijnenlaan 200D B-3001 Leuven, Belgium

Abstract

We investigate the production of gravitinos in a cosmological background. Gravitinos can be produced during preheating after inflation due to a combined effect of interactions with an oscillating inflaton field and absence of conformal invariance. In order to get insight on conformal properties of gravitino we reformulate phenomenological supergravity in )-symmetric way. The Planck mass and F- and D-terms appear via the gauge-fixed value of a superfield that we call conformon. We find that in general the probability of gravitino production is not suppressed by the small gravitational coupling. This may lead to a copious production of gravitinos after inflation. Efficiency of the new non-thermal mechanism of gravitino production is very sensitive to the choice of the underlying theory. This may put strong constraints on certain classes of inflationary models. Onderzoeksdirecteur, FWO, Belgium

## 1 Introduction

The possibility of excessive production of gravitinos is one of the most complicated problems of cosmological models based on supergravity. Such particles decay very late, and lead to disastrous cosmological consequences unless the ratio of the number density of gravitinos to the entropy density is extremely small. For example, the ratio of the number density of gravitinos to the entropy density should be smaller than for gravitinos with mass GeV [1, 2]. The standard thermal mechanism of gravitino production involves scattering of particles at high temperature in the early universe. To avoid excessive production of gravitinos one must assume that the reheating temperature of the universe after inflation was smaller than GeV [1, 2].

However, gravitinos can be produced not only in the thermal bath after reheating, but even earlier, during the oscillations of the inflaton field at the end of inflation. We already know that bosons as well as spin 1/2 fermions can be copiously produced by the coherently oscillating inflaton field. Quite often this effect occurs in a non-perturbative way during the stage of preheating [3, 4]. Similar effect may occur for gravitinos. According to [5], nonthermal gravitino/moduli production may rule out certain classes of inflationary models which otherwise would be quite legitimate.

The theory of the cosmological gravitino production is very
complicated. Recently production of transversal gravitino
components (helicity 3/2) was studied in Ref. [6] (see also
[7] where an attempt has been made to study this
question using perturbation theory). In this paper we will
investigate production of all gravitino components, transversal
(helicity 3/2) and longitudinal (helicity 1/2). As we will show,
the production rate of the longitudinal
gravitino component can be much greater than that of the
transversal gravitino components.^{1}^{1}1In flat spacetime,
transversal components, which exist even if supersymmetry is
unbroken, correspond to the helicity 3/2, while longitudinal
components with non-vanishing and
correspond to the helicity 1/2. Although the helicity concept has
a less precise meaning in FRW metrics, we still will use this
loose definition as a shortcut to the transversal and longitudinal
components.

Since gravitino is a part of the gravitational multiplet, one could expect that their production must be strongly suppressed by the small gravitational coupling. Indeed, usually the production of particles occurs because their effective mass changes nonadiabatically during the oscillations of the inflaton field [3]. This is the main effect responsible for the production of the gravitinos with helicity 3/2. The gravitino mass at small values of the inflaton field is proportional to , where is a superpotential. Thus the amplitude of the oscillations of the gravitino mass is suppressed by . That is why production of the gravitino components with helicity 3/2 is relatively inefficient [6, 7].

There is another mechanism to be considered, which is related to breaking of conformal invariance [5]. It is well known, for example, that expansion of the universe does not lead to production of massless vector particles and massless fermions of spin 1/2 because the theory of such particles is conformally invariant and the Friedmann universe is conformally flat. Meanwhile, massless scalar particles minimally coupled to gravity (as well as gravitons) are created in an expanding universe because the theory describing these particles is not conformally invariant. The rate of scalar particle production is determined by the Hubble constant , where is the energy density. If similar effects are possible for gravitinos, they could be much stronger than the effects discussed in the previous paragraph. Indeed, is suppressed only by the first degree of . As a result typically is much greater than after inflation, so its time-dependence may lead to a more efficient particle production.

The issue of conformal invariance of gravitinos is rather nontrivial, and until now it has not been thoroughly examined. For gravitinos with helicity 3/2 the effects proportional to do not appear, and therefore the violation of conformal invariance for such particles is very small, being proportional to their mass [6]. However, as we will show, the theory of gravitinos with helicity 1/2 is not conformally invariant, and therefore such particles will be produced during the expansion of the universe even if one neglects their mass.

But the most surprising effect which we have found is that the production of the gravitinos of helicity 1/2 by the oscillating scalar field in general is not suppressed by any powers of , and therefore their production can be very efficient. The magnitude of the effect is model-dependent. For example, this effect does not occur in the simplest model of a single chiral multiplet with a quadratic effective potential . Meanwhile this effect is very strong in the theory with the effective potential . Gravitinos in this theory are produced very quickly, within about ten oscillations of the inflaton field, with occupation numbers over a large range of momenta , which is very different from usual perturbative production.

This result may have important cosmological implications, since it may allow to rule out certain classes of cosmological theories. The nature of this effect resembles the well known fact that the longitudinal components of massive vector bosons at high energy behave in the same way as the Goldstone boson that was eaten by the vector field [8]. A similar effect is known to exist in the theory of technicolor [9]. A more direct analogy is the transmutation of gravitational interactions of gravitinos with helicity 1/2 to weak interactions found by Fayet [10]. In our case the effect is nonperturbative, and its adequate interpretation is achieved by finding solutions of the gravitino equations in a nontrivial self-consistent cosmological background.

In order to study conformal properties of gravitinos we reformulated the standard N=1 phenomenological supergravity in an SU-invariant way, which makes the conformal properties of the theory manifest and explains how the conformal symmetry is broken. Our formulation describes arbitrary number of chiral and vector multiplets and is flexible enough to allow investigation of regimes where the superpotential vanishes.

In application to the theory of gravitino production, we concentrate on the simplest models with one chiral multiplet and arbitrary superpotential. We present classical equations of motion and constraints for the transverse and longitudinal gravitino in the expanding Friedmann universe interacting with the moving inflaton field. We use the gauge where the goldstino is absent. Then we solve classical equations for gravitino. This solution confirms the generic prediction from the -symmetric theory that the longitudinal gravitinos are not conformal.

We represent equations describing gravitino components with helicities 3/2 and 1/2 in a form analogous to the equations for the usual spin 1/2 fermions with time-dependent mass. This allows to reduce, to a certain extent, the problem of gravitino production to the problem of production of particles with spin 1/2 after preheating [4].

Finally, we estimate the number density of gravitinos produced by the oscillating scalar field in several inflationary models, and show that in some models the ratio may substantially exceed the bound .

A detailed account of our investigation will be given in a separate publication [11]. Here we will only outline the main points of our study and present the most interesting results.

## 2 Supergravity Lagrangian and conformal properties of gravitino

Fundamental M-theory, which should encompass both supergravity and string theory, at present experiences rapid changes. One may still expect that the low-energy physics will be described by the N=1 d=4 supergravity [12] and address the issues of the early universe cosmology in the context of the most general phenomenological N=1 supergravity–Yang–Mills-matter theory [13].

We are interested in conformal properties of supergravity fields, which include various spin fields, in the conformally flat FRW metric describing the early universe:

(2.1) |

In particular, we will be interested in conformal properties of gravitino. Supergravity is not a conformally invariant theory, despite the fact that it has a long history of being derived using the superconformal tensor calculus [14] as a technical tool. Therefore it is difficult even to address this issue as the supergravity fields do not have specific conformal weights. To solve this problem, the idea is to view the supergravity theory as a gauge-fixed version of the conformally invariant theory describing the most general gauge theory superconformally coupled to supergravity [11]. The derivation invariant Lagrangian of supergravity coupled to chiral multiplets (with complex scalars and fermions ) and Yang–Mills vector multiplets (with gauginos and vectors ) superconformally will be presented in [11]. It has no dimensional parameters. It consists of 3 parts, depending, respectively, on a real function , a holomorphic function and a gauge group 2-tensor . Each of them is conformally invariant by itself.

(2.2) |

The statement of the symmetry of the action (2.2) includes, among others, the symmetries under the following set of local dilatations, with parameter , for the metric and gravitino, for the scalars and spinors of the chiral multiplets, and for the vectors and spinors of the gauge multiplet, respectively:

(2.3) |

The function of scalars codifies the information on Kähler manifold. The holomorphic function of scalars codifies the superpotential. They transform as follows under local dilatations:

(2.4) |

The important term in the conformal action which allows us to distinguish between conformal properties of helicity and gravitino is the following:

(2.5) |

The gauge fixing of the local dilatation of the conformally
invariant action presented in [11] leads to the
standard^{2}^{2}2 In fact two different gauges can be used to fix
the -part of the superconformal symmetry. With the first
choice we get the N=1 phenomenological
supergravity [13] depending on (with
), while the
second one (see notation below) gives the version which is non-singular
in the limit of the vanishing superpotential .
This version is closer to the one in [15] which was obtained
by the superspace methods. The limit from the
first version has been discussed at the end of [16].
However, we find the second version more suitable for cosmology.
It will be presented below. Poincaré supergravity theory.

The dimensionful constants, Plank mass and F- and D-terms, appear via the gauge fixed value of the conformal compensator superfield, which we call conformon. The original complex variables are split into one complex scalar conformon field , and physical complex scalars , which are hermitian coordinates for parametrizing the Kähler manifold in the Poincaré theory. One defines

(2.6) |

and the local dilatation takes the form in which only the conformon transforms and the physical scalars do not transform under the local dilatation

(2.7) |

In these variables the gauge-fixing of the dilatational invariance (2.3), (2.7) is given by

(2.8) |

where the first equation defines , which is the Kähler potential, and the second is the gauge fixing. Here GeV. Thus the conformon field is frozen to , i.e. it becomes a functional of the Kähler potential and not an independent field. The theory becomes that of the Poincaré supergravity theory with

(2.9) |

Here , where is a covariant derivative. The local dilatation of the metric and gravitino in (2) is not compensated anymore by the local dilatation of scalars,

(2.10) |

The same happens with the part of the theory:

(2.11) |

We have chosen here to give mass dimension 3.

Thus, after the freezing of the conformon field some part of the transformation cannot be performed and therefore some parts of the phenomenological supergravity Lagrangian are not invariant under dilatations. One can try to change the dilatational weight for these fields to compensate the appearance of powers of . However, this does not help, since the terms with derivatives on the conformon field are absent after the gauge fixing. The gravitino field equation which follows from the superconformal action is

(2.12) |

In the FRW cosmological problems only time derivatives of the scalar fields are important, therefore in only the term is relevant. After gauge fixing the conformal symmetry will be broken for configurations for which either

(2.13) |

Only such terms will be sensitive to the absence of the terms
due to gauge
fixing^{3}^{3}3In [17], where an attempt to study conformal
properties of gravitino has been made, it was assumed that
and conformal symmetry of gravitino
was
deduced in the context of pure supergravity. Without scalars,
however, pure supergravity does not support a cosmological
background. In the presence of matter the assumption that
is not valid
and conformal symmetry is broken. when . The
gravitino in the general theory with spontaneously broken
supersymmetry will be massive. The states of a free massive spin
3/2 particle were studied by Auvil and Brehm in [18] (see
also [2] for the nice review). A free massive gravitino
has . Helicity states are given
by transverse space components of gravitino, . Helicity
states are given by the time component of the gravitino
field . In cases when gravitino interacts with gravity and
other fields, we will find that
. It will be a function of . Thus the
consideration of superconformal symmetry lead us to a conclusion
that helicity states of gravitino are not conformally
coupled to the metric. When these states are absent, the
helicity states are conformally coupled (up to the mass terms, as
usual). Thus the conformal properties of gravitino are simple, as
it is known for scalars: if the action has an additional term
, the massless scalars are conformal. If
this term is absent, the scalars are not conformal. Note that both
these statements are derivable from the superconformal action. We
will see the confirmation of this prediction in the solutions of
the gravitino equations below.

As we already explained, our formulation starting with superconformal action [11] provides flexibility in the choice of the form of phenomenological supergravity. If we take gauge for -symmetry, we get the action [13] depending on the combination of the Kähler potential and the superpotential, called . Here we use the gauge for -symmetry and present the form of the phenomenological Lagrangian in which the Kähler potential and the superpotential are not combined in one function. This allows to avoid problems which sometimes appear when the superpotential vanishes. The action can be written as

The and denote left and right chirality, e.g. , while for the , the chirality is indicated by the position of the index: is left chiral, while is right chiral. For the scalars, is the complex conjugate of . The Kähler metric is , which is used also for covariant derivatives and Kähler curvature

(2.15) |

Extra indices on quantities, e.g. denote derivatives, here the derivative of with respect to . Other covariant derivatives and notations are (antisymmetrization with weight 1, metric signature , notation and )

(2.16) | |||||

where is a symbol for the transformation under the gauge group for all fields. For the conformon field and for the rest of the scalars we have

(2.17) |

where and are holomorphic functions for every symmetry, such that the quantities in (2.2) are invariant. That determines also

(2.18) |

See [11] for details.

The appearance of in various places in this Lagrangian shows that the conformal symmetry is broken. One can rescale the fields with so that they have standard kinetic terms. For our purpose it will be convenient to replace the scalar field by , chiral fermions by , and similar for the gravitino, .

## 3 Gravitino equations

In general background metrics in the presence of complex scalar fields with non-vanishing VEV’s, the starting equation for the gravitino has in the left hand side the kinetic part and a rather lengthy right hand side which will be given in [11]. Apart of varying gravitino mass , the right hand side contains a chiral connection (see (2.16)) and various mixing terms like those in the 4th, 5th and 6th lines of the phenomenological Lagrangian (2). For a self-consistent setting of the problem, the gravitino equation should be supplemented by the equations for the fields mixing with gravitino, as well as by the equations determining the gravitational background and the evolution of the scalar fields.

Let us make some simplifications. We consider the supergravity
multiplet and a single chiral multiplet
containing
a complex scalar field with a superpotential
and a single chiral fermion .
This is a simple non-trivial extension which allows to study
gravitino in the non-trivial FRW cosmological metric
supported by
the scalar field.
A nice feature of this model is that
the chiral fermion
can be gauged to zero so that the mixing between and
in (2) is absent. We also can choose the
non-vanishing VEV of the scalar field in the real direction, , , so that . The field plays the role of the
inflaton field.^{4}^{4}4Typical time evolution of the homogeneous
inflaton field
starts with the regime of inflation when slowly
rolls down.
One can construct a superpotential which provides
chaotic inflation for .
When drops below , it begins to oscillate
coherent oscillations around the minimum
of its the effective potential .
Then from (2) we can obtain the master
equation for the gravitino field

(3.1) |

where gravitino mass is given by

(3.2) |

Gravitino equation (3.1) is a curved spacetime generalization of the familiar gravitino equation in a flat metric, where is a constant gravitino mass.

The generalization of the constraint equations and reads as

(3.3) | |||

(3.4) |

where stands for a conformal time derivative . It is important that the covariant derivative in these equations must include both the spin connection and the Christoffel symbols, otherwise equation used for the derivation of these equations is not valid.

The last equation will be especially important for us. Naively, one could expect that in the limit , gravitinos should completely decouple from the background. However, this equation implies that this is not the case for the gravitinos with helicity 1/2. Indeed, from (3.4) one can find an algebraic relation between and :

(3.5) |

Here is a matrix which will play a crucial role in our description of the interaction of gravitino with the varying background fields. If and are the background energy-density and pressure, we have , , and one can represent the matrix as follows:

(3.6) |

Note that in the limit with fixed in , one has . If does not blow up in this limit, this matrix is given by

(3.7) |

where stands for derivative , and the relation between physical and conformal times is given by . In the limit of flat case without moving scalars, .

For definiteness, we will consider the minimal Kähler potential . In the models where the energy-momentum tensor is determined by the energy of a classical scalar field and depends only on time we have

(3.8) |

We will use the representation of gamma matrices where
. Then in (3.6) the
combination emerges. For a single
chiral multiplet we obtain . (One can show that for
the theories with one chiral multiplet even if the Kähler is
not minimal.)
Therefore
can be represented as^{5}^{5}5Initial conditions at
inflation at correspond to ,
and ,
which gives . Alternatively, we can start with
inflaton oscillations at , which defines the phase up to
some constant.
The final results depend only on .

(3.9) |

Using the Einstein equations, one obtains for (for minimal Kähler potential, and real scalar field):

(3.10) | |||||

The expression for becomes much simpler and its interpretation is more transparent if the amplitude of oscillations of the field is much smaller than . In the limit one has

(3.11) |

This coincides with the mass of both fields of the chiral multiplet (the scalar field and spin 1/2 fermion) in rigid supersymmetry. When supersymmetry is spontaneously broken, the chiral fermion, goldstino, is ‘eaten’ by gravitino which becomes massive and acquires helicity states in addition to helicity states of the massless gravitino.

The matrix does not become constant in the limit . The phase (3.9) rotates when the background scalar field oscillates. The amplitude and sign of change two times within each oscillation. Consequently, the relation between and also oscillates during the field oscillations. This means that the gravitino with helicity 1/2 (which is related to ) remains coupled to the changing background even in the limit . In a sense, the gravitino with helicity 1/2 remembers its goldstino nature. This is the main reason why the gravitino production in this background in general is not suppressed by the gravitational coupling. The main dynamical quantity which is responsible for the gravitino production in this scenario will not be the small changing gravitino mass , but the mass of the chiral multiplet , which is much larger than . As we will see, this leads to efficient production of gravitinos in the models where the mass of the ‘goldstino’ nonadiabatically changes with time.

We shall solve the
master equation (3.1) using the constraint equations
in the form (3.4) and (3.3). We use plane-wave
ansatz for the
space-dependent part. Then can be decomposed^{6}^{6}6We
use now with for the space components of , while for gamma matrices are space components
of flat , and similarly for the index.
[19] into
its transverse part
, the trace and the trace :

(3.12) |

where , so that . We will relate with and with , so that, after use of the field equations there are two degrees of freedom associated with the transverse part , which correspond to helicity ; and two degrees of freedom associated with (or ) which correspond to helicity .

For the helicity states we have to derive the equation
for . We apply decomposition (3.12) to the master
equation (3.1) for and obtain^{7}^{7}7A similar
equation obtained in [6] has a different coefficient in the
term since they have omitted the
Christoffel symbols in the covariant derivative. One can still use
their equation if one replaces the curved space gravitino vector
for which the equation was derived by the tangent space
vector .

(3.13) |

In the limit of vanishing gravitino mass, the transverse part is conformal with a weight . The transformation reduces the equation for the transverse part to the free Dirac equation with a time-varying mass term . It is well known how to treat this type of equations (e.g. see [4]). The essential part of is given by the time-dependent part of the eigenmode of the transversal component , which obeys second-order equation

(3.14) |

where the effective mass is .

The corresponding equation for gravitino with helicity 1/2 is more complicated. We have to find and . The equation for the components can be obtained from the constraint equation (3.3):

(3.15) |

Combining all terms together, we obtain the on-shell decomposition for the longitudinal part

(3.16) |

Now we can derive an equation for . From the zero component of (3.1) we have

(3.17) |

This equation does not contain the time derivative of . Substituting from (3.5) into (3.17), we get an equation for

(3.18) |

where

(3.19) |

and

(3.20) |

We can split the spinors in eigenvectors of , and . From the Majorana condition it follows that where is the charge conjugation matrix. In a representation with diagonal the components correspond to the -eigenvalues . Acting on (3.18) with the hermitian conjugate operation gives us a second-order differential equation on the . We choose for each a spinor basis for the two components of , and two independent solutions of the second-order differential equations . The general solution is given by

(3.21) |

The last equality determines reality properties of the coefficients. Here we represented as and defined , by analogy with the definitions for the matrix . By the substitution , with equation for the functions is reduced to the final oscillator-like equation for the time-dependent mode function :

(3.22) |

Here

(3.23) |

In the derivation of (3.22) it was essential that has the form (3.9).

Finally we give the expression for the energy density of the longitudinal mode

(3.24) |

In the flat spacetime limit , . From we can define the occupation number of gravitinos of energy at given mode , where is expressed through a bilinear combination of mode functions .

## 4 Gravitino production

One could expect that the gravitino production may begin already at the stage of inflation, due to the breaking of conformal invariance. However, there is no massive particle production in de Sitter space (i.e. as long as one can neglect the motion of the scalar field during inflation). Indeed, expansion in de Sitter space is in a sense fictitious; one can always use coordinates in which it is collapsing or even static. An internal observer living in de Sitter space would not see any time-dependence of his surroundings caused by particle production; he will only notice that he is surrounded by particle excitations at the Hawking temperature .

Gravitino production may occur at the stage of inflation due to the (slow) motion of the scalar field, but the most interesting effects occur at the end of inflation, when the scalar field rapidly rolls down toward the minimum of its effective potential and oscillates there. During this stage the vacuum fluctuations of the gravitino field are amplified, which corresponds to the gravitino production (in a squeezed state).

Production of gravitinos with helicity 3/2 is described in terms of the mode function . This function obeys the equation (3.14) with , which is suppressed by . Non-adiabaticity of the effective mass results in the departure of from its positive frequency initial condition , which can be interpreted as particle production. The theory of this effect is completely analogous to the theory of production of usual fermions of spin 1/2 and mass [4]. Indeed, Eq. (3.14) coincides with the basic equation which was used in [4] for the investigation of production of Dirac fermions during preheating.

The description of production of gravitinos with helicity 1/2 is similar but somewhat more involved. The wave function of the helicity 1/2 gravitino is a product of the factor and the function . The factor does not depend on momenta and controls only the overall scaling of the solution. It is the function that controls particle production which occurs because of the non-adiabatic variations of the effective mass parameter . The function obeys the equation (3.22) with the effective mass , which is given by t