Map Calculation

Each pixel in the map encompasses a range of 20 arc seconds in RA and 20 arc seconds in declination. Using the RA and dec for each frame and each bolometer we determine which data are included in each pixel. To determine the value of the pixel we perform a weighted average of the data within each pixel, given by equation 6.1.

mp = PN i=0di/σ 2 i PN i=01/σ2i (6.1)

where the sum is over the data samples assigned to pixel p,di is a time ordered data

sample, and σi is the noise for that data sample. The noise for each sample, σi, is

calculated from the PSD of the data. The PSD for each bolometer and each sub-scan is calculated after the sky subtraction is performed. These PSD’s are multiplied by the Gaussian beam in Fourier space, and the result is integrated to give a measure of the variance for that data. Each data sample in the given sub-scan for that bolometer is assigned the same noise estimate.

6.1.3

Map Noise

The overall noise of the map is calculated in two ways. The first and easiest is the root mean square (RMS) of the pixel values. This gives an estimate of the noise of the map if we assume there is no signal. Any signal in the map would increase the RMS, a fact we take advantage of in the jack-knife tests described in the next section. The second method for measuring the noise of the map is more accurate. We fit a

Field RMSave [mKCM B] RMSdif f [mKCM B] q RM S2 ave−RM Sdif f2 Lynx 0.744 0.722 0.180 SDS1 2.45 2.40 0.492

Table 6.1: RMS comparison of jack-knifed data

Gaussian to the histogram of the pixel values, and take the standard deviation to be the noise of the map. This method reduces contamination from outlying pixels and any signal in the map.

6.2

Jack-Knife Tests

The first question we ask of the data is whether any of the structure is real. The easiest way to do this is to split the data into two halves, map the data, and compare the RMS of the average and difference of the two maps. The difference map must be multiplied by 0.5 so that the weighting is the same as the average. If there is real structure then the RMS of the difference should be lower than that of the average. If the maps are entirely noise, then the RMS of the two maps should be identical. We followed this procedure for the two blank fields observed by Bolocam, Lynx and SDS1. The data were divided in half by taking the data from every other observation. This is because the weather steadily worsened over the course of the observing run, and the first half of the run would have a lower RMS for this reason, unrelated to the structure in the map. The data for each half of the observation were cleaned of sky noise using PCA sky subtraction, described in chapter 5, before they were mapped. The resulting maps are shown in in figures 6.1 and 6.2. The RMS of the average and difference maps for each blank field are shown in table 6.1.

The RMS for the average and difference maps for both blank regions of the sky were similar. This implies that the data is primarily noise.

Figure 6.1: The average and difference maps of the two halves of the Lynx blank sky observations.

02 20 24 02 19 12 02 18 00 02 16 48 02 15 36 Right Ascension (HH MM SS) −05 12 00 −05 06 00 −05 00 00 −04 54 00 Declination (DD MM SS) SDS1 Difference Map Epoch 2000 Coordinates 02 20 24 02 19 12 02 18 00 02 16 48 02 15 36 Right Ascension (HH MM SS) −05 12 00 −05 06 00 −05 00 00 −04 54 00 Declination (DD MM SS) SDS1 Average Map Epoch 2000 Coordinates

Figure 6.2: The average and difference maps of the two halves of the SDS1 blank sky observations.

6.3

Source Extraction

We can improve the signal-to-noise ratio for compact sources by taking advantage of our knowledge of the spatial structure of the expected sources. This technique is known as spatial filtering.

6.3.1

Optimal Filter

The filter employed on the Bolocam data is an optimal or matched filter. Herranz et al. (2002a) provides an overview of this filter as compared to scale-adaptive filters. If we consider a generic symmetric filter ψ(~x;R,~b) where R defines the scale of the filter, and~b the translation from the origin:

ψ(~x;R, b) = 1 Rnψ ~x−~b R ! (6.2)

we can define the filtered time stream as the convolution of the original map, m(~x) with the the filter

w(R,~b) = Z

d~xm(~x)ψ(~x;R,~b). (6.3) In order for ψ(~x;R,~b) to be an optimal filter for the detection of a source s(~x) at the origin characterized by a scale R0, it must satisfy two conditions:

1. < w(R0,0) >= s(0) ≡ A, the amplitude of the source, or the filtered time

stream is an unbiased estimator of the amplitude of the source

2. The variance of w(R,~b) has a minimum at the scale of the source, R0

A filter with these conditions generates the largest gain in signal to noise ratio (SNR) when convolved with the original map. If we apply these constraints to the generic filter we defined in equation 6.2, and assume that we know the profile of the source we expect to find, we can show that the optimal filter in Fourier space is given by

ψ(q) = 1

a τ(q)

where τ(q) = s(q)/A is the profile of the source in Fourier space, P(q) is the noise spectrum, and a is the normalization required to make the filter unbiased. Equa- tion 6.4 assumes that the source profile is spherically symmetric. For a source with amplitude of 1 at the origin we know

Z

dqτ(q)ψ(q) = 1, (6.5)

which yields a value for the normalization constant a for a given source profile and noise power spectrum:

a= Z dqτ 2_(q) P(q). (6.6)

6.3.2

Source Profile

The most common profile used to describe galaxy cluster is the β-model. This model provides a simple cluster profile that has been seen to fit the available SZ data well (Mohr et al. 1999).

τcluster(x) =

1

[1 + (x/rc)2]3β/2−1/2

. (6.7)

where rc is the core radius of the cluster, and β is usually taken to be 2/3. Unfortu-

nately this model has a diverging Fourier transform. (Hobson and McLachlan 2003) describe a modified β-model that avoids this problem.

τcluster(x) = rcrv rv −rc 1 p r2 c +x2 − _p 1 r2 v +x2 ! , (6.8)

where rc is the core radius as before, and rv is a so called virial radius. This virial

radius is not the actual virial radius of the cluster, but is used to truncate the cluster model at large radii. The authors use rv = 3rc, and we follow their convention. The

a Gaussian. τ(q) = τcluster(q)b(q) = rcrv rv −rc e−rcq−_e−rvq q θ2e−(qθ)2/2 (6.9)