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Cosmological numerical simulations show that model galaxies tend to reach a subtle stationary state where the gas accretion rate from the cosmic web balances the star-formation rate (SFR) once outflows are taken into account (e.g., Finlator & Davé, 2008; Schaye et al., 2010; Fraternali & Tomassetti, 2012; Davé et al., 2012; Dekel et al., 2013; Bothwell et al., 2013; Feldmann, 2013; Altay et al., 2013; Forbes et al., 2014a; see also the contribution by Kereš in this Book). The balance is set because the time-scale to transform gas into stars is significantly shorter than the Hubble time and, thus, galaxies must rely on external gas accretion to maintain star-formation for a long period of time (e.g., Kennicutt, 1983; Sancisi et al., 2008; and also Sect. 3.1).

Numerical simulations reveal an intimate connection between SFR and gas accretion rate, and so, provide the rationale to study the relation observationally. In order to identify the physical parameters that have to be measured, one can resort to the toy galaxy model often referred to as bathtub model or self-regulator model. It is amply described in the literature (Tinsley, 1980; Edmunds, 1990; Bouché et al., 2010; Peeples & Shankar, 2011; Brisbin & Harwit, 2012; Davé et al., 2012; Lilly et al., 2013; Dayal et al., 2013; Dekel & Mandelker, 2014; Forbes et al., 2014b; Peng & Maiolino, 2014; Harwit & Brisbin, 2015; Rodríguez-Puebla et al., 2016; Somerville & Davé, 2015; Ascasibar et al., 2015; and also see the contribution by Lilly in this Book), and it provides the physical insight to understand the self-regulation of the star-formation (SF) process in galaxies. In this simple model, galaxies are described as structureless entities characterized by a single mass of gas Mg, a SFR, an outflow rate dot{M}out, and an inflow rate dot{M}in. We take the nomenclature and the equations from the particular implementation by Sánchez Almeida et al. (2014a). If the model galaxy is isolated and does not receive any external gas supply, then the initial mass of gas Mg(0) drops exponentially in time t due to star formation (SF),

Equation 1


with a characteristic time-scale, τin, given by,

Equation 2


which depends on the so-called gas depletion time-scale τg,

Equation 3


and on the mass loading factor η,

Equation 4


defined to be the scale factor between the SFR and the mass outflows that the SF drives. R in Eq. (2) stands for the fraction of stellar mass that returns to the interstellar medium (ISM) rather than being locked into stars and stellar remnants. If rather than being isolated our toy galaxy is fed at a gas accretion rate dot{M}in(t), then after a transient that lasts τin, it reaches a stationary state where,

Equation 5


Even if oversimplified, the above equations include all the essential ingredients giving rise to the expected relationship between gas accretion rate and SFR. Equation (5) indicates that the stationary-state SFR is set by the gas infall rate, becoming zero when the accretion rate goes to zero. Often η ≫ 1, and so SFR ≪ dot{M}in and τin ≪ τg. In this case, only a minor fraction of the accreted gas is used to form stars. The rest is returned unused to the circum-galactic medium (CGM) and inter-galactic medium (IGM). When this happens, the time-scale to consume the gas τin becomes much shorter than the already short gas depletion time-scale (Sect. 3.1).

Therefore, in order to provide an observational overview of the relation between gas infall rate and SFR, it is essential to keep in mind and constrain the key parameters characterizing the relation, namely, the gas depletion time-scale, the mass loading factor, and the returned mass fraction. Thus, the first section of the paper collects observational constraints on these parameters (Sect. 2). Section 3 constitutes the main body of the work, and it describes observational evidence for a relationship between SFR and metal-poor gas accretion. Unfortunately, despite the large volume of circumstantial evidence for feeding from external gas accretion, we still lack direct evidence. Several factors explain the difficulty. They are pointed out and discussed in Sects. 4 and 5, where we also mention future lines of research.

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