, which is a hypersurface of inhomogeneity [3], [6], [9]. Homogeneity of an frw universe is implied indirectly by the specific forms of the observed quantities. Any deviation from these forms indicates real inhomogeneity (i.e. on spatial hypersurfaces). A precise observational characterisation of homogeneity within the ltb class of universes has been given [3], [8]: This characterisation may be slightly weakened in the parabolic (k = 0) case. In the fundamental equation (28), we may transform the independent variable to N: It is well known [5] that T = const. reduces the metric to frw. The first order ode This result is consistent with [3], but it follows more directly via our approach. Note that the result does not extend to k = 0, essentially since in that case 2 arbitrary functions (T and f ) need to be fixed by 2 independent observational relations.
We emphasize the 'global' nature of these characterisations, i.e. the fact that d A (z) and N(z) must be frw for all z. Local observational parameters derived from d A and N are insufficient to characterise homogeneity. For example, the Hubble constant, derived from d A as in (15), cannot in itself distinguish between inhomogeneous and homogeneous universes. The dependence of H 0 on the spatial geometry (i.e. on k) and on the density ρ 0 at the observer, follows from (12), (23), (24) and (26) as The frw case has f ′ (0) = 1 and B 0 = a 0 (see (32) and (33) below), and for a given ρ 0 and k, the frw H 0 is observationally indistinguishable from the general ltb H 0.
A particular consequence of (31) arises when an ltb under-dense region is matched to an frw region. The low value for ρ 0 implies a low value for H 0 if k = 0, 1. Thus cosmological models with a locally high Hubble constant arising from an under-dense inhomogeneous local region within a global frw universe (see for example [10]), must have k = −1 in the local ltb region. The conditions for smooth matching at the boundary (see [11]) then require k = −1 in the frw region, i.e. an open frw universe.
We now return to the problem of characterising homogeneity, and derive an alternative condition (for all k). As pointed out in [12], the assertion in [8] that the coordinate freedom (4) could be used to set is erroneous, and in fact (32) implies restrictions on the spacetime geometry. This choice made in [8] automatically over-restricts the solutions obtained by that procedure to be homogeneous (frw) solutions, as we show below. As long as y is specified to be comoving, one can set B = A only on one past light cone, i.e. B(w 0, y) = A(w 0, y), and not generally. Even though the ww and wy components of the Lie derivative of the metric g µν along u µ are equal, the time derivatives of A and B are not given by the Lie derivatives. And there is no further freedom to specify them arbitrarily -as that freedom has already been used up by specifying y to be comoving. If, however, y is not chosen comoving, then one could set B(w, y) = A(w, y) without loss of generality.
This correction provides another characterisation of frw universes within the ltb class: an ltb universe is frw if and only if there exists a choice of radial comoving coordinate such that g wy = g ww in observational coordinates.
The proof proceeds simply and directly from the formalism set up in sections 2 and 3. The assumption (32) allows us to integrate (14) and (19), to get A(w, y) = a(η), C(w, y) = a(η) where we have used the central condition (3) for C. Then it follows immediately from (33) that the shear (16) vanishes, so that the spacetime is frw.
For completeness, we present the frw metric and observations in observational coordinates. By (33) we can write the metric (2) as showing that η is conformal time. It follows that y is conformal proper radial distance Now (32), (33) and (34) reduce the field equation (see [8]) where α is constant and the big bang occurs at w = χ.

Observational generalisation of the Einstein-de Sitter universe
The ltb solution (25) allows one to write down any number of explicit exact inhomogeneous generalisations of frw solutions. One simply chooses T (y). However, if we want an observationally based generalisation, then the ltb solution is not appropriate, since t is not an observational variable, and deriving the observational variables requires integration of the null geodesics. In the observational coordinates, the key equation (28) or (29) is far more complicated, but it has the advantage that it is already in observational variables. Thus any explicit solution will be a readily interpreted observational generalisation of an frw universe.
We have found one such solution in the case k = 0, i.e. an observational inhomogeneous generalisation of an Einstein-de Sitter universe. The equation (30) can be simplified by defining the observational variable Then (30) becomes By (40), (23), (24) and (17), we get Now it is clear that the ode (41) is separable for ∂T /∂N = 1 3 mβ, β = const. If β = 0, then T = const., which is the frw case. In the observational variables, the Einstein-de Sitter solution following from (41) and (42) is where P (w) is an arbitrary function, subject to P (w 0 ) = 2(mH 0 ) −1/3, which ensures that ρ 0 = 3H 2 0. The (area distance, number count) relation is d A (N) = C(w 0, N). Note that by (40), (43) and (17), the observational variable D on {w = w 0 } determines the variation in number counts in terms of the variation in density: The inhomogeneous case T = 1 3 mβN, β = 0, leads to the solution where Q is an arbitrary function. For small number counts [equivalently, by (3), (21) and (27), for large D], the solution (45) on {w = w 0 } has the form where Q(w 0 ) has been chosen so that the leading term in (46) (17) and (18) that so that small number counts are equivalent to small redshift. This solution represents a universe that is locally homogeneous, but for larger redshift develops real inhomogeneity. If the Copernican principle is not invoked, and if large-scale observations of the galactic distribution are taken at face value, then such a model of the late universe is not automatically ruled out.
For completeness, we give the explicit form of the integral in (45), which depends on the roots α i of the quartic in the integrand [14]:, j < k < l and i is not equal to j, k or l.
Further calculations show that the qualitative behaviour of the solutions (47) and (48) falls into two types, according to whether or not the past light cone reconverges.

Concluding remarks
Motivated by the direct observational evidence for isotropy, and by the need to identify indirect observational evidence for homogeneity, we have completed an exact observational analysis of isotropic dust cosmologies, i.e. ltb spacetimes. Such an analysis, based on the past light cone (which is the hypersurface on which observational data is given), was begun in [3] and continued in [8]. An important error in [8] has been corrected here, allowing us to properly complete the integration scheme, as described i.e. the condition that B = A in observational coordinates. Finally, in Section 5 we used (28) to construct an exact observational generalisation of the Einstein-de Sitter universe, given by (45), (46), (47) and (48). This spacetime is locally homogeneous but inhomogeneous on large scales. We are not proposing it as a realistic model of the universe, which is more generally believed to be locally inhomogeneous but homogeneous on large enough scales, but it is not ruled out by current galactic observations, and the method of deriving it could be extended to construct models with different behaviour of the inhomogeneity.Mood: Scared at myself right now.
This is the shortest thing I have ever written. Word tells me its 40 words long. I broke someone's mind with this before I posted it. OMG. Love the Crack! If you aren't mentally prepared, do not read this.
Armstrong rolled over, moonlight gleaming off of his chest. He looked to the woman at his side.
“So… when are we going too...?” The huge man blushed as he made a crude motion.
Jantra has left the room, in tears.Don't Buy A Dell!!!
I’m sure many of you who read this post have purchased a Dell computer and are extremely happy with it. Unfortunately for me, that has not been my experience. For a company that prides itself on customer service they have fallen short of my expectations on 2 different occasions. Allow me to give some background information. I purchased my Dell Inspiron 600m on March 15th 2005 and received my NACHI discount. That part I had not problems with.