Retrofitting Models of Inflation

I use the method of retrofitting, developed by Dine, Feng and Silverstein, to generate the scale of inflation dynamically, allowing it to be naturally small. This is a general procedure that may be performed on existing models of supersymmetric inflation. I illustrate this idea on two such models, one an example of F-term inflation and the other an example of D-term inflation.


Introduction
The results of WMAP [1] have put inflation on firm ground. However, inflation model building still faces many challenges. One in particular is the scale of inflation. To understand this, let me review how the inflation scale is determined in slow roll inflation [2]. During inflation, the potential energy is roughly constant, inf is the inflation scale. As the inflaton rolls down its potential, V , it produces a curvature perturbation with a spectrum given by where M p = 2.4 × 10 18 GeV is the reduced Planck mass and ǫ is one of the two standard slow roll parameters, with a prime denoting differentiation with respect to the inflaton. In (1.2) the right hand side is to be evaluated at the scale of horizon exit, which for most models of inflation corresponds to 50-60 e-folds before the end of inflation [3]. The number of e-folds, N (φ * ), from an inflaton value φ = φ * to the end of inflation at φ = φ e is given by In practice, ones uses (1.4) to determine the inflaton value for 50-60 e-folds before the end of inflation and then plugs this value into (1.6). In other words, (1.6) is to be evaluated during inflation, when V = V inf . Since a necessary condition for inflation is ǫ ≪ 1, which requires ǫ 1/4 < 1, we are lead to the bound V 1/4 inf < 0.027M p = 6.6 × 10 16 GeV.
Inflation models are often built such that V inf is made up of dimensionful parameters placed into the scalar potential by hand. In this paper I will take the fundamental scale to be the (reduced) Planck scale. Under this assumption, once these parameters are fit to the CMB constraint they can take on unnaturally small values. 2 I will use the method of retrofitting, developed by Dine, Feng and Silverstein [6], to generate the small parameters dynamically. This is a general procedure that may be applied to previously constructed models of inflation. It works by introducing a new supersymmetric sector, for example a pure SU (n) supersymmetric gauge theory, that at a scale Λ becomes strongly coupled. By coupling specific fields to this new supersymmetric sector and integrating out the massive gaugino condensates below the scale of condensation, a nonperturbative contribution to the superpotential is induced, introducing the naturally small scale Λ. I will demonstrate how to retrofit models of inflation with two specific examples, the first, in section 2, is an example of F -term inflation, and the second, in section 3, is an example of D-term inflation. Both of these models have an inflation scale around 10 15 GeV, and so are models that have parameters only a few orders of magnitude below the Planck scale. Regardless, they are simple, representative models of supersymmetric inflation that act as good examples for demonstrating the retrofitting technique. In both cases I will start by reviewing the models with the small parameters put in by hand and then I will retrofit them.
2 Retrofitting a Model of F -term Inflation

The Model
The model of F -term inflation I will retrofit was first constructed in [4,5] and is an example of hybrid inflation. The superpotential is where Φ is a singlet superfield whose scalar component is the inflaton, X and X are conjugate superfields oppositely charged under all symmetries so that XX is invariant and λ and µ are constants. Since there is only one dimensionful parameter in the model, µ, we should expect it to set the scale of inflation. We will see shortly that this is the case. The scalar potential is where I have used lowercase symbols for the scalar components of the superfields. If X and X are charged under a gauge symmetry there will also be a D-term. However, since we will be considering the D-flat direction |x| = |x|, it does not need to be considered. The second term in (2.2) may be written λxx − µ 2 2 = |λxx| − e iθ |µ| 2 2 , from which it is easy to see that θ = 0, to which it will henceforth be set. Without loss of generality I will take λ and µ to be real and positive. The scalar potential can now be written as The supersymmetric vacuum is at When the inflaton is above its critical value, |φ| > |φ c | = µ/ √ λ, then x =x = 0 is a minimum and the potential in this minimum is which causes inflation. In the notation of the introduction, V 1/4 inf = µ is the scale of inflation. A potential for the inflaton may be introduced by including the one-loop correction [5], given by the well known formula where m j,i is the i-th eigenvalue from the mass matrix for particles of spin j and Q is a momentum cutoff. The scalar fields have squared eigenmasses and the superpartner fermion squared eigenmasses are all equal to λ 2 |φ| 2 . All other contributions vanish during inflation since x =x = 0, leading to the one-loop scalar potential [5] If during inflation we assume |φ| ≫ |φ c | = µ/ √ λ then the potential reduces to Now that we have an inflaton potential, following the outline in the introduction we can use the CMB constraint (1.6) to determine the value of µ. This is done for potentials of the form (2.10) in the appendix. Borrowing the result (A.3) and taking λ = 1, I find (2.11)

Retrofitting
I will use the method of retrofitting [6] to generate the small parameter in (2.2), µ, dynamically. 3 This method begins by introducing a new supersymmetric sector. I will take this sector to be a pure SU (n) supersymmetric gauge theory with superfield strength W α . The superpotential (2.1) is then written without the µ term, W 0 = λΦXX, (2.12) and the new supersymmetric sector is included in the Lagrangian through the terms [6,8] Once this supersymmetric sector becomes strongly coupled and confines it induces a nonperturbative contribution to the superpotential, where b 0 is the β-function for the SU (n) gauge theory. The complete superpotential is the sum of (2.12) and (2.14) and is given by The constant term, Λ 3 , does not play a role in global supersymmetry since it does not enter the scalar potential. Comparing this superpotential with (2.1) we find and thus the dimensionful parameter that was placed in by hand in section 2.1 has been dynamically generated and can be naturally small.

Retrofitting a Model of D-term Inflation
The retrofitted model of D-term inflation will turn out to be rather unsatisfying. One of the primary motivations for D-term inflation is that the vacuum energy causing inflation comes entirely from the D-term of the scalar potential, and in so doing solves the supergravity η-problem. Retrofitting the model requires introducing a new dimensionful parameter and effectively trading the vacuum energy coming from the D-term for vacuum energy coming from the F -term, defeating the original motivation. This immediately introduces the η-problem. As I will explain, the η-problem can be solved for certain choices of the Kähler potential, and is no longer solved for a general Kähler potential.

The Model
The model of D-term inflation that I will retrofit is the original model [9] and, like the F -term model above, is an example of hybrid inflation. The superpotential is where Φ is a singlet superfield whose scalar component is the inflaton, X + and X − are charged under a U (1) gauge symmetry with charges +1 and −1 and λ is a constant. The scalar potential is where the final term is the D-term, a Fayet-Iliopoulos term, ξ, has been included and g is the U (1) coupling constant. Since the Fayet-Iliopoulos term is the only dimensionful parameter in the theory, we will find that it sets the scale of inflation. Without loss of generality I will take λ and ξ to be real and positive. The supersymmetric vacuum is at 4 When the inflaton is above its critical value, |φ| > |φ c | = g √ ξ/λ, then x + = x − = 0 is a minimum and the potential in this minimum is which causes inflation. In the notation of the introduction, V 1/4 inf = √ gξ/2 1/4 is the scale of inflation. A potential for the inflaton may be introduced by including the one-loop correction, using (2.7). The scalar fields have squared eigenmasses m 2 ± = λ 2 |φ| 2 ± g 2 ξ (3.5) 4 If I take ξ to be negative then the VEVs for x+ and x− switch. and the superpartner fermion squared eigenmasses are all equal to λ 2 |φ| 2 . All other contributions vanish during inflation since x + = x − = 0, leading to the one-loop scalar potential (3.6) If during inflation we assume |φ| ≫ |φ c | = g √ ξ/λ then the potential reduces to Now that we have an inflaton potential, following the outline in the introduction we can use the CMB constraint (1.6) to determine the value of gξ. This is done for potentials of the form (3.7) in the appendix. Borrowing the result (A.3) and taking g = 1, I find ξ ∼ 10 −3 M p ∼ 10 15 GeV. (3.8)

Modifying the Model for Retrofitting
Before the above model can be retrofitted, it must be modified [6]. A new dimensionful parameter, m, will be introduced which will effectively replace the parameter ξ in the inflation scale (though, as we will see, it will not amount to a complete replacement). This new parameter will then be retrofitted. The modification begins by adding two new superfields, A + and A − , with charges +1 and −1 under the U (1) gauge symmetry. These new superfields enter the superpotential through a mass term, 5 The scalar potential is where, without loss of generality, I have taken m to be real and positive. The supersymmetric vacuum is at For the inflaton above its critical value, |φ| > |φ c | = m/λ, and for ξ > m 2 /g 2 , there is a minimum at 12) and the potential in this minimum is If we assume ξ ≫ m 2 /2g 2 , then the scalar potential reduces to V = m 2 ξ, (3.14) which comes entirely from the F -term of the scalar potential. The fact that the F -term of the scalar potential is contributing to the vacuum energy during inflation introduces the supergravity η-problem, which I will explain in the next subsection. As before, to introduce an inflaton potential we include the one loop correction using (2.7). From (3.12) we can see that not all of the charged fields have vanishing VEVs. This allows for contributions from gauginos and the gauge boson. However, since the VEVs in (3.12) are all φ-independent, so too are the new contributions. Ignoring these constant, φ-independent contributions, the one loop potential is If during inflation we assume |φ| ≫ |φ c | = m/λ then the potential reduces to 6 If we take ξ ∼ O(M 2 p ), we can see from (3.14) and (3.16) that the addition of the new fields has effectively replaced the inflation scale being determined by gξ (see (3.4)) with it being determined by m. The retrofitting technique used in section 2 can now be used to generate m dynamically, which will be described in section 3.4.

Supergravity and the η-Problem
In the modified model the F -term of the scalar potential is contributing to the vacuum energy during inflation. For this reason retrofitting this model is rather unsatisfying since one of the primary motivations for D-term inflation is that the vacuum energy comes entirely from the Dterm. To see why a contribution from the F -term is problematic we must consider supergravity corrections to the scalar potential.
For convenience, in this subsection I will set the reduced Planck mass equal to one: M p = 1. In supergravity 7 the F -term contribution to the scalar potential is given by where an unbarred subscript denotes differentiation with respect to a chiral superfield and a barred subscript denotes differentiation with respect to an antichiral superfield and K mn is the inverse Kähler metric. Plugging in the superpotential (3.9) and a minimal Kähler potential, 18) 6 In the appendix I mention the problem of Planckian field values in these models. This problem is remedied if the loop correction is suppressed (leading to a small value for C in (A.2)). Once m is retrofitted it can be naturally small, which from (3.16) leads to a suppression of the loop correction. 7 When there is a fixed Fayet-Iliopoulos term, a nonstandard supergravity generalization of global supersymmetry should be used [10]. However, I will use the standard supergravity generalization to explain the η-problem. and placing the fields at their VEVs (3.12), gives V = e K (m 2 ξ) = e ξ m 2 ξ 1 + |φ| 2 + O(|φ| 4 ) , (3.19) where I have assumed, as before, that ξ ≫ m 2 /2g 2 . This potential does not lead to inflation. In particular, inflation requires |η| ≪ 1, where η is the slow roll parameter in (1.3), but from (3.19) we find |η| ∼ 1. This is a generic consequence of the F -term contributing vacuum energy during inflation 8 and is known as the η-problem.
One way to solve this problem is to use a non-minimal Kähler potential. For example, a Kähler potential of the no-scale form that comes from string theory [4,11,12] (3.20) The F -term contribution to the scalar potential can be calculated using (3.17) and is (3.21) Plugging in the superpotential (3.9) and placing the fields at their VEVs (3.12) we can immediately see that only the first term survives and reproduces the F -term from global supersymmetry, solving the η-problem.
More generally, some fields could enter the Kähler potential minimally, so that we could have For this to still solve the η-problem we we need at least the inflaton, φ, and a + to enter the Kähler potential in the form (3.20), i.e. as φ a fields [11]. I will take the rest of the fields to enter the Kähler potential minimally, i.e. as φ b fields. The F -term can again be calculated using (3.17), and plugging in the superpotential (3.9) and placing the fields at their VEVs (3.12), I find where, as before, I have assumed ξ ≫ m 2 /2g. The exponential comes from a − entering the Kähler potential minimally . The D-term contribution to the scalar potential is given by where the q m are the U (1) charges and in the second line I plugged in the Kähler potential (3.22). Note that for the choice of the inflaton, φ, and a + entering the Kähler potential as φ a fields and the rest of the fields entering minimally as φ b fields, when the fields are placed at their VEVs (3.12) the D-term (3.24) gives the same result as in global supersymmetry (remembering, of course, that the inflaton is uncharged). Thus, when using (3.22) as the Kähler potential, the η-problem is solved and the analysis of inflation at the supergravity level is the same as in global supersymmetry, along with the slight modification that the inflation scale is now given by (3.23). Finally, the Kähler potential (3.20) leads to a noncanonical kinetic term for the inflaton. If we ignore the phase, a canonically normalized inflaton can be included in the scalar potential through the substitution

Retrofitting
Retrofitting the parameter m begins by introducing a new supersymmetric sector. As before I will take this to be a pure SU (n) gauge theory with superfield strength W α . The superpotential (3.9) is then written without the m term, Once the supersymmetric sector becomes strongly coupled and confines it induces a nonperturbative contribution to the superpotential, The complete superpotential is the sum of (3.26) and (3.28) and is given by The constant term, Λ 3 , does not play a role in global supersymmetry since it does not enter the scalar potential. 10 Comparing this superpotential with (3.9) we find that From (3.23) this means that the vacuum energy during inflation is now For ξ ∼ O(M 2 p ) the scale of inflation is effectively set by the dynamically generated scale Λ. In this way the Fayet-Iliopoulos term, that was placed in by hand in section 3.1, has been effectively replaced by a dynamically generated one.

Conclusion
Dimensionful parameters in models of inflation are often put in by hand. Some combination of these parameters set the scale of inflation and, when fit against the CMB constraint, can turn out to be unnaturally small. I have used the method of retrofitting, developed by Dine, Feng and Silverstein [6], to generate these parameters dynamically, allowing them to be naturally small. This is a general procedure that may be applied to existing models of inflation. I have illustrated this idea with two such models, one an example of F -term inflation and the other an example of D-term inflation. The retrofitted model of D-term inflation is rather unsatisfying since the vacuum energy during inflation no longer comes entirely from the D-term, introducing the η-problem, which can be solved for certain choices of the Kähler potential. In the course of retrofitting the D-term model I have ignored the production of cosmic strings [14], which leads to various bounds on the parameters of the model. It might be interesting to consider these bounds within the context of retrofitting.

Acknowledgments
I cannot thank enough Linda M. Carpenter for her comments and for reading a draft of this paper. I am also grateful to Rich Barber and Dennis Smolarski.

A Scale of Inflation
In this appendix I will determine the general form of the scale of inflation for the models considered above. In both models, the scalar potential is of the form where C is a dimensionless constant and coefficients of φ have been absorbed into Q. Inflation ends when either of the slow roll parameters in (1.3), ǫ and |η|, are equal to one or when φ reaches its critical value, whichever occurs first. However, we do not need to know when this occurs becuase the integral (1.4) is dominated by φ * . Evaluating (1.4) for the potential (A.1) I find Notice that φ * is near the Planck scale for 50-60 e-folds . This is problematic since these models are built at the level of an effective field theory where field values must be much lower than the Planck scale. Using this value for the inflaton and applying the CMB constraint (1.6) I find for the scale of inflation.