Fractal geometry in an expanding, one-dimensional, Newtonian universe

Miller, Bruce N.; Rouet, Jean‐Louis; Guirriec, E. Le

doi:10.1103/physreve.76.036705

Cited by 21 publications

(35 citation statements)

References 46 publications

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“…[8,9]). A few groups of authors [10][11][12][13][14][15][16][17][18][19] have proposed different variants of the simple one-dimensional (1D) sheet model to mimic these equations. We have underlined in a recent paper [20] that a fundamental question about any such model, just as in three dimensions, is whether the gravitational force term -which is simply the infinite sum representing the force exerted by all other particles on the given particle -is well defined in the class of infinite distributions one wishes to study (which will represent the initial conditions for structure formation in cosmology).…”

Section: Introductionmentioning

confidence: 99%

Gravitational force in an infinite one-dimensional Poisson distribution

Gabrielli

Joyce

2010

Phys. Rev. E

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We consider the statistical properties of the gravitational field F in an infinite one-dimensional homogeneous Poisson distribution of particles, using an exponential cut-off of the pair interaction to control and study the divergences which arise. Deriving an exact analytic expression for the probability density function (PDF) P (F ), we show that it is badly defined in the limit in which the well known Holtzmark distribution is obtained in the analogous three-dimensional case. A well defined P (F ) may, however, be obtained in the infinite range limit by an appropriate renormalization of the coupling strength, giving a Gaussian form. Calculating the spatial correlation properties we show that this latter procedure has a trivial physical meaning. Finally we calculate the PDF and correlation properties of differences of forces (at separate spatial points), which are well defined without any renormalization. We explain that the convergence of these quantities is in fact sufficient to allow a physically meaningful infinite system limit to be defined for the clustering dynamics from Poissonian initial conditions.

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Section: Introductionmentioning

confidence: 99%

Gravitational force in an infinite one-dimensional Poisson distribution

Gabrielli

Joyce

2010

Phys. Rev. E

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“…This range is restricted by the intrinsic limitations of these N -body simulations (even of state-of-the-art simulations). The limitations are much less stringent for one-dimensional cosmological dynamics, which is simulated by Miller et al [20], finding multifractal structure in a very long range of scales. One last but not least argument for scale invariance is that the cosmic web produced by the adhesion model [21] is found to have multifractal features [22,23].…”

Section: Introductionmentioning

confidence: 99%

Statistics and geometry of cosmic voids

Gaite¹

2009

J. Cosmol. Astropart. Phys.

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We introduce new statistical methods for the study of cosmic voids, focusing on the statistics of largest size voids. We distinguish three different types of distributions of voids, namely, Poisson-like, lognormal-like and Pareto-like distributions. The last two distributions are connected with two types of fractal geometry of the matter distribution. Scaling voids with Pareto distribution appear in fractal distributions with box-counting dimension smaller than three (its maximum value), whereas the lognormal void distribution corresponds to multifractals with box-counting dimension equal to three. Moreover, voids of the former type persist in the continuum limit, namely, as the number density of observable objects grows, giving rise to lacunar fractals, whereas voids of the latter type disappear in the continuum limit, giving rise to non-lacunar (multi)fractals. We propose both lacunar and non-lacunar multifractal models of the cosmic web structure of the Universe. A nonlacunar multifractal model is supported by current galaxy surveys as well as cosmological N -body simulations. This model suggests, in particular, that small dark matter halos and, arguably, faint galaxies are present in cosmic voids.

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“…3D simulations, however, can show such behaviour over a very limited spatial range, making it difficult to establish if it is associated with a true scale invariance of the non-linear clustering. The study of 1D models in the family we are considering (Miller & Rouet 2006;Miller et al 2007;Miller & Rouet 2010a;Joyce & Sicard 2011), with much greater spatial resolution and numerical accuracy, shows very convincing evidence that these models do indeed give rise to power-law clustering indicative of truly scale-invariant clustering.…”

Section: Stable Clustering Predictionsmentioning

confidence: 99%

“…Γ can simply be considered as a control parameter for understanding the role of expansion in the determination of the properties of the non-linear clustering. While most previous studies have considered either the EdS model (also known as the "quintic" model for reasons which will be recalled below), or the static model, one other model in this family, corresponding to the case Γ = √ gn0, introduced in Miller & Rouet (2002), has been studied in Miller & Rouet (2006);Miller et al (2007);Miller & Rouet (2010a) as well as in Joyce & Sicard (2011).…”

Section: A Family Of 1d Scale-free Modelsmentioning

confidence: 99%

“…Different initial conditions and variants of the model have been discussed by a number of other authors (Yano & Gouda 1998;Miller & Rouet 2002;Aurell & Fanelli 2002;Miller & Rouet 2006;Valageas 2006;Miller et al 2007;Gabrielli et al 2009;Miller & Rouet 2010a). There is also an extensive literature on the statistical mechanics of finite one dimensional self-gravitating systems (see e.g.…”

Section: Introductionmentioning

confidence: 99%

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Exponents of non-linear clustering in scale-free one-dimensional cosmological simulations

Benhaiem

Joyce

Sicard

2013

Monthly Notices of the Royal Astronomical Society

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One dimensional versions of dissipationless cosmological N-body simulations have been shown to share many qualitative behaviours of the three dimensional problem. Their interest lies in the fact that they can resolve a much greater range of time and length scales, and admit exact numerical integration. We use such models here to study how non-linear clustering depends on initial conditions and cosmology. More specifically, we consider a family of models which, like the three dimensional EdS model, lead for power-law initial conditions to self-similar clustering characterized in the strongly nonlinear regime by power-law behaviour of the two point correlation function. We study how the corresponding exponent γ depends on the initial conditions, characterized by the exponent n of the power spectrum of initial fluctuations, and on a single parameter κ controlling the rate of expansion. The space of initial conditions/cosmology divides very clearly into two parts: (1) a region in which γ depends strongly on both n and κ and where it agrees very well with a simple generalisation of the so-called stable clustering hypothesis in three dimensions, and (2) a region in which γ is more or less independent of both the spectrum and the expansion of the universe. The boundary in (n, κ) space dividing the "stable clustering" region from the "universal" region is very well approximated by a "critical" value of the predicted stable clustering exponent itself. We explain how this division of the (n, κ) space can be understood as a simple physical criterion which might indeed be expected to control the validity of the stable clustering hypothesis. We compare and contrast our findings to results in three dimensions, and discuss in particular the light they may throw on the question of "universality" of non-linear clustering in this context.

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Fractal geometry in an expanding, one-dimensional, Newtonian universe

Cited by 21 publications

References 46 publications

Gravitational force in an infinite one-dimensional Poisson distribution

Gravitational force in an infinite one-dimensional Poisson distribution

Statistics and geometry of cosmic voids

Exponents of non-linear clustering in scale-free one-dimensional cosmological simulations

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