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

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3. Generalisation to a sample of structures

Future CMB (space and balloon borne) experiments will measure the temperature fluctuations with very high accuracy ([FORMULA]) at small angular scales. In our attempts to foresee what the CMB maps would look like and what would be the spurious contributions due to the various astrophysical foregrounds, we investigate the generalisation of the computations made above to a sample of structures. This is done in order to address the questions of the cumulative effect and contamination to the CMB.

Some work has already been done by Tuluie & Laguna 1995 and Tuluie, Laguna & Anninos 1996 who pointed out the moving lens effect in their study of the varying potential effects on the CMB. In their study, they used N-body simulations to evolve the matter inhomogeneities, from the decoupling time until the present, in which they propagated CMB photons. They have estimated the anisotropies generated by three sources of time-variations of the potential: intrinsic changes in the gravitational potential, decaying potential effect from the evolution of gravitational potential in [FORMULA] models, and peculiar bulk motions of the structures across the sky. They evaluated the contribution of the latter effect for rather large angular scales ([FORMULA]) due to the lack of numerical resolution (about [FORMULA] Mpc) and gave estimates of the power spectrum of these effects.

With another approach, we make a similar analysis in the case of the moving lens effect extended to angular scales down to a few tens of arcseconds. We also simulate attempts at the detection and subtraction of the moving lens effect. Our approach is quite different from that of Tuluie, Laguna & Anninos, in that it is semi-empirical and apply the formalism developed for an individual structure (Sect. 2.) to each object from a sample of structures. The predicted number of objects in the sample being derived from the Press-Schechter formalism for the structure formation (Press & Schechter 1974).

3.1. Predicted population of collapsed objects

An estimate of the cumulative effect of the moving lenses requires a knowledge of the number of objects of a given mass that will contribute to the total effect at a given epoch. We assume that this number is accurately predicted by the abundance of collapsed dark matter halos as a function of their masses and redshifts, as derived using the Press-Schechter formalism. This approach was used in a previous paper (Aghanim et al. 1997) which predicted the SZ contribution to the CMB signal in a standard CDM model. In addition to the "traditional" standard Cold Dark Matter (CDM) model ([FORMULA]), in this paper we also address the question of a generalised moving lens effect in other cosmological models. We extend the Press-Schechter formalism to an open CDM model (OCDM) with no cosmological constant ([FORMULA]), and also a flat universe with a non zero cosmological constant ([FORMULA] CDM model) ([FORMULA] and [FORMULA]). Here [FORMULA] is the density parameter, [FORMULA] is the cosmological constant given in units of [FORMULA] and [FORMULA] is the Hubble constant. We take [FORMULA] km/s/Mpc, and assume [FORMULA] throughout the paper.

In any case, the general analytic expression for the number density of spherical collapsed halos in the mass range [FORMULA] can be written as (Lacey & Cole 1993):



where [FORMULA] is the mean background density at redshift z and [FORMULA] is the overdensity of a linearly evolving structure. The mass variance [FORMULA] of the fluctuation spectrum, filtered on mass scale M, is related to the linear power spectrum of the initial density fluctuations [FORMULA] through:


where W is the Fourier transform of the window function over which the variance is smoothed (Peebles 1980) and R is the scale associated with mass M. In the assumption of a scale-free initial power spectrum with spectral index n, the variance on mass scale M can be expressed in terms of [FORMULA], the rms density fluctuation in sphere of [FORMULA] Mpc size. The relationship between these two quantities is given by (Mathiesen & Evrard 1997):


with [FORMULA]. It has been shown that [FORMULA] varies with the cosmological model and in particular with the density parameter [FORMULA]. A general empirical fitting function ([FORMULA]) was derived from a power spectrum normalisation to the cluster abundance with a rather good agreement in the values of the parameters A and B (White, Efstathiou & Frenk 1993, Eke, Cole & Frenk 1996, Viana & Liddle 1996). In our work, we use the "best fitting values" from Viana & Liddle (1996) which are [FORMULA] and [FORMULA] for an open CDM universe ([FORMULA] and [FORMULA]) or [FORMULA] for a flat universe with a non zero cosmological constant ([FORMULA]). We use [FORMULA] for the spectral index in the cluster mass regime which is the theoretically predicted value. Some local constraints on the temperature abundance of clusters favour [FORMULA] (Henry & Arnaud 1991, Oukbir, Bartlett & Blanchard 1997) but we did not investigate this case.

3.2. Peculiar velocities

On the scale of clusters of galaxies, typically 8 [FORMULA] Mpc, one can assume that the density fluctuations are in the linear regime. Therefore the fluctuations are closely related to the initial conditions from which the structures arise. In fact, in the assumption of an isotropic Gaussian distribution of the initial density perturbations, the initial power spectrum [FORMULA] gives a complete description of the velocity field through the three-dimensional rms velocity ([FORMULA]) predicted by the linear gravitational instability for an irrotational field at a given scale R (Peebles 1993). This velocity is given by:


where [FORMULA] is the expansion parameter, the Hubble constant H and the density parameter [FORMULA] vary with time (Caroll, Press & Turner 1992). The function [FORMULA] is accurately approximated by [FORMULA] (Peebles 1980) even if there is a non zero cosmological constant (Lahav et al. 1991). Furthermore, under the assumptions of linear regime and Gaussian distribution of the density fluctuations, the structures move with respect to the global Hubble flow with peculiar velocities following a Gaussian distribution [FORMULA] which is fully described by [FORMULA]. This prediction is in agreement with numerical simulations (Bahcall et al. 1994, Moscardini et al. 1996).

The present observational status of peculiar cluster velocities puts few constraints on the cosmological models. Results from the Hudson (1994) sample using [FORMULA] - [FORMULA] and IRTF distance estimators give respectively [FORMULA] and [FORMULA] km/s, a composite sample gives [FORMULA] km/s (Moscardini et al 1996). Giovanelli's (1996) sample gives a smaller value, [FORMULA] km/s.

In our paper we compute the three-dimensional rms peculiar velocity on scale 8 [FORMULA] Mpc (typical virial radius of a galaxy cluster) using Eq. 6for the three cosmological models. This is because large scale velocities are mostly sensitive to long wavelength density fluctuations. This smoothing allows us to get rid of the nonlinear effects on small scales but it also tends to underestimate the peculiar velocities of the smallest objects that we are interested in. Nevertheless, with regard to the rather important dispersion in the observational values (320 [FORMULA] 780 km/s), we use the predicted theoretical values, which range between 400 and 500 km/s, and are hence in general agreement with the observational data.

3.3. Simulations

For each cosmological model, we generate a simulated map of the moving lens effect in order to analyse the contribution to the signal in terms of temperature fluctuations. The simulations are essentially based on the studies of Aghanim et al. (1997). In the following, we describe briefly the main hypothesis that we make in simulating the maps of the temperature fluctuations induced by the moving lens effect associated with small groups and clusters of galaxies ([FORMULA] and [FORMULA]). The predicted number of massive objects is derived from a distribution of sources using the Press-Schechter formalism normalised (Viana & Liddle 1996) using the X-ray temperature distribution function derived from Henry & Arnaud (1991) data. This normalisation has also been used by Mathiesen & Evrard (1997) for the ROSAT Brightest Clusters Sample compiled by Ebeling et al. (1997). The position and direction of motion of each object are random. Their peculiar velocities are also random within an assumed Gaussian distribution. Here again, the correlations were neglected because the effect is maximum very close to the central part of the structure (about 100 kpc) whereas the correlation length is between 5 and 20 Mpc (Bahcall 1988). The final maps account for the cumulative effect of the moving lenses with redshifts lower than [FORMULA]. We refer the reader to Aghanim et al. (1997) for a detailed description of the simulation.

In this paper, some changes and improvements have been made to our previous study (Aghanim et al. 1997). In this paper, the predicted source counts (Eq. 5, Sect. 3.1) are in agreement with more recent data. They are also adapted to the various cosmological models that we have assumed. The standard deviation of the peculiar velocity distribution is computed using Eq.  6and is in reasonable agreement with the data. The advantage of using this equation is that the variations with time and cosmology are directly handled in the expression. As we pointed out in Sect. 2, the secondary effects we study here are associated with the whole mass of the structure, not only the gas mass. Therefore, the gas part of structures are modelled using the [FORMULA] -profile (as in the previous case) to simulate the SZ effect. Whereas the density profile (Eq. 4) is used to simulate the potential well of the moving lens effect. We note that the results of the N-body simulations of Navarro, Frenk & White (1996) are consistent with the assumption of an intra-cluster isothermal gas in hydrostatic equilibrium with a NFW halo.

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Online publication: May 15, 1998