Professor, Department of Astronomy and Astrophysics
University of Chicago

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Gravitational Lensing

The gravitational potentials of large-scale structure also lens the CMB photons. Since lensing conserves surface brightness, it only affects anisotropies and hence is second order in perturbation theory [Blanchard & Schneider, 1987]. The photons are deflected according to the angular gradient of the potential projected along the line of sight with a weighting of $2 (D_* - D)/(D_* D)$. Again the cancellation of parallel modes implies that it is mainly the large-scale potentials that are responsible for deflections. Specifically, the angular gradient of the projected potential peaks at a multipole $\ell \sim 60$ corresponding to scales of a $k \sim$ few $10^{-2}$ Mpc$^{-1}$ [Hu, 2000b]. The deflections are therefore coherent below the degree scale. The coherence of the deflection should not be confused with its rms value which in the model of Plate 1 has a value of a few arcminutes.

This large coherence and small amplitude ensures that linear theory in the potential is sufficient to describe the main effects of lensing. Since lensing is a one-to-one mapping of the source and image planes it simply distorts the images formed from the acoustic oscillations in accord with the deflection angle. This warping naturally also distorts the mapping of physical scales in the acoustic peaks to angular scales §3.8 and hence smooths features in the temperature and polarization [Seljak, 1996a]. The smoothing scale is the coherence scale of the deflection angle $\Delta \ell \approx 60$ and is sufficiently wide to alter the acoustic peaks with $\Delta \ell \sim 300$. The contributions, shown in Plate 5a are therefore negative (dashed) on scales corresponding to the peaks.

For the polarization, the remapping not only smooths the acoustic power spectrum but actually generates $B$-mode polarization (see Plate 1 and [Zaldarriaga & Seljak, 1998]). Remapping by the lenses preserves the orientation of the polarization but warps its spatial distribution in a Gaussian random fashion and hence does not preserve the symmetry of the original $E$-mode. The $B$-modes from lensing sets a detection threshold for gravitational waves for a finite patch of sky [Hu, 2001b].

Gravitational lensing also generates a small amount of power in the anisotropies on its own but this is only noticable beyond the damping tail where diffusion has destroyed the primary anisotropies (see Plate 5a). On these small scales, the anisotropy of the CMB is approximately a pure gradient on the sky and the inhomogeneous distribution of lenses introduces ripples in the gradient on the scale of the lenses [Seljak & Zaldarriaga, 2000]. In fact the moving halo effect of §4.2.2 can be described as the gravitational lensing of the dipole anisotropy due to the peculiar motion of the halo [Birkinshaw & Gull, 1983].

Because the lensed CMB distribution is not linear in the fluctuations, it is not completely described by changes in the power spectrum. Much of the recent work in the literature has been devoted to utilizing the non-Gaussianity to isolate lensing effects [Bernardeau, 1997,Bernardeau, 1998,Zaldarriaga & Seljak, 1999,Zaldarriaga, 2000] and their cross-correlation with the ISW effect [Goldberg & Spergel, 1999,Seljak & Zaldarriaga, 1999]. In particular, there is a quadratic combination of the anisotropy data that optimally reconstructs the projected dark matter potentials for use in this cross-correlation [Hu, 2001c]. The cross correlation is especially important in that in a flat universe it is a direct indication of dark energy and can be used to study the properties of the dark energy beyond a simple equation of state [Hu, 2001b].

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Wayne Hu 2001-10-15