1. Measuring the Hubble Constant Using Gravitational Lenses
Roger Blandford
KIPAC
Stanford
Sherry Suyu, Phil Marshall, Chris Fassnacht,
Tommaso Treu, Leon Koopmans, Matt Auger,
Stefan Hilbert, Tony Readhead, Steve Myers,
Gabriela Surpi, Frederic Courbin, George Meylan…
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3. Light “travels slower”
Refraction of Light
Light travels
slower in glass
Lens
Light travels
faster in air
Wave crests
Light rays
Light “travels slower”
in glass
and is refracted
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4. Deflection of Light
Newton: Opticks, Query1: Do not bodies act upon Light at a distance, and by their action bend its rays; and is not this action (caeteris paribus) strongest at the least distance?
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5. Einstein’s General Theory of Relativity
1915: Spacetime is curved around a massive body. Light follows straight lines (geodesics) which appear to be curved. This doubles the effect.
1919: Eclipse measurements confirm that solar deflection is twice Newtonian expectation and makes Einstein a household name. Now measured to 1/1000.
1919: Eddington realizes that relativistic problem just like the Newtonian problem. Light travels slower in a gravitational field
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Eddington

6. Clusters of galaxies: a ~ 10 arcsec Surface density ~ 1 g cm-2
Source
Lens
Observer
Stars: a ~ microarcsec
Galaxies: a ~ arcsec
Clusters of galaxies: a ~ 10 arcsec
Surface density ~ 1 g cm-2
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7. Which way shall I go?
Light makes the shortest (or the longest) journeys.
(Fermat)
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8. Gravitational Lenses and the Hubble Constant S D
H0=V/d ~t -1 2 O
Direct measurement
Insensitive to world model
Lens model dependence
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9. Q
Walsh,
Carswell
& Weymann
(1979)
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10. John Bahcall (1934-2005) Moderated debate between
Tammann and van den Berg
in 1996
H0 features prominently in
“Unsolved Problems”
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11. Standard candles, rulers, timers etc
Type Ia supernovae: standard candles
Fluctuations in the Cosmic Microwave Background radiation
Baryon Acoustic Oscillations in the galaxy clustering
power spectrum
Periods of Cepheid
variable stars in local
galaxies
Something else?
(sound speed x age of universe) subtends ~1 degree
gas density fluctuations from CMB era are felt by dark matter - as traced by galaxies in the local(ish) universe
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12. The Measure of the Universe
Historically, h= (H0/100 km s-1 Mpc-1) ~ 0.3-~5
10 x Error!
Recent determinations:
HST KP (Freedman et al)
<h>=0.72+/-0.02+/-0.07
Masers (Macri et al)
h=0.74+/-0.03+/-0.06
WMAP (Komatsu et al)
h=0.71+/ (FCDM)
BAO (Percival et al 2010)
h=0.70+/ (FCDM)
Distance Ladder (Riess et al)
h=0.74+/-0.04
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13. B (Myers, CLASS 1995)
Dust
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14. Data Compact radio source (CLASS) Relative magnifications
VLBI Astrometry to 0.001”
Relative magnifications
m A,C,D =2, 1, 0.35
Time delays (Fassnacht)
tA,C,D = 31.5, 36, 77 d (+/-1.5)
Elliptical galaxy lenses (Fassnacht, Auger)
G1: z=0.6304, s=260(+/-15) km s-1; G2
K+A galaxy source (Myers)
z=1.394
HST imaging
V, I, H bands
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15. Modeling Gravitational Lenses
Surface brightness (flux per solid angle) changes along ray ~ a-3
Unchanged by lens
Images of same region of source have same surface brightness
Complications
Deconvolution (HST blurring)
Deredenning (dust)
Decontamination (source + lens)
Image
Source
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16. Results H0=71+/-3 km s-1 Mpc-1 Iterative modeling Bayesian analysis
Potential residuals ~ 2%
Adopt fixed world model
Major sensitivity is to zL
Assume lens model correct
Assume propagation model correct
H0=71+/-3 km s-1 Mpc-1
Suyu et al (2010)
If relax world model, dh~0.05;
If combine with WMAP5 (+flatness), dw~0.2
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17. Limits to the accuracy Lens Model Time delays
Mass sheet degeneracy
Velocity dispersion
Measuring width of ring
Time delays
Not now limiting accuracy
More monitoring
Structure along line of sight
Distorts images of source and lens
Current effort
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18. “Mass-sheet” model degeneracy
κext
[Courbin et. al. 2002]
To break this degeneracy,
we need more information about the mass distributions:
Stellar dynamics
Slope g from arc thickness
Structures along the LOS
Lens mass, profile slope and
line of sight mass distribution
are all degenerate:
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19. Geodesic deviation equation
Sachs
Zel’dovich
Feynman
Refsdal
Gunn
Penrose\
Alcock
Anderson O x
Proper transverse
Separation vector q
Angle at observer
G=c=H0=1
Null geodesic congruence backward from observer
Convergence k and shear g
First focus, tangent to caustic, multiple imaging
Distance measure is affine parameter
dxa ~ ka dl where ka is a tangent vector along the geodesic
Choose where a =w0/w is the local scale factor
errors O(f) relative to homogeneous reference universe
For pure convergence,
enthalpy density
x=d q
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angular diameter distance

20. Homogeneous Cosmology
For FLCDM universe w=rM
No contribution from L
Introduce h=x/a, comoving distance, radius dr=dl/a2 and RW line element to obtain
Current separation
For k=-1, h = qR0sinh (r/R0)
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21. Time delays Single deflector a h Multi-sheet propagation
Deviation relative to
undeflected ray h
Multi-sheet propagation
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22. Inhomogeneous matter distribution
Group
Galaxy
Void
<r> rb x
Simple Model
Background density rb(a)
halos modeled by spherical profiles centered on galaxy/group centers
amplitude and size scaled to luminosity
incorporate bias?
NFW better than isothermal
Use simulations, GGL to calibrate test convergence and estimate error
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23. Multi-screen Propagation
Treat screens as “weak deflectors”
Potential: Y ~ L.x+x.Q.x/2+… ; deflections, linear
Distort appearance of source and lens
Many screens – multiply matrices
Model lens in lens plane not on sky
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24. B1608+656: Statistical approach
Modeled external shear ~0.1; need k for H0
B has twice the average galaxy number density (Fassnacht et al. 2009)
Find κext along all LOS in MS that have 2x ‹ngal›
Ray-trace through Millennium S
Identify LOS where SL occurs
Find κext along LOS, excluding the SL plane (Hilbert et al. 2007)
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25. B1608+656-Particular Approach
Groups (Fassnacht et al)
z=0.265
Off center => 
z=0.63 (G1, G2)
 =150+/-60 km s-1
z=0.426, 0.52
Centered lens =>  ~0
Photometry
1500 ACS galaxies over 10sm
1700 P60 galaxies over 100 sm
Redshifts
100 zs
Experimenting with different prescriptions for assigning halos
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26. Additional Lenses
Courbin
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27. Future lens cosmography (Marshall et al)
: ~3000 new lensed quasars with PS1, DES, HSC
About 500 of these systems will be quads
A significant monitoring follow-up task!
A larger statistical sample of doubles would provided added value, once calibrated by the quads
The spectroscopic follow-up is not demanding given rewards
Intensive modeling approach seems unavoidable
100 lenses observed to B1608’s level of detail could
yield Hubble’s constant to percent precision
LSST, WFIRST…
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28. Summary Lens H0 is competitive Promising results with B1608+656
~4% with strong priors; ~7% after relaxing world model
Promising results with B
h=0.71+/-0.03 with strong priors
Limited by understanding of line of sight
External convergence and shear
New formalism for multi-path propagation
Distortion not delay – matrix formalism
Observations show overdense line of sight
Imaging and spectroscopy
Other good candidates
Existing and future options
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29. Thanks to: Sherry Suyu, Phil Marshall, Chris Fassnacht,
Tommaso Treu, Leon Koopmans, Matt Auger,
Stefan Hilbert, Tony Readhead, Steve Myers, Gabriela Surpi, Frederic Courbin, George Meylan…
HST
John Bahcall
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