Supersymmetry. Theory, Experiment, and Cosmology

480 An introduction to cosmology

the Hubble radius; they freeze out and continue to evolve classically. The fluctuation spectrum produced is given by

where the subscript k = aH means that the quantities are evaluated at Hubble radius crossing, as expected. We also note that H sets the scale of quantum fluctuations in the de Sitter phase.

The scalar spectral index nS (k) is computed to be:

Thus, because of the slow-roll , the fluctuation spectrum is almost scale invariant, a result that we have alluded to when we discussed the origin of CMB fluctuations.

Besides scalar fluctuations, inflation produces fluctuations which have a tensor structure, i.e. primordial gravitational waves. They can be written as perturbations of

We note that the ratio PT /PS depends only on (V /V ) and thus on ε, which yields a consistency condition: PT /PS = −8 nT .

We conclude this discussion by reviewing briefly the three main classes of inflation models:

•Large field models (0 < η < 2ε). This corresponds to the chaotic inflation scenario

[274] where a field φ of value of the order of a few MP slowly rolls down a power law potential typically V (φ) = λφ4. One drawback is the large value of the field which makes it necessary to include all nonrenormalizable corrections of order

(φ/MP )n.

•Small field models (η < 0). In this class, illustrated first by the new inflation

scenario, the field φ starts at a small value and rolls along an almost flat plateau (where V (φ) < 0) before falling to its ground state.

•Hybrid models (0 < 2ε < η). The field rolls down to a minimum of large vacuum energy (where V (φ) < 0) from a small initial value. Inflation ends because, close to this minimum, another direction in field space takes over and brings the system to a minimum of vanishing energy [276]. We will see examples of such a scenario in Chapter 11.

Cosmic strings 481

r

Large field:

0.4– < η <

0.2

Small field: η < –

0

Fig. D.6 Regions corresponding to the di erent inflation models in the plot

r = C2tensor/C2scalar versus ns [122].

These models make di erent predictions for scalar and tensor perturbations, as can be seen from Fig. D.6.

D.5 Cosmic strings

Among the topological defects, cosmic strings have the most interesting cosmological evolution: because of their interactions they may disappear with time and thus do not necessarily overclose the Universe.

Let us first illustrate what is a cosmic string on the simplest example of a quartic

which is invariant under the global U (1) transformation: Φ → e−iαΦ. Let us consider the field configuration (in cylindrical coordinates z, r, θ) away from the region where r 0:

Obviously, at r = 0, one must have Φ(z, 0, θ) = 0. Thus some potential energy V λη4 is localized around r = 0, typically on a distance ρ m−1 = λ−1/2η−1 (the Compton wavelength of the scalar field).

This finite energy configuration extending along the z axis is called a global string. Its energy per unit length is accordingly:

482 An introduction to cosmology

where we have introduced a cut-o distance R which is the distance between two strings or the diameter of the loop, in the case of a closed string. We deduce that there is an interaction at long distance due to the presence of a Goldstone boson: the force per unit length is simply dµ/dR η2/R. This is analogous to the case of vortex lines in liquid helium.

In the case of a gauge symmetry, the structure is richer since the solution (D.121) has to be associated with a gauge configuration. We take the example of an abelian gauge symmetry (identified with electrodynamics):

which ensures both DµΦ = 0 and Fµν = 0. Thus, the energy density vanishes outside the core of the string: this local string corresponds truly to a localized configuration of energy. The potential energy per unit length is then simply

Hence the string carries n units of elementary magnetic flux 2π/g. The size of the surface through which B is nonvanishing is typically its Compton wavelength MA−1 = (gη)−1. Thus, using (D.127), we have

hence of the same order as µΦ: the total energy per unit length µ scales as η2. Cosmic strings interact [249]: if two strings cross each other, the four ends have a

probability (close to one for the structureless strings that we have considered so far) to change partners, leaving a pair of sharply kinked strings which straighten with time through gravitational wave emission. Occasionally, a string crosses itself, which leads

Exercises 483

to the formation of a loop. This loop shrinks until it vanishes. This has the e ect of reducing the size of strings (and loops).

In realistic models, scalar fields couple to fermions, for example through Yukawa couplings:

Since Φ vanishes in the core of the string, the fermions become massless. We thus have zero modes which may move along the string: currents propagate along the string which is thus called a superconducting string [376]. Such currents may stabilize the loops: it remains then to check how the stabilized loops, sometimes called vortons, contribute to the energy density of the Universe. Supersymmetric models provide examples of such configurations, as we see in Section 11.4 of Chapter 11.

Further reading

•S. Weinberg, Gravitation and cosmology: principles and applications of the general theory of relativity, John Wiley and Sons 1972.

•J. Rich, Fundamentals of Cosmology, Springer 2001.

Exercises

Exercise 1 We detail here the derivation of Einstein’s equations by varying the action (D.14) with respect to the metric gµν .

"

(c)Prove that |g|gµν δRµν is a total derivative.

(d)Vary the action (D.14) with respect to the metric gµν to obtain Einstein’s equations (D.16).

Hints:

(a)gµν gνρ = δρµ yields δgµν gνρ + gµν δgνρ = 0.

(b)Use det A = exp (Tr ln A) to obtain δg = ggµν δgµν .

(c)Check that δRµν = Dα (δΓαµν ) − Dν (δΓαµα). Then the metric tensor in gµν δRµν

Exercise 2 In the case of the Robertson–Walker metric (D.18):

(a)compute the nonvanishing Christo el symbols;

(b)using the fact that the Ricci tensor associated with the three-dimensional metric γij is simply Rij (γ) = 2kγij , compute the components of the Ricci tensor and the scalar curvature;

(c)deduce the components of the Einstein tensor (D.25) and (D.26).

Then integrate and use (D.37).

Exercise 4 We compute exactly the luminosity distance dL = a0r(1 + z) or angular

from which (D.133) can be inferred.

Appendix E

Renormalization group equations

We gather here a set of renormalization group equations [102] which are scattered in the book. The notations are the ones used in the main text. In the case of Yukawa couplings, soft scalar masses, and A-terms, we give the third family couplings. To obtain the corresponding formulas for the other two families, one should just neglect the Yukawa coupling terms on the right-hand side.

Throughout this appendix, we define t ≡ ln(µ/µ0) where µ is the renormalization scale.

E.1 Gauge couplings

In the supersymmetric case,

where, for NF families and ND Higgs doublets (the minimal model has ND = 2), we have

The U (1) gauge coupling has the standard SU (5) normalization (g12 = 5g 2/3).

E.2 µ parameter

We note that the µ parameter does not appear in the evolution equations of the other mass parameters.

486 Renormalization group equations

E.3 Anomalous dimensions

At one loop, we have the following anomalous dimensions γi = d ln Zi/dt:

E.4 Yukawa couplings

For the third generation, we have, in the case of R-parity conservation,

E.5 Gaugino masses

We have, for i = 1, 2, 3,

where the one-loop coe cients are given in (E.2)–(E.4): b(1)1 = −33/5, b(1)2 = −1 and b(1)3 = 3 for NF = 3 and ND = 2.

488 Renormalization group equations

E.7 A-terms

As for the Yukawa couplings, we give only the equations for the third family:

E.8 Dimensional reduction

All preceding equations are understood in the so-called DR renormalization, where DR stands for Dimensional Reduction. Let us explain how this scheme is defined.

In usual gauge theories, the standard renormalisation scheme is based on dimensional regularization: divergent integrals are computed by performing an analytic continuation to D = 4 − 2 spacetime dimensions. Since Ward identities do not depend on the dimension of spacetime, this procedure does not spoil gauge symmetries. The most remarkable consequence of such a continuation is that the gauge coupling is no longer dimensionless: one may write the bare gauge coupling g0 in terms of the (dimensionless) renormalized one gR

where Z is the wave function renormalization constant and the scale µ plays the rˆole of renormalization scale in dimensional regularization. When computing integrals over loop momenta, divergences appear as poles 1/ k.

In the minimal subtraction (noted MS), renormalized quantities are obtained by subtracting these poles (in the M S scheme, one also subtracts a generic additive constant ln 4π − γ, where γ is the Euler constant).

It turns out that dimensional regularization is not compatible with the supersymmetry algebra. A straightforward way to see this is to note that, in D = 4 dimensions, the number of on-shell degrees of freedom of a gauge field (D − 2) does not match the one of a gaugino Majorana field (2(D−2)/2).

One illustration of the fact that the M S scheme does not respect supersymmetry is the fact that the renormalized gauge coupling g measured in gauge boson interactions no longer coincides with the coupling gˆ of a gaugino to the fields of a chiral supermultiplet (cf. (3.43)):

This has led to modify the regularization prescription in the following way: keep the algebra of fields four dimensional while still performing the dimensional continuation