Indirect detection of dark matter

An annihilation cross section on the order of ${\displaystyle \left\langle \sigma v\right\rangle \sim 10^{-26}cm^{3}s^{-1}}$ is consistent with the measured cosmological density of DM.[2] Thus, the objects of indirect searches are the secondary products that are expected from the annihilation of two dark matter particles.[2] When observations of those secondary products reveal cross sections on the order of the expected ${\displaystyle \sim 10^{-26}}$ (or near that order of magnitude, with some expected or known discrepancy) the source of those products may become a dark matter candidate, or an indication of dark matter (an indirect signal). In general, the DM is expected to be ${\displaystyle 5\ MeV\lesssim m_{X}\lesssim 120\ TeV}$ for the cross section given above.[2]

Note that the "J-factor" of a given potential source of dark matter interaction products is the energy spectrum integrated along the line of sight, taking only the term dependent on the distribution of the DM mass density.[1] For annihilation, that J-factor is commonly given as,

${\displaystyle J_{ann}={\frac {1}{8\pi }}\int drd\Omega \rho ({\bar {r}})^{2}}$

where ${\displaystyle \rho ({\bar {r}})}$ is the mass density of DM.[1] The J-factor is essentially a predictive measurement of a potential annihilation signal.[1] The J-factor depends on the density, so if the density of a given region is not well-known or well-defined, then it can be difficult to determine the size of the expected signal. For example, since it is difficult to distinguish and remove backgrounds near the galactic center the calculated J-factor for that region varies by several orders of magnitude, depending on the density profile used.[1]

However, if DM is unstable, it would decay and produce decay products that could be observed. Since decay only involves one DM particle (while annihilation requires two), the flux of DM decay products is proportional to the DM density, ${\displaystyle \rho _{X}}$, rather than ${\displaystyle \rho _{X}^{2}}$ in the case of annihilation.[2] There have been efforts to search for DM decay products in gamma rays, X-rays, cosmic rays, and neutrinos.[2] For unstable dark matter of mass in the GeV–TeV range, the decay products are high-energy photons.[4] These photons contribute to the extragalatic gamma ray background (EGRB). Studies of the EGRB using the Fermi satellite have revealed constraints on the lifetime of dark matter as ${\displaystyle \tau _{X}>10^{28}}$ s, for masses between about 100 GeV and 1 TeV.[4] The constraints derived from the EGRB are relatively unaffected by additional astrophysical uncertainties.[4] NuSTAR observations have been used to search for X-ray lines to further constrain decaying DM for masses in the 10 to 50 keV range.[5] For sterile neutrinos, there are several existing constraints based on X-ray limits.[5] For DM masses ${\displaystyle m_{X}\lesssim 10}$ keV and ${\displaystyle m_{X}\gtrsim 50}$ keV, there are well-defined constraints on the mixing angle, ${\displaystyle \theta }$.[5] Neutrinos have been used to derive constraints for DM masses in the range ${\displaystyle 10^{3}\lesssim m_{X}\lesssim 10^{6}}$ GeV.[6] Combined data from Fermi gamma-ray observations and IceCube neutrino observations give constraints depending on energy and defined by the criterion, ${\displaystyle N_{sig}/{\sqrt {N_{sig}+N_{bkg}}}<\delta }$, with ${\displaystyle N_{sig}}$ defined as the given signal, ${\displaystyle N_{bkg}}$ as the muon neutrino background, and ${\displaystyle \delta }$ as the Gaussian significance.[6] For low energies, the constraints improve with time as ${\displaystyle \sim {\sqrt {t}}}$.[6] For high energies, the constraints are not well-defined, as neutrino flux is no longer dominant.[6] Thus, there are constraints on the properties of decaying DM for masses ${\displaystyle m_{X}}$ ranging from keV to TeV. Additionally, in the case of decay, the signal strength (like J-factor for the case of annihilation) is dependent only on density, rather than density squared: ${\displaystyle \int \rho ({\bar {r}})drd\Omega }$.[1] For sufficiently distant sources, the signal strength can then be approximated as ${\displaystyle M/R^{2}}$, where ${\displaystyle M}$ is the source mass.[1]

In order to detect or constrain the properties of dark matter, observations of dwarf galaxies have been carried out. Limits may be placed on the annihilation cross section of WIMPs based on analysis of either gamma-rays or cosmic rays.[1] The VERITAS, MAGIC, Fermi, and H.E.S.S. telescopes are among those that have been involved in the observation of gamma-rays. The air Cherenkov telescopes (H.E.S.S., MAGIC, VERITAS) are most effective at constraining the annihilation cross section for high energies (${\displaystyle E_{\gamma }>100}$ GeV).

For energies below 100 GeV, Fermi is more effective, as this telescope is not constrained to a view of only a small portion of the sky (as the ground-based telescopes are).[1] From six years of Fermi data, which observed dwarf galaxies in the Milky Way, the DM mass is constrained to ${\displaystyle m_{X}\gtrsim 100}$ GeV (masses both this threshold are not allowed).[1] Then, combining data from Fermi and MAGIC, the upper limit of the cross section is found to be ${\displaystyle \left\langle \sigma v\right\rangle \sim 10^{-25}cm^{3}s^{-1}}$ (that is, with no uncertainties in ${\displaystyle J}$.[7] This collaboration produced constraints for DM masses in the range ${\displaystyle 10\ GeV\lesssim m_{X}\lesssim 100\ TeV}$.[7] Note that Fermi data dominates for the low mass end of the range, while MAGIC dominates for the high masses.[7]

VERITAS has been used to observe high energy gamma-rays in the range 85 GeV to 30 TeV, for the mass range ${\displaystyle 10\ GeV\lesssim m_{X}\lesssim 10\ TeV}$.[8]

Cosmic ray analyses primarily observe positrons and antiprotons. The AMS experiment is one such project, providing data on cosmic ray electrons and positrons in the 0.5 GeV to 350 GeV range.[9] AMS data allows for constraints on DM masses ${\displaystyle m_{X}\lesssim 300}$ GeV.[9] Results from AMS constrain the annihilation cross section to ${\displaystyle \left\langle \sigma v\right\rangle \lesssim 1.1\times 10^{-24}cm^{3}s^{-1}}$ for DM masses ${\displaystyle \sim 100}$ GeV (with the thermally averaged cross section noted as ${\displaystyle \left\langle \sigma v\right\rangle _{therm}\equiv 3\times 10^{-26}}$ ${\displaystyle cm^{3}s^{-1}}$).[9] The upper limit for the annihilation cross section can also be used to find a limit for the decay width of a DM particle.[9] These analyses are also subject to substantial uncertainty, particularly pertaining to the Sun's magnetic field, as well as the production cross section for antiprotons.[1]

The galactic center is hypothesized to be a source of large amounts of dark matter annihilation products.[2] However, the background at the galactic center is both bright and not yet well understood (based on the model of the Milky Way in use, the flux of annihilation products can vary by several orders of magnitude).[2] The Galactic center is a unique source of high mass dark matter, which cannot be replicated in colliders.[1] Thus, telescopes like Fermi and H.E.S.S. have observed the excess of gamma-rays coming from the galactic center, as backgrounds are lower for gamma-rays (and unknown backgrounds at the galactic center typically cause large uncertainties for dark matter searches).[1] The annihilation cross section is consistent with the expected ${\displaystyle \sim 10^{-26}cm^{3}s^{-1}}$, and thus, In the case that those excess gamma-rays are products of dark matter annihilation, they must originate from dark matter with a mass ${\displaystyle 40\ GeV\lesssim m_{X}\lesssim 70\ GeV}$.[2]

Full-sky image of gamma-rays greater than 1 GeV.

H.E.S.S., an imaging atmospheric Cherenkov telescope, has been used to observe this excess of very high energy gamma-rays emanating from the galactic center.[10] Probing energies in the range ${\displaystyle E_{\gamma }\sim 500}$ GeV to ${\displaystyle 25}$ TeV, H.E.S.S. data allowed for limits on internal bremsstrahlung processes to be determined, which then allowed for upper limits on DM annihilation flux to be defined.[10]

Overall, the galactic center is a focus for indirect searches due to its excess of gamma-rays. That excess has ${\displaystyle \left\langle \sigma v\right\rangle \sim 10^{-26}cm^{3}s^{-1}}$ which is on the order of the thermally averaged annihilation cross section, making the gamma-ray excess a potential dark matter candidate.[2]

Heavy DM has ${\displaystyle m_{X}\gg TeV}$.[1] Dark matter with mass in this regime is expected to result in high-energy photons that, through pair production, create a cascade of electrons and photons, eventually leading to low energy gamma-rays.[1] Those low-energy gamma-rays can be observed by telescopes like Fermi, and then constrain the annihilation rate accordingly.[1] Additionally, for decaying DM with masses greater than the TeV range, the lifetime is constrained to ${\displaystyle \tau _{X}\gtrsim 10^{27-28}}$ s.[1]

Contrastingly, light DM has ${\displaystyle m_{X}\ll GeV}$, and it becomes difficult to observe products for these lower masses and energies. Fermi is limited by its angular resolution, and cannot observe products below ${\displaystyle \sim 1\ GeV}$.[1] To observe products at the lower mass limit, either a low-energy gamma-ray telescope or an X-ray telescope is required.[1]

An excess of positrons (in the flux ratio of positrons to electron and positron pairs) was found by PAMELA, in observing cosmic rays.[1] Fermi and AMS-02 later confirmed this excess.[1] One possible explanation for this excess of positrons is annihilating dark matter.[1] For energies ${\displaystyle \sim 10}$ GeV to ${\displaystyle 300}$ GeV, the ratio of positrons to electron-positron pairs continues to increase, indicating that the annihilating dark matter is producing positrons (and the flux increases with the DM mass).[1] There are alternative explanations for this excess of positrons, including pulsars or supernova remnants.[1] In 2017, data from the HAWC Collaboration indicated that the increase in flux of positrons from the two nearest pulsars (Geminga and Monogem) is roughly equivalent to the excess originally observed by PAMELA.[2]

Dark matter shown in the Bullet Cluster.

In 2014, a spectral line an energy of ${\displaystyle \sim 3.5}$ keV was found in the observation of galaxy clusters.[1] Further investigation of this spectral line by Chandra and XMM-Newton failed to find such a line, and thus, there is debate about whether the spectral line is evidence of dark matter.[1] There are several explanations: (1) the source is a decaying sterile neutrino, with a mass ${\displaystyle \sim 7}$ keV (cold dark matter), and thus, is not subject to the constraints on warm dark matter. This explanation is consistent with observation of the spectral line at 3.5 keV, as expected, in both the cosmic X-ray background and the Galactic center, but inconsistent with the results from Chandra and XMM-Newton;[1] (2) the source is heavier than 3.5 keV, but has a "metastable excited state" at 3.5 keV and a decay emits a photon of that same energy;[1] (3) the DM source decays, producing a 3.5 keV axion-like particle, which could turn into a photon under some external magnetic field.[1] The actual explanation cannot yet be confirmed. Thus, the 3.5 keV line remains as evidence of a potential DM candidate.

The cosmic microwave background (CMB) can also be analyzed in order to constrain dark matter annihilation products.[2] If the number of dark matter annihilations is given as, ${\displaystyle N_{ann}={\frac {1}{2}}{\frac {\rho _{X}^{2}\left\langle \sigma v\right\rangle V_{c}}{m_{X}^{2}H}}}$

where ${\displaystyle H}$ is the expansion rate, ${\displaystyle V_{c}}$ is the comoving volume, and ${\displaystyle \left\langle \sigma v\right\rangle }$ is the averaged annihilation cross section, then the number of dark matter annihilations during both the period of matter-radiation equality and matter domination can be determined.[2] From the above equation for number of dark matter annihilations, and based on a typical dark matter mass of ${\displaystyle m_{X}=100}$ GeV, that dark matter would ionize a significant portion of hydrogen atoms (~${\displaystyle 10^{-3}}$) at the time of recombination.[2] Thus, dark matter would have a noticeable effect on the CMB, as observed today.[2]

Because the anisotropies found in the CMB are sensitive to any increase in energy, those anisotropies can be calculated under the assumption that the energy increase is due to some DM annihilation, in an effort to determine constraints on that DM annihilation. The Planck Collaboration used the relation

${\displaystyle p_{ann}\equiv f_{eff}{\frac {\left\langle \sigma v\right\rangle }{m_{X}}}}$

(where ${\displaystyle f_{eff}}$ is the energy released into the intergalactic medium by a DM annihilation process) to determine a parameter, ${\displaystyle p_{ann}}$, to constrain DM annihilations based on CMB anisotropies and polarization.[11] The Planck Collaboration found that CMB-constraints were more reliable than other methods for smaller masses (below ~10 GeV).[2] CMB-constraints are also most reliable for any DM annihilation that results in either protons or electrons (that is, excluding annihilation into neutrinos).[2]