ER

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Aug 2, 2007 ... two key review articles: ... R.J. Gaitskell (experiment review) in Ann. Rev. Nucl. .... Emax. Emin. dR(Ei). Ei r. For Emax use ∞ or vesc (later. . . ) .... e−v2 esc/v2. 0. ] Fortunately, average value well approximated by numerical fit.
Direct Detection of WIMP Dark Matter: Part 1 - a primer Dan Akerib Case Western Reserve University CDMS Collaboration SLAC Summer Institute Aug 2, 2007

Standard Cosmology Colley, Turner & Tyson

from Perlmutter, Phys. Today (units of critical density)

WMAP

WMAP Dan Akerib

SSI 2007

Tytler & Burles Case Western Reserve University

Non-Baryonic Dark Matter Matter = 0.30 ± 0.04





Nature of dark matter Non-baryonic Large scale structure predicts DM is ‘cold’



WIMPs – Weakly Interacting Massive Particle   

Freeze out: H > A~ n A v 

Baryons = 0.05 ± 0.005





Production suppressed (T= OmegaCDMmin) Guidice and Romanino, 2004, mu < 0 A. Pierce, Finely Tuned MSSM Guidice and Romanino, 2004, mu > 0 Chattopadhyay et. al Theory results - post WMAP Baltz and Gondolo, 2004, Markov Chain Monte Carlos (1 sigma) Baer et. al 2003 Kim/Nihei/Roszkowski/de Austri 2002 JHEP Ellis et. al Theory region post-LEP benchmark points Baltz and Gondolo 2003 Baltz and Gondolo, 2004, Markov Chain Monte Carlos 060227062601

Dan Akerib

10

Case Western Reserve University

WIMPs in the Galactic Halo detector

Exploit movements of Earth/ Sun through WIMP halo 



Direction of recoil -- most events should be opposite Earth/Sun direction (Spergel 1988) Annual modulation -- harder spectrum when Earth travels with sun (Drukier, Freese, & Spergel 1986)

galactic center Sun 230 km/s

June Dec.

energy transferred appears in ‘wake’ of recoiling nucleus

WIMP-Nucleus Scattering

Scatter from a Nucleus in a Terrestrial Particle Detector

Log(rate)



WIMP

~2% seasonal effect

Dec

June

Erecoil Dan Akerib

SSI 2007

Case Western Reserve University

Defining the Signal



Kinematics   



Rate  

• •

n(v ) MW

halo density cross section

• •



SD/SI coherence & form factors

ER

dR log dER

annual modulation of rate diurnal modulation of direction

ER

Backgrounds Experimental methods & results Dan Akerib

vE

MN

σW N

Primary signal Secondary features 

• •

halo potential WIMP mass target mass & velocity

} mostly tomorrow

SSI 2007

Case Western Reserve University

References and further reading



References and notation - generally following the treatment of two key review articles:  



A very nice pedagogic reference 



J.D. Lewin and P.F. Smith, Astroparticle Physics 6 (1996) G. Jungman, M. Kamionkowski and K. Griest, Physics Reports 267 (1996) S. Golwala, Ph.D. thesis, UC Berkeley (2000) (http://cdms.berkeley.edu)

• •

10th CDMS dissertation - first astrophysics results! winner of first APS Tanaka dissertation prize!

See also  

Dan Akerib

R.J. Gaitskell (experiment review) in Ann. Rev. Nucl. Part. Sci. 54 (2004) G. Heusser (low background techniques) in Ann. Rev. Nucl. Part. Sci. 45 (1995)

SSI 2007

Case Western Reserve University

Differential energy spectrum (simplified) dR R0 −ER /E0 r = e dER E0 r R ER R0 E0 r MW MN

dR log dER ER

= = = =

event rate per unit mass recoil energy  ∞ total event rate dR dER = R0 dER most probable incident 0 energy (Maxwellian)  ∞ 4MW MN dR = ER  = ER dER (MW + MN )2 dER 0 = mass of WIMP = E0 r = mass of target nucleus

Dan Akerib

SSI 2007

Case Western Reserve University

WIMP search - c.1988 Germanium ionization detector (UCSB/UCB/LBL) http://dmtools.brown.edu/ Gaitskell,Mandic,Filippini

log cross section (arb. units)



070802080001

0

1

2

10 WIMP Mass [GeV]

10

from Jungman et al. (D.O. Caldwell et al.)

Dan Akerib

SSI 2007

Case Western Reserve University

3

Typical numbers For: MW

β ER 

Dan Akerib

=

E0

= MN = 100 GeV/c ⇒r=1

2

∼ 0.75 × 10−3 = 220 km/s MW β02 c2 GeV 1 = 2 100 2 (0.75 × 10−3 )2 c2 c = 30 keV =

1 2

SSI 2007

Case Western Reserve University

Refinements  dR  = R0 S(ER ) F 2 (ER ) I  dER OBS S(ER ) = spectral function — masses and kinematics F 2 (ER ) = form factor correction, with ER = q 2 /2MW I = interaction type

Dan Akerib

SSI 2007

Case Western Reserve University

Kinematics DM particle with velocity v and incident KE Ei = 12 MW v 2 scattered at angle θ in CM frame gives recoil energy in lab frame θ θ

where

(1 − cos θ) E R = Ei r 2 4 M W MN 4m2r = r= MW MN (MW + MN )2

and

MW MN mr = MW + MN

is the reduced mass Dan Akerib

SSI 2007

Case Western Reserve University

Kinematics Isotropic scattering: uniform in cos θ Incident WIMP with energy Ei gives recoil energies uniformly in ER = 0 → Ei r

log

dR dER

Recall “familiar” case for equal masses (r = 1), target at rest, head-on collision ER = Ei dR(Ei) Overall spectrum? —sample incident spectrum In each interval Ei → Ei + dEi contribution to spectrum in ER → ER + dER at rate dR(Ei ) of ER  dR(E )  dR Ei r i (ER ) = d dER Ei r Dan Akerib

SSI 2007

Case Western Reserve University

Kinematics Need to integrate over range of incident energies dR (ER ) = dER



Emax

Emin

dR(Ei ) Ei r

For Emax use ∞ or vesc (later. . . )

ER Ei ≥ ≡ Emin r

dR dER

dR(Ei) log

For Emin , to get recoil of energy ER need incident energy

ER

Ei r and also need differential rate. . . Dan Akerib

SSI 2007

Case Western Reserve University

Differential rate In a kilogram of detector of nuclear mass number A N0 dR = σ v dn A where the differential density dn is taken as a function v n0 dn = f (v , vE ) d3v k

volume v swept per unit time contains dn(v) particles with velocity v

with n0 = ρDM /MW and normalization  k = f d3v

Dan Akerib

SSI 2007

Case Western Reserve University

Coordinate system collision kinematics

v

=

vW,E vE,G vW,G

= = =

WIMP velocity in the target/Earth frame Earth velocity in the Galaxy frame WIMP velocity in the Galaxy frame

galaxy dynamics

vW,G

vE,G

galaxy

vW,G

= vW,E + vE,G = v + vE

−( v + vE )2 /v02

f (v , vE ) = e

Maxwellian velocity distribution Dan Akerib

SSI 2007

Case Western Reserve University

Differential rate For simplified case of vE = 0 and vesc = ∞ 1 3 v f (v, 0) d v dR = R0 4 2πv0 with R0 =

− 12 N0 ρDM 2π A MW

σ0 v0 −v 2 /v02

For Maxwellian f (v, 0) = e , isotropic d3 v → 4πv 2 dv, Ei = 12 MW v 2 and E0 = 12 MW v02 : dR (ER ) = dER = Dan Akerib



∞ ER r

dR(Ei ) R0 = 1 Ei r r( 2 MW v02 )2

R0 −ER /E0 r e E0 r SSI 2007





vmin

vmin

−v 2 /v02

e

v dv

 = 2ER /rMW

Case Western Reserve University

Corrections: escape velocity

For finite vesc 2 dR k0 R0 −ER /E0 r −vesc /v02 = −e ) (e dER k1 (vesc , 0) E0 r

but

k0 k1

= 0.9965 for vesc = 600 km/s,

and for MW = MN = 100 GeV/c2 , maximum ER = 200 keV ⇒ cutoff energy ER  = 30 keV

Dan Akerib

SSI 2007

Case Western Reserve University

Corrections: earth velocity Clearly vE = 0 — but ∼ v0 = 230 km/s. Full calculation yields: √   

2 2 vmin + vE vmin − vE π v0 k0 R0 dR(vesc , vE ) erf −erf −e−vesc /v0 = dER k1 E0 r 4 vE v0 v0  x −t2 3 2 where erf(x) = √π 0 e dt, vmin = v0 ER /E0 r, k0 = (πv02 ) 2 and  

2 2 vesc 2 vesc k1 = k0 erf −√ − e−vesc /v0 v0 π v0 Fortunately, average value well approximated by numerical fit R0 −c2 ER /E0 r dR(vesc = ∞, vE ) = c1 e dER E0 r

~30% increase in integrated rate = c1/c2 = 0.751/0.561 and harder spectrum Dan Akerib

SSI 2007

Case Western Reserve University

Signal modulations: annual effect June

June Dec

Dec

Dec

WIMP Signal

±2%

June

Dec

June

Background Dec

Dec



t − 152.5 vE (t)[km/s] = 232 + 15 cos 2π 365.25 galactic center Sun 230 km/s Dan Akerib

Log(rate)

t in days after January 1 June Dec. SSI 2007



Dec

~2% seasonal effect need ~1000 events June

Erecoil

Case Western Reserve University

Signal modulations: recoil direction



Differential angular spectrum:

d2 R 1 R0 −(vE cos ψ−vmin )2 /v02 = e dER d(cos ψ) 2 E0 r

• •

Asymmetry  more recoils in forward direction by 5x: ~10 events Orientation of lab frame rotates relative to forward direction  

eg, definition of forward/backward in lab frame changes as earth rotates ⊥ versus reduces asymmetry to 20% effect: ~300 events

ψ galactic center Sun 230 km/s Dan Akerib

me a r f g n rotati SSI 2007

θ

Case Western Reserve University

Refinements  dR  = R0 S(ER ) F 2 (ER ) I  dER OBS S(ER ) = spectral function — masses and kinematics time dependence

F 2 (ER ) = form factor correction, with ER = q 2 /2MW I = interaction type in zero-velocity limit (v