Understanding the radio luminosity function of star-forming galaxies and its cosmological evolution

The galform version that we use (Lacey et al., 2016, see also Cole et al. 2000; Baugh et al. 2005; Bower et al. 2006) incorporates theoretically and empirically motivated models for complex physical processes (star formation, supernova, and AGN feedback, various dynamical processes, the evolution of the gas, stars, and dust, etc.) that baryons undergo in DM haloes. The assembly histories of haloes are described by halo merger trees extracted from $N$-body simulations (Guo et al., 2013). The model successfully reproduces the observed K-band, optical, near-infrared, and far-UV luminosity functions of SFGs, far-IR number counts and H i and stellar mass functions over a wide range of redshifts along with the Tully–Fisher, metallicity–luminosity and size–luminosity relations at $z=0$. Nearly all radio-bright SFGs are spirals (Sadler et al., 2002; Condon et al., 2002). Therefore, as in 108, we only consider disc galaxies in the output of galform, which are defined as galaxies where the bulge mass is less than half the total stellar mass.

galform outputs several global properties of galaxies at every snapshot of the $N$-body simulation, like the SFR, stellar mass $M_{\ast}$, the cold gas masses of the disc and the bulge, and the disc half-mass radius ($r_{1/2}$), which is assumed to be the same for stars and gas. Several ISM parameters for magnetizer are derived from these galform outputs in the same way as in 108, as briefly discussed below, but we refer the reader to that paper for further details.

108 focused on the galactic mean fields and therefore selected galaxies with a large disc that can host the mean-field dynamo in a relatively large volume. However, as we discuss below, random magnetic fields generated independently of the mean-field dynamo action make a significant contribution to the total-intensity RLF, and central parts of galaxies can be very bright in the synchrotron. Therefore, we have changed the sample selection criterion and include all galaxies that have a gas disc, either large or small. The motivation is that a higher local gas density in the disc would lead to star formation, turbulence, and, hence, significant magnetic and cosmic ray energy densities.

Large-scale (mean) $\boldsymbol{\overline{B}}$ and small-scale (random) $\boldsymbol{b}$ magnetic fields contribute comparably to the galactic synchrotron emission (here and elsewhere, a bar above a variable denotes ensemble or volume average). In nearby galaxies, the ratio of their mean energy densities, $\overline{b^{2}}/\overline{B}^{2}$, often significantly exceeds unity (Beck, 2015a; Beck et al., 2019), but this is subject to a range of assumptions involved in the interpretation of Faraday rotation and synchrotron observations, including the assumption of energy equipartition between cosmic rays and magnetic fields. Beyond semi-quantitative observational and theoretical estimates, the absolute and relative strengths of the two parts of an interstellar magnetic field remain somewhat uncertain.

The results of the mean-field dynamo theory used in the text are introduced in Appendix A; further details and references can be found in Shukurov & Subramanian (2021), among a number of other books and reviews (Brandenburg & Subramanian, 2005, e.g.,).

The mid-radio ($1\text{--}10\,\text{GHz}$) continuum emission of an SFG consists of the non-thermal (synchrotron) emission due to relativistic (cosmic ray) electrons and the free–free emission from thermal electrons. The radio emission at $1.4\,\text{GHz}$ is dominated by the synchrotron, with the thermal emission contributing about 10 per cent (Tabatabaei et al., 2017), and we neglect the thermal contribution to the RLF (see Schober et al. 2023 for a discussion of the thermal contribution at other frequencies).

We assume that the relativistic electrons have an isotropic energy spectrum

with a factor $K_{E}$, where $N(E)\,\mathrm{d}E$ is the number density of electrons with energy between $E$ and $E+\mathrm{d}E$ and $s$ is the spectral index. The synchrotron emissivity (the energy emitted by the unit volume of the source per unit time and unit frequency interval within the unit solid angle) in a homogeneous magnetic field $\boldsymbol{B}$ is then given by (Ginzburg & Syrovatskii, 1964; Rybicki & Lightman, 1979, e.g.,)

where we use the standard notation for physical constants, $\nu$ is the emission frequency ($=1.4\,\text{GHz}$ in our case), $B_{\perp}$ is the total magnetic field strength in the sky plane perpendicular to the line of sight, and

where $\Gamma$ is the gamma-function (see chapter 3 of Shukurov & Subramanian 2021 for details). This expression is applicable to a partially ordered magnetic field, $\boldsymbol{B}=\boldsymbol{\overline{B}}+\boldsymbol{b}$ in our case, unless the field is inhomogeneous on very small scales: the synchrotron emission pulse from a relativistic electron is formed over a distance of order $r_{B}/\gamma\simeq 10^{9}\,\text{cm}(B/1\,\text{{\textmu}G})^{-1}$ (where $r_{B}$ is the electron Larmor radius and $\gamma$ is the particle Lorentz factor), and magnetic fluctuations at larger scales do not affect the applicability of equation (9). We adopt $s=3$ for the electron spectrum, close to the observed value (Bisschoff & Potgieter, 2014; Bisschoff et al., 2019). For $s=3$, the statistically averaged value of $\varepsilon$ involves the average of $b_{\perp}^{2}$, a quantity predictable theoretically as discussed in Section 2.2.1.

To derive the synchrotron luminosity, the coefficient $K_{E}$ in equation (8) has to be determined. An often-adopted assumption regarding the energy density of cosmic rays (dominated by cosmic ray protons) is their energy equipartition with the magnetic field,

where $\kappa_{\text{cr}}$ is the ratio of the energy densities of the relativistic protons and electrons. For $E_{2}\to\infty$, this yields $K_{E}=(s-2)B^{2}E_{1}^{s-2}/(8\pi\kappa_{\text{cr}})$ and, for $s=3$ and $\kappa_{\text{cr}}=100$ (Beck & Krause, 2005, e.g.,), we have

For $E_{1}$, we adopt the energy above which the electron spectrum measured in the Solar vicinity has the spectral index $s\approx 3$, $E_{1}\approx 8\,\text{GeV}$ (see fig. 10.10 of Shukurov & Subramanian, 2021, and references therein). Since cosmic rays have a very high diffusivity (Longair, 1994, e.g.,), we assume that their energy density depends on the spatially averaged magnetic field and use $B^{2}=\overline{B}^{2}+b_{\text{rms}}^{2}$ in equation (12).

The assumption of the local energy equipartition between cosmic rays and magnetic fields lacks any general and convincing evidence. Both the observed synchrotron intensity fluctuations in galaxies (Stepanov et al., 2014) and simulations of cosmic ray propagation rather suggest a weak anticorrelation between their energy densities (Tharakkal et al., 2023c; Tharakkal et al., 2023b; Tharakkal et al., 2023a). However, there are no readily available, simple alternative approaches to estimate the energy density of cosmic rays in galaxies. For the exploratory study of this paper, we adopt the equipartition assumption but a refinement of our model would include an estimate based on the SFR and a cosmic ray propagation model.

Using the large-scale and small-scale magnetic fields computed for each galaxy, we derive $B_{\perp}$ and obtain the synchrotron luminosity assuming that the galaxies in the sample have random orientations with respect to the line of sight, (we use the uniform distribution of $\cos i$, where $i$ is the inclination angle of a galaxy to the line of sight),

where $\Omega$ is the solid angle. The method for computing the integral in equation (13) is described in Appendix B.

The turbulent speed $v$ is among the model parameters that affect directly the RLF because the strength of the magnetic fields on both large and small scales increases in proportion to the turbulent kinetic energy via their dependence on $B_{\mathrm{eq}}$ through equation (3), and the large-scale field strength has additional dependence on $v$ via the $\alpha$ effect and turbulent diffusion. The models $v$25-R19 and $v(z)$-J23 are designed to explore the role of this parameter.

Apart from the fiducial model results, Fig. 6 also shows the prediction of the $v$25-R19 model (dotted black) which has the constant turbulent speed $v=25\,\,\text{km}\,\text{s}^{-1}$ in all galaxies, in contrast to the fiducial model where $v$ is larger in galaxies with $\text{SFR}\geq 3\,\,\text{M}_{\odot}\,\text{yr}^{-1}$ (Section 2.1.2). The two models differ little in the predicted RLFs at low luminosities, especially at low redshifts where $\text{SFR}<3\,\text{M}_{\odot}\,\text{yr}^{-1}$ and hence $v=25\,\,\text{km}\,\text{s}^{-1}$ in most galaxies. However, the increase in the SFR up to $z\simeq 2$ makes the $v$25-R19 model predictions differ from those of the fiducial model, and they become less and less adequate with increasing $z$, severely under-predicting the number density of galaxies at the high luminosity end. Even if the overall luminosity were increased by a factor by adjusting model parameters (as discussed in Section 5), the shape of the $v$25-R19 RLF does not match the data as well as the fiducial model.

Figure 7 compares the fiducial model with the model $v(z)$-J23 (dotted black), where $v$ is an explicit function of the redshift taken from the bottom panel of Fig. 1. Neither the overall shape nor the magnitude of the resulting RLFs agree with the observations: this model is inferior to the fiducial one.

In fact, neither the fiducial nor the $v(z)$-J23 model is fully justified because model $v(z)$-J23 does not capture the SFR-dependence of turbulence at constant $z$, while the fiducial model does not allow for such effects as the gravitational instability (Krumholz et al., 2018) and gas accretion and mergers (Ginzburg et al., 2022) which are likely to result in an additional dependence of the turbulent intensity on redshift independently of the SFR. A more realistic model for turbulence will be among the refinements of our model to follow.

Under the assumption of energy equipartition between cosmic rays and magnetic field adopted here, the contributions of the mean and random magnetic fields to the RLF are not additive because the radio luminosity includes the integrals of $(\boldsymbol{\overline{B}}_{\perp}\cdot\boldsymbol{b}_{\perp})$, $\overline{B}^{2}b_{\text{rms}}^{2}$ and $\overline{B}_{\perp}^{2}b_{\text{rms}}^{2}$. Therefore, to assess the significance of the mean and random magnetic fields, we consider results obtained when one of them is ignored. In Fig. 7, we show the predictions with only the random field (dash-dotted cyan line) or the mean field (dashed blue line) retained. Both random and mean fields are clearly important. However, the RLF that relies solely on the mean magnetic field decreases with the luminosity too rapidly and by $z\approx 1$ the contribution from the random field dominates that of the mean-field at least at those high luminosities where observational data are available. On the other hand, at $z\approx 0$, we obtain a significantly better fit to the data by neglecting the random field and including only the mean field, particularly at low luminosities where the fiducial model over-predicts the RLF.

We stress that the relation between the contributions of the random and mean magnetic field components to the RLF depends on poorly constrained parameters like $f_{B}$ for the random field and the magnitude of the mean helicity of interstellar turbulence for the mean field. Since the radio luminosity scales as the fourth power of the magnetic field (Section 4), the relative influence of the random and mean magnetic fields is rather sensitive to such parameters. Clearly, the same magnitude of the luminosity function could be obtained by changing the parameter values so as to boost one field component and reduce the other. Therefore, the relative importance of mean and random field is not well-constrained by the model and for realistic parameter values, neither can be neglected.

The current interpretations of the observations of the synchrotron emission in nearby galaxies suggest that the energy density of the random magnetic field exceeds that of the large-scale field by a factor of three or even larger (Beck et al., 2019). However, the interpretations rely on the assumption of the local energy equipartition between cosmic rays and magnetic fields, and so are model-dependent.

To conclude, there are several plausible reasons for the apparent complexity in the relative roles of the mean and random magnetic fields at different redshifts. It is likely that a better model for cosmic rays is necessary (e.g. because their number density might be more sensitive to the mean magnetic field than to its random part). Another reason could be our limited understanding of the interaction of the fluctuation and mean-field dynamos (Shukurov & Subramanian, 2021, chapter 8 of). Both types of dynamo action are sensitive to the multi-phase ISM structure but in a poorly understood manner while it is clear that the ISM structure depends on both the SFR and redshift. Remarkably, the RLF can shed light on rather subtle aspects of both the ISM structure and galactic dynamos. The relative importance of mean and random fields thus deserves careful analysis beyond the scope of this paper.

The redshift evolution of the observed RLF for bright star-forming galaxies ($L_{1.4}\gtrsim 10^{21}\,\text{W}\,\text{Hz}^{-1}$) is discussed by Novak et al. (2017) and van der Vlugt et al. (2022); van der Vlugt et al. (2023); their observational data are shown in Figs 6 and 7 together with our results. Both the fiducial model (where the turbulent speed increases with the SFR) and the observations agree that the range of the radio luminosities extends to larger $L_{1.4}$ as $z$ increases. The galaxy luminosity functions observed in far infrared (Gruppioni et al., 2013; Koprowski et al., 2017; Lim et al., 2020) and K-band (Cirasuolo et al., 2010; Mortlock et al., 2017) evolve similarly with the redshift. However, the RLF shows a slightly stronger evolution compared to that in the infrared as the infrared-to-radio luminosity ratio is observed to decrease mildly with increasing redshift (Delhaize et al., 2017; Calistro Rivera et al., 2017; Magnelli et al., 2015; Ivison et al., 2010).

Both the size of the emission region and the strength of the magnetic field contribute to the increase of the synchrotron luminosity with $z$. This is illustrated in the left-hand panels of Fig. 5, which presents the colour-gradient scatter plot of the radio luminosity of a thousand randomly selected galaxies with $L_{1.4}\geq 10^{21}\,$WHz^-1 from our sample at different redshifts as a function of their disc volume and $B^{4}_{0}$ (as $L_{1.4}\propto B^{4}$ under the assumption of energy equipartition between cosmic rays and magnetic fields). It is clear that the brightest galaxies typically have both large emitting volume and volume-averaged magnetic field strength. For example, the volume-averaged magnetic field strength of the brightest galaxies at $z=1.5$ is about a factor of two larger than at $z=0$

Certain factors that contribute to stronger magnetic fields at high redshifts are clarified in the right-hand panels of Fig. 5. Galaxies with larger $M_{\ast}$ tend to contain stronger magnetic fields, for reasons explained in Section 3. The SFR density has a maximum at $z\simeq 1.0$ (Fig. 2), indicating high SFR in galaxies at high redshifts. This results in a higher turbulent speed in galaxies via equation 2 leading to a systematic increase of the magnetic field strength with $z$ at $0\lesssim z\lesssim 1.5$.

Thus, our fiducial model finds a shift to higher luminosities with increasing $z$, in agreement with the observations up to $z\approx 1.8$. However, as mentioned at the beginning of Section 4, the $z$-dependent shift in the model RLF is weaker than that inferred from observations. The median predictions of the model seem to underestimate the observed RLF at $z\gtrsim 1$, and the difference between them increases with increasing $z$, and at $z\gtrsim 2$ they clearly disagree for reasons discussed in the next section.

The energy density of the random magnetic field is proportional to the energy density of the turbulence with the proportionality factor $f_{b}^{2}$ of equation (3) only known to be of order unity. Adopting $f_{b}=1$, we find a reasonable agreement between the predicted RLF and observations at the redshifts $z\lesssim 1.2$ (see Fig. 6). Our knowledge of the strength and structure of turbulent magnetic fields largely relies on numerical simulations severely limited to modest Reynolds numbers. For example, adopting $f_{b}=1.5$ improves significantly the agreement at higher redshifts. Values of $f_{b}$ exceeding unity can be due to a stronger magnetic helicity since partially helical fields are close to the force-free state, so their back-reaction on the flow is reduced, which allows them to have a higher energy density. Thus, our model might fit the observations much better, had we assumed that $f_{b}$ increases slightly with $z$. Moreover, $f_{b}$ can well depend on various galactic parameters and be sensitive to the multi-phase ISM structure rather than be a constant or a simple function of $z$. Our results highlight the need to understand better how random magnetic fields are maintained in galaxies. The theory of the mean magnetic fields is developed better although many aspects of the action of the mean-field dynamo in the multi-phase, evolving ISM remain speculative. Since the energy density of both the turbulent and large-scale magnetic fields depends on the turbulent energy density, the gas density and turbulent speed are identified as major factors affecting the RLF. However, the mean magnetic field has a more complicated dependence on the galactic parameters than its random part, including the differential rotation and the gaseous disc thickness in addition to the turbulent energy density. The energy density of the mean magnetic field (unlike the random field in our model) can exceed the turbulent energy density. As a result, the total magnetic energy density in a significant fraction of galaxies (especially massive ones) exceeds the turbulent energy density (Rodrigues et al., 2019, see also fig. 6 of). The RLF in polarised emission would help to isolate the contribution of the mean magnetic field to the synchrotron luminosity and thus assess rather subtle properties of evolving galaxies. Our predictions for the polarised RLF will be published elsewhere.

The predictions of our fiducial model, which assumes that the structure of a galaxy is broadly similar to a spiral galaxy in the local Universe, are reasonable until $z\approx 2$, the redshift at which galaxies appear to develop persistent thin discs dominated by rotation similar to those at $z\approx 0$ (Ginzburg et al., 2022; Jiménez et al., 2023, and references therein). Our models implicitly assume that the disc is formed instantaneously whereas the history of the disc formation can affect the outcome of the evolution in the strongly nonlinear galactic systems. Certain effects of enhanced star formation on the galactic structure and environment, such as outflows (fountains and winds) and the emergence of radio haloes, are not captured by our model which only includes magnetic fields in galactic discs. Such effects can affect high-redshift galaxies more profoundly and may help to explain why our model does worse at matching observations as $z$ increases. For example, galaxies with a high SFR may have radio haloes with vertical extent 5–$10\,\text{kpc}$ and magnetic field strength comparable to that in the disc. Thus, future improvements to the magnetic field model should include the radio halo and its contribution to the synchrotron luminosity.

To obtain the number density of cosmic ray electrons, we have adopted a widely used (albeit poorly justified) assumption of energy equipartition between cosmic rays and magnetic fields. However, the energy density of cosmic rays (including the number density of relativistic electrons) is likely to be an increasing function of the supernova rate directly related to the SFR. We shall explore elsewhere an alternative model where the abundance of cosmic rays is controlled by their sources and propagation in the evolving magnetic field.

The predicted RLF is also sensitive to the slope $s$ and amplitude K_E of the energy spectrum of cosmic ray electrons given by equation (8). Under the assumption of energy equipartition between cosmic rays and magnetic field, changes to the value of $s$ also affect $K_{E}$ (in proportion to $s-2$) and hence the predicted luminosity of all galaxies, shifting the RLF along the horizontal ($\log L$) axis. The shape of the RLF also can be mildly modified as the synchrotron emissivity is proportional to $B_{\perp}^{(s+1)/2}$ (see equation 9). Since the emissivity changes as $\nu^{-(s-1)/2}$, multi-wavelength radio observations can constrain $s$ and resolve the degeneracy between K_E and $s$.

The estimate of the synchrotron emissivity of equation (9) needs to be refined at high redshifts because the energy losses of relativistic electrons to the inverse Compton scattering off the CMB photons increase as $(1+z)^{4}$. The ratio of the rates at which a relativistic electron loses energy to the inverse Compton scattering and synchrotron is given by $8\pi\,w_{\text{CMB}}/B^{2}$, where $w_{\text{CMB}}$ is the CMB energy density (Shukurov & Subramanian, 2021, section 3.1.4 of). For electrons with the Lorentz factor $\gamma$, the frequency of the CMB photons is boosted by the factor $\gamma^{2}$, so $1\,\text{GeV}$ electrons emit in the X-ray range. Schleicher & Beck (2013a) suggest that the inverse Compton losses dominate over the synchrotron emission at $z\gtrsim 2$ if the typical strength of the galactic magnetic fields $B$ does not increase with $z$ but at a larger $z$ if $B$ increases with the redshift. The effect of the inverse Compton losses on our results at $z\leq 2$ is likely to be only modest, especially for the radio-bright galaxies.

It appears that the mechanisms of magnetic field generation, and the ISM structure on which they rely, experience a significant change at $z\simeq 2$. This is suggested, in particular, by the increase in the gas velocity dispersion with the SFR (Fig. 1) above the sound speeds in the diffuse ISM phases, $c_{\text{s}}\simeq 10\,\,\text{km}\,\text{s}^{-1}$ in the warm gas and $c_{\text{s}}\simeq 130\,\,\text{km}\,\text{s}^{-1}$ in the hot phase. Meanwhile, simulations of the supernova-driven multi-phase ISM show that the fractional volume of the hot gas at the galactic mid-plane increases only slightly (as $\text{SFR}^{0.36}$ assuming that the supernova rate is proportional to the SFR) from about 0.2 to 0.28 when the supernova rate increases by a factor of four in comparison with the Milky Way rate and does not exceed 0.5 when the supernova rate is 16 times the local rate, while the abundance of the cooler phases reduces as the supernova rate increases (de Avillez & Breitschwerdt, 2004; Breitschwerdt & de Avillez, 2021, and references therein). The main consequence of the increase in the SFR is a more vigorous outflow of the hot gas in the form of the galactic fountain or wind. Although the off-planar gas is hot and highly ionised, significant amounts of neutral hydrogen are present, entrained from the disc, or cooled down in situ). Its H i velocity dispersion is about $60\,\,\text{km}\,\text{s}^{-1}$ in the Milky Way (Kalberla, 2003, table 1 of) while the velocity dispersion of the hot ionised gas is $75\,\,\text{km}\,\text{s}^{-1}$ (Savage et al., 1997, section 8.3 of). A review of relevant observations and further references can be found in section 10.2.2 of Shukurov & Subramanian (2021). Meanwhile, the turbulent velocity in the disc hardly changes as the supernova rate varies by a factor 512, remaining close to the sound speed in the warm gas while the H i line width is larger than the (one-dimensional) turbulent velocity dispersion (Joung et al., 2009, see also Dib et al. 2006). Therefore, an increase of the galactic outflow intensity and an enhanced contribution of turbulence in the off-planar gas appear to be the dominant effects of the increased SFR on a spiral galaxy. In this context, the increase of the turbulent speed with the SFR adopted in the fiducial model can be interpreted as a reflection of the increasing importance of the off-planar gas with its more intense turbulence.

Similarly to 108, we assume that the rms turbulent speed is independent of the galactocentric distance $r$. Some observations of the H i velocity dispersion in spiral galaxies show that it decreases with the galactocentric radius $r$ from often supersonic values in the inner parts of galaxies (Boulanger & Viallefond, 1992; Petric & Rupen, 2007; Tamburro et al., 2009; Mogotsi et al., 2016). On the other hand, there is significant observational evidence for a very weak variation of the H i velocity dispersion with $r$ in the discs of spiral galaxies (Kamphuis, 1993, Dickey et al. 1990; Blitz & Spergel 1991, Section 12.2.3 of). It is plausible that the turbulent speed is indeed independent of the galactocentric radius and the spatial variation of the gas velocity dispersion is due to variations in the outflow intensity. This aspect of the model requires further analysis.

The models and observational estimates of the H i and H$\alpha$ velocity dispersions discussed in Section 2.1.2 are often interpreted as suggesting strongly supersonic turbulence, especially at high SFR. However, the turbulence in the diffuse (warm and hot) interstellar gas is likely to be transonic (or subsonic, depending on the kinetic energy injection rate) because of the self-regulation of the supersonic turbulence (Shukurov & Subramanian, 2021, Section 2.10.2 of). The kinetic energy of supersonic turbulence efficiently dissipates at shock fronts to heat the gas until an equilibrium state is reached in which the turbulent speed is comparable to the sound speed. Thus, turbulent flows with sufficiently strong energy injection rates are likely to be transonic. Such a self-regulation of a turbulent system can be affected by radiative cooling which can prevent the gas heating by removing the dissipated turbulent energy via radiation (Enrique Vázquez-Semadeni, private communication). The cooling is especially efficient in dense regions (like molecular clouds which occupy a negligible fraction of the disc volume), where supersonic turbulence can be maintained, but perhaps not in the warm and hot ISM phases which dominate the radio luminosity. The self-regulation of supersonic turbulence is not included in the turbulence models of Krumholz et al. (2018) and Ginzburg et al. (2022), who assume that the disc is marginally stable concerning the gravitational instability at all times and consider only the energy balance of turbulence driven by supernovae and accretion flows onto and through the disc. The sample of SFGs where Varidel et al. (2020) measured the vertical velocity dispersion in the H$\alpha$ spectral line deliberately includes only nearly face-on galaxies ($0<i<60^{\circ}$ for the inclination angle), so not only the turbulence in the galactic discs but also outflows and turbulence of the off-planar gas are likely to contribute to the velocity dispersion obtained.

The dependence of $v$ on the SFR, equation (2), is rather poorly constrained by the data available, and other fits can be equally acceptable. For example, adopting $\text{SFR}_{0}=1\,\text{M}_{\odot}\,\text{yr}^{-1}$ and $c=1/3$ in our fiducial model produces an RLF that is in reasonable agreement with observations, indicating a degeneracy between these parameters (lower values of the scaling exponent $c$ would require lower values of $\text{SFR}_{0}$). Furthermore, there are alternative prescriptions to derive the velocity dispersion from the star formation surface density (Niklas & Beck, 1997; Chyży et al., 2011; Schleicher & Beck, 2013b; Chamandy et al., 2024). Finally, the velocity dispersion may increase faster with redshift than expected from the SFR–redshift dependence alone (Wisnioski et al., 2015; Übler et al., 2019b), so a better understanding of the interstellar turbulence and the role of the evolving multi-phase ISM structure appears to be essential for further progress in modelling the RLF, which we will address in the future.