Figure 1

The quantity $E(l_N|\mathrm{d}E/\mathrm{d}x{|}_c{)}^{-1}$ as a function of nucleon energy $E$, where $l_N$ is the nucleon mean free path and $|\mathrm{d}E/\mathrm{d}x{|}_c$ is the nucleon energy loss per unit path length due to “continuous” energy losses such as multiple Coulomb scattering. When $E(l_N|\mathrm{d}E/\mathrm{d}x{|}_c{)}^{-1}\gtrsim 1$ nucleons predominantly lose energy due to nucleon-nucleon scattering and spallation processes, whereas in the opposite case nucleons lose their energy predominantly continuously via multiple electromagnetic scatterings on $e^{\pm }$ and photons [cf. Eq. (b1)]. Only in the former limit nuclear cascades occur. The quantity is shown for neutrons [dashed (green) line] at temperatures $T=90$ and 30 keV and protons [solid (red) line] at temperatures $T=30$, 10, 1 and 0.01 keV as labeled in the figure.Reuse & Permissions

Figure 2

Probability $P_{{}^6\mathrm{Li}}$ of energetic ${}^3\mathrm{H}$ nuclei [solid (red) line] and energetic ${}^3\mathrm{He}$ [dotted (blue) line] of initial energy $E$ to fuse on ambient ${}^4\mathrm{He}$ nuclei to form ${}^6\mathrm{Li}$ via the reactions ${}^3\mathrm{H}(\alpha ,n){}^6\mathrm{Li}$ and ${}^3\mathrm{He}(\alpha ,p){}^6\mathrm{Li}$, respectively. The figure shows this probability for ${}^3\mathrm{H}$ nuclei at temperatures $T=5$, 10, 5, 1 and 0.1 keV and for ${}^3\mathrm{He}$ nuclei at temperature $T=0.1\mathrm{keV}$, respectively. Survival of the freshly formed ${}^6\mathrm{Li}$ nuclei against thermal nuclear reactions is also taken into account as evident by the comparatively low $P_{{}^6\mathrm{Li}}$ at $T=15\mathrm{keV}$. The figure illustrates that $P_{{}^6\mathrm{Li}}$ dramatically increases at $T\sim 5-10\mathrm{keV}$ due to a decrease of the efficiency of Coulomb stopping in this narrow temperature interval (see text).Reuse & Permissions

Figure 3

Yields of (from top to bottom) neutrons (red), deuterium nuclei (green), tritium nuclei (blue), and ${}^3\mathrm{He}$ nuclei (magenta) as a function of cosmic temperature $T$ per hadronically decaying particle $X\to q\overline{q}$ of mass $M_x=1\mathrm{TeV}$, where $q$ denotes a quark. Note that initially after hadronization of the $q\mathrm{\text{−}}\overline{q}$ state on average only $1.56$ neutrons result. The remainder of the created neutrons at lower temperatures $T\lesssim 90\mathrm{keV}$ are resulting from the thermalization of the injected neutrons (and protons) due to inelastic nucleon-nucleon scattering processes and ${}^4\mathrm{He}$ spallation processes. Similarly, all the ${}^2\mathrm{H}$, ${}^3\mathrm{H}$, and ${}^3\mathrm{He}$ nuclei are due to ${}^4\mathrm{He}$ spallation processes and $np$ nonthermal fusion reactions (for ${}^2\mathrm{H}$) induced by the thermalization of the injected energetic nucleons. The figure does note include the electromagnetic yields (cf. Fig. 4) due to photodisintegration which is inevitable even for a hadronic decay since approximately 45% of the rest mass energy of the decaying particle is converted into energetic $\gamma$ rays and energetic $e^{\pm }$ after hadronization and pion decays.Reuse & Permissions

Figure 4

Yields of (from top to bottom) ${}^3\mathrm{He}$ (blue), ${}^3\mathrm{H}$ (green), ${}^2\mathrm{H}$ (red), and ${}^6\mathrm{Li}$ (magenta) nuclei per TeV of electromagnetically interacting energy injected (in form of energetic $\gamma$ rays and energetic $e^{\pm }$) due to photodisintegration reactions (${}^2\mathrm{H}$, ${}^3\mathrm{H}$, ${}^3\mathrm{He}$, and ${}^6\mathrm{Li}$) and fusion reactions (${}^6\mathrm{Li}$) as a function of cosmic temperature.Reuse & Permissions

Figure 5

Number of destroyed (from top to bottom) ${}^4\mathrm{He}$ (blue), ${}^2\mathrm{H}$ (red), ${}^3\mathrm{He}$ (green), and ${}^7\mathrm{Li}$ (magenta) (${}^7\mathrm{Be}$ light blue) nuclei per TeV of electromagnetically interacting energy injected into the primordial plasma at temperature $T$. For the purpose of illustration we have taken the ${}^7\mathrm{Li}$ and ${}^7\mathrm{Be}$ abundances equal at ${}^7\mathrm{Li}/\mathrm{H}={}^7\mathrm{Be}/\mathrm{H}\approx 4.34\times {10}^{-10}$. In reality it is the sum of both isotopes which is synthesized at $({}^7\mathrm{Li}+{}^7\mathrm{Be})/\mathrm{H}\approx 4.34\times {10}^{-10}$ in a SBBN scenario at ${\Omega }_b{h}^2\approx 0.02233$ with ${}^7\mathrm{Be}$ being converted to ${}^7\mathrm{Li}$ by electron capture at cosmic temperatures $T\approx 0.1–1\mathrm{keV}$.Reuse & Permissions

Figure 6

Conservative BBN constraints on the abundance of relic decaying neutral particles as a function of their life time for an $M_x=1\mathrm{TeV}$ particle with hadronic branching ratio $B_h=1$. Limits are given on the contribution the decaying particles would have made to the present critical density, ${\Omega }_X{h}^2$ (with $h$ the Hubble parameter), if they would have not decayed. For a conversion to constraints on, for example $M_X{n}_X/n_{\gamma }$ the reader is referred to Appendix . The shaded regions are excluded and correspond to the constraints imposed by the observationally inferred upper limit on ${}^4\mathrm{He}$ (orange) [Eq. (1)], upper limit on ${}^2\mathrm{H}$ (blue) [Eq. (2)], upper limit on ${}^3\mathrm{He}/{}^2\mathrm{H}$ (red) [Eq. (3)], and lower limit on ${}^7\mathrm{Li}$ (light blue) [Eq. (4)]. Conservative constraints derived from ${}^6\mathrm{Li}/{}^7\mathrm{Li}$ [Eq. (6)] are shown by the green region. The region indicated by yellow violates the less conservative ${}^6\mathrm{Li}/{}^7\mathrm{Li}$ [Eq. (5)] constraint but should not be considered ruled out. Rather, this region may be cosmologically interesting as a putative source of ${}^6\mathrm{Li}$ in low-metallicity stars by relic decaying particles.Reuse & Permissions

Figure 7

As Fig. 6 but for hadronic branching ratio $B_h=3.333\times {10}^{-2}$.Reuse & Permissions

Figure 8

As Fig. 6 but for hadronic branching ratio $B_h={10}^{-3}$. The region excluded by the lower limit on ${}^2\mathrm{H}/\mathrm{H}$ [Eq. (2)] (magenta) is also apparent.Reuse & Permissions

Figure 9

Constraints on the abundance of decaying neutral particles as a function of life time for an $M_X=1\mathrm{TeV}$ particle with varying hadronic branching ratios. Constraints are shown for hadronic branching ratios (from bottom to top) ${\log }_{10}{B}_h=0$, $-0.5$, $-1$, $-1.5$, $-2$, $-2.5$, $-3$, $-3.5$, $-4$, $-4.5$, and $-5$, respectively, as labeled. The case $B_h=0$ is also shown. The labels always correspond to the solid (red) line above it. For each $B_h$ the region above this solid (red) line is ruled out when conservative constraints are applied [i.e. Eqs. (1, 2, 3, 4, 6)]. When the ${}^6\mathrm{Li}/{}^7\mathrm{Li}$ constraint Eq. (6) is replaced by the less conservative Eq. (5) the dotted (blue) constraint lines result. These dotted lines coincide for small and large ${\tau }_X$ with the solid (red) lines.Reuse & Permissions

Figure 10

As Fig. 9 but for $M_X=100\mathrm{GeV}$.Reuse & Permissions

Figure 11

Compilation of the available ${}^4\mathrm{He}$ spallation data as a function of nucleon kinetic energy (with ${}^4\mathrm{He}$ at rest) for the spallation reactions 74, 75, 76, and 78. The lines joining independent data points (74, solid line; 75, long dashed line; 76, short dashed line; 78, dotted line) represent the fit used in the present analysis. Refer to the text for references.Reuse & Permissions