Stability of winding cosmic wall lattices with X type junctions

This work confirms the stability of a class of domain wall lattice models that can produce accelerated cosmological expansion, with pressure to density ratio $w=-1/3$ at early times, and with $w=-2/3$ at late times when the lattice scale becomes large compared to the wall thickness. For walls of tension $T_{I}$, the relevant X type junctions could be unstable (for a sufficiently acute intersection angle $\alpha$) against separation into a pair of Y type junctions joined by a compound wall, only if the tension $T_{II}$ of the latter were less than $2T_{I}$ (and for an approximately right-angled intersection if it were less that $\sqrt{2} T_{I}$) which can not occur in the class considered here. In an extensive category of multicomponent scalar field models of forced harmonic (linear or non-linear) type it is shown how the relevant tension -- which is the same as the surface energy density $U$ of the wall -- can be calculated as the minimum (geodesic) distance between the relevant vacuum states as measured on the space of field values $\Phi^i$ using a positive definite (Riemannian) energy metric $dU^2=\tilde G_{ij} d\Phi^i d\Phi^j$ that is obtained from the usual kinetic metric (which is flat for a model with ordinary linear kinetic part) by application of a conformal factor proportional to the relevant potential function $V$. For suitably periodic potential functions there will be corresponding periodic configurations -- with parallel walls characterised by incrementation of a winding number -- in which the condition for stability of large scale bunching modes is shown to be satisfied automatically. It is suggested that such a configuration -- with a lattice lengthscale comparable to intergalactic separation distances -- might have been produced by a late stage of cosmological inflation.