Annu. Rev. Astron. Astrophys. 1999. 37: 127-189
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The measurement of CMB fluctuations is a major goal for cosmology in the next decade (see White et al 1994, and references therein). The shape of the power spectrum over the whole spectral range contains a huge amount of information which permits one to constrain the cosmological scenario with an incredibly high accuracy. However, the reliability of the interpretation of the features visible on the spectrum requires a complete and detailed understanding of all the physical mechanisms responsible for its final shape. Gravitational lensing induced by foreground systems along the line of sight may play a role, so it is important to predict in advance whether it can modify the signal from the CMB and, if so, what the expected amplitudes of the effects are.

Because surface brightness is conserved by the gravitational lensing effect (Etherington 1933), only fluctuations of the CMB temperature maps can be affected by lensing. However, even for strong lenses, no significant modification of the power spectrum is expected on large scales (Blanchard & Schneider 1987) and therefore there is no hope of detecting positive signals of the coupling between CMB and gravitational lensing in the Cosmological Background Explorer (COBE) maps. Nevertheless, since the COBE-Differential Microvawe Radiometer (DMR) experiment has demonstrated that fluctuations exist (Smooth et al 1992), the study of the lensing effect on smaller scales than COBE resolution is potentially interesting and has some advantages with respect to weak lensing on distant galaxies. First, contrary to lensed galaxies, the redshift of the source, namely the last scattering surface, is well known and spreads over a very small redshift range. Second, with the on-going and the coming of high-resolution ground-based and balloon observations as well as the two survey satellites MAP and Planck-Surveyor, observation of low-amplitude temperature distortions on small scales will become possible and will permit investigation of possible lensing effects.

Early theoretical expectations from Blanchard & Schneider (1987) or Cole & Efstathiou (1989) show that the shape of the small-scale temperature fluctuations can be modified by lensing effects; in particular they can redistribute the power in the power spectrum. In contrast, the amplitude of the temperature anisotropy on medium and small scales has been a matter of debates during the last decade (see the review by Blandford & Narayan 1992, and more recently Fukushige & Makino 1994, Fukushige et al 1994, Cayón et al 1993a, b, 1994). The conclusions of these works showed strong discrepancies, depending on the assumptions used to explore the deflection of photons and to model inhomogenous universes. Furthermore, the expectation values also depend on the cosmological models. Indeed, the most recent critical studies show that the effect of lensing on large scales is negligible (Seljak 1996, Martínez-González et al 1997). In particular, the non-linear evolution of the power spectrum does not significantly increase the amplitude on these scales. On the other hand, the gravitational lensing effect reduces the power spectrum on small scales, and eventually can smooth out acoustic peaks on scales below l approx 2000 (Seljak 1996). Martínez-González et al (1997) obtained conclusions similar to those of Seljak - that is, the contribution of lensing is small but not negligible and should be taken into account in the detailed analysis of future CMB maps. Furthermore, the transfer of power from large to small scales induces an important increase of power in the damping tail, which results in a decrease of very small scale amplitudes at a smaller rate than expected without lensing (Metcalf & Silk 1997, 1998). According to Metcalf & Silk (1997), 30% of the additional power at l = 3000 comes from l < 1000 scales, and 8% from l < 500 in the case of a sigma8 = 0.6, h = 0.6 model.

Zaldarriaga & Seljak (1998) pointed out that gravitational lensing not only smoothes the temperature anisotropy, but can also change the polarization. The polarization spectra are more sensitive to gravitational lensing effects than the power spectrum of the temperature because the acoustic oscillations of polarization spectra have sharper oscillations and can be smoothed out more efficiently by lensing than temperature fluctuations. The effect is small but can reach amplitudes of approximately 10% for l < 1000 scales. More remarkably, because of the coupling between E-type and B-type polarizations (Seljak 1997b), gravitational lensing can generate low amplitude B-type polarization, even if none is predicted from primary fluctuations (for instance, for scalar perturbations).

Because the signal is weak and only concerns the small scales, temperature and polarization fluctuations induced by gravitational lensing will be difficult to measure with high accuracy and seems a hopeless task before the Planck-Surveyor mission. It is therefore valuable to explore alternatives which could provide better or complementary information which couples lensing and CMB. An interesting idea is to analyze the non-Gaussian features induced by the displacement fields generated by gravitational lensing on the CMB maps (Bernardeau 1997, 1998b). As for weak lensing on distant galaxies, the CMB temperature map can be sheared and magnified. The resulting distortion patterns are direct signatures of the coupling between the CMB and the foreground lenses. Bernardeau (1998b) argued that the distortion map produced by lensing can be decoupled from other fluctuation patterns because it generates similar magnification and deformation on close temperature patches, which therefore can be correlated. He also explored the consequences of the non-Gaussian signal on the four-point correlation function. Unfortunately, the signal is very small, and it is even not clear on which scale the signal is highest, in particular because the non-linear evolution of the power spectrum was not considered by Bernardeau. The weakness of the signal and the fact that the four-point correlation function could be contaminated by other non-Gaussian features are strong limitations of Bernardeau's suggestion. Therefore, Bernardeau (1998b) preferred to focus on the modification of the ellipticity distribution function of the temperature patches induced by lensing. From his simulated lensed maps, a clear change of the topology of the temperature maps is visible: the shapes of the structures are modified and their contours look sharper than for the unlensed maps. However, the signal is still marginally detectable on a 10°×10° map, even with Planck-Surveyor.

From these investigations it is clear that weak lensing on the CMB has small effects on the spectrum of the temperature and polarization power spectra and on the non-Gaussianity of the CMB temperature maps. However, with typical amplitude of 1% to 10% percent they can be detected with future missions, so they must be taken into account for detailed investigations of the CMB anisotropy on small scales. This is an important prediction since the detection of gravitational lensing perturbations of the CMB will be possible with Planck-Surveyor. Its high sensitivity and spatial resolution are sufficient to permit one to break the geometrical degeneracy expected from linear theory, and to disentangle different (Omega, lambda) common models (Metcalf & Silk 1998, Stompor & Efstathiou 1999). It is worth noting that these analyses can be used jointly with the weak lensing maps of large-scale structures on background galaxies which will also provide (Omega, lambda) with a very good accuracy.

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