The Significance of the Concept `Chaos'

Tomita, Katsutoshi

doi:10.1143/ptps.79.1

Cited by 46 publications

(61 citation statements)

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“…In an earlier study using inhomogeneity models derived from N -body simulations, Tomita (1998) has found the best-fit value of α to be close to one in most cases, with a dispersion in α dependent on the cosmological model, the physical radius of the inhomogenities, the redshift and separation of light-rays. This paper is complementary in the respect that we study the effect of including pointlike compact objects on the propagation of infinitesimal light-rays.…”

Section: Introductionmentioning

confidence: 83%

The Dyer-Roeder distance-redshift relation in inhomogeneous universes

Mörtsell

2002

A&A

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Abstract. Using Monte-Carlo methods, we determine the best-fit value of the homogeneity parameter α in the Dyer-Roeder distance-redshift relation for a variety of redshifts, inhomogeneity models and cosmological parameter values. The relation between α and the fraction of compact objects, fp, is found to be approximately linear. This relation can be parametrized with reasonable accuracy for all cases treated in this paper by 1 − α = a · fp, where a ≈ 0.6.

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Section: Introductionmentioning

confidence: 83%

The Dyer-Roeder distance-redshift relation in inhomogeneous universes

Mörtsell

2002

A&A

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“…30) ) In the latter paper, 26) examples of strong image deformations and their statistics were studied. Using the same approach, Tomita 31) - 33) has recently studied the statistical behavior of optical scalars in more realistic inhomogeneous models with the CDM spectrum (n = 1). All particles are assumed to have the same galactic mass, while their radii depend on the model used.…”

Section: Direct Integration Methodsmentioning

confidence: 99%

Various Approaches to Cosmological Gravitational Lensing in Inhomogeneous Models

Tomita

Premadi

Nakamura

1999

Prog. Theor. Phys. Suppl.

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Gravitational lensing of distant objects caused by gravitational tidal forces from inhomogeneities in the universe is weak in most cases, but it is noticed that it gives a great deal of information about the universe, especially regarding the distribution of dark matter. The statistical values of optical quantities such as convergence, amplification and shear have been derived by many people using various approaches, which include the linear perturbational treatment in the weak limit and the nonlinear treatment considering small-scale matter distribution.In this review paper we compare the following three main approaches: (a) the approach in the multi-lens-plane theory; (b) the approach due to the direct integration method; and (c) the perturbational approach.In the former two approaches inhomogeneous matter distributions are produced in the CDM model using N -body simulations (the P 3 M code and the tree-code, respectively). In (c) the power spectrum corresponding to the CDM model is used for the large-scale matter distribution. §1. IntroductionThe propagation of light from distant objects like galaxies and QSOs is deflected by the gravitational tidal forces caused by inhomogeneous matter distribution between the objects and us. The so-called lens effect creates conspicuous images such as multiple QSOs and arcs in clusters of galaxies, owing to special positional relations among sources, lens objects and the observer, but in most cases it causes small deformations and amplification of optical images, which result from the superposition of deflections from many lens objects on cosmological scales. This so-called weak lensing gives us valuable information on the structure and evolution of the universe, especially regarding the distribution of dark matter, not only on large scales, but also on small scales around galaxies.The statistical behavior of optical quantites such as convergence, amplification and shear due to weak lensing has been studied by many people since the pioneering papers by Gunn, 1) -2) Weinberg 3) and Blandford and Jaroszyński. 4) For the derivations of these quantities there have been various approaches which consist of the multi-lens-plane method, direct integration methods solving the

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“…The length of the shadow of the interval between two rays in the observer plane is given by 17) where e α ≡ δ ⊥ x α /δl and g αβ e α e β = 1. Next, let us consider the area of the crosssection of a ray bundle given by…”

Section: Definition Of Distancesmentioning

confidence: 99%

Distances in Inhomogeneous Cosmological Models

Tomita

Asada

Hamana

1999

Prog. Theor. Phys. Suppl.

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Distances play important roles in cosmological observations, especially in gravitational lens systems, but there is a problem in determining distances because they are defined in terms of light propagation, which is influenced gravitationally by the inhomogeneities in the universe. In this paper we first give the basic optical relations and the definitions of different distances in inhomogeneous universes. Next we show how the observational relations depend quantitatively on the distances. Finally, we give results for the frequency distribution of different distances and the shear effect on distances obtained using various methods of numerical simulation. §1. IntroductionIn optical relations among observed quantities, distances such as the luminosity distance and the angular diameter distances play an important role. They are clearly defined in the homogeneous Friedmann-Lemaitre-Robertson-Walker model (Weinberg, 1) Schneider et al. 2) ) owing to the simple nature of light propagation in this case. In inhomogeneous universes, however, their behavior is complicated, due to gravitational lens effect which implies that light rays are deflected gravitationally by an inhomogeneous matter distribution. On the other hand, we also use distances to interpret the structure of gravitationally lensed systems.To correctly treat distances in inhomogeneous universes, it is necessary first to have a reasonable formulation for the dynamics describing local matter motion and optics and clarify the validity condition of the formulation. A set of fluid dynamical equations and the Poisson equation in the cosmological Newtonian approximation was introduced and discussed by Nariai 3) and Irvine 4) under the conditionswhere Φ, v, L and L H are the Newtonian gravitational potential, matter velocity, the characteristic size of inhomogeneities and the horizon size ≈ ct, respectively, and the spacetime is expressed aswhere a(t) is the scale-factor, σ(χ) = sin χ, χ, sinh χ for the background curvature * )

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The Significance of the Concept `Chaos'

Cited by 46 publications

References 0 publications

The Dyer-Roeder distance-redshift relation in inhomogeneous universes

The Dyer-Roeder distance-redshift relation in inhomogeneous universes

Various Approaches to Cosmological Gravitational Lensing in Inhomogeneous Models

Distances in Inhomogeneous Cosmological Models

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