The scalar field associated with fluctuations in the positions of the two branes, the ``radion'', plays an important role determining the cosmology and collider phenomenology of the Randall-Sundrum solution to the hierarchy problem. It is now well known th

1Introduction

Theories with extra dimensions have received tremendous attention within the last a few years. One of the most interesting incarnations was formulated by Randall and Sundrum(RS1)[1],who postulated a universe with two4-d surfaces(“branes”)bounding a slice of5-d AdS spacetime.The Standard Model(SM)fields are assumed to be located on one brane(the“TeV brane”).Gravity lives on the other brane(the“Planck brane”)and in the bulk as well as the TeV brane.Both branes have equal but opposite tension,while the bulk has a(negative)cosmological constant. By carefully tuning the brane tensions against the bulk cosmological constant,one can achieve a low energy effective theory that hasflat4-d spacetime.The RS1metric takes the form1

ds2=e−2kLyηµνdxµdxν−L2dy2(1) where L is the size of the extra dimension and0≤y≤1.All mass scales in the full5-d theory are of order the Planck scale.By placing the SMfields at y=1,all mass terms must be rescaled by an exponential suppression factor(“warp factor”)e−kL that can bring them down to the TeV scale.This merely requires that L∼35/k,and thus roughly35times the fundamental Planck length.This is a dramatic improvement over the original hierarchy problem between the electroweak scale and the4-d Planck scale M Pl.

Thefirst obvious difficulty with this scenario is to arrange that the extra dimension stabilizes to a size of about an order of magnitude larger than the Planck length.In the original proposal, the potential for the size of the extra dimension isflat,so that all sizes are classically equivalent. The actual size of the extra dimension was tuned appropriately to solve the hierarchy problem.

A more serious concern wasfirst identified by Ref.[2],in which enforcing dL(t)/dt=0in a cosmological context implied a nontrivial relationship between the TeV and Planck brane energy densities.Ultimately this was shown to be a direct result of assuming the potential for the size of the extra dimension isflat.Hence,the scalarfield associated withfluctuations of the size of the extra dimension,the“radion”,is massless.If,on the other hand,bulk dynamics setup a potential whose minimum determined the distance separating the two branes,then the radion acquires a mass.

Goldberger and Wise(GW)[3]noticed that a potential could be setup for the radion,by adding a5-d bulk scalar to RS1arranged so that it acquires an y-dependent vacuum expectation value(vev).Furthermore,the classical potential is stable against quantum corrections[4].In their analysis,the5-d metric was generalized to

ds2=e−2kL(x)yηµνdxµdxν−L(x)2dy2,(2) where L has a vev and an xµ-dependentfluctuation.This generalization with radialfluctuations, however,does not satisfy Einstein’s equations,and so a different ansatz for the metric is needed to