Distances in Inhomogeneous Cosmological Models

Abstract

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 * )

New stochastic approach to cumulative weak lensing

“…The effect of matter clumping into isolated halos, or isolated cores, was already considered by Kantowski in 1969 [3], and more recent analyses of gravitational lensing by statistically distributed inhomogeneities have been carried out, for example, in Refs. [4][5][6][7][8][9][10][11][12][13][14][15]. Lately many authors have studied exactly solvable models for large-scale inhomogeneities, such as swiss-cheese, onion and meatball models [3,[16][17][18][19][20][21][22][23][24].…”

Tidal forces inf(R)theories of gravity

“…Note that we defined an area distance which is valid for isotropic universe with no shear. A linear angular diameter distance can also be defined, which gives the same results for FLRW spacetimes [35].…”

“…At this stage let us stress that the usual GR result is recovered from (35) and that a cosmological constant term in the gravitational Lagrangian with equation of state p Λ ¼ −ρ Λ does not affect the focusing of null geodesics [11]. Nevertheless, in the realm of modified gravity theories, Eq.…”

“…(35). Writing η a ¼ ηe a , e a e a ¼ 1, 0 ¼ e a u a ¼ e a k a , and using a basis which is both parallel propagated and aligned, i.e., δe a =δv ¼ k b ∇ b e a ¼ 0, one can rearrange (35) as…”

Gravitational Lens Statistics and the Density Profile of Dark Halos

“…We shall adopt the so-called Ðlled-beam distance (Dyer & Roeder 1973), since the ray shooting in an inhomogeneous universe created by N-body simulations is consistent with it (Tomita 1998 ;Tomita, Asada, & Hamana 1999).…”