A. Challinor

Efficient computation of CMB anisotropies in closed FRW models

We implement the efficient line-of-sight method to calculate the anisotropy and polarization of the cosmic microwave background for scalar and tensor modes in almost Friedmann-Robertson-Walker models with positive spatial curvature. We present new results for the polarization power spectra in such models.

Weak gravitational lensing of the CMB

Weak gravitational lensing has several important effects on the cosmic microwave background (CMB): it changes the CMB power spectra, induces non-Gaussianities, and generates a B-mode polarization signal that is an important source of confusion for the signal from primordial gravitational waves. The lensing signal can also be used to help constrain cosmological parameters and lensing mass distributions. We review the origin and calculation of these effects. Topics include: lensing in General Relativity, the lensing potential, lensed temperature and polarization power spectra, implications for constraining inflation, non-Gaussian structure, reconstruction of the lensing potential, delensing, sky curvature corrections, simulations, cosmological parameter estimation, cluster mass reconstruction, and moving lenses/dipole lensing.

The linear power spectrum of observed source number counts

We relate the observable number of sources per solid angle and redshift to the underlying proper source density and velocity, background evolution and line-of-sight potentials. We give an exact result in the case of linearized perturbations assuming general relativity. This consistently includes contributions of the source density perturbations and redshift distortions, magnification, radial displacement, and various additional linear terms that are small on sub-horizon scales. In addition we calculate the effect on observed luminosities, and hence the result for sources observed as a function of flux, including magnification bias and radial-displacement effects. We give the corresponding linear result for a magnitude-limited survey at low redshift, and discuss the angular power spectrum of the total count distribution. We also calculate the cross-correlation with the CMB polarization and temperature including Doppler source terms, magnification, redshift distortions and other velocity effects for the sources, and discuss why the contribution of redshift distortions is generally small. Finally we relate the result for source number counts to that for the brightness of line radiation, for example 21-cm radiation, from the sources.

Relativistic Corrections to the Sunyaev-Zeldovich Effect

We present an extension of the Kompaneets equation which allows relativistic effects to be included to any desired order. Using this, we are able to obtain simple analytic forms for the spectral changes due to the Sunyaev-Zeldovich effect in hot clusters, correct to first and second order in the expansion parameter θe = kBTe/mc2. These analytic forms agree with previous numerical calculations of the effect based upon the multiple scattering formalism and are expected to be very accurate over all regions of the cosmic microwave background spectrum for kBTe up to ~10 keV. Our results confirm previous conclusions that the result of including relativistic corrections in the Sunyaev-Zeldovich effect is a small reduction in the amplitude of the effect over the majority of the spectrum: specifically, we find ΔT/T = -2y(1 - 17/10θe + 123/40θ2e) (correct to second order) in the Rayleigh-Jeans region, where y is the usual Comptonization parameter. For a typical cluster temperature of 8 keV, this amounts to a correction downward to the value of the Hubble constant derived using combined X-ray and Rayleigh-Jeans Sunyaev-Zeldovich information by about 5%.

Relativistic cosmology and large-scale structure

General relativity marked the beginning of modern cosmology and it has since been at the centre of many of the key developments in this field. In the present review, we discuss the general-relativistic dynamics and perturbations of the standard cosmological model, the Friedmann–Lemaitre universe, and how these can explain and predict the properties of the observable universe. Our aim is to provide an overview of the progress made in several major research areas, such as linear and non-linear cosmological perturbations, large-scale structure formation and the physics of the cosmic microwave background radiation, in view of current and upcoming observations. We do this by using a single formalism throughout the review, the 1+3 covariant approach to cosmology, which allows for a uniform and balanced presentation of technical information and physical insight.

Fast estimation of polarization power spectra using correlation functions

We present a fast method for estimating the cosmic microwave background polarization power spectra using unbiased estimates of heuristically weighted correlation functions. This extends the O(N 3/2 pix) method of Szapudi et al. to polarized data. If the sky coverage allows the correlation functions to be estimated over the full range of angular separations, they can be inverted directly with integral transforms and clean separation of the electric (E) and magnetic (B) modes of polarization is obtained exactly in the mean. We assess the level of E-B mixing that arises from apodized integral transforms when the correlation function can only be estimated for a subset of angular scales, and show that it is significant for small-area observations. We introduce new estimators to deal with this case on the spherical sky that preserve E-B separation; their construction requires an additional integration of the correlation functions but the computational cost is negligible. We illustrate our methods with application to a large-area survey with parameters similar to Planck, and the small-area Background Imaging of Cosmic Extragalactic Polarization experiment. In both cases we show that the errors on the recovered power spectra are close to theoretical expectations.

CMB power spectrum parameter degeneracies in the era of precision cosmology

Cosmological parameter constraints from the CMB power spectra alone suffer several well-known degeneracies. These degeneracies can be broken by numerical artefacts and also a variety of physical effects that become quantitatively important with high-accuracy data e.g. from the Planck satellite. We study degeneracies in models with flat and non-flat spatial sections, non-trivial dark energy and massive neutrinos, and investigate the importance of various physical degeneracy-breaking effects. We test the CAMB power spectrum code for numerical accuracy, and demonstrate that the numerical calculations are accurate enough for degeneracies to be broken mainly by true physical effects (the integrated Sachs-Wolfe effect, CMB lensing and geometrical and other effects through recombination) rather than numerical artefacts. We quantify the impact of CMB lensing on the power spectra, which inevitably provides degeneracy-breaking information even without using information in the non-Gaussianity. Finally we check the numerical accuracy of sample-based parameter constraints using CAMB and COSMOMC. In an appendix we document recent changes to CAMBs numerical treatment of massive neutrino perturbations, which are tested along with other recent improvements by our degeneracy exploration results.

Lensed CMB power spectra from all-sky correlation functions

Weak lensing of the CMB changes the unlensed temperature anisotropy and polarization power spectra. Accounting for the lensing effect will be crucial to obtain accurate parameter constraints from sensitive CMB observations. Methods for computing the lensed power spectra using a low-order perturbative expansion are not good enough for percent-level accuracy. Nonperturbative flat-sky methods are more accurate, but curvature effects change the spectra at the 0.3%\char21{}1% level. We describe a new, accurate, and fast, full-sky correlation-function method for computing the lensing effect on CMB power spectra to better than 0.1% at

The shape of the CMB lensing bispectrum

l\ensuremath{\lesssim}2500

Asymmetric beams and CMB statistical anisotropy

(within the approximation that the lensing potential is linear and Gaussian). We also discuss the effect of nonlinear evolution of the gravitational potential on the lensed power spectra. Our fast numerical code is publicly available.