Physical Foundations of Cosmology

186 The very early universe

instanton “connects” the metastable vacuumφ(x)=0 to some (classically allowed)
configurationφxwithV(φx)≤ 0 ,and satisfies the equation

φ ̈x(τ)+

δ(−V)

δφx

= 0 , (4.155)

where ̈φx≡∂^2 φ/∂τ^2 and

δ(−V)

δφx

=φ−V,φ

is the functional derivative of the inverted potential. Equation (4.155) is an analog
of (4.147) and it is obtained from the usual scalar field equation in Minkowski space
under Wick rotationt→τ=it.
The Euclidean action is finite for those solutions describing tunneling in which
the fieldφchanges its value fromφ=0toφ=0 only within a bounded region in
space. On symmetry grounds one expects the most favorable emerging configuration
of the scalar field to be a bubble withφc=0 at its center andφ→0 far away from
the center (Figure 4.14). To find the corresponding instanton relating the original
metastable vacuum configurationφ(x)=0 to a bubble filled with a new phase we
can again rely on symmetry. That is, we adopt the most symmetricalO( 4 )-invariant
solution of the Euclidean equation (4.155), which describes thefour-dimensional
spherical “bubble”in Euclidean “spacetime.”The scalar field then depends only
on the radial coordinate

r ̃=

√

x^2 +τ^2.

R−∆l R R+∆l r

wall

φ

φ→ (^0) φc
φc
Fig. 4.14.