# MAGNETIC FIELDS IN ELLIPTICAL GALAXIES:
AN OBSERVATIONAL PROBE OF THE FLUCTUATION DYNAMO ACTION

Amit Seta  Research School of Astronomy and Astrophysics, Australian National
University, Canberra, ACT 2611, Australia  School of Mathematics, Statistics
and Physics, Newcastle University, Newcastle Upon Tyne, NE1 7RU, UK  Luiz
Felippe S. Rodrigues Department of Astrophysics, Radboud University,
Heyendaalseweg 135, 6525 AJ Nijmegen, Netherlands  School of Mathematics,
Statistics and Physics, Newcastle University, Newcastle Upon Tyne, NE1 7RU, UK
Christoph Federrath  Research School of Astronomy and Astrophysics, Australian
National University, Canberra, ACT 2611, Australia  Christopher A. Hales
National Radio Astronomy Observatory, P.O. Box 0, Socorro, NM 87801, USA
School of Mathematics, Statistics and Physics, Newcastle University, Newcastle
Upon Tyne, NE1 7RU, UK

###### Abstract

Fluctuation dynamos are thought to play an essential role in magnetized galaxy
evolution, saturating within $\sim 0.01$ Gyr and thus potentially acting as
seeds for large-scale mean-field dynamos. However, unambiguous observational
confirmation of the fluctuation dynamo action in a galactic environment is
still missing. This is because, in spiral galaxies, it is difficult to
differentiate between small-scale magnetic fields generated by a fluctuation
dynamo and those due to the tangling of the large-scale field. We propose that
observations of magnetic fields in elliptical galaxies would directly probe
the fluctuation dynamo action. This is motivated by the fact that in
ellipticals, due to their lack of significant rotation, the conventional
large-scale dynamo is absent and the fluctuation dynamo is responsible for
controlling the magnetic field strength and structure. By considering
turbulence injected by Type Ia supernova explosions and possible magnetic
field amplification by cooling flows, we estimate expected magnetic field
strengths of $0.2~{}\text{--}~{}1\,\mu{\rm G}$ in the centers of quiescent
elliptical galaxies. We use a semi-analytic model of galaxy formation to
estimate the distribution and redshift evolution of field strengths,
tentatively finding a decrease in magnetic field strength with decreasing
redshift. We analyse a historical sample of radio sources that exhibit the
Laing-Garrington effect (radio polarization asymmetry in jets) and infer
magnetic field strengths between $0.14~{}\text{--}~{}1.33\,\mu{\rm G}$ for a
uniform thermal electron density and between $1.36~{}\text{--}~{}6.21\,\mu{\rm
G}$ for the thermal electron density following the King profile. We examine
observational techniques for measuring the magnetic field saturation state in
elliptical galaxies, focusing on Faraday rotation measure grids, the Laing-
Garrington effect, synchrotron emission, and gravitational lensing, finding
appealing prospects for future empirical analysis.

dynamo – magnetic fields – galaxies: magnetic fields – ISM: magnetic fields –
galaxies: elliptical and lenticular, cD – techniques: polarimetric

##  1 Introduction

Magnetic fields are ubiquitous in galaxies and their presence is usually
explained by the turbulent dynamo theory, a process by which the turbulent
kinetic energy in the medium is converted into the magnetic energy
(Brandenburg & Subramanian, 2005; Federrath, 2016; Rincon, 2019). Fluctuation
dynamo, a type of magnetohydrodynamic (MHD) dynamo, is produced by a turbulent
flow (characterized by the hydrodynamic Reynolds number $\text{Re}=\,u_{\rm
rms}\ell_{0}/\nu$, where $\,u_{\rm rms}$ is the root-mean square velocity
field, $\ell_{0}$ is the driving scale of the turbulence and $\nu$ is the
viscosity) and therefore fluctuates in time and space. A weak seed magnetic
field in a turbulent electrically conducting fluid either grows exponentially
or decays. The field grows only when the magnetic Reynolds’ number
($\text{Re}_{\mathrm{M}}=\,u_{\rm rms}\ell_{0}/\eta$, where $\eta$ is the
magnetic resistivity) is greater than its critical value,
$\text{Re}_{\mathrm{M}}^{\text{(crit)}}$. For fluctuation dynamos, the
$\text{Re}_{\mathrm{M}}^{\text{(crit)}}$ is approximately around $100$ (can be
higher for highly compressible and supersonic flows; Federrath et al., 2014)
whereas the estimated $\text{Re}_{\mathrm{M}}$ in spiral galaxies is of the
order of $10^{18}$ (Brandenburg & Subramanian, 2005). Thus, the fluctuation
dynamo is expected to be active in galaxies. Initially, the field is weak and
the effect of the magnetic field on the velocity flow is negligible (this is
referred to as the kinematic fluctuation dynamo). When the field grows, the
magnetic field becomes strong enough to react back on the flow via the Lorentz
force. This alters the properties of the turbulent flow. Eventually, the
magnetic field saturates due to the back reaction of the Lorentz force on the
flow.

MHD theories predict that a fluctuation dynamo will arise practically in any
turbulent medium (as long as
$\text{Re}_{\mathrm{M}}>\text{Re}_{\mathrm{M}}^{\text{(crit)}}$). The
negligible seed field in protogalaxies is amplified quickly (in about
$10^{7}\,{\rm yr}$ in spiral galaxies) to magnetic field strengths (of the
order of $1\text{--}10\,\mu{\rm G}$) in near equipartition with the turbulent
kinetic energy. The fluctuation dynamo amplified field exists at small scales
(with largest scale of the order of the driving scale of turbulence,
$\ell_{0}\approx 100\,{\rm pc}$ in disks of spiral galaxies) and it requires
the mean-field dynamo (Ruzmaikin et al., 1988; Beck et al., 1996; Brandenburg
& Subramanian, 2005) to further amplify and order the field over the observed
$\,{\rm kpc}$ scales in spiral galaxies (e.g. in M51; Fletcher et al., 2011).
Besides turbulence, the mean field dynamo also requires large-scale galaxy
properties such as rotation, velocity shear and density stratification.
Fluctuation dynamo plays an important role in seeding the mean field dynamo.
The fluctuation dynamo action is described in detail in Brandenburg &
Subramanian (2005).

Besides fluctuation dynamo, small-scale magnetic fields in spiral galaxies can
also be generated by tangling of the mean (or large-scale) field and
compression by shocks (for further discussion of these mechanisms, see
Appendix A in Seta et al., 2018). The magnetic field generated by compression
due to shocks is possibly correlated with the gas density (Federrath et al.,
2011, 2014). The small-scale magnetic field due to a fluctuation dynamo is
spatially intermittent, whereas that due to the tangling of the mean field is
presumably Gaussian in nature due to pervasive Gaussian turbulence. Thus, the
fluctuation dynamo generated magnetic field also decides the small-scale
structure of magnetic fields in galaxies. In this paper, we propose how
studying magnetic fields in quiescent elliptical galaxies can serve as a probe
of the fluctuation dynamo action and discuss ways about pursuing the same.

Considering the driving scale of the turbulence in the interstellar medium
(ISM) of spiral galaxies within a range of $50\text{--}100\,{\rm pc}$ (Ohno &
Shibata, 1993; Gaensler et al., 2005; Fletcher et al., 2011; Houde et al.,
2013)111Some observational studies report even smaller values of $\ell_{0}$ in
the range $1\text{--}20\,{\rm pc}$ (Minter & Spangler, 1996; Haverkorn et al.,
2008; Iacobelli et al., 2013)., it would require a spatial resolution of
$\lesssim 10\,{\rm pc}$ to study the structure of small-scale magnetic fields
in spiral galaxies. Such a resolution is extremely difficult to achieve with
present-day telescopes and thus it is difficult to observationally
differentiate between the magnetic field generated by a fluctuation dynamo and
that due to the tangling of the mean field. In a magnetized turbulent
environment like the ISM of spiral galaxies, the fluctuation dynamo action is
physically the most plausible scenario for magnetic field amplification but we
are yet to confirm this in a galactic environment. The existence of a
fluctuation dynamo in galaxies is fundamental to the galactic dynamo theory.
In the absence of a fluctuation dynamo, the magnetic field in protogalaxies
would be seeded with a much weaker field and then the mean-field dynamo would
take much longer to amplify the field up to the value observed today
(Ruzmaikin et al., 1988; Arshakian et al., 2009). Furthermore, the field
generated by a fluctuation dynamo is important for low-energy cosmic ray
propagation (Shukurov et al., 2017) and for interpretations of radio
observations. Thus, it is important to find a clear observational probe of the
fluctuation dynamo action.

With a motivation to probe the fluctuation dynamo action in a galactic
environment, we consider magnetic fields in elliptical galaxies (slow
rotators, as defined by the ATLAS3D Project in Emsellem et al. 2011;
Cappellari et al. 2011 222http://www-astro.physics.ox.ac.uk/atlas3d/) . Unlike
spirals, elliptical galaxies have a very small rotation rate (with rotation
velocity of about $20\,{\rm km}\,{\rm s}^{-1}$ as compared to $200\,{\rm
km}\,{\rm s}^{-1}$ in a typical spiral galaxy) and so a mean-field dynamo
(conventional $\alpha\omega$ dynamo) action is unlikely. Moreover, these
systems are not differentially rotating, so a magnetorotational instability
generated large-scale field is also unlikely. However, a weak seed magnetic
field can still be amplified by the fluctuation dynamo action due to the
interstellar turbulence driven by supernova explosions and the random motions
of stars and gas induced by galactic dynamics (Moss & Shukurov, 1996). The
resulting magnetic field can be further amplified by compression produced by
cooling flows (Fabian, 1994) in elliptical galaxies (Mathews & Brighenti,
1997). This leads to a magnetic field which is non-Gaussian and spatially
intermittent.

Thus, the detection and characterization of magnetic fields in elliptical
galaxies would be a direct observational confirmation of the fluctuation
dynamo action. Moreover, the observations can be compared with the theories of
fluctuation dynamos to further understand the physics of magnetic field
saturation, which is an unsolved problem (a few possible saturation mechanisms
are discussed in Tobias et al., 2011; Seta et al., 2015; Rincon, 2019; Seta et
al., 2020). It is also unclear whether the dominant scale (at which the
magnetic power is at its maximum) lies close to the resistive scale
(Schekochihin et al., 2002, 2004) or whether it is in fact much larger
(Subramanian, 1999). The spatially intermittent structure of magnetic fields
generated by a fluctuation dynamo has been studied both analytically
(Zeldovich et al., 1990) and numerically (Wilkin et al., 2007; Seta et al.,
2020) but lacks observational confirmation.

The structure of the paper is as follows. In Sec. 2 we discuss turbulence in
elliptical galaxies and predict the magnetic field strength generated by that
turbulence. In Sec. 3 we predict properties of magnetic fields in elliptical
galaxies obtained from a semi-analytical cosmological simulation. In Sec. 4 we
predict properties of the rotation measure distribution and present results
from numerical simulations to verify these predictions. In Sec. 5 we estimate
the magnetic field strengths using the Laing-Garrington effect and also
examine observational prospects for measuring the magnetic field properties in
elliptical galaxies using existing and forthcoming radio telescopes. Finally,
we conclude in Sec. 6.

##  2 Magnetic fields in elliptical galaxies

###  2.1 Turbulence in the hot gas of elliptical galaxies

The source of hot gas in ellipticals can be internal or external to the
galaxy. Internally, the hot gas can be due to outflows from evolved stars and
Type Ia supernova explosions. Externally, some of the circumgalactic gas
(expelled from the galaxy by winds driven by frequent Type II supernovae
explosions at earlier stages of galactic evolution) can fall back into the
galaxy, as well as the gas accreted from the galaxy group or cluster. The hot
gas in elliptical galaxies is observed via its X-ray emission. The
relationship between the X-ray luminosity $L_{X}$ and optical luminosity (in B
band) $L_{B}$ of bright elliptical galaxies is observationally found to be
$L_{X}\propto L_{B}^{2.2}$ (O’Sullivan et al., 2001). This confirms the non-
stellar origin of the X-ray emission, otherwise the relationship would have
been linear. From X-ray observations, gas mass can be inferred. The total mass
of the hot gas is about $1\text{--}2\%$ of the total stellar mass (Mathews &
Brighenti, 2003).

The turbulence in spiral galaxies which merge to form an elliptical galaxy
decays. The numerical simulations of decaying three-dimensional isotropic
turbulence in an isothermal gas suggests that the turbulent velocity $\,u_{\rm
rms}$ decays as a power-law, $(t/t_{0})^{-\eta_{u}}$, with the exponent
${\eta_{u}}$ in the range $0.4\text{--}0.6$ and $t_{0}$ being the eddy
turnover time (Mac Low et al., 1998; Stone et al., 1998; Subramanian et al.,
2006; Sur, 2019). Considering $\eta_{u}$ as $0.6$, the rms turbulent velocity
in the parent spiral galaxy, $\,u_{\rm rms}\simeq 10\,{\rm km}\,{\rm s}^{-1}$
decreases to $1\,{\rm km}\,{\rm s}^{-1}$ in $5\times 10^{8}\,{\rm yr}$ (where
the eddy turnover time is calculated as follows:
$t_{0}\approx\ell_{0}/\,u_{\rm rms}\approx 100\,{\rm pc}/10\,{\rm km}\,{\rm
s}^{-1}\simeq 10^{7}\,{\rm yr}$). However, we aim to look for magnetic fields
in quiescent elliptical galaxies, which probably had a major merger a few Gyr
earlier, so the turbulence due to parent spirals must have significantly
decayed by then. Also, there is a lower level of turbulence (as compared to
the parent spiral galaxies) maintained in quiescent elliptical galaxies due to
continuous energy injection by Type Ia supernovae explosions. This is also
seen in observations. Each explosion of a Type Ia supernova enriches the
medium with roughly $0.7\,{\rm M}_{\odot}$ of iron and iron abundance in the
hot gas of ellipticals increases from the outskirts to the centre of the
galaxy, where the density is highest, so the supernova activity is highest
(Mathews & Brighenti, 2003). Further, the observed iron abundance can be used
to estimate the supernova explosion rate in ellipticals ($10^{-3}\,{\rm
yr}^{-1}$). The rates are a factor of $\sim 3$ lower than in spiral galaxies
($3\times 10^{-2}\,{\rm yr}^{-1}$). Thus, the turbulence decays immediately
after the merger of spiral galaxies but then is maintained at a certain level
by continuous energy injection by Type Ia supernova explosions in quiescent
elliptical galaxies. Type Ia supernova explosions will drive irrotational
turbulence (negligible vorticity) in the ISM of elliptical galaxies. But this
irrotational turbulence interacts with the multiphase ISM (see Li et al. 2020,
and references therein, for reasons justifying the presence of inhomogeneous,
multiphase gas in ellipticals) to generates vortical motions required for the
fluctuation dynamo action.

To estimate the properties of turbulence in elliptical galaxies, we closely
follow Moss & Shukurov (1996)333The method is also very similar to that in
Shukurov (2004) and Mac Low & Klessen (2004) where turbulent properties of the
interstellar medium are estimated for supernova explosions in spiral galaxies.
with two major changes. First, they consider two drivers of turbulence:
supernova explosions and random star motions, whereas we only consider Type Ia
supernova explosions since the length scales associated with random star
motions would be much smaller than those due to supernova explosions.
Secondly, they consider acoustic turbulence but we consider hydrodynamic
turbulence, because Mathews & Brighenti (1997) showed that the acoustic modes
will quickly decay and cannot maintain the pervasive turbulence.

###  2.2 Estimating the properties of the ISM turbulence in elliptical
galaxies

X-ray observations suggest a temperature $T$ of the order of $10^{7}\,{\rm K}$
for the diffuse interstellar gas in elliptical galaxies (Mathews & Brighenti,
2003). This implies a sound speed
$c_{s}\approx\sqrt{k_{B}T/m_{\text{p}}}\approx 300\,{\rm km}\,{\rm s}^{-1}$,
where $k_{B}=1.38\times 10^{-16}\,{\rm erg}\,{\rm K}^{-1}$ is the Boltzmann
constant and $m_{\text{p}}=1.67\times 10^{-24}\,{\rm g}$ is the proton mass
(assuming that most of the hot gas mass is dominated by protons) . We now use
the sound speed and Sedov–Taylor blast wave self-similarity solution (Ostriker
& McKee, 1988) to calculate the driving scale of the turbulence. The radius of
a supernova remnant $R$ as a function of time $t$ is then

$R=\left(\kappa\frac{E_{\rm SN}}{\rho_{0}}\right)^{1/5}t^{2/5},$ (1)

where $E_{\rm SN}$ is the energy of a supernova explosion, $\rho_{0}$ is the
average gas density of the ambient medium and $\kappa\sim 2$ for a monoatomic
ideal gas. Now, the radius $R$ is roughly equal to the driving scale of the
turbulence $\ell_{0}$, when the velocity of the shock front is equal to
$c_{s}$. Thus, $t\approx R/c_{s}$. This gives

$\ell_{0}\simeq\left(\kappa\frac{E_{\rm
SN}}{\rho_{0}}\right)^{1/3}\frac{1}{c_{s}^{2/3}}.$ (2)

Using $E_{\rm SN}=10^{51}\,{\rm erg}$,
$\rho_{0}=m_{\text{p}}\,n_{0}={1.67\times 10^{-27}\,{\rm g}\,{\rm cm}^{-3}}$
(assuming an ambient number density, $n_{0}=10^{-3}\,{\rm cm}^{-3}$, Forman et
al. 1985) and $c_{s}=300\,{\rm km}\,{\rm s}^{-1}$, we obtain $\ell_{0}\simeq
300\,{\rm pc}$. This scale is slightly larger than that in spiral galaxies
($\simeq 100\,{\rm pc}$) because the ambient medium is less dense.

The supernovae survive longer in ellipticals as compared to the spirals, but
their frequency is much lower and so they inject less turbulent kinetic energy
in the medium. To calculate the turbulent velocity, we assume that about $1\%$
of the total supernovae energy is converted to the kinetic energy of the
turbulence (Dyson & Williams, 1997). The rate at which supernovae inject
energy into the medium can be balanced with the rate of gain of kinetic energy
of the medium as

$\frac{v_{0}^{2}}{\ell_{0}/v_{0}}\approx f\nu_{\rm SN}E_{\rm SN}M_{\rm
gas}^{-1},$ (3)

where $v_{0}$ is the turbulent velocity, $f$ is the fraction of energy that is
injected into the medium, $\nu_{\rm SN}$ is the frequency of supernova
explosions and $M_{\rm gas}$ is the mass of the hot gas. Using
$\ell_{0}=300\,{\rm pc}$, $f=0.01$, $\nu_{\rm SN}=10^{-3}\,{\rm
yr}^{-1}$(Cappellaro et al., 1999), $E_{\rm SN}=10^{51}\,{\rm erg}$ and
$M_{\rm gas}=10^{10}\,{\rm M}_{\odot}$, we obtain $v_{0}\simeq 2.5\,{\rm
km}\,{\rm s}^{-1}$. This is lower than that in spiral galaxies ($\simeq
10\,{\rm km}\,{\rm s}^{-1}$).

The estimated driving scale of turbulence ($300\,{\rm pc}$) is roughly similar
to that obtained in Moss & Shukurov (1996) ($350\,{\rm pc}$), but the
turbulent velocity we get ($2.5\,{\rm km}\,{\rm s}^{-1}$) is smaller than
their estimate ($60\,{\rm km}\,{\rm s}^{-1}$). This is because we consider
that Type Ia supernova explosions drive hydrodynamic turbulence rather than
the acoustic turbulence, which they assume. The acoustic turbulence decays
very quickly due to relatively high viscosity in the ISM of elliptical
galaxies. For a fully ionised gas, with electron number density $n_{\rm e}$
and temperature $T$, the viscosity $\nu$ based on the kinetic theory is
estimated as (Brandenburg & Subramanian, 2005)

$\displaystyle\nu\simeq 6.5\times 10^{22}\left(\frac{T}{10^{6}\,{\rm
K}}\right)^{5/2}\left(\frac{n_{\rm e}}{1\,{\rm cm}^{-3}}\right)\,{\rm
cm}^{2}\,{\rm s}^{-1},$ (4)

where we assumed Coulomb logarithmic factor of order unity. For the core for
an elliptical galaxy, $n_{\rm e}=0.1\,{\rm cm}^{-3}$ and $T=10^{7}\,{\rm K}$,
we obtain $\nu\simeq 2\times 10^{26}\,{\rm cm}^{2}\,{\rm s}^{-1}$. The
diffusive length scale $l_{\nu}=(\nu/t)^{1/2}$ in an eddy turnover time
$t_{0}=300\,{\rm pc}/2.5\,{\rm km}\,{\rm s}^{-1}\simeq 10^{7}\,{\rm yr}$ is
$3\times 10^{5}\,{\rm cm}$, which is much smaller than the driving scale
($\sim 300\,{\rm kpc}$). Thus, high viscosity dissipates acoustic waves within
a very short distance. Plasma effects due to the presence of magnetic fields
can reduce the viscosity calculated via Eq. 4, but not significantly to
account for such a large difference. Furthermore, the dissipation of acoustic
waves might provide an additional heating mechanism for the medium (details of
mechanism in Fabian et al., 2005; Zweibel, 2020).

The driving scale of turbulence $\ell_{0}\simeq 300\,{\rm pc}$ and the rms
velocity $v_{0}\simeq 2.5\,{\rm km}\,{\rm s}^{-1}$ are global values
calculated from average quantities close to the core of a typical elliptical
galaxy444The high values of turbulent velocities of around $100\,{\rm
km}\,{\rm s}^{-1}$ reported using X-ray observations (de Plaa et al., 2012;
Ogorzalek et al., 2017) are for a much larger halo of the elliptical galaxy.
Such high values of turbulent velocities can be due to interaction with the
AGN at the center of an elliptical galaxy or the surrounding environment.. The
eddy turnover time can be calculated as $t_{0}\approx\ell_{0}/v_{0}\simeq
1.2\times 10^{8}\,{\rm yr}$. The typical value of magnetic Reynolds numbers
can also be calculated. Spitzer resistivity $\eta$ for a plasma at a
temperature $T$ can be estimated from (Brandenburg & Subramanian, 2005):

$\displaystyle\eta$ $\displaystyle\simeq 10^{4}\left(\frac{T}{10^{6}\,{\rm
K}}\right)^{-3/2}\,{\rm cm}^{2}\,{\rm s}^{-1},$ (5)

where we assumed the Coulomb logarithmic factor of order unity. Using
$T=10^{7}\,{\rm K}$, we obtain $\eta\simeq 3\times 10^{2}\,{\rm cm}^{2}\,{\rm
s}^{-1}$. For $\ell_{0}=300\,{\rm pc}$ and $v_{0}=2.5\,{\rm km}\,{\rm
s}^{-1}$, we obtain $\text{Re}_{\mathrm{M}}=\ell_{0}v_{0}/\eta\simeq 10^{24}$.
The estimated $\text{Re}_{\mathrm{M}}$ is considerably greater than
$\text{Re}_{\mathrm{M}}^{\text{(crit)}}$ ($\approx 10^{2}$), which is required
for the fluctuation dynamo action. Even though the flow is very viscous, such
a high magnetic Reynolds number ensures fluctuation dynamo action in random
flows. Thus, it is reasonable to expect fluctuation dynamo-generated magnetic
fields in elliptical galaxies.

Since the gas and stellar densities in elliptical galaxies vary with radius
(usually the variation is described by the King profile with a central core
radius, see Statler 2012) the properties of the ISM turbulence would also
change. We, however, are interested mostly in the core region of the
elliptical galaxies where the density of the hot gas and supernova activity
are higher and thus the expected magnetic field strength and the probability
for an unambiguous detection is also higher. The magnetic field strength may
be further enhanced close to the core due to the compression by cooling flows.

###  2.3 Fluctuation dynamo action and further amplification by cooling flows

The rms value of fluctuation dynamo generated magnetic field $\,b_{\rm rms}$
is in near equipartition with the turbulent kinetic energy (Haugen et al.,
2004; Federrath et al., 2011; Tzeferacos et al., 2018; Seta et al., 2020),

$\,b_{\rm rms}\approx 0.5(4\pi\rho)^{1/2}v_{0},$ (6)

where $\rho$ is the density of the medium and $v_{0}$ is the turbulent
velocity. This is probably representative of magnetic field strengths in the
core of elliptical galaxies. In the core of a typical elliptical, with
$\rho=0.1m_{\text{p}}=1.67\times 10^{-26}\,{\rm g}\,{\rm cm}^{-3}$ (Forman et
al., 1985) and $v_{0}=2.5\,{\rm km}\,{\rm s}^{-1}$ (as obtained in Sec. 2.1),
we obtain $\,b_{\rm rms}\simeq 0.2\,\mu{\rm G}$.

The gas in the centers of elliptical galaxies quickly loses its energy via
X-ray emission. Since the central core cools quickly, the weight of the outer
region will cause the surrounding material to flow inwards. This is referred
to as the cooling flow (Fabian, 1994). They can also be inferred from the
observations of O $\rm VI$ line emission from elliptical galaxies (Bregman et
al., 2001, 2005). The magnetic field would be further enhanced by such cooling
flows due to compression. By flux conservation under spherically symmetric
compression, the field strength $b$ grows as $\rho^{2/3}$, where $\rho$ is the
gas density. In cooling flows, the density can be enhanced by a factor of $10$
or more (Mathews & Brighenti, 1997). Thus, the magnetic field with magnitude
approximately equal to $0.2\,\mu{\rm G}$ can be further amplified to be around
$1\,\mu{\rm G}$. Mathews & Brighenti (1997), using a detailed spherical
cooling flow model, suggest a radial dependence for the magnetic field
strength of the form $b(r)\propto(1\text{--}10)\,r^{-1.2}\,\mu{\rm G}$. The
radial dependence of thermal electron number density is $n_{\rm e}(r)\propto
0.1r^{-1.5}\,{\rm cm}^{-3}$ (Mathews & Brighenti, 2003). Thus, we do expect a
significant Faraday rotation of the polarized light from the background radio
sources passing through the elliptical galaxy in the front (further discussed
in Sec. 4 and Sec. 5.1).

##  3 Predictions from Cosmological Simulations

We used the galform semi-analytic model of galaxy formation (Lacey et al.,
2016) to estimate the expected distribution of magnetic field strengths in
elliptical galaxies and its redshift evolution. We classify any galaxy as
elliptical in galform’s output where the bulge-to-total luminosity ratio in
the $r$-band $(B/T)_{r}>0.5$ (see §4.2.3 of Lacey et al. 2016 and Benson &
Devereux 2010 for details). For the purposes of the calculations in this
section, the contents of the remainder disk components of the selected
galaxies were ignored.

As earlier, we assume that the magnetic field is amplified by a fluctuation
dynamo (no further amplification due to the cooling flows is considered in the
present model), with a steady-state rms strength given by Eq. (6). For the
sake of definiteness we compute the equipartition strength at the half-mass
radius, $r_{1/2}$, of each individual elliptical galaxy. Thus,

$\displaystyle\left.b_{\text{rms}}\right|_{r=r_{1/2}}$ (7)
$\displaystyle=11.6\,\mu{\rm G}\left(\frac{M_{\text{g}}}{10^{10}\,{\rm
M}_{\odot}}\right)^{1/2}$ (8)
$\displaystyle\times\left(\frac{\sigma_{1/2}}{50\,{\rm km}\,{\rm
s}^{-1}}\right)\left(\frac{r_{1/2}}{5\,{\rm kpc}}\right)^{-3/2}$

where we have assumed that the density follows a Hernquist (1990) profile,
with $M_{g}$ corresponding to the total gas mass of the elliptical galaxy, and
$\sigma_{1/2}$ is the velocity dispersion at the half-mass radius.

Figure 1: Redshift evolution of the distribution of $b_{\text{rms}}$ at the
half-mass radius of simulated elliptical galaxies with non-negligible gas
content. Colors show the number of galaxies in each $b_{\text{rms}}$ per
comoving volume (where $h$ is the is the Hubble parameter).

Fig. 1 shows the predicted distribution of rms strength of the magnetic field
at the half-mass radius of elliptical galaxies as a function of redshift
(ignoring galaxies without any gas). For the same comoving volume, we find
that at a larger redshift the typical rms strength of the magnetic fields in
elliptical galaxies is larger than at lower redshifts, whereas the number of
counts is expected to decrease. These trends follow the hierarchical assembly
history of galaxies: the total number of elliptical galaxies increases with
time, but the amount of available gas (and therefore, the typical field
strength) decreases.

##  4 Properties of Faraday Rotation Measure Distribution from fluctuation
dynamo simulations

###  4.1 Theoretical expectations for standard deviation of rotation measure
fluctuations

The polarization angle of the linearly polarized synchrotron radiation rotates
when the radiation passes through a magnetized medium containing thermal
electrons. The amount of rotation is a function of the wavelength of the
radiation and the Faraday rotation measure, RM, which in terms of convenient
units is written as

$\frac{\text{RM}}{1\,{\rm rad}\,{\rm m}^{-2}}=K\int_{0}^{L/1\,{\rm
pc}}\frac{n_{\rm e}}{1\,{\rm cm}^{-3}}\frac{b_{\parallel}}{1\,\mu{\rm
G}}\mathrm{d}l,$ (9)

where $n_{\rm e}$ is the thermal electron number density, $b_{\parallel}$ is
the magnetic field parallel to the line of sight, $L$ is the path length, and
$K=0.81$ when the units are scaled as indicated. For a cosmologically distant
source at a redshift $z$, the RM in the observer frame is further reduced by a
factor of $(1+z)^{2}$ than that given by Eq. 9 (which is the RM in the rest
frame).

The standard deviation of fluctuations in rotation measure
$\sigma_{\text{RM}}$ can be estimated from the two-point correlation function
of RM, which in turn can be estimated from the two-point correlation of the
magnetic field. For the simplest case, this can be done by assuming that the
number density of thermal electrons is uniform. Assuming a constant $n_{\rm
e}$ in Eq. 9, Shukurov & Sokoloff (2007) demonstrated that

$\frac{\sigma_{\text{RM}}}{1+z^{2}}=\frac{(2\pi)^{1/4}}{3^{1/2}}Kn_{e}\,b_{\rm
rms}(L\ell_{b})^{1/2},$ (10)

where $\,b_{\rm rms}$ is the root mean square magnetic field strength in the
region, $L$ is the path length and $\ell_{b}$ is the correlation length of the
random magnetic field. In the central region, assuming that the fluctuation
dynamo is active (see Sec. 2.3), $n_{\rm e}=0.1\,{\rm cm}^{-3}$ (Mathews &
Brighenti, 2003), $\,b_{\rm rms}=1\,\mu{\rm G}$, $\ell_{b}=100\,{\rm pc}$
($\ell_{0}/\ell_{b}\simeq 3\text{--}4$ and $\ell_{0}\simeq 300\,{\rm pc}$),
and $L=10\,{\rm kpc}$ (since the emission is considered close to the central
region). For $z\approx 0$, these values yield $\sigma_{\text{RM}}\simeq
75\,{\rm rad}\,{\rm m}^{-2}$. We expect that this can be observed in case of
good signal to noise ratio. However, the number density decreases with the
radius and this might further decrease the $\sigma_{\text{RM}}$ at larger
radii.

The analysis can be extended to include the contribution from a varying
thermal electron number density (see Felten (1996) and Appendix A in Bhat &
Subramanian (2013)). Assuming the King profile for electrons in an elliptical
galaxy $n_{\rm e}=n_{\rm e}(0)\left(1+(r/a)^{2}\right)^{-3/4}$, where $n_{\rm
e}(0)$ is the electron number density at $r=0$, $a$ is the core radius, and
also assuming that $\,b_{\rm rms}\propto n_{\rm e}^{\gamma}$,
$\sigma_{\text{RM}}$ at $r=0$ is

$\frac{\sigma_{\text{RM}}(0)}{1+z^{2}}=\frac{(2\pi)^{1/4}}{3^{1/2}}Kn_{\rm
e}(0)\,b_{\rm
rms}(a\ell_{b})^{1/2}\left(\frac{\Gamma\left(\frac{3}{2}(\gamma+1)-0.5\right)}{\Gamma\left(\frac{3}{2}(\gamma+1)\right)}\right)^{1/2}.$
(11)

Using $n_{\rm e}(0)=0.1\,{\rm cm}^{-3},\,b_{\rm rms}=1\,\mu{\rm G},a=3\,{\rm
kpc},\ell_{b}=100\,{\rm pc},$ and $\gamma=2/3$ (motivated by the flux freezing
or cooling flow constraint, i.e., $b\propto\rho^{2/3}$), we obtain
$\sigma_{\text{RM}}(0)\simeq 35\,{\rm rad}\,{\rm m}^{-2}$ at $z\approx 0$. The
$\sigma_{\text{RM}}$ as a function of the radial distance in the plane of the
sky $r_{\perp}$ is

$\sigma_{\text{RM}}(r_{\perp})=\frac{\sigma_{\text{RM}}(0)}{1+z^{2}}\left[1+(r_{\perp}/a)^{2}\right]^{-\displaystyle\left(\frac{1}{4}\Gamma\left(\frac{3}{2}(\gamma+1)-1\right)\right)}.$
(12)

For $\sigma_{\text{RM}}(0)/(1+z^{2})=35\,{\rm rad}\,{\rm m}^{-2}$ and
$\gamma=2/3$, we obtain $\sigma_{\text{RM}}(r_{\perp})=35\,{\rm rad}\,{\rm
m}^{-2}(1+(r_{\perp}/a)^{2})^{-0.22}$.

Therefore, we expect significant fluctuations in Faraday rotation from at
least the core of an elliptical galaxy (usually defined as the radius at which
the slope of the surface brightness profile changes dramatically from low
values).

###  4.2 Numerical simulations of fluctuation dynamo

Figure 2: Two dimensional RM maps obtained using the intermittent magnetic
field from a fluctuation dynamo simulation and three different electron number
density distributions: (a) constant $n_{\rm e}$, (b) $n_{\rm e}$ following the
King profile, $n_{\rm e}(r)=n_{\rm e}(0)/(1+(r/a)^{2})^{-3/4}$, where $n_{\rm
e}(0)$ is the number density at the center, $a$ is the core radius ($L/4$),
and $r=((x-L/2)^{2}+(y-L/2)^{2}+(z-L/2)^{2})^{1/2}$ is the distance from the
center, and (c) $n_{\rm e}$ from the flux-freezing or cooling-flow condition,
$n_{\rm e}\propto b^{3/2}$, where $b$ is the field strength. The dotted, black
circle in the case (b) shows the core radius $a=L/4$ in the King profile. For
all three cases, over the entire numerical domain, the magnetic field is
normalized to have a root mean square value of unity ($b/\,b_{\rm rms}=1$),
and the mean thermal electron density is also normalized to unity ($\langle
n_{\rm e}\rangle=1$). The structures in the cases (a) and (b) are very
similar, except that the intensity of RM decreases as a function of $r$ in the
latter case. In case (c), the structures are different at smaller scales. The
dynamo-generated magnetic field is exactly the same in all three cases, and
the difference in the amplitude and the small-scale RM structures is due to
different thermal electron distributions.

We confirm our expectations for $\sigma_{\text{RM}}$ using numerical
simulations of the nonlinear fluctuation dynamo (numerically solving Navier-
Stokes with a prescribed random forcing and induction equation for an
isothermal gas in a periodic domain as described in Haugen et al. (2004)). The
complete details of the simulations are described in Section II of Seta et al.
(2020). The forcing is solenoidal, nearly incompressible and the Mach number
of the flow is less than $0.1$ We note that in numerical simulations of (Type
Ia) supernova-driven turbulence, the overall Mach number is typically $<0.1$
(Li et al., 2019). The initial conditions of our simulations are as follows: a
uniform density, zero velocity, and a very weak random magnetic field with no
net flux. As the turbulence is driven numerically within the domain, the
density remains roughly the same (since the gas is isothermal, the turbulent
driving is near incompressible, and the Mach number is less then unity), and
the velocity reaches a statistically steady state after about 2 turbulent eddy
turnover times (e.g., Federrath et al., 2009; Price et al., 2011). The
magnetic field first grows exponentially (kinematic stage) and then saturates
(saturated stage) due to the back reaction of the flow via the Lorentz force
induced by the magnetic field (Fig. 1 in Seta et al., 2020). We use the
magnetic field from the saturated stage of the simulation with the forcing
wavenumber $k_{\mathrm{F}}=5$, Reynolds number $\text{Re}=283$ and magnetic
Reynolds number $\text{Re}_{\mathrm{M}}=2261$ in a periodic box of non-
dimensional size $L=2\pi$ with $512^{3}$ points (Seta et al., 2020). This
means that there are $5$ velocity correlation cells along each direction in
the domain and a total of $125$ cells in the whole cube. The simulations are
for an isothermal gas with incompressible forcing. Thus, the electron number
density is roughly uniform within the domain. To consider the effect of
thermal electron number density distribution on the rotation measure
distribution, we consider the following physically motivated distributions:

  * •

a uniform $n_{\rm e}$ as suggested by isothermal simulations,

  * •

$n_{\rm e}$ following the King profile similar to the gas density, i.e.,
$n_{\rm e}(r)=n_{\rm e}(0)(1+(r/a)^{2})^{-3/4}$, where $a$ is the core radius,

  * •

$n_{\rm e}$ proportional to $b^{3/2}$ as suggested by the magnetic flux
freezing or cooling flow condition (extreme situation, ignores the turbulent
nature of the medium).

We assume that the background polarized source density is uniform and the only
effect the radiation has while passing through the simulation box is the
rotation of its polarization angle, which is quantified by RM. With the
fluctuation dynamo generated magnetic field $b$ (normalized to $\,b_{\rm
rms}=1$ over the numerical domain) for each $n_{\rm e}$ (normalized to
$\langle n_{\rm e}\rangle=1$ over the numerical domain) given above, we
calculate the RM at each point within a face of the domain as

$\text{RM}(x_{i},y_{i})=K\sum_{i=1}^{512}n_{\rm
e}(x_{i},y_{i},z_{i})\,b(x_{i},y_{i},z_{i})\,\mathrm{d}z,$ (13)

where $(x_{i},y_{i},z_{i})$ is a coordinate on the grid and
$\mathrm{d}z=2\pi/512$ is the grid spacing. This involves looking along the
sightlines through the core of an elliptical galaxy and gathering statistics
from the periodic box. We show the generated RM maps in Fig. 2 for all three
cases: $n_{\rm e}$ constant (a), $n_{\rm e}$ King profile (b), and $n_{\rm
e}\propto b^{3/2}$ (c). The shape and size of structures for a constant
$n_{\rm e}$ ((a) in Fig. 2) and for $n_{\rm e}$ following the King profile
((b) in Fig. 2) are very similar and for the King profile case, the intensity
of RM decreases as a function of the distance from the center. For the third
case, $n_{\rm e}\propto b^{3/2}$ ((c) in Fig. 2), the structures are larger
than in the other two cases, with very high RM values. On larger scales, the
structures look similar for all three cases because of the same magnetic
field, but the amplitude and small-scale RM structures vary depending on the
$n_{\rm e}$ distribution.

###  4.3 Spectra and correlations function of rotation measure maps

Figure 3: One-dimensional (angle-integrated) power spectrum $M_{k}(\text{RM})$
(a), and correlation function ${\rm C}_{\text{RM}}(r)$ (b) of the RM maps
shown in Fig. 2, for the same intermittent magnetic field obtained from a
fluctuation dynamo simulation and three different thermal electron density
distributions: constant $n_{\rm e}$ (red), $n_{\rm e}$ following the King
profile (blue), and $n_{\rm e}\propto b^{3/2}$ (cyan). In the range $3\leq
kL/2\pi\leq 20$ (in this range the magnetic field roughly follows a $k^{-5/3}$
power-law spectrum; see Fig. 2 of Seta et al., 2020), the constant $n_{\rm e}$
and $n_{\rm e}$ following the King profile cases have very similar slope,
$k^{-0.8}$, whereas for the third case ($n_{\rm e}\propto b^{3/2}$), the
spectrum is flatter, $k^{-0.5}$ (dashed lines of the same color show the
corresponding power-law approximations). The right-hand panel shows the
correlation function for all three cases, and the dashed vertical lines show
the break scale (slope of structure function changes first at this scale, thus
referred to as a break scale). This break scale is usually related to the
driving scale of the turbulence. However, the driving scale of the turbulence
($0.2L$) is the same for all three cases, and thus, the difference in break
scales is due to the different thermal electron distributions.

Magnetic fields generated by nonlinear fluctuation dynamo simulations tend to
follow a power-law spectrum with slope $3/2$ in the kinematic stage and $-5/3$
in the saturated stage (Fig. 2 in Seta et al., 2020). In Fig. 3(a), we show
the one-dimensional (angle-integrated) power spectrum $M_{k}$ of RM using the
magnetic field in the saturated stage for the three different electron density
distributions discussed above. The slope of the power spectra is similar for
the constant $n_{\rm e}$ and $n_{\rm e}$ following the King profile case (red
and blue lines in Fig. 3a), $k^{-0.8}$ and the power spectrum is much flatter
for the case with $n_{\rm e}\propto b^{3/2}$, $M_{k}\propto k^{-0.5}$. Even
though all three cases have the same magnetic field, the RM power spectrum can
be different. Then we calculate the second-order structure function ${\rm
SF}_{\text{RM}}(r)$ as (Monin & Yaglom, 1971):

${\rm
SF}_{\text{RM}}(r)=\langle|\text{RM}(\mathbf{x}+\mathbf{r})-\text{RM}(\mathbf{x})|^{2}\rangle,$
(14)

where $\mathbf{x}$ is the position in the two-dimensional RM plane and
$r=|\mathbf{r}|$ is the length of the displacement vector. Since we have
periodic boundary conditions, we consider only half of the periodic domain,
i.e., $r$ varies from $0$ to $L/2$. The scale at which the slope of the
structure function of RM changes, referred to as the break scale, is usually
related to the driving (or outer) scale of the turbulence (Haverkorn et al.,
2008; Anderson et al., 2015). The structure function of RM can be used to
calculate the RM correlation function ${\rm C}_{\text{RM}}(r)$ as (Hollins et
al., 2017):

${\rm C}_{\text{RM}}(r)=1-\frac{{\rm SF}_{\text{RM}}(r)}{2\sigma^{2}},$ (15)

where $2\sigma^{2}$ is the value of ${\rm SF}_{\text{RM}}$ at which RM is no
longer correlated, which would also be twice the standard deviation of RM, if
there are a sufficient number of correlation cells in the numerical domain.
The break scale can also be determined from the correlation function of RM. It
is the length scale at which the correlation function crosses zero for the
first time starting from $r=0$. Fig. 3(b) shows the correlation function of RM
for different electron density distributions. Even though the turbulence is
driven on the same scale ($0.2L$) for all three cases, the break scale is
different. The correlation length of RM can be calculated by integrating the
correlation function as

$\ell_{b}=\int C(r)\,\mathrm{d}r.$ (16)

Using Eq. 16, we calculate the correlation length of RM for all three cases
and find $\ell_{b}/L=0.125\pm 0.07,0.080\pm 0.006$, and $0.052\pm 0.07$. For
this analysis, we see that the correlation length depends not only on the
thermal electron density distribution, but is also different from the break
scale.

###  4.4 Probability distribution function of rotation measure maps

Figure 4: (a) PDFs of RM distribution for the cases where the electron number
density $n_{\rm e}$ is not correlated to the magnetic field strength: $n_{\rm
e}$ constant (red) and $n_{\rm e}$ following the King distribution (blue) with
$a=L/4=1.25\ell_{0}$, where $\ell_{0}$ is the driving scale of turbulence.
Both distributions are roughly Gaussian (dashed lines of the same color show
the corresponding fitted Gaussian distribution) with mean $\text{RM}\approx 0$
(since a mean magnetic field is absent). The standard deviation $\sigma$ is
lower for the King profile. (b) PDF of RM for the case where $n_{\rm e}$ is
related to magnetic field strength $b$ via the flux freezing or cooling flow
condition, i.e., $n_{\rm e}\propto b^{3/2}$. The distribution (cyan) is
clearly non-Gaussian (the dashed-magenta line shows a Gaussian distribution
with the same standard deviation as the fitted distribution) and can be
approximated by a product of a Cauchy–Lorentz distribution and a Gaussian
function (the dashed cyan line shows the fitted function). For the same
magnetic field, the standard deviation and the shape of the RM distribution
differs significantly due to different (physically motivated) thermal electron
density distributions.

Fig. 4 shows the probability density function (PDF) of RM for the three
electron number density distributions. In all cases, the mean
$\text{RM}\approx 0$ and the distribution is symmetric, because there is no
large-scale field in the domain. Fig. 4(a) shows the PDFs for two
distributions of $n_{\rm e}$: uniform and following a King profile with
$a=L/4=1.25\ell_{0}$. Both distributions are close to a Gaussian distribution
($\mathcal{N}$) with different standard deviations $\sigma_{\text{RM}}$. The
one with the King profile has a smaller standard deviation. The distributions
agree well with the analytical expressions, Eq. 10 and Eq. 11, respectively.
Fig. 4(b) shows the RM distribution when $n_{\rm e}\propto b^{3/2}$. The
distribution is clearly non-Gaussian and can be approximated by a product of a
Cauchy–Lorentz distribution and a Gaussian function ($\mathcal{LN}$). The
Gaussian or non-Gaussian nature of the RM distribution in these cases follows
from Eq. 9 or Eq. 13. In an isotropic random magnetic field, the polarization
angle performs a random walk as it rotates randomly along the path length.
When $n_{\rm e}$ does not depend on $b$ directly, RM depends on the first
power of the magnetic field and the random walk is a Brownian motion which
gives rise to a Gaussian distribution. For the case where $n_{\rm e}\propto
b^{3/2}$, RM depends on a higher power of the magnetic field and since the
underlying magnetic field is intermittent, the resulting RM distribution is
non-Gaussian. In the case of a non-Gaussian distribution, it is difficult to
associate a single number (for example $\sigma_{\text{RM}}$) to the
distribution but the calculated standard deviation for a non-Gaussian
distribution would be far higher than that for a Gaussian distribution
(compare the $x$-axis in Fig. 4(a) and Fig. 4(b)). The calculated standard
deviation of the RM distribution shown in Fig. 4(b) is approximately equal to
$170\,{\rm rad}\,{\rm m}^{-2}$, whereas the deviation for both the Gaussian
distributions in Fig. 4(a) is less than $100\,{\rm rad}\,{\rm m}^{-2}$. We
show that the standard deviation of fluctuations in RM are significantly
different for different thermal electron number density distributions for a
given fixed magnetic field.

Figure 5: Standard deviation of $\text{RM},\,\sigma_{\text{RM}}$, as a
function of beam size $l_{\rm beam}$ for all three electron density
distributions. The standard deviation decreases as the beam size increases,
and the decrease is exponential for large $l_{\rm beam}$. Even when the beam
size is one-tenth of the numerical domain, the standard deviation between the
three cases still differs significantly.

Furthermore, to account for the effects of beam smearing, we perform a
Gaussian smoothing of the RM maps with a beam size (or smoothing scale)
$l_{\rm beam}$ ranging from $0.0$ (no smoothing) to $L/10$ (beam size one-
tenth of the numerical domain). The dependence of the standard deviation of RM
on $l_{\rm beam}$ is shown in Fig. 5. For all three cases of the thermal
electron density distribution, $\sigma_{\text{RM}}$ decreases (exponentially
for high $l_{\rm beam}$), but the difference between the three cases still
persists even for strong smoothing of the RM maps.

Thus, the fluctuations in rotation measure should not be directly associated
with the fluctuations in the magnetic fields without considering the
fluctuations in thermal electron density. This sets an expectation for future
observations.

##  5 Observational Prospects: Magnetic fields using the Laing-Garrington
effect

Observations of magnetic fields in elliptical galaxies are needed to assess
the true magnetic field saturation state resulting from fluctuation dynamo
action. Several possible approaches are outlined below. Selection effects are
discussed in Section 5.5.

###  5.1 Rotation Measure Grid

The most direct approach would be to measure RM variance using a grid of
polarized background radio sources observed through the ISM of a foreground
elliptical galaxy. This requires a sufficient number of background sources to
enable separation of RM fluctuations arising within and external to the target
galaxy. This approach is unlikely to be feasible for any individual galaxy,
even if using future highly sensitive radio telescopes, because the sky
density of background polarized sources (e.g. Hales et al. 2014 and Fig. 3
from Hales 2013) will be too low to yield a sufficient statistical sample
behind even an arcmin-sized galaxy (see Section 5.4 in Seta, 2019, for an
estimate to observe RMs from background extragalactic sources seen through the
nearest elliptical galaxy Maffei 1). A more promising approach could be to
identify elliptical galaxies located in the foreground of extended polarized
radio sources (e.g. lobes), through which spatially-resolved RM fluctuations
may even be discerned. Depolarization in this scenario would also yield
constraints on $\sigma_{\textrm{RM}}$.

Similarly, polarized fast radio bursts (FRBs) could be used to gather RM
statistics. The event rate of FRBs (e.g. Petroff et al., 2019; Cordes &
Chatterjee, 2019) is probably insufficient to sample many independent
sightlines through any individual foreground galaxy within a realistic
timeframe. Furthermore, it may be unrealistic to consider gathering statistics
of host galaxy RM contributions to study ellipticals; while the emission
mechanism(s) of FRBs and their identification with any particular astronomical
object or phenomenon remain unknown, their compelling association with young
neutron stars (Metzger et al., 2019) and with environments that scatter radio
waves (CHIME/FRB Collaboration et al., 2019) appear to argue against an origin
within old stellar populations. Instead, a plausible approach might be to
compare a control sample of RMs from FRBs with those located behind elliptical
galaxies.

The approaches above could also be attempted without prior knowledge of the
foreground elliptical galaxies by searching for correlations with various line
tracers. For example, a similar correlation has been identified between
polarization properties of background sources and Mg II absorption line
systems, indicating a likely association with magnetic outflows from
intervening star forming galaxies (Kim et al., 2016).

###  5.2 Laing-Garrington effect

Table 1: List of sources, their redshifts $z$, their projected linear size $\rm LS$, the depolarization fraction for jets $\text{DP}_{\rm j}$ and counter jets $\text{DP}_{\rm cj}$ with their respective uncertainties (Garrington et al., 1991). The calculated standard deviation of the rotation measure in jets $\sigma_{\text{RM}_{\rm j}}$, counter jets $\sigma_{\text{RM}_{\rm cj}}$, elliptical hosts $\sigma_{\text{RM}_{\rm ellip}}$, and the rms magnetic field in ellipticals with their respective uncertainties for the uniform $n_{\rm e}$ $\,b_{\rm rms}(\rm uni)$ and for the $n_{\rm e}$ following the King profile $\,b_{\rm rms}({\rm KP},r=0)$ are also given. Name | $z$ | $\rm LS$ | $\text{DP}_{\rm j}$ | $\text{DP}_{\rm cj}$ | $\sigma_{\text{RM}_{\rm j}}$ | $\sigma_{\text{RM}_{\rm cj}}$ | $\sigma_{\text{RM}_{\rm ellip}}$ | $\,b_{\rm rms}(\rm uni)$ | $\,b_{\rm rms}({\rm KP},r=0)$
---|---|---|---|---|---|---|---|---|---
– | # | $(\,{\rm kpc})$ | # | # | $(\,{\rm rad}\,{\rm m}^{-2})$ | $(\,{\rm rad}\,{\rm m}^{-2})$ | $(\,{\rm rad}\,{\rm m}^{-2})$ | $(\,\mu{\rm G}\,)$ | $(\,\mu{\rm G}\,)$
0017+15 | $2.012$ | $88$ | $0.61\pm 0.02$ | $0.21\pm 0.07$ | $12.48\pm 0.41$ | $22.17\pm 2.37$ | $18.33\pm 2.88$ | $0.29\pm 0.05$ | $1.82\pm 0.29$
0123+32 | $0.794$ | $192$ | $0.96\pm 0.04$ | $0.89\pm 0.03$ | $3.59\pm 1.83$ | $6.06\pm 0.88$ | $4.88\pm 1.73$ | $0.15\pm 0.05$ | $1.36\pm 0.48$
0225-01 | $2.037$ | $131$ | $0.85\pm 0.03$ | $0.08\pm 0.04$ | $7.16\pm 0.78$ | $28.21\pm 2.79$ | $27.29\pm 2.89$ | $0.35\pm 0.04$ | $2.66\pm 0.28$
0232-04 | $1.436$ | $110$ | $0.81\pm 0.03$ | $0.16\pm 0.05$ | $8.15\pm 0.72$ | $24.03\pm 2.05$ | $22.60\pm 2.19$ | $0.49\pm 0.05$ | $3.42\pm 0.33$
0838+13 | $0.684$ | $89$ | $0.73\pm 0.04$ | $0.25\pm 0.02$ | $9.96\pm 0.87$ | $20.90\pm 0.60$ | $18.37\pm 0.83$ | $0.93\pm 0.04$ | $5.83\pm 0.26$
0850+58 | $1.322$ | $129$ | $0.57\pm 0.04$ | $0.15\pm 0.03$ | $13.31\pm 0.83$ | $24.45\pm 1.29$ | $20.51\pm 1.63$ | $0.45\pm 0.04$ | $3.42\pm 0.27$
1023+06 | $1.699$ | $112$ | $0.45\pm 0.03$ | $0.25\pm 0.06$ | $15.86\pm 0.66$ | $20.90\pm 1.81$ | $13.61\pm 2.88$ | $0.24\pm 0.05$ | $1.68\pm 0.36$
1115+53 | $1.235$ | $77$ | $0.86\pm 0.03$ | $0.18\pm 0.04$ | $6.89\pm 0.80$ | $23.24\pm 1.51$ | $22.20\pm 1.60$ | $0.68\pm 0.05$ | $4.00\pm 0.29$
1218+33 | $1.519$ | $79$ | $0.56\pm 0.02$ | $0.07\pm 0.04$ | $13.52\pm 0.42$ | $28.94\pm 3.11$ | $25.60\pm 3.52$ | $0.61\pm 0.08$ | $3.63\pm 0.50$
1226+10 | $2.296$ | $40$ | $0.69\pm 0.04$ | $0.04\pm 0.02$ | $10.81\pm 0.84$ | $31.85\pm 2.47$ | $29.95\pm 2.65$ | $0.59\pm 0.05$ | $2.48\pm 0.22$
1241+16 | $0.557$ | $116$ | $0.90\pm 0.06$ | $0.41\pm 0.04$ | $5.76\pm 1.82$ | $16.76\pm 0.92$ | $15.74\pm 1.18$ | $0.81\pm 0.06$ | $5.84\pm 0.44$
1258+40 | $1.659$ | $172$ | $0.84\pm 0.04$ | $0.11\pm 0.05$ | $7.41\pm 1.01$ | $26.37\pm 2.72$ | $25.31\pm 2.84$ | $0.37\pm 0.04$ | $3.22\pm 0.36$
1318+11 | $2.171$ | $37$ | $0.97\pm 0.04$ | $0.06\pm 0.06$ | $3.10\pm 2.10$ | $29.77\pm 5.29$ | $29.61\pm 5.32$ | $0.65\pm 0.12$ | $2.65\pm 0.48$
1323+65 | $1.618$ | $71$ | $0.73\pm 0.03$ | $0.18\pm 0.03$ | $9.96\pm 0.65$ | $23.24\pm 1.13$ | $21.00\pm 1.29$ | $0.49\pm 0.03$ | $2.75\pm 0.17$
1634+17 | $1.897$ | $55$ | $0.77\pm 0.08$ | $0.27\pm 0.07$ | $9.07\pm 1.80$ | $20.31\pm 2.01$ | $18.17\pm 2.42$ | $0.39\pm 0.05$ | $1.95\pm 0.26$
1634+58 | $0.985$ | $73$ | $0.84\pm 0.10$ | $0.25\pm 0.04$ | $7.41\pm 2.53$ | $20.90\pm 1.21$ | $19.54\pm 1.61$ | $0.78\pm 0.06$ | $4.46\pm 0.37$
1656+57 | $1.281$ | $33$ | $0.90\pm 0.04$ | $0.10\pm 0.10$ | $5.76\pm 1.22$ | $26.93\pm 5.85$ | $26.31\pm 5.99$ | $1.19\pm 0.27$ | $4.55\pm 1.04$
1709+46 | $0.806$ | $103$ | $0.56\pm 0.02$ | $0.15\pm 0.03$ | $13.52\pm 0.42$ | $24.45\pm 1.29$ | $20.37\pm 1.57$ | $0.83\pm 0.06$ | $5.62\pm 0.43$
1732+16 | $1.270$ | $132$ | $0.68\pm 0.05$ | $0.08\pm 0.04$ | $11.02\pm 1.05$ | $28.21\pm 2.79$ | $25.97\pm 3.07$ | $0.59\pm 0.07$ | $4.53\pm 0.53$
1816+47 | $2.225$ | $49$ | $0.82\pm 0.06$ | $0.06\pm 0.03$ | $7.91\pm 1.46$ | $29.77\pm 2.65$ | $28.70\pm 2.77$ | $0.53\pm 0.05$ | $2.48\pm 0.24$
0107+31 | $0.689$ | $364$ | $0.65\pm 0.03$ | $0.39\pm 0.02$ | $11.65\pm 0.62$ | $17.22\pm 0.47$ | $12.69\pm 0.86$ | $0.31\pm 0.02$ | $4.00\pm 0.27$
0712+53 | $0.064$ | $49$ | $0.34\pm 0.03$ | $0.28\pm 0.03$ | $18.44\pm 0.75$ | $20.03\pm 0.84$ | $7.82\pm 2.80$ | $1.33\pm 0.48$ | $6.21\pm 2.22$
0824+29 | $0.458$ | $129$ | $0.86\pm 0.05$ | $0.44\pm 0.02$ | $6.89\pm 1.33$ | $16.08\pm 0.45$ | $14.53\pm 0.80$ | $0.81\pm 0.04$ | $6.15\pm 0.34$
0903+16 | $0.411$ | $280$ | $0.65\pm 0.02$ | $0.38\pm 0.03$ | $11.65\pm 0.42$ | $17.46\pm 0.71$ | $13.00\pm 1.03$ | $0.53\pm 0.04$ | $5.87\pm 0.46$
1001+22 | $0.974$ | $576$ | $0.50\pm 0.04$ | $0.36\pm 0.02$ | $14.78\pm 0.85$ | $17.94\pm 0.49$ | $10.17\pm 1.51$ | $0.15\pm 0.02$ | $2.35\pm 0.35$
1055+20 | $1.111$ | $399$ | $0.88\pm 0.07$ | $0.52\pm 0.11$ | $6.35\pm 1.97$ | $14.35\pm 2.32$ | $12.87\pm 2.77$ | $0.20\pm 0.04$ | $2.60\pm 0.56$
1354+19 | $0.720$ | $376$ | $0.59\pm 0.05$ | $0.23\pm 0.03$ | $12.89\pm 1.04$ | $21.52\pm 0.95$ | $17.23\pm 1.42$ | $0.41\pm 0.03$ | $5.24\pm 0.43$
1548+11 | $1.901$ | $379$ | $0.89\pm 0.13$ | $0.33\pm 0.04$ | $6.06\pm 3.80$ | $18.69\pm 1.02$ | $17.68\pm 1.69$ | $0.15\pm 0.01$ | $1.89\pm 0.18$
1618+17 | $0.555$ | $430$ | $0.84\pm 0.05$ | $0.62\pm 0.03$ | $7.41\pm 1.27$ | $12.27\pm 0.62$ | $9.78\pm 1.24$ | $0.26\pm 0.03$ | $3.64\pm 0.46$
1830+28 | $0.594$ | $217$ | $0.81\pm 0.03$ | $0.75\pm 0.04$ | $8.15\pm 0.72$ | $9.52\pm 0.88$ | $4.92\pm 2.08$ | $0.18\pm 0.07$ | $1.74\pm 0.74$
2325+29 | $1.015$ | $425$ | $0.83\pm 0.05$ | $0.60\pm 0.04$ | $7.66\pm 1.24$ | $12.69\pm 0.83$ | $10.11\pm 1.40$ | $0.16\pm 0.02$ | $2.24\pm 0.31$

Laing (1988) and Garrington et al. (1988) studied radio polarization from
Fanaroff–Riley Class II (FRII) sources and observed that the level of
fractional polarization was significantly higher on the jet side closer to us
than on the counter-jet side pointing away from us. This is referred to as the
Laing-Garrington effect. Assuming that both jets have similar intrinsic
properties, the difference in fractional polarization can be attributed to
depolarization by a foreground screen arising from the X-ray emitting halo of
the elliptical galaxy, where the radiation from the more distant side travels
more through the screen and is thus more depolarized.

If the elliptical galaxy acts as a Faraday screen, i.e. a medium with magnetic
fields and thermal electrons but lacking relativistic electrons (this is a
justified assumption since an elliptical galaxy has a scarcity of cosmic ray
electrons), and if it is assumed that the jet does not significantly disturb
the ISM, we can use historical observations of the Laing-Garrington to probe
magnetic fields in elliptical galaxies. We investigate this observational
approach below using available historical data.

For a Faraday screen, assuming a Gaussian distribution of Faraday depths, the
degree of polarization $p$ at wavelength $\lambda$ is given by (Burn, 1966;
Sokoloff et al., 1998) 555This also assumes no foreground RM structure; see
Enßlin & Vogt (2003); Murgia et al. (2004); Laing et al. (2008), for a
discussion on its effect.

$p=p_{0}\exp(-2\sigma_{\text{RM}}^{2}\lambda^{4}),$ (17)

where $p_{0}$ is the maximum degree of polarization and $\sigma_{\text{RM}}$
is the standard deviation of the fluctuations in rotation measure due to the
screen. The depolarization DP (see Fig. 7 in Basu et al., 2019, for DP
calculated from MHD simulations) between the wavelengths $\lambda_{1}$ and
$\lambda_{2}$ with $\lambda_{1}>\lambda_{2}$ is given by

$\text{DP}=\exp(-2\sigma_{\text{RM}}^{2}(\lambda_{1}^{4}-\lambda_{2}^{4})).$
(18)

Using Eq. 18, we obtain the standard deviation in the rotation measure
distribution for the jet $\sigma_{\text{RM}_{\rm j}}$ and counter jet
$\sigma_{\text{RM}_{\rm cj}}$. Then, assuming that the difference is due to
the halo of an elliptical galaxy, we derive the standard deviation in the
rotation measure distribution for the galaxy $\sigma_{\text{RM}_{\rm ellip}}$
from the standard deviation of rotation measure for the jet and counter jet as
follows:

$\sigma_{\text{RM}_{\rm ellip}}=\sqrt{\sigma_{\text{RM}_{\rm
cj}}^{2}-\sigma_{\text{RM}_{\rm j}}^{2}},$ (19)

Finally, the $\sigma_{\text{RM}_{\rm ellip}}$ is used to estimate the magnetic
field strength $\,b_{\rm rms}$ using the equation Eq. 10 for uniform thermal
electron number density and Eq. 11 for the King profile.

Figure 6: (a) Histogram of rms magnetic field strength $\,b_{\rm rms}(\rm
uni)\,[\,\mu{\rm G}\,]$ obtained from the asymmetry in polarization properties
of jets from sources associated with elliptical galaxies assuming a uniform
thermal electron density $n_{\rm e}=10^{-2}\,{\rm cm}^{-3}$. (b) Two
dimensional histogram of the redshift of the sources $z$ and $\,b_{\rm
rms}(\rm uni)$. At lower $z$, $\,b_{\rm rms}$ is somewhat higher compared to
that at the higher $z$ values.

Figure 7: (a) Histogram of rms magnetic field strength $\,b_{\rm rms}({\rm
KP},r=0)\,[\,\mu{\rm G}\,]$ obtained from the asymmetry in polarization
properties of jets from sources associated with elliptical galaxies assuming
the King profile for the thermal electron density with the core radius
$a=3\,{\rm kpc}$ and the number density at $r=0$ as $n_{\rm
e}(r=0)=10^{-2}\,{\rm cm}^{-3}$. (b) Two-dimensional histogram of the redshift
of the sources $z$ and $\,b_{\rm rms}({\rm KP},r=0)$. At lower $z$, $\,b_{\rm
rms}$ tends to be somewhat (factor 2–3) higher than at higher $z$, although
there is significant scatter in the intermediate redshift range, $0.5\leq
z\leq 1.75$.

Garrington & Conway (1991) did a similar estimate for a typical single source
at low redshift and another source at a high redshift assuming uniform thermal
electron density. In Table 1, we do this calculation for all the 31 sources
studied by Garrington et al. (1991) with varying path lengths ($L\sim$
projected linear size, $\rm LS$ in Table 1) and two different thermal electron
density models (uniform $n_{\rm e}$ and $n_{\rm e}$ following the King
profile). For the uniform $n_{\rm e}$ case, we assume $n_{\rm e}$ to be
$10^{-2}\,{\rm cm}^{-3}$ and for the King profile case, we assume $n_{\rm e}$
at radius $r=0$ to be $10^{-2}\,{\rm cm}^{-3}$ and the core radius $a=3\,{\rm
kpc}$ (see Tab. 4 in Garrington & Conway, 1991). For both cases, we assume the
magnetic field correlation length $\ell_{b}=100\,{\rm pc}$ (as expected from
the fluctuation dynamo action). The estimated magnetic field for the uniform
$n_{\rm e}$ case $\,b_{\rm rms}(\rm uni)$ and that at around $r=0$ for the
King profile case $\,b_{\rm rms}({\rm KP},r=0)$ is given in the last two
columns of Table 1. The rms magnetic field for the uniform $n_{\rm e}$ case
ranges from $0.14\,\mu{\rm G}$ to $1.33\,\mu{\rm G}$ and for the King profile
case ranges from $1.36\,\mu{\rm G}$ to $6.21\,\mu{\rm G}$. The magnetic field
strengths in the sample roughly agrees with the estimated value in Sec. 2.3.
We propagate the uncertainties reported in the observed values of the
depolarization fraction (Garrington et al., 1991) to magnetic field strengths.
The uncertainties in field strengths ranges from $5\%$ to $42\%$. The
uncertainty reported here is due to the measurement error, there can be
additional systematics due to variations in $\ell_{b}$. Fig. 6 (a) and Fig. 7
(a) shows a histogram of magnetic fields strengths in host ellipticals
obtained using $\,b_{\rm rms}(\rm uni)$ and $\,b_{\rm rms}({\rm KP},r=0)$
(last two columns of Table 1). Fig. 6 (b) and Fig. 7 (b) show two-dimensional
histograms of the calculated magnetic field versus redshift. From the two-
dimensional histograms, it appears that the magnetic field is somewhat
stronger in elliptical galaxies at lower redshifts. However, we caution that
the sample size is small and has not been controlled for any selection biases.
The typical strengths are broadly consistent with the simulated values shown
in Fig. 1 (also in agreement with small-scale simulations in Sec. 4.), which
is an encouraging sign. However, a direct comparison of the redshift trends is
not possible, as the observed sample is small and its volume completeness is
difficult to judge.

The Laing-Garrington effect can only be employed for elliptical galaxies with
an active galactic nucleus (AGN). The jets might contaminate the radio signal
as well as the turbulence and magnetic fields in such ellipticals. Thus, it
may not be trivial to disentangle the influence of the jet from the magnetic
field properties that would arise purely under the influence of supernova
driven turbulence in the hot ISM of elliptical galaxies. In general for
ellipticals with an AGN at the center, the galaxy+jet system can give rise to
large-scale RM gradients and asymmetric RM distributions with non-zero mean RM
(as shown in Fig. 8 of Laing et al., 2008, in comparison to our Fig. 4).
Although the magnetic field strength obtained from the fluctuation dynamo
analysis in quiescent ellipticals (Sec. 2) and that using the Laing-Garrington
effect for ellipticals with jets are similar, the correlation scale of the
magnetic field can be widely different for the two cases. Moreover, the
depolarization asymmetry in jets can also be due to an extended disk of
magneto-ionic medium surrounding elliptical galaxies (Gopal-Krishna & Nath,
1997), absence of a spherically symmetric halo (Laing et al., 2008) or
difference in Faraday rotation due to shocked and un-shocked medium (Guidetti
et al., 2012). To robustly study fluctuation dynamos in elliptical galaxies
using the Laing-Garrington effect, a statistically large sample of sources may
be required.

###  5.3 Synchrotron Emission

The detection of radio synchrotron radiation could be attempted. We are
interested in estimating the synchrotron flux in quiescent elliptical galaxies
for which the contribution due to star formation is negligible and that due to
the central AGN is absent. Assuming that there is a significant population of
relativistic electrons, the flux density from the central region of an
elliptical galaxy can be estimated by considering the relationship between
minimum energy magnetic field strength and radio luminosity for a spherical
radio source. Following Pacholczyk (1970), assuming an unresolved source with
spectral index $S_{\nu}\propto\nu^{-0.8}$ over frequencies between $10\,{\rm
MHz}$ to $100\,{\rm GHz}$, and assuming an ion to electron energy density
ratio of 100, the predicted spectral flux density is approximately

$12\,\textrm{nJy}\left(\frac{\,b_{\rm rms}}{1\,\mu{\rm
G}}\right)^{3.5}\left(\frac{R}{\textrm{1
kpc}}\right)^{3}\left(\frac{\nu}{\textrm{1
GHz}}\right)^{-0.8}\left(\frac{D_{L}}{\textrm{100 Mpc}}\right)^{-2}\;,$ (20)

where $R$ is the volume radius, $\nu$ is the observing frequency, and $D_{L}$
is the luminosity distance. Assuming $\nu=100\,{\rm MHz}$, $R=1\,{\rm kpc}$
from earlier, $\,b_{\rm rms}\approx 0.5\,\mu{\rm G}$ from Sec. 2.3, and a
distance of $50\,{\rm Mpc}$ ($z\approx 0.01$), this equates to approximately
$25$ nJy which is unlikely to be feasible with even the next generation of
radio telescopes. However, it is plausible that this could be detected by
stacking the radio emission over a large sample of suitably selected
elliptical galaxies.

In apparent contrast with the flux density estimate above, Nyland et al.
(2017) found that cores of early-type galaxies are bright in synchrotron
because stars are still being formed in the central regions of the galaxies.
They also found a deficiency in radio emission compared to the infrared
emission. One possible explanation for this deficiency is the presence of
significant magnetic field strengths in the medium666Another proposed
explanation was the quick escape of cosmic ray electrons. However, there is no
sign of an outflow or wind in such galaxies. Thus, the cosmic rays transport
must be dictated by magnetic fields.. Nyland et al. (2017) estimated magnetic
fields in the range $4\text{--}85\,\mu{\rm G}$, with median of $15\,\mu{\rm
G}$ (an order of magnitude higher than that in the nearby spiral galaxies),
but to estimate the magnetic field strength they assumed that those early type
galaxies are very similar to present day spiral galaxies. Their assumptions
included energy equipartition between cosmic rays and magnetic fields (see
Seta & Beck (2019) for pros and cons of the equipartition assumption), ratio
of number density of cosmic ray protons to cosmic ray electrons of $100$, and
disc-like or cylindrical geometry of the synchrotron emitting region with gas
scale heights similar to a spiral galaxy. Additionally, the emission due to
accretion by the central black hole might also contaminate the total
synchrotron intensity. Based on our simple estimates from Sec. 2.3 and the
indirect observational result from Section 5.2, we argue that magnetic field
strengths in elliptical galaxies are probably an order of magnitude lower than
that in spirals and certainly not higher.

###  5.4 Gravitational Lensing

Strong gravitational lens systems could be used to measure differences in RM
or depolarization between multiple (and/or possibly spatially resolved) images
of a background polarized radio source that have been lensed by a foreground
elliptical galaxy (Greenfield et al., 1985; Patnaik et al., 1999; Narasimha &
Chitre, 2004)777Such an approach is shown to be successful for spiral galaxies
in Mao et al. (2017).. A mass model for the lensing system is required. A
complication with this approach is that the local group or cluster environment
will also contribute toward $\sigma_{\textrm{RM}}$.

Greenfield et al. (1985) studied the radio polarization signal from two
gravitationally lensed images of the quasar 0957+561 ($z\approx 1.41$) at
multiple wavelengths and found that the rotation measures of the two images
differ by $100\,{\rm rad}\,{\rm m}^{-2}$. This difference can be attributed to
the magnetic field in the ISM of the lensing (cD) galaxy at $z\approx 0.36$.
So, to estimate the magnetic field in the galaxy using Eq. 9 for the quasar at
a redshift $z$, we can write RM in terms of mean quantities as

$\text{RM}\approx\left(0.81\frac{\langle n_{\rm e}\rangle}{\,{\rm
cm}^{-3}}\frac{\langle b\rangle}{\,\mu{\rm G}}\frac{L}{\,{\rm
pc}}\right)\bigg{/}\left(1+z\right)^{2},$ (21)

where $\langle\cdots\rangle$ denotes the average over the path length.
Assuming $\langle n_{\rm e}\rangle\approx 0.01\,{\rm cm}^{-3}$ and $L\approx
30\,{\rm kpc}$, for $\text{RM}=100\,{\rm rad}\,{\rm m}^{-2}$ and $z=1.41$, we
obtain $\langle b\rangle\simeq 2.5\,\mu{\rm G}$. This is comparable in
strength to magnetic fields observed in the Milky Way and nearby spiral
galaxies (Fletcher, 2010; Haverkorn, 2015; Beck, 2016). Also, since the
$\langle b\rangle$ is not close to zero, it refers to the presence of a large-
scale or ordered field. Furthermore, such strong ordered fields are not
expected in cD galaxies, and thus, the RM difference cannot be completely
associated with the galaxy. The non-zero strong ordered field could be due to
the contribution from the cluster’s magnetic field, which even if random (on
scales of $10\,{\rm kpc}$, Subramanian et al., 2006), can act as a coherent
field over the scale of the galaxy. This argument is supported by the mass
modelling of the system, which suggests that the lens must also have some
contribution from the cluster to explain the lensing observations (Greenfield
et al., 1985). Thus, it then becomes difficult to differentiate between the
contribution to the RM difference due to the galaxy’s and cluster’s magnetic
fields.

Alternatively, rather than examining RMs directly, RM variance could be probed
by observing depolarization similar to Section 5.2 but over the spatial extent
of the foreground lensing elliptical galaxy within an extended background
radio source (Strom & Jaegers, 1988; Laing, 1988; Garrington et al., 1988).
More recent attempts to statistically measure RM excess due to intervening
galaxies (Farnes et al., 2014; Malik et al., 2020; Lan & Prochaska, 2020) can
also be extended to study magnetic fields in elliptical galaxies by
considering the morphology of the intervening galaxies in the statistical
sample.

###  5.5 Selection Effects

Observations must carefully consider selection effects so as to minimize
potential bias from coherent magnetic fields and from unrelated RM
fluctuations. The observations should ideally target giant (core) ellipticals
that as a class exhibit negligible rotation (Kormendy et al., 2009). These
galaxies are typically located within the central regions of clusters, so it
will be important to discriminate between RM fluctuations arising in the
interstellar and intracluster gas (Baldry et al., 2006; Kraljic et al., 2018).
Selection should also ideally target ellipticals in hydrostatic equilibrium
(Statler, 2012; Penoyre et al., 2017). Note that a simple selection on red
galaxies could lead to misleading results (e.g. Evans et al., 2018). Evolution
with redshift in the source sample (van Dokkum et al., 2010; Penoyre et al.,
2017) will need to be taken into account when compiling statistics on
saturation RM variance.

Some of the approaches described in the sections above could be tested now
using a combination of data from surveys such as the SDSS (York et al., 2000)
and NVSS (Condon et al., 1998), or in the near future, for example using
forthcoming data from the VLA Sky Survey (Lacy et al., 2019) and eventually
from the Square Kilometre Array (Taylor et al., 2015).

##  6 Conclusions

We propose that observations of magnetic fields in quiescent elliptical
galaxies would provide a direct confirmation of the operation of a fluctuation
dynamo. We demonstrate that the turbulence driven by Type Ia supernova
explosions in elliptical galaxies would generate magnetic fields of a
sufficient strength ($0.2~{}\text{--}~{}1\,\mu{\rm G}$) to be observable. We
examine observational prospects for measuring the magnetic saturation state of
fluctuation dynamos using elliptical galaxies. We take the first steps toward
characterizing the properties of magnetic fields in elliptical galaxies and
analyzing historical observations of the Laing-Garrington effect to estimate
magnetic field strengths in a small sample of host galaxies (Table 1). We
obtain magnetic field strengths in those host galaxies lies in the range
$0.14~{}\text{--}~{}1.33\,\mu{\rm G}$ if a uniform thermal electron density is
assumed and $1.36~{}\text{--}~{}6.21\,\mu{\rm G}$ if the King profile for the
thermal electron density is assumed. We also examine the redshift dependence
of the observed magnetic field strengths (Fig. 6(b) and Fig. 7(b)) and find
that on average, the magnetic field increases by a factor of 2–3 as the
redshift decreases, but it is difficult to draw robust conclusions from the
small sample. More generally, we also show that fluctuations in rotation
measures from background polarized sources must not directly be associated
with the fluctuations in the magnetic fields as that holds true only for
uniform or very weakly varying thermal electron number density.

If the proposed observations of magnetic fields in elliptical galaxies reveal
much lower saturation magnetic field strengths than our estimate, this would
imply gaps in our theoretical understanding of fluctuation dynamos, and would
also imply that the mean-field dynamo in spiral galaxies are probably seeded
with a weaker magnetic field. This in turn would require a much more efficient
mean-field dynamo mechanism than presently known to explain the observed
magnetic fields in the Milky Way and nearby spiral galaxies.

We thank the anonymous referee for a thorough review of this work. AS thanks
Andrew Fletcher, Sui Ann Mao, Anvar Shukurov, Kandaswamy Subramanian, and
Rainer Beck for very useful discussions. AS thanks Govind Nandakumar for a
useful discussion on error analysis. We thank Sharanya Sur for providing
valuable comments on the paper. LFSR acknowledges funding from the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation programme (grant agreement No 772663). CF acknowledges funding
provided by the Australian Research Council (Discovery Project DP170100603 and
Future Fellowship FT180100495), and the Australia-Germany Joint Research
Cooperation Scheme (UA-DAAD). CAH acknowledges support from the European
Union’s Horizon 2020 research and innovation programme under the Marie
Skłodowska-Curie grant agreement No 705332. We further acknowledge high-
performance computing resources provided by the Leibniz Rechenzentrum and the
Gauss Centre for Supercomputing (grants pr32lo, pr48pi and GCS Large-scale
project 10391), the Australian National Computational Infrastructure (grant
ek9) in the framework of the National Computational Merit Allocation Scheme
and the ANU Allocation Scheme.

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