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Over the past 20 years, a marked dichotomy has emerged that distinguishes between how dark matter halos and galaxies grow. While halos do much or most of their growing by merging with other halos, star-forming galaxies with stellar masses below ∼ 1011 M acquire most of their baryons through relatively smooth gas flows. Theoretical models predict that these inflows leave a number of observable signatures in galaxies' gas-phase metallicities. Mergers leave a statistically-detectable imprint both on galaxies' central gas-phase metallicities and on their radial metallicity gradients, but they are not the primary mechanism for delivering fresh gas to galaxies.

Radial metallicity gradients are a promising probe of inflows because they are expected to flatten once previously-ejected gas begins to re-accrete. This is because metals that form and are ejected in a galaxy's core are “spun up” by the halo and re-distributed to large radii. To date, however, radial metallicity gradients in nearby galaxies have not been demonstrated to require inflows. Instead, they have been attributed either to radial gradients in ISM properties such as the strength of outflows, or to a radially-varying “evolutionary state” in which annuli at large radii are simply less evolved. These conflicting interpretations point to the need for further theoretical inquiry into how low-redshift observations ought to be interpreted.

At high redshifts, simulations predict that metallicity gradients may have been stronger than in the local Universe, possibly because of the weaker role of recycled gas and mergers at early times. Meanwhile, observations seem split between lensed galaxies with generally strong gradients and unlensed galaxies with weak ones. Further observational work is evidently required in order to clarify the level of agreement with theoretical models.

Moving from radial gradients to ensemble statistics, the interplay between inflows and gas processing leads to a rich phenomenology in which M*, Z, SFR, and gas fraction are tightly coupled. The resulting correlations are observed out to at least z = 2, and they are being disentangled via a wide variety of theoretical studies.

Analytical treatments have shown that the observed M*-Z-SFR correlation follows naturally from the assumption that gas processing is nearly in equilibrium with inflows, and that enrichment is nearly in equilibrium with dilution. Models that do not explicitly assume equilibrium tend to recover it once their parameters are tuned to match observations. Consequently, they can be used to infer inflow rates (in addition to other physical parameters) from the M*-Z-SFR relation. They have further shown that there is a tight connection between Z, SFR, and gas fraction in the sense that tuning the model to match one observable often yields agreement with another, free of charge. Detailed measurement of inflows and outflows in high-resolution numerical simulations support the importance of quasi-equilibrium behavior governed by outflows whose efficiency decreases with mass. Meanwhile, cosmological simulations (and at least one semi-analytical model) find that a realistic M*-Z-SFR-gas fraction relation arises naturally within sufficiently realistic frameworks. This ensemble statistical behavior has been interpreted as support for predominantly quiescent, equilibrium-like galaxy growth.

The progress that has been made in understanding the M*-Z-SFR relation, while promising, does not yet explain everything. Even after removing the dependence of Z on specific star formation rate SFR / M*, the residual observed scatter in Z at a given M* remains substantial (Salim et al., 2014, Salim et al., 2015). If the SFR (or gas fraction) dependence can largely be accounted for by relaxing the assumption of gas processing equilibrium, then does the residual scatter indicate departure from enrichment equilibrium or intrinsic scatter in η or Z0? This will be a useful question to take up in future theoretical studies.

In summary, it is now widely-recognized that star-forming galaxies do most of their growing in a quiescent mode where the global SFR, Z, and gas fraction constantly adjust on a relatively short timescale to reflect the influence of inflows. Further study of the flow of gas and metals in numerical simulations is required in order to improve our understanding of equilibrium growth, but its broad role is now beyond dispute. From the beginning, measurements of metallicities have driven the development of this paradigm. There is no doubt that they will continue to do so.

Acknowledgements The author thanks R. Davé and A. Fox for offering him the opportunity to contribute this chapter, and for their patience as he drafted it. Additional thanks go to A. Klypin for detailed and honest comments on an early version.

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