Position-Space Description of the Cosmic Microwave Background and Its Temperature Correlation Function

Abstract

We suggest that the cosmic microwave background (CMB) temperature correlation function C(θ) as a function of angle provides a direct connection between experimental data and the fundamental cosmological quantities. The evolution of inhomogeneities in the prerecombination universe is studied using their Green's functions in position space. We find that a primordial adiabatic point perturbation propagates as a sharp-edged spherical acoustic wave. Density singularities at its wavefronts create a feature in the CMB correlation function distinguished by a dip at θ ≈ 1.2• . Characteristics of the feature are sensitive to the values of cosmological parameters, in particular to the total and the baryon densities.The cosmic microwave background (CMB) radiation provides the best probe today of the early universe and a number of fundamental astrophysical constants [1][2][3][4][5][6][7]. Dynamical evolution of primordial perturbations manifests itself in the form of "acoustic peaks" in the CMB temperature angular power spectrum C l . After over a decade of studies, the physical content of the peaks has become qualitatively understood [8][9][10][11]. Nevertheless, one has to rely on standard numerical codes [12,13] to establish a quantitative connection between the CMB anisotropy pattern and cosmological parameters. This connection is particularly important now that highprecision CMB measurements have become reality.In this letter we show that previously unnoticed interesting phenomena are unraveled by considering the radiation-matter dynamics in position rather than Fourier space.Traditional CMB anisotropy analyses begin with the Fourier expansion of spatial inhomogeneities and studying time evolution of individual Fourier modes [16][17][18]. The position space approach, while offering a formally equivalent description, differs in its physical interpretation, associated calculational methods, and its implications for data analysis. As we show here, a new side of CMB physics is revealed in position space. One goal of our work is a deeper physical understanding of both the CMB power spectrum C l and the angular correlation function C(θ). We demonstrate that acoustic evolution of perturbations before recombination, as viewed in position space, produces sharp features in C(θ). These signatures not only provide a simple interpretation of the acoustic peaks, but they may also enable fast and accurate extraction of cosmological parameters directly from C(θ).As a simplified model for the essential physical processes, we consider the evolution of potential and density perturbations in gravitationally interacting photonbaryon and cold dark matter fluids. We use the conformal Newtonian gauge [18,19] to describe gravity and assume adiabatic (isentropic) primordial fluctuations as preferred by the present data [14,15] and the simplest inflationary models. The photons are assumed to be tightly coupled to electrons and baryons by Thomson scattering until recombination at redshift z rec ∼ 1100. Neutrinos are treated like photons. The disre...

Physics of the cosmic microwave background anisotropy

“…The discussion here goes beyond the Sachs-Wolfe treatment to include a discussion of the dominant contributions to anisotropy on small angular scales. The presentation given here is based in part on the PhD thesis of Sergei Bashinsky (Bashinsky 2001) and Bashinsky & Bertschinger (2001, 2002. More elementary treat ments of CMB anisotropy are given by Chapter 18 of Peacock and online by Wayne Hu at http://background.uchicago.edu/).…”

Large-scale structure observations

“…When the photons and gas decouple, a spherical shell of baryons is left around a central concentration of dark matter. As the perturbation evolves through gravity, the density profiles of the baryons and dark matter grow together, and the final perturbation profile is left with a small increase in density in a spherical shell at a radial location corresponding to the sound horizon at the end of the Compton drag epoch r d : this is the radius of the spherical shell (Bashinsky and Bertschinger 2001;Bashinsky and Bertschinger 2002). In order to understand the effect of this process on a field of perturbations, one can imagine many of these superimposed "waves" propagating simultaneously, resulting in a slight preference for perturbations separated by the scale of the sound horizon (perturbations at the original location, and at the spherical shell).…”

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

“…The most important difference between the two expansions can be grasped by comparing Eqs. (15), (32) and (34). Basically, when going from the Fourier basis to the Fourier-Bessel basis, the angular dependence is still expressed in terms of spherical harmonics, but the radial coordinate is expressed by a sum, not an integral:…”

“…These expressions should still be coupled to the Einstein, continuity and Euler equations through the temperature dipole and quadrupole, but those are local equations so their causal structure is trivial (nevertheless, the Green's function in position space for the cosmological matter and metric perturbations can also reveal very interesting features [34,35]. )…”

“…In particular, our results show that the transfer functions of [29,30] have an invariant (or geometrical) piece which is described by our spacetime window functions, so that the transfer functions can be obtained by integration of these window functions over time with some visibility function. Another work close in spirit to ours was done by Bashinsky and Bertshinger [34,35], who calculated the Green's function for the evolution of linear cosmological perturbations in position space but did not compute the Green's functions for the anisotropies.…”