It is well known that energetic electrons can boost the CMB photons to higher energy through inverse Compton scattering creating a distortion in the CMB blackbody spectrum. If the energy distribution of the electrons is non-relativistic and thermal, then the distortion has y-distortion shape (Zeldovich & Sunyaev 1969) (see Section 6 for further discussions on corrections to y-distortion). For relativistic electrons, with a Lorentz factor γ, a photon with energy ϵ gets boosted to γ2ϵ. In this case, the spectral distortion shape will be a function of the electron energy distribution. The intensity of the CMB spectrum per frequency is given by,
(9)$$\begin{eqnarray*}
I_{\nu }(x)=2\frac{(k_{\rm {B}}T_{\rm {CMB}})^3}{({hc})^2}\frac{x^3}{e^x-1}=I_0I(x),
\end{eqnarray*}$$where $I(x)=\frac{x^3}{e^x-1}$, I(x) is the dimensionless intensity, x is the dimensional frequency which is given by $x=\frac{E_{\gamma }}{k_{\rm {B}}T_{\rm {CMB}}}$, where Eγ is the energy of photon, kB is the Boltzmann constant, TCMB is the CMB temperature, and other symbols have usual meanings. The intensity of distorted CMB spectrum is independent of redshift for a given population of electrons. The CMB distortion in the optically thin limit can be written as (Zeldovich & Sunyaev 1969; Birkinshaw 1999),
(10)$$\begin{eqnarray*}
\Delta I(x)=(j(x)-i(x))\tau ,
\end{eqnarray*}$$where j(x) is the spectral intensity of photons at frequency x after being upscattered while i(x) is the intensity of photons at frequency x before upscattering, τ = σT∫nedl where ne is the electron number density and dl is the line-of-sight width of this electron population. i(x) is non-zero over the range of x where the CMB photons are present. Equation (10) can be recast to include a y-parameter as ΔI(x) = yg(x), where $y=\frac{\sigma _{\rm {T}}}{m_{\rm {e}}\rm {c}^2}\int n_{\rm {e}}k_{\rm {B}}\overset{\sim }{T_{\rm {e}}}\mathrm{ d}l$ with $k_{\rm {B}}\overset{\sim }{T_{\rm {e}}}=\frac{P_{\rm {e}}}{n_{\rm {e}}}$. Here $\overset{\sim }{T_{\rm {e}}}$ refers to a fictitious temperature scale for a non-thermal distribution to make the equations look suggestively similar to thermal SZ (more details can be found in Enßlin & Kaiser 2000). In order to distinguish non-thermal spectral distortion shape from y-type distortion (the well-known thermal $\rm {SZ}$ effect), we will refer to non-thermal distortion amplitude as yNT, such that,
(11)$$\begin{eqnarray*}
\Delta I_{\rm {NT}}(x)=y_{\rm {NT}}g_{\rm {NT}}(x),
\end{eqnarray*}$$where gNT(x) is the spectral distortion function resulting from scattering of the CMB in a non-thermal population of clusters. We use the formalism of Enßlin & Kaiser (2000) to compute gNT(x) and we refer the reader to the paper for details. The pressure for a distribution of relativistic electrons is given by,
(12)$$\begin{eqnarray*}
P_{\rm {e}}=n_{\rm {e}}\int \mathrm{ d}p f_{\rm {e}}(p)\frac{1}{3}p{v}(p)m_{\rm {e}}c,
\end{eqnarray*}$$where fe(p) is the normalized electron spectrum i.e. ∫fe(p)dp = 1 with electron dimension-less momentum $p=\sqrt{(\gamma ^2-1)}$, v = βc, where γ is the Lorentz factor and β is the boost factor of energetic electrons. The number of CMB photons which get upscattered from energy x′ to x is given by,
(13)$$\begin{eqnarray*}
N(x^{\prime }-\gt x)= P(t,p)\times 2\frac{(k_{\rm {B}}T_{\rm {CMB}})^2}{({hc})^2}\frac{x^{\prime 2}\mathrm{ d}x^{\prime }}{e^{x^{\prime }}-1},
\end{eqnarray*}$$where P(t, p) is the kernel of the inverse Compton scattering which captures the kinematics of photon scattering with the electrons with electron energy p, $t=\frac{x}{x^{\prime }}$. The number of CMB photons within energy x′ and x′ + dx′ is $2\frac{(k_{\rm {B}}T_{\rm {CMB}})^2}{({hc})^2}\frac{x^{\prime 2}dx^{\prime }}{e^{x^{\prime }}-1}$ and ∫dtP(t, p) = 1, which conserves the number of photons. The formula for P(t, p) is given by (Enßlin & Kaiser 2000),
(14)$$\begin{eqnarray*}
P(t;p)&amp;=&amp;\frac{-3\left|(1-t)\right|}{32p^6t}[1+(10+8p^2+4p^4)t+t^2]+\frac{3(1+t)}{8p^5}\nonumber\\
&amp;&amp;\times \left[\frac{3+3p^2+p^4}{\sqrt{1+p^2}}-\frac{3+2p^2}{2p}(2\operatorname{arcsinh}{p}-\left| \ln {t}\right|)\right],\nonumber\\
\end{eqnarray*}$$The spectral intensity of upscattered photons per frequency, in frequency bin x and x + Δx, is given by,
(15)$$\begin{eqnarray*}
j(x)=\frac{\int \int f_{\rm {e}}(p)\mathrm{ d}p P(t,p)\frac{x^{\prime 2}\mathrm{ d}x^{\prime }}{e^{x^{\prime }}-1}x}{\Delta x}.
\end{eqnarray*}$$
Citation: Enßlin & Kaiser 2000
Citation function: Uses
Citation: Enßlin & Kaiser (2000)
Citation function: Uses; Uses
Citation: Enßlin & Kaiser 2000
Citation function: Uses; Uses