Using an AKARI multi-wavelength mid-infrared (IR) survey, we identify luminous starburst galaxies at z> 0.5 based on the PAH luminosity, and investigate the nature of these PAH-selected starbursts. An extragalactic survey with AKARI towards the north ecliptic pole (NEP), the NEP-Deep survey, is unique in terms of a comprehensive wavelength coverage from 2 to 24um using all 9 photometric bands of the InfraRed Camera (IRC). This survey allows us to photometrically identify galaxies whose mid-IR emission is clearly dominated by PAHs. We propose a single colour selection method to identify such galaxies, using two mid-IR flux ratios at 11-to-7um and 15-to-9um (PAH-to-continuum flux ratio in the rest-frame), which are useful to identify starburst galaxies at z~0.5 and 1, respectively. We perform a fitting of the spectral energy distributions (SEDs) from optical to mid-IR wavelengths, using an evolutionary starburst model with a proper treatment of radiative transfer (SBURT), in order to investigate their nature. The SBURT model reproduces observed optical-to-mid-IR SEDs of more than a half of PAH-selected galaxies. Based on the 8um luminosity, we find ultra luminous infrared galaxies (ULIRGs) among PAH-selected galaxies. Their PAH luminosity is higher than local ULIRGs with a similar luminosity, and the PAH-to-total IR luminosity ratio is consistent with that of less luminous starburst galaxies. They are a unique galaxy population at high redshifts and we call these PAH-selected ULIRGs PAH-luminous galaxies. Although they are not as massive as submillimetre galaxies at z~2, they have the stellar mass of >3x10^{10} Msun and therefore moderately massive.
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