Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.mrao.cam.ac.uk/ppeuc/astronomy/papers/perivolaropoulos.ps Äàòà èçìåíåíèÿ: Fri Jul 25 14:45:51 1997 Äàòà èíäåêñèðîâàíèÿ: Tue Oct 2 12:14:38 2012 Êîäèðîâêà:

A New Statistic for the Detection of Long Strings in
Microwave Background Maps
Leandros Perivolaropoulos 1
Department of Physics 2 , University of Crete,
71003 Heraklion, GREECE
A new statistic is discussed, designed to detect isolated coherent step­like discontinuities
produced by cosmic strings present at late times. As a background I superpose a scale
invariant Gaussian random field which could have been produced by a superposition of
seeds on all scales and/or by inflationary quantum fluctuations. The effects of uncorrelated
Gaussian random noise are also included. The statistical variable considered is the Sample
Mean Difference (SMD) between large neighbouring sectors of CMB maps, separated by
straight lines in two dimensional maps and points in one dimensional maps. I find that
the SMD statistics can detect at the 1oe level the presense of a long string with G¯(v s fl s ) =
1
8ú
( ffiT
T ) rms ' 0:5 \Theta 10 \Gamma7 while more conventional statistics like the skewness or the kurtosis
require a value of G¯ almost an order of magnitude larger for detectability at a comparable
level.
1 Introduction
The purpose of this talk is to introduce a new statistic Perivolaropoulos (1997) which is optimized
to detect the large scale non­Gaussian coherence induced by late long strings on CMB maps. The
statistical variable to use is the Sample Mean Diffrence that is the difference of the mean ffiT
T between
two large neighbouring regions of CMB maps. I will first briefly review the main predictions of models
for CMB fluctuations. Models based on inflation predict generically the existence of scale invariant
CMB fluctuations with Gaussian statistics which emerge as a superposition of plane waves with random
phases. On the other hand in models based on defects (for a pedagogic reviews see e.g. Brandenberger
(1992), Perivolaropoulos (1994)), CMB fluctuations are produced by a superposition of seeds and are
scale invariant (Perivolaropoulos (1993a), Allen et al. (1996)) but non­Gaussian (Perivolaropoulos
(1993b), Perivolaropoulos (1993c), Gangui g96(1996), Gott et al. (1990), Moessner, Perivolaropoulos
& Brandenberger (1994)). Observations have indicated that the spectrum of fluctuations is scale
invariant Smoot et al. (1992) on scales larger than about 2 ffi , the recombination scale, while there
seem to be Doppler peaks on smaller scales. These results are consistent with predictions of both
inflation (Bond et al. (1994)) and defect models (Perivolaropoulos (1993b), Allen et al. (1996) and
references therein) even though there has been some debate about the model dependence of Doppler
peaks in the case of defects.
Inflation also predicts Gaussian statistics in CMB maps for both large and small scales and this
is in agreement with Gaussianity tests made on large scale data so far. On small angular scale maps
where the number of superposed seeds per pixel is small topological defect models predict non­Gaussian
statistics. This non­Gaussianity however depends sensitively on both, the details of the defect network
at the time of recombination t rec and on the physical processes taking place at t rec . This large scale
coherence can induce specific non­Gaussian features even on large angular scales. The question of how
Gaussian are the topological defect fluctuations on large angular scales will be the focus of this talk.
The reason that the defect induced fluctuations appear Gaussian in maps with large resolution
angle is the large number of seeds superposed on each pixel of the map. This, by the Central Limit
Theorem, leads to a Gaussian probability distribution for the temperature fluctuations ffiT
T . Non­
Gaussianity can manifest itself on small angular scales comparable to minimum correlation length
between the seeds.
1 mailto:leandros@physics.uch.gr
2 http://www.physics.uch.gr/
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These arguments have led most efforts for the detection of defect induced non­Gaussianity towards
CMB maps with resolution angle less than 1 ffi (Gott et al. (1990), Ferreira & Magueijo (1997), Pe­
rivolaropoulos (1993b)). There is however a loophole in these arguments. They ignore the large scale
coherence induced by the latest seeds. Such large scale seeds must exist due to the scale invariance
and they induce certain types of large scale coherence in CMB maps. This coherence manifests itself
as a special type of non­Gaussianity which can be picked up only by specially optimized statistical
tests. Thus a defect induced CMB fluctuations pattern can be decomposed in two parts. A Gaussian
contribution ffiT
T g produced mainly by the superposition of seeds on small scales and possibly by infla­
tionary fluctuations and a coherent contribution ( ffiT
T ) c induced by the latest seeds. The question that
we want to address is: What is the minimum ratio ( ffiT
T ) c
( ffiT
T ) g
of the last seed contribution on ffiT
T over the
corresponding Gaussian contribution that is detectable at the 1oe to 2oe level.
2 Cosmic Strings
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 step­like 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 2ff
(Perivolaropoulos (1997)).
A statistical variable designed to pick up the presence of this step is the Sample Mean Difference
(SMD) Y k which assigns to each pixel of the map the difference of the mean of pixels 1:::k minus the
mean ffiT
T of pixels k+1:::n. It is straightforward to show (Perivolaropoulos (1997)) that
Y k = \Delta ¯
X k +2ff n \Gamma i 0
n \Gamma k
k 2 [1; i 0 ] (1)
Y k = \Delta ¯
X k +2ff i 0
k
k 2 [i 0 ; n \Gamma 1] (2)
where k labels the k th out of the n random variables of the pixel map and i 0 is the location of the
superposed coherent discontinuity. The SMD average statistic Z is defined as the average of Y k over
all k
Z = 1
n \Gamma 1
n\Gamma1 X
k=1
Y k : (3)
It is straightforward to show that the mean ¯
Z over many realizations and locations of the step
function is ¯
Z = ff and the variance of ¯
Z depends both on the number of pixels n and on the step
function amplitude ff
oe 2
Z = 2lnn
n
+ 1
3 ff 2 : (4)
The condition for detectability of the coherent step discontinuity at 1oe level is that ¯
Z is larger than
the standard deviation of Z which implies that ff ? 0:2 for n 'O(10 3 ) where ff 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 ff detectable at the 1oe level is about an order of magnitude
larger. It is therefore clear that SMD statistical variable is particularly effective in detecting coherent
step­like 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 one­dimensional Monte­Carlo 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.
Figures 1 and 2 show 30 \Theta 30 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 ff = 0:5.
2

0 5 10 15 20 25 30
0
5
10
15
20
25
30
Figure 1: A standardized two dimensional pixel array of scale invariant Gaussian fluctuations. No
step function has been superposed.
3

0 5 10 15 20 25 30
0
5
10
15
20
25
30
Figure 2: The two dimesnsional array of Figure 2 with a superposed coherent step­discontinuity of
amplitude ff = 0:5 defined by the random points (x 1 ; y 1 ) = (18:6;13:9) and (x 2 ; y 2 ) = (15:6;21:6).
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Table 1: A comparison of the effectiveness of the statistics considered in detecting the presence of a
coherent step discontinuity with amplitude 2ff 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.
ff Skewness Kurtosis SMD Average
0.00 0:01 \Sigma 0:10 2:96 \Sigma 0:15 0:01 \Sigma 0:24
0.25 0:01 \Sigma 0:09 2:95 \Sigma 0:15 0:28 \Sigma 0:26
0.50 0:02 \Sigma 0:14 2:94 \Sigma 0:18 0:63 \Sigma 0:29
1.00 0:03 \Sigma 0:30 2:78 \Sigma 0:30 1:21 \Sigma 0:46
Uncorrelated noise has also been included with noise to signal ratio of 0.5. The scale invariant
background X(i;j) was constructed in the usual way by taking its Fourier transform g(k 1 ; k 2 ) 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 ff. The results are shown on Table 1 and indicate that the statistics skewness and kurtosis can not
identify a coherent discontinuity of amplitude ff ! 1 but would require a much larger amplitude for
such identification.
On the other hand the SMD statistic can identify a coherent discontinuity at the 1oe to 2oe level
with ff = 0:5. For ffiT
T rms ' G¯v s fl s ? 4 \Theta 10 \Gamma7 where ¯ is the mass per unit length of the string, v s is
its velocity and fl s is the relativistic Lorenz factor.
The main points I wanted to stress in this talk are the following:
ffl The detection of non­Gaussianity induced by cosmic strings on large angular scales is possible.
ffl For this a special statistic is needed optimized to pick up the large scale structure of the latest
seeds.
ffl Such a statistic is the average of the Sample Mean Difference which can pick up non­Gaussian
features of long strings with G¯v s fl s ? 4 \Theta 10 \Gamma7 in maps with about 10 3 pixels and with a noise
to signal ratio of about 0.5 in a Gaussian background with ffiT
T rms ' 10 \Gamma5 .
Acknowledgements
I wish to thank Anne Davis for useful discussions and the DAMTP of Cambridge University for
hospitality and support during the period when this work was in progress. This work was supported
by the E.U. under the HCM programs (CHRX \Gamma CT94 \Gamma 0423, CHRX \Gamma CT93 \Gamma 0340 and CHRX \Gamma
CT94\Gamma0621) as well as by the Greek General Secretariat of Research and Technology grants 95E\Delta1759
and \PiENE \Delta 1170/95.
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