degeneracies in lens models no h0 from gravitational...

Degeneracies in lens modelsNo H0 from gravitational lenses?

Olaf Wucknitz

Hamburger Sternwarte / JBO

[email protected]@jb.man.ac.uk

Jodrell Bank, 20. June 2001

Degeneracies and scaling relations in power-lawmodels for gravitational lenses

• This is the real title

• Expect many lovely equations!

• Results of recent work in Hamburg and JB

• Paper to be submitted soon

titlepage introduction discussion contents back forward previous next fullscreen 1

Introduction (1)

• Introduction

• Time delays

• Spherical power-law models

• External shear

• Mass-sheet degeneracy

• Previous work on 2237+0305 (with interpretation)

• Generalized power-law models


Introduction (2)

• Linear formalism to study H0 and scaling relations

• The “critical shear”

• Special cases

• Application to the systems Q2237+0305, PG 1115+080,

RX J0911+0551 and B 1608+656

• Discussion


Lens configuration

source plane

observer

lens

true source

Dds

Dd

Ds lens plane

α

α∗

α∗

Positions in source plane zs

and lens plane (image plane)

z measured as angles as seen

by the observer.

Dsα = Ddsα∗

Apparent deflection angle α

shifts position.

zs = z −α(z) lens equation


Deflection angle and potential

Point mass passed in a distance of r:

α∗ =4GMc2 r

Deflection angle is conservative field.

α(z) = ∇ψ(z)

Analogy to Newtonian gravitational field in two dimensions


Poisson equation

Surface mass density σ = Σ/Σc in units of critical surface mass

density Σc

∇2ψ(z) = 2σ(z)

Σc =1

4πc2

G

Ds

DdDds

Invert to get potential for arbitrary mass distribution

ψ(z) =1π

∫d2z′ ln |z − z′|σ(z′)


Light travel time

Light travel time of two images of one source can be different!

tz =1c

DdDs

Dds(1 + zd)

(12|α(z)|2 − ψ(z)

)+ const

Cosmology determines the distances:

Di =c

H0di(zi,Ω, λ, α, . . . )

Split constant factor into H0 and

deff = (1 + zd)dd ds

dds


Scaled Hubble constant

This deff only depends on cosmological model and redshifts.

h =H0

deff

h tz =12|α(z)|2 − ψ(z) + const

Time delay between images i and j:

h∆tij = h (ti − tj)

=12(|αi|2 − |αj|2

)− (ψi − ψj)

=12

(αi −αj) · (αi +αj)− (ψi − ψj)


Time delay

h∆tij =12

(αi −αj) · (αi +αj)− (ψi − ψj)

Now use lens equation zs = zi −αi

h∆tij =12

(zi − zj) · (αi +αj)− (ψi − ψj)

This is linear in ψ (resp. α). But: Cannot directly be split into

contributions from two images.

Use lens equation again to transform αi to αj and vice versa in

mixed terms, split delay into light travel times.

h ti = −12r

2i + zi ·∇ψi − ψi − C


Spherical power-law models

Spherically symmetric models

ψ(g)(r) = f rβ

α(r) = β f rβ−1

σ(r) =β2

2rβ−2

Special cases:

• β → 0 has ψ ∝ ln r and α ∝ 1/r. This is a point mass

• β = 1 has α = const. “Singular isothermal sphere” (SIS). Good

approximation for real galaxies!

• β = 2 is constant surface mass density σ


External shear

Influence of external masses (field galaxies, clusters) approximated

by Taylor expansion. First important terms in potential are

quadratic.

ψ(γ)(z) =12ztΓz

Γ =(−γx −γy−γy +γx

)α(z) = Γ z

γ = γ ( cos 2θγ , sin 2θγ )


Constant mass sheet

An additional constant surface mass density σ(z) ≡ κ contributes

with

ψ(κ)(z) = κr2

2,

α(z) = κ z .


The mass-sheet degeneracy

Multiply lens equation by 1− κ.

(1− κ)zs = (1− κ)z − (1− κ)∇ψ(z)

This is again a lens equation with (1− κ)-scaled zs and ψ with an

additional convergence κ. The same is true for the time delays:

(1− κ)h ti =− 12

(1− κ) r2i + (1− κ)zi ·∇ψi

− (1− κ)ψi − (1− κ)C

A potential ψ is equivalent to one of (1− κ)ψ+ κ r2/2, but source

position, potential and time delays (or H0) are scaled with 1− κ.

For 0 6 κ < 1 we only can determine upper limit of H0.


Previous work for 2237+0305 (1)

Wambsganß & Paczynski (1994): Numerical modelfitting (image

positions as constraints) for a range of 0 < β < 2.


Previous work for 2237+0305 (2)

Result: χ2 ≈ const, γ ∝ 2− β, ∆t ∝ 2− β


Interpretation: mass-sheet degeneracy (1)

We learned before: The potentials

ψ and ψ = (1− κ)ψ + κ r2/2

are equivalent. Deflection angles are:

α and α = (1− κ)α+ κ z

0

1α

0 1 2

r/r0

....................................................................................................................................................................................................................................................................................................................................................................................................................................... β = 1 α = 1

..................................

..................................

..................................

..................................

..................................

..................................

..................................

..................................

..................................

..................................

..................................

..................................

..................................

...............................β = 1 α = (1− κ) + κ r

........

........

........

...............................................................................................................................................................................................

................................

..................................

...................................

.......................................

........................................

..........................................

.............................................

...............................................

........................ β = 1.5 α = r0.5


Interpretation: mass-sheet degeneracy (2)

Alternative potential is equal to modified power law near the

images (first order in α).

α(r) = r0 reference model

α(r) = (1− κ) r0 + κ r equivalent model

α(r) = f rβ equivalent power-law model

2− β = 1− κ

Remember from mass-sheet degeneracy:

γ , zs , H0 ∝ 1− κ∝ 2− β


The general power-law model (1)

• radial dependence: power-law

• angular dependence: arbitrary

ψ(g) = rβ F (θ)

With Poisson equation and

∇2 = ∂2r + r−1 ∂r + r−2 ∂2

θ

we find density

σ =rβ−2

2

(β2F (θ) + F ′′(θ)

)


The general power-law model (2)

Examples are elliptical mass distributions, elliptical potentials, . . .

Radial derivative:

z ·∇ψ(g) = β ψ(g)

Shear and convergence are special case with β = 2 and

F (γ)(θ) =κ− γ cos 2(θ − θγ)

2.


Magnification and amplification (1)

Surface brightness is preserved!

But mapping zs→ z does not preserve area. Sources are

magnified and thus amplified.

µ(z) =image area

source area

=∣∣∣∣ ∂z∂zs

∣∣∣∣=∣∣∣∣∂zs

∂z

∣∣∣∣−1

=∣∣∣∣1− ∂α∂z

∣∣∣∣−1

=(

(1− ψxx)(1− ψyy)− ψ2xy

)−1


Magnification and amplification (2)

µ(z) =(

(1− ψxx)(1− ψyy)− ψ2xy

)−1

Magnifications µ (or µ−1)

• are not linear in ψ in the general case,

• depend on second derivatives of ψ,

• are difficult to determine observationally!


Parameters

parameters number

h Hubble constant 1

γ external shear 2

β power-law exponent 1

Fi angular part F (θi) n

F ′i dF/dθ(θi) n

zs source position 2

C constant in light times 1

total without fluxes 2n+ 7F ′′i for fluxes n


Constraints

constraints number

zi lens equations 2nti light travel times n

total without fluxes 3nµi/µj flux ratios n− 1

n = 4: 15 parameters with 12 constraints.

Fix β for all calculations, γ for most.

Include flux ratios? This adds n parameters but only n− 1constraints!


Time delay for power-law model

Remember general equation

h ti = −12r2i + zi ·∇ψi − ψi − C

With z ·∇ψ(g) = β ψ(g), we obtain

h ti = −12r

2i − (1− β)ψ(g)

i + ψ(γ)i − C

ψ(g)i = rβi Fi Fi = F (θi)

How to determine Fi? (later: lens equations!)


Isothermal case

Special case: isothermal! (β = 1)

h ti = −12r2i + ψ

(γ)i − C

Now we do not need lens equations at all.

(cf. Witt, Mao & Keeton 2000)

Four light travel times can be used to determine h, C and γ.


Including the lens equations (1)

Remember lens equation

zs = z −∇ rβ F (θ)− Γz

Transformation from polar to cartesian coordinates(∂x∂y

)=(

cos θ − sin θ/rsin θ cos θ/r

)(∂r∂θ

)In terms of observables zi:

zs =

(1− rβ−2

i

(β Fi −F ′iF ′i βFi

)− Γ

)zi


Including the lens equations (2)

In terms of the unknown parameters:

zs = zi − rβ−2i

(xi −yiyi xi

)(βFiF ′i

)+(xi yi−yi xi

)(γxγy

)Use lens equations to determine Fi and F ′i :

βFi = r−βi

(r2i − xixs − yiys + γx (x2

i − y2i ) + 2 γy xiyi

)F ′i = r−βi

(yixs − xiys − 2 γx xiyi + γy (x2

i − y2i ))


Finally: the general set of equations

Use Fi from lens equation to express ψ(g) in time delay equations.

β

2− βh ti −

1− β2− β

zs · zi +β

2− βC

= − r2i

2− x2

i − y2i

2γx − xiyi γy

h ∝ 2− ββ

Remember 2237+0305 with spherical models:

h ∝ 2− β


Explicit solution for h (1)

Use Cramer’s rule to find solution.

In the shearless case (γ = 0):

h0 := h(γ = 0)

h0 = −2− β2β

g0

∣∣∣∣∣∣∣∣∣t1 x1 y1 1t2 x2 y2 1t3 x3 y3 1t4 x4 y4 1

∣∣∣∣∣∣∣∣∣−1

g0 =

∣∣∣∣∣∣∣∣∣r21 x1 y1 1r22 x2 y2 1r23 x3 y3 1r24 x4 y4 1

∣∣∣∣∣∣∣∣∣titlepage introduction discussion contents back forward previous next fullscreen 29

Explicit solution for h (2)

For arbitrary shear:

h = h0

(1 +

gxg0γx +

gyg0γy

)

g =

∣∣∣∣∣∣∣∣∣x2

1 − y21 x1 y1 1

x22 − y2

2 x2 y2 1x2

3 − y23 x3 y3 1

x24 − y2

4 x4 y4 1

∣∣∣∣∣∣∣∣∣ , 2

∣∣∣∣∣∣∣∣∣x1y1 x1 y1 1x2y2 x2 y2 1x3y3 x3 y3 1x4y4 x4 y4 1

∣∣∣∣∣∣∣∣∣

External shear can change the result significantly.


Introducing the “critical shear” (1)

This equation deserves attention!

h = h0

(1 +

gxg0γx +

gyg0γy

)

In one dimension, define critical shear γc so, that

h

h0= 1− γ

γc

holds, analogous to scaling in mass-sheet degeneracy

h ∝ 1− σ

σc


Introducing the “critical shear” (2)

Include direction here:

h

h0= 1− γ · γc

γ2c

γc = −g0

g2g

Hubble constant vanishes for γ = γc or more generally for

γ · γc = γ2c

If H0 is fixed, all time delays become 0.


Shifting the galaxy

We assumed lens centre z0 = 0 as known.

It can be shown, that h for fixed γ and also γc do not change

when the galaxy is shifted.


Spherical models (1)

Time delay and lens equations are overdetermined. Assume they

are valid for one β. Here r2−β0 = β f .

h ti = −12zt(1− Γ)z − 1− β

βr2−β0 rβ − C

zs =(

1− Γ− r02−β rβ−2

i 1)zi

Eliminate Γ from the first using the second equation and find

solution with Cramer’s rule:

hsph

h0=r2−β0

g0

∣∣∣∣∣∣∣∣∣rβ1 x1 y1 1rβ2 x2 y2 1rβ3 x3 y3 1rβ4 x4 y4 1

∣∣∣∣∣∣∣∣∣titlepage introduction discussion contents back forward previous next fullscreen 34

Spherical models (2)

hsph

h0=r2−β0

g0

∣∣∣∣∣∣∣∣∣rβ1 x1 y1 1rβ2 x2 y2 1rβ3 x3 y3 1rβ4 x4 y4 1

∣∣∣∣∣∣∣∣∣This vanishes for point mass models (β = 0). Remember:

h

h0= 1− γ · γc

γ2c

It can be shown, that even γ = γc.


Nearly Einstein ring systems

h0 ∝2− ββ

general

hsph ∝ 2− β spherical, as in 2237+0305

hsph

h0=β

2= 1− γsph · γc

γ2c

γsph =2− β

2γc


The Einstein cross 2237+0305

• Previous resultsconfirmed

• spherical:hsph ∝ 2− β

• shearless:h0 ∝ (2− β)/β

• hsph/h0 ≈ β/2

• γc = 0.13

• spherical:γ = 0.07 forβ = 1


PG 1115+080

• Nearly equal ri

• Two time delays available

• Group of galaxies

• H0 = 40–80 km s−1 Mpc−1

The external shear in published models is 0.06–0.2. Critical shear

is γc = 0.22. Uncertainties in real γ therefore important!

Estimates for isothermal models: H0 = 47–58 km s−1 Mpc−1 with

errors of ca. 20 %.


RX J0911+0551

• Very different ri

• High shear from cluster

• obs.: γ ≈ 0.1

• mod.: γ ≈ 0.3

Very high critical shear of γc = 0.56, but uncertainties in real shear

also high. No time delay yet.


B 1608+656

• Three time delays!

• Highly accurate positions

• Two lensing galaxies

Critical shear: γc = 0.10

For shearless models: H0 = (37± 5) km s−1 Mpc−1

For isothermal models: γ = 0.34 and

H0 = (130± 15) km s−1 Mpc−1

Models in Koopmans & Fassnacht (1999): Both galaxies have

about same mass.


Discussion (1)

• Simple analytical considerations for general family of models

ψ = rβ F (θ) with external shear

• Include time delays for quadruple lenses

• For constant γ: H0 ∝ (2− β)/β

• Scaling independent of geometry, shear or time delay ratios. Error

will be the same for all lenses!

• 10 % error in β leads to 20 % error in H0.

• Spherical models with varying shear: H0 ∝ 2− βThis is shearless value ×β/2.


Discussion (2)

• Effect of shear quantified by new concept of “critical shear” γc.

• Conclusion for future work

? Shear effects: Choose systems with low uncertainties (large γc)

? β effects: Use other means to measure β (dynamical studies

of galaxies, lenses with extended sources)


H0 from lenses and other methods

(from Koopmans & Fassnacht, 1999)


Contents

0 Title page1 Degeneracies and scaling relations in power-law models for gravitational

lenses2 Introduction (1)3 Introduction (2)4 Lens configuration5 Deflection angle and potential6 Poisson equation7 Light travel time8 Scaled Hubble constant9 Time delay

10 Spherical power-law models11 External shear12 Constant mass sheet13 The mass-sheet degeneracy14 Previous work for 2237+0305 (1)


15 Previous work for 2237+0305 (2)16 Interpretation: mass-sheet degeneracy (1)17 Interpretation: mass-sheet degeneracy (2)18 The general power-law model (1)19 The general power-law model (2)20 Magnification and amplification (1)21 Magnification and amplification (2)22 Parameters23 Constraints24 Time delay for power-law model25 Isothermal case26 Including the lens equations (1)27 Including the lens equations (2)28 Finally: the general set of equations29 Explicit solution for h (1)30 Explicit solution for h (2)31 Introducing the “critical shear” (1)32 Introducing the “critical shear” (2)33 Shifting the galaxy34 Spherical models (1)


35 Spherical models (2)36 Nearly Einstein ring systems37 The Einstein cross 2237+030538 PG 1115+08039 RX J0911+055140 B 1608+65641 Discussion (1)42 Discussion (2)43 H0 from lenses and other methods44 Contents


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