Ciclo iterativo de diseño y microanálisis

From an effective field theory viewpoint, for any gauge singlet or non-singlet field φ, non- renormalizable higher dimensional operators like

Veff(φ) =V0 X n (φ∗_φ)n M2n P , (5.1)

cannot be forbidden in the potential. Here V0 denotes the vacuum energy. Already the first term in such an expansion (n = 1) induces a large contribution to the inflaton mass proportional to the vacuum energy density, i.e., V′′ ∼V0. Plugged in the formula for the slow-roll parameter η defined in (2.17), this generically spoils inflation due to a leading contribution η≈1.

Within SUGRA theories, this so-called η-problem typically appears, since gravity cou- ples to everything and thus also induces a coupling of all the fields to the vacuum energy densityV0. Especially in the F-term contribution to the scalar potential given by Eq. (3.67) this is obvious, since for a minimal K¨ahler potentialK =φ∗_{φgiving rise to canonical kinetic}

terms, an expansion of the exponential in the tree level potential leads to the form

VF ∼ 1 + φ∗φ M2 P +. . . V0. (5.2)

Comparing (5.2) to (5.1), we note that it is exactly these dangerous terms that reap- pear in the F-term potential of a SUGRA theory. This states the η-problem of SUGRA inflation [31, 32].

Note that for small field inflation models1 _{with φ < M}

P, the n = 1 contribution of Eq. (5.1) is bad enough to spoil inflation. The η-problem becomes a lot more severe when dealing with large field models with φ > MP. This is due to the fact that then, an expansion as in Eq. (5.1) breaks down, since every higher order contributes an even larger mass making the η-problem a catastrophe.

In order to control or even solve the η-problem altogether, there are several different approaches proposed in the literature. First of all, if the scalar potential is not dominated by the F-term contribution, but instead arises mainly from the D-term part (3.76), theη- problem is simply not present. The reason is that in this scenario ofD-term inflation [103], the exponential factor eK _{does not appear. Nevertheless, we are interested in building F-}

term models of inflation in SUGRA and thus do not further elaborate on the idea of D-term inflation.

With the focus on the F-term part of the scalar potential Eq. (3.67), we are basically left with two possible ways to solve theη-problem. Lacking an ultraviolet complete description, the K¨ahler potential can be an arbitrary real function of the scalar components of the chiral superfields. One proposal is to stick to the most general non-minimal K¨ahler potential as

5.1 The η- Problem 55

an expansion in terms of all scalar fields present [44, 104, 105]

K =φ∗_iφi+X n=2 κ(_in) (φ∗iφ i₎n M_P2n−2 + X k=m+n κ(_ijk)(φ ∗ iφi)n(φ∗jφj)m M_P2k−2 +. . . , (5.3)

where the leading terms give rise to canonical kinetic terms andφ∗

i =δi¯jφ∗

¯

j_{. For the higher}

order terms with expansion parametersκ(_in), κ(_ijk), . . ., one obtains off-diagonal contributions to the K¨ahler metric. This Ansatz solves the η-problem if the expansion parameters are tuned in such a way that the scalar potential is flat enough for slow-roll inflation.

The second possibility is to apply some fundamental symmetry on the K¨ahler potential to forbid operators as in Eq. (5.2) for the inflaton direction which give rise to theη-problem. One common feature of a solution by symmetry arguments is that only the invariant field combination ρ under the symmetry appears explicitly in the K¨ahler potential

K =k(ρ), (5.4)

where ρ is a DOF different from the inflaton direction which protects the latter from ob- taining large SUGRA mass corrections. Within the remainder of the thesis, we extensively study such symmetry solutions. If one applies these, one should make sure that the sym- metry allows for canonical normalization of the kinetic terms in a not too complicated manner.

The simplest candidate symmetry which can account for the aforementioned properties is a Nambu–Goldstone-like shift symmetry [106, 107, 108] under which the complex scalar containing the inflaton direction transforms as

φ→φ+ iµ . (5.5)

Here, µdenotes a real transformation parameter. The invariant combination in the K¨ahler potential is then given by

φ+φ∗ = 2 Re(φ). (5.6)

Therefore, the imaginary part Im(φ) is a good inflaton direction since it gets protected by the shift symmetry Eq. (5.5).

Another more involved symmetry is called Heisenberg symmetry and is based on non- compact Heisenberg group transformations of two or more complex scalar fields as

T _→T + iµ , T →T +α_i∗φi+ α∗iα i 2 , φi →φi+αi, (5.7)

where µ is again a real transformation parameter and the αi _{are complex transformation}

parameters. The complex scalar fieldT, belonging to a chiral supermultiplet, is a modulus field associated with the Heisenberg symmetry. It was first discussed in the context of

string-inspired models in Ref. [109] and its use for inflation model building has been studied in Refs. [110, 111], however, lacking an explicit model and its predictions. The invariant DOF under the transformations in Eq. (5.7) is given by

ρ =T +T∗−φ∗_iφi. (5.8)

If the K¨ahler potential satisfies the symmetry and thus depends on ρ only, the _|φi_{| are}

viable inflaton directions as we show in the next two chapters. Also, as we shall discuss in more detail, the Heisenberg symmetry has another advantage. Going to a basis where instead of the fundamental DOFs _{φ,Re(T)_}, we treat the DOFs _{φ, ρ_} as independent, the K¨ahler metric and thus the kinetic terms are diagonal. This facilitates the task of canonical normalization.