Dark Gamma Ray Bursts
Many theories of dark matter (DM) predict that DM particles can be captured by stars via scattering on ordinary matter. They subsequently condense into a DM core close to the center of the star and eventually annihilate. In this work, we trace DM capture and annihilation rates throughout the life of a massive star and show that this evolution culminates in an intense annihilation burst coincident with the death of the star in a core collapse supernova. The reason is that, along with the stellar interior, also its DM core heats up and contracts, so that the DM density increases rapidly during the final stages of stellar evolution. We argue that, counterintuitively, the annihilation burst is more intense if DM annihilation is a -wave process than for -wave annihilation because in the former case, more DM particles survive until the supernova. If among the DM annihilation products are particles like dark photons that can escape the exploding star and decay to Standard Model particles later, the annihilation burst results in a flash of gamma rays accompanying the supernova. For a galactic supernova, this “dark gamma ray burst” may be observable in CTA.
Introduction. A series of pioneering papers from the 1980s has established that stars are likely to capture large amounts of dark matter (DM) particles from the galactic halo [1, 2, 3, 4, 5, 6]. This happens because DM particles scattering on atomic nuclei or on previously captured DM particles inside the star may lose enough energy to become gravitationally bound to it. Subsequently, as they orbit its center, they experience additional scatterings and eventually settle down into a thermalized DM core extending over the inner regions of the star. There, they can self-annihilate, and if some of their annihilation products, in particular neutrinos, are able to leave the star, observable signals are expected from the Sun [1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37]. Indeed, some of the strongest limits on DM scattering on nuclei are obtained this way [38, 39, 40, 41]. The potential impact of light DM particles on supernova cooling rates has been studied in .
The aim of this work is to follow the evolution of the DM core of a massive () star from the formation epoch all the way through its death in a core collapse supernova. As the star burns first hydrogen and then heavier and heavier elements at its center , the density and temperature of the baryonic matter increase, and as long as DM scattering rates are sufficiently large, the DM core heats up and contracts in the same way. As we will show in this paper, this leads to strongly enhanced DM annihilation rates at the end of the stellar life cycle, and, in some DM models, to observable signals. We dub these signals “dark gamma ray bursts”, (not to be confused with the scenario from ref. ).
Here, the sum runs over the different isotopes in the star, is its total radius, and is the capture rate per volume element . The latter is proportional to the number density of nuclei of type , to the galactic DM density , and to the DM–nucleon scattering cross section .
We take the radial density profile of the star and its chemical composition from a simulation of a star by Heger et al. [46, 47, 48, 49]. depends on the distance from the Galactic Center, and we model this dependence with an Einasto profile [50, 51, 52]. For the velocity distribution of galactic DM particles, we assume a Maxwell–Boltzmann form with velocity dispersion . This number is almost independent of the location of the star, except for stars very close ( kpc) to the Galactic Center . The DM–nucleon scattering cross section can include both spin-independent (SI) contributions (for instance from scalar or vector interactions) and spin-dependent (SD) contributions (for instance from axial vector interactions): . We allow for both types of interactions and, unless otherwise noted, choose and , in accordance with constraints [54, 55, 56]. For these values, DM scattering on hydrogen is entirely dominated by , while for heavier nuclei, is usually more relevant. The reason is that most of the nuclei abundant in stars do not carry spin. The exception is , which has an unpaired proton and an unpaired neutron. Lacking numerical values for the corresponding spin matrix elements and , we approximate () by its value for () . For spin-independent scattering, nuclear form factors are modeled following ref. [5, 58]. We verified that the non-zero temperature of the nuclei in the star is negligible as long as we consider only DM masses GeV .
In addition to scattering on nuclei, DM particles may also scatter on previously captured DM particles if the underlying particle physics model admits DM self-interactions . Self-interacting DM is motivated for instance by discrepancies between observed and predicted DM distributions on small scales . Though they are not crucial for our conclusions, we allow for self-interactions mediated by a scalar or vector mediator of mass and with coupling . The DM self-capture rate is computed in analogy to the capture rate on ordinary matter . We compute the DM–DM scattering cross section in the classical limit, where (with the DM velocity) [61, 62], and in the perturbative regime, where [35, 63]. Between the two limiting regimes, we use linear interpolation. Numerically, we find that the self-capture rate can be comparable to the capture rate on nuclei, especially when DM annihilation is a -wave process. As we will see below, the number of DM particles in the stellar core is larger in this case.
After the initial DM–nucleon or DM–DM scattering that captures a DM particle by reducing its velocity below the escape velocity, the particle scatters multiple more times, eventually thermalizing with the baryonic matter. The DM core of the star can therefore be assigned a temperature , which we take equal to the baryon temperature at its average radius ,
The DM number density is parameterized as 
with the gravitational potential at radius .
Scattering of a previously captured DM particles on nuclei or other DM particles can also increase its energy and lead to evaporation [4, 6, 26, 64], but we have confirmed numerically that for the DM masses of interest to us (), this effect is negligible.
Here, is the annihilation cross section, multiplied by the relative velocity of the annihilating DM particles and averaged over their thermal velocity distribution. We have here defined the annihilation coefficient in terms of the total number of DM particles in the stellar core, .
Below, we distinguish between models in which is approximately independent of (-wave annihilation) and models where (-wave annihilation) [63, 65]. As a benchmark model for -wave (-wave) annihilation, we consider DM annihilation into dark photons (dark sector scalars ) through -channel and -channel diagrams. We choose the annihilation cross section such that at , characteristic for thermally produced Dirac DM particles [67, 68]. In the star, at temperatures much lower than in the early Universe, the annihilation cross section is then much smaller for -wave annihilation than for -wave annihilation. Therefore, more DM can accumulate inside the star in the -wave case. If or is much lighter than , also Sommerfeld enhancement needs to be taken into account [69, 70, 63].
Evolution of the DM population inside the star. The number of DM particles gravitationally bound to the star obeys the differential equation
with and from eqs. 4 and 1, respectively, and with the self-capture coefficient . The time dependence of , and arises from the time-dependent density, temperature, and chemical composition of the baryonic matter. Our input data for these quantities from ref.  consists of 52 non-linearly spaced snapshots between the beginning of hydrogen burning in the stellar core and the pre-supernova stage, 250 ms before core bounce. Between snapshots, we interpolate linearly. We assume quasi-instantaneous thermalization of captured DM particles because during most of the star’s life, the time between DM–nucleus collisions is much shorter than the time scales over which the properties of the baryonic matter change significantly. Note that DM particles captured during the last yrs do not thermalize, but their number is negligibly small. Particles that were captured earlier and are already thermalized follow the evolution of the baryonic matter even shortly before the supernova. For instance, in the hot iron core just before collapse (temperature MeV, density ), the collision time of a 100 GeV DM particle via is of order 0.001 sec.
The evolution of the stellar DM core is illustrated in fig. 1. We see that the DM core grows during hydrogen burning and then contracts along with the baryonic matter. The reason is that capture is very efficient at early times (see fig. 1 (b)), thanks to the large spin-dependent scattering rate on hydrogen. Later, when hydrogen is depleted, the capture rate drops.
The time evolution of several radius-integrated parameters of the DM core is shown in fig. 2. Panel (a) illustrates again that the capture rate rapidly decreases after core hydrogen burning ceases. This decrease is accompanied by an increase in the DM annihilation rate, caused by contraction of the DM core. (The slight increase of shortly before the supernova is due to contraction of residual hydrogen.) For -wave annihilation (red curves), the number of DM particles in the star decreases by several orders of magnitude once core hydrogen burning is over, see fig. 2 (b). For velocity-suppressed -wave annihilation (blue curves), the qualitative behavior of the annihilation rate is similar, but since its magnitude is much smaller, remains larger.
Increasing leads to a moderate decrease in . For -channel annihilation it also leads to a significant decrease of the annihilation rate because of the smaller . For -wave annihilation, the annihilation rate increases with because of more efficient Sommerfeld enhancement.
It is interesting to observe from fig. 2 (a) that, in contrast to the more widely studied DM core of the Sun, the DM core of a massive star evolves so fast that capture and annihilation never reach a stable equilibrium. is larger during hydrogen burning, while the annihilation rate becomes dominant later. The cross-over between the two occurs much later for -wave annihilation than for -wave models.
We briefly comment on the dependence of our results on the DM scattering and annihilation cross sections. If (or ) is lowered by a factor 10–100, the number of captured DM particles decreases correspondingly if DM annihilation is a -wave process, but only slightly for -wave annihilation. The reason is that already a slight decrease in leads to an appreciable decrease in the annihilation rate, which counteracts the smaller capture rate. This feedback effect is relevant mostly in the -wave case, while for -wave annihilation, it becomes appreciable only for heavy , where Sommerfeld enhancement is strong. For the same reason, increasing (decreasing) the DM annihilation rate by a factor of 10 in -wave models leads to a decrease (increase) of by a similar factor, while doing the same in -wave scenarios has only a small impact on . Annihilation cross sections larger than the canonical value are possible for example in models of multi-component DM [71, 72, 73], while annihilation cross sections smaller than the canonical value are required in scenarios with late time entropy production [74, 75, 76].
The annihilation burst. We see from fig. 2 (a) that the DM annihilation rate increases sharply just prior to the supernova. This trend continues when the iron core of the star collapses and eventually reaches nuclear density . The result is a sudden burst of DM annihilations. To compute the annihilation rate during this burst, we assume that DM particles located within the newly forming neutron star ( km)  thermalize instantaneously with the baryonic matter. This is justified because in the proto-neutron star, collisions via occur every sec. To be conservative, DM particles at larger radii are discarded at this stage. We assume for simplicity that throughout the supernova and the early cooling stage of the newborn neutron star (first seconds), its temperature is constant at MeV. Since according to eq. 3, cooling leads to further contraction of the DM distribution and thus to more efficient annihilation, this assumption is conservative. The annihilation rate during the burst is given by eq. 4, where now 
and evolves as
(see eq. 5, neglecting the capture terms). Here, is the number of DM particles in the star just prior to the supernova. Equation 7 also determines the duration of the DM annihilation burst . For -wave annihilation, varies between – sec for the DM masses and cross sections considered in this paper, while for -wave annihilation, can be as small as 10 sec or as large as sec. The reason is the weaker dependence of on the model parameters for -wave annihilation, caused by the stronger feedback of annihilation on throughout the star’s life.
Note that the annihilation rate during the burst depends on the location of the star in the galaxy through . For a star located about 1 kpc from Earth in the direction of the galactic center, it is plotted in fig. 3. We see that, with increasing , the intensity of the burst increases at first thanks to the increase in , cf. eq. 6. At GeV, the intensity decreases again because of the lower galactic DM density, because capture of particles much heavier than the nuclei in the star is kinematically inefficient, and because of Sommerfeld-enhanced annihilation already at early times. The wiggles in the blue and red curves at GeV are due to Sommerfeld enhancement.
Dark gamma ray bursts. If the DM annihilation products are SM particles, they cannot escape the newly born neutron star and are therefore unobservable. This is true even for neutrinos. Phenomenological prospects are brighter in the benchmark models discussed above, with DM annihilating to dark photons or dark scalars [12, 14, 15, 16, 35]. Dark photons can decay to SM particles through a kinetic mixing interaction of the form , where () is the QED () field strength tensor. Similarly, dark scalars could decay through a Higgs portal coupling of the form , where is the SM Higgs doublet. We assume the coupling constant or to be such that the decay length of or is much larger than the radius of the star, but much smaller than its distance from Earth. Among the secondary particles produced in the decays will be photons from final state radiation or from meson decays. These provide an observable signal, which we dub a “dark gamma ray burst”. The energy distribution of burst photons is found along the lines of ref.  and plotted in fig. 4. In the plot, we also compare to the point source sensitivities of the Fermi Large Area Telescope (Fermi-LAT) and of the future Čerenkov Telescope Array (CTA) [80, 81]. For the latter, we show a conservative estimate ( excess and signal events per bin) and a more optimistic one ( excess and signal events per bin). Taking into account that the dark gamma ray burst signal is spread over several energy bins, we find that the topmost curves for GeV are detectable at the , , and levels, respectively. For GeV and p-wave annihilation, a dark gamma ray burst at 7 kpc from the galactic center can be detectable at . We conclude that, for a sizeable range of DM masses, CTA may hope to observe dark gamma ray bursts if annihilation is a -wave process and if the supernova occurs during the 10% duty cycle of CTA. The fact that -wave annihilation is easier to observe than -wave annihilation sets dark gamma ray bursts apart from other indirect DM signatures. It is also interesting to observe that, for -wave annihilation of not too heavy DM particles, the expected signal from a star close to the Galactic Center (GC) is larger than for a nearby star because of the strong dependence of and on in -wave models.
Constraints. Models with a light force mediator or are constrained by other indirect DM searches. Due to Sommerfeld enhancement, DM annihilation is particularly strong in environments with low DM velocities, therefore the strongest constraints come from the Cosmic Microwave Background (CMB)  and from gamma ray observations of dwarf galaxies . For -wave annihilation, these constraints are shown in fig. 5. If GeV, only DM masses between GeV and GeV are allowed, while for GeV, no open parameter space remains. For -wave annihilation (not shown in fig. 5), Sommerfeld enhancement at low velocity is counteracted by smaller tree level cross sections. Therefore, in the -wave case, all DM masses are allowed if the annihilation cross section at freeze-out is at the level required to explain the DM abundance today and if GeV. For lighter mediators, bound state formation may lead to stronger limits .
Summary. We computed the properties of the DM core of a massive star throughout its life, concluding that the death of the star in a supernova explosion is accompanied by an intense DM annihilation burst. If the DM annihilation products are able to leave the exploding star and decay to SM particles later, this may lead to a burst of secondary photons (“dark gamma ray burst”), potentially observable in CTA.
Acknowledgements. We are indebted to Alexander Heger for providing the data from [46, 47, 48, 49] in machine-readable form and for discussing it with us. We are also grateful to Jatan Buch for useful discussions. The authors have received funding from the German Research Foundation (DFG) under Grant Nos. KO 4820/1–1, FOR 2239 and GRK 1581, and from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 637506, “Directions”).
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