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

Charles Jose1, Luke Chamandy2,3, Anvar Shukurov4, Kandaswamy Subramanian5,6,
Luiz Felippe S. Rodrigues7 and Carlton M. Baugh8
1Department of Physics, CUSAT, Cochin, 682022, India
2National Institute of Science Education and Research, An OCC of Homi Bhabha
National Institute, Bhubaneswar 752050, Odisha, India
3Department of Physics and Astronomy, University of Rochester, Rochester NY
14627, USA
4School of Mathematics, Statistics and Physics, Newcastle University,
Newcastle upon Tyne, NE1 7RU, UK
5IUCAA, Post Bag 4, Ganeshkhind, Pune 411007, India
6Department of Physics, Ashoka University, Rajiv Gandhi Education City, Rai,
Sonipat 131029, Haryana, India
7HAL24K Agri, Uitmeentsestraat 19, 6987 CX Giesbeek, Netherlands
8Institute for Computational Cosmology, Department of Physics, University of
Durham, South Road, Durham DH1 3LE, UK
charles.jose@cusat.ac.inlchamandy<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We explore the redshift evolution of the radio luminosity function (RLF) of
star-forming galaxies using galform, a semi-analytic model of galaxy formation
(SAMGF) and magnetizer, a dynamo model of the magnetic field evolving in a
galaxy through its formation history. Assuming energy equipartition between
the magnetic field and cosmic rays, we derive the synchrotron luminosity of
each sample galaxy. In a model where the turbulent speed is correlated with
the star formation rate, the RLF is in fair agreement with observations in the
redshift range $0\leq z\leq 2$. At larger redshifts, the structure of
galaxies, their interstellar matter and turbulence appear to be rather
different from those at $z\lesssim 2$, so that the turbulence and magnetic
field models applicable at low redshifts become inadequate. The strong
redshift evolution of the RLF at $0\leq z\leq 2$ can be attributed to an
increased number, at high redshift, of galaxies with large disc volumes and
strong magnetic fields. On the other hand, in models where the turbulent speed
in galaxies is a constant or assumed to be an explicit function of the
redshift, the redshift evolution of the RLF is poorly captured. The evolution
of the interstellar turbulence and outflow parameters appear to be major (but
not the only) drivers of the RLF changes. We find that both the small- and
large-scale magnetic fields contribute significantly to the RLF but the small-
scale field dominates at high redshifts. Polarisation observations will
therefore be important to distinguish these two components and understand
better the evolution of galaxies and their nonthermal constituents.

###### keywords:

galaxies: evolution – galaxies: spiral – radio continuum: galaxies – galaxies:
luminosity function – galaxies: magnetic fields – dynamo

††pubyear: 2024††pagerange: 1–C

##  1 Introduction

The galaxy luminosity function (LF) – the comoving number density of galaxies
as a function of their magnitude at a given wavelength – is one of the primary
observables used to probe the physics of formation and evolution of galaxies.
The observed LFs and their redshift evolution have been measured, to various
degrees of detail and reliability, for a wide range of wavelengths from the UV
to the radio. When assessed in the framework of a theoretical model of galaxy
formation, the LFs provide key insights into the evolution of star formation
and feedback processes, the nature of and conditions in the ISM (consisting of
gas, dust, magnetic fields, and cosmic rays), and dynamical processes such as
galaxy mergers. Two major approaches used to model galaxy formation are: (i)
semi-analytic models (SAMGFs) that account for the complex baryonic processes
in evolving dark matter (DM) haloes derived from $N$-body simulations using a
combination of analytic approximations and numerical prescriptions, (ii)
hydrodynamic or magnetohydrodynamic (MHD) simulations based on fundamental
dynamical equations, that include subgrid models to account for unresolved
physical processes(often similar to those used in SAMGFs). There are several
galaxy formation models (Trayford et al., 2015; Trčka et al., 2022; Croton et
al., 2006; Samui et al., 2007; Jose et al., 2013; Lacey et al., 2016) that
interpret the ultra-violet (UV) (Wyder et al., 2005; Page et al., 2021),
optical (Blanton et al., 2003; Loveday et al., 2012) and infrared (IR) (Dunne
et al., 2011; Gruppioni et al., 2013; Marchetti et al., 2016) LFs and shed
light on the evolution of obscured star formation rate (SFR), stellar mass
($M_{\ast}$), ISM, dust and feedback processes in galaxy populations.

Complementary to the UV, optical and IR LFs, is the radio LF (RLF) of star-
forming galaxies (SFGs) at the emission frequency $1.4\,\text{GHz}$ obtained
with the Very Large Array (VLA) and the Westerbork Synthesis Radio Telescope
(WSRT) (Adams & van Leeuwen, 2019; Condon et al., 1998). Recent estimates of
RLFs of SFGs at $1.4\,\text{GHz}$ include those of Sadler et al. (2002);
Condon et al. (2002); Best et al. (2005); Mauch & Sadler (2007); Padovani et
al. (2011) and Condon et al. (2019) for the local Universe and from Smolčić et
al. (2009); Pracy et al. (2016); Novak et al. (2017); Bonato et al. (2021);
Malefahlo et al. (2022); Enia et al. (2022) and van der Vlugt et al. (2022)
for redshifts up to $z\simeq 4.5$. Moreover, several ongoing and upcoming deep
surveys using present and next-generation radio telescopes, including the
Square Kilometre Array (SKA), will improve our knowledge of the RLFs of SFGs
to higher redshifts and fainter luminosities with unprecedented accuracy
(Norris et al., 2013; Adams & van Leeuwen, 2019). For example, deep surveys
using SKA Phase-I will detect sub-$\upmu\text{Jy}$ SFGs up to $z\simeq 6$
which will help to measure RLFs down to luminosities more than an order of
magnitude lower than what is available now (Jarvis et al., 2015).

Synchrotron emission dominates the radio continuum at $1.4\,\text{GHz}$
(Ginzburg & Syrovatskii, 1965), so models of the galactic magnetic field and
cosmic ray propagation should be the starting points in a prediction of the
RLF. MHD simulations of evolving galaxies (e.g., Liu et al., 2022; Pfrommer et
al., 2022) cannot provide the statistically significant galaxy samples needed
to compute RLFs. Moreover, limited spatial resolution prevents such
simulations from probing scales as small as 1–$100\,\text{pc}$, which is
crucial for capturing dynamo processes at the turbulent and global scales, and
hence realistic magnetic fields and, consequently, any realistic distribution
of cosmic rays controlled by them (see section 13.14 of Shukurov &
Subramanian, 2021, for a review). Turbulent magnetic fields, produced by the
fluctuation dynamo, supernova shock fronts, and tangling of the large-scale
magnetic field, require a spatial resolution of order $1\,\text{pc}$ to be
realistically reproduced (Gent et al., 2021, 2023), whereas the mean-field
dynamo, that generates large-scale galactic magnetic fields, relies, apart
from overall rotation, on the density stratification at scales of order
$100\,\text{pc}$ (chapter 11 of Shukurov & Subramanian, 2021). MHD simulations
at an adequate resolution are only available for local ISM regions of order a
few kiloparsecs in size. (e.g. Gressel et al., 2008; Gent et al., 2013b, a;
Hollins et al., 2017; Gent et al., 2024), and subgrid dynamo models are not
yet available despite an effort to relate the statistical properties of
magnetic fields in nearby galaxies to theories (Van Eck et al. 2015; Chamandy
et al. 2016; see Beck et al., 2019, for a review).

Therefore, at present, combining SAMGFs with a model for the evolution of
magnetic fields and cosmic rays is arguably the only viable theoretical
approach for modelling RLFs.

The earliest effort towards incorporating galactic dynamo theory into a
hierarchical model for galaxy formation in cold DM cosmology to produce
galactic magnetic fields in a cosmological volume-sized sample of galaxies was
by Rodrigues et al. (2015). Rodrigues et al. (2019) (hereafter R19) extended
this model into a computational framework called magnetizer (Rodrigues &
Chamandy, 2020) which simulates magnetic fields in individual galaxies as they
evolve. In particular, these authors coupled the magnetizer with the output of
the galform SAMGF of Lacey et al. (2016) and Gonzalez-Perez et al. (2014) to
obtain simulated magnetic fields for a large sample of galaxies in the
redshift range $0\leq z\leq 6$ and compared the predictions of mean-field
strengths and pitch angles with observations of local galaxies, finding
reasonable agreement.

The present work extends the model of R19, to derive the RLF of SFGs at the
rest-frame frequency of $1.4\,\text{GHz}$ over cosmic history. We calculate
the total radio luminosity due to the synchrotron emission based on our
magnetic field and cosmic ray models. Any significant differences of this work
from the model of R19 are mentioned in the text.

Key quantities that affect the amplification and sustenance of galactic
magnetic fields are the gas density and root-mean-square (rms) turbulent
speed, as well as the galactic rotation and stratification (the gas scale
height). Some observational studies indicate that the gas velocity dispersion
in galaxies (which includes a contribution of the turbulence) is correlated
with the SFR surface density (SFRD) (Green et al., 2010; Lehnert et al., 2013;
Moiseev et al., 2015) while other authors find a correlation with the global
SFR (Genzel et al., 2011; Varidel et al., 2016; Varidel et al., 2020; Zhou et
al., 2017; Übler et al., 2019a).

We investigate how the predicted shape and redshift evolution of the RLF are
affected by various phenomenological ISM turbulence models and compare the
results with observations. By contrast, past works have probed the radio
luminosity of isolated template galaxies and used the radio-FIR correlation to
deduce the RLF (e.g. Werhahn et al., 2021; Pfrommer et al., 2022; Vollmer et
al., 2022; Schober et al., 2023).

The RLFs obtained from observations contain contributions from both active
galactic nuclei (AGN) and SFGs (Mauch & Sadler, 2007; van der Vlugt et al.,
2022). The physical nature and processes responsible for the radio emission in
these objects are rather different and should be considered separately. These
contributions can be separated, with varying degrees of confidence, using
multi-wavelength observations to estimate the RLF of each. In this work, we
model the RLF of star-forming disc galaxies only.

This paper is organized as follows. In Section 2, we discuss our models to
compute the RLF of galaxies at $1.4\,\text{GHz}$. We present the predictions
for the magnetic field in Section 3, and for the RLF in Section 4 along with a
comparison with observational data and a discussion of the roles of various
physical processes involved. Our results are put into a wider context in
Section 5 and conclusions are formulated in Section 6. Throughout this work,
we use cosmological parameters based on seven-year WMAP data (Komatsu et al.,
2011), which are also used in the galform model of Lacey et al. (2016). We use
the same parameter values in magnetizer as R19 unless otherwise stated.

##  2 The Model

The RLFs presented in this work are the products of a three-level model. In
the first level, the output of the galform SAMGF of Lacey et al. (2016) is
used to derive the properties of a representative galaxy population of about
$7\times 10^{5}$ disc galaxies in a comoving volume of $6\times 10^{6}$ Mpc3
at each redshift in the redshift range $0\leq\ z\leq 6$ (corresponding to 3.5%
of the total volume of the N-body simulation used to provide halo merger
histories), with stellar masses between about $10^{8}\,\text{M}_{\odot}$ and
$10^{12}\,\text{M}_{\odot}$. In the second level, magnetizer is used to couple
the galactic dynamo theory and the galaxy formation model to predict the
evolution of both turbulent and global magnetic fields as a function of time
and galactocentric radius in each galaxy. To avoid excessive and unnecessary
complications, all galaxy properties are assumed to be axisymmetric; the
dominance of axially symmetric large-scale magnetic fields in nearby galaxies
is well established (e.g., Beck et al., 2019). In the final third stage, the
synchrotron emission at $1.4\,\text{GHz}$ is computed for each galaxy to
derive the RLF over a wide redshift range.

###  2.1 Galaxy formation model

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 R19, 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. Several ISM parameters for magnetizer are derived from
the galform output in the same way as in R19 as briefly discussed below, but
we refer the reader to that paper for further details.

####  2.1.1 Derived galactic quantities

To obtain the RLF, we need to deduce the distribution of the gas density and
the magnetic field along the galactocentric distance $r$ and distance $Z$ from
the midplane in each galaxy, as well as its rotation curve and the gas scale
height.

The galaxy rotation curve, $V(r)$, is computed by assuming that galaxies have
a thin stellar disc with an exponential surface mass density profile, a bulge
with the Hernquist (1990) profile and a DM halo with an adiabatically
contracted Navarro–Frenk–White density profile (Navarro et al., 1997). The
rotation curve is then used to determine the galaxy angular velocity,
$\Omega(r)$, and the rotational shear rate,
$S(r)=r\,\mathrm{d}\Omega/\mathrm{d}r$.

As in R19, we adjust the half-mass radius $r_{1/2}$ of the galform galaxies to
reproduce the observed stellar mass–$r_{1/2}$ relation of Lange et al. (2016)
and $r_{\text{d}}=2.7r_{1/2}$ is adopted as the maximum gas disc radius for
computing quantities like the volume-averaged magnetic field strength.

The surface mass densities of the stars and gas in the disc are both assumed
to have an exponential radial profile with the scale length
$r_{\mathrm{s}}=r_{1/2}/a$, where $a\approx 1.68$.111To obtain $a$, we note
that
$M(r)=2\pi\Sigma_{0}r_{\mathrm{s}}^{2}\int_{0}^{r/r_{\mathrm{s}}}xe^{-x}dx$
for the mass of either stars or gas within the cylindrical radius $r$, where
$\Sigma_{0}$ is the surface density at $r=0$. As noted in the text, the scale
length $r_{\mathrm{s}}$ is assumed to be the same for both stars and total
gas.Since $\int_{0}^{\infty}xe^{-x}\,\mathrm{d}x=1$, we have
$M_{\infty}\equiv\lim_{r\rightarrow\infty}M(r)=2\pi\Sigma_{0}r_{\mathrm{s}}^{2}$.Now,
$M(r_{1/2})=M_{\infty}/2$ by definition, which leads to
$\int_{0}^{a}x\mathrm{e}^{-x}\,\mathrm{d}x=1/2$ and $a$ follows. As in R19,
the disc gas mass is separated into diffuse and ‘molecular’ phases. The ratio
of the surface densities of the diffuse and molecular gas components is
computed using the empirical relation from Blitz & Rosolowsky (2004, 2006).
The density distributions of diffuse gas, molecular gas and stars
perpendicular to the galactic disc are assumed to be exponential with
different scale heights. Thus, for example, for the diffuse gas density we
have

$\rho_{\mathrm{d}}(r,Z)=\rho_{\mathrm{d0}}(r)\mathrm{e}^{-|Z|/h_{\mathrm{d}}(r)}\,,$
(1)

where $Z$ is the distance from the galaxy midplane, $\rho_{\mathrm{d0}}(r)$ is
the midplane density and $h_{\text{d}}(r)$ is the density scale height, both
functions of the galactocentric radius $r$, determined assuming that the ISM
is in hydrostatic equilibrium with a total pressure that includes the thermal,
turbulent, magnetic and cosmic-ray contributions. We use empirical relations
for the scale heights of the molecular gas ($0.032r_{\mathrm{s}}$) and stars
($0.1r_{\mathrm{s}}$) (Kregel et al., 2002; Licquia & Newman, 2016).

####  2.1.2 Interstellar turbulence

Galactic magnetic fields are amplified and sustained by a turbulent dynamo.
Thus, magnetic field models depend on certain statistical properties of the
turbulence, like the correlation length $l$, correlation time $\tau$, and rms
turbulent velocity $v$. As in R19, we adopt
$l(r)=\min[100\,\text{pc},h_{\mathrm{d}}(r)]$ and $\tau=l/v$ (for estimates of
interstellar turbulence parameters, see Chamandy & Shukurov, 2020).

Thus far, the magnetizer model described is the same as that of R19. However,
R19 assumed that the rms turbulent speed is
$v=10\,\,\text{km}\,\text{s}^{-1}$, close to the sound speed in the warm gas
and independent of the redshift and SFR. However, the gas velocity dispersion
(which includes an uncertain contribution from the turbulence) appears to
depend on the intensity of star formation (see Fig. 1 and the text below).
Given this uncertainty, we considered three alternative prescriptions for
determining the turbulent speed.

Figure 1:  Top panel: the observed three-dimensional velocity dispersion as a
function of SFR (Krumholz et al. 2018 and references therein, circles) and the
best-fitting curves of the form of equation (2) to the median (solid/green)
and the 16th (dash-dot/red) and 84th (dashed black) percentiles. The dotted
blue line corresponds to $v=25\,\,\text{km}\,\text{s}^{-1}$. Bottom panel: The
turbulent speed of the cold interstellar gas ($T\leq 10\,\text{K}$) as a
function of redshift from the EAGLE cosmological simulations for $z\leq 4$
(solid line: Jiménez et al., 2023), extrapolated to $z>4$ as a constant
(dotted).

Interstellar turbulence has various drivers, including star formation and the
associated supernova activity, gravitational instability (Krumholz & Burkhart,
2016; Krumholz et al., 2018), cosmological gas accretion, and galaxy mergers
(Jiménez et al., 2023; Ginzburg et al., 2022). Some models (e.g., Krumholz et
al., 2018; Ginzburg et al., 2022) predict that the turbulent speed is
independent of the SFR when $\text{SFR}<\text{SFR}_{0}$ with a certain
threshold $\text{SFR}_{0}$ and increases with the SFR otherwise. The velocity
dispersion data, consistent with more recent observations (Varidel et al.,
2020; Yu et al., 2021), are shown in the upper panel of Fig. 1, and we use the
form

$v=v_{0}\begin{cases}1\,,&\text{if }\text{SFR}\leq\text{SFR}_{0}\,,\\\
(\text{SFR}/\text{SFR}_{0})^{c}\,,&\text{otherwise},\end{cases}$ (2)

with certain constants $v_{0}$ and $c$. As suggested by fig. 3 of Yu et al.
(2021), we adopt $\text{SFR}_{0}=3\,\text{M}_{\odot}\,\text{yr}^{-1}$ and
estimate $v_{0}=25\,\,\text{km}\,\text{s}^{-1}$ and $c=0.50$ by fitting the
median of the relation shown in Fig. 1. The fitted dependence is shown as a
solid curve in Fig. 1, where the one-dimensional velocity dispersion $\sigma$
has been converted to the three-dimensional speed $v=\sigma\sqrt{3}$ assuming
the turbulence to be isotropic. Further, $v$ is assumed to be independent of
$r$ for simplicity.

The data points in Fig. 1 have a large scatter. To account for this, we fit
equation (2) to the $16^{\text{th}}$ and $84^{\text{th}}$ percentiles of $v$
as a function of SFR, to obtain $v_{0}=17\,\,\text{km}\,\text{s}^{-1}$ and
$c=0.45$ for the $16^{\text{th}}$ percentile and
$v_{0}=40\,\,\text{km}\,\text{s}^{-1}$ and $c=0.55$ for the $84^{\text{th}}$
percentile, also shown in Fig. 1. These measures of the scatter are used to
estimate the degree of uncertainty of the predicted luminosity functions. We
note that the SFR-$v$ relation is assumed to be independent of redshift in
this model, which we adopt as the fiducial model.

The turbulent speed of $v_{0}=25\,\,\text{km}\,\text{s}^{-1}$ can be
understood as the volume average of the rms turbulent speed in the multi-phase
ISM. As an illustration, if the turbulence is transonic in both warm and hot
phases, and their respective fractional volumes and sound speeds are
$f_{\text{w}}=0.9$, $c_{\text{w}}=10\,\,\text{km}\,\text{s}^{-1}$ and
$f_{\text{h}}=0.1$, $c_{\text{h}}=100\,\,\text{km}\,\text{s}^{-1}$, the volume
average follows as
$(f_{\text{w}}c_{\text{w}}^{2}+f_{\text{h}}c_{\text{h}}^{2})^{1/2}\approx
33\,\,\text{km}\,\text{s}^{-1}$. We note, however, that the fractional volumes
of the ISM phases are likely to vary between galaxies with different SFRs and
between various locations within a given galaxy.

As an alternative, we consider a model with a constant turbulent speed
$v=25\,\,\text{km}\,\text{s}^{-1}$ suggested at lower SFRs by Fig. 1,
independently of the SFR and redshift (whereas R19 used
$v=10\,\,\text{km}\,\text{s}^{-1}$). For the gas number density
$0.1\,\text{cm}^{-3}$ of the warm interstellar gas, the magnetic field
strength at energy equipartition with the turbulence is then about
$4\,\text{{\textmu}G}$, close to the strength of the random magnetic field
observed in nearby spiral galaxies (see Section 2.2.1). The increase in the
velocity dispersion at large SFR in Fig. 1 can be interpreted to arise partly
from chaotic galactic fountains and winds, which are especially vigorous in
galaxies with high SFR. This model is called $v$25-R19 below.

The velocity dispersion is observed to be correlated with other parameters in
addition to the SFR. These include the gas fraction, DM halo mass and the
stellar mass (e.g., Krumholz et al., 2018; Ginzburg et al., 2022). This
implies that the SFR–$v$ correlation evolves with the redshift, which has been
verified by the observations of Übler et al. (2019a) and also found by Jiménez
et al. (2023) using the EAGLE hydrodynamic simulations (Schaye et al., 2015).
Therefore, we also consider a third model, referred to as $v(z)$-J23, where
$v$ is an explicit function of $z$, as given by the REF-L100 model of Jiménez
et al. (2023) and plotted in the bottom panel of Fig. 1. In this model, $v$
does not depend on the SFR explicitly.

Figure 2: Top panel: the best-fitting dependence of the observed SFRD on the
redshift from Hopkins & Beacom (2006) (dashed/blue) and the SFRD from Lacey et
al. (2016) (solid/red). Bottom panel: the ratio of the SFRDs shown in the top
panel.

####  2.1.3 Correction to the SFR from used in galform

In the version of galform described in Lacey et al. (2016), stars form out of
cold gas in the disc and spheroid, and the corresponding SFR is calculated
using the prescription of Blitz & Rosolowsky (2006). The comoving SFR density
(SFRD), which is the SFR per unit comoving volume, from Lacey et al. (2016) is
shown as a function of redshift in the top panel of Fig. 2. Also shown in the
figure is the best fit to the observed redshift evolution of the SFRD given by
Hopkins & Beacom (2006). Both the theoretical and observed SFRDs are computed
after assuming the IMF of Kennicutt (1983) (see section 6.3 of Lacey et al.,
2016, for details). It is clear from Fig. 2 that, while observations and
predictions agree quite well at $z=0$, the predicted SFRD lies below the fit
to the observations at all redshifts, with their ratio shown in the lower
panel. To account for this discrepancy and reconcile the SFR with that
observed, we multiply the SFR of galaxies from galform at a given redshift
with this ratio, taken at the same redshift.

###  2.2 Magnetic field

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.

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

####  2.2.1 The random (small-scale) field

Random (turbulent) interstellar magnetic fields are amplified by the
fluctuation (or small-scale) dynamo, the tangling of the large-scale (or mean)
magnetic field by turbulence and compression, in particular at random shock
fronts.

The fluctuation dynamo amplifies the random component of the magnetic field on
time scales comparable to the turbulent eddy turnover time. Order-of-magnitude
estimates suggest that the random field reaches energy equipartition with
turbulence after a time of the order of $10\,\text{Myr}$ (e.g., Beck et al.
1994, and sections 6.7 and 13.3 of Shukurov & Subramanian, 2021). Since this
time is short compared to the time scales on which galaxies evolve, we assume
that at any given time and position within the galaxy the random field has
already saturated and its rms strength is proportional to $B_{\mathrm{eq}}$,
the field strength corresponding to equipartition with the local turbulent
kinetic energy density,

$b_{\text{rms}}=f_{b}B_{\mathrm{eq}}\,,$ (3)

where

$B_{\mathrm{eq}}=(4\pi\rho)^{1/2}v\,,$ (4)

$\rho$ is the gas density (a function of $r$ and $Z$ as well as of the
redshift) and $f_{b}$ is a constant.

The value of $f_{b}$ remains uncertain. The fluctuation dynamo produces
intense magnetic ropes and filaments that do not fill the volume and the rms
field strength depends on both the field strength within the filaments and
their fractional volume. Most existing simulations of the fluctuation dynamo
suggest $f_{b}\simeq 0.5$ (Brandenburg & Subramanian 2005, see also section
13.3 of Shukurov & Subramanian, 2021, and references therein) in
incompressible turbulence and even lower in supersonic flows (e.g., Federrath
et al., 2014). However, the simulations are severely limited to modest values
of the kinetic and magnetic Reynolds numbers, while $f_{b}$ can depend on them
and is likely to do so. Much stronger, super-equipartition magnetic fields
with $f_{b}>1$ cannot be excluded. For example, this would be the case as long
as the fluctuation dynamo produces locally helical magnetic fields thus
diminishing the associated Lorentz force and their back-reaction on the
velocity field (see section 4.2 of Belyanin et al., 1994, for a heuristic
model). The ISM multi-phase structure further complicates the action of the
fluctuation dynamo in galaxies (Gent et al., 2023).

The tangling of the mean (large-scale) magnetic field by the turbulence (an
integral part of the mean-field dynamo action) is thought to produce a volume-
filling random magnetic field of a strength comparable to that of the mean-
field (section 13.3 of Shukurov & Subramanian, 2021). Since the mean magnetic
field can be stronger than $B_{\mathrm{eq}}$ if the mean-field dynamo is
strong, e.g., in the central parts of galaxies where the differential rotation
is especially strong (section 13.7 of Shukurov & Subramanian, 2021), this may
lead to $f_{b}>1$.

Compression of magnetic fields at random shock fronts driven by supernovae can
produce very strong local random magnetic fields but their fractional volume
is likely to be small; nevertheless, the resulting rms field strength can
correspond to $f_{b}\simeq 1$ (Bykov & Toptygin, 1987).

We adopt $f_{b}=1$, whereas R19 used $0.5$. As the random field is taken to be
isotropic,
$\overline{b^{2}_{r}}=\overline{b^{2}_{\phi}}=\overline{b^{2}_{Z}}=b_{\text{rms}}^{2}/3$
for the cylindrical field components.

####  2.2.2 The mean (large-scale) field

The mean-field dynamo, by contrast, can amplify a large-scale magnetic field
$\boldsymbol{\overline{B}}$ on length scales of the order of a kiloparsec or
larger, on time scales that depend on the galactic rotation period, velocity
shear due to the differential rotation and the time scale of the turbulent
magnetic diffusion across the gas layer. The relevant parameters can be
combined into the dynamo number of equation (16). The amplification time of
the mean field ranges from a fraction of a gigayear in the inner parts of
spiral galaxies to a few gigayears in the outer parts. We simulate the
evolution of the mean component of the magnetic field in galactic discs by
solving numerically the mean induction equation supplemented by a dynamical
equation that models the non-linear back-reaction from the Lorentz force on
the velocity field that quenches the dynamo to establish a stationary magnetic
field. The resulting strength of the large-scale magnetic field is
proportional to $B_{\mathrm{eq}}$ and also depends on the dynamo number and
other parameters (Shukurov & Subramanian, 2021). The equations solved and the
numerical implementation can be found in R19, and they rely on the thin-disc
approximation in the mean-field dynamo equations applicable for $h/r\lesssim
0.1$ (Baryshnikova et al., 1987). The upper left panel of Fig. 3 shows that
this ratio is less than $0.4$ at the half-mass radius for the vast majority of
galaxies, and even smaller for the galaxies in the larger stellar mass range
and at a high redshift. To ensure that the solutions of the dynamo equations
are computed as fast as required with a large sample of galaxies, we employ
well-tested (chapters 11 and 13 of Shukurov & Subramanian, 2021, see also
Appendix A) approximations applicable to a thin disc (R19 and references
therein), including the no-$Z$ approximation, which replaces $Z$ (the distance
from the midplane) derivatives with divisions by the appropriately scaled gas
scale height, allowing the equations to depend on only one spatial dimension
($r$).

The initial (seed) magnetic field required to launch the mean-field dynamo is
the large-scale component of the random magnetic field produced by the
fluctuation dynamo: it does not vanish when the random field is averaged over
a finite volume of the galactic disc. This provides an initial kiloparsec-
scale random magnetic field of the order of $10^{-9}\,\text{G}$ in strength,
stronger than any plausible primordial magnetic field (Ruzmaikin et al. 1988;
Poezd et al. 1993; Bhat et al. 2016; Gent et al. 2024; see chapter 13 of
Shukurov & Subramanian, 2021, for a review). Thus, we used a Gaussian random
variable with zero mean and standard deviation $10^{-2}B_{\mathrm{eq}}(r)$ as
the seed for each mean magnetic field component ($\overline{B}_{r}$ and
$\overline{B}_{\phi}$) at each radius. This seed is somewhat larger than that
used in R19; however, we use the same prescription as R19 for preventing the
mean field from decaying to very low values when the dynamo is subcritical.

When an evolving galaxy experiences a major merger (where the ratio of the
combined stellar and cold gas masses involved exceeds $0.3$), the mean
magnetic field is reset to the seed field assuming conservatively that such a
merger destroys the large-scale field or distorts it to such an extent that it
is so different from an eigenfunction of the dynamo equations that most of it
decay rapidly because of the turbulent diffusion and the dynamo starts again
with a weak seed magnetic field described above.

Figure 3: The probability distributions of the relative gas disc thickness
$h_{\text{d}}(r_{1/2})/r_{1/2}$ (upper left), volume-averaged mean (upper
right) and random (lower left) magnetic field strengths, and the ratio of
$\overline{B}_{0}$ to $B_{\text{eq,0}}$ (lower right). The magnetic field
parameters presented are defined in equations (6) and (7). At each redshift,
the galaxies are separated into two stellar mass bins defined in the legend.
The area under each curve is proportional to the number of spiral galaxies in
the given mass bin at the given redshift. Figure 4: The galaxy number density
(in comoving coordinates) as a function of various volume-averaged magnetic
field properties defined in equations (6) and (7) for the mean (solid) and
random (dotted) fields (upper panel) and the ratio of $\overline{B}_{0}$ to
$B_{\text{eq,0}}$ (lower panel) for various redshifts.

####  2.2.3 Field distribution across the disc

The procedures described above specify the variation of the random and mean
magnetic fields along the galactocentric distance in each galaxy of the
sample, but the field distribution across the gas layer (in $Z$) also has to
be specified. In our fiducial model, the scale heights, $h_{B}$, of both the
random and the mean magnetic field are assumed to be the same as that of the
diffuse gas density $h_{\text{d}}$. Thus, for example, the distribution in $Z$
of the mean field is adopted as

$\boldsymbol{\overline{B}}(r,Z)=\boldsymbol{\overline{B}}_{\text{M}}(r)\mathrm{e}^{-|Z|/h_{B}}\,,$
(5)

where $h_{B}=h_{\text{d}}$ and $\boldsymbol{\overline{B}}_{\text{M}}(r)$ is
the solution of the mean-field dynamo equations produced using magnetizer.

To provide a simple measure of the magnetic field strength, we use its
strength $B_{0}$ corresponding to the volume-averaged total magnetic energy
density,

$B_{0}=\left[\dfrac{\int_{0}^{r_{\text{d}}}B^{2}h_{\text{d}}\,r\,\mathrm{d}r}{\int_{0}^{r_{\text{d}}}h_{\text{d}}\,r\,\mathrm{d}r}\right]^{1/2},\;$
(6)

for the total magnetic field $\boldsymbol{B}$ (where
$B\equiv|\boldsymbol{B}|$) and similarly $\overline{B}_{0}$ for its mean part
and $b_{0}$ for the random part. We also introduce a similar quantity for the
equipartition magnetic field of equation (4),

$B_{\text{eq,0}}=\left[\dfrac{\int_{0}^{r_{\text{d}}}B_{\mathrm{eq}}^{2}h_{\text{d}}\,r\,\mathrm{d}r}{\int_{0}^{r_{\text{d}}}h_{\text{d}}\,r\,\mathrm{d}r}\right]^{1/2}.$
(7)

###  2.3 The synchrotron emission

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

$N(E)\,\mathrm{d}E=K_{E}E^{-s}\,\mathrm{d}E\,,$ (8)

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
(e.g., Ginzburg & Syrovatskii, 1964; Rybicki & Lightman, 1979)

$\varepsilon=K_{E}a(s)\frac{e^{3}}{m_{\text{e}}c^{2}}\left(\frac{3e}{4\pi
m_{\text{e}}^{3}c^{5}}\right)^{(s-1)/2}B_{\perp}^{(s+1)/2}\nu^{-(s-1)/2}\,,$
(9)

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

$a(s)=\frac{\sqrt{3}}{4\pi(s+1)}\,\Gamma\left(\frac{3s-1}{12}\right)\Gamma\left(\frac{3s+19}{12}\right),$
(10)

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,

$\kappa_{\text{cr}}\int_{E_{1}}^{E_{2}}EN(E)\,\mathrm{d}E=\frac{B^{2}}{8\pi}\
,$ (11)

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
$\kappa_{\text{cr}}=100$ (e.g., Beck & Krause, 2005), we have

$K_{E}\simeq 6.3\times
10^{-19}\frac{\text{particles}}{\text{cm}^{3}\,\text{\,\text{erg}}^{-2}}\left(\frac{B}{1\,\text{{\textmu}G}}\right)^{2}\frac{E_{1}}{1\,\text{GeV}}\,.$
(12)

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 (e.g., Longair,
1994), 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, b, a). 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),

$\displaystyle L$
$\displaystyle=\int\mathrm{d}\Omega\int_{V}\varepsilon(\boldsymbol{r})\,\mathrm{d}^{3}\boldsymbol{r}$
$\displaystyle=8\pi\int_{0}^{r_{\text{d}}}r\,\mathrm{d}r\int_{0}^{2\pi}\mathrm{d}\phi\int_{0}^{h_{\text{d}}}\mathrm{d}Z\,\varepsilon(r,\phi,Z)\,,$
(13)

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

##  3 Magnetic fields in evolving galaxies

The top panel of Fig. 4 shows the number density of galaxies in the fiducial
model as a function of the volume-averaged field strengths of the mean and
random magnetic fields, $\overline{B}_{0}$ and $b_{0}$ defined in equation (6)
in each galaxy at various redshifts. At $z=0$, galaxies with
$1\,\text{{\textmu}G}\lesssim b_{0}\lesssim 5\,\text{{\textmu}G}$ are more
common than galaxies with $\overline{B}_{0}$ in the same range, while galaxies
with $\overline{B}_{0}<0.4\,\text{{\textmu}G}$ or
$\overline{B}_{0}>10\,\text{{\textmu}G}$ are far more common than those with
$b_{0}$ in that range. With redshift, these limits shift to higher values of
magnetic field strength. At lower redshifts, the number density of galaxies as
a function of $b_{0}$ has two distinct maxima. The distribution of
$\overline{B}_{0}$ is wider than that of $b_{0}$ because $b_{\text{rms}}$ is
directly related to $B_{\mathrm{eq}}$ through equation (4), whereas
$\overline{B}$ depends on other galactic parameters as well. The lower panel
of Fig. 4 shows the number density of galaxies as a function of
$\overline{B}_{0}/B_{\mathrm{eq,0}}$ and since
$b_{\text{rms}}=B_{\mathrm{eq}}$ for $f_{b}=1$ (Section 2.2.1) and hence
$b_{0}=B_{\text{eq,0}}$, it is clear that most galaxies in our model have a
stronger random magnetic field compared to the mean field. As a larger volume-
averaged field strength typically results in a higher synchrotron luminosity,
magnetic fields in radio-bright galaxies are predominantly random, especially
at high redshift (see also Section 4).

Figure 3 shows the evolution of the probability density of the mean and random
magnetic field components in the redshift range $0<z<3$ with galaxies grouped
into two stellar mass bins, $8.5\leq\log(M_{\ast}/\\!\,\text{M}_{\odot})<10.5$
and $\log(M_{\ast}/\\!\,\text{M}_{\odot})\geq 10.5$. The probability density
is normalized such that the total probability over all redshifts is unity in
each mass bin, separately. This allows us to visualize the relative number of
galaxies with a given magnetic field both at each redshift and between
different redshifts. The highest mass bin contains about $3\%$ of the total
number of galaxies in our sample, and the number of galaxies in a given mass
bin changes with $z$ owing to star formation, mergers, etc. The probability
distribution of $\overline{B}_{0}$ has a single peak, with most galaxies
having $\overline{B}_{0}>0.1\,\text{{\textmu}G}$ at all redshifts. This is
different from the results of R19 their fig. 6, where a significant fraction
of galaxies have very weak magnetic fields of the order of
$10^{-4}\,\text{{\textmu}G}$ (the strength of the seed field) because the
mean-field dynamo is subcritical in some galaxies, particularly at higher
redshifts. The mean-field dynamo is significantly stronger in our fiducial
model because of the higher turbulent speed $v$. Stronger turbulent flows
promote the mean-field amplification despite the increased turbulent diffusion
because the dynamo number in the kinematic regime scales as
$h_{\text{d}}^{2}/v^{2}$ (see Appendix A) and $h_{\mathrm{d}}$ increases with
$v$ faster than linearly. Indeed, under hydrostatic equilibrium, the disc
scale height is proportional to the gas pressure, which includes the turbulent
contribution $\rho v^{2}$, so $h_{\text{d}}$ increases with $v$ approximately
as $v^{2}$, and therefore, the dynamo number increases with $v$.

Figures 4 and 3 also show that the mean magnetic field across a sample of
galaxies decreases with time because of the depletion of the interstellar gas
(although individual galaxies may have monotonically increasing field strength
depending on their formation history), which is consistent with the results of
R19 and Rodrigues et al. (2015). We also find, as R19 do, that the mean field
strength shows a stronger scatter for galaxies with higher stellar masses.
However, a new feature of the present model is that galaxies with larger
stellar masses have stronger mean magnetic fields, on average, whereas in R19
there was less of a distinction. Galaxies with higher stellar mass generally
have higher SFR, as seen in the right-hand panels of Fig. 5, where we show the
scatter plot of $M_{\ast}$ versus SFR, with data points coloured according to
the volume-averaged total magnetic field strength. The correlation between the
SFR and $M_{\ast}$ (the so-called galaxy main sequence) is consistent with
several observational studies and galaxy formation models (Davé, 2008; Davies
et al., 2017; Bauer et al., 2011; Whitaker et al., 2014; Sparre et al., 2015;
Katsianis et al., 2016; Cowley et al., 2017). In our fiducial model, $v$
increases with SFR. The strength of the saturated mean field increases with
$v$ because of the increased dynamo number and the enhancement of
$B_{\mathrm{eq}}$ of equation (4). This explains the increase of
$\overline{B}_{0}$ with the stellar mass seen in Fig. 3. The increase of the
total magnetic field strength $B_{0}$ with the stellar mass, evident in Fig.
5, is explained by the increase of both $\overline{B}_{0}$ and $b_{0}$
($\equiv B_{\mathrm{eq,0}}$) with $M_{*}$.

Figure 3 also shows the probability density of the volume-averaged random
magnetic field strength $b_{0}$. Similarly to $\overline{B}_{0}$, galaxies of
a higher stellar mass have larger $b_{0}$. This can be explained by the higher
SFR of such galaxies, hence higher $v$ according to equation (2) and larger
$B_{\mathrm{eq}}$ according to equation (4) (we have confirmed that the
correlation of $\rho$ with $M_{*}$ is very weak). The distribution of $b_{0}$
is bimodal for low-mass galaxies, but not for galaxies of a high mass, for
reasons not yet quite clear to us.

The lower panel of Fig. 3 shows the distribution of the ratio of the
characteristic field strengths $\overline{B}_{0}$ and $B_{\text{eq,0}}$
introduced in equations (6) and (7). The ratio
$\overline{B}_{0}/B_{\text{eq,0}}$ tells us how the volume-averaged magnetic
energy density of a galaxy $\overline{B}_{0}^{2}/8\pi$ compares to its
averaged turbulent energy density $B_{\text{eq,0}}^{2}/8\pi$ and how close is
the mean-field dynamo to the saturated state.

The ratio $\overline{B}_{0}/B_{\text{eq,0}}$ is sensitive to the distribution
of the magnetic field within the disc. The energy density of the turbulence,
given by $B_{\mathrm{eq}}^{2}/8\pi$ from equation (4), decreases with the
galactocentric radius $r$ because of the reduction in the gas density $\rho$
alone as we assume that the turbulent speed $v$ is independent of $r$. The
strength of the mean magnetic field, while generally decreasing with $r$, can
have a different radial distribution as it depends, in particular, on the
angular velocity and shear rate of the galactic rotation. As an illustration,
consider the radial dependencies $\overline{B}=\tilde{B}\exp(-r/r_{B})$,
$B_{\mathrm{eq}}=\tilde{B}_{\mathrm{eq}}\exp(-r/r_{\mathrm{eq}})$ and
$h_{\text{d}}=\tilde{h}_{\text{d}}\mathrm{e}^{r/r_{h}}$ with certain length
scales $r_{B}$, $r_{\mathrm{eq}}$ and $r_{h}$ for $0\leq r<\infty$ (and the
variables with the tilde denoting the central values, those at $r=0$), so that

$\frac{\overline{B}_{0}}{B_{\text{eq,0}}}=\frac{\tilde{B}}{\tilde{B}_{\mathrm{eq}}}\,\left|\frac{2r_{h}/r_{\mathrm{eq}}-1}{2r_{h}/r_{B}-1}\right|\,.$
(14)

At all redshifts, the probability distributions of
$\overline{B}_{0}/B_{\text{eq,0}}$ have extended tails, especially in the case
of massive galaxies, where this ratio exceeds unity (up to three) suggesting
that most galaxies have $\tilde{B}_{0}/\tilde{B}_{\mathrm{eq}}\gtrsim 1$
and/or $r_{B}\gtrsim r_{\mathrm{eq}}$. The mean magnetic field strength likely
exceeds $B_{\mathrm{eq}}$ in the central parts of galaxies where the dynamo is
the strongest. A very slow decrease in the strength of the mean magnetic field
with $r$, $r_{B}\gg r_{\mathrm{eq}}$, has been observed in nearby galaxies
(section 4.2 of Beck, 2015b, and references therein). A large radial extent of
magnetised region in spiral galaxies is explained by the slow decrease of the
dynamo number with $r$ because of the disc flaring which compensates for the
relatively slow decrease of the rotation rate in a flat rotation curve, as can
be seen from equation (17). The radial extent of the magnetised region is
controlled by the decrease of the gas density and $B_{\mathrm{eq}}$ with $r$.

Figure 5: Left: the scatter plots of the colour-coded radio luminosity
$\log(L_{1.4})$ of $10^{3}$ randomly selected galaxies with $L_{1.4}\geq
10^{21}\,$WHz-1 at $z=0.0,0.5,1.0$ and $1.5$ versus the fourth power of the
volume-averaged magnetic field strength and the volume of the emission region
(disc volume). Right: the colour-coded volume-averaged magnetic field strength
of galaxies shown in the left-hand panels versus their star formation rate and
stellar mass.

##  4 The radio luminosity function

Figure 6: The 1.4-GHz radio luminosity function of star-forming galaxies at
various redshifts $z$ specified within each frame. The median outcome of the
fiducial model is shown as a red solid line with the shaded region
representing the 16–84 percentile range of the turbulent speed shown in Fig.
1. The $v$25-R19 model (with the constant turbulent speed
$v=25\,\,\text{km}\,\text{s}^{-1}$) is represented with the dotted black line.
The observational data are taken from Condon et al. (2002) (black crosses),
Condon et al. (2019) (blue crosses), Mauch & Sadler (2007) (red circles),
Novak et al. (2017) (black circles) and van der Vlugt et al. (2022) (green
triangles).

Figure 6 shows the variation with the redshift of the RLF $\Phi(L_{1.4})$ of
SFGs at the rest-frame frequency $1.4\,\text{GHz}$, where $\Phi(L_{1.4})$ is
the number of galaxies per unit comoving volume per decade in luminosity, at
luminosity $L_{1.4}$. For comparison, we note that the Milky Way luminosity at
$1.4\,\text{GHz}$ is $L_{0}=3\times 10^{21}\,\text{W}\,\text{Hz}^{-1}$
(Berkhuijsen, 1984). The predictions are computed at the redshifts (shown on
each panel) closest to the median of the redshift bins utilized for RLF
measurements by Condon et al. (2002) and van der Vlugt et al. (2022) where
simulation snapshots are available. We show the RLFs obtained from two models
described in Section 2.1.2: the fiducial model (where the turbulent speed $v$
depends on the SFR) and the model $v$25-R19 where
$v=25\,\,\text{km}\,\text{s}^{-1}$ in all galaxies. The observed luminosity
functions are from Condon et al. (2002, 2019) and Mauch & Sadler (2007) for
the local Universe and Novak et al. (2017) and van der Vlugt et al. (2022) for
the higher redshifts.

The agreement of the fiducial model with the data is quite satisfactory, if
not perfect, over a wide redshift range. In particular, for $z\lesssim 1.2$,
the high-luminosity end of the RLF is reproduced remarkably well. Although the
median luminosity is under-predicted at higher redshifts for all values of
$L_{1.4}$, the model and observations still marginally agree for $z\lesssim
2$, including the changes in the slope around
$L_{1.4}=10^{23}\,\text{W}\,\text{Hz}^{-1}$ for $z\lesssim 0.5$. Moreover,
when extrapolated to higher luminosities, the median outcome of the fiducial
model is in agreement with the data point at the highest luminosity at small
$z$ (our galaxy sample is not large enough to probe such bright, rare galaxies
at certain redshifts including $z\approx 0$). However, the number density of
galaxies in the luminosity range
$20.5\leq\log(L_{1.4}/\,\text{W}\,\text{Hz}^{-1})\leq 21.5$ in the local
Universe is significantly higher than what is observed. Also, for $z>2$, the
model number densities are systematically and significantly smaller than those
observed, and the discrepancy between the predictions and the data increases
with redshift.

Figure 7: As Fig. 6 but for the fiducial model, presented as in Fig. 6,
compared with the $v(z)$-J23 model (black dotted) where the turbulent speed is
an explicit function of the redshift. The dashed blue and dash-dotted cyan
lines show the contributions to the synchrotron luminosity solely due to the
mean and random magnetic fields, respectively, in the fiducial model.

###  4.1 The role of the turbulent speed

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.

###  4.2 Relative importance of the random and mean magnetic fields

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
(chapter 8 of Shukurov & Subramanian, 2021). 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.

###  4.3 The redshift evolution of the RLF

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, 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.

##  5 Discussion

Our goal in this paper is, firstly, to explore the applicability of the theory
and models of galactic magnetic fields to young and evolving galaxies and,
secondly, to identify the most important galactic properties and parameters
that can be deduced and understood with the help of their radio luminosity
function. The synthetic RLFs of star-forming galaxies obtained from their
modelled synchrotron emission at $1.4\,\text{GHz}$ are in broad agreement with
the available observations, reproducing the number density of galaxies in the
luminosity range $10^{19}\lesssim L_{1.4}\lesssim
10^{25}\,\text{W}\,\text{Hz}^{-1}$ ($3\times 10^{-3}$ to $3\times 10^{3}$ in
terms of the Milky Way luminosity at $1.4\,\text{GHz}$ – Berkhuijsen, 1984) in
the redshift range $0\lesssim z\lesssim 2$, within the uncertainties in the
turbulent speed discussed in Section 2.1.2. Moreover, the form (e.g., changes
in the slope) of the RLF agrees well with what is observed. We do not consider
$z\gtrsim 2$, because the synchrotron luminosity predicted by the model
becomes significantly lower than what is observed, continuing the trend with
$z$ already visible in Fig. 6.

We have made no persistent attempt to achieve any better agreement with the
observational results even though this would be possible given that many
properties of the ISM in evolving galaxies leave much freedom for adjustment.
We avoid making heuristic assumptions that do not have explicit observational
or theoretical justification.

We have identified the turbulent speed $v$ as one of the predominant factors
affecting the RLF. Our results firmly indicate that $v$ must be an increasing
function of the star formation rate for the SFR exceeding a certain threshold
(here adopted as $\text{SFR}_{0}=3\,\text{M}_{\odot}\,\text{yr}^{-1}$), and
the model with $v=\text{const}{}$ ($v$25-R19) cannot explain the increases in
the galactic luminosity with $z$ even though its results are close to those
with the SFR-dependent turbulent speed (the fiducial model) at $z\approx 0$
(Fig. 6).

We also find that the shape and redshift-dependence of the RLF cannot be
reproduced by making $v$ depend explicitly on $z$ instead of on the SFR. This
is elucidated by the model $v(z)$-J23, which predicts a decrease of the RLF
with luminosity that is too steep, failing badly to reproduce observations.

We appreciate, however, that the variation of the turbulent speed with the SFR
adopted in the fiducial model may just serve as a proxy for other effects
discussed below.

###  5.1 Interstellar magnetic fields

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 (see also fig. 6 of
Rodrigues et al., 2019). 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.

###  5.2 Galactic structure

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.

###  5.3 Cosmic rays

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.

###  5.4 Interstellar turbulence and galactic outflows

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 midplane 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 (table 1 of Kalberla, 2003) while the velocity dispersion of
the hot ionised gas is $75\,\,\text{km}\,\text{s}^{-1}$ (section 8.3 of Savage
et al., 1997). 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 R19, 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 (Dickey et al. 1990; Blitz & Spergel 1991, Section 12.2.3 of
Kamphuis, 1993). 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 (Section 2.10.2 of
Shukurov & Subramanian, 2021). 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 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.

##  6 Conclusions

The magnetic field and dynamo models presented here lead to a satisfactory
agreement, within uncertainties, of the predicted RLF of star-forming galaxies
with observations up to $z\simeq 2$. At higher redshifts, although the form of
the theoretical luminosity function is similar to what is observed, the theory
predicts a smaller number of galaxies of high luminosity, and the discrepancy
increases with the redshift. While the observational data at higher redshifts
still lack galaxies of faint and moderate luminosity, our findings enable us
to identify a range of effects that need to be understood and included to
improve the agreement with the data.

We have identified the strength of interstellar turbulence as one of the major
factors affecting the RLF. A better understanding of the interstellar
turbulence, primarily the turbulent speed (which makes only a limited and
uncertain contribution to the observed spectral line widths) at high redshifts
and the effects of the evolving multi-phase structure of the interstellar
medium on magnetic fields and cosmic rays are required to improve our models.

Models presented here are based on a well-tested galaxy formation model;
however, we rely on an implicit assumption that the overall structure of the
interstellar medium does not change as the galaxies evolve. This is an
oversimplification but little is known about some aspects of galactic
evolution which are relevant to magnetic fields and cosmic rays (the turbulent
parameters, gas disc thickness, the multi-phase ISM structure, etc.). It is
then not surprising that our model becomes inapplicable beyond the redshift
$z\simeq 2$ at which galaxies undergo a significant structural change, the
development of pronounced gas discs in particular. Among distinct structural
features of young galaxies, important for their nonthermal constituents, might
be widespread radio haloes associated with stronger star formation and
correspondingly vigorous galactic fountain flows. Our limited knowledge of the
properties of turbulent magnetic fields, especially in the evolving multi-
phase ISM, is another question that requires a better answer.

We have shown that the galactic luminosity function in the total radio
intensity is a sensitive indicator of the state of the interstellar gas,
especially its turbulence, density, and galactic fountains and winds. Our
results are consistent with the notion that the structure of star-forming
galaxies, and their magnetic fields, are different at $z\lesssim 2$ and the
higher redshifts. The luminosity function in polarised radio emission would
provide even richer information on galactic rotation, gas stratification, and
anisotropy of interstellar turbulence.

## Acknowledgements

CJ is supported by the University Grant Commission, India through a BSR start-
up grant (F.30-463/2019(BSR)) and the Rashtriya Uchchatar Shiksha Abhiyan
(RUSA) scheme (No.CUSAT/PL(UGC).A1/2314/2023, No:T3A). CJ also acknowledges
the utilization of the high-performance cluster at IUCAA (Pune, India)
facilitated through the associateship program. CMB is supported by the UK
Science and Technology Funding Council (STFC) through grants ST/T000244/1 and
ST/X001075/1.

## Data Availability Statement

The model data presented in this paper may be available upon request from the
lead author.

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##  Appendix A Galactic mean-field dynamo

The interstellar magnetic field can be separated into mean and random
components,

$\boldsymbol{B}=\overline{\boldsymbol{B}}+\boldsymbol{b},$ (15)

where the overbar represents ensemble or volume averaging. The mean magnetic
field is amplified by the joint inductive action of the galactic differential
rotation (at an angular velocity $\omega$) and helicity of interstellar
turbulence (the $\alpha$-effect) in what is known as the $\alpha\omega$-dynamo
(Shukurov & Subramanian, 2021, and references therein). The intensity of these
effects relative to the turbulent magnetic diffusion, which destroys the mean
magnetic field by tangling it, is quantified by the dimensionless dynamo
number,

$D=\frac{\alpha Sh_{\text{d}}^{3}}{\beta^{2}}\,,$ (16)

where $\alpha$ is the magnitude of the $\alpha$ effect, $\beta$ is the
turbulent magnetic diffusivity and $S=r\,\mathrm{d}\omega/\mathrm{d}r$. The
induction effects that amplify the magnetic field can overcome the destructive
action of the turbulent diffusion if the magnitude of the dynamo number
exceeds a certain critical value $|D_{\text{cr}}|$, $|D|>|D_{\text{cr}}|$, and
we note that both $D<0$ and $D_{\text{cr}}<0$ because $S<0$ in most parts of
galactic discs. In a thin, rotating gas layer, $D_{\text{cr}}\approx-10$.
Using the estimates $\alpha\simeq l_{0}^{2}\omega/h_{\text{d}}$ and
$\beta\simeq\tfrac{1}{3}l_{0}v$, where $l_{0}$ and $v$ are the turbulent scale
and speed, the dynamo number can be expressed in terms of directly observable
galactic parameters,

$D\simeq 9\frac{\omega Sh_{\text{d}}^{2}}{v^{2}}\,.$ (17)

The magnetic field thus generated in a thin disc has the quadrupolar parity
with respect to the galactic midplane $Z=0$:
$\overline{B}_{r}(-Z)=\overline{B}_{r}(Z)$,
$\overline{B}_{\phi}(-Z)=\overline{B}_{\phi}(Z)$ and
$\overline{B}_{z}(-Z)=-\overline{B}_{z}(Z)$ in cylindrical coordinates
$(r,\phi,Z)$.

As the magnetic field strength increases to become comparable to
$B_{\mathrm{eq}}$, the Lorentz force becomes comparable to the Coriolis and
other forces, and the field amplification slows down until the dynamo reaches
a saturated, steady state where the magnetic field strength is estimated as
(Chamandy et al., 2014, and sections 12.3 and 13.7.3 of Shukurov & Subramanian
2021)

$\overline{B}^{2}\simeq\uppi^{2}B_{\mathrm{eq}}^{2}\left(\frac{l}{h_{\text{d}}}\right)^{2}\left(\frac{D}{D_{\text{cr}}}-1\right)R_{\kappa}\,.$
(18)

where $R_{\kappa}$ is a parameter of order unity, and we have set the outflow
speed to zero. magnetizer includes the nonlinearity associated with the
magnetic helicity balance and solves for the equivalent strength of the
magnetic field as a function of the galactocentric radius $r$ using the so-
called no-$Z$ approximation applicable to a thin disc, $h_{\text{d}}/r\lesssim
0.1$. The magnetic field components in this solution can be thought of as the
midplane values if the magnetic field distribution in $Z$ is exponential (and
the average values otherwise).

##  Appendix B Synchrotron luminosity

Figure 8: The schematic diagram of a galaxy disc plane showing the line of
sight unit vectors ${\widehat{\boldsymbol{n}}}$ passing through $r=0$ and an
arbitrary location $(r,\phi)$ as red arrows. The vector
${\widehat{\boldsymbol{n}}}_{\text{d}}$ is the projection of
${\widehat{\boldsymbol{n}}}$ onto the disc plane.

The synchrotron luminosity given in equations (9) and (13) is proportional to
the integral
$\int_{V}K_{E}B_{\perp}^{2}\,\mathrm{d}^{3}\boldsymbol{r}^{\prime}\,,$ where
$K_{E}\propto B^{2}$ under the assumption of the energy equipartition between
cosmic rays and the magnetic field. magnetizer solves the mean-field dynamo
equations for the cylindrical components of the axisymmetric mean magnetic
field
$\boldsymbol{\overline{B}}=(\overline{B}_{r},\overline{B}_{\phi},\overline{B}_{z})$
for each galaxy in the sample as a function of the distance to the galactic
centre $r$ in the galactic disc plane (corresponding to $Z=0$), and we augment
the solution assuming its exponential distribution across the disc (in $|Z|$)
with the scale height $h_{B}$ (Section 2.2.3). The random magnetic field is
assumed to have the same scale height, with the strength obtained from the
turbulent energy density (Section 2.2.1).

The synchrotron luminosity depends on the inclination angle $i$ ($0\leq
i\leq\pi$) between the observer’s line of sight (LoS) and the $Z$-axis
(perpendicular to the galactic disc). The unit vector along any LoS
(${\widehat{\boldsymbol{n}}}$) is resolved into
$\boldsymbol{n}_{\text{d}}={\widehat{\boldsymbol{n}}}_{\text{d}}\sin{i}$ along
the galactic disc plane and
$\boldsymbol{n}_{Z}={\widehat{\boldsymbol{n}}}_{Z}\cos{i}$ along the $Z$-axis
(see Fig. 8). The $\phi=0$ direction is determined by the direction of
${\widehat{\boldsymbol{n}}}_{\text{d}}$ as it passes through $r=0$. Hence, the
$r$ and $\phi$ components of ${\widehat{\boldsymbol{n}}}_{d}$ for an arbitrary
LoS passing through a location $(r,\phi,0)$ in the disc plane are $\cos{\phi}$
and $-\sin{\phi}$ and correspondingly, the components of the unit vector
${\widehat{\boldsymbol{n}}}$ passing through $(r,\phi,0)$ are

${\widehat{\boldsymbol{n}}}\equiv(n_{r},n_{\phi},n_{Z})=(\sin i\cos\phi,-\sin
i\sin\phi,\cos i)\,,$ (19)

which is used to derive the magnetic field component in the sky plane
$\boldsymbol{\overline{B}}_{\perp}=\boldsymbol{\overline{B}}-{\widehat{\boldsymbol{n}}}(\boldsymbol{\overline{B}}\cdot{\widehat{\boldsymbol{n}}})$.

Assuming that the turbulent magnetic field $\boldsymbol{b}$ is isotropic,
$\overline{b_{\perp}^{2}}=\tfrac{2}{3}b_{\text{rms}}^{2}$, and
$b_{\text{rms}}$ is estimated in Section 2.2.1. Since we have adopted
$K_{E}\propto\overline{B}^{2}=\overline{B}^{2}+b_{\text{rms}}^{2}$, the
synchrotron luminosity of a galaxy is proportional to the following integral
over the volume $V$ of the magnetised region:

$J=\int_{V}(\overline{B}^{2}+b_{\text{rms}}^{2})(\overline{B}_{\perp}^{2}+\tfrac{2}{3}b_{\text{rms}}^{2})\,\mathrm{d}^{3}\boldsymbol{r}^{\prime}\,.$
(20)

When both parts of the magnetic field have an exponential distribution in $Z$
with a scale height $h_{B}$ and integration over $Z$ is extended over
$|Z|<\infty$, we have

$J=\dfrac{1}{2}\int_{0}^{r_{\text{d}}}r^{\prime}\mathrm{d}r^{\prime}\int_{0}^{2\pi}\mathrm{d}\phi\,h_{B}(r^{\prime})(\overline{B}^{2}+b_{\text{rms}}^{2})(\overline{B}_{\perp}^{2}+\tfrac{2}{3}b_{\text{rms}}^{2})\,,$
(21)

where we adopt $h_{B}=h_{\text{d}}$; $h_{\text{d}}$ and $r_{\text{d}}$ are
defined in Sections 2.1.1 and 2.2.3 and $\overline{B}_{\perp}$ depends on
$\phi$ via the unit LoS vector (19). We note that the scale height of the
magnetic field can be larger than that of the gas. For example,
$h_{B}=2h_{\text{d}}$ for $B\propto B_{\mathrm{eq}}\propto\rho^{1/2}$. Such an
increase in $h_{B}$ would have led to an increase in the RLF by a factor of
two, so the values of the RLF presented are conservative.

##  Appendix C Testing the synchrotron emission model: M51

To verify the evaluation of the synchrotron luminosity, we apply the procedure
described above to the nearby spiral galaxy M51. The radio emission of this
galaxy is thoroughly explored and interpreted in terms of the galactic
magnetic field by Fletcher et al. (2011). To derive the magnetic field
strength, these authors use the assumption of the local energy equipartition
between cosmic rays and magnetic fields, similar to equation (11). The total
magnetic field strength derived from the radio intensity at the wavelength
$\lambda=6\,\text{cm}$ ($\nu=5\,\text{GHz}$ in terms of the frequency)
assuming the path length of $S=1\,\text{kpc}$ through the synchrotron-emitting
region is $30\,\text{{\textmu}G}$ in the central part of the galaxy,
20–$25\,\text{{\textmu}G}$ in the spiral arms and 15–$20\,\text{{\textmu}G}$
between the arms in the main part within $5\,\text{kpc}$ of the centre
(section 4.1 and fig. 8 of Fletcher et al., 2011). The mean magnetic field
strength derived from the Faraday rotation does not exceed
$3\,\text{{\textmu}G}$ (section 6.2 of Fletcher et al., 2011). The intensity
of the radio emission at $\lambda=6\,\text{cm}$ for distances
$1.6<r<4.8\,\text{kpc}$ from the centre, given in Table 2 of Fletcher et al.
(2011), ranges from $0.5\,\text{mJy/beam}$ between the arms to
$1.1\,\text{mJy/beam}$ within them, with a beam of $W=15\,\text{arcsec}$ in
diameter. It includes the thermal emission, and its fraction at
$\lambda=6\,\text{cm}$ is estimated as 25 per cent (section 4.1 of Fletcher et
al., 2011).

Our goal is to test our choice of various parameters involved in the
calculation of the synchrotron intensity rather than to achieve a precise
agreement with the observations of M51. Therefore we only include isotropic
and homogeneous magnetic field (with no $Z$ dependence) with a scale-height of
$0.5\,\text{kpc}$ (half the path length of the synchrotron-emitting region)
that is independent of the distance from the galactic centre. The rest of the
parameters are the same as in the fiducial model presented in this paper. The
synchrotron intensity is derived using equations (9) and (13) with
$b_{\perp}^{2}=\tfrac{2}{3}b^{2}$ neglecting both the anisotropy of the random
field and the mean magnetic field since $b^{4}\gg\overline{B}^{4}$ in M51;
this makes our estimate of the radio intensity quite conservative.

The total synchrotron intensity for the solid angle $\pi(W/2)^{2}$ of a flat
beam is then

$I(\nu)=\pi\left(\frac{W}{2}\right)^{2}\frac{L(\nu)}{4\pi}\epsilon_{\text{th}}\,,$
(22)

where the factor $\epsilon_{\text{th}}=1.25$ accounts for the contribution of
the thermal radio emission. The total synchrotron intensity from (22) at
$\nu=5\,\text{GHz}$ in the main part of the disc of M51 within
$r=5\,\text{kpc}$ for a field strength of $25\,\text{{\textmu}G}$, and a flat
beam of $W=15\,\text{arcsec}$ in diameter is $0.7\,\text{mJy/beam}$, which is
in good agreement with Fletcher et al. (2011), who measures the same to be
$0.8\,\text{mJy/beam}$. If we instead use the magnetic field strength of $20$
$\mu$G in our model, the resulting synchrotron intensity is
$0.3\,\text{mJy/beam}$.