WIMP Velocity Distribution and Mass from Direct Detection Experiments

WIMP Velocity Distribution and Mass from Direct Detection Experiments Manuel Drees Bonn University WIMP Distribution and Mass p. 1/33

Contents 1 Introduction WIMP Distribution and Mass p. 2/33

Contents 1 Introduction 2 Determining the Local WIMP Velocity Distribution WIMP Distribution and Mass p. 2/33

Contents 1 Introduction 2 Determining the Local WIMP Velocity Distribution 3 Determining the WIMP mass WIMP Distribution and Mass p. 2/33

Contents 1 Introduction 2 Determining the Local WIMP Velocity Distribution 3 Determining the WIMP mass 4 Summary WIMP Distribution and Mass p. 2/33

Contents 1 Introduction 2 Determining the Local WIMP Velocity Distribution 3 Determining the WIMP mass 4 Summary Based on MD, C. L. Shan, astro-ph/0703651, JCAP 0706, 011 (2007), and arxiv:0803.4477 [hep-ph] (JCAP, to appear). WIMP Distribution and Mass p. 2/33

Introduction: WIMPs as Dark Matter Several observations indicate existence of non-luminous Dark Matter (DM) (more exactly: missing force) WIMP Distribution and Mass p. 3/33

Introduction: WIMPs as Dark Matter Several observations indicate existence of non-luminous Dark Matter (DM) (more exactly: missing force) Galactic rotation curves imply Ω DM h 2 0.05. Ω: Mass density in units of critical density; Ω = 1 means flat Universe. h: Scaled Hubble constant. Observation: h = 0.72 ± 0.07 (?) Models of structure formation, X ray temperature of clusters of galaxies,... WIMP Distribution and Mass p. 3/33

Weakly Interacting Massive Particles (WIMPs) Exist in well motivated extensions of the SM: SUSY, (Little Higgs with T Parity), ((Universal Extra Dimension)) WIMP Distribution and Mass p. 4/33

Weakly Interacting Massive Particles (WIMPs) Exist in well motivated extensions of the SM: SUSY, (Little Higgs with T Parity), ((Universal Extra Dimension)) Can also (trivially) write down tailor made WIMP models In standard cosmology, roughly weak cross section automatically gives roughly right relic density for thermal WIMPs! (On logarithmic scale) WIMP Distribution and Mass p. 4/33

Direct WIMP detection WIMPs are everywhere! WIMP Distribution and Mass p. 5/33

Direct WIMP detection WIMPs are everywhere! Can elastically scatter on nucleus in detector: χ + N χ + N Measured quantity: recoil energy of N WIMP Distribution and Mass p. 5/33

Direct WIMP detection WIMPs are everywhere! Can elastically scatter on nucleus in detector: χ + N χ + N Measured quantity: recoil energy of N Detection needs ultrapure materials in deep underground location; way to distinguish recoils from β,γ events; neutron screening;... WIMP Distribution and Mass p. 5/33

Direct WIMP detection: theory Counting rate given by dr dq = AF 2 (Q) v max v min Q: recoil energy f 1 (v) v dv A= ρσ 0 /(2m χ m r ) = const.: encodes particle physics F(Q): nuclear form factor v: WIMP velocity in lab frame v 2 min = m NQ/(2m 2 r ) v max : Maximal velocity if WIMPs bound to galaxy f 1 (v): normalized one dimensional WIMP velocity distribution Note: Q 2 v 2 (1 cosθ ) dσ 1 dσ. dq v 2 d cos θ WIMP Distribution and Mass p. 6/33

Why bother? Allows to test models of galaxy formation: might shed new light on CDM crises WIMP Distribution and Mass p. 7/33

Why bother? Allows to test models of galaxy formation: might shed new light on CDM crises Might teach us something about merger history of our own galaxy (e.g. if tidal stream is detected) WIMP Distribution and Mass p. 7/33

Direct reconstruction of f 1 f 1 (v) = N { 2Q d dq [ 1 F 2 (Q) ]} dr dq Q=2m 2 rv 2 /m N WIMP Distribution and Mass p. 8/33

Direct reconstruction of f 1 f 1 (v) = N { 2Q d dq [ 1 F 2 (Q) ]} dr dq Q=2m 2 rv 2 /m N N : Normalization ( 0 f 1 (v)dv = 1). WIMP Distribution and Mass p. 8/33

Direct reconstruction of f 1 f 1 (v) = N { 2Q d dq [ 1 F 2 (Q) ]} dr dq Q=2m 2 rv 2 /m N N : Normalization ( 0 f 1 (v)dv = 1). Need to know form factor = stick to spin independent scattering. WIMP Distribution and Mass p. 8/33

Direct reconstruction of f 1 f 1 (v) = N { 2Q d dq [ 1 F 2 (Q) ]} dr dq Q=2m 2 rv 2 /m N N : Normalization ( 0 f 1 (v)dv = 1). Need to know form factor = stick to spin independent scattering. Need to know m χ, but do not need σ 0,ρ. WIMP Distribution and Mass p. 8/33

Determining the logarithmic slope of dr/dq Good local observable: Average energy transfer Q i in i th bin WIMP Distribution and Mass p. 9/33

Determining the logarithmic slope of dr/dq Good local observable: Average energy transfer Q i in i th bin Stat. error on slope (bin width) 1.5 = need large bins WIMP Distribution and Mass p. 9/33

Determining the logarithmic slope of dr/dq Good local observable: Average energy transfer Q i in i th bin Stat. error on slope (bin width) 1.5 = need large bins To maximize information: use overlapping bins ( windows ) WIMP Distribution and Mass p. 9/33

Recoil spectrum: prediction and simulated measurement 0.004 χ 2 /dof = 0.73 500 events, 5 bins, up to 3 bins per window 0.003 f 1 (v) [s/km] 0.002 0.001 input distribution 0 0 100 200 300 400 500 600 700 v [km/s] WIMP Distribution and Mass p. 10/33

Recoil spectrum: prediction and simulated measurement 0.004 5,000 events, 10 bins, up to 4 bins per window χ 2 /dof = 0.98 f 1 (v) [s/km] 0.003 0.002 input distribution 0.001 0 0 100 200 300 400 500 600 700 v [km/s] WIMP Distribution and Mass p. 11/33

Statistical exclusion of constant f 1 0.1 Average over 1,000 experiments Probability 0.01 0.001 0.0001 1e-05 1e-06 1e-07 1e-08 1e-09 median mean 1e-10 100 1000 N ev WIMP Distribution and Mass p. 12/33

Statistical exclusion of constant f 1 0.1 Average over 1,000 experiments Probability 0.01 0.001 0.0001 1e-05 1e-06 1e-07 1e-08 1e-09 median mean 1e-10 100 1000 N ev Need several hundred events to begin direct reconstruction! WIMP Distribution and Mass p. 12/33

Determining moments of f 1 v n 0 v n f 1 (v)dv WIMP Distribution and Mass p. 13/33

Determining moments of f 1 v n 0 v n f 1 (v)dv 0 Q (n 1)/2 1 F 2 (Q) dr dq dq WIMP Distribution and Mass p. 13/33

Determining moments of f 1 v n 0 v n f 1 (v)dv 0 Q (n 1)/2 1 F 2 (Q) events a Q(n 1)/2 a F 2 (Q a ) dr dq dq WIMP Distribution and Mass p. 13/33

Determining moments of f 1 v n 0 v n f 1 (v)dv 0 Q (n 1)/2 1 F 2 (Q) events a Q(n 1)/2 a F 2 (Q a ) dr dq dq Can incorporate finite energy (hence velocity) threshold WIMP Distribution and Mass p. 13/33

Determining moments of f 1 v n 0 v n f 1 (v)dv 0 Q (n 1)/2 1 F 2 (Q) events a Q(n 1)/2 a F 2 (Q a ) dr dq dq Can incorporate finite energy (hence velocity) threshold Moments are strongly correlated! High moments, and their errors, are underestimated in typical experiment: get large contribution from large Q WIMP Distribution and Mass p. 13/33

Determination of first 10 moments v n / v n exact 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 100 events 0 2 4 6 8 10 n WIMP Distribution and Mass p. 14/33

Constraining a late infall component 1000 25 events, fit moments n = -1, 1, 2 800 χ 2 = 1 χ 2 = 4 v esc [km/s] 600 400 200 0 0 0.2 0.4 0.6 0.8 1 N l WIMP Distribution and Mass p. 15/33

Constraining a late infall component 1000 800 100 events, fit moments n = -1, 1, 2, 3 χ 2 = 1 χ 2 = 4 v esc [km/s] 600 400 200 0 0 0.2 0.4 0.6 0.8 1 N l WIMP Distribution and Mass p. 16/33

Determining the WIMP mass Needed to check if DM WIMP is same as collider WIMP WIMP Distribution and Mass p. 17/33

Determining the WIMP mass Needed to check if DM WIMP is same as collider WIMP Method described above yields normalized f 1 (v) for any assumed m χ WIMP Distribution and Mass p. 17/33

Determining the WIMP mass Needed to check if DM WIMP is same as collider WIMP Method described above yields normalized f 1 (v) for any assumed m χ cannot determine m χ from single recoil spectrum, unless f 1 (v) is (assumed to be) known Can determine m χ model independently from two (or more) measurement, by demanding that they yield the same (moments of) f 1! WIMP Distribution and Mass p. 17/33

Formalism v n = α n (n + 1) I n I 0 α = m N 2m 2 red,n, I n = 0 Q (n 1)/2 F 2 (Q) dr dq dq WIMP Distribution and Mass p. 18/33

Formalism v n = α n (n + 1) I n I 0 α = m N 2m 2 red,n, I n = 0 Q (n 1)/2 F 2 (Q) dr dq dq m χ = mx m Y m X R n R n m X /m Y, R n α Y α X = I n,xi 0,Y I n,y I 0,X WIMP Distribution and Mass p. 18/33

Formalism v n = α n (n + 1) I n I 0 α = m N 2m 2 red,n, I n = 0 Q (n 1)/2 F 2 (Q) dr dq dq m χ = mx m Y m X R n R n m X /m Y, R n α Y α X = I n,xi 0,Y I n,y I 0,X σ(m χ ) vn R n mx /m Y m X m Y (R n ) 2 m X /m Y (m χ + m X )(m χ + m Y ) m X m Y κ WIMP Distribution and Mass p. 18/33

Selecting target materials Contours of constant κ/m χ 10 κ/m χ = 2 m Y / m χ 1 8 4 m Y / m X = 76/28 0.1 16 32 0.01 0.01 0.1 1 10 m X / m χ WIMP Distribution and Mass p. 19/33

Selecting target materials Contours of constant κ/m χ 10 κ/m χ = 2 m Y / m χ 1 8 4 m Y / m X = 76/28 0.1 16 32 0.01 0.01 0.1 1 10 m X / m χ Target nuclei should have quite different masses, preferably bracketing WIMP mass WIMP Distribution and Mass p. 19/33

Systematic errors Equality of moments of f 1 holds only if integrals run over identical ranges of v, e.g. v min = 0, v max =. WIMP Distribution and Mass p. 20/33

Systematic errors Equality of moments of f 1 holds only if integrals run over identical ranges of v, e.g. v min = 0, v max =. Real experiments have finite acceptance windows for Q, and hence for v Ensuring v min,x = v min,y and v max,x = v max,y only possible if m χ is known WIMP Distribution and Mass p. 20/33

Effect of Q min 0 m χ,rec / m χ,in 2 1.5 1 solid: including r(q min ) terms dashed: without r(q min ) terms black: Q min = 10 kev red: Q min = 3 kev 0.5 0 10 100 m χ,in [GeV ] WIMP Distribution and Mass p. 21/33

Effect of Q min 0 m χ,rec / m χ,in 2 1.5 1 solid: including r(q min ) terms dashed: without r(q min ) terms black: Q min = 10 kev red: Q min = 3 kev 0.5 0 10 100 m χ,in [GeV ] Use Q min = 0 from now on. WIMP Distribution and Mass p. 21/33

Effect of finite Q max (Higher) moments are very sensitive to high Q region, even to region with N ev < 1 WIMP Distribution and Mass p. 22/33

Effect of finite Q max (Higher) moments are very sensitive to high Q region, even to region with N ev < 1 Imposing finite Q max can alleviate this problem, WIMP Distribution and Mass p. 22/33

Effect of finite Q max (Higher) moments are very sensitive to high Q region, even to region with N ev < 1 Imposing finite Q max can alleviate this problem, but introduces systematic error unless Q max values of two targets are matched; matching depends on m χ. WIMP Distribution and Mass p. 22/33

Median reconstructed WIMP mass: no cut on Q 50 + 50 events, Si and Ge, standard halo, no cut on Q 1000 n = 1 n = 2 m χ,rec, hi, lo [GeV] 100 10 σ tot n = -1 1 10 100 1000 m χ,in [GeV] WIMP Distribution and Mass p. 23/33

edian reconstructed WIMP mass: optimal Q max matching 50 + 50 events, Si and Ge, standard halo, optimally matched Q max < 50 kev 1000 m χ,rec, hi, lo [GeV] 100 10 tot n = 2 n = 1 σ n = -1 1 10 100 1000 m χ,in [GeV] WIMP Distribution and Mass p. 24/33

Median reconstructed WIMP mass: equal Q max 50 + 50 events, Si and Ge, standard halo, Q max < 50 kev 1000 n = 1 m χ,rec, hi, lo [GeV] 100 10 σ tot n = 2 n = -1 1 10 100 1000 m χ,in [GeV] WIMP Distribution and Mass p. 25/33

Matching procedures Iterative: m χ,0 used for matching m χ,rec,1, used as new input... : converges on average Ge and Si, true m χ = 100 GeV 1000 2 x 50 events, Q max,ge = 50 kev 2 x 500 events, Q max,ge = 100 kev median m χ,rec [GeV] 100 10 10 100 1000 m χ,in [GeV] WIMP Distribution and Mass p. 26/33

Matching procedures Iterative: m χ,0 used for matching m χ,rec,1, used as new input... : converges on average Ge and Si, true m χ = 100 GeV 1000 2 x 50 events, Q max,ge = 50 kev 2 x 500 events, Q max,ge = 100 kev median m χ,rec [GeV] 100 Unfortunately, in given experiment often leads to endless loop! 10 10 100 1000 m χ,in [GeV] WIMP Distribution and Mass p. 26/33

Matching procedures Iterative: m χ,0 used for matching m χ,rec,1, used as new input... : converges on average Ge and Si, true m χ = 100 GeV 1000 2 x 50 events, Q max,ge = 50 kev 2 x 500 events, Q max,ge = 100 kev median m χ,rec [GeV] 100 Unfortunately, in given experiment often leads to endless loop! 10 10 100 1000 m χ,in [GeV] Instead developed matching procedure based on total χ 2 fit WIMP Distribution and Mass p. 26/33

Median reconstructed WIMP mass: χ 2 matching 50 + 50 events, Si and Ge, standard halo, matched Q max < 50 kev 1000 n = 2 m χ,rec, hi, lo [GeV] 100 10 σ tot n = 1 n = -1 1 10 100 1000 m χ,in [GeV] WIMP Distribution and Mass p. 27/33

Median reconstructed WIMP mass 50 + 50 events, Si and Ge, standard halo, Q max < 100 kev m χ,rec, hi, lo [GeV] 1000 100 10 algorithmic Q max matching optimal Q max matching no Q max matching 1 10 100 1000 m χ,in [GeV] WIMP Distribution and Mass p. 28/33

Median reconstructed WIMP mass: non standard halo 50 + 50 events, Si and Ge, halo with 25% late infall, Q max < 100 kev m χ,rec, hi, lo [GeV] 1000 100 10 algorithmic Q max matching optimal Q max matching no Q max matching 1 10 100 1000 m χ,in [GeV] WIMP Distribution and Mass p. 29/33

Comparison of corresponding recoil spectra 0.1 Ge target, m χ = 50 GeV 0.01 1/R dr/dq [1/keV] 0.001 standard halo 25% late infall 0.0001 1e-05 0 50 100 150 Q [kev] Difference is smaller for larger m χ WIMP Distribution and Mass p. 30/33

Median reconstructed WIMP mass 500 + 500 events, Si and Ge, standard halo, Q max < 100 kev 1000 m χ,rec, hi, lo [GeV] 100 algorithmic Q max matching optimal Q max matching no Q max matching 10 10 100 1000 m χ,in [GeV] WIMP Distribution and Mass p. 31/33

Distribution of measurements 0.08 2 x 50 events, Si and Ge, standard halo, Q max < 100 kev, m χ = 50 GeV 2 x 500 events, Si and Ge, standard halo, Q max < 100 kev, m χ = 200 GeV 0.1 0.07 fraction of expts in bin 0.06 0.05 0.04 0.03 0.02 no Q max matching optimal Q max matching algorithmic Q max matching fraction of expts in bin 0.08 0.06 0.04 0.02 no Q max matching optimal Q max matching algorithmic Q max matching 0.01 0-5 -4-3 -2-1 0 1 2 3 4 5 δm 0-5 -4-3 -2-1 0 1 2 3 4 δm WIMP Distribution and Mass p. 32/33

Summary: Learning from direct WIMP detection Learning about our galaxy: WIMP Distribution and Mass p. 33/33

Summary: Learning from direct WIMP detection Learning about our galaxy: Direct reconstruction of f 1 (v) needs several hundred events WIMP Distribution and Mass p. 33/33

Summary: Learning from direct WIMP detection Learning about our galaxy: Direct reconstruction of f 1 (v) needs several hundred events Non trivial statements about moments of f 1 possible with few dozen events Needs to be done to determine ρ χ : required input for learning about early Universe! Learning about WIMPs: Can determine m χ from moments of f 1 measured with two different targets. Issues regarding Q max remain. WIMP Distribution and Mass p. 33/33