Lensing effects in inhomogeneous cosmological models
📝 Abstract
Concepts developed in the gravitational lensing techniques such as shear, convergence, tangential and radial arcs maybe used to see how tenable inhomogeneous models proposed to explain the acceleration of the universe models are. We study the widely discussed LTB cosmological models. It turns out that for the observer sitting at origin of a global LTB solution the shear vanishes as in the FRW models, while the value of convergence is different which may lead to observable cosmological effects. We also consider Swiss-cheese models proposed recently based on LTB with an observer sitting in the FRW part. It turns out that they have different behavior as far as the formation of radial and tangential arcs are concerned.
💡 Analysis
Concepts developed in the gravitational lensing techniques such as shear, convergence, tangential and radial arcs maybe used to see how tenable inhomogeneous models proposed to explain the acceleration of the universe models are. We study the widely discussed LTB cosmological models. It turns out that for the observer sitting at origin of a global LTB solution the shear vanishes as in the FRW models, while the value of convergence is different which may lead to observable cosmological effects. We also consider Swiss-cheese models proposed recently based on LTB with an observer sitting in the FRW part. It turns out that they have different behavior as far as the formation of radial and tangential arcs are concerned.
📄 Content
arXiv:0901.0340v1 [astro-ph.CO] 3 Jan 2009 Lensing effects in inhomogeneous cosmological models Sima Ghassemi1∗, Salomeh Khoeini Moghaddam2†, and Reza Mansouri1,3‡ 1School of Astronomy and Astrophysics, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5531, Tehran, Iran, 2Physics Dept., Faculty of Science, Tarbiat Moalem University, Tehran, Iran, 3Department of Physics, Sharif University of Technology, Tehran 11365-9161, Iran Concepts developed in the gravitational lensing techniques such as shear, convergence, tangential and radial arcs maybe used to see how tenable inhomogeneous models proposed to explain the ac- celeration of the universe models are. We study the widely discussed LTB cosmological models. It turns out that for the observer sitting at origin of a global LTB solution the shear vanishes as in the FRW models, while the value of convergence is different which may lead to observable cosmological effects. We also consider Swiss-cheese models proposed recently based on LTB with an observer sitting in the FRW part. It turns out that they have different behavior as far as the formation of radial and tangential arcs are concerned. I. INTRODUCTION There is an on-going debate in the community wether inhomogeneities on scales up to several hundred mega parsecs could account, at least partially, for the observed dimming of the SNe Ia [Celerier [1], Sarkar [2]]. Irre- spective of successes of these models to explain the dark energy, it is desirable to have some independent cosmo- logical tests of these models. The obvious effect of the inhomogeneities is the light propagation which may differ significantly in a clumpy universe compared with a ho- mogeneous FRW model. Recently Vanderveld et al [3], being interested in the possibility of explaining the su- pernova data, have studied light propagation in one of the Swiss cheese models recently proposed [4] using weak field gravitational lensing techniques. We are interested in the potential gravitational lensing effects in inhomo- geneous models to see if there are crucial effects which could constrain or rule out some of the models. In this paper we focus on LTB-based inhomogeneous models. The LTB solution of Einstein equations de- scribes an inhomogeneous spherically symmetric dust filled cosmological model with a distinguished center of symmetry. Having a center of symmetry, thus not re- flecting the homogeneity of the universe at large, these models are mainly used as toy models to study the lo- cal inhomogeneities of the universe. After a short review of LTB inhomogeneous solutions, and Swiss-cheese mod- els based on it, we look at the general question of how different does a LTB model behave relative to a FRW model as far as gravitational lensing concepts are con- cerned (section III). For an observer at the origin of a flat LTB metric, the shear vanishes as in the case of a FRW metric. However, the convergence is different from that of the FRW due to the r-dependence of metric coeffi- cients and the corresponding scale factor. The vanishing of the shear is only true for an observer at the origin of LTB. The more general and interesting case of an off- center observer has to be treated separately. This case, however, maybe treated in Swiss cheese models based on LTB which is the subject of the section IV, where arcs in strong lensing regime are investigated for two differ- ent Swiss-cheese models, assuming the observer and the source are sitting in the cheese (FRW metric). While the onion model [5] turns out to produce neither tangen- tial nor radial arc, the model proposed by Marra, Kolb, Matarrese and Riotto [4] do produce both radial and tan- gential arcs. The models we have considered could be looked at as providing different density profiles of a lens, to be compared with the profiles published so far and mainly based on simulations. The Swiss cheese models, e.g., provide us with density profiles which maybe com- pared for special parameters to the NFW one which is a phenomenological density profile well supported by the simulations [6]. II. A REVIEW ON INHOMOGENEOUS LTB MODELS The LTB metric (c = 1) in co-moving and proper time coordinates with zero cosmological constant is given by ds2 = −dt2 + S2(r, t)dr2 + R2(r, t)(dθ2 + sin2 θdφ2). (1) Considering dust stress-energy tensor, Einstein’s equa- tions imply the following constraints: S2(r, t) = R ′2(r, t) 1 + 2E(r), (2) 1 2 ˙R(r, t) −GM(r) R(r, t) = E(r), (3) 4πρ(r, t) = M ′(r) R ′(r, t)R2(r, t), (4) where dot and prime denote partial derivatives with re- spect to t and r respectively. Function ρ(r, t) is energy density of the matter. Functions E(r) and M(r) are left arbitrary. M(r) can be interpreted as the mass inside of co-moving sphere with coordinate radius r. Assuming M ′(r) > 0, M(r) can be chosen as M(r) = 4π 3 M 4 0 r3. (5) Now, taking this as the definition of the coordinate ra- dius, we may write solutions of Einstein’s equations de- pending on E(r) in the following ways:
- For E > 0 th
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