Astron. Astrophys. 323, 305-311 (1997)

1. Introduction

The origin and development of structure in the universe needs to be probed by means of observations of distant sources. Not only the direct properties of those sources, however, but also the characteristics induced in the radiation propagating through the gravitational inhomogeneities of the structure provide clues to the nature and formation of structure. Within linear theory, i.e. when the deviations from homogeneity are still perturbations, there are simple and intimate relations between the important cosmological descriptors of the matter, such as the large scale velocity flow field or the energy density fluctuations, and of radiation propagation characteristics such as temperature anisotropies (redshifts) and gravitational lensing deflections (momentum changes). These interrelationships constrain theories of structure formation.

This paper concentrates on the statistical effects of the inhomogeneities on observations, developing an analytic approach that requires only that the physical processes behind the observable and lensing quantities be gaussian in nature. This avoids sensitivity of results to a specific geometry or density perturbation model and is both physically relevant and surprisingly powerful. Although here we are primarily concerned with temperature anisotropies in the cosmic microwave background radiation (CMB), the method is easily generalised to investigations of the velocity flow field or density perturbations.

Before primordial CMB anisotropies were first detected by the Cosmic Background Explorer satellite (COBE; Smoot et al. 1992) there was much discussion in the literature concerning the effects of inhomogeneities and their gravitational lensing on the anisotropies (Blandford & Narayan 1992 provide a review with references). More recently Fukushige, Makino, and Ebisuzaki (1994) investigated this by calculating scattering with an N-body code, and Tomita (1996) considered the effects of superhorizon scale inhomogeneities. For the statistical distribution of the lensing deflections, in particular the correlation of neighboring lines of sight, crucial to the key property of geodesic deviation, various assumption have been made, ranging from a diffusion approximation to nearest neighbor correlations.

No previous work, however, dealt with the case where the deflections are not superimposed independently on the temperature fluctuations but are physically correlated with them. For example, consider a hot spot in the CMB. This corresponds to a region where the primordial gravitational potential deviates strongly from the mean but that potential also determines the strength of the lensing caused by the primordial density field from that region. Thus one can imagine that extremes of the temperature field are preferentially strongly lensed relative to the milder deviations, leading to a significant distortion of the intrinsic temperature correlation function.

This paper investigates the properties of cosmologically important gaussian fields including the possibility of such a cross correlation. The joint probability formalism allows for concise derivations of familiar results, e.g. beam smearing, from gravitational lensing while showing new effects from the cross correlation. By estimating the magnitude of these on anisotropy observations we can probe large scale structure in the universe by placing constraints on the density power spectrum . The conditional probability formalism also points out ways to test the underlying gaussian nature of the primordial cosmological perturbations.

In Sect. 2 we review the mathematical basis and derive properties of joint and conditional probability distributions for the variables entering the problem - the relative deflections of the null geodesics and the intrinsic source sky characteristics. These are combined in Sect. 3 to form expressions for the observable correlation functions, especially under the condition of coherence between the source and propagation conditions. Section 4 relates the mathematical results to the underlying physics in terms of the density fluctuation power spectrum and Sect. 5 presents quantitatively the effects on CMB anisotropy measurements.

© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998

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