3 The SZ effect generated from a non-thermal electron population

Using the same general analytical approach derived in Sect. 2 above we derive here the exact spectral features of the SZ effect produced by a single non-thermal population of electrons. Such an exact derivation has been not done so far. For such a population we consider here two different phenomenological spectra: i) a single power-law energy spectrum, $n_{\rm rel} = n_0 E^{-\alpha}$, like that which is able to fit the radio-halo spectra of many clusters, and ii) a double power-law spectrum which is able to fit both the radio halo spectrum and the EUV and hard X-ray excess spectra observed in some nearby clusters (see Petrosian 2001). Our formalism is, nonetheless, so general that it can be applied to any electron distribution so far considered to fit both the observed radio-halo spectra and those of the EUV/Hard X-ray excesses (see, e.g., Sarazin 1999; Blasi & Colafrancesco 1999; Colafrancesco & Mele 2001; Petrosian 2001).

The single power-law electron population is described by the momentum spectrum

$$f_{\rm e,rel}(p;p_1,p_2,\alpha)=A(p_1,p_2,\alpha) p^{-\alpha} ~; \qquad p_1 \leq p \leq p_2$$ (32)

where the normalization term $A(p_1,p_2,\alpha)$ is given by:

$$A(p_1,p_2,\alpha) = \frac{(\alpha-1)} {p_1^{1-\alpha}-p_2^{1-\alpha}}$$ (33)

with $\alpha \approx 2.5$. This is the simplest electron distribution which is consistent with the spectra of the radio halos observed in many galaxy clusters (see, e.g., Feretti 2001 for a recent review). In the calculation of the non-thermal SZ effect we consider the minimum momentum, p₁, of the electron distribution as a free parameter since it is not constrained by the available observations, while the specific value of $p_2 \gg 1$ is irrelevant for power-law indices $\alpha > 2$ which are required by the radio halo spectra observed in galaxy clusters.

The distribution in Eq. (32) can be however affected by a crucial problem: if the electron spectrum is extended at energies below 20 MeV (i.e., at $p\ \raise -2.truept\hbox{\rlap{\hbox{$\sim$ }}\raise5.truept \hbox{$<$ }\ }40$) with the same spectral slope $\alpha \approx 2.5$ of the high energy tail, the heating rate of the IC gas produced by Coulomb collisions of the relativistic electrons becomes larger than the bremsstrahlung cooling rate of the IC gas (see, e.g., Petrosian 2001). This would produce unreasonably and unacceptably large heating of the IC gas.

For this reason, we consider also a double power-law electron spectrum which has a flatter slope below a critical value $p_{\rm cr}$:

 

$$f_{\rm e,rel}(p;p_1,p_2,p_{\rm cr},\alpha_1,\alpha_2) = K(p_1,p_2... ...m cr}^{-\alpha_1+\alpha_2}~ p^{-\alpha_2} & p_{\rm cr}<p<p_2 \end{array}\right.$$ (34)

with a normalization factor given by:

 

$$K(p_1,p_2,p_{\rm cr},\alpha_1,\alpha_2)= \left[ \frac{p_1^{1-\alp... ...~ \frac{p_{\rm cr}^{1-\alpha_2}-p_2^{1-\alpha_2}}{\alpha_2-1} \right]^{-1}\cdot$$ (35)

If $p_{\rm cr} \sim 400$, the electron distribution of Eq. (34) with $\alpha_1 \sim 0.5$ at $p \ \raise -2.truept\hbox{\rlap{\hbox{$\sim$ }}\raise5.truept \hbox{$<$ }\ }p_{\rm cr}$ can be extended down to very low energies without violating any constraint set by the IC gas heating. Hence, we assume here the following parameter values: $\alpha _1=0.5$, $\alpha _2=2.5$, $p_{\rm cr}=400$ e $p_2\rightarrow \infty$. We again consider p₁ as a free parameter.

A crucial quantity in the calculation of the non-thermal SZ effect is given by the number density of relativistic electrons,  $n_{\rm e,rel}$. The quantity $n_{\rm e,rel}$ in galaxy clusters can be estimated from the radio halo spectrum intensity, but this estimate depends on the assumed value of the IC magnetic field and on the model for the evolution of the radio-halo spectrum (see, e.g. Sarazin 1999 for time-dependent models and Blasi & Colafrancesco 1999; Colafrancesco & Mele 2001 for stationary models). For the sake of illustration, the radio halo flux $J_{\nu}$ for a power-law spectrum is, in fact, given by $J_{\nu} \propto n_{\rm e,rel} B^{\alpha_{\rm r}-1} \nu ^{-\alpha_{\rm r}}$, where $\alpha_{\rm r}$ is the radio halo spectral slope. Because of the large intrinsic uncertainties existing in the density of the relativistic electrons which produce the cluster non-thermal phenomena, a value $n_{\rm e,rel}=10^{-6}$ cm^-3 for p₁=100 has been assumed here.

Our results on the amplitude of the non-thermal SZ effect can be easily rescaled to different values of $n_{\rm e,rel}$: in fact, decreasing (increasing) $n_{\rm e,rel}$will produce smaller (larger) amplitudes of the SZ effect, as can be seen from Eqs. (22), (24), (28), (29).

The density $n_{\rm e,rel}$ increases for decreasing values of p₁. In fact, multiplying the electron distribution in Eq. (32) by the quantity  $n_{\rm e,rel}(p_1)$ one obtains

$$N_{\rm e}(p;p_1)\equiv n_{\rm e,rel}(p_1) A(p_1) p^{-\alpha}$$ (36)

where the function A(p₁) is given by Eq. (33). Thus, the electron density scales as

$$n_{\rm e,rel}(p_1)=n_{\rm e,rel}( \tilde{p_1}) \frac{A(\tilde{p_1})}{ A( p_1)},$$ (37)

where we normalized the electron density at a fixed value of $\tilde{p_1}=100$.

The value of p₁ sets the value of the electron density as well as the value of the other relevant quantities which depend on it, namely the optical depth $\tau_{\rm rel}$ and the pressure $P_{\rm rel}$ of the non-thermal population. In particular, the pressure $P_{\rm rel}$ for the case of an electron distribution in Eq. (32), is given by

 

$$% P_{\rm rel} n_{\rm e} \int_0^\infty {\rm d}p f_{\rm e}(p) \frac{1}{3} p v(p) m_{\rm e} c$$ =

$$\frac{n_{\rm e} m_{\rm e} c^2 (\alpha -1)}{6[p^{1-\alpha}]_{p_2}^... ...}{1+p^2}}\left(\frac{\alpha-2}{2}, \frac{3-\alpha}{2}\right)\right]_{p_2}^{p_1}$$ = (38)

(see, e.g., Ensslin & Kaiser 2000), where B_x is the incomplete Beta function

$$B_x(a,b)=\int_0^x t^{a-1} (1-t)^{b-1}{\rm d}t$$ (39)

(see, e.g., Abramowitz & Stegun 1965). The optical depth of the non-thermal electron population is given by

$$\tau_{\rm rel}(p_1) = 2\times 10^{-6} ~\frac{A(\tilde{p_1})}... ... cm}^{-3}}\right] \left[ \frac{\ell}{1~{\rm Mpc}}\right] \cdot$$ (40)

For an electron population as in Eq. (34) we obtain analogous results. The electron density with the same normalization as before is given by

$$n_{\rm e,rel}(p_1)=n_{\rm e,rel}( \tilde{p_1}) \frac{K(\tilde{p_1})}{ K( p_1)}\cdot$$ (41)

The optical depth is again given by Eq. (40), where we substitute the ratio $K(\tilde{p_1})/K(p_1)$ to $A(\tilde{p_1})/A(p_1)$. The electron pressure of this last population has instead a more complicated expression given by:

 

$$P_{\rm rel}=\frac{K(p_1) n_{\rm e}(p_1) m_{\rm e} c^2}{6} \left \... ...right) \right]^{p_{\rm cr}}_{p_2} p_{\rm cr}^{-\alpha_1+\alpha_2} \right\}\cdot$$ (42)

To evaluate the SZ effect induced by a single non-thermal electron population we use the relativistically correct formalism described in Sect. 2 which also takes into account the effects of multiple scattering. Again, we are working here in the Thomson limit. The general expression for the distorted spectrum I(x) is given again by Eq. (8). The only change in our general formulation consists in inserting the non-thermal distribution $f_{\rm e,non-th}$ in the function P₁(s). The functions P_n(s), which determine the probabilities to have a frequency shift s due to n scattering are still given by the convolution of ndistributions P₁(s) (see Eq. (6)). Figure 3 shows the function P₁(s) for different values of p₁ in the range 0.5-1000.

Figure 3: The distribution P1(s) for a non-thermal population as in Eq. (32) with $\alpha =2.5$, $p_2\rightarrow \infty$ and p1= 0.5, 1, 10, 100, 1000, as indicated.

For low values of p₁, the function P₁(s) is centered around s=0 and does not produce, hence, large frequency changes. For higher values of p₁ (i.e., increasing values of the minimum energy of the electrons) the function P₁(s) becomes wider and it is centered at higher and higher values of s. For p₁=1000 the redistribution function P₁(s) is centered around s=15, which corresponds to a frequency change $\nu' / \nu = {\rm e}^s \sim 3\times 10^6$. Thus, the SZ effect caused by electronic populations with very high values of p₁ moves the photon frequencies from the CMB region up to very different regions causing a depletion of scattered CMB photons at sub-mm wavelengths where the SZ effect is usually studied. We report in Table 4 the values of the pressure and of the density of a single power-law population as a function of p₁.

Figure 4: The distributions P1(s) (solid line), P2(s) (dashed) and P3(s) (dotted) evaluated for non-thermal electron populations with a single power-law spectrum given in Eq. (32) with $\alpha =2$, p1=1 and p2=100 (left panel) and with a double power-law spectrum given in Eq. (34) (right panel). Note that these distributions are wider and more skewed towards larger values of s than the corresponding distributions evaluated in the case of a thermal electron population.

 

 

Table 4: Values of the pressure and of the density of a single power-law population as in Eq. (32) with $\alpha =2.5$, $p_2\rightarrow \infty$ and $n_{\rm e,rel}( \tilde{p}_1=100)=1\times 10^{-6}$ cm^-3 for different values of p₁.

p1	$P_{\rm rel}$ (keV cm-3)	$n_{\rm e,rel}$ (cm-3)
0.5	0.59	$2.83\times 10^{-3}$
1	0.47	$1\times 10^{-3}$
10	0.16	$3.16\times 10^{-5}$
100	$5.11\times 10^{-2}$	$1\times 10^{-6}$
1000	$1.61\times 10^{-2}$	$3.16\times 10^{-8}$

To emphasize the changes in the photon redistribution function produced by a non-thermal electron distribution, we show in Fig. 4 the functions P₁(s), P₂(s) and P₃(s) for a single and double power-law electron distributions. It is evident that these functions are wider and more a-symmetrically skewed towards large and positive values of s with respect to the same distributions evaluated in the case of a single thermal electron population. Once the exact redistribution function P(s) is known, it is possible to evaluate the exact distorted spectrum I(x) through Eq. (8).

It is also possible to evaluate the spectral distortion using Eq. (16) which gives the series expansion up to the nth order in $\tau$ and to compare the exact and approximated shapes for I(x). We show in Fig. 5 the first three terms b₁,b₂, b₃ of the series expansion in Eq. (16) as a function of the a-dimensional frequency x. The approximated expression for I(x) up to third order in $\tau$ is an excellent fit of its exact behaviour at frequencies around the minimum of the effect ( $x \approx 2.3$) and it shows a difference $\sim$3% at $x \approx 6.5$ and a difference $\approx$0.1% at high frequencies, $x \approx 15$ (see Table 5). The precision of the approximation depends also on the value of $\tau$ and we verified that the precision of the second and third order approximations does not increase for values $\tau \ \raise -2.truept\hbox{\rlap{\hbox{$\sim$ }}\raise5.truept \hbox{$<$ }\ }10^{-3}$ at any frequency between $x \approx 2$ and $x \approx 15$.

 

 

Table 5: Fractional difference (in units of 10^-2) between the exact spectral distortion and that evaluated at first, second and third order in $\tau$. We report here the cases for values of the optical depth $\tau =0.01$ and $\tau =0.001$.

	First order	Second order	Third order
$\tau =0.01$
x=2.3	0.82	0.44	0.44
x=6.5	1.42	2.90	2.90
x=15	0.83	0.12	0.12
$\tau =0.001$
x=2.3	0.49	0.45	0.45
x=6.5	2.73	2.88	2.88
x=15	0.19	0.12	0.12

Figure 5: The functions b1(x) (solid), b2(x) (dashed) and b3(x) (dotted) evaluated for a non-thermal electron population with a single power-law spectrum as in Eq. (32).

3.1 The function $\tilde{{\mathsfsl g}}({\mathsfsl x})$ for a single non-thermal population

Also for a non-thermal electron population it is possible to write the spectral distortion in the general form of Eq. (22) as:

$$\Delta I_{\rm non-th}(x)=2\frac{(k_{\rm B} T_0)^3}{(hc)^2}y_{\rm non-th} ~\tilde{g}(x) ,$$ (43)

where the Comptonization parameter is given by the general expression

$$y_{\rm non-th}=\frac{\sigma_{\rm T}}{m_{\rm e} c^2}\int P_{\rm rel} {\rm d}\ell ,$$ (44)

in terms of the electronic pressure $P_{\rm rel}$ which, for a single and double power-law populations, is given by Eq. (38) and by Eq. (42), respectively.

At first order in $\tau$ we can write the spectral function $\tilde{g}(x)$ of the non-thermal SZ effect as

$$\tilde{g}(x)=\frac{\Delta i}{y}=\frac{\tau [j_1-j_0]}{\frac... ...{m_{\rm e} c^2}{\langle k_{\rm B} T_{\rm e} \rangle} [j_1-j_0]$$ (45)

where we defined the quantity

 

$$\langle k_{\rm B} T_{\rm e} \rangle \equiv \frac{\sigma_{\rm T}}{... ...m d}\ell} =\int_0^\infty {\rm d}p f_{\rm e}(p) \frac{1}{3} p v(p) m_{\rm e} c ,$$ (46)

which is the analogous of the average temperature for a thermal population. In fact, comparing Eq. (46) with Eq. (25), $\langle k_{\rm B} T_{\rm e} \rangle = k_{\rm B} T_{\rm e}$ obtains. For a non-thermal population with a single power-law distribution one gets

$$\langle k_{\rm B} T_{\rm e} \rangle= \frac{ m_{\rm e} c^2 (\a... ...{\alpha-2}{2}, \frac{3-\alpha}{2}\right)\right]_{p_2}^{p_1} ,$$ (47)

while in the case of a double power-law distribution we find instead

$$\langle k_{\rm B} T_{\rm e} \rangle =\frac{K(p_1) m_{\rm e} c^2}{... ...2} \right) \right]^{p_{\rm cr}}_{p_2} p_{\rm cr}^{-\alpha_1+\alpha_2} \right\}.$$ (48)

It is possible to write the Comptonization parameter $y_{\rm non-th}$ as a function of the quantity $\langle k_{\rm B} T_{\rm e} \rangle$ and of the optical depth $\tau$:

 

$$% y_{\rm non-th} \frac{\sigma_{\rm T}}{m_{\rm e} c^2}\int P_{\rm rel} {\rm d}\ell=... ...ngle k_{\rm B} T_{\rm e} \rangle}{m_{\rm e} c^2} \int n_{\rm e,rel} {\rm d}\ell$$ =

$$\frac{\langle k_{\rm B} T_{\rm e} \rangle}{m_{\rm e} c^2} \tau$$ = (49)

(note that here $\tau = \tau_{\rm rel}$ holds).

We can also write the nth order approximations of the distorted spectrum using Eq. (16). For example, at third order in $\tau$, the following explicit expression founds:

$$\tilde{g}(x)=\frac{m_{\rm e} c^2}{\langle k_{\rm B} T_{\rm e} \ra... ...rac{1}{2}\tau(j_2-2j_1+j_0) +\frac{1}{6}\tau^2 (j_3-3j_2+3j_1-j_0)\right] \cdot$$ (50)

Finally, the exact form of the function $\tilde{g}(x)$ for a non-thermal electron population writes as:

$$\tilde{g}(x)=\frac{m_{\rm e} c^2}{\langle k_{\rm B} T_{\rm e... ... i_0(x{\rm e}^{-s}) P(s) {\rm d}s- i_0(x)\right] \right\}\cdot$$ (51)

We show in Fig. 6 a comparison between the function g(x)obtained under the Kompaneets approximation (see Eq. (2)) and the function $\tilde{g}(x)$, approximated to first order in $\tau$, and evaluated for a single power-law population. A major difference between the two functions is the different position of the zero of the SZ effect which is moved to higher frequencies in the case of the non-thermal population with respect to the case of the thermal one. In Fig. 7 we compare the function $\tilde{g}(x)$ for a non-thermal population evaluated at first order in $\tau$(which is actually independent from the value of $\tau$) with the exact function for values $\tau =1$, $\tau =0.1$ and $\tau =0.01$. We notice that the first order approximation is a good approximation of the exact case for values $\tau < 10^{-3}$.

Figure 6: The spectral function g(x) for a thermal population in the Kompaneets limit (solid line) is compared with the function $\tilde{g}(x)$ approximated at first order in $\tau$ for a non-thermal population with single power-law spectrum with parameters p1=4, p2=1010 and $\alpha =2.5$ (dashed line), evaluated from Eq. (45). The dotted line represents the zero reference value.

Figure 7: The function $\tilde{g}(x)$ for a non-thermal population with single power-law spectrum with parameters p1=4, p2=1010 and $\alpha =2.5$. We show the first order approximation in $\tau$ (solid line) and the exact case for $\tau =1$ (dashed line), $\tau =0.1$ (dotted line) and $\tau =0.01$ (dot-dashed line). Note that the first order approximation is a reasonable approximation of the exact case only for small values of $\tau$.

Figure 8 shows the function $\tilde{g}(x)$ (evaluated at first order in $\tau$) for a single power-law population and considering different values of p₁. For increasing values of p₁ the maximum of the function $\tilde{g}(x)$ moves towards higher and higher frequencies. As a consequence, also the zero of the non-thermal SZ effect moves to higher values of x as shown also in Fig. 9. The location of x₀changes rapidly with increasing values of p₁ and reaches a value $x_0 \approx 22$ for $p_1 = 2 \times 10^3$. At such values of p₁ the pressure $P_{\rm rel}$takes values $\approx$0.02 keV cm^-3, as shown in Fig. 9.

Figure 8: In the left panel we show the function $\tilde{g}(x)$ (see Eq. (45)) for a non-thermal population as in Eq. (32) with p1= 0.5 (solid line), 1 (dashed) and 10 (dotted). The right panel shows the same function for p1= 10 (solid line), 100 (dashed) and 1000 (dotted).

Figure 9: The behaviour of the zero of the SZ effect for a non-thermal population given in Eq. (32) as a function of p1 (left panel) and of the pressure $P_{\rm rel}$ (right panel) expressed in keV cm-3. Note that the pressure $P_{\rm rel}$ decreases for increasing values of p1, as discussed in Sect. 3

An important point to notice is that the electron density of the non-thermal population decreases for increasing values of p₁ as was already shown in Table 4. For this reason the pressure $P_{\rm rel}$ and the resulting spectral distortion decreases in amplitude for increasing values of p₁. This result is shown more specifically in Fig. 10 where we plot the quantity $\Delta i(x) = \Delta I(x)[(hc)^2/2(k_{\rm B}T_0)^3]$ for values of p₁ in the range 0.5-10³.

For an electron population with a double power-law spectrum as in Eq. (34) the values of the density and pressure are also reported in Table 6 as a function of p₁ (note that the case p₁=1000 is the same of the single power-law spectrum). The values of $P_{\rm rel}$ and $n_{\rm e,rel}$ for low values of p₁ are now smaller than those of the single power-law case (see Table4), thus avoiding the problem of an excessive heating of the cluster IC gas for low values of p₁.

 

 

Table 6: Values of the pressure and of the density of a double power-law population as in Eq. (34) with $\alpha _1=0.5$, $\alpha _2=2.5$, $p_{\rm cr}=400$, $p_2\rightarrow \infty$ and $n_{\rm e,rel}( \tilde{p}_1=100)=1\times 10^{-6}$ cm^-3 for different values of p₁.

p1	$P_{\rm rel}$ (keV cm-3)	$n_{\rm e,rel}$ (cm-3)
0.5	$1.26\times10^{-3}$	$1.56\times10^{-6}$
1	$1.26\times10^{-3}$	$1.54\times10^{-6}$
10	$1.23\times10^{-3}$	$1.41\times10^{-6}$
100	$9.88\times10^{-4}$	$1\times 10^{-6}$

In Fig. 11 we show the function P₁(s) evaluated for an electron population with double power-law spectrum (see Eq. (34)) and its dependence on the lower momentum cutoff p₁. The behaviour of P₁(s) in this case is quite different with respect to the single power-law case (see Fig. 3) and depends mainly on the flatness of the electron spectrum below the momentum cutoff $p_{\rm cr}$. We notice that i) even for low values of $p_1 \sim 0.5$ small frequency changes are unlikely, as shown by the strongly a-symmetric shape of P₁(s); ii) there are small differences in the function P₁(s) for different values of $p_1 \ \raise -2.truept\hbox{\rlap{\hbox{$\sim$ }}\raise5.truept \hbox{$<$ }\ }p_{\rm cr}$because the electron density in this range is lower than in the single power-law case and its variation is mild.

Figure 10: The spectral distortion $\Delta i(x)$ in units of $2(k_{\rm B} T_0)^3/(hc)^2$ for a non-thermal population as in Eq. (32) evaluated for p1= 0.5 (solid line), 1 (dashes), 10 (dotted), 100 (long dashes) and 1000 (dot-dashes).

Figure 11: The function P1(s) (see Eq. (5)) is shown for non-thermal populations given in Eq. (34) with p1= 0.5 (solid), 1 (dashes), 10 (dotted), 100 (dot-dashes).

We also show in Fig. 12 the function $\tilde{g}(x)$ evaluated at first order in $\tau$ for a double power-law electron population as in Eq. (34). In this case, there is no well defined maximum of the SZ effect since low frequency changes are unlikely. Moreover, the function $\tilde{g}(x)$ is less sensitive to different values of the lower momentum cutoff p₁ for $p_1 < p_{\rm cr}$ because of the flatness of the electron spectrum in this region. This fact is also confirmed by the variation of the zero location of the non-thermal SZ effect (see Fig. 13). The location of x₀ changes now less rapidly than in the case of a single power-law distribution (see Fig. 11) due to the different shape of $\tilde{g}(x)$, and hence of P(s), caused by the lower density of the low-energy electrons with respect to the case of the single power-law spectrum. The zero of the non-thermal SZ effect produced by a single non-thermal population is also shifted to frequencies $x_0 \ \raise -2.truept\hbox{\rlap{\hbox{$\sim$ }}\raise5.truept \hbox{$>$ }\ }10$ (much higher than the value $x_{\rm0,th}=3.83$) due to the large frequency shifts experienced by the CMB photons scattering the high energy non-thermal electrons.

We finally show in Fig. 14 the spectral distortion $\Delta i(x)$ for different values of p₁. Note, finally, that the amplitude of $\Delta I(x)$ produced by an electron population with a double power-law spectrum as in Eq. (34) is much lower than that produced by a population with a single power-law spectrum as in Eq. (32).

Figure 12: The function $\tilde{g}(x)$ (see Eq. (45)) for non-thermal populations as in Eq. (34) with p1= 0.5 (solid line), 1 (dashes), 10 (dotted) and 100 (dot-dashes).

The case of a system dominated by non-thermal electrons is not applicable straightforwardly to galaxy clusters but rather to radio galaxies whose jets inject in the ICM large quantities of relativistic non-thermal electrons. We want to stress that our general approach is able to describe the Compton scattering distortions even in the cases of the radio lobes and of the jets of radio galaxies or AGNs with large optical depths and ultra-relativistic electron energies. We will address more specifically this issue in a forthcoming paper.

Figure 13: The behaviour of the zero of the SZ effect for a non-thermal population with spectrum given by Eq. (34) as a function of p1 (left panel) and of the pressure $P_{\rm rel}$ (right panel) in units of keV cm-3. As in Fig. 9, the pressure $P_{\rm rel}$ decreases for increasing values of p1 but with a different rate depending on the different spectrum of the electron distribution.