Axionic dark matter signatures in various halo models

In the present work we study possible time signatures Axion Dark Matter searches employing resonant cavities for various halo models. We study in particular the time dependence of the resonance width (modulation) and possible asymmetries in directional experiments.


Introduction
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]. Its importance to Cosmology has also been recognized [4][5][6] long time ago and a bit later it has been considered as a prominent dark matter candidate [7]. In fact, realizing an idea proposed a long time ago by Sikivie [8], various experiments such as ADMX and ADMX-HF collaborations [9][10][11][12] are now planned to search for them, using SQUID and HFET technologies [13]. In addition, the newly established center for axion and physics research (CAPP) has started an ambitious axion dark matter research program [14], using high volume geometry and high field magnet technology. The allowed parameter space [15], containing information for all the axion like particles, defines a region for invisible axions, which can be dark matter candidates.
In the present work we will assume that the axion is non-relativistic with mass in µeV-meV scale, moving with an average velocity which is ≈ 0.8 × 10 −3 c. It is expected to be observed in resonance cavities with a width that depends on the axion mean square velocity in the local frame. The latter depends on the assumed halo model. We will expand and improve our previous work [16,17] by including various popular halo models. We will study the time variation of the width due to the motion of the Earth around the sun and the motion of the sun around the center of the galaxy, an important signature needed if indeed the predicted axion to photon coupling becomes weaker as some recent models predict [18].

Brief summary of the formalism
The photon axion interaction is dictated by the Lagrangian: where E and B are the electric and magnetic fields, g γ a model dependent constant of order one [9,19,20] and f a the axion decay constant. Axion dark matter detectors [19] employ an external magnetic field, B → B 0 in 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. The power produced, see e.g. [9], is given by: Q L is the loaded quality factor of the cavity. Here we have assumed Q L is smaller than the axion width Q a , see below. More generally, Q L should be substituted by min (Q L , Q a ). This power depends primarily on the axion density, but, as we shall see, the width of the expected resonance depends on the assumed velocity distribution. The axion power spectrum [19,21], which is of great interest to experiments, is written as a Breit-Wigner shape [19]: Our findings, namely the time variation of the width due to the motion of the Earth around the sun holds even when there is an overlap between the two resonances [19]. Since in the axion DM search case the cavity detectors have reached such a very high energy resolution [22,23], one should try to accurately evaluate the width of the expected power spectra in various theoretical models.

Evaluation of the width in the local frame
We will derive the expression for the width assuming in the galactic frame a Maxwell Boltzmann axion velocity distribution: We will use the relation In the galactic frame the number of axions in the with frequency between ω and ω + dω in a solid angle d is Introducing the variable x = 2 ω−m a m a υ 2 0 we find: The maximum of the distribution occurs at x = 1/2. The width at half maximum is δx = x 2 − x 1 with x i the roots of the equation We thus find δx = 1.8. Thus the width at half maximum in frequency space is: where the last quantity is the average of the square of the axion velocity in the galactic frame. Our next task is to transform the velocity distribution from the galactic to the local frame. The needed equation, see e.g. [24], is: with γ ≈ π/6, υ s a unit vector in the Sun's direction of motion, x a unit vector radially out of the galaxy in our position and ŷ =υ s ×x. The last term in Eq. (3.9) corresponds to the motion of the Earth around the Sun with δ being 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. υ 0 ≈ 220 km/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. (3.9) α is the phase of the Earth (α = 0 around the beginning of June). 2 We will consider the effect of the Earth's motion in the standard experiments, usually known as modulation effect, which has been recognized as important in axion searches [25]. The azimuthal angle (φ) dependence, averages out to zero. Thus the distribution takes the form: The maximum of the distribution g(x, cos α) occurs at the root r 1 (cos α) of the equation: which is obtained graphically. Then δx(cos α) = x 2 (cos α) − x 1 (cos α) where x 1 (cos α) and x 2 (cos α) are the roots of the equation: The obtained results are exhibited in Fig. 3.1. The existing resonators in the axion dark matter search have a quality factor which is more than one order of magnitude larger than the natural axion width. Nonetheless the axion width can be resolved as a narrow line, narrower than the resonator width. This can be accomplished by recording the data with adequate frequency resolution as is currently done by the ADMX and ADMX-HF experiments [9][10][11][12]. Even though the cavity width is larger than the axion width the induced electromagnetic field due to axion → photon conversion in the presence of the B-field preserves the axion width. The expected width modulation due to the Earth's motion around the sun is roughly ±5%, which requires a frequency resolution of better than 1% or better than about 10 Hz for axions at the GHz range. This requirement is well within present technology.

More complicated velocity distributions
In the context of dark matter other velocity distributions have also been considered, e.g. completely phase-mixed DM, dubbed "debris flow" (Kuhlen et al. [26]), and caustic rings (Sikivie [27][28][29], Vergados [30]). We are now going to investigate what is the effect of these distributions on the axion width.

The case of debris flows
In the case of debris flows one assumes a dark matter distribution which is of the form (see Fig. 4.2): The introduction of debris flows has two effects: • It shifts the maximum of the distribution, without changing the location of the maximum r 1 (see Fig. 4.3).
• The width at half maximum is affected: We thus find: • In the galactic frame: The modulation of the width is exhibited in Fig. 4.4.

The case of caustic rings
Our study of the phase space structure of the Milky Way halo is motivated in large part by the ongoing searches for dark matter on Earth using axion, see e.g. [13,11], and WIMP detectors, see e.g. [31][32][33][34][35]. The signal in such detectors depends on the velocity distribution of dark matter in the solar neighborhood. A search for non-virialized axionic dark matter has been previously considered [36]. The caustic ring halo model predicts, see e.g. [29], that most of the local dark matter is in discrete flows and provides the velocity vectors and densities of the first forty flows at the Earth's location in the Galaxy, which are essential when interpreting a signal in a dark matter detector on Earth. In principle the halo model can be treated as phase-mixed DM leading to a linear combination of a M-B distribution and one appropriate to caustic rings, in a fashion similar to that with debris flow discussed above. We will, however, concentrate here in caustic rings.
The relevant information for our purposes can be extracted from table V of Duffy and Sikivie [29], which has improved and updated earlier versions, and for the reader's convenience is summarized in Table 4.1. One then writes the velocity distribution as where υ x , υ y , υ z are the components of the velocity components in our notation. η υ n± φ , η υ n± ρ , η υ n± z are suitable normalization factors extracted from the corresponding densities of the last columns of Table 4.1. With these ingredients we compute the modulated widths, see Fig. 4.5. We observe that the difference between the maximum and the minimum is 0.08. Note, however, that the maximum is shifted from α = 0 (June third) α = 0.079π , i.e. to approximately two weeks later. This is expected due to the asymmetries in the caustic velocity distribution.

Discussion
In the present work we studied possible effects on the width of the axion to photon resonance cavities involved in Axion Dark Matter Searches, by considering a number of popular halo models. We also examined the annual variation of the width due to the motion of the Earth around the sun. The relative width, i.e. the width divided by its time average, can attain significant differences 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 searches of WIMPs. It does not depend on the geometry of the cavity or other details of the apparatus. It Table 4.1 The velocities caustic ring velocities in our vicinity of our galaxy. The components are given in Sikivie's notation. In our notation for the galactic axes we use φ →ẑ,ρ →x,ẑ →ŷ. only depends on the assumed velocity distribution. Furthermore the predicted time variation of the width (modulation) is observable with the present technology.
In conclusion in this work we have elaborated on signatures that might aid the analysis of axion dark matter searches. Eventually, if such an observation is made, one may be able to exploit the results obtained here to gain information about the velocity distribution associated with the various halo models.