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The evidence discussed in Sect. 3 is all indirect. We have to rely on models to interpret the observables as evidence for gas accretion. The toy model setting up the scene in Sect. 1, and then used in many of the above arguments, oversimplifies many aspects of the accretion process that are important to identify in real galaxies the observational signatures of ongoing cosmological gas accretion. The purpose of this section is to point out some of the obvious complications of a more realistic modeling, and also to point out new observational pathways that may reveal connections between SF and the gas of the IGM.

The CGM and the galactic fountain. The CGM is a complex region where gas ejected from the galaxy and gas falling into the galaxy coexist. The gas inflow is not only of cosmological origin. Part of the metal-rich materials ejected in previous SF episodes fall back to the disk in what is called galactic fountain (e.g., Fraternali & Binney 2006; 2008; Hobbs et al. 2015). The mixing with metal-rich gas from SF processes speeds up the cooling of the hot metal-poor CGM, and the fountain also returns gas that was never ejected (Melioli et al. 2008; Marinacci et al. 2010; Marasco & Fraternali 2011; Marasco et al. 2012).

The structure of the CGM is extremely complicated according to the current numerical simulations. The morphology of the gas streams becomes increasingly complex at higher resolution, with large coherent flows revealing density and temperature structure at progressively smaller scales, and with no evidence that the substructure is properly captured in the simulations (Nelson et al. 1402016). Multiple gas components co-exist at the same radius within the halo, making radially averaged analyses misleading. This is particularly true where the hot quasi-static halo interacts with cold rapidly-inflowing IGM accretion. Some of the resulting complications are revealed in the study of metal-poor DLAs carried out by Yuan & Cen (2016). The majority of the metal-poor DLAs are far from the central galaxy (≥ 20 kpc) and result from the cold gas streams from the IGM. In the migration inwards to the galaxy, they mix up with high-metallicity gas from stellar outflows, removing themselves from the metal-poor category. The change from metal-poor to metal-rich complicates the observational identification of the gas coming from the IGM. IGM gas clouds that get mixed with the CGM and become metal-rich are also found in the simulation by Gritton et al. (2014). The difficulties of interpretation are also put forward by Ford et al. (2014). Using cosmological simulations, Ford et al. examine how H I and metal absorption lines trace the dynamical state of the CGM around low-redshift galaxies. Recycled wind material is preferentially found close to galaxies, and is more dominant in low-mass halos. Typical H I absorbers trace unenriched ambient material that is not participating in the baryon cycle, but stronger H I absorbers arise in cool, metal-enriched inflowing gas. Instantaneous radial velocity measurements are generally poor at distinguishing between inflowing and outflowing gas, except in the case of very recent outflows.

Galactic fountains and metal-rich gas produced in previous starbursts are very important because their presence complicates the search for metal-poor gas inflows in the CGM of galaxies. However, their role in sustaining SF should not be overestimated. Most of the gas used to produce stars at any time is pristine. It was never pre-processed by a star. This issue has been recently quantified by Segers et al. (2016) using galaxies from the EAGLE numerical simulation (Schaye et al. 2015). For MW-like galaxies, recycled stellar ejecta account for only 35 % of the SFR and 20 % of the stellar mass. The contribution was even less important in the past. The toy model in Sect. 1 provides the right order of magnitude for this estimate (see Sánchez Almeida et al. 2014a).

Star formation generated by gas accretion. The current cosmological numerical simulations produce model galaxies that look impressively realistic, and follow most of the well known scaling relations (e.g., Vogelsberger et al. 2014; Schaye et al. 2015). They provide the theoretical framework to understand the formation of galaxies and the role played by cosmological gas accretion in maintaining SF. However, their limited resolution and the dependence of many predictions on the adopted sub-grid physics make them less reliable to study how individual starbursts grow out of the gas that arrives to the galaxy disk. (Predictions on individual star-formation events are discussed in Sec. 3.3.) Improving this aspect is critically important to secure the interpretation of many observables currently used as evidence for cosmological gas accretion.

Imaging the cosmic web. Much of our knowledge on the CGM and IGM comes from observing absorption lines against background sources that happen to be next to galaxies. However, the observation and analysis of these absorption systems is extremely time-consuming, and even the best cases only provide a very sparse sampling of the CGM and IGM around individual galaxies. A complementary approach is observing the cosmic web gas in emission. The mechanisms to produce such emission are varied. Lyα (as well as Hα) can be produced by electron collisions within a gas stream that releases the gravitational energy gained as gas flows from the IGM into the galaxy halo (Dijkstra & Loeb 2009; Goerdt et al. 2010; Faucher-Giguère et al. 2010). Emission also results from fluorescence induced by an intense UV radiation field such as that produced by a large nearby starburst or a QSO (e.g., Hogan & Weymann 1987; Cantalupo et al. 2012; Ao et al. 2015).

Radio emission is also expected to trace the cosmic web. In this sense, the search for dark galaxies (i.e., objects emitting in 21 cm without optical counterpart) is a very revealing and active field of research (e.g., Cannon et al. 2014; Serra et al. 2015; Janesh et al. 2015; Janowiecki et al. 2015). Gas filaments associated with star-forming galaxies are very interesting too (Sancisi et al. 2008; Lelli et al. 2012; Filho et al. 2013). Radio data easily provide kinematical information, which is so important when investigating flows. In this context, a large filament of molecular gas accreting onto a group of massive high redshift galaxies has been recently discovered by Ginolfi et al. (2016, private communication). This observation is intriguing, but it may open up a new way of addressing the search for IGM gas.

Extremely promising is the recent discovery of an extended Lyα blob connected to a QSO (Cantalupo et al. 2014). The emission extends beyond the virial radius of the host galaxy so that it traces gas in the IGM. The existence of extended Lyα emission around QSOs seems to be very common when the observation is deep enough (Borisova et al. 2016). The sensitivity is very much improved using spectrographic observations, which also have the capability of providing the eagerly needed kinematical information (see the chapter by Cantalupo in this Book). Extended Lyα halos are common around all kinds of galaxies (e.g., Rauch et al. 2016, Momose et al. 2016).

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