4 research outputs found
A completeness-like relation for Bessel functions
Completeness relations are associated through Mercer's theorem to complete
orthonormal basis of square integrable functions, and prescribe how a Dirac
delta function can be decomposed into basis of eigenfunctions of a
Sturm-Liouville problem. We use Gegenbauer's addition theorem to prove a
relation very close to a completeness relation, but for a set of Bessel
functions not known to form a complete basis in
CMB and Random Flights: temperature and polarization in position space
The fluctuations in the temperature and polarization of the cosmic microwave
background are described by a hierarchy of Boltzmann equations. In its integral
form, this Boltzmann hierarchy can be converted from the usual Fourier-space
base into a position-space and causal description. We show that probability
densities for random flights play a key role in this description. The integral
system can be treated as a perturbative series in the number of steps of the
random flights, and the properties of random flight probabilities impose
constraints on the domains of dependence. We show that, as a result of these
domains, a Fourier-Bessel decomposition can be employed in order to calculate
the random flight probability densities. We also illustrate how the H-theorem
applies to the cosmic microwave background: by using analytical formulae for
the asymptotic limits of these probability densities, we show that, as the
photon distribution approaches a state of equilibrium, both the temperature
anisotropies and the net polarization must vanish.Comment: Minor revisions; matches version published in JCAP06(2013)04
Redshift-space distortions with wide angular separations
35 pages, 5 figuresRedshift-space distortions are generally considered in the plane parallel limit, where the angular separation between the two sources can be neglected. Given that galaxy catalogues now cover large fractions of the sky, it becomes necessary to consider them in a formalism which takes into account the wide angle separations. In this article we derive an operational formula for the matter correlators in the Newtonian limit to be used in actual data sets, both in configuration and in Fourier spaces without relying on a plane-parallel approximation. We then recover the plane-parallel limit not only in configuration space where the geometry is simpler, but also in Fourier space, and we exhibit the first corrections that should be included in large surveys as a perturbative expansion over the plane-parallel results. We finally compare our results to existing literature, and show explicitly how they are related