The Energy Spectrum of Primary Cosmic-Ray Electrons in Clusters of Galaxies and Inverse Compton Emission

Models for the evolution of the integrated energy spectrum of primary cosmic-ray electrons in clusters of galaxies have been calculated, including the effects of losses due to inverse Compton (IC), synchrotron and bremsstrahlung emission, and Coulomb losses to the intracluster medium (ICM). The combined timescale for these losses reaches a maximum of ~3 × 109 yr for electrons with Lorentz factor γ ~ 300. A variety of models for the time evolution of particle injection are considered, including models in which the electrons are all produced at a single epoch in the past, models with continuous particle acceleration, and combinations of these. Analytical solutions are given for a number of limiting cases. Numerical solutions are given for more general cases. Only a cluster in which there has been a substantial injection of relativistic electrons since z ≲ 1 will have any significant population of primary cosmic-ray electrons at present. For models in which all of the electrons were injected in the past, there is a high-energy cutoff γmax to the present electron distribution. At low energies, at which Coulomb losses dominate, the electron distribution function N(γ) tends to a constant value, independent of γ. On the other hand, if electrons are being accelerated at present, the energy distribution at high and low energies approaches steady state. If the electrons are injected with a power-law distribution, the steady state distribution is one power steeper at high energies and one power flatter at low energies. In models with large initial populations of particles, but also with significant rates of current particle injection, the electron distributions are a simple combination the behavior of the initial population models and the steady injection models. There is a steep drop in the electron population at γmax, but higher energy electrons are present at a rate determined by the current rate of particle injection. Increasing the ICM thermal gas density decreases the number of low-energy electrons (γ ≲ 100). If the magnetic field is greater than generally expected, (B ≳ 3 μG), synchrotron losses will reduce the number of high-energy electrons. A significant population of electrons with γ ~ 300, associated with the peak in the particle loss time, is a generic feature of the models, as long as there has been significant particle injection since z ≲ 1. The IC and synchrotron emission from these models was calculated. In models with steady particle injection with a power-law exponent p, the IC spectra relax into a steady state form. At low energies, the spectrum is a power law with α ≈ -0.15, while at high energies α ≈ -1.15. These two power laws meet at a knee at ν ~ 3 × 1016 Hz. In models with no current particle injection, the cutoff in the electron distribution at high energies (γ ≥ γmax) results in a rapid drop in the IC spectrum at high frequencies. In models in which the current rate of particle injection provides a small but significant fraction of the total electron energy, the spectra show an extended hump at low frequencies (ν ≲ 1017 Hz), with a rapid fall off above ν ~ 1016 Hz. However, they also have an extended hard tail of emission at high frequencies, which has a power-law spectrum with a spectral index of α ≈ -1.15. In the models, EUV and soft X-ray emission are nearly ubiquitous. This emission is produced by electrons with γ ~ 300, which have the longest loss times. The spectra are predicted to drop rapidly in going from the EUV to the X-ray band. The IC emission also extends down the UV, optical, and IR bands, with a fairly flat spectrum (-0.6 ≳ α ≳ +0.3). At hard X-ray energies ≳20 keV, IC emission should become observable against the background of thermal X-ray emission. Such hard X-ray (HXR) emission is due to high-energy electrons (γ ~ 104). The same electrons will produce diffuse radio emission (cluster radio halos) via synchrotron emission. Because of the short loss times of these particles, HXR and diffuse radio emission are expected only in clusters that have current (or very recent) particle acceleration. Assuming that the electrons are accelerated in ICM shocks, one would expect diffuse HXR/radio emission only in clusters that are currently undergoing large mergers. The luminosity of HXR emission is primarily determined by the current rates of particle acceleration in the clusters. The spectra in most models with significant HXR or radio emission are approximately power laws with α ≈ -1.1. The IC spectra from the EUV to HXR are not generally fit by a single power law. Instead, there is a rapid falloff from EUV to X-ray energies, with a power-law tail extending into the HXR band.