Harmonics in the Dark-Matter Sky: Directional Detection in the Fourier-Bessel Basis

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by  Samuel K. Lee

Harmonics in the Dark-Matter Sky:
Directional Detection in the Fourier-Bessel Basis
Samuel K. Lee
Princeton Center for Theoretical Science,
Princeton University, Princeton, NJ 08544∗
(Dated: January 27, 2014)
4
Details about thevelocity distribution of weakly interactingmassive particle (WIMP) dark mat-
1
ter in our galaxy may be revealed by nuclear-recoil detectors with directional sensitivity. Previous
0
studies have assumed that the velocity distribution takes a simple functional form characterized
2
by a small number of parameters. More recent work has shown that basis-function expansions
n may allow for more general parameterization; such an approach has been considered for both the
a
one-dimensional speed and momentum distributions, and also for three-dimensional velocity distri-
J
butionsobeying certain equilibrium conditions. In this work, I extendthis basis-function approach

3 to allow for arbitrary velocity distributions by working in the Fourier-Bessel basis, deriving an an-
2 alytic expression for the directional recoil spectrum. Such an approach is completely general, and

may be useful if the velocity distribution is too complex to be characterized by simple functional
]
O formsoris notcompletely virialized. Resultsconcerningthethree-dimensional Radontransform of
theFourier-Bessel basis functions may be of general interest for tomographic applications.
C
.
h
I. INTRODUCTION centannualmodulationinthesignalthatcanbe
p
usedbynon-directionalexperimentstodiscrim-
-
o inate against background [3]. Yet despite this
ThedirectdetectionofWIMPdarkmatterby
r advantage, it is unlikely that directional exper-
t experimentssensitivetotheenergyspectrumof
s iments will be the first to detect andcharacter-
a WIMP-induced nuclear recoils may be on the
ize the particle properties of dark matter, since
[ near horizon. Although there have been sev-
their lower target densities and higher recoil-
eral tantalizing hints of signals, a definitive de-
1
energy thresholds result in relatively smaller
v tection has yet to be achieved. Nevertheless,
event rates. However, directional detectors will
9 some theoretical and experimental efforts are
beinauniquepositiontoprobethedark-matter
7 already looking ahead to a new generation of
1 direct-detection experiments,which employ de- velocity distribution in the post-discovery era.
6 tectors sensitive to the directional recoil spec- Theymayallowforthestudyof“WIMPastron-
. omy,” possibly revealing insights about struc-
1 trum–i.e.,abletomeasurenotonlythe energy
ture formation on galactic scales via observa-
0 but also the direction of nuclear recoils (for a
4 review, see [1]). tions of the local dark-matter sky. Further-
1 more, a detailed characterization of the dark-
A primary advantage of directional detectors
: matter velocity distribution may be crucial for
v istheirabilitytodiscriminatesignaleventsfrom
Xi background events; the former should have a using direct-detection experiments to precisely
constrain the dark-matter particle properties.
distinct directionality as a result of the WIMP
r
a “wind” induced by the sun’s motion through To this end, understanding how the WIMP
the galaxy, whereas the latter should generally velocity distribution affects the observed di-
have an isotropic distribution [2]. The ampli- rectional recoil spectrum is of primary impor-
tude ofthe directionalasymmetry canbe of or- tance. Previous studies have been conducted
der unity, and is much larger than the few per- along these lines, albeit with some simplifying
assumptions. For example, earlier work con-
sidered only velocity distributions with param-
eterized functional forms [4–7]. The strength
∗Electronicaddress: [email protected] of this parametric, functional-form approach is
2
perhaps the same as its weakness – it reduces can still achieve an analytic expression for the
theamountofinformationtobe constrainedby directional recoil spectrum.
directional experiments to only a few numbers, InthenextSection,wewillbrieflyreviewba-
allowing for only simple models to be tested. sic expressions for the directional recoil spec-
Recent studies on non-directional exper- trum of WIMP-induced nuclear recoils. In
iments have sought to move beyond the Sec. III, we will discuss the Fourier-Bessel ba-
functional-form approach. For example, sis and its mathematical properties. Sec. IV
Refs.[8,9]exploredthepossibilityofconstrain- will elucidate the application of the Fourier-
ingeithertheone-dimensionalspeedormomen- Bessel expansion to directional detection, and
tum distribution in a series of bins, assuming will derive an analytic expression for the pre-
that the distribution is constant in each bin. dicted directional recoil spectrum in terms of
An alternative approach is to perform a basis- the velocity-distribution Fourier-Bessel coeffi-
function expansion of the speed distribution, cients. Appendicesdefiningmathematicalnota-
which is then completely characterized by its tion and addressing some additional points fol-
expansion coefficients [10–12]. low the concluding discussion in Sec. V.
This latter approach can also be applied to
the full three-dimensional velocity distribution
II. DIRECTIONAL DETECTION:
in studies of directional experiments. In this
BASICS
context,Ref.[13]examinedabasisconsistingof
functions of the integrals of motion E, L, and
We are ultimately interested in determining
L . Such a basis is convenient for two reasons:
z
the Galactic-frame velocity distribution func-
1) if the velocity distribution is at equilibrium
tion by a measurement of the directional recoil
(virialized) andcan hence be written as a func-
spectrumofWIMP-inducednuclearrecoils. Let
tion of integrals of motion, measuring the de-
usfirstexaminehowthesetwoquantitiesarere-
pendenceofthelocaldistributionontheseinte-
lated.
gralsofmotionallowsus toinfer theglobaldis-
The Galactic-frame velocities of the WIMP
tribution(throughoutthe galaxy),and2)ifthe
and the lab are related to the lab-frame WIMP
velocity distribution is separable in E, L, and
velocity v by a simple Galilean transformation,
L , the basis functions can be chosen so that
z
such that the Galactic-frame WIMP velocity is
the resulting directionalrecoilspectrum can be
v + v . Hence, the Galactic-frame velocity
found analytically. However,we see that we re- lab
distributiong and the lab-framevelocity distri-
quire the assumptions of equilibrium and sepa-
bution f are simply related by a translation,
rability in order to make use of this integrals-
of-motion basis. f(v)=g(v+v ). (1)
lab
One motivation of such basis-function ap-
proacheshasbeentoremovethebiasesincurred Assuming elastic WIMP-nucleus scattering,
by the choice of the form of the velocity distri- Ref. [4] showed that the directional recoil spec-
butioninthefunctional-formapproach,moving trum is given by
towards a more agnostic description of the ve-
dR ρ σ S(q)
locitydistribution. Assuch,thenaturalcontin- dEdΩ = 40πmN µ2 f(vq,q). (2)
uationistoconsiderabasisthatallowsforcom- q χ N
pletely general velocity distributions. In this Here, R is the number of evebnts pber exposure
paper, we explore an expansion of the veloc- (detector mass multiplied by time), ρ is the
0
itydistributionintheFourier-Besselbasis. The local WIMP density, µ =m m /(m +m )
N χ N χ N
Fourier-Bessel basis indeed allows for complete is the reduced mass of the WIMP-nucleus sys-
generality, so that we can drop the assump- tem, q=qq is the lab-frame nuclear-recoilmo-
tions of equilibrium and separability required mentum, E =q2/2m isthe lab-framenuclear-
N
by Ref. [13]. Furthermore, we shall see that we recoil energy, and v =q/2µ is the minimum
b q N
3
lab-frame WIMP speed required to yield a re- III. FOURIER-BESSEL
coil energy E.1 Furthermore, we have writ- DECOMPOSITION OF THE VELOCITY
ten the WIMP-nucleus elastic cross section as DISTRIBUTION
dσ/dq2 =σ S(q)/4µ2v2 (where S(q)is the nu-
N N
clear form factor), and Ifour goalis to determine the velocity distri-
bution g, we must first understand how it may
f(vq,q)= d f(v) be characterized. Most direct-detection studies
P
ZP (3) assume that g can be approximated by a trun-
b b = d3vδ(v v q)f(v) cated Maxwellian; such a velocity distribution
q
− · is isotropic and is appropriate for collisionless
Z
is the Radontransformofthe lab-frameWIMP particles with a halo density profile that falls
b
velocity distribution.2 These relations follow as ρ r−2. This “Standard Halo Model” ve-
∝
from the kinematics; the rate of nuclear recoils locitydistributionmaybespecifiedbyonlytwo
withagivenenergyE anddirectionqispropor- parameters: the velocity dispersion σH and the
tional to the number of WIMPs with velocities truncating escape velocity vesc. Likewise, more
v that can induce such recoils – i.e., that sat- complex velocity distributions – which may be
b
isfy the non-relativistic elastic-scattering kine- non-Maxwellian[15, 16], anisotropic[17–20], or
matics determined by v . Such velocities lie on include the presence of a dark-matter disk [21–
q
theplane inthelab-framevelocityspacethat 28],streams[29–32],debrisflows[33,34],orex-
is definedPby v q=v , andare selectedby the tragalactic components [35, 36] – may be mod-
q
delta function i·n the Radon transform.3 eled by using similarly manageable functional
From the perspective of particle physics, the forms with a few additional parameters. These
b
primary quantities of interest to be determined parameters may then be constrained by mea-
by direct-detection experiments are the WIMP surement of the directional recoil spectrum.
particle properties m and σ . However, from However, it is clear that such studies are de-
χ N
the perspective of astrophysics, the determina- pendent on the a priori functional form chosen
tion of the velocity distribution g is also of in- for g, which may be oversimplified. It might
terest; for example, galactic-scale N-body sim- thenbeadvantageoustoinsteadconsideranal-
ulations may give predictions for g. Further- ternative approach in which we use a complete
more, an understanding of g may be critical basis for functions of velocity to decompose g.
in correctly interpreting the results of direct- We canthenstudy completely generaldistribu-
detectionexperimentsandaccuratelyconstrain- tion functions.
ing the WIMP particle properties. In what fol-
lows, our primary goalwill be to examine what
A. The Fourier-Bessel Basis
can be learned about g from the directional re-
coil spectrum dR/dEdΩ .
q
In this paper, we will show that a convenient
basisisgivenbyageneralizationoftheFourier-
Bessel basis. We can define basis functions for
1 Inelasticscatteringiseasilyaccountedforbychanging functions of velocity g(v) simply given by the
therelationbetweenvq andE appropriately. productofasphericalBesselfunctionandareal
2 The Radon transform is widely used in tomographic spherical harmonic
applicationsinvolvingimagingbysections(planes,in
thecaseofthreedimensions);e.g.,magneticresonance Ψu (v)=4πj (uv)S (v), (4)
lm l lm
imaging. Foranoverview,seeRef.[14].
3 In general, the presence of this delta function makes such that the basis functions are labeled by a
difficultthecalculation of thedirectional recoilspec- b
real number u and integers l and m. For the
trumforarbitraryvelocitydistributionsvianumerical
purposesofpresentation,therealsphericalhar-
integration of Eq. (3). We shall see that use of the
Fourier-Besselbasisallowsforananalyticresult. monics Slm (defined in Appendix A) are used
4
instead of the usual spherical harmonics Y , Note that the integral includes negative values
lm
since we are primarily concerned with expand- of v here.
q
ing the real velocity distribution. Furthermore, examining Eqs. (8) and (9),
The functions Ψu (v) are solu- taking note of the plane-wave expansion
lm
tions to the scalar Helmholtz equation
( 2+u2)Ψu (v)=0, and obey the complete- eiuv = 4πilj(uv)S (u)S (v)
∇ lm · l lm lm
ness relation lm
X (12)
(u22πd)u3Ψulm(v)Ψulm(v′)=δ3(v−v′) (5) = lm ilΨulm(v)Slm(ub), b
X
lm Z
X b
andusingtheorthonormalityrelationofEq.(6),
and the orthonormality relation
it can be shown that the Fourier-Bessel and
d3vΨulm(v)Ψul′′m′(v)= (2uπ2)3δ(u−u′)δll′δmm′. Fourier coefficients are simply related by
Z
(6) gu =il dΩ S (u)g(u). (13)
lm u lm
Z
The Ψu (v) thus constitute a complete or-
thonormalml basis for velocity distributions, any Thatis,theFourier-Besselcoebfficeientisgivenby
multiplying the Fourier coefficient by the cor-
of which can be decomposed as
responding spherical harmonic and performing
u2du the angular integral.
g(v)= Ψu (v)gu . (7)
(2π)3 lm lm
lm Z
X
We may also invert this relation to obtain an C. Boundary Conditions
expression for the expansion coefficients,
The Galactic-frame WIMP velocity distribu-
gu = d3vΨu (v)g(v). (8) tion is typically truncated at the escape veloc-
lm lm
Z ity vesc. Thus, we now consider the decompo-
sition for the case in which the function g(v)
vanishes for v >v [37]. This boundary con-
B. Relation to Fourier Basis esc
dition constrains the allowed radial eigenfunc-
tions and generates a discrete radial spectrum
Alternatively, one might use the standard
with eigenvalues u that satisfy
ln
Fourier basis to decompose the velocity distri- { }
bution into plane waves,
j (u v )=0. (14)
l ln esc
d3u
g(v)= (2π)3 eiu·vg(u), (9) Weseethatuln =xln/vesc,wherexln isthenth
Z zero of j.
l
with expansion coefficients e Functions that obey this boundary condition
can be expanded using the basis functions
g(u)= d3ve−iu·vg(v). (10) Ψn (v)=4πc j (u v)S (v)θ(v v)
Z lm ln l ln lm esc−
For our pureposes, this is of special interest be- =clnΨulmln(v)θ(vesc−v),
cause of the Fourier slice theorem that relates b (15)
theFouriertransformg totheRadontransform
whicharelabeledbyintegersn,l,andm. Here,
g via
e x j (x )
b g(u)= dvqe−iuvqg(vq,u). (11) c−ln1 ≡ ln l√+1π ln (16)
Z
e b b
5
is a normalization factor, such that these func- assumption of equilibrium allows one to infer
tions obey the completeness relation the global velocity distribution from measure-
mentsofthelocalvelocitydistribution,itisnot
u(2l2nπv)e−3s1cΨnlm(v)Ψnlm(v′)=δ3(v−v′) (17) cplreoabredif bthyisdiarsescutm-dpettieocntioisnveaxlpideroimnetnhtes.scTalhees
nlm
X generality of the Fourier-Bessel basis allows us
and the orthonormality relation to remain completely agnostic about the form
of the velocity distribution.
Z d3vΨnlm(v)Ψnl′′m′(v)= u(2l2nπv)e−3s1cδnn′δll′δmm′. disOcruerteuFltoimuraietre-Bgeosasleliscotehffiecnientots rgelncmovferromthae
(18) directional experiment. In an ideal experi-
ment with nearly continuous data and com-
A velocity distribution truncated at vesc can plete spectral coverage, one could use tradi-
therefore be expanded as tional tomographic methods to simply invert
the Radon transform and deduce the under-
g(v)= u2lnve−s1cΨn (v)gn , (19) lying glnm (see Appendix C). Realistically, we
(2π)3 lm lm are limited to using maximum-likelihood meth-
nlm
X ods to scan over the parameter space of the
with the inverse relation giving the discrete ex- coefficients gn , searching for those coefficients
lm
pansion coefficients that yield the best-fit predicted directional re-
coilspectrum for a givendata set; similar anal-
gn = d3vΨn (v)g(v) yses were carried out for simulated data sets
lm lm (20) for both non-directional and directional exper-
Z
=c guln. iments in Refs. [10–13]. Since we must limit
ln lm
the parameter space according to the available
Comparing with the expressions in Sec. IIIA, computational power by truncating at some n
it is easy to see that the discrete and con- and l, it is possible that only large-scale fea-
tinuous expressions are related by Ψn Ψu , turesing maybe reconstructedatfirst. Thisis
u u, and v 1 du in thelmlim→it thlmat an unfortunate drawback of the basis-function
ln → n e−sc →
v . Working with the discrete expres- and maximum-likelihood approach. Neverthe-
esc
sions→ha∞s the oPbvious advRantage of limiting the less, in order to construct the likelihood func-
expansion coefficients gn to a countable set, tion, one only needs to know how to calculate
lm
which can be truncated to give a finite number the predicted directional recoil spectrum given
of quantities to be determined. the coefficients gn , andcansimply reconstruct
lm
g from the maximum-likelihood coefficients us-
ing Eq. (19).
D. Advantages and Disadvantages As we shall demonstrate shortly, the Fourier
slice theorem in Eq. (11) and the unique ap-
OneclearadvantageoftheFourier-Besselba- pearanceoftheFourier-Besselbasisfunctionsin
sis is that it allows the representation of gen- theplane-waveexpansioninEq.(12)ultimately
eral functions of velocity. By moving beyond a allow us to find an analytic expression for the
priori assumptions of simple functional forms, Radon transform of the basis functions. The
we can avoid biasing inferences drawn from upshot of this is that given the expansion co-
direct-detectionexperiments. Furthermore,un- efficients gn for an arbitrary velocity distribu-
lm
like with the integrals-of-motion basis used in tiong,thereisasimpleanalyticexpressionthat
Ref. [13], we are not limited to equilibrium ve- yieldsthepredicteddirectionalrecoilspectrum,
locitydistributionsthatareseparableinthein- allowing us to easily construct the desired like-
tegrals of motion. Although it is true that the lihood function. This avoids the need for com-
6
putingtheRadontransformnumerically,apro- implies that
cessmadeungainlybythe presenceofthedelta
function in Eq. (3). This is the primary reason f(v ,u)=g(v +v u,u). (21)
q q lab
the Fourier-Besselbasis might be a unique and ·
useful basis for directional-detection analyses. Thus, forba gibven vblab, we canbebasily find f
However, it could be argued that numerical
once we have first calculated g. Noting that
integrationisstillrequiredtofindtheexpansion
the Radontransformis linear,wecanexpandgb
coefficients gn for a given distribution g via
lm as in Eq. (19) and find b
Eq.(20). Forexample,onemaybeinterestedin
cthaelcudliasttrinibgutthioencogepffirceideincttsedglnmbycaonrreNs-pboonddyinsgimto- g(vq,u)= u(2l2nπv)e−3s1cΨnlm(vq,u)glnm. (22)
ulation (in order to compare with those coeffi- nlm
X
cients recovered from a directional experiment, b b b b
We then see that the problemreduces to calcu-
perhaps). Fortunately, the Fourier-Bessel ba-
lating the RadontransformΨn of the Fourier-
sis is well studied, so that efficient routines to lm
Besselbasis functions. However,recallthat the
calculate Fourier-Bessel coefficients are readily
available(e.g.,seeRefs.[38,39]). Alternatively, Fourier slice theorem relatesbthe Radon trans-
form Ψn to the Fourier transform Ψn via
recent work has shown that the Fourier-Bessel lm lm
Eq. (11), the inverse of which yields
expansioncanbesimplyrelatedtoananalogous
Fourier-Laguerre expansion. Fortunately, there b e
1
exists an exact quadrature rule for the evalua- Ψn (v ,u)= dueiuvqΨn (u). (23)
lm q 2π lm
tionoftheintegralsfortheFourier-Laguerreco-
Z
efficients (i.e., integrals analogous to Eq. (20)),
b b e
and a corresponding sampling theorem can be Thus, the crux of the problem of finding f
found [40, 41]. Thus, if the velocity distribu- ultimatelyreducestofirstcalculatingthethree-
tionisband-limitedintheFourier-Laguerreba- dimensional Fourier transform Ψn and thenb
lm
sis, one can easily find the Fourier-Bessel coef- taking an additional one-dimensional Fourier
ficients using these results. In any case, such transform to yield Ψn . The firest step is ac-
lm
concerns are secondary if we are primarily con- complished by using the plane-wave expansion
cerned with computing the directional recoil of Eq. (12), b
spectrum, which we shall now do.
Ψn (u)= d3ve iuvΨn (v)
lm − · lm
Z
SPEIVC.TRTUHME IDNIRTEHCETFIOONUARLIERRE-BCEOSILSEL e =16π2i−lclnjl−1(xln) (24)
BASIS ulnve2sc j(uv )S (u),
× u2 u2 l esc lm
− ln
Our goal is to calculate the predicted direc-
tional recoil spectrum dR/dEdΩ given an ar- wherewehavehaveusedaresultforthebintegral
bitraryGalactic-framedark-matteqrvelocitydis- 0vescdvv2j(ulnv)jl(uv) and the orthonormality
tribution g truncated at vesc. We shall see that oRfthe Slm tosimplify. Thesecondstep,accom-
utilization of the Fourier-Bessel basis results in plished by substituting Eq. (24) into Eq. (23),
a straightforward expression for dR/dEdΩ in then yields
q
terms of the expansion coefficients gn .
First, because the lab-frame distlrmibution f Ψnlm(vq,u)=8πve2sci−lclnxlnjl−1(xln) (25)
and the Galactic-frame distribution g are re- Kn(v /v )S (u),
× l q esc lm
lated by a translation as in Eq. (1), the behav- b b
ior of the Radon transform under translation where we have defined the integrals
b
7
l=0 l=1 l=2
a) 10
nK(l10 5
)n 5
xl
j(1l−5 0 n=1
3xln 0 -5 n=2
n n=3
cl
li−0 -5 n=4
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
a a a
n
FIG. 1: The functions Kl (a) (normalized by the n- and l-dependent factors present in Eq. (27)) are
plotted for a few valuesof n andl. Notethat theKln are simply sinusoidal for l=0and are combinations
of sinusoidal and polynomial terms for higher l. The vq dependence of the directional recoil spectrum is
given bya linear combination of these functions evaluated at a=(vq+vlab·qb)/vesc.
eiaxj (x)
Kn(a)= dx l
l x2 x2
Z − ln
(26)
l+1
1
= [(α isβ )ReI(a+s,r,x )+(iα +sβ )ImI(a+s,r,x )] .
lr lr ln lr lr ln
2 −
r=1s= 1
X X±
The calculation of the integrals Kn(a) requires simplynormalizethefunctionandareeasilycal-
l
contour integration and is somewhat involved, culable. Itisdifficultto thinkofamoregeneric
sowerelegatethederivationofthisresulttoAp- and convenient angular dependence than that
pendix B. The coefficients α and β are also givenbythesphericalharmonics. Furthermore,
lr lr
defined there, as is the function I(a+s,r,x ). as shown in Appendix B, Kn is actually com-
ln l
However, we will mention here that α , β , posed of a finite sum of elementary functions,
lr lr
and I(a+s,r,x ) are such that i lKn is al- so the dependence on v can be easily calcu-
ln − l q
ways real,so that Ψn is also real,as expected. lated. Again, that we are able to achieve a
lm
The functions Kn are plotted for a few values relatively simple analytic result for the Radon
l
of n and l in Fig. 1b. transform of the Fourier-Bessel basis functions
Although these equations may appear rel- is partly due to the unique appearance of these
atively complicated, they are actually sur- basis functions in the plane-wave expansion, as
prisingly straightforward. The dependence of advertised in Sec. IIID.
Ψn (v ,u) on v and u is wholly contained in
lm q q
the functions Kn(v /v ) and S (u), respec- The key result of this paper is then given by
l q esc lm
tbively; thbe other factorbs appearing in Eq. (25) combining Eqs. (2), (21), (22), (25), and (26),
b
dR ρ σ S(q)
dEdΩq = 4π30mNχµ2Nvesc nlmi−lclnx3lnjl−1(xln)Kln[(vq+vlab·q)/vesc]Slm(q)glnm. (27)
X
b b
8
Thisexpressiongivesthedirectionalrecoilspec- basis. Unlike other bases explored in previ-
trum dR/dEdΩ expected from an arbitrary ous work, the Fourier-Bessel basis allows for
q
Galactic-frame velocity distribution g charac- the representation of general velocity distribu-
terized by the Fourier-Bessel expansion coeffi- tions,withoutrequiringassumptionsofequilib-
cients gn . Given the coefficients gn , we see riumorseparability. Moreimportantly,anana-
lm lm
that the energy dependence of the directional lyticexpressionforthepredicteddirectedrecoil
recoilspectrumis thendeterminedbythe func- spectrum in terms of the Fourier-Bessel coef-
tions Kn and the nuclear form factor S(q). ficients of the velocity distribution can be de-
l
Alternatively, one may instead choose to rived, which allows for the calculation of event
write Eq. (27) in terms of the expansion co- rateswithoutrequiringnumericalintegrationof
efficients fn for the lab-frame velocity distri- the Radon transform.
lm
bution f. Besides the replacement of gn with As previously mentioned, one way to explore
lm
fn , the rest of the expression is similar except the power of this formalism is to consider a
lm
fortwodifferences: 1)basisfunctionstruncated fiducialdark-mattervelocitydistribution,simu-
at ve′sc =vesc+vlab instead of at vesc should latenuclear-recoileventsandthecorresponding
be used throughout, and 2) Kln(vq/ve′sc) ap- directional recoil spectrum, and then attempt
pears instead of Kln[(vq+vlab·q)/vesc]. The to recover the Fourier-Bessel coefficients of the
seconddifferenceis especiallynoteworthy,since distribution via maximum-likelihood methods.
the angular dependence of dR/dbEdΩq is then One might consider fiducial distributions mo-
completely contained in the real spherical har- tivated by the results of galactic-scale N-body
monics Slm(q). This suggests that a spherical- simulations. We leave such studies to future
harmonicdecompositionofthedirectionalrecoil work.
spectrum might be useful, since the moments
b Furthermore, the harmonic decomposition of
(dR/dEdΩ ) are then generated only by the
q lm theangularandenergydependenceoftheevent
coefficients fn with the same l and m indices.
lm rate we have studied here might also be com-
One might then try to choose a coordinate sys-
bined with recent work studying the harmonic
tem in whichsome of either the coefficients fn
lm decomposition of the time dependence of the
ortheobservedmoments(dR/dEdΩ ) vanish
q lm rate [42, 43]. This is simply accomplished
or are relatively small. For example, if the dis-
within the framework we have constructed by
tribution f is symmetric about the axis defined allowing v to be a time-dependent quantity.
lab
by v , then f contains only zonal harmonics
lab It may then be important to also account for
in the coordinate system with z v , in which
lab thegravitationalfocusingofWIMPsbythesun,
||
casethecoefficientsandobservedmomentswith
whichaffectsallthreeofthesedependences[44].
m=0 vanish.
6 b In practice, it is likely that analyses of data
in the post-discovery era will begin by assum-
ing that the velocity distribution takes a sim-
V. DISCUSSION
ple functional form, progressing to more gen-
eralbasis-functionmethods onlyafter the basic
Directional detectors will play a key role features of the velocity distribution have been
in the post-discovery era of WIMP direct- discerned. Itmaythen be sometime before the
detection experiments. They may allow for the power of the integrals-of-motion and Fourier-
mapping of the local WIMP velocity distribu- Besselbases,orotheryetunexploredbases,can
tion, which may yield both insights on galactic be fully utilized in directional analyses. Never-
structure formation and improved constraints theless, with this paper we have prepared the
on the WIMP particle properties. necessary framework for such future studies,
We haveshownthat aconvenientparameter- whichmayultimatelybecrucialinrevealingthe
ization of the velocity distribution is provided natureofthe darkmatteronscalesbothmicro-
by an expansion in the familiar Fourier-Bessel scopic and galactic.
9
Acknowledgments and
1 m 0
This work was initially inspired by Ref. [45], Bm = ≥ (A4)
which explored the use of the Fourier-Bessel (−1 m<0
functionsandtheirhigher-spinanaloguesincos-
are constants we define here to condense the
mology; a prepublication draft of that paper
notation that follows.
was kindly provided by Liang Dai, Donghui
TheS provideacomplete,orthonormalba-
lm
Jeong, and Marc Kamionkowski, as were help-
sis for square-integrable real functions on the
ful comments. The author also thanks Adam sphere. Such a function f(n) can then be de-
Burrows for pointing out Ref. [38], as well as
composed as
Mariangela Lisanti and Annika Peter for useful
b
remarks on various drafts of this paper. f(n)= α S (n),
lm lm (A5)
lm
X
where the reabl-spherical-harmobnic expansion
Appendix A: Real Spherical Harmonics
coefficients α = dΩ S (n)f(n) are re-
lm n lm
lated to the usual spherical-harmonic ex-
Since the velocity distribution function g(v) pansion coefficientsRa = dΩ Y (n)f(n) by
lm bn l∗mb
is real, we choose our Fourier-Bessel eigenfunc-
α =A a +A a .
tions inEqs.(4) and(15)tocontainreal spher- lm m ∗lm ∗m lm R
Consider the Gaunt coefficient given by the
b b
ical harmonics S (as opposed to the usual
lm angular integral of the product of three spher-
spherical harmonics Y ). This results in real
lm ical harmonics, which may be written in terms
expansion coefficients gu and gn .
lm lm of Wigner-3j symbols
We define our real spherical harmonics to be
√2N Pm(cosθ)cosmφ m>0 G(LL′L′′)= dΩuYlm(u)Yl′m′(u)Yl′′m′′(u)
lm l Z
Slm(n)=Nl0Pl(cosθ) m=0 = [l][l′][l′′] lbl′ l′′ b l l′ bl′′ ,
√2NlmPlm(cosθ)sinmφ m<0 √4π (cid:18)0 0 0(cid:19)(cid:18)m m′ m′′(cid:19)
b (A6)
√12[Ylm(n)+Yl∗m(n)] m>0
= Y (n) m=0 where [l] √2l+1 and L (l,m). We may
 l0 ≡ ≡
−√i2[Ylmb(n)−Yl∗mb(n)] m<0 dofefithnreeeanreaanlaslpohgeoruicsacloheaffirmcioennticfso,r the integral
b
=Am[Ylm(n)+BmYl∗m(n)] ,
b b
(A1) H(LL′L′′)= dΩuSlm(u)Sl′m′(u)Sl′′m′′(u)
wherePm areassociabtedLegendrebpolynomials, Z
l =AmAm′Am′b′(1+BmbBm′Bm′′b)
[G(LLL )
(2l+1)(l m)! × ′ ′′
Nlm =s 4π (l+−m)! (A2) +(−1)mBmG(L¯L′L′′)
is the usual normalization factor (such that +(−1)m′Bm′G(LL¯′L′′)
the Slm obey the orthonormality relation +(−1)m′′Bm′′G(LL′L¯′′)],
dΩnSlm(n)Sl′m′(n)=δll′δmm′), and (A7)
R 1/√2 m>0 wherewehaveintroducedL¯ (l, m)andused
b b ≡ −
the selection rules governing the Gaunt coeffi-
A = 1/2 m=0
m 
(A3) cientstosimplifythefinalexpression. Notethat
 i/√2 m<0
− H(LL′L′′) is non-zero only for m+m′+m′′
=BmA∗m even.
10
Appendix B: Contour Integration tegrand,
Ourgoalherewillbetocalculatetheintegral
l+1
α cosx+β sinx
j (x)= lr lr , (B2)
eiaxj (x) l xr
Kn(a)= dx l (B1) r=1
l x2 x2 X
Z − ln
where the expansion coefficients α and β
that appears in Eq. (25) and determines the v lr lr
q
can be found by using the standard re-
dependence of Ψn (v ,u) Kn(v /v ). We
shallfollowthemelmthodqsuse∝dinlReqf.[4e6sc],which wcuitrhrenjc(exr)e=latsioinnxjal(nxd)j=(2xl)x−=1jls−in1(xx)−cojslx−.2(x),
investigated thebuse of tbhe Fourier-Bessel basis 0 x 1 x2 − x
inmodelingpseudopotentialsforcalculationsin Writing the exponential appearing in the
density function theory. numerator of the integrand of Eq. (B1) as
We start by expanding the spherical Bessel eiax =cosax+isinax,we see thatthe integral
function appearing in the numerator of the in- becomes
l+1
α cosaxcosx+β cosaxsinx+i(α sinaxcosx+β sinaxsinx)
Kn(a)= dx lr lr lr lr
l xr(x2 x2 )
r=1Z − ln
X (B3)
l+1
1 (α isβ )cos[(a+s)x]+(iα +sβ )sin[(a+s)x]
lr lr lr lr
= dx − .
2 xr(x2 x2 )
r=1s= 1Z − ln
X X±
Here,wehaveusedtrigonometricadditioniden- Eq.(B3)shouldthenidenticallyyieldKn,since
l
tities to simplify the numerator of the inte- the singular contributions would have canceled
grand. in the sum regardless.
Examining Eq.(B3), we see that the integral Thus,we mustthen calculatenonsingularin-
Kn is itself a sum of terms proportional to the tegrals of the form
l
realandimaginarypartsofintegralsoftheform eiAx r−2 (iAx)p
eiAx I(A,r,xln)=P dx xr−(x2p=x02 )p! .
dxxr(x2 x2 ), (B4) Z P− ln (B5)
Z − ln
with A=a+s. However, such integrals are TheCauchyprincipalvalueofthisimproperin-
clearlysingular,generallyhavingapoleoforder tegral can be found using contour integration
r at the origin. It must then be that the singu- and the residue theorem. We let x be a com-
larities cancel in the sum of terms in Eq. (B3), plex number, and then integrate the integrand
since Kn must be nonsingular (following from ofEq.(B5)overacontourthatincludesthereal
l
the fact that it is proportional to the Radon line – avoiding the simple poles at x=0, x
ln
transform of the bounded function Ψn ). – and closes in either the upper or lower±half-
lm
We can therefore subtract the singular con- plane, depending on the sign of A. Done ap-
tributions from each of the integrals of the propriately, this contour encloses no poles and
form in Eq. (B4) to yield nonsingular integrals; hence the contour integral vanishes; further-
substituting these for the singular integrals in more, since the integral over the portion of the