How isotropic can the UHECR flux be?
MMNRAS , 1–9 (2015) Preprint 10 September 2018 Compiled using MNRAS L A TEX style file v3.0
How isotropic can the UHECR flux be?
Armando di Matteo (cid:63) and Peter Tinyakov † Service de Physique Th´eorique, Universit´e Libre de Bruxelles (ULB), CP225 Boulevard du Triomphe, B-1050 Bruxelles, Belgium
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
Modern observatories of ultra-high energy cosmic rays (UHECR) have collected over10 events with energies above 10 EeV, whose arrival directions appear to be nearlyisotropically distributed. On the other hand, the distribution of matter in the nearbyUniverse – and, therefore, presumably also that of UHECR sources – is not homoge-neous. This is expected to leave an imprint on the angular distribution of UHECRarrival directions, though deflections by cosmic magnetic fields can confound the pic-ture. In this work, we investigate quantitatively this apparent inconsistency. To thisend we study observables sensitive to UHECR source inhomogeneities but robust touncertainties on magnetic fields and the UHECR mass composition. We show, in arather model-independent way, that if the source distribution tracks the overall matterdistribution, the arrival directions at energies above 30 EeV should exhibit a sizeabledipole and quadrupole anisotropy, detectable by UHECR observatories in the verynear future. Were it not the case, one would have to seriously reconsider the presentunderstanding of cosmic magnetic fields and/or the UHECR composition. Also, weshow that the lack of a strong quadrupole moment above 10 EeV in the current dataalready disfavours a pure proton composition, and that in the very near future mea-surements of the dipole and quadrupole moment above 60 EeV will be able to provideevidence about the UHECR mass composition at those energies. Key words: cosmic rays – astroparticle physics – ISM: magnetic fields
The last generation of ultra-high energy cosmic ray(UHECR) observatories — the Pierre Auger Observatory(hereinafter Auger) (Aab et al. 2015) in the Southern hemi-sphere and the Telescope Array (TA) (Abu-Zayyad et al.2013) in the Northern one — has accumulated over 10 cos-mic ray events with energies larger than 10 eV. Theseevents typically have angular resolution ∼ ◦ and energyresolution and systematic uncertainty (cid:46) E (cid:38)
10 EeV,a small deviation from isotropy has been found — a 6 . E >
39 EeV, the Auger collaboration recently reported evidencefor a correlation of ∼
10% of the flux with the positions ofstarburst galaxies on 13 ◦ angular scales (Giaccari 2017). Atthe highest energies ( E >
57 EeV), there is an indication ofan overdensity in a ∼ ◦ -radius region (the “TA hotspot”,Abbasi et al. 2014) as well as possible spectrum variations (cid:63) E-mail: [email protected] † E-mail: [email protected] with the arrival direction (Nonaka 2017), but at these en-ergies all angular power spectrum coefficients up to l = 20(i.e. down to ∼ ◦ scales) are less than 0 . (cid:46)
100 Mpc, so the distribution of UHECR sourcescan be presumed to be inhomogeneous as well. It thereforeappears puzzling that the observed event distribution doesnot bear an imprint of these inhomogeneities, albeit possiblydistorted by magnetic fields.At a closer look, however, the situation is not sostraightforward. At the highest energies, where the magneticdeflections are smaller, the experimental sensitivity to pos-sible anisotropies is still poor, due to the steeply decreasingenergy spectrum and consequently the limited statistics. Atlower energies where the statistics is larger, the magneticdeflections are larger as well. Worse, there is a large uncer-tainty in the estimate of these deflections related to boththe poorly known charge composition of UHECRs and poor c (cid:13) a r X i v : . [ a s t r o - ph . H E ] F e b A. di Matteo and P. Tinyakov knowledge of magnetic fields. Are these deflections enough toerase the traces of the inhomogeneous source distribution?If no major anisotropy is detected with further accumula-tion of data, at which point one should start to worry thatsomething is fundamentally wrong in our understanding ofUHECRs and/or their propagation conditions? In this paperwe attempt to give a quantitative answer to this question.Unfortunately not much can be done about the poorlyknown chemical composition at present: estimating it is onlypossible through indirect means and needs to rely on ex-trapolations of the properties of hadronic interactions tovery high energies. Several different models of these interac-tions tuned to LHC data are available. Auger results (Bellido2017) indicate that the average mass of cosmic rays above2 EeV increases with energy (roughly as A ∝ E . ), but theestimates at any given energy are strongly dependent on thechoice of hadronic interaction model and, to a lesser extent,systematic uncertainties on the measurements (Figure 2),ranging e.g. from helium to silicon at 43 EeV. TA data havelarger uncertainties, and are compatible with either Augerresults or a pure-proton composition (de Souza 2017). It hasbeen claimed that the differences between hadronic interac-tion models may even understate the actual relevant uncer-tainties (Abbasi & Thomson 2016). In order to be conserva-tive, we will consider several different hypotheses bracketingall the most recent estimates.Large uncertainties also plague the models of magneticfields. These can be divided into the extragalactic (EGMF)and Galactic field (GMF), the latter in turn being composedof regular and random components. We will argue below insubsection 2.3 that the regular part of the GMF is likelyto dominate the deflections. Unfortunately, the GMF as afunction of position in 3D cannot be directly measured, butmust be estimated from observables integrated along the lineof sight, potentially leading to degeneracies; several differ-ent models of GMF can fit these observables (see Haverkorn(2015) and references therein). In addition, an independentknowledge of the 3D electron density is required to recon-struct the magnetic field value. Different models of GMFfitted to the same data can result in rather different predic-tions about cosmic ray deflections, especially at low rigidities(Unger & Farrar 2017). This is the main problem that wewill have to deal with.The approach we take is to find an observable that is theleast affected by the existing uncertainties. It has been ar-gued (Tinyakov & Urban 2015) that the angular power spec-trum C l of the CR flux has little sensitivity to the details ofthe GMF model, as it does not carry information on whereprecisely on the sky the flux has minima or maxima butonly about the overall magnitude and angular scale of thevariations. Rather model-independent predictions for thesecoefficients can thus be made. Another observable that hasbeen studied in the literature is the auto-correlation func-tion N ( θ ). The angular power spectrum at small l is mostlysensitive to large-scale anisotropies ( ∼ π/l rad), whereas theauto-correlation function at small θ is mostly sensitive tosmall-scale anisotropies ( ∼ θ ).Early works performed on this subject include Sigl et al.(2003, 2004), which computed predictions for both C l and N ( θ ), but they were restricted to a pure proton composition,used distributions of sources based on cosmological largescale structure simulations rather than the observed galaxy distribution, and neglected the effects of the GMF. Morerecent studies include Harari et al. (2014, 2015); Mollerachet al. (2016), but those studies concentrated on the tran-sition between the diffusive and the ballistic propagationregime (occurring at E/Z ∼ N ( θ ) andnot for C l . Our approach is very similar, but we adopted afew simplifications, and computed C l instead.We present our results for a variety of energy thresh-olds E min ranging from 60 EeV down to 10 EeV, thoughour approximations are not as reliable towards the low endof this range. On the other hand, unlike for higher energythresholds, measurements of C l from joint full-sky analysesby the Auger and TA collaborations for E min = 10 EeV (Aabet al. 2014; Deligny 2015) are already reasonably precise.The paper is organised as follows. In section 2 we as-semble the ingredients necessary to calculate the expectedUHECR flux distribution over the sky. We discuss the sourcedistribution in subsection 2.1, set up a generic source modelin subsection 2.2, and summarise the existing knowledge onthe UHECR deflections in cosmic magnetic fields in sub-section 2.3. In section 3 we present the results of the fluxcalculation and multipole decomposition. Finally, section 4summarizes our conclusions. The matter distribution in the Universe is inhomogeneousat scales of several tens of Mpc, consisting of clusters ofgalaxies, filaments and sheets, and becomes nearly homoge-neous at scales of a few hundred Mpc and larger. If UHECRsources are ordinary astrophysical objects, their distributioncan be expected to follow these inhomogeneities.If we assume that the ratio of UHECR sources to to-tal galaxies is approximately the same in every galaxy clus-ter, we can approximate the distribution of sources in thenearby Universe (up to a harmless overall normalisation fac-tor) from a complete catalogue of galaxies by simply treatingeach galaxy as an UHECR source and all sources as identi-cal. There are a number of possible subtler effects (such assource evolution and clustering properties of different typesof galaxies) which may cause the actual UHECR source dis-tribution to deviate from being exactly proportional to theoverall galaxy distribution, but we will neglect any such ef-fects in what follows.We obtain the galaxy distribution from the 2MASSGalaxy Redshift Catalog (XSCz) derived from the 2MASSExtended Source Catalog (XSC) (see Skrutskie et al. (2006)for the published version). A complete flux-limited sample ofgalaxies with observed magnitude in the Ks band m < . MNRAS , 1–9 (2015) ow isotropic can the UHECR flux be? which assumes that the spatial distribution and absolutemagnitude distribution of galaxies are statistically indepen-dent. The objects further than 250 Mpc are cut away and re-placed by the uniform flux normalised as to correctly repro-duce the combined contribution of sources beyond 250 Mpc,as described below. The resulting catalogue contains 106 218galaxies, which is sufficient to accurately describe the fluxdistribution at angular scales down to ∼ ◦ (see Koers &Tinyakov (2010); Abu-Zayyad et al. (2012) for further de-tails). It is not our goal here to construct a realistic modelof sources, but rather to understand, in a best model-independent way, what minimum anisotropy of UHECR fluxmust be present. Therefore, we consider three different mod-els of sources (for E ≥
10 EeV) corresponding to extremeassumptions about the UHECR mass composition (compati-ble with the observed lack of a sizeable fraction of very heavynuclei), expecting that the resulting predictions will bracketthose from any realistic scenario. We consider: (i) a pure proton injection with a power-law spectrum N ( ≥ E min ) ∝ E − γ min at all energies, with spectral index γ =2 . γ = 2 . γ = 1 . at 280 EeV.The spectral indices were chosen to match the expectedUHECR spectra at Earth at E ≥
10 EeV to the observa-tions, as shown in Figure 1. In all three cases, we neglect anypossible evolution of sources with cosmological time, becauseat these energies the vast majority of detected particles canbe presumed to originate from sources at redshift z (cid:28) A in-jected with E > A EeV immediately disintegrate into A protons each with energy 1 /A times the initial energy ofthe nucleus, as at these energies the energy loss lengths forspallation by cosmic microwave background photons are afew Mpc or less, as is the beta-decay length of neutrons. Inthe case of a power-law injection of nuclei with no cutoff, Scenarios (ii) and (iii) are qualitatively similar to the “secondlocal minimum”and the“best fit”scenarios from Aab et al. (2017),respectively. The choice of shape of the cutoff function ( f cut such thatd N/ d E ∝ E − γ f cut ( E )) is irrelevant, provided f cut ( E ) ≈ E (cid:46)
200 EeV and f cut ( E ) ≈ E ≥ E min , becausenuclei with energies in between will fully disintegrate but none oftheir secondaries will reach Earth with E ≥ E min .
19 19.2 19.4 19.6 19.8 20 20.2 20.4 E J [ e V k m − y r − s r − ] log ( E /eV)p inj., γ = 2.6, no cuto ff O inj., γ = 2.1, no cuto ff Si inj., γ = 1.5, w. cuto ff TA 2017 ( E TA /0.95)Auger 2017 ( E Auger /0.87)
Figure 1.
Energy spectra predicted at Earth in the three compo-sition scenarios described in the text, from
SimProp v2r4 simula-tions (Aloisio et al. 2017) assuming a uniform source distribution.TA data (Tsunesada 2017) and Auger data (Fenu 2017) are shownwith energy scales shifted as recommended by Dembinski et al.(2017). The oxygen injection scenario includes secondary protons,whereas the silicon injection scenario does not due to the injectioncutoff. ⟨ l n A ⟩ E [EeV] p inj.O inj.Si inj. w. cut Auger 2017:QGSJET II-04EPOS-LHCSIBYLL 2.3 Figure 2.
Average mass composition as a function of energyin our three scenarios, compared to Auger data as interpretedthrough three different hadronic interaction models (Bellido 2017;bars and shaded areas denote statistical and systematic uncertain-ties respectively). this results in a number of secondary protons above a giventhreshold N p ( ≥ E min ) = A − γ N A ( ≥ E min ) . (1)The result is equivalent, in this approximation, to injectingdirectly a mixture of nuclei and protons in the proportionset by eq. (1). In the case A = 16 and γ = 2 .
1, the injectednumber fraction of protons is ≈
57% of the total.Next, we have to take into account energy losses ex-perienced by lower-energy nuclei and protons. These in-clude the adiabatic energy loss due to the expansion ofthe Universe (redshift), interactions with background pho-tons such as electron-positron pair production and pion pro-duction, and again spallation, this time mostly on infraredbackground photons. We do so via an attenuation func-tion a A ( E min , D, γ ) such that the number of nuclei (otherthan secondary nucleons) reaching our Galaxy with E ≥ E min from a source at a distance D that emits nuclei ofmass A is a A ( E min , D, γ ) times as much as if there were no MNRAS000
57% of the total.Next, we have to take into account energy losses ex-perienced by lower-energy nuclei and protons. These in-clude the adiabatic energy loss due to the expansion ofthe Universe (redshift), interactions with background pho-tons such as electron-positron pair production and pion pro-duction, and again spallation, this time mostly on infraredbackground photons. We do so via an attenuation func-tion a A ( E min , D, γ ) such that the number of nuclei (otherthan secondary nucleons) reaching our Galaxy with E ≥ E min from a source at a distance D that emits nuclei ofmass A is a A ( E min , D, γ ) times as much as if there were no MNRAS000 , 1–9 (2015)
A. di Matteo and P. Tinyakov a A ( E m i n , γ ) E min [EeV]protons, γ = 2.6oxygen, γ = 2.1protons, γ = 2.1silicon, γ = 1.5 Figure 3.
Fraction of nuclei (excluding secondary protons) hav-ing originated from within 250 Mpc among all those reachingEarth with E ≥ E min , computed from SimProp v2r4 simulations(Aloisio et al. 2017) assuming a uniform source distribution energy losses. The attenuation of protons can likewise be de-scribed by a function a p ( E min , D, γ ).We need not take intoaccount the secondary nucleons produced in the spallationof parent nuclei with E < A EeV, because those will allreach our Galaxy with energies below 10 EeV, which is thelowest energy we consider in this work. Therefore, in the sil-icon case no secondary nucleons are present. Also, most ofthe nuclei that undergo spallation but still reach our Galaxywith E ≥
10 EeV only lose a few nucleons at most, so weapproximate them as all having the same electric charge asthe primaries. The closeness of the (cid:104) ln A (cid:105) line in Figure 2for the silicon scenario to a constant ln 28 ≈ . A nuclei attenuated by a factor a A ( E min , D, γ ),and secondary protons, initially A − γ times as many as thenuclei but then attenuated by a factor a p ( E min , D, γ ). Inthe proton and silicon case, only the protons or only thenuclei are present, respectively. We use parametrizationsfor a A ( E min , D, γ ) and a p ( E min , D, γ ) fitted to results from SimProp v2r4 (Aloisio et al. 2017) simulations.Finally, we have to estimate the contribution to the totalflux at Earth of sources outside the catalog (
D >
250 Mpc),which we assume to be isotropic. We do so by defining afunction f A ( E min , γ ) as follows, f A ( E min , γ ) = N A ( E Earth ≥ E min , D ≤
250 Mpc) N A ( E Earth ≥ E min , all D )in the hypothesis that sources emitting nuclei with massnumber A are uniformly distributed per unit comov-ing volume; we define an analogous function f p ( E min , γ )for protons. We use parametrizations for f A ( E min , γ )and f p ( E min , γ ) fitted to results from SimProp v2r4 (Aloisioet al. 2017) simulations (Figure 3).The total directional flux just outside the Galaxy isthen the sum of four contributions (nuclei from catalogsources, nuclei from isotropic far sources, protons from cat-alog sources, protons from isotropic far sources), which we compute asΦ cat A ( ˆn , ≥ E min ) = N p ( ≥ E min ) (cid:88) s w s a A ( D s )4 πD s S ( ˆn , ˆn s ) , Φ far A ( ˆn , ≥ E min ) = 1 − f A f A (cid:82) π Φ cat A ( ˆn ) dΩ4 π , Φ catp ( ˆn , ≥ E min ) = N A ( ≥ E min ) (cid:88) s w s a p ( D s )4 πD s S ( ˆn , ˆn s ) , Φ farp ( ˆn , ≥ E min ) = 1 − f p f p (cid:82) π Φ catp ( ˆn ) dΩ4 π , where the index s runs over sources, w s is a weight thattakes into account the observational bias in the flux-limitedcatalog (see Koers & Tinyakov (2009) for details), the de-pendence of f i and a i on γ and E min is omitted for brevity,and S ( ˆn , ˆn s ) ∝ exp( ˆn · ˆn s /σ ) is a smearing function totake into account the finite detector angular resolution anddeflections in random magnetic fields. In the proton case N A ( ≥ E min ) = 0, in the silicon case N p ( ≥ E min ) = 0, andin the oxygen case they are related by eq. (1).We compute such fluxes at several different values of E min , so that by subtracting them we can find the direc-tional flux in each energy bin, and then compute magneticdeflections for each energy bin as described below. The present constraints on the magnitude B of the extra-galactic field are at the level of B (cid:46) l c ∼ E/Z = 10 EV(where Z is the electric charge) would be deflected by ap-proximately2 π eBE/Z √ l c D rad ≈ ◦ (cid:18) D
50 Mpc (cid:19) / ,D being the distance to the source (Lee et al. 1995). Likely,for most of the directions the deflections are even smaller(Dolag et al. 2005).The GMF is usually assumed to have regular and tur-bulent components. This field may be inferred from theFaraday rotations of galactic and extragalactic sources, syn-chrotron emission of relativistic electrons in the Galaxy, andpolarized dust emission (see Haverkorn (2015) for a review).Two recent regular field models can be found in Pshirkovet al. (2011); Jansson & Farrar (2012); in what follows thesewill be referred to as PT2011 and JF2012, respectively.These models use different input data and a different globalstructure of the GMF. The predicted deflections are consis-tent in magnitude, having typical values 20 ◦ –40 ◦ for nucleiwith rigidity E/Z = 10 EV, but often differ in direction.The PT2011 model was only fitted to Faraday rotation datawhich are not sensitive to magnetic field components perpen-dicular to the line of sight, which are the most important forUHECR deflections, so the fact that even this model resultsin dipole and quadrupole magnitudes not very different fromthose from JF2012 is a strong indication of the robustnessof our approach to uncertainties in the details of the GMF. For both regular and random magnetic fields, deflections areinversely proportional to the magnetic rigidity of the particles.MNRAS , 1–9 (2015) ow isotropic can the UHECR flux be? The turbulent component of GMF has a larger magni-tude than the regular one, but a small ( (cid:46)
100 pc) coherencelength makes the effect of this field subdominant when av-eraged over the particle trajectory. Quantitative estimates(Tinyakov & Tkachev 2005; Pshirkov et al. 2013) indicatethat the contribution of the turbulent field into the UHECRdeflections is at least a factor ∼ (cid:18) E/Z
40 EV (cid:19) − ◦ sin b + 0 . , (2)which for nuclei with rigidity E/Z = 10 EV ranges from3 . ◦ at the Galactic poles to 27 ◦ along the Galactic plane,implemented as described in appendix A. We have calculated the expected angular distribution of thearrival directions of UHECRs at Earth and the correspond-ing harmonic coefficients for the three scenarios describedin subsection 2.2 and energy thresholds E min between 10and 60 EeV. We do not consider higher thresholds, becauseabove the cutoff in the UHECR spectrum the statistics dropsrapidly which makes the measurement of multipoles difficult.All calculations were repeated with the two GMF models.We first present the flux sky maps of events above60 EeV. The maps assuming a pure proton or silicon in-jection and the PT2011 GMF model are shown in Figure 4.The stronger large-scale anisotropy is obtained from the sil-icon injection with a cutoff, because the short energy losslengths imply that the most of the flux originates fromsources within a few tens of Mpc, where the matter dis-tribution is dominated by a few large structures. Protonshave longer energy loss lengths, so a larger number of struc-tures, up to a couple hundred Mpc, can contribute. Themagnetic deflections somewhat smear the picture at smallangular scales, especially near the Galactic plane, being of afew degrees for protons and a few tens of degrees for silicon.The case of oxygen injection with no cutoff is similar to thatof protons, because the flux at high energies is dominatedby the secondary protons.At lower energies, the effect of magnetic deflections be-comes much more important, up to several tens of degreesfor protons and even larger for silicon. As shown in Figure 5,the structures visible in the previous plots get smeared anddisplaced. The overall contrast of these maps is smaller thanin the previous case (note that the color codings for theseplots are different, in order to make smaller flux variationsmore visible), especially for silicon, whose flux only varies This upper bound was determined assuming there is no strongrandom field in regions with low total electron density, but weverified that the precise values used for the smearing angles haveno major effect on our results.
Figure 4.
The expected angular distribution of UHECR arrivaldirections at Earth with energies above 60 EeV for pure proton(top) or silicon (bottom) injection, assuming the PT2011 GMFmodel, in Galactic coordinates. The normalization is such that (cid:82) π Φ( ˆn ) dΩ = 1 (mean value 1 / π ≈ . by ∼
10% around the mean value 1 / π ≈ . To characterize the anisotropy quantitatively, we use theangular power spectrum C l = 12 l + 1 + l (cid:88) m = − l | a lm | , where a lm are the coefficients of the spherical harmonic ex-pansion of the directional fluxΦ( ˆn ) = + ∞ (cid:88) l =0 + l (cid:88) m = − l a lm Y lm ( ˆn ) . The angular power spectrum C l quantifies the amount ofanisotropy at angular scales ∼ ( π/l ) rad and is rotationallyinvariant.Explicitely, retaining only the dipole ( l = 1) and MNRAS000
10% around the mean value 1 / π ≈ . To characterize the anisotropy quantitatively, we use theangular power spectrum C l = 12 l + 1 + l (cid:88) m = − l | a lm | , where a lm are the coefficients of the spherical harmonic ex-pansion of the directional fluxΦ( ˆn ) = + ∞ (cid:88) l =0 + l (cid:88) m = − l a lm Y lm ( ˆn ) . The angular power spectrum C l quantifies the amount ofanisotropy at angular scales ∼ ( π/l ) rad and is rotationallyinvariant.Explicitely, retaining only the dipole ( l = 1) and MNRAS000 , 1–9 (2015)
A. di Matteo and P. Tinyakov
Figure 5.
Sky maps of the expected UHECR directional fluxabove 10 EeV for pure proton (top) or silicon (bottom) injection,assuming the PT2011 GMF model, normalized to (cid:82) π Φ( ˆn ) dΩ =1 (mean value 1 / π ≈ . quadrupole ( l = 2) contributions, the flux Φ( ˆn ) can be writ-ten as Φ( ˆn ) = Φ (1 + d · ˆn + ˆn · Q ˆn + · · · ) , where the average flux is Φ = a / √ π (Φ = 1 / π ifwe use the normalization (cid:82) π Φ( ˆn ) dΩ = 1), the dipole d isa vector with 3 independent components, which are linearcombinations of a m /a , and the quadrupole Q is a rank-2 traceless symmetric tensor (i.e., its eigenvalues λ + , λ , λ − sum to 0 and its eigenvectors ˆq + , ˆq , ˆq − are orthogonal)with 5 independent components, which are linear combina-tions of a m /a . The rotationally invariant combinations | d | = 3 (cid:112) C /C and (cid:113) λ + λ − + λ = 5 (cid:112) C / C charac-terize the magnitude of the corresponding relative flux vari-ations over the sphere. The dipole and quadrupole momentsquantify anisotropies at scales ∼ ◦ and ∼ ◦ respec-tively, and are therefore relatively insensitive to magneticdeflections except at the lowest energies.In Figure 6 and Figure 7, we present the energy depen-dence of the dipole amplitude | d | and the quadrupole ampli-tude ( λ + λ − + λ ) / respectively in the various scenarioswe considered. The first thing we point out is that, whereasthere are some differences between predictions using the twodifferent GMF models with the same injection model, theyare not so large as to impede a meaningful interpretationof the results in spite of the GMF uncertainties. Conversely,the results from the three injection models do differ signif-icantly, with heavier compositions resulting in larger dipole d i po l e a m p lit ud e E min [EeV]p inj., no cut, PT11p inj., no cut, JF12 O inj., no cut, PT11O inj., no cut, JF12 Si inj., w. cut, PT11Si inj., w. cut, JF12 Auger 2017Auger + TA 201599.9% sensitivity Figure 6.
The magnitude of the dipole as a function of the en-ergy threshold E min for the three injection models and two GMFmodels we considered. The points labelled “Auger + TA 2015”and “Auger 2017” show the dipole magnitude reported in Deligny(2015) and Taborda (2017) respectively. The dotted lines showthe 99.9% C.L. detection thresholds using the current and near-future Auger and TA exposures (see the text for details). s q r t ( λ + + λ + λ − ) E min [EeV]p inj., no cut, PT11p inj., no cut, JF12 O inj., no cut, PT11O inj., no cut, JF12 Si inj., w. cut, PT11Si inj., w. cut, JF12 Auger + TA 201499.9% sensitivity Figure 7.
The magnitude of the dipole as a function of the energythreshold E min (same notation as in Figure 6). The point labelled“Auger + TA 2014” is the quadrupole magnitude computed fromthe a m coefficients reported in Aab et al. (2014). and quadrupole moments for high energy thresholds (due tothe shorter propagation horizon) but smaller ones for lowerthresholds (due to larger magnetic deflections).Increasing the energy threshold, the expected dipoleand quadrupole strengths increase, but at the same time theamount of statistics available decreases due to the steeplyfalling energy spectrum, making it non-obvious whether theoverall effect is to make the detection of the dipole andquadrupole easier with higher or lower E min . To answerthis question, we have calculated the 99.9% C.L. detectionthresholds, i.e., the multipole amplitudes such that largervalues would be measured in less than 0.1% of random re-alizations in case of a isotropic UHECR flux. The detectionthresholds scale like ∝ / √ N with the number of events N . Since below the observed cutoff ( ∼
40 EeV) the inte-gral spectrum at Earth N ( ≥ E min ) is close to a power law ∝ E − , the detection threshold is roughly proportional to E min . At higher energies, the experimental sensitivity de-grades faster as the result of the cutoff.In order to compute the detection thresholds, we as-sumed the energy spectrum measured by Auger (Fenu 2017) MNRAS , 1–9 (2015) ow isotropic can the UHECR flux be? and: (i) the sum of the exposures used in the most recentAuger (Giaccari 2017) and TA (Nonaka 2017) analyses (lineslabelled “2017”); (ii) the sum of the exposures expected ifanother 3 yr of data are collected with 3 000 km effectivearea by each observatory, as planned following the fourfoldexpansion of TA (Sagawa 2015) (lines labelled “2020”). Thesensitivity is less than what it would be if we had uniformexposure over the full sky, as the actual exposure is currentlymuch larger in the southern than in the northern hemi-sphere. Also, we neglected the systematic uncertainty dueto the different energy scales of the two experiments, whichmainly affects the z -component of the dipole. We find thatthe dipole and quadrupole strengths increase with the en-ergy threshold faster than the statistical sensitivity degradesin the case of a heavy composition but slower in the case ofa medium or light composition, making higher thresholdsmore advantageous in the former case, and lower thresholdsin the latter.At the highest energies (where there cannot be largeamounts of intermediate-mass nuclei, due to photodisinte-gration), a heavy composition would result in a dipole andespecially quadrupole moment large enough to be detectedin the very near future; failure to do so would be stronglyindicative of a proton-dominated composition at those ener-gies.At intermediate energies ( E min ∼
30 EeV), the dipoleand quadrupole are guaranteed to be above the near-futuredetection threshold regardless of the mass composition. Un-fortunately the model predictions do not vary dramaticallyat these energies, so while a lack of dipole or quadrupolewould imply that some of our assumptions must be wrong,a successful detection will not be particularly useful in dis-criminating between the various injection scenarios.At even lower energy thresholds, the sensitivity of thedipole and quadrupole moment to the UHECR mass compo-sition is again stronger; in particular, the combined Augerand TA dataset (Aab et al. 2014) is already able to dis-favour a pure proton composition, as it would result ina much stronger quadrupole moment than observed, asshown by the corresponding data point in Figure 7. Wealso show the dipole magnitudes reported by TA and Augerfor E min = 10 EeV (Deligny 2015) and by Auger only for E min = 8 EeV (Taborda 2017) in Figure 6. The latter hassmaller error bars, but it is the result of an analysis rely-ing on the hypothesis that the angular distribution is purelydipolar with zero quadrupole or higher moments, contraryto our model predictions of a quadrupole moment similarin magnitude to the dipole in all cases. The combined TA–Auger analysis, due to its full-sky coverage, does not requiresuch a hypothesis. To summarize, we have calculated a minimum level ofanisotropy of the UHECR flux that is expected in a genericmodel where the sources trace the matter distribution in thenearby Universe, under the assumption that UHECRs above10 EeV have a light or medium (but otherwise arbitrary)composition. To this end we calculated the expected angulardistribution of UHECR arrival directions resulting from thecorresponding distribution of sources in the case of a pure silicon injection (which maximizes the magnetic deflections)and a pure proton injection (which maximizes the contribu-tion of distant, near-homogeneous sources), as well as an in-termediate case (oxygen injection with secondary protons).To quantify the resulting anisotropy we have chosen the lowmultipole power spectrum coefficients C l , namely the dipoleand quadrupole moments, which are the least sensitive tothe coherent magnetic deflections and the most easily mea-sured. The uncertainties in the magnetic field modeling wereroughly estimated by comparing two different GMF models.We also calculated the smallest dipole and quadrupole mo-ments that could be unambiguously detected in present ornear-future data.Several conclusions follow from our results. First, thereis a minimum amount of anisotropy that the UHECR fluxmust exhibit, regardless of the composition and the GMF de-tails, provided our assumptions are correct: the dipole andquadrupole amplitudes above 30 EeV are expected to be | d | (cid:38) .
13 and ( λ + λ − + λ ) / (cid:38) .
24 in all cases. Sec-ond, larger statistics at low energies does not give a majoradvantage in the anisotropy detection (except for a proton-dominated composition) in the ideal situation, because theexpected signal strength increases with energy roughly pro-portionally to the experimental sensitivity. (In the case ofsilicon injection, anisotropies at higher energies are even eas-ier to detect than at lower energies.) Finally, in terms of de-tectability the quadrupole is about as good as the dipole,again in the ideal situation we have considered.In reality, the terrestrial UHECR experiments do nothave a complete sky coverage. To unambiguously measurethe multipole coefficients one has to combine the TA andAuger data. Because of a possible systematic energy shiftbetween the two experiments, a direct cross-calibration offluxes is required (Aab et al. 2014). The cross-calibration in-troduces additional errors that affect more the dipole thanthe quadrupole measurement. Taking into account our re-sults, this makes the quadrupole moment a more promisingtarget to search for anisotropies resulting from an inhomoge-neous distribution of matter in the Universe with the currentconfiguration of the UHECR experiments. In particular, thenon-observation of a strong quadrupole moment (Aab et al.2014) already disfavors a pure proton composition.The TA experiment is now being expanded to ∼ ACKNOWLEDGEMENTS
We thank Olivier Deligny, Sergey Troitsky, Grigory Rubtsovand Michael Unger for fruitful discussions about the topic
MNRAS , 1–9 (2015)
A. di Matteo and P. Tinyakov of this paper. This work is supported by the IISN project4.4502.16.
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APPENDIX A: IMPLEMENTATION OFDEFLECTIONS IN THE TURBULENT GMF
If the random diffusion of particles in the turbulent GMFwere homogeneous and isotropic and its typical magnitude σ were independent of the position in the sky, it could be sim-ply be modelled by applying a Gaussian blurΦ new ( ˆn ) = (cid:90) π πσ exp (cid:18) − | ˆn − ˆn (cid:48) | σ (cid:19) Φ old ( ˆn (cid:48) ) dΩ (cid:48) (A1)to each flux map; but actually the deflections are larger nearthe Galactic plane than far away from it, as described byEquation 2.As a result, it is not immediately obvious how to imple-ment them; for example, if Equation A1 is used, either thesmearing magnitude as a function of the undeflected direc-tion σ ( ˆn (cid:48) ) or of the deflected direction σ ( ˆn ) might be used.At a closer look, the former procedure is clearly unphysicalbecause it does not leave an isotropic flux map unchanged.On the other hand, the latter has the apparently counter-intuitive property that the image of a point source is notGaussian. The exact motivation for the latter choice is alsonot clear, particularly at large deflection angles.The correct procedure is obtained by noticing that,in the case of constant smearing angle, the full one-stepsmearing is equivalent to N successive smearings with thesmaller angle σ/ √ N . This is readily generalized to the angle-dependent case. Namely, we successively apply locally Gaus-sian smearingsΦ i +1 ( ˆn ) = 1 πσ i ( ˆn ) (cid:90) π Φ i ( ˆn (cid:48) ) exp (cid:18) − | ˆn − ˆn (cid:48) | σ i ( ˆn ) (cid:19) dΩ (cid:48) with a smearing angle σ i ( ˆn ) = σ ( ˆn ) / √ N , where σ ( ˆn ) isgiven by Equation 2 and N is the number of iterations. Thismimics the actual physical process where the random de-flections are not a one-step process and particles initiallydeflected towards the Galactic plane will likely end up de-flected more than particles initially deflected away from it.We found that the result is independent of N providedit is large enough, and similar but not identical to smearingthe map only once using the smearing angle as a functionof the deflected direction σ ( ˆn ). The results we show in ourplots were obtained with N = 9. This also allowed us tocompute each smearing via Monte Carlo integration (usingfor Φ i +1 ( ˆn ) in each pixel the average of Φ i ( ˆn (cid:48) ) for 500 val-ues of ˆn (cid:48) sampled from a Gaussian distribution around ˆn )rather than a more time-consuming full numerical integra-tion, after verifying in one case that the two methods givenear-identical results for large N .In Figure A1 we show the flux maps for E min = 10 EeV MNRAS , 1–9 (2015) ow isotropic can the UHECR flux be? in the silicon injection scenario (the one with the largest de-flections) after zero, one, four and nine of the nine smearingsteps used in Figure 5b. This paper has been typeset from a TEX/L A TEX file prepared bythe author.
Figure A1.
Flux maps as in Figure 5b, before the random smear-ing and after one, four and nine of the nine smearing steps used,using the same color scaleMNRAS000