Approximate self-similarity and universality classes

2.6 Approximate self-similarity and universality classes

The field equations for the massless scalar field coupled to the Einstein equations are scale-free. Realistic matter models introduce length scales, and the field equations then do not allow for exactly self-similar solutions. They may however admit solutions which are CSS or DSS asymptotically on small spacetime scales as the dimensionful parameters become irrelevant, including type II critical solutions [49

, 34

, 52

]. This can be explored by a formal expansion in powers of the small parameter $Le −τ$ , where L is a parameter with dimensions length in the evolution equations. The zeroth order of the expansion is the self-similar critical solution of the system with L = 0. A similar ansatz can be made for the linear perturbations of the resulting background. The zeroth order of the background expansion determines $Δ$ exactly and independently of L, and the zeroth order term of the linear perturbation expansion determines the critical exponent $1∕λ0$ exactly, so that there is no need in practice to calculate any higher orders in L to make predictions for type II critical phenomena where they occur. (With $L ⁄= 0$ , the basin of attraction of the type II critical solution will depend on L, and type I critical phenomena may also occur; see Section 2.4.) A priori, there could also be more than one type II critical solution for L = 0, although this has not been observed.)

This procedure has been carried out for the Einstein–Yang–Mills system [97 ] and for massless scalar electrodynamics [106 ]. Both systems have a single length scale 1/e (in geometric units c = G = 1), where e is the gauge coupling constant. All values of e can be said to form one universality class of field equations [112 ] represented by e = 0. This notion of universality classes is fundamentally the same as in statistical mechanics. Other examples include modifications to the perfect fluid equation of state (EOS) that do not affect the limit of high density [161 ]. A simple example is that any scalar field potential $V(ϕ )$ becomes dynamically irrelevant compared to the kinetic energy $2 |∇ ϕ|$ in a self-similar solution [49 ], so that all scalar fields with potentials are in the universality class of the free massless scalar field. Surprisingly, even two different models like the SU(2) Yang–Mills and SU(2) Skyrme models in spherical symmetry are members of the same universality class [22 ].

If there are several scales $L0$ , $L1$ , $L2$ etc. present in the problem, a possible approach is to set the arbitrary scale in Equation (29 ) equal to one of them, say $L0$ , and define the dimensionless constants $li = Li∕L0$ from the others. The scope of the universality classes depends on where the $li$ appear in the field equations. If a particular $Li$ appears in the field equations only in positive integer powers, the corresponding $li$ appears only multiplied by $−τ e$ , and will be irrelevant in the scaling limit. All values of this $li$ therefore belong to the same universality class. From the example above, adding a quartic self-interaction $λ ϕ4$ to the massive scalar field gives rise to the dimensionless number $λ ∕m2$ but its value is an irrelevant (in the language of renormalisation group theory) parameter.

Contrary to the statement in [106 ], we conjecture that massive scalar electrodynamics, for any values of e and m, is in the universality class of the massless uncharged scalar field in a region of phase space where type II critical phenomena occur. Examples of dimensionless parameters which do change the universality class are the k of the perfect fluid, the $κ$ of the 2-dimensional sigma model or, probably, a conformal coupling of the scalar field [47 ] (the numerical evidence is weak but a dependence should be expected).