arXiv:2005.09480v2 [hep-ph] 9 Mar 2021 bv hc Mcnecp [ escape can DM which above one u:nneaiitcD a ebotdb cos- by [ boosted (CRs) be rays can DM mic nonrelativistic out: pointed of velocity the DM the in on limit confined upper results Galaxy an DM the in of nonrelativistic potential gravitational The of Galaxy. existence the h estv asrgo.TeCD a loproduce also can CRDM The region. mass sensitive the spans [ mass understood: magnitude possible of a poorly orders with 80 is unknown nearly DM is identity of DM the How- physical interactions. the electromagnetic ever, via invisible but ing aaee pc fsbGVD htwspeiul in- previously was that [ DM accessible sub-GeV certain a of of space explorations to parameter the allows compared it of fraction DM, tiny flux nonrelativistic a the the is Although (CRDM) DM CRs. is and CR-boosted energetic it scattered the be matter, by to with DM boosted interactions nonrelativistic finite the for has inevitable DM as long As [ recoils DM-electron of detection 23 direct [ the Bremsstrahlung effects, the via as well typically is which threshold, ergy [ way particles direct DM most of the nature be the nuclei to unveil target believed to of is which recoil electrons, the or via interactions their measure [ tion aswno fdrc eeto xeiet a only can experiments to detection down direct extend of window mass incpblt i o hehl ooee [ bolometer threshold low detec- a recoil via nuclear capability been the tion have expanding approaches including many DM, explored, sensitivity the sub-GeV enhance detecting to years, of recent In channel. recoil fdr atr(M [ (DM) matter existence dark the of supports observations cosmological and ical nte neetn osblt a enrecently been has possibility interesting Another assume experiments detection direct Conventionally, ,advrosnvldtcinpooas[ proposals detection novel various and ], Introduction 3 ist eiyteeitneo Mprilsand particles DM of existence the verify to aims ] 4 colo srnm n pc cec,Uiest fScienc of University Science, Space and Astronomy of School 36 intr o h ietdtcino ose u-e M in DM, sub-GeV boosted of detection diurnal energy. direct This the modulation. for diurnal i signature strong DM flux a DM nonrelativistic to the leading the large, Earth, sufficiently boost are and of sections with cross detection scattering scatter direct to the rays allow cosmic only not interactions nucleon , ira ffc fSbGVDr atrBotdb omcRays Cosmic by Boosted Matter Dark Sub-GeV of Effect Diurnal 40 epitotanwtp fdunleetfrtecsi a boos ray cosmic the for effect diurnal of type new a out point We vrhligeiec rmastrophys- from evidence Overwhelming – – O 35 43 upeMuti bevtr,CieeAaeyo Sciences, of Academy Chinese Observatory, Mountain Purple 1 e i h ovninlnuclear conventional the via GeV (1) 1 hoFn Ge, Shao-Feng , sn-a e nttt,Saga ioTn nvriy Sh University, Tong Jiao Shanghai Institute, Lee Tsung-Dao ndrc eeto,tu expanding thus detection, direct in ] 36 1 e aoaoyfrPril hsc n omlg,Shangha Cosmology, and Physics Particle for Laboratory Key e aoaoyfrPril srpyisadCsooy(MO Cosmology and Astrophysics Particle for Laboratory Key 5 ,wihi rvttoal interact- gravitationally is which ], etrfrHg nryPyis eigUiest,Beijing University, Peking Physics, Energy High for Center rteslrrflcin[ reflection solar the or ] 2 colo hsc n srnm,Saga ioTn Univers Tong Jiao Shanghai Astronomy, and Physics of School 6 2 3 , .TeD ietdetec- direct DM The ]. e aoaoyo akMte n pc Astronomy, Space and Matter Dark of Laboratory Key 10 7 ,2, 1, .Bcueo h en- the of Because ]. O n idl[ Migdal and ] kV,tesensitive the (keV), 4 ∗ , inliLiu, Jianglai 5 v ]. χ 24 . – 34 0 km/s 600 8 ]. 37 , 11 9 – – as ] 39 18 ,1, 2, 17 ]. – ] † in Yuan, Qiang inwt otc neato a eqieszbe e.g., sizable, quite 10 be as can large interaction as contact a with tion inl nlrenurn xeiet [ experiments neutrino large in signals i rmdsrbtos sgvnb [ spec- by and given The is spatial unaffected. distributions, its trum is describes distribution which emissivity, CR CRDM the that assume we for mainly is that effect DM. diurnal nonrelativistic conventional the from where detec- [ direct expected at is modulation experiments flux tion diurnal [ a Earth anisotropic, trav- the when through attenuated eling become experience and can scatterings particles multiple [ DM case strength, mediator interaction light lowed the to contrast in T u Reeg eurdt os h Mkntcenergy kinetic DM the boost to to required energy CR mum h ffc sipratol o eylrescattering large very a for ( only section important cross is effect is the flux CRDM re- the sky. Galaxy the scattering, inner over isotropic the to anisotropic for in highly likely DM Even more the much gion. boost are and with CRs scatter Therefore, Galaxy (GC). the center toward concentrated Galaxy the more in DM the locations becoming the their Galaxy, in with Both vary intensities distributions processes. CR scattering and CR density CRDM the and as de- DM well flux as the CRDM the on of pend distributions spectral and spatial i n h ose Mwt mass with DM boosted the and o u-e M h Mncensatrn rs sec- cross scattering DM-nucleon the DM, sub-GeV For u-e akMte ose yCsi Rays Cosmic by Boosted Matter Dark Sub-GeV lhuhteCD cteigas ffcsteCRs, the affects also scattering CRDM the Although T [ 36 χ n ehooyo hn,Hfi202,China 230026, Hefei China, of Technology and e ζ and , .Teeaetremi nrdet nE.( Eq. in ingredients main three are There ]. χ teutda tpoaae hog the through propagates it as attenuated s T Mwt ula eol,btas allow also but recoils, nuclear with DM ( ouainpoie nte prominent another provides modulation i r and T , diint inl ihhge recoil higher with signals to addition ohge nris fteDM-nuclei the If energies. higher to T ,4 5, 4, 3, χ χ max = ) − T e akmte D) h DM- The (DM). matter dark ted χ 31 σ χp stemxmmD iei nrygiven energy kinetic DM maximum the is ‡ ajn 103 China 210033, Nanjing r h iei nriso h Rspecies CR the of energies kinetic the are cm nhi204,China 200240, anghai ρ χ n igZhou Ning and m 020 China 200240, i ( 2 > | )&Shanghai & E) 081 China 100871, χ r se[ (see | 10 ) i = − X 49 p, 47 27 ity, He – n h eeecsi [ in references the and ] cm 52 Z 53 T 36 .I h RMflxis flux CRDM the If ]. 2 ∞ i min , [ ) 2, ] × m 54 § 35 d v χ .Ti sdifferent is This ]. i T , σ 44 .Frsimplicity, For ]. i 48 T χi n i – min T .Wt hsal- this With ]. CR G 46 χ max i 2 ,i ]. ( stemini- the is Q ( r ( 2 T T , ) i , ) i 1 ) The – :the ): 35 (1) ]), 2

σ -32 2 -4 GC direction, χp=10 cm 10 ) -1 -7 sr 10 -1 -5 -4 10 -4 -8 halo × 10 mχ=10 GeV 10 (cm s β

) -6 -9

/d 10

-1 10 10-3 GeV Φ )d -10 sr χ 10

-7 ρ -2 / -1

10 10 GeV β s χ 10-11 -2 (m -8 10-4 10-3 10-2 10-1 100 10 β 10-1 GeV (cm -9 χ 10

/dT -10 100 GeV

Φ 10 d

χ -11 101 GeV T 10 10-12 10-13 10-3 10-2 10-1 100 101 102 103 Tχ (GeV)

FIG. 2. The CRDM energy spectra at the GC direction for DM masses 10−4, 10−3, 10−2, 0.1, 1.0, and 10 GeV from top to bottom. The scattering cross section σχp is assumed to be 10−32 cm2. The inset is the distribution of DM velocities, β = v/c, compared to the Maxwellian distribution of the Standard DM Halo. For a clear comparison, we rescale the Standard DM Halo curve by 10−4 (labeled as halo ×10−4 in the inset) so that all curves have a similar height.

ing CR spatial distribution, please see the Supplemental Material [67]. FIG. 1. Relative sky maps of CRDM fluxes in the Galactic The DM-nucleus interaction is the least known part in coordinates with amplitude in the GC direction set to unity. Eq. (1). For simplicity, we assume that the DM-nucleus The upper and lower panels are for the NFW and Isothermal cross section σχA has a coherent enhancement, DM density profiles, respectively. m (m + m ) 2 σ = σ A2 A χ p , (2) χA χp m (m + m )  p χ A  DM density ρχ(|r|) at location r, the CR density nCR,i times its velocity vi, and the scattering cross section σχi. where σχn = σχp is the constant DM-nucleon cross sec- 2 2 2 2 tion, while m and m are the proton and nuclear masses The form factor Gi(Q ) ≡ 1/(1 + Q /Λi ) [55] is a func- p A tion of the momentum transfer Q with Λp ≈ 770 MeV for the CR. For mχ ≪ mp,mA, the enhancement mainly 2 and ΛHe ≈ 410 MeV [56] for proton and helium, respec- comes from the A factor. Extra enhancement may come 2 2 tively. from (mχ + mp) /mp when mχ goes beyond mp. The dipole hadronic form factor G (Q2) in Eq. (1) suppresses For the DM density ρχ(|r|), we adopt the i nfw the interaction at large momentum transfer Q. Navarro-Frenk-White (NFW) [57] profile, ρχ (r) = 2 The CRDM flux arriving at the Earth along a given ρs/[(r/rs)(1 + r/rs) ] with rs = 20 kpc and −3 direction nˆ is a line-of-sight integral of all contributions ρs = 0.35 GeV cm , as the benchmark DM mass distribution. For comparison, a cored isothermal distri- along the way, iso 2 bution, ρχ (r) = ρs/[1 + (r/rs) ] with rs = 5 kpc and dΦ 1 −3 ˆn r ρ = 1.56 GeV cm , is also studied. These parameters ( ,Tχ)= ζχ( ,Tχ) dl. (3) s dT 4π correspond to a local DM density of 0.4 GeV cm−3 in χ Z our Solar System [58] for both profiles. The difference Fig. 1 shows the relative all-sky maps of the CRDM fluxes between the two profiles and more details are given in in the Galactic coordinate, a spherical coordinate with the Supplemental Material [67]. The amplitudes of the the Sun as its center, the latitude measuring the angle diurnal modulation vary by only around 7% for different above/below the galactic plane, and the longitude mea- density profiles. suring the azimuth angle from the GC. The peak value at For the CR contribution in Eq. (1), we employ the the GC is set to 1. The top (bottom) panel presents the GALPROP [59] code (version 54) to simulate its distri- NFW (isothermal) profile. The CRDM fluxes are clearly bution. In this Letter, we only consider the dominating anisotropic, with the maximum (the GC direction) and proton and helium species of CRs, and leave the rest, in the minimum differing by about two orders of magnitude. particular electrons and positrons, for future discussions. To match the grid resolution of GALPROP, we set the For the detailed CR model parameters and the result- NFW density within 0.5 kpc of the GC to ρ(0.5 kpc). 3

10−6 This approximation has a negligible effect on the diurnal ] m = 10 MeV 1 −3 χ − 10 σ = 10−32 cm2 −4 modulation, as shown in the Supplemental Material [67]. χp s sr) 10 2 10−5

Fig. 2 shows the CRDM spectra from the GC direction [(cm −6 ]

χ 10 10−7

−1 −7

/dT 10 min for different DM masses. The number density ρχ/mχ in T Φ Standard Halo DM d −8

χ χ 10 −11 −10 −9 −8 −7

T 10 10 10 10 10

Eq. (1) accounts for the decrease of CRDM flux for larger s sr)

2 Tχ [GeV] DM masses. On the other hand, on average the maxi- 10−8 [(cm mum boost occurs when mχ approaches the mass of the χ dT

incident proton or helium, manifesting in the change of Φ/

d −9 χ 10 spectrum shape for different energies. At the high energy T Original ◦ 2 θnadir = 30 (11034 km) end, the spectra are suppressed by the form factor Gi(Q ) ◦ θnadir = 60 (6373 km) 2 θ = 90◦ with Q =2mχTχ. We also show the nonrelativistic DM nadir (160 km) 10−10 velocity distribution predicted by the Standard DM Halo 10−3 10−2 10−1 100 101 T [GeV] model (labeled as halo ×10−4) in Fig. 2 for comparison. χ We find that the CRDM spectra depend very weakly on directions, mainly due to the similar CR spectral shapes FIG. 3. The attenuated CRDM spectra for the nadir angles ◦ ◦ ◦ θnadir = 30 (red), 60 (green), and 90 (blue) with σχp = throughout the Galaxy. For simplicity, in the following −32 2 discussion we will separate the energy and angular dis- 10 cm , mχ = 10 MeV, and the detector at a depth of 2 km. For comparison, we also show the original Standard tributions of the CRDM fluxes. Halo DM flux distribution in the inset. Earth Attenuation – With a large enough scattering cross section, the DM can frequently scatter with matter max max when traveling through the Earth [19, 49–52], transfer- dσ = σdTr/Tr = σd ln Tχ(Tχ/Tr ). The attenuated ring its kinetic energy to matter nuclei. Although the DM flux can be obtained by integrating Eq. (4) step decelerated DM particle may still reach the detector, the by step over the traversed distance. Fig. 3 shows the DM energy spectrum is shifted lower, leading to fewer attenuated CRDM fluxes with different nadir angles to events above the detector energy threshold. For simplic- the underground detector. To be realistic, we consider ¯ ◦ ity, we use the average nucleon numbers, Am = 24 in a detector 2 km underground. Then for θnadir = 90 , the Earth mantle and A¯c = 54 in the Earth core, to DM needs to travel 160 km before reaching the detector, approximate the matter compositions [60]. As a con- corresponding to 9 mean free paths in the mantle. The −32 2 crete example, for σχp = 10 cm , the mean free path, CRDM flux at medium energy is largely reduced first Lfree ≡ mN /(ρN σχA), is around 2.7/17 km in the Earth and then goes back up at high energy. The limited at- core/mantle omitting the form factor effects. Similar at- tenuation at high energy is due to the highly suppressed tenuation happens in the atmosphere, but due to the 3 weight factor wFF(Tχ) in Eq. (4). Consequently, the orders lower density, the effect is only visible at much CRDM is much more energetic than the nonrelativistic larger cross sections. DM (see the inset of Fig. 3) and can produce recoil The differential CRDM flux dΦ(ˆn,l,Tχ)/d ln Tχ, at the events with much higher energy. This makes direct distance l through the Earth, is a combination of the loss detection experiments sensitive to sub-GeV DMs. of DM particles to an energy lower than Tχ and the gain Boosted Diurnal Effect – The two anisotropies from the ′ from a higher energy Tχ to Tχ. For an incoming DM Earth and the Galaxy lead to the diurnal effect. First, the ′ particle with a higher energy Tχ, the nuclear recoil energy path lengths that DM particles traverse are anisotropic ′ ′ Tr is evenly distributed in the range 0 ≤ Tr ≤ Tχ(Tχ + since the underground lab is close to the Earth surface ′ max ′ 2mχ)/(Tχ + mµ) ≡ Tr (Tχ) with reduced mass mµ ≡ and its depth is typically much smaller than the Earth ra- 2 (mN +mχ) /2mN . Because of energy conservation, Tχ is dius. Second, the CRDM flux is strongly peaked toward ′ ′ also evenly distributed: Tχ(mµ−2mχ)/(Tχ+mµ) ≤ Tχ ≤ the GC due to both the DM and the CR distributions. ′ ′ Tχ. For a given Tχ, the DM particles with energy Tχ in The CRDM flux is thus significantly attenuated by the ′ the range Tχ ≤ Tχ ≤ mµTχ/(mµ − 2mχ − Tχ) increases Earth when the GC and the detector are on opposite the flux at Tχ. The CRDM flux evolution contains two sides of the Earth but much less affected if they are on contributions [44]: the same side. To avoid confusion with the usual diur- nal effect for nonrelativistic DM [53, 54], we call this the ∂ dΦ(l,Tχ) ρN (l) dΦ(l,Tχ) = σ − wFF(T ) “boosted diurnal effect”. ∂l d ln T m χN d ln T χ χ N  χ Fig. 4 shows the diurnal modulation of the CRDM at a ′ ′ N ◦ dΦ l,T Tχ(T + m ) direct detection experiment located at a latitude of 28 N + χ χ µ G2 (Q2)d ln T ′ .(4) ′ ′ ′ N χ (approximate location of the China Jinping Underground d ln Tχ  Tχ(Tχ +2mχ) # Z Laboratory) and a depth of 2 km underground. Within The weight factor is defined as, wFF ≡ one sidereal day, the underground lab rotates around the 2 2 2 2 max GN (Q )dQ /Qmax, and the factor Tχ/Tr in Earth axis and its position is parameterized by the side- the second term comes from the differential cross section real hour in the range between [0, 24] hours. We define a R 4

100 50 101 ◦ θnadir = 30 ] 45 θ = 60◦ T T min −1 nadir χ ≥ χ ◦ 80 −32 2 θ = 90 (σχp = 10 cm ) nadir 40 100 ◦ θnadir = 180

35 [(ton yr) 60 T T min T ≥ 3 r χ ≥ χ r keV −32 2 σ = 10−32 2 (σχp = 3 × 10 cm ) ( χp cm ) 30 10−1 dlogT / 40 Tr ≥ 3keV 25 dN −32 2 (σχp = 3 × 10 cm ) 20 10−2 Survival Probability [%] Survival Probability 20 mχ = 10 MeV −32 2 σχp = 10 cm m = 10 15 ◦ χ MeV Event Rate Lattitude = 28 Lattitude = 28◦ 0 10 10−3 0 2 4 6 8 10 12 14 16 18 20 22 24 10−5 10−4 10−3

Sidereal Hour Tr [GeV]

FIG. 4. The survival probability of CRDM arriving at an ◦ FIG. 5. The nuclear recoil spectrum, including the 3 keV underground lab at latitude 28 N and a depth of 2km vs the detector threshold, for a detector with 1 ton·year ex- sidereal hour, relative to the number of DM particles arriving posure. To illustrate the attenuation effect, each curve cor- at the Earth for two different cross sections σχp = 1 (3) × responds to the integrated DM flux at a given nadir angle 10−32 cm2. The red curves correspond to the total CRDM min θnadir. arriving at the detector with Tχ ≥ Tχ , and the blue curves are those above the detector threshold (Tr > 3 keV for a liquid xenon detector). coil part as illustrated in Fig. 3. For two years of data at a benchmark liquid xenon detector PandaX-4T (5.6 ton×year exposure) [62], on average 8.1 (55) events are −32 2 survival probability as the ratio between the attenuated expected for σχp = 1 (3) × 10 cm and mχ = 10 MeV, CRDM flux in the underground lab and the one arriv- which is quite significant compared to the background −32 2 ing the Earth. At a cross section of 1 × 10 cm , we level [63]. For the same detector, the event rate and hence observe significant “boosted diurnal modulation” with the the sensitivity is roughly independent of the DM mass for survival probability varying in the range of 64% ∼ 95%. mχ . 0.1 GeV. In addition to a quadratic scaling with For comparison, we also show the curves for a cross sec- the cross section, one from the CRDM production and −32 2 tion of 3 × 10 cm where a larger modulation can be the other from its detection, the event rate is suppressed observed. Given the DM energy Tχ, the nuclear recoil once the attenuation from the Earth becomes dominating max has a wide distribution, 0 ≤ Tr ≤ Tr (Tχ), and hence for a sufficiently large cross section (∼ 10−28 cm2)[36]. max only a fraction, 1−Tth/Tr (Tχ), can pass the detection The cross section region that this technique can probe threshold, leading to a reduction from the red curve to spans roughly 4 orders of magnitude. the blue one in Fig. 4. Another factor is the scattering angle, which leads to Instead of performing numerical integration of Eq. (4), deflection [19]. For the relativistic CRDM with typical the curves in Fig. 4 are obtained by Monte Carlo sim- 1GeV kinetic energy, mass mχ = 10 MeV, and typical ulations. Since the spectrum of the CRDM is almost momentum transfer Q ≈ Λ ≈ 200 MeV [56], the scatter- independent of its direction, it is a good approximation ing angle is 3◦ ∼ 5◦. Although not completely negligible, to first sample the direction of the incoming DM parti- the scattering angle does not affect the diurnal modu- cles according to the sky map in Fig. 1 and then sample lation effect due to the following arguments. For the the boosted DM kinetic energy Tχ according to the spec- peak region of Fig. 4, the DM from the GC only needs trum in Fig. 2. The incident DM particle would then to penetrate O(1) km. With a mean free path of around experience multiple scatterings when crossing the Earth. 17km, most CRDMs experience only one scattering at For each interaction step, we first sample the distance most. Therefore, the peak region would not be affected that the DM particle travels before the next scattering significantly. Multiple scatterings will further suppress based on the mean free path and then sample the reduced the valley region of the curve and therefore enhance the kinetic energy. The simulation stops when the DM parti- modulation effect. cle reaches the underground detector or drops below the The recoil energy spectra for incident CRDMs along detection threshold. different nadir angles in a liquid xenon detector are shown Imposing the detection threshold on the nuclear re- in Fig. 5. Since the recoil energy can reach O(1 MeV), coil energy, Tr ≥ 3 keV for a liquid xenon detector [61], observing a high energy recoil event is a smoking gun for would reduce the event rate but still keep the modula- the CRDM, especially when the detector and the GC are tion behavior as illustrated in Fig. 4. This is because on the same side of the Earth. However, these energetic the diurnal modulation mainly comes from the high re- recoils may excite target isotopes and therefore may no 5 longer be simple nuclear recoils. 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Boron by the Alpha Magnetic Spectrometer on the In- SUPPLEMENTAL MATERIAL ternational Space Station,” Phys. Rev. Lett. 120, no.2, 021101 (2018). Dark matter density profile [71] G. L. Case and D. Bhattacharya, “A new sigma-d relation and its application to the galactic supernova remnant distribution,” Astrophys. J. 504, 761 (1998) Two kinds of density profiles, a cuspy NFW one and a [arXiv:astro-ph/9807162 [astro-ph]]. cored isothermal one are employed in this work. Fig. 6 [72] A. Cummings, E. Stone, B. Heikkila, N. Lal, W. Web- shows the density distributions of both profiles. To match ber, G. J´ohannesson, I. Moskalenko, E. Orlando and with the grid precision of the GALPROP propagation T. Porter, “Galactic Cosmic Rays in the Local Interstel- model, we employ a cut radius for the calculation of the lar Medium: Voyager 1 Observations and Model Results,” Astrophys. J. 831, no.1, 18 (2016). NFW density profile, within which the density is taken [73] M. Aguilar et al. [AMS], “Precision Measurement of the as a constant. We have tested that for rc =0.2, 0.5, and Proton Flux in Primary Cosmic Rays from Rigidity 1 1.0 kpc, the final results differ very little. GV to 1.8 TV with the Alpha Magnetic Spectrometer on the International Space Station,” Phys. Rev. Lett. 114, 102 171103 (2015). Isothermal [74] M. Aguilar et al. [AMS], “Precision Measurement of the NFW Helium Flux in Primary Cosmic Rays of Rigidities 1.9 101 GV to 3 TV with the Alpha Magnetic Spectrometer on )

the International Space Station,” Phys. Rev. Lett. 115, -3 0 no.21, 211101 (2015). 10 [75] Y. Yoon et al. [CREAM], “Proton and Helium Spectra

from the CREAM-III Flight,” Astrophys. J. 839, no.1, 5 (GeV cm 10-1 DM

(2017) [arXiv:1704.02512 [astro-ph.HE]]. ρ [76] Q. An et al. [DAMPE], “Measurement of the cosmic- -2 ray proton spectrum from 40 GeV to 100 TeV with 10 the DAMPE satellite,” Sci. Adv. 5, no.9, eaax3793 [arXiv:1909.12860 [astro-ph.HE]]. 10-3 [77] L. Gleeson and W. Axford, “Solar Modulation of Galactic 10-1 100 101 102 Cosmic Rays,” Astrophys. J. 154, 1011 (1968). r (kpc)

FIG. 6. The NFW and Isothermal density profiles of DM in the Milky Way. The vertical dotted line shows the cut radius of 0.5 kpc within which the DM density is taken to be constant.

Cosmic ray modeling

Charged CRs propagate diffusively in the random mag- netic field of the Milky Way. Besides the diffusion ef- fect, particles may get re-accelerated by randomly mov- ing magnetic turbulence, advected from the Galactic disk to the halo, and interact with the medium. It has been shown that the diffusion plus reacceleration propagation framework can best match the current data [68]. The propagation parameters are obtained by fit- ting the most recent AMS-02 measurements of the sec- ondary and primary nuclei [69, 70]. The propagation pa- rameters adopted are: the diffusion coefficient D(R) = η δ 28 2 −1 β D0(R/4 GV) with D0 = 7.13 × 10 cm s , δ = 0.353, η =0.0, the half-height of the propagation cylinder −1 zh =5.4 kpc, and the Alfven velocity vA = 35.4 km s that characterizes the reacceleration effect. The spatial distribution of CR sources is assumed to follow that of supernova remnants [71]. For the source spectra of CRs, we employ a spline-interpolation ap- proach as in Ref. [68]. The spectral parameters are de- 9

are measured in a wide energy range from outside of the Solar System by Voyager-1 [72], and at the top-of- Proton density at 10 GeV (10-15 MeV-1cm-3) atmosphere by AMS-02 [73, 74], CREAM-III [75], and 5.0 1.8 1.6 DAMPE [76]. To connect the local interstellar spectra of 1.4 CRs with the measured ones around the Earth, a force- 2.5 1.2 field solar modulation model is applied [77]. 1.0 Fig. 7 shows the distribution of protons at energies of 0.0 0.8 10 GeV, calculated with the GALPROP tool. It shows z (kpc) 0.6 that the CR density is not uniform, and higher in the -2.5 0.4 inner Galaxy. The product of the CR density and the DM 0.2 -5.0 density thus gives a strong anisotropy of the expected 0.0 CRDM fluxes, as shown in Fig. 1. 0.0 5.0 10.0 15.0 20.0 R (kpc) Effect on Diurnal Effect

Fig. 8 compares the effects of different density profiles FIG. 7. Density distribution of 10 GeV CR protons in the Milky Way as a function of radius R and height z. and smooth scales of the DM distribution on the diur- nal modulation effect. We can see that for different DM profiles, the basic features of diurnal modulation is not affected with only slight change. The largest difference between Isothermal and NFW profiles is around 7% at rived through fitting to the proton and helium data which the modulation valley. 10

100 Isothermal NFW (rc = 0.2 kpc) NFW (rc = 0.5 kpc) 90 NFW (rc = 1.0 kpc)

80

70 Survival Probability [%] Survival Probability

mχ = 10 MeV −32 2 σχp = 10 cm 60 0 2 4 6 8 10 12 14 16 18 20 22 24 Sidereal Hour

FIG. 8. The effect of DM profiles on the diurnal modulation.