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Far-Ultraviolet and Far-Infrared Bivariate Luminosity Function of Galaxies: Complex Relation between Stellar and Dust Emission

Far-ultraviolet (FUV) and far-infrared (FIR) luminosity functions (LFs) of galaxies show a strong evolution from $z = 0$ to $z = 1$, but the FIR LF evolves much stronger than the FUV one. The FUV is dominantly radiated from newly formed short-lived OB stars, while the FIR is emitted by dust grains heated by the FUV radiation field. It is known that dust is always associated with star formation activity. Thus, both FUV and FIR are tightly related to the star formation in galaxies, but in a very complicated manner. In order to disentangle the relation between FUV and FIR emissions, we estimate the UV-IR bivariate LF (BLF) of galaxies with {\sl GALEX} and {\sl AKARI} All-Sky Survey datasets. Recently we invented a new mathematical method to construct the BLF with given marginals and prescribed correlation coefficient. This method makes use of a tool from mathematical statistics, so called "copula". The copula enables us to construct a bivariate distribution function from given marginal distributions with prescribed correlation and/or dependence structure. With this new formulation and FUV and FIR univariate LFs, we analyze various FUV and FIR data with {\sl GALEX}, {\sl Spitzer}, and {\sl AKARI} to estimate the UV-IR BLF. The obtained BLFs naturally explain the nonlinear complicated relation between FUV and FIR emission from star-forming galaxies. Though the faint-end of the BLF was not well constrained for high-$z$ samples, the estimated linear correlation coefficient $ρ$ was found to be very high, and is remarkably stable with redshifts (from 0.95 at $z = 0$ to 0.85 at $z = 1.0$). This implies the evolution of the UV-IR BLF is mainly due to the different evolution of the univariate LFs, and may not be controlled by the dependence structure.