We present a spectral analysis of the e+e- annihilation emission from the Galactic Centre region based on the first year of measurements made with the spectrometer SPI of the INTEGRAL mission. We have found that the annihilation spectrum can be modelled by the sum of a narrow and a broad 511 keV line plus an ortho-positronium continuum. The broad line is detected (significance 3.2 sigma ) with a flux of (0.35 +/- 0.11) * 10-3 photons s-1 cm-2. The measured width of 5.4+/-1.2 keV FWHM is in agreement with the expected broadening of 511 keV photons emitted in the annihilation of positroniums that are formed by the charge exchange process of slowing down positrons with hydrogen atoms. The flux of the narrow line is (0.72 +/- 0.12) * 10-3 photons s-1 cm-2 and its width is 1.3+/-0.4 keV FWHM. The measured ortho-positronium continuum flux yields a fraction of positronium of (96.7+/-2.2)%. To derive in what phase of the interstellar medium positrons annihilate, we have fitted annihilation models calculated for each phase to the data. We have found that 49+2_-23 % of the annihilation emission comes from the warm neutral phase and 51+3_-2% from the warm ionized phase. While we may not exclude that less than 23% of the emission might come from cold gas, we have constrained the fraction of annihilation emission from molecular clouds and hot gas to be less than 8% and 0.5%, respectively. We have compared our knowledge of the interstellar medium in the bulge (size, density, and filling factor of each phase) and the propagation of positrons with our results and found that they are in good agreement if the sources are diffusively distributed and if the initial kinetic energy of positrons is lower than a few MeV. Despite its large filling factor, the lack of annihilation emission from the hot gas is due to its low density, which allows positrons to escape this phase.
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