Cosmic Microwave Background Primary Temperature Anisotropies Polarization Secondary Anisotropies Scott Dodelson PASI 2006

Coherent Picture Of Formation Of Structure In The Universe Photons freestream: Inhomogeneities turn into anisotropies t ~100,000 years Quantum Mechanical Fluctuations during Inflation Perturbation Growth: Pressure vs. Gravity Matter perturbations grow into non-linear structures observed today m, r , b , f Scott Dodelson PASI 2006

Goal: Explain the Physics and Ramifications of this Plot Scott Dodelson PASI 2006

Notation Scale Factor a(t) Conformal time/comoving horizon Gravitational Potential  Photon distribution  Fourier transforms with k comoving wavenumber Wavelength k-1 Scott Dodelson PASI 2006

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 … Scott Dodelson PASI 2006

We see photons today from last scattering surface at z=1100  accounts for redshifting out of potential well D* is distance to last scattering surface Scott Dodelson PASI 2006

Can rewrite  as integral over Hubble radius (aH)-1 Perturbations outside the horizon Scott Dodelson PASI 2006

Inflation produces perturbations Quantum mechanical fluctuations <(k) (k’)> = 33(k-k’) P(k) Inflation stretches wavelength beyond horizon: (k,t) becomes constant Infinite number of independent perturbations w/ independent amplitudes Scott Dodelson PASI 2006

Perturbations in metric induce photon, dark matter perturbations To see how perturbations evolve, need to solve an infinite hierarchy of coupled differential equations Perturbations in metric induce photon, dark matter perturbations Scott Dodelson PASI 2006

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) Scott Dodelson PASI 2006

Before recombination, electrons and photons are tightly coupled: equations reduce to Temperature perturbation Very similar to … Displacement of a string Scott Dodelson PASI 2006

What spectrum is produced by a stringed instrument? C string on a ukulele Scott Dodelson PASI 2006

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! Scott Dodelson PASI 2006

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 Scott Dodelson PASI 2006

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 Scott Dodelson PASI 2006

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? Scott Dodelson PASI 2006

Without synchronization: First “Trough” First “Peak” Scott Dodelson PASI 2006

Inflation synchronizes all modes  All modes remain constant until they re-enter horizon. Scott Dodelson PASI 2006

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 Scott Dodelson PASI 2006

The spectrum at last scattering is: Scott Dodelson PASI 2006

Anisotropy spectrum today Fourier transform of temperature at Last Scattering Surface Anisotropy spectrum today Scott Dodelson PASI 2006

One more effect: Damping on small scales But So Scott Dodelson PASI 2006

On scales smaller than D (or k>kD) perturbations are damped Scott Dodelson PASI 2006

When we see this, we conclude that modes were set in phase during inflation! Bennett et al. 2003 Scott Dodelson PASI 2006

Polarization Polarization field decomposes into 2-modes E-mode B-mode B-mode smoking gun signature of tensor perturbations, dramatic proof of inflation... We will focus on E. Scott Dodelson PASI 2006

Three Step argument for <TE> Polarization proportional to quadrupole Quadrupole proportional to dipole Dipole out of phase with monopole Scott Dodelson PASI 2006

Isotropic radiation field produces no polarization after Compton scattering Modern Cosmology Adapted from Hu & White 1997 Scott Dodelson PASI 2006

Radiation with a dipole produces no polarization Scott Dodelson PASI 2006

A quadrupole is needed Scott Dodelson PASI 2006

Quadrupole proportional to dipole Scott Dodelson PASI 2006

Dipole is out of phase with monopole Roughly, Scott Dodelson PASI 2006

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. Scott Dodelson PASI 2006

DASI initially detected TE signal Kovac et al. 2002 Scott Dodelson PASI 2006

WMAP provided indisputable evidence that monopole and dipole are out of phase Kogut et al. 2003 This is most remarkable for scales around l~100, which were not in causal contact at recombination. Scott Dodelson PASI 2006

Parameter I: Curvature Same wavelength subtends smaller angle in an open universe Peaks appear on smaller scales in open universe Scott Dodelson PASI 2006

Parameters I: Curvature As early as 1998, observations favored flat universe DASI, Boomerang, Maxima (2001) WMAP (2006) Scott Dodelson PASI 2006

Parameters II Reionization 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 Scott Dodelson PASI 2006

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 Scott Dodelson PASI 2006

E.g.: Baryon density Here, F is forcing term due to gravity. As baryon density goes up, frequency goes down. Greater odd/even peak disparity. Scott Dodelson PASI 2006

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 Scott Dodelson PASI 2006

What have we learned from WMAP III Polarization map Later Epoch of Reionization Scott Dodelson PASI 2006

Secondary Anisotropies in the Cosmic Microwave Background Limber Sunyaev-Zel’dovich Gravitational Lensing Scott Dodelson PASI 2006

Secondary Anisotropies Get Contributions From The Entire Line Of Sight Temperature anisotropy angular distance  from z-axis Weighting Function Comoving distance  Position dependent Source Function; e.g., Pressure or Gravitational Potential Scott Dodelson PASI 2006

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) Scott Dodelson PASI 2006

Derivation 2D 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: Scott Dodelson PASI 2006

Do the l’ integral; use the delta function to do the ’ integral Fourier transform S and use to get Scott Dodelson PASI 2006

Do the  integral to get a delta function and then d2k Invoke physics Mode with large kz Mode with small kz Scott Dodelson PASI 2006

… Leading To The Correct Answer Of all modes with magnitude k, only those with kz small contribute A ring of volume 2kdk/ contributes This is a small fraction of all modes (4k2dk): Secondary Anisotropies are suppressed by a factor of order l Scott Dodelson PASI 2006

Courtesy Frank Bertoldi Scott Dodelson PASI 2006

This is of the standard form with Pressure So we can immediately write the power spectrum: Scott Dodelson PASI 2006

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 8 ( 87) GADGET: Borgani et al. Scott Dodelson PASI 2006

There is a tantalizing hint from CBI of a detection! Geisbuesch et al. 2004 Bond et al. 2002 Scott Dodelson PASI 2006

If this high signal is due to SZ, amplitude of perturbations is high and matter density is low Lensing CBI (1- and 2- sigma) Contaldi, Hoekstra, Lewis 2003 Komatsu & Seljak 2002 Scott Dodelson PASI 2006

WMAPIII has made thing worse Amplitude σ8 is currently the most contentious cosmological parameter Scott Dodelson PASI 2006

There’s more … Notice the difference between these 2 maps Bennett et al. 2003 Da Silva et al. SZ simulation Secondary anisotropies are non-Gaussian! Scott Dodelson PASI 2006

One way to probe non-Gaussianity: Peaks (aka Cluster Counts) m 8 Battye & Weller 2003 Scott Dodelson PASI 2006

This will provide tight constraints on Dark Energy parameters South Pole Telescope SNAP Scott Dodelson PASI 2006

Gravitational Lensing 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? Einstein 1912! Scott Dodelson PASI 2006

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 Scott Dodelson PASI 2006

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!) Scott Dodelson PASI 2006

The obvious effect of redirection is to smooth out the peaks Seljak 1996 Scott Dodelson PASI 2006

Leading to percent level changes of the acoustic peaks Lewis 2005 Scott Dodelson PASI 2006

Again non-Gaussianities lead to new possibilities: Consider the 2D Fourier transform of the temperature Recall that ‘ 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 Scott Dodelson PASI 2006

There have been improvements in reconstruction techniques Hirata & Seljak 2003 Projected potential Optimal Quadratic Likelihood Requires 1K/pixel noise w/beam smaller than 4’ Scott Dodelson PASI 2006

What can we do with this? Hirata & Seljak 2003 Scott Dodelson PASI 2006

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

We have only scratched the surface Patchy Reionization Cross-Correlating w/ Other probes Data Analysis: Generalizing to non-Gaussianities Observations on the horizon Scott Dodelson PASI 2006

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 Scott Dodelson PASI 2006

Upcoming Experiments ACT 100 square degrees; 2muK/pixel; 2’ pixels Scott Dodelson PASI 2006

Available at www.amazon.com Scott Dodelson PASI 2006