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Bayesian Constraints on Supersymmetric Neutralino Dark Matter

Bayesian Constraints on Supersymmetric Neutralino Dark Matter Roberto Ruiz de Austri IFIC In collaboration with: R. Trotta, L. Roszkowski, M. Hobson, F. Feroz, J. Silk, C. P. de los Heros, A. Casas, M. E. Cabrera. Outline.

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Bayesian Constraints on Supersymmetric Neutralino Dark Matter

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  1. Bayesian Constraints on Supersymmetric Neutralino Dark Matter Roberto Ruiz de Austri IFIC In collaboration with: R. Trotta, L. Roszkowski, M. Hobson, F. Feroz, J. Silk, C. P. de los Heros, A. Casas, M. E. Cabrera

  2. Outline There are three kinds of lies: lies, damned lies, and statistics. (Mark Twain, quoting Benjamin Disraeli)‏ • Bayesian approach and SUSY searches • CMSSM forecast for the LHC

  3. The model & data • The general Minimal Supersymmetric Standard Model (MSSM): 105 free parameters! • Need some simplifying assumption: i.e. the Constrained MSSM (CMSSM) reduces the free parameters to just 4 continous variables plus a discrete one (sign(μ))‏ • Present-day data: collider measurements of rare processes, CDM abundance (WMAP), sparticle masses lower limits, EW precision measurements. Soon, LHC sparticle spectrum measurements • Astrophysical direct and indirect detection techniques might also be competitive: neutrino (IceCUBE), gamma-rays (Fermi), antimatter (PAMELA), direct detection (XENON, CDMS, Eureca, Zeplin)‏ • Goal: inference of the model parameters but it is difficult problem

  4. Why is this a difficult problem? • Inherently 8-dimensional: reducing the dimensionality over-simplifies the problem. Nuisance parameters (in particular mt) cannot be fixed! • Likelihood discontinuous and multi-modal due to physicality conditions • RGE connect input parameters to observables in highly non-linear fashion: only indirect (sometimes weak) constraints on the quantities of interest (-> prior volume effects are difficult to keep under control)‏ • Mild discrepancies between observables (in particular, g-2 and b→sγ) tend to pull constraints in different directions

  5. Impact of nuisance parameters

  6. Random scans • Points accepted/rejected in a in/out fashion (e.g., 2-sigma cuts)‏ • No statistical measure attached to density of points • No probabilistic interpretation of results possible • Inefficient/Unfeasible in high dimensional parameters spaces (N>3)‏

  7. The accessible “surface” Scan from the prior with no likelihood except physicality constraints

  8. Bayesian parameter inference

  9. The Bayesian approach • Bayesian approach led by two groups (early work by Baltz & Gondolo, 2004): • Ben Allanach (DAMPT) et al (Allanach & Lester, 2006 onwards, Cranmer, and others) • RdA, Roszkowski & Roberto Trotta (2006 onwards) SuperBayeS public code (available from: superbayes.org) + Feroz & Hobson (MultiNest), + Silk (indirect detection), + de los Heros (IceCube)‏, + Casas et al. (Naturalness) + Bertone et al. (pmssm) ‏

  10. Bayes’ theorem H: hypothesis d: data I: external information • Prior: what we know about H (given information I) before seeing the data • Likelihood: the probability of obtaining data d if hypothesis H is true • Posterior: our state of knowledge about H after we have seen data d • Evidence: normalization constant (independent of H), crucial for model comparison

  11. Continuous parameters Bayesian evidence: average of the likelihood over the prior For parameter inference it is sufficient to consider posterior ∝ likelihood x prior

  12. ... • Ignoring the prior and identifying • implicitly amounts to • But e. g.

  13. But • If data are good enough to select a small region of {θ} then the prior p(θ) becomes irrelevant

  14. Priors • There is a vast literature on priors: Jeffreys’, conjugate, non-informative, ignorance, reference, ... • In simple problems, “good” priors are dictated by symmetry properties • Flat: All values of θ equally probable • Logarithmic: All magnitudes of θ equally probable

  15. Key advantages • Efficiency: computational effort scales ~ N rather than k^N as in grid-scanning methods. Orders of magnitude improvement over previously used techniques. • Marginalisation: integration over hidden dimensions comes for free Suppose we have and are interested in • Inclusion of nuisance parameters: simply include them in the scan and marginalise over them. Notice: nuisance parameters in this context must be well constrained using independent data. • Derived quantities: probabilities distributions can be derived for any function of the input variables (crucial for DD/ID/LHC predictions).

  16. Analysis pipeline

  17. Posterior Samplers • MCMC:A Markov Chain is a list of samples θ1, θ2, θ3,... whose density reflects the (unnormalized) value of the posterior • Crucial property: a Markov Chain converges to a stationary distribution, i.e. one that does not change with time. In our case, the posterior • Different algorithms: MH, Gibbs... all need a proposal distribution => difficult to find a good one in complex problems • Nested: New technique for efficient evidence evaluation (and posterior samples) (Skilling 2004)‏ • MultiNest: Also an extremely efficient sampler for multi-modal likelihoods ! Feroz & Hobson (2007), RT et al (2008), Feroz et al (2008)‏

  18. The Multinest Algorithm • MultiNest: Also an extremely efficient sampler for multi-modal likelihoods! Feroz & Hobson (2007), RT et al (2008), Feroz et al (2008)‏

  19. Marginal Posterior vs Profile likelihood

  20. The SuperBayeS package (superbayes.org)‏ • Supersymmetry Parameters Extraction Routines for Bayesian Statistics • Implements the CMSSM, but can be easily extended to the general MSSM • Currently linked to SoftSusy 2.0.18, DarkSusy 4.1, MICROMEGAS 2.2, FeynHiggs 2.5.1, Hdecay 3.102. New release (v 1.5)‏ • Includes up-to-date constraints from all observables • Fully parallelized, MPI-ready, user-friendly interface a la cosmomc (thanks Sarah Bridle & Antony Lewis)‏, plotting routines • Bayesian MCMC, MULTI-MODAL NESTED SAMPLING or grid scan mode • MULTI-MODAL NESTED SAMPLING (Feroz & Hobson 2008), efficiency increased by a factor 200. A full 8D scan now takes 3 days on a single CPU (previously: 6 weeks on 10 CPUs)‏

  21. Global CMSSM constraints

  22. Variables of the scan From Robertos' talk

  23. Samples from priors only • No data in the likelihood, non-physical points discarged priors • => flat prior on log means

  24. Samples from priors only • Volume effect from the non-physical regions

  25. Priors distributions from observables • Priors are quite informative regardless the quantities being constrained !!!

  26. Data included Indirect observablesSM parameters

  27. Parameter inference (all data included)‏ Flat priors Log priors

  28. 2D posterior vs profile likelihood Posterior Profile likelihood

  29. The model Universality at High Scale supported for FCNC constraints

  30. Goal • Scan the model evaluating Forecast map for LHC

  31. Towards a more refineanalysis • Recall an usual assumption should be • In order to get a Natural Electroweak symmetry Breaking (with no fine-tunings)

  32. Bayesian and Naturalness

  33. ... • Instead solving in terms of and the other soft-terms and, treat as another exp. Data • Approximate the likelihood as

  34. ... • Use to marginalize (Barbieri Gudice measure of fine-tunning)‏ • Probability of cancellation between the various contributions to get

  35. At practical level • Besides, we have done a similar analysis for the fermion masses • And traded

  36. Putting all the pieces together

  37. Finally • For the prior we take the two basic possibilities: flat logarithmic

  38. brings SUSY to the LHC region • We may vary up to the results do not depend on the range choosen • This suggests that large soft-masses are disfavoured

  39. Data Included

  40. Adding g-2 using (e+ e- data) Preferred region clearly within the LHC reach

  41. Adding [and notg-2]

  42. Adding all Events with more or equal to 2 jets[Baer et al 0907.1922]

  43. Comparison with Likelihood based inference Buchmueller et al. (2009) Cabrera et al. (2009)‏

  44. ATLAS will solve the prior dependency • Projected constraints from ATLAS, (dilepton and lepton+jets edges, 1 fb-1 luminosity)‏

  45. Residual dependency on the statistics • Marginal posterior and profile likelihood will remain somewhat discrepant using ATLAS alone. Much better agreement from ATLAS+Planck CDM determination.

  46. ... • For the results are similar with an important difference: • Now SUSY produces contributions to in the wrong direction • So adding (using e+ e- data) does not help in this case to bring the preferred region to the LHC • And, of course, gets strongly disfavoured with respect to

  47. Astrophysical probes • Direct detection: underground detectors looking for nuclear recoils from WIMP scattering. It is fundamental to account for the uncertainty in the local WIMP distribution. • Indirect detection: detection of annihilation products from WIMP-WIMP annihilation. • Gamma ray(galactic centre, galactic halo, diffuse extragalactic sources, nearby dwarf galaxies). • Antimatter (positrons, anti-proton) from local clumps. • Neutrinosfrom the center of the Sun/Earth.

  48. Direct and indirect detection prospects

  49. Direct detection prospects

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