Linear dispersion relation for longitudinal modes of two temperature electron-positron plasma in relativistic regime

The dispersion relations of superluminal waves for the longitudinal collective modes in unmagnetized, collisionless and homogeneous relativistic electron-positron plasmas are derived. Using the standard technique of successive approximations, an analytic study of longitudinal dispersion equations in two limiting cases is presented. The full dispersion equations are transformed so that they are well suited for numerical evaluation in the temperature range where a fully relativistic treatment is needed. Using the numerical simulation method, we obtain the full dispersion curve which can't get from the analytic method.