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In the standard ΛCDM model of the cosmos, the broad structures we see today (galaxies clusters) are thought to have formed from primordial quantum fluctuations in the early (observable) universe. These fluctuations were random variations in density so that over-dense locations formed galaxies and under-dense areas formed what is called 'voids'. The over-dense regions became increasingly over-dense and voids became increasingly under-dense as the cosmos expanded. On top of the normal cosmic expansion, matter inside a void would tend to move outward due to the gravitational pull of the surrounding denser regions.
Even if the cosmos at large was not expanding, an observer near the center of a void would still have noticed something like Hubble's law, i.e. more distant galaxies receding faster than nearer ones. Hence, this concept is called the "Hubble Bubble".1 It has been shown that under certain conditions (e.g. our region with all its superclusters, is near the center of a 'super-sized void'), it could even give an apparent accelerated expansion in the larger universe - hence mimicking dark energy.
The main problems with this idea are: (i) it seemingly requires our galaxy to be near the center of a void, which is apparently not the case; (ii) the rate at which galaxies would recede from us would be too the low for what we observe today, unless the void was unrealistically large (some billion years across) (iii) We are much more likely to live inside a "void wall", the dense structures that surround voids. Surprisingly, this location would give the same type of apparently accelerated expansion.
For the relatively nearby "Hubble galaxies",2 it would give redshifts according to Hubble's law, but when we look at the range of the supernovae where the accelerated expansion was observed, we observe on the far side of a void. Due to the void characteristics, those galaxies would be farther away than they should be for a matter-only cosmos - hence apparent acceleration of expansion. There were still quite a few problems with this idea, amongst others, it would still require some dark energy to fit observations.
Since late 2007, a New Zealand cosmologist, David Wiltshire, has been advocating an improvement to this scheme, apparently solving most (plus a few extra) of the Hubble Bubble problems. It has not been convincingly refuted up to now. He calls it the 'Fractal Bubble' (FB) 3.1 or 'Timescape'3.2 cosmic model, which includes a few very interesting, but entirely reasonable ideas. The main three are: (i) the voids and walls of the cosmos forms a fractal structure (i.e. scale-independent); (ii) the cosmic time of the voids and walls are different, because they comply strictly to Einstein's gravitational energy and time dilation; (iii) the cosmos is about one billion years older (on average) than what the standard ΛCDM model predicts, which also solves some additional problems.
The standard ΛCDM model works only on the average density of the cosmos (i.e. homogeneous matter), giving a surprisingly good fit to observations; but - many of the interpretations and values derived from the data using the homogeneous ΛCDM model as departure point. Hence, the results are model dependent. For one thing, it uses a 'cosmic time' that ticks at the same rate everywhere in the model. As can be seen in NASA's 'cosmic map of local structure',4 the roughly 500 million light years radius around us is not homogeneous at all. At larger distances, the voids are apparently even larger (not shown on this diagram).

Wiltshire has shown that when he uses a non-homogeneous density model and assumes that the large voids comprise more than 50% of the volume of the universe, there may be up to a 43% difference between 'void wall clocks' and 'void center clocks'. This effect is due to Einstein's standard gravitational time dilation and the combined inhomogeneous density and time progression [Wiltshire3.2 Fig 1(b), right] cause effects that appear to be an accelerating cosmic expansion.
If he factors this into his FB model (with voids expanding faster than walls), the predictions are an equal or even better fit to observations than the standard model. Wiltshire's Table 2 shows a comparison of the ΛCDM and FB models.

If his sums are right, this table is very impressive. What is even more interesting, it requires no dark energy! He labels the FB a 'conservative' model, because it requires no new physics. Further, it needs only half as much dark matter as the ΛCDM model, meaning that normal matter makes up at least one third of the total energy of the cosmos (against the mere 4% in the standard model). On average, it means we may live in a standard flat Einstein-de Sitter universe that will not expand forever, but neither will it ever collapse. A sort of relief…
Personally, I do not understand his model very well (I doubt if very many cosmologists fully understand it, or have even seriously looked at it), but I did not see any devastating critiques against it. Some of the critiques seem to confuse it with 'Hubble Bubble models', where the observer is near the center of a void. In the FB model, bubbles play a very important role, but they do not have to be excessively large. The only requirement is that they make up more half of the cosmos by volume, which is apparently the case (counting in the mini-voids, the FB model requires a void-to-wall volume ratio of about 70:30 presently).
Jorrie
Notes
1. The pretty bubble picture top left is from David Wiltshire's home page: http://www2.phys.canterbury.ac.nz/~dlw24/. 'Fractal Bubbles" are not quite the same as "Hubble Bubbles". I combined them in the title for effect... :)
2. 'Hubble galaxies' refer to those on which Edwin Hubble originally based his linear redshift-distance law. They were all well within 100 million light years from us. He was a bit out on the value of the constant Ho, but that was due to distance measurement errors, which were corrected much later.
3. David L. Wiltshire, (3.1) Dark energy without dark energy, Dec, 2007 and (3.2) Gravitational energy as dark energy: Cosmic structure and apparent acceleration, Feb, 2011.
4. I want to acknowledge the stimuli given by jonmtkisco and Cedar in my Blog Alternative cosmologies. Jon originally drew my attention to it, but I thought it an improbable theory at that time. Cedar's ideas made me look at it again and I now no longer think it is that unlikely... 
-J
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