Appendix A: The total SZ effect for a combination of two electron populations: analytic approximations

In this appendix, we derive detailed expressions for $I_{\rm tot}(x)$ which are approximated at first and second order in $\tau$ according to the general expression:

$$I(x)=I_0(x)+\tau[J_1(x)-J_0(x)]+\frac{1}{2}\tau^2[J_2(x)-2J_1(x)+J_0(x)].$$ (A.1)

We remind the reader that the function J₁(x) is given by Eq. (18):

$$J_1(x)=\int_{-\infty}^{+\infty} I_0(x{\rm e}^{-s}) P_1(s) {\rm d}s,$$ (A.2)

where P₁(s) is given by Eq. (5):

$$P_1(s)=\int_0^\infty {\rm d}p f_{\rm e}(p) P_{\rm s}(s;p).$$ (A.3)

Inserting Eq. (52) in the previous equation for P₁(s), we obtain:

 

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

$$\equiv c_{\rm A} P_{\rm 1A}(s) + c_{\rm B} P_{\rm 1B}(s).$$ (A.4)

Inserting Eq. (A.4) in Eq. (A.2), we also obtain the expression:

 

$$c_{\rm A}\int_{-\infty}^{+\infty} I_0(x{\rm e}^{-s}) P_{\rm 1A}(s... ...s+ c_{\rm B} \int_{-\infty}^{+\infty} I_0(x{\rm e}^{-s}) P_{\rm 1B}(s) {\rm d}s$$ J₁(x) =

$$\equiv c_{\rm A} J_{\rm 1A}(x) + c_{\rm B} J_{\rm 1B}(x).$$ (A.5)

Inserting Eqs. (A.5), (55) and (54) in Eq. (A.1), we derive, at first order in $\tau$, the expression:

 

$$\Delta I_{\rm tot}^{(1)}(x) (\tau_{\rm A} + \tau_{\rm B})[c_{\rm A} J_{\rm 1A}(x)+c_{\rm B} J_{\rm 1B}(x)-J_0(x)]$$ =

$$\tau_{\rm A} J_{\rm 1A}(x)+ \tau_{\rm B} J_{\rm 1B}(x)-\tau_{\rm A} J_0(x)-\tau_{\rm B} J_0(x)$$ =

$$\tau_{\rm A}[J_{\rm 1A}(x)-J_0(x)]+\tau_{\rm B}[J_{\rm 1B}(x)-J_0(x)].$$ = (A.6)

As expected, the SZ effect at first order in $\tau$ is given by the sum of the separate SZ effects produced by the two electron distributions, respectively.

To evaluate the effect at second order in $\tau$, it is necessary to calculate the expression:

$$J_2(x)=\int_{-\infty}^{+\infty} I_0(x{\rm e}^{-s}) P_2(s) {\rm d}s.$$ (A.7)

The function P₂(s) can be derived by Eq. (6) using Eq. (A.4):

 

$$P_1(s)\otimes P_1(s)$$ P₂(s) =

$$[c_{\rm A} P_{\rm 1A}(s)+c_{\rm B} P_{\rm 1B}(s)] \otimes [c_{\rm A} P_{\rm 1A}(s)+c_{\rm B} P_{\rm 1B}(s)]$$ =

$$c_{\rm A}^2 P_{\rm 1A}(s)\otimes P_{\rm 1A}(s)+2c_{\rm A}c_{\rm B} P_{\rm 1A}(s) \otimes P_{\rm 1B}(s)$$ =

$$+ c_{\rm B}^2 P_{\rm 1B}(s) \otimes P_{\rm 1B}(s)$$

$$\equiv c_{\rm A}^2 P_{\rm 2AA}(s)+2c_{\rm A}c_{\rm B} P_{\rm 2AB}(s)+c_{\rm B}^2 P_{\rm 2BB}(s).$$ (A.8)

Inserting this expression in Eq. (A.7), we obtain:

 

$$c_{\rm A}^2 \int_{-\infty}^{+\infty} I_0(x{\rm e}^{-s}) P_{\rm 2AA}(s) {\rm d}s$$ J₂(x) =

$$+2c_{\rm A} c_{\rm B} \int_{-\infty}^{+\infty} I_0(x{\rm e}^{-s}) P_{\rm 2AB}(s) {\rm d}s$$

$$+ c_{\rm B}^2 \int_{-\infty}^{+\infty} I_0(x{\rm e}^{-s}) P_{\rm 2BB}(s) {\rm d}s$$

$$\equiv c_{\rm A}^2 J_{\rm 2AA} (x) +2c_{\rm A}c_{\rm B} J_{\rm 2AB}(x)+c_{\rm B}^2 J_{\rm 2BB}(x).$$ (A.9)

From Eq. (A.1) we derive the expression of the second order correction to the total distorted spectrum

 

$$\Delta I_{\rm tot}^{(2)}(x) \frac{1}{2}(\tau_{\rm A}+\tau_{\rm B})^2 [c_{\rm A}^2J_{\rm 2AA}+2c_{\rm A}c_{\rm B}J_{\rm 2AB}$$ =

$$+c_{\rm B}^2J_{\rm 2BB} -2c_{\rm A}J_{\rm 1A}-2c_{\rm B} J_{\rm 1B}+J_0]$$

$$\frac{1}{2}\{\tau_{\rm A}^2 J_{\rm 2AA} +2\tau_{\rm A}\tau_{\rm B} J_{\rm 2AB} -2\tau_{\rm A}(\tau_{\rm A}+\tau_{\rm B}) J_{\rm 1A}$$ =

$$-2\tau_{\rm B}(\tau_{\rm A}+\tau_{\rm B}) J_{\rm 1B}+(\tau_{\rm A}^2+2\tau_{\rm A}\tau_{\rm B}+\tau_{\rm B}^2)J_0\}$$

$$\frac{1}{2}\{\tau_{\rm A}^2[J_{\rm 2AA}-2J_{\rm 1A}+J_0]+ \tau_{\rm B}^2[J_{\rm 2BB}-2J_{\rm 1B}+J_0]$$ =

$$+2\tau_{\rm A}\tau_{\rm B}[J_{\rm 2AB}-J_{\rm 1A}-J_{\rm 1B}+J_0]\}.$$ (A.10)

Notice that at second order in $\tau$ there is an additional term describing the probability that a CMB photon can suffer first a scattering from the electrons of the distribution $f_{\rm A}$ and then another scattering from the electrons of the distribution $f_{\rm B}$.

In a similar approach, we can derive the further corrections to the total distorted spectrum evaluated at any order in $\tau$; these will contain a larger and larger number of cross-scattering terms. In a completely analogous way, we can also consider the case of three and more electron distributions.