The first COSMOSOMAS

In Hancock et al.[1] our most sensitive scans at declination $+40.0^\circ$ at 10, 15 and 33 GHz were analyzed. In the final stacked scans the average of the two channels (A+B)/2 was used, but also the difference was produced (A-B)/2 and denoted the noise scan as common sky structure in both should be remove leaving just noise. Even to the unaided eye there seems to be a clear signal present just by comparing final scans with the noise scans in figure . It is is perhaps not surprising that a simple excess variance test $\sigma_{CMB}^2 = \sigma_{(A+B)/2}^2 - \sigma_{(A-B)/2}^2$ gives sensible and consistent results as shown in table .

  

Figure: The highly sensitive declination $+40^\circ$ scans showing the first individual features (COSMOSOMAS) in the CMB.

A signal of approximately $40\mu$K seems to be present at all frequencies judging by the differences in the probabilities associated with the calculated $\chi^2$. Although the signal at 10 GHz is at this level it is not very significant due to the poor sensitivity of this experiment (notice that the scale on the 10 GHz scans is twice that of the others in order to accommodate the extra noise). Therefore we only combine 15 and 33 GHz to give the definitive scan of a feature in the CMB, which generates an RMS signal of $42\pm 9\mu$K within our triple beam configuration. To carry out a more rigorous analysis one must turn to methods which includes correlations between data points, which means using correlation analysis or likelihood analysis.

 

Table: Results of the excess noise test

$\nu$	$\sigma_{CMB}$	$\sigma_{A+B}$	$\sigma_{A-B}$	$\chi^2_{A+B}$	PA+B	$\chi^2_{A-B}$	P(A-B)
(GHz)	($\mu$K)	($\mu$K)	($\mu$K)
10	36$\pm$116	190$\pm$21	186$\pm$21	19.9	0.22	10.7	0.86
15	41$\pm$24	94$\pm$10	85$\pm$10	29.6	0.02	11.9	0.7
33	49$\pm$10	66$\pm$6	44$\pm$6	45.7	$1\times10^{-4}$	22.0	0.14
15+33	42$\pm$9	58$\pm$5	40$\pm$5	55.9	$3\times10^{-6}$	18.0	0.32

The weighted cross-correlation between two frequency channels can be found by $C(\theta) = (\sum_{i,j} \Delta T_i\Delta T_j w_i w_j)/(\sum_{i,j}w_i w_j)$, where i and j are indices on scans pixels at different frequencies and w is the weight calculated from the reciprocal variance of that pixel (ie. $w_i = 1/\sigma_i^2$).

  

Figure: The weighted cross-correlations of the 15 and 33 GHz signal (A+B)/2 and noise scans (A-B)/2.

As can be seen in figure  there is strong signal in the cross-correlation of the 15 and 33 GHz scans (A+B)/2 and absence of signal in cross-correlation of the noise scans (A-B)/2. The signal is consistent with the presence of a Harrison-Zeldovich spectrum normalized to an RMS power spectrum quadrupole $Q_{rms-ps} = 26\mu$K which is shown as a thick line. Monte Carlo simulations were carried out to calculate the 68% error bounds for the observing scheme for that model and no significant deviations are found from it. It should be pointed out that due to the triple beam configuration of the Tenerife experiment it only has sensitivity to a limited band of spherical harmonics, with the $8^\circ$ switch cutting out harmonics $\ell < 10$ and the $5^\circ$ beam-size smearing out harmonics $\ell > 30$. Consequently it has poor discriminating power as regards the angular spectrum shape and only samples that part falling within the above band, which is usually referred to as its window function. Therefore the correlation function above is dominated by the beam profile, but important in confirming that the signal is in the sky and not systematic.

More quantitative comparisons between different models are possible calculating the likelihood of the results for over a range of model parameters. The likelihood is calculated via equation .

$$L(\Delta T\vert{\rm sky model}) \propto \frac{\exp{-\frac{1}{2}\Delta T^TV^{-1}\Delta T}}{(\vert V\vert)^{\frac{1}{2}}},$$ (2)

where $\Delta T$ refers to the vector of measured temperatures differences within the scan, V is the covariance matrix which is the sum of the expect covariances between data points for the model under consideration and diagonal matrix of the instrumental variances. Two models for sky signal were used; a Gaussian Autocorrelation Function (GACF) and a power-law spectrum. The GACF is a simple-minded model no longer in favor, which assumes a Gaussian form for the autocorrelation function for sky variance $C(\theta) = C_0 \exp[\frac{-\theta^2}{2\theta_c^2}]$, where C₀ is the intrinsic sky variance and $\theta_c$ is the characteristic coherence angle. The main advantage of this model is the ease with which one could make inter-comparisons with other experiments by allowing for filtering with a Gaussian beam of dispersion $\sigma$ to give a modified covariance function $C_M(\theta,\sigma) = \frac{C_0 \theta_c^2}{2\sigma^2+\theta_c^2} \exp[\frac{-\theta^2}{2(2\sigma^2+\theta_c^2)}]$, which one can use to calculate the expected signal in experiment after taking into account the switch pattern. The above method is only really acceptable when there is no known form for the true sky ACF. A much better approximation is that of the power-law spectrum. Here the correlation function is calculated using spherical harmonics $C(\theta) = \frac{1}{4\pi}\sum_{\ell>2}^{\infty} C_\ell(2\ell+1)P_\ell(\cos(\theta))$, where $P\ell(\cos(\theta))$ are Legendre polynomials. Again filtering for a Gaussian beam can be done and the power-law dependence of $C_\ell$ on n and be substituted to give equation .

$$\begin{array}{rl} C_M(\theta,\sigma)= \sum_{l \geq 2}^{\infty... ...{3+n}{2}\right)} \right) \frac{Q_{rms-ps}^2}{5} \\ \end{array}$$ (3)

 

Table: Results of the likelihood tests

$\nu$(GHz)	Detected $\sqrt{C_o}$($\mu$K)	Detected Qrms-ps
10	29+20-30	15+9-12
15	48+16-16	23+10-8
33	60+16-16	29+8-7
15+33	54+14-10	26+6-6