degeneracies in lens models no h0 from gravitational...
TRANSCRIPT
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Degeneracies in lens modelsNo H0 from gravitational lenses?
Olaf Wucknitz
Hamburger Sternwarte / JBO
[email protected]@jb.man.ac.uk
Jodrell Bank, 20. June 2001
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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
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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
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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
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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
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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
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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′)
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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
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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)
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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
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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
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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 )
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Constant mass sheet
An additional constant surface mass density σ(z) ≡ κ contributes
with
ψ(κ)(z) = κr2
2,
α(z) = κ z .
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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.
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Previous work for 2237+0305 (1)
Wambsganß & Paczynski (1994): Numerical modelfitting (image
positions as constraints) for a range of 0 < β < 2.
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Previous work for 2237+0305 (2)
Result: χ2 ≈ const, γ ∝ 2− β, ∆t ∝ 2− β
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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
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...............................β = 1 α = (1− κ) + κ r
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........................ β = 1.5 α = r0.5
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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
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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)
)
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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.
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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
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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!
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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
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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!
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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!)
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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.
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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
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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 ))
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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
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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
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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.
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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
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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.
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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.
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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
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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.
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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
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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
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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 %.
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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.
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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.
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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.
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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)
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H0 from lenses and other methods
(from Koopmans & Fassnacht, 1999)
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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)
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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)
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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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