Gravitational force distribution in fractal structures

We study the (newtonian) gravitational force distribution arising from a fractal set of sources. We show that, in the case of real structures in finite samples, an important role is played by morphological properties and finite size effects. For dimensions smaller than d-1 (being $d$ the space dimension) the convergence of the net gravitational force is assured by the fast decaying of the density, while for fractal dimension D>d-1 the morphological properties of the structure determine the eventual convergence of the force as a function of distance. We clarify the role played by the cut-offs of the distribution. Some cosmological implications are discussed.Comment: 9 pages, latex, 2 postscript figures, also available at http://www.phys.uniroma1.it/DOCS/PIL/pil.html Accepted for Publication in Europhysics Letters. Minor modifications adde

The fractal structure of the universe : a new field theory approach

While the universe becomes more and more homogeneous at large scales, statistical analysis of galaxy catalogs have revealed a fractal structure at small-scales (\lambda < 100 h^{-1} Mpc), with a fractal dimension D=1.5-2 (Sylos Labini et al 1996). We study the thermodynamics of a self-gravitating system with the theory of critical phenomena and finite-size scaling and show that gravity provides a dynamical mechanism to produce this fractal structure. We develop a field theoretical approach to compute the galaxy distribution, assuming them to be in quasi-isothermal equilibrium. Only a limited, (although large), range of scales is involved, between a short-distance cut-off below which other physics intervene, and a large-distance cut-off, where the thermo- dynamic equilibrium is not satisfied. The galaxy ensemble can be considered at critical conditions, with large density fluctuations developping at any scale. From the theory of critical phenomena, we derive the two independent critical exponents nu and eta and predict the fractal dimension D = 1/nu to be either 1.585 or 2, depending on whether the long-range behaviour is governed by the Ising or the mean field fixed points, respectively. Both set of values are compatible with present observations. In addition, we predict the scaling behaviour of the gravitational potential to be r^{-(1 + eta)/2}. That is, r^{-0.5} for mean field or r^{- 0.519} for the Ising fixed point. The theory allows to compute the three and higher density correlators without any assumption or Ansatz. We find that the N-points density scales as r_1^{(N-1)(D-3)}, when r_1 >> r_i, 2 leq i leq N . There are no free parameters in this theory.Comment: Latex, 20 pages, no figures, to be published in the Astrophysical Journa

The complex universe: recent observations and theoretical challenges

The large scale distribution of galaxies in the universe displays a complex pattern of clusters, super-clusters, filaments and voids with sizes limited only by the boundaries of the available samples. A quantitative statistical characterization of these structures shows that galaxy distribution is inhomogeneous in these samples, being characterized by large-amplitude fluctuations of large spatial extension. Over a large range of scales, both the average conditional density and its variance show a nontrivial scaling behavior: at small scales, r<20 Mpc/h, the average (conditional) density scales as 1/r. At larger scales, the density depends only weakly (logarithmically) on the system size and density fluctuations follow the Gumbel distribution of extreme value statistics. These complex behaviors are different from what is expected in a homogeneous distribution with Gaussian fluctuations. The observed density inhomogeneities pose a fundamental challenge to the standard picture of cosmology but it also represent an important opportunity which points to new directions with respect to many cosmological puzzles. Indeed, the fact that matter distribution is not uniform, in the limited range of scales sampled by observations, rises the question of understanding how inhomogeneities affect the large-scale dynamics of the universe. We discuss several attempts which try to model inhomogeneities in cosmology, considering their effects with respect to the role and abundance of dark energy and dark matter.Comment: 30 pages, 10 figure

The Non-Uniform Distribution of Galaxies from Data of the SDSS DR7 Survey

We have analyzed the spatial distribution of galaxies from the release of the Sloan Digital Sky Survey of galactic redshifts (SDSS DR7), applying the complete correlation function (conditional density), two-point conditional density (cylinder), and radial density methods. Our analysis demonstrates that the conditional density has a power-law form for scales lengths 0.5-30 Mpc/h, with the power-law corresponding to the fractal dimension D = 2.2+-0.2; for scale lengths in excess of 30 Mpc/h, it enters an essentially flat regime, as is expected for a uniform distribution of galaxies. However, in the analysis applying the cylinder method, the power-law character with D = 2.0+-0.3 persists to scale lengths of 70 Mpc/h. The radial density method reveals inhomogeneities in the spatial distribution of galaxies on scales of 200 Mpc/h with a density contrast of two, confirming that translation invariance is violated in the distribution of galaxies to 300 Mpc/h, with the sampling depth of the SDSS galaxies being 600 Mpc/h.Comment: 22 page

Scale Dependent Dimension of Luminous Matter in the Universe

We present a geometrical model of the distribution of luminous matter in the universe, derived from a very simple reaction-diffusion model of turbulent phenomena. The apparent dimension of luminous matter, $D(l)$, depends linearly on the logarithm of the scale $l$ under which the universe is viewed: $D(l) \sim 3\log(l/l_0)/\log(\xi/l_0)$, where $\xi$ is a correlation length. Comparison with data from the SARS red-shift catalogue, and the LEDA database provides a good fit with a correlation length $\xi \sim 300$ Mpc. The geometrical interpretation is clear: At small distances, the universe is zero-dimensional and point-like. At distances of the order of 1 Mpc the dimension is unity, indicating a filamentary, string-like structure; when viewed at larger scales it gradually becomes 2-dimensional wall-like, and finally, at and beyond the correlation length, it becomes uniform.Comment: 6 pages, 2 figure

Invariant measures of the 2D Euler and Vlasov equations

We discuss invariant measures of partial differential equations such as the 2D Euler or Vlasov equations. For the 2D Euler equations, starting from the Liouville theorem, valid for N-dimensional approximations of the dynamics, we define the microcanonical measure as a limit measure where N goes to infinity. When only the energy and enstrophy invariants are taken into account, we give an explicit computation to prove the following result: the microcanonical measure is actually a Young measure corresponding to the maximization of a mean-field entropy. We explain why this result remains true for more general microcanonical measures, when all the dynamical invariants are taken into account. We give an explicit proof that these microcanonical measures are invariant measures for the dynamics of the 2D Euler equations. We describe a more general set of invariant measures, and discuss briefly their stability and their consequence for the ergodicity of the 2D Euler equations. The extension of these results to the Vlasov equations is also discussed, together with a proof of the uniqueness of statistical equilibria, for Vlasov equations with repulsive convex potentials. Even if we consider, in this paper, invariant measures only for Hamiltonian equations, with no fluxes of conserved quantities, we think this work is an important step towards the description of non-equilibrium invariant measures with fluxes.Comment: 40 page

Tapping into rhythm generation circuitry in humans during simulated weightlessness conditions

An ability to produce rhythmic activity is ubiquitous for locomotor pattern generation and modulation. The role that the rhythmogenesis capacity of the spinal cord plays in injured populations has become an area of interest and systematic investigation among researchers in recent years, despite its importance being long recognized by neurophysiologists and clinicians. Given that each individual interneuron, as a rule, receives a broad convergence of various supraspinal and sensory inputs and may contribute to a vast repertoire of motor actions, the importance of assessing the functional state of the spinal locomotor circuits becomes increasingly evident. Air-stepping can be used as a unique and important model for investigating human rhythmogenesis since its manifestation is largely facilitated by a reduction of external resistance. This article aims to provide a review on current issues related to the ‘locomotor’ state and interactions between spinal and supraspinal influences on the central pattern generator circuitry in humans, which may be important for developing gait rehabilitation strategies in individuals with spinal cord and brain injuries

Conceptual Problems of Fractal Cosmology

This report continues recent Peebles-Turner debate "Is cosmology solved?" and considers the first results for Sandage's program for "Practical cosmology". A review of conceptual problems of modern cosmological models is given, among them: the nature of the space expansion; recession velocities of distant galaxies more than velocity of light; cosmological Friedmann force; continuous creation of gravitating mass in Friedmann's equation; cosmological pressure is not able to produce a work; cosmological gravitational frequency shift; Friedmann-Holtsmark paradox; the problem of the cosmological constant; Einstein's and Mandelbrot's Cosmological Principles; fractality of observed galaxy distribution; Sandage's 21st problem: Hubble - de Vaucouleurs paradox; quantum nature of gravity force.Comment: 17 pages, no Figures, report presented at Gamow Memorial Conference, August 1999, St.-Petersburg, Russi

Initial conditions, Discreteness and non-linear structure formation in cosmology

In this lecture we address three different but related aspects of the initial continuous fluctuation field in standard cosmological models. Firstly we discuss the properties of the so-called Harrison-Zeldovich like spectra. This power spectrum is a fundamental feature of all current standard cosmological models. In a simple classification of all stationary stochastic processes into three categories, we highlight with the name ``super-homogeneous'' the properties of the class to which models like this, with $P(0)=0$, belong. In statistical physics language they are well described as glass-like. Secondly, the initial continuous density field with such small amplitude correlated Gaussian fluctuations must be discretised in order to set up the initial particle distribution used in gravitational N-body simulations. We discuss the main issues related to the effects of discretisation, particularly concerning the effect of particle induced fluctuations on the statistical properties of the initial conditions and on the dynamical evolution of gravitational clustering.Comment: 28 pages, 1 figure, to appear in Proceedings of 9th Course on Astrofundamental Physics, International School D. Chalonge, Kluwer, eds N.G. Sanchez and Y.M. Pariiski, uses crckapb.st pages, 3 figure, ro appear in Proceedings of 9th Course on Astrofundamental Physics, International School D. Chalonge, Kluwer, Eds. N.G. Sanchez and Y.M. Pariiski, uses crckapb.st

Light propagation in statistically homogeneous and isotropic universes with general matter content

We derive the relationship of the redshift and the angular diameter distance to the average expansion rate for universes which are statistically homogeneous and isotropic and where the distribution evolves slowly, but which have otherwise arbitrary geometry and matter content. The relevant average expansion rate is selected by the observable redshift and the assumed symmetry properties of the spacetime. We show why light deflection and shear remain small. We write down the evolution equations for the average expansion rate and discuss the validity of the dust approximation.Comment: 42 pages, no figures. v2: Corrected one detail about the angular diameter distance and two typos. No change in result