Fractal Cosmology in an Open Universe

– The clustering of galaxies is well characterized by fractal properties, with the presence of an eventual cross-over to homogeneity still a matter of considerable debate. In this letter we discuss the cosmological implications of a fractal distribution of matter, with a possible cross-over to homogeneity at an undetermined scale Rhomo. Contrary to what is generally assumed, we show that, even when Rhomo → ∞, this possibility can be treated consistently within the framework of the expanding universe solutions of Friedmann. The fractal is a perturbation to an open cosmology in which the leading homogeneous component is the cosmic background radiation (CBR). This cosmology, inspired by the observed galaxy distributions, provides a simple explanation for the recent data which indicate the absence of deceleration in the expansion (qo ≈ 0). Correspondingly the ‘age problem’ is also resolved. Further we show that the model can be extended back from the curvature dominated arbitrarily deep into the radiation dominated era, and we discuss qualitatively the modifications to the physics of the anisotropy of the CBR, nucleosynthesis and structure formation. One of the most extraordinary findings of the last two decades in observational cosmology has been the existence of a network of voids and structures in the distribution of galaxies in space. The enormous scales of these structures were completely unsuspected in earlier extensive observations of galaxy distributions, in which only angular coordinates were measured, obscuring the richness subsequently revealed in the third coordinate. These findings have become increasingly difficult to reconcile with standard cosmological theories, in which the approach to homogeneity at large scales is a central element [1]. Observationally, however, not (∗) Current Address: LPT, Université Paris-XI, Bâtiment 211, F-91405 Orsay Cedex, France Typeset using EURO-LTEX 2 EUROPHYSICS LETTERS only the scale at which the matter distribution approaches an average density, but the very existence of such a scale, remains the subject of intense debate [2]-[6]. At small scales it is well established that the distribution of galaxies is fractal, and the debate can be phrased in terms of the deviation from this behaviour towards homogeneity. Some consensus has been achieved about the optimal statistical methods to use in the analysis of three dimensional data, with disagreement remaining on details of the treatment of some data sets [5, 6]. The case for homogeneity still rests primarily on indirect observational evidence, such as the angular data. Independently of the data, however, resistance to the fractal picture is certainly to a considerable degree due to the conviction that it is incompatible with the framework of the standard theories (see e.g. [7]), and in particular with the high degree of isotropy of the microwave background radiation [8, 9]. In this respect one should note that in standard models the origin of radiation and baryonic matter is completely separate, with the latter being created in a dynamical process (‘baryogenesis’) completely distinct from the origin of the primordial radiation bath. The isotropy of the latter is therefore not fundamentally tied to the distribution of the matter, and the only real constraint is how much any such distribution actually perturbs the radiation. In this Letter we do not discuss the details of the evidence for or against homogeneity, but rather consider the grounds for these theoretical biases against the possible continuation of a fractal distribution to arbitrarily large scales. Our central result is that a fractal distribution for matter, even when there is no upper cut-off to homogeneity, can in fact be treated in the framework of an expanding universe Friedmann cosmology. A fractal [10, 11] is a self-similar and intrinsically fluctuating distribution of points at all scales, which appears to preclude the description of its gravitational dynamics in the framework of the Friedmann-Robertson-Walker (FRW) solutions to general relativity [1]. The problem is often stated as being due to the incompatibility of a fractal with the Cosmological Principle, where this principle is identified with the requirement that the matter distribution be isotropic and homogeneous[7]. This identification is in fact very misleading for a non-analytic structure like a fractal, in which all points are equivalent statistically, satisfying what has been called a Conditional Cosmological Principle [10, 11, 12]. The obstacle to applying the FRW solutions has in fact solely to do with the lack of homogeneity. One of the properties of a fractal of dimension D, however, is that the average density of points in a radius r about any occupied point decreases as rD−3, so that asymptotically the mass density goes to zero [10, 11]. An approximation which therefore may describe the large scale dynamics of the universe in the case that the matter has such a distribution continuing to all scales is given by neglecting the distribution of matter at leading order, relative to the small but homogeneous component coming from the cosmic microwave background. We will now show that is indeed a good perturbation scheme, and calculate the physical scale characterizing its validity. Consider first the standard FRW model with contributions from matter and radiation, for which the expansion rate is