Hmm. I was under the impression that Big World theories would be relatively accepted. At least Bostrom and Tegmark seem to argue as if they were:

Self-Locating Belief in Big Worlds: Cosmology’s Missing Link to Observation (Bostrom 2002):

Space is big. It is very, very big. On the currently most favored cosmological theories, we are living in an infinite world, a world that contains an infinite number of planets, stars, galaxies, and black holes. This is an implication of most “multiverse theories”, according to which our universe is just one in a vast ensemble of physically real universes. But it is also a consequence of the standard Big Bang cosmology, if combined with the assumption that our universe is open or flat, as recent evidence suggests it is. An open or flat universe – assuming the simplest topology[1] – is spatially infinite at any time and contains infinitely many planets etc.[2] [...]

[1] I.e. that space is simply connected. There is a recent burst of interest in the possibility that our universe might be multiply connected, in which case it could be both finite and hyperbolic. A multiply connected space could lead to a telltale pattern consisting of a superposition of multiple images of the night sky seen at varying distances from Earth (roughly, one image for each lap around the universe that the light has traveled). Such a pattern has not been found, although the search continues. For an introduction to multiply connected topologies in cosmology, see M. Lachièze-Rey and J.-P. Luminet, J.-P., “Cosmic Topology,” Physics Reports, 254(3) (1995): 135-214.

[2] A widespread misconception is that the open universe in the standard Big Bang model becomes spatially infinite only in the temporal limit. The observable universe is finite, but only a small part of the whole is observable (by us). One fallacious intuition that might be responsible for this misconception is that the universe came into existence at some spatial point in the Big Bang. A better way of picturing things is to imagine space as an infinite rubber sheet, and gravitationally bound groupings (such as stars and galaxies) as buttons glued on to it. As we move forward in time, the sheet is stretched in all directions so that the separation between the buttons increases. Going backwards in time, we imagine the buttons coming closer together until, at “time zero”, the density of the (still spatially infinite) universe becomes infinite everywhere. See e.g. J. L. Martin, General Relativity (London: Prentice Hall, 1995).

Parallel Universes (Tegmark 2003):

How large is space? Observationally, the lower bound has grown dramatically (Figure 2) with no indication of an upper bound. We all accept the existence of things that we cannot see but could see if we moved or waited, like ships beyond the horizon. Objects beyond cosmic horizon have similar status, since the observable universe grows by a light-year every year as light from further away has time to reach us. Since we are all taught about simple Euclidean space in school, it can therefore be difficult to imagine how space could not be infinite - for what would lie beyond the sign saying "SPACE ENDS HERE - MIND THE GAP"? Yet Einstein's theory of gravity allows space to be finite by being differently connected than Euclidean space, say with the topology of a four-dimensional sphere or a doughnut so that traveling far in one direction could bring you back from the opposite direction. The cosmic microwave background allows sensitive tests of such finite models, but has so far produced no support for them - flat infinite models fit the data fine and strong limits have been placed on both spatial curvature and multiply connected topologies. In addition, a spatially infinite universe is a generic prediction of the cosmological theory of inflation (Garriga & Vilenkin 2001b). The striking successes of inflation listed below therefore lend further support to the idea that space is after all simple and infinite just as we learned in school.

How uniform is the matter distribution on large scales? In an "island universe" model where space is infinite but all the matter is confined to a finite region, almost all members of the Level I multiverse would be dead, consisting of nothing but empty space. Such models have been popular historically, originally with the island being Earth and the celestial objects visible to the naked eye, and in the early 20th century with the island being the known part of the Milky Way Galaxy. Another nonuniform alternative is a fractal universe, where the matter distribution is self-similar and all coherent structures in the cosmic galaxy distribution are merely a small part of even larger coherent structures. The island and fractal universe models have both been demolished by recent observations as reviewed in Tegmark (2002). Maps of the three-dimensional galaxy distribution have shown that the spectacular large-scale structure observed (galaxy groups, clusters, superclusters, etc.) gives way to dull uniformity on large scales, with no coherent structures larger than about 10^24m. More quantitatively, imagine placing a sphere of radius R at various random locations, measuring how much mass M is enclosed each time, and computing the variation between the measurements as quantied by their standard deviation
M. The relative fluctuations
M/M have been measured to be of order unity on the scale R ~ 3 X 10^23m, and dropping on larger scales. The Sloan Digital Sky Survey has found
M/M as small as 1% on the scale R ~ 10^25m and cosmic microwave background measurements have established that the trend towards uniformity continues all the way out to the edge of our observable universe (R ~ 10^27m), where
M/M ~ 10^(-5). Barring conspiracy theories where the universe is designed to fool us, the observations thus speak loud and clear: space as we know it continues far beyond the edge of our observable universe, teeming with galaxies, stars and planets.

Though those papers are from 2002 and 2003 - have the theories in question been disproven since then? If so, I'd be curious to read about it.