Primordial nongaussianities I: cosmic microwave background

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Primordial nongaussianities I: cosmic microwave background Uros Seljak, UC Berkeley Rio de Janeiro, August 2014

Transcript of Primordial nongaussianities I: cosmic microwave background

Page 1: Primordial nongaussianities I: cosmic microwave background

Primordial nongaussianities I: cosmic microwave background

Uros Seljak, UC Berkeley Rio de Janeiro, August 2014

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Outline •  Primordial nongaussianity •  Introduction and basic physics •  CMB temperature power spectrum and

observables •  Primordial perturbations •  CMB Polarization: E and B modes

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Source: NASA/WMAP Science Team

Observations

Theory

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Gaussian  random  field,  ζ(x)  •  normal  distribu6on  of  values  in  real  space,  Prob[ζ(x)]  

•  defined  en6rely  by  power  spectrum  in  Fourier  space  

•  bispectrum  and  (connected)  higher-­‐order  correla6ons  vanish    

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( ) ( ) ( )'2 33' kkkPkk

+= δπζζ ζ

0''' =kkk ζζζ

( ) ⎟⎠⎞

⎜⎝⎛−= 2

2

2exp21Prob

σζ

σπζ

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non-­‐Gaussian  random  field,  ζ(x)  

anything  else  

David Wands 5  

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Rocky  Kolb  

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non-Rocky Kolb

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Primordial  Gaussianity  from  infla2on  •  Quantum  fluctua2ons  from  infla2on  

–  ground  state  of  simple  harmonic  oscillator  –  almost  free  field  in  almost  de  Si5er  space  –  almost  scale-­‐invariant  and  almost  Gaussian  

•  Power  spectra  probe  background  dynamics  (H,  ε,  ...)      –  but,  many  different  models,  can  produce  similar  power  spectra  

•  Higher-­‐order  correla?ons  can  dis?nguish  different  models  

–  non-­‐Gaussianity  ←  non-­‐linearity  ←  interac?ons  =  physics+gravity  

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Wikipedia: AllenMcC

( ) ( ) ( ) ( ) 421

33 ,221

−∝+= nkk kkPkkkP ζζ δπζζ

( ) ( ) ( )3213

3213 ,,2

321kkkkkkBkkk ++= δπζζζ ζ

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Many  sources  of  non-­‐Gaussianity  Initial vacuum Excited state

Sub-Hubble evolution Higher-derivative interactions e.g. k-inflation, DBI, Galileons

Hubble-exit Features in potential

Super-Hubble evolution Self-interactions + gravity

End of inflation Tachyonic instability

(p)Reheating Modulated (p)reheating

After inflation Curvaton decay

Magnetic fields

Primary anisotropies

Last-scattering

Secondary anisotropies

ISW/lensing + foregrounds

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primordial non-Gaussianity

infla

tion

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Many  shapes  for  primordial  bispectra  

•  local  type  (Komatsu&Spergel  2001)  –  local  in  real  space  –  max  for  squeezed  triangles:  k<<k’,k’’  

 •  equilateral  type  (Creminelli  et  al  2005)  

–  peaks  for  k1~k2~k3    

•  orthogonal  type  (Senatore  et  al  2009)  –  independent  of  local  +  equilateral  shapes  

•  separable  basis  (Ferguson  et  al  2008)   9  

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛++∝ 3

133

33

32

32

31

321111,,kkkkkk

kkkBζ

( ) ( )( )( )⎟⎟⎠

⎞⎜⎜⎝

⎛ −+−+−+∝ 3

332

31

213132321321

3,,kkk

kkkkkkkkkkkkBζ

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

++∝ 3

321321321

81,,kkkkkk

kkkBζ

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Hu & White, Sci. Am., 290 44 (2004)

Evolution of the universe

Opaque

Transparent

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Black body spectrum observed by COBE

- close to thermal equilibrium: temperature today of 2.726K ( ~ 3000K at z ~ 1000 because ν ~ (1+z))

Residuals Mather et al 1994

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Source: NASA/WMAP Science Team

O(10-5) perturbations (+galaxy)

Dipole (local motion)

(almost) uniform 2.726K blackbody

Observations: the microwave sky today

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Can we predict the primordial perturbations?

•  Maybe..

Quantum Mechanics “waves in a box” calculation

vacuum state, etc…

Inflation make >1030 times bigger

After inflation Huge size, amplitude ~ 10-5

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Perturbation evolution – what we actually observe CMB monopole source till 380 000 yrs (last scattering), linear in conformal time

scale invariant primordial adiabatic scalar spectrum

photon/baryon plasma + dark matter, neutrinos

Characteristic scales: sound wave travel distance; diffusion damping length

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Observed ΔT as function of angle on the sky

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Perturbations O(10-5) Simple linearized equations are very accurate (except small scales) Can use real or Fourier space Fourier modes evolve independently: simple to calculate accurately

Calculation of theoretical perturbation evolution

• Thomson scattering (non-relativistic electron-photon scattering) - tightly coupled before recombination: ‘tight-coupling’ approximation (baryons follow electrons because of very strong e-m coupling) • Background recombination physics (Saha/full multi-level calculation) • Linearized General Relativity • Boltzmann equation (how angular distribution function evolves with scattering)

Physics Ingredients

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CMB power spectrum Cl •  Theory: Linear physics + Gaussian primordial fluctuations

〉〈= 2|| lml aCTheory prediction

- variance (average over all possible sky realizations) - statistical isotropy implies independent of m

Cl

∫ ΔΩ= *lmlm YTda

CMBFAST: cmbfast.org CAMB: camb.info CMBEASY: cmbeasy.org COSMICS, etc..

Initial conditions + cosmological parameters

linearized GR + Boltzmann equations

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Sources of CMB anisotropy Sachs Wolfe: Potential wells at last scattering cause redshifting as photons climb out Photon density perturbations: Over-densities of photons look hotter Doppler: Velocity of photon/baryons at last scattering gives Doppler shift Integrated Sachs Wolfe: Evolution of potential along photon line of sight: net red- or blue-shift as photon climbs in an out of varying potential wells Others: Photon quadupole/polarization at last scattering, second-order effects, etc.

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Hu & White, Sci. Am., 290 44 (2004)

CMB temperature power spectrum Primordial perturbations + later physics

diffusion damping acoustic oscillations

primordial power spectrum

finite thickness

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Why Cl oscillations? Think in k-space: modes of different size

•  Co-moving Poisson equation: (k/a)2 Ф = κ δρ / 2 - potentials approx constant on super-horizon scales - radiation domination ρ ~ 1/a4 à δρ/ρ ~ k2 a2 Ф à since Ф ~ constant, super-horizon density perturbations grow ~ a2

•  After entering horizon pressure important: perturbation growth slows, then bounces back à series of acoustic oscillations (sound speed ~ c/√3)

•  CMB anisotropy (mostly) from a surface at fixed redshift: phase of oscillation at time of last scattering depends on time since entering the horizon à k-dependent oscillation amplitude in the observed CMB

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Challinor: astro-ph/0403344

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Contributions to temperature Cl

Challinor: astro-ph/0403344

+ other

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What can we learn from the CMB? •  Initial conditions

What types of perturbations, power spectra, distribution function (Gaussian?); => learn about inflation or alternatives. (distribution of ΔT; power as function of scale; polarization and correlation)

•  What and how much stuff Matter densities (Ωb, Ωcdm);; neutrino mass (details of peak shapes, amount of small scale damping)

•  Geometry and topology global curvature ΩK of universe; topology (angular size of perturbations; repeated patterns in the sky)

•  Evolution Expansion rate as function of time; reionization - Hubble constant H0 ; dark energy evolution w = pressure/density (angular size of perturbations; l < 50 large scale power; polarizationr)

•  Astrophysics S-Z effect (clusters), foregrounds, etc.

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∑+=

m lmobsl a

lC 2||

121

•  Cosmic Variance: only one sky

)|( obsll CCP

122||

22

+≈ΔlCC lobs

l

“Cosmic Variance”

Use estimator for variance:

- inverse gamma distribution (+ noise, sky cut, etc).

WMAP low l

l

d.o.f. 12 with ~ 2 +lC obsl χ

Cosmic variance gives fundamental limit on how much we can learn from CMB

Assume alm gaussian:

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How to generate CMB polarization?

25 (Wayne Hu, CMB Tutorials)

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How to generate CMB polarization?

26 (Wayne Hu, CMB Tutorials)

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How to generate CMB polarization?

27 (Wayne Hu, CMB Tutorials)

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Polarization: Stokes’ Parameters

- -

Q U Q → -Q, U → -U under 90 degree rotation

Q → U, U → -Q under 45 degree rotation

Spin-2 field Q + i U or Rank 2 trace free symmetric tensor

θ

sqrt(Q2 + U2)

θ = ½ tan-1 U/Q

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WMAP Polarization

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E and B mode patterns

Unchanged if seen through the mirror

Pattern reverses if seen through the mirror

E-mode

B-mode

Seljak and Zaldarriaga, 1998

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E and B harmonics

•  Expand scalar PE and PB in spherical harmonics •  Expand Pab in tensor spherical harmonics

Harmonics are orthogonal over the full sky:

E/B decomposition is exact and lossless on the full sky

Zaldarriaga, Seljak: astro-ph/9609170 Kamionkowski, Kosowsky, Stebbins: astro-ph/9611125

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CMB Polarization Signals

Parity symmetric ensemble:

Average over possible realizations (statistically isotropic):

•  E polarization from scalar, vector and tensor modes

•  B polarization only from vector and tensor modes (curl grad = 0) + non-linear scalars

Power spectra contain all the useful information if the field is Gaussian

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Scalar adiabatic mode

E polarization only correlation to temperature T-E

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Primordial Gravitational Waves (tensor modes)

•  Well motivated by some inflationary models - Amplitude measures inflaton potential at horizon crossing - distinguish models of inflation

•  Observation would rule out other models - ekpyrotic scenario predicts exponentially small amplitude - small also in many models of inflation, esp. two field e.g. curvaton

•  Weakly constrained from CMB temperature anisotropy

Look at CMB polarization: ‘B-mode’ smoking gun Has BICEP2 observed it?

- cosmic variance limited to 10% - degenerate with other parameters (tilt, reionization, etc)

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CMB polarization from primordial gravitational waves (tensors)

Adiabatic E-mode

Tensor B-mode

Tensor E-mode

Planck noise (optimistic)

Weak lensing

•  Amplitude of tensors unknown •  Clear signal from B modes – there are none from scalar modes •  Tensor B is always small compared to adiabatic E

Seljak, Zaldarriaga: astro-ph/9609169

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Reionization Ionization since z ~ 6-20 scatters CMB photons

Measure optical depth with WMAP T-E correlation

Temperature signal similar to tensors

Quadrupole at reionization implies large scale polarization signal

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Other non-linear effects

•  Thermal Sunyaev-Zeldovich Inverse Compton scattering from hot gas: frequency dependent signal

•  Kinetic Sunyaev-Zeldovich (kSZ) Doppler from bulk motion of clusters; patchy reionization; (almost) frequency independent signal

•  Ostriker-Vishniac (OV) same as kSZ but for early linear bulk motion

•  Rees-Sciama Integrated Sachs-Wolfe from evolving non-linear potentials: frequency independent

•  Gravitational lensing discussed on friday

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Liguori, Matarrese and Moscardini (2003)

Newtonian potential a Gaussian random field Φ(x) = φG(x)

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Liguori, Matarrese and Moscardini (2003)

fNL=+3000

Newtonian potential a local function of Gaussian random field Φ(x) = φG(x) + fNL ( φG

2(x) - <φG2> )

ΔT/T ≈ -Φ/3, so positive fNL ⇒ more cold spots in CMB

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Liguori, Matarrese and Moscardini (2003)

fNL=-3000

Newtonian potential a local function of Gaussian random field Φ(x) = φG(x) + fNL ( φG

2(x) - <φG2> )

ΔT/T ≈ -Φ/3, so negative fNL ⇒ more hot spots in CMB

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Newtonian potential a local function of Gaussian random field Φ(x) = φG(x) + fNL ( φG

2(x) - <φG2> )

⇒  Large-scale modulation of small-scale power

split Gaussian field into long (L) and short (s) wavelengths φG (X+x) = φL(X) + φs(x)

two-point function on small scales for given φL

< Φ(x1) Φ(x2) >L = (1+4 fNL φL ) < φs (x1) φs (x2) > +...

X1 X2 i.e., inhomogeneous modulation of small-scale power

P ( k , X ) -> [ 1 + 4 fNL φL(X) ] Ps(k)

but fNL <5 so any effect must be 10-4

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Optimal estimation from CMB data

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Planck Paper XXIV constraints: no evidence for primordial nongaussianity

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Conclusions

•  CMB contains lots of useful information! - primordial perturbations + well understood physics (cosmological parameters)

•  Precision cosmology - constrain many cosmological parameters + primordial perturbations

•  Currently no evidence for any deviations from standard near scale-invariant purely adiabatic primordial spectrum

•  E-polarization and T-E measure optical depth, constrain reionization; constrain isocurvature modes

•  Large scale B-mode polarization from primordial gravitational waves: - energy scale of inflation - rule out most ekpyrotic and pure curvaton/ inhomogeneous reheating models and others

•  No evidence for primordial nongaussianity from CMB