Context. Star formation rate (SFR), metallicity, and stellar mass are among the most important parameters of star-forming (SF) galaxies characterizing their formation and evolution. They are known to be related to each other both at low and high redshift in the mass-metallicity, mass-SFR, and metallicity-SFR relations. Aims. We demonstrate the existence of a plane in a 3D parameter space defined by the axes SFR [log (SFR)(M yr −1 )], gas metallicity [12 + log (O/H)], and stellar mass [log (M star /M )] of SF galaxies. Methods. We used SF galaxies from the "main galaxy sample" of the Sloan Digital Sky Survey-Data Release 7 (SDSS-DR7) in the redshift range 0.04 < z < 0.1 and r-magnitudes between 14.5 and 17.77. Metallicities, SFRs, and stellar masses were taken from the Max-Planck-Institute for Astrophysics-John Hopkins University (MPA-JHU) emission-line analysis database. Results. From a final sample of 32 575 galaxies, we find for the first time a fundamental plane for field galaxies relating the SFR, gas metallicity, and stellar mass for SF galaxies in the local universe. One of the applications of this plane would be to estimate stellar masses from SFR and metallicity. High redshift data from the literature at redshift ∼0.85, 2.2, and 3.5, do not show evidence of evolution in this fundamental plane.

Sensitive Herschel far-infrared observations can break degeneracies that were inherent to previous studies of star formation in high-z AGN hosts. Combining PACS 100 and 160 μm observations of the GOODS-N field with 2 Ms Chandra data, we detect ∼20% of X-ray AGN individually at >3σ. The host far-infrared luminosity of AGN with L 2−10 keV ≈ 10 43 erg s −1 increases with redshift by an order of magnitude from z = 0 to z ∼ 1. In contrast, there is little dependence of far-infrared luminosity on AGN luminosity, for L 2−10 keV < ∼ 10 44 erg s −1 AGN at z > ∼ 1. We do not find a dependence of far-infrared luminosity on X-ray obscuring column, for our sample which is dominated by L 2−10 keV < 10 44 erg s −1 AGN. In conjunction with properties of local and luminous high-z AGN, we interpret these results as reflecting the interplay between two paths of AGN/host coevolution. A correlation of AGN luminosity and host star formation is traced locally over a wide range of luminosities and also extends to luminous high-z AGN. This correlation reflects an evolutionary connection, likely via merging. For lower AGN luminosities, star formation is similar to that in non-active massive galaxies and shows little dependence on AGN luminosity. The level of this secular, non-merger driven star formation increasingly dominates over the correlation at increasing redshift.

We compare the average star formation (SF) activity in X-ray selected AGN hosts with a mass-matched control sample of inactive galaxies, including both star forming and quiescent sources, in the 0.5 < z < 2.5 redshift range. Recent observations carried out by PACS, the 60−210 μm photometric camera on board the Herschel Space Observatory, in GOODS-S, GOODS-N and COSMOS allow us to obtain an unbiased estimate of the far-IR luminosity, and hence of the SF properties, of the two samples. Accurate AGN host stellar mass estimates are obtained by decomposing their total emission into the stellar and the nuclear components. We report evidence of a higher average SF activity in AGN hosts with respect to the control sample of inactive galaxies. The level of SF enhancement is modest (∼0.26 dex at ∼3σ confidence level) at low X-ray luminosities (L X 10 43.5 erg s −1 ) and more pronounced (0.56 dex at >10σ confidence level) in the hosts of luminous AGNs. However, when comparing to star forming galaxies only, AGN hosts are found broadly consistent with the locus of their "main sequence". We investigate the relative far-IR luminosity distributions of active and inactive galaxies, and find a higher fraction of PACS detected, hence normal and highly star forming systems among AGN hosts. Although different interpretations are possible, we explain our findings as a consequence of a twofold AGN growth path: faint AGNs evolve through secular processes, with instantaneous AGN accretion not tightly linked to the current total SF in the host galaxy, while the luminous AGNs co-evolve with their hosts through periods of enhanced AGN activity and star formation, possibly through major mergers. While an increased SF activity with respect to inactive galaxies of similar mass is expected in the latter, we interpret the modest SF offsets measured in low-L X AGN hosts as either a) generated by non-synchronous accretion and SF histories in a merger scenario or b) due to possible connections between instantaneous SF and accretion that can be induced by smaller scale (non-major merger) mechanisms. Far-IR luminosity distributions favour the latter scenario.