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arXiv:1511.08516v1 [astro-ph.GA] 26 Nov 2015

Modulation and diurnal variation in axionic dark

matter searches

Y. Semertzidis1andJ.D.V ergados1,2

1Center for Axion and Precision Physics Research, IBS, Daejeon 305-701, Republic of

Korea

2Centre for the Subatomic Structure of Matter (CSSM), University of Adelaide, Adelaide

SA 5005, Australia#1

Abstract

In the present work we study possible time dependent eﬀects in Axion Dark

Matter searches employing resonant cavities. We ﬁnd that the width of the

resonance, which depends on the axion mean square velocity in the local frame,

will show an annual variation due to the motion of the Earth around the sun

(modulation). Furthermore, if the experiments become directional, employing

suitable resonant cavities, one expects large asymmetries in the observed widths

relative to the sun’s direction of motion. Due to the rotation of the Earth around

its axis, these asymmetries will manifest themselves as a diurnal variation in the

observed width.

1. Introduction

In the standard model there is a source of CP violation from the phase in

the Kobayashi-Maskawa mixing matrix. This, however, is not large enough to

explain the baryon asymmetry observed in nature. Another source is the phase

in the interaction between gluons (θ-parameter), expected to be of order unity.

The non observation of elementary electron dipole moment limits its value to

be θ≤10−9this has been known as the strong CP problem. A solution to

this problem has been the P-Q (Peccei-Quinn) mechanism. In extensions of the

S-M, e.g. two Higgs doublets, the Lagrangian has a global P-Q chiral symmetry

UP Q(1), which is spontaneously broken, generating a Goldstone boson, the axion

(a). In fact the axion has been proposed a long time ago as a solution to the

strong CP problem [1] resulting to a pseudo Goldstone Boson [2, 3, 4, 5, 6].

QCD eﬀects violate the P-Q symmetry and generate a potential (maa)2/2 for the

#1Theoretical Physics,University of Ioannina, Ioannina, Gr 451 10, Greece.

E-mail:Vergados@uoi.gr

Preprint submitted to Elsevier November 30, 2015

Figure 1.1: The axion decay to photons, axion photon coupling, (a) and the

axion to photon conversion in the presence of a magnetic ﬁeld, the Primakoﬀ

eﬀect, (b).

axion ﬁeld a=θfawith axion mass ma = (Λ2

QCD )/fawith minimum at θ= 0.

Axions can be viable if the SSB (spontaneous symmetry breaking) scale is large

fa≥100 GeV. Thus the axion becomes a pseudo-Goldstone boson An initial

displacement ai=θifaof the axion ﬁeld causes an oscillation with frequency

ω=ma and energy density ρD= (θifama)2/2 The production mechanism

varies depending on when SSB takes place, in particular whether it takes before

or after inﬂation.

The axion ﬁeld is homogeneous over a large de Broglie wavelength, oscillating

in a coherent way, which makes it ideal cold dark matter candidate in the mass

range 10−6eV≤ma≤10−3eV. In fact it has been recognized long time ago as

a prominent dark matter candidate by Sikivie [7], and others, see, e.g, [8]. The

axions are extremely light. So it is impossible to detect dark matter axions via

scattering them oﬀ targets. They are detected by their conversion to photons in

the presence of a magnetic ﬁeld (Primakoﬀ eﬀect), see Fig. 1.1. The produced

photons are detected in a resonance cavity as suggested by Sikivie [7].

In fact various experiments2such as ADMX and ADMX-HF collaborations

[11, 12], [13],[14] are now planned to search for them. In addition, the newly es-

tablished center for axion and physics research (CAPP) has started an ambitious

axion dark matter research program [15], using SQUID and HFET technologies

2Heavier axions with larger mass in the 1eV region produced thermally ( such as via the

aπππ mechanism), e.g. in the sun, are also interesting and are searched by CERN Axion

Solar Telescope (CAST) [9]. Other axion like particles (ALPs), with broken symmetries not

connected to QCD, and dark photons form dark matter candidates called WISPs (Weakly

Interacting Slim Particles) , see, e.g.,[10], are also being searched.

2

[16]. The allowed parameter space has been presented in a nice slide by Raﬀelt

[17] in the recent Multidark-IBS workshop and, focusing on the axion as dark

matter candidate, by Stern [11] (see Fig. 1.2, derived from Fig. 3 of ref [11]).

ma−→ µeV

Figure 1.2: The parameter space relevant for axion as dark matter candidate.

In the present work we will take the view that the axion is non relativistic with

mass in µeV-meV scale moving with an average velocity which is ≈0.8×10−3c.

The width of the observed resonance depends on the axion mean square veloc-

ity in the local frame. Thus one expects it to exhibit a time variation due to

the motion of the Earth. Furthermore in directional experiments involving long

cavities, one expects asymmetries with regard to the sun’s direction of motion

as it goes around the center of the galaxy. Due to the rotation of the Earth

around its axis these asymmetries in the width of the resonance will manifest

themselves in their diurnal variation. These two special signatures, expected to

be sizable, may aid the analysis of axion dark searches in discriminating against

possible backgrounds.

2. Brief summary of the formalism

The photon axion interaction is dictated by the Lagrangian:

Laγγ =gaγ γ aE·B, gaγγ =αgγ

πfa

,(2.1)

3

E/N →

Figure 1.3: The ratio g2

aγγ

m2

a

, in units ofGeV −2/eV2, as a function of E

N, where

Eis the axion electromagnetic anomaly and Nis the color anomaly number.

where Eand Bare the electric and magnetic ﬁelds, fathe axion decay constant

and gγa model dependent constant of order one [11],[18],[19],[20] given by:

gγ=1

2E

N−2

3

4 + z

1 + z(2.2)

where zis the ratio of the up and down quark masses, Nis the color axion

anomaly and Nis the axion EM anomaly. The second z-dependent term [20]

is 1.95. In grand uniﬁable models, like the DFZX axion, E/N is 8/3. In the

case of the KSFZ axion E/N = 0, while in the ﬂavor Peccei-Quin (ﬂavored PQ)

symmetry model to be discussed below, E/N = 112/51. We thus see that the

g2

γrange is given by:

g2

γ=

DFSZ: 0.13

KSVZ: 0.94

Flavored PQ: 0.061

(2.3)

Axion dark matter detectors [18] employ an external magnetic ﬁeld, B→B0in

the previous equation, in which case one of the photons is replaced by a virtual

photon, while the other maintains the energy of the axion, which is its mass

plus a small fraction of kinetic energy.

4

10-8

10-7

10-6

10-5

10-4

0.001

10-27

10-26

10-25

10-24

10-23

10-22

ma−→ eV

Figure 1.4: The ratio g2

aγγ

main units of GeV−2/eV as a function of ma. The curves

correspond to the current experimental limit (solid line) and three theoretical

predictions extracted from Fig. 3 of ref [21]. The solid, thick solid and dashed

curve correspond to E/N=0, 8/3 and 112/51 resp ectively. The ﬁrst almost co-

incides with that extracted from experiment, while the last one is proposed as

theoretically favored. It is seen that for ma= 10−6eV the power produced can

drop by almost two orders of magnitude below the current experimental limit,

if the theoretically favored curve (dashed line).

The power produced, see e.g. [11], is given by:

Pmnp =g2

aγγ

ρa

ma

B2

0V CmnpQL(2.4)

QLis the loaded quality factor of the cavity. Here we have assumed QLis

smaller than the axion width Qa, see below. More generally, QLshould be

substituted by min (QL,Qa). This power depends on the axion particle density

na=ρa/mawith density ρaassumed to be the same with that used for WIMPs

(Weakly Interacting Massive Particles), inferred from the rotational curves, i.e.

the axion particle density is much larger than that expected for WIMPs. In any

case the power produced is pretty much independent of the velocity and the

velocity distribution.

In addition to the axion density in our position, which can somehow be

determined from the rotation curves, it is a function of the theoretical parameter

g2

aγγ /ma. The study of this parameter has been the subject of many theoretical

studies (see e.g. the recent works [18],[19] and references there in). Recently,

5

however, an unconventional but economical extension of the Standard Model of

particle physics has been proposed [21], which attempts to deal with the fermion

mass hierarchy problem and the strong CP problem, in a way that no domain

wall problem arises, in a supersymmetric framework involving the A4×U(1)X

symmetry. The global U(1)Xsymmetry, which can tie the above together is

referred to as ﬂavored Peccei-Quinn (Flvored PQ) symmetry. In this model

the ﬂavon ﬁelds, which are responsible for the spontaneous symmetry breaking

of quarks and leptons, charged under U(1)X, connect the various expectation

values. Thus the Peccei-Quinn symmetry is estimated to be located around 1012

GeV through its connection to the fermion masses obtained in the context of

A4. They are thus able to obtain the ratio of g2

aγγ /m2

a, as a function of E/N

(see Eq. (2.3) and Fig. 1.3, based on Fig. 3 of Flavored PQ symmetry [21] ).

From this ﬁgure we can extract a relationship between g2

aγγ /maand ma. This

is shown in Fig. 1.4. From this ﬁgure we see that the power produced may be

almost two orders of magnitude below the current experimental limit.

Admittedly this A4×U(1)Xsymmetry is introduced ad hoc and especially

the introduction U(1)Xis not particularly otherwise motivated. Furthermore

the fermion masses and paricularly the neutrino mass are not fully understood.

Anyway the experiments may have to live with such a pessimistic prospect and

we may have to explore other signatures of the process. The power spectrum

comes to mind. .

The axion power spectrum, which is of great interest to experiments, is

written as a Breit-Wigner shape [18], [22]:

|A(ω)|2=ρD

m2

a

Γ

(ω−ωa)2+ (Γ/2)2,Γ = ωa

Qa

(2.5)

with ωa=ma1 + (1/6) ≺υ2≻≈magiving the location of the maximum of

the spectrum and Qa= 1/(≺υ2≻/3),Γ = ma(≺υ2≻/3).

Since in the axion DM search case the cavity detectors have reached such a very

high energy resolution [23, 24], one should try to accurately evaluate the width

of the expected power spectra in various theoretical models.

The width explicitly depends on the average axion velocity squared, ≺υ2≻.

With respect to the galactic frame ≺υ2≻takes the usual value of (3/2)υ2

0. In

the laboratory frame, taking the sun’s motion into account, we ﬁnd ≺υ2≻=

(5/2)υ2

0, i.e. the width in the laboratory is aﬀected by the sun’s motion. If we

take into account the motion of the Earth around the sun ≺υ2≻depends on

the phase of the Earth and, correspondingly, the width becomes time dependent

(modulation) as described below (see section 3.2).

The situation becomes more dramatic as soon the experiment is directional. In

this case the width depends strongly on the direction of observation relative

to the sun’s direction of motion. Directional experiments can, in principle, be

performed by changing the orientation of a long cavity [11],[25],[26], provided

that the axion wavelength is not larger than the length of the cylinder, λa≤h.

6

In the ADMX [11] experiment h= 100cm, while from their Fig. 3 one can see

that the relevant for dark matter wavelengths λaare between 1 and 65 cm.

3. Mo diﬁcation of the width due to the motion of the Earth and the

sun.

From the above discussion it appears that velocity distribution of axions

may play a role in the experiments.

3.1. The velocity distribution

If the axion is going to be considered as dark matter candidate, its density

should ﬁt the rotational curves. Thus for temperatures Tsuch that ma/T ≈

4×106the velocity distribution can be taken to be analogous to that assumed

for WIMPs. In the present case we will consider only a M-B distribution in the

galactic frame with a characteristic velocity which equals the velocity of the sun

around the center of the galaxy, i.e. υ0≈220km/s. Other possibilities such as,

e.g., completely phase-mixed DM, dubbed “debris ﬂow” (Kuhlen et al. [30]), and

caustic rings (Sikivie [31],[32][33], Vergados [36] etc are currently under study

and hey will appear elsewhere[37]. So we will employ here the distribution:

f(~υ) = 1

(√πυ0)3e

−υ2

υ2

0(3.6)

In order to compute the average of the velocity squared entering the power

spectrum we need to ﬁnd the local velocity distribution by taking into account

the velocity of the Earth around the sun and the velocity of the sun around the

center of the galaxy. The ﬁrst motion leads to a time dependence of the observed

signal in standard experiments , while the latter motion leads to asymmetries

in directional experiments.

3.2. The annual modulation in non directional experiments

The modiﬁcation of the velocity distribution in the local frame due to annual

motion of the Earth is expected to aﬀect the detection of axions in a time

dependent way, which, following the terminology of the standard WIMPs, will

be called the modulation eﬀect [27] (the corresponding eﬀect due to the rotation

of the Earth around its own axis is too small to be observed). Periodic signatures

for the detection of cosmic axions were ﬁrst considered by Turner [28].

So our next task is to transform the velocity distribution from the galactic

to the local frame. The needed equation, see e.g. [29], is:

y→y+ ˆυs+δ(sin αˆx−cos αcos γˆy+ cos αsin γˆυs), y =υ

υ0

(3.7)

with γ≈π/6, ˆυsa unit vector in the Sun’s direction of motion, ˆxa unit vector

radially out of the galaxy in our position and ˆy= ˆυs×ˆx. The last term in the

7

ﬁrst expression of Eq. (3.7) corresponds to the motion of the Earth around the

Sun with δbeing the ratio of the modulus of the Earth’s velocity around the Sun

divided by the Sun’s velocity around the center of the Galaxy, i.e. υ0≈220km/s

and δ≈0.135. The above formula assumes that the motion of both the Sun

around the Galaxy and of the Earth around the Sun are uniformly circular.

The exact orbits are, of course, more complicated but such deviations are not

expected to signiﬁcantly modify our results. In Eq. (3.7) αis the phase of the

Earth (α= 0 around the beginning of June)3.

The velocity distribution in the local frame is aﬀected by the motion of the

Earth as exhibited in Fig. 3.5 at four characteristic periods. Appropriate in the

y2f(y)−→

1

2

3

4

0.1

0.2

0.3

0.4

0.5

0.6

0.7

y=υ/υ0−→

Figure 3.5: The axion velocity distribution in the local frame. It is changing

with time due to the motion of the Earth and the sun. We exhibit here the

distribution in June (solid line), in December (thick solid line) and in September

or March (dotted line). The last curve coincides with that in which the motion

of the Earth is ignored.

analysis of the experiments is the relative modulated width, i.e. the ratio of the

time dependent width divided by the time averaged with, is shown in Fig. 3.6.

The results shown here are for spherically symmetric M-B distribution as well

as an axially symmetric one with asymmetry parameter β= 0.5 with

β= 1 −hυ2

ti

2hυ2

ri(3.8)

with υrthe radial, i.e. radially out of the galaxy, and υtthe tangential com-

ponent of the velocity. Essentially similar results are obtained by more exotic

models, like a combination of M-B and Debris ﬂows considered by Spegel and

3One could, of course, make the time dependence of the rates due to the motion of the

Earth more explicit by writing α≈(6/5)π(2(t/T )−1), where t/T is the fraction of the year.

8

collaborators [30]. We see that the eﬀect is small, around 15% diﬀerence be-

tween maximum and minimum in the presence of the asymmetry, but still larger

than that expected in ordinary dark matter searches. If we do detect the axion

frequency, then we can determine its width with high accuracy and detect its

modulation as a function of time.

Γ(α)/≺Γ≻−→

1

2

3

4

5

6

0.94

0.96

0.98

1.00

1.02

1.04

1.06

α−→

Figure 3.6: The ratio of the modulated width divided by the time averaged

width as a function of the phase of the Earth. The solid line corresponds to

the standaed M-B distribution and the dotted line to an axially symmetric M-B

distribution with asymmetry parameter β= 0.5 (see text).

3.3. Asymmetry of the rates in directional experiments

Consideration of the velocity distribution will give an important signature,

if directional experiments become feasible. This can be seen as follows:

•The width will depend speciﬁed by two angles Θ and Φ.

The angle Θ is the polar angle between the sun’s velocity and the direction

of observation. The angle Φ is measured in a plane perpendicular to the

sun’s velocity, starting from the line coming radially out of the galaxy and

passing through the sun’s location.

•The axion velocity, in units of the solar velocity, is given as

y=yˆxp1−ξ2cos φ+ ˆyp1−ξ2sin φ+ ˆzξ(3.9)

9

•Ignore the motion of the Earth around the sun, i.e. δ= 0 Then the ve-

locity distribution in the local frame is obtained by the substitution:

υ2→υ2

0y2+ 1 + 2yξcos Θ + p1−ξ2sin Θ (cos Φ cos φ+ sin Φ sin φ)

(3.10)

One then can integrate over ξand φ. The results become essentially

independent of Φ, so long as the motion of the Earth around the sun is

ignored4.Thus we obtain ≺υ2≻from the axion velocity distribution for

various polar angles Θ.

We write the width observed in a directional experiment as:

Γ = r(Θ)Γst (3.11)

where Γst is the width in the standard experiments. Ignoring the motion of

the Earth around the sun the factor rdepends only on Θ. Furthermore, if for

simplicity we ignore the upper velocity bound (cut oﬀ) in the M-B distribution,

i.e. the escape velocity υesc = 2.84υ0, we can get the solution in analytic form.

We ﬁnd:

r(Θ) = 2

5

e−1

2ecos2Θ(cos 2Θ + 4) erfc(cos Θ) −2 cos Θ

√π,(sense known),

(3.12)

with erfc(z) = 1 −erf(z),erf(z) = Rz

0dte−t2( error function)

r(Θ) = 2

5

1

2e−sin2Θ(cos 2Θ + 4),(sense of direction not known),(3.13)

The adoption of an upper cut oﬀ has little eﬀect. In Fig. 3.7 we present the exact

results. The above results were obtained with a M-B velocity distribution5.

Our results indicate that the width will exhibit diurnal variation! For a

cylinder of Length Lsuch a variation is expected to be favored [26] in the

regime of maL= 10 −25 ×10−4eV-m. This diurnal variation will be discussed

in the next section.

4. The diurnal variation in directional experiments

The apparatus will be oriented in a direction speciﬁed in the local frame,

e.g. by a point in the sky speciﬁed, in the equatorial system, by right ascension

4The annual modulation of the expected results due to the motion of the Earth around the

sun will show up in the directional experiments as well, but it is going to be less important

and it will not be discussed here.

5Evaluation of the relevant average velocity squared in some other models [34],[35], which

lead to caustic ring distributions, can also be worked out for axions as above in a fashion

analogous to that of WIMPs [36], but this is not the subject of the present paper

10

r(Θ,Φ) = Γdir/Γst →

0.5

1.0

1.5

2.0

2.5

3.0

0.5

1.0

1.5

Θ−→ radians

Figure 3.7: The ratio of the width expected in a directional experiment divided

by that expected in a standard experiment. The solid line is expected, if the

sense of direction is known, while the dotted will show up, if the sense of direction

is not known.

˜αand inclination ˜

δ6. This will lead to a diurnal variation7of the event rate

[38]. This situation has already been discussed in the case of standard WIMPs

[39, 40]. We will brieﬂy discuss the transformation into the relevant astronomical

coordinates here.

A vector oriented by (˜α, ˜

δ) in the laboratory is given in the galactic frame

by a unit vector with components:

y

x

z

=

−0.868 cos ˜αcos ˜

δ−0.198 sin ˜αcos ˜

δ+ 0.456 sin ˜

δ

0.055 cos ˜αcos ˜

δ+ 0.873 sin ˜αcos ˜

δ+ 0.4831 sin ˜

δ

0.494 cos ˜αcos ˜

δ−0.445 sin ˜αcos ˜

δ+ 0.747 sin ˜

δ

.(4.14)

where ˜αis the right ascension and ˜

δthe inclination.

Due to the Earth’s rotation the unit vector (x, y, z), with a suitable choice of

the initial time, ˜α−˜α0= 2π(t/T ), is changing as a function of time

6We have chosen to adopt the notation ˜αand ˜

δinstead of the standard notation αand

δemployed by the astronomers to avoid possible confusion stemming from the fact that αis

usually used to designate the phase of the Earth and δfor the ratio of the rotational velocity

of the Earth around the Sun by the velocity of the sun around the center of the galaxy

7This should not be confused with the diurnal variation expected even in non directional

experiments due to the rotational velocity of the Earth, which is expected to be too small.

11

x= cos γcos ˜

δcos 2πt

T−sin γcos δcos θPsin 2πt

T+ sin ˜

δsin θP,

(4.15)

y= cos (θP) sin ˜

δ−cos ˜

δsin 2πt

Tsin θP,(4.16)

z= cos 2πt

Tcos ˜

δsin γ+ cos γcos ˜

δcos θPsin 2πt

T+ sin ˜

δsin θP,

(4.17)

where Tis the period of the Earth’s rotation, γ≈330was given above and

θP= 62.60is the angle the Earth’s north pole forms with the axis of the galaxy.

Thus the angles Θ, which is of interest to us in directional experiments, is given

by

Θ = cos−1z, (4.18)

An analogous, albeit a bit more complicated, expression dependent on x, y, z

can be derived for the angle Φ.

The angle Θ scanned by the direction of observation is shown, for various

inclinations ˜

δ, in Fig. 4.8. We see that for negative inclinations, the angle Θ can

take values near π, i.e. opposite to the direction of the sun’s velocity, where the

rate attains its maximum (see Fig. 4.8).

The equipment scans diﬀerent parts of the galactic sky, i.e. observes diﬀerent

angles Θ. So the rate will change with time depending on whether the sense of

of observation. We assume that the sense of direction can be distinguished in

the experiment. The total ﬂux is exhibited in Fig. 4.9.

5. Discussion

In the present work we discussed the time variation of the width of of the

axion to photon resonance cavities involved in Axion Dark Matter Searches. We

ﬁnd two important signatures:

•Annual variation due to the motion of the Earth around the sun. We ﬁnd

that in the relative width, i.e. the width divided by its time average, can

attain diﬀerences of about 15% between the maximum expected in June

and the minimum expected six months later. This variation is larger than

the modulation expected in ordinary dark matter of WIMPs. It does not

depend on the geometry of the cavity or other details of the apparatus. It

does not depend strongly on the assumed velocity distribution.

•A characteristic diurnal variation in of the width in directional experiments

with most favorable scenario in the range of meL= 1.0−2.5×10−3eV m.

This arises from asymmetries of the local axion velocity with respect to the

sun’s direction of motion manifested in a time dependent way due to the

12

Θ−→ radians

0.2

0.4

0.6

0.8

1.0

0.5

1.0

1.5

2.0

2.5

3.0

˜α−→

Figure 4.8: Due to the diurnal motion of the Earth diﬀerent angles Θ in galactic

coordinates are sampled as the earth rotates. The angle Θ scanned by the

direction of observation is shown for various inclinations ˜

δ. We see that, for

negative inclinations, the angle Θ can take values near π, i.e. opposite to the

direction of the sun’s velocity, where the rate attains its maximum. For an

explanation of the curves see Fig. 4.9

rotation of the Earth around its own axis. Admittedly such experiments

are much harder, but the expected signature persists, even if one cannot

tell the direction of motion of the axion velocity entering in the expression

of the width. Anyway once such a device is operating, data can be taken as

usual. Only one has to bin them according the time they were obtained.

If a potentially useful signal is found, a complete analysis can be done

according the directionality to ﬁrmly establish that the signal is due to

the axion.

In conclusion in this work we have elaborated on two signatures that might aid

the analysis of axion dark matter searches.

Acknowledgments: IBS-Korea partially supported this pro ject under sys-

tem code IBS-R017-D1-2014-a00.

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15

Γ/Γst (sense known)

0.2

0.4

0.6

0.8

1.0

2

4

6

8

t

T−→

Γ/Γst (sense unknown)

0.2

0.4

0.6

0.8

1.0

0.5

1.0

1.5

2.0

t

T−→

Figure 4.9: The time dependence (in units of the Earth’s rotation period) of the

ratio of the directional width divided by the non directional width for various in-

clinations ˜

δ, when the sense can be determined (top) or both senses are included

(bottom). In The curves indicated by intermediate thickness solid, the short

dash, thick solid line, long dashed, dashed, ﬁne solid line, and the long-short

dashed correspond to inclination ˜

δ=−π/2,−3π/10,−π/10,0, π /10,3π/10 and

π/2 respectively. We see that, for negative inclinations, the angle Θ can take

values near π, i.e. opposite to the direction of the sun’s velocity, where the

rate attains its maximum if the sense of direction is known. There is no time

variation, of course, when ˜

δ=±π/2.

16