216 Supergravity and cosmology
6.4 Super-Higgs effect in cosmology
We would like to choose a gauge in which a goldstino equals zero. The question
is which field is this goldstino: we start with the gravitinoψμand some number
of left- and right-handed chiral fermionsχi,χi. In the past, this has been sought
for constant backgrounds [2], but in cosmological applications the scalar fields
are time-dependent in the background. Therefore we need a modification.
In the action there are a few terms where gravitinos mix with the other
fermions, and these as well as the supersymmetry transformations should give us
the possibility of finding the correct goldstino in the cosmological time-dependent
background. We want to obtain a combination whose variation is always non-zero
for spontaneously broken supersymmetry. This leads to the following definition
of a goldstino:
υ=ξ†iχi+ξi†χi+^12 iγ 5 Dαλα, (6.10)where theλαare gauginos, theDαare auxiliary fields from the vector multiplets
and
ξ†i≡eK/^2 DiW−γ 0 gjiφ ̇j,ξi†≡eK/^2 DiW−γ 0 gijφ ̇j. (6.11)The goldstino defined here differs from the one in the flat background by the
presence of the time-dependent derivatives of the scalar fields.
Goldstino is non-vanishing in the vacuum supersymmetry transformation:
δυ=−^32 (H^2 +m^23 / 2 ). (6.12)HereHis the Hubble ‘constant’:
(
a ̇
a) 2
=H^2 =
ρ
3 MP^2. (6.13)
This has important implications. First of all, it shows that, in a
conformally flat universe (6.1), the parameterαis strictly positive. To avoid
misunderstandings, we should note that, in general, one may consider situations in
which the energy densityρis negative. The famous example is anti-de Sitter space
with a negative cosmological constant. However, in the context of inflationary
cosmology, theenergy density never can turn negative, so anti-de Sitter space
cannot appear. The reason is that inflation makes the universe almost exactly flat.
As a result, the termk/a^2 drops out from the Einstein equation for the scale factor
independently of whether the universe is closed, open or flat. Then gradually the
energy density decreases, but it can never become negative even if a negative
cosmological constant is present, as in anti-de Sitter space. Indeed, the equation
(
a ̇
a) 2
=
ρ
3 MP^2