cosmic microwave background

Scott Dodelson PASI 2006 1

Cosmic Microwave Background

Primary Temperature Anisotropies Polarization Secondary Anisotropies


Coherent Picture Of Formation Of Structure In The Universe

3410 sect −=

Quantum Mechanical Fluctuations

during Inflation

( )V φ

Perturbation Growth: Pressure

vs. Gravity

t ~100,000 years

Matter perturbations grow into non-

linear structures observed today

, ,reion dez wΩ

Photons freestream: Inhomogeneities turn into anisotropies

Ωm, Ωr , Ωb , f


Goal: Explain the Physics and Ramifications of this

Plot


Notation Scale Factor a(t) Conformal time/comoving horizon Gravitational Potential Photon distribution Fourier transforms with k

comoving wavenumber

Wavelength k-1

),()2(

),( 3

3

ηπ

η kekdx xkirr rr

Ψ=Ψ •∫

∫ ′′

≡t

tatd

0 )(η


Photon Distribution Distribution depends on position x (or wavenumber

k), direction n and time t. Moments

Monopole:

Dipole:

Quadrupole:

You might think we care only about at our position because we can’t measure it anywhere else, but …


We see photons today from last scattering surface at

z=1100

D* is distance to last scattering surface

accounts for redshifting out of

potential well


Can rewrite η as integral over Hubble radius (aH)-1

Perturbations outside the

horizon


Inflation produces perturbations

Quantum mechanical fluctuations (k)k’=πdk-k’Pk

Inflation stretches wavelength beyond horizon: k,tbecomes constant

Infinite number of independent perturbations w/ independent amplitudes


To see how perturbations evolve, need to solve an infinite hierarchy of coupled

differential equations

Perturbations in metric induce photon, dark matter perturbations


Evolution upon re-entry Pressure of radiation acts against

clumping If a region gets overdense,

pressure acts to reduce the density Similar to height of an instrument

string (pressure replaced by tension)


Before recombination, electrons and photons are tightly coupled:

equations reduce to

Displacement of a string

Temperature perturbation

Very similar to …


What spectrum is produced by a stringed

instrument?

C string on a ukulele

C:\Program Files\Audacity\audacity.exe


CMB is different because …

Fourier Transform of spatial, not temporal, signal

Time scale much longer (400,000 yrs vs. 1/260 sec)

No finite length: all k allowed!


Why peaks and troughs? Vibrating String:

Characteristic frequencies because ends are tied down

Temperature in the Universe: Small scale modes enter the horizon earlier than large scale modes


Interference could destroy peak structure

There are many, many modes with similar

values of k. All have different

initial amplitude. Why all are in

phase?

First Peak Modes


An infinite number of violins are synchronized

Similarly, all modes

corresponding to first trough are in phase: they all have

zero amplitude at

recombination. Why?


Without synchronization:First “Peak” First “Trough”


Inflation synchronizes all modes

All modes remain constant until they re-enter horizon.


How do inhomogeneities at last scattering show up as anisotropies

today?

Perturbation w/ wavelength k-1

shows up as anisotropy on angular scale ~k-

1/D* ~l-1

Cl simply related to [0+]RMS(k=l/D*)

Since last scattering surface is so far away, D* ≈ η0


The spectrum at last scattering is:


Fourier transformof temperature atLast Scattering Surface

Anisotropy spectrum

today


One more effect: Damping on small scales

But

So


On scales smaller than D (or k>kD) perturbations are

damped


When we see this,

we conclude

that modes

were set in phase during

inflation!

Bennett et al. 2003


Polarization

E-mode B-mode

B-mode smoking gun signature of tensor perturbations, dramatic proof of inflation... We will focus on E.

Polarization field decomposes into 2-modes


Three Step argument for <TE>

Polarization proportional to quadrupole

Quadrupole proportional to dipole

Dipole out of phase with monopole


Isotropic radiation

field produces

no polarization

after Compton scattering

Modern CosmologyAdapted from Hu & White 1997


Radiation with a dipole produces no polarization


A quadrupole is needed


Quadrupole proportional to dipole

( )ee MFP

vvx∂

−∂

( )ee MFP

vvx∂

− +∂


Dipole is out of phase with monopole

1 00vtρ ρ

•∂+ ∇• = ⇒ ∝

∂

r r

0 * *

1 * *

( , ) cos[ ( )]( , ) sin[ ( )]

s

s

k A krk B kr

η η η η

==

Roughly,


The product of monopole and dipole is initially positive (but small, since dipole vanishes as k goes to

zero); and then switches signs several times.


DASI initially detected TE signal

Kovac et al. 2002


WMAP provided indisputable evidence that monopole and dipole

are out of phase

This is most remarkable for scales around l~100, which were not in causal contact at recombination.

Kogut et al. 2003


Parameter I: Curvature Same wavelength subtends smaller angle in an open universe Peaks appear on smaller scales in open universe


Parameters I: Curvature

As early as 1998, observations favored flat universe

DASI, Boomerang, Maxima (2001)

WMAP (2006)


Parameters IIReionization lowers

the signal on small scales

A tilted primordial spectrum (n<1) increasingly reduces signal on small scales

Tensors reduce the scalar normalization, and thus the small scale signal


Parameters III Baryons accentuate

odd/even peak disparity

Less matter implies changing potentials, greater driving force, higher peak amplitudes

Cosmological constant changes the distance to LSS


E.g.: Baryon density

2x x Fω+ =&&Here, F is forcing term due to gravity.

3(1 3 / 4 )sb

kkcγ

ωρ ρ

= =+

As baryon density goes up, frequency goes down. Greater odd/even peak disparity.


Bottom line

Baryon Density agrees with BBN There is ~5-6 times more dark matter than baryons There is dark energy [since the universe is flat] Primordial slope is less than one


What have we learned from WMAP III

n<1 Polarization

map Later Epoch

of Reionization


Secondary Anisotropies in the Cosmic Microwave

Background

Limber Sunyaev-Zel’dovich Gravitational Lensing


Secondary Anisotropies Get Contributions From The Entire Line

Of Sight

Temperature anisotropy angular distance from

z-axis

Comoving distance

Weighting Function

Position dependent Source Function; e.g., Pressure or

Gravitational Potential


What is the power spectrum of a secondary

anisotropy? This is different from primary anisotropies; there Cl depended on at last scattering

Many modes do not contribute to the power because of cancellations along the line of sight

A wonderful approximation is the Limber formula (1954)


Derivation2D Fourier Transform of temperature field

In the small angle limit variance of the Fourier transform is Cl

Integrate both sides over l’ and plug in:


Do the l’ integral; use the delta function to do the ’ integral

Fourier transform S and use

to get


Do the integral to get a delta function and then d2k

Invoke physics

Mode with large kz Mode with small kz


… Leading To The Correct Answer

Of all modes with magnitude k, only those with kz small contribute A ring of volume 2πkdk/ contributes

Secondary Anisotropies are suppressed by a factor of order l

11 ~21 −− lkχ

This is a small fraction of all modes (4πk2dk):

Scott Dodelson PASI 2006 49Courtesy Frank Bertoldi


This is of the standard form with

)()(;)()( xPxSm

aWe

T rr==

σ

So we can immediately write the power spectrum:

Pressure

)/()(2

2

2

2

σ Padm

C Pe

T ∫=


Computing the pressure power spectrum requires large

N-Body/hydro simulations Need good

resolution to pick up low-mass objects and large volume to get massive clusters

Simulations have not yet converged

However, groups agree at l~2000 and agree that Cl very sensitive to σ8( σ8

7)

GADGET: Borgani et al.


There is a tantalizing hint from CBI of a detection!

Bond et al. 2002 Geisbuesch et al. 2004


If this high signal is due to SZ, amplitude of perturbations is high and matter

density is low

Komatsu & Seljak 2002

CBI (1- and 2- sigma)

Lensing

Contaldi, Hoekstra, Lewis 2003


WMAPIII has made thing worse

Amplitude σ8 is currently the most contentious cosmological parameter


There’s more … Notice the difference between these 2

maps

Da Silva et al. SZ simulation

Secondary anisotropies are non-Gaussian!

Bennett et al. 2003


One way to probe non-Gaussianity: Peaks (aka Cluster

Counts)

Battye & Weller 2003

Ωm

σ8


This will provide tight constraints on Dark Energy

parameters

SNAP

South Pole Telescope


Gravitational Lensing

Einstein 1912!

We are used to discrete objects (galaxies, QSOs) being lensed.

How do we study the lensing of the temperature (a Gaussian field) at last scattering?


Gravitational Lensing of the Primordial CMB

Primordial unlensed temperature u is re-mapped to

where the deflection angle is a weighted integral of the gravitational potential along the line of sight


Taylor expand …

leading to a new term

This has a very familiar form (don’t worry about the prefactor: we know its RMS extremely well!)


The obvious effect of redirection is to smooth out the peaks

Seljak 1996


Leading to percent level changes of the acoustic peaks

Lewis 2005


Again non-Gaussianities lead to new possibilities:

Consider the 2D Fourier transform of the temperature

Now though different Fourier modes are coupled! The quadratic combination

would vanish w/o lensing. Because of lensing, it serves as an estimator for the projected potential

Recall that

‘


There have been improvements in reconstruction techniques

Hirata & Seljak 2003

Requires 1mK/pixel noise w/beam smaller than 4’

Projected potential Optimal Quadratic Likelihood


What can we do with this?Hirata & Seljak 2003

Scott Dodelson PASI 2006 66Knox & Song; Kesden, Cooray, & Kamionkowski 2002

Can clean up B-mode contamination and measure even small tensor component

Probe inflation even if energy scale is low


We have only scratched the

surface Patchy Reionization Cross-Correlating w/ Other probes Data Analysis: Generalizing to

non-Gaussianities Observations on the horizon


Conclusions Primary temperature anisotropies

well measured; determine cosmological parameters to unprecedented accuracy

E-mode of Polarization measured; improvements upcoming; B-mode is holy grail

Secondary anisotropies will dominate the upcoming decade


Upcoming Experiments ACT 100 square degrees;

2muK/pixel; 2’ pixels

Scott Dodelson PASI 2006 70Available at www.amazon.com

cosmic microwave background

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