GLYPH<B9>GLYPH<BH>GLYPH<BC> GLYPH<B9>GLYPH<BG>GLYPH<BC> GLYPH<B9>GLYPH<BF>GLYPH<BC> GLYPH<B9>GLYPH<BE>GLYPH<BC> GLYPH<B9>GLYPH<BD>GLYPH<BC> GLYPH<BC> GLYPH<BD>GLYPH<BC> GLYPH<BE>GLYPH<BC> GLYPH<BC> GLYPH<BC>GLYPH<BA>GLYPH<BH> GLYPH<BD> GLYPH<BD>GLYPH<BA>GLYPH<BH> GLYPH<BE> E ( πR ) [ M ∗ ] πR [ M − 1 ∗ ] φ$_{kink}$ φ$_{tri}$$_{v}$ πR$_{1}$ πR$_{∗}$ πR$_{2}$ GLYPH<B9>GLYPH<BE> GLYPH<B9>GLYPH<BD> GLYPH<BC> GLYPH<BD> GLYPH<BE> GLYPH<BF> GLYPH<BG> GLYPH<BC> GLYPH<BC>GLYPH<BA>GLYPH<BH> GLYPH<BD> GLYPH<BD>GLYPH<BA>GLYPH<BH> φ ( y ) [ M 3 / 2 ∗ ] y [ M − 1 ∗ ] φ$_{kink}$ φ$_{triv}$ πR = 1 . 368 M − 1 ∗ πR FIG. 3: Profiles (left panel) in the extra dimension interval [0 , πR ] of the two possible stable static configurations of the Dirichlet scalar field φ , defined by the scalar potential V ( φ ) = − 1 2 | µ $^{2}$| φ 2 − 1 4 | λ | φ 4 + 1 6 | ξ | φ 6 (with µ 2 = 4 M 2 $_{∗}$, λ = 4 M − 1 ∗ , ξ = 0 . 6 M − 4 ∗ , and πR = 1 . 368 M − 1 ∗ ). In the right panel, we show the energy of the two stable solutions as a function of πR , and it is seen how the absolute stability of coexisting static configurations is determined by the size of the radius R . The dots show critical points where the scalar perturbations contain a massless mode. are perturbatively stable. At the radius R$_{∗}$ the two solutions are degenerate, marking the transition from one true vacuum to another ( φ$_{triv}$ for R < R$_{∗}$ and φ$_{kink}$ for R > R$_{∗}$ ). From this we see that the inverse length scale 1 /R plays the role of an order parameter of a phase transition, much like temperature T in finite temperature field theory. For a very small radius R (analogous to high T ) the system is stable only around its trivial solution, with all symmetries restored. As the radius increases (analogous to T decreasing) the system can undergo a phase transition, which could be of either first or second order. The analogy with temperature, however, is not meant to be taken literally. For whereas the temperature in any 4D effective cosmology must be monotonically decreasing for most of its history, the orbifold radius R could in principle increase, decrease or oscillate on very long time scales, depending on the dynamics of the stabilization mechanism (which we have so far ignored). III. KINKS ON A WARPED BACKGROUND We now extend previous investigations to the case of a scalar field in a warped extra dimension, while neglecting any backreaction on the warping from the scalar field itself. 7