# Likelihood analysis of CMB temperature and polarization power spectra

## Abstract

Microwave background temperature and polarization observations are a powerful way to constrain cosmological parameters if the likelihood function can be calculated accurately. The temperature and polarization fields are correlated, partial-sky coverage correlates power spectrum estimators at different l, and the likelihood function for a theory spectrum given a set of observed estimators is non-Gaussian. An accurate analysis must model all these properties. Most existing likelihood approximations are good enough for a temperature-only analysis, however they cannot reliably handle temperature-polarization correlations. We give a new general approximation applicable for correlated Gaussian fields observed on part of the sky. The approximation models the non-Gaussian form exactly in the ideal full-sky limit and is fast to evaluate using a precomputed covariance matrix and set of power spectrum estimators. We show with simulations that it is good enough to obtain correct results at l > or approx. 30 where an exact calculation becomes impossible. We also show that some Gaussian approximations give reliable parameter constraints even though they do not capture the shape of the likelihood function at each l accurately. Finally we test the approximations on simulations with realistically anisotropic noise and asymmetric foreground mask.

- Authors:

- Institute of Astronomy, Madingley Road, Cambridge, CB3 0HA (United Kingdom)

- Publication Date:

- OSTI Identifier:
- 21204960

- Resource Type:
- Journal Article

- Journal Name:
- Physical Review. D, Particles Fields

- Additional Journal Information:
- Journal Volume: 77; Journal Issue: 10; Other Information: DOI: 10.1103/PhysRevD.77.103013; (c) 2008 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0556-2821

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; AMBIENT TEMPERATURE; ANISOTROPY; APPROXIMATIONS; ASYMMETRY; CAPTURE; CORRELATIONS; FIELD THEORIES; FUNCTIONS; GAUSSIAN PROCESSES; MAXIMUM-LIKELIHOOD FIT; MICROWAVE RADIATION; NOISE; POLARIZATION; SIMULATION; SPECTRA

### Citation Formats

```
Hamimeche, Samira, and Lewis, Antony.
```*Likelihood analysis of CMB temperature and polarization power spectra*. United States: N. p., 2008.
Web. doi:10.1103/PHYSREVD.77.103013.

```
Hamimeche, Samira, & Lewis, Antony.
```*Likelihood analysis of CMB temperature and polarization power spectra*. United States. doi:10.1103/PHYSREVD.77.103013.

```
Hamimeche, Samira, and Lewis, Antony. Thu .
"Likelihood analysis of CMB temperature and polarization power spectra". United States. doi:10.1103/PHYSREVD.77.103013.
```

```
@article{osti_21204960,
```

title = {Likelihood analysis of CMB temperature and polarization power spectra},

author = {Hamimeche, Samira and Lewis, Antony},

abstractNote = {Microwave background temperature and polarization observations are a powerful way to constrain cosmological parameters if the likelihood function can be calculated accurately. The temperature and polarization fields are correlated, partial-sky coverage correlates power spectrum estimators at different l, and the likelihood function for a theory spectrum given a set of observed estimators is non-Gaussian. An accurate analysis must model all these properties. Most existing likelihood approximations are good enough for a temperature-only analysis, however they cannot reliably handle temperature-polarization correlations. We give a new general approximation applicable for correlated Gaussian fields observed on part of the sky. The approximation models the non-Gaussian form exactly in the ideal full-sky limit and is fast to evaluate using a precomputed covariance matrix and set of power spectrum estimators. We show with simulations that it is good enough to obtain correct results at l > or approx. 30 where an exact calculation becomes impossible. We also show that some Gaussian approximations give reliable parameter constraints even though they do not capture the shape of the likelihood function at each l accurately. Finally we test the approximations on simulations with realistically anisotropic noise and asymmetric foreground mask.},

doi = {10.1103/PHYSREVD.77.103013},

journal = {Physical Review. D, Particles Fields},

issn = {0556-2821},

number = 10,

volume = 77,

place = {United States},

year = {2008},

month = {5}

}