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Statistics of the CMB. From Boltzmann equation of photons to power spectra 2-point statistics on the sphere: ML, quadratic estimators, polarisation-specifics. Outline of the lectures. Boltzmann equation of CMB. FLAT SCLICING GAUGE. NEWTONIAN GAUGE. Homogeneous solution.

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statistics of the cmb
Statistics of the CMB
  • From Boltzmann equation of photons to power spectra
  • 2-point statistics on the sphere: ML, quadratic estimators, polarisation-specifics

Outline of the lectures

boltzmann equation of cmb
Boltzmann equation of CMB

FLAT SCLICING GAUGE

NEWTONIAN GAUGE

Homogeneous solution

Perturbed metric reads

Perturbed photon energy-momentum

boltzmann equation of photons
Boltzmann equation of photons

Geodesic parametrization

Geodesic equation of particles (interacting gravitationally only)

Unperturbed

========

background

Homogeneous evolution

perturbed boltzmann equation
Perturbed Boltzmann equation

Geodesic equation for the energy, in perturbed metric

Boltzmann equation for perturbed distribution, in perturbed background

Collisional cross-section is frequency independent: can integrate over frequency:

STF TENSORS

svt stf spherical harmonics
SVT, STF, Spherical harmonics…

Vector field: potential plus solenoidal:

STF tensor of rank 2:

Generalization:

Fourier space, with k=e3

NORMAL

MODES

thomson scattering term temperature
Thomson scattering term (temperature)

COLLISION TERM

THOMSON PHASE FUNCTION

Energy as seen by

observer comoving with

baryons/photons fluid

gauge invariant phase space density perturbation
Gauge-invariant phase-space density perturbation

Gauge-invariant Boltzmann equation reads

(Newtonian gauge)

temperature hierarchy scalar modes
Temperature hierarchy, scalar modes

Monopole is unaffected by scattering

Forward photons are scattered away

Baryon-photon drag

Anisotropic pressure

USING THE FOLLOWING

GRAVITY

THOMSON SCATTERING

TRANSPORT

einstein and conservation equations
Einstein and conservation equations

Scalar modes, Einstein equations

Constraint equations

(Poisson)

Evolution equations

Scalar modes, conservation equations

Energy

Momentum (Euler)

Tensor modes, Einstein equation

do you really think this is it
DO YOU (REALLY) THINK THIS IS IT ??

NOT YET….

POLARISATION

OK, I’ll make it soft !

polarisation
Polarisation
  • Due to quadrupolar anisotropy in the electron rest frame
  • Linked to velocity field gradients at recombination
e and b modes of polarisation
E and B modes of polarisation

Scalar quantity

Pseudo-scalar quantity

Scalar perturbations cannot produce B modes

B modes are model-independent tracers of tensor perturbations

normal modes
Normal modes

are gauge-invariant (Stewart-Walker lemma)

As for temperature, we have normal modes for polarisation

Temperature and polarisation get decomposed on these modes

boltzmann equation for stokes q u
Boltzmann equation for Stokes Q,U

Redefining

SIMPLE, ISN’T IT ?

Stokes parameters are absent in unperturbed background

Their evolution does not couple to metric perturbations at linear order

polarized scattering term
Polarized scattering term

SCATTERING GEOMETRY

boltzmann polarization hierarchy
Boltzmann polarization hierarchy

ONLY E-MODES COUPLE

TO TEMPERATURE

QUADRUPOLE

SCALARS DO NOT

PRODUCE B MODES

As for the temperature case, express gradient term in terms of spherical harmonics

Using the following recurrence formula:

normal modes and integral solutions
Normal modes and integral solutions

Interpretation

State of definite total angular momentum results in a weighted sum of

SOURCE DEPENDANCE

PLANE WAVE MODULATION

Using recurrence relations of spherical Bessels:

Develop the plane wave into radial modes

normal modes and integral solutions1
Normal modes and integral solutions

Linear dependance in the primordial perturbations amplitudes

These normal modes are the solutions of the equations of free-streaming !!

(Boltzmann equation without gravity and collisions)

  • Line-of-sight integration codes
  • Sources depend on monopole, dipole and quadrupole only
cmb imaging scanning experiments
CMB imaging: scanning experiments

Time-response of the instrument

(detector + electronics)

Simplified linear model (pixelized sky)

EM filters band-pass

Detector noise

Angular response: beam

and scanning strategy

BICEP focal plane

Spider web bolometer

Archeops, Kiruna

imagers map making
Imagers: map-making

Huge linear system to solve: use iterative methods (PCG) + FFTs

BAYES theorem

Linear data model

Uniform signal prior

Sufficient statistics

Covariance matrix of the map

imagers power spectrum
Imagers: power spectrum

Signal covariance matrix

BAYES again…

Marginalize over the map

TO BE MAXIMIZED WITH RESPECT TO POWER SPECTRUM

imagers power spectrum cont
Imagers: power spectrum (cont.)

PSEUDO-NEWTON (FISHER)

Second order

Taylor expansion

For each iteration and each band, Npix3 operation scaling !!

imagers too many pixels
Imagers: too many pixels !

Quite ugly at first sight !!

 New (fast) analysis methods needed

  • Fast harmonic transforms
  • Heuristically weighted maps
imagers cont
Imagers (cont.)

Power spectrum expectation value

…simplifies, after summation over angles (m):

imagers master method
Imagers: “Master” method

Finite sky coverage  loss of spectral resolution  need to regularize inversion

Spectral binning of the kernel

Unbiased estimator

MC estimation of covariance matrix of PS estimates

Works also for polarization (easier regularization on correlation function)

imagers polarised map making
Imagers: polarised map-making

One polarised detector (i)

Let us consider n measurements of the same pixel, indexed by their angle 

ML solution

polarisation optimal configurations
Polarisation: optimal configurations

General expression of the covariance matrix

Assume uncorrelated and equal variance measurements, look for optimal configuration of angles :

  • Stokes parameters errors are uncorrelated
  • Covariance determinant is minimized
imagers polarised spectrum estimation
Imagers: polarised spectrum estimation

Stokes parameter in the great circle basis

polarisation correlation functions
Polarisation: correlation functions

Polynomials in cos(): integrate exactly with Gauss-Legendre quadrature

polarisation fast cf estimators
Polarisation: (fast) CF estimators

for m=n=2 involves

Using

with

Weighted polarization field

Using

We get

Heuristic weighting (wP,wT):

Normalization: correlation function of the weights

polarisation fast cf and ps estimators
Polarisation: (fast) CF and PS estimators

Define the pseudo-Cls estimates:

These can be computed using fast SPH transforms in O(npix3/2) (compare to o(npix3) scaling of ML…)

If CF measured at all angles:

integrate with GL quadrature

Assuming parity invariance

polarisation cf estimators on finite surveys
Polarisation: CF estimators on finite surveys

Results in E/B modes leakage

Incomplete measurement of correlation function: apodizing function f():

Normalization of the window functions

polarisation e b coupling of cut sky
Polarisation: E/B coupling of cut-sky

Leakage window functions (not normalized)

Recovered BB spectra (dots)

No correlation function information over max=20±

polarisation e b coupling of cut sky1
Polarisation: E/B coupling of cut-sky

No correlation function over

Gaussian apodization

Leakage window functions (not normalized)

Recovered BB spectra (dots)

polarisation e b leakage correction
Polarisation: E/B leakage correction

Define:

Then:

We have obtained pure E and B spectra (in the mean)

As a function of +

quadratic estimators covariances
Quadratic estimators: covariances

RAPPELS

Edge-corrected estimators covariances in terms of pseudo-Cls covariances

As long as Mll’ is invertible, same information content in edge-corrected Cls and pseudo-Cls

pseudo cls estimators cosmic variance
Pseudo-Cls estimators: cosmic variance

Forget noise for the moment, consider signal only:

Case of high ells and/or almost full sky

If simple weighting (zeros and ones)

the case of interferometers
The case of interferometers

CBI – Atacama desert

interferometers data model
Interferometers: data model

Visibilities: sample the convolved UV space:

Relationship between (Q,U) and (E,B) in UV (flat) space

Idem for Q and U Stokes parameters

RL and LR baselines give (Q§iU)

Visibilities correlation matrix

UV coverage of a single pointing of CBI (10 freq. bands)

( Pearson et al. 2003)

pixelisation in uv pixel space
Pixelisation in UV/pixel space
  • Redundant measurements in UV-space
  • Possibility to compress the data ~w/o loss

Use in conjonction with an ML estimator

Newton-like iterative maximisation

Least squares solution

  • Hobson and Maisinger 2002
  • Myers et al. 2003
  • Park et al. 2003

Fisher matrix

For an NGP pointing matrix:

Covariance derivatives for one visibility

Resultant noise matrix