Supersymmetry. Theory, Experiment, and Cosmology

where αψ and αF are real normalization constants.

The supersymmetry transformations (C.110) read in terms of component fields:

√

δS φ = αψ 2 η ψ

√ √

αψ δS ψα = 2 αF F ηα − i 2 ρQ (σµη¯)α ∂µφ

or in four-component notation (see Exercise 2 for notation and Appendix B for our conventions on gamma matrices):

452 Superfields

Thus the action for the scalar field reads (up to total derivatives which do not contribute)

One may write the preceding action using four-component spinors (see Exercise 2 for notation). For example, in the case where

This can be shown directly to be invariant under the supersymmetry transformations (C.119).

9Note that, for ρQ = −1 (as in Wess and Bagger [362]), we obtain an unusual form for the Dirac equation: iγµ∂µΨ = −mΨ. It is to recover the standard Dirac equation that we have chosen ρQ = +1,

and thus real γQ , γQ¯ (see (C.109)).

Exercises 453

whereas the complete supersymmetry transformations are

The conventions followed by various authors are summarized in the following table.

Elements of general relativity 455

The situation is reminiscent of the one encountered in gauge theories. An ordinary derivative ∂µΨ of a gauge nonsinglet field Ψ does not transform covariantly under a gauge transformation. One has to introduce the covariant derivative DµΨ = ∂µΨ − igAaµtaΨ which has the same gauge transformation as Ψ (see Section A.1.4 of Appendix Appendix A).

Similarly here one defines the covariant derivatives

where Γρµν , the analog of the gauge field, is called a Christo el symbol or spin connection (it does not transform as a tensor under general coordinate transformations) and is defined in terms of the metric as:

In the same way that one defines the field strength by di erentiating the gauge field (Fµν = ∂µAν − ∂ν Aµ − ig [Aµ, Aν ]), one introduces the Riemann curvature tensor:

By contracting indices, one also defines the Ricci tensor Rµν and the curvature scalar R

which is reminiscent of a similar equation in gauge theories (see (A.51) of Appendix Appendix A).

An important notion connected with covariant di erentiation is the notion of parallel transport. A vector is said to be parallel-transported along a curve parametrized by τ if its covariant derivative projected along the curve is zero:

For example, the velocity U µ = dxµ/dτ (τ proper time: dτ 2 = gµν dxµdxν ) of a freely falling object is parallel-transported along the free fall trajectory (this generalizes the notion of zero acceleration in flat space). Such a path is called a geodesic. In other words, a geodesic is a curve that parallel transports its tangent vector along itself. On a sphere the geodesics are the great circles.

456 An introduction to cosmology

Let us now restrict to a 2n-dimensional compact manifold K with metric gij and consider a small loop γ in the (xi, xj ) plane at a point P . Take a vector V k at P and parallel transport it along γ. If the space is flat, when one reaches P again the vector recovers its previous value. This is not true in curved space and V k is changed by the amount (cf. (D.10)):

Let us restrict our attention to vectors tangent to K and note TP the tangent space at point P . The holonomy group at point P is the set of transformations hγ that associate to V TP the tangent vector V TP obtained by parallel transporting V along the loop γ through P (each loop determines one element of HP ).

Using the equation of parallel transport (D.11) together with (D.5), one can write (δV k = −dxiΓkij V j ), using Stokes’ theorem,

where S is a surface delimited by the loop γ. It is straightforward to check the group structure of HP (take two loops γ1 and γ2 through P : hγ1 hγ2 = hγ1γ2 ). Let us note that (D.13) shows that the spin connection can be interpreted as the gauge field associated with the holonomy group (Γkij = Ai kj ).

Generally, for a complex manifold, HP SO(2n), independently of P . Take the example of the sphere S2 and consider first the loops γ which consist of a triangle made of (parts of) three great circles. Since these are geodesics it is easy to parallel transport a tangent vector V . By changing the angles of this triangle, one may generate any rotation in the tangent plane, hence any transformation of SO(2).

The action of general relativity in D dimensions has the form

where κ2D = 8πG(D) (κ2 = 8πGN in the four-dimensional case), g is the determinant of gµν , Ψ stands for the matter fields and λ is the cosmological constant. Using

one derives (see Exercise 1) from the Lagrangian (D.14) the equations of motion of the metric components, i.e. Einstein’s equations:

where the tensor Gµν thus defined is called the Einstein tensor. In four dimensions,

458 An introduction to cosmology

One now obtains from the (0, 0) and (i, j) components of the Einstein equations (D.17):

D.2.1 Friedmann equation

The first of the preceding equations can be written as the Friedmann equation, which gives an expression for the Hubble parameter H ≡ a/a˙ measuring the rate of the expansion of the Universe:

Note that the cosmological constant appears as a constant contribution to the Hubble parameter.

This equation should be supplemented by the conservation of the energy–momentum tensor which simply yields:

Hence a component with equation of state (D.24) has its energy density scaling as ρ a(t)−3(1+w). Thus nonrelativistic matter (often referred to as matter) energy density scales as a−3. In other words, the energy density of matter evolves in such a way that ρa3 remains constant. Radiation scales as a−4 and a component with equation of state p = −ρ (w = −1) has constant energy density. The latter case corresponds to a cosmological constant as can be seen from (D.26)–(D.27) where the cosmological constant can be replaced by a component with ρΛ = −pΛ = λ/(8πGN ).

The Friedmann equation allows us to define the Hubble constant H0, i.e. the present value of the Hubble parameter, which sets the scale of our Universe at present time. Because of the troubled history of the measurement of the Hubble constant, it has become customary to express it in units of 100 km s−1 Mpc−1 which gives its order of magnitude. Present measurements give

H0

h0 ≡ 100 km s−1 Mpc−1 = 0.7 ± 0.1.

The corresponding length and time scales are: