MODERN COSMOLOGY

130 Cosmological models

3.5.2.3 Spatially inhomogeneous universes

These models haves≤2. TheLRS cases(q= 1 ⇒s= 2 ,r= 3 )are the
spherically symmetric family with metric form:

ds^2 =−C^2 (t,r)dt^2 +A^2 (t,r)dr^2 +B^2 (t,r)(dθ^2 +f^2 (θ)dφ^2 ), ua=δ 0 a,
(3.79)
where f(θ)is given by (3.76). In the dust case, we can setC(t,r)=1and
can integrate the EFE analytically; fork=+1, these are the (‘LTB’) spherically
symmetric models [5,87]. They may have a centre of symmetry (a timelike world-
line), and can even allow two such centres, but they cannot be isotropic about a
general point (because isotropy everywhere implies spatial homogeneity).
Solutions with no symmetries at all haver = 0 ⇒ s = 0, q =

0. The real universe, of course, belongs to this class; all the others are
   intended as approximations to this unique universe. Remarkably, we know
   some exact solutions without any symmetries, specifically (a) the Szekeres quasi-
   spherical models [121, 122], (b) Stephani’s conformally flat models [84, 116],
   and (c) Oleson’s type-N solutions (for a discussion of these and all the other
   inhomogeneous models, see Krasi ́nski [85] and Krameret al[83]). One further
   interesting family without global symmetries are the ‘Swiss-cheese’ models,
   made by cutting and pasting segments of spherically symmetric models [23, 112].
   Because of the nonlinearity of the equations, it is helpful to have exact
   solutions at hand as models of structure formation as well as studies of linearly
   perturbed FL models (briefly discussed later). The dust (Tolman–Bondi) and
   perfect fluid spherically symmetric models are useful here, in particular in terms
   of relating the time evolution to self-similar models. However, in the fully
   nonlinear regime numerical solutions of the full equations are needed.

3.6 Friedmann–Lemaˆıtre models

The FL models are discussed in detail in other chapters, so here I will only briefly
mention some interesting properties of these solutions (and see also [33]). These
models are perfect fluid solutions with metric form (3.75), characterized by

u ̇= 0 =ω=σ= 0 ,θ= 3

S ̇

S

(3.80)

⇒∇ ̃eμ=∇ ̃ep=∇ ̃eθ= 0 , Eab=Hab= 0. (3.81)

They are isotropic about every point (q = 3 )and consequently are spatially
homogeneous(s = 3 ). The equations that apply are the covariant equations
(3.80), (3.83) with restrictions (3.80). The dynamical equations are the energy
equation (3.29)

μ ̇=− 3

S ̇

S

(μ+p), (3.82)