Almost all massive galaxies today are understood to contain supermassive black holes (SMBH) at their centres. SMBHs grew by accreting material from their surroundings, emitting X-rays as they did so. X-ray luminosity functions (XLFs) of active galactic nuclei (AGN) have been extensively studied in order to understand the AGN populationβs cosmological properties and evolution. We present a new fixed rest-frame method to achieve a more accurate study of the AGN XLF evolution over cosmic time. Normally, XLFs are constructed in a fixed observer-frame energy band, which can be problematic because it probes different rest-frame energies at different redshifts. In the new method, we construct XLFs in the fixed rest-frame band instead, by varying the observed energy band with redshift. We target a rest-frame 2β8 keV band using XMM-Newton and HEAO 1 X-ray data, with seven observer-frame energy bands that vary with redshift for 0 < z < 3. We produce the XLFs using two techniques; one to construct a binned XLF, and one using a maximum likelihood (ML) fit, which makes use of the full unbinned source sample. We find that our ML best-fitting pure luminosity evolution results for both methods are consistent with each other, suggesting that performing XLF evolution studies with the high-redshift data limited to high-luminosity AGN is not very sensitive to the choice of fixed observer-frame or rest-frame energy band, which is consistent with our expectation that high-luminosity AGN typically show little ABSORPTION. We have demonstrated the viability of the new method in measuring the XLF evolution
