C. Scaling laws in physics

The discovery of scaling laws and symmetries in natural phenomena is a fundamental part of the methodology of physics. This is not new: we can think of Galileo's observations of the oscillations of a pendulum, Kepler's discovery of the equal area law for planetary motion and Newton's inverse square law of gravitation. Some authors claim that the actual discovery of the scaling laws is attributable to Galileo in the context of the strength of materials as discussed in his book Two New Sciences (Peterson, 2002).

The establishment of a scaling relationship between physical quantities reveals an underlying driving mechanism. It is the task of Physics to understand and to provide a formalism for that mechanism.

The self-affine Brownian motion is a good example for visual illustration of a scaling process (see Fig. 1). In this case scaling is non-uniform, because different scaling factors have to be applied to each coordinate to keep the same visual appearance.

 Figure 1. Scaling relations in one-dimensional Brownian motion x(t). In successive zooms the vertical coordinate (x) is multiplied by 2, while the horizontal coordinate (the time t) is multiplied by 4 to properly rescale the curve.

The breaking of symmetries and of scaling laws is equally important and has played a key role in 20th century physics. Scale invariance is typically broken when some new force or phenomenon comes into play, and the result can look far more significant than it really is. Dubrulle and Graner, 1994; Graner and Dubrulle, 1994 have suggested that this may be the case for the Titius-Bode law (which is, of course, not a law, and can be traced back before Titius and Bode at least to David Gregory in 1702). Their point is that, if the primordial proto-planetary disk had a power-law distribution of density and angular momentum then any process that forms planets will give them something like the Titius-Bode distribution of orbit sizes. Thus the distribution cannot be used as a test for any particular formation mechanism.

Within cosmology, some of the examples of quantized redshifts reported over the years (Tifft, 1976; Burbidge and Burbidge, 1967; Burbidge, 1968) may have been analogous cases, where the "new phenomenon of physics" was observational selection effects resulting when strong emission lines passed into and out of the standard observed wavelength bands.

As we shall see, there are important scaling relationships in the spatial distribution of galaxies. This scaling is almost certainly a consequence of two factors: the nature of the initial conditions for cosmic structure formation and the fact that the gravitational force law is itself scale-free.

This scaling is observed to break down at very large distance. This breakdown is a consequence of the large-scale homogeneity of the Universe and of the fact that the Universe has a finite age: gravitational agglomeration of matter has only been able to spread over a limited domain of scales, leaving the largest scales unaffected.

The scaling is also expected to break down for small objects where non-gravitational forces have played a role: gas-dynamic processes play an important role in the later stages of galaxy formation. There are important scaling relationships among the properties of galaxies which provide clues to the mechanisms of their formation. We do not deal with these in detail here, although the main scaling laws in the galaxy properties are summarized in Sect. VII.A.5.