statistics of the cmb from boltzmann equation of photons to power spectra 2-point statistics on the...
TRANSCRIPT
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