© G. Peron and F. Aharonian 2022

1 Introduction

The paradigm of Galactic cosmic rays (CRs) assumes that the locally measured CR density (ρ_⊙(1GeV) ~ 1 eV cm⁻³), represents the average level of CRs in the Galactic disk (GD) (see, e.g., Amato & Casanova 2021). During their confinement in the GD, CRs mix and lose track of their production sites, creating the so-called CR sea. The spatial distribution of CRs in the Milky Way depends on the distributions of CR sources and the diffusion coefficient characterizing the CR propagation in GDs. It is believed that the mixture of CRs caused by diffusion is so effective that one should expect a uniform distribution of CRs throughout the Galaxy with almost constant level of the CR sea. However, significant deviations of the density cannot be excluded both on small (tens of parsecs) scales, because of the concentration of active or recent CR accelerators, and on large (kiloparsec) scales, due to the spatial variations of the CR diffusion coefficient.

The locally measured CR fluxes give direct information about the CR sea level only at a single point in the Milky Way. Meanwhile, the measurements of the CR sea level throughout the Galaxy is of paramount importance. Low-energy (MeV/GeV) CRs play a significant role in the regulation of the ionization, chemistry, and dynamics of the gas and dust, and consequently on star and planet formation (Padovani et al. 2020). Moreover, CRs might have a non-negligible impact on the habitability of planets around other stars (Atri et al. 2014). At very high energies (TeV/PeV), the influence of CRs on these processes is less pronounced. However, the information about the distribution of the highest energy CRs in the GD is essential when searching for CR TeVatrons and PeVatrons, such as TeV and PeV particle accelerators, in the Milky Way.

γ-ray astronomy provides a unique channel for investigating the distribution of CRs far from the Solar System. Of particular interest is the γ-ray emission produced at interactions of CRs with the interstellar medium (ISM) which provide straightforward information on the CR content at the location of the interaction. The observations with the Fermi-Large Area Telescope (LAT) demonstrated the feasibility of this method: CR densities have been extracted both from studies of the diffuse γ-ray emission (Acero et al. 2016; Yang et al. 2016; Pothast et al. 2018) and from giant molecular clouds (GMCs) (Yang et al. 2015; Neronov et al. 2017; Aharonian et al. 2020; Peron et al. 2021). The latter, being small regions of enhanced gas density, provide localized information on the CR content with accuracy better than 100 pc.

Fermi-LAT is the only instrument that succeeded in measuring the γ-ray flux from “passive” GMCs. The latter are clouds that do not have a recent or currently operating CR accelerator(s) in their vicinity. Yet, the detection is limited to exceptionally massive (≳ 10⁶ M_⊙) or nearby (d ≲ 1 kpc) clouds, if the illuminating CR flux coincides with the local flux of CRs, J_⊙. The soft power-law spectrum of the CR sea (α_⊙~ 2.7) makes the studies at TeV and higher energies very difficult. The High-altitude Water Cherenkov (HAWC) collaboration reported flux upper limits from several local MCs (Abeysekara et al. 2021). Meanwhile, the High-energy StereoscopisSystem (H.E.S.S.) collaboration reported preliminary results on the detection at TeV energies of γ-rays from a GMC located in the Galactic plane characterized by an enhanced CR density (Sinha et al. 2021). The advent of new and improved γ-ray instruments opens up new possibilities for the exploration of the sea of Galactic CRs in the near future. The Cherenkov Telescope Array (CTA) is designed to reach a sensitivity better than the current IACT systems by an order of magnitude. This should allow the detection of at least a few “passive” MCs. More optimistic are the predictions in the ultra-high-energy (UHE; E_γ ≥ 100 TeV) band, thanks to the dramatic improvement of the flux sensitivity at these energies by the Large High Altitude Air Shower Observatory (LHAASO; Cao et al. 2021). In the next section, we discuss the perspectives for the detection of γ-rays from GMCs in high-, very-high-, and ultra-high-energy bands.

2 Cosmic-ray interaction in molecular clouds

The inelastic interactions of CRs with the interstellar gas result in the production and decay of secondary unstable products, mostly π-mesons, resulting in γ-rays, neutrinos, and electrons. The γ-ray emissivity induced in MCs by penetrating CRs depends on (i) the energy distribution of CR protons J(E_p), (ii) the density of the ambient hydrogen, and (iii) the content of heavier elements both in the projectile CRs and in ambient gas, quantified by the nuclear enhancement factor, ξ_N. The resulting flux on Earth is given by (1)

where (M₅ ≡ M∕10⁵ M_⊙, d_kpc = d∕1 kpc) is related to the column density of the targeted material, and dσ_pp→γ∕dE_γ is the differential γ-ray cross-section of proton-proton interactions as calculated by Kafexhiu et al. (2014) for the broad energy interval from the threshold of pion production ≈ 280 MeV to PeV energies. In the last part of Eq. (1), we define , the emissivity per target hydrogen atom, which only depends on the CR flux.

The local spectrum of protons, J_⊙, has been measured with great precision in the Earth’s vicinity (e.g., AMS - Aguilar et al. 2015 and DAMPE - Amenomori et al. 2021), from Earth (e.g., KASCADE Apel et al. 2013, Icetop Aartsen et al. 2019), and now also outside the Solar System thanks to the Voyager mission (Stone et al. 2019), where the CR spectrum does not suffer solar modulation. Recent measurements revealed that the CR proton spectrum does not have a single power-law shape; the spectral index Γ changes from 2.8 below ~700 GeV to 2.6 up to 15 TeV and steepens again (Γ ≈ 2.85 at higher energies (Lipari & Vernetto 2020). Above 1 PeV, CR measurements are carried out by ground-based detectors that measure the air showers produced by the CR interactions in the atmosphere. The interpretation of these measurements depends on the interaction models, which remain, to a certain extent, controversial. Moreover, the lack of CR measurements between 100 TeV and 1 PeV introduces additional uncertainties both in the proton spectrum and composition of heavier nuclei. At high energies, different experiments show discrepancies (see Fig. 1), as comprehensively discussed in Lipari & Vernetto (2020). Figure 1 shows the recent local measurements of the CR proton fluxes. The black line was obtained using the fitting parameters derived by Lipari & Vernetto (2020) above 100 GeV, while below 100 GeV, to account for the solar modulation, the curve matches the data of Voyager to the data of AMS, as was done in Vos & Potgieter (2015). This effect needs to be taken into account in order not to underestimate the γ-ray flux below ~3 GeV.

At low energies, for the standard compositions of the interstellar medium and CRs, the contribution of nuclei to the γ-ray production is comparable to the contribution from the pp-interactions, namely ξ_N ≈ 1.8 (Mori 2009). At higher energies, especially around the knee at PeV energies, CRs become “heavier”; consequently, the nuclear enhancement factor ξ_N increases with energy.The significant uncertainty in the CR composition in the knee region introduces non-negligible uncertainty in ξ_N. Our calculations based on the available CR data showed that ξ_N progressively increases from 1.8 at 10 GeV to ≃2.6 at 1 PeV.

The third parameter that determines the cloud’s flux is which is the measure of the column density of the gas embedded in the cloud. The gas column density can be traced by the ¹²CO (J = 1 →0) line, in the case of molecular gas, and by the HI line, in the case of atomic hydrogen. Alternatively, it can be derived by measuring the interstellar dust opacity, which traces both the molecular and the atomic phase. Given that M = N_colθd², we have A ∝ N_colθ, where θ is the angular area of the considered region. It can be presented in the following form: (2)

where dl and db are the pixel size of the gas tracer map. For the given CR spectrum, the only uncertainty in the prediction of the γ-ray flux of a GMC is related to A. On the other hand, A is independent of uncertainties of the cloud mass and of its distance. The only significant uncertainty of A is linked to the column density, N_col, and it depends on the different tracers of gas. MCs are usually identified in CO maps, which bring an uncertainty on the column density of ~ 30% (Bolatto et al. 2013). As a consequence, the relative uncertainty in their A factor and on their γ-ray flux is 30%.

Fig. 1 Local spectrum of CR protons as measured by different experiments (see the figure legend). The black line is the interpolation of the experimental points using the fitting parameters reported by Vos & Potgieter (2015) and Lipari & Vernetto (2020) below and above 100 GeV, respectively. The dotted part represents protons below the energy threshold of 280 MeV, which do not participate in the π-meson production.

3 Molecular clouds in the Milky Way

From the recent analysis of the all-Galaxy CO survey of Dame et al. (2000), Miville-Deschênes et al. (2016) identified more than 8000 MCs distributed all throughout the galactic plane. When inspecting the clouds of Miville-Deschênes et al. (2016), hereafter MD16, we see that most of the clouds have a low A parameter (see Fig. 2) below 0.4. This was determined in Aharonian et al. (2020) to be a safe threshold for spectral measurements with Fermi-LAT of clouds of different extensions, located both in the inner and outer parts of the Galaxy. These considerations were based on the assumption that the level of CRs that illuminate the cloud is coincident with the local level of CR, J_⊙, which was taken as a reference value. Among the MD16 catalog clouds, fewer than 1% are above the detection threshold of Fermi-LAT; the fraction is even lower when considering only the inner (|l| < 60°) Galactic regions (~0.3%). For the outer Galaxy, the threshold can be lowered by a factor of 2. However, in this part of the Galaxy the clouds are less dense, with A parameters in most cases lower than 0.6. Clouds in this region overcome the detection threshold only for the ~1% of the cases, even when lowering the threshold to A = 0.2. This means that in the most part of the GD the “CR sea” cannot be explored by Fermi-LAT. Remarkably, ~15% of the MCs, corresponding to more than 1000 MCs, have an A factor between 0.1 and 0.4, just below the Fermi-LAT detection threshold which could be reached with a small improvement of the detectorsensitivity.

In addition to the emissivity, source confusion affects the detectability of clouds in γ-rays. Confusion can arise both due to the proximity of known γ-ray sources, anddue to other clouds located along the same line of sight.

3.1 Overlaps with other γ-ray sources

We took into consideration all reported GeV and TeV γ-ray sources from the Fermi 4th source catalog (4FGL); The Fermi-LAT collaboration 2019), the HGPS (HESS Galactic Plane Survey; Abdalla et al. 2018), and the third HAWC catalog (3HWC; Albert et al. 2020), which lie within the radius of 1.1 θ, where θ is the angular size of the cloud: 75% of clouds do not have an overlapping source; 3% of clouds have at least one overlapping known source; 22% of clouds have only unidentified overlapping sources; 63% of the clouds do not have nearby sources within 0.5°. Clouds without nearby sources are ideal for testing the CR sea, even though this does not exclude possible contributions of yet unresolved γ-ray sources.

3.2 Fraction of gas along the line of sight

Differently from the smoothly distributed atomic gas, the molecular component of the interstellar medium (ISM) is clumpy and mostly concentrated in dense clouds. Miville-Deschênes et al. (2016) pointed out that the line of sight column densities, N_col, in most (≈60%) directions are contributed by three or fewer MCs; in the 20% of cases, the column density is dominated by a single cloud. Following the approach proposed by Peron et al. (2021), one can derive a relation between the fraction of column density that belongs to the cloud, ρ_cloud ≡ N_col,cloud∕N_col,tot, and the level of excess, x, with respect to the local γ-ray emissivity (ϕ_γ(J_⊙)) that can be detected: (3)

For example, if a cloud has an emissivity larger than the nominal value, by a factor x = 4, it would be detected if the fraction of column density belonging to the cloud is at least 32% (ρ_cloud = 0.32). For detection of the local CR sea in MCs (x = 1), the fraction of gas in the cloud has to be at least 65%. The condition expressed by Eq. (3) is calculated accounting only for variations in the normalization and not in the spectral index, and it requires that the cloud exceed over the background of at least 30% of the flux. The latter is the typical magnitude of the uncertainty of the gas column density and therefore of the flux normalization. This guarantees the distinction of the cloud above the background gas, even without subtracting the contribution of the latter. Otherwise, the flux of the cloud, even if enhanced, would be masked by the γ-ray flux of the back- and foreground gas. To avoid this, it is necessary to model the background and foreground gas as a separate source, as is done, for example, in Aharonian et al. (2020). This approach, however, is subject to the large uncertainties of the CO and HI measurements, which are the only tracers that can be used for 3D decomposition. We nevertheless draw attention to the fact that, even without a 3D decomposition, measuring a flux similar to J_⊙ coming from a column of gas is a strong indication that the entire column is emitting at a similar level as the local CR sea. If any exist, deviations from the nominal flux, J_⊙, would appear as an enhancement compared to the locally measured level of CRs. The local flux can be considered a minimum level, as no lower flux has been recorded so far, except for the outermost part of the Galaxy, which are far from the highest concentration of supernova remnants (SNRs) and pulsar wind nebulae (PWNe), while enhancements can be caused by nearby CR accelerators.

We calculated the fraction of gas belonging to each cloud of the MD16 catalog relatively to the total gas in the line of sight (l.o.s.) included in the area of the cloud from the brightness of the CO, W_CO: (4)

where θ is the angular size of the cloud, v is the radial velocity, and σ_v is the dispersion of the velocity profile assumed to be Gaussian. The results are shown in Fig. 3 for clouds not overlapping (within 1.1 θ) with any cataloged Fermi-LAT source. It appears that only one cloud satisfies all the conditions of detectability. The limiting factor is given by the threshold of A > 0.4 imposed by the sensitivity of Fermi-LAT. If the threshold is lowered to 0.2, there are more than 50 clouds that are the dominant objects on the l.o.s., having ρ > 0.65. This number rises to 200 if we lower the threshold to A = 0.1.

Fig. 2 Histograms of the A parameter and angular extension, θ, of MCs fromthe MD16-catalog. White bars include all the clouds in the catalog, while the gray bars correspond to MCs in the inner Galaxy (|l| < 60°), and hatched bars correspond to MCs in the outer Galaxy (|l| > 60°). Top panel:number fraction with respect to the total is also reported as percentages, and the inset panel shows a zoomed-in view of the parameter range below 0.4 determined as the detection threshold of MCs by Fermi-LAT (Aharonian et al. 2020). Lower panel: average angular resolutions of the currently operating γ-ray telescopes are reported.

Fig. 3 Distribution of the ratio of the cloud’s column density to the total column density in the direction of the cloud for different intervals of distances from the Galactic center (white solid bars). Only the clouds not overlapping with Fermi-LAT γ-ray sources are shown. The fraction of clouds characterized by the parameter A > 0.4 (solid gray) and 0.1 < A < 0.4 (hatched) are highlighted.

4 The potential of current γ-ray instruments

Fermi-LAT is a powerful, large, field-of-view γ-ray detector that performs best at GeV energies and has been observing the whole sky for more than 13 yr (since 2008). The sensitivity of the LAT for observations of the outer Galaxy (|l| > 60°) is two times better than for observations in the inner Galaxy. Fermi-LAT is well designed to explore extended Galactic sources, particularly SNRs and PWNe. This also concerns GMCs; however, the sensitivity of Fermi-LAT is at the margin of detection of γ-rays from only a handful GMCs, unless the CR density in the vicinity of the clouds substantially exceeds the local level. Another problem is the energy coverage. Because of the steep spectrum and the limited detection area of Fermi-LAT, the detection of γ-rays, even from the most favorable “passive” GMCs with A ~ 1, cannot be extended beyond 0.1 TeV. The range of TeV energies is the domain of ground-based detectors: imaging Atmospheric Cherenkov Telescopes (IACTs) and air shower particle arrays. Currently operating IACT detectors, the Very Energetic Radiation Imaging Telescope Array System (VERITAS), in Arizona, the Major Atmospheric γ-ray Imaging Cherenkov Telescopes (MAGIC) in the Canary islands, and the H.E.S.S., in Namibia, operate in the energy range between ~50 GeV and ~100 TeV. Located in the southern hemisphere, the H.E.S.S. is more sensitive to inner Galactic regions, while MAGIC and VERITAS are more sensitive to sources of the outer Galaxy. HAWC and LHAASO, located in Mexico and China, respectively,observe the northern sky. LHAASO in particular, with its unprecedented collection area, is sensitive to energies up to 1 PeV. For current VHE detectors, particularly H.E.S.S. and HAWC, GMCs illuminated by J_⊙ are not accessible. Only LHAASO will be able to detect “passive” GMCs, after a few years of observations.

This can be seen in the left panel of Fig. 4, where the flux sensitivities of the currently operating detectors, calculated for typical exposure times, are shown together with the flux induced by interactions of the CR sea with a cloud with A = 1. The following are shown in the plot: the sensitivity achieved after ten years of observations with Fermi-LAT for the inner (l, b = (0, 0); dark red) and outer (l, b = (0, 30); light red) Galaxy (Maldera et al. 2019); the H.E.S.S. sensitivity for 100 hours of observation with the four-telescope configuration (solid yellow) (Funk & Hinton 2013) and the preliminary calculation of the sensitivity for the five-telescope configuration (yellow dashed; Holler et al. 2015); the HAWC sensitivity for five years of observations (green;¹); and the LHAASO sensitivity for one-year observations (magenta; Di Sciascio 2016). The γ-ray flux of the given cloud exceeds the sensitivity of current instruments only at GeV energies.

The condition for visibility of a MC can be determined by imposing that the flux of a cloud is higher or equal than the sensitivity, S(t, E), calculated for a certain exposure time, t:

here, S₀ is the sensitivity calculated at a specific exposure t₀. In this sense represents acondition for visibility as it is the minimum ratio to detect a cloud of A = 1, which is characterized by a emissivity ϕ_γ with an instrument of sensitivity S₀(E) calculated for a t₀-long exposure.

The values for for the current γ-ray instruments are plotted (dotted curves) in Fig. 5, for the assumption of the local γ-ray emissivity ϕ_γ = ϕ_γ(J_⊙). One can see that in order to measure a similar emissivity as the local one with the current TeV instruments, of ~3 should be obtained. No single cloud in the Galaxy is characterized by A ~ 3, except for some of the Gould Belt’s clouds. However, these nearby clouds are very (~ 1°) extended; thus, the sensitivity is significantly reduced. The sensitivity for a source of extension θ compared to the point-like source is worsened by the following factor: (8)

where σ_PSF is the instrument’s point spread function (PSF). This results in a stricter condition on the visibility factor: (9)

As a consequence, the larger the source, the larger A should be to compensate the loss in sensitivity due to the higher background rate: . The worsening isespecially significant for imaging air Cherenkov telescopes (IACTs) with the best angular resolution of 0.05–0.1° or better (see the lower panel of Fig. 6). The effect is less dramatic for water Cherenkov (WC) detectors, which have a PSF of 0.1–0.3°, comparable to the Fermi-LAT one and to the typical angular extensions of most of the clouds in the Galactic plane.

The exposure time is another important factor for the detectability of GMCs. The visibility factor, , increases with time as. IACTs are pointed telescopes with a small (a few degrees) FoV, while WC detectors cover a significant fraction of the sky simultaneously. The typical exposure time for specific segments of the Galactic plane during the survey of the latter by IACTs over several years could be as large as 100 hours. H.E.S.S. in particular intensively surveyed the Galactic plane between l = 60° and l = 260°, reaching 300 hours of exposure in some regions. This is still not sufficient to detect TeV γ-rays from “passive” GMCs if illuminated by the local CR flux ().

The exposure time of a large fraction of clouds in the Galaxy by large-FoV ground-based air shower particle detectors is much larger; it can be as large as 2000 h per year (approximately 6 h per night). Nevertheless, to observe passive clouds with HAWC, at least a factor of is needed, which is too large to be reached only with the exposure time increase. This would in fact require a 100-times-larger exposure time.

The recently completed Water Cherenkov Detector Array (WCDA) of LHAASO will be able to detect a limited number (~ 0.06%) of passive GMCs between 1 and 10 TeV. A breakthrough is expected at higher energies, thanks to the superior sensitivity of the KM2A (square kilometer array) of LHAASO. For KM2A, after five years of observations, making the ultra-high energies as effective as Fermi-LAT at GeV energies, for searches of γ-ray-emitting clouds (see Figs. 4 and 5). Therefore, the ~1% of clouds of the MD16, corresponding to approximately 80 objects, will be accessible in the coming years through ultra-high energy observations.

Fig. 4 γ-ray fluxes expected from a GMC with A = 1 compared to the point-like source sensitivities of currently operating (left) and future (right) γ-ray instruments. On the left: the ten-year Fermi-LAT sensitivity for the outer (light red) and inner Galaxy (dark red); the H.E.S.S. sensitivity for 100-h observations with the four-telescope system configuration (solid yellow) and for 50 h with the five-telescope configuration (dashed yellow); the HAWC five-year sensitivity (green); and the LHAASO one-year sensitivity (solid violet). On the right: the sensitivity for a hypothetical Very Large Area Telescope (VLAT) with a three-times-larger effective area (blue curves); the expected sensitivity of CTA from the northern (light orange) and southern (orange) sites, for 50 h of observations; the sensitivity of SWGO (cyan) after five years of observations; and the sensitivity of LHAASO (violet) for five years of observations. The black solid line is the flux of a MC of A = 1 illuminated by the CR sea. The two spectra above 104 GeV represent two options of CR proton spectra based on the spectra reported above 106 GeV by the KASCADE (dashed line) and ICEtop (dotted line) collaborations (see main text). As a reference, the fluxes representing 10, 1, and 0.1% of the γ-ray flux from the Crab Nebula (Cao et al. 2021) are shown.

Analysis technique

The 3D-likelihood analysis technique is an essential tool to analyze faint and extended sources such as GMCs. This technique consists of defining a spatial and a spectral model of the sources in the field of view and fitting their parameters simultaneously. This method has largely been used in the analysis of GeV data, and it has recently been extended to the VHE regime, in particular to the analysis of H.E.S.S. data (Mohrmann et al. 2019). One of the main differences between space-based (GeV) and ground-based (TeV) observations is in the hadronic background rejection. In the first case, hadrons are already excluded from the detector, and in the second case their signal is detected and needs to be rejected in the analysis process. Traditional analysis methods (Berge et al. 2007) are based on the subtraction of the background calculated in an “off” region. While this is effective for observations of bright sources, this method is not sensitive to the large-scale diffuse emission, which is subtracted along with the background. The employment of the 3D analysis technique instead allows us to define specific regions where we can calculate the background and therefore efficiently separate it from the diffuse emission, which can be modeled as an independent component. Preliminary results on VHE observations of GMCs obtained with the 3D analysis method have been presented in Sinha et al. (2021). There, the authors applied an exclusionmask on the diffuse gas of the Galactic plane as traced by dust and evaluated the truly hadronic contributionelsewhere. The separation of the background caused by the hadronic showers from the large-scale diffuse emission makes the analysis more sensitive to low-surface-brightness sources with fluxes comparable to the diffuse γ-ray emission of the Galactic disk. Besides this, the simultaneous fitting of all sources in the region of interest and the possibility of defining a spatial template based on observations on other wavelengths help to resolve the cases of crowded sky regions.

Fig. 5 values calculated for the current (uppermost panel) and next-generation (middle panel) instruments, for different exposure times as functions of energy and of the angular extension (lowermost panel). In the two upper plots, the curves refer to the point-source hypothesis (dash-dotted lines) and to a 0.5°-wide source (solid lines). The percentage of the MD16 clouds that overcome an value of 0.05, 0.3, and 1, assuming the local CR spectrum, is indicated in the figure. As a reference, the values of Aquila rift (AqR) and Sagittarius B2 (SgrB) are also plotted. Those were computed from the spectra reported in Baghmanyan et al. (2020) and Aharonian et al. (2020), respectively. In the lowermost panel, the physical properties (A, θ) of the MCsof the MD16 catalog are compared to the detection thresholds of the considered gamma instruments, (see Eq. (7)). The same exposure as in the upper panels is assumed.

5 The prospects

The next generation of instruments will include CTA and the Southern wide-field γ-ray observatory (SWGO). These will be ten times more sensitive than the currently operating H.E.S.S. and HAWC. The CTA will have two sites: one in the northern and one in the southern sky, while SWGO will cover the southern hemisphere. Such an improved sensitivity would be promising to the detection of a few passive MCs, because in the case of the CTA, at 1 TeV. For typical clouds’ angular extensions, this factor increases to 1 with 50 h of observations. That can easily be improved by enlarging the exposure time on specific clouds. For instance, with a 300-h exposure, . The improved sensitivity, together with the better angular resolution, make the CTA an ideal instrument to study not only the spectral energy distribution, but also to investigate the spatial distribution of CRs inside the cloud itself. SWGO will also reach adequate sensitivity levels of after five years. Therefore, it could be a valid counterpart of WCDA-LHAASO to observe the southern sky, where the most massive clouds are located. Nevertheless, even with the improved sensitivity of the forthcoming γ-ray telescopes, the measurements will remain limited to a handful of clouds in the VHE regime.

Meanwhile, even a relatively moderate, by a factor of Σ = 2−3, improvement of the Fermi-LAT sensitivity at GeV energies would dramatically increase the number of detectable clouds and thus provide a probe of the CR energy density throughout a substantial fraction of the Galactic disk. We define Σ as the overall improvement in sensitivity of a future GeV instrument compared to Fermi-LAT.

Achieving such an improvement in sensitivity is not a trivial task. Given that GeV γ-rays are detected in the background-dominated regime, the minimum detectable flux (sensitivity) decreases with the exposure time as t^−1∕2. Therefore, the resource of 12-yr old Fermi-LAT in this regard is rather limited. Even assuming that Fermi-LAT will continue observations for another decade, the gain in the sensitivity cannot exceed 40%. Clearly, one needs a new, more sensitive detector of GeV γ-rays. The improvement of sensitivity for the specific task of detecting γ-rays from extended GMCs cannot be realized by improving the angular resolution. Taking Fermi-LAT’s ten-year sensitivity as reference, Fig. 6 shows how enlarging the exposure time by a factor of τ or the size of the detector by a factor of Λ affects the sensitivity. A breakthrough can be achieved only through an order of magnitude (Σ > 3) increase of the detection area. Thus, we would need a new Very large area telescope (VLAT) to achieve our goals. This is a challenging yet feasible task for the space-based instruments (see, e.g., the recent proposal for the Advanced Particle-astrophysics Telescope (APT; Buckley et al. 2019). An improvement in sensitivity of a Σ factor of ~3 can already be achieved through a 15-yr observation with a Fermi-like instrument of (3 × 3) m² in size. The APT aims to have an effective area ~10 times largercompared to the Fermi-LAT one. Two designs have been proposed: a (3 × 3) m² instrument and a (3 × 6) m² one. It is clear that, if this project is approved, it will be an ideal instrument for our scopes.

With its current sensitivity, the Fermi-LAT can map at most 1% ( 0.3) of the MCs identified in the MD16-catalog between 1 and 10 GeV. This corresponds to 80 objects, of which only 15 belong to the inner (<4 kpc) Galaxy. Lowering the sensitivity by Σ = 3 would increase the detectable clouds to more than 1100, of which ~150 are in the inner Galaxy. Moreover, it will allow us to extend the sampled energy range for most clouds. The spatial distribution of the detectable MCs from the MD16-catalog is plotted in Fig. 7. We considered 0.7 (0.1) and 0.25 (0.04) as a threshold in order to have a detection up to 10 GeV with ten years of observations with LAT and VLAT, respectively, of clouds in the inner (outer) Galaxy. The angular size of the clouds was taken into consideration to draw the plot, and the PSF value of 0.15°, which is characteristic of Fermi-LAT at 10 GeV, is assumed. This can be regarded as a lower limit of the number of clouds that can be detected; with a larger PSF, such as the one that characterizes the LAT at lower energy (~0.5°), the detection is further favored. With an improvement of Σ =3, all galactocentric distances will be sampled with sufficient clouds, especially the 2–4 kpc ring, which is the most difficult to analyze because it isprojected in a small range of longitudes, and therefore several sources may overlap. Finally, while Fermi-LAT is limited to the observation of MCs relatively close to the Galactic plane, an advanced detector with improved sensitivity would allow access to several locations up to 400 parsecs above the plane (see the lower panel of Fig. 7). The combined knowledge of the CR density at different distances from the Galactic center and at different heights from the Galactic plane would improve our knowledge regarding the propagation properties of these high-energy particles in the radial and perpendicular directions. The same evaluation was conducted for LHAASO observations at 100 TeV, considering a = 0.2 and a PSF of 0.2° and plotted in the right panels of Fig. 7. We assumed the same performance in the entire Galaxy, even though the sensitivity in the inner Galaxy should be worse.

Fig. 6 Dependence of the sensitivity on different factors. Upper panel: combined effect of extending (Λ) the size of the telescope and increasing (τ) the observation time with regard to the total improvement(Σ) of the sensitivity, taking Fermi-LAT as a reference. Lower panel: worsening factor of the sensitivity due to the source extension asa function of the extension for different angular resolutions.

Runaway cosmic rays

While the above considerations involved the hypothesis of a uniform CR sea coincident with the local CR flux, the detectability of a cloud in γ-rays is significantly improved when the cloud is located in an environment with enhanced CR density caused by the proximity of recent and currently operating CR accelerator(s). Runaway CRs, meaning particles that already have left the accelerator and injected into the circumstellar medium have been registered both at GeV and TeV energies in the vicinity of some middle-aged supernova remnants (e.g., W44 Peron et al. 2020, W28 Aharonian et al. 2008). The spectrum of runaway particles close to or inside the clouds is hard to predict as it depends on different conditions such as the age of the accelerator, the distance of the clouds, and the diffusion coefficient (Aharonian & Atoyan 1996). Meanwhile, the flux can be enhanced in the surroundings (~25 pc), for example, of a middle-aged SNR such as W44 (Peron et al. 2020), by an order of magnitude compared to the local CR flux. If the injection occurs in a continuous regime, as in the case of massive star clusters, the CR density is expected to strongly peak towardthe accelerator; therefore, it could be enhanced around the latter by orders of magnitude (Aharonian et al. 2019). VHE instruments may play a important role in investigating the CR contents in these, “active”, clouds. The visibility of GMC with H.E.S.S. and CTA when illuminated by CR escaping SNRs under certain acceleration and propagation hypotheses is discussed in detail by Mitchell et al. (2021). Observations of the escaped particles are fundamental to understanding the entire acceleration power of a source (Gabici et al. 2007), and new and future γ-ray instruments will help in constraining the spectrum of escaped particles at the highest energies.

6 Conclusions

γ-ray-emitting GMCs play a unique role as CR barometers, allowing deep probes of the energy density of CRs throughout the Galactic disk, with long-lasting astrophysical implications. The γ-ray fluxes from GMCs are faint and extended, which makes their detection difficult. Yet, the analysis of Fermi-LAT observations of the Galactic disk revealed γ-rays from a limited number of GMCs in the energy interval between 1 and 100 GeV (Aharonian et al. 2020; Peron et al. 2021; Baghmanyan et al. 2020). These results convincingly demonstrated the feasibility for GMC detection in γ-rays. At the same time, they indicated that the potential of Fermi-LAT with regard to the studies of GMCs is almost saturated. To deeply probing CRs in different, more remote parts of the Galactic disk, we need a new and advanced γ-ray detector in the GeV band (a “VLAT”) with improved sensitivity compared to Fermi-LAT by a factor of a few. Hopefully, such a detector will come to light in the foreseeable future. Such an instrument will be beneficial not only for effectively probing the CR sea but also for searching for dark matter, investigating the nature of γ-ray bursts, and resolving other faint sources.

Although with the increase of energy, the detection of GMCs in γ-rays becomes more challenging, the CTA, as well as the water Cherenkov detectors like the proposed SWGO and currently operating WCDA-LHAASO, should be able to detect γ-rays in the energy interval between 1 and 10 TeV from GMCs characterized by the parameter A ≳ 0.5. The domain of ultra-high-energy γ-rays from 30 TeV up to 1 PeV looks even more promising. One may predict that the recently completed and presently working in its full power KM2A-LHAASO will detect GMCs in ultra-high energies in the coming years and thus contribute significantly to uncovering the origin of highest energy CRs around the knee and beyond.

Fig. 7 Spatial distribution of the clouds from the MD16-catalog in the (Xgal, Ygal) plane (upper panel) and in the (Rgal, Zgal) plane (lower panel). On the left, the MCs that overcome the detection threshold of Fermi-LAT and of an advanced detector with improved sensitivity Σ = 3 are indicated in pink and blue, respectively. On the right, the clouds visible by LHAASO after five (pink) and ten (blue) years of observations are indicated. The catalog size of the clouds is considered, and angular resolutions of 0.15° and 0.2° are assumedfor the LAT and LHAASO, respectively.

Acknowledgements

We would like to thank G. Di Sciascio for the discussion on the LHAASO sensitivity, and the anonymous referee for the very helpful comments and suggestions which helped us to improve the manuscript.

References

All Figures

Fig. 1 Local spectrum of CR protons as measured by different experiments (see the figure legend). The black line is the interpolation of the experimental points using the fitting parameters reported by Vos & Potgieter (2015) and Lipari & Vernetto (2020) below and above 100 GeV, respectively. The dotted part represents protons below the energy threshold of 280 MeV, which do not participate in the π-meson production.

In the text

Fig. 2 Histograms of the A parameter and angular extension, θ, of MCs fromthe MD16-catalog. White bars include all the clouds in the catalog, while the gray bars correspond to MCs in the inner Galaxy (|l| < 60°), and hatched bars correspond to MCs in the outer Galaxy (|l| > 60°). Top panel:number fraction with respect to the total is also reported as percentages, and the inset panel shows a zoomed-in view of the parameter range below 0.4 determined as the detection threshold of MCs by Fermi-LAT (Aharonian et al. 2020). Lower panel: average angular resolutions of the currently operating γ-ray telescopes are reported.

In the text

	Fig. 3 Distribution of the ratio of the cloud’s column density to the total column density in the direction of the cloud for different intervals of distances from the Galactic center (white solid bars). Only the clouds not overlapping with Fermi-LAT γ-ray sources are shown. The fraction of clouds characterized by the parameter A > 0.4 (solid gray) and 0.1 < A < 0.4 (hatched) are highlighted.
In the text

Fig. 4 γ-ray fluxes expected from a GMC with A = 1 compared to the point-like source sensitivities of currently operating (left) and future (right) γ-ray instruments. On the left: the ten-year Fermi-LAT sensitivity for the outer (light red) and inner Galaxy (dark red); the H.E.S.S. sensitivity for 100-h observations with the four-telescope system configuration (solid yellow) and for 50 h with the five-telescope configuration (dashed yellow); the HAWC five-year sensitivity (green); and the LHAASO one-year sensitivity (solid violet). On the right: the sensitivity for a hypothetical Very Large Area Telescope (VLAT) with a three-times-larger effective area (blue curves); the expected sensitivity of CTA from the northern (light orange) and southern (orange) sites, for 50 h of observations; the sensitivity of SWGO (cyan) after five years of observations; and the sensitivity of LHAASO (violet) for five years of observations. The black solid line is the flux of a MC of A = 1 illuminated by the CR sea. The two spectra above 104 GeV represent two options of CR proton spectra based on the spectra reported above 106 GeV by the KASCADE (dashed line) and ICEtop (dotted line) collaborations (see main text). As a reference, the fluxes representing 10, 1, and 0.1% of the γ-ray flux from the Crab Nebula (Cao et al. 2021) are shown.

In the text

Fig. 5 values calculated for the current (uppermost panel) and next-generation (middle panel) instruments, for different exposure times as functions of energy and of the angular extension (lowermost panel). In the two upper plots, the curves refer to the point-source hypothesis (dash-dotted lines) and to a 0.5°-wide source (solid lines). The percentage of the MD16 clouds that overcome an value of 0.05, 0.3, and 1, assuming the local CR spectrum, is indicated in the figure. As a reference, the values of Aquila rift (AqR) and Sagittarius B2 (SgrB) are also plotted. Those were computed from the spectra reported in Baghmanyan et al. (2020) and Aharonian et al. (2020), respectively. In the lowermost panel, the physical properties (A, θ) of the MCsof the MD16 catalog are compared to the detection thresholds of the considered gamma instruments, (see Eq. (7)). The same exposure as in the upper panels is assumed.

In the text

Fig. 6 Dependence of the sensitivity on different factors. Upper panel: combined effect of extending (Λ) the size of the telescope and increasing (τ) the observation time with regard to the total improvement(Σ) of the sensitivity, taking Fermi-LAT as a reference. Lower panel: worsening factor of the sensitivity due to the source extension asa function of the extension for different angular resolutions.

In the text

Fig. 7 Spatial distribution of the clouds from the MD16-catalog in the (Xgal, Ygal) plane (upper panel) and in the (Rgal, Zgal) plane (lower panel). On the left, the MCs that overcome the detection threshold of Fermi-LAT and of an advanced detector with improved sensitivity Σ = 3 are indicated in pink and blue, respectively. On the right, the clouds visible by LHAASO after five (pink) and ten (blue) years of observations are indicated. The catalog size of the clouds is considered, and angular resolutions of 0.15° and 0.2° are assumedfor the LAT and LHAASO, respectively.

In the text