Stellar feedback
Stellar feedback is a key component of cosmological simulations of structure formation. Among its many crucial roles, it prevents overcooling at high redshift, distributes energy in the ISM, drives metals out of the star formation sites, and triggers galactic outflows, which guarantee a continuous interaction between the forming galaxy and its surrounding CGM. Since the understanding of the pc-scale physics responsible for launching and driving winds is still partial and however far from being implemented directly in cosmological simulations, these simulations have to resort to phenomenological prescriptions to capture the effects of galactic outflows. As for the sources of energy that power outflows and govern their kinematic, there are two commonly pursued approaches: winds can be either energy-driven or momentum-driven. For details about these two possible scenarios, we refer the reader to 星系形成章节, which provides complementary information on the specific topic of this sub-section. This outline is far from being complete: deeper insight can be gained e.g. from [213] ,[155] ,[214] ,[215] .
A variety of stellar feedback schemes has been proposed in cosmological simulations: the numerical description of the energy injection from SN explosions in the ISM has indeed a strong impact on final results. How reliable different sub-resolution models are in capturing an effective description of feedback energy injection is still debated.
SN energy can be distributed by star-forming gas elements to the surroundings in the form of thermal or kinetic energy. Injecting feedback energy as thermal, as in the pioneer model of [17] , may have the limitation that the feedback usually does not result effective in driving energy far from regions where it is released. Gas elements surrounding those that provide energy have a typical temperature of $\sim 10^5$ K and density high enough to radiate energy almost immediately, their cooling time being short. As a result, feedback is not effective in counterbalancing cooling and in preventing excessive star formation. To overcome this weakness, a stochastic thermal feedback model has been proposed [216] , where gas receiving thermal feedback energy is heated up to a temperature high enough to guarantee the effectiveness of feedback. Specifically, the aforementioned model assumes that thermal energy released by SN explosions is injected in the surrounding medium using a selection criterion: the gas particles that experience feedback have to be heated up to a threshold temperature, which is a parameter of the model (whose value is usually close to $10^7$ K). Such a temperature increase ensures that the cooling time of a heated particle is longer than its sound-crossing time, so that heated gas can effectively leave the star formation site before it radiates all the energy away. It is worth noting that stellar feedback is actually implemented as a thermal channel in this model: however, the effective outcome is a galactic wind, because thermal energy is converted into momentum and hence outflows originate. This model for stellar feedback is for instance adopted by the EAGLE and FLAMINGO simulations [41] ,[6] .
Otherwise, stellar feedback energy is provided in the form of kinetic energy and used to boost the velocity of surrounding gas elements (e.g. [131] ,[217] ), which are kicked from their original position. They can eventually thermalise and radiate energy away, but later than in the thermal scenario by construction. To enforce the effectiveness of kinetic feedback schemes, particles that receive energy and sample galactic outflows (usually referred to as {\it{wind particels}}) are often decoupled from hydrodynamic interactions: in this way, they are prevented from being halted and from thermalising energy soon after they have been provided with.
An alternative stellar feedback model is represented by the so-called blast-wave feedback [218] ,[219] . Within this scheme, eligible particles are provided with thermal feedback energy, but are then prevented from cooling for a short period of time (typically few tens of Myr). The physical motivation behind this prescription stems from the evolution of a SN remnant in the ISM [220] . As soon as an exploding SN drives a blast wave, this undergoes a first phase of free expansion, followed by an adiabatic stage (the Sedov-Taylor phase) where radiative losses are negligible, before entering the radiative phase. Temporarily disabling cooling mimics the unresolved adiabatic phase, whose duration is estimated to be of order $30$ Myr; afterwards, gas is allowed to cool again. Alternatively, the switch off of cooling can be explained by assuming that the energy released by SN explosions generates turbulence at unresolved scales and is partially dissipated over few tens of Myr, thus hindering gas cooling [221] ,[222] . When implemented in cosmological simulations, this feedback prescription results effective in avoiding excessive star formation and also succeeds at producing a two-phase ISM, whose gas components are not in pressure equilibrium locally.
Feedback processes through which massive stars can affect the reservoir of gas of a galaxy stem not only from the energy deposition and momentum injection following SN explosions, but also from the ionizing effect that massive stars have before exploding (usually referred to as early stellar feedback; e.g. [179] ) and from stellar winds (e.g. [223] ,[224] ). Specifically, the early stellar feedback by [179] represents a UV ionization source, that increases the surrounding gas temperature and supplies heating and pressure support. This feedback channel pre-processes the star-forming ISM and facilitates the effectiveness of SNe at regulating star formation. In their simulations, [179] show how early stellar feedback helps in suppressing SF at high z (see also the NIHAO simulations to appreciate how early stellar feedback operates [137] ,[225] ).
In addition, the energy released by SNe II is not expected to be constant across cosmic time nor in all the star-forming regions, and theoretical models predict that SNe exploding in almost pristine or weakly enriched environments provide the ISM with a larger amount of stellar feedback energy. The idea of differentiating the outcome of stellar feedback according to the physical properties of the star-forming ISM has been already pursued in cosmological simulations, and a few effective prescriptions have been proposed. The basic idea consists in adopting a non-constant value of the stellar feedback efficiency (i.e. the fraction of energy provided by each SN that is actually coupled to the surrounding gas as feedback energy).
A metallicity- and density-dependent stellar feedback efficiency has been introduced in the EAGLE simulations [41] ,[226] . There, the aforementioned efficiency decreases with gas metallicity while increasing with gas density. The adoption of a similar parametrization has a twofold reason: $i)$ radiative losses are expected to increase with increasing metallicity; $ii)$ energy losses in high-density, star-forming regions can make the stellar feedback too inefficient, and have to be counterbalanced. Being the gas metallicity lower at higher redshifts, when also higher densities are usually reached in the star-forming ISM, such a prescription yields a redshift-dependent stellar feeback efficiency. The latter ranges between asymptotic values, which are parameters of the model tuned to reproduce low-redshift observables, e.g. the galaxy stellar mass function.
A similar parametrization is adopted in the Illustris-TNG simulation [139] , too. They assume that the wind energy available to a star-forming resolution element is metallicity-dependent. SNe II release indeed a feedback energy which spans the range $(0.9 - 3.6) \times 10^{51}$ erg, according to the metallicity of the star-forming gas cell in which they are expected to explode -- the lower the metallicity, the larger the energy budget. [139] demonstrate how the metallicity-dependent stellar feedback energy modulation has an impact both on the stellar-to-halo mass relation and on the galaxy stellar mass function, final results strongly depending on the tuning of the parameters. They show that the stellar mass of a $\sim 10^{12}$ M$_{ \odot}$ halo at $z=0$ can vary by up to a factor of $\sim 2$.
At a higher level of complexity, though in simulations targeting smaller volumes, [227] first modelled the simultaneous evolution of different stellar populations in cosmological simulations. Their runs feature stellar feedback dependent on the underlying gas metallicity and stellar population, where the injected energy and the assumed stellar yields depend on Population III versus Population II regimes, i.e. pair-instability SNe versus SNe. Interestingly, [228] included the effect of self-consistently coupled radiative transfer, concluding that radiative feedback from massive Population III stars could be a viable justification for powerful feedback in pristine environments.
Motivated by the aforementioned findings and by theoretical studies (e.g., [229] ,[230] for reviews) that suggest that hypernovae and SNe II of Population III stars can release energy higher than in the present-day Universe by more than a factor of $\sim10$, [160] introduced an effective low-metallicity feedback. In this implementation of metallicity-dependent stellar feedback, they accounted for the effect of the explosion of hypernovae and SNe II in weakly-enriched environments, by boosting the energy released by SNe exploding in an almost pristine ambient medium (i.e. in an ISM with average metallicity below a threshold).
The impact of different sub-grid prescriptions accounting for the same physical process (stellar feedback and SN-triggered galactic outflows in this specific case) on final results is striking, though often overlooked. [135] investigate the impact of stellar feedback modelling on the formation and evolution of a disc galaxy, by performing a suite of cosmological zoom-in simulations of a Milky Way-size halo. They show how sensitive the general properties of the simulated galaxy are to the way in which stellar feedback triggered outflows are implemented, comparing results obtained by adopting different state-of-the-art stellar feedback models (see Fig. 10). We refer the reader to [114] ,[115] for more extended comparison campaigns, i.e. the Aquila and the Agora comparison projects (see also Fig. 8). Interestingly, [215] also show how making different assumptions as for the stellar feedback efficiencies in the EAGLE and Illustris-TNG simulations reflects on the predicted galaxy stellar mass functions (see also [226] ,[139] ).
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