The stationary non-equilibrium plasma of cosmic-ray electrons and positrons

Keywords

• Stationary non-equilibrium distributions;
• Cosmic-ray electron–positron plasma;
• Relativistic statistical ensembles;
• Power-law densities with exponential cutoff;
• Nonthermal ensemble averaging;
• Classical & quantum partitions with extensive entropy

1. Introduction

We study the statistical mechanics of the cosmic-ray electron–positron plasma, based on high-precision spectra obtained with the Alpha Magnetic Spectrometer (AMS-02)  [1] and [2]. We demonstrate that this plasma can be treated, in close analogy to the photon gas of the cosmic microwave background radiation, as a primordial electron gas in stationary non-equilibrium. A primordial origin is also suggested by the high isotropy of the observed fluxes. Primordiality of the cosmic-ray electron–positron plasma is tantamount to the material realization of a universal cosmic reference frame as the rest frame of a relativistic gas of massive particles. An alternative approach is to use relativistic kinetic theory, but to arrive at quantitative densities suitable for spectral fitting, one has to specify production mechanisms and interaction processes with cosmic matter and radiation fields which are uncertain  [3].

We perform spectral fits to the AMS-02 electron and positron fluxes and reconstruct, from the inferred spectral densities, the distribution functions of the electronic and positronic plasma components. The partial number densities are relativistic non-equilibrium distributions, exponentially cut power-law densities  [4] and [5] which converge to the Maxwell–Boltzmann distribution for low particle velocities, the Coulomb interaction being negligible. We calculate the classical partition function, by ensemble averaging in phase space, and the entropy function which is an extensive variable. We then quantize the grand partition function and show, based on the fugacity obtained from the spectral fits, that the classical limit is realized. Finally we calculate the positron fraction and give estimates of the thermodynamic parameters of this low-density high-temperature plasma.

In Section  2, we consider a relativistic plasma in stationary non-equilibrium and relate the classical spectral number density to the empirical flux density obtained from spectral fits to the measured electron and positron fluxes (performed in Section  5). In Section  3, we derive the thermodynamic variables of the electronic and positronic plasma components in phase space, based on probability distributions inferred from the spectral fits, in particular the electronic/positronic energy and entropy densities.

In Section  4, we explain the quantization of the classical nonthermal partition function and demonstrate that the classical limit applies to the cosmic-ray electron–positron plasma due to the small fugacity. The formalism developed in Sections  2, 3 and 4 is based on relativistic dispersion relations and designed to be practically suitable for spectral fits in any energy range. Foundational aspects of relativistic statistical mechanics and thermodynamics such as the arrow of time and entropy are discussed in Refs.  [6], [7], [8] and [9]. Recent applications of relativistic statistical mechanics include the quark–gluon plasma  [10] and [11], plasmas in gravitational fields  [12], gases of neutral particles with long-range spin–spin interaction  [13] and condensation effects in relativistic Bose–Einstein gases  [14]. Kinetic theory with relativistic dispersion relations, in particular the relativistic Boltzmann equation in Grad’s moment expansion and Chapman–Enskog approximation, is discussed in Refs.  [15] and [16] and a relativistic version of the Fokker–Planck equation in Ref.  [17].

In Section  5, we perform least-squares fits to the AMS-02 and HESS  [18] electron and positron spectra. The measured spectra are located in the GeV and low TeV range, where we can use the ultra-relativistic approximation of the spectral densities derived in Sections  2, 3 and 4, which means to drop the mass term in the electronic dispersion relation. The ultra-relativistic electronic/positronic flux densities obtained in this way are then used to assemble the positron fraction, which has been measured by the AMS-02 Collaboration in the GeV range  [19], thus providing a test of the spectral number densities on which the thermodynamic variables are based. Extrapolation of the analytic formula for the positron fraction into the TeV range suggests an extended spectral peak centered around 1 TeV and exponential decay above 10 TeV.

As mentioned, this primordial approach to the cosmic-ray electron–positron plasma is alternative to the use of kinetic equations which require the assumption of specific electron/positron production mechanisms  [20]. A possible mechanism is electron/positron emission caused by interaction of high-energy protons or heavier cosmic-ray nuclei with the interstellar medium  [3] and [21]. Quantitative models thereof turn out to be inconsistent with the pronounced rise of the positron fraction, so that additional positron sources have to be invoked, for instance, discrete high-energy sources such as pulsar magnetospheres or the shocked plasmas of supernova remnants  [22], even if isotropy is somewhat compromised. Other authors prefer positron production via decay or annihilation of a variety of hypothetical dark matter particles at TeV energies  [23] which could exist in galactic halos.

In Section  5, we also give estimates of the specific energy and number densities, entropy production, temperature, fugacity and chemical potential of the electronic/positronic plasma components. We calculate the specific densities of the actually measured electrons/positrons in the energy range above 1 GeV up to low TeV energies where an exponential cutoff occurs. We then restore the electron mass in the dispersion relation and extrapolate the analytic spectral densities obtained from the spectral fits to lower energies. In this way, we can predict the specific energy, number and entropy densities of the complete statistical ensemble comprising all gas particles by integrating the electronic/positronic spectral densities down to the electron mass. In Section  6, we present our conclusions.

2. Relativistic plasma in stationary non-equilibrium

We start with the classical number density of a free electron plasma,

equation(2.1)

$\frac{d{\rho }_B}{dE}=\frac{s}{2{\pi }^2}{E}^2{e}^{-\beta E-f\left(E\right)-\alpha }\sqrt{1-m^2/E^2}\text{,}$

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$e^{-\alpha }$

$\beta =1/\left(k_{B}T\right)$

The number density (2.1) is assembled with the integration measure $s{d}^{3}p/{\left(2\pi \right)}^3$ and the dispersion relation for free relativistic particles $p=\sqrt{E^2-m^2}$, so that $d^{3}p=4\pi \sqrt{1-m^2/E^2}{E}^{2}dE$. The normalization factor s/(2π)³$s/{\left(2\pi \right)}^3$ arises from box quantization in the continuum limit, see (4.4). The quantized spectral density of which (2.1) is the classical limit is discussed in Section  4. Density (2.1) is thus assembled as $d{\rho }_B={\left(2\pi \right)}^{-3}sG\left(E\right)e^{-\beta E-\alpha }{d}^{3}p$, with the normalized spectral function G(E)=g(E)/g(m)$G\left(E\right)=g\left(E\right)/g\left(m\right)$. The non-relativistic limit of $d{\rho }_B\left(E\right)$ is thermal since $g\left(E\right)=g\left(m\right)+\text{O}\left({\upsilon }^2\right),E=m/\sqrt{1-{\upsilon }^2}$, where υ≪1$\upsilon \ll 1$ is the particle speed parametrizing the non-relativistic Maxwell–Boltzmann distribution, whose chemical potential is obtained from the relativistic potential $\mu \left[\text{GeV}\right]=-\alpha /\beta$ by subtraction of the rest mass m$m$.

The spectral flux density is found by multiplying the number density with the group velocity, $\upsilon =dE/dp=\sqrt{1-m^2/E^2}$. The flux density per steradian is thus

equation(2.2)

${\Phi }_E\left[{\text{GeV}}^{-1}{\text{sr}}^{-1}{\text{m}}^{-2}{\text{s}}^{-1}\right]=\frac{1}{4\pi }\sqrt{1-\frac{m^2}{E^2}}\frac{d{\rho }_B}{dE}\text{,}$

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equation(2.3)

$E^3{\Phi }_E\left[{\text{GeV}}^2{\text{sr}}^{-1}{\text{m}}^{-2}{\text{s}}^{-1}\right]=\frac{s}{8{\pi }^3}{E}^5\frac{g\left(E\right)}{g\left(m\right)}{e}^{-\beta E-\alpha }(1-\frac{m^2}{E^2})\text{.}$

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equation(2.4)

$E^3{\Phi }_E=A_{\Phi }{E}^{{\alpha }_1}\frac{1+{(E/c_1)}^{{\gamma }_1}}{1+{(E/b_1)}^{{\beta }_1}}{e}^{-\beta E}(1-\frac{m^2}{E^2})\text{,}$

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$p=E\sqrt{1-m^2/E^2}$

$dE/dp$

$d^{3}p$

In flux density (2.4), we specified the spectral function g(E)$g\left(E\right)$ in (2.3) by the ansatz

equation(2.5)

$g\left(E\right)=E^{{\alpha }_1-5}\frac{1+{(E/c_1)}^{{\gamma }_1}}{1+{(E/b_1)}^{{\beta }_1}}\text{,}$

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equation(2.6)

$A_{\Phi }\left[{\text{GeV}}^{2-{\alpha }_1}{\text{sr}}^{-1}{\text{m}}^{-2}{\text{s}}^{-1}\right]=\frac{s}{{(2\pi ħ)}^3{c}^2}\frac{e^{-\alpha }}{g\left(m\right)}\approx 3.146\times {10}^{53}\frac{e^{-\alpha }}{g\left(m\right)}\text{.}$

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The number of particles in a volume V$V$ with energies exceeding $E_{\mathrm{cut}}\ge m$ is calculated as

equation(2.7)

$N=-\frac{\partial }{\partial \alpha }log{Z}_B(\beta ,\alpha ,V)=V{\int }_{E_{\mathrm{cut}}}^{\infty }d{\rho }_B\left(E\right)\text{,}$

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$E_{\mathrm{cut}}$

equation(2.8)

$U=-\frac{\partial }{\partial \beta }log{Z}_B(\beta ,\alpha ,V)=V{\int }_{E_{\mathrm{cut}}}^{\infty }Ed{\rho }_B\left(E\right)\text{.}$

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$E_{\mathrm{cut}}=1\mathrm{GeV}$

$d{\rho }_B\left(E\right)$

$E_{\mathrm{cut}}$

$4.95\mathrm{keV}/m^3$

$4.17\mathrm{keV}/m^3$

$261\mathrm{keV}/m^3$

$E_{\mathrm{cut}}=1\mathrm{GeV}$

$E_{\mathrm{cut}}=m$

Entropy is an extensive quantity proportional to V$V$,

equation(2.9)

$S(\beta ,\alpha ,V)=k_B(log{Z}_B+\beta U+\alpha N)\text{,}$

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$log{Z}_B$

To demonstrate thermodynamic stability, we first show that the inequalities 0≤C_V≤C_P$0\le C_V\le C_P$ hold for the isochoric and isobaric heat capacities. To this end, we note, cf. (2.7) and (2.8),

equation(2.10)

$U(\beta ,N)=-N\frac{\partial logK}{\partial \beta }\text{,}K\left(\beta \right)={\int }_{E_{\mathrm{cut}}}^{\infty }{E}^2{e}^{-\beta E-f\left(E\right)}\sqrt{1-m^2/E^2}dE$

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equation(2.11)

$C_V=\frac{\partial }{\partial T}U(\beta ,N)=k_{B}N{\beta }^2\frac{{\partial }^{2}logK}{\partial {\beta }^2}\text{,}{C}_P=C_V+k_{B}N\text{.}$

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We also have to show mechanical stability, 0≤κ_S≤κ_T$0\le {\kappa }_S\le {\kappa }_T$, where κ_S${\kappa }_S$ and κ_T${\kappa }_T$ denote adiabatic and isothermal compressibility. Since $N=log{Z}_B$, cf. (2.7), the thermal equation of state reads P=N/(βV)$P=N/\left(\beta V\right)$, and the isothermal compressibility is thus κ_T=−(1/V)∂V(β,P,N)/∂P=1/P${\kappa }_T=-(1/V)\partial V(\beta ,P,N)/\partial P=1/P$. The adiabatic compressibility is related to the heat capacities by κ_S=κ_TC_V/C_P${\kappa }_S={\kappa }_T{C}_V/C_P$, which implies the stated inequalities since κ_T>0${\kappa }_T>0$ and C_P>C_V$C_P>C_V$. This holds true for the electronic and positronic plasma components alike.

3. Thermodynamic variables in phase space

We consider the phase space of n$n$ electrons with measure

$d^{3n}(p,q)=\frac{1}{{\left(2\pi \right)}^{3n}}{d}^3{p}_1{d}^3{q}_1{d}^3{p}_2{d}^3{q}_2\cdots d^3{p}_n{d}^3{q}_n\text{,}$

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equation(3.1)

$d^3{p}_k=4\pi \sqrt{1-m^2/E_k^2}{E}_k^{2}d{E}_k\text{,}$

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$d^3{q}_k$

$d{E}_k$

$E_{\mathrm{cut}}\le E_k\le \infty$

$P_n(p,q)=\frac{s^n}{n!}{b}_n(p,q)h_n(p,q)\text{,}$

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equation(3.2)

$b_n=exp(-\xi -\beta \sum _{k=1}^n{E}_k-\alpha n)\text{,}{h}_n=exp(-\sum _{k=1}^{n}f\left(E_k\right))\text{.}$

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The phase-space probability is normalized to one,

equation(3.3)

$\int \sum _{n=0}^{\infty }{P}_n(p,q)d^{3n}(p,q)=1\text{,}$

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$log{Z}_B=N$

equation(3.4)

$Z_B=\sum _{n=0}^{\infty }\frac{s^n}{n!}\int exp(-\beta \sum _{k=1}^n{E}_k-\alpha n)h_n(p,q)d^{3n}(p,q)\text{.}$

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equation(3.5)

$\sum _{n=0}^{\infty }\int n{P}_n(p,q)d^{3n}(p,q)=-\frac{d\xi }{d\alpha }=N\text{,}$

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equation(3.6)

$\sum _{n=0}^{\infty }\int \left(\sum _{k=1}^n{E}_k\right)P_n(p,q)d^{3n}(p,q)=-\frac{d\xi }{d\beta }=U\text{.}$

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equation(3.7)

$S=-k_B\int \sum _{n=0}^{\infty }\frac{s^n}{n!}{h}_n(p,q)b_n(p,q)log{b}_n(p,q)d^{3n}(p,q)\text{,}$

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equation(3.8)

$S=k_B\int \sum _{n=0}^{\infty }(\xi +\beta \sum _{k=1}^n{E}_k+\alpha n)P_n(p,q)d^{3n}(p,q)\text{.}$

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4. Quantized partition function of the electronic and positronic plasma components

The quantized version of the classical partition function $Z_B$ in (2.7) is obtained by a trace calculation in fermionic occupation number representation  [25] and [27],

equation(4.1)

$Z_F=\text{Tr}[exp(-\alpha \hat{N}-\beta \hat{H}-f\left(\hat{H}\right))]\text{.}$

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$\hat{N}={\sum }_{\left|k\right|\ge k_{\mathrm{cut}}}{\hat{N}}_k,{\hat{N}}_k=b_k^{+}{b}_k$

$b_k^{(+)}$

$\hat{H}={\sum }_{\left|k\right|\ge k_{\mathrm{cut}}}{E}_k{\hat{N}}_k$

$f\left(\hat{H}\right)={\sum }_{\left|k\right|\ge k_{\mathrm{cut}}}f\left(E_k\right){\hat{N}}_k$

${\hat{N}}_k|n〉=n_k|n〉$

$n_k$

$k$

$k=p(ħ=c=1)$

$E_k=\sqrt{k^2+m^2}$

$k_{\mathrm{cut}}=m\sqrt{E_{\mathrm{cut}}^2/m^2-1}$

$k=2\pi n/L$

$k$

$n$

With these prerequisites, trace (4.1) can be evaluated as

equation(4.2)

$Z_F=\sum _{n_k=0,1}exp\sum _{\left|k\right|\ge k_{\mathrm{cut}}}(-\alpha -\beta E_k-f\left(E_k\right))n_k=\prod _{\left|k\right|\ge k_{\mathrm{cut}}}[1+exp(-\alpha -\beta E_k-f\left(E_k\right))]\text{.}$

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equation(4.3)

$log{Z}_F=\sum _{\left|k\right|\ge k_{\mathrm{cut}}}log(1+e^{-\alpha -\beta E_k-f\left(E_k\right)})\text{.}$

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$sV{\int }_{\left|k\right|\ge k_{\mathrm{cut}}}{d}^{3}k/{\left(2\pi \right)}^3$

equation(4.4)

$log{Z}_F=\frac{sV}{{\left(2\pi \right)}^3}{\int }_{\left|k\right|\ge k_{\mathrm{cut}}}log(1+e^{-\alpha -\beta E_k-f\left(E_k\right)})d^{3}k\text{.}$

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$d^{3}k$

$d^{3}p$

equation(4.5)

$log{Z}_F=\frac{Vs}{2{\pi }^2}{\int }_{E_{\mathrm{cut}}}log(1+e^{-\alpha -\beta E-f\left(E\right)})\sqrt{1-m^2/E^2}{E}^{2}dE\text{.}$

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$d{\rho }_F\left(E\right)$

equation(4.6)

$N=-\frac{\partial }{\partial \alpha }log{Z}_F=V{\int }_{E_{\mathrm{cut}}}^{\infty }d{\rho }_F\left(E\right)\text{,}$

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equation(4.7)

$d{\rho }_F\left(E\right)=\frac{s}{2{\pi }^2{(ħc)}^3}\frac{\sqrt{1-m^2/E^2}}{e^{\alpha +\beta E+f\left(E\right)}+1}{E}^{2}dE\text{.}$

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$d{\rho }_B\left(E\right)$

$e^{\alpha +\beta E+f\left(E\right)}\gg 1$

$e^{\alpha }=1/z$

5. Spectral fits to high-energy cosmic-ray electron and positron fluxes

The spectral fits in Fig. 1 and Fig. 2 are performed in the GeV and TeV range, the ultra-relativistic regime, so that we can drop the factor (1−m²/E²)$(1-m^2/E^2)$ in the flux density, see the discussion following Eq. (2.4),

equation(5.1)

$E^3{\Phi }_E\left[{\text{GeV}}^2{\text{m}}^{-2}{\text{sr}}^{-1}{\text{s}}^{-1}\right]\sim A_{\Phi }{E}^{{\alpha }_1}\frac{1+{(E/c_1)}^{{\gamma }_1}}{1+{(E/b_1)}^{{\beta }_1}}{e}^{-\beta E}\text{.}$

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${\Phi }_E\left(e^{+}\right)/({\Phi }_E\left(e^{-}\right)+{\Phi }_E\left(e^{+}\right))$

$\sim 0.52E^{-0.066}{e}^{-7.4\times 10^{-5}E}$

The classical spectral number density reads, cf. (2.1) and Refs.  [33] and [34],

equation(5.2)

$d{\rho }_B\left(E\right)\left[{\text{m}}^{-3}\right]\sim \frac{s}{2{\pi }^2{(ħc)}^3}\frac{g\left(E\right)}{g\left(m\right)}{E}^2{e}^{-\beta E-\alpha }\sqrt{1-\frac{m^2}{E^2}}dE\text{,}$

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$k_{B}T\left[\text{GeV}\right]=1/\beta$

$z=e^{-\alpha }$

equation(5.3)

$\hat{n}\left[{\text{m}}^{-3}\right]\sim {\int }_{E_{\mathrm{cut}}}^{\infty }d{\rho }_B\left(E\right)\text{,}\hat{u}[\text{GeV}/{\text{m}}^3]\sim {\int }_{E_{\mathrm{cut}}}^{\infty }Ed{\rho }_B\left(E\right)\text{,}$

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$d{\rho }_B\left(E\right)$

$\hat{s}=S/V$

equation(5.4)

$\frac{\hat{s}}{k_B}\left[{\text{m}}^{-3}\right]=(1+\alpha )\hat{n}+\beta \hat{u}\text{.}$

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$E_{\mathrm{cut}}=m$

$E_{\mathrm{cut}}=1\mathrm{GeV}$

6. Conclusion

We developed a statistical description of the cosmic-ray electron–positron plasma based on stationary non-equilibrium distributions. The latter are inferred from spectral fits to the electron and positron spectra measured by the AMS-02 experiment in the GeV interval and the Cherenkov telescope HESS in the low TeV range. In particular, we do not need to make hypothetical assumptions on sources and production mechanisms of high-energy cosmic-ray electrons/positrons. Their spectral number densities are semi-empirically reconstructed from the measured electron and positron fluxes as classical ensemble averages in phase space.

The cosmic-ray electron–positron plasma is relativistic and nonthermal, the Coulomb interaction can be neglected due to the low particle density and high temperature. The criterion for this approximation is a small ratio $e^2/\left(4\pi d{k}_{B}T\right)\ll 1$, where $d={\hat{n}}^{-1/3}$ is the interparticle distance and e²/(4π)≈1/137$e^2/\left(4\pi \right)\approx 1/137$ the fine-structure constant  [35] and [36]. In the case of cosmic-ray electrons and positrons, this ratio is of order 10⁻²³. Temperature and the specific number density $\hat{n}$ of the non-equilibrated plasma components are listed in Table 2. Quantum corrections to the classical partition function are also negligible, cf. after (4.7).

The entropy of the cosmic-ray electron–positron plasma is an extensive quantity, even though the underlying electronic/positronic probability distributions are nonthermal. We also checked the positivity of the heat capacities and compressibility, which ensures the thermodynamic stability of each plasma component implying positive root mean squares of the thermodynamic variables. The thermal Maxwell–Boltzmann distribution is recovered in the non-relativistic limit of the spectral number density (2.1).

To summarize the approximations, cosmic-ray electrons and positrons at GeV energies and beyond constitute a relativistic gas, so that one has to use relativistic dispersion relations. As the particles are charged, they constitute a two-component plasma, but since the gas is dilute and thus the interparticle distances large according to the number densities in Table 2, one can ignore Coulomb interaction and Debye screening, see the estimate above. As the fugacity is low, the quantum distribution (4.7) can be approximated by its classical limit. In the spectral fits in Section  5, the ultra-relativistic approximation is used, because the electronic mass/energy ratio is negligible in the GeV band, cf. after (2.4). In the calculation of the specific thermodynamic densities of the electronic and positronic gas components, cf. Table 2, we restored this ratio in the electronic dispersion relation, since the fitted spectral densities determining the specific densities are extrapolated to lower energies, see the remarks after (5.4).

We calculated the positron fraction using the parameters obtained from the spectral fits in Fig. 1 and Fig. 2, and compared it with the AMS-02 fraction measured in the GeV range. As the analytic representation (5.1) of the electronic/positronic flux densities also applies at TeV energies and since the temperature of the positronic plasma component is lower than of the electrons, cf. Table 1, we predict a broad spectral peak of the positron fraction at around 1 TeV which is followed by exponential decay, see after (5.1) and Fig. 3.

References