<figure><img src="image_148.png" /></figure>

**Figure 29.1:** Theoretical CMB anisotropy power spectra, using the best-fitting ΛCDM model from Planck, calculated using CAMB. The panel on the left shows the theoretical expectation for scalar perturbations, while the panel on the right is for tensor perturbations, with an amplitude set to $r = 0.01$ for illustration. Note that the horizontal axis is logarithmic here. For the well-measured scalar TT spectrum, these regions, each covering roughly a decade in $\ell$, are labeled as in the text: the ISW rise; Sachs-Wolfe plateau; acoustic peaks; and damping tail. The TE cross-correlation power spectrum changes sign, indicated here by plotting the absolute value, but switching color for the negative parts.

## 29.4.2 Background Cosmology Parameters

The FRW cosmology requires an expansion parameter (the Hubble constant, $H_0$, often represented through $H_0 = 100 \text{ h km s}^{-1}\text{Mpc}^{-1}$) and several parameters to describe the matter and energy content of the Universe. These are usually given in terms of the critical density, i.e., for species 'x,' $\Omega_x \equiv \rho_x/\rho_{crit}$, where $\rho_{crit} \equiv 3H_0^2/8\pi G$. Since physical densities $\rho_x \propto \Omega_x h^2 \equiv \omega_x$ are what govern the physics of the CMB anisotropies, it is these $\omega$s that are best constrained by CMB data. In particular, CMB observations constrain $\Omega_b h^2$ for baryons and $\Omega_c h^2$ for cold dark matter (with $\rho_m = \rho_c + \rho_b$ for the matter sum).

The contribution of a cosmological constant $\Lambda$ (or other form of dark energy, see Dark Energy—Sec. 28) is usually included, together with a parameter that quantifies the curvature, $\Omega_K \equiv 1 - \Omega_{tot}$, where $\Omega_{tot} = \Omega_m + \Omega_\Lambda$. The radiation content, while in principle a free parameter, is precisely enough determined by the measurement of $T_\gamma$ that it can be considered fixed, making a $< 10^{-4}$ contribution to $\Omega_{tot}$ today.

Astrophysical processes at relatively low redshift can also affect the $C_\ell$s, with a particularly significant influence during reionization. The Universe became reionized at some redshift $z_i$, long after recombination, affecting the CMB when it passed through the integrated Thomson-scattering optical depth:

$$ \tau = \int_{0}^{z_i} \sigma_T n_e(z) \frac{dt}{dz} dz, \qquad (29.4) $$

where $\sigma_T$ is the Thomson cross-section, $n_e(z)$ is the number density of free electrons (which depends on astrophysics), and $dt/dz$ is fixed by the background cosmology. In principle, $\tau$ could be determined from the small-scale matter power spectrum, together with the physics of structure formation and radiative feedback processes; however, because this is a sufficiently intractable calculation, in practice $\tau$ needs to be considered as a free parameter.

Thus, we have eight basic cosmological parameters, namely $A_s$, $n_s$, $r$, $h$, $\Omega_b h^2$, $\Omega_c h^2$, $\Omega_{tot}$, and $\tau$. One can add additional parameters to this list, particularly when using the CMB in combination with other data sets. The next most relevant ones might be: $\Omega_b h^2$, the massive neutrino contribution; $w$ ($\equiv p/\rho$), the equation of state parameter for the dark energy; and $dn_s/d\ln k$, measuring deviations from a constant spectral index. To these 11 one could

of course add further parameters describing additional physics, such as details of the reionization process, features in the initial power spectrum, a sub-dominant contribution of isocurvature modes, etc.

As well as these underlying parameters, there are other (dependent) quantities that can be obtained from them. Such derived parameters include the actual $\Omega$s of the various components (e.g., $\Omega_m$), the variance of density perturbations at particular scales (e.g., $\sigma_8$), the angular scale of the sound horizon ($\theta_*$), the age of the Universe today ($t_0$), the age of the Universe at recombination, reionization, etc. (see The Cosmological Parameters—Sec. 25).

## 29.5 Physics of Anisotropies

The cosmological parameters affect the anisotropies through the well understood physics of the evolution of linear perturbations within a background FRW cosmology. There are very effective, fast, and publicly-available software codes for computing the CMB temperature, polarization, and matter power spectra, e.g., CMBFAST [32], CAMB [33], and CLASS [34]. These have been tested over a wide range of cosmologies and are considered to be accurate to much better than the 1% level [35], so that numerical errors are less than 10% of the parameter uncertainties for Planck [6].

For pedagogical purposes, it is easiest to focus on the temperature anisotropies, before moving to the polarization power spectra. A description of the physics underlying the $C_\ell^{TT}$s can be separated into four main regions (the first two combined below), as shown in the top left part of Fig. 29.1.

### 29.5.1 The ISW Rise, $\ell \lesssim 10$, and Sachs-Wolfe Plateau, $10 \lesssim \ell \lesssim 100$

The horizon scale (or more precisely, the angle subtended by the Hubble radius) at last scattering corresponds to $\ell \sim 100$. Anisotropies at larger scales have not evolved significantly, and hence directly reflect the 'initial conditions.' Temperature variations are $\delta T/T = -(1/5)\mathcal{R}(x_{LSS}) \simeq (1/3)\delta\phi/c^2$, where $\delta\phi$ is the perturbation to the gravitational potential, evaluated on the last-scattering surface (LSS). This is a result of the combination of gravitational redshift and intrinsic temperature fluctuations, and is usually referred to as the Sachs-Wolfe effect [36].

Assuming that a nearly scale-invariant spectrum of curvature (and corresponding density) perturbations was laid down at early times (i.e., $n_s \simeq 1$, meaning equal power per decade in $k$), then $\ell(\ell+1)C_\ell \simeq$ constant at low $\ell$s. This predicted near-flatness is hard to see unless the multipole axis is plotted logarithmically (as in Fig. 29.1, and part of Fig. 29.2).

Time variation of the potentials (i.e., time-dependent metric perturbations) at late times leads to an upturn in the $C_\ell$s in the lowest several multipoles; any deviation from a total equation of state $w = 0$ has such an effect. So the dominance of the dark energy at low redshift (see Dark Energy—Sec. 28) makes the lowest $\ell$s rise above the plateau. This is usually called the integrated Sachs-Wolfe effect (or ISW rise), since it comes from the line integral of $\dot{\phi}$; it has been confirmed through correlations between the large-angle anisotropies and large-scale structure [37]. Specific models can also give additional contributions at low $\ell$ (e.g., perturbations in the dark-energy component itself [38]), but typically these are buried in the cosmic variance.

In principle, the mechanism that produces primordial perturbations could generate scalar, vector, and tensor modes. However, the vector (vorticity) modes decay with the expansion of the Universe. The tensors (transverse trace-free perturbations to the metric) generate temperature anisotropies through the integrated effect of the locally-anisotropic expansion of space. Since the tensor modes also redshift away after they enter the horizon, they contribute only to angular scales above about 1° (see Fig. 29.1). Hence, some fraction of the low-$\ell$ signal could be due to a gravitational wave contribution, although small amounts of tensors are essentially impossible to discriminate from other effects that might raise the level of the plateau. Nevertheless, the tensors can be distinguished using polarization information (see Sec. 29.7).
### 29.5.2 The Acoustic Peaks, $100 \lesssim l \lesssim 1000$

On sub-degree scales, the rich structure in the anisotropy spectrum is the consequence of gravity-driven acoustic oscillations occurring before the atoms in the Universe became neutral [39]. Perturbations inside the horizon at last scattering were able to evolve causally and produce anisotropy at the last-scattering epoch, which reflects this evolution. The frozen-in phases of these sound waves imprint a dependence on the cosmological parameters, which gives CMB anisotropies their great constraining power.

The underlying physics can be understood as follows. Before the Universe became neutral, the proton-electron plasma was tightly coupled to the photons, and these components behaved as a single ‘photon-baryon fluid.’ Perturbations in the gravitational potential, dominated by the dark-matter component, were steadily evolving. They drove oscillations in the photon-baryon fluid, with photon pressure providing most of the restoring force and baryons giving some additional inertia. The perturbations were quite small in amplitude, $O(10^{-5})$, and so evolved linearly. That means each Fourier mode developed independently, and hence can be described as a driven harmonic oscillator, with frequency determined by the sound speed in the fluid. Thus, the fluid density underwent oscillations, giving time variations in temperature. These combine with a velocity effect, which is $\pi/2$ out of phase and has its amplitude reduced by the sound speed.

After the Universe recombined, the radiation decoupled from the baryons and could travel freely towards us. At that point, the (temporal) phases of the oscillations were frozen-in, and became projected on the sky as a harmonic series of peaks and troughs in power. The main peak is the mode that went through $1/4$ of a period, reaching maximal compression. The even peaks are maximal under-densities, which are generally of smaller amplitude because the rebound has to fight against the baryon inertia. The troughs, which do not extend to zero power, are partially filled by the Doppler effect because they are at the velocity maxima.

The physical length scale associated with the peaks is the sound horizon at last scattering, which can be straightforwardly calculated. This length is projected onto the sky, leading to an angular scale that depends on the geometry of space, as well as the distance to last scattering. Hence, the angular position of the peaks is a sensitive probe of a particular combination of cosmological parameters. In fact, this characteristic angular scale, $\theta_*$, is the most precisely measured observable, and hence is usually treated as an element of the cosmological parameter set.

One additional effect arises from reionization at redshift $z_i$. A fraction of photons ($\tau$) will be isotropically scattered at $z < z_i$, partially erasing the anisotropies at angular scales smaller than those subtended by the Hubble radius at $z_i$. This corresponds typically to $\ell$s above about 10, depending on the specific reionization model. The acoustic peaks are therefore reduced by a factor $e^{-2\tau}$ relative to the plateau.

These acoustic peaks were a clear theoretical prediction going back to about 1970 [40]. One can think of them as a snapshot of stochastic standing waves. Since the physics governing them is simple and their structure rich, one can see how they encode extractable information about the cosmological parameters. Their empirical existence started to become clear around 1994 [41], and the emergence, over the following decade, of a coherent series of acoustic peaks and troughs is a triumph of modern cosmology. This picture has received further confirmation with the detection in the power spectrum of galaxies (at redshifts $z \lesssim 1$) of the imprint of these same acoustic oscillations in the baryon component [42], as well as through detection of the expected oscillations in CMB polarization power spectra (see Sec. 29.7).

### 29.5.3 The Damping Tail, $\ell \gtrsim 1000$

The recombination process is not instantaneous, and this imparts a thickness to the LSS. This leads to a damping of the anisotropies at the highest $\ell$s, corresponding to scales smaller than that subtended by this thickness. One can also think of the photon-baryon fluid as having imperfect coupling, so that there is diffusion between the two components, and hence the amplitudes of the oscillations decrease with time. These effects lead to a damping of the $C_{\ell s}$, sometimes called ‘Silk damping’ [43], which cuts off the anisotropies at multipoles above about 2000. So, although

in principle it is possible to measure to ever smaller scales, this becomes increasingly difficult in practice.

### 29.5.4 Gravitational Lensing Effects

CMB gravitational lensing is caused by structures at lower redshift along the line of sight to the LSS. Photon paths are deflected by the lensing potential $\phi$, such that $T(\hat{n}) \rightarrow T(\hat{n} + \nabla\phi)$. Typical deflections are around 2 arcmin, but coherent over scales of a degree or so. Lensing preserves surface brightness, which means that a uniform temperature field would be unaffected; however, since there are anisotropies, then several distinct effects can be identified. The $C_{\ell s}$ are convolved with a smoothing function in a calculable way, partially flattening the peaks and troughs, generating a power-law tail at the highest multipoles, and complicating the polarization signal [44] (see Sec. 29.7.3). Additionally, the effect of lensing on the CMB can be detected through the 4-point function, which correlates temperature gradients and small-scale anisotropies, enabling a map of the lensing potential to be constructed [45],

Lensing is important because it gives an independent estimate of $A_s$, breaking the parameter combination $A_s e^{-2\tau}$ that is largely degenerate in the temperature anisotropy power spectra. Lensing is an example of a ‘secondary effect,’ i.e., the processing of anisotropies due to relatively nearby structures (see Sec. 29.9.2). Galaxies and clusters of galaxies give several such effects; all are expected to be of low amplitude, but are increasingly important at the highest $\ell$s. Such effects carry additional cosmological information (about evolving gravitational potentials in the low-redshift Universe) and are receiving more attention as experiments push to higher sensitivity and angular resolution. The lensing power spectrum (see Sec. 29.8) can potentially constrain dark-energy evolution, while future measurements at high $\ell$ are a particularly sensitive probe of the sum of the neutrino masses [46].

<figure><img src="image_149.png" /><figcaption>**Figure 29.2:** CMB temperature anisotropy band-power estimates from the Planck, WMAP, ACT, and SPT experiments. Note that the widths of the multipole bands vary between experiments and have not been plotted. This figure represents only a selection of the most recent available experimental results, and some points with large error bars have been omitted. At the higher multipoles these band-powers involve subtraction of particular foreground models, and so proper analysis requires simultaneous fitting of CMB and foregrounds over multiple frequencies. The horizontal axis here is logarithmic for the lowest multipoles, to show the Sachs-Wolfe plateau, and linear for the other multipoles. The acoustic peaks and damping region are very clearly observed, with no need for a theoretical line to guide the eye; however, the curve plotted is the best-fit Planck $\Lambda$CDM model.</figcaption></figure>

## 29.6 Current Temperature Anisotropy Data

Steady improvement in CMB data quality has led to the present-day cosmological model. The most robust constraints currently available come from Planck satellite [47, 48] data (together with constraints from non-CMB cosmological data sets), although
smaller-scale results from the ACT [49] and SPT [50] experiments are beginning to add useful constraining power. We plot power spectrum estimates from these experiments in Fig. 29.2, along with WMAP data [5] for comparison (see previous versions of this review for data from earlier experiments). Independent experimental data show consistency, both in maps and in derived power spectra (up to systematic uncertainties in the overall calibration for some experiments). This makes it clear that systematic effects are largely under control.

The band-powers shown in Fig. 29.2 are in very good agreement with a ‘ACDM’ model. As described earlier, several (at least seven) of the peaks and troughs are quite apparent. The original papers present the details on how these estimates were made, on the band-power correlation strengths, and on the information needed for their proper interpretation.

## 29.7 CMB Polarization

Thomson scattering of an anisotropic radiation field also generates linear polarization and the CMB is predicted to be polarized, at the level of roughly 5% of the temperature anisotropies [51]. Polarization is a spin-2 field on the sky, and the algebra of the modes in multipole space is strongly analogous to spin-orbit coupling in quantum mechanics [52]. The linear polarization pattern can be decomposed in a number of ways, with two quantities required for each pixel in a map, often given as the Q and U Stokes parameters. However, the most intuitive and physical decomposition is a geometrical one, splitting the polarization pattern into a part that comes from a divergence (often referred to as the ‘E mode’) and a part with a curl (called the ‘B mode’) [53]. More explicitly, the modes are defined in terms of second derivatives of the polarization amplitude, with the Hessian for the E modes having principal axes in the same sense as the polarization, while the B-mode pattern can be thought of as a 45° rotation of the E-mode pattern. Globally one sees that the E modes have $(-1)^\ell$ parity (like the spherical harmonics), while the B modes have $(-1)^{\ell+1}$ parity.

The existence of this linear polarization allows for six different cross-power spectra to be determined from data that measure the full temperature and polarization anisotropy information. Parity considerations make two of these zero, and we are left with four potential observables, $C_\ell^{TT}$, $C_\ell^{TE}$, $C_\ell^{EE}$, and $C_\ell^{BB}$ (see Fig. 29.1). Because scalar perturbations have no handedness, the B-mode power spectrum can only be sourced by vectors or tensors. Moreover, inflationary scalar perturbations give only E modes, while tensors generate roughly equal amounts of E and B, therefore the determination of a non-zero B-mode signal is a way to measure the gravitational-wave contribution (and thus potentially derive the energy scale of inflation). However, because the signal is expected to be rather weak, one must first eliminate the foreground contributions and other systematic effects down to very low levels. In addition, CMB lensing creates B modes from E modes, further complicating the extraction of a tensor signal.

As with temperature, the polarization $C_\ell$s exhibit a series of acoustic peaks generated by the oscillating photon-baryon fluid. The main ‘EE’ power spectrum has peaks that are out of phase with those in the ‘TT’ spectrum because the polarization anisotropies are sourced by the fluid velocity. The ‘TE’ part of the polarization and temperature patterns comes from correlations between density and velocity perturbations on the last-scattering surface, which can be both positive and negative, and is of larger amplitude than the EE signal. There is no polarization Sachs-Wolfe effect, and hence no large-angle (low-$\ell$) plateau. However, scattering during a recent period of reionization can create a polarization ‘bump’ at large angular scales.

Because the polarization anisotropies have only a small fraction of the amplitude of the temperature anisotropies, they took longer to detect. The first measurement of a polarization signal came in 2002 from the DASI experiment [54], which provided a convincing detection, confirming the general paradigm, but of low enough significance that it lent no real constraint to models. Despite dramatic progress since then, it is still the case that polarization data mainly support the basic paradigm, while reducing error bars on parameters by only around 20%. However, there are exceptions to this, specifically in the reionization optical depth, and the potential to constrain primordial gravitational waves. Moreover, the situation is expected to change dramatically as more of the available polarization modes are measured.

<figure><img src="image_150.png" /><figcaption>**Figure 29.3:** Cross-power spectrum band-powers of the temperature anisotropies and E-mode polarization signal from Planck (the low multipole data have been binned here), as well as WMAP, BICEP2/Keck, ACTPol, and SPTPol/SPT-3G. The curve is the best fit to the Planck temperature, polarization, and lensing band-powers. Note that each data point is an average over a band of multipoles, and hence to compare in detail with a model one has to integrate the theoretical curve through the band.</figcaption></figure>

### 29.7.1 T-E Power Spectrum

Because the T and E skies are correlated, one has to measure the TE power spectrum, as well as TT and EE, in order to extract all the cosmological information. This TE signal has now been mapped out extremely accurately by Planck [48], and these band-powers are shown in Fig. 29.3, along with those from WMAP [55] and BICEP2/Keck [56], with ACTPol [57] and SPTPol/SPT-3G [58, 59] extending to smaller angular scales. The anti-correlation at $\ell \simeq 150$ and the peak at $\ell \simeq 300$ were the first features to become distinct, but now a whole series of oscillations is clearly seen in this power spectrum (including six or seven peaks and troughs [10]). The measured shape of the cross-correlation power spectrum provides supporting evidence for the general cosmological picture, and also directly constrains the thickness of the last-scattering surface. Because the polarization anisotropies are generated in this scattering surface, the existence of correlations at angles above about a degree demonstrates that there were super-Hubble fluctuations at the recombination epoch. The sign of this correlation also confirms the adiabatic paradigm.

The overall picture of the source of CMB polarization and its oscillations has also been confirmed through tests that average the maps around both temperature hot spots and cold spots [60]. One sees precisely the expected patterns of radial and tangential polarization configurations, as well as the phase shift between polarization and temperature. This leaves no doubt that the oscillation picture is the correct one and that the polarization is coming from Thomson scattering at $z \simeq 1100$.

### 29.7.2 E-E Power Spectrum

Experimental band-powers for $C_\ell^{EE}$ from Planck, WMAP, BICEP2/Keck Array [56], ACTPol [57], and SPTPol/SPT-3G [58, 59] are shown in Fig. 29.4. Without the benefit of correlating with the temperature anisotropies (i.e., measuring $C_\ell^{TE}$), the polarization anisotropies are very weak and challenging to measure. Nevertheless, the oscillatory pattern is now well established and the data closely match the TT-derived theoretical prediction. In Fig. 29.4 one can clearly see the ‘shoulder’ expected at $\ell \simeq 140$, the first main peak at $\ell \simeq 400$ (corresponding to the first trough in $C_\ell^{TT}$), and the series of oscillations that is out of phase with those of the temperature anisotropy power spectrum (including six or seven peaks and troughs [10]).
<figure><img src="image_151.png" /></figure>

**Figure 29.4:** Power spectrum of *E*-mode polarization from Planck, together with WMAP, BICEP2/Keck, ACTPol, and SPTPol/SPT-3G. Note that some band-powers with larger uncertainties have been omitted and that the unbinned Planck low-ℓ data have been binned here. Also plotted is the best-fit theoretical model from Planck temperature + polarization + lensing data.

Perhaps the most unique result from the polarization measurements is at the largest angular scales ($\ell < 10$) in $C_{\ell}^{TE}$ and $C_{\ell}^{EE}$, where there is evidence for an excess signal (not visible in Fig. 29.4) compared to that expected from the temperature power spectrum alone. This is precisely the signal anticipated from an early period of reionization, arising from Doppler shifts during the partial scattering at $z < z_i$. The amplitude of the signal indicates that the first stars, presumably the source of the ionizing radiation, formed around $z \simeq 8$ (although the uncertainty is still quite large). This corresponds to a scattering optical depth $\tau \simeq 0.06$, so roughly 6% of CMB photons were re-scattered at the reionization epoch, with the other 94% last scattering at $z \simeq 1100$. However, estimates of the amplitude of this reionization excess have come down since the first measurements by WMAP (indicating that this is an extremely difficult measurement to make) and the latest Planck results have reduced the value further [11, 61].

### 29.7.3 B–B Power Spectrum

The expected amplitude of $C_{\ell}^{BB}$ is very small, so measurements of this polarization curl-mode are extremely challenging. The first indication of the existence of the *BB* signal came from the detection of the expected conversion of *E* modes to *B* modes by gravitational lensing, through a correlation technique using the lensing potential and polarization measurements from SPT [62]. However, the real promise of *B* modes lies in the detection of primordial gravitational waves at larger scales. This tensor signature could be seen either in the ‘recombination bump’ at around $\ell = 100$ (caused by an ISW effect as gravitational waves redshift away at the last-scattering epoch) or the ‘reionization bump’ at $\ell \lesssim 10$ (from additional scattering at low redshifts).

Results from the BICEP2 experiment [63] in 2014 suggested a detection of the primordial *B*-mode signature around the recombination peak. BICEP2 mapped a small part of the CMB sky with the best sensitivity level reached at that time (below 100 nK), but at a single frequency. Higher frequency data from Planck indicated that much of the BICEP2 signal was due to dust within our Galaxy, and a combined analysis by the BICEP2, Keck Array, and Planck teams [64] indicated that the data are consistent with no primordial *B* modes. The current constraint from Planck data alone is $r < 0.069$ (95% at $k = 0.05$ Mpc$^{-1}$ [11, 61]) using all CMB power spectra, and this limit is reduced to $r < 0.044$ with the inclusion of BICEP2/Keck Array data [61, 65]. The most constraining limit is $r < 0.036$ from a combination of BICEP2, Keck Array, and BICEP3 data, using WMAP and Planck maps to help remove foregrounds [66].

Several experiments are continuing to push down the sensitivity

<figure><img src="image_152.png" /></figure>

**Figure 29.5:** Power spectrum of *B*-mode polarization, including results from the BICEP2/BICEP3/Keck Array combined analysis, Planck, POLARBEAR, SPT, and ACT. Note that some of the measurements are direct estimates of *B* modes on the sky, while others are only sensitive to the lensing signal and come from combining *E*-mode and lensing potential measurements. Several earlier experiments reported upper limits, which are all off the top of this plot. A logarithmic horizontal axis is adopted here and the *y*-axis has been divided by a factor of $\sqrt{\ell}$ in order to show all three theoretically expected contributions: the low-ℓ reionization bump; the $\ell \simeq 100$ recombination peak; and the high-ℓ lensing signature. The dotted line is for a tensor (primordial gravitational wave) fraction $r = 0.05$, simply as an example, with all other cosmological parameters set at the best Planck-derived values, for which model the expected lensing *B* modes have also been shown with a dashed line.

of *B*-mode measurements, motivated by the enormous importance of a future detection of this telltale signature of inflation (or other physics at the highest energies). A compilation of experimental results for $C_{\ell}^{BB}$ is shown in Fig. 29.5, coming from a combination of direct estimates of the *B* modes (BICEP2/BICEP3/Keck Array [66], POLARBEAR [67], SPTPol [68], and ACTPol [57]) and indirect determinations of the lensing *B* modes based on estimating the effect of measured lensing on measured *E* modes (Planck [69], SPT [62], and ACT [70]). Additional band-power estimates are expected from these and other experiments in the near future, with the Simons Observatory [71], the so-called ‘Stage 4’ CMB project [72] and the LiteBIRD satellite [73], holding great promise for pushing down to the $r \sim 0.001$ level.

## 29.8 CMB Lensing Power Spectrum

One further CMB observable is the gravitational lensing deflection, leading to the construction of a map of the lensing potential. The latest Planck results [74] give a map that is detected at the $> 40\sigma$ level using a minimum-variance procedure from the 4-point function of temperature and polarization data. From these maps, estimates can be constructed of $C_{\ell}^{\phi\phi}$, the lensing-potential power spectrum, which is found to be consistent with predictions from the best-fit temperature and polarization model. Recent results from ACT give a power spectrum that has a similar overall signal-to-noise ratio [75] and there are also interesting measurements from SPT [76]. Figure 29.6 plots the Planck, ACT, and SPT estimates of $C_{\ell}^{\kappa\kappa}$, the lensing convergence power spectrum, which is proportional to $\ell^2(\ell + 1)^2$ times the potential power spectrum $C_{\ell}^{\phi\phi}$.

We can think of each sky pixel as possessing three independent quantities that can be measured, namely *T*, *E*, and *φ* (and potentially *B*, if that becomes detectable). Determining the constraining power comes down to counting $Y_{\ell m}$ modes [77], as well as appreciating that some modes help to break particular parameter degeneracies. We have only scratched the surface of CMB lensing so far, and it is expected that future small-scale experiments will
<figure><img src="image_153.png" /><figcaption>**Figure 29.6:** Power spectrum measurements for CMB lensing, including selected results from Planck, ACT, and SPT. The quantity plotted is the (dimensionless) potential power spectrum, scaled by $\ell^2(\ell + 1)^2$ and multiplied by a factor of $10^8$ to make the numbers more manageable. Some less reliable measurements have not been plotted. The best-fit Planck $\Lambda$CDM spectrum is plotted as a gray line.</figcaption></figure>

be able to extract more of the cosmological information. Further constraints can also be derived on the lower-redshift Universe by cross-correlating CMB lensing with other cosmological tracers of large-scale structure. Additionally, small-scale lensing, combined with E-mode measurements, can be used to 'delens' CMB B-mode data, which will be important for pushing down into the $r \lesssim 0.01$ regime [78].

## 29.9 Complications

There are a number of issues that complicate the interpretation of CMB anisotropy data (and are considered to be *signal* by many astrophysicists), some of which we sketch out below.

### 29.9.1 Foregrounds

The microwave sky contains significant emission from our Galaxy and from extragalactic sources [79]. Fortunately, the frequency dependence of these various sources is in general substantially different from that of the CMB anisotropy signals. The combination of Galactic synchrotron, bremsstrahlung, and dust emission reaches a minimum at a frequency of roughly 100 GHz (or wavelength of about 3 mm). As one moves to greater angular resolution, the minimum moves to slightly higher frequencies, but becomes more sensitive to unresolved (point-like) sources.

At frequencies around 100 GHz, and for portions of the sky away from the Galactic plane, the foregrounds are typically 1 to 10% of the CMB anisotropies. By making observations at multiple frequencies, it is relatively straightforward to separate the various components and determine the CMB signal to the few per cent level. For greater sensitivity, it is necessary to use the spatial information and statistical properties of the foregrounds to separate them from the CMB. Furthermore, at higher $\ell$s it is essential to carefully model extragalactic foregrounds, particularly the clustering of infrared-emitting galaxies and scattering due to galaxy clusters, which dominate the measured power spectrum as we move into the damping tail.

The foregrounds for CMB polarization follow a similar pattern to those for temperature, but are intrinsically brighter relative to CMB anisotropies. WMAP showed that the polarized foregrounds dominate at large angular scales, and that they must be well characterized in order to be discriminated [80]. Planck has shown that it is possible to characterize the foreground polarization signals, with synchrotron dominating at low frequencies and dust at high frequencies [81]. On smaller scales there are no strongly-polarized foregrounds, and hence at high multipoles it is in principle easier to measure foreground-free modes in polarization than in temperature. Although foreground contamination will no doubt become

more complicated as we push down in sensitivity, making analysis more difficult, for the time being, foreground contamination is not a fundamental limit for CMB experiments.

### 29.9.2 Secondary Anisotropies

With increasingly precise measurements of the primary anisotropies, there is growing theoretical and observational interest in 'secondary anisotropies,' pushing experiments to higher angular resolution and sensitivity. These secondary effects arise from the processing of the CMB due to ionization history and the evolution of structure, including gravitational lensing (which was already discussed) and patchy reionization effects [82]. Additional information can thus be extracted about the Universe at $z \ll 1000$. This tends to be most effectively done through correlating CMB maps with other cosmological probes of structure. Secondary signals are also typically non-Gaussian, unlike the primary CMB anisotropies.

A secondary signal of great current interest is the Sunyaev-Zeldovich (SZ) effect [83], which is Compton scattering ($\gamma e \rightarrow \gamma'e'$) of the CMB photons by hot electrons in intergalactic plasma. This creates spectral distortions by transferring energy from the electrons to the photons. It is particularly important for clusters of galaxies, through which one observes a partially Comptonized spectrum, resulting in a decrement at radio wavelengths and an increment in the submillimeter.

The imprint on the CMB sky is of the form $\Delta T/T = y f(x)$, with the $y$ parameter being the integral of Thomson optical depth times $kT_e/m_e c^2$ through the cluster, and $f(x)$ describing the frequency dependence. This is simply $x \coth(x/2) - 4$ for a non-relativistic gas (the electron temperature in a cluster is typically a few keV), where the dimensionless frequency $x \equiv \hbar v/kT_e$. As well as this 'thermal' SZ effect, there is also a smaller 'kinetic' effect due to the bulk motion of the cluster gas, giving $\Delta T/T \sim \tau(v/c)$, with either sign, but having the same spectrum as the primary CMB anisotropies.

A significant advantage in finding galaxy clusters via the SZ effect is that the signal is largely independent of redshift, so in principle clusters can be found to arbitrarily large distances. The SZ effect can be used to find and study individual clusters, and to obtain estimates of the Hubble constant. There is also the potential to constrain other cosmological parameters, such as the clustering amplitude $\sigma_8$ and the equation of state of the dark energy, through counts of detected clusters as a function of redshift. The promise of the method has been realized through detections of clusters purely through the SZ effect by SPT [84], ACT [85], and Planck [86]. Results from Planck clusters [87] suggest a somewhat lower value of $\sigma_8$ than inferred from CMB anisotropies, but there are still systematic uncertainties that might encompass the difference, and a more recent analysis of SPT-detected clusters shows better agreement [88]. Further analysis of scaling relations among cluster properties should enable more robust cosmological constraints to be placed in future, so that we can understand whether this 'tension' might be a sign of new physics.

### 29.9.3 Higher-order Statistics

Although most of the CMB anisotropy information is contained in the power spectra, there will also be weak signals present in higher-order statistics. These can measure any primordial non-Gaussianity in the perturbations, as well as non-linear growth of the fluctuations on small scales and other secondary effects (plus residual foreground contamination of course). There are an infinite variety of ways in which the CMB could be non-Gaussian [26]; however, there is a generic form to consider for the initial conditions, where a quadratic contribution to the curvature perturbations is parameterized through a dimensionless number $f_{\text{NL}}$. This weakly non-linear component can be constrained in several ways, the most popular being through measurements of the bispectrum (or 3-point function).

The constraints depend on the shape of the triangles in harmonic space, and it has become common to distinguish the 'local' or 'squeezed' configuration (in which one side is much smaller than the other two) from the 'equilateral' configuration. Other configurations are also relevant for specific theories, such as 'orthogonal' non-Gaussianity, which has positive correlations for
$k_1 \simeq 2k_2 \simeq 2k_3$, and negative correlations for the equilateral configuration. The constraints from the Planck team [89] are
$f_{\text{NL}}^{\text{local}} = 1 \pm 5$, $f_{\text{NL}}^{\text{equil}} = -26 \pm 47$, and $f_{\text{NL}}^{\text{ortho}} = -38 \pm 24$.

These results are consistent with zero, but are at a level that is now interesting for model predictions. The amplitude of $f_{\text{NL}}$ expected is small, so that a detection of $f_{\text{NL}} \gg 1$ would rule out all single-field, slow-roll inflationary models. It is still possible to improve upon these Planck results, and it certainly seems feasible that a measurement of primordial non-Gaussianity may yet be within reach. Non-primordial detections of non-Gaussianity from expected signatures have already been made. For example, the bispectrum and trispectrum contain evidence of gravitational lensing, the ISW effect, and Doppler boosting. For now the primordial signal is elusive, but should it be detected, then detailed measurements of non-Gaussianity will become a unique probe of inflationary-era physics. Because of that, much effort continues to be devoted to honing predictions and measurement techniques, with the expectation that we will need to go beyond the CMB (e.g., 3D galaxy surveys) to dramatically improve the constraints.

### 29.9.4 Anomalies

Several features seen in the Planck data [29,60,90] confirm those found earlier with WMAP [28], showing mild deviations from a simple description of the sky; these are often referred to as 'anomalies.' One such feature is the lack of power in the multipole range $l \simeq 20-30$ [11, 48]. Other examples involve the apparent breaking of statistical anisotropy, caused by alignment of the lowest multipoles, as well as a somewhat excessive cold spot and a power asymmetry between hemispheres. No such feature is significant at more than the roughly $3\sigma$ level, and the importance of 'a posteri-ori' statistics here has been emphasized by many authors. Since these effects are at large angular scales, where cosmic variance dominates, the results will not increase in significance with more data, although there is the potential for more sensitive polarization measurements to provide independent tests.

## 29.10 Constraints on Cosmological Parameters

The most striking outcome of the last couple of decades of experimental results is that the standard cosmological paradigm continues to be in very good shape. A large amount of high-precision data on the CMB power spectrum is adequately fit with fewer than 10 free parameters (and only six need non-trivial values). The framework is that of FRW models, which have nearly flat geometry, containing dark matter and dark energy, and with adiabatic perturbations having close to scale-invariant initial conditions.

Within this basic picture, the values of the cosmological parameters can be constrained. Of course, more stringent bounds can be placed on models that cover a restricted parameter space, e.g., assuming that $\Omega_{\text{tot}} = 1$ or $r = 0$. More generally, the constraints depend upon the adopted prior probability distributions, even if they are implicit, for example by restricting the parameter freedom or their ranges (particularly where likelihoods peak near the boundaries), or by using different choices of other data in combination with the CMB. As the data become even more precise, these considerations will be less important, but for now we caution that restrictions on model space and choice of non-CMB data sets and priors need to be kept in mind when adopting specific parameter values and uncertainties.

There are some combinations of parameters that fit the CMB anisotropies almost equivalently. For example, there is a nearly exact geometric degeneracy, where any combination of $\Omega_{\text{m}}$ and $\Omega_{\Lambda}$ that provides the same angular-diameter distance to last scattering will give nearly identical $C_{\ell}$s. There are also other less exact degeneracies among the parameters. Such degeneracies can be broken when using the CMB results in combination with other cosmological data sets. Particularly useful are complementary constraints from baryon acoustic oscillations, galaxy clustering, the abundance of galaxy clusters, weak gravitational lensing measurements, and Type Ia supernova distances. For an overview of some of these other cosmological constraints, see The Cosmological Parameters—Sec. 25 of this Review.

Within the context of a 6-parameter family of models (which fixes $\Omega_{\text{tot}} = 1$, $dn_s/d\ln k = 0$, $r = 0$, and $w = -1$) the

Planck results for TT, together with TE, EE, and CMB lensing, yield [11]: $\ln(10^{10}A_s) = 3.044 \pm 0.014$; $n_s = 0.965 \pm 0.004$; $\Omega_b h^2 = 0.02237 \pm 0.00015$; $\Omega_c h^2 = 0.1200 \pm 0.0012$; $100\theta_* = 1.04092 \pm 0.00031$; and $\tau = 0.054 \pm 0.007$. Other parameters can be derived from this basic set, including $h = 0.674 \pm 0.005$, $\Omega_{\Lambda} = 0.685 \pm 0.007$ ($= 1 - \Omega_{\text{m}}$) and $\sigma_8 = 0.811 \pm 0.006$ (see also Astrophysical Constants and Parameters—Sec. 2). Somewhat different (although consistent) values are obtained using other data combinations, such as including BAO, supernova, $H_0$, or weaklensing constraints (see Sec. 25 of this Review). However, the Planck results quoted above are currently the best available from CMB data alone. Results from other CMB experiments (e.g., SPT-3G [59]) are consistent and becoming competitive.

The standard cosmological model still fits the data well, with the error bars on the parameters continuing to shrink. Improved measurement of higher acoustic peaks has dramatically reduced the uncertainty in the $\theta_*$ parameter, which is now detected at $> 3000\sigma$. The evidence for $n_s < 1$ is now at the $8\sigma$ level from Planck data alone. The value of the reionization optical depth has decreased compared with earlier estimates; it is convincingly detected, but still not at very high significance.

The inferred value of $H_0$ is smaller than the most precise values derived from the cosmic distance ladder. This parameter tension is discussed more fully in other sections of this Review. The CMB anisotropies also provide the most precise estimate of the age of the Universe, with Planck giving the value $t_0 = 13.797 \pm 0.023$ Gyr.

Constraints can also be placed on parameters beyond the basic six, particularly when including other astrophysical data sets. Relaxing the flatness assumption, the constraint on $\Omega_{\text{tot}}$ is $1.011 \pm 0.006$. Note that for $h$, the CMB data alone provide only a very weak constraint if spatial flatness is not assumed. However, with the addition of other data (particularly powerful in this context being a compilation of BAO measurements; see Sec. 25 of this Review), the constraints on the Hubble constant and curvature improve considerably, leading to $\Omega_{\text{tot}} = 0.9993 \pm 0.0019$ [11].

For $\Omega_b h^2$ the CMB-derived value is generally consistent with completely independent constraints from Big Bang nucleosynthesis (see Sec. 24 of this Review). Related are constraints on additional neutrino-like relativistic degrees of freedom, which lead to $N_{\text{eff}} = 2.99 \pm 0.17$ (including BAO), i.e., no evidence for extra neutrino species.

The tightest published limit on the tensor-to-scalar ratio is $r < 0.036$ (measured at $k = 0.05 \text{ Mpc}^{-1}$) from BICEP/Keck Array [66]. The detailed limit depends on how other parameters, especially $A_s$, $n_t$, and $dn_s/d\ln k \neq 0$ are restricted. The joint constraints on $n_s$ and $r$ allow specific inflationary models to be tested [30,91]. Looking at the $(n_s, r)$ plane, this means that $m^2\phi^2$ (mass-term quadratic) inflation is disfavored by the data, as well as $\lambda\phi^4$ (self-coupled) inflation.

The addition of the dark-energy equation of state $w$ adds the partial degeneracy of being able to fit a ridge in $(w, h)$ space, extending to low values of both parameters. This degeneracy is broken when the CMB is used in combination with other data sets, e.g., adding a compilation of BAO and supernova data gives $w = -1.028 \pm 0.031$ [11]. Constraints can also be placed on more general dark energy and modified-gravity models [92]. However, when extending the search space, one needs to be careful not to over-interpret some tensions between data sets as evidence for new physics.

For the reionization optical depth, a reanalysis of Planck data in 2016 resulted in a reduction in the value of $\tau$, with the tightest result giving $\tau = 0.055 \pm 0.009$, and the newest analyses giving similar numbers. This corresponds to $z_i = 7.8-8.8$ (depending on the functional form of the reionization history), with an uncertainty of $\pm 0.9$ [93]. This redshift is only slightly higher that that suggested from studies of absorption lines in high-$z$ quasar spectra [94] and Ly $\alpha$-emitting galaxies [95], perhaps hinting that the process of reionization was not as complex as previously suspected. The important constraint provided by CMB polarization, in combination with astrophysical measurements, thus allows us to investigate how the first stars formed and brought about the end of the cosmic dark ages.
## 29.11 Particle Physics Constraints

CMB data place limits on parameters that are directly relevant for particle physics models. For example, there is a limit on the sum of the masses of the neutrinos, $\Sigma m_\nu < 0.12$ eV (95%) [11] coming from Planck together with BAO measurements (although limits are weaker when considering both $N_{\text{eff}}$ and $\Sigma m_\nu$ as free parameters). This assumes the usual number density of fermions, which decoupled when they were relativistic. The limit is tantalizingly only a factor of a few higher than the minimum value coming from neutrino mixing experiments (see Neutrino Mixings—Secs. 14 and 26). As well as being an indirect probe of the neutrino background, Planck data also require that the neutrino fluid has perturbations, i.e., that it possesses a sound speed $c_s^2 \simeq 1/3$, as expected [9].

The current suite of data suggests that $n_s < 1$, with a best-fitting value about 0.035 below unity. This is already quite constraining for inflationary models, particularly along with $r$ limits. There is no current evidence for running of the spectral index, with $dn_s/d\ln k = -0.004 \pm 0.007$ from Planck alone [11] (with a similar value when BAO data are included), although this is less of a constraint on models. Similarly, primordial non-Gaussianity is being probed to interesting levels, although tests of simple inflationary models will only come with significant reductions in uncertainty.

The large-angle anomalies, such as the hemispheric modulation of power and the dip in power at $\ell \simeq 20-30$, have the potential to be hints of new physics. Such effects might be expected in a Universe that has a large-scale power cut-off, or anisotropy in the initial power spectrum, or is topologically non-trivial. However, cosmic variance and a posteriori statistics limit the significance of these anomalies, absent the existence of a model that naturally yields some of these features (and ideally also predicting other phenomena that can be tested).

Constraints on 'cosmic birefringence' (i.e., rotation of the plane of CMB polarization that generates non-zero TB and EB power) can be used to place limits on theories involving parity violation, Lorentz violation, or axion-photon mixing [96].

It is possible to place limits on additional areas of physics [97], for example annihilating dark matter [9, 9], primordial magnetic fields [98], and time variation of the fine-structure constant [99], as well as the neutrino chemical potential, a contribution of warm dark matter, topological defects, or physics beyond general relativity. Further particle physics constraints will follow as the smaller-scale and polarization measurements continue to improve.

The CMB anisotropy measurements precisely pin down physics at the time of last-scattering, and so any change of physics can be constrained if it affects the relevant energies or timescales. Future, higher sensitivity measurements of the CMB frequency spectrum will push the constraints back to cover energy injection at much earlier times ($\sim 1$ year). Comparison of CMB and BBN observables extend these constraints to timescales of order seconds, and energies in the MeV range. And to the extent that inflation provides an effective description of the generation of perturbations, the inflationary observables may constrain physics at GUT-type energy scales.

More generally, careful measurement of the CMB power spectra and non-Gaussianity can in principle put constraints on physics at the highest energies, including ideas of quantum gravity, string theory, extra dimensions, colliding branes, etc. At the moment any calculation of predictions appears to be far from definitive. However, there is a great deal of activity on implications of fundamental theories for the early Universe, and hence a chance that there might be observational implications for specific scenarios.

## 29.12 Fundamental Lessons

More important than the precise values of parameters is what we have learned about the general features that describe our observable Universe. Beyond the basic hot Big Bang picture, the CMB has taught us that:

* the (observable) Universe is very close to isotropic;

* the Universe recombined at $z \sim 1000$ and started to become ionized again at $z \sim 10$;

* the geometry of the Universe is close to flat;

* both dark matter and dark energy are required;

* gravitational instability is sufficient to grow all of the observed large structures in the Universe;

* topological defects were not important for structure formation;

* there were 'synchronized' super-Hubble modes generated in the early Universe;

* the initial perturbations were predominantly adiabatic in nature;

* the primordial perturbation spectrum has a slightly red tilt;

* the perturbations had close to Gaussian (i.e., maximally random) initial conditions.

These features form the basis of the cosmological standard model, ΛCDM, for which it is tempting to make an analogy with the Standard Model of particle physics (see earlier Sections of this Review). In comparison, the cosmological model is much further from any underlying 'fundamental theory,' which might ultimately provide the values of the parameters from first principles. Nevertheless, any genuinely complete 'theory of everything' must include an explanation for the values of these cosmological parameters in addition to the parameters of the Standard Model of particle physics.

## 29.13 Future Directions

Given the significant progress in measuring the CMB sky, which has been instrumental in tying down the cosmological model, what can we anticipate for the future? There will be a steady improvement in the precision and confidence with which we can determine the appropriate cosmological parameters. Ground-based experiments operating at smaller angular scales will continue to place tighter constraints on the damping tail, lensing, and cross-correlations. New polarization experiments at small scales will probe further into the damping tail, without the limitation of extragalactic foregrounds. And polarization experiments at large angular scales will push down the limits on primordial B modes.

Planck, the third generation CMB satellite mission, was launched in May 2009, and produced a large number of papers, including a set of cosmological studies based on the first two full surveys of the sky (accompanied by a public release of data products) in 2013, a further series coming from analysis of the full mission data release in 2015 (eight surveys for the Low Frequency Instrument and five surveys for the High Frequency Instrument), and a third series derived from a final analysis of the 2018 data release, including full constraints from polarization data. Planck data currently dominate constraints on models, but that situation will change soon.

A set of cosmological parameters is now known to percent-level accuracy, and that may seem sufficient for many people. However, we should certainly demand more of measurements that describe the entire observable Universe! Hence a lot of activity in the coming years will continue to focus on determining those parameters with increasing precision. This necessarily includes testing for consistency among different predictions of the cosmological Standard Model, and searching for signals that might require additional physics.

A second area of focus will be the smaller-scale anisotropies and 'secondary effects.' There is a great deal of information about structure formation at $z \ll 1000$ encoded in the CMB sky. This may involve higher-order statistics and cross-correlations with other large-scale structure tracers, as well as spectral signatures, with many experiments targeting the galaxy cluster SZ effect, for example. The current status of CMB lensing is similar (in terms of total signal-to-noise) to the quality of the first CMB anisotropy measurements by COBE, and thus we can expect that experimental probes of lensing will improve dramatically in the coming years. All of these investigations can provide constraints on the dark-energy equation of state, for example, which is a major area of focus for several future cosmological surveys at optical wavelengths. CMB lensing also promises to yield a measurement of the sum of the neutrino masses.

A third direction is increasingly sensitive searches for specific signatures of physics at the highest energies. The most promising
of these may be the primordial gravitational wave signals in $C_l^{BB}$, which could be a probe of the ~$10^{16}$ GeV energy range. There are several experiments underway or being developed that are designed to search for the polarization B modes, with the most ambitious being CMB-S4 on the ground and LiteBIRD in space. Additionally, non-Gaussianity holds the promise of constraining models beyond single-field slow-roll inflation.

Anisotropies in the CMB have proven to be the premier probe of cosmology and the early Universe. Theoretically the CMB involves well-understood physics in the linear regime, and is under very good calculational control. A substantial and improving set of observational data now exists. Systematics appear to be under control and are not currently a limiting factor. And so for the next several years we can expect an increasing amount of cosmological information to be gleaned from CMB anisotropies, with the prospect also of some genuine surprises.

### References

[1] A. A. Penzias and R. W. Wilson, Astrophys. J. 142, 419 (1965); R. H. Dicke et al., Astrophys. J. 142, 414 (1965).

[2] M. White, D. Scott and J. Silk, Ann. Rev. Astron. Astrophys. 32, 319 (1994); W. Hu and S. Dodelson, Ann. Rev. Astron. Astrophys. 40, 171 (2002), [arXiv:astro-ph/0110414]; A. Challinor and H. Peiris, in M. Novello and S. Perez, editors, "American Institute of Physics Conference Series," volume 1132, 86-140 (2009), [arXiv:0903.5158].

[3] G. F. Smoot et al., Astrophys. J. Lett. 396, L1 (1992).

[4] C. L. Bennett et al., Astrophys. J. Supp. 148, 1 (2003), [arXiv:astro-ph/0302207].

[5] G. Hinshaw et al., Astrophys. J. Supp. 208, 19 (2013), [arXiv:1212.5226].

[6] Planck Collab. 2013 Results XVI, Astron. Astrophys. 571, A16 (2014), [arXiv:1303.5076].

[7] J. A. Tauber et al., Astron. Astrophys. 520, A1 (2010); Planck Collab. 2013 Results I, Astron. Astrophys. 571, A1 (2014), [arXiv:1303.5062].

[8] Planck Collab. 2015 Results I, Astron. Astrophys. 594, A1 (2016), [arXiv:1502.01582].

[9] Planck Collab. 2015 Results XIII, Astron. Astrophys. 594, A13 (2016), [arXiv:1502.01589].

[10] Planck Collab. 2018 Results I, Astron. Astrophys. 641, A1 (2020), [arXiv:1807.06205].

[11] Planck Collab. 2018 Results VI, Astron. Astrophys. 641, A6 (2020), [arXiv:1807.06209].

[12] D. S. Swetz et al., Astrophys. J. Supp. 194, 41 (2011), [arXiv:1007.0290].

[13] J. E. Carlstrom et al., Proc. Astron. Soc. Pacific 123, 568 (2011), [arXiv:0907.4445].

[14] V. V. Klimenko et al., Astronomy Letters 46, 11, 715 (2020), [arXiv:2106.00119]; D. A. Riechers et al., Nature 602, 7895, 58 (2022), [arXiv:2202.00693].

[15] D. J. Fixsen et al., Astrophys. J. 734, 5 (2011), [arXiv:0901.0555].

[16] J. Singal et al., Proc. Astron. Soc. Pacific 130, 985, 036001 (2018), [arXiv:1711.09979].

[17] A. Kogut et al., J. Cosmology Astropart. Phys. 2011, 7, 025 (2011), [arXiv:1105.2044]; P. André et al., J. Cosmology Astropart. Phys. 2014, 2, 006 (2014), [arXiv:1310.1554]; J. Delabrouille et al., Experimental Astronomy 51, 3, 1471 (2021), [arXiv:1909.01591].

[18] V. Desjacques et al., Mon. Not. R. Astron. Soc. 451, 4460 (2015), [arXiv:1503.05589].

[19] D. J. Fixsen, Astrophys. J. 707, 916 (2009), [arXiv:0911.1955].

[20] J. C. Mather et al., Astrophys. J. 512, 511 (1999), [arXiv:astro-ph/9810373].

[21] Y. Hoffman, H. M. Courtois and R. B. Tully, Mon. Not. R. Astron. Soc. 449, 4494 (2015), [arXiv:1503.05422].

[22] D. J. Fixsen et al., Astrophys. J. 420, 445 (1994).

[23] Planck Collab. 2013 Results XXVII, Astron. Astrophys. 571, A27 (2014), [arXiv:1303.5087].

[24] S. Seager, D. D. Sasselov and D. Scott, Astrophys. J. Supp. 128, 407 (2000), [arXiv:astro-ph/9912182].

[25] L. Knox, Phys. Rev. D52, 4307 (1995), [arXiv:astro-ph/9504054].

[26] N. Bartolo et al., Phys. Rep. 402, 103 (2004), [arXiv:astro-ph/0406398].

[27] Planck Collab. 2013 Results XXIV, Astron. Astrophys. 571, A24 (2014), [arXiv:1303.5084].

[28] C. L. Bennett et al., Astrophys. J. Supp. 192, 17 (2011), [arXiv:1001.4758].

[29] Planck Collab. 2013 Results XXIII, Astron. Astrophys. 571, A23 (2014), [arXiv:1303.5083].

[30] Planck Collab. 2018 Results X, Astron. Astrophys. 641, A10 (2020), [arXiv:1807.06211].

[31] A. R. Liddle and D. H. Lyth, Cosmological Inflation and Large-Scale Structure, Cambridge University Press, Cambridge (2000).

[32] U. Seljak and M. Zaldarriaga, Astrophys. J. 469, 437 (1996), [arXiv:astro-ph/9603033].

[33] A. Lewis, A. Challinor and A. Lasenby, Astrophys. J. 538, 473 (2000), [arXiv:astro-ph/9911177].

[34] D. Blas, J. Lesgourgues and T. Tram, J. Cosmology Astropart. Phys. 7, 034 (2011), [arXiv:1104.2933].

[35] U. Seljak et al., Phys. Rev. D68, 083507 (2003), [arXiv:astro-ph/0306052].

[36] R. K. Sachs and A. M. Wolfe, Astrophys. J. 147, 73 (1967).

[37] R. G. Crittenden and N. Turok, Phys. Rev. Lett. 76, 575 (1996), [arXiv:astro-ph/9510072]; Planck Collab. 2015 Results XXI, Astron. Astrophys. 594, A21 (2016), [arXiv:1502.01595].

[38] W. Hu et al., Phys. Rev. D59, 2, 023512 (1998), [arXiv:astro-ph/9806362].

[39] W. Hu, N. Sugiyama and J. Silk, Nature 386, 37 (1997), [arXiv:astro-ph/9604166].

[40] P. J. E. Peebles and J. T. Yu, Astrophys. J. 162, 815 (1970); Sunyaev, R. A. and Zeldovich, Ya. B., Astron. Astrophys. Supp. 7, 3 (1970).

[41] D. Scott, J. Silk and M. White, Science 268, 829 (1995), [arXiv:astro-ph/9505015].

[42] D. J. Eisenstein, New Astron. Rev. 49, 360 (2005).

[43] J. Silk, Astrophys. J. 151, 459 (1968).

[44] M. Zaldarriaga and U. Seljak, Phys. Rev. D58, 023003 (1998), [arXiv:astro-ph/9803150].

[45] Planck Collab. 2013 Result XVII, Astron. Astrophys. 571, A17 (2014), [arXiv:1303.5077].

[46] M. Kaplinghat, L. Knox and Y.-S. Song, Phys. Rev. Lett. 91, 24, 241301 (2003), [arXiv:astro-ph/0303344].

[47] Planck Collab. 2013 Results XV, Astron. Astrophys. 571, A15 (2014), [arXiv:1303.5075].

[48] Planck Collab. 2018 Results V, Astron. Astrophys. 641, A5 (2020), [arXiv:1907.12875].

[49] S. Das et al., J. Cosmology Astropart. Phys. 4, 014 (2014), [arXiv:1301.1037].

[50] K. T. Story et al., Astrophys. J. 779, 86 (2013), [arXiv:1210.7231].

[51] W. Hu and M. White, New Astron. 2, 323 (1997), [arXiv:astro-ph/9706147].

[52] W. Hu and M. J. White, Phys. Rev. D56, 596 (1997), [arXiv:astro-ph/9702170].
[53] M. Zaldarriaga and U. Seljak, Phys. Rev. D55, 1830 (1997), [arXiv:astro-ph/9609170]; M. Kamionkowski, A. Kosowsky and A. Stebbins, Phys. Rev. D55, 7368 (1997), [arXiv:astro-ph/9611125].

[54] J. M. Kovac et al., Nature 420, 772 (2002), [arXiv:astro-ph/0209478].

[55] D. Larson et al., Astrophys. J. Supp. 192, 16 (2011), [arXiv:1001.4635].

[56] Keck Array and BICEP2 Collabs. V, Astrophys. J. 811, 126 (2015), [arXiv:1502.00643].

[57] S. K. Choi et al., J. Cosmology Astropart. Phys. 2020, 12, 045 (2020), [arXiv:2007.07289].

[58] A. T. Crites et al., Astrophys. J. 805, 36 (2015), [arXiv:1411.1042].

[59] L. Balkenhol et al., Phys. Rev. D108, 2, 023510 (2023), [arXiv:2212.05642].

[60] Planck Collab. 2018 Results VII, Astron. Astrophys. 641, A7 (2020), [arXiv:1906.02552].

[61] M. Tristram et al., Astron. Astrophys. 647, A128 (2021), [arXiv:2010.01139].

[62] D. Hanson et al., Phys. Rev. Lett. 111, 14, 141301 (2013), [arXiv:1307.5830].

[63] BICEP2 Collab., Phys. Rev. Lett. 112, 24, 241101 (2014), [arXiv:1403.3985].

[64] BICEP2/Keck and Planck Collabs., Phys. Rev. Lett. 114, 10, 101301 (2015), [arXiv:1502.00612].

[65] BICEP2 and Keck Array Collab., Phys. Rev. Lett. 121, 22, 221301 (2018), [arXiv:1810.05216].

[66] BICEP/Keck Collab., Phys. Rev. Lett. 127, 151301 (2021), [arXiv:2110.00483].

[67] POLARBEAR Collab., Astrophys. J. 848, 121 (2017), [arXiv:1705.02907].

[68] J. T. Sayre et al., Phys. Rev. D101, 12, 122003 (2020), [arXiv:1910.05748].

[69] Planck Collab. 2015 Results XV, Astron. Astrophys. 594, A15 (2016), [arXiv:1502.01591].

[70] A. van Engelen et al., Astrophys. J. 808, 7 (2015), [arXiv:1412.0626].

[71] P. Ade et al. (Simons Observatory Collab.), J. Cosmology Astropart. Phys. 2019, 2, 056 (2019), [arXiv:1808.07445].

[72] K. N. Abazajian et al., ArXiv e-prints (2016), [arXiv:1610.02743].

[73] LiteBIRD Collaboration, Progress of Theoretical and Experimental Physics 2023, 4, 042F01 (2023), [arXiv:2202.02773].

[74] Planck Collab. 2018 Results VIII, Astron. Astrophys. 641, A8 (2020), [arXiv:1807.06210].

[75] F. J. Qu et al., arXiv e-prints arXiv:2304.05202 (2023), [arXiv:2304.05202].

[76] W. L. K. Wu et al., Astrophys. J. 884, 1, 70 (2019), [arXiv:1905.05777].

[77] D. Scott et al., J. Cosmology Astropart. Phys. 2016, 6, 046 (2016), [arXiv:1603.03550].

[78] L. Knox and Y.-S. Song, Phys. Rev. Lett. 89, 1, 011303 (2002), [arXiv:astro-ph/0202286]; M. Kesden, A. Cooray and M. Kamionkowski, Phys. Rev. Lett. 89, 011304 (2002), [arXiv:astro-ph/0202434]; C. M. Hirata and U. Seljak, Phys. Rev. D68, 8, 083002 (2003), [arXiv:astro-ph/0306354].

[79] Planck Collab. 2013 Results XII, Astron. Astrophys. 571, A12 (2014), [arXiv:1303.5072]; Planck Collab. 2015 Results X, Astron. Astrophys. 594, A10 (2016), [arXiv:1502.01588]; Planck Collab. 2018 Results IV, Astron. Astrophys. 641, A4 (2020), [arXiv:1807.06208].

[80] B. Gold et al., Astrophys. J. Supp. 192, 15 (2011), [arXiv:1001.4555].

[81] Planck Collab. Interm. Results XXX, Astron. Astrophys. 586, A133 (2016), [arXiv:1409.5738].

[82] M. Millea et al., Astrophys. J. 746, 4 (2012), [arXiv:1102.5195].

[83] Sunyaev, R. A. and Zeldovich, Ya. B., Ann. Rev. Astron. Astrophys. 18, 537 (1980).

[84] R. Williamson et al., Astrophys. J. 738, 139 (2011), [arXiv:1101.1290].

[85] M. Hilton et al., Astrophys. J. Supp. 253, 1, 3 (2021), [arXiv:2009.11043].

[86] Planck Collab. Early Results VIII, Astron. Astrophys. 536, A8 (2011), [arXiv:1101.2024].

[87] Planck Collab. 2013 Results XX, Astron. Astrophys. 571, A20 (2014), [arXiv:1303.5080].

[88] T. de Haan et al., Astrophys. J. 832, 95 (2016), [arXiv:1603.06522].

[89] Planck Collab. 2018 Results IX, Astron. Astrophys. 641, A9 (2020), [arXiv:1905.05697].

[90] Planck Collab. 2015 Results XVI, Astron. Astrophys. 594, A16 (2016), [arXiv:1506.07135].

[91] Planck Collab. 2013 Results XXII, Astron. Astrophys. 571, A22 (2014), [arXiv:1303.5082]; Planck Collab. 2015 Results XX, Astron. Astrophys. 594, A20 (2016), [arXiv:1502.02114].

[92] Planck Collab. 2015 Results XIV, Astron. Astrophys. 594, A14 (2016), [arXiv:1502.01590].

[93] Planck Collab. Interm. Results XLVI, Astron. Astrophys. 596, A107 (2016), [arXiv:1605.02985].

[94] X. Fan, C. L. Carilli and B. Keating, Ann. Rev. Astron. Astrophys. 44, 415 (2006), [arXiv:astro-ph/0602375].

[95] C. A. Mason et al., Astrophys. J. 856, 1, 2 (2018), [arXiv:1709.05356].

[96] Planck Collab. Interm. Results XLIX, Astron. Astrophys. 596, A110 (2016), [arXiv:1605.08633]; E. Komatsu, Nature Reviews Physics 4, 7, 452 (2022), [arXiv:2202.13919].

[97] M. Kamionkowski and A. Kosowsky, Ann. Rev. Nucl. Part. Sci. 49, 77 (1999), [arXiv:astro-ph/9904108].

[98] Planck Collab. 2015 Results XIX, Astron. Astrophys. 594, A19 (2016), [arXiv:1502.01594].

[99] Planck Collab. Interm. Results XXIV, Astron. Astrophys. 580, A22 (2015), [arXiv:1406.7482].
# 30. Cosmic Rays

Revised March 2024 by J. Alvarez-Muñiz (Santiago de Compostela U.), Z. Cao (IHEP Beijing), U.F. Katz (Erlangen U.), P. Mertsch (TTK, RWTH) and C. Spiering (DESY, Zeuthen).

## 30.1 Theory

Cosmic rays (CRs) are a non-thermal population of particles that pervade the Universe. Their salient characteristics can be inferred from their major observational properties: spectrum, composition and arrival directions. For charged CRs, the energies extend from tens of MeV to close to 1 ZeV, the intensity is ~$10^4 \text{ m}^{-2} \text{ s}^{-1} \text{ sr}^{-1}$ above 1 GeV, but the differential spectrum falls steeply with energy *E*, following a power-law dependence $E^{-\gamma}$. The most striking spectral features are the “knee” at a few PeV where the spectral index $\gamma$ changes from ~2.7 to ~3, the “second knee” at ~100 PeV with a change to ~3.3 and the “ankle” at a few EeV where $\gamma$ changes to ~2.5. The flux is largely suppressed above a few tens of EeV. (More detailed discussion of spectral features can be found below in Secs. 30.2.1 and 30.2.2.) Charged CRs are composed mostly of protons, helium, and other nuclei, as well as electrons, positrons and anti-protons. The arrival directions are mostly isotropic, but below and around the knee interesting $O(10^{-4}, \dots, 10^{-3})$ anisotropies due to the distribution of sources and properties of the Galactic magnetic fields have been observed, reaching ~$O(10^{-1})$ at the highest energies. Gamma-rays can be resolved into those from astrophysical sources (~6660 [36] above 50 MeV, ~300 [37, 38] at TeV energies), plus diffuse fluxes of galactic and extra-galactic origin, predominantly showing power-law dependence on energy. The observation of high-energy neutrinos has opened a new window; while the distribution is largely isotropic, evidence for two extra-galactic sources as well as for a contribution from the galactic plane has been found. The energy spectra of charged CRs, diffuse gamma-rays and neutrinos are shown in Fig. 30.1. Combined observations of charged CRs, gamma-rays and neutrinos as well as gravitational waves (see Sec. 21.2.3) allow for valuable insights into the most extreme astrophysical environments and is referred to as multi-messenger astrophysics. Adding the contribution from all species results in the all-particle spectrum. While it was believed for a long-time that it was a featureless power law up to the knee at a few PeV, it has now been recognized that it has much more structure, mirroring the features in the individual species. These features carry important information on the acceleration and transport of CRs.

The energy variables used are kinetic energy *E*, kinetic energy per nucleon, $E_n = E/A$ for a particle of mass number *A*, or rigidity $R \equiv pc/(Ze)$ (given in units of volt) for a particle of charge number *Z*, *p* being the momentum of the particle; the term rigidity refers to the resistance against deflections in a magnetic field *B*: particles of low (high) rigidity have small (large) gyroradii $r_g = R/B$. Kinetic energy is closely related to the experimental signatures of a calorimetric instrument, while rigidity is the most natural one for a spectrometric one. Note also that relativistic nuclei suffer little energy losses and their transport is prescribed by magnetic fields, thus it can only depend on rigidity. The intensity *J* of CRs, also called the diffuse flux, is defined through the differential number $dN$ of particles with energy in the interval $[E, E+dE]$, crossing the area $dA$ from a solid angle $d\Omega$ in the time $dt$: $dN = J dE dA d\Omega dt$. Its isotropic part is related to the differential density $\psi = (4\pi/v)J$, *v* being the particle speed and to the phase-space density *f* as $J = p^2 f$. Note that the intensity can be also defined in reference to particle energy per nucleon or rigidity. To stress this, often the intensity is written as $dJ/dE$, $dJ/dE_n$ or $dJ/dR$.

In the detection of CRs, two classes of techniques are distinguished [39]. Direct observations, see Sec. 30.2.1, make use of the interactions of CRs in particle physics detectors (e.g. trackers, spectrometers and calorimeters). Given the limited exposures of such instruments and the steeply falling spectra, this is currently only realistic below ~100 TeV. In indirect observations, see Sec. 30.2.2, showers of secondary particles initiated by charged CR interactions in natural materials (e.g. air, water or ice) are detected through fluorescence, air-Cherenkov, water-Cherenkov or scintillation effects. This allows for large instrumented surfaces or volumes. Historically, the two classes of techniques have been

employed by two separate communities and we structure the review of charged CRs accordingly. For detection of gamma-rays, direct observations are relevant mostly below hundreds of GeV; for gamma-rays of higher energies and neutrinos of essentially all energies, indirect observations are required.

Charged CRs are deflected by magnetic fields and so generally speaking the observed events do not point back to sources. CRs can however reach the earth from galactic and even cosmological distances. Between hundreds of MeV and at least a few PeV, CRs are believed to be of galactic origin; above a few EeV, the sources are most likely extra-galactic. If CRs with energies in excess of ~1 EeV came from sources in the Galactic disk, the angular distribution at Earth would be very anisotropic, which is at variance with observations. And if CRs below ~1 PeV were dominantly of extra-galactic origin, this would result in a gamma-ray flux from objects like the Large Magellanic Cloud exceeding observations [40]. Direct and indirect observations therefore also largely refer to galactic/extra-galactic sources, respectively. CRs with energies in excess of 1 EeV are referred to as ultra-high energy cosmic rays (UHECRs).

Sources of non-thermal electromagnetic radiation are routinely observed, see Sec. 30.3, yet the dominant source of locally measured CRs have not been unambiguously identified. Source candidates are typically associated with endpoints of stellar evolution or supermassive black holes that release large amounts of gravitational or rotational energy [41]. In the Galaxy, the prime candidates are supernova remnants [42] where particles can be shock-accelerated by their blast waves [43]. Other candidates are star cluster winds [44,45], stellar wind binaries, micro quasars [46] (a source powered by accretion from a donor star onto a stellar mass black hole) or even the Galactic Center [47]. The candidate sources for extra-galactic CRs are Active Galactic Nuclei (AGN) [48], specifically blazars and radio galaxies, gamma-ray bursts [49], starburst galaxies [50, 51], pulsars [52] and magnetars [53]. These objects exhibit power-law spectra in electromagnetic radiation, albeit in limited wavelength ranges. The acceleration mechanisms considered in the literature are shock acceleration [43], stochastic acceleration [54], and reconnection [55]. A fundamental constraint on the maximum energy, the so-called Hillas criterion [56], follows from the requirement that the gyroradius must be smaller than the source size.

In the Milky Way, the dominant process in CR transport is diffusion as evidenced by the small anisotropies in CR arrival directions and by certain abundance ratios of nuclear species. This diffusive transport bears some resemblance with heat transport in that it smooths the spatial distribution of CRs. However, in contrast to heat transport, CR diffusion is not due to collisions, but interactions with turbulent magnetic fields. Generally, CRs interact “resonantly” with plasma waves, that is they get affected only by waves with a wavelength similar to the gyroradius of the CR particle. If this condition is satisfied, a CR particle will be deflected by the Lorentz force. Many random deflections lead to a random walk in space, that is diffusion [57].

There is a number of other processes contributing to the transport of charged CRs: momentum losses, i.e. radiative losses for electrons and positrons [59], ionization and Coulomb losses for nuclei, electrons and positrons [60]; spallation, that is production of (mostly) lighter nuclei by inelastic collisions of heavier ones; and other inelastic collisions, e.g. the production of light mesons. We note that progress in the study of CR transport is often limited by the nuclear interactions since many cross-sections are poorly known, if at all [61]. The various processes are encoded in the CR transport equation [62], also referred to as the Parker transport equation [63] in the space physics community. This partial differential equation is supplemented by the boundary condition of free escape on the surface of the (often cylindrical) confinement volume. Only in simplified cases can this equation be solved analytically. The most instructive case is the 1D approximation where only the direction perpendicular to the Galactic disk is retained. The solution differs for primaries, that is species present and accelerated in the CR sources, and secondaries, which are not present in CR sources, but produced by inelastic collisions of primaries in the gaseous disk of
<figure><img src="image_154.png" /><figcaption>**Figure 30.1:** The spectrum of cosmic rays (CRs). Shown are measurements of the intensity of charged and neutral CRs, multiplied by kinetic energy squared. The data for the charged CRs [1–30] have been extracted from the Cosmic Ray Database (CRDB) [31]. Below $10^4$ GeV, the all-particle spectrum is the sum of spline fits of the most important nuclear species. The diffuse γ-ray fluxes have been extracted from Refs. [32–34], measurements of the diffuse neutrino background from Ref. [35]. Energy-integrated intensities are indicated by the various diagonal lines.</figcaption></figure>

the Galaxy. Diffusion in the Galaxy, characterized by a diffusion coefficient $\kappa(\mathcal{R})$, modifies the source spectrum $q(\mathcal{R})$, resulting in the steady-state spectrum $\psi(\mathcal{R}) \propto q(\mathcal{R})/\kappa(\mathcal{R})$. As the production spectrum of secondaries follows the steady-state spectrum of primaries, $q_2(\mathcal{R}) \propto \psi_1(\mathcal{R})$, the secondary steady-state spectrum is $\psi_2(\mathcal{R}) \propto q_2(\mathcal{R})/\kappa(\mathcal{R}) \propto \psi_1(\mathcal{R})/\kappa(\mathcal{R})$. As $\kappa(\mathcal{R})$ grows with $\mathcal{R}$, secondary spectra $\psi_2(\mathcal{R})$ fall more quickly with rigidity than primary spectra $\psi_1(\mathcal{R})$. Unstable secondaries provide additional constraints on the gas density and residence time of CRs. The solution of the transport equation in more realistic setups requires numerical codes [64–67].

Protons (iron nuclei) with energies in excess of ~1 EeV (~26 EeV) in a micro-Gauss magnetic field have gyroradii larger than the typical kiloparsec scales of the Galaxy, thus they cannot be confined. Their directions are however affected by so-called small-angle scattering, both in the Milky Way and possibly outside. As far as their spectrum and composition is concerned, protons and heavier nuclei suffer from inelastic collisions with the CMB and the extra-galactic background light at the highest energies. For protons above 10 EeV, photo-pion production limits the propagation distances to ~100 Mpc; for nuclei, heavier ones photo-disintegrate to lighter ones, also limiting their spatial reach. Both processes can lead to a suppression of the flux known as the

GZK effect [68, 69] and produce secondary gamma-rays and neutrinos. Other secondary particles include neutral and charged pions, which ultimately decay to photons, electrons, positrons and neutrinos, and electron-positron pairs which can also initiate a cascade of photons and lower energy electrons and positrons. The stable decay products, in particular gamma-rays and neutrinos can be used as an additional diagnostic tool for the study of the origin and transport of UHECRs.

In many astrophysical environments, the energy density of CRs constitutes a significant fraction of the total or is comparable to that of other ingredients. For instance, on galactic scales, the CR energy density is of the same order as that of magnetic fields, radiation fields or the turbulent gas. Due to their pressure, CRs can be dynamically important and shape their environments: Specifically, CRs contribute to the gravitational support of galaxies and can drive galaxy-scale outflows. They ionize neutral gas, thus determining coupling of gas and magnetic fields. CRs can generate turbulence by a variety of streaming instabilities [70, 71], which play a central role in shock acceleration and galactic transport [72]. Finally, CRs also produce diffuse emission through interactions with gas and radiation fields, an additional handle for CR studies.

CRs are also important probes of fundamental physics. If dark
<figure><img src="image_155.png" /><figcaption>Figure 30.2: CR abundances compared with solar system ones [58]. Modified from [31].</figcaption></figure>

matter has interactions with the Standard Model, the products of self-annihilation or decay of DM particles can potentially be observed (DM, see Sec. 27.7). This is called DM indirect observation. Generically, one would expect similar rates of production of particles and antiparticles whereas astrophysical sources predominantly accelerate matter particles. It is therefore advisable to search for signals in CRs positrons, anti-protons and anti-nuclei [73, 74]. In fact, excesses have been reported of positrons and anti-protons, yet astrophysical pollutions and/or experimental systematic uncertainties can not be excluded. There have also been searches in gamma-rays and neutrinos and a number of studies have identified an excess of gamma-rays from the Galactic Center that is compatible with DM particles of weak-scale mass and weak annihilation cross-section [75]. Another target for exotic searches in CRs is anti-matter of primordial origin [76]. Most interestingly, the AMS-02 collaboration has claimed $O(10)$ He candidate events, but whether those are of primordial origin as opposed to instrumental effects is at this point not clear.

## 30.2 Charged cosmic rays

### 30.2.1 Direct observations

Direct observations [77] cover energies between a few MeV and a few tens of TeV, where CRs must be of Galactic origin. At energies lower than a few MeV, solar energetic particles [78] are dominating the flux; above a few tens of TeV, the fluxes are too low to be observed directly and instead indirect detection techniques [79] need to be used. Besides spectral information, direct observations enable identification of species on an event-by-event basis, certain experiments even provide isotopic discrimination. We present a compilation of direct observations of the most important primary and secondary nuclei in Fig. 30.3. These show broken power-law spectra in rigidity with $J \propto E^{-\gamma}$ in different rigidity ranges. The various changes in spectral slope contain valuable information on the rigidity-dependence of the diffusion coefficient $\kappa$ and the primary source spectra $q_1$.

Protons are dominating the CR flux by number and by energy below ~10 TeV and their spectrum has been measured with excellent precision. Measurements of the proton spectrum and other primaries are shown in the left panel of Fig. 30.3. Data from the AMS-02 experiment [8] show a $R^{-2.8}$ spectrum above ~10 GeV with a break to $R^{-2.7}$ at ~300 GV. This feature had been observed by PAMELA earlier [2], and also with CREAM and ATIC for heavier nuclei [80, 81]. As a reminder, at energies above a few GeV, the measured spectrum can be interpreted as the ratio of the source spectrum $q(R) \propto R^{-\Gamma}$ divided by the diffusion coefficient $\kappa(R) \propto R^{\delta}$. Thus a spectral index of $\gamma = \Gamma + \delta \simeq 2.8$ can describe the data. On the basis of proton data alone, the degen-

eracy between the source spectral index $\Gamma$ and the spectral index of the diffusion coefficient $\delta$ cannot be broken. Also, the break at ~300 GV could be a break in the source spectrum or in the diffusion coefficient, but see below. At energies lower than ~10 GeV, the spectrum turns over because of energy losses and solar modulation. At energies beyond the reach of AMS-02, both DAMPE and CALET data show a softening at rigidity ~10 TV [7, 10], confirming earlier indications [80, 81].

Most of the other primary species, that is helium, carbon, nitrogen, oxygen, neon, magnesium and silicon with the notable exception of iron, also show a spectral break at ~300 GV, but their spectra are generally harder than the proton one, e.g. $\gamma_{He} < \gamma_{p}$, both below and above the break. Suggested explanations for this difference in spectrum include gradients in the relative circumstellar abundances of protons and heavier nuclei or invoking different source populations for these two groups. At ~10 TV a softening break mirrors the feature observed in protons [82]. Above an energy of ~100 TeV, elemental information is scarse, but a combined analysis of proton and helium by DAMPE extends to ~300 TeV and indicates another spectral hardening [83]. (Note that this data are not included in Fig. 30.3 though.) Such a feature was to be expected to a certain degree in order to connect to the indirect measurements of proton and helium which start at ~1 PeV. Note that there exist some 20% discrepancies between the oxygen and carbon data from AMS-02 [8], CALET [13] and PAMELA experiments [84] in normalization, possibly also in spectral shapes.

Iron constitutes an exception to the otherwise rather similar slopes of primary species heavier than proton. As is also the case for other nuclei, AMS-02 [15] and CALET [16] disagree on the normalization by ~20%, but both measurements of the iron spectrum exhibit spectral shapes different from the oxygen ones. While the ratio of Fe/O could be accommodated if all nuclei traversed the same amount of matter (grammage) between source and observer (the so-called slab model), more realistic models cannot explain the observations [85]. Also, the AMS-02 data on Fe/O are in tension with earlier measurements [86, 87]. Most interestingly, if extrapolated to lower energies, the Fe/O could only be accommodated if solar modulation was negligible.

Measurements of stable secondary nuclei contribute two important pieces of information to the standard scenario of galactic CRs: First, their over-abundance in CRs (with respect to solar system values, see Fig. 30.2) indicates diffusion as the primary transport process; and second, their spectral shapes, or rather the difference in spectral shapes between secondaries and primaries allows inference of the diffusion coefficient. Specifically, at energies where energy losses can be ignored for nuclei, that is above a
few GeV, the propagated spectra of primaries $\psi_1(\mathcal{R})$ are approximately $\propto \mathcal{R}^{-2.8...-2.7}$, which is explained as a softening of the sources spectrum $q_1(\mathcal{R})$. Secondaries, however, are produced with a spectrum proportional to the propagated primary spectrum, $q_2(\mathcal{R}) \propto \psi_1(\mathcal{R})$ which also gets further softened by the diffusion coefficient. The difference in spectral index can be clearly seen when comparing primary with secondary spectra. A compilation of such secondary spectra from AMS-02 [8, 88, 89], CALET [90] and PAMELA [84] is shown in the right panel of Fig. 30.3. Lithium, beryllium, boron and fluorine can be identified as almost pure secondaries by their soft spectra. Aluminium, sodium and neon, however, have harder spectra, somewhere intermediate between primaries and secondaries. This can be understood as being due to mixture of softer secondary contribution from spallation in the interstellar medium (ISM) and a harder primary source component. Secondaries also exhibit a hardening break at $\sim 300$ GV, but the change in spectral slope is about twice as big [91] when compared to that of the primaries. This fact serves as evidence for the break being due to a change in the rigidity-dependence of the diffusion coefficient, as opposed to a break in the source spectrum [92, 93]. Recently, there have been some claims of excesses in low-energy intensities of Li, F, Al [94], however these claims are somewhat controversial. We note the importance of composition studies for shedding light on cosmic ray origin.

Unstable isotopes with a lifetime similar to the residence time of CRs can be used to study galactic propagation. Among these species, $^{10}$Be, $^{26}$Al and $^{60}$Fe have been studied the most as their rest lifetime is close to the residence time of CRs at GV rigidities. Roughly speaking, the flux ratio of an unstable and a stable secondary that are both produced from similar primary species, reflects the ratio of production cross-section at high energies (where decay is suppressed by time-dilation). On the other hand, at low energies, this ratio is suppressed due to the decay of the unstable species. The transition takes place at energies where the decay time equals the residence time in the Galactic disk. At present, there is limited information on ratios at energies beyond 1 GeV/nucleon where the transition for the ratios of the above mentioned unstable nuclei is expected to take place.

CR electrons are also believed to be accelerated in the preferred candidate sources of nuclei, that is, SNRs. Other source candidates are pulsars and pulsar wind nebulae (PWNe); in the simplest models those also accelerate similar amounts of CR positrons. We show a compilation of the most recent and precise data from AMS-02 [8], CALET [19], DAMPE [17], Fermi-LAT [18], H.E.S.S. [95] and PAMELA [96] on CR electrons, positrons and the sum of electrons and positrons in Fig. 30.4. In addition to the processes at work in the propagation of nuclei, electrons suffer significant radiative losses due to synchrotron emission and inverse Compton scattering. In fact, for the parameters determined in the standard scenario, the cooling time is shorter than the residence time above $\sim 10$ GeV. For the same source spectral index, the spectra are predicted to be softer than those of primary nuclei. Indeed, the observed spectrum is roughly $\propto E^{-3.1}$.

If positrons were only produced by spallation in the interstellar medium, the positron spectrum should be softer than the electron one, however, on the contrary it is markedly harder. Between $\sim 10$ GeV and $\sim 100$ GeV its spectral index $\gamma_{e+}$ transitions from 3 to 2.7 before a spectral break or cut-off at a few hundred GeV as shown in Fig. 30.4. The fact that the positron spectrum is harder than the electron one above a few GeV had first been detected by PAMELA [97] after earlier indications. AMS-02 has now measured the spectrum with great precision up to $\sim 1$ TeV [8]. The origin of the harder positron spectrum remains unclear today. After the discovery by PAMELA, there had been hope that this was the long sought for signature of self-annihilation or decay of weakly interacting massive particles (WIMP) dark matter [98]. However, such interpretations are now severely constrained by observations of gamma-rays, anti-protons and the cosmic microwave background (CMB, see Chap. 29). Production of $e^+$ in pulsars or PWNe is an astrophysical explanation that had been hypothesized long before the PAMELA observations (e.g. [99]). Recently, there have been observations of extended emission of high-energy gamma-rays, so-called gamma-ray halos

around PWNe [100], interpreted as evidence of the presence of high-energy electrons and positrons around these objects. Other astrophysical explanations include the production and subsequent acceleration of secondaries in old SNRs [101, 102].

Measurements of the spectra of electrons and positrons are particularly interesting at high energies. Due to the increasingly important radiative losses, the distance to the sources which could significantly contribute to the total flux decreases with energy. Estimates at 1 TeV of this diffusion-loss length are only $\simeq 0.3$ kpc. The predicted spectrum therefore becomes rather sensitive to the exact distances and ages of young, nearby sources which manifest as individual bumps in the spectrum. Turning this around, observations can be used to search for young, nearby sources of CRs. At energies above $\sim 1$ TeV, only calorimetric observations are currently available which cannot discriminate between electrons and positrons. Observations of the all-electron flux, that is the sum of electron and positron fluxes, find a break at $\sim 1$ TeV, displayed in Fig. 30.4. This was first seen by H.E.S.S. [95] and later confirmed by DAMPE [17] and CALET [19]. The break by about one power in energy has been interpreted either as due to a break in the spectrum of a large number of sources, e.g. [104] or as a stochasticity effect from a small population of sources [105].

In the standard scenario of galactic cosmic rays, anti-protons are also produced as secondary particles, but unlike secondary nuclei or electrons and positrons, their production is kinematically suppressed at lower energies. The spectrum of anti-protons observed by PAMELA (Fig. 3 in [106]) and AMS-02 (Fig. 62 in [8]) is close to $E^{-2.8}$, which is somewhat harder than predicted by earlier models. More recent models can accommodate the observations, in part due to a re-evaluation of the production cross-section. Note that there have been claims that the transport parameters obtained when fitting to proton and anti-proton data differ from those obtained by fitting to heavier nuclei [107]. An alternative explanation for the harder anti-proton spectrum is the acceleration of secondaries in old supernova remnants [102, 108]. At energies of a few GeV, there have been claims of an excess in the measured anti-proton spectrum. If interpreted as a sign of dark matter annihilation, such an excess could be explained by a weak-scale particle of mass of a few tens of GeV and annihilation cross-sections close to the thermal relic value. However, the significance of the signal is relatively low if correlations in the cross-section uncertainties and in the measurements are taken into account. Considering the spectra of anti-protons together with protons and positrons, it appears that all three have similar spectral indices between $\sim 10$ GV and $\sim 300$ GV. This is rather surprising, given that as secondaries positrons and anti-protons should have spectra softer than the primary protons, due to energy losses and diffusive losses, respectively. In the standard scenario this cannot be explained and thus must be considered a mere coincidence.

The only astrophysical mechanism for production of anti-nuclei like anti-deuterons ($\overline{D}$) or anti-helium ($^3\overline{He}$ or $^4\overline{He}$) is coalescence of anti-nucleons produced in the spallation of primary nuclei in the interstellar medium. Due to kinematics, however, this is significantly suppressed at energies below a few GeV, thus rendering this an interesting channel for searches of dark matter, in particular for relatively light WIMPs with masses of a few GeV or less. The AMS-02 collaboration has reported some ten candidate events of anti-helium, yet the rate of $^3\overline{He}$ or $^4\overline{He}$ is incompatible with models of coalescence (e.g. [109]). Currently, the most constraining limits on $\overline{D}$ are from the BESS balloon flights [110]. This should be improved on by the upcoming GAPS experiment, and also an upgrade of the AMS-02 experiment increasing the acceptance by a factor of three could help clarifying the situation.

Another important observable in the study of CRs are anisotropies, most commonly defined as the relative directional deviation of the CR intensity from the angular average. As the typical anisotropies are at the level of one part in $10^3$ or $10^4$, high statistics are required. At TeV and PeV energies, only ground-based experiments have enough statistics, yet they cannot constrain the component of the anisotropy that is aligned with the Earth's rotation axis. Traditionally, therefore, only the projection of the true 2D dipole into the equatorial plane has been reported in terms of an amplitude and phase. We show a
<figure><img src="image_156.png" /><figcaption>**Figure 30.3:** Rigidity spectra for a selection of CR nuclei. Primary species are shown in the left panel, (mostly) secondary species in the right panel. The error bars denote the statistical uncertainties, the shaded bars represent the systematic uncertainties. The spectra have been shifted and multiplied by $R^{2.7}$ to better bring out spectral features. Here, we have constrained ourselves to data from space-borne experiments of the last ten years, that is AMS-02 [8, 15, 88, 89, 103], CALET [10, 16, 82, 90], DAMPE [7, 13], NUCLEON [5] and PAMELA [2, 84]. Data have been extracted from [31].</figcaption></figure>

compilation [31] in Fig. 30.5. The amplitude is at the level of $\simeq 10^{-4}$ at energies of hundreds of GeV and increases to $\simeq 10^{-3}$ at tens of TeV. At $\sim 100$ TeV there is a dip before the amplitude rises again. Beyond a few PeV upper limits exist at the level of $10^{-2}$. The Compton-Getting effect [111, 112], that is a dipole anisotropy due to the relative motion of the solar system with respect to the Galactic rest frame has been experimentally confirmed [113]. The phase is relatively constant at a direction of 45° and switches to 250°, which is close to the Galactic Center direction, also around 100 TeV. For a description of anisotropies at the highest energies see Section 30.2.2. In addition, observations of anisotropies on scales smaller than the dipole have been reported (see Ref. [114]). These are not expected in standard diffusion theories that only predict a dipole. The leading explanation is the generation due to the specific realization of the turbulent magnetic field in our Galactic neighbourhood.

### 30.2.2 Indirect observations

In the energy range above $\simeq 1$ PeV the intensity of cosmic rays is so low that it prevents the collection of a statistically significant sample of these particles in a direct manner with detectors carried on balloons or satellites. Instead, cosmic rays are detected indirectly through the cascade of particles they produce

in the atmosphere, the so-called extensive air showers (EAS). In this way, cosmic rays up to and even above 100 EeV have been observed [115, 116], corresponding to about a hundred million times larger energy than that achieved by particles accelerated in human-made experiments.

**Extensive Air Showers.** Extensive air showers (EAS) are cascades of millions to billions of secondary particles initiated by a single primary particle (cosmic ray: proton, neutron or heavier nucleus; photon; or neutrino) of energy $E$ in the atmosphere. Due to the small density of air $\rho \lesssim 10^{-3}$ g cm$^{-3}$, the shower develops throughout the whole of the atmosphere, and for a primary particle of 10 EeV has a lateral spread of $\sim 10$ km$^2$ at ground level, hence the name EAS.

When an energetic, $E \gtrsim 1$ PeV, cosmic proton or nucleus interacts with a nucleus of air, it generates between a few and hundreds of secondary particles depending on its energy. On average $\sim 30\% - 40\%$ of the energy of the primary particle is carried by a leading baryon or nucleus. The remaining energy is employed in the creation of ultra-relativistic secondary particles, most of them charged ($\pi^{\pm}$) and neutral ($\pi^0$) pions, with a smaller number of hadrons and heavier mesons such as charged and neutral kaons, $\rho$-mesons, etc. The energy of the primary particle
<figure><img src="image_157.png" /><figcaption>**Figure 30.4:** Intensities of cosmic ray electrons (squares), positrons (diamonds) and their sum (dots) as a function of kinetic energy *E*. We have included data from the most recent space experiments AMS-02 [8], CALET [19], DAMPE [17], Fermi-LAT [18] and PAMELA [96] as well as from the ground-based H.E.S.S. experiment [95]. The spectra have been multiplied by *E*³ to enhance the visibility of the spectral features. Data have been extracted from [31].</figcaption></figure>

<figure><img src="image_158.png" /><figcaption>**Figure 30.5:** Phase and amplitude of the dipole anisotropy in CRs. Modified from Ref. [31].</figcaption></figure>
rapidly degrades into a large number of secondaries that can subsequently interact or decay depending on the nature and energy of the particle and the density of the medium, further contributing to the shower development. If $\pi^{\pm}$ live long enough they can interact with an air nucleus before decaying. At altitudes of a few km where the atmosphere is less dense than at sea level, most $\pi^{\pm}$ with $E_{\pi^{\pm}} \gtrsim 100$ GeV interact while lower energy $\pi^{\pm}$ decay. Interacting $\pi^{\pm}$ along with the leading baryon and other hadrons, form a penetrating core of shower particles constituting the hadronic component of the cascade (Fig. 30.6). Charged pions decaying into muons, $\pi^{+} \rightarrow \mu^{+} + \nu_{\mu}$ and $\pi^{-} \rightarrow \mu^{-} + \bar{\nu}_{\mu}$, represent the main contribution to the muonic component of the shower (Fig. 30.6), being also responsible for the production of the bulk of the so-called atmospheric neutrinos. Most muons of energies above a few GeV travel through the atmosphere without decaying and reach ground, constituting a penetrating component of the shower. On the other hand, most $\pi^{0}$s, continuously produced in the hadronic core of the shower, do not travel long enough distances to interact with an air nucleus unless their energy is well above 1 EeV. Instead, they decay almost immediately $\pi^{0} \rightarrow \gamma\gamma$, initiating electromagnetic showers. Neutral pions play a significant role in generating the electromagnetic component that forms the majority of the shower (Fig. 30.6). This is composed of $e^{-}$ and $e^{+}$ with a mean energy of 10 MeV as a result of photon pair production, as well as secondary photons from bremsstrahlung. Decaying muons also contribute to the electromagnetic component of the shower. The longitudinal development of the distinct and intertwined shower components, electromagnetic, muonic, and hadronic, is shown in Fig. 30.6. High-energy primary or secondary photons can occasionally photoproduce $\pi^{\pm}$ and initiate a hadronic subcascade resulting in an additional muonic component.

As a cosmic ray shower develops in the atmosphere, the number of particles increases in the longitudinal direction along the shower axis. The amount of matter traversed, $X$, is measured in units of $g \text{ cm}^{-2}$. The shower reaches a maximum size of the electromagnetic component $N_{\text{max}}$ at a depth known as $X_{\text{max}}$ (Fig. 30.6). The number of electrons and positrons declines rapidly after maximum when their energy reaches the critical energy in air, ~80 MeV, and they lose their remaining energy dominantly through ionization loss. Since neutrinos do not suffer electromagnetic interactions, and high energy muons reach ground level after releasing only a portion of their energy, a fraction of the energy of the primary initiating the shower is not deposited in the atmosphere. This is the so-called invisible energy that amounts to ≈ 10% – 20% of the shower energy for energies above 0.1 EeV, decreasing with the energy of the primary particle initiating the shower and increasing with its mass number [118]. At the same time the shower develops in the transverse or lateral dimension to the shower axis, as shown in Fig. 30.6, mainly due to the transverse momentum in hadronic interactions and multiple Coulomb scattering of $e^{-}$ and $e^{+}$.

Shower development is driven by the particles carrying most of the energy in the forward region of the kinematical phase space of the collisions with air nuclei. Particle production is dominated by non-perturbative QCD which is necessarily treated by using phenomenological approaches. Moreover, particle collisions occurring in the atmosphere at 0.1 EeV are equivalent to center-of-mass energies presently achieved at the LHC, and this energy scales up by a factor of approximately 30 at the highest cosmic ray energies observed. These two facts result in the need for extrapolations both in energy and phase space when modeling hadronic interactions. The cosmic-ray community has developed sophisticated simulation packages such as AIRES [117] and CORSIKA [119] to predict EAS observables as a function of primary energy and mass. These programs adopt models of hadronic interactions such as SIBYLL [120], QGSJET [121] and EPOS [122], tuned to a large data set including recent LHC data. Inferring fundamental properties of the primary particle initiating the shower, most notably its mass, requires the use of these air shower simulations. Large uncertainties remain, both due to theoretical limitations and the lack of data from existing collider experiments. A toy model to describe electromagnetic showers is due to Heitler [123], while for hadronic showers a model was developed by Matthews [124]. These oversimplified models shed light on the

relation between the main shower observables and the nature of the primaries and predict that for the electromagnetic component, $N_{\text{max}} \propto E$; $X_{\text{max}} \propto \log E$ and $X_{\text{max}} \propto \log A^{-1}$ with A the mass number of the primary nucleus; while for the muonic component, $N_{\mu} \propto A^{(1-\beta)}E^{\beta}$ with $\beta \simeq 0.85...0.95$. These important features of air showers constitute the basis for the identification of the primary particle with EAS (see next section). Similar features are also obtained in detailed Monte Carlo simulations of EAS development (Fig. 30.6), and have been observed experimentally.

**Shower detection and cosmic-ray reconstruction.** EAS have been detected with several techniques. Arrays of conventional particle detectors such as scintillators, water-Cherenkov stations or underground muon detectors, measure the lateral distribution of the shower front at ground, i.e., at a fixed depth. Due to the low cosmic-ray intensity especially at EeV energies (Figs. 30.1 and 30.7), the particle detectors are spread over large areas to compensate for the low flux, separated by distances that range from hundreds of meters to above 1 km. From the times at which the shower front hits several of these particle detectors, the arrival direction of the cosmic ray is measured with an accuracy typically better than ~1°. Shower arrays measure a shower size parameter, proportional to the number of secondary particles, that can be related to the energy E of the primary particle. With several types of detectors working together, it becomes possible to effectively separate the electromagnetic and muonic components of the shower, thereby providing an estimate of the mass of the primary. The main arrays of particle detectors for EAS observation currently in operation are the surface detector of water-Cherenkov stations of the Pierre Auger Observatory in Argentina [125] spread over a total area of ~3000 km², and the Telescope Array (TA) of scintillators in USA (~700 km²) [126]. They typically work with almost 100% duty cycle and have measured many properties of the cosmic-ray flux (see next section).

Other arrays can measure the radiation emitted when the shower develops in the atmosphere, namely, optical Cherenkov radiation, fluorescence light, or radio emission in the MHz to GHz frequency range. Cherenkov detectors measure the forward-beamed incoherent emission at optical wavelengths. This can be achieved with arrays of individual elements such as at the Yakutsk [127] and Tunka-133 [128] observatories. A fluorescence telescope can monitor a large volume of the atmosphere and detect the isotropic fluorescence radiation emitted by the nitrogen molecules of air excited by the passage of the charged particles of the shower. The fluorescence radiation is produced in proportion to the energy dissipation and the shower size $N(X)$, allowing a reconstruction of the longitudinal profile of the energy deposit $dE/dX$ of the shower as a function of atmospheric depth $X$ and, in particular, providing a measurement of $X_{\text{max}}$. With the fluorescence detection technique, the atmosphere is used as a calorimeter, with the integral $E_{\text{cal}} = \int (dE/dX)dX$ called the calorimetric energy of the shower. An estimation of the primary energy with the fluorescence detection technique is obtained by adding to $E_{\text{cal}}$ a correction to account for the invisible energy introduced above. Fluorescence and optical Cherenkov detectors are limited by operation during clear and moonless nights which reduces their duty cycle to ≈ 15%. The main fluorescence detectors currently in operation are those of the Pierre Auger Observatory [125] and TA [126].

Arrays of detectors of radio emission employ cost-effective elements (antennas) to detect the radiation in the 10's of MHz to GHz produced by the charged particles in the EAS. Radio emission is mainly due to two mechanisms: the transverse current induced by the magnetic field of the Earth that travels at the speed of light along shower axis, and the Askaryan effect i.e. the radiation emitted by the excess of negative charge ($e^{-}$) that develops in the shower through Compton scattering of shower photons on $e^{-}$ in the medium, as well as Möller and Bhabha scatterings. The measurement of the radiated energy in radio waves at ground also provides an estimate of the primary energy [129]. In a radio-quiet environment, these detectors can operate with a close to 100% duty cycle, only limited by thunderstorms and heavy rain. The principal arrays of radio antennas currently operating include the Auger Engineering Radio Array (AERA) and the ongoing deploy-
<figure><img src="image_159.png" /><figcaption>**Figure 30.6:** Left: Average number of $e^+ + e^-$, $\mu^+ + \mu^-$, and hadrons (lines), and their $\pm 1\sigma$ deviation (bands), as a function of matter (in $g cm^{-2}$) crossed in the atmosphere along the shower axis. The profiles are obtained with AIRES simulations [117] of proton, iron and photon-induced extensive air showers of energy $E = 1$ EeV $= 10^{18}$ eV and zenith angle $\theta = 45^\circ$ w.r.t. the vertical to ground (at depth $870$ $g cm^{-2}$). Right: Lateral development (perpendicular to shower axis) at ground level of the same components as in the left panel as a function of distance to shower axis.</figcaption></figure>

ment of the AugerPrime Radio Detector at the site of the Pierre Auger Observatory in Argentina [130], the Tunka-Rex [131] and Yakutsk [132] radio extensions in Russia, the LOw Frequency AR-ray (LOFAR) with its core detector in The Netherlands [133], and the Square Kilometer Array (SKA) in Australia [134].

Combining several types of detectors in the same array has proven very fruitful for increasing the accuracy in event reconstruction and energy calibration. Complementary measurements of the same event are expected to improve the determination of mass composition of the primary flux on an event-by-event basis. Projects under construction exploiting the simultaneous use of several detector kinds include, AugerPrime (water-Cherenkov stations, scintillators, underground muon detectors, fluorescence detectors and radio antennas) [135], TAx4 (scintillators and fluorescence detectors) [136], and the IceCube-Gen2 surface array (scintillators and radio antennas) [137]. Other arrays in the planning stages utilizing radio antennas are GRAND [138] and BEACON [139].

**Experimental results.** Cosmic rays in the PeV to EeV energy range have been regularly detected for decades, with energies up to > 100 EeV. Despite the progress made in the field in the last 20 years, the nature and identity of their sources as well as the acceleration mechanisms that boost the particles to these extreme energies remain open questions. The understanding of CRs is mainly derived from analyzing their energy spectrum, several observables sensitive to their primary nature, and their arrival direction distribution. This is a challenging task mainly due to the small flux, with less than 1 particle per km² per year above 30 EeV (see Fig. 30.1), but also because the properties of CRs in this energy range can only be inferred indirectly from measurements of the EAS whose description relies on extrapolations of particle physics properties at energies several orders of magnitude above those achieved in terrestrial accelerators. Last but not least, since CRs are charged they are deflected by the galactic and intergalactic magnetic fields and their arrival directions at Earth point back only approximately to their actual sources.

*Energy spectrum:* The cosmic-ray spectrum observed at Earth (Fig. 30.1, with Fig. 30.7 highlighting the highest energies) can be characterized by a series of power-law functions proportional to $E^{-\gamma}$ with $\gamma \in (2.5, 5.0)$ the spectral index that changes with energy. The production mechanism of CRs, mainly the unknown maximum CR energy achievable, CR composition and spectral index at the sources; the source type and distribution in the Universe; and the propagation through the Galaxy and the

intergalactic space, all leave distinct imprints on the spectrum giving rise to several observable features. The differential flux over 5 orders of magnitude in energy is shown in Fig. 30.7. The flux has been multiplied by $E^3$ to highlight the deviations from a pure power law with a single value of $\gamma$. Several features are apparent, namely: a steepening at $\sim 5$ PeV known as the *knee* (Fig. 30.1); another steepening around 100 PeV dubbed the *second knee*; the *ankle*, a flattening of the spectrum at around 5 EeV; a recently discovered feature corresponding to a further flattening starting at 10 EeV dubbed the *instep*; and the *suppression* of the spectrum starting around 50 EeV. These features are thought to be correlated with the observed changes on the average CR composition (see *Primary composition* below).

Cosmic-ray experiments have systematic differences in their energy scales and this manifests itself in differences in the measured spectrum, not only on the energy axis but also on the normalization when the flux is scaled with a power of the energy. The diagonal uncertainty bars in Fig. 30.7 serve to visualize the impact of these systematic differences on the spectrum. In the energy range above $\sim 100$ PeV the fluorescence technique is used to determine the primary cosmic-ray energy, with the atmosphere where the EAS develops functioning as a calorimeter. In the two state-of-the-art experiments for UHECRs, the Pierre Auger Observatory and the Telescope Array, a sub-sample of events is recorded simultaneously with the fluorescence (FD) and the surface detector (SD) arrays. With the SD, a shower size parameter is measured and calibrated against the energy measured with the FD. This approach provides a more direct method for determining cosmic-ray energy without relying on simulations. Differences in the energy scale between Auger and TA remain, however, primarily due to the use of different measurements of the amount fluorescence light emitted per unit of energy deposited, and to the different models for the invisible energy in EAS adopted. When these systematic uncertainties are accounted for, the spectra of the Pierre Auger Observatory and TA have been shown to agree within 5%, except in the energy region near the end of the instep feature and at the suppression.

*Primary composition:* The main observable sensitive to CR composition is the depth in the atmosphere along shower axis ($X_{max}$) at which the number of particles in the shower is maximum. Observatories capable of detecting Cherenkov and/or fluorescence light induced by the passage of the EAS through the atmosphere, can measure $X_{max}$ on an event-by-event basis. With sufficient statistics, the distributions of $X_{max}$ can be determined from which the average value $\langle X_{max} \rangle$ and its fluctuations
<figure><img src="image_160.png" /><figcaption>**Figure 30.7:** Measurements of the all-particle CR flux from Telescope Array (TA) [140] (blue triangles) and the Pierre Auger Observatory [29, 141] (black circles). The direction and magnitude of the systematic uncertainty on the flux due to the energy scale ($\sigma_E^{sys}$) for TA and Auger is indicated by the corresponding arrows. For a comprehensive compilation of measurements see [142].</figcaption></figure>

<figure><img src="image_161.png" /><figcaption>**Figure 30.8:** Measurements of $\langle X_{max} \rangle$ (top) and $\sigma(X_{max})$ (bottom) obtained by the Telescope Array [143, 144], Pierre Auger Observatory [145, 146], Yakutsk [127] and Tunka [147] UHECR detectors, compared to predictions for proton and iron nuclei using the hadronic models SIBYLL2.3c [148], EPOS-LHC [122] and QGSJET-II.04 [121]. Detection techniques: fluorescence (FD), Cherenkov (Yakutsk, Tunka, and TA below $\simeq$ 1 EeV), and using the surface detector array of the Auger Observatory (SD). For a compilation including experiments using the radio technique see [142] and also [149].</figcaption></figure>
$\sigma(X_{\text{max}})$ are obtained. A comparison with the $\langle X_{\text{max}} \rangle$ and $\sigma(X_{\text{max}})$ as predicted in simulations of EAS for different primaries and energies, as well as fits to the $X_{\text{max}}$ distributions lead to the determination of the mean fractions of primary protons, helium, carbon-nitrogen-oxygen and iron in the cosmic-ray flux at Earth. However, the interpretation of any mass-sensitive observable relies on modeling hadronic interactions up to the highest energies where there are no data from terrestrial accelerators, and this introduces a considerable uncertainty in the determination of the mass. Also, due to intrinsic shower-to-shower fluctuations (see Fig. 30.6), an event-by-event determination of the mass of the primaries is not currently possible. Despite these limitations, the measurements presented in Fig. 30.8 alongside the predictions of EAS simulations for proton and iron primaries, reveal a broad trend toward a lighter composition at the knee, within the energy range of a few PeV. This is followed by a gradual rise in the average logarithm of the primary cosmic-ray mass, eventually leading to a heavier composition at around 100 PeV, although notable differences between experiments exist. As the energy increases from 100 PeV up to ~2 EeV the measurements point consistently to a predominantly light composition with a large fraction of primary protons. Above that energy, the data from the largest statistical sample of events collected with the FD of the Pierre Auger Observatory indicate that the composition is mixed with the mean mass steadily growing. This observation is further supported by the shower-to-shower fluctuations of $X_{\text{max}}$ also shown in Fig. 30.8. Though the data of the Telescope Array seem in apparent tension with this picture, they have been shown to be compatible with the results of the Pierre Auger and Yakutsk Observatories [150] within the current levels of statistics and understanding of systematic uncertainties. In particular, TA data are compatible with the mixed composition inferred by the Pierre Auger Collaboration [142].

The small duty cycle of the fluorescence technique combined with the steeply falling spectrum above ≃ 50 EeV prevents a statistically significant determination of the composition in this energy range. Information on the distribution of nuclear masses can be obtained with the surface arrays alone that work with a high duty cycle. Exploiting the arrival times of particles in the SD stations, that are known to contain information on the penetration of the shower in the atmosphere, measurements of $X_{\text{max}}$ with the Pierre Auger Observatory were extended up to 100 EeV as shown in Fig. 30.8, after calibration of the mass-sensitivity of the SD with the measurements of $X_{\text{max}}$ with FD at lower energy. Other novel methods such as those based on deep learning analysis techniques are also being explored [151]. Additionally, the arrays of radio detectors are also sensitive to $X_{\text{max}}$ through the measurement of the shape of the radio emission footprint on the ground induced mainly by the electromagnetic component of the EAS. The radio technique demonstrates considerable potential in this respect and scalability, making it a cost-effective possibility for future UHECR detectors [149,152].

**Arrival directions:** Anisotropies in the arrival direction of CRs are a key to discovering their sources. This is in principle feasible if the charges of the CRs and the galactic and extragalactic magnetic fields are well-enough determined. UHECRs are subject to the GZK effect, the energy loss of protons and heavier nuclei in interactions with the cosmic background. This imposes a horizon to possible sources of ~250 Mpc for protons and Fe nuclei, but shorter for nuclei in between these two, for instance ~5 Mpc for He and ~100 Mpc for Si at energies above ≃ 50 EeV [153]. Having only a limited range of source distances contributing to the signal makes it potentially feasible to identify sources or infer their properties statistically.

The Pierre Auger Collaboration has discovered a large-scale dipolar anisotropy at energies above 8 EeV with $6.9\sigma$ significance [154]. The amplitude of the dipolar anisotropy is $d = 0.073_{-0.008}^{+0.010}$ and the dipole points at a direction that is ~115° away from the Galactic Center, evidencing the extragalactic origin of UHECRs above this energy threshold. The dipole direction is not aligned with any obvious individual source, or with the expected CMB dipole, and is instead consistent with the local distribution of stellar mass (2MASS Redshift Survey), after accounting for the deflections expected in the Galactic magnetic

field. A compelling feature is the growth of the amplitude of the dipole with energy in agreement with the expectation that particles of higher rigidity are less deflected by magnetic fields. Combining the Pierre Auger Observatory data set in the southern hemisphere and the smaller statistical sample of events from Telescope Array in the north, a full-sky coverage is achieved. The combined results regarding the dipolar anisotropy are compatible with the Pierre Auger Observatory-only results with no evidence of the existence of a quadrupole component. At energies between ~10 PeV and ~1 EeV the amplitude of the dipole is smaller (Fig. 30.5, bottom panel), but there is a trend for its direction to change from pointing close to the Galactic Center (GC) at low energy towards a direction away from the GC (Fig. 30.5, top panel) in the EeV energy range. This suggests that there is a transition between the galactic and extragalactic origin of UHECRs around the ankle in the spectrum, at a few EeV.

Hints of correlation of arrival direction of UHECRs with potential sources have been confirmed in several analyses [142,154,155]. With data from the Pierre Auger Observatory a $4\sigma$ excess of events above 38 EeV was found in a region of angular radius ≃ 27° centered on the radio galaxy Centaurus A (CenA) [154]. The Telescope Array Collaboration has also reported an excess of events with energies above 57 EeV (the so-called *hot spot*) in the northern sky in the direction of R.A. = 144.0°, Dec. = 40.5°, with a $2.8\sigma$ significance and in an angular window of 25° [155]. Correlations with catalogs of potential UHECR sources have also been found. The sky viewed from the Pierre Auger Observatory is better modeled with a ≃ 10% flux excess in the directions of nearby starburst galaxies with a significance of 3.8$\sigma$ and an energy threshold of 38 EeV with the correlation driven by the region of CenA where two prominent starburst galaxies are also located [154].

Despite the comprehensive observations made, it remains challenging to definitively ascertain the origin of UHECRs, primarily due to their deviation caused by the insufficiently constrained galactic and extragalactic magnetic fields. However, the constructed CR observatories designed to detect extensive air showers have the potential to also identify neutral particles of comparable energies. These include neutrinos, neutrons, and photons. Such searches can be made provided that methods are devised to separate photon and neutrino-induced EAS from those produced by the more abundant charged CRs, or in the case of neutrons if excesses in the sky are found. No photon or neutrino candidates have been clearly identified in the energy range around and above 100 PeV and upper limits to their flux have been obtained (see Fig. 30.13). UHECR observatories are thus also multi-messenger observatories of neutral particles [156]. These can be combined with gravitational wave detection and conventional radio, optical, X-ray and gamma-ray astronomy to greatly improve our knowledge about the most powerful objects in the Universe, as happened with the discovery of gravitational and electromagnetic radiation from the binary neutron star merger system in 2017 [157], with no neutrinos identified in temporal and spatial coincidence with it [158].

**Astrophysical interpretation.** A coherent picture of the nature and sources of CRs is emerging from the observations of the spectrum, composition and arrival directions, with many open questions. Cosmic rays below ≃ 10 PeV most likely have a galactic origin. The observed dipolar anisotropy in arrival directions imply that above 8 EeV they have their origin in extragalactic sources. One likely scenario is that the transition takes place at a few EeV, a hypothesis supported by the observed phase of the dipolar anisotropy changing around that energy from pointing to near the Galactic Center towards away from it (Fig. 30.5). Cosmic-ray rigidity $Z_e$ determines the maximum energy at which different particle species can be accelerated in the sources $E_{\text{max}}(Z) = Z \times E_{\text{max}}(Z = 1)$. As a consequence, protons would cutoff first in the spectrum, followed by helium, carbon ... up to iron. The observed increase with energy of the mass below the *second knee* could be interpreted as the signature of the end of the Galactic contribution dominating below the *second knee* [159]. As energy increases above a few times 100 PeV, the composition starts to become lighter. This could signify the
emergence of an extragalactic component because no correlation with the densely populated Galactic plane is observed. A combined fit of the Pierre Auger Observatory measurements of the UHECR energy spectrum and composition [160] is compatible with the existence of this extragalactic component along with a second one at higher energy responsible for the sharp change in spectral index at the *ankle* feature in the spectrum at ≈ 5 EeV. Observations above the ankle suggest that each accelerated nuclear species from protons to iron, dominates in a narrow energy band of the UHECR spectrum produced through a combination of the individual composition peaks. In particular, the *instep* reflects the interplay between the flux contributions of the helium and carbon-nitrogen-oxygen components injected at the source with their distinct cutoff energies, all shaped by nuclear photo-disintegration on background light during the propagation. This is consistent with the *suppression* above ≈ 50 EeV attributable to the $E_{max}$ reachable at the cosmic accelerators along with the GZK and photodisintegration propagation effects. This comprehensive and unified understanding is continually evolving as additional data accumulates. Multi-messenger observations of the expected secondary neutrino and gamma-ray fluxes, or the lack thereof, will provide valuable constraints.

### Hadronic interactions and Extensive Air Showers.
EAS detectors probe energy and phase-space regions of hadronic interactions currently not accessible with terrestrial accelerators, although only in a very indirect manner. The tail of the distribution of $X_{max}$ is sensitive to the attenuation length of primary p in the atmosphere, and the inelastic p-air cross-section can be inferred. In this manner, measurements of the p-air cross-section have been obtained at equivalent center-of-mass pp collision energy, $\sqrt{s_{pp}} > 40$ TeV, and are in agreement with model predictions within systematic uncertainties [142]. The muonic component in the air shower is generally used as a probe of the hadronic interactions during the shower development. Various measurements of muons with energies in the ≈ 100 MeV to ≈ 10 GeV range produced in EAS, have revealed that hadronic models of multiparticle production fall short in predicting the observed number of muons above cosmic-ray energies of ≈ 100 PeV [161]. On the other hand, the simulated shower-to-shower fluctuations in the muon number that are particularly sensitive to the first hadronic interactions in the shower, are in good agreement with the measurements [162]. This indicates that the shortage in the number of simulated muons builds-up in multiple soft-QCD processes throughout the EAS development rather than being driven by only the first few and highest-energy interactions of the EAS. This constrains possible explanations of the so-called *Muon Puzzle* based on new physics beyond the Standard Model.

## 30.3 Gamma Rays

There is a well-studied flux of cosmic gamma rays (defined here as having energy greater than 1 MeV) present at the top of the atmosphere. Known as the diffuse gamma-ray flux, it has been measured with multiple space and ground-based instruments across a broad range of energy. On top of the diffuse one, many discrete sources have been found with either a steady gamma-ray flux or variability of many types.

### 30.3.1 Observational Instruments: Space borne and Ground based

Space borne γ-ray detectors, including AGILE [163], DAMPE [164], CALET [165], mainly Fermi-LAT [166], cover the energy range above 10 MeV up to 100 GeV. With the wide Field-of-View (FoV), they effectively survey the whole sky for steady γ-ray sources, and monitor all types of transient phenomena such as flares of Active Galactic Nuclei (AGN) and γ-ray bursts (GRBs). Above 100 GeV, ground-based Imaging Air Cherenkov telescope (IACT) experiments H.E.S.S [167], MAGIC [168], VERITAS [169] and CTA-LST [170] are very effective observational instruments. Using the stereo measurement technique, the IACT arrays achieve the angular resolution of the arrival direction of γ-rays at the level of 3 arc minute which is necessary for the high precision measurements of discrete sources. The small Field-of-View (FoV) of the telescopes and low-operation duty cycle make the all-sky

surveys for new sources difficult. Only H.E.S.S scanned the region along the Galactic plane around the Galactic Center [171], as shown in Figure 30.9. Non-imaging Cherenkov light technique is also developed trying to cover wider FoV, e.g. in the TAIGA experiment [172]. The most suitable detectors for all-sky surveys are extensive air shower detector arrays at high altitudes, such as ARGO-YBJ [173], ASγ [174], HAWC [175] and LHAASO [176]. They are operated with very high duty cycle, typically > 95%, have compatible sensitivity for discrete sources to IACT telescopes in much wider FoV in the same energy band, and higher sensitivity at higher energies above 10 TeV. In Table 30.1, the main features of the major facilities in γ-ray astronomy are listed.

### 30.3.2 Diffuse Gamma Rays

The majority of photons detected at high energies are characterized as diffuse emission that is not resolved as discrete sources. The observations of the diffuse emission are mainly carried out by wide FoV detectors, including space missions, OSO-3 [180], SAS-2 [181], COS-B [182], CGRO/EGRET [183], and Fermi-LAT [184] at energies below 1 TeV, and ground-based experiments Milagro [185], H.E.S.S [171], ARGO-YBJ [186], ASγ [187] and LHAASO [33] at energies up to 1 PeV. Evident in Figure 30.9, the dominant component comes from the plane of the Galaxy (i.e., $|b| \lesssim 10^°$, and is referred as the Galactic Diffuse Emission (GDE). Precise measurement of the GDE flux strongly depends on the subtraction of the contamination from discrete sources most of which are found spatially extended. With significantly improved sensitivity of source detection, Fermi-LAT and LHAASO largely reduce the contamination by removing the regions associated with known sources. In case of LHAASO, the flux in the Galactic plane with longitudes of $15^° < l < 235^°$ and latitudes of $|b| < 5^°$ is well measured [33].

Three components are expected in GDE, namely γ-rays from the decay of neutral pions produced via inelastic collisions between energetic cosmic ray nuclei and the interstellar medium (ISM); bremsstrahlung radiation of electrons and positrons in ISM; and inverse Compton scattering of electrons and positrons off the interstellar radiation field. Possible annihilation or decay of dark matter (DM) particles might also give rise to diffuse γ-rays particularly in the densest region of the Galaxy. Therefore, GDE serves not only as a very important tool to probe the production, propagation, and interaction of cosmic rays, but also as a route to search for DM in any excess over the expectation, which is based on assumptions of the cosmic-ray spatial distribution, composition, energy spectrum and on the ISM column density along the line-of-sight. The search for DM also depends on the fraction of the contribution by the unresolved dim discrete sources in the GDE. Unfortunately, all of the assumptions have their own large uncertainties at present.

Evidence of an isotropic component of diffuse γ-ray emission, presumably of extragalactically originated, was obtained with the measurement of its spectrum by CGRO/EGRET [188] and Fermi-LAT [189]. The Fermi-LAT spectrum covers the energy range from 0.1 GeV to 820 GeV and is well described by a power-law with an exponential cutoff. It is consistent with a scenario of a discrete source population, such as blazars with power-law spectral shape, dominating the emission and experiencing attenuation due to the extragalactic background light (EBL) [189]. Contributions from the cascade of very-high-energy gamma rays and UHECRs are not ruled out as a fraction of the extragalactic diffuse emission.

### 30.3.3 Discrete Gamma Ray Sources

As a neutral component of cosmic rays, γ-rays directly reach the earth from their sources, thus allowing us to identify the origin of the photons by association with known celestial objects or events. As shown in Figure 30.9, the source-associated contributions are revealed directly by the bright spots whose sizes are due to the point-spread-function (PSF) of the instruments and the intrinsic spatial extensions of the sources. It is also clearly shown that the sources are divided into two groups: Galactic and extragalactic.

**Galactic Sources** More than 6600 sources have been found emitting γ-rays in the Milky Way. Most of them have steady photon emissions over a wide energy range above 0.1 GeV up to
<table><caption>**Table 30.1:** Facilities for γ-ray detection and their specifications in reference a) [166], b) [177], c) [178], d) [179], e) [170], f) [174], g) [173], h) [175], i) [176]. † milli-CU = $10^{-3}$ Crab Unit.</caption><thead><tr><th>Instrument</th><th>energy range</th><th>Effective area (m²)</th><th>sensitivity (milli-CU†)</th><th>energy resolution (%)</th><th>PSF (°)</th><th>FoV (sr)</th><th>duty cycle (%)</th></tr></thead><tbody><tr><td>Fermi-LAT<sup>a)</sup></td><td>20 MeV - 300 GeV</td><td>~ 0.95</td><td>10 - 30</td><td>8 - 18</td><td>0.15 - 3.5</td><td>~ 2.4</td><td>~ 60</td></tr><tr><td>H.E.S.S.<sup>b)</sup></td><td>10 GeV - 50 TeV</td><td>~ 10<sup>5</sup></td><td>~ 5</td><td>15</td><td>&lt; 0.1</td><td>~ 0.03</td><td>~ 10</td></tr><tr><td>MAGIC<sup>c)</sup></td><td>≥ 30 GeV</td><td>~ 10<sup>5</sup></td><td>~ 7</td><td>16</td><td>≤ 0.07</td><td>0.02</td><td>~ 18</td></tr><tr><td>VERITAS<sup>d)</sup></td><td>50 GeV - 50 TeV</td><td>~ 10<sup>5</sup></td><td>~ 5</td><td>10 - 15</td><td>0.08 - 0.13</td><td>0.02</td><td>10-14</td></tr><tr><td>CTA-LST1<sup>e)</sup></td><td>20 GeV - 300 GeV</td><td>~ 10<sup>4</sup></td><td>~ 10</td><td>10 - 30</td><td>0.05 - 0.1</td><td>0.02</td><td>~ 10</td></tr><tr><td>ASγ<sup>f)</sup></td><td>10 TeV - 1 PeV</td><td>~ 7 × 10<sup>4</sup></td><td>~ 200</td><td>20 - 40</td><td>~ 0.8</td><td>~ 2.0</td><td>90</td></tr><tr><td>ARGO-YBJ<sup>g)</sup></td><td>50 GeV - 10 TeV</td><td>~ 0.8 × 10<sup>4</sup></td><td>~ 300</td><td>&gt;13</td><td>~ 0.5</td><td>~ 2.0</td><td>86</td></tr><tr><td>HAWC<sup>h)</sup></td><td>100 GeV - 100 TeV</td><td>~ 3 × 10<sup>4</sup></td><td>~ 50</td><td>20 - 50</td><td>~ 0.69</td><td>&gt; 1.5</td><td>95</td></tr><tr><td>LHAASO-WCDA<sup>i)</sup></td><td>100 GeV - 20 TeV</td><td>~ 0.8 × 10<sup>5</sup></td><td>~ 12</td><td>~33</td><td>0.2 - 0.84</td><td>~ 2.0</td><td>95</td></tr><tr><td>LHAASO-KM2A<sup>i)</sup></td><td>10 TeV - 4 PeV</td><td>~ 10<sup>6</sup></td><td>~ 12</td><td>15 - 40</td><td>0.2 - 0.6</td><td>~ 2.0</td><td>95</td></tr></tbody></table>

<figure><img src="image_162.png" /><figcaption>**Figure 30.9:** The GeV γ-ray sky maps surveyed by Fermi-LAT [184], LHAASO [38] and H.E.S.S [171] from 1 GeV to 100 TeV and above.</figcaption></figure>

2 PeV. 43 of them have emission of γ-rays above 0.1 PeV referred as PeVatrons since the parent particles that emitted the photons must be accelerated above 1 PeV on average in the source regions. Variability of the photon fluxes has also been observed from known objects such as pulsars and binary systems or transient phenomena.

**A. Sources with steady emissions** are the majority of Galactic γ-ray emitters which are well documented in reviews [190], [191], [192] and in γ-ray source catalogs [193] (for < 100 MeV), 4FLGT (for 0.1-300 GeV) by Fermi-LAT [194], TeV-Catalog available on-line (http://tevcat.uchicago.edu) and the newly released 1LHAASO (for 1-2000 TeV) by LHAASO [38]. All associations with known celestial objects, such as SNRs, PWNe, young massive-star clusters (YMC) are discussed in the corresponding catalog papers (e.g. individual catalogs collected in http://tevcat.uchicago.edu). Most of the sources, particularly for their high-energy emission above tens of TeV, remain unidentified. Among the identified ones, the Crab [195] and RX J1713-3946 [196] are the most extensively studied objects and constitute paradigmatic examples of PWNe and SNRs.

The Crab has a compact PWN powered by the rotation energy of the pulsar in its heart. Electrons and positrons in the wind are accelerated at the termination shocks and emit γ-rays through synchrotron radiation at energies lower than 0.1 GeV. Simultaneously, accelerated electrons and positrons boost soft background photons, such as those in the cosmic microwave background, to energies well above 1 GeV via inverse Compton scattering. Figure 30.10 shows the spectral energy density (SED) of the Crab

nebula with a fit using a simple one-zone leptonic model. It generally describes the radiation of the nebula over energies up to $E_γ \sim 1.1$ PeV, implying that electrons and positrons are confined in a small region of $\simeq 0.2$ pc by a magnetic field of intensity $B \simeq 110$ µG and are accelerated to energies as high as $\simeq 2.3$ PeV. This requires a surprisingly high acceleration rate, $\eta = \bar{\epsilon}/B = 0.14(B/100\mu G)(E_γ/1$ PeV)$^{1.54}$ PeV, at the level of 15%, where $\bar{\epsilon}$ is projection of the electric field averaged over the particle trajectory. This rate is 3 orders of magnitude higher than that in the normal shock environment such as in SNRs, revealing the existence of a so-called ‘extreme accelerator’ in the middle of the nebula [195]. The flux of photons above 0.8 PeV indicates some deviation from the pure leptonic scenario implying that a hadronic component might be responsible. It could be a hint of the super PeVatron which may contribute to the Galactic cosmic rays above the knee [195] (see Fig. 30.1).

RX J1713.7-3946 is one of the brightest objects in the TeV sky, and is the first SNR shell to be confirmed as a TeV gamma-ray source [196]. The detailed TeV morphology reveals a shell structure similar to the X-ray observations, indicating that particles are accelerated to very high energy therein. The TeV gamma-ray spectrum extends to nearly 100 TeV. It is not determined yet whether the gamma-rays are produced via Inverse Compton scattering of relativistic electrons (leptonic process), or via neutral pion decay with the pions produced in the inelastic scattering of CR protons with ambient gas (hadronic process) [200]. Nevertheless, the spectral information demonstrates the efficient acceleration of charged particles to energies beyond 100 TeV in this
<figure><img src="image_163.png" /><figcaption>**Figure 30.10:** Spectral energy density (SED) of photons from the Crab from sub-μeV to 1.1 PeV. Above 1 TeV, data are collected from 7 experiments listed in legend in the upper panel. The lower panel shows the deviation between data and the model based on a widely accepted one-zone leptonic origin in the nebula. Only Fermi-LAT and LHAASO data are shown for simplicity. Inset is the X-ray image of the nebula by Chandra [197].</figcaption></figure>

object. The latest H.E.S.S observations also found that the TeV gamma-ray emission region extends significantly beyond the X-ray emitting shell [201], which may be due to either the escape of particles from the shell or to particle acceleration in the shock precursor region.

**B. Variable sources** Crab flares, binaries and pulsars are observed in the Galactic γ-ray sky. Their activity varies on timescales from a fraction of one second (pulsars), to days (Crab Nebula, Novae), weeks, months or even years (γ-ray binaries). Various analysis pipelines are developed mainly by the LAT collaboration to search for and monitor γ-ray transients.

The Crab Nebula shows flare activities on a timescale of days [202]. The instability is observed only in GeV band by AGILE and Fermi/LAT. The flaring mechanism is still unclear with a couple of models being under debate.

Each γ-ray binary is composed of a compact object (pulsar or black hole) and a massive star. Their γ-ray emissions characterized by the orbital modulation, with time scales ranging from 3.9 days (LS 5039) to ~ 47 years (PSR J2032+4127). Microquasars are also binary systems that launch powerful jets from the compact objects by accreting matter from their companions. Three microquasars have been detected namely, Cyg X-1, Cyg X-3, and SS 433. They have shown γ-ray activities correlated with orbital motion [203], jet precession [204], X-ray state changes [205], and flares [206]. Eta Carinae is the only colliding wind binary firmly detected, showing orbital modulation in the γ-ray emission in a period of 5.54 years [207].

Pulsars show steady pulsation on their spin periods ranging from several milliseconds to seconds in GeV band [208], thereby

qualifying as γ-ray pulsars. Only the Crab and Vela pulsar are pulsating in TeV [209] [210]. PSR J2021+4026 is observed for the γ-ray flux variability on time scales of several years [211], making it the only variable γ-ray pulsar. Many millisecond pulsars themselves are γ-ray pulsars showing periodic variability in milliseconds [212]. γ-ray eclipses and orbital modulations have been detected in several binaries hosting millisecond pulsars, usually on the time scale of hours to days, arising from the intra binary shock between the pulsar wind and stellar wind [213] [214]. Millisecond pulsars could transit between spin-down-powered state and accretion-powered state. Variability in the γ-ray flux is observed accompanying these changes [215].

Novae outbursts have been detected in γ-rays, arising from the particle acceleration at shocks and last for months [216]. RS Ophiuchi is the first nova with its outburst detected in TeV up to 1 month [217].

**Extragalactic Sources** The majority of extragalactic γ-ray sources are active galactic nuclei (AGN). The fourth catalog of AGNs (4LAC) by LAT for 8 year observations contains 2863 objects located at high Galactic latitudes (|b| > 10°) while 344 others were found at lower latitudes [218]. 98% of them are blazars. The blazar population consists of Flat Spectrum Radio Quasars (FSRQs), BL Lac-type objects and unknown types [218]. The Data Release 3 adds 587 more blazars and 4 radio galaxies to the catalog [219]. More than 70 blazars and a few radio galaxies are detected in the very high-energy (VHE) band, namely photon energy greater than 0.1 TeV, by imaging atmospheric Cherenkov
<figure><img src="image_164.png" /><figcaption>**Figure 30.11:** The multi wavelength analysis of Mrk 421 involving measurements over wide energy range from radio to TeV γ-ray during 4.5 years when ARGO-YBJ and LAT were operated [198]. The discrete correlation function between the light curves in X-ray band and TeV γ-ray band is calculated and clearly shows no time lag between emissions in the two bands [198] (left panel). The SED significantly changes from steady states (grey solid lines in middle and right panels) to flaring states depending on the strength of flares. Comparing a small flare in the middle panel and a larger one in right panel, the later has harder SED in VHE band [199].</figcaption></figure>

telescopes (IACT) (http://tevcat.uchicago.edu).

Blazars belong to a class of AGN where the jets are oriented at a relatively small angle with respect to the line of sight. This specific viewing angle allows us to observe the emission from those jets with significant Doppler boosting. The photon flux in the jets may enhance significantly with respect to that in quiet states, a phenomenon referred to as blazar flares. They are simultaneously observed over a wide energy band from radio to VHE γ-rays with timescales ranging from a few minutes to year [220] [221]. The broad-band SED of blazars typically shows two humps of non-thermal radiation. The one in the lower energy range from radio to X-rays is generally attributed to synchrotron emission from relativistic electrons and positrons in the jet, while the origin of the hump at higher energies is still under debate. Possible emission mechanisms are the IC scattering of accelerated leptons off low energy photons, which are inside jets or from outside regions [222], or decay of neutral pions produced in collisions of accelerated protons against ambient matter [223], or some combination of the two processes.

The ground-based EAS experiments are suitable γ-ray detectors for monitoring the flares of AGNs with the advantages of large FoV and full duty cycle of operation. The ARGO-YBJ collaboration reported a long-term continuous monitoring of Mrk 421 and Mrk 501 [198, 199, 224], the two brightest blazars in the TeV sky. The γ-ray flux shows a good long-term correlation with that in the X-ray band. HAWC detected the persistent emission from those two AGNs [225]. Recently, the LHAASO collaboration reported eight sources detected at Galactic latitude $|b| > 12°$, among which some are AGNs, including well-known emitters, such as the radio galaxy M87, VER J0521+211 and 1ES 1215+303 [38].

Using data for 4.5 years when ARGO-YBJ and Fermi-LAT simultaneously operated from 2008 to 2013, the ARGO-YBJ collaboration realized the long-term multi wavelength observation of Mrk 421 in all bands of radio, optical, X-ray, GeV, and TeV gamma-rays [198]. The well-known correlation [226] between the photon flux in X-ray band and TeV γ-ray band is investigated in a quantitative way, namely the discrete correlation function between the light curves in two energy bands being calculated. No time lag in the scale of days is found(see Figure 30.11). This is regarded as evidence of favoring the one-zone Self-Synchrotron Compton scattering scenario [199].

Flares happened in that time window are systematically analyzed and clear correlation between the strength of flares and features of the SEDs are observed as an example shown in Figure 30.11. Detailed investigation of SEDs implies that various particle acceleration mechanisms, such as relativistic diffusive shock acceleration, might be responsible for different flares. A similar analysis is carried out with data during outbursts of Mrk 501. The SEDs in the VHE band are observed to be harder during flares than that in the quite period. The detection of energetic photons above 10 TeV and the hardness of the SED hint some exceeds over the expectation of one-zone SSC model [224].

**Gamma-Ray Bursts (GRBs)** GRBs are violent explosions observed in distant galaxies, characterized by a rapid flash of γ-rays lasting from a fraction of a second up to several hundreds of seconds. The emission from GRBs occurs in two stages, namely the prompt emission phase and the afterglow phase, as can be seen in Figure 30.12.

In the second catalog of Fermi/LAT-detected GRBs, covering the first 10 yr of operations, 169 GRBs are detected above 100 MeV [231]. The high-energy emission generally lasts much longer than the prompt keV-MeV emission. The widely discussed scenario for the extended GeV emission is synchrotron radiation by relativistic electrons that are accelerated in shocks resulting from the interaction of the jet with the surrounding medium [232,233].

VHE emission has been detected from just a few GRBs by IACTs during the afterglow decay phase. MAGIC slewed to the direction of GRB 190114C only $\simeq$ 60 seconds after the trigger and detected photons above 300 GeV from the burst for the first 20 minutes [234]. This VHE emission is explained by inverse Compton up-scattering of synchrotron photons by high energy electrons in the external shock [229], as shown in the right-upper panel of Figure 30.12. Two other events, GRB 180720B and GRB 190829A, were also detected during the afterglow decay phase by H.E.S.S [230,235], the unexpected SED is shown in the right-lower panel of Figure 30.12.

The brightest-of-all-time GRB 221009A serendipitously occurred within the FoV of LHAASO. More than 64,000 photons (above 0.2 TeV) were detected within the first 3000 seconds by LHAASO-WCDA [228]. The detection covers both the prompt emission phase and the early afterglow in the TeV band, revealing the onset of afterglow emission in the TeV band for the first time, as shown in the left panels of Figure 30.12. In addition, the unprecedented photon statistics of the TeV emission enabled identifying a jet break in the light curve, indicating an opening angle of GRB 221009A of $\sim$ 0.8°. This is consistent with the brightest core of a structured jet, explaining the unexpected high isotropic equivalent energy of this GRB. LHAASO-KM2A registered more than 140 photons with energies above 3 TeV from GRB 221009A (to be published). The intrinsic energy spectrum of gamma-rays is well fitted by a power law after correcting for EBL absorption, with the highest energy photon above 13 TeV traveling over a distance with red shift $z \sim 0.152(\text{TBP})$.

**Tests on Lorentz Invariance Violation (LIV)** Tests on LIV are feasible and thus, limits can be set using the highest energetic photons collected from remote sources. Any tiny violation of the invariance leads the fundamental dissipation relation to be rewritten [236] as $E^2 = m^2c^4 + p^2c^2(1 + A_1pc/M_p c^2 + A_2 pc/M_p c^2)^2 + \dots$), where E, p are the energy and momentum of a particle with the mass m, the $M_p$ is the Plank scale and $A_1, A_2$ are theoretic parameters of the violation at the first and second order levels. This must lead instability of photons that may decay on their way to the Earth [237]. The first order violation effect due to the decay channel $\gamma \rightarrow e^+ + e^-$ has not been found at energies $\sim 5$ orders
<figure><img src="image_165.png" /><figcaption>**Figure 30.12:** The typical multi-wavelength measurement of the light curves of the γ-ray burst GRB 221009A in the GeV band [227] (top left) and in the TeV band [228] (bottom left). The clear different structures of the brightness variation patterns indicate the different origins of the synchrotron radiation and inverse Compton scattering of leptons accelerated by the internal and external shock waves, respectively. The spectral energy distributions of the GRB 190114C and GRB 1908129A, respectively measured by MAGIC [229] (top right) and H.E.S.S [230] (bottom right). The Self-Synchrotron Compton scattering model of the afterglow is marginally supported in the observations covering a wide energy band from X-rays to TeV γ-rays.</figcaption></figure>

of magnitudes higher than the Planck scale, regarded as the null evidence of the quantum gravitational effect. However, the limit due to second-order effect caused by decay channel $γ → 3γ$ is still found still ~3 orders of magnitudes below the Planck scale, using the latest observations of photons up to 1.4 PeV from the Cygnus X direction [238].

## 30.4 Neutrinos

In contrast to charged particles, neutrinos are not deflected by electromagnetic fields and thus point back to their origin, making them appealing messengers for astronomical observations. In addition, neutrinos of cosmic origin, or produced in the atmosphere of the Earth, allow us to study particle physics in a kinematic regime not accessible to date to accelerator experiments. In the following, we will consider neutrinos with energies at the GeV energy scale and above.

### 30.4.1 Production of high-energy neutrinos

High-energy neutrinos can be produced in hadronic interactions in a variety of astrophysical objects (astrophysical or cosmic neutrinos), in scattering of extremely energetic protons (p) with the cosmic microwave background (cosmogenic neutrinos) and by cosmic-ray interactions in the Earth's atmosphere (atmospheric neutrinos). The main production chain is (A denoting atomic nuclei and X a hadronic system)

$$ \begin{gathered} p(A) + p(A, \gamma) \rightarrow \pi^{\pm} + X \\ \pi \rightarrow \mu \nu_{\mu} \text{ and } \mu \rightarrow e \nu_{\mu} \nu_{e}, \end{gathered} \quad (30.1) $$

resulting in a ratio $\nu_e : \nu_\mu : \nu_\tau = 1 : 2 : 0$ (not distinguishing here between $\nu$ and $\bar{\nu}$). If the source density is high, muons can lose most of their energy before decaying, yielding a ratio 0 : 1 : 0

in the extreme case. If protons were kept in the source region by magnetic fields without interacting, and only neutrons could escape and then decay via $n \rightarrow p + e^{-} + \bar{\nu}_{e}$, the ratio would be 1 : 0 : 0. Over cosmic distances, flavor mixing turns these ratios to 0.30:0.36:0.34 for the first source scenario (1:2:0), 0.17:0.45:0.37 for the second (0:1:0) and 0.55:0.17:0.28 for the third scenario (1:0:0) [239].

### 30.4.2 Measurement methods

Most experiments targeting high-energy cosmic neutrinos are based on the detection of Cherenkov light induced by charged secondary particles produced in neutrino interactions. This includes underground detectors like Super-Kamiokande with its fiducial volume of $22.5 \times 10^3 \text{ m}^3$ (see Sect. 36.3.1) and underwater/ice detectors like ANTARES, KM3NeT or IceCube, the latter with a volume of $10^9 \text{ m}^3$ (see Sect. 36.3.2). Neutrinos with energies exceeding some tens of PeV can be also detected via the coherent emission of Cherenkov radiation at radio wavelengths in ice or other radio-transparent media (e.g. ANITA or RNO-G, see Sect. 36.3.3). Neutrinos with EeV energies could be identified via extensive air showers starting too deep in the atmosphere to be due to particles with strong or electromagnetic interaction. Alternatively, they could be detected via slightly upward-directed air showers from the decay of tau leptons generated in interactions of Earth-skimming or mountain-traversing $\nu_{\tau}$ [249]. These signatures could be observed by cosmic-ray experiments such as the Pierre Auger Observatory [250].

### 30.4.3 Diffuse fluxes

**Atmospheric neutrinos** The flux of atmospheric neutrinos as measured by different experiments is shown in Fig. 30.13. The production chain for conventional atmospheric neutrinos follows eq. (30.1), with an observed ratio $\nu_e : \nu_\mu = 1 : 2$ at energies of
<figure><img src="image_166.png" /><figcaption>**Figure 30.13:** Measured energy spectra of atmospheric and cosmic diffuse neutrino fluxes. Experimental limits at highest energy are compared to model predictions for cosmogenic neutrinos [160]. All fluxes are normalized to one flavor ($\bar{\nu} : \nu = 1$ and, for cosmic neutrinos, a ratio $\nu_e : \nu_\mu : \nu_\tau = 1 : 1 : 1$ at Earth. Data sources: SuperK 2016 [240]; ANTARES 2021 [241]; IceCube 2015 $\nu_e$ [242]; IceCube 2015 $\nu_\mu$ [243]; IceCube 2021 astrophysical $\bar{\nu}_e$ ($\nu + \bar{\nu}$ flux derived from a single candidate event for the Glashow resonance) [244]; IceCube 2023 astrophysical all flavor [35]. The limits at the highest energies are taken from [245] (IceCube) and [246] (Auger). They are shown in the energy bins used for the corresponding analyses and have been adjusted to bin widths of one decade. The region allowed for the expected cosmogenic neutrino fluxes is taken from [247] (see also Fig. 36.10).</figcaption></figure>

a few GeV (where almost all muons decay in the atmosphere), and a ratio below 1 : 10 at TeV energies, where muons reach the ground before decaying. The flux of muon neutrinos with energies below 100 GeV from the lower hemisphere is reduced due to neutrino oscillations, reducing the $\nu_e : \nu_\mu$ ratio at the lowest energies. The flux of $\nu_e$ is obtained by subtracting the number of neutral-current muon neutrino interactions (as deduced from the measured charged-current muon neutrino reactions) from the number of contained events without a muon track and converting the resulting $E_{\nu_e}$ distribution into the $\nu_e$ flux using the differential cross sections for CC and NC $\nu_e$-nucleon scattering. The conventional atmospheric neutrino flux dominates in the GeV to TeV region, following an $E_{\nu}^{-3.7}$ power law.

*Prompt* atmospheric neutrinos are produced by the decay of hadrons containing a charm or bottom quark, the production of which is strongly suppressed. Since they are produced early in the air shower and decay before losing energy, however, the energy spectrum of prompt neutrinos is expected to be harder ($E_\nu^{-2.7}$) and to dominate the atmospheric neutrino flux at its highest energies. The corresponding theoretical prediction has large uncertainties related to the cosmic-ray spectrum and mass composition, the model for heavy-flavor production, and the parton distribution functions [251]. At present, experimental measurements only provide upper limits for the prompt flux (see e.g. Fig. 30.14 where the prompt component was fitted to zero).

**Diffuse extragalactic neutrino flux** In 2013, a diffuse and isotropic flux of neutrinos of astrophysical origin was discovered with the IceCube Neutrino Observatory. The flux was identified with the so-called HESE (high-energy starting events) selection as an excess of neutrino events at high energies over atmospheric neutrinos and a background of misreconstructed atmospheric muons (see Fig. 30.14, for the 7.5-year HESE sample [252]). Similar high-energy excesses were also identified for tracks entering the detector from outside [253], and for fully contained cascade events. Evidence for this flux has meanwhile also been

reported by ANTARES [254] and Baikal-GVD [255], albeit with much lower significance. The measured fluxes are consistent with a parameterization as power-law functions proportional to $E_\nu^{-\gamma}$, with the normalization factor and the spectral index $\gamma$ as parameters. The resulting $\gamma$ values vary from 2.2 to 2.5 for different event classes covering different energy ranges.

Different production mechanisms result in different neutrino flavor compositions at sources and, after neutrino oscillations over astrophysical distances, at the Earth. An IceCube study of the event topologies [239] slightly favors $\nu_e : \nu_\mu : \nu_\tau = 1 : 2 : 0$ at the source (pion decay according to eq. (30.1) versus 0 : 1 : 0 (damped muons) and disfavors a 1 : 0 : 0 scenario (neutron decay).

A single event with energy $6.05 \pm 0.72$ PeV has been detected in IceCube [244]. It is compatible with having been generated by the resonant process $\bar{\nu}_e + e^- \to W^- \to$ hadrons at $E_\nu = 6.3$ PeV (Glashow resonance) [256]. A statistically significant observation of the Glashow resonance would reveal the presence of electron antineutrinos in the astrophysical flux and could provide information about their production process: the expected ratio $\bar{\nu}_e : \nu_e$ depends on the mass composition of cosmic-rays and also on the photon density and the magnetic field strength of the source [257].

**Cosmogenic neutrinos** Cosmogenic neutrinos stem from the decay of charged pions generated in interactions of ultra-high-energy cosmic rays with cosmic microwave (a) and infrared (b) background radiation ($p+\gamma \to n+\pi^+$) [258] and from the decays of neutrons produced in photodisintegration processes [259, 260] (c). The neutrino flux at EeV energies is expected to be dominated by (a) and (c).

Measurement methods and experimental data are described in Sect. 36.3.3. and Fig. 36.10. The expected flux can be estimated from the measured spectrum and composition of charged cosmic rays, using the CMB density and the well measured properties of proton and nuclear interactions with photons at a center-of-mass energy in the GeV range. The current uncertainty of these pre-
<figure><img src="image_167.png" /></figure>

**Figure 30.14:** Measured (data points) and expected (histograms) distributions of the deposited energy measured in the IceCube detector for events passing the HESE selection, which strongly suppresses down-going atmospheric neutrinos by vetoing accompanying muons. The cosmic neutrino flux has been assumed to follow an unbroken power law in $E_ν$. Its normalization and spectral index, as well as the normalizations of the conventional and prompt atmospheric neutrino and the atmospheric muon contributions were fitted to the data, where only the region above 60 TeV was used to constrain the cosmic neutrino flux.

dictions exceeds one order of magnitude. Figure 30.13 reproduces the presently tightest limits and compares them to the region for which cosmogenic neutrino fluxes would be consistent with those limits (as of 2019) and cosmic-ray analyses [247].

**Neutrinos from the Galactic Plane** The Milky Way is an emitter across the electromagnetic spectrum ranging from radio to gamma-rays. A large part of the observed gamma-ray flux is believed to consist of photons from the decays of π⁰s produced by cosmic rays colliding with the interstellar medium. The TeV diffuse neutrino flux from the corresponding π± decays along the Galactic plane has been estimated with different models, ranging from a simple extrapolation of Fermi-LAT gamma-ray data to higher energies, to more sophisticated approaches (see e.g. [261]).

First hints for a TeV neutrino emission from the Galactic Ridge have been reported by the ANTARES collaboration [262]. Ice-Cube has measured the flux of galactic neutrinos with a significance of 4.5σ against the only-background-hypothesis [248]. Figure 30.15 shows a map of the Galactic plane with the point-by-point pre-trial significance of the IceCube neutrino observations. While the overall signal is consistent with the modeled diffuse emission from the Galactic plane, a significant part could also arise from a population of unresolved point sources. All individual "warm spots", however, are statistically consistent with background fluctuations.

### 30.4.4 Neutrinos from astrophysical point sources

Most point source candidates include Active Galactic Nuclei (AGN). In particular, blazars – AGN that have jets pointing

towards the observer – are thought to be likely sources. In 2017, a spatial and temporal correlation was found between a flare of the electromagnetic emission from blazar TXS 0506+056 and the high-energy neutrino event IC170922A, detected by IceCube. An alert was sent to other observatories and a follow-up campaign in multiple wavelengths was started. At the moment of neutrino detection, the blazar was flaring in gamma rays, with an accidental coincidence excluded at the 3σ level [263]. Subsequent analyses of archival data revealed an excess in neutrino events with a significance of 3.5σ with respect to the expected atmospheric background, coincident with the position of TXS 0506+056, albeit not correlated with an electromagnetic flare [264].

In another analysis, the positions of 110 gamma-ray sources selected a priori were analyzed individually for a possible surplus of neutrinos over atmospheric and cosmic background expectations. An excess of $79_{-20}^{+22}$ neutrinos was found in the direction of the nearby AGN NGC 1068 (not a blazar, but a Seyfert-2 galaxy), at a significance of 4.2σ (see figure 30.16). The excess is spatially consistent with the direction of the strongest clustering of neutrinos in the Northern sky. The inferred flux exceeds the potential TeV gamma-ray flux by at least an order of magnitude [265], suggesting that the sight to the gamma-ray generation region is obscured, as expected for Seyfert-2 galaxies.

Other potential sources include starburst galaxies, galaxy clusters, supernovae remnants (steady emission) as well as Gamma Ray Bursts (GRBs) and Tidal Disruption Events, TDEs (transient emission). Indeed, using data from IceCube and the Zwicky Transient Facility ZTF, hints for a correlation of high-energy muon neutrinos with TDEs have been reported [266].
<figure><img src="image_168.png" /><figcaption>**Figure 30.15:** The Milky Way plane in neutrinos (Galactic coordinates, with the origin being at the Galactic Center). Shown is the pre-trial significance of the IceCube observations using a cascade neutrino event sample, calculated from an all-sky scan for point-like sources [248]. Contours indicate the regions that contain 20% and 50% of the predicted diffuse neutrino emission according to an extrapolation of Fermi-LAT results. Grey lines indicate the Northern-Southern sky horizon line.</figcaption></figure>

<figure><img src="image_169.png" /><figcaption>**Figure 30.16:** Left: The sky region around the most significant spot in the Northern Hemisphere and NGC 1068. Shown is a fine scan of the region around the hottest spot, marked by a white cross. The red circle shows the position of NGC 1068. The solid and dashed contours show the 68% and 95% confidence regions of the hot spot localization. Right: Distribution of the squared angular distance between NGC 1068 and the reconstructed event direction. Figures taken from [265].</figcaption></figure>

### 30.4.5 Results on particle physics

Investigating the attenuation of high-energy neutrinos as a function of energy and their path length through the Earth, the first measurement of the neutrino-nucleon cross section at energies between a few tens of TeV and ≈ 1 PeV (corresponding to center-of-mass energies between a few 100 GeV and about 1.4 TeV) has been achieved [267], giving access in deep-inelastic scattering in a kinematic region currently not accessible to accelerator-based experiments.

The study of atmospheric high-energy neutrinos allows for the investigation of oscillation physics (see Sect. 14) at higher energies and over longer baselines than Earth-bound experiments. Constraints ($\sin^2 \theta_{23} = 0.51 \pm 0.050$ and $\Delta m^2_{32} = (2.41 \pm 0.07) \times 10^{-3} \text{ eV}^2$) have been obtained with IceCube and are compatible with corresponding results from accelerator experiments [268]. Both IceCube and KM3NeT/ORCA are also sensitive to $\nu_\tau$ appearance [269]. The KM3NeT/ORCA detector (under construction, see Sect. 36.3.2) is furthermore expected to provide sensitivity to the neutrino mass hierarchy.

Also, Beyond-Standard-Model (BSM) physics is being tested. For instance, constraints by IceCube on 2/4 mixing (the fourth

state being a sterile neutrino) are comparable to constraints from accelerator experiments [270]. Neutrinos may have BSM interactions, and if the coupling strengths are weak or if heavy mediators are involved, these interactions may only manifest themselves in the HE and UHE neutrino sector. Corresponding constraints from IceCube have been published in [271]. Also, possible oscillation-like effects induced by quantum gravity are investigated [272].

**Acknowledgement** Many thanks to team members, Prof. Q. Yuan, R.Z. Yang, J. Li, X.Y. Wang, S.Z. Chen and Dr. S.Q. Xi, L. Nie, R.F. Xu, who made substantial contributions to the γ-ray part of this section. We are grateful to C. Haack and L. Schumacher for their invaluable help in producing Fig. 30.13.

### References

[1] Y. S. Yoon et al., Astrophys. J. 728, 122 (2011), [arXiv:1102.2575].

[2] O. Adriani et al. (PAMELA), Science 332, 69 (2011), [arXiv:1103.4055].

[3] O. Adriani et al., Adv. Space Res. 51, 219 (2013).

[4] Y. S. Yoon et al., Astrophys. J. 839, 1, 5 (2017), [arXiv:1704.02512].
[5] N. Gorbunov et al., Adv. Space Res. 64, 12, 2546 (2019), [arXiv:1809.05333].

[6] M. G. Aartsen et al. (IceCube), Phys. Rev. D 100, 8, 082002 (2019), [arXiv:1906.04317].

[7] Q. An et al. (DAMPE), Sci. Adv. 5, 9, eaax3793 (2019), [arXiv:1909.12860].

[8] M. Aguilar et al. (AMS), Phys. Rept. 894, 1 (2021).

[9] G. H. Choi et al., Astrophys. J. 940, 2, 107 (2022).

[10] O. Adriani et al. (CALET), Phys. Rev. Lett. 129, 10, 101102 (2022), [arXiv:2209.01302].

[11] F. Alemanno et al., Phys. Rev. Lett. 126, 201102 (2021), [arXiv:2105.09073].

[12] H. S. Ahn et al., Astrophys. J. 707, 593 (2009), [arXiv:0911.1889].

[13] O. Adriani et al., Phys. Rev. Lett. 125, 251102 (2020), [arXiv:2012.10319].

[14] F. Aharonian et al. (H.E.S.S.), Phys. Rev. D 75, 042004 (2007), [arXiv:astro-ph/0701766].

[15] M. Aguilar et al. (AMS), Phys. Rev. Lett. 126, 4, 041104 (2021).

[16] O. Adriani et al. (CALET), Phys. Rev. Lett. 126, 24, 241101 (2021), [arXiv:2106.08036].

[17] G. Ambrosi et al. (DAMPE), Nature 552, 63 (2017), [arXiv:1711.10981].

[18] S. Abdollahi et al. (Fermi-LAT), Phys. Rev. D 95, 8, 082007 (2017), [arXiv:1704.07195].

[19] O. Adriani et al., Phys. Rev. Lett. 120, 26, 261102 (2018), [arXiv:1806.09728].

[20] T. Antoni et al. (KASCADE), Astropart. Phys. 24, 1 (2005), [arXiv:astro-ph/0505413].

[21] M. G. Aartsen et al. (IceCube), Phys. Rev. D 88, 4, 042004 (2013), [arXiv:1307.3795].

[22] V. V. Prosin et al., Nucl. Instrum. Meth. A 756, 94 (2014).

[23] S. Schoo et al. (KASCADE-Grande), PoS ICRC2015, 263 (2016).

[24] K. Rawlins and T. Feusels (IceCube), PoS ICRC2015, 334 (2016).

[25] D. Ivanov, PoS ICRC2015, 349 (2016).

[26] A. Aab et al. (Pierre Auger), in "34th Int. Cosmic Ray Conf.", (2015), [arXiv:1509.03732].

[27] R. Alfaro et al. (HAWC), Phys. Rev. D 96, 12, 122001 (2017), [arXiv:1710.00890].

[28] M. G. Aartsen et al. (IceCube), Phys. Rev. D 102, 122001 (2020), [arXiv:2006.05215].

[29] P. Abreu et al. (Pierre Auger), Eur. Phys. J. C 81, 11, 966 (2021), [arXiv:2109.13400].

[30] J. A. Morales-Soto et al. (HAWC), PoS ICRC2021, 330 (2021), [arXiv:2108.04748].

[31] D. Maurin et al. (2023), [arXiv:2306.08901].

[32] M. Ackermann et al. (Fermi-LAT), Astrophys. J. 750, 3 (2012), [arXiv:1202.4039].

[33] Z. Cao et al. (LHAASO) (2023), [arXiv:2305.05372].

[34] M. Ackermann et al. (Fermi-LAT), Astrophys. J. 799, 86 (2015), [arXiv:1410.3696].

[35] R. Abbasi et al. (IceCube), PoS ICRC2023, 1064 (2023).

[36] S. Abdollahi et al. (Fermi-LAT), Astrophys. J. Supp. 260, 2, 53 (2022), [arXiv:2201.11184].

[37] S. P. Wakely and D. Horan, in "30th Int. Cosmic Ray Conf.", volume 3, 1341-1344 (2007).

[38] Z. Cao et al. (LHAASO) (2023), [arXiv:2305.17030].

[39] T. K. Gaisser, R. Engel and E. Resconi, *Cosmic Rays and Particle Physics: 2nd Edition*, Cambridge University Press (2016), ISBN 978-0-521-01646-9.

[40] P. Sreekumar et al., Phys. Rev. Lett. 70, 127 (1993).

[41] W. Baade and F. Zwicky, Proceedings of the National Academy of Science 20, 259 (1934).

[42] D. Ter Haar, Reviews of Modern Physics 22, 119 (1950).

[43] L. O. Drury, Rept. Prog. Phys. 46, 973 (1983).

[44] C. J. Cesarsky and T. Montmerle, Space Sci. Rev. 36, 2, 173 (1983).

[45] A. M. Bykov, Astron. Astrophys. Rev. 22, 1, 77 (2014), [arXiv:1511.04608].

[46] V. Bosch-Ramon, F. A. Aharonian and J. M. Paredes, Astron. Astrophys. 432, 609 (2005), [arXiv:astro-ph/0411508].

[47] V. S. Ptuskin and Y. M. Khazan, Soviet Astronomy 25, 547 (1981).

[48] F. M. Rieger, Universe 8, 11, 607 (2022), [arXiv:2211.12202].

[49] M. Milgrom and V. Usov, Astrophys. J. Lett. 449, L37 (1995), [arXiv:astro-ph/9505009].

[50] L. A. Anchordoqui, G. E. Romero and J. A. Combi, Phys. Rev. D 60, 103001 (1999), [arXiv:astro-ph/9903145].

[51] G. E. Romero, A. L. Müller and M. Roth, Astron. Astrophys. 616, A57 (2018), [arXiv:1801.06483].

[52] P. Blasi, R. I. Epstein and A. V. Olinto, Astrophys. J. 533, L123 (2000), [arXiv:astro-ph/9912240].

[53] J. Arons, Astrophys. J. 589, 871 (2003), [arXiv:astro-ph/0208444].

[54] V. Petrosian, Space Sci. Rev. 173, 535 (2012), [arXiv:1205.2136].

[55] E. G. Zweibel and M. Yamada, ARA&A 47, 1, 291 (2009).

[56] A. M. Hillas, Ann. Rev. Astron. Astrophys. 22, 425 (1984).

[57] J. R. Jokipii, Astrophys. J. 146, 480 (1966).

[58] K. Lodders, Astrophys. J. 591, 1220 (2003).

[59] G. B. Rybicki, *Radiative Processes in Astrophysics*, Wiley-VCH (2004), ISBN 978-0-471-82759-7, 978-3-527-61817-0.

[60] K. Mannheim and R. Schlickeiser, Astron. Astrophys. 286, 983 (1994), [arXiv:astro-ph/9402042].

[61] Y. Genolini et al., Phys. Rev. C 98, 3, 034611 (2018), [arXiv:1803.04686].

[62] V. L. Ginzburg and S. I. Syrovatskii, *The Origin of Cosmic Rays*, Pergamon Press (1964).

[63] E. N. Parker, Planet. Space Sci. 13, 1, 9 (1965).

[64] A. E. Vladimirov et al., Comput. Phys. Commun. 182, 1156 (2011), [arXiv:1008.3642].

[65] C. Evoli et al., JCAP 10, 018 (2008), [Erratum: JCAP 04, E01 (2016)], [arXiv:0807.4730].

[66] R. Kissmann, Astropart. Phys. 55, 37 (2014), [arXiv:1401.4035].

[67] D. Maurin, Comput. Phys. Commun. 247, 106942 (2020), [arXiv:1807.02968].

[68] K. Greisen, Phys. Rev. Lett. 16, 748 (1966).

[69] G. T. Zatsepin and V. A. Kuzmin, JETP Lett. 4, 78 (1966).

[70] R. Kulsrud and W. P. Pearce, Astrophys. J. 156, 445 (1969).

[71] A. R. Bell and S. G. Lucek, Mon. Not. Roy. Astron. Soc. 321, 3, 433 (2001).

[72] C. Evoli et al., Phys. Rev. Lett. 121, 2, 021102 (2018), [arXiv:1806.04153].

[73] J. Silk and M. Srednicki, Phys. Rev. Lett. 53, 624 (1984).

[74] F. Donato, N. Fornengo and P. Salati, Phys. Rev. D 62, 043003 (2000), [hep-ph/9904481].

[75] S. Murgia, Ann. Rev. Nucl. Part. Sci. 70, 455 (2020).

[76] G. Steigman, Ann. Rev. Astron. Astrophys. 14, 339 (1976).

[77] J. W. Mitchell and T. Hams, *Astrophysics and Space Instrumentation*, 559-592 (2012).
[78] D. V. Reames, Lect. Notes Phys. **978**, pp. (2021), [arXiv:2010.08517].

[79] R. Engel, *Indirect Detection of Cosmic Rays*, 593–632 (2012).

[80] A. D. Panov et al., Bull. Russ. Acad. Sci. Phys. **71**, 494 (2007), [arXiv:astro-ph/0612377].

[81] H. S. Ahn et al., Astrophys. J. Lett. **714**, L89 (2010), [arXiv:1004.1123].

[82] O. Adriani et al. (CALET), Phys. Rev. Lett. **130**, 17, 171002 (2023), [arXiv:2304.14699].

[83] F. Alemanno et al. (DAMPE) (2023), [arXiv:2304.00137].

[84] O. Adriani et al., Astrophys. J. **791**, 2, 93 (2014), [arXiv:1407.1657].

[85] B. Schroer, C. Evoli and P. Blasi, Phys. Rev. D **103**, 123010 (2021), [arXiv:2102.12576].

[86] K. A. Lave et al., Astrophys. J. **770**, 2, 117 (2013).

[87] J. J. Engelmann et al., Astron. Astrophys. **233**, 96 (1990).

[88] M. Aguilar et al. (AMS), Phys. Rev. Lett. **126**, 8, 081102 (2021).

[89] M. Aguilar et al. (AMS), Phys. Rev. Lett. **127**, 2, 02101 (2021), [Erratum: Phys.Rev.Lett. 127, 159901 (2021)].

[90] O. Adriani et al. (CALET), Phys. Rev. Lett. **129**, 25, 251103 (2022), [arXiv:2212.07873].

[91] M. Aguilar et al. (AMS), Phys. Rev. Lett. **120**, 2, 021101 (2018).

[92] A. E. Vladimirov et al., Astrophys. J. **752**, 68 (2012), [arXiv:1108.1023].

[93] P. Blasi, E. Amato and P. D. Serpico, Phys. Rev. Lett. **109**, 061101 (2012), [arXiv:1207.3706].

[94] M. J. Boschini et al., Astrophys. J. **889**, 167 (2020), [arXiv:1911.03108].

[95] F. Aharonian et al. (H.E.S.S.), Phys. Rev. Lett. **101**, 261104 (2008), [arXiv:0811.3894].

[96] O. Adriani et al. (PAMELA), Phys. Rev. Lett. **111**, 081102 (2013), [arXiv:1308.0133].

[97] O. Adriani et al. (PAMELA), Nature **458**, 607 (2009), [arXiv:0810.4995].

[98] M. Klasen, M. Pohl and G. Sigl, Prog. Part. Nucl. Phys. **85**, 1 (2015), [arXiv:1507.03800].

[99] F. A. Aharonian, A. M. Atoyan and H. J. Volk, Astron. Astrophys. **294**, L41 (1995).

[100] A. U. Abeysekara et al. (HAWC), Science **358**, 6365, 911 (2017), [arXiv:1711.06223].

[101] P. Blasi, Phys. Rev. Lett. **103**, 051104 (2009), [arXiv:0903.2794].

[102] P. Mertsch, A. Vittino and S. Sarkar, Phys. Rev. D **104**, 103029 (2021), [arXiv:2012.12853].

[103] M. Aguilar et al. (AMS), Phys. Rev. Lett. **124**, 21, 211102 (2020).

[104] C. Evoli et al., Phys. Rev. D **103**, 8, 083010 (2021), [arXiv:2010.11955].

[105] P. Mertsch, JCAP **11**, 045 (2018), [arXiv:1809.05104].

[106] O. Adriani et al., Pisma Zh. Eksp. Teor. Fiz. **96**, 693 (2012).

[107] G. Jóhannesson et al., Astrophys. J. **824**, 1, 16 (2016), [arXiv:1602.02243].

[108] P. Blasi and P. D. Serpico, Phys. Rev. Lett. **103**, 081103 (2009), [arXiv:0904.0871].

[109] V. Poulin et al., Phys. Rev. D **99**, 2, 023016 (2019), [arXiv:1808.08961].

[110] H. Fuke et al., Phys. Rev. Lett. **95**, 081101 (2005), [arXiv:astro-ph/0504361].

[111] A. H. Compton and I. A. Getting, Phys. Rev. **47**, 11, 817 (1935).

[112] L. J. Gleeson and W. I. Axford, Astrophys. Sp. Sc. **2**, 4, 431 (1968).

[113] M. Aglietta et al. (EAS-TOP), Astrophys. J. **470**, 501 (1996).

[114] M. Ahlers and P. Mertsch, Prog. Part. Nucl. Phys. **94**, 184 (2017), [arXiv:1612.01873].

[115] A. Abdul Halim et al. (Pierre Auger), Astrophys. J. Suppl. **264**, 2, 50 (2023), [arXiv:2211.16020].

[116] R. U. Abbasi et al. (Telescope Array), Science **382**, 6673, ab05095 (2023), [arXiv:2311.14231].

[117] S. Sciutto, AIRES User's Manual and Reference Guide; version 19.04.00http://aires.fisica.unlp.edu.ar/.

[118] A. Aab et al. (Pierre Auger), Phys. Rev. D **100**, 8, 082003 (2019), [arXiv:1901.08040].

[119] D. Heck and T. Pierog, Extensive Air Shower Simulation with CORSIKA: A User's Guide (Version 7.7420) https://www.iap.kit.edu/corsika/70.php.

[120] F. Riehn et al., Phys. Rev. D **102**, 6, 063002 (2020), [arXiv:1912.03300].

[121] S. Ostapchenko, EPJ Web Conf. **208**, 11001 (2019).

[122] T. Pierog et al., Phys. Rev. C **92**, 3, 034906 (2015), [arXiv:1306.0121].

[123] W. Heitler, *The quantum theory of radiation*, volume 5 of *International Series of Monographs on Physics*, Oxford University Press, Oxford (1936).

[124] J. Matthews, Astropart. Phys. **22**, 387 (2005).

[125] A. Aab et al. (Pierre Auger), Nucl. Instrum. Meth. A **798**, 172 (2015), [arXiv:1502.01323].

[126] H. Kawai et al. (Telescope Array), Nucl. Phys. B Proc. Suppl. **175-176**, 221 (2008).

[127] S. Knurenko and I. Petrov, Adv. Space Res. **64**, 12, 2570 (2019), [arXiv:1908.01508].

[128] N. M. Budnev et al., Astropart. Phys. **117**, 102406 (2020), [arXiv:2104.03599].

[129] A. Aab et al. (Pierre Auger), Phys. Rev. Lett. **116**, 24, 241101 (2016), [arXiv:1605.02564].

[130] J. Pawlowski (Pierre Auger), PoS ICRC2023, 016 (2023), [arXiv:2312.14673].

[131] P. A. Bezyazeekov et al., Phys. Rev. D **97**, 12, 122004 (2018), [arXiv:1803.06862].

[132] I. Petrov and S. Knurenko, PoS ICRC2019, 385 (2020).

[133] P. Schellart et al., Astron. Astrophys. **560**, A98 (2013), [arXiv:1311.1399].

[134] S. Buitink et al., PoS ICRC2023, 503 (2023).

[135] A. Aab et al. (Pierre Auger) (2016), [arXiv:1604.03637].

[136] E. Kido (Telescope Array), EPJ Web Conf. **210**, 06001 (2019).

[137] M. G. Aartsen et al. (IceCube) (2014), [arXiv:1412.5106].

[138] J. Álvarez-Muñiz et al. (GRAND), Sci. China Phys. Mech. Astron. **63**, 1, 219501 (2020), [arXiv:1810.09994].

[139] D. Southall et al., Nucl. Instrum. Meth. A **1048**, 167889 (2023), [arXiv:2206.09660].

[140] D. Ivanov (Telescope Array), PoS ICRC2019, 298 (2020).

[141] A. Aab et al. (Pierre Auger), Phys. Rev. D **102**, 6, 062005 (2020), [arXiv:2008.06486].

[142] A. Coleman et al., Astropart. Phys. **149**, 102819 (2023), [arXiv:2205.05845].

[143] R. U. Abbasi et al. (Telescope Array), Astrophys. J. **858**, 2, 76 (2018), [arXiv:1801.09784].

[144] R. U. Abbasi et al. (Telescope Array), Astrophys. J. **909**, 2, 178 (2021), [arXiv:2012.10372].

[145] A. Yushkov (Pierre Auger), PoS ICRC2019, 482 (2020).
[146] C. J. Todero Peixoto (Pierre Auger), PoS ICRC2019, 440 (2020).

[147] N. M. Budnev et al. (TAIGA), PoS ICRC2021, 731 (2021).

[148] A. Fedynitch et al., Phys. Rev. D 100, 10, 103018 (2019), [arXiv:1806.04140].

[149] A. Abdul Halim et al. (Pierre Auger), Phys. Rev. Lett. 132, 2, 021001 (2024), [arXiv:2310.19963].

[150] A. A. Watson, JHEAp 33, 14 (2022), [arXiv:2112.06525].

[151] A. Aab et al. (Pierre Auger), JINST 16, 07, P07019 (2021), [arXiv:2101.02946].

[152] F. G. Schröder, Prog. Part. Nucl. Phys. 93, 1 (2017), [arXiv:1607.08781].

[153] D. Hooper, S. Sarkar and A. M. Taylor, Astropart. Phys. 27, 199 (2007), [arXiv:astro-ph/0608085].

[154] G. Golup et al. (Pierre Auger), PoS ICRC2023, 252 (2023).

[155] J. Kim et al. (Telescope Array), PoS ICRC2023, 244 (2023).

[156] A. Aab et al. (Pierre Auger), Front. Astron. Space Sci. 6, 24 (2019), [arXiv:1904.11918].

[157] B. P. Abbott et al., Astrophys. J. Lett. 848, 2, L12 (2017), [arXiv:1710.05833].

[158] A. Albert et al., Astrophys. J. Lett. 850, 2, L35 (2017), [arXiv:1710.05839].

[159] S. Mollerach and E. Roulet, JCAP 03, 017 (2019), [arXiv:1812.04026].

[160] A. A. Halim et al. (Pierre Auger), JCAP 05, 024 (2023), [arXiv:2211.02857].

[161] J. C. Arteaga Velazquez, PoS ICRC2023, 466 (2023).

[162] A. Aab et al. (Pierre Auger), Phys. Rev. Lett. 126, 15, 152002 (2021), [arXiv:2102.07797].

[163] M. Tavani et al. (AGILE), Astron. Astrophys. 502, 995 (2009), [arXiv:0807.4254].

[164] J. Chang et al. (DAMPE), Astropart. Phys. 95, 6 (2017).

[165] S. Torii, C. Collaboration et al., J. Jpn. Soc. Microgravity Appl. 24, 1, 120 (2007).

[166] W. B. Atwood et al. (Fermi-LAT), Astrophys. J. 697, 1071 (2009).

[167] F. Aharonian et al. (H.E.S.S.), Astron. Astrophys. 457, 899 (2006), [arXiv:astro-ph/0607333].

[168] J. Sitarek et al. (MAGIC), in “33rd Int. Cosmic Ray Conf.”, 0074 (2013), [arXiv:1308.0141].

[169] D. B. Kieda (VERITAS), in “33rd Int. Cosmic Ray Conf.”, 0700 (2013), [arXiv:1308.4849].

[170] F. Di Pierro (CTA LST Project), J. Phys. Conf. Ser. 2429, 1, 012020 (2023).

[171] A. Abramowski et al. (H.E.S.S.), Phys. Rev. D 90, 12, 122007 (2014).

[172] N. Budnev et al., Nucl. Instrum. Meth. A 958, 162113 (2020).

[173] G. Aielli et al., Nucl. Instrum. Meth. A 661, S50 (2012).

[174] M. Takita et al. (TIBET ASγ), in “30th Int. Cosmic Ray Conf.”, volume 2, 575–578 (2007).

[175] B. M. Baughman, in “32nd Int. Cosmic Ray Conf.”, volume 9, 123 (2011).

[176] Z. Cao, Nature Astron. 5, 8, 849 (2021).

[177] J. A. Hinton (H.E.S.S.), New Astron. Rev. 48, 331 (2004).

[178] C. Baixeras, Nuclear Physics B - Proceedings Supplements 114, 247 (2003), ISSN 0920-5632.

[179] H. B. T.C Weekes et al., Astroparticle Physics 17, 2, 221 (2002), ISSN 0927-6505.

[180] G. W. Clark et al., Astrophysical Journal, Letters 153, L203 (1968).

[181] C. E. Fichtel et al., Astrophysical Journal 198, 163 (1975).

[182] H. A. Mayer-Hasselwander et al., Astronomy and Astrophysics 105, 1, 164 (1982).

[183] S. D. Hunter et al., Astrophys. J. 481, 205 (1997).

[184] M. Ackermann et al., The Astrophysical Journal 750, 1, 3 (2012).

[185] A. A. Abdo et al., Astrophys. J. 688, 1078 (2008).

[186] B. Bartoli et al. (ARGO-YBJ), Astrophys. J. 806, 20 (2015).

[187] M. Amenomori et al. (Tibet ASgamma), Phys. Rev. Lett. 126, 14, 141101 (2021).

[188] P. Sreekumar et al. (EGRET), Astrophys. J. 494, 523 (1998).

[189] M. Ackermann et al. (Fermi-LAT), Astrophys. J. 799, 86 (2015).

[190] Z. Cao et al., Annual Review of Nuclear and Particle Science 73, in press, 341 (2023).

[191] S. Funk, Ann. Rev. Nucl. Part. Sci. 65, 245 (2015).

[192] J. A. Hinton and W. Hofmann, Ann. Rev. Astron. Astrophys. 47, 523 (2009), [arXiv:1006.5210].

[193] V. Schoenfelder (COMPTEL), Astron. Astrophys. Suppl. Ser. 143, 145 (2000), [arXiv:astro-ph/0002366].

[194] S. Abdollahi et al. (Fermi-LAT), Astrophys. J. Suppl. 247, 1, 33 (2020).

[195] Z. Cao et al. (LHAASO), Science 373, 6553, 425 (2021).

[196] F. A. Aharonian et al. (H.E.S.S.), Nature 432, 75 (2004).

[197] M. C. Weisskopf et al., Astrophys. J. Lett. 536, L81 (2000).

[198] B. Bartoli et al. (ARGO-YBJ), Astrophys. J. Suppl. 222, 1, 6 (2016).

[199] B. Bartoli et al. (ARGO-YBJ), Astrophys. J. 734, 110 (2011).

[200] F. Aharonian (H.E.S.S.), Astron. Astrophys. 464, 235 (2007).

[201] H. Abdalla et al. (H.E.S.S.), Astron. Astrophys. 612, A6 (2018).

[202] R. Bühler and R. Blandford, Rept. Prog. Phys. 77, 066901 (2014).

[203] A. A. Abdo et al., Science 326, 5959, 1512 (2009).

[204] J. Li et al., Nature Astron. 4, 12, 1177 (2020).

[205] R. Zanin et al., Astron. Astrophys. 596, A55 (2016), [arXiv:1605.05914].

[206] J. Albert et al. (MAGIC), Astrophys. J. Lett. 665, L51 (2007).

[207] G. Martí-Devesa and O. Reimer, Astron. Astrophys. 654, A44 (2021).

[208] A. A. Abdo et al., VizieR Online Data Catalog J/ApJS/208/17 (2013).

[209] S. Ansoldi et al. (MAGIC), Astron. Astrophys. 585, A133 (2016), [arXiv:1510.07048].

[210] A. Djannati-Ataï, in “31st Texas Symposium on Relativistic Astrophysics,” 33 (2022).

[211] A. Allafort et al. (Fermi-LAT), Astrophys. J. Lett. 777, L2 (2013).

[212] A. A. Abdo et al. (Fermi-LAT), Astrophys. J. Suppl. 208, 17 (2013).

[213] H. An, Astrophys. J. 924, 2, 91 (2022).

[214] C. J. Clark et al., Nature Astron. 7, 4, 451 (2023).

[215] D. F. Torres et al., Astrophys. J. 836, 1, 68 (2017).

[216] A. Franckowiak et al., Astron. Astrophys. 609, A120 (2018).

[217] S. De Wolf et al. (H.E.S.S.), Science 376, 6588, abn0567 (2022).

[218] M. Ajello et al. (Fermi-LAT), Astrophys. J. 892, 105 (2020).

[219] M. Ajello et al. (Fermi-LAT), Astrophys. J. Suppl. 263, 2, 24 (2022).
[220] M. Hayashida et al., Astrophys. J. 807, 1, 79 (2015).

[221] V. A. Acciari et al., Astrophys. J. 738, 25 (2011), [arXiv:1106.1210].

[222] M. Sikora, M. C. Begelman and M. J. Rees, Astrophys. J. 421, 153 (1994).

[223] F. A. Aharonian, New Astron. 5, 377 (2000).

[224] B. Bartoli, r. Xu et al. (ARGO-YBJ), Astrophys. J. 758, 2 (2012).

[225] A. Albert et al. (HAWC), Astrophys. J. 929, 2, 125 (2022).

[226] K. Katarzynski et al., Astron. Astrophys. 433, 479 (2005), [arXiv:astro-ph/0412405].

[227] Z.-H. An et al. (Insight-HXMT, GECAM) (2023), [arXiv:2303.01203].

[228] Z. Cao et al. (LHAASO), Science 380, 6652, 9328 (2023).

[229] V. A. Acciari et al. (MAGIC), Nature 575, 7783, 459 (2019).

[230] H. Abdalla et al., Nature 575, 7783, 464 (2019).

[231] M. Ajello et al., Astrophys. J. 878, 1, 52 (2019).

[232] P. Kumar and R. B. Duran, Mon. Not. Roy. Astron. Soc. 400, 75 (2009).

[233] X.-Y. Wang et al., Astrophys. J. 712, 1232 (2010).

[234] V. A. Acciari et al. (MAGIC), Nature 575, 7783, 455 (2019).

[235] H. Abdalla et al. (H.E.S.S.), Science 372, 6546, 1081 (2021).

[236] S. R. Coleman and S. L. Glashow, Phys. Rev. D 59, 116008 (1999), [hep-ph/9812418].

[237] F. W. Stecker and S. L. Glashow, Astropart. Phys. 16, 97 (2001), [arXiv:astro-ph/0102226].

[238] Z. Cao et al. (LHAASO), Phys. Rev. Lett. 128, 5, 051102 (2022).

[239] R. Abbasi et al. (IceCube), Eur. Phys. J. C 82, 11, 1031 (2022), [arXiv:2011.03561].

[240] E. Richard et al. (Super-Kamiokande), Phys. Rev. D 94, 052001 (2016), [arXiv:1510.08127].

[241] S. Adrian-Martinez et al. (ANTARES), Eur. Phys. J. C 73, 2606 (2013), [arXiv:1308.1599].

[242] M. G. Aartsen et al. (IceCube), Phys. Rev. D 91, 122004 (2015), [arXiv:1504.03753].

[243] M. Börner et al. (IceCube), PoS ICRC2015, 1098 (2016).

[244] M. Aartsen et al. (IceCube), Nature 591, 220 (2021), [arXiv:2110.15051].

[245] M. G. Aartsen et al. (IceCube), Phys. Rev. D 98, 6, 062003 (2018), [arXiv:1807.01820].

[246] M. Niechciol et al. (Pierre Auger), PoS ICRC2023, 1488 (2023).

[247] A. van Vliet, R. Alves Batista and J. R. Hörandel, Phys. Rev. D 100, 2, 021302 (2019), [arXiv:1901.01899].

[248] R. Abbasi et al. (IceCube), Science 380, 6652 (2023), [arXiv:2307.04427].

[249] R. Mammen Abraham et al., J. Phys. G 49, 11, 110501 (2022), [arXiv:2203.05591].

[250] J. Abraham et al. (Pierre Auger), Phys. Rev. Lett. 100, 211101 (2008), [arXiv:0712.1909].

[251] S. Ostapchenko, M. V. Garzelli and G. Sigl, Phys. Rev. D 107, 2, 023014 (2023), [arXiv:2208.12185].

[252] R. Abbasi et al. (IceCube), Phys. Rev. D 104, 02202 (2021), [arXiv:2011.03545].

[253] R. Abbasi et al. (IceCube), Astrophys. J. 928, 50 (2022), [arXiv:2111.10299].

[254] L. A. Fusco and F. Versari (ANTARES), PoS ICRC2019, 891 (2020).

[255] V. Allakhverdyan et al. (Baikal-GVD), Phys. Rev. D 107, 042005 (2023), [arXiv:2211.09447].

[256] S. L. Glashow, Phys. Rev. 118, 316 (1960).

[257] D. Biehl et al., JCAP 01, 033 (2017), [arXiv:1611.07983].

[258] V. Berezinsky and G. Zatsepin, Phys. Lett. B 28, 423 (1969).

[259] M. Ave et al., Phys. Rev. Lett. 85, 2244 (2000), [arXiv:astro-ph/0007386].

[260] D. Hooper, A. Taylor and S. Sarkar, Astropart. Phys. 23, 11 (2005), [arXiv:astro-ph/0407618].

[261] D. Gaggero et al., Phys. Rev. D 91, 083012 (2014), [arXiv:1411.7623].

[262] A. Albert et al. (ANTARES), Phys. Lett. B 481, 137951 (2023), [arXiv:2212.11876].

[263] M. Aartsen et al. (IceCube, Fermi-LAT, MAGIC, AGILE, ASAS-SN, HAWC, H.E.S.S., INTEGRAL, Kanata, Kiso, Kapteyn, Liverpool Telescope, Subaru, Swift NuSTAR, VERITAS, VLA/17B-403), Science 361, eaat1378 (2018), [arXiv:1807.08816].

[264] M. Aartsen et al. (IceCube), Science 361, 147 (2018), [arXiv:1807.08794].

[265] R. Abbasi et al. (IceCube), Science 378, 538 (2022), [arXiv:2211.09972].

[266] R. Stein et al., MNRAS 521, 4, 5046 (2023), [arXiv:2203.17135].

[267] M. G. Aartsen et al. (IceCube), Nature 551, 596 (2017), [arXiv:1711.08119].

[268] R. Abbasi et al. (IceCube), Phys. Rev. D 108, 1, 012014 (2023), [arXiv:2304.12236].

[269] N. Geisselbrecht (KM3NeT), PoS ICRC2023, 1107 (2023).

[270] M. G. Aartsen et al. (IceCube), Phys. Rev. Lett. 125, 14, 141801 (2020), [arXiv:2005.12942].

[271] R. Abbasi et al. (IceCube), Phys. Rev. Lett. 129, 1, 011804 (2022), [arXiv:2201.03566].

[272] A. Addazi et al., Prog. Part. Nucl. Phys. 125, 103948 (2022), [arXiv:2111.05659].
Experimental Methods and Colliders

<table><tr><td>31. Accelerator physics of colliders (rev.)</td><td>559</td></tr><tr><td>32. High-energy collider parameters (rev.)</td><td>569</td></tr><tr><td>33. Neutrino beam lines at high-energy proton synchrotrons (rev.)</td><td>574</td></tr><tr><td>34. Passage of particles through matter (rev.)</td><td>575</td></tr><tr><td>35. Particle detectors at accelerators (rev.)</td><td>591</td></tr><tr><td>36. Particle detectors for non-accelerator phys. (rev.)</td><td>649</td></tr><tr><td>37. Radioactivity and radiation protection (rev.)</td><td>679</td></tr><tr><td>38. Commonly used radioactive sources (rev.)</td><td>685</td></tr></table>
<figure><img src="image_170.png" /></figure>
# 31. Accelerator Physics of Colliders

Revised July 2023 by V. Shiltsev (FNAL) and F. Zimmermann (CERN).

This article provides background for the High-Energy Collider Parameter Tables that follow and some additional information; see in-depth review and a comprehensive list of references in [1]; citations below are limited to widely used textbooks and open access seminal papers and reviews.

## 31.1 Energy and Luminosity

Collisions of two beams of particles accelerated to high energies $E_{1,2}$ provide access to center-of-mass energies (c.m.e.) $E_{cme} \approx 2\sqrt{E_1E_2}$, assuming a typically small or zero crossing angle. Most of the 31 colliders that have ever reached the operational stage (seven are operational now) used equal masses and energies of colliding particles, with c.m.e. equal to twice the beam energy $E_{cme} = 2E_b$. Other machines collide beams of unequal energies, such as electron-proton or electron-ion colliders, or asymmetric B-factories, that produce new short-lived particles, whose decays are more easily detected and analyzed with a Lorentz boost.

In an accelerator, charged particles gain energy from an electric field, which usually varies in time at a high frequency ranging from 100s of kHz to 10s of GHz. With proper phasing to the RF field over distance $l$, the energy gain of a particle with charge $Ze$ is proportional to the average accelerating gradient $G$, i.e. $\Delta E_b = ZeGl$. In principle, the highest beam accelerating gradients achieved to date in operational machines or beam test facilities ($G \approx 100$ MV/m in 12 GHz normal-conducting RF cavities and 31.5 MV/m in 1.3 GHz superconducting ones) allow accessing high energies over reasonably long linear accelerators (linacs), but cost considerations often call for minimization of RF acceleration via repeated use of the same RF system which, in that case, would boost the energy in small portions $\Delta E_b = ZeV_{RF}$ per turn every time a particle passes through the total cavity voltage $V_{RF}$. Such an arrangement can be realized either in the form of storage-ring circular colliders or also through novel schemes based on, e.g., recirculating linear accelerators (RLAs) with or without energy recovery. Circular colliders are by far the most common; here, the momentum and energy of ultra-relativistic particles are determined by the bending radius inside the dipole magnets, $\rho$, and by the average magnetic field $B$ of these magnets:

$$p = ZeB\rho \quad \text{or} \quad E_b \, [\text{GeV}] = 0.3Z(B\rho) \, [\text{Tm}] \quad (31.1)$$

Such synchrotron condition assures approximately constant radius of the beam orbit during acceleration. Transverse focusing by quadrupole magnets is needed to keep particles inside the rather limited space provided by the accelerator beam pipe passing through the magnet apertures. The maximum field of normal-conducting (NC) magnets is about 2 T, due to the saturation of ferromagnetic materials, and, while this is sufficient for lower energy colliders, such as most $e^+e^-$ storage rings, it is not adequate for frontier-energy hadron (or muon) beams, because of the implied need for excessively long accelerator tunnels and prohibitively high total magnet power consumption. The development of superconducting (SC) magnets that employ high electric current carrying Nb-Ti wires cooled by liquid helium below 5 K, opened up the way towards higher fields and to hadron colliders at record energies [2]. For example, the 14 TeV c.m.e. LHC at CERN, uses double-bore SC magnets with a maximum field of 8.3 T at a temperature of 1.9 K, in a tunnel of $C = 26.7$ km circumference (dipole-magnet bending radius $\rho = 2800$ m). The double-bore design allows acceleration of the same particle type in opposite directions and also the operation with different particle species (e.g., protons and heavy ions) in the two apertures, while a single bore magnet implies the use of particles and antiparticles for the collider application. As the production of anti-particles is energy consuming and therefore limited, this concept opens the door to high-performance hadron colliders.

The exploration of rare nuclear and high energy particle physics phenomena requires not only an appropriately high energy, but also a sufficiently large number of detectable reactions. The number of events of interest $N_{exp}$ is given by the product of the cross section of the reaction under study, $\sigma_{exp}$, and the time integral

over the instantaneous luminosity, $\mathcal{L}$:

$$N_{\text{exp}} = \sigma_{\text{exp}} \cdot \int \mathcal{L}(t) dt. \quad (31.2)$$

In the Tables, luminosity is stated in the units of $cm^{-2}s^{-1}$. The integral on the right is referred to as integrated luminosity $\mathcal{L}_{int}$, and, reflecting the smallness of typical particle-interaction cross-sections is often reported in units of inverse femto- or attobarn, e.g., $1 ab^{-1}=10^{42} cm^{-2}$. Colliders usually employ bunched beams of particles with approximately Gaussian distributions, and for two bunches containing $N_1$ and $N_2$ particles colliding head-on with frequency $f_{coll}$, a basic expression for the luminosity is

$$\mathcal{L} = f_{\text{coll}} \frac{N_1 N_2}{4\pi \sigma_x^* \sigma_y^*} \mathcal{F} \quad (31.3)$$

where $\sigma_x^*$ and $\sigma_y^*$ characterize the rms transverse beam sizes in the horizontal and vertical directions at the interaction point, and $\mathcal{F}$ is a factor of order 1, that takes into account inefficient geometric overlapping of the beams due to a crossing angle and finite bunch length, and dynamic effects, such as the mutual focusing of the two beam during the collision (see below). Having $n_b$ bunches per beam increases the frequency of collisions $f_{coll} = n_b f_0$ where $f_0$ is either the revolution frequency of a circular collider or the repetition rate of a linear one. To achieve a high luminosity, one, therefore, has to maximize the population and number of bunches, either producing these narrowly or focusing them tightly, and colliding them at high frequencies at dedicated locations, where products of their reactions can be registered by particle detectors.

Subsequent sections in this report briefly expand on the beam dynamics behind collider design, comment on the realization of collider performance in a selection of today's facilities, and end with some remarks on future possibilities.

## 31.2 Beam Dynamics

Given the enormous and highly concentrated power carried by modern high energy particle beams, the main concern of beam dynamics in colliders is stability of motion of i) individual particles in accelerators, ii) single high-intensity beams of many particles moving together, and iii) colliding beams [3–5].

### 31.2.1 Single Particle Dynamics

While a reference particle at the nominal energy proceeds along the design trajectory (reference orbit) mostly determined by transverse magnetic dipole fields, other particles in the bunch are kept close by through the focusing effect of quadrupole fields. Assume that the reference particle carries a right-handed Cartesian coordinate system, with the co-moving z-coordinate pointed in the direction of motion along the reference trajectory, $z = s - vt$ (with $v$ the reference particle velocity, and $t$ time). The independent variable is the distance $s$ of the reference particle along this trajectory, rather than time $t$, and for simplicity this reference path is taken to be planar. The transverse coordinates are $x$ (horizontal) and $y$ (vertical), where $\{x, z\}$ defines the plane of the reference trajectory.

Several time scales are involved, and this is reflected in the approximations used in formulating the equations of motion. All of today's high-energy colliders are alternating gradient synchrotrons or, respectively, storage rings and the shortest time scale is set by so-called betatron oscillations. The linearized equations of motion of a particle displaced from the reference trajectory are:

$$x'' + K_x(s)x = 0, y'' + K_y(s)y = 0, z' = -x/\rho(s), \\ \text{with } K_x \equiv \frac{Ze}{p} \frac{\partial B_y}{\partial x} + \frac{1}{\rho^2} \text{ and } K_y \equiv -\frac{Ze}{p} \frac{\partial B_y}{\partial x} \quad (31.4)$$

where $\rho = p/ZeB_y$ is the radius of curvature due to the field on the reference orbit. The prime denotes $d/ds$ and the Maxwell equation in vacuum $\nabla \times \mathbf{B} = 0$ helps to eliminate $B_x(s)$ using the relation $\partial B_x/\partial y = \partial B_y/\partial x$. In this linear approximation, the vertical magnetic field $B_y(s)$ in the $(x, z)$-plane contains only