We present a detailed measurement of charged two-pion correlation functions in 0%-30% centrality $sqrt{s_{_{NN}}}=200$ GeV Au$+$Au collisions by the PHENIX experiment at the Relativistic Heavy Ion Collider. The data are well described by Bose-Einstein correlation functions stemming from Levy-stable source distributions. Using a fine transverse momentum binning, we extract the correlation strength parameter $lambda$, the Levy index of stability $alpha$ and the Levy length scale parameter $R$ as a function of average transverse mass of the pair $m_T$. We find that the positively and the negatively charged pion pairs yield consistent results, and their correlation functions are represented, within uncertainties, by the same Levy-stable source functions. The $lambda(m_T)$ measurements indicate a decrease of the strength of the correlations at low $m_T$. The Levy length scale parameter $R(m_T)$ decreases with increasing $m_T$, following a hydrodynamically predicted type of scaling behavior. The values of the Levy index of stability $alpha$ are found to be significantly lower than the Gaussian case of $alpha=2$, but also significantly larger than the conjectured value that may characterize the critical point of a second-order quark-hadron phase transition.
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