Markov chain Monte Carlo analyses of the flux ratios of B, Be and Li with the DRAGON2 code
P. De La Torre Luque, M. N. Mazziotta, F. Loparco, F. Gargano, D. Serini
PPrepared for submission to JCAP
Markov chain Monte Carlo analysesof the flux ratios of B, Be and Liwith the DRAGON2 code
P. De La Torre Luque a,b
M. N. Mazziotta a F. Loparco a,b
F. Gargano a D. Serini a,b a Istituto Nazionale di Fisica Nucleare, Sezione di Bari, via Orabona 4, I-70126 Bari, Italy b Dipartimento di Fisica “M. Merlin" dell’Università e del Politecnico di Bari, via Amendola173, I-70126 Bari, ItalyE-mail: [email protected], [email protected]
Abstract.
Understanding the transport of cosmic rays is challenging our models of propa-gation in the Galaxy. A good characterization of the secondary cosmic rays (B, Be, Li andsub-iron species) is crucial to constrain these models and exploit the precision of modern CRexperiments. In this work, a Markov chain Monte Carlo analysis has been implemented tofit the experimental flux ratios between B, Be and Li and their flux ratios to the primarycosmic-ray nuclei C and O. We have fitted the data using two different parametrizations forthe spallation cross sections. The uncertainties in the evaluation of the spectra of these sec-ondary cosmic rays, due to spallation cross sections, have been taken into account introducinga scale factor as a nuisance parameter in the fits, assuming that this uncertainty is mostlydue to the normalization of the cross sections parametrizations. We have also tested twodifferent kind of diffusion coefficients, which differ in the origin of the high energy hardening( ∼
200 GeV / n ) of cosmic rays. Additionally, two different approaches are used to scale thecross sections, one based on a combined analysis of all the species (“combined” analysis) andthe other reproducing the high energy spectra of the secondary-to-secondary flux ratios ofBe/B, Li/B, Li/Be (“corrected” analysis). This allows us to make a better comparison betweenthe propagation parameters inferred from both cross sections parametrizations. This novelanalysis has been successfully implemented using the numerical code DRAGON2 dedicatedto cosmic-ray propagation to reproduce the cosmic-ray nuclei data up to Z = 14 from theAMS-02 experiment. In general, it is found that the ratios of Li favor a harder spectral indexof the diffusion coefficient, but compatible with the other ratios inside the observed uncer-tainties. In addition, it is shown that, including these scale factors, the secondary-to-primaryflux ratios can be simultaneously reproduced, obtaining that the scale factor associated to thecross sections of boron production is the lowest one, whereas that associated to Li productionis the largest one. a r X i v : . [ a s t r o - ph . H E ] F e b ontents The high precision measurements provided by recent cosmic-ray missions as AMS-02 [1–3] andDAMPE [4, 5] together with the increasing quality and quantity of gamma-ray data, thanksto the Fermi Large Area Telescope [6, 7], HAWC [8–10] and H.E.S.S. [11–13], are requiringimproved models on particle acceleration and transport for their interpretation [14]. Thetransport of Galactic cosmic rays (CRs) is governed by their interactions with the magneto-hydrodynamical turbulence generated in the plasma of the interstellar medium (ISM). Thismechanism is responsible for the observed isotropy in their arrival directions at Earth [15, 16]and is usually described with a diffusive transport equation.Injection and acceleration of primary CRs is believed to take place mainly in supernovaremnants (SNRs) [17] (although also pulsars [18, 19] and OB stars [20] are likely CR sources)and it is explained, at least in general principles, by the diffusive shock acceleration (DSA)model [21–24]. In turn, the study of secondary CRs (i.e. those generated from the spallationreactions of primary CRs with gas nuclei in the ISM) provides crucial information about thepropagation of CRs; the amount of matter traversed (grammage) or path length [25, 26], thesize of the magnetised halo [27–29] or the spectrum of self-generated turbulence [30, 31] can,in fact, be inferred from the measured fluxes of secondary CR nuclei.In particular, the flux ratio of secondary to primary CRs is sensitive to the propagationparameters [32, 33], whereas it is mostly insensitive to the source spectrum of primary species.As a first order approximation, the flux ratio between the k -th species of secondary CRs andthe i -th species of primaries can be written as: J k J i ( R ) ≈ τ diff ( E ) Q k ( E ) J i ( R ) ∼ σ i → k ( R ) τ diff ( R ) ∝ σ i → k ( R ) /D ( R ) (1.1)where R is the rigidity of CR particles. In the previous equation we indicate with Q k = n ISM σ i → k J i the source term for secondary CRs, where σ i → k is the production cross sectionof the secondary CR species k from the interaction of the primary CR species i with the– 1 –as in the ISM (spallation reactions) and n ISM is the density of gas in the ISM (consistingof H:He = 1:0.1 and traces of heavier elements [34]). Finally, τ diff ∝ /D ( R ) is the meandiffusion time of CRs in the Galaxy, where D ( R ) is the diffusion coefficient used to describeCR propagation in the Galaxy.The spallation cross sections are one of the main ingredients in CR propagation com-putations, since they are crucial for evaluating the fluxes of different secondary CR species.However, due to the lack of experimental data and to the large number of interaction chan-nels involved in the production of each isotope, the uncertainties on the cross sections are stillquite large. These uncertainties propagate to the predicted fluxes of secondary CRs, whichare therefore poorly accurate. In order to reduce the impact of the lack of data, we make useof parametrizations and extrapolations of the spallation cross sections involved in the CR net-work [35]. This is due to the fact that the energy dependence of the cross sections is generallywell known and causes that most of the uncertainties are still present in the normalization ofthese parametrizations, as argued in ref. [36].Recently, some authors have proposed to use other secondary-to-primary flux ratios tocross check the validity of the cross sections parametrizations [37], concluding that this crosscheck will help reducing the systematic uncertainties related to the determination of thepropagation parameters. In this work, we have implemented two different and well knowncross sections parametrizations in a customised version [38] of the DRAGON2
CR propagationcode [39, 40]. We have then optimized the propagation parameters to reproduce the AMS-02experimental data on secondary-to-primary flux ratios of B, Be and Li to C and O as well asthe corresponding secondary-to-secondary flux ratios. In this way we study how spallationcross sections parametrizations affect the determination of the propagation parameters forthe different secondary CRs. This work is the follow-up of our recent paper [41], where weinvestigated how the choice of cross sections affects the predictions on the fluxes of the lightsecondary CRs B, Be and Li, and demonstrated that a simultaneous fit of the fluxes of thesesecondary CRs can be achieved above a few GeV/n taking into account the uncertainties inthe normalization of the cross sections used.There are very few works involving the ratios of secondary Li and Be nuclei (see [42,43]). This is mainly due to the fact that experimental fluxes of Li and Be were poorlymeasured before AMS-02 (for a comparison, see ref. [44]). In addition, the uncertainties onthe predicted fluxes of Li and Be [43] are larger than those on the flux of B. In this work, twodifferent strategies have been adopted to reduce the systematic uncertainties related to thethe spallation cross sections parametrizations using combined analyses of the secondary-to-primary flux ratios with a nuisance parameter to account for the cross sections normalizationuncertainties and the secondary-to-secondary flux ratios between B, Be and Li.This paper is organised as follows: in section 2 we illustrate the details of the Markovchain Monte Carlo (MCMC) algorithm and the analyses performed for the flux ratios of B,Be and Li to C and O (B/C, B/O, Be/C, Be/O, Li/C and Li/O). Then, in section 3, wereport and discuss the results on the determination of the diffusion parameters using what wecall “standard analyses” for the different diffusion coefficient and cross sections parametriza-tions adopted, taking into account the uncertainties related to the normalization of spallationcross sections for Be, B and Li production by means of the use of nuisance parameters. Insection 3.2, we employ another approach that allows us to compare the diffusion parame-ters found for different secondary CRs after tuning the normalization of the cross sections This version is public at https://github.com/tospines/Customised-DRAGON2_beta/ – 2 –arametrizations in a similar way as explored in [41]. Finally, in sect. 4 we discuss the mainconclusions of this work.
A spectral break (the rigidity where the CR spectrum changes its slope) at around
300 GV ,has been observed by many experiments in primary and secondary species [1, 45–48]. Al-though a quantitative analysis of B/C data favours a scenario with a break due to a diffusionmechanism (“diffusion hypothesis”) [42, 49], the scenario of a break in the injection spectra(“source hypothesis”) or a combination of both mechanisms (supported by the fact that differ-ent primary species point to different break positions [50]) can not be discarded. In this workwe have therefore tested two well known parametrizations of the diffusion coefficient, thatdiffer in the physical interpretation of the high energy part of the CR spectra, as explainedabove, and are given by the following equations: D = D β η (cid:18) RR (cid:19) δ Source hypothesis (2.1) D = D β η ( R/R ) δ (cid:104) R/R b ) ∆ δ/s (cid:105) s Diffusion hypothesis (2.2)where the reference rigidity is set to R = 4 GV .As in [41], where we refer to for full details about the set-up of the CR propagationsimulations, we solved the complete propagation equation using the DRAGON2 code for adiffusion-reacceleration model, adopting a 2D ISM gas distribution. In both diffusion modelswe use the values of the Alfvén velocity (that accounts for CR reacceleration), V A , the nor-malization of the diffusion coefficient D , the spectral index, δ , and the β (particle’s speedin speed of light units) exponent, η , as the parameters that we will include in our analyses,and we refer to them as diffusion parameters. In equation 2.2 there are three additional pa-rameters: the rigidity break ( R b ), the change in spectral index ( ∆ δ = δ − δ h , where δ h is thespectral index for rigidities above the break), and a smoothing parameter s , used to allow asoft transition around the hardening position. In our analyses, we fix these parameters to thevalues found in ref. [49], which were determined from a detailed analysis of the AMS-02 fluxesof protons and helium (with less experimental uncertainties than heavier nuclei at high ener-gies). These values are: ∆ δ = 0 . ± . , R b = (312 ±
31) GV and s = 0 . ± . . Thefact that these parameters are fixed from the primary fluxes allows us to avoid degeneracieswith the main diffusion parameters, leading to a more rigid determination.The mathematical model used to parametrize the injection spectra of primary nuclei is adoubly broken power law, with a high-energy break at
335 GeV for protons and at
200 GeV / n for heavier elements, and a low-energy break at / n for all primary CRs, when we treatthe diffusion parameters in the source hypothesis, whereas we use a simple broken powerlaw when we treat the diffusion hypothesis, with low-energy break in the injection spectra at / n .Basically, our strategy consists of an iterative procedure, which starts taking a set ofpropagation parameters ( D , V A , η and δ ) as first guess, then, a fit of the ratios of Li, Beand B to C and O to the AMS-02 data is performed to determine these parameters andtheir associated uncertainties. Before the fit, the injection spectra are adjusted by fitting– 3 –he AMS-02 data for C, N, O, Ne, Mg and Si. If the guessed diffusion parameters are verydifferent (out of 1 σ uncertainty) from those predicted by the fit, the injection spectra areadjusted again with the new predicted set of diffusion parameters (as primary spectra slightlydepend on them too) and the fit is performed again to obtain a more refined prediction on thediffusion parameters. Convergence is reached when the output diffusion parameters after thelast iteration are consistent (within 1 σ uncertainty) with those used in the previous iterationto adjust the injection spectra. We remark that in this analyses it is crucial the use of thecurrent AMS-02 data, including the recently published fluxes of Mg, Si and Ne [51], thusproviding accurate measurements of the fluxes of the main primary CR species producing B,Be and Li [1, 46, 47, 52–54].In order to take into account the solar modulation effect, we make use of the Force-fieldapproximation [55], which is characterized by the value of the Fisk potential. We set the Fiskpotential to the value found in [41], corresponding to φ = 0 .
61 GV (obtained from the NEWKneutron monitor experiment neutron monitor experiment (see [56, 57]) in combination withVoyager-1 data [58]) for the period between May, 2011 to May, 2016, that corresponds to theperiod of collected B, Be and Li data by the AMS-02 collaboration.In order to perform the fit of the ratios, a Markov chain Monte Carlo (MCMC) proce-dure, relying on Bayesian inference, has been implemented to get the probability distributionfunctions for a set of diffusion parameters ( D , V A , η and δ ) to describe the data (in this casethe AMS-02 data) and their confidence intervals. The technical details are presented in thenext section. Bayesian inference is used to get the posterior probability distribution function (PDFs), P ,for a set of diffusion parameters to explain the AMS-02 data. To evaluate it, the prior PDF, Π , and the likelihood, L , must be defined. These three terms are related by: P ( (cid:126)θ | (cid:126)D ) ∝ L ( (cid:126)D | (cid:126)θ )Π( (cid:126)θ ) (2.3)where (cid:126)θ = { θ , θ , ..., θ m } is the set of parameters, (cid:126)D is the data set used (CR flux ratiosmeasured by AMS-02) and P ( (cid:126)θ | (cid:126)D ) is the posterior PDF for the parameters.The posterior probability is calculated from equation 2.3 by a MCMC algorithm whichuses, instead of the classical Metropolis-Hastings algorithm, a modified version of the
Goodman& Weare algorithm [59]. This method is directly implemented in the emcee module of Python(see ref. [60] for full technical information).In this procedure, as it is computationally impossible to have a simulation for each of allpossible combinations of propagation parameters, a “grid” of simulations is built for ∼ regularly spaced combinations of parameters. For each of these combinations, we computethe flux ratios among different CR species and carry out interpolations for other combinationsof propagation parameters by using the python tool RegularGridInterpolator from the
Scipy module. In other words, we create a sort of four-dimensional “matrix” (a dimension for eachdiffusion parameter) and, to each of the points of the matrix, we associate a vector containingthe simulated flux ratios at different energies. The diffusion equation is integrated in 171points with equal spacing in a logarithmic scale, from
10 MeV / n to ∼
100 TeV / n .The errors coming from this interpolation were estimated comparing different simulatedfluxes of different nuclei with the interpolated fluxes, obtaining errors always much smallerthan 1%. Errors in the interpolation of fluxes in different energies are completely negligible. – 4 –he prior PDF is defined as a uniform distribution for all the parameters: Π( (cid:126)θ ) = (cid:81) i θ i,max − θ i,min for θ i,min < θ i < θ i,max elsewhere (2.4)The likelihood L is set to be a Gaussian function (as in many other studies, e.g. [61]or [62]): L ( (cid:126)D | (cid:126)θ ) = (cid:89) i,j (cid:113) πσ i,j exp (cid:34) − (Φ i,j ( (cid:126)θ ) − Φ i,j,data ) σ i,j (cid:35) (2.5)where Φ i,j ( (cid:126)θ ) is the flux ratio computed in the simulation, Φ i,j,data is the corresponding fluxratio measured by AMS-02 and σ i,j is its associated error for the i -th flux ratio and for the j -th energy bin. In this approach we do not take into account possible correlations amongdata, as the public data from the AMS-02 Collaboration do not include the full covariancematrix [42, 63].Then, in order to take into account cross sections uncertainties, we make use of a nuisanceparameter for each of the secondary CRs analysed (i.e. B, Be and Li), which enables arenormalization (or scaling) of the parametrization used for their production cross sections.The associated nuisance term is usually defined as (eq. 4 of ref. [42]): N X = (cid:88) i ( y i,X − ˆ y i,X ) σ i,X (2.6)where ˆ y i,X is the experimental cross section value at the i -th point of energy and σ i,X itsassociated experimental uncertainty referred to the secondary CR species X , that can be B,Be or Li. The term y i,X is defined as y i,X ≡ S X · ˆ y i,X , where S X is the nuisance parameterto be adjusted, referred to the species X, and independent of energy.As the nuisance parameters are independent of energy, eq. 2.6 can be rewritten as: N X = ( S X − (cid:88) i (cid:18) ˆ y i,X σ i,X (cid:19) = ( S X − N (cid:104) ˆ y/σ (cid:105) X (2.7)Here N stands for the number of experimental data points and (cid:104) σ/ ˆ y (cid:105) X is the average relativeerror for a given channel of production of X . Given the lack of cross sections experimentaldata, we use the reaction channels with C and O as projectiles. Then, the final expressionfor the nuisance term associated to a secondary CR species X is: N X = ( S X − (cid:88) X (cid:48) ( N C → X (cid:48) (cid:104) ˆ y/σ (cid:105) − C → X (cid:48) + N O → X (cid:48) (cid:104) ˆ y/σ (cid:105) − O → X (cid:48) ) (2.8)where X (cid:48) stands for all possible isotopes of the CR species X (for example, for B it means thesum for the isotopes B and B). The nuisance terms are therefore originated from Gaussiandistributions with mean 1 and sigma given by (cid:104) σ/ ˆ y (cid:105) , and can be interpreted as penalty termsthat prevent the scaling factors to be far from 1.With this strategy we are including the normalization uncertainty of the cross sectionsin the analysis, thus enabling the cross sections to be scaled up or down, with each secondary– 5 –eing more penalized as the scaling associated to its production cross section deviates from1. The effect of scaling all the production cross sections causes the fluxes to be scaled by thesame factor. Therefore the fluxes of the CR species X in the likelihood will be scaled of afactor S X .Nevertheless, when studying each ratio independently, the cross sections uncertainties aremostly absorbed by the normalization of the diffusion coefficient, as also reported in ref. [42],which means that S X will remain equal to 1 and no conclusions on these uncertainties can beobtained. Combining the secondary-to-primary ratios of different species allows accessing tothis information. In order to include the full correlation between cross sections and CR fluxes,the secondary-over-secondary ratios are included in this combined analysis. As we have seen,they are a perfect tool to study cross sections parametrizations with very small uncertaintyat high energies. Therefore including these ratios in the analysis can make the determinationof the diffusion coefficient more robust, providing information about the correlations betweenthe secondary species. In this way, to avoid biasing the results (since we do not include thehalo size), we use the likelihood function defined using the secondary-over-secondary ratiosabove
30 GeV .In conclusion, the logarithm of the likelihood function used to carry out the combinedanalysis of Li, B and Be is given by: ln L T otal = Li,Be,B/ ( C,O,Li,Be,B ) (cid:88) F ln( L ( F )) + B,Be,Li (cid:88) X N X (2.9)where F indicates the flux ratios (six secondary-over-primary ratios and three secondary-over-secondary ratios) and X indicates the cross sections channels included (those coming from C and O projectiles only).
The algorithm explained above has been applied to find the optimal diffusion parametersthat best fit the spectra of each of the ratios of B, Be and Li to C and O (B/C, B/O, Be/C,Be/O, Li/C, Li/O) independently, for both diffusion models (eqs. 2.1 and 2.2). In section 3.1,we report the results of this analysis for each of these fluxes independently. Additionally,in this section we report the results for a combined analysis of the spectra of the secondaryCRs B, Be and Li. In this analysis we combine the studied secondary-to-primary flux ratiosand the secondary-over-secondary ratios Be/B, Li/B and Li/Be (at
E >
30 GeV , in orderto avoid the uncertainties at low energy, largely studied in [41]) as well as the nuisanceparameters accounting for uncertainties related to the normalization of the cross sectionsparametrizations.Then, in section 3.2, we employ another analysis in order to consider, in a differentway, the uncertainties in the normalization of the cross sections parametrizations and obtainthe best-fit diffusion parameters from the secondary-to-primary flux ratios. This strategyconsist of re-scaling the cross sections of production of B, Be and Li by reproducing thehigh energy part of the spectra of the secondary-to-secondary flux ratios Be/B, Li/B andLi/Be, provided that this part of their spectra is intimately related to their cross sectionsand rather independent of the diffusion parameters used, as discussed in [41]. After re-scalingthe spallation cross sections used, we apply the MCMC analysis to each of the secondary-to-primary ratios. In this way, we can compare individually the parameters inferred from each– 6 –econdary CR after “correcting” their cross sections. Although similar analysis have beenpresented in the past based on refitting the cross sections parametrizations using updatedcross sections measurements (see, e.g., ref. [64]), the novelty here consists of refitting theparametrizations to a combination of available cross sections data and the high energy dataof secondary-to-secondary flux ratios. In fact, this strategy has been successfully implementedfor the study of antiprotons in ref. [65], where the author shows that, considering the scalefactors calculated in this way, the antiproton-to-proton spectrum from the AMS-02 experimentcan be reproduced with a high precision.These analyses are carried out using two different spallation cross sections parametriza-tions: the
DRAGON2 cross sections [40, 41, 66] and the GALPROP [67, 68] cross sectionsparametrizations. It must be stressed here that we used slightly different values of the halosize for the simulations with the two cross sections parametrizations. In particular, we usedthe values of .
76 kpc and .
93 kpc for the
DRAGON2 and
GALPROP parametrizations, respec-tively, as found in section 5 of [41].
The results of the standard analyses are summarized in Figure 1. The posterior PDF of eachparameter is usually very close to a Gaussian distribution function, with the exception of the V A parameter in the Be/C and Be/O ratios and, in a few cases, of the η parameter in the GALPROP cross sections, whose PDFs look more similar to an inverse log-normal distribution.An explicit summary of the median values and of the ranges of propagation parameterswithin the 68 and 95% of their PDFs is given in form of tables in the appendix A for eachof the cross sections and diffusion coefficient parametrizations. In addition, the corner plotswith the PDFs for the combined analyses are also shown in the appendix, for both the sourceand diffusion hypotheses.Very similar results (always inside 2 σ ) are found for the propagation parameters ana-lyzed in both hypotheses for the CR hardening. In general, the diffusion hypothesis favorssmaller values of D , V A and η and larger values of δ . This is something expected, sincethe slope (controlled by the δ parameter) in the source hypothesis tends to be smaller, tocompensate the hardening observed in CR spectra at the highest energies. Therefore, if δ islarger, D must be smaller to balance this change (see the contour plots in appendix A wherethe correlations between parameters are explicitly shown). In the following, some importantremarks are discussed for these results. In general, the relative uncertainties in the determination of propagation parameters are largerin the η and V A parameters, mainly due to their degeneracy and the need of more data pointsat the low energy region.The propagation parameters obtained for each pair of ratios involving the same secondary(e.g. B/C and B/O) are always compatible with each other. Small discrepancies can arise,mainly due to the production of secondary carbon, which is important at low energies.Each of the ratios (also in the combined analysis and with both cross sections sets used)favors a negative value of η , theoretically motivated by dissipation of MHD waves [69] orappearance of non-resonant interactions at low energies [70]. The Be ratios show clearly Publicly available as ASCII files at https://raw.githubusercontent.com/cosmicrays/DRAGON2-Beta_version/master/data/crxsecs_fragmentation_Evoli2019_cumulative_modified.dat Available as ASCII files at https://dmaurin.gitlab.io/USINE/input_xs_data.html – 7 – C log ( D [10 cm s ] H [ kpc ] ) V A [km/s] BOBeCBeOLiCLiO C o m b i n e d Source DRAGON2Diff. DRAGON2Source GALPROPDiff. GALPROP
Box-plot propagation parameters
Figure 1 . Box-plots representing the PDFs resulting from the MCMC algorithm used to find thebest propagation parameters to describe AMS-02 results with diffusion coefficient parametrizationsof eq. 2.1, and with the
DRAGON2 and
GALPROP cross sections. The orange solid lines within each boxindicate the medians of the corresponding PDFs, while the dashed cyan lines indicate the mean values;the colored boxes correspond to the interquartile ranges (IQR) from the th (Q1) to the th (Q3)percentile; finally, the black lines correspond to the range [ Q − . IQR, Q . IQR ] which, in caseof gaussian PDFs contains the . of the probability. smaller values of this parameter for every cross sections set used, but this seems to be due tothe halo size used in the simulations. The degeneracy between these two parameters makesone compensate the defects of the other, thus requiring much lower η values.Interestingly, the value of δ is usually around 0.42-0.45 for the B and Be ratios, while alower value is favored for the Li ratios (between 0.36 and 0.40). The fact that the Li spectrumis harder at high energies (i.e. smaller δ value) than those of other species was pointed out inref. [71] and lead the authors to suggest the presence of primary Li. Nevertheless, providedthat the uncertainties related to the spallation cross sections for Li production are speciallylarge, this is not clear at all (and a proof is the large difference in the inferred D valuebetween both cross sections parametrizations for the Li flux ratios).The values of V A obtained are smaller than
30 km / s for the B ratios ( ∼
26 km / s ), whilethe Li ratios yields larger values (around
35 km / s ). On the other hand, the V A values for Be– 8 – Energy (GeV/n) F l u x r a t i o Source hypothesisDiffusion hypothesisB/C AMS-02 dataBe/C AMS-02 dataLi/C AMS-02 data
LiBeB/C spectra - DRAGON2 Energy (GeV/n) F l u x r a t i o Source hypothesisDiffusion hypothesisB/O AMS-02 dataBe/O AMS-02 dataLi/O AMS-02 data
LiBeB/O spectra - DRAGON2
Figure 2 . Simulated secondary-over-primary ratios with the propagation parameters determinedindependently in the MCMC analysis for the
DRAGON2 cross sections parametrizations and for bothhypothesis of the CR fluxes hardening. The parametrizations of the diffusion coefficients allow perfectreproduction of the shapes of the ratios. Fits with same quality are achieved for the
GALPROP crosssections too. ratios are much smaller, usually compatible with V A = 0 . This is due to the degeneracy withthe halo size employed in the simulations. Therefore, excluding the Be ratios, an averagevalue of V A ∼
30 km / s is found, which seems to be in agreement with the expected values(see refs. [72–74]).Finally, as we see from Figure 2, the secondary-over-primary flux ratios are perfectly re-produced when making each fit individually. This means that the degrees of freedom availablein the parametrizations of the diffusion coefficient are enough to suitably match observations,although there is a clear improvement on the predictions when using the break in the diffusion(eq 2.2). On the other hand, it should be mentioned that the level of uncertainty appreciablychanges when taking into account correlated errors from the AMS-02 experiment. In general, the different cross sections give very similar predictions. One of the main differ-ences are the η and V A values obtained for the Be ratios with the DRAGON2 cross sections,which are larger (and more consistent with the Li and B ratios) than those obtained withother cross sections. However, this might be also related to the degeneracy with the halo sizevalue used and suggests that the determination of the halo size performed with the
DRAGON2 cross sections seems to be more accurate than that performed with the
GALPROP cross sections.The most meaningful difference is that, as discussed above, for both cross sections sets,the value of δ determined by the Li ratios is significantly smaller (often more than σ ) than theone determined by the other ratios. This points to the fact that, in both, parametrizations,the overall slope of the cross sections for Li production at high energies tends to be softerthan it should be. Nonetheless, the best fit δ value for the Li ratios with the DRAGON2 crosssections is closer to the value inferred from the Be and B ratios, which might indicate thatupdating cross sections measurements can help to solve this issue.Finally, it is worth noticing that the D values obtained are slightly different in theindependent analyses of different secondary CRs, which mostly depends on the normalizationof the spallation cross sections parametrizations in each channel. This is prevented by adding– 9 – ross sections scale factors * $ / 3 5 2 3 ' 5 $ * 2 1 Figure 3 . Corner plot of the scaling (nuisance) parameters determined by the MCMC combinedanalysis. A dashed line indicating scaling = 1 is added as reference for each of the panels. an overall scaling factor ( S ) on the cross sections in the combined analysis. The relative dif-ferences of D obtained and the secondary-over-secondary ratios for each of the cross sectionssets reveal the relative degree of scaling on the production cross sections needed for each ofthe secondary CRs. These scaling factors, determined as nuisance parameters, are shown inFigure 3 and discussed in detail down below and in section 3.2. The most remarkable fact seems to be that the values inferred from the combined analysis tendto compensate the defects of the parametrization describing the Li cross sections, leading tosmaller δ values in both parametrizations and larger V A values for the GALPROP cross sections,which suggests that the most constraining ratios are those involving Li. Only the
DRAGON2 cross sections show a median value of δ higher than 0.40, but also in this case it is belowthe median for B and Be ratios. This is likely related to the lack of experimental data oncross sections of Li production (implying a poorer parametrization of the Li production crosssections) and may cause an important bias in our determination of the propagation parameters(and also in the determination of the nuisance parameters) in the combined analysis. Ingeneral, it seems that the parameters found in the combined and independent analyses for DRAGON2 cross sections are more compatible than with the
GALPROP cross sections.The results of the combined analysis are shown in Figs. 4 and 5 for the
DRAGON2 and
GALPROP cross sections, respectively. As we see, the results of the combined analysis with the
DRAGON2 cross sections show an almost perfect fit of all the ratios, with the largest discrepan-– 10 – F l u x r a t i o &