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Statistics Case Study Sam Richardson

Literature Review

DAMA/LIBRA is an experiment created with the aim of direct detection of the dark

matter candidates WIMPS in the Galactic halo. The WIMPS being searched for are

slow moving so predicted to cluster together in the galactic halo. This means a signalfrom these WIMPS would modulate annually as the Earth moved around the Sun. In

order to prove this statistical tests were preformed showing the following

hypothesis are valid;

y A cosine modulation is present in the signaly The modulation has a period of one yeary A definite phase corresponding to the 2nd of Julyy The modulation is only present at low energies

The detector collected residual rates of single hit events. The residual rates given by

ijk-flatjk> where flat was the constant component of this signal, this gave data

centered around 0. Single hit events were used due to the probability of a WIMPinteracting with more than one detector being negligible. The combined data from

both detectors found evidence for a model independent annual modulation signal, (1)

Where S0 is the constant component of the signal, Sm is the modulation amplitude,

is the angular frequency (

), t is the time and t0 is the phase.

The study shows the data from both DAMA\LIBRA and DAMA\NaI in the 2-6 KeVregion being fit to equation (1) at a confidence of 8.2 with 2/d.o.f = 53.4/64. The

maximum likelihood was used to obtain the modulation amplitude obtained by the

maximum likelihood method peaked between 2-6KeV. The likelihood function was

defined as,

(2)

This test showed that a modulation was only present in the 2-6KeV region. The runand 2 tests were used to show that with 90% confidence the data from both

DAMA\LIBRA and DAMA\NaI were modulating around their best fit values furtherproving the modulation was present. From reviewing other papers on this

experiment it seems to be agreed that there is a modulation detected by DAMA but

not that it is definite proof of DM in the galactic halo. Many papers are particularly

concerned that similar experiments such as XENON and CDMS found no results,

proposing new DM models that would allow this.

Practical

y The data was binned with a 60 day width using matlabs histogram functiony This binned data was then fitted to the function using

matlabs curve fitting function

y The values found for a, b and c were used to create a points in a fitted curveon excel

y A 2 test was used to compare these fitted points and the observed data usingexcels =CHITEST function

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y The maximum likelihood was evaluated using the equation

y The probability was maximised by finding the optimum value for eachvariable with excels goal seek function.

yConclusions

Equation (3) was fitted to the data with a=2.813, b=-1.325 and c=11.900 at 2 =0.995. The cosine was fitted with an error of 0.12 rads. The maximum likelihood

produced a almost straight line disproving the hypothesis that the data is modulated

as the likelihood function gave a probability of 0.999 that a data point was most

likely to be on the fit line.

Freq

uency

Cosine fit

Maximum likelihood fit