Flux predictions in the transition region incorporating the effects from propagation of cosmic rays in the Galactic magnetic field

. Galactic cosmic rays (GCRs) and (anisotropically injected) extragalactic cosmic rays (EGCRs) are both a ff ected by the Galactic magentic field (GMF) on their voyage to Earth at energies pertaining to the transition from GCRs to EGCRs, such that their flux, composition and arrival directions are modified. GCRs increasingly leak from the Galaxy with rising energy, leading to a flux suppression. The flux modification imposed on EGCRs is more complex, but may exhibit (subtle) spectral breaks depending on the direction and nature of the injected anisotropy. Using a full Monte Carlo approach with CRPropa and making realistic and minimal assumptions about the injected GCR and EGCR fluxes, we derive a prediction of the total all-particle flux in the transition region. We find that it cannot account for the flux measured by various cosmic ray experiments in this energy range. This calls for the need of an additional component to the flux in the transition region.


Introduction
The energy at which the dominant contribution to the cosmic ray (CR) flux transitions from Galactic to extragalactic cosmic rays (GCRs and EGCRs, respectively) remains an open question to this day. Numerous models have been shown to account for certain or most features of CR observables, such as the flux, composition and anisotropy (e.g. [1][2][3]). Two features of the energy spectrum, the knee 1 [4] and the ankle 2 [5], are of particular importance, as they set the upper and lower bound of the energy range within which this transition is believed to occur for virtually all models 3 ([1-3] mentioned above). This energy range will henceforth be referred to as the transition region (or shin when referring to the energy spectrum [4]). The rather wide range of proposed origins of these features, however, is exemplary of the substantial uncertainties that still govern our understanding of the processes included in the creation & acceleration, propagation (including deflections in magnetic fields and interactions), and even the detection & identification of CRs.
Major progress has been made on all fronts, however, constraining models, and even strongly disfavouring certain, once popular, ones [1]. On top of that, the development of improved numerical methods and increased computing power have lead to unequalled precision in the description of various processes the CR undergoes from source to observer. One aspect that is particularly relevant for the transition region is the propagation in the Galactic magnetic field (GMF).
While the GMF itself is still poorly constrained, recent models have, for the first time, been fit to observational data, that additionally employ a more complex topology and structure [6]. Especially noteworthy is the separation of the GMF into several components, which are each relevant for different propagation regimes. They include a coherent field which is responsible for large-scale deflections in the ballistic regime, as well as a random component that leads to turbulent motion in the diffusive regime. Propagation studies with such GMF models are particularly useful in the energy range pertaining to transition region, as CR propagation changes from diffusive to ballistic within this range.
This work is based on results from simulations studying the propagation of GCRs and EGCRs in the GMF [7,8]. Therein, significant features were found for CR observables, such as the flux, composition and anisotropy, that could be traced back to the change in propagation regimes from diffusive to ballistic propagation. The objective here is to make a flux prediction based on these features that can be compared to data (i.e. the energy spectrum). The outline is as follows: In section 2, we will outline the propagation effects and resulting spectral features for both GCRs and EGCRs. In section 3, further considerations and assumptions that are needed to make a flux prediction are discussed. In section 4, the resulting total (i.e. involving both GCR and EGCR) all-particle (i.e. consisting of a mixed composition CR nuclei) energy spectrum is pre-sented and discussed. Finally, conclusions are drawn and an outlook is given in section 5.

Propagation of GCRs and EGCRs in the GMF in the transition region
The objective of this work is to derive a realistic prediction for the total 4 all-particle 5 CR flux in the transition region, by extrapolating two individual contributions, one tuned to the flux up to the knee (representing GCRs), the other to the flux beyond the ankle (representing EGCRs). The defining feature of this prediction is its incorporation of propagation effects that the GMF imposes on GCRs and well as EGCRs, based on [7,8]. With this prediction, we wish to make inferences about the prospects of describing the shin without invoking any additional contributions. For the latter objective, an outline of said studies is in order. On top of giving the reader an overview of the observed features in CR observables, particularly the flux, their framing in the context of the change in propagation regimes is crucial for drawing valid conclusions from the comparison of our prediction within the CR spectrum.

Summary of earlier simulation studies
Our propagation studies were performed with the established and commonly used Monte-Carlo-based software framework for CR propagation CRPropa3 [9]. We performed forward-tracking studies, where we simulated in the order of 10 9 particles with rigidities ranging from 10 16 -10 20 V, by tracking them individually through the GMF, modeled by the JF12 field 6 [10]. For all simulations, particles are injected with a flat injection spectrum 7 R −1 . The range of injection positions (and directions) that define the source varies depending on the intended class of CRs (i.e. of Galactic or extragalactic origin). The sources of GCRs all lie in the Galactic plane (GP) 8 . The distances of EGCR sources to the Galaxy exceed Galactic distances by a factor of 100 or more, meaning that the bulk of the voyage of EGCRs to Earth takes place outside our region of interest. On the one hand, this means that we can neglect this part of their propagation in the context of our studies, such that their points of injection are located at the edge of the volume that defines the Galaxy 9 . On the other 4 In this work, this term always refers to the combination of both GCR and EGCR contributions. 5 We assume a mixed composition of CRs, i.e. contributions from several different CR species (i.e. nuclei), across the entire measured energy range. Nuclear fractions vary with energy owed to both their differing charges (that affect their rigidities and, hence, nature of propagation) and masses (that affect they interaction cross sections). All contributions cumulatively give rise to the total flux. 6 The JF12 field is to-date the default GMF model, owing its prominence to the fact that it results from a fit to actual data of relevant observables. In addition, it is made up of both a regular (or ordered), as well as two random components (striated and turbulent), making it particularly well-suited for our purposes. 7 This choice is primarily owed to the fact that arising features in CR observables are best visually discernible this way. 8 The GP is parameterised by a disk centered at Galactic latitude b = 0 with a radius of 20 kpc 9 Following the default volume within which the JF12 field is defined, this constitutes a 20 kpc shell centered around the Galactic centre.
hand, such a projection of the source to the Galactic shell requires an adaptation of the injection direction range, depending on the desired properties of the sources. One goal of our studies was to be able to easily rescale our results across as wide a range of parameters as possible. As a result, the distribution of sources was to mimic isotropy as closely as possible. This entails a continuous and uniform distribution of sources, as well as an isotropic injection direction distribution 10 . Particles are tracked until they satisfy certain break conditions (i.e. diverging propagation times, or leaving the propagation volume), or if they enter or leave the boundary of some specific predetermined region that defines the observer. Depending on the nature of our analysis, two classes of observers are defined. In the context of illustrating the propagation effects arising from the change in propagation regimes, the GP 11 , which plays an instrumental role in giving rise to said effects, constitutes the observer. For determining the effects this has on CR observables, the observer represents Earth 12 .
The theoretical starting point of our studies is the fact that the transition region also marks a shift in propagation regimes from diffusive to ballistic for CRs propagating within the GMF. This shift can best be illustrated using the particle's gyro radius r g = R B . R refers to the CR's rigidity, and is defined as the ratio between its total energy E and electric charge q(= Z e, Z constituting its atomic number) 13 . B denotes the strength of (the perpendicular component of) the magnetic field. The typical scale of B for the GMF (particularly in the GP region) is ≈ 3 µG. Translating the energy range of the transition region into its corresponding rigidity range, the lower bound is set by the heaviest CR nucleus, iron, at knee energies, whose gyro radius is r g ≲ 1 pc. The size of the Galaxy is in the order tens of kpc. Hence, the particle can undergo a very large number of gyrations, and in the presence of strong turbulent fields (which the GMF has), the motion is chaotic. The upper end of the rigidity range is set by the proton energy beyond the ankle, yielding r g ∼ 10 kpc, i.e. of similar scale as the size of the Galaxy. Propagation is no longer turbulent in this case, as the particle escapes the volume before it can undergo a large number of gyrations. All of these facts are illustrated in Fig. 1, depicting the gyro radius r g as a function of rigidity R for various magnetic field strengths B, along with relevant length scales (horizontal lines). 10 For EGCRs, with the Galactic edge being a shell, the injection directions follow a Lambertian distribution to account for the aforementioned projection of the sources to the shell. 11 The GP as observer is to be modified w.r.t to its parametrisation as a source. On the one hand, it has a non-zero width (100 pc) to account for the condition that the observer be a volume. On the other hand, in the study of EGCRs, its radius is reduced to 19 kpc to eliminate an artefact that occurs due to the discontinuous cut-off of the GMF to zero at the edge of the Galaxy (which is most pronounced at the GP). 12 Earth is parametrised via a shell centered at Earth's position in the Galaxy. Several shells of varying radius, ranging from 5 pc to 1 kpc, are included in each simulation, to be sensitive to artefacts arising from their non-zero size. 13 Technically, this definition, R = E q , is only valid in the ultrarelativistic limit, where γ = E m c 2 ≫ 1, and the particle's total energy is essentially equal to its kinetic energy T .  Figure 1: Gyro radius r g as a function of rigidity R for various magnetic field strengths B (indicated in legend), along with relevant length scales (horizontal lines).
The dependence of r g on the rigidity R, rather than just E, exemplifies one way in which the nature of propagation varies for mono-energetic CRs across nuclear species. It also shows that inferences on the deflections in a magnetic field of one CR species are transferable to any other species, by scaling the energy via the ratio of their charges to match the respective rigidity. In fact, we made use of this fact in our studies and performed our simulations to the propagation of protons 14 .
The shift from highly turbulent/chaotic motion (diffusive propagation) to largely ordered deflections of a few degrees (ballistic propagation) leads to large-scale propagation effects concerning the evolution the spatial CR distribution. A decisive role here is played by the GP, as it encompasses the region with the strongest fields, while at the same time being confined to a narrow volume of the Galaxy, particularly in north/south direction (≲ 1 kpc height versus ≈ 20 kpc radius [10]). GCRs that originate therein, will experience confinement within the GP at lowest rigidities with propagation being diffusive. EGCRs approaching the GP from outside the Galaxy will observe a large positive gradient in the field strength, and consequently experience strongly increasing deflections; once they start pointing away from the GP, they observe the same gradient in reverse, and their deflection quickly weakens. Effectively, they are reflected, or shielded, from the GP. If they do manage to penetrate into the GP, they are trapped therein, i.e. experiencing similar confinement as GCRs do. With other words, EGCRs experience two counteracting effects.
The aforementioned effects are qualitatively illustrated in Fig. 2, which show the evolution of the number of GCRs (left, a) and EGCRs (right, b) residing in the GP for various rigidity bins, which are colour-coded. Confinement is discernible via the "depletion times" (the intersection with 14 This is the most straightforward choice of nuclear species, as the determination of the CR rigidity from its energy is the simplest with Z = 1 (i.e. rigidity and energy are identical, save the differing dimension between the two quantities of [R] [E] = 1/e) the x-axis where all CRs have escaped), and shielding via the initial (EG)CR count (the intersection of the lines with the y-axis). Both effects decrease as rigidity increases. The resulting effect on the CR flux depends on their origin. A decreasing confinement as CR rigidity increases, i.e. a quicker escape from the GP, results in a decreasing probability of reaching a designated region of the GP, including Earth. Hence, the flux of GCRs, which only experience confinement due to the GMF, decreases with rigidity. This can be seen in their rigidity spectrum depicted in Fig. 3. The spectrum flattens at highest rigidities, indicating that propagation becomes ballistic and the probability of reaching Earth no longer depends on rigidity, and is determined by the geometric fractions of the observer w.r.t the source. For EGCRs, the same considerations apply for the EGCRs confined in the GP. Since shielding leads to a lower probability of reaching Earth, the combination of both effects determines the resulting flux. In fact, their contributions cancel exactly, meaning that the flux is conserved across the entire rigidity range. This result is a consequence of Liouville's theorem [11,12], which our studies effectively serve to confirm.
While the flux of isotropically injected EGCRs is unaffected by the GMF, modifications to the spectrum may occur in the case of anisotropic injection. Since the EGCR flux is not necessarily isotropic upon arrival at the Galaxy, our propagation studies included the injection of anisotropic EGCR fluxes. Two scenarios were explored, one representing the largest-scale, the other the smallestscale anisotropy: a dipole and a point-source. With the help of the Galactic lensing scheme [13,14] and using custom lenses 15 , several iterations of these scenarios were studied, each varying the phase (of the dipole) or direction (of the point source), respectively. For both scenarios, the injected EGCR flux is modified by the GMF. In the case of an injected dipole, this takes the form of a subtle change in the spectral slope. For point sources, the flux modification is significantly more complex, ranging from changes in the slope, similar to the dipole scenario, to the occurrence of spectral breaks, depending on the source direction. An exemplary plot of such a break is given in Fig. 4, showing the modified rigidity spectrum for an injection from the direction of Centaurus A. To understand this complexity, consider that modifications to the shape of an injected anisotropic EGCR flux come about due to rigidity-dependent changes in the magnetic transparency of the GMF. The transparency of the GMF regarding some injection direction, which we henceforth call "directional magnetic transparency", is determined by the trajectory that the CR undergoes. Due to the turbulent fields, small deviations from that trajectory quickly diverge, such that the CR does not hit Earth in the first place, making the GMF opaque concerning said injection direction. That the directional transparency exhibits such strong rigidity- 15 The custom lenses were generated from the inverted injection and arrival directions of 10 8 anti-protons that are injected isotropically from Earth and backtracked to the edge of the Galactic shell. Timereversibility of trajectories guarantees that these directions represent the arrival and injection directions of isotropically injected protons into the Galaxy.     dependence is reflective of the complex structure of the GMF.

Adaptation towards a concrete description of propagation effects
The key goal of this work is to ascertain in what way the aformentioned propagation effects affect the CR flux in the transition region. To this end, the source distribution for both GCRs and EGCRs must be specified, such that we obtain a concrete description of the flux modifications that the GMF imposes on both classes of CRs. For GCRs, this implies a re-weighting of the uniform source distribution used in our studies to a more realistic distribution. Since supernova remnants (SNRs) are considered to be the prime source candidate of GCRs, performing our re-weighting according to their (galactocentric) distribution is a natural choice. Following [15], the re-weighting function becomes: The distribution of EGCR sources follows the positions of the ten brightest galaxies of the "starburst catalogue", a catalogue of galaxies the Pierre Auger Collaboration compiled for analysis of possible source candidates of ultra-high energy CRs (or UHECRs) [16]. We distribute our sources according to the positions of the 10 brightest galaxies from said catalogue, with their relative contributions matching the relative brightness assigned in the analysis. On top of that, we added a distance-and rigiditydependent smearing ∆ ∝ d R to the flux to account for the diffusive propagation in the intergalactic magnetic fields. A skymap of the injected flux for a fixed (arbitrary) rigidity in given in Fig. 5.  Figure 5: Skymap of the injected EGCR flux following the distribution of the 10 brightest galaxies from the starburst catalogue, with their relative contributions matched to their given brightness, as well as rigidity-and distancedependent smearing.
The resulting rigidity spectra, indicating the concreting flux modifications imposed by the GMF, are shown in Fig. 6 for both GCRs (left, a), and EGCRs (right, b). Besides larger fluctuations, due to larger contributions of further-away sources, the radially dependent SNR distribution hardly affects the spectral shape of GCR flux. The EGCR spectrum takes on a rather complex structure. At lowest and highest rigidities, the spectrum is flat, i.e. it has the same spectral index as the injected spectrum, but undergoes both a hardening, occurring at roughly 10 18 V, followed by a steepening at about 10 18.5 V. Assuming the accuracy of the choice of distribution of EGCR sources and of the JF12 field to model the GMF, this indicates that the ankle may partly arise from propagation effects. While being an intriguing result, the determination of both the EGCR sources and the GMF is subject to large uncertainties, so this assumption is rather spurious. Nevertheless, we have obtained a quantitative description of the rigidity-dependent modifications that the GMF introduces to the flux of GCRs and EGCRs originating from a fixed, but at the same time realistic and generic, distribution of sources. These shall be captured in the quantity m i (R), where i = GCR, EGCR. This lays the foundation for obtaining the final flux prediction.

Towards a flux prediction
The propagation studies outlined in section 2 show that the fluxes of GCRs and (anisotropically injected) EGCRs are modified by the GMF, the origin of which can be traced to a shift in propagation regimes. By specifying the source distributions, we have arrived at a full quantitative prediction of the modifications that the GMF imposes on the flux of GCRs and EGCRs as a function rigidity.
For a complete flux prediction, further parameters that impact the CR flux have to be considered. The ones we in-cluded all specify source properties: an intrinsic rigiditydependent cut-off in the source output (e.g. via a maximum acceleration rigidity of the sources) which we denote by c i (R); the fraction of each injected CR species, signified by f j i ; the injected spectral slope of each nucleus γ j i . As for the flux modification m i , i = GCR, EGCR, and j denotes the individual nuclei contributing to the flux (specified below). These quantities are tuned in order to reproduce the spectral data outside of the transition region, with the GCR flux to match the energy range around the knee, and the EGCR flux to match the energy range beyond the ankle. The two individual energy spectra J i (E) are then combined to obtain the total, combined all-particle energy spectrum J(E). This spectrum extrapolated to the transition region is our flux prediction that we set out for, and that can be compared with spectral data.
The motivation to introduce a rigidity-dependent cutoff c i (R) to the spectrum is similar for both classes of CRs, namely to account for a specific spectral feature. A large consensus exists that GCR flux is tightly intertwined with the knee of the spectrum, though its origin is still disputed. One popular description of the knee is that it marks the maximum rigidity that Galactic sources, e.g. the shock front of (SNRs) can accelerate CRs to. Since the propagation-related flux suppression of GCRs occurs for rigidities exceeding the energy of the knee for protons, including an additional cut-off of this kind is necessary to be able to describe the knee feature in the first place. A cutoff to the EGCR spectrum is invoked to recover the highenergy cut-off in the energy spectrum at highest energies. Its origin is also not settled, though evidence that the composition is becoming heavier at highest energies points to a underlying rigidity-dependent process, like a maximum acceleration rigidity of cosmic accelerators [17]. Since the knee exhibits a gradual change of the spectral index, as opposed to a hard break for the high-energy cut-off, we introduced a smooth cut-off (c GCR (R) = exp(1 − R/R GCR cut ) to the GCR spectrum, and a broken cut-off (c EGCR (R) = exp(R/R EGCR cut ) for R > R EGCR cut , c EGCR (R) = 1 otherwise) to the EGCR spectrum.
While the position of the break can be determined via the aforementioned cut-offs, the shape that the spectrum takes immediately following said break is primarily determined by the injected composition 16 . Thus, the spectral trend directly following the knee and the high-energy cut-off, respectively, serves to determine the injected compositions of the GCR and EGCR sources. We modeled the composition via a four-component approach where the flux is made up of the nuclei Hydrogen ( j = 1, Z = 1, A = 1), Helium ( j = 2, Z = 2, A = 4), Oxygen ( j = 3, Z = 8, A = 16) and Iron ( j = 4, Z = 26, A = 56), each representing individual mass groups (protons, light, intermediate, heavy) [18]. This categorisation has the benefit of keeping the composition simple, yet approximately capturing the main features CR transport that affect the composition 17 . For both GCRs and EGCRs, all mass groups 16 Since the cut-off is rigidity-dependent, its onset is shifted further to the right in energy, the heavier the injected composition is. 17 The most noteworthy of these is that interactions and deflections are sensitive to different properties of the nucleus and hence modify the flux  are injected with the same spectral index and are set equal to the slope of the measured spectrum prior to the knee, i.e. γ ≡ γ j i = −2.7 ∀i, j [4]. The parameters we alter to reproduce the aforementioned regions of the spectrum are the fractions of each mass group, by which their spectra are scaled.
All in all, the total, combined energy spectrum that we wish to match, and later compare, with spectral data has the following functional form: J 0 is a normalisation factor, and f i j , R i cut are free variables, which we tune to match the spectrum up to and around the knee for J GCR and the region beyond the ankle for J EGCR .

Results and discussion
The individual energy spectra J GCR and J EGCR are shown in Fig. 7. The spectra of the individual components J i, j are also depicted. The rather smooth change in slope around the knee is partly reproduced via roughly equal fractions of light, intermediate and heavy nuclei, which cumulatively roughly match the proton fraction. Matching the shape of of a mixed composition of CRs differently. While deflections depend on rigidity and scale with the charge number Z, interactions depend on the energy per nucleon, thereby scaling with the mass number A. Since A ≈ Z/2 for most nuclei except Hydrogen, interaction-related modifications to the spectrum that occur at the same energy as some rigidity-dependent modification for protons will be shifted to the right in energy by a factor of two compared to the latter. In the standard log-log-representation of the spectrum, this shift is most striking for Helium, where we have a fourfold increase of A compared to protons, whereas a two-fold increase is less pronounced for the intermediate to heavy mass range whose shift in energy with Z is already sizeable. the spectrum above the ankle and particularly the slope of the high-energy cut-off requires a dominant light component, with a small but significant proton fraction. Intermediate to heavy mass groups influence the spectral shape only at the highest energies of the cut-off, where the flux is the lowest, and the spectral data carries the highest degree of uncertainty. Hence, their contribution is negligible when describing the CR flux in the transition region. Such fractions are reasonably in line with other composition studies, given the qualitative nature of their determination.
The combined spectrum, overlain on the measured energy spectrum, is depicted in Fig. 8. As already suggested in the individual spectral plots, the fluxes of both GCRs and EGCRs, do not follow the trend of the measured flux into the transition region. Each individual spectrum undershoots the measured spectrum from their respective side, such that their combined contribution is consistently too small. The GCR flux drops off too rapidly, whereas the break in the EGCR spectrum is not sufficiently pronounced. The difference between the measured and predicted flux, the "missing" flux, is indicated in black. This suggests the necessity of some additional component to account for this difference, a notion that has already been discussed in other contexts [1,19].
When considering the robustness of this finding, prospects of improving our prediction, particularly m i (R), are worth discussing. The steep suppression of the GCR flux appears to be a necessary consequence of the propagation effects in the GMF outlined in section 2. Their gradual escape from the GP and the resulting suppression of their flux is inexorably linked to the rigidity range in question and the strength of the (turbulent) field through which they propagate. The propagation effects involving EGCRs are subject to more uncertainty. Since the flux that is injected into the Galaxy strongly affects the nature of the flux modification after propagation in the GMF, the parameters determining said flux are worth further scrutinising. They  include the source distribution, and the smearing of the fluxes originating from each. Concerning the sensitivity of m i (R) to the latter, we varied the smearing to twice and half the value chosen for this prediction, and found that this has no large effect on the overall trend of the resulting flux modification, and mainly affects statistics 18 . Besides the nature of the incoming flux, the turbulent component of the GMF plays an instrumental role in the modification of the EGCR flux. It does so by governing the amount of smearing of the incoming EGCR flux and, effectively, the Galactic transparency. Both the angular scale of regions of higher or lower transparency, as well as their amplitudes, are affected.
Since the flux modification of both GCRs and EGCRs depends chiefly on the turbulent field strength, exploring the effect of higher or lower turbulence levels naturally presents itself. Such a discussion is further motivated due 18 Additional considerations such as these to appear in a publication that is still in preparation at the time of writing these proceedings.
to the general uncertainty of our current GMF. With regard to the JF12 field specifically, there is, in fact, a longstanding suspicion that the strength of its turbulent component is overestimated. Other publications have found hints of this, pertaining to the deflection of EGCRs [20], average grammage of GCRs [3], or different treatments of relevant observables [21,22]. Further, two observations from our studies support this notion. For one, the Galactic residence times of GCRs (in our case estimated where the count has decreased by one half) is slightly high at lowest rigidities with t res ≈ 100 Myr [8,23]. The residence time scales with the degree of confinement in the GP which, in turn, are determined by the strength of the turbulent component. The second observation is found in our study of anisotropically injected EGCRs. They also contained dipole anisotropy studies, where the dipole amplitude was determined for several rigidity bins. At highest rigidities, where a significant dipole has been measured by the Pierre Auger Observatory [24,25], even the amplitude of the most significant dipole among the point sources can barely account for the experimental measured value of 8 %. This tension only seems more pertinent in light of the injected fluxes in our studies being purely anisotropic (i.e. no isotropic fraction). The observed amplitudes, therefore, are better understood as upper limits and are, thus, expected to be even smaller for non-zero isotropic fractions to the injected EGCR flux. This tension can be remedied, if the random field strength is in fact overestimated, since this would imply that the smearing of directions by the GMF that lowers the dipole amplitude is too strong.
Smaller turbulence levels weaken the confinement of GCRs and decrease the smearing of anisotropically injected EGCRs. The flux suppression of GCRs would commence at lower rigidities, potentially eliminating the need for a rigidity-dependent cut-off from a phenomenological point of view. The suppression of the GCR flux is thence less steep than our prediction suggests, and part of the "missing" flux we find may be accounted for by it. Anisotropies in the EGCR flux are better preserved, which may also increase the angular resolution and amplitude of the directional transparency of the GMF. While the sensitivity to the injected anisotropy would likely increase, spectral breaks arising in the EGCR flux may become more pronounced, improving the description of the transition region also at highest energies.
Whether a reduction of the turbulent field strength would eliminate the need of an additional component altogether is not clear. For this to be the case, however, the rigidity-dependent escape of GCRs would have to be sufficiently slow to adequately flatten the resulting flux suppression, and the Galactic transparency of EGCRs would have to exhibit a rigidity-dependence (from the injection directions of interest), such that the modified EGCR flux possesses a spectral break at roughly the same position as does our prediction, while, at the same time, being sufficiently steep left of the break. Further, Galactic sources would be required to accelerate CRs to energies comfortably exceeding the knee, such that the lack of a cut-off is justified. In this context, we also wish to refer to [3], where a weaker turbulence level of the JF12 was analysed and found to fully describe the transition region.
For completeness, we briefly outline the effects of larger turbulence levels. On top of being poorly motivated, this would result in the GCR flux overshooting around the knee due to their increased confinement in the GP, which would have to be compensated by a smaller cutoff rigidity R cut . While this would loosen the requirements on the energy output of Galactic sources, the suppression of the Galactic flux beyond the knee would only become steeper. Anisotropic EGCR fluxes would be smeared out more strongly and flux modifications would likely become even less pronounced.

Conclusion
In this work, we detail the procedure for obtaining a flux prediction of GCRs and EGCRs based on propagation studies of both classes of CRs in the GMF [7,8]. Said prediction is constructed with the objective of relying on a minimal set of additional assumptions pertaining to the distribution and properties of the respective sources. To this end, the GCR sources are distributed according to the galactocentric distribution of SNRs [15], the EGCRs sources follow the distribution of ten brightest galaxies in the "Starburst catalogue" from [16]. For both classes of CRs, a four-component composition is injected from the sources with equal spectral indices for all mass groups that set to the value of the measured energy spectrum below the knee. In addition, a rigidity-dependent cutoff is introduced for both classes with differing shapes in order to match the smoothness of the breaks in the measured spectrum that the are aimed to produce (knee, and high-energy cutoff, respectively). The cut-off rigidity R i cut , i = GCR, EGCR, and the nuclear fractions f i j , j = H, He, O, Fe, are free parameters that are varied to match the position and subsequent spectral shape following the aforementioned breaks.
We find that the obtained flux underpredicts the measured flux across the entire transition region, suggesting the need for an additional component to fill the unaccounted for flux. We argue that this discrepancy between prediction and data follows rather strictly from the propagation effects themselves. Motivated by the presumed overestimation of the strength of the turbulent component of the JF12, the effects of lower turbulence levels are briefly explored. We conclude that this could alleviate some of the aforementioned tension that we find. The prospect of this eliminating the need to invoke additional components entirely is addressed by alluding to the rather strict set of requirements this places on the nature of the propagation effects in the GMF (and on the energy output of Galactic sources).
As a final point, it is worth noting that our prediction makes an important contribution, despite being based on a presumably systematically biased GMF model (i.e. one overestimating the strength of the random component of the GMF). This holds true because the JF12 model constitutes a GMF model that results from fits to observational data. Modifying the random component ad hoc, as was done in [3], has the drawback of not fulfilling said criterion anymore. Given the ongoing work on creating updated and improved GMF models (see e.g. [22]), an outlook on further research presents around this results itself, namely re-investigating the propagation we identified using on an alternative GMF model with an improved determination of the random field component, ideally one that is specifically based on the JF12 model.