MEASURING THE GALAXY CLUSTER BULK FLOW FROM WMAP DATA

The SZ effect can be used to measure the deviations of galaxy clusters from the Hubble flow (Sunyaev & Zeldovich 1980). Approximately 1% of CMB photons traveling through galaxy clusters are scattered by electrons trapped in the gravitational potential of the cluster. If the galaxy cluster is moving with respect to the CMB rest frame, the scattered photons are redshifted or blueshifted. This is the kinetic Sunyaev–Zeldovich (kSZ) effect (Sunyaev & Zeldovich 1980) and results in a fractional temperature change in the radiation of

$$\begin{equation} \frac{\Delta T_{\rm KSZ}}{T} = - \int \textbf {dl} \cdot \frac{\textbf {v}_p}{c} n_e \sigma _T, \end{equation} \tag{ 1 }$$

where $\textbf {dl}$ is the line-of-sight distance through the cluster, $\textbf {v}_p$ is the cluster peculiar velocity, n_e is the electron density, and σ_T is the Thompson scattering cross section. The quantity ∫n_eσ_Tdl is the optical depth to Thompson scattering, τ, with typical value ∼0.01 for the clusters we observe. By observing the temperature increment or decrement in the direction of a galaxy cluster, the line-of-sight cluster peculiar velocity can be estimated. For clusters in our sample we expect ΔT_KSZ/T ∼ 10⁻⁵. To date no detection of a cluster velocity has been made using the kSZ effect.

A bulk motion of mass in the universe will cause a dipole pattern in temperature at the cluster positions. The kSZ signal from each cluster that is part of a bulk motion is given by

$$\begin{equation} \frac{\Delta T_{{\rm KSZ}}}{T} = \frac{v_{{\rm bulk}}}{c} \tau \cos \theta, \end{equation} \tag{ 2 }$$

where τ is the optical depth of the cluster and θ is the angle between the cluster position and the bulk flow direction. Cluster motions with more structure than monopole and dipole terms can be described by decomposing the cluster velocity field into spherical harmonics.

The electrons within massive clusters have temperatures that are typically a few keV. The thermal motion of these electrons causes an increase in the energy of the scattered photons, which is the thermal Sunyaev–Zeldovich (tSZ) effect (Sunyaev & Zeldovich 1970). The fractional temperature change in the radiation in the non-relativistic limit is

$$\begin{equation} \frac{\Delta T_{\rm TSZ}}{T} = \left(x \frac{e^x + 1}{e^x - 1} - 4 \right) \int dl \; \frac{k_B T_e}{m_e c^2} n_e \sigma _T, \end{equation} \tag{ 3 }$$

where x = hν/k_BT_CMB, ν is the radiation frequency, dl is the line-of-sight distance through the cluster, and T_e is the electron temperature. To determine the importance of relativistic corrections to Equation (3), we have calculated the thermal SZ spectrum of all 736 clusters in our cluster sample (we describe the cluster modeling we use in Section 4.2), using Equation (3) to calculate the tSZ emission and also including relativistic corrections using the model of Nozawa et al. (2000). We combine the spectra, weighting the spectrum from each cluster with the same weight that we use when we measure the bulk flow signal (we describe how we calculate the bulk flow signal in Section 4.4). We find that the difference between the flux calculated using Equation (3) and the flux from the relativistic formula is less than 5% at all of the frequencies we use, and so for our purposes we can safely ignore the relativistic corrections.

For typical clusters ΔT_TSZ/T ∼ 10⁻⁴ at 90 GHz, and the ratio of the kinetic signal to the thermal signal is (assuming the clusters are isothermal)

$$\begin{equation} \frac{\Delta T_{\rm KSZ}}{\Delta T_{\rm TSZ}} \sim \frac{v_p m_e c}{k_B T_e } \sim 0.1 \left(\frac{v_p}{300 {\rm \;km\;s^{-1}}} \right) \left(\frac{T_e}{5 \rm \;keV} \right)^{-1}. \end{equation} \tag{ 4 }$$

The thermal signal is therefore a significant contaminant to any measurement of the kinetic signal. However, since there are not expected to be intrinsic large-scale moments in the tSZ signal, the tSZ dipole amplitude is expected to be smaller than the monopole amplitude.

Unlike the tSZ signal, the kSZ signal has an identical frequency spectrum to the CMB. The kSZ signal can therefore only be distinguished from CMB fluctuations by the different spatial properties of the two signals. The CMB power peaks on degree scales, whereas the SZ emission from galaxy clusters typically varies on arcminute scales. The signal we detect is smeared out by the instrumental beams of the experiment, 306 in the WMAP Q band, 21' in the V band, and 132 in the W band.

On scales larger than ∼10 h⁻¹ Mpc, fluctuations in the matter density of the universe are Gaussian distributed (Rimes & Hamilton 2005, and references therein). The inhomogeneities in the matter density cause galaxy clusters to have peculiar velocities, with typical values of ∼300 km s⁻¹ (e.g., Bhattacharya & Kosowsky 2008). If a sample of galaxy clusters within a region is chosen, that sample will have a non-zero bulk motion due to the non-uniform matter density on even larger scales. Although the direction of this bulk motion is not determined by the ΛCDM model, the rms amplitude can be calculated given a set of cosmological parameters. We now calculate this amplitude.

We can estimate the expected cluster bulk flow velocity by expanding the line-of-sight peculiar velocity distribution in spherical harmonics. The rms bulk flow velocity will be the power in the first multipole. We follow the derivation for the density distribution given in Peebles (1973). We decompose the line-of-sight peculiar velocity into spherical harmonics, $Y_{\ell m}(\hat{\textbf {r}})$, with the amplitude of the ℓ, m mode given by

$$\begin{equation} a_{\ell m} = \int dr r^2 \phi (r) \int d \Omega _r Y^{\ast }_{\ell m}(\hat{\textbf {r}}) \; \textbf {v}(\textbf {r}) \cdot \hat{\textbf {r}}, \end{equation} \tag{ 5 }$$

where $\textbf {r}$ is the comoving radial distance, ϕ is the comoving number density of objects in the sample, $\textbf {v}$ is the object peculiar velocity, and Ω_r is the solid angle. We approximate ϕ(r) by an isotropic function, which we estimate by calculating the number density of clusters in our sample within radius r. We normalize ϕ such that

$$\begin{equation} \int _0^R dr r^2 \phi (r) = 1, \end{equation} \tag{ 6 }$$

where R is the comoving distance within which the a_ℓm are calculated. Figure 1 shows the redshift distribution of our cluster sample used to calculate ϕ.

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The power in multipole ℓ is defined as C_ℓ = 〈|a_ℓm|²〉 where the average is over all values of m within multipole ℓ. If the line-of-sight peculiar velocity over the whole sky is a Gaussian random field, as expected in the ΛCDM model, then it is completely described by C_ℓ. The power in the dipole (ℓ = 1) is (the details of the derivation are given in Appendix A):

$$\begin{eqnarray} C_1 &=& \langle | a_{1m} |^2 \rangle = \frac{2}{9 \pi } f^2 H_0^2 \int dk P(k) \nonumber\\ &&\times\left(\int dr r^2 \phi (r) \left(j_1 (kr) - 2j_2 (kr) \right) \right)^2, \end{eqnarray} \tag{ 7 }$$

where P(k) is the matter power spectrum, f = (a/D) dD/da, a is the scale factor, D is the growth function, H₀ is the Hubble constant, and j₁ and j₂ are spherical Bessel functions. Since the clusters we observe are all at redshift less than one, we approximate f by its value today, taking it to be equal to Ω^0.6_m. We obtain a prediction of the rms dipole velocity shown in Figure 2. Since our cluster sample contains few clusters with redshifts greater than ∼0.3, the bulk velocity of our entire cluster sample is largely determined by clusters with redshifts less than this. The effect of the selection function is therefore to increase the expected dipole velocity in shells extending to redshifts greater than 0.3. The shaded area in Figure 2 is the uncertainty from cosmic variance. The additional uncertainty on C₁ from sample variance is inversely proportional to the number of clusters that are used to measure the dipole. For redshift shells extending beyond z = 0.05 our cluster sample contains over 100 clusters and so the sample variance in Figure 2 will be more than 10 times smaller than the cosmic variance. We therefore do not include it in the figure.

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A measurement of a kSZ bulk flow with an amplitude larger than that expected from cosmic variance would be an important result if confirmed, requiring modifications of inflation to explain it (e.g., Turner 1991; Kashlinsky et al. 1994). A measurement of a kSZ monopole with a velocity greater than that expected by sample variance would be a violation of the Copernican principle, which states that we do not live in a specially favored place in the universe. However, a measurement of the kSZ monopole is more susceptible to contamination. The thermal SZ signal, and the radio point-source signal from clusters, are both expected to be isotropic, with any dipole signal coming from sample variance. Any dipole signal from tSZ or radio point sources is therefore expected to be weaker than any corresponding monopole signal. The radio source monopole at the cluster locations could be removed by observing and subtracting the emission, or by utilizing the different spectra of the radio source and kSZ signals. For thermal SZ the monopole can be suppressed spectrally.

Following KAKE we derive our cluster sample from the REFLEX (Böhringer et al. 2004), BCS (Ebeling et al. 2000b), extended BCS (Ebeling et al. 2000a), and CIZA (Ebeling et al. 2002; Kocevski et al. 2007) cluster catalogs. These catalogs contain 447, 206, 107, and 130 clusters, respectively. After removing 20 overlapping clusters we obtain a sample of 870 clusters. After removing clusters whose center lies behind the WMAP galactic mask we find 736 clusters (when using the WMAP Kp0 galactic mask that was used in the KAKE analysis we find 771 clusters in the sample). The redshift distribution of the cluster sample is shown in Figure 1. We convert the luminosities in the BCS, BCSe, and CIZA catalogs to the values for a ΛCDM universe. For the REFLEX sample we use a catalog with the luminosities already converted to an h = 0.7, ΛCDM cosmology so no correction is necessary for these clusters.

To derive average electron density and temperature profiles for our cluster sample we use the Archive of Chandra Cluster Entropy Profile Tables (ACCEPT; Donahue et al. 2006; Cavagnolo et al. 2009). This is a sample of 239 clusters observed by the Chandra X-ray Observatory (Weisskopf et al. 2000), with electron density and temperature profiles for each cluster derived from a deprojection analysis. One hundred and forty-five of the ACCEPT clusters overlap with our sample.

To test for contamination from galactic emission, thermal SZ, and unresolved radio sources, we use simulated maps produced using the Planck Sky Model⁵ (PSM).

The PSM is a set of programs and data used for the simulation of full sky microwave maps that includes galactic synchrotron, dust and free–free emission, kinetic and thermal SZ, radio and infrared point sources, and CMB. The PSM simulations do not include a kinetic SZ monopole or cluster bulk flow velocity. We use one realization of the sky generated at the WMAP Q-, V-, and W-band center frequencies.

To test the effect of radio point sources we use 100 simulated maps of the unresolved radio background produced by Colombo & Pierpaoli (2010). They combine sources from the NRAO-VLA Sky Survey (NVSS) catalog (Condon et al. 1998) with higher frequency surveys to extrapolate the radio point-source emission at 1.4 GHz to the WMAP frequencies. The radio source flux distribution in the maps at each frequency therefore reflects the probability distribution for the flux of each NVSS source.

The bulk motion of many galaxy clusters relative to the CMB rest frame creates a signal in the WMAP temperature maps that has a dipole pattern at the location of clusters. We now describe the method we use to recover the amplitude and direction of any dipole motion using the WMAP CMB maps.

In addition to kSZ the dipole signal at the position of clusters includes contributions from CMB, instrumental noise, tSZ, galactic emission, and infrared and radio point-source emission. The first step is to filter the maps to enhance the kSZ signal relative to the other terms. We use two filters, one of which enhances the cluster signal relative to the CMB and instrument noise, the other filter also removes the thermal SZ term. The galactic term is suppressed by masking the WMAP maps with the WMAP extended temperature analysis mask and only fitting for a cluster dipole outside of this region. In Section 5.2 we give upper limits on the radio point-source contamination and in Section 5.3 we give upper limits on the galactic signal. Using the PSM simulations we find a negligible contribution from infrared point-source emission.

The next step is to calculate the monopole and dipole at the locations of the ROSAT observed clusters in the filtered maps. We use simulated kinetic SZ maps to convert the amplitude of the dipole at the cluster locations to a bulk flow velocity using a method we describe in Section 4.5. Our simulated kinetic and thermal SZ maps include signal only from the ROSAT observed clusters that we use, and will be described in Section 4.2.

There are two ways we could calculate the cluster velocity dipole. We could calculate the optical depth through each cluster and use Equation (1) to estimate the velocity of each cluster. We could then fit for a dipole in the cluster velocity distribution. Alternatively, we could calculate the dipole at the cluster positions in the WMAP maps and convert the dipole signal to a velocity using an estimate of the average optical depth of the clusters in our sample. The two methods differ in the weight given to each cluster in the dipole fit. We expect the signal to noise of the latter method to be higher since all clusters are given equal weight in the fit, whereas in the former method clusters with a larger optical depth (and hence a greater kSZ signal in the map) are given less weight. We follow KAKE and choose the latter method. We split the cluster sample into different redshift bins (some of which overlap) and calculate the cluster dipole in each.

To calculate the errors on the monopole, dipole, and higher order modes, we generate 100 realizations of the CMB, convolve the maps with the beams from each of the eight channels, and add instrument noise using the procedure described in Limon et al. (2006). We pass these maps through our pipeline to find the distribution of monopole, dipole, and higher order modes.

Our measurement of the WMAP temperature dipole does not depend on any cluster simulations. However, in order to convert the cluster temperature dipole measurement into a velocity dipole we require simulated maps of the kSZ emission from our cluster sample. In addition, we require tSZ realizations to estimate the level of tSZ contamination in our results.

The clusters in our sample have typical angular sizes of ∼1', which is small compared with the WMAP beam FWHM, 306 in Q band and 132 in W band. For this reason we model the clusters as point sources convolved by the WMAP beams.

4.2.1. Optical Depth Determination

In order to generate kSZ realizations with a simulated bulk flow velocity we require the optical depth to Thompson scattering for every cluster in our sample. We calculate this using average electron density and temperature profiles that we derive from the ACCEPT catalog (described in Section 3.1).

To calculate the electron density and temperature profiles we select 145 ACCEPT clusters that overlap with our sample. We average the electron density and temperature profiles of the overlapping clusters within radial bins. We use bins with a smaller size nearer to the center of the cluster where the error on the electron density and temperature is lower (the error is provided in the catalog). The mean bin size is 10 kpc, and the largest radial bin extends to 1 Mpc from the cluster center. Each profile is normalized so that it has unit value at the center of the cluster. The densities and temperatures of all of the clusters are then averaged within each radial bin. We compute the kSZ and tSZ signals using these profiles, giving the profiles a different normalization for each cluster, as described in the following paragraph. Extrapolating the density profile to larger radii using a β-model (Cavaliere & Fusco-Femiano 1976), we find a negligible contribution to the optical depth from beyond this region. The profiles are not significantly changed when all of the clusters in the ACCEPT sample are used to calculate them.

Since we use the same electron density profile for each cluster, we only need to calculate the normalization to that profile to estimate the optical depth. The optical depth is calculated by integrating the electron density along the line of sight,

$$\begin{equation} \tau (R) = \sigma _T N_e \int \frac{f_e(r) r dr}{\sqrt{r^2-R^2}}, \end{equation} \tag{ 8 }$$

where R = d_Aθ is the distance from the cluster center perpendicular to the line of sight, d_A is the angular diameter distance, σ_T is the Thompson scattering cross section, f_e(r) is the normalized electron density profile, and N_e is the normalization to that profile.

We calculate the normalization to the electron density profile by requiring that the cluster bolometric luminosity in the ROSAT catalogs is equal to the luminosity from bremsstrahlung emission, which depends on N_e (Gronenschild & Mewe 1978),

$$\begin{eqnarray} L &= &\frac{32 \pi }{3} \left(\frac{2\pi k_B}{3 m_e} \right)^{\frac{1}{2}} \frac{Q^6}{m_e c^3 h} Z^2 g N_e N_i \nonumber\\ &&\times\int d^2 r \; 4 \pi r^2 \; f_e (r) f_i (r) T(r)^{\frac{1}{2}} \;\; {\rm erg\;s^{-1}}, \end{eqnarray} \tag{ 9 }$$

where k_B is the Boltzmann constant, m_e is the electron mass, Q is the electron charge, c is the speed of light, h is Planck's constant, Z is the number of protons per nucleus, g is the frequency averaged Gaunt factor (of order unity), T(r) is the electron temperature, f_e(r) and f_i(r) are the normalized electron and ion density profiles (which we take to be equal), and N_e and N_i are the normalizations to those profiles. We normalize the temperature profile such that the X-ray emission weighted temperature is equal to T_X, which we calculate from the X-ray luminosity of each cluster using the relation T_X = (2.76 ± 0.08)L^{0.33 ± 0.01}_X (White et al. 1997).

Using this procedure we find that the mean central optical depth of our cluster sample is (4.9 ± 0.9) × 10⁻³. The mean central optical depth of the 145 clusters that overlap with the ACCEPT sample is τ₀ = (6.1 ± 1.0) × 10⁻³ which is in agreement with the value calculated using each individual electron density profile in the ACCEPT catalog, τ₀ = (6.7 ± 0.9) × 10⁻³. This gives us confidence that we can calculate accurate values of the optical depth for clusters that are not in the ACCEPT sample. The dominant source of uncertainty in our optical depth calculation is the cluster X-ray luminosity, from the X-ray catalogs. Propagating all errors through to the optical depth gives uncertainties of ∼15%. These errors are propagated through to the simulated kSZ maps which allows us to calculate the precision of our method.

We have not accounted for any anisotropy in the optical depth distribution of our clusters or accounted for the different methods used to calculate the X-ray luminosities in the different X-ray catalogs we use. Both of these steps would require an independent analysis of the X-ray data. However, since our dipole fit does not depend on the X-ray luminosities our null detection is unaffected. Any error in the X-ray luminosities will only affect the conversion of the dipole amplitude from μK to km s⁻¹. Since the X-ray luminosities are used to calculate the optical depth any asymmetry in the optical depths will be accounted for when converting from μK to km s⁻¹.

4.2.2. kSZ and tSZ Simulations

The kSZ signal is calculated from Equation (1). We give each cluster in our simulated maps a bulk velocity using Equation (2) and a random velocity drawn from a Gaussian distribution, with variance (e.g., Gorski 1988)

$$\begin{equation} \sigma ^2_{{\rm v}} = \frac{f^2 H_0^2}{6\pi ^2} \int _0^{\infty } d k \; P(k) \; | W(kR) |^2, \end{equation} \tag{ 10 }$$

where |W(kR)|² is the Fourier transform of the window function, and again we use f ≈ Ω^0.6_m. To select cluster scales we use a top hat window function with R = 10 h⁻¹ Mpc and find σ_v = 272 km s⁻¹. We ignore correlations between the velocities of different clusters, which we expect to be small for our cluster sample (e.g., Gorski 1988; Bhattacharya & Kosowsky 2008).

The expected tSZ signal is calculated by integrating Equation (3) through the cluster using the same electron density and temperature profiles calculated in Section 4.2.1. Since WMAP is sensitive enough to detect the tSZ signal in stacked images we have checked that the average tSZ signal in our simulations agrees with WMAP observations. Following the method in Atrio-Barandela et al. (2008) (also Diego & Partridge 2010), we have stacked maps centered on each cluster to produce a co-added map in the Q, V, and W bands from the WMAP maps and our simulated tSZ maps. We find that the residual after subtracting the two is consistent with CMB and instrument noise.

In Table 1, we show the simulated thermal SZ monopole in maps filtered by the matched filter we use in our analysis. We compare this to the monopole found in the WMAP maps filtered by the same filter. Although the monopole in our simulated maps is biased high, it is within 2σ of the measured monopole in all except the two lowest redshift shells. This gives us confidence that our optical depth calculation is reasonable.

Table 1. Simulated Thermal SZ Monopole and Measured Monopole

zmin	zmax	Ncl	Simulated tSZ Monopole, a0 (μK)	Measured Monopole, a0 (μK)
0.0	0.04	95	−78.7	−22 ± 33
0.0	0.05	139	−84.2	−41 ± 27
0.0	0.06	192	−97.0	−62 ± 24
0.0	0.08	294	−97.9	−63 ± 18
0.0	0.12	445	−99.5	−73 ± 13
0.0	0.16	546	−103.4	−81 ± 12
0.0	0.20	619	−109.1	−85 ± 12
0.0	1.0	736	−123.9	−91 ± 11
0.05	0.30	578	−128.6	−99 ± 12
0.12	0.30	271	−152.2	−112 ± 17

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The distribution of our cluster sample on the sky is not isotropic and so measurements of the peculiar velocity multipoles will be contaminated by higher order CMB multipoles. The higher order CMB multipoles couple to the signal we are trying to measure through the mask we use to cover non-cluster regions of the WMAP maps. To reduce the contamination from the CMB we filter the maps.

Two filters are used and described here. The first is an optimal matched filter that removes low-order CMB multipoles by utilizing information at locations other than the cluster position to distinguish a CMB multipole from a kSZ multipole. The second filter uses both spatial information and the spectral difference between the thermal and kinematic SZ effects to remove the contaminating tSZ signal.

An optimal filter is constructed for each channel such that when the filtered maps from each channel are combined, the kSZ signal-to-noise ratio is maximized. In order to construct our filters we require a matrix of cross spectra, $\textbf {C}_{\ell }=\textbf {C}^{\rm CMB}_{\ell } + \textbf {C}^{\rm noise}_{\ell }$, that describes the statistical properties of the maps to be filtered. We assume that the noise in map ν, n_ν(θ), is a homogeneous and isotropic random field. The cross power spectrum of maps ν₁ and ν₂ is equal to $\langle n_{\ell m,\nu _1}n_{\ell m,\nu _2}^\ast \rangle$, where n_{ℓm, ν} are the coefficients in the spherical harmonic expansion of n_ν(θ), and the average is over m. We generate $\textbf {C}_{\ell }$ by averaging the cross power spectra of 100 CMB and noise realizations.

The maps in each channel are filtered and combined to produce the spherical harmonic coefficients of the filtered map as follows:

$$\begin{equation} f_{\ell m} = \sum _{i=1}^{n_{{\rm channels}}} \Phi ^i_{\ell } a^i_{\ell m}, \end{equation} \tag{ 11 }$$

where aⁱ_ℓm are the spherical harmonic coefficients of the WMAP channel i map and Φⁱ_ℓ is the filter for channel i. We include the factor of $\sqrt{4\pi /2\ell +1}$ from spherical convolution in Φ_ℓ. The first filter we use has a filter function of the form (McEwen et al. 2008)

$$\begin{equation} \Phi ^{{\rm m}}_{\ell } = \frac{\textbf {C}_{\ell }^{-1}}{\gamma } \textbf {B}_{\ell }, \end{equation} \tag{ 12 }$$

where m indicates that the filter is a matched filter, $\gamma = (1/n_{{\rm pix}})\sum _{\ell } (2\ell +1) \textbf {B}^T_{\ell } \textbf {C}_{\ell }^{-1} \textbf {B}_{\ell }$, n_pix is the number of map pixels, and $(4\pi /n_{{\rm pix}}^2) \textbf {B}_{\ell }^2$ is an estimate of the kSZ power spectrum from a single cluster, in each channel (we assume the beam convolved sources are symmetric). Since the WMAP beams are much larger than galaxy clusters we approximate $\textbf {B}_{\ell }$ by the WMAP beam function. We have calculated the filter using the WMAP beam profile convolved with an average cluster profile and also a cluster profile modeled as a β-model (Cavaliere & Fusco-Femiano 1976) and find no significant difference in the results.

A simple modification to this filter allows the thermal SZ bias to be removed (e.g., Herranz et al. 2005; Schäfer et al. 2006, for the flat and curved sky cases, respectively),

$$\begin{equation} \Phi ^{{\rm u}}_{\ell } = \frac{\textbf {C}_{\ell }^{-1}}{\Delta } \left(\alpha \textbf {B}_{\ell } - \beta \textbf {F}_{\ell } \right), \end{equation} \tag{ 13 }$$

where u indicates that the filter is an unbiased filter, $\textbf {F}_{\ell }$ is an estimate of the thermal SZ signal in each channel, Δ = αγ − β² is a normalization factor, $\alpha = (1/n_{{\rm pix}})\sum _{\ell } (2\ell +1) \textbf {F}^T_{\ell } \textbf {C}_{\ell }^{-1} \textbf {F}_{\ell }$, $\beta = (1/n_{{\rm pix}})\sum _{\ell } (2\ell +1) \textbf {B}^T_{\ell } \textbf {C}_{\ell }^{-1} \textbf {F}_{\ell }$, and again $\gamma = (1/n_{{\rm pix}})\sum _{\ell } (2\ell +1) \textbf {B}^T_{\ell } \textbf {C}_{\ell }^{-1} \textbf {B}_{\ell }$. We are using maps with units of thermodynamic temperature and so $\textbf {F}_{\ell } = \left[x (e^x+1)/(e^x-1) - 4.0 \right] \textbf {B}_{\ell }$.

Figures 3 and 4 show the functions Φ^m_ℓ and Φ^u_ℓ for each channel. These filters have very different behaviors and we now discuss why this arises.

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The matched filter (hereafter MF) suppresses the input map at multipoles less than ∼300 (large scales) due to the strong CMB signal at these multipoles. The filter function also suppresses the map at multipoles greater than ∼1000 (small scales), where the expected kSZ signal-to-noise ratio is low. The W-band filter is non-zero up to the highest multipoles since the W-band beams are smaller, and so the signal extends to higher multipoles. Between multipoles 300 and 1000 the signal-to-noise ratio is highest and so these scales are retained in the filtered map. The bumps near multipoles 500 and 700 in the MF correspond to the troughs in the CMB spectrum—the filter amplifies scales where the contaminating CMB emission is weaker.

The tSZ bias removing filter (hereafter UF) in Figure 4 is less intuitive. At low multipoles the CMB signal in each channel is almost identical because each channel observes the same CMB sky, and at low multipoles the beam functions are all close to unity. At low multipoles the CMB can therefore be removed by subtracting the Q- and W-band maps. Since there are two Q-band channels and four W-band channels the CMB is removed by giving the Q-band maps double the weight of the W-band maps in the subtraction. This explains why the two Q-band filters have an absolute value double that of the four W-band filters at low multipoles. This is reflected in Equation (13) by the off-diagonal elements being almost equal to the diagonal elements in the cross-power spectra matrix at low multipoles. If the CMB sky in each channel were different then the filter would look similar to the MF, but with different amplitudes for each channel. In Figure 5, we show what the tSZ bias removing filters would look like for an experiment with four frequency channels: 100 GHz, 143 GHz, 217 GHz, and 353 GHz, with beam FWHM of 10', 7', 5', and 5', respectively, and white noise levels: 25, 15, 25, and 75 μK/K arcmin. The fact that there are no peaks in the WMAP filters is a reflection of the limited frequency coverage and large beams.

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Since the UF combines channels in a way that removes the tSZ signal, the filters do not increase the kSZ signal-to-noise ratio as much as the MF. The signal to noise of the cluster temperature dipole measurement is therefore lower.

Figure 6 shows the spectra of a simulated map containing only the kSZ signal at the ROSAT cluster locations with and without a bulk flow component of 1000 km s⁻¹, and convolved by the WMAP beams. The maps with and without a bulk flow both have spectra that are identical to the WMAP beam function at high multipoles. The map with the bulk flow component contains more power at all multipoles as well as additional power at the dipole and octupole modes. Although the difference between the spectra with and without a bulk flow looks predominantly like an amplitude shift, the bulk flow signal is recoverable by information contained in the spherical harmonic m modes, which are averaged to create the spectra. Our filters are designed to optimize the detection of sources shaped like the WMAP beams. The similarity of the simulated kSZ spectrum with a bulk flow component to the beam profile indicates that our filter will have the desired effect of increasing the signal to noise of the cluster dipole measurement.

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Figures 7 and 8 show the WMAP maps after filtering with the MF and UF, respectively. Both filters have suppressed the CMB signal and the only visible structure in the map is the low noise region around the ecliptic poles. The maps are noise dominated; the larger values of the map in Figure 8 reflects the lower signal-to-noise ratio obtained with the UF.

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Since we are only calculating the dipole in a small region at the center of the clusters, we could filter the maps only in this region. This would save computation time but care would need to be taken to avoid ringing effects around the edge of the mask. For this reason we filter the maps in all regions outside the WMAP galactic mask.

4.3.1. Wiener Filters

We now compare our filter to the Wiener filter used by KAKE. A Wiener filter (hereafter WF) minimizes the squared difference between the filtered map and an estimate of the signal. Although a multi-frequency WF can be constructed, we follow KAKE and create a filter for each map that is designed to suppress the CMB component. The KAKE filter is (Kashlinsky et al. 2009)

$$\begin{equation} \Phi ^{{\rm w}}_{\ell } = \frac{C_{\ell }^{\rm sky}/f_{\rm sky} - C_{\ell }^{\rm CMB} B_{\ell }^2}{C_{\ell }^{\rm sky}/f_{\rm sky}}, \end{equation} \tag{ 14 }$$

where C^CMB_ℓ is the CMB power spectrum, B_ℓ is the WMAP beam function, C^sky_ℓ is the spectrum of the WMAP map estimated outside of the galactic mask, and f_sky is the fraction of the sky outside of the mask. The full sky spectra could alternatively be calculated by deconvolving the mask: $C_{\ell }^{\rm unmasked} = \sum _{\ell ^{\prime }} M^{-1}_{\ell \ell ^{\prime }} C_{\ell }^{\rm masked} \approx C_{\ell }^{\rm masked}/f_{\rm sky}$, where $M_{\ell \ell ^{\prime }}$ is the multipole mixing matrix which accounts for the cut sky and is calculated from the galactic mask (see, e.g., Appendix A of Hivon et al. 2002 for details). The KAKE filter we use is shown in Figure 9. We have determined that this is the same Wiener filter that was used in the KAKE analysis and give details of our pipeline in Appendix B. The KAKE filter is constructed from the spectrum of the map that is to be filtered, and maps filtered with it suffer from low multipole fluctuations, caused by cosmic variance. We create our Wiener filter in such a way that it does not suffer from this problem (Tegmark & Efstathiou 1996),

$$\begin{equation} \Phi ^{{\rm w}}_{\ell } = \frac{C_{\ell }^{{\rm noise}}/f_{\rm sky}}{C_{\ell }^{{\rm CMB}} B_{\ell }^2 + C_{\ell }^{{\rm noise}}/f_{\rm sky}}, \end{equation} \tag{ 15 }$$

where C^noise_ℓ is the power spectrum of the instrument noise plus all foreground components in the masked map. This filter is not normalized and so pixels in the filtered map are not equal to the estimated amplitudes of the kSZ signal in the unfiltered map pixels. Since we model our clusters as point sources convolved by the beam the normalization factor would be n_pix/∑_ℓ(2ℓ + 1)B_ℓΦ_ℓ. The normalization is instead applied later when the dipole amplitude is converted from kelvin to km s⁻¹. Our Wiener filter is shown in Figure 10.

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Since the noise in the WMAP maps has a flat spectrum and the CMB signal decreases at high multipoles, the WF acts like a high pass filter for each of the WMAP channels. At low multipoles the shape of this filter is similar to our MF (although the relative normalizations of the channels is not), but at high multipoles the shapes differ since the WF does not take into account the fact that the cluster signal decays exponentially with increasing multipole, while the noise spectrum remains flat. We find that both our Wiener filter and the KAKE filter suppress the cluster signal approximately equally as shown in Figure 11. At the multipoles where the signal peaks (∼400 in the V band) the two filters suppress power by a similar amount. We find that our Wiener filter reduces the noise in the a_1m by a factor of ∼2 relative to the KAKE filter by reducing CMB noise at multipoles less than ∼100 that is correlated between the WMAP channels (we give a more detailed comparison of the filters in Appendix B), and is therefore more sensitive to the cluster signal.

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For the matched filter and unbiased filter we calculate the cluster dipole in the pixels that overlap with the centers of each cluster. For our Wiener filter we instead use a 15' aperture around each cluster, in order to increase the signal to noise of the measurement. We have tried fitting for the dipole using different aperture sizes out to 30' as well as different weighting schemes for the pixels within the aperture: uniform weighting, weighting by the inverse noise variance, and weighting by the expected cluster signal-to-noise ratio. The results presented in Table 4 have a uniform weighting within the aperture, which was chosen to give a good signal-to-noise measurement while keeping our results free from any bias caused by incorrect signal or noise estimates.

The WF that we use is not a multi-frequency filter (although in principle a multi-frequency WF could be used) and so there is a filtered map for each of the WMAP channels. We filter each map and compute the cluster dipole in each. We then average the dipoles from the different channels with uniform weights:

$$\begin{equation} a_{1m} = \frac{1}{8} \sum _{i=1}^8 a_{1m}^{(i)}. \end{equation} \tag{ 16 }$$

The signal to noise could be increased by weighting the results from each channel by the expected signal-to-noise ratio of the kSZ signal. In this scheme, the weights given to each channel are approximately equal—the noise at higher frequencies is compensated for by the decreased beam FWHM, and therefore larger cluster signal.

To measure the cluster dipole we perform a weighted least-squares fit to a dipole function at the locations of the clusters in the filtered map. We also fit for the monopole and, separately, we perform the fit for modes up to ℓ = 2 and up to ℓ = 3, 4, 5. Our dipole fitting method is based on the Healpix IDL procedure remove_dipole (Górski et al. 2005). We fit for a vector of coefficients, β, of the real spherical harmonics using the least-squares formula,

$$\begin{equation} \beta = (X^T W X)^{-1} X^T W y, \end{equation} \tag{ 17 }$$

where y is the data map, W is a diagonal matrix with diagonal values equal to a weights map, and X is a matrix giving the contribution of the fitting function to each pixel. If a monopole and three dipole coefficients are fit to the data, then X is an n_pix × 4 matrix. We choose the weights map to have non-zero values only in the central cluster pixels and equal to the inverse noise variance of the filtered map. The noise variance is calculated from 100 filtered WMAP CMB and noise realizations. The vector β are the coefficients of the real spherical harmonics, R_lm, defined by

$$\begin{equation} R_{lm} = \left\lbrace \begin{array}{@{}l@{\quad }l@{}}Y_{{\ell } 0} & \mbox{if $m=0$} \\ { \frac{1}{ \sqrt{2}} } \left(Y_{\ell m}+(-1)^m \, Y_{\ell -m} \right) & \mbox{if $m>0$} \\ { \frac{1}{i\sqrt{2}} } \left(Y_{\ell -m}-(-1)^m \, Y_{\ell m} \right) & \mbox{if $m<0.$} \end{array}\right. \end{equation} \tag{ 18 }$$

The matrix X^TWX is a mixing matrix that couples different spherical harmonic modes together. The effect of the mask W is that a least-squares fitting process can then be used on the masked map to determine the dipole from clusters alone. Caution is needed because information is lost when the map is masked. Consequently if too few modes are fitted, power will "leak" from higher order modes into the fit parameters, corrupting the result.

Since the kSZ signal is proportional to the optical depth, in principle the signal-to-noise ratio of the measurement can be increased by weighting each pixel by the optical depth. We find that the increase in signal to noise is approximately 10%. However, when filtering with the MF this weighting scheme increases the contamination from thermal SZ, since the thermal SZ signal is also proportional to the optical depth. We therefore do not weight by the optical depth to reduce the risk of contaminating our result.

In principle, the correlated errors on the three dipole m values can be calculated, with covariance matrix

$$\begin{equation} N = (X^T W X)^{-1} X^T W C W^T X (X^T W X)^{-1}, \end{equation} \tag{ 19 }$$

where C is the pixel–pixel noise covariance matrix. Since it is not practical to compute a matrix as large as C, we calculate the covariance using simulations. The dominant source of error is CMB and instrument noise, which we estimate by passing 100 CMB and noise realizations through our pipeline, performing the least-squares fit on each. There are additional sources of error from our uncertainty in the optical depth, and from the random component of the galaxy cluster peculiar velocities, which are calculated from Equation (10). We estimate the errors from both of these terms by passing simulations of the kSZ signal through our analysis pipeline, performing the least-squares fit on each realization. The scatter in the derived values of β then provides an estimate of the noise correlations between the dipole directions, which is then used to calculate the χ² significance of the measured dipole in each redshift shell. We find that the errors from the latter two terms are negligible compared to the CMB and noise error, and that the errors on the dipole directions are Gaussian with a narrower distribution in the direction perpendicular to the galactic plane. This is due to clusters lying within the galactic mask being removed from the fit.

As a check on this process, we also estimate the errors from the WMAP maps by re-calculating the dipole fit after rotating our weights map away from the clusters. We rotate the weights' map in increments of 10' in the galactic longitudinal direction. We remove points within 1° of the cluster center to reduce any residual SZ emission. This leaves us with 2147 different directions. This process allows us to preserve the angular distribution of the cluster sample in the fit. When either filter is used we find errors that are consistent with those found from the CMB noise realizations. The errors we find by rotating the weights map are less reliable than those calculated from the CMB and noise realizations, since when we rotate our weights map some regions get rotated into the galactic mask, which we then exclude from the fit. The dipole is therefore calculated from fewer clusters in this scheme, and so the errors we quote are from the CMB and noise realizations.

We find that 100 CMB and noise realizations is sufficient to estimate our errors. Taking a typical error for one of the dipole coefficients in the Wiener filtered maps to be 0.8 μK we find that the standard deviation of the sample variance is 0.09 μK for 100 noise realizations. Even if our errors are 10% too large we would still not detect a significant cluster dipole with our filters.

We create a conversion matrix to calculate the velocity dipole from the temperature dipole,

$$\begin{equation} \textbf {a}_{{\rm v}} = \textbf {M} \; \textbf {a}_{{\rm T}}, \end{equation} \tag{ 20 }$$

where $\textbf {a}_{{\rm v}}$ are the velocity monopole and dipole coefficients, $\textbf {a}_{{\rm T}}$ are the temperature coefficients, and $\textbf {M}$ is a 4 × 4 conversion matrix. We calculate $\textbf {M}$ by creating four simulated maps of kSZ signal alone from our cluster sample. In one map all of the clusters are given a monopole velocity. Each of the other three maps has a bulk flow velocity in one of three basis directions. These maps are passed through our analysis pipeline to use the same filtering and velocity fitting processes that are used for the real data. The four fit coefficients from each of the input maps are the elements of each row of $\textbf {M}^{-1}$. The matrix $\textbf {M}$ is close to diagonal in all redshift shells, and the velocity and temperature dipoles are close to alignment, as they should be. To convert from μK to 1000 km s⁻¹ we use the following values for the diagonal entries of M: 4.07 (our Wiener filter), 3.41 (KAKE Wiener filter), 0.075 (matched filter), and 0.12 (unbiased matched filter). For the dipole contribution:

$$\begin{equation} \textbf {M} = \frac{10^{-12}}{T_{\rm CMB}} \frac{c}{\tau _{\rm eff}} \left(\begin{array}{@{}ccc@{}}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array} \right), \end{equation} \tag{ 21 }$$

where τ_eff is the effective optical depth after filtering and we have used the same normalization as KAKE (which makes $\textbf {M}$ a factor of $\sqrt{3/4\pi }$ smaller than for the conventional definition of a_1m). For our Wiener filter we find τ_eff = 2.8 × 10⁻⁵. This value is smaller than the average optical depth of our cluster sample (τ = 4.9 × 10⁻³) due to the filtering, as well as the fact that the signal is diluted by beam smoothing and averaging over the aperture used to calculate the cluster dipole. We find a lower effective optical depth than the value used in the KAKE analysis of τ_eff ∼ 10⁻⁴, which could explain why we find velocity limits ∼10 times larger. We are confident that we correctly recover the cluster velocities and show in Figure 13 that we correctly recover the velocities in simulations.

We have performed tests to check that the results from our filtering and dipole fitting procedures are not contaminated by systematic effects. First, we verify that we can correctly recover a known cluster bulk flow from simulated kSZ maps alone, then we consider maps containing kSZ, CMB, and instrument noise. In Section 5, we describe the effects of thermal SZ, unresolved sources, and galactic emission on this process.

Using simulated maps with a kSZ component only, but that include a cluster bulk flow we have verified that we recover the input bulk flow velocity using our method. We assign all of the clusters in the simulated maps a kSZ signal from both the bulk flow and a random velocity drawn from the expected ΛCDM distribution in Equation (10). For each choice of input bulk velocity we generate 25 realizations, filter the maps, and fit for a cluster dipole using the same pipeline that is used for the WMAP data. We repeat the process with three different input bulk flow directions: (galactic latitude, longitude) = (0°, 0°), (0°, 90°), and (90°, 0°) and bulk flow velocities of 0 km s⁻¹, 500 km s⁻¹, and 1000 km s⁻¹ (7 cases with 25 realizations each). Figure 12 shows the results of the fits. We do not plot the recovered velocity for the cases where the simulated bulk velocity is non-zero because we use our kSZ simulations to calibrate the recovered velocity, and so the mean recovered velocity is exact by design. The scatter in the recovered dipole direction is due to the random component of the cluster line-of-sight velocity (with variance given by Equation (10)) and uncertainty in the cluster optical depth. The recovered amplitude in the maps with no bulk flow velocity has significantly larger scatter when it is calculated using the UF. This reflects the lower signal to noise of this filter caused by the extra degree of freedom.

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We have repeated the test using simulations that additionally contain CMB and instrument noise. We create 100 realizations for each WMAP channel and add a bulk flow signal to the kSZ component, with a range of velocities logarithmically spaced between 100 and 100,000 km s⁻¹ in the direction of galactic latitude −11° and longitude 103°, which is the bulk flow direction found by KAKE in the redshift shell extending to z = 1. Figure 13 shows the recovered bulk flow amplitude and Figure 14 shows the error in the direction of the recovered dipole when all clusters are included in the fit. The lower signal to noise of the UF is apparent in both of these figures. Figure 15 shows the recovered monopole velocity in a different set of 100 simulated WMAP realizations.

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There are not expected to be intrinsic large-scale moments in the tSZ signal, but due to the limited size of our cluster sample the tSZ dipole signal caused by random scatter is not negligible. We have examined the effects of thermal SZ on our results using our own simulations of the tSZ emission from our cluster sample, which are described in Section 4.2, as well as a simulated tSZ map produced using the PSM, described in Section 3.2.

We find that when we fit for the cluster dipole amplitude in our tSZ simulated maps filtered with the MF, using all clusters in our sample, there is a 3σ bias in the y- and z-directions, which equates to a dipole amplitude of approximately 4000 ± 400 km s⁻¹ in the direction of l, b = (88°, 54°) ± 40°, which is 66° from the KAKE bulk flow direction in the same redshift shell. However, in the PSM tSZ map we find a reduced dipole of 2500 km s⁻¹, in the direction of l, b = (327°, 10°), which is 137° from the KAKE bulk flow direction. These simulations suggest that contamination to the cluster dipole measurement is at the 1σ level and so cannot be ignored. As expected, we find that the tSZ monopole in the simulated maps is large and equivalent to a kSZ signal from clusters with velocities of 12,000 km s⁻¹ in our simulations, and 10,000 km s⁻¹ in the PSM maps.

The simulated tSZ maps filtered with the UF have a cluster monopole and dipole amplitude that are more than an order of magnitude lower. In our simulated maps containing only tSZ we find a recovered cluster dipole amplitude of approximately 200 km s⁻¹. In the PSM simulated tSZ map we find a recovered cluster dipole amplitude of 70 km s⁻¹, significantly less than the measured cluster dipole and noise. The monopole amplitude is 200 km s⁻¹ in our simulated tSZ map and 100 km s⁻¹ in the PSM tSZ map.

Although the UF removes the tSZ signal sufficiently for our purposes, it does not remove it to the ΛCDM limit. The residual tSZ in the filtered maps, and the cause of the non-zero cluster monopole and dipole is source confusion. The emission from clusters with small angular separation overlaps more strongly in the Q band where the beam is large, than in the W band. When the channels are subtracted the tSZ signal does not subtract perfectly in the central cluster pixel and there is residual tSZ in the map. Since this only affects a small number of clusters, it leaves a dipole signal in the map.

At frequencies below 100 GHz extragalactic radio sources can be a significant contribution to CMB maps. We use simulations of the unresolved extra-galactic radio point-source emission produced by Colombo & Pierpaoli (2010) and summarized in Section 3.2, to verify that our bulk flow measurement is not significantly affected by sources. Because the model is based on sources observed by NVSS, the simulated maps retain information about the distribution of sources on the sky and account for the increased radio emission at galaxy cluster positions, caused by clustering of radio galaxies.

The uncertainty on the radio point-source monopole and dipole in the maps recovered using the UF is larger than in the maps filtered with the MF. In the maps filtered with the MF we find a non-zero dipole introduced by radio sources with a signal of amplitude 0.2σ in each of the dipole directions. In the redshift shell encompassing all clusters we find signals of 0.2σ, 0.1σ, and −0.3σ in the x-, y-, and z-directions previously defined. We find a monopole signal of −2σ, equivalent to a kSZ monopole velocity of approximately −2000 km s⁻¹. In the maps filtered with the UF we find a larger signal with an amplitude of −0.9σ, 1σ, and 1.1σ in each of the dipole directions. The monopole signal is −6σ, as expected it is large because the radio sources all add signal to the maps whereas the dipole is caused by sample variance.

Because the UF is designed to remove tSZ by a specific weighting of channels, the radio source contribution is actually amplified by the filter. In the UF, the Q-band channel is given relatively more weight, so that emission with a tSZ spectrum is canceled when the channels are combined. The radio point-source signal is strongest in the Q band and weakest in the W band. The UF therefore has a large contribution from the Q-band radio signal that is not offset by the W band signal, where the radio point-source signal is much weaker. The resulting map generated using the UF therefore has a greater residual point-source signal than the map generated with the MF. The radio point-source signal and the tSZ signal could in principle both be suppressed simultaneously by using a filter designed to remove both signals (e.g., Tegmark & Efstathiou 1996). However, the increase in the number of degrees of freedom that this would introduce would further reduce the signal to noise of the measurement.

For our analysis we use the WMAP foreground reduced maps which have been processed by the WMAP team to suppress the galactic signal outside of the WMAP galactic mask (Jarosik et al. 2010). We test the effect that any residual galactic foreground would have on our results by repeating the analysis using maps that have not had the galactic components suppressed. The cluster monopole and dipole that we obtain from these maps is therefore an upper limit on any residual galactic contamination in our results.

Figures 16 and 17 show the sum of the filtered PSM simulated maps. Both maps show residual galactic emission that is not visible in the WMAP maps (Figures 7 and 8) due to their foreground reduction procedure. The larger visible foreground signal in the PSM simulated map filtered with the UF is caused by galactic synchrotron. The synchrotron signal is smaller in the map filtered with the MF because the large-scale galactic signal is removed at each frequency by the MF, but remains in the maps filtered by the UF. When the filtered maps at each frequency are combined, the signals from the galactic components are not canceled since the filters are designed to only cancel a signal that has a thermal SZ spectrum.

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Because the maps containing only the individual contribution of a particular foreground component are available for the PSM simulations, we have also run our analysis pipeline on each component separately to identify the source of any bias. The results when all clusters are included in the dipole fit are shown in Table 2.

Table 2. Monopole and Dipole in the PSM Simulations^a

Component	Filter	Monopole		Dipolec						$\sqrt{C_1}$ (μK)	95% Confidence Limit (μK)
		a0 (μK)	σb	x (μK)	σb	y (μK)	σ	z (μK)	σ
Synchrotron	Matched	0.40	0.038	0.054	0.0020	0.16	0.0079	−0.086	−0.0048	0.19	62
Dust	Matched	0.36	0.034	0.034	0.0013	0.38	0.019	0.018	0.0010	0.38	62
Free–free	Matched	0.042	0.0040	−0.46	−0.017	0.38	0.019	−0.042	−0.0023	0.60	62
All galactic	Matched	0.80	0.077	−0.37	−0.014	0.92	0.046	−0.11	−0.0061	1.00	62
Synchrotron	tSZ bias removing	−469	−5.7†	−717	−4.0††	156	1.0	−120	−1.0	743	432
Dust	tSZ bias removing	58	0.70	170	0.94	−159	−1.1	70	0.58	244	432
Free–free	tSZ bias removing	−101	−1.2	188	1.0	177	1.2	−29	−0.24	260	432
All galactic	tSZ bias removing	−513	−6.2	−357	−2.0	174	1.2	−80	−0.66	405	432

Notes. ^aIn the redshift 0.00–1.00 shell. ^bThe uncertainty estimates are obtained from CMB and noise realizations. ^cThe dipole basis directions are x (l = 0°, b = 0°), y (l = 90°, b = 0°), and z (l = 0°, b = 90°).

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We find no significant bias in the monopole or dipole of the maps filtered by the MF. The combined map filtered by the UF has the largest galactic contribution from synchrotron emission, with a −5.7σ signal in the monopole (labeled † in the table) and a −4σ signal in the dipole direction that points toward the center of the galaxy (labeled ^†† in the table). The dust and free–free monopole signals largely cancel resulting in a −6.2σ monopole signal. The significant dipole from synchrotron emission is partially canceled by the dust and free–free signals, resulting in a dipole signal that is not significant and would not be considered a detection.

As a further check, we repeat the cluster dipole analysis with the cluster coordinates rotated by one degree away from the true cluster locations. We find that the emission from the galactic components is not significantly changed. Since the emission is not localized at galaxy cluster locations, we expect the galactic signals we find to be suppressed by the WMAP foreground reduction procedure, and so we expect the galactic signals in Table 2 to be an upper limit on residual galactic emission in the WMAP maps.

Table 3 and Figure 18 show our KAKE filter pipeline results, Table 4 and Figure 19 show the WF results, Table 5 and Figure 20 show the MF results and Table 6 and Figure 21 show the UF results. The points in Figures 18–21 are from the WMAP seven-year data. The green line is the noise bias, the red and blue lines are the 95% and 99.7% confidence limits that there is no bulk flow, which are estimated from our realizations containing CMB and instrumental noise. We find smaller errors for our Wiener filter than for the KAKE filter since our filter suppresses more power at ℓ ≲ 100 as shown in Figure 24. The noise at these multipoles is dominated by the CMB and is correlated between channels (Keisler 2009). We do not find a significant dipole in any of the redshift shells using any of the filters. In Figure 21, some of the points are close to the 99.7% confidence limit. Since this filter is much less sensitive than the other filters which have results that are consistent with noise, this result cannot be due to a bulk flow. Instead it is likely that it is at least partly due to radio point-source contamination which in Section 5.2 we estimated to cause a ∼1σ bias in each of the a_1m components in maps filtered by the UF. We have repeated our analysis for the WMAP five-year data and find no significant cluster dipole. We find that for the MF the CMB and instrument noise contribute approximately equally. For the UF the noise is dominated by the instrument noise, with approximately 90% contribution.

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Table 3. Results from the KAKE Filter^a

zmin	zmax	〈z〉	zmedian	zσ	Ncl	Monopole, a0 (μK)b	a1x (μK)	a1y (μK)	a1z (μK)	Dipole Amplitude, $\sqrt{C_1}$ (μK)
0.0	0.02	0.014	0.015	0.0048	28	−1.4 ± 3.8	−12. ± 10.	−12. ± 8.7	−6.6 ± 8.7	18.
0.0	0.025	0.016	0.016	0.0057	36	−6.2 ± 3.5	−8.6 ± 8.7	−11. ± 7.6	−5.6 ± 5.5	15.
0.0	0.03	0.019	0.019	0.0075	50	−5.8 ± 3.1	−1.8 ± 6.3	−4.3 ± 6.3	−2.4 ± 4.4	5.2
0.0	0.04	0.027	0.030	0.0100	95	−1.6 ± 2.2	−3.2 ± 4.9	−4.9 ± 4.3	−5.6 ± 3.1	8.1
0.0	0.05	0.033	0.035	0.012	139	−2.3 ± 2.1	−2.1 ± 4.9	−6.0 ± 3.8	−1.5 ± 3.0	6.6
0.0	0.06	0.039	0.041	0.015	192	−3.9 ± 1.9	−2.4 ± 3.8	−4.9 ± 3.5	0.017 ± 2.8	5.5
0.0	0.08	0.050	0.052	0.019	294	−4.6 ± 1.7	−1.2 ± 3.0	−3.9 ± 3.0	−1.9 ± 2.5	4.5
0.0	0.12	0.067	0.066	0.029	445	−4.4 ± 1.5	−0.82 ± 2.5	−2.7 ± 2.3	−1.1 ± 2.2	3.0
0.0	0.16	0.080	0.076	0.039	546	−3.9 ± 1.5	−0.77 ± 2.2	−2.7 ± 2.1	−0.86 ± 1.9	3.0
0.0	0.20	0.092	0.083	0.049	619	−3.6 ± 1.4	0.024 ± 1.9	−2.6 ± 2.0	−0.31 ± 1.7	2.6
0.0	1.0	0.12	0.097	0.079	736	−3.6 ± 1.2	0.55 ± 1.7	−2.1 ± 1.8	−0.26 ± 1.6	2.2
0.05	0.30	0.13	0.12	0.064	578	−3.9 ± 1.3	1.2 ± 1.7	−1.4 ± 1.9	−0.098 ± 1.7	1.9
0.12	0.30	0.19	0.18	0.048	271	−2.1 ± 1.5	2.9 ± 2.2	−1.8 ± 2.4	0.56 ± 2.0	3.4

Notes. ^aThe monopole and dipole are calculated within 15' of the cluster centers. In Tables 5 and 6 the monopole and dipole are calculated in the central pixel of the clusters. ^bThis filter is not normalized (see discussion in text).

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Table 4. Results from Our Wiener Filter^a

zmin	zmax	〈z〉	zmedian	zσ	Ncl	Monopole, a0 (μK)b	a1x (μK)	a1y (μK)	a1z (μK)	Dipole Amplitude, $\sqrt{C_1}$ (μK)
0.0	0.02	0.014	0.015	0.0048	28	0.40 ± 2.5	−4.8 ± 5.3	0.73 ± 5.6	−0.97 ± 5.0	5.0
0.0	0.025	0.016	0.016	0.0057	36	−1.9 ± 2.2	−3.8 ± 4.7	−2.1 ± 5.0	−3.2 ± 3.7	5.4
0.0	0.03	0.019	0.019	0.0075	50	−3.0 ± 1.9	0.066 ± 3.9	0.15 ± 3.9	−1.4 ± 2.8	1.4
0.0	0.04	0.027	0.030	0.0100	95	−2.5 ± 1.3	−0.58 ± 2.3	−1.7 ± 2.5	−1.6 ± 1.9	2.4
0.0	0.05	0.033	0.035	0.012	139	−2.2 ± 1.0	−0.92 ± 2.1	−1.3 ± 2.0	0.096 ± 1.5	1.6
0.0	0.06	0.039	0.041	0.015	192	−3.4 ± 0.91	−2.3 ± 1.6	−1.8 ± 1.7	2.1 ± 1.3	3.6
0.0	0.08	0.050	0.052	0.019	294	−3.8 ± 0.79	−2.3 ± 1.4	−1.9 ± 1.5	−0.59 ± 1.0	3.0
0.0	0.12	0.067	0.066	0.029	445	−3.5 ± 0.62	−1.6 ± 1.1	−1.1 ± 1.1	−0.29 ± 0.78	2.0
0.0	0.16	0.080	0.076	0.039	546	−3.7 ± 0.55	−1.3 ± 0.91	−1.1 ± 0.96	−0.41 ± 0.68	1.7
0.0	0.20	0.092	0.083	0.049	619	−3.7 ± 0.51	−0.91 ± 0.85	−0.66 ± 0.91	−0.59 ± 0.63	1.3
0.0	1.0	0.12	0.097	0.079	736	−3.7 ± 0.45	−0.87 ± 0.80	−0.95 ± 0.82	−0.47 ± 0.61	1.4
0.05	0.30	0.13	0.12	0.064	578	−4.1 ± 0.50	−1.1 ± 0.96	−0.93 ± 0.91	−0.74 ± 0.68	1.6
0.12	0.30	0.19	0.18	0.048	271	−4.0 ± 0.66	−0.19 ± 1.6	−1.2 ± 1.4	−0.99 ± 1.2	1.5

Notes. ^aThe monopole and dipole are calculated within 15' of the cluster centers. In Tables 5 and 6 the monopole and dipole are calculated in the central pixel of the clusters. ^bThis filter is not normalized (see discussion in text).

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The UF is an order of magnitude less sensitive than the MF. This is due to the channels being combined in a way that is not optimal for maximizing the kSZ signal, but results in cancellation of the tSZ signal. The reduction in sensitivity of the UF is largely due to the limited frequency coverage of the maps we use, and so an experiment with greater frequency coverage, such as Planck, would perform better with this filter.

Although the dipole amplitude we find is consistent with zero, our limits to the bulk flow velocity are tighter in some directions than in others. Figure 22 shows the 95% confidence upper limit to the bulk flow over the whole sky using the results from the MF with all clusters included in the fit. Table 7 gives the upper limit to the flow in the direction of other well-known dipoles. The upper limit in all directions is above the limit expected by cosmic variance in the ΛCDM model and is above the measured low redshift flow.

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Table 5. Results from the Matched Filter

zmin	zmax	〈z〉	zmedian	zσ	Ncl	Monopole, a0 (μK)a	a1x (μK)	a1y (μK)	a1z (μK)	Dipole Amplitude, $\sqrt{C_1}$ (μK)
0.0	0.02	0.014	0.015	0.0048	28	59 ± 65	−39 ± 117	−10 ± 140	−56 ± 101	69
0.0	0.025	0.016	0.016	0.0057	36	55 ± 57	−111 ± 110	−79 ± 126	−68 ± 70	152
0.0	0.03	0.019	0.019	0.0075	50	−15 ± 44	−82 ± 97	−110 ± 101	−1.8 ± 60	137
0.0	0.04	0.027	0.030	0.0100	95	−22 ± 33	−20 ± 60	−107 ± 70	−16 ± 40	110
0.0	0.05	0.033	0.035	0.012	139	−41 ± 27	−45 ± 58	−44 ± 55	13 ± 36	65
0.0	0.06	0.039	0.041	0.015	192	−62 ± 24	−16 ± 49	−23 ± 43	33 ± 29	43
0.0	0.08	0.050	0.052	0.019	294	−63 ± 18	−39 ± 37	−38 ± 32	−7.1 ± 24	55
0.0	0.12	0.067	0.066	0.029	445	−73 ± 13	−22 ± 28	−6.1 ± 28	−0.059 ± 20	23
0.0	0.16	0.080	0.076	0.039	546	−81 ± 12	−18 ± 28	1.8 ± 25	−1.4 ± 18	19
0.0	0.20	0.092	0.083	0.049	619	−85 ± 12	−14 ± 26	5.7 ± 23	−9.3 ± 17	18
0.0	1.0	0.12	0.097	0.079	736	−91 ± 11	−5.7 ± 24	−1.3 ± 21	−8.7 ± 17	11
0.05	0.30	0.13	0.12	0.064	578	−99 ± 12	−2.2 ± 26	9.5 ± 24	−16 ± 19	19
0.12	0.30	0.19	0.18	0.048	271	−112 ± 17	11 ± 37	−0.13 ± 37	−30 ± 27	32

Note. ^aThe monopole is contaminated by foregrounds with levels consistent with simulations.

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Table 6. Results from the tSZ Bias Removing Filter

zmin	zmax	〈z〉	zmedian	zσ	Ncl	Monopole, a0 (μK)	a1x (μK)	a1y (μK)	a1z (μK)	Dipole Amplitude, $\sqrt{C_1}$ (μK)
0.0	0.02	0.014	0.015	0.0048	28	118 ± 425	−242 ± 869	−1045 ± 866	−29 ± 746	1073
0.0	0.025	0.016	0.016	0.0057	36	−156 ± 326	269 ± 756	−1268 ± 862	−449 ± 537	1373
0.0	0.03	0.019	0.019	0.0075	50	73 ± 267	1256 ± 598	−29 ± 610	−93 ± 420	1259
0.0	0.04	0.027	0.030	0.0100	95	−223 ± 193	938 ± 410	−281 ± 384	159 ± 314	992
0.0	0.05	0.033	0.035	0.012	139	−164 ± 169	616 ± 394	−310 ± 298	170 ± 241	710
0.0	0.06	0.039	0.041	0.015	192	−22 ± 140	694 ± 344	−200 ± 263	−74 ± 224	726
0.0	0.08	0.050	0.052	0.019	294	−60 ± 122	660 ± 280	−230 ± 228	15 ± 166	699
0.0	0.12	0.067	0.066	0.029	445	−38 ± 102	607 ± 225	−200 ± 180	−131 ± 133	653
0.0	0.16	0.080	0.076	0.039	546	−56 ± 95	591 ± 199	−147 ± 159	−182 ± 135	636
0.0	0.20	0.092	0.083	0.049	619	−14 ± 89	549 ± 183	−123 ± 154	−214 ± 128	603
0.0	1.0	0.12	0.097	0.079	736	40 ± 77	486 ± 165	−68 ± 140	−178 ± 119	522
0.05	0.30	0.13	0.12	0.064	578	98 ± 92	367 ± 185	66 ± 150	−320 ± 134	492
0.12	0.30	0.19	0.18	0.048	271	189 ± 129	230 ± 272	343 ± 230	−418 ± 184	588

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The monopole result from the MF is expected to be strongly contaminated with thermal SZ emission, and we find a result consistent with tSZ contamination: −6868 ± 838 km s⁻¹. In the maps filtered with the UF we find a monopole consistent with zero: 4692 ± 8947 km s⁻¹. We find no moments between multipoles 2 and 5 that are significantly different from zero.

The SuZIE II experiment has placed limits on the bulk flow using pointed observations of galaxy clusters (Benson et al. 2003; Benson 2004). SuZIE II made simultaneous measurements of the SZ effect in three frequency bands, centered at 145 GHz, 221 GHz, and 355 GHz, from the Caltech Submillimeter Observatory, with a 15 beam FWHM at all frequencies. A significant detection of the thermal SZ signal was made in 15 clusters; no significant detection of the kinetic SZ signal was made in any of the clusters observed.

Benson et al. (2003) fit a thermal and kinetic SZ spectrum to the measured cluster temperature at each frequency. No significant cluster dipole was found in the SuZIE cluster sample. We find that the SuZIE bulk flow limits are more sensitive than our results with the MF. We have combined the cluster dipole from the SuZIE data with the dipole limit we find in the WMAP data. When combining the data we weight the dipole amplitude in each of the x, y, and z directions by the inverse noise variance. We find that the 95% confidence limit in the direction of the CMB dipole is increased from the SuZIE result of 1500 km s⁻¹ to 1800 km s⁻¹ with the combined data, because our best-fit dipole points are in a different direction to the SuZIE dipole. With the UF our result is given less than 1% weight, and the combined result is unchanged from the SUZIE dipole measurement. The cluster monopole velocity in the SuZIE data using all of the clusters is −570 ± 320. The result with the UF is not sensitive enough to tighten this constraint.

We have verified that we can reproduce the KAKE pipeline by comparing our filtered maps with the publicly available maps used for the Kashlinsky et al. (2011) analysis. We find that the mean difference between the W-band maps passed through our pipeline and the filtered maps from the KAKE pipeline is −1.5 × 10⁻¹⁰ mK indicating that our pipeline can reproduce their results. As a further check we calculate the spectrum of the filtered W1 channel map, and the absolute value of the difference between our filtered map spectra and the spectra from the KAKE maps in Figure 23. In Figure 24, we show the spectra of the WMAP maps passed through the two pipelines. The signal to noise is low at ℓ < 100, which is why both filters suppress these multipoles. The KAKE filter is larger at ℓ < 100 because the filter does not remove the fluctuations in the map that are due to cosmic variance. This leaves noise in the maps that is correlated between channels (Keisler 2009). By further suppressing the map at ℓ < 100 our filter reduces the noise in our dipole measurement by a factor of ∼2 while not significantly affecting the signal, as can be seen in Figure 11.

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Since we model our clusters as point sources we can calculate the profile of the cluster signal in the filtered maps. In Figure 25, we show the integrated signal out to a given radius in the Q1, V1, and W1 channels after filtering with the Wiener filter. The signal is normalized so that if no filtering were applied the result would approach unity at large radius. The fact that the signal peaks at 10'–15' justifies our choice of a 15' aperture within which to calculate the monopole and dipole.

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In this section, we remove clusters with bolometric luminosity L_X < 2 × 10⁴⁴ erg s⁻¹ from our sample and use the WMAP Kp0 mask instead of the seven-year extended temperature analysis mask to better compare with the KAKE results. As before we remove the monopole and dipole of the map outside of the mask before fitting for the cluster monopole and dipole. We find that the sample of clusters satisfying the luminosity cut and the redshift cut of z ⩽ 0.16, 0.2, 0.25 is almost identical to the sample used in the KAKE analysis. There are some differences in the REFLEX, BCS, and BCSe catalogs for clusters with luminosities close to L_X = 2 × 10⁴⁴ erg s⁻¹, and some differences for clusters in the CIZA catalog that are near to the WMAP galactic mask boundary. We show the differences we find in Table 8.

To check that these differences do not change the results by a large amount we have calculated the cluster dipole using the KAKE filter with both the KAKE cluster sample and our own sample. We find the results shown in Table 9 when averaging over the Q, V, and W bands.

All of the results have less than 3σ significance with some of the a_1y components having ∼2.5σ significance. The results averaged over the W-band channels only (but with the same filter and cluster samples used to produce Table 9) are shown in Table 10.

These results have greater significance than the combined channel results. We still find nothing with more than 3σ significance, although the z = 0.0–0.20 shell a_1y value has 2.9σ significance. The result from Kashlinsky et al. (2011) is shown in Table 11 which is similar to our result above. These results are more significant than those obtained using the full cluster sample. This cannot be fully explained by the higher luminosity clusters having larger optical depths. In Table 12, we show the average optical depth and number of clusters in the sample both with and without the luminosity cut.

Although the average optical depth is higher with the luminosity cut, the noise (which is roughly proportional to $1/\sqrt{N_{\rm clusters}}$) is also higher and so we expect a similar sensitivity both with and without the luminosity cut. However, when we calculate the cluster dipole in maps filtered with our Wiener filter with our cluster sample we find much lower significance as shown in Table 13.

We conclude from these results that we can accurately reproduce the KAKE pipeline and do not detect a bulk flow with greater than 3σ significance in the WMAP maps. The significance of the results is lower when we use our Wiener filter and the full cluster sample than when using the KAKE filter and the cluster sample that has low luminosity clusters removed. By comparing with Table 3 we see that the results with the smaller cluster sample are more significant than those with the full cluster sample. However, since we would expect any bulk flow to be of similar significance with either of the cluster samples we conclude that there is no significant detection of a bulk flow.