Dynamical systems approach

As implied in Chapter 1, usage of the term “asymptotic self-similarity” is most prevalent in a subfield of cosmology that employs general methods in dynamical systems theory to deduce the qualitative behaviour of cosmological models. Application of such methods to homogeneous models was initiated by Collins [48, 49], then developed extensively by Wainwright, Hsu and Hewitt [50, 51, 52] in their comprehensive analysis of Bianchi cos- mologies. The dynamical systems approach has been generalised (with limited success) to inhomogeneous and anisotropic models with a G2 isometry group [53, 54, 55], and

more recently to fully inhomogeneous models that admit no isometries [56].

Central to the dynamical systems approach is the orthonormal frame formalism pio- neered by Ellis and MacCallum [57, 58]. Instead of choosing the usual coordinate basis

{∂/∂xa}for a spacetime(M,g)such that g ∂ ∂xa, ∂ ∂xb =gab, ds2 =gabdxadxb, (2.37)

§2.3 Self-similarity in cosmology 23

we switch to an orthonormal basis{ea}that satisfies

g(ea,eb) =ηab, ds2=ηabωaωb, (2.38) where [ηab] = diag (−1,1,1,1) and{ωa} is the corresponding dual basis of one-forms. Expanding the Lie bracket relative to{ea}such that[ea,eb] = γcabecyields the 24 inde- pendent commutation functionsγc_ab, and the Jacobi identities

[[ea,eb],ec] + [[ec,ea],eb] + [[eb,ec],ea] = 0 (2.39) may be written as

e_[dγc_ab]−γc_e[dγe_ab]= 0. (2.40) Equations (2.2), (2.7) and (2.40) essentially allow the EFE to be cast as a system of first-order evolution equations for the commutation functions, as opposed to second- order partial differential equations for the metric components. If a fundamental four- velocity fielduis specified, the commutation functions may be decomposed into various geometric and kinematic variables (including the generalised Hubble parameterH), and the EFE recast in terms of these variables.

For a homogeneous cosmological model(M,g,u), the vector fielde0is typically cho-

sen to be u = ∂/∂T, where T is a cosmic time function determined by u. The EFE simplify to a system of first-order ordinary differential equations in T, which may be analysed qualitatively using the methods of dynamical systems theory. If (M,g,u) is further assumed to be exactly self-similar, the EFE become purely algebraic; this implies that exactly self-similar models correspond to equilibrium points in the state space of the system. The number of commutation function variables describing the state of a cosmo- logical model is also reduced by the assumption of homogeneity. In the case of Bianchi cosmologies, the state vector(x, H)is six-dimensional, i.e. it comprises three curvature variables, two shear variables, and the expansion variableH.

Now, defining dimensionless timeτ = ln|a/a0|(whereais the generalised scale fac-

tor) and five dimensionless, expansion-normalised variablesy = x/H, it follows from (2.14) and (2.15) (witha˙ =da/dT) that

dT dτ = 1 H, (2.41) dH dτ =−(1 +q)H. (2.42)

Equation (2.42) gives the evolution ofH, which is decoupled in that the evolution equa- tions foryare an autonomous system of ordinary differential equationsdy/dτ = f(y). Hence the effects of expansion are essentially scaled away by this transformation to di- mensionless variables, and the dynamical evolution of a Bianchi model may be analysed in a five-dimensional state space.

The notion of asymptotic self-similarity in the dynamical systems approach follows naturally from this framework: a cosmological model is asymptotically self-similar in the past (future) if there exists a past (future) attractor for its dynamical evolution in state space. Such attractors (i.e. equilibrium points wheredy/dτ = 0) represent exactly self-similar models in state space. For example, the open and closed FLRW models with

γ ∈ (2/3,2]are asymptotically self-similar in the past to the corresponding flat FLRW models; in the future, however, the open models approach the Milne universe while the

W =₁ W =₀

W

dWdΤ

Figure 2.1:Graph ofdΩ/dτagainstΩfor (2.48), withγ∈(2/3,2]. All stable (solid) and unstable (open) equilibrium points wheredΩ/dτ= 0have also been plotted.

closed models asymptote to the time-reversed, contracting flat “models” [59, 60].

These are the simplest results in the dynamical systems approach, as the state space for the FLRW models is reduced to just one dimension by the high degree of symmetry. The first three may be derived in a few short steps; a different analysis is required for closed models in the future, since they recollapse and the expansion-normalised variables diverge at the point of maximal expansion (H = 0). First we define the density parameter

Ω = µ

3H2, (2.43)

which allows (2.12) and (2.15) to be written jointly as

q = 1

2(3γ−2) Ω. (2.44)

From (2.13), (2.14) and (2.43), the value ofΩis related to the geometry of the model by Ω>1⇔k= 1, Ω = 1⇔k= 0, Ω<1⇔k=−1. (2.45) Next, we eliminateafrom the Friedmann equations (2.12) and (2.13) such that

˙

µ=−3Hγµ, (2.46)

which yields (via (2.41) and (2.42))

dΩ

dτ = (2q−(3γ−2)) Ω. (2.47)

Substituting (2.44) into (2.47), we arrive at

dΩ

dτ = (3γ−2) (Ω−1) Ω, (2.48)

which describes the (expansion-normalised) dynamical evolution of a single-component perfect fluid FLRW model in terms of its density. Equation (2.48) has an unstable equilib- rium point atΩ = 1and a stable one atΩ = 0(see Figure 2.1). Hence the open and closed models are asymptotically self-similar in the past to the flat models, and the open models are asymptotically self-similar in the future to the Milne universe (which hasΩ = 0as it is empty). We also note that all FLRW models withγ = 2/3are exactly self-similar, since they correspond to equilibrium points of (2.48) as well.

§2.3 Self-similarity in cosmology 25

Asymptotic self-similarity in the dynamical systems approach agrees with the notion of approximate self-similarity at early and late times, but only up to the effects of ex- pansion. This is not necessarily undesirable, as a similar caveat exists in the conformal framework of QC–WCH: spacetimes that admit an isotropic past/future state are kine- matically isotropic up to expansion effects as well (see (2.17)). There are some problems with fitting the dynamical systems definition into the conformal framework, however.

Firstly, due to the difficulty of extending dynamical systems theory to systems of par- tial differential equations such as the unsimplified EFE, the dynamical systems approach is restricted mainly to homogeneous models for now. On the other hand, the conformal framework is far more general as it focuses on the geometric properties of a given space- time instead of its dynamical behaviour. Another obstacle is that the dynamical systems definition for an asymptotically self-similar model depends on the existence of an ex- actly self-similar counterpart for it to asymptote to; it is not immediately clear how this is reflected in the properties of the asymptotic model itself. The definition of asymptotic self-similarity that we are after is ideally an intrinsic one, since spacetimes are dealt with individually in the conformal framework.

A successful generalisation of the dynamical systems definition to the conformal framework is likely to use the fact that all dimensionless variables are preserved by the flow of an HVF (see Section 3.1.3), and/or incorporate some form of normalisation with respect to expansion (see Section 5.1). Even if such ideas are not adopted, it is still desir- able for the eventual working definition to demonstrate good agreement with the numer- ous asymptotic self-similarity results that exist within the dynamical systems approach.