Axion astronomy with microwave cavity experiments

4.1 MB·

by  Ciaran A. J. O'Hare

Axion astronomy with microwave cavity experiments
Ciaran A. J. O’Hare∗ and Anne M. Green
School of Physics and Astronomy, University of Nottingham,
University Park, Nottingham, NG7 2RD, United Kingdom
(Dated: March 29, 2017)
Terrestrial searches for the conversion of dark matter axions or axion-like particles into photons
inside magnetic fields are sensitive to the phase space structure of the local Milky Way halo. We
simulate signals in a hypothetical future experiment based on the Axion Dark Matter eXperiment
(ADMX)thatcouldbeperformedoncetheaxionhasbeendetectedandafrequencyrangecontain-
ing the axion mass has been identified. We develop a statistical analysis to extract astrophysical
parameters, such as the halo velocity dispersion and laboratory velocity, from such data and find
7 thatwithonlyafewdaysintegrationtimealevelofprecisioncanbereachedmatchingthatofastro-
1 nomicalobservations. Forlongerexperimentslastinguptoayearindurationwefindthatexploiting
0 the modulation of the power spectrum in time allows accurate measurements of the Solar peculiar
2 velocity with an accuracy that would improve upon astronomical observations. We also simulate
r signalsbasedonresultsfromN-bodysimulationsandfindthatfinersubstructureintheformoftidal
a streamswouldshowupprominentlyinfuturedata, evenifonlyasubdominantcontributiontothe
M
local dark matter distribution. In these cases it would be possible to reconstruct all the properties
ofadarkmatterstreamusingthetimeandfrequencydependenceofthesignal. Finallyweconsider

8 the detection prospects for a network of streams from tidally disrupted axion miniclusters. These
2 features appear much more prominently in the resolved spectrum than suggested by calculations
based on a scan over a range of resonant frequencies, making the detection of axion minicluster
] streams more viable than previously thought. These results confirm that haloscope experiments in
O
a post-discovery era are able to perform “axion astronomy”.
C
. PACSnumbers: 14.80.Va;95.35.+d
h
p
-
o I. INTRODUCTION tests for axions predominantly rely on their coupling to
r electromagnetismresultinginthePrimakoffconversionof
t
s Axions are light pseudoscalar particles that appear in axions into photons inside strong magnetic fields. Cav-
a ity resonators can exploit this effect if the resonant fre-
[ thesolutionofPecceiandQuinn[1,2]toexplaintheun-
natural absence of CP-violation in quantum chromody- quency is chosen to match the axion mass [18]. As the
2 axion mass is unknown experiments must be designed
namics (QCD). More modern motivation from the land-
v to scan over a range of resonant frequencies correspond-
scape of axion-like particles (ALPs) appearing in string
8 ing to a range of axion masses. Experiments include
theory[3–5],inspiresthegeneralisationofaxionstolight
1
the helioscope CAST [19] (and the planned IAXO [20])
1 pseudoscalarswithatheoreticaloriginoutsideoftheorig-
searching for axions produced inside the Sun, and halo-
3 inal Peccei-Quinn solution. Such ALPs can cover an ex-
scopes such as ADMX searching for Galactic dark mat-
0 tremely wide range of masses and couplings to the stan-
1. dard model [6]. Axions and ALPs are an attractive cold teraxions[21]. Theseexperimentsoperateoveranarrow
range of frequencies and hence make constraints on the
0 dark matter candidate and can be produced in the early
axion mass in small bands, where the smallest accessible
7 Universe through a variety of non-thermal mechanisms
photon coupling is controlled by the signal-to-noise level
1 such as vacuum misalignment or the decay of topolog-
: of the experiment. Planned dark matter axion experi-
v ical defects [7, 8] in ways that are consistent with the
ments such as QUAX [22–24], CULTASK [25] and the
i known cosmological abundance and phenomenology of
X layered dielectric haloscope MADMAX [26, 27] are be-
dark matter [9, 10]. For a recent review of axion cos-
r mology see e.g., Ref. [11]. In the following we use the ingdesignedtoproberangesofaxionmassesinaccessible
a duetothetechnicalrestrictionsoftheADMXdesign. As
term ‘axion’ to refer to both the QCD axion and generic
well as axion ‘observation’ experiments there exist solely
axion-like particles.
lab-based experiments such as the Any Light Particle
Certain axion mass and coupling ranges can be ruled
Search [28] using the technique known as light-shining-
out with various astrophysical observations such as the
through-a-wall [29]. In haloscopes beyond ADMX, it
cooling of white dwarfs [12, 13], neutron star interac-
may be possible to search for lighter axions with broad-
tions[14],thelifetimesofhorizontalbranchstarsinglob-
band readout circuits [30] or LC circuits [31] as well as
ular clusters [15] and supernovae neutrinos [16, 17]. Ax-
heavier masses in the meV range with Josephson junc-
ions may also be observable in the lab. Experimental
tions[32,33]. Othercouplingssuchasthosetonuclei[34]
canbeprobedinnuclearmagneticresonanceexperiments
suchasCASPEr[35,36]andthecouplingtoelectronscan
beconstrainedusingconventionaldarkmatterdirectde-
∗Electronicaddress: [email protected]
2
tection searches for weakly interacting massive particles describeourmockexperimentandexplaintheprocedure
(WIMPs) [37]. For a recent review of axion experiments used to extract astrophysical information from the sim-
see for example Ref. [38]. ulated data. The first results applying these methods to
The goal of a haloscope experiment is to tune the fre- thereconstructionbasicsetsofinputparametersarepre-
quency of a cavity mode to the axion mass resulting in sentedinSec.VandthenextendedtoN-bodydatafrom
the resonant enhancement of the axion-photon conver- the Via Lactea II (VL2) [50] simulation in Sec. VI. Fi-
sion. ADMX achieves this with the use of movable tun- nallyweextendthissimulationtotidalstreamsfromdis-
ing rods placed inside the cavity itself to modulate the ruptedaxionminiclustersinSec.VII,beforesummarising
resonantfrequencyoverarangeofseveralhundredMHz. in Sec. VIII.
Althoughusuallyunimportantwhenscanningoverarel-
atively large range of resonant frequencies, the velocity
distributionofaxionsinthehalowouldcauseasmallfre- II. AXIONS AND ALPS
quency spread in the resonance [39]. Furthermore there
arealso0.5%and1%modulationsintimeduetotherel- First we outline some of the essential steps in calcu-
ative velocity of the Earth and Sun with respect to the lating the resonantly enhanced axion-photon conversion
halo dark matter ‘wind’ [40–42]. There is also a poten- powerinsideamicrowavecavity. Fulldetailsofthesecal-
tially exploitable O(1) modulation dependent on cavity culations can be found in Refs. [39, 51, 52]. Importantly
orientation with respect to the incoming axion direction we wish to make the connection to realistic halo velocity
foraxionmasses(andhaloscopevolumes)experiencinga distributions so we depart from an often used approxi-
loss of coherence over the cavity dimensions [43]. mation that the axion power spectrum can be described
Here we consider a scenario in which the axion mass with a Breit-Wigner function.
has been determined down to a level of precision dic- The axion to two photon coupling is given by the for-
tated by the scanning approach of ADMX and a ded- mula,
icated high spectral resolution experiment is then per-
g α
formed at a single resonant frequency. In such a situ- g = γ , (1)
ation, the shape of the spectrum of axion-photon con- aγγ πfa
version will be accessible. This power spectrum is re-
which includes the fine structure constant α and the ax-
lated to the velocity distribution of the local dark mat-
ion decay constant f . The dimensionless coupling g
ter halo, hence the precise functional form and parame- a γ
is,
ters which arise from astrophysics are important. Past
axion searches with ADMX have incorporated some of
(cid:18) (cid:19)
1 E 24+z
these astrophysical uncertainties, for example by search- g = − . (2)
ing for discrete flows of axions [44–46] or applying con- γ 2 N 31+z
straintstodifferenthalomodels[42,47]. Inthesituation
In which E/N is the ratio of the colour axion anomaly
we consider here however it will be possible to perform
to the electromagnetic axion anomaly and z is the ratio
“axion astronomy” in the sense that a measurement can
oftheupanddownquarkmasses. Thevalueofthiscon-
be made directly of the axion power spectrum to learn
stant is model dependent: E/N = −0.97 for the KSVZ
about the structure of dark matter in the Galaxy. For
model [53, 54] and 0.36 for the DFSZ model [55, 56] for
this reason we develop an analysis that shares similarity
example. In the interest of model independence and to
with well-established methods for extracting astrophys-
generalisetoALPsweexpresstheinteractionintermsof
ical information in the case of WIMP direct detection
the coupling g .
experiments e.g., Ref. [48]. Since axion haloscopes effec- aγγ
tively observe the axions directly - as opposed to WIMP TheeffectiveLagrangianforaxionscoupledtoelectro-
direct detection experiments which observe the WIMP magnetism is,
flux convolved with a stochastic scattering process - the
prospects for sensitive measurements of the dark matter L= 21∂µa∂µa−V(a)+ 14gaγγaFµνF˜µν
halo are much greater. Here, we show how powerful fu- −1F Fµν +jµA +aρ , (3)
4 µν µ q
ture ADMX-like experiments might be for doing axion
astronomy. We discuss this idea in the context of simple where F is the electromagnetic field strength tensor,
µν
analytic halo models, distributions from N-body simula- andF˜µν = 1(cid:15)µνρσF itsdual. TheaxionpotentialV(a)
2 ρσ
tions,andminiclusterstreams-aphenomenonuniqueto isprovidedbyQCDinstantoneffectsandcanbeapprox-
axion dark matter [49]. imated with a simple mass term 1m2a2. The axionic
2 a
To begin in Sec. II we review some of the basic the- charge density and the electromagnetic current density
oryforaxionsandALPs, aswellasthelaboratoryframe arewrittenasρ andjµ. WritingF F˜µν =−4E·Bwe
q µν
speed distribution relevant for calculating signals inside thenseetheaxion-photoninteractionintermsofelectric
a haloscope experiment. In Sec. III we outline the steps and magnetic field strengths,
in calculating the expected power inside a magnetic cav-
ity resonating at a given frequency. Then in Sec. IV we L =−g aE·B. (4)
aγγ aγγ
3
This interaction modifies Maxwell’s equations to include The convention in dark matter detection literature is
an additional axion current, to use a velocity distribution to describe the kinematics
of the local halo. In this context we must blur the dis-
∇·E =ρ +g ∇a·B, (5)
q aγγ tinction between the interpretation of axionic dark mat-
∇·B =0, (6) ter as a classically oscillating field and as a collection of
∂B particles. The velocity distribution is related to the ax-
∇×E =− , (7)
∂t ion power spectrum in the following way. First we write
∂E ∂a down the distribution of axion velocities f (v) in the
lab
∇×B =µ j+ −g B −g ∇a×E. (8)
0 ∂t aγγ 0∂t aγγ laboratory frame (i.e., flab(v) = fgal(v + vlab)) by
temporarily introducing a number density,
However these equations simplify for the setup we con-
sider here. Firstly we assume the axion field has no spa- dn=n f (v)d3v, (17)
tial dependence on laboratory scales (∇a = 0). We can 0 lab
do this because the size of ADMX is around the 1 meter
wherednisthenumberdensityof“particles”withspeeds
scale and is well below the de Broglie wavelength of the
between v and v +dv. The constant n is found from
0
axion field for the mass ranges we consider (>100 m).
integratingdnoverallvelocitiesandisusedtodefinethe
This allows us to assume that there is no spatial depen-
local axion number density n ≡ρ /m . This allows the
0 a a
denceintheaxionfieldoverthedimensionsofthecavity
connection to a classical field oscillating at m , which
a
and hence no additional modulations due to the chang- should have (cid:104)a2(t)(cid:105)=n/m , to be made [39].
a
ing orientation of the cavity with respect to the axion An expression for the axion power spectrum |A(ω)|2
wind. We also assume that there is no axionic charge
can be obtained by satisfying Parseval’s relation and
and no electromagnetic current inside the cavity: ρ =0
q changing variables from ω to v,
and jµ = 0. This results in the following simple set of
equations, d(cid:104)a2(t)(cid:105) dv
|A(ω)|2 =2π , (18)
∇·E =0, (9) dv dω
∇·B =0, (10) we can then substitute for d(cid:104)a2(t)(cid:105)/dv using,
∂B
∇×E =− , (11) dn (cid:90)
∂t = n v2f (v)dΩ (19)
∂E ∂a dv 0 lab
∇×B = −g B . (12)
∂t aγγ 0∂t = n0flab(v), (20)
Under the above assumptions the equation of motion
where this expression clarifies the distinction between
for the axion field is,
a 3-dimensional velocity distribution f(v) and its 1-
∂2a dimensional speed distribution f(v). Hence, the formula
(cid:3)a(cid:39) =−V(cid:48)(a)−g E·B. (13)
∂t2 aγγ for the axion power spectrum on Earth can be written
as,
Dark matter axions in the Milky Way undergo es-
sentially no interactions, so in a quadratic potential
ρ dv
V(a) (cid:39) 1m2a2, the field oscillates coherently at the |A(ω)|2 =2π a f (v) . (21)
2 a m2 lab dω
axion mass a(t) = a0eimat ≡ a0eiωt. However due to a
thermalisation in the Milky Way the coherence of the The simplest assumption for a dark matter halo is the
oscillationsisspoiledslightlybyadispersioninaxionve- Standard Halo Model (SHM) which is spherically sym-
locities: ω = ma(1 + 21v2 + O(v4)). One can account metric with a 1/r2 density profile and yields a Maxwell-
for this by moving to a Fourier description of the field, Boltzmann velocity distribution,
written as A(ω),
1 1
a(t) = √T (cid:90) +∞ dωA(ω)e−iωt, (14) fgal(v)= π√πv3e−|v|2/v02. (22)
2π 0
−∞
1 (cid:90) T/2 To simplify the following expressions we use the peak
A(ω) = √ dta(t)eiωt, (15) velocity v for the shape of the distribution, however
T 0
−T/2 it can also be written in terms of a velocity dispersion,
where T is some large reference time used to take the v2 = 2σ2. The speed distribution follows from an inte-
0 v
averages. Thequantity|A(ω)|2 isreferredtoastheaxion gral over all directions,
power spectrum. The rms of the axion field squared is
creolnantieocnte,d to the axion power spectrum by the Parseval fgal(v)=4ππ√1πvv23e−v2/v02. (23)
0
(cid:104)a2(t)(cid:105)= 1 (cid:90) T/2 dta2(t)=(cid:90) +∞ dω|A(ω)|2. (16) However the power spectrum should reflect the fact that
T 2π weobservef notf soweneedtocomputethespeed
−T/2 −∞ lab gal
4
distributioninthelaboratoryframe. Todothiswemake cavity of radius R and length L, with radial, azimuthal
the transformation v → v−v (t) which yields for the and vertical co-ordinates labelled (ˆr,φˆ,zˆ) respectively.
lab
velocity and speed distributions, Themagneticfieldisgeneratedbyasolenoidwithcurrent
density in the φˆ-direction. The electric and magnetic
1 1
flab(v,t)= π√πv3e−(v−vlab(t))2/v02, (24) fields we write as,
0
E = 0 (28)
and, 0
B = n IΘ(R−r)zˆ, (29)
0 L
(cid:16) (cid:17)
f (v,t)= 2ve−(v2+vlab√(t)2)/v02sinh 2vlavb02(t)v . (25) whereΘ(r)istheHeavisidestepfunction,I isthecurrent
lab π πv0vlab(t) and nL is the number of wire turns in the solenoid per
unitlength. Forconvenienceweusethemagnitudeofthe
Since we are now in the moving laboratory frame a time
magnetic field B =n I in the following expressions.
dependence appears in the speed distribution. Finally 0 L
In the cylindrical cavity design the important cavity
after changing variables to ω = m (1+v2/2) we arrive
a modeorientationsaretheTM modeswhichhavetrans-
attheaxionpowerspectrum. Theaxionpowerspectrum 0l0
verse magnetic fields (in the φˆ-direction and hence have
mustbe0forω <m whichisenforcedbyrequiringthat
a associated electric fields in the zˆ-direction). It is use-
f(v) be real.
ful to write these induced fields in terms of their Fourier
For use in later examples we also define the velocity
components,
distribution for streams fstr(v) which are spatially and
lab
kinematically localised substructure components of the (cid:18)√ (cid:90) +∞ dω (cid:19)
darkmatterhalo. Theirvelocitydistributioncanalsobe E = Ez(r,t)zˆ= T E (r,ω)e−iωt zˆ,
a a 2π a
described with a Maxwellian, −∞
(cid:18)√ (cid:90) +∞ dω (cid:19)
1 B = Bφ(r,t)φˆ = T B (r,ω)e−iωt φˆ.
flsatbr(v,t)= (2πσ2 )3/2e−(v−(vlab−vstr(t))2/2σs2tr, (26) a a −∞ 2π a
str
In this case, Amp`ere’s law from Maxwell’s equations re-
where the stream is parameterised by its Galactic
frame velocity v ∼ O(100 km s−1) and dispersion duces to,
str
σ ∼ O(1 km s−1). We will assume that the stream
str ∂ ∂a
comprises some fraction ρ /ρ of the local dark matter ∇×(B +B )= (E +E )−g (B +B ) . (30)
str a 0 a ∂t 0 a aγγ 0 a ∂t
density.
The description of the lab velocity is well known in Solvingthisequationinsideandoutsidethecavityand
the context of WIMP direct detection but is not usu- matchingboundaryconditionsleadsonetoasolutionfor
ally considered for axion detection. Since we are reliant the Fourier components of the axion generated magnetic
on its precise description we will briefly elaborate on its and electric fields. The solutions are resonances at par-
contents. The lab velocity is written, ticular frequencies corresponding to the zeros of a Bessel
function(althoughwewillonlybeinterestedinthelowest
v (t)=v +v +v (t)+v (t). (27)
lab 0 pec rev rot resonance which we label ω ). Following the derivation
0
of Ref. [51], the axion power is calculated by evaluating
containing respectively, the bulk rotation velocity of the
the following integral over the volume of the cavity V,
stellardisk(thelocalstandardofrest,LSR),thepeculiar
velocity of the Sun with respect to the LSR, the orbital
(cid:42) (cid:43)
speedoftheEartharoundtheSunandtherotationspeed P = ω0U = ω0 (cid:90) d3r E2a+B2a , (31)
of the Earth about its axis. The latter two velocities are Q Q 2
V
responsible for the annual and diurnal modulations re-
spectively and are known theoretically with effectively where U is the energy stored in the electric and mag-
perfect precision (see the Appendix of Ref. [57] for a re- netic fields inside the cavity. This expression introduces
view of these calculations). In the SHM the velocity of the quality factor Q which is a number that quantifies
the LSR is usually written as v0 = (0,v0,0) in Galactic how well the cavity stores energy and depends on the
co-ordinates, with v0 = 220 km s−1. An often quoted material properties of the cavity wall. Evaluating the
value for the peculiar velocity of the Sun from Schoen- above formula with the solution for the Fourier compo-
rich et al. [58] is vpec = (11.1,12.24,7.25) km s−1 with nentsoftheaxionelectricandmagneticfields(whichare
roughly 1 km s−1 sized systematic errors. expressed in terms of |A(ω)|2) one arrives at,
4 (cid:90) +∞ dω
P =g2 B2Vω Q3 T(ω)|A(ω)|2. (32)
III. RESONANCE POWER aγγ 0 0 χ2 2π
0l −∞
Wemodelamicrowavecavityexperimentwithastatic where χ is the l-th zero of the 0th Bessel function of
0l
uniformmagneticfieldB maintainedinsideacylindrical the first kind. We have also defined T(ω), which is a
0
5
Lorentzianthatdescribesthelossinpoweroffresonance,
1
T(ω)= . (33)
(cid:16) (cid:17)2
1+4Q2 ω −1
ω0
0.03
Usually the haloscope power is written in terms of a
cavityformfactor. Forthetransversemagneticfieldcon- W]0.025
sidered here (TM0l0) this is written C0l0 = 4/χ20l1. We −22 0.02
aCre pr=inc0i.p6a9l.lyAinDteMreXstecdaninttuhneeTthMe01T0Mmodemwohdiechfrhoams [10N0.015
010 010 P
−
rtroiucgfihelyld5o0f0thMeHTzMto 900moMdHezca[2n1b].eIwnrigtetnenera[5l9t]h,e elec- wer 0.01
nlm Po0.005
(cid:16)x (cid:17) (cid:16)nπz(cid:17) Jan 2018
E (r,φ,z,t)=E(t)J mlr e±imφcos . (34) 0
z m R L
Jan 2017
In which, E(t) is the time dependent component of the 0 νmax 2
4 Jan 2016
field, Jm is a Bessel function, xml is the lth root of ν− m2πa [kHz]
J (x) = 0, R is the cavity radius and L is the cavity
m
length. Modes with n (cid:54)= 0 and m (cid:54)= 0 have very small
form factors.
Our simulation is based upon the calculation of
Eq. (32) so for our purposes it would be sufficient to
stop here. But in the interest of comparison with pre-
0.1
vious calculations we will calculate the power on reso-
W]
nuasencae.BrTeoit-dWoigthniesrwapepsriomxpimlyatsieotnωfo0r=thωeaax(cid:39)ionmpaoawnedr −22 0.08
0
spectrum with an analogous Q-factor Q ∼ω/∆ω ∼106 [10.06
a N
P
(this permits an analytic evaluation of the integral in −
0.04
Eq. (32)). We also introduce the axion density by writ- wer
ing (cid:104)a2(t)(cid:105)=ρa/m2a. Resulting ultimately in, Po0.02 Jan 2018
ρ
P =(cid:126)2c5ε g2 VB2C a min(Q,Q ), (35) 0
a 0 aγγ nlmma a 1 νmax 1.5 Jan 2017
where we have restored the factors of (cid:126), c and ε0 for ν− ma [kHz] 2 Jan 2016
completeness. If the quality factor of the resonant cav- 2π
ity is very high (i.e., the cavity is very good at storing
energy and the dissipation is very slow) then the axion FIG. 1: Example simulated power spectra as a function of
conversionpowerislimitedbythespreadinaxionkinetic time. Eachlineistheaveragepowerspectrumobservedovera
10dayperiod. Thetoppanelshowsthespectraforasmooth
energy. Thefactormin(Q,Q )arisesfromtheintegralof
a
Maxwellian halo and bottom for a pure tidal stream with
two Breit-Wigner functions and indicates how the total
parameter values displayed in Table I. The purple line in the
power received on resonance is set by the wider of the
frequency-timeplaneshowstheevolutionofthefrequencyat
two power spectra. whichthepowerismaximised: 2πν =m (1+v2 /2)and
Inputting typical values for the experimental parame- 2πν = m (1+|v −v |2/2)mfaoxr theaMaxwelallbian halo
max a lab str
ters we arrive at a total power of the order 10−22 W as and stream respectively.
is usually quoted,
P = 6.3×10−22W
a
IV. MOCK EXPERIMENT
(cid:18) g (cid:19)2(cid:18) V (cid:19)(cid:18) B (cid:19)2
× aγγ
10−15GeV−1 220l 8T
Our simulation is an approximation of the current
(cid:18) (cid:19)(cid:18) (cid:19)
× Cnlm ρa ADMX setup. We list a set of benchmark experimen-
0.69 0.3GeV cm−3 tal parameters in Table I. The magnetic field strength,
(cid:18) (cid:19)(cid:18) (cid:19) quality factor and noise temperature are roughly in line
3µeV Q
× . (36) with what is currently achievable. For calculating the
m 70,000
a time dependence we also include the latitude and longi-
tude of the experiment.
Inthissectionwewillconsiderahypotheticalscenario
in which the axion has been discovered after a success-
1 Other mode orientations, the transverse electric (TE ) and
nlm ful low resolution scan over a wider mass range. Once
transverseelectromagnetic(TEM )modesbothhavenoaxial
nlm
electricfieldmeaningtheyhavenegligibleformfactors. the resonance has been found then an experiment can
6
Axion: m 3.4671 µeV Our mock experiment consists of a long total running
a
gaγγ 10−15 GeV−1 time τtot which is divided into separate time integrated
Experiment: B 8 T bins of length τ. Inside a given time bin we calculate a
0
Q 70,000 power spectrum which would correspond to the average
V 220 l of N Fourier transformed timestream samples of dura-
∆τ 0.2 s tion ∆τ. The Fourier transform of a given sample is a
τ 10 days power spectrum with frequency resolution ∆ν = 1/∆τ.
τ 2 years
tot The noise we simulate as Johnson white noise which has
T 4 K
S rms power P = k T ∆ν inside a given frequency bin
Latitude 47.6553◦ N B S
with an exponential distribution [44]. The noise power
Longitude −122.3035◦
spectrum of the average of N = τ/∆τ individual ex-
Maxwellian halo: ρ 0.3 GeV cm−3
a
v (0,220,0) km s−1 ponential power spectra corresponding to the N Fourier
0
v (11.1,12.24,7.25) km s−1 transformed timestream samples then approaches Gaus-
pec
Stream: v 400×(0,0.233,−0.970) km s−1 sianwhitenoiseinaccordancewiththecentrallimittheo-
str
σ 20 km s−1 rem. Henceoursimulatednoiseinsidethelargertimebin
str
ρstr 0.05 ρ0 τ is Gaussian white noise with√mean value PN and stan-
dard deviation P /N = P / τ∆ν. The full dataset
N N
then consists ofa total number of N =τ /τ time inte-
τ tot
TABLE I: Benchmark axion, halo, experimental and stream grated power spectra each of which consists of the axion
parameters. power spectrum averaged over the time τ added to the
Gaussian white noise.
The major motivation for a long running time, aside
be performed at a single frequency. The running time of from simply reducing noise, is to utilise the annual mod-
the experiment needs to be long enough to ensure that ulationduetov (t)whichprovidesaGalacticperspec-
rev
the signal-to-noise ratio is high but for our purposes also tive to the signal. It has previously been shown that the
needstobecomprisedoflongtimestreamsamplestoob- annual modulation signal allows astrophysical parame-
tain high frequency resolution in the resulting spectrum. ters to be measured more accurately using WIMP direct
For now we pick a benchmark set of particle parame- detection data, as it breaks degeneracies [64]. Below we
ters that lie in the QCD axion band: νa = 842.0 MHz show that this is also the case for axion searches. We
(= 3.4671 µeV) and gaγγ = 10−15 GeV−1. This choice testoursimulationbyfirstgeneratingamockdatasetand
evades existing constraints but is easily within the reach thenattemptingtoreconstructtheinputparticleandas-
of ADMX given a long enough running time at the cor- trophysicalparameterswithamaximumlikelihoodanal-
rect frequency. We use only a single particle benchmark ysis. Two examples of such data are displayed in Fig. 1
in this study as we are placing the focus on the under- corresponding to two halo models, a smooth isotropic
lying astrophysical parameters. This is justified however Maxwelliandistributionandapurestream(withparam-
because many of the conclusions are either independent eter values listed in Table I). The annual modulation of
of the choice in mass and coupling (provided the run- the signal is indicated by the purple line labelled ν .
max
ning time and resonant frequency are suitably adjusted) Webaseourlikelihoodonaχ2statisticwhichmeasures
orhavedependenciesthataresimpletoexplainfromthe the offset between the observed value of power Pij , and
obs
scaling of the axion power. We discuss how one might the expected power (signal + rms noise) Pij + P in
a N
extendourconclusionstootheraxionmassandcoupling each bin, where i and j label frequency and time bins
ranges in the Summary Sec. VIII. respectively,
Thesensitivityofahaloscopeexperimentislimitedby
the strength of the axion conversion power compared to (cid:16) (cid:17)2
thenoiselevel. Therearetwomainsourcesofbackground (cid:88)Nν (cid:88)Nt Poibjs−Paij −PN
χ2 = , (38)
noiseinresonantcavityexperiments: thesignalamplifier σ2
and the cavity walls. The cavity walls produce thermal i=1j=1 N
blackbody photons or Johnson noise whereas the ampli-
where the sums run over N = (ν − ν )/∆ν fre-
fiers produce electrical noise which depends on the pre- ν max min
quency bins and N = τ /τ time bins. The error σ
cise technology, however both can be modelled as white τ tot √ N
is given by the suppressed rms noise power P / τ∆ν.
noise [60–62]. The signal-to-noise ratio for a haloscope N
We then construct a likelihood based on this statistic.
experiment of duration τ, is set by the Dicke radiometer
Mathematically the likelihood as a function of a set of
equation [63]
parameters given data D is,
(cid:114)
S P τ
= a , (37) L(m ,g ,P ,Θ|D)=e−χ2/2L (Θ)L (P ),(39)
N k T ∆ν a aγγ N astro N N
B S a
where ∆ν is the bandwidth of the axion signal and T where we assume m , g and P are free parameters.
a S a aγγ N
is the noise temperature. We also use the generic Θ to label a set of astrophysical
7
1.01 166
τ = 10 days
tot
τ = 0.5 yr
1.008 tot
τ = 1 yr
164 tot
1]1.006
−
V
Ge1.004 −1]162 i
s n
1.002 p
−150 1 [km160 utva
[1 σv lue
γ0.998
aγ 158
g
0.996
0.994 156 input value
0 2 4 6 225 230 235
∆m [Hz] |v | [km s−1]
a lab
FIG. 2: Reconstructed axion mass and coupling as well astrophysical parameters, v and σ , for a smooth Maxwellian halo
lab v
model. We show sets of 68% and 95% confidence level contours in the m −g and |v |−σ planes (left and right panels
a aγγ lab v
respectively). Weexpresstheaxionmassas∆m whichhasthetrue(input)valuesubtracted. Theblue,greenandredsetsof
a
contourscorrespondtotheestimateswithexperimentsofdifferentdurations: 10days,halfayearand1yearrespectively. The
maximum likelihood values are indicated by triangles and the input values for the parameters are indicated by dashed lines
and a yellow star.
parameters as we will perform tests with varying num-
bersoffreeparameters. WeuseL toincorporatethe
astro
optionaluncertaintyintheknowledgeofanastrophysical
parameter (it can be set to unity if no prior knowledge
is assumed about a given parameter). The final term
LN(PN) parametrises the likelihood of the noise power −1]223450
s
which can be measured externally (although we set this m230
k225
to unity unless otherwise stated). [220
ramOeutrerlikvealliuheosotdhaatnmalyasxiismciosnestihstesliokfelfiihrsotofidnodfinEgq.th39e,pwae- yvv+0pec22240110505 τττtttooottt===1010.5ydryarys
interpretthissetofparametersasthebestfitpoints. We −1] 20
then construct 68% and 95% confidence regions around s
m
these points which are either 1-dimensional intervals [k 0
when we are only interested in the reconstruction of one zvpec−20
parameter or 2-dimensional contours when we are inter- −40
−40 −20 0 20 210 220 230 −40 −20 0 20 40
ested in the reconstruction of two parameters and their vpxec [kms−1] v0+vpyec [kms−1] vpzec [kms−1]
correlation. We do this by first profiling over all other
parameters other than those of interest and then calcu- FIG. 3: Reconstructed lab velocity components (vx , v +
pec 0
late a likelihood ratio between the maximum likelihood vy , vz ) at 68% and 95% confidence for three datasets
pec pec
foreachpointθintheremaininglikelihoodfunction. The of length 10 days, half a year and 1 year, indicated by blue,
likelihood ratio we define as, λ(θ)=−2(lnL(θ)−L(θˆ)), green and red sets of contours respectively. The maximum
likelihood values are indicated by triangles and the true (in-
where θˆ are the maximum likelihood estimators. Ac-
put) values are indicated by dashed lines with a yellow star.
cording to Wilks’ theorem [65] this is asymptotically χ2
k
distributed for k parameters. We then find intervals or
contoursaroundthebestfitpointswhichencloseregions
V. RECONSTRUCTING PARAMETERS
of parameter values with λ less than a certain critical
value. Thecriticalvalueofλisthatforwhichthecumu-
lative distribution of χ2 is the desired confidence level. In this section we use the simulation and analysis
k
For example for k = 1 the 68% interval encloses values methodology described in Sec. IV to attempt to recon-
of λ<1 and the 95% encloses values of λ<4. structsetsofinputparticleandastrophysicsparameters.
8
10 andσ (right). Weshowthreesetsofcontourswhichcor-
v
respond to experiments of different durations: 10 days,
z] 0 half a year and 1 year. The 10 day long experiment cor-
H
[ responds to a single time integrated bin of the 0.5 and
a
m−10 1 year long experiments. The annual modulation signal
∆
does not play a large role in constraining these param-
eters, hence the effect of increasing the experiment du-
1]1.04 ration is to shrink the confidence intervals by a factor
−
eV1.02 (cid:112)1 year/10 days. The axion mass and coupling can be
G
measured to a high level of precision even with only 10
15 1 days of data taking, however there is some bias in the
−
10 best fit values since the dataset consists of a single real-
[0.98
γ isation of stochastic noise. The shapes of the contours
γ
ga are roughly one sided for masses m > ma due to the
1] 260 fact that the axion power spectrum is only non-zero for
−
s 240 ω >ma. The astrophysical parameters can be measured
m
to a high level of accuracy too. With a 1 year duration
k 220
[ the level of precision would reach around the 1 km s−1
|
ab200 level which roughly matches the accuracy of current as-
v|l180 tronomical observations [58].
Withafullannualmodulationsignalwecanalsoaccess
180
1] the 3-dimensional components of vlab. However since v0
− 170
s and vpec are summed in the Galactic frame we can only
m160 measure directly the x and z components of v . The
k pec
[150 y component (i.e., that which lies along the direction of
v
σ140 the rotation of the Milky Way) can only be measured in
combination with the LSR speed v . In Fig. 3 we show
0
4.0005 the measurement of these parameters for the same three
K]
[ experiment durations of 10 days, 0.5 and 1 year. Since
∆ν 4 the 10 day duration experiment consists only of a single
B time integrated bin we have no annual modulation sig-
k
/N3.9995 nalandonlythereconstructionofthelargestcomponent
P (v +vy )ispossibleasthishasthegreatestinfluenceon
0 pec
the shape of the spectrum. The remaining two compo-
1 day 10 days 1 yr
τ nents have essentially flat likelihoods as the single time
tot
binspectrumisnotsensitivetotheirvalues. Howeverfor
longer durations with modulation in time, the measure-
FIG. 4: Reconstructed parameters for multiple stochastic ment of all three components becomes possible. Even
data realisations. The 1 and 2 sigma error bars are shown
with only half a year of the annual modulation signal we
for five parameters, from top to bottom, m , g , |v |, σ
a aγγ lab v can still make a measurement of the three components
and the noise (which we express as P /k ∆ν). There are
N B ofv . However, asthesignal-to-noiseislowerthemea-
30setsofrepeatedmeasurementsfor3differentexperimental lab
surementisbiasedbyparticularlargefluctuations,which
durationsτ =1day,10daysand1year(fromlefttoright).
tot
in this example leads to the input values lying outside of
the95%contour. Withafullyearofdatahoweveravery
Theaimistoquantifyhowaccuratelyandwithwhatcor- accuratemeasurementcanbemadewith95%confidence
relationsanddegeneraciesafutureADMX-likehaloscope intervals smaller than 5 km s−1 and the true values (in-
experiment would measure the local axionic dark matter dicated by dashed lines and stars) lying within the 95%
distribution. In the following results we show 1- and 2- interval in all cases.
dimensional 68% and 95% confidence intervals/contours Finally in Fig. 4 we show the 1 and 2 sigma error bars
calculated using the profile likelihood, along with best for various parameter measurements as a function of the
fit parameters values which maximise the likelihood. To total experiment duration τ . We use three experiment
tot
explore the likelihood function we use nested sampling durations from 1 day to 1 year and for each we repeat
algorithmsprovidedbytheMultiNestpackage[66–68], the experiment 30 times with different randomly gener-
and set a tolerance of 10−3 and use between 2×103 and ated noise in each to demonstrate the sensitivity to the
104 live points depending on the number of parameters individual data realisation. As shown in Figs. 2 and 3
being reconstructed. the short duration experiments as well as setting much
In Fig. 2 we show the reconstructed axion parameters weaker measurements are also biased by the particular
m and g (left) and the astrophysical parameters v data causing some reconstructions to lie further than 2
a aγγ lab
9
icalparametersthanstandardaxionsearchesandWIMP
direct detection.
WeusedatafromtheViaLacteaII(VL2)[50]simula-
tion and select 200 analogue Earth locations at a Galac-
tic radius of 8 kpc and calculate a velocity distribution
1) #3 from all particles contained within 1 kpc spheres cen-
n
a tred on each of these locations (we also enforce that no
J
= spheres overlap). Although there are more recent hy-
v,t drodynamicsimulationswhichwillbetterreflectaMilky
(b #2,#4 Way-like dark matter distribution, the VL2 data is suffi-
a
fl cient for the illustrative examples we show here and will
not change the general conclusions.
#4
HellotoJasonIsaacs #1 inWFiegd.i5spwlaiyththceerrtaanignesoafmthpelesse2la0b0evlleeldocwithyidchistcroibnutatiinonas
significantsubstructurecomponent. Welabelthesesam-
0 100 200 300 400 500 600 700 800
v [kms−1] ples from #1 - #4. The substructure appearing promi-
nently here are types of tidal stream which are present
FIG.5: Setoflaboratoryframespeeddistributionsofthe200 in real galaxies due to the orbits of satellite galaxies. As
sampleschosenfromtheVL2simulation. Theshadedregions these smaller galaxies orbit close to their host halo both
indicatetherangeoff(v)valuesforagivenv. Thesolidpur- the stellar and dark matter components can be tidally
ple lines indicate the maximum and minimum values of f(v) strippedleavingalongtrailofmaterialaroundthegalaxy
andthedashedlineisthemeandistributionoverallsamples. which may intersect the main galactic disk. In our own
The black line is the SHM Maxwellian with parameters from Milky Way there has been evidence from observations
TableI.Welabelparticularsampleswhichcontainprominent
and simulations that a particular tidal stream from the
streams with arrows and the sample number.
Sagittarius dwarf galaxy could pass very close to the lo-
cation of the Solar System [69]. Tidal streams are of
particularinteresthereastheyareverykinematicallylo-
sigma away from their input values. In the case of the
calised. Inparticular,streamsincomingwithvelocitiesat
axionmassweexpectone-sidedmeasurementsduetothe
an angle with respect to the motion of the Solar System
one-sidednatureofthepowerspectrum. Thisisthecase
would give rise to unique annual modulation signals.
for 10 day and 1 year durations, however for the 1 day
We calculate the axion conversion power spectrum in
durationweseemultipleexperimentsreconstructamass
the same way as before but we substitute the analytic
smaller than the input mass due to large noise fluctua-
f(ω)withadiscretisedversioncalculatedbybinningpar-
tions in bins slightly below the axion mass. Interestingly
ticle velocities with a bin size roughly corresponding to
forthelongerdurationexperimentstheconstraintonthe
the frequency resolution of the experiment. Importantly
axionmassreachesalevelsmallerthanasinglefrequency
foreachtimebinattwerotateallparticlevelocitiesinto
bin (5 Hz), this is because the shape of the power spec-
the laboratory frame with the time dependent Galactic
trum and the annual modulation signal also provide ad-
tolaboratorytransformationdetailedinAppendixA.We
ditionalinformationaboutm . Thesizeoftheerrorbars
a √ mustalsoboostallparticlevelocitiesbyv→v−v (t).
fortheremainingparametersdecreaseroughlyas1/ τ lab
tot In Fig. 6 we show a selection of four axion conversion
andfordurationslongenoughtoexploittheannualmod-
power spectra for a range of sample VL2 velocity distri-
ulation signal we see a significant decrease in the scatter
butions (the same selection as labelled in Fig. 5). The
in the reconstructed values over different realisations of
four examples are selected because they contain signif-
the experiment. This means that a future experiment
icant substructure components in the form of streams.
such as this would be able to make fine measurements of
Theseshowupinthepowerspectraassinusoidallymod-
the axion particle parameters in conjunction with astro-
ulating features in time, some examples such as #2 and
physical parameters and with no significant biases.
#3havingsingledominatingstreamswhereasotherssuch
as#4possessmultiplestreamswithdifferentamplitudes
and phases.
VI. N-BODY DATA We can parameterise the frequency dependence of the
modulating streams with the function,
We can source more realistic examples of dark matter
(cid:18) (cid:18) (cid:19)(cid:19)
distributionsfromN-bodysimulationsofMilkyWay-like ν(t)=ν sin 2π t−t0 +ν . (40)
halos. These might more accurately reflect the inhomo- 1 1 year 0
geneities and anisotropies that will likely be present in a
real dark matter halo. This is of particular interest for In principle the three parameters of this function are
a high resolution axion experiment because, as shown in related to the three Galactic frame components of the
theprevioussection,itisfarmoresensitivetoastrophys- stream velocity, although this will not be a one-to-one
10
Jan 2018 Jan 2018
Jan 2017 Jan 2017
Jan 2016 Jan 2016#1 #2
Jan 2018 Jan 2018
0.06
Jan 2017 0.05P− Jan 2017
0.04NP
00..0023[10W22−
0.01]
Jan 2016 0 Jan 2016#3 #4
−1 0 1 2 3 −1 0 1 2 3 −1 0 1 2 3 −1 0 1 2 3
ν−ma [kHz] ν−ma [kHz]
2π 2π
FIG. 6: Left: Axion conversion power spectra for a selection of four Earth-radius dark matter velocity distributions from the
VL2 simulation. In each of the four examples the power spectrum has the amplitude of the noise power (P ) subtracted and
N
isdisplayedasafunctionoftime(runningover2yearsfrom2016-2018). Thefrequencydependenceisshownasthedifference
between the photon frequency and the axion mass. Right: The same set of power spectra after performing the various cuts
detailedinthetext. Theremainingpointsshowfluctuationsintheaxionpowerspectraafterthetimeindependentcomponents
have been subtracted. The best fit to Eq.(40) is shown as a red line, the power spectrum lying along this best fit line is
then extracted to measure the properties of the stream. For clarity in the left hand power spectra we have divided the noise
amplitudeby4sothatthesubstructureisclearer,howevertherighthanddata(usedtodothereconstruction)retainsthefull
noise amplitude with T =4K.
S
f(vx) f(vy) f(vz) 0.022
ρ
#1 str0.0215
ρ
a
0.021
−500 0 500−500 0 500−500 0 500 1 1.02 1.04 1.06
x 10−3
5.2
f(vx) f(vy) f(vz)
5.15
ρ
#2 ρstr 5.1
a
5.05
5
−500 0 500−500 0 500−500 0 500 2.062.08 2.1 2.122.142.162.18
f(vx) f(vy) f(vz) 0.0114
#3 ρstr 0.0113
ρ
a 0.0112
0.0111
−500 0 500−500 0 500−500 0 500 4.35 4.4 4.45 4.5 4.55
x 10−3
f(vx) f(vy) f(vz) 1.65
1.64
ρ
#4 str1.63
ρa 1.62
1.61
1.6
−500 0 500−500 0 500−500 0 500 1.7 1.75 1.8
vx [km s−1] vy [km s−1] vz [km s−1] σstr [km s−1]
FIG. 7: Measurements of stream velocity (vertical black lines) and intervals at the 68% and 95% confidence level (dotted and
dashedlinesrespectively)foreachofthefoursampleVL2velocitydistributions. The1-dimensionalspeeddistributionsineach
Galactic co-ordinate (v ,v ,v ) correspond to the first three columns. Each row corresponds to the four sample distributions
x y z
chosen. The final column is the reconstruction in the plane of the stream density fraction and dispersion.