The Astrophysical Journal, 706:482–496, 2009 November 20 doi:10.1088/0004-637X/706/1/482
C
2009. The American Astronomical Society. All rights reserved. Printed in the U.S.A.
THE FAR-INFRARED–RADIO CORRELATION AT HIGH REDSHIFTS: PHYSICAL CONSIDERATIONS AND
PROSPECTS FOR THE SQUARE KILOMETER ARRAY
Eric J. Murphy
Spitzer Science Center, California Institute of Technology, MC 314-6, Pasadena, CA 91125, USA; emurphy@ipac.caltech.edu
Received 2009 July 22; accepted 2009 October 1; published 2009 November 2
ABSTRACT
I present a predictive analysis for the behavior of the far-infrared (FIR)–radio correlation as a function of redshift
in light of the deep radio continuum surveys which may become possible using the Square Kilometer Array
(SKA). To keep a fixed ratio between the FIR and predominantly non-thermal radio continuum emission of
a normal star-forming galaxy, whose cosmic-ray electrons typically lose most of their energy to synchrotron
radiation and inverse Compton (IC) scattering, requires a nearly constant ratio between galaxy magnetic field
and radiation field energy densities. While the additional term of IC losses off of the cosmic microwave
background (CMB) is negligible in the local universe, the rapid increase in the strength of the CMB energy density
(i.e., ∼(1 + z)
4
) suggests that evolution in the FIR–radio correlation should occur with infrared (IR; 8–1000 μm)/
radio ratios increasing with redshift. This signature should be especially apparent once beyond z ∼ 3 where the
magnetic field of a normal star-forming galaxy must be ∼50 μG to save the FIR–radio correlation. At present,
observations do not show such a trend with redshift; z ∼ 6 radio-quiet quasars appear to lie on the local FIR–radio
correlation while a sample of z ∼ 4.4 and z ∼ 2.2 submillimeter galaxies exhibit ratios that are a factor of
∼2.5 below the canonical value. I also derive a 5σ point-source sensitivity goal of ≈20 nJy (i.e., σ
rms
∼ 4nJy)
requiring that the SKA specified sensitivity be A
eff
/T
sys
≈ 15,000 m
2
K
−1
; achieving this sensitivity should enable
the detection of galaxies forming stars at a rate of 25 M
yr
−1
, such as typical luminous infrared galaxies
(i.e., L
IR
10
11
L
), at all redshifts if present. By taking advantage of the fact that the non-thermal component
of a galaxy’s radio continuum emission will be quickly suppressed by IC losses off of the CMB, leaving only
the thermal (free–free) component, I argue that deep radio continuum surveys at frequencies 10 GHz may
prove to be the best probe for characterizing the high-z star formation history of the universe unbiased by dust.
Key words: galaxies: evolution – infrared: galaxies – magnetic fields – radio continuum: galaxies
Online-only material: color figures
1. INTRODUCTION
Radio continuum emission from galaxies arises due to a
combination of thermal and non-thermal processes primarily
associated with the birth and death of young massive stars,
respectively. The thermal (free–free) radiation of a star-forming
galaxy is emitted from H ii regions and is directly proportional to
the photoionization rate of young massive stars. Since emission
at GHz frequencies isopticallythin, the thermal radio continuum
emission from galaxies is a very good diagnostic of a galaxy’s
massive star formation rate.
Massive (8M
) stars which dominate the Lyman continuum
luminosity also end their lives as supernovae (SNe) whose
remnants (SNRs) are responsible for the acceleration of cosmic-
ray (CR) electrons into a galaxy’s general magnetic field,
resulting in diffuse synchrotron emission. Thus, in a more
complicated manner, the non-thermal radio continuum emission
also traces the most recent star formation activity in a galaxy.
Ideally, one would like to isolate the thermal component as it is
a more direct measure of the most recent massive star formation
activity. However, at GHz frequencies, the non-thermal fraction
typically dominates the total radio continuum emission (i.e.,
∼10:1 at ∼1 GHz; Condon & Yin 1990) making the isolation
of the thermal fraction difficult.
These same massive stars are often the primary sources of
dust heating in the interstellar medium (ISM) as their starlight
is absorbed and reradiated at far-infrared (FIR) wavelengths by
interstellar grains. This common origin between the FIR dust
emission and thermal + non-thermal radio continuum emission
from galaxies is thought to be the dominant physical process
driving the FIR–radio correlation on global (e.g., de Jong
et al. 1985; Helou et al. 1985; Niklas 1997; Niklas & Beck
1997; Yun et al. 2001) and local (e.g., Beck & Golla 1988;
Xu et al. 1992; Marsh & Helou 1995; Hoernes et al. 1998;
Hippelein et al. 2003; Murphy et al. 2006, 2008; Hughes et al.
2006) scales. While massive star formation provides a shared
origin between emission at these two wavelengths, a number of
physical processes must conspire to yield such a remarkably
constant ratio among galaxies spanning nearly 5 orders of
magnitude in luminosity (e.g., Yun et al. 2001), among other
properties (e.g., Hubble type, FIR color, and FIR/optical ratio).
For example, since the non-thermal synchrotron emissivity of
a single electron is roughly proportional to the electron density
times the square of the magnetic field strength, it would seem
that the FIR–radio correlation should be highly dependent on
the propagation of CR electrons and a galaxy’s magnetic field
distribution and strength. If magnetic fields in galaxies are
built-up over time and therefore were weaker in the past, the
correlation should be different at higher redshifts. However, all
indications suggest that the FIR–radio correlation holds out to
moderate redshifts (e.g., Garrett 2002; Gruppioni et al. 2003;
Appleton et al. 2004; Frayer et al. 2006; Murphy et al. 2009a;
Sargent et al. 2009), suggesting that the magnetic field strength
and structure in these galaxies are similar to what is observed
for galaxies in the local universe. Consequently, comparing the
FIR and radio emission characteristics of galaxies can illuminate
482
No. 1, 2009 FIR–RADIO CORRELATION AT HIGH REDSHIFTS 483
properties that are typically inaccessible by other observational
means.
At present, detailed studies of the FIR–radio correlation
among star-forming galaxies has been limited to only mod-
erate redshifts at z 1. With the sensitivity of existing FIR
and radio capabilities, studies at higher redshifts only probe the
most extreme objects. For example, deep (σ
rms
∼ 0.55 mJy)
70 μm observations (Frayer et al. 2006) from the Far-Infrared
Deep Extragalactic Legacy (FIDEL; P.I.: M. Dickinson) sur-
vey are able to probe luminous infrared galaxies (LIRGs;
10
11
L
L
IR
< 10
12
L
) and ultraluminous infrared galax-
ies (ULIRGs; 10
12
L
) out to redshifts of z ∼ 1 and z ∼ 2,
respectively. With Herschel, surveys such as GOODS-Herschel
should improve this situation by measuring the FIR properties
of normal star-forming galaxies out to z ∼ 1, and that of LIRGs
and ULIRGs out to redshifts of z ∼ 2 and z ∼ 4, respectively.
ALMA will also play a significant role by measuring the peak
of the FIR spectral energy distribution (SED) for all LIRGs at
z 5.
These populations of dusty star-forming galaxies appear to
dominate the stellar mass assembly at increasing redshifts; the
star formation rate density increases by a factor of ∼5 − 10
between z ∼ 0 and z ∼ 1, becoming increasingly obscured
(i.e., 60%) by dust (Elbaz et al. 2002; Chary & Elbaz 2001;
Le Floc’h et al. 2005; Magnelli et al. 2009). Thus, LIRGs and
ULIRGs appear to dominate the luminosity density at increasing
redshifts, and optically thin measures of star formation and
active galactic nuclei (AGNs) activity are critical for the proper
quantification of stellar mass build-upover cosmic time. At radio
wavelengths, however, detecting such high-redshift galaxies
remains extremely difficult. Even with a fully operational
EVLA, IR-bright star-forming galaxies (e.g., M 82; L
IR
≈
4 × 10
10
L
) and moderate LIRGs (i.e., L
IR
≈ 3 × 10
11
L
)
will not be detectable beyond redshifts of z ∼ 1 and z ∼ 2,
respectively.
A next-generation radio facility such as the Square Kilometer
Array (SKA) should easily remedy this disparity between the
depth of FIR and radio continuum surveys. While the SKA
was initially proposed solely on the basis of H i science, deep
continuum imaging is critical for the realization of nearly all
of the five established Key Science Projects (KSPs), especially
studies of “The Origin and Evolution of Cosmic Magnetism”
and “Galaxy Evolution and Cosmology.” Essential to these two
science goals is the proper measurement of the star formation
and AGN history over cosmic time for which deep radio
continuum studies may provide an excellent advantage over
other wavelengths.
In this paper, I present physically motivated expectations for
the behavior of the FIR–radio correlation at increasing redshift
and discuss their implications. I also discuss how the SKA, when
combined with future FIR observations to be obtained with next
generation facilities both prior to, and commensurate with, fi-
nal SKA science operations, will be able to tackle interesting
problems associated with these KSPs. This paper is organized as
follows. In Section 2, I discuss the FIR–radio correlation and in-
troduce the physical processes for which it depends on. Then, in
Section 3, I present the results for the expected evolution in the
FIR–radio correlation at increasingly high redshifts and report
on the technical specifications for detecting high-z galaxies in
deep radio continuum surveys. In Section 4, I discuss the phys-
ical implications of the results for characterizing the properties
of galaxies at high z and compare these theoretical expectations
with existing observations. Finally, in Section 5, I summarize my
conclusions.
2. THE FIR–RADIO CORRELATION: PHYSICAL
CONSIDERATIONS
A major result of the Infrared Astronomical Satellite (IRAS;
Neugebauer et al. 1984) all-sky survey was the discovery
of a correlation between the globally measured far-infrared
(FIR; 42–122 μm) dust emission and the optically thin radio
continuum emission of normal late-type star-forming galaxies
without AGNs (de Jong et al. 1985; Helou et al. 1985). The
most remarkable feature of this correlation is that it displays
such little scatter (i.e., ≈0.26 dex) among galaxies spanning 5
orders of magnitude in luminosity (Yun et al. 2001). While the
FIR emission is due to the thermal re-radiation of interstellar
starlight by dust grains, the radio emission is primarily non-
thermal synchrotron emission from CR electrons that propagate
in a galaxy’s magnetic field after initially being accelerated
by SN shocks or other processes. The physics that maintains
a strong correlation between these two quantities over such
a wide range of galaxies remains unclear, however, departures
from the nominal ratio can shed insight on a number of typically
inaccessible galaxy properties.
While the FIR emission is dependent upon the chemical
make-up and size distribution of dust grains, which is hard
to decipher, a galaxy’s non-thermal radio emission is relatively
simple to interpret. What is not simple, however, is determining
which energy-loss processes may dominate a galaxy’s popula-
tion of CR electrons, as well as the role of escape. CR electrons
will cool via a number of mechanisms as they propagate through
the ISM of galaxies. These energy-loss processes include syn-
chrotron radiation and inverse Compton (IC) scattering, as well
as ionization, bremsstrahlung, and adiabatic expansion losses.
Below, I briefly describe the role that each of these energy-loss
processes may play for the cases of normal star-forming galaxies
and compact starbursts following Longair (1994).
2.1. Essential Physics: Normal Star-forming Galaxies
For a typical star-forming galaxy, CR electrons primarily lose
their energy due to synchrotron and IC processes (e.g., Condon
1992), although escape also plays a role (e.g., Helou & Bicay
1993). Let us assume that CR electrons propagating with a pitch
angle φ in a magnetic field of strength B have isotropically
distributed velocities such that <
sin
2
φ> =
2
3
, leading to
B
⊥
≈ 0.82B. A CR electron having energy E will emit most of
its energy at a critical frequency ν
c
where
ν
c
GHz
= 1.3 × 10
−2
B
μG
E
GeV
2
. (1)
The energy loss of CR electrons by synchrotron radiation
goes as dE/dt ∝ U
B
E
2
, where U
B
= B
2
/(8π) is the magnetic
field energy density of the galaxy. Using Equation (1), the
synchrotron cooling timescale, τ
syn
≡ E/|dE/dt |
syn
,forCR
electrons can be expressed as
τ
syn
yr
∼ 5.7 × 10
7
ν
c
GHz
−1/2
B
μG
1/2
×
U
B
10
−12
erg cm
−3
−1
. (2)
Naturally, the synchrotron cooling timescale for CR electrons
can be rewritten as
τ
syn
yr
∼ 1.4 × 10
9
ν
c
GHz
−1/2
B
μG
−3/2
. (3)
484 MURPHY Vol. 706
Similarly, the energy loss of CR electrons due to IC scattering
goes as dE/dt ∝ U
rad
E
2
, where U
rad
is the radiation field energy
density of the galaxy. Equation (1) can therefore be used again
to write the IC cooling timescale, τ
IC
≡ E/|dE/dt|
IC
,as
τ
IC
yr
∼ 5.7 × 10
7
ν
c
GHz
−1/2
B
μG
1/2
×
U
rad
10
−12
erg cm
−3
−1
. (4)
For GeV electrons considered here, the bulk of losses arise from
interactions with IR/optical photons, which dominate U
rad
.
The effective cooling timescale for CR electrons due to
synchrotron and IC losses is
τ
−1
cool
= τ
−1
syn
+ τ
−1
IC
, (5)
which, by combining Equations (2) and (4), we can express as
τ
cool
yr
∼ 5.7 × 10
7
ν
c
GHz
−1/2
B
μG
1/2
×
U
B
+ U
rad
10
−12
erg cm
−3
−1
. (6)
If CR electrons do not lose all of their energy as they
propagate through the ISM of a galaxy, they may eventually
escape the system. In simple diffusion models, the propagation
of CR electrons is usually characterized by an empirical,
energy-dependent diffusion coefficient, D
E
(e.g., Ginzburg et al.
1980). Values of D
E
have been found to be around (4–6) ×
10
28
cm
2
s
−1
for GeV CRs by fitting diffusion models with
direct measurements of CR nuclei (i.e., secondary-to-primary
ratios like boron-to-carbon) within the solar neighborhood (e.g.,
Jones et al. 2001; Moskalenko et al. 2002; Maurin et al. 2002).
While this empirically measured value is for CR nuclei within
the Milky Way, it has been found to be consistent with inferred
diffusion coefficients for CR electrons both radially along
field lines (∼10
29
cm
2
s
−1
) and vertically across field lines
(∼10
28
cm
2
s
−1
) in the thin disks of galaxies (i.e., NGC 891
and NGC 4631; Dahlem et al. 1995). Heesen et al. (2009)
also report a diffusion coefficient of ∼2 × 10
29
cm
2
s
−1
in
the halo of NGC 253. Similar values near ∼10
29
cm
2
s
−1
are also found in hydrodynamic simulations of bubble/super-
bubble induced galaxy outflows (e.g., Rebusco et al. 2005;
Roediger et al. 2007). Using the average value of the diffusion
coefficient obtained through direct measurements of CR nuclei,
which is quite similar to that inferred for radial and vertical
diffusion coefficients for CR electrons, the diffusion coefficient
is expressed as
D
E
cm
2
s
−1
∼
5 × 10
28
,E<1GeV
5 × 10
28
E
GeV
1/2
,E 1GeV.
(7)
Then, for the case of random walk diffusion, such that τ
esc
=
l
2
esc
/D
E
, where l
esc
is the escape scale-length, CR electrons will
escape a galaxy in
τ
esc
yr
∼
⎧
⎨
⎩
6.0 ×10
6
l
esc
kpc
2
,E<1 GeV
2.0 ×10
6
l
esc
kpc
2
ν
GHz
−1/4
B
μG
1/4
,E 1 GeV,
(8)
where Equation (1) has been used to express the electron energy
dependence in terms of its synchrotron emitting frequency for a
given magnetic field strength.
Most edge-on galaxies possess thick synchrotron disks with
scale heights ranging between 1 and 3 kpc (Lisenfeld et al.
2004). In the case of equipartition between the energy density
of CRs and the magnetic field, the CR scale height will be
(3 + α)/2 ≈ 2 times that of the synchrotron scale height for a
synchrotron spectral index of α ≈ 0.8 (e.g., Niklas et al. 1997).
Then, assuming a magnetic field strength of 10 μGandaCR
scale height of ∼3 kpc (synchrotron scale height of 1.5 kpc),
the typical escape time of a CR electron emitting at 1.4 GHz is
∼3 ×10
7
yr. This is comparable to the expected cooling time of
such electrons to synchrotron and IC processes, assuming that
U
rad
= 10
−12
erg cm
−3
, indicating that normal galaxies do not
behave like calorimeters.
Vertical diffusion of CRs, eventually leading to their escape
into intergalactic space due to either open magnetic field lines
or Parker instabilities (Parker 1966), was looked at in detail by
Helou & Bicay (1993). Their proposed physical model for the
FIR-radio correlation includes a leaky-box confinement scheme
in which the escape scale-length of CR electrons is nearly
independent of ISM density and magnetic field strength, but
scales with disk scale height to keep a constant IR/radio ratio.
While escape may play a role, injecting scatter into the observed
IR/radio ratios in larger spirals, it is noteworthy that for the
case of irregular galaxies, which generally lack dense ISM and
magnetic field to keep CR electrons trapped once leaving their
initial clouds around SNRs, escape appears to play a much more
critical role pushing such galaxies off of the correlation (e.g.,
Cannon et al. 2005, 2006; Murphy et al. 2008).
2.2. Essential Physics: Starbursting Galaxies
Additional energy-loss terms that may become increasingly
important in the case of galaxies hosting strong starbursts
are now considered. In such systems, whose energetics and
ISM may be vastly different, ionization, bremsstrahlung, and
adiabatic cooling through advection out of a galaxy by galactic-
scale winds can all play a significant role in cooling CR
electrons. Ionization losses begin to dominate over synchrotron
and IC losses for CR electron energies below about ∼1.3 GeV
such that
τ
ion
yr
∼ 4.1 × 10
9
E
GeV
n
ISM
cm
−3
−1
×
3ln
E
GeV
+42.5
−1
.
(9)
Combining this with Equation (1) yields
τ
ion
yr
∼ 3.6 × 10
10
ν
GHz
1/2
B
μG
−1/2
n
ISM
cm
−3
−1
×
3
2
ln
ν
GHz
− ln
B
μG
+49
−1
.
(10)
Thus, if magnetic fields are sufficiently large in starbursting sys-
tems (i.e., 1 mG), as suggested by Thompson et al. (2006),
electrons radiating GHz synchrotron continuum emission will
have energies 0.3 GeV, and ionization losses should domi-
nate over synchrotron cooling, likely leading to depressed syn-
chrotron emission (see Figure 1).
No. 1, 2009 FIR–RADIO CORRELATION AT HIGH REDSHIFTS 485
Figure 1. Left: cooling timescales of CR electrons as a function of electron energy due to a number of physical processes: synchrotron, inverse Compton (IC),
ionization, and bremsstrahlung. For these calculations typical parameters for normal star-forming galaxies are assumed: B = 10 μG, U
rad
= 10
−12
erg cm
−3
,and
n
ISM
= 1cm
−3
. A timescale due to the escape of CR electrons from a galaxy, assuming random walk diffusion and an escape scale-length of l
esc
= 3 kpc, is also
included. The top axis indicates the synchrotron emitting frequency of the CR electrons. The shaded region indicates the electron energies emitting predominantly in
the 0.5–2 GHz passband. Right: same as what is plotted in the left panel, but for values that may be more typical of galaxies hosting compact starbursts, namely: B = 1
mG and n
ISM
= 10
4
cm
−3
, which assume the usual B ∝
√
n
ISM
scaling. The additional term from adiabatic expansion losses due to a starburst-driven galactic wind,
having a velocity of v
w
= 300 km s
−1
and assuming a disk scale height h = 1 kpc, are also shown. U
rad
was also scaled up to 10
−8
erg cm
−3
. For normal star-forming
galaxies, synchrotron and IC losses dominate CR electron energy losses, along with escape, for radiation observed at GHz frequencies. However, in the case where
an mG magnetic field is considered, synchrotron and IC losses are no longer the dominant cooling processes for CR electrons emitting at GHz frequencies.
Furthermore, bremsstrahlung losses are probably negligible
in normal galaxies given that the CR electron mean free path is
∼5gcm
−2
(Garcia-Munoz et al. 1977), an order of magnitude
smaller than the ∼60 g cm
−2
for the radiation length of the ISM.
However, for a sufficiently dense ISM, bremsstrahlung losses
may exceed ionization losses for electrons at energies above
∼1.1 GeV such that, for neutral hydrogen,
τ
brem
yr
∼ 8.6 × 10
7
n
ISM
cm
−3
−1
. (11)
Additionally, if these starbursts are compact, the adiabatic
expansion losses to CR electrons may become more rapid due
to the advection out of each system with a galactic wind, having
speed v
w
, where the adiabatic lifetime goes as
τ
ad
yr
∼ 1.0 × 10
9
h
kpc
v
w
km s
−1
−1
. (12)
In either case, if the physical conditions of galaxies at higher
redshifts are typically well described by compact starbursts, the
above-mentioned energy-loss processes will become important
relative to synchrotron losses for GHz radiating CR electrons.
To demonstrate this, I plot the cooling timescales of CR
electrons associated with each of the above-mentioned energy-
loss processes versus electron energy in Figure 1.Intheleft
panel of Figure 1, parameters typical for normal star-forming
galaxies are assumed: B = 10 μG, U
rad
= 10
−12
erg cm
−3
,
n
ISM
= 1cm
−3
, and l
esc
= 3 kpc. Losses due to advection by a
galactic wind are ignored. At GHz frequencies, synchrotron
and IC losses are clearly the dominant energy-loss terms
along with escape. In the right panel of Figure 1,thesame
cooling timescales are plotted as in the left panel; however, the
case of a compact starburst having a 1 mG magnetic field is
considered. Assuming a flux freezing scaling of B ∝
√
n
ISM
(e.g., Ruzmaikin et al. 1988; Helou & Bicay 1993; Niklas &
Beck 1997; Crutcher 1999), it follows that n
ISM
= 10
4
cm
−3
.
The same ratio of U
B
/U
rad
as was used in the left panel is
also assumed, setting U
rad
= 10
−8
erg cm
−3
. Energy losses via
advection by a starburst-driven galactic wind, having a velocity
of v
w
= 300 km s
−1
(i.e., typical of z ∼ 3 Lyman break galaxies;
Shapley et al. 2003) and assuming a disk scale height h = 1 kpc,
are also shown. At GHz frequencies, it is found that synchrotron
radiation and IC scattering are no longer the dominant energy-
loss mechanisms. Ionization and bremsstrahlung losses now
dominate the cooling of CR electrons contributing to the GHz
emission from galaxies. While such strong fields are present in
starbursts on small scales, as Zeeman splitting measurements of
OH megamasers in ULIRGs show line-of-sight field strengths
ranging from ≈0.5 to 18 mG in individual masing components
(Robishaw et al. 2008), given that local starburst galaxies follow
the FIR–radio correlation, this may suggest that their average
magnetic fields never reach ∼mG strengths since the depressed
synchrotron emission at GHz frequencies should yield larger
than nominal IR/radio ratios, which is not observed.
2.3. Radio Spectra of Galaxies
Let us now see how the combination of these energy-loss
processes translates into the radio spectra of galaxies. For an
electron injection spectrum Q(E) = κE
−p
, the number density
of particles per unit energy is expressed as (e.g., Longair 1994)
N(E) =
κE
−(p−1)
(p − 1)b(E)
, (13)
where b(E) =−dE/dt is the total energy losses for electrons
including escape. The injected electrons, if shock-accelerated
by SNRs, will have a spectrum typically characterized by
p
inj
≈ 2.4. However, as the CR electrons interact with the ISM,
diffusive losses will modify the spectrum to have an index of
p ≈ 2.8 (e.g., Biermann & Strom 1993; Becker et al. 2009),
which is assumed for the model. Then, the non-thermal radio
continuum emission (i.e., the energy losses to CR electrons by
synchrotron radiation relative to the total energy losses) is given
by
S
NT
ν
∝ E/τ
sync
N(E), (14)
486 MURPHY Vol. 706
Figure 2. Model radio–FIR spectrum for a galaxy having a FIR flux of
F
FIR
= 10
−13
Wm
−2
. The FIR portion of the spectrum is given using the
IR dust emission component of the Dale & Helou (2002) SED libraries. The
thermal radio spectrum, which goes as S
T
ν
∝ ν
−0.1
, is normalized assuming
that the ionizing photon rate estimated by the thermal radio continuum and
IR luminosity are the same, and that the thermal fraction at 1.4 GHz is 10%.
Under these assumptions the nominal FIR–radio correlation (i.e., q
IR
≈ 2.64)
is achieved. The non-thermal spectrum includes energy losses from all of those
mechanisms considered in deriving the CR cooling timescales shown Figure 1
(see Section 2.1 for details).
where emission at frequency ν depends on CR electron energy E
according to Equation (1). For a constant electron temperature,
the amount of thermal radio continuum emission goes as
S
T
ν
∝ ν
−0.1
. (15)
Assuming B ≈ 10 μG, U
B
≈ U
rad
, and n
ISM
≈ 1cm
−3
,the
expected radio continuum spectrum is plotted in Figure 2 along
with an IR dust emission spectrum scaled such that its infrared
(IR; 8–1000 μm) flux is 10
−13
Wm
−2
. The dust emission
spectrum is taken from the SED libraries of Dale & Helou (2002)
using the template which best fit typical star-forming galaxies in
the Spitzer Infrared Nearby Galaxies Survey (SINGS; Kennicutt
et al. 2003). The thermal radio spectrum was then normalized
by equating the relations between the ionizing photon rate and
the thermal radio continuum (Condon 1992) and IR luminosity
(Kennicutt 1998) resulting in
S
T
ν
Wm
−2
Hz
−1
∼ 6.6 × 10
−17
T
e
10
4
K
0.45
×
ν
GHz
−0.1
F
IR
Wm
2
,
(16)
where T
e
= 10
4
K is the assumed electron temperature for an
H ii region.
Next, the non-thermal component was scaled such that the
thermal radio fraction is ∼10% at 1.4 GHz, which is typically
found for star-forming systems (Condon & Yin 1990; Niklas
et al. 1997). The combination of these assumptions naturally
leads to the nominal FIR–radio correlation where
q ≡ log
F
FIR
3.75 × 10
12
Wm
−2
− log
S
1.4 GHz
Wm
−2
Hz
−1
(17)
is ≈2.34 ± 0.26 dex (Yun et al. 2001). Replacing the FIR flux
with the total IR flux in Equation (17) leads to an average
q
IR
≈ 2.64 ± 0.26 dex (Bell 2003). For these parameters, it
is also found that a least squares fit to the non-thermal radio
spectrum yields a spectral index of α
NT
∼ 0.8, similar to what
is observed in normal star-forming galaxies (e.g., Niklas et al.
1997).
To keep a fixed ratio between the FIR and radio emission
of galaxies requires that the ratio of synchrotron to total energy
losses for CR electrons to be nearly constant among each system.
Or, in the case of normal star-forming galaxies, where CR
electron energy losses are dominated by synchrotron losses and
IC scattering off a galaxy’s interstellar radiation field (ISRF), the
FIR–radio correlation is preserved under the condition where
U
B
U
rad
+ U
CMB
1, (18)
where U
CMB
is the radiation field energy density of the cosmic
microwave background (CMB). In the local universe, IC losses
off of the CMB are negligible to those occurring off the ISRF
within a galaxy; however, this is not necessarily the case at
higher redshifts.
3. RESULTS: EXPECTATIONS FOR DEEP RADIO
CONTINUUM SURVEYS
I now explore implications of the above physical arguments
for evolution in the observed radio continuum emission from
galaxies at increasing redshifts. I also report on how this relates
to an expected evolution in the FIR–radio correlation; at present,
it is assumed that any change in the FIR–radio correlation
arises solely from changes to the radio continuum emission of
galaxies. Variations arising from differences in the IR emission
of galaxies are discussed below in Section 4.4.2. Implications
for how various galaxy properties, including their star formation
activity and magnetic field properties, can be inferred from
the discrepancies between the expected and observed IR/radio
ratios are discussed in Section 4.
3.1. Suppression of Non-thermal Emission by the CMB
As already shown, the FIR–radio correlation relies on a
fixed ratio between synchrotron and the total energy losses
of CR electrons. At z = 0, U
CMB
∼ 4.2 × 10
−13
erg cm
−3
,
which is significantly smaller than the radiation field energy
density of the Milky Way (i.e., U
MW
∼ 10
−12
erg cm
−3
). Thus,
CR electron energy losses from IC scattering off the CMB are
negligible at low redshifts. However, U
CMB
∝ (1 + z)
4
, mak-
ing such losses increasingly important with redshift. For in-
stance, by z ∼ 3, U
CMB
∼ 1.1 × 10
−10
erg cm
−3
; equating
this to U
B
= B
2
/(8π) results in a corresponding magnetic field
strength of ∼50 μG, nearly an order of magnitude larger than the
ambient field strength in the solar neighborhood. Consequently,
the non-thermal component of a galaxy’s radio continuum
emission will be increasingly suppressed with increasing red-
shift, eventually resulting in only the thermal component being
detectable.
Assuming an intrinsic FIR–radio correlation for galaxies with
q
IR
≈ 2.64 (Bell 2003), the expected observed-frame 1.4 GHz
flux density for star-forming galaxies having a range of IR (8–
1000 μm) luminosities as a function of redshift are estimated
(Figure 3). These estimates rely on the results of the previous
section where, on a theoretical basis, a realistic radio spectrum
was constructed. It is assumed that the radio continuum emission
is comprised of two components, thermal (free–free) and non-
thermal (synchrotron) emission, both of which can be expressed
