Gas-Regulator Model in Galaxy Evolution

• Gas-Regulator Model is a continuity-based framework that treats galaxies as open systems with a dynamic gas reservoir regulating star formation.
• It integrates key processes such as gas inflow, stellar recycling, and outflows using physically interpretable parameters like star-formation efficiency and mass-loading factor.
• The model explains galaxy scaling relations and CFIRB anisotropies, emphasizing the role of the equilibration timescale in governing star formation and metal enrichment.

The gas-regulator model, also called the “bathtub” model, is a simple physical framework for galaxy evolution in which the instantaneous star-formation rate is set by the mass of gas in a reservoir, while inflow, star formation, outflow, stellar mass return, and metal enrichment are coupled through continuity equations for gas, stars, and metals [(Lilly et al., 2013); (Peng et al., 2014); (Wu et al., 2016)]. In this formulation, the same minimal set of processes is used to connect the cosmic time evolution of the specific star-formation rate, the gas-phase metallicity and its dependence on star-formation rate, the stellar-to-halo mass relation, and, in an extended implementation, the anisotropies of the cosmic far-infrared background (CFIRB) [(Lilly et al., 2013); (Wu et al., 2016)].

1. Conceptual basis

In the regulator picture, a galaxy is treated as an open system containing a gas reservoir. Gas arrives at a rate $\Phi$ or $\dot M_a$, stars form out of the reservoir, some fraction of the newly formed stellar mass is rapidly recycled, and some gas is ejected in winds. The core premise is that the formation of stars is instantaneously regulated by the mass of gas in the reservoir, rather than being specified independently of the gas supply (Lilly et al., 2013).

The model is usually written in terms of a small set of physically interpretable parameters. These include the star-formation efficiency $\varepsilon$, the mass-loading factor $\lambda$ or $\eta$, the return fraction $R$, the nucleosynthetic yield $y$, the metallicity of infalling gas $Z_0$, and a baryon penetration or retention factor such as $f_{\rm gal}$ or $p$ that relates halo accretion to the inflow reaching the galaxy [(Lilly et al., 2013); (Wu et al., 2016)]. In this sense, the regulator is not merely a phenomenological star-formation law; it is a continuity-based description of the baryon cycle.

Lilly et al. used this framework to link together the evolving galaxy population’s specific star-formation rates, metallicities, and stellar halo content (Lilly et al., 2013). Later work derived the general analytic form of the evolution of gas mass, star formation rate, stellar mass, specific star-formation rate, gas fraction, gas-phase metallicity, and stellar metallicity without assuming equilibrium (Peng et al., 2014). Wu et al. then embedded the same physical picture in a halo-model treatment of CFIRB anisotropies, thereby extending the regulator formalism from galaxy scaling relations to infrared background clustering (Wu et al., 2016).

2. Governing equations

A standard formulation begins with the relations

$\dot M_a$0

$\dot M_a$1

$\dot M_a$2

and

$\dot M_a$3

For the gas-phase metal mass,

$\dot M_a$4

with $\dot M_a$5 (Peng et al., 2014).

In the formulation used to connect the regulator to halo growth,

$\dot M_a$6

where $\dot M_a$7 is the cosmic baryon fraction and $\dot M_a$8 is the fraction of halo-accreted baryons that reaches the galaxy reservoir (Peng et al., 2014). In the CFIRB implementation, the baryonic inflow is written as

$\dot M_a$9

where $\varepsilon$0 is a penetration factor and $\varepsilon$1 is the dark matter halo accretion rate (Wu et al., 2016).

The same structure can also be expressed directly in terms of gas, stars, and metals:

$\varepsilon$2

and, in quasi-steady state,

$\varepsilon$3

with gas-phase metallicity

$\varepsilon$4

In this notation, $\varepsilon$5 is the fraction of accreted baryons in gas and $\varepsilon$6 is the mass loading factor (Wu et al., 2016).

These equations encode the partition of inflow among star formation, outflows, and reservoir growth. In the notation of Lilly et al., the gas-to-stars ratio is

$\varepsilon$7

and the instantaneous fraction of incoming baryons turned into stars is

$\varepsilon$8

with corresponding expressions for the fractions ejected and stored in the reservoir (Lilly et al., 2013).

3. Equilibrium, quasi-equilibrium, and the central timescale

A central result of the non-equilibrium analysis is that the timescale

$\varepsilon$9

is the parameter that is essentially in control of the evolution of all key galaxy properties (Peng et al., 2014). Here $\lambda$0 is the gas depletion time.

For constant or slowly varying parameters, the time-dependent solutions are

$\lambda$1

$\lambda$2

and

$\lambda$3

In equilibrium, $\lambda$4, the star-formation rate becomes

$\lambda$5

The specific star-formation rate for the general solution, with $\lambda$6, is

$\lambda$7

and in equilibrium

$\lambda$8

The equilibrium gas-phase metallicity is

$\lambda$9

These analytic forms are valid both in and out of equilibrium (Peng et al., 2014).

The physical interpretation of $\eta$0 is direct. Short $\eta$1 corresponds to rapid equilibration, typical of massive galaxies with low gas fractions; long $\eta$2 corresponds to slow evolution, typical of gas-rich low-mass galaxies (Peng et al., 2014). The same paper emphasizes that the scatters in most key scaling relations, such as the mass–SFR relation and mass–metallicity relation, are primarily governed by $\eta$3, and that low-mass, gas-rich galaxies are very unlikely to live around the equilibrium state at any epoch (Peng et al., 2014).

4. Metallicity, specific star-formation rates, and the halo connection

A defining strength of the regulator model is that it links star formation, metallicity, and halo growth in the same set of equations. In the idealized case of constant $\eta$4 and $\eta$5, the specific star-formation rate of star-forming galaxies self-adjusts to match the galaxy’s specific gas inflow rate, written as

$\eta$6

Under plausible assumptions, $\eta$7 tracks the specific halo mass increase rate $\eta$8 (Lilly et al., 2013).

When the fraction of accreted gas converted to stars increases with mass, the reduced specific star-formation rate is boosted above the specific accretion rate. Lilly et al. summarize this as

$\eta$9

where $R$0 is the log-slope of $R$1 (Lilly et al., 2013). In the non-equilibrium treatment, the predicted specific star-formation rate history simply mimics that of the specific mass increase rate of the dark matter haloes, with the typical value of the specific star-formation rate around $R$2, and the difference between the minimum and maximum specific star-formation rate at any epoch less than a factor of four (Peng et al., 2014).

For metallicity, the regulator yields

$R$3

or, equivalently,

$R$4

This produces a $R$5 relation in which star-formation rate acts as a second parameter in the mass–metallicity relation, and it will be the same at all epochs unless the efficiency and mass-loading change with time, naturally producing a so-called “fundamental metallicity relation” (Lilly et al., 2013).

The observed $R$6 relation of SDSS galaxies implies a strong dependence of stellar mass on halo mass that reconciles the different faint-end slopes of the stellar and halo mass-functions in standard $R$7CDM (Lilly et al., 2013). In the same fit, the model finds $R$8 nearly constant gives the correct normalization of $R$9, while the fitted efficiency is consistent with observed molecular gas-depletion timescales and the fitted mass-loading is also plausible (Lilly et al., 2013). A central implication is that the declining stellar-to-halo mass ratio at the low-mass end arises from internal, baryonic processes, namely low efficiency and high mass loading, rather than primarily from suppression of halo feeding (Lilly et al., 2013).

5. Non-equilibrium behavior and recognized limitations

The regulator model is often described as an equilibrium framework, but one of the main technical lessons of the later analytic treatment is that non-equilibrium evolution cannot be neglected. The evolution of gas mass, star-formation rate, gas fraction, gas-phase metallicity, and stellar metallicity all contain explicit factors of $y$0, and these terms remain important whenever $y$1 (Peng et al., 2014). This is especially relevant for low-mass, gas-rich galaxies, for which $y$2 can be comparable to or larger than the Hubble time (Peng et al., 2014).

The principal tension identified in this literature concerns the cosmic specific star-formation-rate history. The predicted history is smooth and monotonic, closely tied to $y$3 and the Hubble rate, whereas the observed specific star-formation-rate history shows a sharp rise to $y$4 and a turnover (Peng et al., 2014). The model cannot prevent excessive early star formation, so by the time $y$5 is reached, galaxies are overmassive and the specific star-formation rate is too low (Peng et al., 2014).

Attempts to repair this discrepancy by tuning $y$6 or $y$7 were argued to require unphysical values, such as extremely high mass-loading or near-zero efficiency, and to create inconsistencies with gas fractions or with outflows never observed (Peng et al., 2014). The conclusion stated in that analysis is that some key process is missing in both typical semi-analytic models and the gas regulator model (Peng et al., 2014). The only viable route identified there is to introduce a time-dependent, highly suppressed baryonic inflow, that is, a reduction in $y$8 in the earliest epochs (Peng et al., 2014). This suggests that the regulator formalism is structurally useful but not, by itself, a complete account of galaxy evolution.

6. Extension to CFIRB anisotropies

Wu et al. applied the gas regulator model to the cosmic far-infrared background, treating the CFIRB as a probe of the history of star formation rate and of the connection between baryons and dark matter across cosmic time (Wu et al., 2016). In that implementation, the galaxy-evolution model is embedded in the halo model, with the specific intensity

$y$9

and the emissivity

$Z_0$0

The infrared luminosity is tied to star formation through

$Z_0$1

with

$Z_0$2

The gas mass is written as

$Z_0$3

and the mass-loading factor is parameterized as a function of halo mass with a characteristic mass $Z_0$4 and slopes $Z_0$5 and $Z_0$6 controlling low- and high-mass feedback (Wu et al., 2016).

The CFIRB angular power spectrum is decomposed into two-halo, one-halo, and shot-noise terms:

$Z_0$7

with

$Z_0$8

$Z_0$9

and

$f_{\rm gal}$0

The effective bias is

$f_{\rm gal}$1

A notable feature of this construction is that galaxy bias is not a free parameter; it emerges from the $f_{\rm gal}$2–halo mass relation and halo bias (Wu et al., 2016).

The model was fit to the Planck CFIRB auto- and cross-power spectra at 217, 353, 545, and 857 GHz, with both the two-halo and shot-noise terms modeled self-consistently, using Markov Chain Monte Carlo in a six-dimensional parameter space $f_{\rm gal}$3 (Wu et al., 2016). Additional tests included CFIRB–CMB lensing cross-correlations, bolometric infrared luminosity functions, submillimetre source counts, and the CFIRB intensity (Wu et al., 2016).

The main physical conclusions from that application are specific. The strong clustering of the CFIRB indicates a large galaxy bias, which corresponds to haloes of mass $f_{\rm gal}$4 at $f_{\rm gal}$5, higher than the mass associated with the peak of the star formation efficiency (Wu et al., 2016). The far-infrared luminosities of haloes above $f_{\rm gal}$6 are higher than the expectation from the star-formation rate observed in ultraviolet and optical surveys (Wu et al., 2016). The model agrees with the compilation of Madau & Dickinson (2014) at $f_{\rm gal}$7, but predicts higher star-formation-rate density at high redshift, and it predicts characteristic dust temperatures increasing towards higher and lower halo masses (Wu et al., 2016).

The same study also states clear limitations. Because of the simplicity of the physical model, the goodness of fit is limited (Wu et al., 2016). The regulator-based CFIRB model underpredicts bright Herschel counts and the bright end of the infrared luminosity function, overpredicts small-scale CFIRB power seen by Herschel, uses a single-temperature modified blackbody spectral energy distribution that may be too simple, and does not explicitly include starburst and lensed sources (Wu et al., 2016). A plausible implication is that while the gas-regulator framework provides a physically transparent mapping from baryon cycling to infrared clustering, more complex feedback, dust physics, and source populations are required for a fully accurate description.