Various CR observatories have revealed anisotropies in the arrival directions of TeV-PeV Galactic CRs on large and small angular scales . The explanation of the strength and phase of the observed dipole anisotropy is challenging, but is qualitatively consistent with the diffusive prediction . Medium and small scale anisotropies could be a combined effect of nearby CR sources and the local interstellar magnetic field structure introducing an energy-dependent magnetic mirror leakage , preferred CR transport directions or magnetic lenses (). The possible influence of the heliosphere via magnetic reconnection in the heliotail , non-isotropic particle transport in the heliosheath or the heliospheric electric field structure have also been considered. More exotic models invoke strangelet production in molecular clouds or in neutron stars .

We will focus in this paper on the power spectrum of small-scale anisotropies in the arrival directions of CRs. This has been studied by IceCube with about 1.5×1011 collected CR events at median energy of 20 TeV and more recently with HAWC including 4.6×1010 events at a median energy of 2 TeV. The results are usually expressed in terms of the relative intensity map δI defined by the local phase space distribution (PSD) f as δI≡(4πf-n)/n, where n is the angular averaged CR density. The data points shown in Fig.  correspond to the multipole power spectra defined as Cℓ=14π∫dp^1∫dp^2Pℓp^1p^2δIp^1δIp^2. An average, both power spectra show a strong contribution of low-ℓ multipoles and a subsequent fall-off of high-ℓ multipole power. Note that the analysis of the power spectra in both experiments are affected by the limited field of view (FoV) and the finite size of the event samples. The limited FoV introduces cross-talk between the power of neighboring multipoles and introduces large uncertainties in the reconstruction of the low-ℓ multipole moments. On the other hand, the finite size of the event sample introduces noise at the level of N≃fsky4π/NCR, where fsky is the effective fraction of the sky that contains NCR collected events. For IceCube we can estimate fsky≃0.3 and NIC≃2.5×10-11 and for HAWC with fsky≃0.7 we have NHAWC≃1.8×10-10. These different noise levels are indicated as horizontal dotted lines in Fig. .

It has been argued by that the appearance of small-scale anisotropies is a natural consequence of CR propagation in the local turbulent magnetic field with a global dipole anisotropy set by diffusion. It was subsequently shown that the behavior of the power spectra inferred from experimental data can be explained by a hierarchical transition between multipole moments in local turbulence . The result of this analytical model is also shown as a dashed line in Fig. . In this paper we will summarize the results of and the recent update of .

We first consider a case of a idealized situation where the initial PSD f̃ is homogeneous but non-isotropic . Using Liouville's theorem, we can simply relate the PSD at a later time to the initial homogenous configuration as f(T,r,p)=f(0,r(-T),p(-T))=f̃(p(-T)). Numerically, this corresponds to the backtracking of CRs with arrival orientation p^ along their trajectory to the initial homogeneous configuration at time -T. The multipole spectrum of the PSD f(T) can then be simply related to the initial configuration f̃ via the identity 14π∑ℓ≥02ℓ+1Cℓ(T)=14π∫dp^1f̃p1(-T)2. This expression is for a particular magnetic field ensemble. However, for the ensemble-average we can make use of the fact that the combination of a regular and average random rotation of an isotropic distribution is eventually isotropic. More precisely, in the case of pure turbulence the average distribution is isotropic at all times, whereas for the combination of turbulent and regular magnetic fields the average distribution at intermediate times can in principle be non-isotropic but is expected to reach an isotropic state eventually. Therefore, we can relate the power spectra at different times as 14π∑ℓ≥0(2ℓ+1)Cℓ(T)=14π∑ℓ≥0(2ℓ+1)C̃ℓ.

Some additional assumptions have to be made to infer the power of individual multipoles. argued that individual multipole powers decay with a rate proportional to ℓ(ℓ+1). However, from the expression () we know that this process also redistributes power between multipole moments. If one makes the assumption that the transition between powers is hierarchical and the decay of Cℓ generates only power in Cℓ+1 one can derive an asymptotic form of the relative power given by the simple expression lim⁡T→∞〈Cℓ〉(T)〈C1〉(T)≃18(2ℓ+1)(ℓ+2)(ℓ+1). The same relative power spectrum should also be followed by the relative intensity map δI. We show the ratio () in Fig. () in comparison with the power spectrum observed with IceCube and HAWC. The scaling of ℓ>5 multipoles is well described by the asymptotic form (Eq. ).

Note, however, that the individual ensemble-averaged contributions 〈C〉(T) of the model of all decay exponentially in time. For the discussion of the expected small-scale anisotropies of a quasi-stationary diffusion state it is necessary to introduce an effective source term in form of the large-scale CR gradient. This was recently discussed by and we summarize this approach in the next sections.

The previous thought experiment was designed to extract the influence of the local turbulence on the redistribution of multipole power. In order to apply these results to the realistic case of cosmic ray diffusion we have to account for source terms of the initial dipole anisotropy in the form of a cosmic ray gradient ∇n. Instead of a homogeneous initial condition we can approximate this situation by assuming a quasi-stationary solution as a result of Fick's law , 4π〈f〉≃n+r-3p^K∇n, where K is the diffusion tensor and n the local (r=0) CR density. The standard dipole prediction of the relative intensity is in this case 14πC1=K∇nn. We can now use again Liouville's theorem to evaluate the sum over multipoles. For large look-back times T the spatial term in the ansatz () dominates over the anisotropy induced by Fick's law. The ensemble-averaged power spectrum of the relative intensity δI can then be evaluated as 14π〈Cℓ〉≃∫dp^14π∫dp^24πPℓp^1p^2×lim⁡T→∞〈r1i(-T)r2j(-T)〉∂in∂jnn2. The sum over dipoles as in Eq. () is not constant in this case, but increases linearly for large backtracking times T. More precisely, the sum over ℓ≥1 multipoles approaches asymptotically

We use the identity δ(x-y)=∑ℓ2ℓ+12Pℓ(x)Pℓ(y) and Pℓ(1)=1.

We follow the approach of and define a turbulent magnetic field as δB(r)=∑n=1NA(kn)ancos⁡αn+bnsin⁡αncos⁡knr+βn, where the N wave vectors kn have random orientations in three dimensions and fixed amplitudes kn=kmin⁡e(n-1)Δln⁡k with Δln⁡k=ln⁡(kmax⁡/kmin⁡)/N, i.e. equal logarithmic spacing. The parameters αn and βn are random phases in [0,2π) and the vectors an and bn are defined as unit vectors in the direction kn×ez and kn×an, respectively. In this way the field obeys ∇B=0. The amplitudes A(kn) are defined as A2(kn)=2σ2B02G(kn)/∑n=1NG(kn) with G(kn)=4πkn2knΔln⁡k1+knλcγ. We assume that for kn≫1/λc the turbulence follows a Kolmogorov-type turbulence with γ=11/3. The scaling factor σ2 is normalized such that σ2=UδB/UB0 and λc is a wave vector corresponding to the coherence scale of the turbulent magnetic field.

In our simulation we assume σ2=1 and choose λmin⁡=0.01λc and λmax⁡=100λc with k=2π/λ. We fix the rigidity of the CRs to rL=0.1λC. We sample over NB=10 different turbulent magnetic field configurations. For each field configuration we uniformly sample Npix= 12 288 CR orientation p^i(0) following the HEALPix parametrization (nside=32) , that we track back through the static magnetic field B(r)=B0e^z+δB(r) to find the initial orientation p^i(-T) and position ri(-T). Figure  shows the power spectrum at different times determined via a multipole expansion of the maps using the HEALPix utilities.

At large times the high-ℓ components in the multipole expansion are dominated by shot noise, due to the finite sample size and the uncorrelated random walk at large back-tracking times. This noise level can be estimated as N≃4πNpix2TKij∂in∂jnn2. Therefore, for large backtracking times T the map is eventually dominated by noise. However, for sufficiently low multipole moments and large sample size Npix the stable asymptotic behavior of low-ℓ multipoles can be recovered.

We have discussed in this paper the generation of small-scale anisotropy from CR propagation in local magnetic turbulence. We have shown that with minimal assumptions about the decay of and transition between multipole powers one can derive an asymptotic solution that is consistent with observations of IceCube and HAWC. We have discussed our recent approach to bring these results into a more realistic setup of CR diffusion with a large-scale CR gradient. One can show analytically that the generation of multipole power is related to properties of relative CR diffusion. This was supported by numerical simulations.