4 The total SZ effect produced by a combination of thermal and non-thermal populations

For galaxy clusters which contain two (or more) different electronic populations, like radio-halo clusters which contain a thermal population emitting X-rays and a non-thermal population producing the radio halo emission, one has to evaluate the spectral distortion produced simultaneously by both populations on the CMB radiation. Such a derivation has been not done so far. We assume here that the two populations are independent and that no change on the thermal population is induced by the non-thermal electrons. This condition is reasonable for non-thermal electrons with energies $\ \raise -2.truept\hbox{\rlap{\hbox{$\sim$ }}\raise5.truept \hbox{$>$ }\ }$150 MeV which do not appreciably loose energy through Coulomb collisions in the ICM (see, e.g., Blasi & Colafrancesco 1999; Colafrancesco & Mele 2001) and hence do not produce a substantial heating of the IC gas. Electrons with $E_{\rm e} \ \raise -2.truept\hbox{\rlap{\hbox{$\sim$ }}\raise5.truept \hbox{$>$ }\ }150$ MeV loose energy mainly through ICS and synchrotron losses. The radio halo emission in galaxy clusters is, in fact, produced by electrons with energy $E_{\rm e} \approx 16.4 ~{\rm GeV}~ ({\rm B/\mu G})^{-1/2} (\nu_{\rm r}/{\rm GHz})^{1/2}$ (see, e.g., Colafrancesco & Mele 2001) which yield $E_{\rm e} \sim 1.6$-52 GeV for a typical IC magnetic field at the level of $B = 1~\mu$G at the typical frequencies, $\nu_{\rm r} \sim 10^{-2}$-10 GHz, at which radio halo spectra are observed. For such energies the non-thermal electrons do not strongly affect the thermal IC gas. However, we discuss in the following the impact of different non-thermal population spectra on both the total SZ effect and on the cluster IC gas.

Under the hypothesis of independence between the thermal and non-thermal electron populations, the probability that a CMB photon is scattered by an electron of population A (say thermal) is not affected by the fact that the same photon has been scattered by an electron of population B (say non-thermal). Thus, at first order in $\tau$ we expect that the total SZ effect is given by the sum of the first order SZ effects due to each single population. At higher order approximation in $\tau$, however, one has to consider the effect of repeated scattering (see Sect.2). It is intuitive to think that in this last case the total SZ effect is not simply given by the sum of the separate SZ effects at higher orders in $\tau$ and the terms describing the cross-scattering between CMB photons and the electrons of populations A and B have to be taken into account. In the following, we will derive in detail the general expression for the total SZ distortion caused by a combination of two electronic populations.

Let us consider two electron populations with momentum distributions $f_{\rm A}(p)$and $f_{\rm B}(p)$ with optical depths $\tau_{\rm A}$ and $\tau_{\rm B}$, respectively. We also assume that the two electron distributions are both normalized to 1. The total distribution of the electron momenta is

$$f_{\rm e}(p)=c_{\rm A}f_{\rm A}(p)+c_{\rm B}f_{\rm B}(p) ,$$ (52)

with appropriate normalization coefficients $c_{\rm A}$ and $c_{\rm B}$. Since the total distribution $f_{\rm e}(p)$ is also normalized to 1, the coefficients $c_{\rm A}$ and $c_{\rm B}$are related by the condition $c_{\rm A}+c_{\rm B}=1$. The coefficients $c_{\rm A}$ and $c_{\rm B}$ yield the weight of each population with respect to the other and thus their ratio is given by

$$\frac{c_{\rm A}}{c_{\rm B}}=\frac{\tau_{\rm A}}{\tau_{\rm B}}\cdot$$ (53)

Such a condition, together with the normalization condition, yields:

$$c_{\rm A}=\frac{\tau_{\rm A}}{\tau_{\rm A}+\tau_{\rm B}} \qquad c_{\rm B}=\frac{\tau_{\rm B}}{\tau_{\rm A}+\tau_{\rm B}}\cdot$$ (54)

The total distribution $f_{\rm e}(p)$ has an optical depth given by

$$\tau=\tau_{\rm A} + \tau_{\rm B}.$$ (55)

In fact, the probability that a CMB photon can suffer n scatterings by the electrons of each distribution is given by the sum of all the possible combinations of the probability to suffer $n_{\rm A}$ scatterings from the electrons of the distribution $f_{\rm A}$ and the probability to suffer $n_{\rm B}$ scattering from the electrons of the distribution $f_{\rm B}$, with $n_{\rm A}+n_{\rm B}=n_{\rm e}$. Since the two distributions are independent, this probability is given by the product of the two separate probabilities:

$$\sum_{n_{\rm A}+n_{\rm B}=n}p_{n_{\rm A}}\cdot p_{n_{\rm B}}=\sum... ...m A}!}\cdot \frac{{\rm e}^{-\tau_{\rm B}} \tau_{\rm B}^{n_{\rm B}}}{n_{\rm B}!}$$ p_n =

$$\sum_{n_{\rm A}+n_{\rm B}=n} \frac{{\rm e}^{-(\tau_{\rm A}+\tau_{... ...)}\tau_{\rm A}^{n_{\rm A}}\tau_{\rm B}^{n-n_{\rm A}}}{n_{\rm A}!(n-n_{\rm A})!}$$ =

$${\rm e}^{-(\tau_{\rm A}+\tau_{\rm B})}\sum_{n_{\rm A}=0}^n \frac{... ...n!}{n_{\rm A}!(n-n_{\rm A})!}\tau_{\rm A}^{n_{\rm A}}\tau_{\rm B}^{n-n_{\rm A}}$$ =

$$\frac{{\rm e}^{-(\tau_{\rm A}+\tau_{\rm B})}(\tau_{\rm A}+\tau_{\rm B})^n}{n!} \equiv\frac{{\rm e}^{-\tau}\tau^n}{n!}$$ = (56)

(we assume here a Poisson probability distribution). This means that the electrons of the distribution $f_{\rm B}$simply add to the electrons of the distribution $f_{\rm A}$. In fact, from the definition of optical depth, one gets:

$$\tau=\sigma_{\rm T} \int n_{\rm e} {\rm d}\ell\equiv\tau_{\rm A}+\tau_{\rm B}=\sigma_{\rm T} \int (n_{\rm A}+n_{\rm B}) {\rm d}\ell.$$ (57)

Figure 14: The spectral distortion in units of $2(k_{\rm B} T_0)^3/(hc)^2$ for a non-thermal population given by Eq. (34) computed for p1= 0.5 (solid), 1 (dashes), 10 (dotted) and 100 (dot-dashes).

The exact expression for the spectral distortion produced by two electron populations can be evaluated from Eqs. (19)-(21), using the appropriate expression for P₁(s). The general expression for the distribution P₁(s) is given by:

$$P_1(s)=\int_0^\infty {\rm d}p f_{\rm e}(p) P_{\rm s}(s;p).$$ (58)

Inserting Eq. (52) in Eq. (58) one obtains the function P₁(s)for two populations as a function of the distributions P₁(s) of each single population:

 

$$c_{\rm A} \int_0^\infty {\rm d}p f_{\rm A}(p) P_{\rm s}(s;p)+ c_{\rm B} \int_0^\infty {\rm d}p f_{\rm B}(p) P_{\rm s}(s;p)$$ P₁(s) =

$$\equiv c_{\rm A} P_{\rm 1A}(s) + c_{\rm B} P_{\rm 1B}(s).$$ (59)

The Fourier transform of Eq. (59) is

$$\tilde{P}_1(k)=c_{\rm A}\tilde{P}_{\rm 1A}(k)+c_{\rm B}\tilde{P}_{\rm 1B}(k)$$ (60)

and from this expression one obtains the Fourier transform of the total redistribution function:

$$\tilde{P}(k) {\rm e}^{-\tau[1-\tilde{P}_1(k)]}$$ =

$${\rm e}^{-(\tau_{\rm A}+\tau_{\rm B})\left[1- \frac{\tau_{\rm A}}... ...)- \frac{\tau_{\rm B}}{\tau_{\rm A}+\tau_{\rm B}}\tilde{P}_{\rm 1B}(k) \right]}$$ =

$${\rm e}^{-\tau_{\rm A}[1-\tilde{P}_{\rm 1A}(k)]-\tau_{\rm B}[1-\tilde{P}_{\rm 1B}(k)]}$$ =

$$\tilde{P}_{\rm A}(k) \tilde{P}_{\rm B}(k).$$ = (61)

The exact total redistribution function $P_{\rm tot}(s)$ is given by the convolution of the redistribution functions of the separate electron populations:

$$P_{\rm tot}(s)=P_{\rm A}(s)\otimes P_{\rm B}(s).$$ (62)

Thus, the exact spectral distortion produced by two electron populations on the CMB radiation is given by:

$$I_{\rm tot}(x)=\int_{-\infty}^{+\infty} I_0(x{\rm e}^{-s}) P_{\rm tot}(s)~ {\rm d}s,$$ (63)

in terms of the exact total redistribution function $P_{\rm tot}(s)$ given in Eq. (62). We also give in the Appendix the analytic expressions for the approximations at first and second order in $\tau$ of the total spectral distortion $I_{\rm tot}(x)$.

   
4.1 The function $\tilde{{\mathsfsl g}}$(x) for a combination of two electron populations

The total SZ effect produced by the combination of two electron populations can be written again in terms of a Comptonization parameter $y_{\rm tot}$ and of a spectral function $\tilde{g}(x)$, in the form of Eq. (22). For two electron populations, we can write the average temperature in Eq. (46) as:

$$% \langle k_{\rm B} T_{\rm e} \rangle \int_0^\infty {\rm d}p f_{\rm e}(p) \frac{1}{3} p v(p) m_{\rm e} c$$ =

$$\int_0^\infty {\rm d}p \left[c_{\rm A}f_{\rm A}(p)+c_{\rm B}f_{\rm B}(p) \right] \frac{1}{3} p v(p) m_{\rm e} c$$ =

$$c_{\rm A} \langle k_{\rm B} T_{\rm e} \rangle_{\rm A} +c_{\rm B} \langle k_{\rm B} T_{\rm e} \rangle_{\rm B}.$$ = (64)

The Comptonization parameter $y_{\rm tot}$ can be evaluated using Eq. (49) as:

$$% y_{\rm tot} \frac{1}{m_{\rm e}c^2}\left[c_{\rm A} \langle k_{\rm B} T_{\rm e}... ... \langle k_{\rm B} T_{\rm e} \rangle_{\rm B}\right] (\tau_{\rm A}+\tau_{\rm B})$$ =

$$\frac{1}{m_{\rm e}c^2}\left[\tau_{\rm A} \langle k_{\rm B} T_{\rm... ... B} \langle k_{\rm B} T_{\rm e} \rangle_{\rm B}\right] = y_{\rm A} + y_{\rm B}.$$ = (65)

At first order in $\tau$, the function $\tilde{g}(x)$ for two populations can be expressed as

$$% \tilde{g}(x) \frac{\Delta i(x)}{y_{\rm tot}}= \frac{\tau_{\rm A} [j_{\rm 1A}(x)-j_0(x)] + \tau_{\rm B} [j_{\rm 1B}(x)-j_0(x)]}{y_{\rm A}+y_{\rm B}}$$ =

$$\frac{y_{\rm A} \tilde{g}_{\rm A}(x)+ y_{\rm B} \tilde{g}_{\rm B}(x)}{y_{\rm A}+y_{\rm B}} ,$$ = (66)

in terms of the functions $\tilde{g}(x)$ for each single electron population. We notice that in this case the function $\tilde{g}(x)$ depends from the Comptonization parameters, and hence from the optical depths, of the two electron populations even at the first order approximation in $\tau$. Moreover, at higher orders in $\tau$, the function $\tilde{g}(x)$ cannot be expressed simply in terms of the spectral functions of the single populations because in the expression for $\Delta i(x)$ (see Eq. (A.10)), cross-correlation terms appear.

   
4.2 The SZ effect produced by a combination of a thermal plus a non-thermal populations

Here we apply the previous derivation of the total SZ effect to a galaxy cluster which contains both a termal and a non-thermal electron populations. In the following computation of the total SZ effect for a Coma-like cluster we choose a thermal population with the following parameters: $k_{\rm B} T_{\rm e}= 8.5$ keV, $n_{\rm e,th}\simeq 3\times 10^{-3}$ cm^-3 and a cluster radius of 1  h₅₀^-1 Mpc. With these parameters the optical depth of the thermal population is

$$\tau_{\rm th}\simeq 6\times 10^{-3} \left[\frac{n_{\rm e,th}}... ...3}}\right] \left[ \frac{\ell}{1~h_{50}^{-1}~{\rm Mpc}}\right],$$ (67)

and its pressure is:

$$P_{\rm th}\simeq 2.55\times10^{-2} ~{\rm keV ~cm}^{-3} \left[... ... \left[ \frac{k_{\rm B} T_{\rm e}}{8.5~ {\rm keV}}\right]\cdot$$ (68)

As for the spectrum of the non-thermal population we consider both the phenomenological cases of a single and double power-law populations (see Eq. (32) and Eq. (34) respectively). The parameters we consider are specifically: $\alpha =2.5$ and $p_2\rightarrow \infty$ for the single power-law population and $\alpha _1=0.5$, $\alpha _2=2.5$, with $p_{\rm cr}=400$ and $p_2\rightarrow \infty$ for the double power-law population. We consider p₁ as a free parameter in both cases. The relativistic electron density has been normalized at $n_{\rm e,rel}( \tilde{p}_1)=10^{-6}$ cm^-3 for $\tilde{p_1}=100$ in both cases. The final SZ effect can be re-scaled simply to different values of $n_{\rm e,rel}$ as discussed in Sect. 3.

The amplitude of the non-thermal contribution to the total SZ effect increases with decreasing value of p₁. For the case of a single non-thermal power-law population it becomes appreciable for $p_1 \ \raise -2.truept\hbox{\rlap{\hbox{$\sim$ }}\raise5.truept \hbox{$<$ }\ }10$ while for $p_1 \ \raise -2.truept\hbox{\rlap{\hbox{$\sim$ }}\raise5.truept \hbox{$>$ }\ }100$ the total SZ effect is indistinguishable from the pure thermal effect. In fact, for $p_1 \ \raise -2.truept\hbox{\rlap{\hbox{$\sim$ }}\raise5.truept \hbox{$>$ }\ }10$the function P₁(s) is nearly coincident with that of a single thermal population as shown in Fig. 15, while for p₁ <10 the function P₁(s) is more a-symmetric towards large and positive values of s. The peaks of the non-thermal distributions at large s are present in the total distribution P₁(s) but their amplitude decreases with $\tau_{\rm rel}$ and becomes lower and lower with increasing values of p₁.

Figure 15: The function P1(s) (see Eq. (59)) for a combination of thermal population and non-thermal population as in Eq. (32) with p1= 0.5 (solid), 1 (dashes) and 10 (dotted).

The amplitude and the shape of the total SZ effect in a cluster which contains non-thermal phenomena depends on the spectral and spatial distribution of the non-thermal population, given that the cluster IC gas properties are well known from X-ray observations. We can take advantage of this fact to use the non-thermal SZ effect as a tool to constrain the physical properties of the non-thermal population. We show in Fig. 16 the total spectral distortion due to the SZ effect in a cluster with non-thermal population as a function of p₁. Low values of $p_1 \ \raise -2.truept\hbox{\rlap{\hbox{$\sim$ }}\raise5.truept \hbox{$<$ }\ }1$ for the non-thermal population induce large distortions with respect to the pure thermal SZ effect. However, for low values of p₁ the pressure of the non-thermal population with single power-law distribution exceeds substantially the thermal pressure $P_{\rm th}$ (see Table 7); as a consequence, the low energy electrons produce an exceedingly large pressure unbalance and also a large heating of the IC gas, which are not acceptable.

Figure 16: The spectral distortion, in units of $2(k_{\rm B} T_0)^3/(hc)^2$, of a single thermal population of a Coma-like cluster (solid) is compared with the spectral distortion evaluated at first order in $\tau$ (see Eq. (A.6)), produced by a combination of thermal and non-thermal population given in Eq. (32) with p1= 0.5 (dashes) and 1 (dotted) (left panel) and for p1=10 (dashes), 100 (dotted) and 1000 (dot-dashes) (right panel).

 

 

Table 7: Values of pressure and density ratios for a combination of thermal population and a non-thermal population with single power-law spectrum for different values of p₁ (see text for details).

p1	$P_{\rm rel}/P_{\rm th}$	$n_{\rm e,th}/n_{\rm e,rel}$
0.5	23.3	1.06
1	18.6	3.00
10	6.33	94.87
100	2.00	3000
1000	0.63	94868

The frequency location of the zero of the total SZ effect, x₀, is also a powerful diagnostic of the presence and of the nature of the non-thermal population. The frequency location of x₀ increases for decreasing values of p₁ as the impact of the non-thermal population becomes larger (see Fig. 17). Consistently, x₀ increases with increasing values of the pressure ratio $P_{\rm rel}/P_{\rm th}$, as also shown in Fig. 17.

Figure 17: The behaviour of the zero of the total SZ effect evaluated for a combination of thermal and non-thermal population given by Eq. (32) is shown as a function of minimum momentum, p1, of the non-thermal population (left) and of the pressure ratio $P_{\rm rel}/P_{\rm th}$ (right).

In the case of a double power-law non-thermal population the ratio $P_{\rm rel}/P_{\rm th}$ always remains of order 10^-2 even for vary low values $p_1 \sim 0.5$ (see Table 8), and in this case there is a negligible dynamical and thermal influence of the non-thermal population on the thermal one.

Figure 18: The spectral distortion, in units of $2(k_{\rm B} T_0)^3/(hc)^2$, of a single thermal population (solid) is compared with the spectral distortion evaluated at first order in $\tau$ produced by a combination of thermal and non-thermal population given in Eq. (34) with $n_{\rm e,rel}( \tilde{p}_1=100)=1\times 10^{-6}$ and p1= 0.5 (dashes), 1 (dotted), 10 (dot-dashes) and 100 (long dashes). We show the enlargement in the regions of the minimum (top), of the zero (mid) and of the maximum (bottom) of the total SZ effect.

 

 

Table 8: Values of pressure and density ratio for a thermal population and a non-thermal population with double power-law spectrum for different values of p₁ (see text for details).

p1	$P_{\rm rel}/P_{\rm th}$	$n_{\rm e,th}/n_{\rm e,rel}$
0.5	$4.94\times10^{-2}$	1926
1	$4.94\times10^{-2}$	1948
10	$4.84\times10^{-2}$	2127
100	$3.87\times10^{-2}$	3000

In this case, also the total distribution P₁(s) and the total spectral distortion are little affected by the value of p₁. We show in Fig. 18 the spectral distortion of a single thermal population compared with the spectral distortion produced by a combination of thermal and non-thermal population with double power-law spectrum. The contribution of the non-thermal population to the total SZ effect depends on the normalization of the density $n_{\rm e,rel}$ (which is highly uncertain). For increasing values of the density of the non-thermal population the non-thermal SZ effect becomes more relevant and produces substantial changes to the SZ distortion produced by a single thermal population at frequencies $x \ \raise -2.truept\hbox{\rlap{\hbox{$\sim$ }}\raise5.truept \hbox{$<$ }\ }10$ (see Fig. 19). Much milder changes are present in the high frequency range $x \ \raise -2.truept\hbox{\rlap{\hbox{$\sim$ }}\raise5.truept \hbox{$>$ }\ }15$ due to the low amplitude of the non-thermal signal in this region (see, e.g., Fig. 14).

 

 

Table 9: Values of pressure and density ratio and the corresponding position of the zero for a thermal population and a non-thermal double power-law population for p₁=0.5 as a function of $n_{\rm e,rel}( \tilde{p}_1=100)$ (see text for details).

$n_{\rm e,rel}( \tilde{p}_1=100)$ (cm-3)	$P_{\rm rel}/P_{\rm th}$	$n_{\rm e,th}/n_{\rm e,rel}$	x0
$1\times 10^{-6}$	$4.94\times10^{-2}$	1926	3.9085
$1\times10^{-5}$	0.49	192	3.9965
$3\times 10^{-5}$	1.48	64	4.1765

Figure 19: The spectral distortion, in units of $2(k_{\rm B} T_0)^3/(hc)^2$, of a single thermal population (solid) is compared with the spectral distortion evaluated at first order in $\tau$ produced by a combination of thermal and non-thermal population given in Eq. (34) with p1= 0.5 and $n_{\rm e}( \tilde{p}_1=100)= 10^{-5}$ (dashes) and $3\times 10^{-5}$ cm-3 (dotted).

We also show in Fig. 20 the behaviour of x₀ as a function of p₁ and of the pressure ratio for a density normalization $n_{\rm e,rel}(\tilde{p}_1=100)=3\times10^{-5}$ cm^-3. High values of the ratio $P_{\rm rel}/P_{\rm th}$ produce a substantial displacement of x₀ up to values $x_0 \ \raise -2.truept\hbox{\rlap{\hbox{$\sim$ }}\raise5.truept \hbox{$>$ }\ }4$ which correspond to frequencies $\ \raise -2.truept\hbox{\rlap{\hbox{$\sim$ }}\raise5.truept \hbox{$>$ }\ }227$ GHz. In Fig. 21 we also show the behaviour of x₀ a a function of $n_{\rm e,rel}( \tilde{p}_1=100)$ for p₁=0.5. We also report in Table 9 the values of the density and the corresponding values of $P_{\rm rel}/P_{\rm th}$ and of $n_{\rm e,th}/n_{\rm e,rel}$ as well as the frequency position of the zero of the total SZ effect. It is interesting to note in this context that the precise detection of the frequency location of x₀ provides a direct measure of the ratio $P_{\rm rel}/P_{\rm th}~$and, in turn, of the energy density and of the properties of the spectrum of the non-thermal population once the pressure of the thermal population is known from X-ray observations. In fact, all these physical information contributes to set the specific amplitude and the spectral shape of the total SZ effect (see Figs. 17 and 20 and Figs. 16, 18 and 20 for the different SZ effect features produced by two different non-thermal electron populations). This is a relevant and unique feature of the non-thermal SZ effect in clusters since it yields a measure of the total pressure in relativistic non-thermal particles in the cluster atmosphere, an information which is not easily accessible from the study of other non-thermal phenomena like radio halos and/or EUV or hard X-ray excesses. Thus, the detailed observations of the frequency shift of x₀ in clusters provides unambiguously relevant constraints to the relativistic particle content in the ICM and to their energy distribution. Such a measurement is also crucial to determine the true amount of kinematic SZ effect due to the cluster peculiar velocity since it is usually estimated from the residual SZ signal at the location of the zero of the thermal (relativistic) SZ effect. Due to the steepness of the SZ spectral shape in the region of the null of the overall SZ effect, the possible additional non-thermal SZ signal must be determined precisely in order to derive reliable limits to the kinematic SZ effect. We will present a more detailed analysis of the kinematic SZ effect in a forthcoming paper.