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Anisotropy of ultra high energy cosmic rays in the dark matter halo model
11 March 1999
Physics Letters B 449 Ž1999. 237–239
Anisotropy of ultra high energy cosmic rays in the dark matter halo model V. Berezinsky
a,b
, A.A. Mikhailov
c
a
c
INFN, Laboratori Nazionali del Gran Sasso, I-67010 Assergi (AQ), Italy b Institute for Nuclear Research, Moscow, Russia Institute of Cosmophysical Research and Aeronomy, 31 Lenin AÕe., 677891 Yakutsk, Russian Federation Received 20 November 1998 Editor: R. Gatto
Abstract The harmonic analysis of anisotropy of Ultra High Energy Cosmic Rays is performed for the Dark Matter halo model. In this model the relic superheavy particles comprise part of the Dark Matter and are concentrated in the Galactic halo. The Ultra High Energy Cosmic Rays are produced by the decays of these particles. Anisotropy is caused by the non-central position of the Sun in the Galactic halo. The calculated anisotropy is in reasonable agreement with the AGASA data. For more precise test of the model a comparison of fluxes in the directions of the Galactic Center and Anticenter is needed. q 1999 Elsevier Science B.V. All rights reserved.
The spectrum of Ultra High Energy Cosmic Rays ŽUHECR. is measured now up to a maximum energy of Ž2–3. = 10 20 eV w1,2x. More than 1000 particles are detected at energies higher than 1 = 10 19 eV w2–5x. The detailed energy spectrum was recently presented in Ref. w6x. No steepening of the spectrum has been observed in the energy range between 10 18 and 2 = 10 20 eV. If extragalactic, the UHE protons must have the Greisen-Zatsepin-Kuzmin ŽGZK. cutoff w7x at energy E1r2 s 6 = 10 19 eV w8x. A similar cutoff should exist if primaries are extragalactic nuclei w8–10x or photons w11,12x. Recently it was suggested that UHECR can be generated by the decay of superheavy relic particles w13–16x. These particles can be effectively produced in the post-inflationary Universe w14,17,18x and can constitute now a small or large part of Cold Dark Matter ŽDM.. As any other form of Cold DM these relic particles are concentrated in the halo of our
galaxy, and thus UHECR produced by their decays do not exhibit the GZK cutoff w14x. Realistic particle candidates for this scenario and possible mechanisms to provide the long lifetime for superheavy particles are discussed in Refs. w13,14,19,20x. The halo model discussed above has three signatures w14,21,22x: the excess of high energy photons in the primary radiation, direct signal from a nearby clump of DM Že.g. Virgo Cluster., and anisotropy caused by the asymmetric position of the Sun in the Galactic halo. These signatures allow to confirm or to reject the DM halo hypothesis by the data of existing arrays. As calculations show w22,23x, anisotropy reveals itself very strongly in the direction of the Galactic Center. This prediction can be reliably examined by the Pierre Auger detector in the southern hemisphere w24x. At present there is no detector which can observe this direction. In this article we present the
0370-2693r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 0 5 3 - 2
V. Berezinsky, A.A. MikhailoÕr Physics Letters B 449 (1999) 237–239
238 Table 1 Anisotropy
ISO model Rc
A1
A2
5 kpc 10 kpc 50 kpc
0.43Ž0.46. 0.32Ž0.35. 0.15Ž0.15.
0.11Ž0.13. 0.06Ž0.07. 0.01Ž0.01.
NFW model
a 2508 2508 2508
calculations of anisotropy for the arrays in the northern hemisphere, taking as an example the geographical position of the Yakutsk and AGASA arrays. We shall assume that primary photons dominate in the decays of SH particles as QCD calculations w25x imply. Then the flux of UHE photons in the direction Ž z , f . per unit solid angle can be written as
H0r
IŽ z . sK
max Ž z
.
dr r X Ž R . ,
Ž 1.
where r and R are the distances to a decaying X-particle from the Sun and the Galactic Center, respectively, z is the angle between the line of observation and the direction to the Galactic Center, f is the azimuthal angle in respect to Galactic plane Žthe flux depends only on z ., r X Ž R . is the mass density of superheavy particles Ž X . at a distance R from the Galactic Center, K is an overall constant, rmax Ž z . s R 2h y r(2 sin2z q r(cos z , r( s 8.5 kpc is a distance between the Sun and Galactic Center, R h is the size of the halo, in our calculations we shall use R h s 50 kpc Žthe values of 100 and even 500 kpc result in similar anisotropy.; the distance R from the Galactic Center to the decaying particle is given by R 2 Ž z . s r 2 q r(2 y 2 rr(cos z . We shall use two distributions of DM in the halo: one given by the Isothermal Model ŽISO. w26x, r0 r Ž R. s , Ž 2. 2 1 q Ž RrR c .
Rs
A1
A2
a
30 kpc 45 kpc 100 kpc
0.41Ž0.45. 0.37Ž0.41. 0.33Ž0.36.
0.10 Ž0.13. 0.09 Ž0.11. 0.07 Ž0.08.
2508 2508 2508
kpc and 100 kpc. The NFW distribution w27x is given in terms of the virial radius r 200 , the rotational velocity at the virial distance Õ 200 , the constant dc and the dimensionless Hubble constant h. We applied this distribution to our Galaxy using the following parameters: the local density of DM r D M Ž r(. s 0.3 GeVrcm3, Õ 200 s 200 kmrs and h s 0.6. As a result we obtain R s f 45 kpc. The flux Ž1. was expressed first in terms of galactic coordinates, longitude l s f and latitude b, which is given by cos b s cos zrcos f , and then transferred into equatorial coordinates, declination d and right ascension a . We calculated the amplitudes
(
and the other following the NFW numerical simulation w27x r0 r Ž R. s . Ž 3. 2 RrR s Ž 1 q RrR s . In the ISO model we shall use for R c the values 5 kpc, 10 kpc and 50 kpc. For the NFW model the calculations are performed for R s equal to 30 kpc, 45
Fig. 1. Amplitude and phase of the first harmonic of anisotropy for the AGASA and Yakutsk arrays. Solid lines are for the ISO distribution of DM and dots ŽNFW. for the NFW numerical simulation. BM and TW show the results of this paper and Ref. w28x, respectively. The three dots of BM Žfrom left to right. are given for R s s 30, 45 and 100 kpc, respectively; five dots of TW for R s s10, 20, 30, 50 and 100 kpc. The AGASA data are taken from Ref. w28x.
V. Berezinsky, A.A. MikhailoÕr Physics Letters B 449 (1999) 237–239
of the first and the second harmonics Ž A1 and A 2 , respectively. and the phase a of the first harmonic for the geographical position of the Yakutsk and AGASA arrays. These quantities are the standard ones used for measured anisotropy. The results are given in Table 1 Žpredictions for the AGASA array are shown in brackets.. Depending on the parameters of the DM distribution, the anisotropy varies from 10% to 45%. The phase of the first harmonic a f 2508 is close to the RA of Galactic Center, a f 2658. After this paper was submitted for publication we received the preprint by Medina-Tanco and Watson w28x, where similar calculations were performed. The results of both calculations are displayed in Fig. 1 for E ) 4 = 10 19 eV together with the data of AGASA ŽAG. and Yakutsk ŽYK. arrays. The AGASA anisotropy is taken from analysis of Ref. w28x. Our calculations ŽBM. agree well with that of Ref. w28x. Both agree with the data of AGASA array and do not contradict to the Yakutsk data. Acknowledgements The authors are grateful to P. Blasi, M. Hillas, B. Hnatyk, M. Nagano, P. Sokolsky, A. Vilenkin and A. Watson for discussions and correspondence. References w1x w2x w3x w4x w5x
N. Hayashida et al., Phys. Rev. Lett. 73 Ž1994. 3491. D.J. Bird et al., Ap. J. 424 Ž1994. 491. G. Cunningham et al., Ap. J. 236 Ž1980. L71. S. Yoshida et al., Astroparticle Physics 3 Ž1995. 105. B.N. Afanasiev et al., in: M. Nagano ŽEd.., Extremely High
w6x w7x w8x
w9x w10x w11x w12x w13x
w14x w15x w16x w17x w18x w19x w20x w21x w22x w23x w24x w25x w26x w27x w28x
239
Energy Cosmic Rays: Astrophysics and Future Observations, ICRR, University of Tokyo, 1996, p. 32. M. Takeda et al., Phys. Rev. Lett. 81 Ž1998. 1163. K. Greisen, Phys. Rev. Lett. 16 Ž1966. 748; G.T. Zatsepin, V.A. Kuzmin, Pisma Zh. Exp. Teor. Fiz. 4 Ž1996. 114. V.S. Berezinsky, S.V. Bulanov, V.A. Dogiel, V.L. Ginzburg, V.S. Ptuskin, Astrophysics of Cosmic Rays, chapter 4, Elsevier, 1990. V. Berezinsky, S. Grigorieva, G.T. Zatsepin, in: Proc. 14th Int. Cosm. Ray Conf., vol. 2, Munich, 1975, p. 711. L.N. Epele, E. Roulet, J. High Energy Phys. 9810 Ž1998. 009. astro-phr9808104. V.S. Berezinsky, Soviet Phys. Nucl. Phys. 11 Ž1970. 399. R.J. Protheroe, P.L. Biermann, Astroparticle Physics 6 Ž1996. 45. V.A. Kuzmin, V.A. Rubakov, Talk at the Workshop Beyond the Desert, Castle Rindberg, 1997; Yadernaya Fisika 61 Ž1998. 1122. V. Berezinsky, M. Kachelriess, A. Vilenkin, Phys. Rev. Lett. 79 Ž1997. 4302. P.H. Frampton, B. Keszthelyi, N.J. Ng, astro-phr9709080. M. Birkel, S. Sarkar, Astrop. Phys. 9 Ž1998. 297. D.J.H. Chung, E.W. Kolb, A. Riotto, Phys. Rev. D 59 Ž1999. 023501. V. Kuzmin, I. Tkachev, JETP Lett. 68 Ž1998. 271. K. Benakli, J. Ellis, D. Nanopoulos, heprphr9803333. K. Hamaguchi, Y. Nomura, T. Yanagida, Phys. Rev. D 58 Ž1998. 103503. S.L. Dubovsky, P.G. Tinyakov, JETP Lett. 68 Ž1998. 107. V. Berezinsky, P. Blasi, A. Vilenkin, Phys. Rev. 58 Ž1998. 103515. A. Benson, A. Smialkowski, A.W. Wolfendale, 1998, submitted to Astrop. Phys. J.W. Cronin, Nucl. Phys. B ŽProc. Suppl.. 28B Ž1992. 213. V. Berezinsky, M. Kachelriess, Phys. Lett. B 434 Ž1998. 61. A.V. Kravtsov, A.K. Klypin, J.S. Bullock, J.R. Primack. astro-phr9708176 to be published in Ap. J. J.F. Navarro, C.S. Frenk, S.D.M. White, Astroph. J. 462 Ž1996. 563; F.F. Navarro astro-phr9801073. G.A. Medina-Tanco, A.A. Watson, 1998, submitted to Astrop. Phys.
Physics Letters B 449 Ž1999. 237–239
Anisotropy of ultra high energy cosmic rays in the dark matter halo model V. Berezinsky
a,b
, A.A. Mikhailov
c
a
c
INFN, Laboratori Nazionali del Gran Sasso, I-67010 Assergi (AQ), Italy b Institute for Nuclear Research, Moscow, Russia Institute of Cosmophysical Research and Aeronomy, 31 Lenin AÕe., 677891 Yakutsk, Russian Federation Received 20 November 1998 Editor: R. Gatto
Abstract The harmonic analysis of anisotropy of Ultra High Energy Cosmic Rays is performed for the Dark Matter halo model. In this model the relic superheavy particles comprise part of the Dark Matter and are concentrated in the Galactic halo. The Ultra High Energy Cosmic Rays are produced by the decays of these particles. Anisotropy is caused by the non-central position of the Sun in the Galactic halo. The calculated anisotropy is in reasonable agreement with the AGASA data. For more precise test of the model a comparison of fluxes in the directions of the Galactic Center and Anticenter is needed. q 1999 Elsevier Science B.V. All rights reserved.
The spectrum of Ultra High Energy Cosmic Rays ŽUHECR. is measured now up to a maximum energy of Ž2–3. = 10 20 eV w1,2x. More than 1000 particles are detected at energies higher than 1 = 10 19 eV w2–5x. The detailed energy spectrum was recently presented in Ref. w6x. No steepening of the spectrum has been observed in the energy range between 10 18 and 2 = 10 20 eV. If extragalactic, the UHE protons must have the Greisen-Zatsepin-Kuzmin ŽGZK. cutoff w7x at energy E1r2 s 6 = 10 19 eV w8x. A similar cutoff should exist if primaries are extragalactic nuclei w8–10x or photons w11,12x. Recently it was suggested that UHECR can be generated by the decay of superheavy relic particles w13–16x. These particles can be effectively produced in the post-inflationary Universe w14,17,18x and can constitute now a small or large part of Cold Dark Matter ŽDM.. As any other form of Cold DM these relic particles are concentrated in the halo of our
galaxy, and thus UHECR produced by their decays do not exhibit the GZK cutoff w14x. Realistic particle candidates for this scenario and possible mechanisms to provide the long lifetime for superheavy particles are discussed in Refs. w13,14,19,20x. The halo model discussed above has three signatures w14,21,22x: the excess of high energy photons in the primary radiation, direct signal from a nearby clump of DM Že.g. Virgo Cluster., and anisotropy caused by the asymmetric position of the Sun in the Galactic halo. These signatures allow to confirm or to reject the DM halo hypothesis by the data of existing arrays. As calculations show w22,23x, anisotropy reveals itself very strongly in the direction of the Galactic Center. This prediction can be reliably examined by the Pierre Auger detector in the southern hemisphere w24x. At present there is no detector which can observe this direction. In this article we present the
0370-2693r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 0 5 3 - 2
V. Berezinsky, A.A. MikhailoÕr Physics Letters B 449 (1999) 237–239
238 Table 1 Anisotropy
ISO model Rc
A1
A2
5 kpc 10 kpc 50 kpc
0.43Ž0.46. 0.32Ž0.35. 0.15Ž0.15.
0.11Ž0.13. 0.06Ž0.07. 0.01Ž0.01.
NFW model
a 2508 2508 2508
calculations of anisotropy for the arrays in the northern hemisphere, taking as an example the geographical position of the Yakutsk and AGASA arrays. We shall assume that primary photons dominate in the decays of SH particles as QCD calculations w25x imply. Then the flux of UHE photons in the direction Ž z , f . per unit solid angle can be written as
H0r
IŽ z . sK
max Ž z
.
dr r X Ž R . ,
Ž 1.
where r and R are the distances to a decaying X-particle from the Sun and the Galactic Center, respectively, z is the angle between the line of observation and the direction to the Galactic Center, f is the azimuthal angle in respect to Galactic plane Žthe flux depends only on z ., r X Ž R . is the mass density of superheavy particles Ž X . at a distance R from the Galactic Center, K is an overall constant, rmax Ž z . s R 2h y r(2 sin2z q r(cos z , r( s 8.5 kpc is a distance between the Sun and Galactic Center, R h is the size of the halo, in our calculations we shall use R h s 50 kpc Žthe values of 100 and even 500 kpc result in similar anisotropy.; the distance R from the Galactic Center to the decaying particle is given by R 2 Ž z . s r 2 q r(2 y 2 rr(cos z . We shall use two distributions of DM in the halo: one given by the Isothermal Model ŽISO. w26x, r0 r Ž R. s , Ž 2. 2 1 q Ž RrR c .
Rs
A1
A2
a
30 kpc 45 kpc 100 kpc
0.41Ž0.45. 0.37Ž0.41. 0.33Ž0.36.
0.10 Ž0.13. 0.09 Ž0.11. 0.07 Ž0.08.
2508 2508 2508
kpc and 100 kpc. The NFW distribution w27x is given in terms of the virial radius r 200 , the rotational velocity at the virial distance Õ 200 , the constant dc and the dimensionless Hubble constant h. We applied this distribution to our Galaxy using the following parameters: the local density of DM r D M Ž r(. s 0.3 GeVrcm3, Õ 200 s 200 kmrs and h s 0.6. As a result we obtain R s f 45 kpc. The flux Ž1. was expressed first in terms of galactic coordinates, longitude l s f and latitude b, which is given by cos b s cos zrcos f , and then transferred into equatorial coordinates, declination d and right ascension a . We calculated the amplitudes
(
and the other following the NFW numerical simulation w27x r0 r Ž R. s . Ž 3. 2 RrR s Ž 1 q RrR s . In the ISO model we shall use for R c the values 5 kpc, 10 kpc and 50 kpc. For the NFW model the calculations are performed for R s equal to 30 kpc, 45
Fig. 1. Amplitude and phase of the first harmonic of anisotropy for the AGASA and Yakutsk arrays. Solid lines are for the ISO distribution of DM and dots ŽNFW. for the NFW numerical simulation. BM and TW show the results of this paper and Ref. w28x, respectively. The three dots of BM Žfrom left to right. are given for R s s 30, 45 and 100 kpc, respectively; five dots of TW for R s s10, 20, 30, 50 and 100 kpc. The AGASA data are taken from Ref. w28x.
V. Berezinsky, A.A. MikhailoÕr Physics Letters B 449 (1999) 237–239
of the first and the second harmonics Ž A1 and A 2 , respectively. and the phase a of the first harmonic for the geographical position of the Yakutsk and AGASA arrays. These quantities are the standard ones used for measured anisotropy. The results are given in Table 1 Žpredictions for the AGASA array are shown in brackets.. Depending on the parameters of the DM distribution, the anisotropy varies from 10% to 45%. The phase of the first harmonic a f 2508 is close to the RA of Galactic Center, a f 2658. After this paper was submitted for publication we received the preprint by Medina-Tanco and Watson w28x, where similar calculations were performed. The results of both calculations are displayed in Fig. 1 for E ) 4 = 10 19 eV together with the data of AGASA ŽAG. and Yakutsk ŽYK. arrays. The AGASA anisotropy is taken from analysis of Ref. w28x. Our calculations ŽBM. agree well with that of Ref. w28x. Both agree with the data of AGASA array and do not contradict to the Yakutsk data. Acknowledgements The authors are grateful to P. Blasi, M. Hillas, B. Hnatyk, M. Nagano, P. Sokolsky, A. Vilenkin and A. Watson for discussions and correspondence. References w1x w2x w3x w4x w5x
N. Hayashida et al., Phys. Rev. Lett. 73 Ž1994. 3491. D.J. Bird et al., Ap. J. 424 Ž1994. 491. G. Cunningham et al., Ap. J. 236 Ž1980. L71. S. Yoshida et al., Astroparticle Physics 3 Ž1995. 105. B.N. Afanasiev et al., in: M. Nagano ŽEd.., Extremely High
w6x w7x w8x
w9x w10x w11x w12x w13x
w14x w15x w16x w17x w18x w19x w20x w21x w22x w23x w24x w25x w26x w27x w28x
239
Energy Cosmic Rays: Astrophysics and Future Observations, ICRR, University of Tokyo, 1996, p. 32. M. Takeda et al., Phys. Rev. Lett. 81 Ž1998. 1163. K. Greisen, Phys. Rev. Lett. 16 Ž1966. 748; G.T. Zatsepin, V.A. Kuzmin, Pisma Zh. Exp. Teor. Fiz. 4 Ž1996. 114. V.S. Berezinsky, S.V. Bulanov, V.A. Dogiel, V.L. Ginzburg, V.S. Ptuskin, Astrophysics of Cosmic Rays, chapter 4, Elsevier, 1990. V. Berezinsky, S. Grigorieva, G.T. Zatsepin, in: Proc. 14th Int. Cosm. Ray Conf., vol. 2, Munich, 1975, p. 711. L.N. Epele, E. Roulet, J. High Energy Phys. 9810 Ž1998. 009. astro-phr9808104. V.S. Berezinsky, Soviet Phys. Nucl. Phys. 11 Ž1970. 399. R.J. Protheroe, P.L. Biermann, Astroparticle Physics 6 Ž1996. 45. V.A. Kuzmin, V.A. Rubakov, Talk at the Workshop Beyond the Desert, Castle Rindberg, 1997; Yadernaya Fisika 61 Ž1998. 1122. V. Berezinsky, M. Kachelriess, A. Vilenkin, Phys. Rev. Lett. 79 Ž1997. 4302. P.H. Frampton, B. Keszthelyi, N.J. Ng, astro-phr9709080. M. Birkel, S. Sarkar, Astrop. Phys. 9 Ž1998. 297. D.J.H. Chung, E.W. Kolb, A. Riotto, Phys. Rev. D 59 Ž1999. 023501. V. Kuzmin, I. Tkachev, JETP Lett. 68 Ž1998. 271. K. Benakli, J. Ellis, D. Nanopoulos, heprphr9803333. K. Hamaguchi, Y. Nomura, T. Yanagida, Phys. Rev. D 58 Ž1998. 103503. S.L. Dubovsky, P.G. Tinyakov, JETP Lett. 68 Ž1998. 107. V. Berezinsky, P. Blasi, A. Vilenkin, Phys. Rev. 58 Ž1998. 103515. A. Benson, A. Smialkowski, A.W. Wolfendale, 1998, submitted to Astrop. Phys. J.W. Cronin, Nucl. Phys. B ŽProc. Suppl.. 28B Ž1992. 213. V. Berezinsky, M. Kachelriess, Phys. Lett. B 434 Ž1998. 61. A.V. Kravtsov, A.K. Klypin, J.S. Bullock, J.R. Primack. astro-phr9708176 to be published in Ap. J. J.F. Navarro, C.S. Frenk, S.D.M. White, Astroph. J. 462 Ž1996. 563; F.F. Navarro astro-phr9801073. G.A. Medina-Tanco, A.A. Watson, 1998, submitted to Astrop. Phys.