It is known since twenty years that the galaxy distribution exhibits fractal behavior on small scales (Peebles 1980, Mandelbrot, 1982). Several recent statistical analyses of three dimensional galaxy catalogues indicate that galaxies are distributed fractally with dimension out to the largest scales for which statistically significant data is available, i.e. , up to about 100-Mpc (Coleman & Pietronero, 1992; Sylos Labini et al., 1998). In the opinion of the authors of the present note, no data is pointing convincingly towards a homogenization of the galaxy distribution (i.e. , ). We believe that the standard analyses which indicate that fluctuations of the galaxy number density decrease on large enough scales are not well suited to study scale invariant structures, since they a priori assume the existence of a well defined average density inside the given sample (Peebles, 1993; Davis 1997). On this point the scientific community has not reached consensus (Wu et al., 1998). In this work we assume without further arguments that the fractal picture is correct. Our main point here is to show that a fractal distribution of galaxies even up to the Hubble scale may be consistent with a homogeneous and isotropic universe.
On the other hand, many observations, most notably the superb isotropy of the cosmic microwave background together with its perfect blackbody spectrum, give strong evidence that the geometry of our Universe is very homogeneous and isotropic on large enough scales, a so-called Friedmann model.
In this letter we argue that this seemingly flagrant contradiction of different pieces of observational data may actually be reconciled in a rather simple way.
Our main point is that a fractal distribution of galaxies need not imply a fractal matter distribution (as it is also pointed out in Wu et al., 1998).
It is often mentioned that galaxies represent peaks in the matter distribution. Let us compare these to the distribution of mountain peaks on the surface of the earth which we know are fractally distributed over a certain range of scales. But it would be false to conclude from this fact that the radius of the surface of the earth from its center is grossly variable; as we know, this number is very well approximated by a constant. A similar effect may actually be at work in the matter distribution of the universe.
We first interpret the COBE DMR results (Bennett et al., 1996) and other observations of CMB anisotropies on smaller angular scales (De Bernardis et al., 1997). They indicate that matter fluctuations are reasonably well described by a Harrison-Zel'dovich spectrum with , where denotes the comoving horizon scale, , and a is the scale factor. But small density perturbations in a Friedmann universe grow only once the universe becomes matter dominated and even then rather slowly (proportional to the scale factor ). Density fluctuations today on a given comoving scale should thus be on the order of . Here denotes the cosmic time at which the scale enters the horizon; and are the present cosmic and conformal times respectively. For example on the scale Mpc matter density fluctuations should not be larger than
today. (Here h parameterizes the uncertainty of relating the recession velocity of a galaxy to its distance, i.e. the uncertainty in the Hubble parameter .)
We now argue that a fractal galaxy distribution may well be compatible with such a homogeneous Universe. This can be seen by the following observation: It is well known that the dark matter distribution around a galaxy leads to flat rotation curves (Rubin et al., 1980). Without cutoffs this yields a matter density distribution
around each galaxy. To obtain the total matter distribution, we actually have to convolve with the number distribution of galaxies, . The fractal dimension means that the average number of galaxies in a sphere of radius r around a given galaxy, denoted by , scales like . This implies that the mean number density of galaxies around an occupied point decays like
This shows that a fractal galaxy distribution with dimension together with flat rotation curves, indicate a smooth matter distribution in agreement with our expectations from a Friedmann universe with small fluctuations. The dark matter distribution of a two dimensional model where the galaxies are distributed with fractal dimension is shown in Fig. 1. Clearly, this dark matter distribution is very homogeneous up to finite size effects.
Note that the essential ingredient for this result is that and with .
It is important to insist that the cutoff is larger than all the scales x considered. Far away from galaxies, for example in a void, many galaxies contribute to the density in a given point and the important message is just that flat rotation curves indicate actually that the total resulting density may be rather constant in voids and has about the same value as it has close to galaxies.
One may argue that the notion of a certain lump of matter `belonging' to a certain galaxy is ill defined sufficiently far away from a galaxy and thus the `halo' density cannot be defined on scales larger than, say, half the distance to the next galaxy. With this objection we agree in practice, it just means that at a sufficient distance from a given galaxy the only measurable density is the total density which contains relevant contributions from many galaxies. But in principle it is possible to assign to each galaxy a density profile , and it is interesting to note that the form of the density profile indicated by measurements, which are possible close to isolated galaxies, is just such as to lead to a constant total density if we convolve it with the number density of galaxies.
Another objection may be, that with this density profile the total mass of a galaxy is infinite. But this is in fact irrelevant, since not only the mass of a single galaxy goes to infinity as , but also the galaxy density goes to zero as and this just in a way that the measurable total density is constant. Besides, the integral of should not be considered as the `mass of the galaxy'. More profoundly, the flat rotation curves are a consequence of the fact that the gravitational potential remains constant during linear clustering in a Friedmann universe, which holds beyond the scale of single galaxies.
Clearly, the amplitudes C and A of and depend on the type of galaxies considered. Galaxies of different absolute luminosities in general have different circular speeds and different abundances.
To quantify Eq. (2) we use the Tully-Fisher relation (Tully & Fisher, 1977), between the circular speed v of a galaxy and its luminosity L, , with . The luminosity corresponds to an absolute magnitude of .
The abundance of galaxies as estimated from Sylos Labini et al. (1998) is
Considering for the time being just galaxies, this gives with .
To determine the amplitude of the halo density, , we use the flatness of galaxy rotation curves with (Rubin et al., 1976)
where c denotes the speed of light. Combining this with gives
The convolution of with then leads to
Comparing this with the critical density of a universe expanding with Hubble parameter ,
Clearly, this estimate is very crude since not all galaxies have the same rotation speeds and the constant B depends on the luminosity of the galaxy. But it is a reassuring non-trivial `coincidence' that the density parameter obtained in this way is of the right order of magnitude. To refine this model we have to find a Tully-Fisher type relation for and integrate over luminosities. But since the abundance of galaxies decays exponentially with luminosity above , we have
and reproduce the result (5).
Our arguments suggest that a fractal galaxy distribution may well be in agreement with a smooth matter distribution. Neglecting log corrections (which may be absent in a more realistic, detailed model and which are certainly not measurable with present accuracy), our model describes a perfectly homogeneous and isotropic universe. We do not specify the process which has induced small initial fluctuations and finally led to the formation of galaxies. We are thus still lacking a specific picture of how the fractal distribution of galaxies may have emerged. Purely Gaussian initial fluctuations are probably not suited to reproduce a fractal galaxy distribution. But cosmic strings or other `seeds' with long range correlation could do it. A working model remains to be worked out.
It may be useful to mention that the view presented here actually redefines the notion of bias. In the standard scenario, the dark matter density field (on large scales) is Gaussian and galaxies form in the peaks of the underlying distribution and their correlation functions are simply related 10 Here we consider the possibility that the galaxy and dark matter distributions may have different correlation properties and hence different fractal dimensions, namely for the galaxies and for dark matter. Especially in view of Eq. (1), we want to warn the reader against interpreting the galaxy distribution as proportional to the matter distribution even on large scales.
A similar idea is the one of a universe with a fractal galaxy distribution but a dominant cosmological constant . This possibility is also discussed in Baryshev et al. (1998), in connection with the linearity of the Hubble law.
More precisely, the fractal dimension of a set of density fluctuations can depend on the threshold (see Fig. 2). In a realistic model, we would expect that also the dark matter, above a certain threshold is fractally distributed. In galaxy catalogues, this tendency is actually indicated. Observations show a slight increase of the fractal dimension with decreasing absolute luminosity of the galaxies in the sample (Sylos Labini et al., 1998), however, still with relatively modest statistics. Such a scenario is naturally formulated within the framework of multi-fractals (Falconer, 1990; Sylos Labini et al., 1998). It is, for example, a well known fact that bright ellipticals lie preferentially in clusters whereas spiral galaxies prefer the field (Dressler, 1984). From the perspective of multi-fractality this implies that ellipticals are more clustered than spirals, i.e. , their fractal dimension is lower (Giovanelli et al., 1986).
Our arguments indicate that the voids might be filled by dark matter. But since this dark matter is relatively smoothly distributed, it cannot be detected by measurements sensible only to density gradients (like, e.g., peculiar velocities). One needs to determine the total density of the universe, for example by measuring the deceleration parameter .
© European Southern Observatory (ESO) 1998
Online publication: October 22, 1998