# Presentation: 004091

## Gravitational lensing of the CMB

**Richard Lieu **
- Jonathan Mittaz
- University of Alabama in Huntsville
- Tom Kibble
- Blackett Laboratory, Imperial College London

## Slide 2

## +ve curvature

- Flat

- -ve curvature

## Positive curvature: parallel rays converge, sources

- appear `larger’.
- Source distance (or *angular size distance *D) is
- `smaller’

- Zero curvature: parallel rays stay parallel, sources
- have `same’ size
- Angular size distance has Euclidean value

- Negative curvature: parallel rays diverge, sources
- appear `smaller’.
- Angular size distance* *D is `larger’

- Angular magnification

## EXAMPLES TO ILLUSTRATE THE BEHAVIOR OF PROPAGATING LIGHT

- The general equation is

- _Non-expanding empty Universe_

- Parallel rays stay parallel

- _Expanding empty Universe_

- where

- Parallel rays diverge;

## Non-expanding Universe with some matter

- _Non-expanding Universe with some matter_

- Parallel rays diverge;

- _Expanding Universe with matter and energy at critical density_

- Parallel rays stay parallel;

- The general equation is

- where

## PROPAGATION THROUGH THE REAL UNIVERSE

- We know the real universe is clumped.  There are three possibilities

- Smooth medium all along, with

- WMAP papers assumed this
- scenario

- At low z smooth medium has

- CLUMPS are small and rare
- Hardly visited by light rays

## CMB lensing by primordial matter

## 2dF/WMAP1 matter spectrum (Cole et al 2005) 

## PROPAGATION THROUGH THE REAL UNIVERSE

- We know the real universe is clumped.  There are three possibilities

- Smooth medium all along, with

- WMAP papers assumed this
- scenario

- Smooth medium has

- CLUMPS are small and rare
- Hardly visited by light rays

## This is known as the Dyer-Roeder empty beam

- Let us suppose that _all_ the matter in            is clumped i.e. the voids are matter free

- The percentage increase in D is given by

- where c=1 and     &      are the Euclidean angular size and angular size distance of the source

- z=zs

- z=0

## What happens if the bundle encounters a gravitational lens 

- where the meanings of the D’s is

- assuming Euclidean distances since mean density is ~ critical.  Also the deflection angle effect
- is

- We can use this to calculate the average

## The magnification by the lenses and demagnification at the voids exactly 

- 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**

## δ θ  ≈  π2  GM √nD 

**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 size
- Fluctuation

- Mass of
- One clump

- Number density
- of clumps

**In the limit of infrequent lensing****,**
**this is >> magnification fluctuation**
**due to the deflection of boundary ray**
**by boundary clumps, viz****.**

- δ θ  ≈ 2π2 n GMRD

- o

- Radius of lens

## Homogeneous

- Returning to the three possibilities

- Source Size

- Source Size

- Source Size

- Inhomogeneous at
- low z

- Clumps are missed
- by most rays

## WHY THE PRIMORDIAL P(k) SPECTRUM DOES NOT ACCOUNT FOR

**LENSING 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 size
- can fluctuate by the amount

- where

## Cluster CMB lensing parameters