Lensing by (the most) massive structures in the universe

Transcription

1 Lensing by (the most) massive structures in the universe today: basics of lensing tomorrow: how can we investigate the matter content of galaxy clusters using lensing (strong and weak) Wednesday: cosmology with galaxy clusters

2 Lecture 1: a concise introduction to gravitational lensing Massimo Meneghetti INAF - Osservatorio Astronomico di Bologna Dipartimento di Astronomia - Università di Bologna

3 Gravitational lensing General relativity (GR): explains gravity in terms of assemblies of mass and energy curving the space-time photons feel gravity similarly to massive particles how can we formalize this?

4 Gravitational lensing General relativity (GR): explains gravity in terms of assemblies of mass and energy curving the space-time photons feel gravity similarly to massive particles how can we formalize this?

5 Gravitational lensing General relativity (GR): explains gravity in terms of assemblies of mass and energy curving the space-time photons feel gravity similarly to massive particles how can we formalize this?

6 Gravitational lensing General relativity (GR): explains gravity in terms of assemblies of mass and energy curving the space-time photons feel gravity similarly to massive particles how can we formalize this?

7 Gravitational lensing General relativity (GR): explains gravity in terms of assemblies of mass and energy curving the space-time photons feel gravity similarly to massive particles how can we formalize this?

8 Gravitational lensing General relativity (GR): explains gravity in terms of assemblies of mass and energy curving the space-time photons feel gravity similarly to massive particles how can we formalize this?

9 Fermat principle n>1

10 Fermat principle n>1 Classical optics: Snell law

11 Fermat principle n>1 Classical optics: Snell law General relativity: deflection angle

12 Deflection angle for a point mass Φ= GM r r = x 2 + y 2 + z 2 = b 2 + z 2 b = x 2 + y 2 Φ= x Φ y Φ = GM r 3 x y ˆα (b) = 2GM c 2 x y = 4GM c 2 = b cos φ sin φ x y x y + dz (b 2 + z 2 ) 3/2 z b 2 (b 2 + z 2 ) 1/2 0 = 4GM c 2 b cos φ sin φ

13 Deflection angle for the general lens For an ensamble of point masses: For a more general three-dimensional distribution of matter, using the thin screen approximation:

14 Lensing potential θ ˆψ = DL ξ ˆψ = 2 D LS c 2 = α D S Φdz α = θ ˆψ

15 Some examples

16 Lens equation The lens equation links the true and the apparent positions of the source when it is lensed by a matter distribution.

17 Point mass Ψ(θ) = D LS D S 4GM ln θ D L c2 multiple images and magnification!

18 More complicated lenses: time delay Let s go back a few slides and consider again the concept of effective refraction index: This concept implies that, when photons travel along a ray, they will appear to travel slower than in vacuum and will take an extra-time to pass by a massive object: t grav = 2 c 3 Φdz In a cosmological context: t grav = 2 c 3 (1 + z d) (Shapiro delay) (1 + z d )Ψ Φdz An additional contribution to the time delay comes from the extra geometrical path followed by light when it gets deflected towards the observer: s = ξ ˆα 2 Again, putting this in a cosmological context: t geom = (1+z d ) D dd ds 2D s c ˆα2 (1 + z d )( θ β) 2

19 More complicated lenses: time delays

20 More complicated lenses: time delays t( θ) = (1 + z L) D L D S c D LS = t geom + t grav 1 2 ( θ β) 2 ψ( θ)

21 More complicated lenses: time delays t( θ) = (1 + z L) D L D S c D LS = t geom + t grav 1 2 ( θ β) 2 ψ( θ) The solutions of the lens equation correspond to the stationary points of the time delay function (maxima, minima, saddle points)

22 More complicated lenses: time delays t( θ) = (1 + z L) D L D S c D LS = t geom + t grav 1 2 ( θ β) 2 ψ( θ) The solutions of the lens equation correspond to the stationary points of the time delay function (maxima, minima, saddle points)

23 Time delay surfaces

24 B Koopmans et al. 2003

25 B Koopmans et al. 2003

26 Lens mapping and distortions Consider the limit of small deflections: in this case the lens equation can be written as: β = θ α ( θ) β = β θ θ = Aθ

27 Lens mapping and distortions convergence Shear Eigenvalues λ t = 1 κ γ λ r = 1 κ + γ

28 Critical lines and caustics The inverse of the determinant Jacobian measures the magnification: The magnification diverges where the eigenvalues of A vanish: λ t = 1 κ γ λ r = 1 κ + γ det A =0 These two conditions define the critical lines These are lines on the lens plane which are mapped on the caustics on the source plane radial eigenvector Critical line tangential eigenvector

29 Examples

30 Convergence

31 Shear

32 Magnification

33 Critical lines and caustics

34 CL/CA vs source redshift z=2 z=4 z=1

35 Tangential and radial distortions x1 x1 x2 x2 Tangential Critical line Radial Critical line λ t =1 κ γ = δ 1 λ t =1 κ + γ = δ 1 A(0,x c )= δ κ + γ A(0,x c )= 1 κ γ 0 0 δ If the source is circular, how is its elliptical image oriented?

36 Tangential and radial distortions x1 x1 x2 x2 Tangential Critical line Radial Critical line λ t =1 κ γ = δ 1 λ t =1 κ + γ = δ 1 A(0,x c )= δ κ + γ A(0,x c )= 1 κ γ 0 0 δ If the source is circular, how is its elliptical image oriented?

37 Tangential and radial distortions x1 x1 x2 x2 Tangential Critical line Radial Critical line λ t =1 κ γ = δ 1 λ t =1 κ + γ = δ 1 A(0,x c )= δ κ + γ A(0,x c )= 1 κ γ 0 0 δ If the source is circular, how is its elliptical image oriented?

38 Second order lensing First flexion Second flexion

39 Second order lensing

40 Image configurations Note that multiple images exist only if the critical lines and the caustics exist! This condition defines a strong lens.

41 Conclusions From the fact that masses perturb the space-time, we expect: 1. deflections (displacements) 2. increased multiplicity 3. time delays 4. distortions (radial, tangential)

42 Now real observations...

43 Now real observations...

44 Now real observations... MACS J

45 Now real observations...