Bayesian Reconstruction of the WIMP Velocity Distribution with a Non-Negligible Threshold Energy

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1 Bayesian Reconstruction of the WIMP Velocity Distribution with a Non-Negligible Threshold Energy Chung-Lin Shan Xinjiang Astronomical Observatory Chinese Academy of Sciences National Center for Theoretical Sciences, National Taiwan University June 8, 2015 Based on arxiv: and JCAP

2 Outline Motivation With a non-negligible threshold energy Summary C.-L. Shan (XAO-CAS) NCTS, NTU, June 8, 2015 p. 1

3 Motivation 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 (XAO-CAS) NCTS, NTU, June 8, 2015 p. 2

4 Motivation 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 (XAO-CAS) NCTS, NTU, June 8, 2015 p. 3

5 Motivation 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 (XAO-CAS) NCTS, NTU, June 8, 2015 p. 4 ]

6 Motivation 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 (XAO-CAS) NCTS, NTU, June 8, 2015 p. 5

7 Motivation 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 (XAO-CAS) NCTS, NTU, June 8, 2015 p. 6

8 With a non-negligible threshold energy Problem! Reconstructed f 1,rec (v s,n ) with a non-negligible threshold energy ( 76 Ge, 2-50 kev, 500 events, m χ = 25 GeV) [CLS, arxiv: ] C.-L. Shan (XAO-CAS) NCTS, NTU, June 8, 2015 p. 7

9 With a non-negligible threshold energy Modification of the estimator for the normalization constant N Consider the non-zero minimal cut-off velocity where N 2 α [ 2Q 1/2 min F 2 (Q min ) ( ) ] 1 dr + I 0 (Q min, Qmax dq ) expt, Q=Q min ( ) dr = r 1 e k 1(Q min Q s,1) r(q min ) dq expt, Q=Q min I n(q min, Q max ) = Q max Q min [ Q (n 1)/2 1 F 2 (Q) ( Qmax min Q max, Q max,kin = v max 2 ) α 2 ( )] dr dq dq [CLS, arxiv: ] C.-L. Shan (XAO-CAS) NCTS, NTU, June 8, 2015 p. 8

10 With a non-negligible threshold energy Modification of the estimator for the normalization constant N Reconstructed f 1,rec (v s,n ) with the input WIMP mass ( 76 Ge, 2-50 kev, 500 events, m χ = 25 GeV) [CLS, arxiv: ] C.-L. Shan (XAO-CAS) NCTS, NTU, June 8, 2015 p. 9

11 With a non-negligible threshold energy Modification of the estimator for the normalization constant N Height of the velocity distribution at the non-zero minimal cut-off velocity [ ] [ ] f 1,rec (vmin ) = N 2Q min r(q min ) d F 2 (Q min ) dq ln F 2 (Q) k 1 N f 1,rec (vmin ) Q=Qmin Consider the contribution below the non-zero minimal cut-off velocity N = 2 α [ f 1,rec (vmin ) Q1/2 min + 2Q1/2 min F 2 (Q min ) ( ) ] 1 dr + I 0 (Q min, Qmax dq ) expt, Q=Q min [CLS, arxiv: ] C.-L. Shan (XAO-CAS) NCTS, NTU, June 8, 2015 p. 10

12 With a non-negligible threshold energy Modification of the estimator for the normalization constant N Reconstructed f 1,rec (v s,n ) with the input WIMP mass ( 76 Ge, 2-50 kev, 500 events, m χ = 25 GeV) [CLS, arxiv: ] C.-L. Shan (XAO-CAS) NCTS, NTU, June 8, 2015 p. 11

13 With a non-negligible threshold energy Modification of the estimator for the normalization constant N Reconstructed f 1,rec (v s,n ) with the input WIMP mass ( 76 Ge, 5-50 kev, 500 events, m χ = 25 GeV) [CLS, arxiv: ] C.-L. Shan (XAO-CAS) NCTS, NTU, June 8, 2015 p. 12

14 With a non-negligible threshold energy Modification of the estimator for the normalization constant N Reconstructed f 1,rec (v s,n ) with the input WIMP mass ( 76 Ge, 5-50 kev, 500 events, m χ = 25 GeV) [CLS, arxiv: ] C.-L. Shan (XAO-CAS) NCTS, NTU, June 8, 2015 p. 13

15 With a non-negligible threshold energy Modification of the estimator for the normalization constant N Reconstructed f 1,rec (v s,n ) with the input WIMP mass ( 28 Si, 5-50 kev, 500 events, m χ = 25 GeV) [CLS, arxiv: ] C.-L. Shan (XAO-CAS) NCTS, NTU, June 8, 2015 p. 14

16 With a non-negligible threshold energy Modification of the estimator for the normalization constant N Reconstructed f 1,rec (v s,n ) with the input WIMP mass ( 28 Si, 5-50 kev, 500 events, m χ = 25 GeV) [CLS, arxiv: ] C.-L. Shan (XAO-CAS) NCTS, NTU, June 8, 2015 p. 15

17 With a non-negligible threshold energy Theoretical bias estimate [ v min 0 ] v min f 0 1 (v) dv / v max f 0 1 (v) dv [CLS, arxiv: ] C.-L. Shan (XAO-CAS) NCTS, NTU, June 8, 2015 p. 16

18 Bayesian reconstruction of the WIMP velocity distribution C.-L. Shan (XAO-CAS) NCTS, NTU, June 8, 2015 p. 17

19 Formalism Formalism 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 (XAO-CAS) NCTS, NTU, June 8, 2015 p. 18

20 Formalism Formalism 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 [CLS, JCAP ] C.-L. Shan (XAO-CAS) NCTS, NTU, June 8, 2015 p. 19

21 Formalism Formalism 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 [CLS, JCAP ] C.-L. Shan (XAO-CAS) NCTS, NTU, June 8, 2015 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 (XAO-CAS) NCTS, NTU, June 8, 2015 p. 21

23 Reconstructed f 1,Bayesian (v) with the 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, JCAP ] C.-L. Shan (XAO-CAS) NCTS, NTU, June 8, 2015 p. 22

24 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, JCAP ] C.-L. Shan (XAO-CAS) NCTS, NTU, June 8, 2015 p. 23

25 Reconstructed f 1,Bayesian (v) with the 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, JCAP ] C.-L. Shan (XAO-CAS) NCTS, NTU, June 8, 2015 p. 24

26 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, JCAP ] C.-L. Shan (XAO-CAS) NCTS, NTU, June 8, 2015 p. 25

27 Reconstructed f 1,Bayesian (v) with the input WIMP mass ( 76 Ge, kev, 500 events, m χ = 100 GeV, f 1,sh,v0 (v) f 1,sh (v), Gaussian-dist.) [CLS, JCAP ] C.-L. Shan (XAO-CAS) NCTS, NTU, June 8, 2015 p. 26

28 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, JCAP ] C.-L. Shan (XAO-CAS) NCTS, NTU, June 8, 2015 p. 27

29 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, JCAP ] C.-L. Shan (XAO-CAS) NCTS, NTU, June 8, 2015 p. 28

30 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, JCAP ] C.-L. Shan (XAO-CAS) NCTS, NTU, June 8, 2015 p. 29

31 Reconstructed f 1,Bayesian (v) with the 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, JCAP ] C.-L. Shan (XAO-CAS) NCTS, NTU, June 8, 2015 p. 30

32 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, JCAP ] C.-L. Shan (XAO-CAS) NCTS, NTU, June 8, 2015 p. 31

33 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, JCAP ] C.-L. Shan (XAO-CAS) NCTS, NTU, June 8, 2015 p. 32

34 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, JCAP ] C.-L. Shan (XAO-CAS) NCTS, NTU, June 8, 2015 p. 33

35 function Reconstructed f 1,Bayesian (v) with the input WIMP mass ( 76 Ge, 2-50 kev, 500 events, m χ = 25 GeV, f 1,sh,v0 (v) f 1,sh,v0 (v), flat-dist.) [CLS, arxiv: ] C.-L. Shan (XAO-CAS) NCTS, NTU, June 8, 2015 p. 34

36 function Reconstructed f 1,Bayesian (v) with the input WIMP mass ( 76 Ge, 5-50 kev, 500 events, m χ = 25 GeV, f 1,sh,v0 (v) f 1,sh,v0 (v), flat-dist.) [CLS, arxiv: ] C.-L. Shan (XAO-CAS) NCTS, NTU, June 8, 2015 p. 35

37 function Reconstructed f 1,Bayesian (v) with the input WIMP mass ( 28 Si, 2-50 kev, 500 events, m χ = 25 GeV, f 1,sh,v0 (v) f 1,sh,v0 (v), flat-dist.) [CLS, arxiv: ] C.-L. Shan (XAO-CAS) NCTS, NTU, June 8, 2015 p. 36

38 function Reconstructed f 1,Bayesian (v) with the input WIMP mass ( 28 Si, 5-50 kev, 500 events, m χ = 25 GeV, f 1,sh,v0 (v) f 1,sh,v0 (v), flat-dist.) [CLS, arxiv: ] C.-L. Shan (XAO-CAS) NCTS, NTU, June 8, 2015 p. 37

39 Summary Summary C.-L. Shan (XAO-CAS) NCTS, NTU, June 8, 2015 p. 38

40 Summary Summary We derived the expression for estimating v max v min =α Q min f 1 (v) dv. We suggested the simple model-independent triangular estimator v min 0 = f 1,rec (vmin ) v min /2 for approximating to v min f 0 1 (v) dv. By adopting the most commonly used shifted Maxiwellian velocity distribution (or another well motivated one), one can estimate the difference: v min 0 v min f 0 1 (v) dv. For vmin O(200) km/s, we could provide precise reconstructed velocity distribution points to match the true WIMP velocity distribution with a < 10% bias. Thank you very much for your attention! C.-L. Shan (XAO-CAS) NCTS, NTU, June 8, 2015 p. 39

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