I will focus on the case of cosmic strings. In this case the contribution of the latest long string comes in the form of a step-like discontinuity (Kaiser & Stebbins (1984), Gott (1985)) coherent on large angular scales. As a toy model we may first consider a one dimensional pixel array of standardized, scale invariant Gaussian fluctuations with a superposed temperature discontinuity of amplitude (Perivolaropoulos (1997)).
A statistical variable designed to pick up the presence of this step is the Sample Mean Difference (SMD) which assigns to each pixel of the map the difference of the mean of pixels minus the mean of pixels . It is straightforward to show (Perivolaropoulos (1997)) that
where labels the out of the random variables of the pixel map and is the location of the superposed coherent discontinuity. The SMD average statistic Z is defined as the average of over all
It is straightforward to show that the mean over many realizations and locations of the step function is and the variance of depends both on the number of pixels and on the step function amplitude
The condition for detectability of the coherent step discontinuity at level is that is larger than the standard deviation of which implies that for where is measured in units of the standard deviation of the underlying Gaussian map. It is straightforward to apply a similar analysis for the more conventional statistics skewness and kurtosis. That analysis (Perivolaropoulos (1997)) shows that the minimum value of detectable at the level is about an order of magnitude larger. It is therefore clear that SMD statistical variable is particularly effective in detecting coherent step-like discontinuities superposed on Gaussian CMB maps.
A detailed understanding of the effectiveness of the SMD statistic requires the use of Monte-Carlo simulations. In order to verify the analytical results for the mean and variance of the SMD variable I first applied this statistic on one-dimensional Monte-Carlo maps of scale invariant Gaussian fluctuations with step function superposed. The results were in good agreement with the analytical predictions shown above and are described in detail in Perivolaropoulos (1997). Here I will only discuss the two dimensional Monte Carlo simulations.
Figure 1: A standardized two dimensional pixel array of scale invariant Gaussian fluctuations. No step function has been superposed.
Figures 1 and 2 show pixel maps of standardised Gaussian scale invariant fluctuations without (1) and with (2) a coherent step function superposed. The amplitude of the superposed coherent seed is .
Figure 2: The two dimesnsional array of Figure 2 with a superposed coherent step-discontinuity of amplitude defined by the random points and .
Uncorrelated noise has also been included with noise to signal ratio of 0.5. The scale invariant background was constructed in the usual way by taking its Fourier transform to be a Gaussian complex random variable. Its phase was taken to be random with a uniform distribution and its magnitude was a Gaussian random variable with 0 mean and variance equal to a scale invariant power spectrum. The SMD was obtained by randomly dividing the map in two sectors and taking the difference of the means of the two sectors. The SMD average was then obtained by averaging over many randomly chosen divisions for each map realization. Using 50 such map realizations I obtained the mean and the standard deviation of the statistics skewness and SMD average for several values of . The results are shown on Table 1 and indicate that the statistics skewness and kurtosis can not identify a coherent discontinuity of amplitude but would require a much larger amplitude for such identification.
Table 1: A comparison of the effectiveness of the statistics considered in detecting the presence of a coherent step discontinuity with amplitude relative to the standard deviation of the underlying Gaussian map. The SMD average was obtained after ignoring 150 pixels on each boundary of the Monte Carlo maps. The discontinuities were also excluded from these 300 pixels. This significantly improved the sensitivity of the SMD test.
On the other hand the SMD statistic can identify a coherent discontinuity at the to level with . For where is the mass per unit length of the string, is its velocity and is the relativistic Lorenz factor.
The main points I wanted to stress in this talk are the following: