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 O
B 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 $rho$ 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.
The relation between the ratio of infrared (IR) and ultraviolet (UV) flux densities (the infrared excess: IRX) and the slope of the UV spectrum (beta) of galaxies plays a fundamental role in the evaluation of the dust attenuation of star forming gala
xies especially at high redshifts. Many authors, however, pointed out that there is a significant dispersion and/or deviation from the originally proposed IRX-beta relation depending on sample selection. We reexamined the IRX-beta relation by measuring the far- and near-UV flux densities of the original sample galaxies with GALEX and AKARI imaging data, and constructed a revised formula. We found that the newly obtained IRX values were lower than the original relation because of the significant underestimation of the UV flux densities of the galaxies, caused by the small aperture of IUE, Further, since the original relation was based on IRAS data which covered a wavelength range of lambda = 42--122mum, using the data from AKARI which has wider wavelength coverage toward longer wavelengths, we obtained an appropriate IRX-beta relation with total dust emission (TIR): log(L_{rm TIR}/L_{rm FUV}) = log [10^{0.4(3.06+1.58beta)}-1] +0.22. This new relation is consistent with most of the preceding results for samples selected at optical and UV, though there is a significant scatter around it. We also found that even the quiescent class of IR galaxies follows this new relation, though luminous and ultraluminous IR galaxies distribute completely differently as well known before.
