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Gravitational lensing of the CMB
Richard Lieu Jonathan Mittaz
University of Alabama in HuntsvilleTom Kibble
Blackett Laboratory, Imperial College London
0
E
E
0
E
E
E
dD
Positive curvature: parallel rays converge, sourcesappear `larger’. Source distance (or angular size distance D) is `smaller’
Zero curvature: parallel rays stay parallel, sourceshave `same’ size Angular size distance has Euclidean value
Negative curvature: parallel rays diverge, sourcesappear `smaller’. Angular size distance D is `larger’
% Angular magnification
EXAMPLES TO ILLUSTRATE THE BEHAVIOR OF PROPAGATING LIGHT
The general equation is DzGDzHd
Ddmm
55202
2
)1(4)1(2
3
Non-expanding empty Universe
Parallel rays stay parallel
1,00,02
2
0 d
dt
d
DdH m
tcttcdtD eobs
t
t
e
obs
)(
Expanding empty Universe
where )1( zd
dt
Parallel rays diverge; 0 orDD E
Dord
Ddm 00
2
2
0 orDD E
Non-expanding Universe with some matter
DGd
DdH m
40
2
2
0
Parallel rays diverge; 0 orDD E
Expanding Universe with matter and energy at critical density
DzHd
Ddm
5202
2
)1(2
31
Parallel rays stay parallel; 0 orDD E
The general equation is DzGDzHd
Ddmm
55202
2
)1(4)1(2
3
where )1( z
d
dt
PROPAGATION THROUGH THE REAL UNIVERSE
We know the real universe is clumped. There are three possibilities
Smooth medium all along, with 1 m
WMAP papers assumed thisscenario
At low z smooth medium has 1
CLUMPS are small and rareHardly visited by light rays
PROPAGATION THROUGH THE REAL UNIVERSE
We know the real universe is clumped. There are three possibilities
Smooth medium all along, with 1 m
WMAP papers assumed thisscenario
Smooth medium has 1
CLUMPS are small and rareHardly visited by light rays
If a small bundle of rays misses all the clumps, it will map back to a demagnified regionLet us suppose that all the matter in is clumped i.e. the voids are matter freeszz
Dord
Dd0
2
2
The percentage increase in D is given by
)20
9
2
11(
822
0022
0
smssms
s
s
s DHDHDHD
DD
where c=1 and & are the Euclidean angular size and angular size distance of the source
This is known as the Dyer-Roeder empty beam
s sD
z=zs z=0
What happens if the bundle encounters a gravitational lens
E
E
db
db
Dz
DDD
sL
LLs )()1(2
)(
where the meanings of the D’s is
assuming Euclidean distances since mean density is ~ critical. Also the deflection angle effectis
b
drbr
rrGbb
22
)(44)(')(
We can use this to calculate the average
Consider a tube of non-evolving randomly placed lenses
ndV
LL bdbdDzn 20 )1(2.
sD
M
bs
s drbr
rrbdbdD
D
DDDzGn
0 0220
)(4)()1(4
)20
9
2
11(
822
0022
0 smssm DHDHDH
Thus
The magnification by the lenses and demagnification at the voids exactly compensate each other.
The average beam is Euclidean if the mean density is critical.
How does gravitational lensing conserve surface brightness?Unlike ordinary magnifying glass, gravitational lens magnifies a central pixel and tangentially shear an outside pixel.
Only rays passing through the gravitational lens are magnified
• The rest of the rays are deflected outwards to make room for the central magnification (tangential shearing)
Before Lensing After Lensing
When lens is "inside"source is magnified
When lens is "outside" the source is distorted but not magnified
Gravitational lensing of
a large source
If there is a Poisson distribution of foreground clumps extending from the observer's neighborhood
to a furthest distance D
δ θ ≈ π² GM √nD o
Source sizeFluctuation Mass of
One clump
Number densityof clumps
In the limit of infrequent lensing,this is >> magnification fluctuation
due to the deflection of boundary rayby boundary clumps, viz.
δ θ ≈ 2π² n GMRDo
Radius of lens
Returning to the three possibilities
Homogeneous1 m Source Size
Source Size
Source Size
1 m
1 m
Inhomogeneous atlow z
Clumps are missedby most rays
WHY THE PRIMORDIAL P(k) SPECTRUM DOES NOT ACCOUNT FORLENSING BY NON-LINEAR GROWTHS AT Z < 1
Homogeneous Universe
Mass Compensation(swiss cheese)
Poisson Limit
While the percentage angular magnification has an average of
Its variance is given by
For a large source (like CMB cold spots), this means the average angular sizecan fluctuate by the amount
ndV
2
0222
22022 )(4
5
3
2
31
3
8)(
bs
f
s
f drbr
rrbdb
D
D
D
DGnndV
N
where lensofarea
sourceofareaN
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