Professor, Department of Astronomy and Astrophysics
University of Chicago

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Modulated Doppler Effects

The Doppler effect can survive cancellation if the optical depth has modulations in a direction orthogonal to the bulk velocity. This modulation can be the result of either density or ionization fluctuations in the gas. Examples of the former include the effect in clusters, and linear as well as non-linear large-scale structures.

CLUSTER MODULATION: The strongly non-linear modulation provided by the presence of a galaxy cluster and its associated gas leads to the kinetic Sunyaev-Zel'dovich effect. Cluster optical depths on order $10^{-2}$ and peculiar velocities of $10^{-3}$ imply signals in the $10^{-5}$ regime in individual arcminute-scale clusters, which are of course rare objects. While this signal is reasonably large, it is generally dwarfed by the thermal Sunyaev-Zel'dovich effect (see §4.3.5) and has yet to be detected with high significance (see [Carlstrom et al, 2001] and references therein). The kinetic Sunyaev-Zel'dovich effect has negligible impact on the power spectrum of anisotropies due to the rarity of clusters and can be included as part of the more general density modulation.

LINEAR MODULATION: At the opposite extreme, linear density fluctuations modulate the optical depth and give rise to a Doppler effect as pointed out by [Ostriker & Vishniac, 1986] and calculated by [Vishniac, 1987] (see also [Efstathiou & Bond, 1987]). The result is a signal at the $\mu$K level peaking at $\ell \sim $ few $\times 10^3$ that increases roughly logarithmically with the reionization redshift (see Plate 5b).

GENERAL DENSITY MODULATION: Both the cluster and linear modulations are limiting cases of the more general effect of density modulation by the large scale structure of the Universe. For the low reionization redshifts currently expected ( $z_{\rm ri} \approx 6-7$) most of the effect comes neither from clusters nor the linear regime but intermediate scale dark matter halos. An upper limit to the total effect can be obtained by assuming the gas traces the dark matter [Hu, 2000a] and implies signals on the order of $\Delta_T \sim $ few $\mu$K at $\ell > 10^3$ (see Plate 5b). Based on simulations, this assumption should hold in the outer profiles of halos [Pearce et al, 2001,Lewis et al, 2000] but gas pressure will tend to smooth out the distribution in the cores of halos and reduce small scale contributions. In the absence of substantial cooling and star formation, these net effects can be modeled under the assumption of hydrostatic equilibrium [Komatsu & Seljak, 2001] in the halos and included in a halo approach to the gas distribution [Cooray, 2001].

IONIZATION MODULATION: Finally, optical depth modulation can also come from variations in the ionization fraction [Aghanim et al, 1996,Gruzinov & Hu, 1998,Knox et al, 1998]. Predictions for this effect are the most uncertain as it involves both the formation of the first ionizing objects and the subsequent radiative transfer of the ionizing radiation [Bruscoli et al, 2000,Benson et al, 2001]. It is however unlikely to dominate the density modulated effect except perhaps at very high multipoles $\ell \sim 10^4$ (crudely estimated, following [Gruzinov & Hu, 1998], in Plate 5b).

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Wayne Hu 2001-10-15