Is there a magnification paradox in gravitational lensing?

Is there a magnification paradox in gravitational lensing? Olaf Wucknitz wucknitz@astro.uni-bonn.de Astrophysics seminar/colloquium, Potsdam, 26 November 2007

Is there a magnification paradox in gravitational lensing? gravitational lenses formalism lens equation distortion, magnification, amplification deflection potential, Fermat potential magnification theorem magnification paradox from planes to spheres summary titlepage introduction summary contents back forward previous next fullscreen 1

Gravitational lenses What are they? titlepage introduction summary contents back forward previous next fullscreen 2

Light deflection Naive Newtonian calculation: α = dz dl = 1 c dl c Φ Φ = GM R α Soldner 1801 (Newton) Einstein 1915 (general relativity) α = 2 G c 2 M r α = 4 G c 2 M r titlepage introduction summary contents back forward previous next fullscreen 3

Gravitational lenses Gallery titlepage introduction summary contents back forward previous next fullscreen 4

Cluster lensing titlepage introduction summary contents back forward previous next fullscreen 5

Lens equation source α ~ lens α observer deflection angle true α apparent α D s D ds D ds α = D s α image position source position θ θ s lens equation/mapping θ s = θ α(θ) all angles small, tangential plane titlepage introduction summary contents back forward previous next fullscreen 6

Distortion/magnification source lensed M θ s θ lens equation θ s = θ α(θ) first derivative dθ s = dθ α θ dθ = M 1 dθ magnification / mapping matrix M(θ) = ( 1 α ) 1 θ (area) magnification = amplification µ = ± det M titlepage introduction summary contents back forward previous next fullscreen 7

Magnification amplification observer s view observer Ω lens source Ω s solid angles measure apparent size magnification = Ω Ω s source s view _ Ω s observer lens _ Ω source flux distributed over solid angle amplification = Ω Ω s titlepage introduction summary contents back forward previous next fullscreen 8

Magnification amplification surface brightness magnification matrix M 1 = 1 D dsd d D s α x source exchange source and observer: D d D sd, D s D so, D ds D do D ds D s α ~ M 1 = 1 (1 + z d)(1 + z s ) D ds D d (1 + z d )(1 + z s ) D s α x = M 1 lens magnification = amplification surface brightness conserved D d α observer θ = x D d, D sd = 1 + z s 1 + z d D ds, D so = (1 + z s )D s, D do = (1 + z d )D d titlepage introduction summary contents back forward previous next fullscreen 9

Potential potential ψ α(θ) = ψ(θ) Poisson equation 2 ψ(θ) =: 2κ(θ) = 2σ(θ) normalized surface mass density examples point mass constant surface mass density singular isothermal sphere σ = Σ Σ c ψ(θ) = m ln θ ψ(θ) = σ 2 θ2 ψ(θ) = 4π D s D ds σ 2 v c 2 θ light is delayed by ψ titlepage introduction summary contents back forward previous next fullscreen 10

Fermat potential, time-delay surface source light-travel time for virtual ray (θ s fixed) t = D dd s cd ds φ(θ) lens Fermat-potential φ(θ) = (θ θ s) 2 2 ψ(θ) θ observer titlepage introduction summary contents back forward previous next fullscreen 11

Fermat s principle light travel time is stationary 0 = φ(θ) [ (θ θs ) 2 = 2 = θ θ s ψ(θ) ] ψ(θ) lens equation θ s = θ α(θ) real images are positions of minima, (e.g. unperturbed image) maxima, or saddle-points of φ titlepage introduction summary contents back forward previous next fullscreen 12

Magnification theorem [ Schneider (1984) ] Hessian of Fermat-potential is inverse magnification matrix µ 1 = 1 α θ = 2 φ θ 2 = 1 κ γ x γ y γ y 1 κ + γ x diagonalise: rotate shear, γ = γ x + γ y µ 1 = 1 κ γ 0 0 1 κ + γ minimum: both eigenvalues positive Poisson: convergence κ = σ 0 sum: 2(1 κ) > 0 κ < 1 0 κ < 1 µ 1 = (1 κ) 2 γ 2 0 < µ 1 1 titlepage introduction summary contents back forward previous next fullscreen 13

Example: point-mass 4 µ + µ 2 0-2 -4-2 -1.5-1 -0.5 0 0.5 1 1.5 2 θ s [ Wambsganss (1998), Liv. Rev. Rel. 1, 12 ] titlepage introduction summary contents back forward previous next fullscreen 14

I ve seen this before... [ Einstein (1936) ] [ Einstein notebooks 1910 1912 ] titlepage introduction summary contents back forward previous next fullscreen 15

An apparent paradox amplification > 1 in all directions integrate over complete sphere total flux amplification! conservation of energy? solution to energy crisis? lensing cannot create photons titlepage introduction summary contents back forward previous next fullscreen 16

The standard explanation lens distorts geometry area of surface shrinks! have to compare with same mean geometry compare with same mean density in Universe [ Weinberg (1976) ] titlepage introduction summary contents back forward previous next fullscreen 17

But... equivalent: refraction or Newtonian deflection does not change geometry same formalism same paradox! so far: tangential plane no problem in the plane now: do it on the sphere! titlepage introduction summary contents back forward previous next fullscreen 18

Deflection angle for the sphere calculation for D s α = 2GM c 2 x 0 0 dz [x 2 0+(z z 0 ) 2] 3/2 α = m 2 cot θ 2 m θ 3 2 m/θ (m/2) cot (θ/2) α θ s α 1 M D d θ z x 0 0 π/2 θ π titlepage introduction summary contents back forward previous next fullscreen 19

Magnification for the sphere curvature of celestial sphere only second-order effects for point-mass µ 1 = m (1 ) 2 = 2.35 10 11 1 α θ = 1 + m 2 m2 θ 4 [ 1 + O ( θ 2)] planar approximation µ 1 = 1 m2 θ 4 10 µ tot µ + µ tot 1 µ + 1 3 2 µ µ 1 1 0-1 -2 [10 11 ] 0.1 1 10 θ s [arcsec] 0 500 1000 1500 2000 2500 θ s [arcsec] -3 titlepage introduction summary contents back forward previous next fullscreen 20

Lensing on the sphere far from optical axis: µ < 1 in this situation: magnification theorem not valid integration over sphere: mean µ is 1 no paradox modified Poisson equation [ ] 2 ψ(θ) =: 2κ(θ) = 2 σ(θ) σ not always κ 0 failure of theorem field lines decay! titlepage introduction summary contents back forward previous next fullscreen 21

Field lines on the sphere FIELD LINES titlepage introduction summary contents back forward previous next fullscreen 22

Back to gravitation short summary flat spacetime with refractive medium (or Newtonian) magnification theorem not valid modified Poisson equation equivalent: gravity with appropriate reference situation constant coordinate distance (e.g. isotropic or Schwarzschild coordinates) constant metric distance magnification theorem invalid, no paradox not equivalent: inappropriate reference situation constant affine distance: magnification theorem valid, seeming paradox titlepage introduction summary contents back forward previous next fullscreen 23

Affine distance, light travel time metric (c = 1) ds 2 = (1 + 2Ψ) dt 2 (1 2Ψ) dx 2 affine distance, light travel time: measured at observer s position light travel time: T = (1 + Ψ 0 ) dx (1 2Ψ) affine distance: L = (1 Ψ 0 ) dx general focusing theorem: µ > 1 for constant affine distance titlepage introduction summary contents back forward previous next fullscreen 24

Lensing by a spherical shell shell of radius (metric) r 0 with mass M σ GM/c2 r 0 for σ 1 limit r 0, M 0 with σ = const no change of global geometry unlensed situation (re-)move sphere or... affine distance Λ = r 1 2σ constant focusing theorem: compare with constant Λ focusing titlepage introduction summary contents back forward previous next fullscreen 25

Surface brightness theorem 1: magnification µ > 1 theorem 2: surface brightness is conserved? previous derivation: did not consider local metric perturbation in terms of photon number density per solid angle: µ (A) (s, o) µ (A) (o, s) = 1 + 2GM c 2 ( 1 1 ) D d D ds reciprocity theorem [ Etherington (1933), Phil. Mag., 15, 761 ] in terms of energy flux density: F obs F 0 = 1 + 4GM c 2 D yx D xy = 1 + z y 1 + z x ( 1 1 ) D d D ds titlepage introduction summary contents back forward previous next fullscreen 26

Summary seeming paradox if µ > 1 everywhere standard solution unlensed reference situation with different total surface area focusing theorem: constant affine distance this talk consider curvature of celestial sphere (necessary!) modified deflection angle modified Poisson equation, no field line paradox keep area unchanged for comparison no magnification theorem, no paradox surface brightness not conserved beware of general lensing theorems! [ Wucknitz (2007), A&A submitted ] titlepage introduction summary contents back forward previous next fullscreen 27

Bonus-pages: Exact magnification on the sphere We want to calculate the magnification matrix for arbitrary functions of the deflection angle. This includes large deflections and multi-plane lenses, where the deflection angle can no longer be written as the gradient of a potential. This generality is not necessary for the main part of this paper but may serve as the basis for future work. In the plane, the total displacement is not relevant, so that the magnification matrix is determined exclusively by the first-order derivatives of the deflection. On the sphere, we have to take into account the curvature, and the lens equation is no longer a vector equation. To determine the source position Θ s from the image position Θ, we have to move along a geodesic (or great circle) in the direction of the negative deflection angle and follow this geodesic for a length corresponding to the absolute deflection angle. The geodesic equation for arbitrary coordinates is ẍ α + Γ α µνẋ µ ẋ ν = 0. (1) The affine parameter λ runs from 0 at Θ to 1 at Θ s. Derivatives with respect to λ are written as dots. In the following, we write the deflection angle as a µ (with a 2 = a µ a µ ) titlepage introduction summary contents back forward previous next fullscreen 28

to avoid confusion with tensor indices. The boundary conditions are x α (0) = Θ α, ẋ α (0) = a α, x α (1) = Θ α s. (2) For the magnification matrix, we have to consider additional geodesics infinitely close to the reference geodesic. The equation for the difference ɛξ α, where ɛ is infinitely small, is the differential equation for the geodesic deviation: D 2 ξ α Dλ 2 = ẋβ ẋ µ ξ ν R α µβν (3) The differential operator D denotes covariant derivatives. The curvature tensor R has a particularly simple form for two-dimensional manifolds. It can be written in terms of the metric g µν as R α µβν := Γ α µν,β Γ α µβ,ν + Γ α ρβγ ρ µν Γ α ρνγ ρ µβ (4) = 1 K 2 ( δ α β g µν δ α ν g µβ ). (5) The curvature radius K is constant (K = 1) on the sphere. The limit of the tangential plane can be found as K. Eq. (3) is valid in any coordinate system. For our titlepage introduction summary contents back forward previous next fullscreen 29

convenience we use the system defined by the coordinates at Θ, which is then parallel-transported along the geodesic. In this way, the covariant derivatives become partial derivatives of the components, and ẋ α (a, 0) as well as the curvature tensor Eq. (5) have constant components. We use a local Cartesian system (with locally vanishing Christoffel symbols) in which ξ is measured parallel to the negative deflection angle and ξ orthogonal to this direction. This leads to ξ = 0, ξ = ω 2 ξ, ω := a K. (6) With the starting condition ξ α = Dξα Dλ = ξµdaα Dx µ (7) from the derivative of Eq. (2b), we can easily solve the differential equation (6) for the two starting vectors (1, 0) and (0, 1) and in this way write the transport equation from Θ to Θ s for arbitrary vectors ξ µ as ξ(1) = M 1 ξ(0). (8) titlepage introduction summary contents back forward previous next fullscreen 30

The inverse magnification matrix of this mapping in (, ) coordinates reads M 1 = 1 a ; a ; a ; sin ω ω cos ω a ; sin ω ω, (9) in terms of the derivatives (covariant or partial in these coordinates) of the deflection function a µ. The magnification depends on the derivatives, but also on the deflection angle itself, just as expected. We notice that finite deflection angles introduce rotation even if the deflection field is rotation-free (a ; = a ; ). Furthermore does the curvature of the sphere lead to a magnification of 1/ cos ω in the perpendicular direction even for (locally) constant deflection fields. In the interpretation of this, one should keep in mind that a covariantly constant deflection does not correspond to a rigid rotation of the sphere, not even locally. We can decompose the inverse magnification matrix into a rotated convergence and a shear part, ( ) A := M 1 cos ϕ sin ϕ = (1 κ) sin ϕ cos ϕ ( ) γ1 γ 2 γ 2 γ 1, (10) titlepage introduction summary contents back forward previous next fullscreen 31

where the parameters are determined by the following equations: tan ϕ = A 21 A 12 A 11 + A 22 = a, a, sin ω/ω 1 + cos ω a, a, sin ω/ω (11) 1 κ = sign(a 11 + A 22 ) (A11 + A 22 ) 2 2 + (A 12 A 21 ) 2 (12) γ 1 = A 11 A 22 2 γ 2 = A 12 + A 21 2 = cos ω 1 + a, a, sin ω/ω 2 = a, + a, sin ω/ω 2 Note that the mapping is invariant under a sign change of 1 κ with a simultaneous shift of π in ϕ. Equations (11) and (12) are consistent for π/2 < ϕ < π/2. In the limit of small ω (corresponding to small deflection angles or K ), the matrix reduces to the standard form ( 1 a M 1 ; a ) ; =. (15) a ; 1 a ; (13) (14) titlepage introduction summary contents back forward previous next fullscreen 32

Contents 1 Is there a magnification paradox in gravitational lensing? 2 Gravitational lenses What are they? 3 Light deflection 4 Gravitational lenses Gallery 5 Cluster lensing 6 Lens equation 7 Distortion/magnification 8 Magnification amplification 9 Magnification amplification surface brightness 10 Potential 11 Fermat potential, time-delay surface 12 Fermat s principle 13 Magnification theorem 14 Example: point-mass 15 I ve seen this before... 16 An apparent paradox titlepage introduction summary contents back forward previous next fullscreen 33

17 The standard explanation 18 But... 19 Deflection angle for the sphere 20 Magnification for the sphere 21 Lensing on the sphere 22 Field lines on the sphere 23 Back to gravitation 24 Affine distance, light travel time 25 Lensing by a spherical shell 26 Surface brightness 27 Summary 28 Bonus-pages: Exact magnification on the sphere 33 Contents titlepage introduction summary contents back forward previous next fullscreen 34