Cosmic Microwave Background  Cosmological Overview/Definitions  Temperature  Polarization  Ramifications  Cosmological Overview/Definitions  Temperature  Polarization  Ramifications Scott Dodelson Academic Lecture V

Goal: Explain the Physics and Ramifications of this Plot

Coherent picture of formation of structure in the universe Quantum Mechanical Fluctuations during Inflation Perturbation Growth: Pressure vs. Gravity t ~100,000 years Matter perturbations grow into non- linear structures observed today Photons freestream: Inhomogeneities turn into anisotropies  m,  r,  b, f

Review of Notation  Scale Factor a(t)  Conformal time/comoving horizon  dt/a(t)  Gravitational Potential   Photon distribution   Will use Fourier transforms  x  d 3 k  k  e ik  x /(2  ) 3 k is comoving wavenumber  Wavelength  k -1  Scale Factor a(t)  Conformal time/comoving horizon  dt/a(t)  Gravitational Potential   Photon distribution   Will use Fourier transforms  x  d 3 k  k  e ik  x /(2  ) 3 k is comoving wavenumber  Wavelength  k -1

Photon Distribution  Distribution  depends on position x (or wavenumber k), direction n and time t (or η).  Moments Monopole:    Dipole:    Quadrupole:    You might think we care only about  at our position because we can’t measure it anywhere else, but …  Distribution  depends on position x (or wavenumber k), direction n and time t (or η).  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  as integral over Hubble radius (aH) -1 Perturbations outside the horizon

Perturbations can be decomposed into Fourier modes + =

Combine Fourier Modes to Produce Structure in our Universe + = + =

In this simple example, all modes have same wavelength/frequency More generally, at each wavelength/frequency, need to average over many modes to get spectrum

Inflation produces perturbations  Quantum mechanical fluctuations in gravitational potential  (k)  k ’      k-k ’  P   k    Inflation stretches wavelength beyond horizon:  k,t  becomes constant  Infinite number of independent perturbations w/ independent amplitudes  Quantum mechanical fluctuations in gravitational potential  (k)  k ’      k-k ’  P   k    Inflation stretches wavelength beyond horizon:  k,t  becomes constant  Infinite number of independent perturbations w/ independent amplitudes

Evolution of Fluctuations

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: restoring force  Similar to height of guitar string (pressure replaced by tension)  Pressure of radiation acts against clumping  If a region gets overdense, pressure acts to reduce the density: restoring force  Similar to height of guitar 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

Compare the ukulele spectrum to CMB spectrum

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!  Fourier Transform of spatial, not temporal, signal  Time scale much longer (400,000 yrs vs. 1/260 sec)  No finite length: all k allowed!

Largest Wavelength/ Smallest Frequency Smallest Wavelength/ Largest Frequency

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  Vibrating String: Characteristic frequencies because ends are tied down  Temperature in the Universe: Small scale modes enter the horizon earlier than large scale modes

The spectrum at last scattering is: Θ 0 + Ψ ~ cos[k r s (η * ) ] Peaks at k = nπ/r s (η * )

One more effect: Damping on small scales But So

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

Fourier transform of temperature at Last Scattering Surface Anisotropy spectrum today C l simply related to [  0 +  ] RMS (k=l/D*)

Remember that at any wavelength, we are averaging over many modes with different direction.

Puzzle: Why are all modes in phase? The perturbation corresponding to each mode can either have zero initial velocity or zero initial amplitude We implicitly assumed that every mode started with zero velocity.

Interference could destroy peak structure There are many, many modes with similar values of k. All have different initial amplitude. But all are in phase. First Peak

An infinite number of ukuleles are synchronized Similarly, all modes corresponding to first trough are in phase: they all have zero amplitude at recombination

Without synchronization: First “Peak” First “Trough”

All modes remain constant until they re-enter horizon. Inflation synchronizes all modes 

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

When we see this, we conclude that modes were set in phase during inflation! Bennett et al NASA/WMAP

Compton Scattering produces polarized radiation field 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  E-Polarization proportional to quadrupole  Quadrupole proportional to dipole  Dipole out of phase with monopole  E-Polarization proportional to quadrupole  Quadrupole proportional to dipole  Dipole out of phase with monopole

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

Radiation with a dipole produces no polarization

A quadrupole is needed

Quadrupole proportional to dipole

Dipole is out of phase with monopole 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 has 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. NASA/ WMAP

Different Geometries Possible

Inflation predicts a flat universe

We now have a solid argument that the total density is flat Object with known physical size

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

Hot/cold spots of known physical size has been observed

Angular size demonstrates flatness

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  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 n is degenerate w/ reionization, but polarization pins down the latter: we now know n to within a few percent.

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  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 Here, F is forcing term due to gravity. As baryon density goes up, frequency goes down. Greater odd/even peak disparity.

Bottom line h often used instead of   CMB data says matter density is only 30% of critical: Need Dark Energy and Dark Matter

Conclusions  Strong evidence for inflation from CMB anisotropy/polarization spectra  Baryon and matter densities tightly constrained, consistent with other determinations. Dark energy & dark matter needed  Connection between particle physics & cosmology (inflation, dark matter, dark energy) more solid than ever. We need new tools …  Strong evidence for inflation from CMB anisotropy/polarization spectra  Baryon and matter densities tightly constrained, consistent with other determinations. Dark energy & dark matter needed  Connection between particle physics & cosmology (inflation, dark matter, dark energy) more solid than ever. We need new tools …

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