Supersymmetry. Theory, Experiment, and Cosmology

Inflation scenarios 319

energy (and modulus mass arises through supersymmetry breaking). It is, however, possible to ease the bound by allowing for a late period of inflation. This requires a rather low reheat temperature, estimated to be smaller than 108 or 109 GeV [248] (or even lower if the gravitino has hadronic decay modes).

11.3Inflation scenarios

Since the central prediction of inflation, namely that the total energy density of the Universe is very close to the critical energy density ρc for which space is flat (i.e. Ω ≡ ρ/ρc 1), seems to be in good agreement with observation, any theory of fundamental interactions should provide an inflation scenario.

Such a scenario was first imagined by Guth [214] in the context of the phase transition associated with grand unification. There is thus an obvious connection with supersymmetry. Let us see indeed why supersymmetry provides a natural setting for inflation scenarios.

We recall in Section D.4 of Appendix D that a standard scenario for inflation involves a scalar field φ evolving slowly in its potential V (φ). This is obtained by requiring two conditions on the form of the potential:

In the de Sitter Phase, i.e. in the phase of exponential growth of the cosmic scale factor, quantum fluctuations of the scalar field value are transmitted to the metric. Because the size of the horizon is fixed (to H−1) in this phase, the comoving scale a/k associated with these fluctuations eventually outgrows the horizon, at which time the fluctuations become frozen. It is only much later when the Universe has recovered a radiation or matter dominated regime that these scales reenter the horizon and evolve again. They have thus been protected from any type of evolution throughout most of the evolution of the Universe (this is in particular the case for the fluctuations on a scale which reenters the horizon now). Fluctuations in the cosmic microwave background provide detailed information on the fluctuations of the metric. In particular, the observation by the COBE satellite of the largest scales puts a important constraint on inflationary models. Specifically, in terms of the scalar potential, this constraint known as COBE normalization, reads:

Using the slow-roll parameter introduced above, this can be written as

In most of the models that we will be discussing, ε is very small. However as long as ε 10−52, we have V 1/4 1 TeV. In other words, the typical scale associated with inflation is then much larger than the TeV, in which case it makes little sense to work outside a supersymmetric context.

322 Supersymmetry and the early Universe

Φ0, Φ+ and Φ− with charges equal to 0, +1 and −1, respectively. The superpotential has the form

which can be justified by several choices of discrete or continuous symmetries and in particular by R-symmetry. The scalar potential in the global supersymmetry limit reads:

where g is the gauge coupling and ξ is a Fayet–Iliopoulos D-term (which we choose to be positive). This system has a unique supersymmetric vacuum with broken gauge symmetry

that for |φ0| > φc ≡ g ξ/λ, the minimum is at φ+ = φ− = 0. Thus, for |φ0| > φc and φ+ = φ− = 0 the tree level potential has a vanishing curvature in the φ0 direction and large positive curvature in the remaining two directions (m2± = λ2|φ0|2 g2ξ). Along the φ0 direction (|φ0| > φc, φ+ = φ− = 0), the tree level value of the potential remains constant: V = g2ξ2/2 ≡ V0. Thus φ0 provides a natural candidate for the inflaton field.

Along the inflationary trajectory all the F -terms vanish and the universe is dominated by the D-term which splits the masses of the Fermi–Bose components in the φ+ and φ− superfields. Such splitting results in a one-loop e ective potential (see Section A.5.3 of Appendix Appendix A). In the present case this potential can

where ϕ ≡ |φ0| and C 1.

Along this potential, the value of ϕ that leads to the right number N 50 of

This is safely of order g in the model that we consider but might be dangerously close to 1 for models which yield a larger value for C (see below). The values of the slow-roll parameters (11.21) and (11.22) are correspondingly

Let us now consider the supergravity extension of our model. For definiteness we will assume canonical normalization for the gauge kinetic function f and the K¨ahler potential (i.e. K = |φ−|2 + |φ+|2 + |φ0|2; note that this form maximizes the problems for the F -type inflation). The scalar potential reads

Again for values of |φ0| > φc, other fields than φ0 vanish and the behavior is much similar to the global supersymmetry case. The zero tree level curvature of the inflaton potential is not a ected by the exponential factor in front of the first term since this term is vanishing during inflation. This solves the problems of the F -type inflation.

Let us now consider the case of a pseudo-anomalous U (1) symmetry. Such symmetries usually appear in the context of string theories and have been discussed in Section 10.4.5 of Chapter 10: the anomaly is cancelled by the Green–Schwarz mechanism [205]. To see how D-term inflation is realized in this case, let us consider the simple example of such a U (1) symmetry under which n+ chiral superfields Φi+ and n− superfields ΦA− carry one unit of positive and negative charges, respectively. For definiteness let us assume that n− > n+, so that the symmetry is anomalous and Tr Q = 0. We assume that some of the fields transform under other gauge symmetries, since the Green–Schwarz mechanism requires nonzero mixed anomalies. Let us introduce a single gauge-singlet superfield Φ0. Then the most general trilinear coupling of Φ0 with the charged superfields can be put in the form:

n+

(for simplicity we assume additional symmetries that forbid direct mass terms). Thus there are n− − n+ superfields Φi− with negative charge that are left out of the superpotential. The potential has the form

(11.43) where summation over A = 1, 2, . . . , n+ and i = 1, 2, . . . , (n− − n+) is assumed. Again, minimizing this potential for fixed values of φ0 we find that for

324 Supersymmetry and the early Universe

|φ0| > φc = ξGS1/2maxA (g/λA), the minimum for all φ+ and φ− fields is at zero. Thus, the tree level curvature in the φ0 direction is zero and inflation can occur.

During inflation masses of 2n+ scalars are m2A± = λ2A|φ0|2 g2ξGS and the remaining n− − n+ negatively charged scalars have masses squared equal to g2ξGS. We see that inflation proceeds much in the same way as for the nonanomalous U (1) example discussed above. The interesting di erence is that in the latter case the scale of inflation is an arbitrary input parameter (although in concrete cases it can be determined by the grand unified scale), whereas in the anomalous case it is predicted by the Green–Schwarz mechanism.

11.4Cosmic strings

In the context of supersymmetry, cosmic strings have a remarkable property: they carry currents. This tends to stabilize them, which turns out to be a mixed blessing, since their relic density might overclose the Universe. As stressed in Section D.5 of Appendix D, which presents a short introduction to cosmic strings, an advantage of strings over other types of defects is that they interact and may thus disappear with time. This is lost in the context of supersymmetry and one must take into account the corresponding constraints on parameters, for each phase transition that may lead to the formation of strings.

Just as for inflation, the form of the scalar potential as a sum of F -terms and D-terms leads to the two main types of supersymmetric cosmic strings: F -strings and D-strings. We review their construction and some of their properties.

11.4.1F -term string

We consider a U (1) supersymmetric gauge theory with two charged superfields Φ± (of respective charge ±1) and a neutral superfield Φ0. The superpotential is chosen to be [98]

W = ρΦ0 Φ+Φ− − η2 , (11.44)

with η and ρ real. The scalar potential (F and D terms) vanishes for φ0 = 0, φ± =

ηe±iα.

We look for a solution of the equations of motion corresponding to a string extending over the z axis. Following Section D.5 of Appendix D, we thus consider

We note that, whereas the ground state of the theory (which coincides here with the state far away from the string) conserves supersymmetry and breaks gauge symmetry, supersymmetry is broken in the core of the string (F0 = ρη2(1 − f 2)) and gauge symmetry is restored (φ± = 0).

which allows us to determine the profile functions f (r) and a(r).

We then turn our attention to the fermionic degrees of freedom: zero-energy solutions Ψi(r, θ) (i = 0, ±) and λ(r, θ) exist in the string core. According to standard index theorems [97,237,358], there are 2n of them. As in the nonsupersymmetric superconducting string [376], such solutions may be turned into more general solutions propagating along the string. Indeed, let us write the ansatz

In the interior of the strings, the Yukawa couplings vanish since all scalar fields are

zero and D/T ΨiL (r, θ) = 0. One infers from the Dirac equation for Ψi

i.e. α(t, z) = f (t ± z).

Thus fermions which are trapped in the transverse zero modes (satisfying (11.50) travel at the speed of light along the string (in the z direction). The presence of these excitations is responsible for turning the string into a superconducting wire.

[For the sake of completeness, we derive here the explicit form of the fermion zero

Cosmic strings 327

11.4.2D-term string

We now turn to the case of a phase transition where the gauge symmetry breaking occurs through a D-term:

For convenience, we disregard other scalar fields than the one responsible for gauge symmetry breaking and we write ξ ≡ η2. The local string ansatz

with the same boundary conditions as in (11.46), leads to the equations of motion:

Again, supersymmetry is broken (D = gη2(1 − f 2)) and gauge symmetry restored in the string core.

The surprise comes from the fermionic zero modes: they only travel in one direction, which expresses the fact that supersymmetry is only half broken. [As above, we find the fermionic zero modes by performing a supersymmetry transformation on the bosonic background (11.58). Using (11.58), we find

We see that, if we set ξ1 = 0, the two expressions vanish. We thus only obtain zero modes by considering the case ξ1 = α and ξ2 = 0:

According to the discussion in the preceding section, only modes moving in one direction are present.]