A gravitational diffusion model without dark matter (galactic rotation curvesyclusters and superclustersyzero dark matteryinitial gravitational collapseygalaxy formation)

In this model, without dark matter, the f lat rotation curves of galaxies and the mass-to-light ratios of clusters of galaxies are described quantitatively. The hypothesis is that the agent of gravitational force is propagated as if it were scattered with a mean free path of '5 kiloparsecs. As a result, the force between moderately distant masses, separated by more than the mean free path, diminishes as the inverse first power of the distance, following diffusion equations, and describes the f lat rotation curves of galaxies. The force between masses separated by <1 kiloparsec diminishes as the inverse square of distance. The excess gravitational force (ratio of 1yr:1yr2) increases with the scale of structures from galaxies to clusters of galaxies. However, there is reduced force at great distances because of the '12 billion years that has been available for diffusion to occur. This model with a mean free path of '5 kiloparsecs predicts a maximum excess force of a few hundredfold for objects the size of galactic clusters a few megaparsecs in size. With only a single free parameter, the predicted curve for excess gravitational force vs. size of structures fits reasonably well with observations from those for dwarf galaxies through galactic clusters. Under the diffusion model, no matter is proposed in addition to the observed baryons plus radiation and thus the proposed density of the universe is only a few percent of that required for closure. The concept of diffusion of gravity arose from recognition that flat rotation curves of galaxies would result from the equations of diffusion (1). The model that has developed includes: (i) an unspecified agent responsible for the force of gravity, probably traveling at the speed of light over small distances; (ii) for distances more than a few kiloparsecs (kpc), the agent propagates following the diffusion equations; (iii) the effective mean free path is '5 kpc, apparently independent of the local matter density; and (iv) the process that causes the propagation according to diffusion equations is probably not scattering of the direction of travel of gravitational elements but something more subtle, involving distortion of the metric. A process for which the continued propagation is proportional to the concentration of elements in local regions follows the diffusion equations. The phrases ‘‘gravitational elements’’ or the ‘‘agents of gravity’’ used in this article are shorthand for an unknown underlying process that amounts to propagation of the curvature of the metric. In the diffusion model, the retardation of the agent of gravity increases the gravitational force (compared with inverse square) at distances from a few kpc to many megaparsecs (Mpc), owing to the effective higher concentration of the agent. This increase is described as ‘‘excess gravitational force.’’ It explains the observed differences in the mass-to-light ratio for structures of various sizes. There is no known reason to propose that the mean free path or diffusion constant varies over space or time or that the propagation is affected by the presence of matter or radiation. Questions about the nature of gravitation and the mechanism of its propagation are bypassed. A scattering process is not favored because of the problem of preservation of the vector of attractive force through scattering events. This article reports the agreement of astronomical observations of excess gravitational force with the quantitative predictions of the diffusion concept. Quantitative Description of Diffusion The standard solution for diffusion in three dimensions from a point source can be transformed to amount per spherical shell (Ps): Ps 5 C r erfc@ry2 ~D t!#, [1] where C is a constant; r is radius; t is time; and D is a diffusion constant. This equation is accurate for distances much greater than the mean free path but does not apply for small distances. To obtain an equation suitable for small distances, Monte Carlo calculations were made for elements traveling at constant speed (c) that are scattered in a totally random direction after traveling an average distance p. The following equation matched the results quite well: Ps 5 ~1 1 bryp!erfc@rya ~pct!#, [2] where Ps again is the number per spherical shell. For the best fit, b 5 3.1 and a 5 1.1. With this definition of the mean free path, the diffusion coefficient is approximated by p c (over a small factor) and thus the gravitational force (F) in this model becomes F 5 Gmm r ~1 1 bryp!erfc@rya ~pct!#, [3] where r is distance; a and b are constants; p is mean free path; c is speed of travel of a gravitational element; and t is the time since the start. This equation is graphed in Fig. 1. The process equivalent to scattering might be represented as many small deflections instead of the large deflections used in this Monte Carlo model, but tests show that this does not affect to a great degree the approximation, although it could affect the constant b, which is uncertain by perhaps 50%. There are three important domains as follows. If r ,, p, the inverse square term dominates and gravitational force is inverse square. If r is greater than p but less than a few Mpc, there is a 1yr relationship, which applies for galaxies and for small clusters. Finally, if r is many Mpc, then as a result of the slowness of diffusion, the gravitational force only partially reaches the distant regions and the decay of the erfc function dominates. Thus, under the diffusion model, the present gravitational force reaches a maximum ratio to inverse square The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked ‘‘advertisement’’ in accordance with 18 U.S.C. §1734 solely to indicate this fact. © 1998 by The National Academy of Sciences 0027-8424y98y953351-5$2.00y0 PNAS is available online at http:yywww.pnas.org. Abbreviations: kpc, kiloparsec; Mpc, megaparsec; MOND, modification of Newtonian gravity. *To whom reprint requests should be addressed. e-mail: rbritten@ etna.bio.uci.edu.