Chapter 1 Cosmology
Z- Spec has been operating as a PI instrument at the Cal- tech Sub-millimeter Observatory (CSO) on Mauna Kea, Hawaii
3.4 Noise Characterization and Point Source Removal
The scaling relation analysis depends critically on accurate noise characterization. This is because a misestimate of the noise will not only affect the derived uncertainty estimates, but will also bias the determination of the best-fit scaling relation. The Bolocam SZE cluster images contain noise from a variety of sources: atmospheric fluctuations, instrument noise, flux calibration, primary CMB anisotropies, and emission from the non-uniform distribution
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Fig. 3.—Images of the best-fit spherical Nagai model for MS 0451.6-0305. The left image is the model and the right image is the model after being processed through our data reduction pipeline, which high-pass filters the image in a complex way. This filtering significantly reduces the peak decrement of the cluster and creates a ring of positive flux atr&2 arcmin. Note that the processed image is not quite azimuthally symmetric.
Fig. 4.—The magnitude of the transfer function for MS 0451.6- 0305 as a function of Fourier wavenumber u = 1/λ. At large scales, or small u, the measurement error is negligible and the error bars provide an indication of the azimuthal variation. At u >0.75 arcmin−1, the measurement error becomes non-negligible and we set the transfer function equal to 1. Note that this az- imuthally averaged transfer function is for display purposes only;
we have used the full two-dimensional transfer function throughout our analysis.
processed cluster image using the transfer function (see Section 5), which has been approximated as 1 at small angular scales, produces an image that is slightly biased compared to the input cluster. The residuals between these two images are approximately white, with an RMS of.0.1µKCMB. This transfer-function-induced bias is negligible compared to our noise, which has an RMS of
≃10µKCMB.
As noted above, the transfer function (weakly) de- pends on the profile of the cluster; larger clusters are more heavily filtered than smaller clusters. Therefore, we determine a unique transfer function for each cluster using the best-fit elliptical Nagai model for that clus- ter (Nagai et al. 2007, hereafter N07). The details of this fit are given in Section 4. Since the transfer func- tion depends on the best-fit model, and vice versa, we determine the best-fit model and transfer function in an iterative way. Starting with a generic cluster profile, we
first determine a transfer function, and then fit an ellip- tical Nagai model using this transfer function (i.e., the Nagai model parameters are varied while the transfer function is held fixed). This process is repeated, using the best-fit model from the previous iteration to calculate the transfer function, until the best-fit model parameters stabilize. This process converges fairly quickly, usually after a single iteration for the clusters in our sample.
The model dependence of the resulting transfer function is quantified in Section 6.1.
3.4. Noise estimation
In order to accurately characterize the sensitivity of our images, we compute our map-space noise directly from the data via 1000 jackknife realizations of our clus- ter images. In each realization, random subsets of half of the≃100 observations are multiplied by −1 prior to adding them into the map. Each jackknife preserves the noise properties of the map while removing all of the as- tronomical signal, along with any possible fixed-pattern or scan-synchronous noise due to the telescope scanning motion4. Since these jackknife realizations remove all as- tronomical signal, we estimate the amount of astronom- ical noise in our images separately, as described below.
After normalizing the noise estimate of each map pixel in each jackknife by the square root of the integration time in that pixel, we construct a sensitivity histogram, in µKCMB-s1/2, from the ensemble of map pixels in all 1000 jackknifes. The width of this histogram provides an accurate estimate of our map-space sensitivity (see Fig- ure 5). We then assume that the noise covariance matrix is diagonal5and divide by the square root of the integra- tion time in each map pixel to determine the noise RMS in that pixel. This method is analogous to the one used in S09, where it is described in more detail.
There is a non-negligible amount of noise in our
4 We show that there is no measurable fixed-pattern or scan- synchronous noise in our data later in this section and in Sec- tion 6.2.
5 This approximation is justified for our processed data maps in Section 6.2. Note that the approximation fails for our deconvolved images (see Section 5), which contain a non-negligible amount of correlated noise. We describe how this correlated noise is accounted for in our results in Section 5.
Figure 3.11 The magnitude of the azimuthally-averaged transfer function for MS 0451.6- 0305 as a function of Fourier wavenumber u = 1/λ. At large scales, the error bars are good indicators of the rms azimuthal variation of the map. At small scales, however, the cluster model has little signal/constraining power and the measurement noise increases due to numerical uncertainty. The transfer function is therefore set equal to 1 atu >0.75arcmin−1. The full 2-dimensional transfer function is used for the Bolocam data analysis. More details can be found in Sayers et al. [255].
of fore- and background galaxies. Section 3.5.1 reviews how additional uncertainties due to the deconvolution of the signal transfer function are accounted for. Section 3.7 characterizes the uncertainties of the YSZ estimates that arise from the uncertainty in the overdensity radius used for integration.
Noise realizations are created for each cluster by multiplying a randomly chosen subset of half of the ∼50-100 observations by −1 prior to coadding them together. A total of 1000 such jackknife noise realizations are created for each cluster. The noise realizations contain no astronomical signal but retain the statistical properties of the atmospheric and instrumental noise. To account for noise from primary CMB fluctuations and unresolved galaxies, a random realization of the 140 GHz astronomical sky is added to each noise realization, using the measured angular power spectrum from the SPT [138, 241] under the assumption that the fluctuations are Gaussian. The resulting noise realizations are statistically indistinguishable fromBolocam maps of blankfields, thereby verifying that this noise model provides an adequate description of the Bolocam data. These noise realizations provide the basis to which all of the modeled astronomical noise, discussed below, is added.
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Dusty star-forming regions and radio-bright point sources will bias YSZ measurements low. Fortunately, the 140 GHz regime is located in a valley of low background contamination between radio and thermal dust emission. The frequency dependence of the flux density, Sν, for radio sources can be approximated as:
Sν ∝ν−α, (3.2)
where 0.5. α.1.4 for the BOXSZ sample. The flux density of dusty thermal sources can be characterized with a gray-body spectrum (not quite in the Raleigh-Jeans limit), peaking at 10’s of Kelvin,
Sν ∝να, (3.3)
where typically alpha&2. Several of the clusters in the BOXSZ sample contain signal from bright radio galaxies that are not accounted for in the SPT power spectrum. In particular, the brightest cluster galaxy (BCG) is often a bright radio emitter, and this emission will systematically reduce the magnitude of the SZE decrement towards the cluster.
A full description of the methodology in which we systematically characterize and sub- tract the flux of these bright radio galaxies is given in Sayers et al. [257], and the general procedure is described below. A total of 6 bright radio sources are detected in the Bolocam 140 GHz maps for the entire cluster sample. These maps are used to constrain the normal- ization of a point-source template, centered on the coordinates of the detected radio source in the 1.4 GHz NVSS radio survey [58], and the best-fit template is subsequently subtracted from the data. In addition to this, NVSS-detected sources near the centers of 11 clusters in the BOXSZ sample have extrapolated 140 GHz flux densities greater than 0.5 mJy. This is the threshold found to produce more than a 1% bias in the SZE signal of the cluster, and an effort is made to remove them from the cluster signal. All of the undetected sources are subtracted using the extrapolated flux densities based on a combination of 1.4 GHz NVSS and 30 GHz OVRO/BIMA/SZA measurements.
Furthermore, the uncertainties of these subtracted point sources are accounted for in the estimated error of theYSZ parameters. This is performed by adding to each noise realization
introduced in Section 3.4 the corresponding point-source template, multiplied by a random value drawn from a Gaussian distribution. The standard deviation of the distribution is equal to either the uncertainty on the normalization of the detected sources, or is based on a fixed 30% uncertainty on the extrapolated flux density for the undetected radio sources.