Bayes and present dark matter direct search status
TTTK-11-45
Bayes and present dark matter direct search status
Chiara Arina
Institut f¨ur Theoretische Teilchenphysik und Kosmologie, RWTH Aachen, 52056 Aachen,GermanyE-mail: [email protected]
Abstract.
Recently there has been a huge activity in the dark matter direct detection field,with the report of an excess from CoGeNT and CRESST along with the annual modulated signalof DAMA/Libra and the strong exclusion bound from XENON100. We analyse these resultswithin the framework of Bayesian inference and evidence. Indeed bayesian methods are wellsuited for marginalizing over experimental systematics and background. We present the resultsfor spin-independent interaction on nucleus with particular attention to the low dark mattermass region and the compatibility between experiments. In the same vein we also investigatethe impact of astrophysical uncertainties on the WIMP preferred parameter space within theclass of isotropic dark matter velocity distributions.
1. Introduction
Dark Matter (DM) direct searches experience a fervent activity with the report of an annualmodulation by the CoGeNT collaboration [1] along with the established DAMA/Libra (DAMAhereafter) signal [2]. The exclusion bound of the XENON100 collaboration [3] (Xe100 hereafter),obtained via a profile likelihood approach [4], puts to test those excesses. The aim of thisproceeding is to update the bayesian analysis of [5] with the CoGeNT data 2011 and toinvestigate the mutual agreement/disagreement between those experiments for spin-independentinteraction. As a novelty, we present the assessment of the need of including astrophysicaluncertainties in the direct detection (DD) analysis, with the use of the Bayesian evidence.
2. Bayesian framework and experimental likelihoods
The bayesian statistics is a well defined framework that allows the inclusion of experimentalsystematics and backgrounds, as well as astrophysical uncertainties common to all experiments.Indeed the posterior probability density function (pdf) is given by Bayes’ theorem: P ( θ | X ) = π ( θ ) L ( X | θ ) Z ( X ) , (1)where the likelihood L describes how the theoretical model, with free parameters θ , describes thedata X . The prior probability density π encodes the state of knowledge on θ before observingthe data and is therefore independent of X . The posterior pdf is sampled via MCMC techniquesusing the package CosmoMC [6, 7] and then marginalized over nuisance parameters. For DDpurposes, the important quantity is P marg ( m DM , σ SIn ), where m DM is the DM mass and σ SIn isthe DM-nucleus cross-section. a r X i v : . [ a s t r o - ph . C O ] O c t e take as nuisance parameters the mean velocity of the DM at the sun position v , theescape velocity from the DM halo v esc and the DM density at the sun position ρ (cid:12) (altogethernamed astrophysical variables). We consider two models for the DM velocity distribution: SMH(Standard Model Halo) – that is a simple Maxwellian halo with astrophysical variables fixed attheir mean value – and NFW – namely we derive the velocity distribution from this DM densityprofile and marginalize over all possible values of v , v esc and ρ (cid:12) as well as profile parameters.For all technical details about the analysis, prior choices and likelihood definitions, we referto [5]. Here we just recall that the main systematic for Xe100 is the scintillation efficiency, whilefor DAMA the quenching factor on Sodium q Na and on Iodine q I are marginalized over all theirexperimental range. Regarding the new CoGeNT data, the likelihood is given by the productof two gaussian distributions, one for the total rate and one for the modulated signal. Thedata binning, analysis and subtraction of cosmogenic peaks follow closely [1] and [8]. A flat andexponential background, which does not modulate in time and which is described by 3 nuisanceparameters, is added on top of the DM signal. The exponential background accounts for theeffect of a bad rejection of surface events near threshold.In Eq. 1 the denominator Z ( X ), called Bayesian evidence, is defined as the average of thelikelihood over the prior for a specific model MZ = (cid:90) L ( X | θ ) π ( θ ) d D θ , (2)where D is the dimensionality of the parameter space. This quantity is of crucial importance inmodel comparison, which is defined as the ratio of posterior probabilities. A model M can becompared with M , which has extra parameters, through the Bayes factor B , see e.g. [9, 10]: P ( M | X ) P ( M | X ) = Z Z π ( M ) π ( M ) = B π ( M ) π ( M ) , (3)where π ( M i ) is the prior probability of each model. The Bayes factor automatically favourssimpler models unless the data justify the complexity of more complicated alternatives, becauseof the marginalization procedure used to calculate the evidence, Eq. 2. In the following wewill compare the model SMH with NFW, the evidence for each model being computed with MULTINEST [11, 12].
3. Results
Figure 1 summarizes the 2D inference in the { m DM , σ SIn } plane for individual experiments(blue dashed curve for Xe100, gray dashed and solid black contours for CoGeNT and DAMArespectively) and for the combined fit (solid red lines), in the SMH and NFW model (left andright panel respectively). Table 1 resumes the preferred values for the DM and astrophysicalobservables in each model (recalling that v , v esc and ρ (cid:12) are fixed for SMH model). Comparingthe left and right panel one notes that the closed regions become wider, while the Xe100 exclusionbound moves slightly to the right, improuving the agreement between various search results. Thisis a volume effect due to the marginalization procedure over the astrophysical variables. In bothmodels, the inclusion of uncertainties on the scintillation energy leads to compatibility betweenCoGeNT, the combined fit and Xe100 exclusion bounds at 90 S % C.L., while leaving a marginalcompatibility with DAMA at 99% C.L.. It is always possible to combine experiments and finda common region, but actually the question is whether the preferred point for { m DM , σ SIn } isa good fit for both experiments and what are the values for the nuisance parameters that areselected. Notice from table 1 that the combined fit chooses values which are in line with the oneof CoGeNT and DAMA alone, within the statistical errors. The large deviation from DAMAalone fit concerns the quenching factor on Sodium: as it has been shown in figure 2 of [5] the 1D igure 1. Left : 2D credible regions for the individual experimental bounds and regions assumingthe SMH, combined in a single plot. For DAMA (black solid), CoGeNT (gray dashed) and thecombined fit (red solid) we show the 90% and 99% contours. The blue dashed line representsthe 90 S % confidence level (C.L.) for Xe100. Right : Same as left for the NFW model.posterior pdf for q Na is flat all along its prior range, while the combined fit is strongly peakedat the value q Na = 0 . ± .
03. This can be understood thinking that the larger q Na the lowerthe DM mass should be to account for the DM modulated signal. The selected value for thecombined fit therefore pushes the DAMA region towards the direction of the CoGeNT region.Table 2 resumes the results for model comparison in the case of individual experiments and forthe DAMA+CoGeNT combined fit. A positive (negative) value for ln B represents an increase(decrease) of the support in favour of the simplest model given the observed data. The strengthof evidence is given by the empirical ‘Jeffreys scale’ (see table 1 in [10]), where threshold valuesare set for inconclusive, moderate or strong evidence: e.g. ln B = 5 states that the odds againstthe most complicated models are 150:1 and corresponds to a probability of 0.993. This holdsin the case where the probability of each model is the same, namely π ( M ) = π ( M ) = 1 / B in table 2 it is clear that there is strong or moderate evidence against NFWmodel for single experiments. Therefore data at present time do not allow a determinationof astrophysical observables. On the contrary there is a very strong evidence for NFW forthe combined fit, namely experiments need to adjust astrophysical observable values to find acommon agreement. Similar results hold for the whole class of spherically symmetric DM halos.
4. Conclusion
We have employed Bayesian inference to study the concordance between various DD experimentsand model comparison to infer the necessity of a more complicated model than the SMH atpresent time. The outcomes of the analysis may be summarized in few main points: • the combined fit of DAMA and CoGeNT requires parameters closely in line with those ofindividual fits, the only exception being a large quenching factor on Sodium; • the CoGeNT region and the combined fit are compatible at 90 S % C.L. with the Xe100exclusion bounds, while DAMA is only marginally allowed at 99% C.L.; • a combination of at least two experiments demonstrates the ability of constrainingastrophysical variables, on the contrary of a single DD experiment. able 1.
1D posterior pdf modes and 90% credible intervals for the circular velocity v , escapevelocity v esc , and the local DM density ρ (cid:12) for the 2 DM halo model considered in this work. Model m DM (GeV) σ SIn (cm ) v (km/s) v esc (km/s) ρ (cid:12) (GeV / cm ) SMH
DAMA 12 2 . × −
230 544 0.4CoGeNT 7.5 1 . × −
230 544 0.4Combined 6.9 1 . × −
230 544 0.4Xenon100 – – 230 544 0.4
NFW
DAMA 12 1 . × − +40 − +19 − . +0 . − . CoGeNT 7.4 9 . × − +39 − +18 − . +0 . − . Combined 7. 1 . × − +32 − +14 − . +0 . − . Xenon100 – – 219 +43 − ±
18 0 . +0 . − . Table 2.
The odds against the NFW model for individual and combined experiments.
DAMA CoGeNT Combinedln B Remarkably the excess region falls in the same ballpark of { m DM , σ SIn } values that couldexplain the excess of events reported by the CRESST experiment [13]. The CoGeNTcollaboration has presented an ongoing analysis on the rejection of surface events nearthreshold [14], which could account for at least 50% of the counts at low energies: the totalrate would therefore be accommodated by smaller DM cross-sections than those in figure 1. Acknowledgments
It’s a pleasure to thank R. Trotta for discussions on model comparison and comments onthe manuscript and the CoGeNT collaboration for making data available for public analysis.This work acknowledges use of the COSMO computing resource at CP3 of Louvain University.Published under licence in
Journal of Physics: Conference Series by IOP Publishing Ltd.
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