2. Formalism for an individual moving structure
One of the first studies of the photon-gravitational potential well interactions is related to the Sachs-Wolfe effect (Sachs & Wolfe 1967). At the recombination time () the photons and matter decouple while they are in potential wells; the photons are redshifted when they leave the potential wells. This generates the large angular scale temperature fluctuations.
Other authors have investigated the effect of time varying potentials on the CMB photons after the recombination, namely the Rees-Sciama effect (Rees & Sciama 1968). If the potential well crossed by the photons evolves between the time they enter and their exit, the extra-time delay they suffer changes the temperature of the CMB and induces an additional anisotropy. The variation of the potential well can have an "intrinsic" or a "kinetic" origin. The first case describes the evolution with respect to the background density distribution. The second case is related to the bulk motion of a gravitational potential well across the line of sight which mimics a time variation of the potential. Photons crossing the leading edge of a structure will be redshifted because of the increasing depth of the potential well during their crossing time; while photons crossing the trailing edge of the same structure are blueshifted. This results in a characteristic spatial signature for the induced anisotropy: a hot-cold temperature spot.
The specific effect of a moving cluster across the sky was first studied by Birkinshaw & Gull (1983) (correction to this paper was made in Birkinshaw 1989) and it was invoked as a method to measure the transverse velocity of massive clusters of galaxies. These authors found that the transverse motion of a cluster across the line of sight induces a frequency shift given by:
Here, is the peculiar velocity in units of the speed of light (), is the Lorenz factor (), and are respectively the angle between the peculiar velocity and the line of sight of the observer and the azimuthal angle in the plane of the sky, and is the deflection angle due to the gravitational lensing by the cluster at a distance equal to the impact parameter . This frequency shift induces a brightness variation which in turn can be expressed as a secondary temperature fluctuation . In their paper, Birkinshaw & Gull derived an expression for in the Rayleigh-Jeans regime, with some specific assumptions on the gravitational potential well associated with the cluster. They assumed that the matter in the galaxy cluster was homogeneously distributed in an isothermal sphere of radius R, where R is the characteristic scale of the cluster.
In our paper, we basically follow the same formalism as Birkinshaw & Gull's using the corrected expression from Birkinshaw 1989. We compute the gravitational deflection angle at the impact parameter , the corresponding frequency shift and then derive the associated temperature fluctuation.
The main difference between our approach in this section and the previous work concerns the physical hypothesis that we adopt to describe the distribution of matter in the structures. In fact, in order to derive the deflection angle, we find the homogeneous isothermal distribution a too simple and rather unrealistic hypothesis and choose another more realistic description. For the structures such as those we are interested in (clusters down to small groups), almost all the mass is "made" of dark matter. In order to study the gravitational lensing of a structure properly, one has to model the gravitational potential well using the best possible knowledge for the dark matter distribution. The corrections, due to the more accurate profile distribution that we introduce, will not alter the maximum amplitude of an individual moving lens effect since it is associated with the central part of the lens. However, when dealing with some average signal coming from these secondary anisotropies, the contribution from the outskirts of the structures appears important and thus a detailed model of the matter profile is needed.
In view of the numerous recent studies on the formation of dark matter halos, which are the formation sites for the individual structures such as clusters of galaxies, we now have a rather precise idea of their formation and density profiles. Specifically, the results of Navarro, Frenk & White (1996, 1997) are particularly important. In fact, these authors have used N-body simulations to investigate the structure of dark matter halos in hierarchical cosmogonies; their results put stringent constraints on the dark matter profiles. Over about four orders of magnitudes in mass (ranging from the masses of dwarf galaxy halos to those of rich clusters of galaxies), they found that the density profiles can be fitted over two decades in radius by a "universal" law (hereafter NFW profile) which seems to be the best description of the structure of dark matter halos (Huss, Jain & Steinmetz 1997). The NFW profile is given by:
where is the scale radius of the halo, its characteristic overdensity, is the critical density of the universe and c is a dimensionless parameter called the concentration. The radius is the radius of the sphere where the mean density is . This is what we refer to as a virialised object of mass .
In addition to the fact that the shape is independent of the halo mass over a wide range, the NFW profile is also independent of the cosmological model. The cosmological model intervenes essentially in the formation epoch of the dark matter halo and therefore in the parameters of the profile, namely c, and .
Using the density profile, one can compute the deflection angle at the impact parameter which gives the shape of the pattern and the amplitude of the induced secondary anisotropy. In our work, we compute the deflection angle following the formalism of Blandford & Kochanek (1987), which is given by the expression:
here, the integral is performed over the length element dl along the line of sight. and are respectively the distances between lens and source and the observer and source. In the redshift range of the considered structures (), the distance ratios range between 1 and 0.68 for the standard CDM model, between 1 and 0.53 for the open CDM and between 1 and 0.74 for the lambda CDM model. These cosmological models will be defined in the next section. In Eq. 3 is the position of the structure and is the associated gravitational potential. In order to get an analytic expression of the deflection angle and hence of the anisotropy, we used a density profile which gives a good approximation to the NFW density profile (Eq. 2), in the central part of the structure. This density profile is given by:
The fitted profile leads to a diverging mass at large radii and we therefore introduce a cut-off radius to the integral. This cut-off should correspond to some physical size of the structure. With regard to the different values of the concentration c, we set which is in most cases equivalent to ,i.e., close to the virial radius. The integral giving the deflection angle is performed on the interval . For , our fit gives a mass which is about 20% lower than the mass derived from NFW profile. This difference is larger for larger , and for we find that the mass is about 33% lower. However, the larger radii the temperature fluctuations are at the level. On the other hand, the Hernquist (1990) profile is also in agreement with the results of N-body simulations. Indeed, both NFW and Hernquist profiles have a similar dependence in the central part of the structure but differ at large radii where the NFW profile is proportional to and the Hernquist profile varies as . However, the amplitude of the anisotropy at large radii is very small and the results that we obtain does are not sensitive to the cut-off.
Given the peculiar velocity of the structure and its density profile, we can calculate the deflection angle (Eq. 3). Then one can determine the relative variation in frequency, , using Eq. 1and thus evaluate the secondary distortion induced by a specific structure moving across the sky. We find that individual massive structures (rich galaxy clusters) produce anisotropies ranging between a few to ; but within a wider range of masses the amplitudes are smaller and these values are only upper limits for the moving lens effect.
© European Southern Observatory (ESO) 1998
Online publication: May 15, 1998