CMB in a box: causal structure and the Fourier-Bessel expansion

Abstract

This paper makes two points. First, we show that the line-of-sight solution to cosmic microwave anisotropies in Fourier space, even though formally defined for arbitrarily large wavelengths, leads to position-space solutions which only depend on the sources of anisotropies inside the past light-cone of the observer. This happens order by order in a series expansion in powers of the visibility $\gamma =e^{-\mu }$, where $\mu$ is the optical depth to Thompson scattering. We show that the CMB anisotropies are regulated by spacetime window functions which have support only inside the past light-cone of the point of observation. Second, we show that the Fourier-Bessel expansion of the physical fields (including the temperature and polarization momenta) is an alternative to the usual Fourier basis as a framework to compute the anisotropies. In that expansion, for each multipole $l$ there is a discrete tower of momenta $k_{i,l}$ (not a continuum) which can affect physical observables, with the smallest momenta being $k_{1,l}l$. The Fourier-Bessel modes take into account precisely the information from the sources of anisotropies that propagates from the initial value surface to the point of observation - no more, no less. We also show that the physical observables (the temperature and polarization maps), and hence the angular power spectra, are unaffected by that choice of basis. This implies that the Fourier-Bessel expansion is the optimal scheme with which one can compute CMB anisotropies. (Abridged)

arXiv comment23 pages, 7 figures

References in corpus (30)

• First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Determination of Cosmological Parameters
• First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Preliminary Maps and Basic Results
• Non-Gaussian features of primordial fluctuations in single field inflationary models
• A Line of Sight Approach to Cosmic Microwave Background Anisotropies
• Cosmological Perturbation Theory in the Synchronous and Conformal Newtonian Gauges
• Cosmological Perturbation Theory in the Synchronous vs. Conformal Newtonian Gauge
• Three Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Polarization Analysis
• An All-Sky Analysis of Polarization in the Microwave Background
• Statistics of Cosmic Microwave Background Polarization
• Signature of Gravity Waves in Polarization of the Microwave Background
• Small Scale Cosmological Perturbations: An Analytic Approach
• A Probe of Primordial Gravity Waves and Vorticity
• CMB Anisotropies: Total Angular Momentum Method
• The Physics of Microwave Background Anisotropies
• Measuring primordial non-Gaussianity in the cosmic microwave background
• Improved measurements of the temperature and polarization of the CMB from QUaD
• Cosmic Microwave Background Polarization
• Constraining the Topology of the Universe
• Measurement of Cosmic Microwave Background Polarization Power Spectra from Two Years of BICEP Data
• A Spherical Harmonic Analysis of Redshift Space
• A Complete Treatment of CMB Anisotropies in a FRW Universe
• Getting Around Cosmic Variance
• Improved simulation of non-Gaussian temperature and polarization CMB maps
• High-Resolution Simulations of Cosmic Microwave Background non-Gaussian Maps in Spherical Coordinates
• Temperature and Polarization CMB Maps from Primordial non-Gaussianities of the Local Type
• From Primordial Quantum Fluctuations to the Anisotropies of the Cosmic Microwave Background Radiation
• Dynamics of Cosmological Perturbations in Position Space
• Position-Space Description of the Cosmic Microwave Background and Its Temperature Correlation Function
• Probing the Universe on Gigaparsec Scales with Remote Cosmic Microwave Background Quadrupole Measurements
• Real space tomography of the primordial Universe with cluster polarization

Cited by in corpus (12)

• Total Angular Momentum Waves for Scalar, Vector, and Tensor Fields
• Exact Wavelets on the Ball
• 3DEX: a code for fast spherical Fourier-Bessel decomposition of 3D surveys
• Effects of Linear Redshift Space Distortions and Perturbation Theory on BAOs: A 3D Spherical Analysis
• The Complementarity of Redshift-space Distortions and the Integrated Sachs-Wolfe Effect: A 3D Spherical Analysis
• Galaxy Clustering in 3D and Modified Gravity Theories
• Slepian Spatial-Spectral Concentration on the Ball
• Two-point Correlation Functions in Inhomogeneous and Anisotropic Cosmologies
• CMB and Random Flights: temperature and polarization in position space
• Flaglets for studying the large-scale structure of the Universe
• Fast expansion into harmonics on the ball
• Fisher matrix for the angular power spectrum of multi-tracer galaxy surveys