The basic hypothesis of a post-Copernican cosmological theory is that all
the points of the universe have to be essentially equivalent. This
hypothesis is required in order to avoid any privileged observer.
This assumption has been implemented by Einstein in the s.c. *cosmological
principle*: all the positions in the universe have to be essentially
equivalent, so that the universe is (at least mathematically) *homogeneous*.
This situation implies the condition of some (spherical) symmetry about
every point, so that the universe is (at least locally) *isotropic*.
But an *hidden* assumption seems to be in the formulation of the
cosmological principle. In fact, the condition that all the points are
(statistically) equivalent (with respect to their environment) corresponds
to the property of a local *isotropy*. And it is generally accepted that the
universe can not be *isotropic* about every point without being also
*homogeneous*.
But local *isotropy* does not necessarily implies *homogeneity*. In fact a
topological theorem states that homogeneity requires (at least local)
isotropy together with the assumption of the *analyticity*. Analyticity was
an usual assumption in any physical problem: before the *fractal* geometry!
Actually a *fractal* structure has some local isotropy but has not
homogeneity. In simple terms one observes the same mix (structures and
vacua) in different directions (statistical isotropy). This means that a
*fractal* structure satisfies the cosmological principle! In the sense that
all the points are essentially equivalent (no center, no special points).
But this does *not* imply that these points are distributed uniformly!
Now astronomy showed some intrinsically *irregular* structures for which the
analyticity assumption might be reconsidered and fractal properties might be
investigated.
The space distribution of galaxies and clusters, the cosmic microwave
background radiation, the linearity of the redshift-distance relation
(Hubble law), the abundance of (light) elements in the universe: each of
these four points provides independent experimental facts. The objective of
a cosmological theory of the universe (fractal or not) should be to provide
a coherent explanation of all these facts together. An important point in
the theoretical investigation concerns the distribution of the gravitational
force inside structures which could be irregular or fractal.
But the recent statistical analysis of the experimental data already shows
also that *the distribution of galaxies is fractal* up to the deepest
observed scales. In the near future one could describe structures in which
intrinsic *self-similar* irregularities develop at *all* scales and
fluctuations cannot be described in terms of *analytical* functions. The
theoretical methods to describe this situation could not be based on
ordinary differential equations because *self-similarity* implies
singularities and the absence of analyticity.
About the fractal universe:
http://pil.phys.uniroma1.it/astro.html
http://pil.phys.uniroma1.it/debate.html
http://pil.phys.uniroma1.it/
http://pil.phys.uniroma1.it/eec1.html
The Nobel laureate (1977) P.W. Anderson is working on this field (now at the
Princeton University and also at the Rome University, La Sapienza).
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