Bayesian Reconstruction of the WIMP Velocity Distribution Function from Direct Dark Matter Detection Data

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1 Bayesian Reconstruction of the WIMP Velocity Distribution Function from Direct Dark Matter Detection Data Chung-Lin Shan National Center for Theoretical Sciences National Tsing Hua University April 18, 2014 Based on arxiv:

2 Outline Review Dark Matter searches Model-independent data analyses Bayesian analysis technique AMIDAS-II package and website Online demonstration - blind analyses Summary C.-L. Shan NCTS, NTHU, April 18, 2014 p. 1

3 Review Dark Matter searches Dark Matter searches C.-L. Shan NCTS, NTHU, April 18, 2014 p. 2

4 Review Dark Matter searches Dark Matter searches DM should have small, but non-zero interactions with SM matter. = Three different ways to detect DM particles p, e DM DM e, ν µ, γ DM DM (, ) p, e + DM ( ) DM e +, p, D q q Colliders Indirect detection Direct detection C.-L. Shan NCTS, NTHU, April 18, 2014 p. 3

5 Review Model-independent data analyses Model-independent data analyses C.-L. Shan NCTS, NTHU, April 18, 2014 p. 4

6 Review Model-independent data analyses Motivation Differential event rate for elastic WIMP-nucleus scattering dr vmax [ ] f1(v) dq = AF 2 (Q) dv v min (Q) v Here v min (Q) = α Astrophysics Q is the minimal incoming velocity of incident WIMPs that can deposit the recoil energy Q in the detector, A ρ0σ0 mn α m 2m χmr,n 2 2mr,N 2 r,n = mχm N 2 m χ + m N Particle physics ρ 0 : WIMP density near the Earth σ 0 : total cross section ignoring the form factor suppression F (Q): elastic nuclear form factor f 1 (v): one-dimensional velocity distribution of halo WIMPs C.-L. Shan NCTS, NTHU, April 18, 2014 p. 5

7 Review Model-independent data analyses Motivation Differential event rate for elastic WIMP-nucleus scattering dr vmax [ ] f1(v) dq = AF 2 (Q) dv v min (Q) v Here v min (Q) = α Q is the minimal incoming velocity of incident WIMPs that can deposit the recoil energy Q in the detector, ρ0σ0 A 2m χmr,n 2 α mn 2m 2 r,n m r,n = mχm N m χ + m N ρ 0 : WIMP density near the Earth σ 0 : total cross section ignoring the form factor suppression F (Q): elastic nuclear form factor f 1 (v): one-dimensional velocity distribution of halo WIMPs C.-L. Shan NCTS, NTHU, April 18, 2014 p. 6

8 Review Model-independent data analyses Reconstruction of the WIMP velocity distribution Normalized one-dimensional WIMP velocity distribution function { [ ( )]} d 1 dr f 1 (v) = N 2Q dq F 2 (Q) dq N = 2 { [ 1 1 α 0 Q F 2 (Q) ( )] } dr 1 dq dq Q=v 2 /α 2 Moments of the velocity distribution function ( α v n n+1 = N (Q thre ) 2 N (Q thre ) = 2 α [ 2Q 1/2 thre F 2 (Q thre ) [ I n(q thre ) = Q (n 1)/2 Q thre ) [ 2Q (n+1)/2 thre F 2 (Q thre ) ( ) dr dq ( ) dr + I 0 (Q thre ) dq Q=Q thre 1 F 2 (Q) ( )] dr dq dq + (n + 1)I n(q thre ) Q=Q thre ] 1 [M. Drees and CLS, JCAP 0706, 011 (2007)] C.-L. Shan NCTS, NTHU, April 18, 2014 p. 7 ]

9 Review Model-independent data analyses Reconstruction of the WIMP velocity distribution Ansatz: the measured recoil spectrum in the nth Q-bin ( ) dr r n e kn(q Qs,n) r n Nn dq expt, Q Q n b n Logarithmic slope and shifted point in the nth Q-bin Q Q n n 1 N n ( ) bn (Q n,i Q n) = coth N n 2 i=1 Q s,n = Q n + 1 [ ] sinh(knbn/2) ln k n k nb n/2 ( knb n 2 ) 1 k n Reconstructing the one-dimensional WIMP velocity distribution [ ] [ 2Qs,nrn d ] f 1 (v s,n) = N F 2 (Q s,n) dq ln F 2 (Q) k n Q=Qs,n [ ] N = v s,n = α Q s,n α Qa F 2 (Q a) a [M. Drees and CLS, JCAP 0706, 011 (2007)] C.-L. Shan NCTS, NTHU, April 18, 2014 p. 8

10 Review Model-independent data analyses Reconstruction of the WIMP velocity distribution Reconstructed f 1,rec (v s,n ) ( 76 Ge, 500 events, 5 bins, up to 3 bins per window) [M. Drees and CLS, JCAP 0706, 011 (2007)] C.-L. Shan NCTS, NTHU, April 18, 2014 p. 9

11 Review Model-independent data analyses Determination of the WIMP mass Estimating the moments of the WIMP velocity distribution v n = α n 2Q1/2 min r 1 min F 2 (Q min ) + I 0 2Q(n+1)/2 r min min F 2 + (n + 1)I n (Q min ) I n = a Q (n 1)/2 a F 2 (Q a) Determining the WIMP mass mx m m χ v Y m X R n n = R n m X /m Y R n = ( ) dr r min = = r 1 e k 1 (Q min Q s,1 ) dq expt, Q=Q min [M. Drees and CLS, JCAP 0706, 011 (2007)] 2Q(n+1)/2 min,x r min,x /FX 2 (Q min,x ) + (n + 1)I n,x (X ) 1/n 1 Y 2Q 1/2 (n 0) min,x r min,x /F X 2 (Q min,x ) + I 0,X Assuming a dominant SI scalar WIMP-nucleus interaction m χ σ = (m X /m Y ) 5/2 m Y m X R σ R σ (m X /m Y ) 5/2 R σ = E Y E X [CLS and M. Drees, arxiv: ] 2Q1/2 min,x r min,x /F 2 X (Q min,x ) + I 0,X 2Q 1/2 min,y r min,x /F 2 Y (Q min,y ) + I 0,Y [M. Drees and CLS, JCAP 0806, 012 (2008)] C.-L. Shan NCTS, NTHU, April 18, 2014 p. 10

12 Review Model-independent data analyses Determination of the WIMP mass χ 2 -fitting ( ) fi,x f i,y C 1 ( ) ij fj,x f j,y χ 2 (m χ) = i,j where f i,x = α i 2Q(i+1)/2 min,x r min,x /FX 2 (Q min,x ) + (i + 1)I ( i,x X 2Q 1/2 min,x r min,x /FX 2 (Q min,x ) + I 0,X f nmax+1,x = E X A 2 X 2Q 1/2 min,x r min,x /FX 2 (Q min,x ) + I 0,X C ij = cov ( f i,x, f j,x ) + cov ( fi,y, f j,y ) km/s ( mx m χ + m X ) ) i Algorithmic Q max matching ( ) 2 αx Q max,y = Q max,x α Y (v cut = α Q max ) [M. Drees and CLS, JCAP 0806, 012 (2008)] C.-L. Shan NCTS, NTHU, April 18, 2014 p. 11

13 Review Model-independent data analyses Determination of the WIMP mass Reconstructed m χ,rec ( 28 Si + 76 Ge, Q max < 100 kev, 2 50 events) [M. Drees and CLS, JCAP 0806, 012 (2008)] C.-L. Shan NCTS, NTHU, April 18, 2014 p. 12

14 Bayesian reconstruction of the WIMP velocity distribution function C.-L. Shan NCTS, NTHU, April 18, 2014 p. 13

15 Aim Reconstructed f 1,rec (v s,n ) with an input WIMP mass ( 76 Ge, kev, 500 events, m χ = 100 GeV, f 1,gen,sh (v)) [CLS, arxiv: ] C.-L. Shan NCTS, NTHU, April 18, 2014 p. 14

16 Bayesian analysis technique Bayesian analysis technique C.-L. Shan NCTS, NTHU, April 18, 2014 p. 15

17 Bayesian analysis technique Bayesian analysis technique Baye s theorem p(a B) = p(b A) p(a) p(b) = p(b A) p(a) p(b A) p(a) + p(b A) p(a) p(a) = 1 p(a) Bayesian statistics p(theory result) = p(result theory) p(result) p(theory) Bayesian analysis p(θ data) = p(data Θ) p(data) p(θ) C.-L. Shan NCTS, NTHU, April 18, 2014 p. 16

18 Bayesian analysis technique Bayesian analysis technique Bayesian analysis p(θ data) = p(data Θ) p(θ) p(data) Θ: { a 1, a 2,, a NBayesian }, a specified (combination of the) value(s) of the fitting parameter(s) p(θ): prior probability, our degree of belief about Θ being the true value(s) of fitting parameter(s), often given in form of the (multiplication of the) probability distribution(s) of the fitting parameter(s) p(data): evidence, the total probability of obtaining the particular set of data p(data Θ): the probability of the observed result, once the specified (combination of the) value(s) of the fitting parameter(s) happens, usually be described by the likelihood function of Θ, L(Θ). p(θ data): posterior probability density function for Θ, the probability of that the specified (combination of the) value(s) of the fitting parameter(s) happens, given the observed result C.-L. Shan NCTS, NTHU, April 18, 2014 p. 17

19 Bayesian analysis technique Bayesian analysis technique Likelihood function for p(data Θ) with Theoretical one-dimensional WIMP velocity distribution function: f 1,th (v; a 1, a 2,, a NBayesian ) Assuming that the reconstructed data points are Gaussian-distributed around the theoretical predictions ) L (f 1,rec(v s,µ), µ = 1, 2,, W ; a i, i = 1, 2,, N Bayesian W ) Gau (v s,µ, f 1,rec(v s,µ), σ f1,s,µ; a 1, a 2,, a NBayesian µ=1 ) Gau (v s,µ, f 1,rec(v s,µ), σ f1,s,µ; a 1, a 2,, a NBayesian [ ] 1 2 / e f 1,rec (v s,µ) f 1,th (v s,µ;a 1,a 2,,a NBayesian ) 2σf 2 1,s,µ 2π σf1,s,µ C.-L. Shan NCTS, NTHU, April 18, 2014 p. 18

20 Bayesian analysis technique Bayesian analysis technique Probability distribution functions for p(θ) Without prior knowledge about the fitting parameter Flat-distributed p i (a i ) = 1 for a i,min a i a i,max, With prior knowledge about the fitting parameter Around a theoretical predicted/estimated or experimental measured value µ a,i With (statistical) uncertainties σ a,i Gaussian-distributed p i (a i ; µ a,i, σ a,i ) = 1 2π σa,i e (a i µ a,i ) 2 /2σ 2 a,i C.-L. Shan NCTS, NTHU, April 18, 2014 p. 19

21 C.-L. Shan NCTS, NTHU, April 18, 2014 p. 20

22 Input and fitting one-dimensional WIMP velocity distribution functions One-parameter shifted Maxwellian velocity distribution f 1,sh,v0 (v) = 1 ( ) v [ ] e (v ve)2 /v0 2 e (v+ve)2 /v0 2 v e = 1.05 v 0 π v 0 v e Shifted Maxwellian velocity distribution f 1,sh (v) = 1 ( ) v [ ] e (v ve)2 /v0 2 e (v+ve)2 /v0 2 π v 0 v e Variated shifted Maxwellian velocity distribution f 1,sh, v (v) = 1 [ ] { } v e [v (v 0+ v)] 2 /v0 2 e [v+(v 0+ v)] 2 /v0 2 π v 0 (v 0 + v) Simple Maxwellian velocity distribution f 1,Gau (v) = 4 ( v 2 ) π v0 3 e v 2 /v0 2 Modified simple Maxwellian velocity distribution f 1,Gau,k (v) = v 2 ( ) e v 2 /kv0 2 e v max 2 /kv 2 k 0 for v v max N f,k C.-L. Shan NCTS, NTHU, April 18, 2014 p. 21

23 Reconstructed f 1,Bayesian (v) with an input WIMP mass ( 76 Ge, kev, 500 events, m χ = 100 GeV, f 1,sh,v0 (v) f 1,sh,v0 (v), flat-dist.) [CLS, arxiv: ] C.-L. Shan NCTS, NTHU, April 18, 2014 p. 22

24 Reconstructed f 1,Bayesian (v) with an input WIMP mass ( 76 Ge, kev, 500 events, m χ = 100 GeV, f 1,sh,v0 (v) f 1,sh,v0 (v), flat-dist.) [CLS, arxiv: ] C.-L. Shan NCTS, NTHU, April 18, 2014 p. 23

25 Distribution of the reconstructed v 0 ( 76 Ge, kev, 500 events, m χ = 100 GeV, f 1,sh,v0 (v) f 1,sh,v0 (v), flat-dist.) [CLS, arxiv: ] C.-L. Shan NCTS, NTHU, April 18, 2014 p. 24

26 Reconstructed f 1,Bayesian (v) with an input WIMP mass ( 76 Ge, kev, 500 events, m χ = 100 GeV, f 1,sh,v0 (v) f 1,sh,v0 (v), Gaussian-dist.) [CLS, arxiv: ] C.-L. Shan NCTS, NTHU, April 18, 2014 p. 25

27 Distribution of the reconstructed v 0 ( 76 Ge, kev, 500 events, m χ = 100 GeV, f 1,sh,v0 (v) f 1,sh,v0 (v), Gaussian-dist.) [CLS, arxiv: ] C.-L. Shan NCTS, NTHU, April 18, 2014 p. 26

28 Reconstructed f 1,Bayesian (v) with an input WIMP mass ( 76 Ge, kev, 500 events, m χ = 100 GeV, f 1,sh,v0 (v) f 1,sh (v), Gaussian-dist.) [CLS, arxiv: ] C.-L. Shan NCTS, NTHU, April 18, 2014 p. 27

29 Distribution of the reconstructed v 0 v e ( 76 Ge, kev, 500 events, m χ = 100 GeV, f 1,sh,v0 (v) f 1,sh (v), Gaussian-dist.) [CLS, arxiv: ] C.-L. Shan NCTS, NTHU, April 18, 2014 p. 28

30 Distribution of the reconstructed v 0 ( 76 Ge, kev, 500 events, m χ = 100 GeV, f 1,sh,v0 (v) f 1,sh (v), Gaussian-dist.) [CLS, arxiv: ] C.-L. Shan NCTS, NTHU, April 18, 2014 p. 29

31 Distribution of the reconstructed v e ( 76 Ge, kev, 500 events, m χ = 100 GeV, f 1,sh,v0 (v) f 1,sh (v), Gaussian-dist.) [CLS, arxiv: ] C.-L. Shan NCTS, NTHU, April 18, 2014 p. 30

32 Reconstructed f 1,Bayesian (v) with an input WIMP mass ( 76 Ge, kev, 500 events, m χ = 100 GeV, f 1,sh,v0 (v) f 1,sh, v (v), Gaussian-dist.) [CLS, arxiv: ] C.-L. Shan NCTS, NTHU, April 18, 2014 p. 31

33 Distribution of the reconstructed v 0 v ( 76 Ge, kev, 500 events, m χ = 100 GeV, f 1,sh,v0 (v) f 1,sh, v (v), Gaussian-dist.) [CLS, arxiv: ] C.-L. Shan NCTS, NTHU, April 18, 2014 p. 32

34 Distribution of the reconstructed v 0 ( 76 Ge, kev, 500 events, m χ = 100 GeV, f 1,sh,v0 (v) f 1,sh, v (v), Gaussian-dist.) [CLS, arxiv: ] C.-L. Shan NCTS, NTHU, April 18, 2014 p. 33

35 Distribution of the reconstructed v ( 76 Ge, kev, 500 events, m χ = 100 GeV, f 1,sh,v0 (v) f 1,sh, v (v), Gaussian-dist.) [CLS, arxiv: ] C.-L. Shan NCTS, NTHU, April 18, 2014 p. 34

36 Reconstructed f 1,Bayesian (v) with an input WIMP mass ( 76 Ge, kev, 500 events, m χ = 100 GeV, f 1,sh,v0 (v) f 1,Gau (v), flat-dist.) [CLS, arxiv: ] C.-L. Shan NCTS, NTHU, April 18, 2014 p. 35

37 Distribution of the reconstructed v 0 ( 76 Ge, kev, 500 events, m χ = 100 GeV, f 1,sh,v0 (v) f 1,Gau (v), flat-dist.) [CLS, arxiv: ] C.-L. Shan NCTS, NTHU, April 18, 2014 p. 36

38 AMIDAS-II package and website AMIDAS-II package and website C.-L. Shan NCTS, NTHU, April 18, 2014 p. 37

39 AMIDAS-II package and website AMIDAS-II package and website A Model-Independent Data Analysis System for direct Dark Matter detection experiments and phenomenology DAMNED Dark Matter Web Tool (ILIAS Project) [CLS, arxiv: , ; ] TiResearch (Taiwan interactive Research, 臺灣互動研究網 ) Online interactive simulation/data analysis system Full Monte Carlo simulations Theoretical estimations Real/pseudo- data analyses C.-L. Shan NCTS, NTHU, April 18, 2014 p. 38

40 AMIDAS-II package and website Online demonstration - blind analyses Online demonstration - blind analyses C.-L. Shan NCTS, NTHU, April 18, 2014 p. 39

41 Summary Summary C.-L. Shan NCTS, NTHU, April 18, 2014 p. 40

42 Summary Summary By using Bayesian analysis/fitting one could reconstruct f 1 (v) very precisely: small or even negligible systematic deviations of v 0, v e and the peak position of f 1 (v) 1σ statistical uncertainties of < 20 km/s on v 0, v e and the peak position of f 1 (v) With prior knowledge about fitting parameters systematic deviations and statistical uncertainties could be reduced (significantly) even when the expected values are (slightly) incorrect Using variated analytic form of the same fitting f 1 (v) systematic deviations could be reduced (significantly) statistical uncertainties might however be (a bit) larger C.-L. Shan NCTS, NTHU, April 18, 2014 p. 41

43 Summary Summary Using an improper fitting f 1 (v) information about the true velocity distribution function (e.g. the peak position of f 1 (v)) could still be reconstructed approximately with clearly 2σ - 6σ deviations of v 0 and v e from the theoretical expections For light m χ : (very) shape recoil energy spectrum only with very few reconstructed-input data points For heavy m χ : large statistical fluctuation of and uncertainty on m χ,rec only with reconstructed-input data points in the low-velocity range With a small fraction of unrejected background events C.-L. Shan NCTS, NTHU, April 18, 2014 p. 42

44 Summary Thank you very much for your attention! C.-L. Shan NCTS, NTHU, April 18, 2014 p. 43

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