An introduction to gravitational lensing and a possible combined - PowerPoint PPT Presentation

An introduction to gravitational lensing and a possible combined analysis with rotation curves Matteo Trudu Universit` a degli Studi di Cagliari High Energy Physics Colloquia December 5, 2017 Matteo Trudu (UniCa) High Energy Physics Colloquia December 5,2017 1 / 28

Summary Introduction to gravitational lensing 1 Deflection angle 2 General properties of a gravitational lens 3 Lensing regimes Combining rotation curves and gravitational lensing Matteo Trudu (UniCa) High Energy Physics Colloquia December 5,2017 2 / 28

Gravitational Lensing An observation taken with the Hubble Space Telescope Matteo Trudu (UniCa) High Energy Physics Colloquia December 5,2017 3 / 28

Gravitational Lensing Historical highlights In 1783, speculating that light consists of corpuscoles, an astronomer, named John Michell (1724-1793) hypothesized the possibility that a gravitational field could bend a light ray. He found out the following result for the deflection angle: α = 2 GM c 2 ξ John Michell With the full equations of General Relativity did Einstein obtain twice the Newtonian value: α = 4 GM = 2 R S c 2 ξ ξ Eddington used the gravitational lens of the sun to test Einstein’s theory of general relativity during the solar eclipse of the 29 th May Sir Arthur 1919. He measured a deflection angle of 1 . 75 ′′ which is compatible Eddington with the Einstein’s prediction! Matteo Trudu (UniCa) High Energy Physics Colloquia December 5,2017 4 / 28

Gravitational Lensing Deflection Angle (I) In weak field approximation: for a photon we have Dk µ = dk µ + Γ αβ dx α k β = 0 µ � 1 + 2Φ N � � 1 − 2Φ N � ds 2 = − x | 2 c 2 dt 2 + | d� c 2 c 2 we have to compute Geodesic equation: dk 2 = − Γ αβ dx α k β 2 du µ αβ u α u β = 0 µ + Γ dx α = ( cdt, dx 1 , 0 , 0) k β = ( ω/c, k 1 , 0 , 0) dτ Matteo Trudu (UniCa) High Energy Physics Colloquia December 5,2017 5 / 28

Gravitational Lensing Deflection Angle (II) The Christofell’s symbols: = 1 2 η µρ ( ∂ α h ρβ + ∂ β h αρ − ∂ ρ h αβ ) µ Γ αβ hence � ω c dx 1 = − 2 ω GMx 2 dk 2 = − 2 2 2 2 ) 3 / 2 dx 1 � Γ + 2Γ + Γ 00 01 11 c 3 ( x 2 1 + x 2 � ∞ ∆ k 2 ≃ − 2 GMξ 1 + ξ 2 ) 3 / 2 dx 1 = − 4 GM 1 k 1 c 2 c 2 ξ ( x 2 −∞ Since the gravitational field is weak we are entitled to assume small deflection angle ∆ k 2 � � � = 4 GM � � tan (ˆ α ) ≃ ˆ α = � � k 1 c 2 ξ � Matteo Trudu (UniCa) High Energy Physics Colloquia December 5,2017 6 / 28

Gravitational Lensing Fermat’s Principle Light deflection can equivalently be described by Fermat’s principle, as in geometrical optics! Fermat’s principle reads: � dt = 1 � δ n [ � x ( l )] dl = 0 c We need a refractive index. Since ds 2 = 0 for photons the light speed in the gravitational field is thus � � 1 + 2Φ N � � � d� x � 1 + 2Φ N � c ′ = � c 2 � � � = c ≃ c � � 1 − 2Φ N dt c 2 � c 2 n = c 1 � ≃ 1 − 2Φ N c ′ = � c 2 1 + 2Φ N c 2 Fermat’s principle suggests us a Lagrangian approach! Considering a generic parametric curve for the light’s path one finds: � + ∞ α = 2 ˆ � ∇ ⊥ (Φ N ) dz c 2 −∞ Matteo Trudu (UniCa) High Energy Physics Colloquia December 5,2017 7 / 28

Gravitational Lensing General Lens Since for a point-like mass the deflection angle is proportional to the mass we can use the superposition principle. N N ξ − � � ξ i ) = 4 G ξ i α ( � ˆ � α i ( � ˆ ξ − � � ξ ) = � � M i c 2 | � ξ − � ξ i | 2 i =1 i =1 The typical length-scales for the lenses are low compared to the distances between observer and source! Thin Screen Approximation Mass projected on the line of sight: � � � Σ( � � ξ ) = ρ ξ, z dz ξ − � � ξ ′ ξ ) = 4 G � ˆ α ( � Σ( � ξ ′ ) ξ ′ | 2 d 2 � ξ ′ � c 2 | � ξ − � Matteo Trudu (UniCa) High Energy Physics Colloquia December 5,2017 8 / 28

Gravitational Lensing Lens Equation � � η = βD S : Source Position � � ξ = θD L : Image Position If ˆ � α is the deflection angle, by a geometrical construction we can write a relation between the angles. Lens Equation � � � βD S + ˆ � � θD S = � α θ D LS D LS � � � � � ˆ � α � θ = � α θ D S � � � � � β = θ − � α θ Let’s introduce the dimensionless quantities � ξ = ξ 0 � x and � η = η 0 � y : Figure from P. Schneider et al. � y = � x − � α ( � x ) Gravitational Lenses , 1992, Spinger-Verlag Matteo Trudu (UniCa) High Energy Physics Colloquia December 5,2017 9 / 28

Gravitational Lensing Lensing Potential An extended distribution of matter is characterized by its effective Lensing Potential � ∞ x ) = 2 D L D LS Ψ ( � Φ N ( � x, z ) dz c 2 ξ 2 0 D S −∞ This lensing potential satisfies two important properties: 1 the gradient of Ψ gives the scaled deflection angle ∇ x Ψ ( � x ) = � α ( � x ) 2 the Laplacian of Ψ gives twice the convergence ∇ 2 x Ψ ( � x ) = 2 κ ( � x ) where c 2 x ) = Σ( � x ) D S κ ( � Σ crit = Σ crit 4 πG D L D LS Matteo Trudu (UniCa) High Energy Physics Colloquia December 5,2017 10 / 28

Gravitational Lensing Magnification and distortion (I) One of the main features of gravitational lensing is the distortion which it introduces into the shape of the sources. The distortion arises because light bundles are deflected differentially. The distortion of images can be described by the Jacobian matrix ∂ 2 Ψ A = ∂� y x = I − Ψ where I ij = δ ij Ψ ij = ∂� ∂x i ∂x j We can define the Shear Matrix: � − 1 A − 1 � � � 2 (Ψ 11 − Ψ 22 ) − Ψ 12 − γ 1 − γ 2 S = 2 tr ( A ) I = = 1 − γ 2 γ 1 − Ψ 12 2 (Ψ 11 − Ψ 22 ) � 1 � γ ( � � x ) = ( γ 1 ( � x ) , γ 2 ( � x )) = 2 (Ψ 11 − Ψ 22 ) , Ψ 12 Shear Vector A = (1 − κ ( � x )) I + S Matteo Trudu (UniCa) High Energy Physics Colloquia December 5,2017 11 / 28

Gravitational Lensing Magnification and distortion (II) Convergence : encodes all the variations of the surface of the source, which possess the same shape. It rescale the surface of the source. (scalar quantity) Shear : encodes all the variations of the shape of the source, which possess the same surface. (vectorial quantity) Effect of convergence and shear on a gaussian PSF Credit: Andrea Enia, Ricostruzione di sorgenti gravitazionalmente lensate selezionate nel sub-millimetrico , Master Degree Thesis Matteo Trudu (UniCa) High Energy Physics Colloquia December 5,2017 12 / 28

Gravitational Lensing Magnification and distortion (III) Credit: Andrea Enia, Ricostruzione di sorgenti gravitazionalmente lensate selezionate nel sub-millimetrico , Master Degree Thesis Matteo Trudu (UniCa) High Energy Physics Colloquia December 5,2017 13 / 28

Gravitational Lensing Occurence of images (I) The deflection of light rays causes a delay in the time between the emission of radiation by the source and the signal reception by the observer. ξ 2 � 1 � x ) = 1 + z L 0 D S y ) 2 − Ψ( � τ ( � 2 ( � x − � x ) c D L D LS this equation implies that images satisfy the Fermat Principle ∇ x τ ( � x ) = 0. Credit: M.Meneghetti, Introduction to Gravitational Lensing Time delay surfaces of an axially symmetric lens for three different source positions. Matteo Trudu (UniCa) High Energy Physics Colloquia December 5,2017 14 / 28

Gravitational Lensing Occurence of images (II) Let’s consider a point-like mass acting as a lens. Deflection angle: α = 4 GM c 2 D L θ Lensing Potential: Ψ ( θ ) = 4 GM D LS D L D S ln | θ | c 2 The lens equation reads: c 2 θD L D S = θ − θ 2 � β = θ − 4 GMD LS 4 GM D LS E θ E = Einstein Radius θ c 2 D S D L solving for θ : θ 2 − βθ − θ 2 E = 0 Two images, if β = 0 → Einstein Ring Matteo Trudu (UniCa) High Energy Physics Colloquia December 5,2017 15 / 28

Gravitational Lensing Occurence of images (III) Simulation: Point lens and extended source aaaaaaaaaaaaaaaUnlensedaaaaaaaaaaaaaaaaaaaaaaaaaaaaaLensed Credit: Joachim Wambsganss, Gravitational Lensing Theory and Applications Matteo Trudu (UniCa) High Energy Physics Colloquia December 5,2017 16 / 28

Gravitational Lensing Strong Lensing multiple images of background sources. The displacement of such images is determined by the mass distribution of the lenses. highly distorted images. If the source is extended, the differential deflection of the light creates distortions in the images. Matteo Trudu (UniCa) High Energy Physics Colloquia December 5,2017 17 / 28

Gravitational Lensing Weak Lensing weak distortions and small magnifications. we need a stastical approach. Matteo Trudu (UniCa) High Energy Physics Colloquia December 5,2017 18 / 28

Gravitational Lensing Micro Lensing microlensing phenomena are produced by lenses whose sizes are small compared to the scale of the lensing system can be thought of as a version of strong gravitational lensing in which the image separation is too small to be resolved. such lenses can be for example planets, stars or any compact object floating in the halo or in the bulge of our or of other galaxies. Matteo Trudu (UniCa) High Energy Physics Colloquia December 5,2017 19 / 28