Ultra High Energy Cosmic Rays Diffusion in an Expanding Universe
330 TH I NTERNATIONAL C OSMIC R AY C ONFERENCE
Ultra High Energy Cosmic Rays Diffusion in an Expanding Universe
R. A
LOISIO , V. B EREZINSKY , , A. G AZIZOV , INFN - Laboratori Nazionali del Gran Sasso, I-67010 Assergi (AQ), Itlay Institute for Nuclear Research of RAS, 60th October Revolution prospect 7a, 117312 Moscow Russia Independence Avenue 68, BY-220072 Minsk, Republic of Belarus [email protected]
Abstract:
We study the solution of the diffusion equation for Ultra-High Energy Cosmic Rays in thegeneral case of an expanding universe, comparing it with the well known Syrovatsky solution obtainedin the more restrictive case of a static universe. The formal comparison of the two solutions with allparameters being fixed identically reveals an appreciable discrepancy. This discrepancy is less importantif in both models a different set of best-fit parameters is used.
Introduction
Diffusive propagation of Ultra High Energy Cos-mic Rays (UHECR) in extragalactic space has beenrecently studied by [1, 2, 8, 3] using the Syrovatskysolution (see [9]) of the diffusion equation. TheSyrovatsky solution is obtained under the restric-tive assumptions of time-independent diffusion co-efficient ( D = D ( E )) and energy losses of par-ticles ( dE/dt = b ( E )) . Recently two papers ap-peared [5, 6] solving the problem of the generaliza-tion of the diffusion equation (and its solution) inthe case of an expanding universe, i.e. in the caseof time dependent diffusion coefficient and energylosses. In these works an analytic solution of thediffusion equation in an expanding universe wasfound, valid in the general case of time-dependentdiffusion coefficient and energy losses, we will re-fer to this solution as the Berezinsky-Gazizov (BG)solution [5]. In the present paper we will compare,following the approach of [6], the spectra com-puted in the generalized case (BG solution) andthe spectra obtained with the Syrovatsky solutionas in the above cited papers. The diffusion equationfor ultra-relativistic particles propagating in an ex-panding universe from a single source, as obtainedin [5], reads ∂n∂t − b ( E, t ) ∂n∂E + 3 H ( t ) n − n ∂b ( E, t ) ∂E − D ( E, t ) a ( t ) ∇ x n = Q s ( E, t ) a ( t ) δ ( (cid:126)x − (cid:126)x g ) , (1)where the coordinate (cid:126)x corresponds to the comov-ing distance and a ( t ) is the scaling factor of theexpanding universe, n = n ( t, (cid:126)x, E ) is the parti-cle number density per unit energy in an expandingvolume, dE/dt = − b ( E, t ) describes the total en-ergy losses, which include adiabatic H ( t ) E as wellas interaction b int ( E, t ) energy losses. Q s ( E, t ) isthe generation function, that gives the number ofparticles generated by a single source at coordinate (cid:126)x g per unit energy and unit time.According to [5], the spherically-symmetric solu-tion of Eq. (1) is n ( x g , E ) = (cid:90) z g dz (cid:12)(cid:12)(cid:12)(cid:12) dtdz ( z ) (cid:12)(cid:12)(cid:12)(cid:12) Q s [ E g ( E, z ) , z ]exp[ − x g / λ ( E, z )][4 πλ ( E, z )] / dE g dE ( E, z ) , (2)where dtdz ( z ) = − H (1 + z ) (cid:112) Ω m (1 + z ) + Λ (3)with cosmological parameters Ω m = 0 . and Λ = 0 . , λ ( E, z ) = (cid:90) z dz (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) dt (cid:48) dz (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) D ( E g , z (cid:48) ) a ( z (cid:48) ) , (4) a r X i v : . [ a s t r o - ph ] J un HECR D
IFFUSION IN AN E XPANDING U NIVERSE dE g ( E, z ) dE = (1 + z )exp (cid:20)(cid:90) z dz (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) dt (cid:48) dz (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) ∂b int ( E g , z (cid:48) ) ∂E g (cid:21) . (5)The generation energy E g = E g ( E, z ) is the solu-tion of the energy-losses equation: dE g dt = − [ H ( t ) E g + b int ( E g , t )] (6)with initial condition E g ( E,
0) = E .In the present paper we will discuss the propa-gation of UHE protons in Intergalactic MagneticFields (IMF) following the approach used by [1, 2],in which the IMF is produced by a turbulent mag-netized plasma. In this picture the IMF is charac-terized by a coherent field B c on scales l > l c ,where l c is the basic scale of turbulence, and onsmaller scales l < l c the IMF is determined by its(assumed) turbulent spectrum. In our estimates wewill keep l c (cid:39) Mpc.The propagation of UHE protons in IMF is char-acterized by two basic scales: an energy scale E c that follows from the condition r L ( E c ) = l c ,with r L Larmor radius of the proton, and the dif-fusion length l d ( E ) , that is defined as the dis-tance at which a proton is scattered by 1 rad. Us-ing l d ( E ) the diffusion coefficient is defined as D ( E ) = cl d ( E ) / .We can easily identify two separate regimes in theparticle propagation in IMF, that follows from thecomparison of the two scale r L and l c . In the case r L ( E ) (cid:29) l c ( E (cid:29) E c ) the diffusion length canbe straightforwardly found from multiple scatter-ing as l d ( E ) = 1 . E B nG Mpc , (7)where E = E/ (10 eV) and B nG = B/ (1 nG).At E = E c , l d = l c . In the opposite scenario when r L < l c ( E < E c ) the diffusion length depends onthe IMF turbulent spectrum. In this case, following[6], we have assumed two different pictures: theKolmogorov spectrum l d ( E ) = l c ( E/E c ) / andthe Bohm spectrum l d ( E ) = l c ( E/E c ) .The strongest observational upper limit on the IMFin our picture is given by [7] as B c ≤ nG on theturbulence scale l c = 10 Mpc. In the calculationspresented here we assume a typical value of B c inthe range (0 . − nG and l c = 1 Mpc. In the present paper we will not perform a de-tailed discussion of the proton diffusion in the gen-eral case of an expanding universe, we will ad-dress this issue in a forthcoming paper [4], ourmain goal here is to perform a detailed compar-ison of the BG solution with the Syrovatsky solu-tion. As already discussed in [6], the difference be-tween these two solutions is substantial at energies E ≤ × eV, where the effect of the universeexpansion (in particular, of the CMB temperaturegrowth with red-shift) is not negligible. The highenergy tail of the UHECR spectrum is less affectedby the expansion of the universe, nevertheless it isinteresting to test the compatibility of the BG andSyrovatsky spectra at these energies where a sub-stantial agreement of the two is expected. Diffusive energy spectra of UHECR
In the present calculations we used a simplifieddescription of the IMF evolution with redshift,namely we parametrize the evolution of magneticconfiguration ( l c , B c ) as l c ( z ) = l c / (1 + z ) , B c ( z ) = B c (1 + z ) − m , where the term (1 + z ) describes the depletion ofthe magnetic field with time due to the magneticflux conservation and (1 + z ) − m due to MHD am-plification of the field. The critical energy E c ( z ) found from r L ( E ) = l c ( z ) is given by E c ( z ) = 0 . × (1 + z ) − m B c nGfor l c = 1 Mpc. The maximum redshift used in thecalculations is z max = 4 .Following [2], we have computed the diffuse fluxassuming a distribution of sources on a lattice withspacing d and an injection spectrum, equal for allsources, given by Q s ( E ) = q ( γ g − E (cid:18) EE (cid:19) − γ g , (8)where E is a normalizing energy (we used E =1 × eV) and q represents the source luminos-ity in protons with energies E ≥ E , L p ( ≥ E ) .The corresponding emissivity L = q /d , i.e. theenergy production rate in particles with E ≥ E TH I NTERNATIONAL C OSMIC R AY C ONFERENCE
Figure 1: Convergence of the diffusive solution tothe universal spectrum when the distance betweensources diminishes from 50 to 10 Mpc shown bynumbers on the curves.per unit comoving volume, will be used to fit theobserved spectrum by the calculated one.In figure 1 we test the BG solution with the helpof the diffusion theorem [1], which states that thediffusive solution converges to the universal spec-trum, i.e. the flux computed with rectilinear propa-gation for an homogeneous distribution of sources,in the limit d → , being d the lattice spacing. Fig-ure 1 clearly shows this convergence even in thecase of a strong magnetic field B c = 100 nG (andKolmogorov diffusion).In the case of a small distance between sourceand observer the diffusive approximation is notvalid. This result follows from a simple argument,the diffusive approximation works if the diffusivepropagation time r /D is larger than the time ofrectilinear propagation, r/c . This condition, us-ing D ∼ c l d , results in r ≥ l d . At distances r ≤ l d the rectilinear and diffusive trajectories inIMF differ by a little quantity and rectilinear prop-agation is a good approximation as far as spec-tra are concerned. The number densities of par-ticles Q/ πcr and Q/ πDr , calculated in recti-linear and diffusive approximations, respectively,are equal at r ∼ l d , where Q is the rate of parti-cle production. We calculated the number densitiesof protons n ( E, r ) numerically for both modes ofpropagations with energy losses of protons taken Figure 2: Equal parameter comparison of the BG(expanding universe) and Syrovatsky (static uni-verse) solutions, for γ g = 2 . , L = 2 . × erg/Mpc yr and d = 30 Mpc. The magneticfield configuration assumed is B c = 0 . nG and l c = 1 Mpc with different diffusion regimes as in-dicated on the plot.into account, and the transition is taken from theequality of the two spectra. We know that thisrecipe is somewhat rough and an interpolation be-tween the two regimes is required [2]. However,this interpolation is somewhat difficult because thediffusive regime sets up at distances not less thansix diffusion lengths l d . At distances l d ≤ r ≤ l d some intermediate regime of propagation is valid.When studied in numerical simulations (e.g. [10]),the calculated number density n ( E, r ) satisfies theparticle number conservation πr nu = Q , where u is the streaming velocity, while with a simple in-terpolated spectrum this condition is not fulfilled apriori. In the present paper we will not address thisproblem, that will be studied in a forthcoming pa-per [4], assuming the rough recipe for the transitionbetween diffusive and rectilinear regimes depictedabove. This computation scheme can produce ar-tificial features in the spectra, that are useful as amark of the transition between the two regimes.The direct comparison of the BG and Syrovatskysolutions of the diffusion equations is not possi-ble because they are embedded in different cos-mological environments. While the BG solution isvalid for an expanding universe, the Syrovatsky so-lution is valid only for a static universe. Using two HECR D
IFFUSION IN AN E XPANDING U NIVERSE
Figure 3: Best fit comparison of the BG (expand-ing universe) and Syrovatsky (static universe) so-lutions, for γ g = 2 . , L = 2 . × erg/Mpc yrand d = 30 Mpc. The magnetic field configura-tion assumed is B c = 0 . nG and l c = 1 Mpc withdifferent diffusion regimes as indicated on the plot.different cosmological models for these solutions,there are two ways of comparison. The first one isgiven by equal values of parameters in both solu-tions. In this method for BG solution we use thestandard cosmological parameters for an expand-ing universe H , Ω m , Λ and maximum red-shift z max up to which UHECR sources are still active,magnetic field configuration ( B c , l c ), separation d and UHECR parameters γ g and L , determined bythe best fit of the observed spectrum. For a staticuniverse with Syrovatsky solution we use the sameparameters H , d , ( B c , l c ), γ g and L . The max-imum red-shift in the BG solution is fixed by theage of the universe which equals to t = H − in the static universe ( z max = 1 . ). This for-mal method of comparison will be referred to as”equal-parameter method”. Physically a better jus-tified comparison is given by the best fit method, inwhich γ g and L are chosen as the best fit parame-ters for both solutions independently.The comparison of the two solutions is given inFigures 2 and 3 in the case of B c = 0 . nGand l c = 1 Mpc with a source spacing d = 30 Mpc. From these figures one can see a reason-ably good agreement between the Syrovatsky solu-tion, embedded in a static universe model, with theBG solution for an expanding universe at energies
E > × eV, at smaller energies appears anoticeable discrepancy between the two solutionsthat is natural and understandable as discussed inthe introduction. We conclude stating that, from aphysical point of view, the second method of com-parison is more meaningful and it gives a substan-tial agreement of the spectra obtained in the twocases. References [1] R. Aloisio and V. Berezinsky. Diffusive prop-agation of UHECR and the propagation theo-rem.
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