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If Dark Matter (DM) is a weakly interacting massive particle (WIMP) and it is a thermal relic, then its self-annihilation cross-section can be determined by its relic density today through the Boltzmann equation (e.g.

Depending on the model, DM particles can annihilate into gauge bosons, charged leptons, neutrinos, hadrons, or more exotic states. These annihilation products then decay or interact with the photon-baryon fluid and produce electrons, positrons, protons, photons, and neutrinos. Neutrinos do not further interact with the photon-baryon fluid, but interact gravitationally and their effects can be observed through the lensing of the CMB. A proton's energy deposition to the fluid is inefficient due to their high penetration length. Hence, the main channels for energy injection are through electrons, positrons, and photons. At high energies, positrons lose energy through the same mechanisms as electrons. High energy electrons lose energy through inverse Compton scattering of the CMB photons, while low energy electrons lose energy through collisional heating, excitation, and ionization. Photons lose energy through photoionization, Compton scattering, pair-production off nuclei and atoms, photon-photon scattering, and pair-production through CMB photons. The rate of energy release per unit volume by a self-annihilating DM particle is given by (e.g.

equation 1, energy release by self-annihilating DM particle

Energy release eqn.png

where rho_c is the critical density of the universe today, Omega_{DM} is the DM density, f is the energy deposition efficiency factor, <sigma v> is the velocity-weighted annihilation cross section, and m_{DM} is the mass of the DM particle, assumed to be a Majorana particle.

Due to these processes, the photon-baryon plasma is heated and the ionization fraction is modified. This leads to modifications of the recombination history (e.g. and, consequently, of the CMB spectra. For details on the energy injection processes, see The energy injection due to DM annihilation broadens the surface of last scattering, but does not slow recombination ( The extra scattering of photons at redshift z ~< 1000 damps power in the CMB temperature and polarization fluctuations at small angular scales (ell ~>100), and adds power in the E-mode polarization signal at large scales (ell < 100). The 'screening' effect at ell ~> 100 goes as an exponential suppression factor Cl --> e^{-2\Delta\tau} Cl, where \Delta\tau is the excess optical depth due to dark-matter annihilation. This exponential factor is partially degenerate with the amplitude of the scalar perturbation power spectrum A_s, and polarization data helps to break this partial degeneracy.

The CMB is sensitive to the parameter p_{ann}, defined as

equation 2, p_ann

Pann eqn.png

For a thermal s-wave annihilation cross-section <sigma v> = 3e-26 cm^3/s/GeV, and a energy deposition efficiency factor f=1, one can extract the 95% CL upper limit on DM particle masses that can introduce the excess optical death for ell ~> 100. For example, the 95% CL upper mass limit of the DM particle is 170 GeV for 3e-26 cm^3/s/GeV thermal cross-section in the nominal 1uK-arcmin noise in T, 3' beam, fsky=0.4 case.

(Text largely taken and modified from, section 5)


In the following, we present constraints on p_ann for varying beam (fixed noise), varying noise (fixed beam), and varying fsky (fixed beam, and fixed 'effort').

The inputs are as defined in ForecastingStep1, specifically I used:

  • the lensed TT, EE, TE spectra (no Cldd)
  • ell range = [30, 3000] in TT and TE, ell range = [30, 5000] in EE (unless otherwise stated)
  • Planck prior ell range = [2, 2500] in TT (fsky=0.8), and ell range = [30, 2500] for TE/EE (fsky=0.2 for varying noise/beam cases and fsky = 0.6-fsky_S4 for varying fsky case)
  • Planck only configuration constraints: sigma(p_ann) = 0.014; Planck published 95% CL limit p_ann < 0.04 [cm^3 s^-1 GeV^-1]

Figure 1: sigma(p_ann), varying beam, fixed noise at 1uK-arcmin, fsky=0.4

Pann varybeam.png
  • a lot of the signal-to-noise comes from ell < 1000 in TT, and EE and TE are there to break degeneracies with A_s. So the beam doesn't matter as much.

Figure 2: sigma(p_ann), varying noise, fixed 3' beam. fsky=0.4

Pann varynoise.png

Figure 3: sigma(p_ann), varying sky, beam=3'

Pann fixed effort.png
  • As is found in, for the sensitivity level we are considering for S4, p_ann is close to sample variance limited and benefits from observing more sky given a fixed number of detector/effort in building an instrument.