2. Testing the SNR paradigm

While as early as the 1930s Baade & Zwicky (Reference Baade and Zwicky1934) suggested that CRs might originate from SNs, the SNR paradigm was later formulated based on energetic arguments: SN explosions in the Galaxy can easily provide the power needed to sustain the CR population (Morrison Reference Morrison1957; Ginzburg & Syrovatskii Reference Ginzburg and Syrovatskii1964). Assuming that the locally measured CR energy density, $w_{\textrm {cr}} \sim 1$ eV/cm$^3$, is representative of the CR energy density everywhere in the Galaxy, the CR production rate is $\dot {W}_{\textrm {CR}} = V w_{\textrm {cr}}/t_{\textrm {conf}}$ with $t_{\textrm {conf}}$ and $V$ the CR confinement time and volume, respectively. Using $t_{\textrm {conf}}\approx 15$ Myr (Yanasak et al. Reference Yanasak, Wiedenbeck, Mewaldt, Davis, Cummings, George, Leske, Stone, Christian and von Rosenvinge2001) and describing the Galaxy as a cylinder of radius $R_d\approx 15$ kpc and height $H_d\approx 5$ kpc (Evoli et al. Reference Evoli, Morlino, Blasi and Aloisio2020b; Morlino & Amato Reference Morlino and Amato2020) we can estimate $\dot W_{\textrm {CR}}\approx \mbox {1--3} \times {10}^{41}$ erg s$^{-1}$.

SN events happen in the Galaxy about every 30 years and typically release 10$^{51}$ erg. Thus, the power needed to sustain the galactic CR population turns out to be about 10 $\%$ of the power provided by SN events. The appeal of the SNR hypothesis was then increased by the formulation of the theory of diffusive shock acceleration (DSA) (Bell Reference Bell1978; Blandford & Ostriker Reference Blandford and Ostriker1978): this acceleration process, expected to be active at most shock waves, predicts the spectrum of accelerated particles to be a power law, with an index close to $-2$ in the case of a strong shock, such as the blast wave of a SN explosion. Such a spectrum perfectly fits what inferred for the sources of Galactic CRs, lending support to the association. An equally remarkable, but much more recent finding, concerns the acceleration efficiency: kinetic simulations of DSA in a regime that is close to representing a SN blast wave have finally become available in the last decade, and they show that for parallel shock waves (magnetic field perpendicular to the shock surface) the acceleration efficiency is 10–20 % (Caprioli & Spitkovsky Reference Caprioli and Spitkovsky2014a), exactly as required for the SNR–CR paradigm to work.

Testing the SNR paradigm for the origin of CRs has long been one of the top priorities in high-energy astrophysics. Since the early days, photons with energies in the range between GeVs and TeVs have provided a unique tracer of the CR population far from Earth: the interactions of CRs with the environment are expected, indeed, to produce radiation in this energy range. The process that most directly reveals the presence of hadrons is the decay of neutral pions produced when CR hadrons collide inelastically with ambient gas in the interstellar medium (ISM): this process is commonly referred to as the hadronic production mechanism. As a rule of thumb, an inelastic nuclear collision will produce $\gamma$-rays with $\sim$10 % of the energy of the parent CR, so, for instance, photons of several tens to hundreds of TeV for CRs close to PeV. Typically, the spectrum of the hadronic radiation at TeV mimics the parent CR spectrum shifted to lower energy by a factor of 20–30 (Kelner, Aharonian & Bugayov Reference Kelner, Aharonian and Bugayov2006; Kafexhiu et al. Reference Kafexhiu, Aharonian, Taylor and Vila2014). In the same energy range, radiation can originate from leptonic production mechanisms, mainly inverse Compton (IC) scattering of CR electrons off ambient radiation fields, and non-thermal electron bremsstrahlung (Blumenthal & Gould Reference Blumenthal and Gould1970), which plays an important role in dense gas regions at sub-GeV to GeV energies. A recurring difficulty when trying to infer the CR population from the GeV and TeV $\gamma$-ray emission from SNRs is to break the hadronic-leptonic degeneracy and unveil the dominant emission process: the spectral and morphological features of the emission are crucial to this task.

The $\gamma$-ray spectrum is also affected by the particles’ energy losses. For $\gamma$-ray-emitting particles, a number of loss mechanisms that are important at lower energies, such as ionisation and bremsstrahlung, have a negligible impact. The most relevant loss processes affect electrons, whose synchrotron and IC emission must be properly taken into account when trying to disentangle the radiative contribution from leptons and hadrons. Electron synchrotron, particularly efficient in the strong magnetic fields believed to be associated with particle accelerators, has a fundamental role in cooling the CR electron population and thus shape the leptonic $\gamma$-ray spectrum. Therefore, observations of non-thermal radio and X-ray synchrotron emission can provide invaluable constraints on the electron population, and clues to solve the hadronic–leptonic degeneracy.

The SNR paradigm implies that young SNRs accelerate CRs up to the knee and that on average each SNR provides roughly $10^{50}$ erg in accelerated particles. In order to test whether SNRs are the main contributors to the Galactic CR population, it is thus crucial to assess the energetics in electrons and protons and the spectra of accelerated particles in young SNRs.

Studies of the GeV-to-TeV spectra of the remnants, of their morphology and of the spatial correlation between TeV $\gamma$-rays and the distribution of ambient gas and of X-ray radiation are the three most powerful instruments to extract the population of accelerated particles in the SNRs. GeV-to-TeV $\gamma$-ray studies of both young SNRs (Drury, Aharonian & Voelk Reference Drury, Aharonian and Voelk1994), and dense molecular clouds close to middle-aged SNRs (Aharonian, Drury & Voelk Reference Aharonian, Drury and Voelk1994) were undertaken to assess the role of SNRs in the origin of Galactic CRs, by understanding and disentangling the dynamics of the coexisting and competing processes of acceleration and escape or release of particles in the ISM. A useful repository of high-energy observations of SNRs is the Manitoba Catalogue (http://snrcat.physics.umanitoba.ca/).

Several young shell-type SNRs have been observed at TeV energies. Among these, the best studied are Cas A (Aharonian et al. Reference Aharonian, Akhperjanian, Barrio, Bernlöhr, Börst, Bojahr, Bolz, Contreras, Cortina and Denninghoff2001; Albert et al. Reference Albert, Aliu, Anderhub, Antoranz, Armada, Baixeras, Barrio, Bartko, Bastieri and Becker2007a; Ahnen et al. Reference Ahnen, Ansoldi, Antonelli, Arcaro, Babić, Banerjee, Bangale, Barres de Almeida, Barrio and Becerra González2017), Tycho (Acciari et al. Reference Acciari, Aliu, Arlen, Aune, Beilicke, Benbow, Bradbury, Buckley, Bugaev and Byrum2011; Archambault et al. Reference Archambault, Archer, Benbow, Bird, Bourbeau, Buchovecky, Buckley, Bugaev, Cerruti and Connolly2017) and RX J1713.7-3946 (Aharonian et al. Reference Aharonian, Akhperjanian, Aye, Bazer-Bachi, Beilicke, Benbow, Berge, Berghaus, Bernlöhr and Bolz2004; Abdalla et al. Reference Abdalla, Abramowski, Aharonian, Benkhali, Akhperjanian, Andersson, Angüner, Arrieta and Arrieta2018). We discuss the latter as an example of how the SNR paradigm is tested with radiation from X-rays to $\gamma$-rays.

As an example of multi-wavelength studies concerning middle-aged SNRs showing the so-called pion bump, a clear signature of the presence of relativistic protons, we discuss the case of the brightest GeV source among these: W44.

2.1. Young SNRs: the case of RX J1713.7-3946

The SNR RX J1713.7-3946, possibly associated with a star explosion seen by Chinese astronomers in the year AD393 (Pfeffermann & Aschenbach Reference Pfeffermann and Aschenbach1996; Wang, Qu & Chen Reference Wang, Qu and Chen1997), is one of the brightest sources of $\gamma$-rays in the TeV energy range. The sky map of RX J1713.7-3946, obtained with the H.E.S.S. telescope, is shown in the top left panel of figure 1 (Abdalla et al. Reference Abdalla, Abramowski, Aharonian, Benkhali, Akhperjanian, Andersson, Angüner, Arrieta and Arrieta2018). The morphology of the TeV shell resembles closely the XMM X-ray shell (Cassam-Chenaï et al. Reference Cassam-Chenaï, Decourchelle, Ballet, Sauvageot, Dubner and Giacani2004), plotted with blue contours in the top left panel of the figure. The other five panels of figure 1 show the radial profiles at keV and TeV photon energies from different regions of the shell: in four cases, the TeV SNR extends, in radius, beyond the X-ray emitting shell (Aharonian et al. Reference Aharonian, Akhperjanian, Aye, Bazer-Bachi, Beilicke, Benbow, Berge, Berghaus, Bernlöhr and Bolz2004; Uchiyama et al. Reference Uchiyama, Aharonian, Tanaka, Takahashi and Maeda2007; Tanaka et al. Reference Tanaka, Uchiyama, Aharonian, Takahashi, Bamba, Hiraga, Kataoka, Kishishita, Kokubun and Mori2008; Abdalla et al. Reference Abdalla, Abramowski, Aharonian, Benkhali, Akhperjanian, Andersson, Angüner, Arrieta and Arrieta2018). The main shock position and extent are visible in the X-ray data and the $\gamma$-ray emission extending further is either due to accelerated particles escaping the acceleration (shock) region or particles in the shock precursor region.

Figure 1. Top left panel: H.E.S.S. $\gamma$-ray skymap of RX J1713.7-3946 (Aharonian et al. Reference Aharonian, Akhperjanian, Aye, Bazer-Bachi, Beilicke, Benbow, Berge, Berghaus, Bernlöhr and Bolz2004; Abdalla et al. Reference Abdalla, Abramowski, Aharonian, Benkhali, Akhperjanian, Andersson, Angüner, Arrieta and Arrieta2018). In blue the contours of the shell as observed by XMM in X-rays (Cassam-Chenaï et al. Reference Cassam-Chenaï, Decourchelle, Ballet, Sauvageot, Dubner and Giacani2004) are shown. The morphology of the TeV shell resembles closely the X-ray shell. Top right panel and following: radial profiles of the emission at selected places along the shell from H.E.S.S. and XMM-Newton. In four out of five regions of the shell, the TeV SNR is more extended than the X-ray SNR as a result of possible escape of high energy particles from the shell. The coordinate ‘Radius’, used for the one-dimensional profiles, refers to the mean radius ‘$r$’ of the shell over which average is performed.

VHE electrons, up to hundreds of TeV, are accelerated in the shell of RX J1713-3946. These particles emit hard X-rays through synchrotron processes and TeV radiation through IC scattering off CMB photons very efficiently. A crucial piece of evidence in support of efficient particle acceleration comes from hard X-ray measurements with Suzaku (Uchiyama et al. Reference Uchiyama, Aharonian, Tanaka, Takahashi and Maeda2007). Suzaku detected X-ray hot-spots brightening and disappearing within a 1 year timescale. If the X-ray variability is associated with the acceleration and immediate synchrotron cooling of the accelerated electrons, then the magnetic field in the shell must be roughly two orders of magnitude higher than the average magnetic field in the Galaxy, about 100 $\mathrm {\mu }$Gauss. Broadband X-ray spectrometric measurements of RXJ1713.7-3946 indicate also that electron acceleration proceeds in the most effective Bohm-diffusion regime. Finally, the presence of strongly amplified magnetic fields lends further support to the idea that not only electrons, but also protons up to 100 TeV are accelerated in the shell, as these amplified magnetic fields would be the result of CR-induced instabilities (Uchiyama et al. Reference Uchiyama, Aharonian, Tanaka, Takahashi and Maeda2007).

The high-energy part of the spectrum of RX J1713.7-3946, from X-rays to TeV photons, can be used to infer the population of particles producing this emission. The particle spectrum of the young SNR in the left panel of figure 2 is characterised by a spectral index close to $-2$, as predicted by DSA, up to a few TeV (=${10}^{12}$ eV). Although the hard spectrum shows that particles are efficiently accelerated in the shell of this young SNR, the cutoff at a few TeV suggests that RX J1713.7-3946 is currently accelerating electrons and protons up to 100 TeV but not up to PeV energies (Abdalla et al. Reference Abdalla, Abramowski, Aharonian, Benkhali, Akhperjanian, Andersson, Angüner, Arrieta and Arrieta2018).

Figure 2. (a) The black data points represent the GeV (Fermi-LAT) and TeV (H.E.S.S.) spectral energy distribution of the radiation from the shell of RX J1713.7-3946. The lines represent the multi-wavelength spectra predicted by NAIMA (Zabalza Reference Zabalza2015), assuming either a leptonic or hadronic origin of the emission. (b) The spectral energy distributions of the electrons and protons producing the leptonic and hadronic emissions shown on the left panel.

As mentioned, protons are likely accelerated in the SN shell through the same mechanism accelerating electrons, but it is not clear what fraction of the energy input by the SN explosion goes into the acceleration of electrons and what fraction goes into the acceleration of protons. Both protons or electrons can be, in fact, responsible for the TeV emission from RX J1713.7-3946 (leptonic–hadronic degeneracy, see e.g. Morlino, Amato & Blasi Reference Morlino, Amato and Blasi2009). If electrons are responsible for the emission from the shell, by IC scattering of the ambient radiation fields, one speaks of the leptonic scenario. On the other hand, if the emission is produced by protons colliding with the ambient gas, one speaks of the hadronic scenario. In figure 2 the measurements of the shell emission carried out by Fermi LAT (Abdo et al. Reference Abdo, Ackermann, Ajello, Allafort, Baldini, Ballet, Barbiellini, Baring, Bastieri and Bellazzini2011), HESS (Abdalla et al. Reference Abdalla, Abramowski, Aharonian, Benkhali, Akhperjanian, Andersson, Angüner, Arrieta and Arrieta2018) and Suzaku (Tanaka et al. Reference Tanaka, Uchiyama, Aharonian, Takahashi, Bamba, Hiraga, Kataoka, Kishishita, Kokubun and Mori2008) are compared with leptonic (left panel) and hadronic (right panel) model predictions by Zabalza (Reference Zabalza2015). The total energetics in accelerated electrons and protons in the relevant leptonic and hadronic models of $\gamma$-rays can be estimated by assuming a distance to the source of about 1 kpc (Pfeffermann & Aschenbach Reference Pfeffermann and Aschenbach1996; Wang et al. Reference Wang, Qu and Chen1997). The required budget in electrons is determined only by the reported $\gamma$-ray fluxes, if the target radiation is the CMB: one finds $W_\textrm {e} \simeq 1.2 \times 10^{47}$ erg. On the other hand, the total energy budget of protons in hadronic models depends on the highly uncertain ambient gas density: $W_\textrm {p} \simeq 5 \times {10}^{49} (n_H/1\ \textrm {cm}^{-3})^{-1}$ erg. The leptonic model in the left panel corresponds to the electron IC scattering off CMB photons and an infrared radiation field with energy density 0.415 eV cm$^{-3}$ and temperature $T = 26.5$ K.

The leptonic model requires a break in the electron spectrum at 2.5 TeV with the index of the electron energy distribution changing from $\varGamma =1.7$ to $\varGamma =3$ beyond the break. If, due to synchrotron cooling, this spectral break implies a magnetic field of about $140\ \mathrm {\mu }$G, which is much larger than what the X-ray flux allows within the same scenario, $B\approx 15\ \mathrm {\mu }$G. Similar difficulties arise if one tries to explain the break as a result of IC cooling, in which case a photon field of energy density of 140 eV cm$^{-3}$ would be required, about 100 times larger than the average galactic value.

If the emission is hadronic in origin, a spectrum of protons with index $\varGamma =1.5$, steepening to $\varGamma =1.9$ at about 1 TeV, and with an exponential cutoff at 79 TeV is required. The break in the proton population spectrum can be explained if the hadronic emission is produced mostly in dense clumpy regions. In fact, low-energy protons can be efficiently excluded from dense gas region during the timescale of $\sim$1000 years since the SNR explosion. The exclusion of low-energy CRs would also explain the hard $\gamma$-rays detected by the Fermi LAT telescope (Fukui et al. Reference Fukui, Sano, Sato, Torii, Horachi, Hayakawa, McClure-Griffiths, Rowell, Inoue and Inutsuka2012; Inoue et al. Reference Inoue, Yamazaki, Inutsuka and Fukui2012; Gabici & Aharonian Reference Gabici and Aharonian2014), and the lack of thermal X-ray emission from the shell. The latter has traditionally been one of the strongest arguments against a possible hadronic origin of the emission from the shell of RXJ1713.7-3946, because it implies an ambient gas density as low as $0.1\ \textrm {cm}^{-3}$ (Ellison et al. Reference Ellison, Patnaude, Slane and Raymond2010; Fukui et al. Reference Fukui, Sano, Sato, Torii, Horachi, Hayakawa, McClure-Griffiths, Rowell, Inoue and Inutsuka2012; Gabici & Aharonian Reference Gabici and Aharonian2014). Finally, hadronic $\gamma$-ray production in gas condensations results in a narrow angular distribution of the radiation, which is beyond the reach of the current generation of Imaging Air Cherenkov Telescopes (IACTs), but could be tested with the upcoming Cherenkov Telescope Array (CTA), which will have an angular resolution of $\sim$1–2 arcmin (Zirakashvili & Aharonian Reference Zirakashvili and Aharonian2010; Ambrogi, Celli & Aharonian Reference Ambrogi, Celli and Aharonian2018).

The conclusion one reaches, after investigating the spectrum and morphology of RX J1713.7$-$3946 as currently known, is that neither the hadronic nor the leptonic scenario is fully satisfactory. Each emission mechanism has strengths and weaknesses when compared with observations, suggesting that the ambient conditions might differ in different parts of the remnant, making one or the other process locally dominant. Although for some specific SNRs one of the two scenarios might indeed be favoured (see e.g. Funk Reference Funk2015), the general conclusion is that no known SNR has proved to accelerate particles beyond 100 TeV.

2.2. Molecular clouds close to middle-aged SNRs: the case of W44

Molecular clouds are regions of the Galaxy, typically a few tens of parsecs in radius, where the density of cold molecular gas is often orders of magnitude higher than elsewhere in the diffuse ISM. Stars are believed to be born in these clouds. Radio observations of the rotational $1\rightarrow 0$ line emission of carbon monoxide are mainly used to trace the distribution of molecular gas (Heyer & Dame Reference Heyer and Dame2015). The cloud Galactocentric distance is usually estimated using a kinematical distance method, through each the radial velocity of the cloud is related to the rotation velocity of the Galaxy. Cross-calibrations with the distance of spiral arms or known objects with precise parallax determination are also carried out (Roman-Duval et al. Reference Roman-Duval, Jackson, Heyer, Johnson, Rathborne, Shah and Simon2009; Reid et al. Reference Reid, Menten, Brunthaler, Zheng, Dame, Xu, Wu, Zhang, Sanna and Sato2014).

Giant molecular clouds, which are typically 5 to 200 parsecs in diameter and have masses of 10 000 to 10 million Solar masses (Murray Reference Murray2011), are excellent laboratories for CR physics. In these clouds, the hadronic channel of $\gamma$-ray production is enhanced by the high target density, and easily dominates over the leptonic production mechanism. In contrast to the warmer atomic gas phase, which is homogeneously distributed in the Galaxy, one or a few giant molecular clouds are essentially the dominant contributions to the gas column density along a given direction in the sky. The emission enhancement associated with the clouds makes it possible to precisely locate along a given line of sight where the CR population produces the emission. Molecular clouds are thus used to perform a sort of tomography, and obtain a three-dimensional view of the CR distribution in the Galaxy (Issa & Wolfendale Reference Issa and Wolfendale1981; Aharonian & Atoyan Reference Aharonian and Atoyan1996; Aharonian Reference Aharonian2001; Casanova et al. Reference Casanova, Aharonian, Fukui, Gabici, Jones, Kawamura, Onishi, Rowell, Sano and Torii2010a; Aharonian et al. Reference Aharonian, Peron, Yang, Casanova and Zanin2020; Baghmanyan et al. Reference Baghmanyan, Peron, Casanova, Aharonian and Zanin2020).

The plasma conditions in molecular clouds are generally different from those in the diffuse ISM. In addition to the gas density, the magnetic field energy density and turbulence level are also enhanced (see e.g.Crutcher (Reference Crutcher2012) for a comprehensive review). This will lead to a suppression of the diffusion coefficient and effective exclusion of lower-energy CRs from the cloud (Gabici, Aharonian & Blasi Reference Gabici, Aharonian and Blasi2007). If CRs can penetrate clouds, the $\gamma$-ray emission from $\pi ^0$-decay depends only upon the total mass of the cloud, $M_{cl}$, its distance from the Earth, $d$, and the CR flux within the cloud, $\varPhi _{\textrm {CR}}$. The latter is thus determined as $\varPhi _{\textrm {CR}} \propto {\varPhi _{\gamma } d^2}/{M_{cl}}$, where ${\varPhi _{\gamma }}$ is the $\gamma$-ray flux from the cloud. Lower-energy CRs can be effectively excluded from penetrating clouds, which results in peculiar features in the $\gamma$-ray spectrum from clouds, such as hardenings with respect to the average interstellar spectrum (Gabici et al. Reference Gabici, Aharonian and Blasi2007).

The role of molecular clouds in testing the SNR paradigm is made particularly crucial by the time evolution of particle acceleration in SNRs. Within a DSA scenario, SNRs accelerate the highest-energy CRs (in principle, up to a few PeV, but see § 3) at the transition between the free expansion and the Sedov phase, which typically happens a few tens to a few hundred years after the SN explosion, depending on the explosion type and properties of the surrounding medium. During the Sedov phase, the SN shock slows down and the magnetic field intensity decreases, so that the most energetic particles cannot be confined any longer and are free to escape. In practice, a SNR can accelerate the highest-energy particles only for a short time. This fact, coupled to the low rate of PeV particles accelerating events (see § 3) makes the chances of observing a SNR when it still acts as a PeVatron very low. The runaway CRs can illuminate molecular clouds located close to the SNRs and this enhanced $\gamma$-ray emission can thus provide crucial insights on the parental population of runaway CRs, which would otherwise escape the SNR without leaving a footprint (Montmerle & Cesarsky Reference Montmerle and Cesarsky1979; Issa & Wolfendale Reference Issa and Wolfendale1981; Aharonian & Atoyan Reference Aharonian and Atoyan1996; Aharonian Reference Aharonian2001; Gabici et al. Reference Gabici, Aharonian and Blasi2007). The emission produced by these runaway CRs is essential to trace back acceleration up to PeV energies.

Gamma-rays in association with dense molecular clouds located close to SNRs have been detected both at GeV and TeV energies. Depending on the location of massive clouds, on the acceleration history and on the timescales of the particle escape into the ISM (which depend, in turn, on the diffusion coefficient), a broad variety of energy distributions of $\gamma$-rays is produced, from very hard (much harder than the spectrum of the SNR itself) to very steep spectra (Aharonian & Atoyan Reference Aharonian and Atoyan1996; Gabici, Aharonian & Casanova Reference Gabici, Aharonian and Casanova2009; Casanova et al. Reference Casanova, Jones, Aharonian, Fukui, Gabici, Kawamura, Onishi, Rowell, Sano and Torii2010b).

In figure 3 we show the spectrum of two bright middle-aged SNRs, IC443 and W44. W44 is a 20  000-year-old SNR located on the Galactic Plane at a distance of roughly 3 kpc from the Sun. W44, which is the brightest middle-aged SNR at GeV energies, is a perfect laboratory to test the coexisting processes of acceleration and escape of CRs from SNRs. Owing to the slow shock speed, high-energy particles are expected to have already escaped from this source. A population of protons with spectral index close to $-$2.3 and a cutoff at about 80 GeV is likely responsible for the $\gamma$-ray emission from the remnant measured by AGILE and Fermi-LAT (Abdo et al. Reference Abdo, Ackermann, Ajello, Baldini, Ballet, Barbiellini, Baring, Bastieri, Baughman and Bechtol2010a; Giuliani et al. Reference Giuliani, Cardillo, Tavani, Fukui, Yoshiike, Torii, Dubner, Castelletti, Barbiellini and Bulgarelli2011). The presence of such a low-energy cutoff might be the effect of the escape of the highest-energy CRs. Indeed, the remnant is thought to be currently a rather poor accelerator and most of its $\gamma$-ray emission can be interpreted as due to re-acceleration of ambient CRs, rather than to acceleration of fresh particles (Uchiyama et al. Reference Uchiyama, Blandford, Funk, Tajima and Tanaka2010; Cardillo et al. Reference Cardillo, Amato and Blasi2016).

Figure 3. Gamma-ray spectra of IC443 (a) and W44 (b) as measured with the Fermi LAT (Abdo et al. Reference Abdo, Ackermann, Ajello, Baldini, Ballet, Barbiellini, Baring, Bastieri, Baughman and Bechtol2010a), in blue, and AGILE (Giuliani et al. Reference Giuliani, Cardillo, Tavani, Fukui, Yoshiike, Torii, Dubner, Castelletti, Barbiellini and Bulgarelli2011), in magenta. The pion bump feature at about 1 GeV is evident in the spectra of both SNRs. For IC433 the data points of MAGIC (Albert et al. Reference Albert, Aliu, Anderhub, Antoranz, Armada, Baixeras, Barrio, Bartko, Bastieri and Becker2007b) and VERITAS (Acciari et al. Reference Acciari, Aliu, Arlen, Aune, Bautista, Beilicke, Benbow, Bradbury, Buckley and Bugaev2009) are also shown. Colour-shaded areas denote the best-fit broadband smooth broken power law (60 MeV to 2 GeV); grey-shaded bands show Fermi LAT systematic errors below 2 GeV. The data points at the highest energy ($4 \times {10}^{10}$–$1 \times {10}^{11}$ eV) suggest a hardening of the spectrum, which might be produced by runaway CRs illuminating a molecular cloud (MC) located in front of the shells.

Owing to their short lives, SNRs are often found within the giant molecular gas complexes where they were born as massive stars. W44 resides within a giant gas complex of $10^6$ Solar masses, homogeneously distributed over an extended region surrounding the remnant (Dame, Hartmann & Thaddeus Reference Dame, Hartmann and Thaddeus2001). North west and south east of the remnant, Uchiyama et al. (Reference Uchiyama, Funk, Katagiri, Katsuta, Lemoine-Goumard, Tajima, Tanaka and Torres2012) discovered two bright sources, which the authors associated to runaway CRs colliding with the dense gas clouds. Peron et al. (Reference Peron, Aharonian, Casanova, Zanin and Romoli2020) re-analysed both the Fermi-LAT data and the gas data from the clouds around W44 and noted that, despite the gas is distributed homogeneously, the GeV emissivity is enhanced only in the two regions which can be associated to regions of enhanced CR density, CR clouds, rather than gas clouds as previously thought. This phenomenon suggests the existence of a preferential path for CR escape, likely linked to the magnetic field structure in the vicinity of this source.

3. Recent developments on CR acceleration in SNRs

Despite all the recent observational developments, providing several hints of efficient CR acceleration in SNRs, a number of unsettled questions remain and challenge the association. In particular, one of the strongest arguments to look for alternative particle accelerators has to do with the difficulties SNRs have at reaching PeV energies. This aspect of the CR–SNR connection has long appeared as the most delicate.

As already mentioned, CR acceleration in SNRs is thought to be well described within the framework of DSA. Acceleration mechanisms alternative to DSA have also been proposed and studied (see e.g. Lazarian et al. Reference Lazarian, Eyink, Jafari, Kowal, Li, Xu and Vishniac2020). In this article, however, we focus on DSA, which is still the best studied and the most promising, in this context, to reach the highest energies. The idea at the basis of the theory is that particles gain energy each time they cross the shock thanks to the discontinuity of the fluid velocity field at the shock, that leaves an unscreened electric field. The energy gain of the particle is a constant fraction ($v_\textrm {sh}/c$ with $v_\textrm {sh}$ the shock velocity) of its energy before the crossing. The particle moves diffusively between crossings, scattering on low-frequency magnetic turbulence and continuously changing its pitch angle. Reaching high energies requires a large number of crossings, which have to occur in a time shorter than the minimum between the duration of the system $t_\textrm {life}$ and the time scale over which energy losses become important $t_\textrm {loss}$. In the case of protons losses are negligible and the energy is limited by the time for which the SNR is active as an accelerator, $t_\textrm {life}<10^4$ yr. Then $t_\textrm {life}$ must be compared with the acceleration time, $t_\textrm {acc}$, which depends on how quickly the particle is able to go back to the shock after each crossing: we can estimate $t_\textrm {acc}(E)\approx D(E)/v_\textrm {sh}^2$, where $D(E)$ is the particle diffusion coefficient, typically an increasing function of the particle energy $E$. For relatively low turbulence levels one can estimate $D(E)$ in quasi-linear theory, writing it as (see e.g. Amato Reference Amato2014)

(3.1)\begin{equation} D(E)\approx \frac{v(E) r_L(E)}{3}\frac{1}{kW(k)},\end{equation}

where $v(E)$ is the particle velocity, $r_L(E)$ its Larmor radius, and $W(k)$ is the spectrum of magnetic fluctuations causing the diffusion, with

(3.2)\begin{equation} \int_{1/L}^\infty W(k')\, \textrm{d} k'=\frac{\delta B^2}{B_0^2}, \end{equation}

and $k$ the wave number resonant with the particle of energy $E$, namely $k=1/r_L(E)$.

Assuming that the scattering is due to turbulence that is injected by SNRs on a typical scale $L\approx 100$ pc with $\delta B/B_0\approx 1$, and then develops a Kolmogorov-type spectrum $W(k)\propto k^{-5/3}$, from (3.1) and (3.2),Footnote ¹ one can estimate the diffusion coefficient as $D(E)=7\times 10^{27} (E/\textrm {GeV})^{1/3}$ cm$^2$ s$^{-1}$, which, as we discuss in the following, is not very different from what CR observations indicate ($D(1\ \mathrm {GeV})\approx 10^{28}$ cm$^2$ s$^{-1}$). If this estimate of the diffusion coefficient were appropriate to describe particle transport in the vicinity of a SNR shock, the maximum achievable energy in these systems would only be $E_\textrm {max}\sim$ few GeV, and could only improve to $\sim$100 TeV if particle transport were described by Bohm diffusion, namely $\delta B/B_0\approx 1$ at all relevant scales (Lagage & Cesarsky Reference Lagage and Cesarsky1983). It is then clear that in order to reach PeV energies, not only is efficient (Bohm-like) scattering needed, but so are largely amplified magnetic fields.

As already mentioned, this is exactly what X-ray observations of young SNRs were found to show: aside from the short variability time scales already discussed in § 2.1, X-ray synchrotron emission is seen to be confined to rims of thickness $\varDelta \sim \textrm {few} \times 0.01$ pc (Vink Reference Vink2012). The extremely small value of $\varDelta$ can be interpreted either as a result of radiative losses, that kill the electron population with increasing distance from the shock, or as a result of magnetic field damping (Rettig & Pohl Reference Rettig and Pohl2012). If we interpret the thickness of the rims as the distance travelled by the emitting electrons in a synchrotron loss time, we can write $\varDelta \approx \sqrt {D(E)\ t_\textrm {sync}(E)}=0.04\ \textrm {pc}\ B_{-4}^{-3/2}$ with $B_{-4}$ the magnetic field downstream of the shock in units of $10^{-4}$ G. It is clear then that a magnetic field amplified by a factor of $\sim$100 with respect to the interstellar value ($B_\textrm {ISM}\approx 3\ \mathrm {\mu }$G) is implied in the rims, if they are to be interpreted as the result of radiative losses. On the other hand, interpreting $\varDelta$ as a result of magnetic damping requires a value of the magnetic field which is still in the same range estimated previously (Pohl, Yan & Lazarian Reference Pohl, Yan and Lazarian2005), and hence largely amplified.

There are several mechanisms by which a magnetic field can be amplified at a shock (Bykov, Ellison & Renaud Reference Bykov, Ellison and Renaud2012): some involve fluid instabilities (Giacalone & Jokipii Reference Giacalone and Jokipii2007; Ohira Reference Ohira2016), some involve magnetohydrodynamic (MHD) instabilities and CR-related effects (Beresnyak, Jones & Lazarian Reference Beresnyak, Jones and Lazarian2009), some others involve CR-induced instabilities, either of resonant type ( see Amato & Blasi Reference Amato and Blasi2006 and references therein) or of non-resonant type (Bell Reference Bell2004; Reville et al. Reference Reville, O'Sullivan, Duffy and Kirk2008; Amato & Blasi Reference Amato and Blasi2009). Except for purely fluid instabilities (Giacalone & Jokipii Reference Giacalone and Jokipii2007), all other classes require efficient CR acceleration, and in this sense the detection of amplified magnetic fields can be considered as indirect evidence of efficient CR acceleration in SNRs. However, the different mechanisms proposed are not equivalent in terms of the spectrum of magnetic fluctuations they produce and, hence, in terms of consequences on particle acceleration. Particle scattering is only efficient with resonant waves. Hence, MFA will increase scattering efficiency and lead to high maximum energies of the accelerated particles only if there is enough power on scales comparable with the particle Larmor radii. The most promising MFA mechanism in this sense turns out to be the non-resonant CR streaming instability (Bell's instability).

The super-Alfvénic streaming of energetic particles has long been known to induce the growth of magnetic fluctuations at wavelengths that are resonant with the gyroradii of the exciting particles (Skilling Reference Skilling1975): this is the so-called resonant streaming instability. While creating fluctuations on the right scales, this instability can only lead to MFA up to a level $\delta B/B_0\lesssim 1$, and hence it is not powerful enough to explain the field strength deduced in SNRs, nor to guarantee particle acceleration up to the knee. In more recent times, however, it has been recognised that a more powerful instability arises if the CR current is large enough to twist the ambient magnetic field on the scale of the Larmor radius ($r_{L,0}$) of the particles that carry the current ($J_\textrm {CR}>(c/4 \pi )(B_0/r_{L,0})$) or, equivalently, if the CR energy density, $U_\textrm {CR}$, is larger than $c/v_D$ times the magnetic energy density (Bell Reference Bell2004):

(3.3)\begin{equation} U_\textrm{CR}>\frac{c}{v_D} \frac{B_0^2}{4 \pi},\end{equation}

where $v_D$ is the bulk velocity of CRs. If this condition is satisfied, the magnetic field grows very rapidly. The basic physical process can be described as follows: the CR current induces a compensating return current in the ISM plasma; the force $\vec J_\textrm {ret}\wedge \vec B$ induces transverse plasma motion; the current associated with this motion acts as a source of magnetic field; the result is that the magnetic field lines associated with right-hand polarised waves are stretched and for these modes the field is amplified.

In the vicinity of a shock that is accelerating particles, (3.3) can be turned into a condition for the acceleration efficiency $\xi _\textrm {CR}$:

(3.4)\begin{equation} \xi_\textrm{CR}>\frac{c B_0^2}{8 \pi \rho_\textrm{ISM} v_\textrm{sh}^3}. \end{equation}

Here $\xi _\textrm {CR}$ is the fraction of incoming flow energy ($\rho _\textrm {ISM} v_\textrm {sh}^2/2$) that is converted into accelerated particles, $\rho _\textrm {ISM}$ is the mass density of the ISM plasma and $v_\textrm {sh}$ is the velocity of the blast wave, also coincident with the CR bulk velocity. Equation (3.4) makes it clear that the possibility for the non-resonant streaming instability to operate strongly depends on the shock velocity: detailed calculations show that it is indeed a viable MFA mechanism in the vicinity of the fast shocks of young SNRs (Amato & Blasi Reference Amato and Blasi2009), but then stops working early during the Sedov–Taylor phase of expansion of the blast wave, typically after a few hundred to a few thousand years, depending on the ambient medium density and magnetic field strength.

The growth initially occurs on very small scales ($k\approx (4\pi /c)(J_\textrm {CR}/B_0 \gg r_\textrm {L,0}$, but quickly the power moves to larger and larger scales (Riquelme & Spitkovsky Reference Riquelme and Spitkovsky2009), possibly due to some mean-field dynamo process (Bykov, Osipov & Ellison Reference Bykov, Osipov and Ellison2011; Rogachevskii et al. Reference Rogachevskii, Kleeorin, Brandenburg and Eichler2012), and numerical simulations show that the final outcome of the instability, when it develops at a shock, is a spectrum of fluctuations with $W(k)\propto k^{-1}$, leading to Bohm diffusion of the particles (Caprioli & Spitkovsky Reference Caprioli and Spitkovsky2014c). When saturation is reached, the magnetic energy density is a substantial fraction of the CR energy density (Caprioli & Spitkovsky Reference Caprioli and Spitkovsky2014b).

The level of MFA that the non-resonant streaming instability can provide is in the correct range to explain the magnetic field strength inferred in SNRs (Schure et al. Reference Schure, Bell, Drury and Bykov2012), and as we just mentioned the associated scattering is efficient (Caprioli & Spitkovsky Reference Caprioli and Spitkovsky2014c), and yet recent studies have cast doubt on the fact that these sources might be able to accelerate particles up to PeV energies (Cardillo, Amato & Blasi Reference Cardillo, Amato and Blasi2015). In fact, assuming that MFA is primarily due to the streaming of particles that leave the acceleration region, it is possible to build a description of the shock as a self-regulating system: when the level of turbulence in the upstream is low, a large fraction of particles can escape from the shock; when this happens, however, the large current in the upstream causes the growth of turbulence and escape is reduced; if the fraction of escaping particles becomes too small, then the turbulence level is reduced again favouring escape. The self-regulation mechanism qualitatively illustrated previously translates into a quantitative prescription for the maximum energy of shock-accelerated particles as a function of the system parameters. The current of escaping particles, which is what determines the level of MFA, is, in turn, determined by the shock velocity and the spectrum of accelerated particles, including its total energy content (which determines the amount of energy available), its slope and the maximum particle energy (which determine the current, once the energy content is fixed). When writing the equation describing $E_\textrm {max}$ as a function of time during the expansion of a SN blast wave, one finds that $E_\textrm {max}$ is an ever-decreasing function of time during the SNR evolution, so that the relevant value of $E_\textrm {max}$ is that reached at the beginning of the Sedov–Taylor phase, namely the highest possible after a sufficient amount of mass has been processed by the blast wave. It is clear then that a higher maximum energy will be achieved in systems that enter the Sedov–Taylor phase earlier in time after the explosion. This occurs for type II explosions expanding in the dense and slow wind of a progenitor red super-giant star. In this case, assuming a $\propto E^{-2}$ spectrum, the maximum achievable energy reads

(3.5)\begin{align} E_{\max}\approx 0.5 \left(\frac{\xi_\textrm{CR}}{0.1}\right) \left(\frac{E_\textrm{SN}}{10^{51}\ \textrm{erg}}\right) \left(\frac{M_\textrm{ej}}{M_\odot}\right)^{{-}1} \left(\frac{\dot M}{10^{{-}5} M_\odot{/}\textrm{yr}}\right)^{{1}/{2}} \left(\frac{v_w}{10\ \textrm{km\ s}^{{-}1}}\right)^{-{1}/{2}}\ \textrm{PeV},\end{align}

where $\xi _\textrm {CR}$ is the CR acceleration efficiency, $E_\textrm {SN}$ and $M_\textrm {ej}$ are the energy released and the mass of material ejected in the SN explosion, and $\dot M$ and $v_w$ are the mass loss rate and speed of the progenitor's wind. If one takes into account that the mass of the ejecta in a type II SN explosion is more likely around $10\ M_\odot$, it is immediately clear that reaching the knee is very challenging in this framework. One is forced to invoke extremely energetic events, or extreme acceleration efficiency, or finally extreme properties of the progenitor. In all cases these must be rare events. In fact, the product of $\xi _\textrm {CR}$, $E_\textrm {SN}$ and the frequency in the Galaxy of the PeV producing SN explosions are constrained by the overall flux of CRs measured flux at Earth. When all of this is taken into account, one determines an event rate that cannot exceed a few in $10^4$ yr (Cristofari, Blasi & Amato Reference Cristofari, Blasi and Amato2020).

In reality the possibility for SNRs to reach the knee becomes even more challenging when taking into account the fact that the accelerated CR spectrum is likely $\propto E^{-p}$ with $p>2$. Such a spectrum corresponds to a smaller current for a given $E_\textrm {max}$ and total energy in accelerated particles, hence requiring even more extreme parameters to reach $E_\textrm {max}\approx$ PeV (e.g.Cardillo et al. Reference Cardillo, Amato and Blasi2015). On the other hand a spectrum steeper than $E^{-2}$ is exactly what $\gamma$-ray observations of the majority of young SNRs require and what recent progress on propagation also seems to require (Evoli, Aloisio & Blasi Reference Evoli, Aloisio and Blasi2019): the CR spectrum released in the ISM will be steeper than $E^{-2}$ only if the spectrum in the source is itself steeper than $E^{-2}$ (Schure & Bell Reference Schure and Bell2014; Cardillo et al. Reference Cardillo, Amato and Blasi2015) and a steep injection spectrum of CRs is exactly what the most recent CR data seem to suggest, as we discuss in § 4.

4. CR transport in the Galaxy

As mentioned in § 1 the last decade has also been rich with observational progress providing interesting constraints on the properties of CR transport throughout the Galaxy. In this section we review the main findings and try to put them in a coherent theoretical framework.

The first important discovery was that of a hardening in the spectrum of protons and He nuclei (Adriani et al. Reference Adriani, Barbarino, Bazilevskaya, Bellotti, Boezio, Bogomolov, Bonechi, Bongi, Bonvicini and Borisov2011; Aguilar et al. Reference Aguilar, Aisa, Alpat, Alvino, Ambrosi, Andeen, Arruda, Attig, Azzarello and Bachlechner2015a,Reference Aguilar, Aisa, Alpat, Alvino, Ambrosi, Andeen, Arruda, Attig, Azzarello and Bachlechnerb) and also, though with somewhat lower statistics, of heavier nuclei (Ahn et al. Reference Ahn, Allison, Bagliesi, Beatty, Bigongiari, Childers, Conklin, Coutu, DuVernois and Ganel2010). Within the framework of diffusive transport, the detection of a break in the spectrum of CRs can be interpreted as a signature of a change either in the injection spectrum or in the diffusion coefficient. During propagation through the Galaxy, CRs primarily loose energy due to adiabatic expansion and ionisation. In addition, particles of a given species can be also lost due to spallation or decay. However, if one focuses on stable primary CRs, with energy above a few tens of GeV, energy losses during propagation can be neglected and a very simple expression for the diffuse steady-state spectrum in the Galaxy can be found. If we assume that CRs are injected in the Galaxy at a constant rate

(4.1)\begin{equation} Q_\textrm{p}(E)\propto E^{-\gamma_\textrm{inj}}, \end{equation}

and that particles then propagate diffusively with an average diffusion coefficient

(4.2)\begin{equation} D(E)\propto E^\delta, \end{equation}

the steady-state spectrum of stable CR nuclei in the Galaxy will be given by the product between injection and confinement time, $Q(E)\times \tau _\textrm {esc}$. This will read, for primary nuclei,

(4.3)\begin{equation} N_\textrm{p}(E)\propto Q(E)\frac{H^2}{D(E)}\propto E^{-\gamma_\textrm{inj}-\delta}, \end{equation}

where the confinement time has been taken to be $\tau _\textrm {esc}=H^2/D(E)$, with $H$ the size of the magnetised halo in which CRs are confined (see e.g. Amato & Blasi Reference Amato and Blasi2018 for a more refined description). It is then clear then that the detected hardening implies a change in $\gamma _\textrm {inj}$ or $\delta$. Several models have been proposed invoking either of the two (Tomassetti Reference Tomassetti2012, Thoudam & Hörandel Reference Thoudam and Hörandel2012, and references therein). A possibility is that this feature is signalling the importance of non-linear effects in CR propagation. An early suggestion (Blasi, Amato & Serpico Reference Blasi, Amato and Serpico2012) was that the break could point to the transition between scattering in self-generated and external turbulence. It had indeed been suggested since the 1970s (Wentzel Reference Wentzel1974; Cesarsky Reference Cesarsky1980) that at scales comparable with the Larmor radius of GeV particles, CRs could be an important source of turbulence in the galaxy through the resonant streaming instability. At larger scales, on the other hand, CRs become too few, given their steep spectrum, and the main source of scattering would become the large-scale turbulence present in the Galaxy. The latter is usually assumed to be injected by SNRs on a scale of order tens to 100 pc and then cascade to smaller scales developing a Kolmogorov-type spectrum $k^{-5/3}$. Such a spectrum translates in a diffusion coefficient with $\delta \approx 1/3$, flatter than the low-energy value $\delta \approx 0.7$ that is appropriate to describe CR self-generated turbulence. A back of the envelope calculation places the transition between self-generated and external turbulence at a CR rigidity in the range 200–300 GV (Blasi et al. Reference Blasi, Amato and Serpico2012), tantalisingly close to the value of 336 GV at which AMS-02 detects the hardening in the proton spectrum.

This explanation of the hardening as due to a change in the properties of galactic transport entails a clear prediction for the spectrum of secondary CR nuclei. These are injected in the Galaxy as a result of spallation of primaries. If we approximate the spallation cross-section as independent of energy, their injection will be

(4.4)\begin{equation} Q_\textrm{sec}(E)\propto N_p(E) \sigma_\textrm{sp}c n_\textrm{ISM}\propto E^{-\gamma_\textrm{inj}-\delta}, \end{equation}

where $n_\textrm {ISM}$ is the target gas density. Their equilibrium spectrum in the Galaxy will then read

(4.5)\begin{equation} N_\textrm{sec}(E)\propto Q_\textrm{sec}(E)\tau_\textrm{esc}\propto E^{-\gamma_\textrm{inj}-2\delta}, \end{equation}

which implies that any hardening $\varDelta$ of the spectrum of primaries due to a change in the slope of the diffusion coefficient will reflect in a hardening of the spectrum of secondaries equal to $2\varDelta$: this expectation is perfectly consistent with the analysis of secondaries performed by AMS-02 (Aguilar et al. Reference Aguilar, Ali Cavasonza, Ambrosi, Arruda, Attig, Aupetit, Azzarello, Bachlechner, Barao and Barrau2018).

Within this interpretation of the observed hardenings, the transport of CRs through the Galaxy becomes a complex non-linear problem, where the diffusion coefficient and the spectra of all nuclei need to be determined self-consistently. Once this complex problem is solved, however, and all available AMS-02 data are reproduced, the low-energy Voyager data (Stone et al. Reference Stone, Cummings, McDonald, Heikkila, Lal and Webber2013) are automatically reproduced (Aloisio, Blasi & Serpico Reference Aloisio, Blasi and Serpico2015). One important result that comes out of this analysis is directly related to the boron-over-carbon (B/C) measurements of AMS-02 at high energies (Aguilar et al. Reference Aguilar, Ali Cavasonza, Ambrosi, Arruda, Attig, Aupetit, Azzarello, Bachlechner, Barao and Barrau2016b). The B/C ratio has traditionally been considered the primary indicator of CR transport: B is the most abundant stable secondary and its mostly produced by the spallation of C, though the contribution by N and O is not negligible. In practice, the B/C ratio provides a direct measurement of the grammage (mass per unit surface, or mass density integrated along the path length) encountered by CRs during their propagation from their sources to Earth. If one compares (4.5) and (4.3), for the spectrum of secondary and primary nuclei, respectively, one immediately sees that the ratio between the two is expected to scale with energy exactly as the diffusion coefficient. In fact this ratio, as measured by AMS-02 (Aguilar et al. Reference Aguilar, Ali Cavasonza, Ambrosi, Arruda, Attig, Aupetit, Azzarello, Bachlechner, Barao and Barrau2016b), well agrees with a high-energy slope of the diffusion coefficient $\delta \approx 0.4$ (Aloisio et al. Reference Aloisio, Blasi and Serpico2015; Evoli et al. Reference Evoli, Aloisio and Blasi2019), but an additional, energy independent contribution seems to be required to well reproduce not only the highest-, but also the lowest-energy (Cummings et al. Reference Cummings, Stone, Heikkila, Lal, Webber, Jóhannesson, Moskalenko, Orlando and Porter2016) available data. The additional grammage needed, $X_s \approx 0.15\ \textrm {g\ cm}^{-2}$, is independent of energy and of the order of what particles can accumulate within a SNR, or any source with an ambient density of order $1\ \textrm {cm}^{-3}$ and a duration $\approx 10^4\ \textrm {yr}$, during acceleration (Aloisio et al. Reference Aloisio, Blasi and Serpico2015; Bresci et al. Reference Bresci, Amato, Blasi and Morlino2019). Within this modelling, namely if hardenings are interpreted as a result of turbulence self-generation at low energies, the contribution $X_s$ turns out to be fundamental also for explaining another recent surprise found in CR data, namely the spectrum of anti-protons, which we discuss in § 6.

Before concluding this section we would like to remark on the most important consequence of a scenario in which CR propagation at high energy is described by a diffusion coefficient $\delta \approx 0.4$: CRs must be injected in the galaxy with a spectrum $E^{-\gamma _\textrm {inj}}$ with $\gamma _\textrm {inj}\approx 2.3$. This combination of injection and propagation parameters and, in particular, the relatively weak energy dependence of the diffusion coefficient, helps to explain the low level of anisotropy detected at TeV energies, as shown by Blasi & Amato (Reference Blasi and Amato2012a,Reference Blasi and Amatob) and Sveshnikova, Strelnikova & Ptuskin (Reference Sveshnikova, Strelnikova and Ptuskin2013). On the other hand, as discussed in § 3, such a steep spectrum makes it very difficult for SNRs to be the primary sources of PeV CRs.

Within a DSA scenario, SNRs accelerate the highest-energy CRs (up to at least a few PeV) at the transition between the free expansion and the Sedov–Taylor phase, which typically happens a few hundred years after the SN explosion. During the Sedov–Taylor phase the SN shock slows down and the magnetic field intensity decreases, so that the shock cannot confine any longer the most energetic particles, which escape the SNR. This means that a SNR can act as a PeVatron for a relatively short time. Considering the rate of SN explosions in the entire Galaxy (about three per century), the chances of observing a SNR when it is still a PeVatron are thus very low, and any proof of emission from PeV CRs even from young SNRs is challenging to find.

5. Alternative CR sources

Although $\gamma$-ray observations have proven that SNRs are efficient accelerators of CR electrons, and possibly protons, up to 100 TeV, the hypothesis that acceleration of CRs proceeds up to PeV energies has been rejected in all known young SNRs because of the clear cutoffs detected at several TeVs in their spectra.

The GeV-to-TeV radiation from young shell-type SNRs, which had been long expected to provide final evidence to settle the question of the origin of CRs, can be either of hadronic or leptonic origin. Typically, a 1 TeV $\gamma$-ray photon is emitted either by an electron or a proton of about 10 TeV. The cooling time for 10 TeV electrons is $\approx 5 \times 10^4$ yr (for IC scattering off the CMB photons) whereas for protons the cooling time is $5 \times 10^7 (n/1 {\rm cm}^{-3})^{-1}$ yr (see e.g.Aharonian Reference Aharonian2004). Thus, the ratio of the luminosity in IC $\gamma$-rays to $\pi ^0$-decay $\gamma$-rays is of the order of $10^3 \times ({W_e}/{W_p}) \times {({n}/{\mathrm {1cm}^{-3}})}^{-1}$. The leptonic contribution to the emission is thus dominant over the hadronic one unless ${W_e}/{W_p} \ll 10^{-3}$ or, alternatively, if the gas density inside the shell is $n \gg 1 \ \textrm {cm}^{-3}$. This latter condition, however, if realised, would have the drawback of slowing down the shock wave very quickly and thus prevent efficient acceleration. On the other hand, a much larger energy density in protons than in electrons could result from rapid cooling of the electron population, especially in the presence of an amplified magnetic field, as inferred for young SNRs. If $B>10\ \mathrm {\mu }$G, only a small fraction of their energy $w_{MBR}/w_{B} \approx 0.1 (B/10\ \mathrm {\mu }\textrm {G})^{-2}$, is released in IC $\gamma$-rays, making the conditions for detection of $\pi ^0$ decay more favourable.

To recap, no known SNR has proved to accelerate particles beyond 100 TeV. In addition, the power in accelerated protons within SNRs derived from $\gamma$-ray observations depends on the highly uncertain local gas density $n$ and on the magnetic and radiation fields within the remnant. As a result, it has so far been impossible to unequivocally prove that the population of SNRs in the Galaxy injects the necessary power (about 10$^{50}$ erg per SN event) to sustain the CR population. In fact simulations of the SNR population show how the choice of acceleration and ISM parameters lead to remarkably different SNR populations as CR sources (Cristofari et al. Reference Cristofari, Gabici, Humensky, Santand er, Terrier, Parizot and Casanova2017).

Some other classes of astrophysical sources, such as superbubbles or star-forming regions (see § 5.2) or remnants of GRBs in our Galaxy (Atoyan, Buckley & Krawczynski Reference Atoyan, Buckley and Krawczynski2006), and, in particular, the centre of our Galaxy (see § 5.1), have long been proposed as alternative accelerators of particles and major contributors to the population of Galactic CRs up to PeV energies (Casse & Paul Reference Casse and Paul1980; Voelk & Forman Reference Voelk and Forman1982; Cesarsky & Montmerle Reference Cesarsky and Montmerle1983; Bykov & Toptygin Reference Bykov and Toptygin2001; Parizot et al. Reference Parizot, Marcowith, van der Swaluw, Bykov and Tatischeff2004; Torres, Domingo-Santamaría & Romero Reference Torres, Domingo-Santamaría and Romero2004; Higdon & Lingenfelter Reference Higdon and Lingenfelter2005; Atoyan et al. Reference Atoyan, Buckley and Krawczynski2006; Reimer, Pohl & Reimer Reference Reimer, Pohl and Reimer2006; Aharonian, Yang & de Oña Wilhelmi Reference Aharonian, Yang and de Oña Wilhelmi2019).

6. Implications of anti-matter data and CR leptons

One final subject we want to address in this paper concerns the implications of recent observational results on the spectrum of CR electrons, positrons and anti-protons. Although electrons can be directly accelerated from the ISM plasma, positrons and anti-protons are extremely low in number in the ordinary ISM and, hence, they have long been considered to be purely secondaries, namely a byproduct of CR interactions during propagation in the Galaxy. If one assumes that $e^+$ and $\bar p$ are pure secondaries, the straightforward expectation is that the ratio between their fluxes and that of protons (their primary parent particles) should monotonically decrease with energy, as is true for the B/C ratio discussed in § 4. In fact, the ratio between $e^+$ and $p$ should decrease even faster, because positrons are subject to radiation losses which further steepen the spectrum at high energies (see e.g.Amato & Blasi (Reference Amato and Blasi2018) for extended discussion). Direct observations of the flux of positrons (Adriani et al. Reference Adriani, Barbarino, Bazilevskaya, Bellotti, Boezio, Bogomolov, Bonechi, Bongi, Bonvicini and Bottai2009; Aguilar et al. Reference Aguilar, Alberti, Alpat, Alvino, Ambrosi, Andeen, Anderhub, Arruda, Azzarello and Bachlechner2013) and anti-protons (Aguilar et al. Reference Aguilar, Ali Cavasonza, Alpat, Ambrosi, Arruda, Attig, Aupetit, Azzarello, Bachlechner and Barao2016a) contradict this expectation, showing that the spectra of $e^+$, $\bar p$ and $p$ are all parallel.

Several different scenarios have been proposed in the literature to explain this finding (e.g.Blum, Katz & Waxman Reference Blum, Katz and Waxman2013; Cowsik, Burch & Madziwa-Nussinov Reference Cowsik, Burch and Madziwa-Nussinov2014; Cowsik & Madziwa-Nussinov Reference Cowsik and Madziwa-Nussinov2016; Eichler Reference Eichler2017; Lipari Reference Lipari2017). An interesting possibility is that CRs accumulate sizeable grammage in the vicinity of their sources, where the diffusion coefficient is reduced with respect to the Galactic average. Here diffusion is energy dependent and this is where most of the boron is produced. Then, in the rest of the Galaxy, diffusion must be energy independent and faster than usually assumed. Most $e^+$ and $\bar p$ would be produced by particles that accumulate most of their grammage in this second region: at a given rigidity their parent particles are $\sim$10 times more energetic than those that produce B nuclei, and hence spend a shorter time in the source vicinity.

Although it is not fully clear that these assumptions would not violate other constraints, such as those provided by anisotropy, a possible physical justification of such a scenario could stem from the effects of self-generated turbulence (see § 3) which could considerably reduce the diffusion coefficient in the vicinity of CR sources. In terms of theory the viability of this scenario is not fully clear, especially due to the unknowns related to the abundance of neutrals, which can effectively damp the CR-induced resonant streaming instability (D'Angelo, Blasi & Amato Reference D'Angelo, Blasi and Amato2016; Nava et al. Reference Nava, Gabici, Marcowith, Morlino and Ptuskin2016). However, there are observational indications that enhanced confinement indeed occurs, at least in the vicinity of some sources. A striking example are the so-called TeV halos (Abeysekara et al. Reference Abeysekara, Albert, Alfaro, Alvarez, Álvarez, Arceo, Arteaga-Velázquez, Avila Rojas, Ayala Solares and Barber2017) surrounding evolved pulsars. Many more cases are expected to be found with next-generation $\gamma$-ray telescopes, such as CTA.

In reality, given the present uncertainties in many of the parameters involved, among which, most notably, the $\bar p$ production cross-section (Korsmeier, Donato & Di Mauro Reference Korsmeier, Donato and Di Mauro2018), the $\bar p/p$ ratio is not in a statistically significant disagreement with the standard description of CR propagation through the Galaxy, nor with the B/C ratio, when all available information is taken into account and the subtle effects of re-acceleration are included in the description (Bresci et al. Reference Bresci, Amato, Blasi and Morlino2019). Indeed the high precision of the currently available data prompts theory to move forward and include second-order effects: one of these is the finite probability that during propagation particles might encounter an active CR accelerator (e.g. a SNR shock) and be re-accelerated. This has no effect on CR primaries, but flattens the spectrum of CR secondaries. Taking this phenomenon into account, together with the hardening of primaries, the energy dependence of the $\bar p$ production cross-section and, finally, the source grammage mentioned in § 4, allows one to reproduce the B/C ratio, the $\bar p/p$ ratio and all available data on stable nuclei (Bresci et al. Reference Bresci, Amato, Blasi and Morlino2019).

Differently from $\bar p$, the spectrum of $e^+$ is not easy to reproduce in the standard CR propagation scenario, unless additional sources of primary positrons are invoked.

The leptonic CR population is a small fraction of the total CR population. For instance at $\sim$1 GeV the ratio of the hadronic versus leptonic fluxes measured at Earth is roughly 100 : 1 (https://lpsc.in2p3.fr/crdb/). Differently from hadrons, leptons are heavily affected by energy losses owing to synchrotron and IC processes in the interstellar magnetic and radiation fields. If positrons were purely secondary products of CR interactions, their spectrum would have to be steeper than that of protons at energies where losses are important. As recently shown by Diesing & Caprioli (Reference Diesing and Caprioli2020), in order for this not to happen, the escape time from the Galaxy would have to be so short that protons would not be able to produce the observed $\bar p$ flux.

The anomalous positron spectrum has been at the origin of a huge number of papers, proposing either astrophysical or particle physics (dark matter annihilation-related) solutions (Aguilar et al. Reference Aguilar, Ali Cavasonza, Ambrosi, Arruda, Attig, Azzarello, Bachlechner, Barao, Barrau and Barrin2019b). Limiting the discussion to astrophysical scenarios, important constraints on the plausibility of the different proposals come from the maximum age and distance of the highest-energy lepton sources. As shown in figure 4, electrons and positrons are now detected up 20 TeV. The electron spectrum shows roughly a constant slope up to an energy of about 1 TeV, above which the spectrum becomes steeper. One possibility to interpret the break at 1 TeV is to relate it to the transition between a regime where a large number of lepton accelerators contribute to the spectrum, to a regime where only a few or even only one single electron source very close to the Sun contributes (Recchia et al. Reference Recchia, Gabici, Aharonian and Vink2019).

Figure 4. (a) Differential spectrum of electrons and positrons multiplied by $E^3$. CR electrons can be measured using spaceborne instruments such as AMS or Fermi-LAT up to 1 $\sim$TeV and CALET or DAMPE up to $\sim$10 TeV. Ground-based Cherenkov telescopes, such as H.E.S.S., MAGIC and VERITAS, which benefit from very large effective areas, have measured the flux up to 20 TeV. The black line shows the proton spectrum multiplied by 0.01 (Shikaze et al. Reference Shikaze, Haino, Abe, Fuke, Hams, Kim, Makida, Matsuda, Mitchell and Moiseev2007). (b) Positron fraction. Not all experiments are able to measure the electrons and positrons separately. Here measurements by Aguilar et al. (Reference Aguilar, Ali Cavasonza, Alpat, Ambrosi, Arruda, Attig, Azzarello, Bachlechner, Barao and Barrau2019a), Adriani et al. (Reference Adriani, Barbarino, Bazilevskaya, Bellotti, Boezio, Bogomolov, Bonechi, Bongi, Bonvicini and Bottai2009) and Beatty et al. (Reference Beatty, Bhattacharyya, Bower, Coutu, Duvernois, McKee, Minnick, Müller, Musser and Nutter2004) are reported, together with a number of model predictions: the black curve is for pure secondary production in the standard scenario (Moskalenko & Strong Reference Moskalenko and Strong1998); the blue curve includes a more sophisticated propagation model (Gaggero et al. Reference Gaggero, Maccione, Di Bernardo, Evoli and Grasso2013); the green curve invokes dark matter decay (Ibarra, Tran & Weniger Reference Ibarra, Tran and Weniger2013); the red curve includes a contribution from pulsars (Yin et al. Reference Yin, Yu, Yuan and Bi2013). Figure reproduced from Particle Data Group et al. (Reference Zyla, Barnett, Beringer, Dahl, Dwyer, Groom, Lin, Lugovsky and Pianori2020).

The number of contributing sources depends, in turn, on the average intensity of the magnetic and radiation fields and on the diffusion coefficient in the Galaxy. The latter, in particular, seems to need some revision in light of the recent measurements of secondary nuclei, both stable and unstable, provided by AMS-02 (Aguilar et al. Reference Aguilar, Ali Cavasonza, Ambrosi, Arruda, Attig, Aupetit, Azzarello, Bachlechner, Barao and Barrau2018).

For the standard average values of the magnetic and radiation fields in the Galaxy, the cooling time of leptons with energies above few tens of GeV can be approximately estimated as

(6.1)\begin{equation} t_\textrm{cool} \approx 10^6 \left[\left(\frac{U_\textrm{rad}}{0.3\ \textrm{eV}\ \textrm{cm}^{{-}3}}\right)+\left(\frac{B}{3\ \mathrm{\mu}\textrm{G}}\right)^2\right]^{{-}1} \left(\frac{E}{1\ \mathrm{TeV}}\right)^{{-}1}. \end{equation}

Losses limit the distance $\lambda$ from which leptons of a given energy can reach us to $\lambda \approx 2\sqrt {D(E)t_\textrm {cool}}(E)$, with $D(E)$ the galactic diffusion coefficient.

Using for $D(E)$ the estimate provided by Evoli et al. (Reference Evoli, Morlino, Blasi and Aloisio2020b), which allows all the available AMS-02 data on both primary and secondary nuclei (both stable and unstable) to be reproduced, one can estimate that the highest-energy leptons detected at the Earth cannot come from further than $\lambda (1~\textrm {TeV})\approx 3.5$ kpc. This is a much larger distance than typically estimated in the past (see e.g. Atoyan, Aharonian & Völk Reference Atoyan, Aharonian and Völk1995) and ensures that many sources contribute to the lepton flux even at the highest energies (Evoli et al. Reference Evoli, Blasi, Amato and Aloisio2020a).

The most promising sources of primary positrons are likely PWNs. These are the nebulae formed by the highly relativistic magnetised wind produced by a fast spinning, highly magnetised neutron star (see e.g. Amato (Reference Amato2019) for a recent review). The magnetosphere of such a star is a very efficient anti-matter factory and the pulsar wind contains electrons and positrons in about equal amounts. At the wind termination shock, these particles are accelerated up to PeV energies with a spectrum in the form of a broken power law, harder than $E^{-2}$ at energies below $\approx$500 GeV and softer than $E^{-2}$ at higher energies. As discussed by Bykov et al. (Reference Bykov, Amato, Petrov, Krassilchtchikov and Levenfish2017), the accelerated particles are released in the ISM after the pulsar leaves its associated SNR to become a bow shock PWN. Once taken into account, the contribution of CR leptons from these sources allows to very well explain both the overall flux of leptons and the positron fraction with very reasonable values of the few free parameters (Evoli et al. Reference Evoli, Blasi, Amato and Aloisio2020a).