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Astron. Astrophys. 334, 409-419 (1998)

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1. Introduction

During the next decade, several experiments are planned to observe the Cosmic Microwave Background (CMB) and measure its temperature fluctuations (Planck surveyor, Map, Boomerang etc.). Their challenge is to measure the small scales anisotropies of the CMB (a few arcminutes up to ten degrees scale) with sensitivities better by a factor 10 than the COBE satellite (Smoot et al. 1992). These high sensitivity and resolution measurements will tightly constrain the value of the main cosmological parameters (Kamionkowski et al. 1994). However, the constraints can only be set if we are able to effectively measure the primary temperature fluctuations. These fluctuations, present at recombination, give an insight into the early universe since they are directly related to the initial density perturbations which are the progenitors to the cosmic structures (galaxies and galaxies clusters) in the present universe; but which are first and foremost the relics of the very early initial conditions of the universe.

Between recombination and the present time, the CMB photons could have undergone various interactions with the matter and structures present along their lines of sight. Some of these interactions can induce additional temperature fluctuations called, secondary anisotropies because they are generated after the recombination. Along a line of sight, one measures temperature fluctuations which are the superposition of the primary and secondary anisotropies. As a result, and in the context of the future CMB experiments, accurate analysis of the data will be needed in order to account for the foreground contributions due to the secondary fluctuations. Photon-matter interactions between recombination and the present time are due to the presence of ionised matter or to variations of the gravitational potential wells along the lines of sight.

The CMB photons interact with the ionised matter mainly through Compton interactions. In fact, after recombination the universe could have been re-ionised globally or locally. Global early re-ionisation has been widely studied (see Dodelson & Jubas 1995 for a recent review and references therein). Its main effect is to either smooth or wipe out some of the primary anisotropies; but the interactions of the photons with the matter in a fully ionised universe can also give rise to secondary anisotropies through the Vishniac effect (Vishniac 1987). This second order effect has maximum amplitudes for a very early re-ionisation. The case of a late inhomogeneous re-ionisation and its imprints on the CMB fluctuations has been investigated (Aghanim et al. 1996) and found to be rather important. In this case, the secondary anisotropies are due to the bulk motion of ionised clouds with respect to the CMB frame. When the re-ionisation is localised in hot ionised intra-cluster media the photons interact with the free electrons. The inverse Compton scattering between photons and electrons leads to the so-called Sunyaev-Zel'dovich (hereafter SZ) effect (Sunyaev & Zel'dovich 1972, 1980). The Compton distortion due to the motion of the electrons in the gas is called the thermal SZ effect. The kinetic SZ effect is a Doppler distortion due to the peculiar bulk motion of the cluster with respect to the Hubble flow. The SZ thermal effect has the unique property of depressing the CMB brightness in the Rayleigh-Jeans region and increasing its brightness above a frequency of about 219 GHz. This frequency dependence makes it rather easy to observe and separate from the kinetic SZ effect. In fact, the latter has a black body spectrum which makes the spectral confusion between kinetic SZ and primary fluctuations a serious problem. The SZ effect has been widely studied for individual clusters and for populations of clusters. For full reviews on the subject we refer the reader to two major articles: Rephaeli 1995 and Birkinshaw 1997. These investigations have clearly shown that the SZ effect in clusters of galaxies provides a powerful tool for cosmology through measurements of the Hubble constant, the radial peculiar velocity of clusters and consequently the large scale velocity fields.

Besides the interactions with the ionised matter, some secondary effects arise when the CMB photons traverse a varying gravitational potential well. In fact, if the gravitational potential well crossed by the photons evolves between the time they enter the well and the time they leave it, the delay between entrance and exit is equivalent to a shift in frequency, which induces a temperature anisotropy on the CMB. This effect was first studied by Rees & Sciama (1968) for a potential well growing under its own gravity. Numerous authors have investigated the potential variations due to collapsing objects and their effect on the CMB (Kaiser 1982, Nottale 1984, Martinez-González, Sanz & Silk 1990, Seljak 1996). Similarly, a gravitational potential well moving across the line of sight is equivalent to a varying potential and will thus imprint secondary fluctuations on the CMB. This effect was first studied for one cluster of galaxies by Birkinshaw & Gull (1983) (Sect. 2). Kaiser & Stebbins (1984) and Bouchet, Bennett & Stebbins (1988) investigated a similar effect for moving cosmic strings. Recent work (Tuluie & Laguna 1995, Tuluie, Laguna & Anninos 1996) based on N-body simulations has pointed out this effect in a study of the effect of varying potential on rather large angular scales ([FORMULA]). A discussion of some of these results and a comparison with ours will follow in the next sections.

In this paper, following the formalism of Birkinshaw & Gull (1983) and Birkinshaw (1989), we investigate the contribution of secondary anisotropies due to a population of collapsed objects moving across the line of sight, these objects range from small groups to rich clusters in scale ([FORMULA] to [FORMULA] [FORMULA]). In Sect. 2., we first study in detail the case of a unique collapsed structure. We use a structure model to compute in particular the deflection angle and derive the spatial signature of the moving lens effect. We then account (Sect. 3.) for the contribution, to the primordial cosmological signal, of the whole population of collapsed objects using predicted counts and we simulate maps of these secondary anisotropies. In Sect. 4., we analyse the simulated maps and present our results. We give our conclusions in Sect. 5.

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© European Southern Observatory (ESO) 1998

Online publication: May 15, 1998