Rich structure of nonthermal relativistic CMB spectral distortions from high energy particle cascades at redshifts $z\lesssim 2\times {10}^5$

It is generally assumed that for energy injection before recombination, all of the injected energy is dissipated as heat in the baryon-photon plasma, giving rise to the $y$-type, $i$-type, and $\mu$-type distortions in the cosmic microwave background (CMB) spectrum. We show that this assumption is incorrect when the energy is injected in the form of energetic (i.e., energy much greater than the background CMB temperature) particles. We evolve the electromagnetic cascades, from the injection of high energy particles, in the expanding Universe and follow the nonthermal component of CMB spectral distortions resulting from the interaction of the electromagnetic shower with the background photons, electrons, and ions. The electromagnetic shower loses a substantial fraction of its energy to the CMB spectral distortions before the energy of the particles in the shower has degraded to low enough energies that they can thermalize with the background plasma. This spectral distortion is the result of the interaction of nonthermal energetic electrons in the shower with the CMB and thus has a shape that is substantially different from the $y$-type or $i$-type distortions. The shape of the final nonthermal relativistic ($ntr$-type) CMB spectral distortion depends upon the initial energy spectrum of the injected electrons, positrons, and photons and thus has information about the energy injection mechanism, e.g., the decay or annihilation channel of the decaying or annihilating dark matter particles. The shape of the spectral distortion is also sensitive to the redshift of energy injection. Our calculations open up a new window into the energy injection at $z\lesssim 2\times {10}^5$ which is not degenerate with, and can be distinguished from the low redshift thermal $y$-type distortions.

Figure 1

Cooling times for the photons at different redshifts are compared with the Hubble time (a) $z=1000$ and (b) $z=20000$.

Figure 2

Cooling times for electrons and positrons are compared with the Hubble time.

Figure 3

Spectral distortion of the CMB after deposition of all of the electron energy for energy injection of ${10}^{-5}\times {\rho }_{\mathrm{CMB}}$, where ${\rho }_{\mathrm{CMB}}=0.26(1+z{)}^4\mathrm{eV}/{\mathrm{cm}}^3$ is the CMB energy density at injection redshift $z=20000$. We have defined dimensionless frequency $x=h\nu /kT$. The $y$-distortion curve is shown just for reference. The $y$-distortion from the energy dissipated as heat is not added to any of the curves.

Figure 4

Comptonization of $y$-distortion to $\mu$-distortion.

Figure 5

Flowchart explaining the working and interfacing of the two codes used in the calculations.

Figure 6

Comparison of spectral distortions of the CMB with varying initial electron and photon energies with Recoil-only code for one-time energy injection. Energy injected is ${10}^{-5}\times {\rho }_{\mathrm{CMB}}$, where ${\rho }_{\mathrm{CMB}}=0.26(1+z{)}^4\mathrm{eV}/{\mathrm{cm}}^3$ at $z=20000$ (a) 10 keV to 100 GeV electrons. (b) 1 MeV to 10 MeV electrons. (c) 4 MeV to 10 MeV electrons and photons. (d) 10 keV to 1 MeV electrons and photons.

Figure 7

Comparison of spectral distortions of the CMB with varying initial electron energies with Kompaneets kernel code for one-time energy injection. Energy injected is ${10}^{-5}\times {\rho }_{\mathrm{CMB}}$ at $z=20000$ (a) 10 keV to 200 keV electrons. (b) 200 keV to 2 MeV electrons. (c) 4 MeV to 10 MeV electrons. (d) 10 MeV to 100 MeV electrons.

Figure 8

Comparison of spectral distortions of the CMB with varying initial electron and photon energies with Kompaneets kernel code for one-time energy injection. Energy injected is ${10}^{-5}\times {\rho }_{\mathrm{CMB}}$ at $z=20000$ (a) 100 MeV to 1 GeV electrons. (b) 1 GeV to 100 GeV electrons. (c) 10 keV to 1 MeV electrons and photons. (d) 100 MeV to 100 GeV electrons and photons.

Figure 9

Comparison of spectral distortions of the CMB with Kompaneets kernel code for one-time injection of ${10}^{-5}\times {\rho }_{\mathrm{CMB}}$ at $z=20000$ (a) 10 MeV electrons and photons. (b) Electrons and electron-positron pairs.

Figure 10

Comparison of spectral distortions of the CMB with varying redshift for one-time energy injection of ${10}^{-5}\times {\rho }_{\mathrm{CMB}}$. Top and bottom panels are for monochromatic electrons with kinetic energy 10 and 100 keV respectively. Left and right panels are the distortions obtained with full Kompaneets kernel and Recoil-only codes respectively.

Figure 11

Comparison of spectral distortions of the CMB with varying redshift for one-time energy injection of ${10}^{-5}\times {\rho }_{\mathrm{CMB}}$. Top and bottom panels are for monochromatic electrons with kinetic energy 1 and 100 GeV respectively. Left and right panels are spectral distortions obtained with full Kompaneets kernel and Recoil-only codes respectively.

Figure 12

Comparison of spectral distortions of the CMB with Recoil-only and full Kompaneets kernel solution for energy injection of ${10}^{-5}\times {\rho }_{\mathrm{CMB}}$ for one-time energy injection.

Figure 13

Fraction of energy injected going to heat and spectral distortions. (a) Fraction of energy going to heat as a function of electron energy. (b) Fraction of energy going to heat as a function of photon energy. (c) Fraction of energy going to heat, and photons with $x\le 20$ and $x>20$ as a function of electron energy. Redshift of injection $z=20000$.

Figure 14

Dark matter energy injection profile and spectral distortions from dark matter decays. (a) Ratio of energy released at a particular redshift to average CMB energy density for decay and annihilation of dark matter. (b) Spectral distortions in the CMB as a function of injected electron kinetic energy and photon energy. $z_X=2\times {10}^4$, $f_X$ is chosen so that if all energy is dissipated as heat, then $y=1.5\times {10}^{-5}$.

Figure 15

Comparison of $ntr$-distortion for decaying dark matter (decay redshift $z_X=2\times {10}^4$) with best fit thermal distortion ($y$-type, $i$-type, $\mu$-type and temperature shift). The residuals can be considered to be the pure nonthermal part of the spectral distortion.