Kinematic Sunyaev-Zeldovich Effect

Peculiar velocities of the hot intra-cluster gas lead to a Doppler shift of the scattered photons which is proportional to the product of the radial peculiar velocity $v_{\rm pec}$ and the electron density $n_{\rm e}$ integrated along the line of sight through the cluster. For small optical depths, the relative change in intensity of the CMB is given by

$${\Delta I \over I} = {\sigma_{\rm T}\over c } \; \int_{{\rm ... ...\over {e^{x}-1}} \Biggl ], \qquad \qquad x = {h\nu \over kT },$$ (12)

where T is the temperature of the microwave background and $\sigma_{\rm T}$is the Thompson cross section. For a typical cluster, the kinematic SZ effect at the cluster center is of order

$${\Delta T} \sim 30 \; \biggl ( {n_{\rm e} \over 3\times 10^{... ...; \biggl ( {v_{\rm pec} \over 500 km s^{-1}} \biggl )\; \mu K,$$ (13)

where n_e is the electron density in the core, r_c is the core radius and we have scaled to the values of the Coma cluster. Since this effect is independent of frequency, the maximum attainable signal-to-noise ratio for a typical cluster is determined primarily by confusion with the primordial CMB fluctuations. It is therefore essential to use information on the statistical properties of the primordial CMB anisotropies and the gas distributions of the individual clusters which will be given by the high precision y determinations derived from the PLANCK mission itself. This knowledge makes it possible to analyze the CMB maps with a spatial filter optimized for individual clusters. An improvement in signal-to-noise by a factor of two is easily achievable with this technique, and even a factor of 10 is possible if the gas mass distribution is well known. The final signal-to-noise ratio depends sensitively on the angular resolution of the instrument and on the temperature power spectrum of the primordial CMB anisotropies (and hence on the cosmological parameters), thus the precise accuracy of peculiar velocity measurements for individual clusters depends on parameters which are poorly known at present (see Figure 1.17). Simulations of the PLANCK data analysis discussed in From Observations to Scientific Information show that the estimates of Figure 1.17 are realistic. Prime candidates for accurate peculiar velocity measurements ($\Delta v_{pec}\lower.5ex\hbox{$\; \buildrel < \over \sim \;$}200$ km/s) are clusters at intermediate redshifts with core radii slightly smaller than the angle subtended by the first Doppler peak in the primordial CMB power spectrum.

  

Figure 1.17: The $1\sigma$ error in the determination of the peculiar velocity as a function of the core radius of the cluster using an axisymmetric optimal filter function for a standard CDM scenario ($\Omega_{\rm tot} =1$, $H_{0}= 50 {\rm km}{\rm s}^{-1}{\rm Mpc}^{-1}$) with varying baryonic fraction. The pixel noise is fixed and corresponds to $7\,\mu{\rm K}$ in a $4^\prime$ (FWHM) beam. Thick curves are for a beam size of $8^\prime$ and thin curves are for a beam size of $4^\prime$.The inset shows the angular power spectra of temperature fluctuations for the three cosmological models.

In addition to estimates of peculiar velocities for individual clusters, it will be possible to extract statistical information on the bulk motion of clusters on scales of $\lower.5ex\hbox{$\; \buildrel \gt \over \sim \;$}100 h^{-1}{\rm Mpc}$. So far, not much is known about bulk velocities on such large scales. The measurement of the bulk velocity of a volume-limited sample of 119 Abell clusters out to a distance of $15 000 {\rm km}{\rm s}^{-1}$ has yielded a high value of $700-800 {\rm km}\;{\rm s}^{-1}$ (Lauer and Postman 1994). This is considerably larger than the $400-500 {\rm km}\;{\rm s}^{-1}$ expected in most cosmological scenarios currently favored. Detailed calculations show that PLANCK is capable of determining bulk motions to an accuracy of between 100 and 300 km/s within a sphere of radius $100 h^{-1}{\rm Mpc}$, depending on the statistical properties of the primordial CMB anisotropies (Haehnelt & Tegmark 1996). PLANCK can therefore provide unique statistical information on bulk motions, which can be used to constrain the power spectrum of matter fluctuations on large scales and to elucidate the relationship between irregularities in the galaxy and mass distribution (Dekel 1994).

[last update: 1 August 1999 by P. Fosalba]