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ever since r. sunyaev and ya .
b. zeldovich predicted the distortion of the spectral density of the cosmic microwave background radiation ( cmbr ) in 1969 @xcite @xcite by the ionized plasma in globular clusters , whose main constituent is a relatively dense electron gas , dozens of papers have been published addressing various aspects of this phenomenon .
this avalanche of works is mainly due to the undeniable importance of such effect , not only because it yields information on deviations from the isotropy and homogeneity of the universe , but also provides an aid in determining other important cosmological parameters such as the baryonic density of matter , hubble s constant , age and velocity of massive clusters , and others .
these features have been thoroughly reviewed in recent literature so we shall not dwell with them here @xcite-@xcite .
although practically every cosmologist or astrophysicist believes that this effect is now `` clearly '' understood , there are reasons that we believe will help clarifying some of the still subtle details that remain unclear in the available treatments .
firstly , the early attempts to study this effect by visualizing the motion of the photons through the hot electron gas in the cluster as a diffusion process which , in the non - relativistic limit is described by the famous kompaneets equation @xcite @xcite .
the analysis of the results obtained through the hot electron gas in the cluster through the use of this equation are well - known @xcite-@xcite .
nevertheless , it was soon realized by many authors , including sunyaev himself , that due to the very small probability that an incoming photon is scattered even once in its passage through the gas , a compton - like scattering of a photon off an electron was a much more suitable mechanism to explain the effect .
this trend of ideas has been recently reviewed by dolgov et al @xcite where the reader may find more literature about the subject .
the somewhat intriguing result is that both approaches lead to identical results , a fact that needs further clarification .
we address this question in this paper .
secondly , the role of the optical depth @xmath1 , related to the distance a photon can travel in the plasma before it is scattered off an electron , plays in the way the broadening of the spectral lines of the scattered photon is modified by the doppler effect , has not yet been justified in a convincing manner . here
we address this question form an entirely different point of view .
and thirdly , we also show that this different interpretation of @xmath1 also leads to the kinematic sunyaev - zeldovich ( sz ) effect in a rather trivial way .
we begin our discussion by establishing a general expression for the full distorted spectrum , @xmath2 of the scattered radiation off the plasma .
let @xmath1 be the optical depth as discussed above .
this quantity is a direct measure of the probability that a photon is scattered off an electron in the gas .
thus , since the effect is small , @xmath3 is the probability that the photon traverses the plasma unscattered .
therefore , @xmath4 \,d\bar{\nu } \end{array } \label{uno}\ ] ] the first term in eq .
( [ uno ] ) is self - explanatory , the lower limit justified since @xmath5 , the incoming flux is zero for negative frequencies .
the second term , the important one , defines a gaussian probability function describing scattering of a photon with incoming frequency @xmath6 off an electron whose average kinetic energy in the gas , in the non - relativistic approximation is @xmath7 .
this function has been thoroughly discussed in earlier work @xcite so we shall not discuss it here further . in eq .
( [ uno ] ) the incoming flux is given , for @xmath8 by : @xmath9 where @xmath10 , the temperature of the cosmic background radiation , @xmath11 is boltzmann s constant , @xmath12 is planck s constant , and @xmath13 the speed of light .
further , @xmath14 is the width of the spectral line at frequency @xmath15 and its squared value reads @xcite : @xmath16 in eq.([dos ] ) @xmath17 is the temperature of the electron scatterers , @xmath18 is the electron mass and @xmath19 for @xmath20 .
notice should be made of the fact that the mathematical expression for @xmath21 corresponds to what in the literature is called the inverse compton scattering @xcite .
in particular , this is the reason of the factor @xmath22 in such equation . to show this fact we take into account the frequency distribution function for photons with incoming frequency @xmath23 isotropically scattered by electrons with velocities @xmath24@xcite , @xmath25 where @xmath26 . the expected value for the photon frequency at fixed @xmath27 is then given by : @xmath28 direct application of the equipartition energy theorem allows us to identify @xmath29 @xmath30 .
physically , this implies that a typical photon scattered off an electron will have a temperature dependent blue shift simply given by @xmath31 .
it is interesting to notice that a similar `` frequency shift approach '' leads to the analytical well known expression of the kinematic sz effect .
this is shown in the appendix .
( [ uno ] ) can be evaluated analytically leading to an expression which is identical to the one reported in the literature . setting in eq .
( [ uno ] ) @xmath32 we get that @xmath33 \,d\alpha \label{cinco}\ ] ] where @xmath34 since @xmath35 is a small number @xcite and @xmath36 is an analytical function of @xmath15 we may expand the integrand in eq .
( [ cinco ] ) in a taylor series : @xmath37 all odd powers of @xmath38 yield zero upon integration so , after a straightforward calculation , we get that @xmath39 to first order in @xmath40 @xmath35  we reproduce _ exactly _ the analytic expression for the distortion curve @xcite-@xcite .
it is also interesting to notice that , taking into account the second order terms in @xmath35 included in eq .
( [ ocho ] ) , the crossover frequency @xmath41 corresponding to the solution of equation @xmath42 is now clearly dependent on the value of @xmath35 namely , on the electron temperature , a feature not ordinarily recognized for a non - relativistic intra - cluster gas . for typical values of @xmath43 , and @xmath44 a numerical calculation yields @xmath45 , while the value obtained by considering only linear terms in @xmath35 yields @xmath46 ghz , a difference of nearly @xmath47 .
1 shows the plot of the analytical non - relativistic expression up to second order in @xmath48 , compared with its first order counterpart for a temperature of @xmath49 .
it is interesting to notice that , although a _ non - relativistic _
maxwellian was used in the convolution integral , the wien side of the distortion curve is practically identical to its relativistic counterpart , as presented in ref .
this remark is confirmed in fig .
2 with @xmath50 .
the result expressed in eq .
( [ ocho ] ) is of significant importance .
( [ ocho ] ) contains the famous kompaneets equation @xcite-@xcite describing the thermal sz effect using a diffusive mechanism to account for the motion of photons in an optically thin electron gas where the electron temperature @xmath51 , the photon temperature ( see specifically ref .
@xcite ) . one must notice that , since most photons _ are not scattered even once _
, a diffusion approximation would then hardly seem adequate @xcite .
nevertheless , the diffusion mechanism is mathematically identical to the absortion - emission process of photons by electrons to first order in @xmath35 .
the problem posed in ref .
@xcite requiring an explanation of why two different mechanisms lead to the same result is now mathematically solved .
finally , we want to stress that it is now clear why the curves describing the thermal sz effect obtained by numerical integration of different mathematical expressions turn out to be identical .
we therefore expect that the relativistic sz effect and other photon scattering problems in hot plasmas can be handled by a similar procedure .
the central idea of the convolution integral approach to cmbr distortions is that , in a dilute gas , a scattering law @xmath52 is given by what in statistical physics is known as the _
dynamic structure factor _ , where @xmath53 and @xmath15 are the corresponding incoming / outcoming photon frequencies .
then , one computes the distorted occupation number through the convolution integral @xmath54 with @xmath55 in the non - relativistic kinetic sz effect , one can easily show that a suitable structure factor is given by :    @xmath56 @xmath57 being the cluster velocity ( _ approaching the observer _ ) .
the optical depth @xmath58 measures essentially the proportion of photons being captured ( scattered ) by electrons . the intergalactic gas cloud in clusters of galaxies
has an optical depth @xmath59 .
the formalism basically states that a fraction @xmath60 of photons passes through the cluster without being scattered , while a @xmath58 fraction of them shifts its frequency through the ordinary doppler effect .
the integrals can be evaluated leading to an expression which is identical to the one reported in the literature .
indeed , since @xmath61 then @xmath62 if we now perform the substitution @xmath63 we may expand the integrand in a taylor series .
straightforward calculation leads to the ordinary expression for the kinetic sz effect : @xmath64 if we now define @xmath65 as the distorted kinetimatic sz spectrum and consider the expressions @xmath66 and @xmath67 , we finally get the desired expression : @xmath68
this paper shows that a simple convolution integral expression based on the mean value of the isotropic frequency distribution corresponding to photon scattering off electrons leads to useful analytical expressions describing the thermal sunyaev - zeldovich effect . 
 the approach , to first order in the compton parameter is able to reproduce the kompaneets equation describing the effect . 
 second order effects in the parameter @xmath0 induce a slight increase in the crossover frequency .