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 and peculiar velocities of
imply signals in the
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 K level peaking at
few
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 ( ) 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
few
K at
(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
(crudely estimated, following [Gruzinov & Hu, 1998], in Plate 5b).