Geometrical models for spheroidal cosmological voids

Transcription

1 Geometrical models for spheroidal cosmological voids talk by: Osvaldo M. Moreschi collaborator: Ezequiel Boero FaMAF, Universidad Nacional de Córdoba, Instituto de Física Enrique Gaviola (IFEG), CONICET, Ciudad Universitaria, (5000) Córdoba, Argentina CosmoSur-III; August 4, 2015 (FaMAF, UNC, CONICET) Geometrical models for voids CosmoSur-III; August, / 34

2 Content 1 Introduction 2 Homogeneous and isotropic spacetimes 3 Weak gravitational lensing of cosmological objects 4 Spherically symmetric compensated void 5 The model for spheroidal voids 6 Final comments (FaMAF, UNC, CONICET) Geometrical models for voids CosmoSur-III; August, / 34

3 Content 1 Introduction 2 Homogeneous and isotropic spacetimes 3 Weak gravitational lensing of cosmological objects 4 Spherically symmetric compensated void 5 The model for spheroidal voids 6 Final comments (FaMAF, UNC, CONICET) Geometrical models for voids CosmoSur-III; August, / 34

4 Introduction: Motivations I The very large scale structure of the Universe is normally represented by an homogeneous isotropic geometry; namely, the Robertson-Walker spacetimes. If we further assume the Einstein equations, then they must satisfy the Friedman equations. However when one considers smaller scales, one encounters the structure of filaments, clusters, voids, etc. (FaMAF, UNC, CONICET) Geometrical models for voids CosmoSur-III; August, / 34

5 Introduction: Motivations II Figure: Picture of the galaxy distribution from the Max Plank Institut für Astrophysic, Millennium Simulation Project. (FaMAF, UNC, CONICET) Geometrical models for voids CosmoSur-III; August, / 34

6 Introduction: Motivations III It is common to make models of these structures as isolated in an otherwise smooth homogeneous and isotropic geometry. Otherwise, one can also consider Swiss-cheese type models in which the holes are spherically symmetric geometries; which do not affect the outside metric. But. (FaMAF, UNC, CONICET) Geometrical models for voids CosmoSur-III; August, / 34

7 Introduction: Motivations IV Figure: In the real world, the voids in a Swiss-cheese Universe, are not exactly spherically symmetric. We concentrate in this case in a model for voids, but with spheroidal symmetry (FaMAF, UNC, CONICET) Geometrical models for voids CosmoSur-III; August, / 34

8 Introduction: Motivations V We propose a complete geometrical model for voids in Robertson-Walker cosmologies. The model provides with the metric with spheroidal symmetry which outside the void region is an exact homogeneous and isotropic Robertson-Walker spacetime. In the interior of the void one can adjust for the mass distribution and peculiar spacelike components of the energy momentum tensor that we have discussed elsewhere. The models are designed for the calculations of gravitational lens effects. (FaMAF, UNC, CONICET) Geometrical models for voids CosmoSur-III; August, / 34

9 Content 1 Introduction 2 Homogeneous and isotropic spacetimes 3 Weak gravitational lensing of cosmological objects 4 Spherically symmetric compensated void 5 The model for spheroidal voids 6 Final comments (FaMAF, UNC, CONICET) Geometrical models for voids CosmoSur-III; August, / 34

10 Homogeneous and isotropic spacetimes The line element of a Robinson-Walker Universe can be expressed by ds 2 = dt 2 A(t) 2 dl 2 k; (1) where A(t) is the expansion parameter and dl 2 k is the line element of a homogeneous and isotropic space with constant curvature k = 1, 0, 1. We can assume that A has units of length, and therefore dl 2 k does not have units. The line element dl 2 k can be expressed in more than one way, namely: dl 2 k = I dr 2 1 kr 2 + r 2 dσ 2, (2) d r 2 + r 2 dσ 2 dl 2 k =, (3) (1+ k r2 )2 4 dl 2 k = dχ 2 + fk 2 (χ)dσ 2 ; (4) where dσ 2 = d θ 2 + sin( θ) 2 d φ 2 is the line element of the unit sphere and f k (χ) = { sinh(χ) for k = 1, χ for k = 0, sin(χ) for k = 1. (5) In this presentation we will make use of version (3). (FaMAF, UNC, CONICET) Geometrical models for voids CosmoSur-III; August, / 34

11 Homogeneous and isotropic spacetimes II In these spacetimes a comoving observer (galaxy) has fixed coordinates ( r, θ, φ). The coordinates for prolate spheroids can be defined, in terms of Cartesian coordinates, by with ξ [0, ], θ [0, π] and φ [0, 2π]. Alternatively, using the definition one has the relations x = r µ sinh(ξ) sin(θ) cos(φ), (6) y = r µ sinh(ξ) sin(θ) sin(φ), (7) z = r µ cosh(ξ) cos(θ); (8) r = r µ sinh(ξ). (9) x = r sin(θ) cos(φ), (10) y = r sin(θ) sin(φ), (11) z = r 2 + rµ 2 cos(θ); (12) (FaMAF, UNC, CONICET) Geometrical models for voids CosmoSur-III; August, / 34

12 Homogeneous and isotropic spacetimes so that Let us note that dx 2 + dy 2 + dz 2 = (r 2 + r 2 µ sin 2 (θ))(dξ 2 + dθ 2 ) III + r 2 sin 2 (θ)dφ 2 ( ) = (r 2 + rµ 2 sin 2 dr 2 (θ)) + dθ 2 r 2 + rµ 2 + r 2 sin 2 (θ)dφ 2. (13) r 2 = x 2 + y 2 + z 2 = r 2 sin(θ) 2 + (r 2 + r 2 µ) cos(θ) 2 = r 2 + r 2 µ cos(θ) 2. (14) Therefore we can write dl 2 k = d r 2 + r 2 dσ 2 (r 2 + rµ 2 sin 2 (θ)) = (1 + k r 2 4 )2 ( dr 2 r 2 +r 2 µ + dθ 2 ) + r 2 sin 2 (θ)dφ 2 (1 + k 4 (r 2 + r 2 µ cos(θ) 2 )) 2. (15) Therefore, the Robertson-Walker line element, can be expressed in spheroidal coordinates as: ( ) ( (r 2 + r 2 ds 2 = dt 2 A(t) 2 µ sin 2 dr (θ)) 2 + dθ 2 + r 2 sin 2 (θ)dφ 2 ) r 2 +rµ 2 (1 + k (r. (16) rµ 2 cos(θ) 2 )) 2 (FaMAF, UNC, CONICET) Geometrical models for voids CosmoSur-III; August, / 34

13 Content 1 Introduction 2 Homogeneous and isotropic spacetimes 3 Weak gravitational lensing of cosmological objects 4 Spherically symmetric compensated void 5 The model for spheroidal voids 6 Final comments (FaMAF, UNC, CONICET) Geometrical models for voids CosmoSur-III; August, / 34

14 Weak gravitational lensing of cosmological objects We have shown elsewhere that the optical scalars produced by some distribution in a cosmological background, are given by I κ = (1 κ c) κ L + κ c, (17) γ 1 = (1 κ c) γ 1L, (18) γ 2 = (1 κ c) γ 2L ; (19) For stationary spherically symmetric objects, we have also shown that dls ( )] κ L (J) = 4πD ls [ϱ(r) + P r (r) + J2 P t(r) P r (r) dy, (20) γ L (J) = D ls dls d l J 2 d l r 2 r 3 r 2 [ ( )] 3M(r) 4π ϱ(r) + P t(r) P r (r) dy; (21) where d l and d ls are the lens and lens-source geometric distances, and D ls = 1 D Asl D Al ; (22) (1 + z l ) D As (FaMAF, UNC, CONICET) Geometrical models for voids CosmoSur-III; August, / 34

15 Weak gravitational lensing of cosmological objects II where D Asl, D Al and D As are the cosmic angular diameter distances. Let us also recall that the cosmic convergence is given by: while the magnification is: κ c = 1 D A(λ) λ ; (23) µ c = where we are using λ to denote the geometric distance. ( ) 2 1 λ (1 κ = ; (24) c) 2 D A (λ) (FaMAF, UNC, CONICET) Geometrical models for voids CosmoSur-III; August, / 34

16 Content 1 Introduction 2 Homogeneous and isotropic spacetimes 3 Weak gravitational lensing of cosmological objects 4 Spherically symmetric compensated void 5 The model for spheroidal voids 6 Final comments (FaMAF, UNC, CONICET) Geometrical models for voids CosmoSur-III; August, / 34

17 Spherically symmetric compensated void Let us consider the simple mass density profile used in the work [AFW99] which is given by: 4π 3 r3 ρ int for r < R M(r) = M(R) + 4π 3 (r3 R 3 )ρ bor for R r < R + d (25) 0 for R + d r; where in the interior of the void, one has and at the border ρ int = ρ δ, with δ 1, (26) ρ bor = ρ δ (1 + d R )3 1. (27) For the spherically symmetric case, this profile is such that the mass at the border compensates for the amount missing in the void. The density ρ is chosen to agree with the outside cosmic density; which according to the Friedman equation, it satisfies ( ) 2 da + kc 2 = 8πG dt 3 ρ A2 ; (28) where we have left, in this case, the universal constants in place. I (FaMAF, UNC, CONICET) Geometrical models for voids CosmoSur-III; August, / 34

18 Spherically symmetric compensated void II The lens optical scalars are: κ L ( J) = 8πδ ϱrd ls ( 1+ d) 2 J2 1 J 2 ( ) d ( ) 1+ d 2 J2 1 J 2 for 0 < J < 1 ( ) 3 1 for 1 < J < 1 + d 1+ d 0 for 1 + d < J ; (29) γ L ( J) = 8πδ ϱr D ls 3 J 2 ( ) 2 J2[ ( ) ] 1+ d 2 1+ d 2+ J2 [ ( ] d) 1 J 2( 2+ J 2)( ) 3 1+ d [ ( ] 3 1 for 0 < 1+ d) J < 1 ( ) 2 J2[ ( ) ] 1+ d 2 1+ d 2+ J2 [ ( ] d) for 1 < J < 1 + d 0 for 1 + d < J (30) 1 The expression agree with those of [AFW99]; except for the redshift contribution: 1+z L within the factor D ls.. (FaMAF, UNC, CONICET) Geometrical models for voids CosmoSur-III; August, / 34

19 Spherically symmetric compensated void III Figure: The optical scalars plotted for the following set of values: δ = 0.3, R = 50 Mpc h, d = 0.1. Left: Our plot for the lens optical scalars. Right: Optical scalars taken from the work [AFW99] (FaMAF, UNC, CONICET) Geometrical models for voids CosmoSur-III; August, / 34

20 Spherically symmetric compensated void IV The complete optical scalars with the cosmological corrections. Figure: Complete optical scalars for the simple compensated spherically symmetric void. (FaMAF, UNC, CONICET) Geometrical models for voids CosmoSur-III; August, / 34

21 Content 1 Introduction 2 Homogeneous and isotropic spacetimes 3 Weak gravitational lensing of cosmological objects 4 Spherically symmetric compensated void 5 The model for spheroidal voids 6 Final comments (FaMAF, UNC, CONICET) Geometrical models for voids CosmoSur-III; August, / 34

22 The model for spheroidal voids I It is not clear that voids can be safely studied en the framework of thin weak gravitational lensing. The assumption of spherical symmetry should be generalized. We need for a complete geometry that allows for these studies in detail of gravitational lens effects. Our model is given by the line element: ( ( (r 2 + rµ 2 sin 2 (θ)) ds 2 = a(r)dt 2 A(t) 2 dr 2 r 2 2M(r)r(1+ r2 µ r 2 sin2 (θ))+r 2 µ (1 + k 4 (r 2 + r 2 µ cos(θ) 2 )) 2 + dθ 2 ) + r 2 sin 2 (θ)dφ 2 (31) ) ; (FaMAF, UNC, CONICET) Geometrical models for voids CosmoSur-III; August, / 34

23 The model for spheroidal voids II where the timelike component is a(r) can be used to introduce peculiar spacelike components of the energy momentum tensor; that we have shown to be useful for the description of dark matter phenomena. The mass function M(r) is the one used in the Amendola, et.al. in their spherical model[afw99] for weak gravitational lensing voids, namely: 4π r 3 ρ 3 int for r < R M(r) = M(R) + 4π (r 3 R 3 )ρ 3 bor for R r < R + d (32) 0 for R + d r; but where now we take for the interior: and for the border: ρ int = A 2 v ρ δ, with δ 1, (33) ρ bor = where A v is the value of the expansion parameter at the void. A 2 v ρ δ (1 + d R )3 1 ; (34) (FaMAF, UNC, CONICET) Geometrical models for voids CosmoSur-III; August, / 34

24 The model for spheroidal voids III Figure: From [SLWW12]. (FaMAF, UNC, CONICET) Geometrical models for voids CosmoSur-III; August, / 34

25 The model for spheroidal voids IV Figure: From [SLWW12]. (FaMAF, UNC, CONICET) Geometrical models for voids CosmoSur-III; August, / 34

26 The model for spheroidal voids V We will consider a void of typical size of R = 20Mpc at redshift z=0,2. Also, let us take δ = 0, 8, d = 0, 1R, and the cosmic parameters to have the values: h = 0.72, Ω m = , Ω r = , Ω Λ = 0.68; which give the value A 0 = 2, m. This means that the physical size of R = 20Mpc corresponds to a comoving coordinate size of r R = Then we take as examples r µ = 0.01r R, r µ = 0.1r R and r µ = 0.8r R. Using the compensated profile used in [AFW99], we show the T 00 component of the energy momentum tensor in the next graph. (FaMAF, UNC, CONICET) Geometrical models for voids CosmoSur-III; August, / 34

27 The model for spheroidal voids VI Figure: Void with prolate spheroidal symmetry. (FaMAF, UNC, CONICET) Geometrical models for voids CosmoSur-III; August, / 34

28 The model for spheroidal voids VII Figure: Void with prolate spheroidal symmetry. (FaMAF, UNC, CONICET) Geometrical models for voids CosmoSur-III; August, / 34

29 The model for spheroidal voids VIII Figure: Void with prolate spheroidal symmetry. (FaMAF, UNC, CONICET) Geometrical models for voids CosmoSur-III; August, / 34

30 Content 1 Introduction 2 Homogeneous and isotropic spacetimes 3 Weak gravitational lensing of cosmological objects 4 Spherically symmetric compensated void 5 The model for spheroidal voids 6 Final comments (FaMAF, UNC, CONICET) Geometrical models for voids CosmoSur-III; August, / 34

31 Final comments I In order to carry out detailed studies of cosmological voids one needs to consider complex geometries, as for example the generalization to spheroidal symmetries. From the study of the weak lensing of spheroidal over density regions we have found that the behaviour of the shear is not simple. (FaMAF, UNC, CONICET) Geometrical models for voids CosmoSur-III; August, / 34

32 Final comments II Figure: The modulus γ of the shear expansion. The contour curves do not follow the projection of the spheroidal distribution. (FaMAF, UNC, CONICET) Geometrical models for voids CosmoSur-III; August, / 34

33 Final comments III We have shown in the past that the extension of geometrical model for over density regions, from spherical to spheroidal symmetry is rather straight forward. Instead, the spheroidal geometrical model we have just presented for cosmic voids, shows that in this case, the construction of these models requires much more care. We expect to improve on these model since, to have control for the construction of geometrical cosmic voids will allow us to carry out detailed studies of the gravitational lens effects. (FaMAF, UNC, CONICET) Geometrical models for voids CosmoSur-III; August, / 34

34 Bib I Luca Amendola, Joshua A. Frieman, and Ioav Waga. Weak Gravitational lensing by voids. Mon. Not. Roy. Astron. Soc., 309:465, P. M. Sutter, Guilhem Lavaux, Benjamin D. Wandelt, and David H. Weinberg. A public void catalog from the SDSS DR7 Galaxy Redshift Surveys based on the watershed transform. Astrophys. J., 761:44, (FaMAF, UNC, CONICET) Geometrical models for voids CosmoSur-III; August, / 34