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As a simple example, consider Eq. m #  .    It has the same Z symmetry as the original potential, corresponding to invariance under  P! But as shown in Fig. 3, the minimum at the origin is replaced by minima at "$(m/( ). (m/( ). Then, if the constant < is chosen appropriately, one has near the minimum  (141) <"mJ  I #A I #B I  ,  where mJ "(2m and we are not interested in the precise values of A and B. The minima represent possible vacuum expectation values (vevs) of the "eld. Each of them represents a possible vacuum of the theory, around which are small oscillations corresponding (after quantization) to particles.

S, where S is evaluated on the last-scattering   surface. ect coming from the curvature perturbation we just calculated, and the "rst term is the anisotropy  / just after last scattering (on a comoving  P P hypersurface). ect, so for a given large-scale density perturbation the isocurvature perturbation gives an anisotropy six times bigger. As a result an isocurvature perturbation with a #at spectrum cannot be the dominant contribution to the cmb, though one could contemplate a small contribution [295].

133)  I The use of a complex "eld is particularly appropriate if the potential depends only on " ". Then Eq. (127) is replaced by (134) <( )"< #m" "# " " .   Complex "elds are, in any case, part of the language of supersymmetry. With two or more real "elds, it is no longer true that the most general Lorentz-invariant Lagrangian density L can be reduced to the above form, Eq. (129), by a "eld rede"nition. For several real "elds , the most general kinetic term involving derivatives is L L " H R RI , (135)  KL I K L K L where H is an arbitrary function of the "elds.

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