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We present a systematic study for ionization state of inter-stellar medium ingalaxies at z=0-3 with ~140,000 SDSS galaxies and 108 intermediate to highredshift galaxies from the literature, using an ionization-parameter sensitiveline ratio of [OIII]5007/[OII]3727 and photoionization models. We confirm thatz=2-3 galaxies show an [OIII]/[OII] ratio significantly higher than a typicalstar-forming galaxy of SDSS by a factor of >~10, and the photoionization modelsreveal that these high-z galaxies have an ionization parameter of log(qion/cms^{-1})~7.6-9.0, a factor of ~4-10 higher than local galaxies. For galaxies atany redshift, we identify a correlation between the [OIII]/[OII] ratio andgalaxy global properties of star-formation rate (SFR), stellar mass (M_star),and metallicity (Z). We extend the fundamental metallicity relation (FMR) anddevelop the fundamental ionization relation (FIR), a four-dimensional relationof ionization parameter, SFR, M_star, and Z. The intermediate and high-zgalaxies up to z~3 follow the FIR defined with the local galaxies, in contrastwith the FMR whose possible evolution from z~2 to 3 is reported. We find thatthe FMR evolution of z~2-3 appears, if one omits ionization parameterdifferences, and that the FMR evolution does not exist for an averagemetallicity solution of z~3 galaxies with a high-ionization parameter.Interestingly, all of two local Lyman-continuum emitting galaxies (LyC leakers)have a high [OIII]/[OII] ratio, indicating a positive correlation between[OIII]/[OII] and ionizing photon escape fraction (fesc), which is successfullyexplained by our photoionization models. Because [OIII]/[OII] ratios of z~2-3galaxies, especially Ly-alpha emitters (LAEs), are comparable to, or higherthan, those of the local LyC leakers, these high-z galaxies are candidates ofLyman-continuum emitting objects. (abridged)

Ionization State of Inter-Stellar Medium in Galaxies: Evolution, SFR-M_star-Z Dependence, and Ionizing Photon Escape