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

Homogeneous solution

Perturbed metric reads

FLAT SCLICING GAUGE

NEWTONIAN GAUGEPerturbed photon energy-momentum

Boltzmann equation of photons

Geodesic parametrization

Geodesic equation of particles (interacting gravitationally only)

Homogeneous evolution

Unperturbed========background

Perturbed Boltzmann equationGeodesic 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…

Vector field: potential plus solenoidal:

STF tensor of rank 2:

NORMAL MODES

Generalization:

Fourier space, with k=e3

Thomson scattering term (temperature)COLLISION TERM THOMSON PHASE FUNCTION

Energy as seen by observer comoving withbaryons/photons fluid

Gauge-invariant phase-space density perturbation

Gauge-invariant Boltzmann equation reads(Newtonian gauge)

USING THE FOLLOWINGMonopole is unaffected by scatteringForward photons are scattered awayBaryon-photon dragAnisotropic pressure

Temperature hierarchy, scalar modes

TRANSPORT GRAVITY THOMSON SCATTERING

Boltzmann hierarchy, tensor modes

Einstein and conservation equationsScalar 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 ??

NOT YET….

POLARISATIONOK, I’ll make it soft !

Polarisation• Due to quadrupolar anisotropy in the electron rest frame• Linked to velocity field gradients at recombination

E and B modes of polarisation

Scalar quantity

Pseudo-scalar quantity

Scalar perturbations cannot produce B modesB modes are model-independent tracers of tensor perturbations

Normal modes

As for temperature, we have normal modes for polarisation

Temperature and polarisation get decomposed on these modes

are gauge-invariant (Stewart-Walker lemma)

Boltzmann equation for Stokes Q,U

Stokes parameters are absent in unperturbed background

Their evolution does not couple to metric perturbations at linear order

Redefining

SIMPLE, ISN’T IT ?

Polarized scattering term

SCATTERING GEOMETRY

Boltzmann polarization hierarchyAs for the temperature case, express gradient term in terms of spherical harmonics

Using the following recurrence formula:

SCALARS DO NOTPRODUCE B MODES

ONLY E-MODES COUPLETO TEMPERATUREQUADRUPOLE

Interpretation

Normal modes and integral solutionsDevelop the plane wave into radial modes Using recurrence relations of spherical Bessels:

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

PLANE WAVE MODULATIONSOURCE DEPENDANCE

Normal modes and integral solutions

These normal modes are the solutions of the equations of free-streaming !! (Boltzmann equation without gravity and collisions)

Line-of-sight integration codesSources depend on monopole, dipole and quadrupole only

Linear dependance in the primordial perturbations amplitudes

CMB power spectra

Wayne Hu

CMB imaging: scanning experiments

Time-response of the instrument(detector + electronics)EM filters band-passAngular response: beam

and scanning strategy

Detector noiseSimplified linear model (pixelized sky)

Archeops, Kiruna BICEP focal plane Spider web bolometer

Imagers: map-making

BAYES theorem

Linear data model

Sufficient statisticsCovariance matrix of the map

Uniform signal prior

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

Imagers: power spectrum

Signal covariance matrix

BAYES again…

Marginalize over the map

TO BE MAXIMIZED WITH RESPECT TO POWER SPECTRUM

Imagers: power spectrum (cont.)

Second orderTaylor expansion

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

PSEUDO-NEWTON (FISHER)

Imagers: too many pixels !

New (fast) analysis methods needed

• Fast harmonic transforms• Heuristically weighted maps

Quite ugly at first sight !!

Imagers (cont.)Power spectrum expectation value

…simplifies, after summation over angles (m):

Imagers: “Master” methodFinite sky coverage loss of spectral resolution need to regularize inversion

MC estimation of covariance matrix of PS estimates

Spectral binning of the kernel Unbiased estimator

Works also for polarization (easier regularization on correlation function)

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

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

General expression of the covariance matrix

• Stokes parameters errors are uncorrelated• Covariance determinant is minimized

Imagers: polarised spectrum estimation

Stokes parameter in the great circle basis

Polarisation: correlation functions

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

Polarisation: (fast) CF estimatorsHeuristic weighting (wP,wT): Normalization: correlation function

of the weights

Using for m=n=2 involves

with Weighted polarization field

Using

We get

Polarisation: (fast) CF and PS estimators

Define the pseudo-Cls estimates:

These can be computed using fast SPH transforms in O(npix

3/2) (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 surveysIncomplete measurement of correlation function: apodizing function f():

Normalization of the window functions

Results in E/B modes leakage

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-sky

Leakage window functions (not normalized) Recovered BB spectra (dots)

No correlation function overGaussian apodization

Polarisation: E/B leakage correction

Define:

Then:

As a function of +

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

Quadratic estimators: covariancesRAPPELS

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

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

CBI – Atacama desert

Interferometers: data model

Visibilities: sample the convolved UV space:

Idem for Q and U Stokes parameters

RL and LR baselines give (Q§iU)

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

Visibilities correlation matrix

UV coverage of a single pointing of CBI (10 freq. bands)( Pearson et al. 2003)

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

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

Least squares solution

For an NGP pointing matrix:

Resultant noise matrix

Use in conjonction with an ML estimator

Newton-like iterative maximisation

Fisher matrix

Covariance derivatives for one visibility