MNRAS 480, 4265–4272 (2018) doi:10.1093/mnras/sty2123
Advance Access publication 2018 August 6
The cosmic microwave background and the stellar initial mass function
Adam S. Jermyn,1‹ Charles L. Steinhardt2 and Christopher A. Tout1
1Institute of Astronomy, University of Cambridge, Madingley Rd, Cambridge CB3 0HA, UK
2Cosmic Dawn Center, Niels Bohr Institute, Blegdamsvej 17, DK-2100 København, Denmark
Accepted 2018 August 2. Received 2018 July 27; in original form 2017 December 13
ABSTRACT
We argue that an increased temperature in star-forming clouds alters the stellar initial mass
function to be more bottom-light than in the Milky Way. At redshifts z  6, heating from
the cosmic microwave background radiation produces this effect in all galaxies, and it is also
present at lower redshifts in galaxies with very high star formation rates (SFRs). A failure to
account for it means that at present photometric template fitting likely overestimates stellar
masses and SFRs for the highest redshift and highest SFR galaxies. In addition, this may
resolve several outstanding problems in the chemical evolution of galactic haloes.
Key words: galaxies: luminosity function, mass function – galaxies: star formation – galaxies:
stellar content – cosmic background radiation – cosmological parameters.
1 IN T RO D U C T I O N
Recent ultradeep surveys (Grogin et al. 2011; Steinhardt et al. 2014;
Bouwens et al. 2015, 2016; Laigle et al. 2016) have measured
the rest-frame ultraviolet (UV) luminosity functions for the most
luminous galaxies to redshifts of 6 < z < 10. These studies find
a substantial population of UV-bright galaxies at high redshift, so
that a survey should expect to find several galaxies per 100 arcmin2
at z = 8 and even one z ≈ 10 galaxy brighter than 26th (AB)
magnitude in the H band (rest-frame UV for 8 < z < 10 galaxies).
This population presents a rich target environment for follow-up
observations over the next few years on the James Webb Space
Telescope (JWST; Gardner et al. 2006).
However, connecting these measurements with theory is a far
more difficult proposition. Theoretical models of galaxy assembly
predominantly describe the dark matter halo (Press & Schechter
1974; Sheth, Mo & Tormen 2001) rather than the baryons that
provide the measured luminosity. Further, even nascent attempts
to include baryons in halo simulations (Vogelsberger et al. 2014;
Somerville & Dave´ 2015) describe galaxies in terms of stellar mass
(MH) and star formation rates (SFRs), which require several addi-
tional assumptions to convert to UV luminosity (LUV). These include
a stellar initial mass function (IMF), dust abundance, composition
and corresponding extinction law, and even a star formation history.
These are difficult to constrain even for local galaxies and, as a
result, there is substantial uncertainty in the M∗/LUV and MH/LUV
ratios at high redshift.
Improving our understanding of these processes at high redshift
has recently become critical because of the remarkable abundance
of UV-bright galaxies at high redshift. Use of abundance matching
(cf. Behroozi & Silk 2015) to find a correspondence between the
 E-mail: adamjermyn@gmail.com
halo mass function produced by the standard cosmological  cold
dark matter (CDM) paradigm and observed luminosity functions,
MH/LUV must decrease sharply for z > 6 from the 0 < z < 4 ratio.
It has been proposed that this might be due to an increased stellar
baryon fraction (Finkelstein et al. 2015), increased star formation
efficiency (Trac, Cen & Mansfield 2015), or additional extinction
(Mashian, Oesch & Loeb 2016) at high redshift, each of which
would change M∗/LUV and therefore the inferred MH/LUV as well.
However, there are currently no theoretical models to explain why
these mass-to-light ratios should decrease sharply at z ≈ 6 after
remaining nearly constant at lower redshifts. If MH/LUV at z > 6
is the same as at z = 4, the existence of these luminous and hence
massive early galaxies would be strongly inconsistent with CDM
(Steinhardt et al. 2016).
A top-heavy (or bottom-light) IMF at z > 6 could also change
the stellar mass-to-light ratio. However, local dwarf galaxies with
less than 1 per cent solar metallicity, lower than expected for z ≈
6 galaxies, have an IMF consistent with the Milky Way (Fagotto
et al. 1994; Dias et al. 2010). Thus, metallicity-driven changes in
the IMF are likely reserved for redshifts well above z = 6.
However, metallicity is not the only relevant variable. Processes
that increase the temperature of star-forming molecular clouds could
also alter the IMF and the stellar mass-to-light ratio. We show
that if the IMF depends on temperature, then cosmic microwave
background (CMB)-driven and cosmic ray (CR)-driven heating of
these clouds should alter the IMF for all galaxies, independently of
properties or environment, at z  6 and for the galaxies with the
highest SFRs at lower redshifts.
Unfortunately, the small-scale physical processes that affect star
formation are still not well understood and so models of the IMF
remain either empirical or phenomenological (see e.g. Bastian,
Covey & Meyer 2010; Oey 2011; Offner et al. 2014). Nevertheless,
such models are sufficient to capture the underlying physics and
C© 2018 The Author(s)
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4266 A. S. Jermyn, C. L. Steinhardt and C. A. Tout
scaling laws and so have proven quite useful. Along these lines, in
Section 2, we develop a temperature-dependent, bottom-light IMF
along with a discussion of the physics of molecular cloud fragmen-
tation. This has been previously explored in the context of the Jeans
mass (Narayanan & Dave´ 2013) with encouraging results, though
we find that a somewhat different scaling is more likely relevant. We
emphasize that this is just one such model, and discuss alternative
scaling laws in the same section. All of these laws are compatible
with our later analysis, but we focus on the fragmentation mass
scaling in the rest of this work because that is one of the better
understood models.
In Section 3, we examine the implications of this IMF for stellar
populations and the mass-to-light ratio of early galaxies, account-
ing for CMB heating. In Section 4, we attempt similar modelling
for CR heating. This model is incomplete because it neglects feed-
back between the shape of the IMF and the SFR–gas temperature
relationship but it suffices to highlight the expected magnitude of
the CR effect. We then discuss various observational tests of this
model, as well as existing evidence, in Section 5, and conclude with
a discussion of possible complications in Section 6.
We emphasize that our intention is to provide a simple model of
these phenomena to emphasize the potential importance and effects
of CMB and CR heating, particularly at high redshift and in extreme
environments. This we hope will motivate more detailed studies of
these phenomena.
2 TEMPER ATU RE DEPENDENCE OF STAR
F O R M AT I O N
Although a full treatment of star formation is very complex and
would require modelling many different baryonic processes (cf.
Larson 1985), the observed qualitative and quantitative features of
the IMF can be reproduced with a much simpler model (Bonnell,
Larson & Zinnecker 2007).
(i) The gas in molecular clouds is characterized by its temperature
and density. These two quantities define a mass above which the
cloud is unstable to gravitational collapse, known as the Jeans mass
(Jeans 1902). Star formation begins when a cloud exceeds its Jeans
mass and begins to collapse.
(ii) In the early stages the cloud is optically thin and efficiently
cools. As a result, it remains isothermal through this collapse, with
its temperature set by that of the ambient radiation field. It is straight-
forward to show that this means the Jeans mass decreases as the
cloud becomes denser and so the collapse continues unimpeded
(Larson 1985).
(iii) However, at some point, the cloud becomes optically thick.
When this occurs the cloud collapses adiabatically rather than
isothermally, causing the Jeans mass to rise to meet the cloud mass
(Low & Lynden-Bell 1976), halting the collapse. This effect is cru-
cial because an isothermal collapse is never halted by gas pressure
(see e.g. Lee & Hennebelle 2018).
(iv) An ultimate cut-off on the final fragment mass distribution is
set by the minimum mass m˜, known as the minimum fragmentation
mass, at which a cloud can cool efficiently (Low & Lynden-Bell
1976).
This description is sufficient to match both observations and sim-
ulations well (Bonnell et al. 2007). Of particular note are that there
is a knee in the IMF at the Jeans mass at which clouds are forced to
be isothermal (Bonnell, Clarke & Bate 2006), initial separations of
single stars are of the order of the Jeans length (Hartmann 2002), and
few stars are seen below the minimum fragmentation mass (Bate &
Bonnell 2005). In addition, the IMF power law above the knee is
consistent with scale-free gravitational collapse and fragmentation
(Klessen 2001) and below the knee it is consistent with the results
of simulations (Bate & Bonnell 2005).
Our goal in this work is to determine how this process should
be affected by changing the cloud temperature T, so that we can
determine the effect of CMB-driven heating at z  6, where the
CMB temperature exceeds the 20 K or so in typical star-forming
molecular clouds in the Milky Way (Schnee et al. 2008). To that
end, a key prediction of this model is that the characteristic mass
scale of the IMF is set not by the initial cloud mass but rather by
this minimum fragmentation mass. That is, the IMF ought to depend
not on the mass m of a star but rather on the dimensionless quantity
m/m˜. Therefore, we expect that the higher temperature IMF ξ (m,
T) behaves as a rescaled function of mass, such that
dN
dm
(m, T ) = ξ (m, T ) = g
(
m
m˜(T )
)
, (1)
where N is the number of stars, or equivalently
ξ (m, T ) = g
(
m
f (T )m˜(T0)
)
, (2)
where
f (T ) ≡ m˜(T )
m˜(T0)
(3)
is the temperature rescaling function and T0 is a reference temper-
ature.
In practice equation (2) gives a formula for turning an observed
IMF into one with a different ambient temperature, as long as the
temperature giving rise to the observed function is also known.
Given m˜(T ) and T0 we may pick our favourite observational IMF,
rescale the mass according to m˜(T ), and obtain a new IMF, which
we expect to be valid at a different temperature. In this work we
choose the IMF of Kroupa (2001) because it is integrable and has
a readily interpreted pair of kinks close together and near the mini-
mum fragmentation mass in the Milky Way. This IMF is
ξ (m) = dN
dm
∝
⎧⎨
⎩
m−0.3, m < a1m˜,
m−1.3, a1m˜ < m < a2m˜,
m−2.3, a2m˜ < m,
(4)
where a1 and a2 are dimensionless constants. These are assumed
to be universal such that the minimum fragmentation mass is the
only relevant mass scale. Matching equation (4) to the z = 0 IMF
of Kroupa (2001) we find that
a2 = 0.50.08a1 = 6.25a1 (5)
and
a1m˜0 = 0.08 M, (6)
where m˜0 is the present-day Milky Way minimum fragmentation
mass. Note that for masses below approximately 0.08 M the ob-
jects are not stars in the sense of fusing but we include them in our
calculations because they still contribute to the condensed baryonic
mass of a stellar population.
When the ambient temperature is large enough it serves to regu-
late the cooling of collapsing clouds. Because cooling is the limiting
factor for fragmentation this ultimately regulates the fragmentation
mass, such that
m˜ = κ
2
(
πkBT
Gμ
)2
= 1.5 × 10−3 M
(
μ
mp
)−2(
κ
κ0
)(
T
K
)2
(7)
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The CMB and the IMF 4267
(Low & Lynden-Bell 1976), where kB is the Boltzmann constant, G
is the gravitational constant, μ is the mean molecular weight, mp is
the proton mass, κ is the opacity of the cloud, and κ0 is the opacity
of ionized hydrogen. This is the case for
T > Tc ≈ 4.1
(κ0
κ
)4/7
K. (8)
Here we take κ = κ0 but Tc is sufficiently smaller than the tempera-
tures of interest, which generally exceed 20 K, that even if this were
not the case, then equation (8) would still be satisfied. In this case
the rescaling function takes on the simple form
f (T ) =
(
T
T0
)2
. (9)
This is in agreement with the simulations performed by Bate (2009).
Putting this all together, the IMF becomes
ξ (m, T ) = dN
dm
(T ) ∝
{
m−0.3, m < 0.08 Mf (T ),
m−1.3, 0.08 Mf (T ) < m < 0.50 Mf (T ),
m−2.3, 0.50 Mf (T ) < m,
(10)
where the proportionality constants are such that ξ (m, T) is
continuous.
A variety of other models have been proposed for the dependence
of the mass scale on temperature. Jappsen et al. (2005) found in
simulations that f(T)∝T3/2. Bate & Bonnell (2005) likewise propose
f(T)∝T3/2 but with an additional dependence on density. Hopkins
(2012) proposes f(T)∝T with an additional dependence on the sonic
radius as defined therein. Yet another scaling relation is provided by
Krumholz (2011), who suggests that f(T)∝T−1/18 with an additional
dependence on density. Neglecting the variation of the density and
sonic radius each of these relations fits well into our formalism and
we are agnostic as to which ought to be preferred.1 For simplicity
we proceed with the model described by equation (10) but note that
our analysis is straightforwardly extended to other models.
3 C MB HEATING
The dependence of m˜ on the ambient temperature T leads to the
remarkable conclusion that, even at modest redshift, the CMB tem-
perature TCMB becomes relevant (Larson 2005; Bailin et al. 2010)
and gives rise to scaling with redshift z of the form
m˜ ∝ T 2 ∝ (1 + z)2 (11)
for
z > zc = Tcloud
TCMB,0
− 1 ≈ 6.3, (12)
where TCMB, 0 is the current temperature of the CMB and Tcloud is
the background temperature the molecular cloud would otherwise
have, here taken to be Tcloud = 20 K. This means that if all other
physics remains the same, we should expect all mass scales to shift
upward with redshift, with the possible exception of the cloud mass,
which is determined by the large-scale dynamics and contents of
the galaxy rather than by smaller scale thermodynamics. So
f (z) ≡ f (T (z)) = min
(
1,
1 + z
6.3
)2
. (13)
1We may expand f(T) to include these other parameters but such an analysis
is left for the future.
Figure 1. The IMF described by equation (10) is shown for several tem-
peratures, each with the same total mass, corresponding to the CMB at
z = 6.3 (20 K), 8.3 (25 K), and 15.5 (45 K). As z increases, the breaks in the
IMF power-law shift to higher masses. Other phenomena that increase the
minimum gas temperature in star-forming regions will have the same effect.
Thus, the redshift dependence of the IMF can be expressed solely
in terms of the redshift dependence of m˜. As z increases, the breaks
in the power-law IMF shift towards higher masses (Fig. 1).
3.1 Effects on inferred quantities
If the IMF is indeed bottom-light at high gas temperatures compared
with the local Universe, every quantity currently inferred for these
galaxies (which include all z  6 galaxies) with the assumption of
a static IMF have been incorrectly estimated. Here, we attempt to
estimate the magnitude of the possible corrections to key quantities
used to describe the first galaxies.
It would be ideal to simply perform a new analysis of high-
redshift photometric catalogues with spectra generated from a vari-
able IMF. The potential effects on inferred quantities could then
be calculated directly from a comparison with the previous cata-
logue. However, current photometric template-fitting codes are not
designed to support this sort of link between the IMF and red-
shift or to track the history of the stellar population in a way that
allows for the effects of a time-dependent IMF. So the full calcu-
lation is not straightforward. Instead we can get a good idea of the
effects by calculating the stellar population and its effect on the
two most widely used inferred quantities (apart from redshift) for
high-redshift galaxies, stellar mass M∗, and SFR.
First, the IMF (equation 10) must be turned into a stellar popu-
lation. Two additional ingredients, namely the history of the SFR
m˙SFR(t) and the lifetime of stars τ s(m), are needed to do this. For
the SFR we use the prescription
log10
˙MSFR
M Gyr−1
=
(
0.84 − 0.026 t
Gyr
)
log10
M∗
M
− 6.51 + 0.11 t
Gyr
(14)
(Steinhardt & Speagle 2014). This gives a mass-independent SFR at
early times, approximately exponential growth later on and finally
quiescence at low redshift.
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4268 A. S. Jermyn, C. L. Steinhardt and C. A. Tout
Next we obtain τ s from scaling relations. With stellar main-
sequence luminosities of (Bo¨hm-Vitense 1992)
Ls(m) ≈ L
(
m
M
)3.5
(15)
and lifetime energy released E∝m (Eggleton, Fitchett & Tout 1989),
we find an effective lifetime of (Bo¨hm-Vitense 1992)
τs(m) ≈ 1010
(
m
M
)−2.5
yr. (16)
This is not quite correct owing to deviations in both the luminosity
and lifetime for high stellar masses (Eggleton et al. 1989; Tout
et al. 1996), but it does a good job of reflecting the fact that the
specific energy released over the lifetime of a star is approximately
independent of mass.
To put the pieces together we integrate star formation over time.
To first order, stellar remnants may be neglected when producing
light curves, so that the relevant stellar population has a mass profile
approximated by
η(m, t) = dN
dm
= ∫ t0 ξ (m, z(t ′)) ˙MSFR(t ′)H (τs(m) + t ′ − t)dt ′ (17)
= ∫ t
max(0,t−τs(m)) ξ (m, z(t ′)) ˙MSFR(t ′)dt ′, (18)
where H is the Heaviside step function.
The stellar mass distribution η(m, t) becomes significantly top-
heavier (or bottom-lighter) within the first Gyr after the big bang
(Fig. 2) because the CMB suppresses the low end of the mass range.
The excess is sharpest at higher masses, because that is where the
IMF slope is most extreme, but drops off at the mass corresponding
to when the stellar lifetime equals the elapsed time since initiation
of star formation in the galaxy. Thus, the peak in the excess moves
to lower masses as time goes on.
Although a full treatment of the effect on the inferred stellar
mass requires a modified photometric template fitting code, it can
be estimated by examination of the mass-to-light ratio. As with the
static IMF, stellar populations produced by the IMF in equation (10)
have a luminosity (Bo¨hm-Vitense 1992)
L =
∫ Mmax
Mmin
η(m, t)m3.5dm, (19)
whereMmin = 0.08 M is the minimum stellar mass and Mmax is the
maximum stellar mass. Importantly, this luminosity is dominated
by the upper end of the mass distribution. In contrast, the mass of
this population,
M∗ =
∫ Mmax
Mmin
η(m, t)m dm, (20)
is dominated by the cut-off 0.08 Mf (z) at which the exponent of
the mass crosses −1. In total, the effects of the CMB produce a
top-heavier IMF and thus stellar population than a static z = 0 IMF
and so produce a higher luminosity for the same amount of mass.
Failure to account for this effect results in an overestimate of the
mass-to-light ratio and so an overestimate of both the masses and
SFRs of high-redshift galaxies.
To estimate this we calculate the correction to the stellar mass-to-
light ratio for monochromatic luminosity at rest-frame wavelengths
of 3000 Å as a function of time for our fiducial cosmological model,
shown in the top panel of Fig. 3.2 The bottom panel of the same
2The correction is nearly independent of wavelength. Although the mass-
to-light ratio varies sharply with wavelength, the flux at all wavelengths is
Figure 2. (a) A sample stellar population mass distribution is shown for
several times, normalized in each case to the low end of the mass range. At
early times the IMF is more top-heavy owing to suppression of the low end
of the mass range. (b) The same distributions, normalized to the population
η0 that would result from a static IMF.
shows the effective slope of the time-dependent IMF computed
between the Milky Way knee of 0.5 and 50 M.
The stellar mass-to-light ratio sharply rises in the first Gyr or so
as the effective IMF slope changes rapidly and the initial stellar
population is established, so that stellar masses may be signifi-
cantly overestimated during this epoch. This is precisely the region
in which the inferred masses of high-redshift galaxies apparently
require either rapid shifts in the stellar baryon fraction (Finkelstein
et al. 2015) or may even be impossible to produce with the CDM
halo mass function (Steinhardt et al. 2016). The predicted shift in
M∗/LUV is in the correct direction to reduce the tension between
theory and observation but this effect alone is insufficient to solve
the problem entirely and an additional effect is required.
4 C OSMI C R AY HEATI NG
Although the CMB provides a universal contribution to all galaxies
independent of environment or stage of evolution, local effects can
further increase the gas temperature in the star-forming regions of
individual galaxies. Likely the strongest effect in high-redshift star-
forming galaxies comes from CRs (Papadopoulos 2010). Even at
dominated by high-mass stars and to leading order the correction just tracks
the change in this population.
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The CMB and the IMF 4269
Figure 3. Top: the ratio of the mass-to-light ratio for the time-dependent
IMF to that for the time-independent IMF is shown as a function of time
for our fiducial cosmology at 3000 Å (UV, proxy for SFR). Bottom: the
effective IMF slope between the Milky Way knee of 0.5 and 50 M is
shown as a function of time for the same cosmology. Note that at high
redshift the discrepancy in mass-to-light ratio and effective slope becomes
quite large and is in the direction needed to resolve the impossibly early
galaxy problem. Both would be overestimated at z 6 if the effects of CMB
temperature are neglected.
z ≤ 2, where the CMB contribution is negligible, dust temperatures
in massive star-forming galaxies are observed to range from 25 to
45 K (Magnelli et al. 2014; Privon et al. 2017). Because dust, and
therefore very likely gas (Bothwell et al. 2017), temperatures are
found to increase towards higher specific SFRs, this is particularly
relevant in starburst galaxies, where there is some evidence of a
top-heavy IMF (Doane & Mathews 1993; Sliwa et al. 2017).
For instance if the gas temperatures in the star-forming regions of
high-redshift galaxies are well approximated by their observed dust
temperatures (Magnelli et al. 2014), at 25–45 K both the stellar mass
and SFRs are overestimated with a static IMF. Correcting both with
the temperature-dependent mass-to-light ratio derived in Section 3,
we find that the star-forming main sequence may be even narrower
than originally believed (Fig. 4).
5 O BSERVATIONA L TESTS
We predict that the IMF, and hence the stellar population, should
contain a significantly larger fraction of massive stars at high red-
shift, or when other conditions drive an increased gas temperature,
than would be predicted by a low-redshift Milky Way IMF. It is
impossible to directly measure the stellar mass distribution of high-
redshift galaxies so a direct test is impossible. However, we have
described many possible indirect tests for the bottom-light IMF.
Figure 4. An idealized set of probability contours for the star-forming main
sequence is shown for the currently inferred population (black) and the IMF-
corrected inference (red). The centre line in each case is the median and the
shaded region is the ±1σ contour. After correcting for the variability of the
IMF with SFR the main sequence shifts both down and to the left, doing so
more strongly the more massive and more rapidly star forming the galaxy
is. This means that the already narrow main sequence is actually narrower
than previously inferred.
The tests with the strongest observational constraints come from
well-measured nearby galaxies. The spectra of these galaxies ex-
hibit discrepancies between the predicted and inferred mass-to-light
ratio (Cappellari et al. 2012), providing indirect evidence for a vari-
able IMF (Narayanan & Dave´ 2012, 2013).
We also expect several chemical signatures because of this phe-
nomenon. Increasing the typical stellar mass also increases the num-
ber of massive stars that explode as supernovae at the expense of
low-mass asymptotic giants (AGB stars). The net result of this is
likely to be an increase in 16O and other heavier α-process isotopes
at the expense of carbon in the combined stellar chemical yield
(Timmes, Woosley & Weaver 1995; Karakas & Lattanzio 2014).
Similarly, the increase in higher mass AGB stars relative to low-
mass stars would mean more suffer hot bottom burning during third
dredge up with the consequence of increasing the 14N yield at the
expense of 12C (Timmes et al. 1995; Karakas & Lattanzio 2014).
Chemical evidence of this sort has been seen in the halo of the Milky
Way (Lucatello et al. 2005; Pols et al. 2012) and it is possible that
more detailed studies of this and the haloes of galaxies at higher
redshift may further constrain the IMF.
Additional indirect evidence comes from observations of the
gamma-ray burst (GRB) population. Because GRBs are thought
to originate only from higher mass stars, they serve as a probe of
that population and hence as a proxy for the IMF. Several observa-
tions suggest an evolving luminosity function (Tanvir et al. 2012;
McGuire et al. 2016). Unfortunately, the effect of the IMF on GRB
observations is largely degenerate with the effect of metallicity and
these observations are currently explained by fitting the metallicity
evolution (Perley et al. 2016). Nevertheless, modern cosmological
simulations are capable of testing this evolution and there has been
recent interest in understanding the effect of the IMF in these sim-
ulations (Guszejnov, Hopkins & Ma 2017), so these observations
may prove useful in the near future.
It may also be possible to test our predictions with observations
of galactic clusters. Indeed Guszejnov et al. (2017) have already
performed such a test by simulating the formation of a galaxy sim-
ilar to the Milky Way with a variety of different IMF temperature
dependences. They find that our model with f(T)∝T2 produces a fac-
tor of several more variation in the IMF of galactic clusters than is
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4270 A. S. Jermyn, C. L. Steinhardt and C. A. Tout
observed, which suggests that the effect of temperature is smaller
than what we have suggested. This is certainly possible. There are
well-motivated models that propose significantly weaker depen-
dences that could well be correct. In which case the effects of the
CMB and CRs should be correspondingly smaller.3 It is also possi-
ble that the molecular clouds that form stars do so in more uniform
environments than observations of present-day clouds suggest. For
instance there is some evidence that star formation in at least some
clouds requires a gravitational trigger (Longmore et al. 2015), which
could serve to reduce the scatter in their initial temperatures. Finally,
it is also possible that the details of the simulation matter, particu-
larly with regards to the criteria for star formation4 and the statistics
of the star particles, which in this case were comparable to or larger
than the clusters from which the variation has been deduced (see
e.g. Bastian et al. 2010, for details of the observed clusters).
Finally, the most direct test would be to compare the spectra
of high-redshift galaxies with those predicted by the IMF (equa-
tion 10). To perform this test at the redshifts of interest is beyond
current observational capabilities but should be possible in the near
future with the JWST. The Near-Infrared Spectrograph (NIRSpec)
should enable detailed galactic spectroscopy at z > zcrit ≈ 6 and
thereby measure high-redshift stellar populations (Gardner et al.
2006; Volonteri et al. 2017).
As an immediate step, in principle it should be possible to fit ex-
isting photometry of high-redshift galaxies and determine whether
this time-variable IMF produces a superior fit. However, we cannot
use models with a time-varying IMF in current codes without gener-
ating a new set of templates and implementing the requirement that
the redshift of the object matches the template redshift rather than
that being a fully independent parameter (see e.g. Conroy, Gunn
& White 2009; Conroy & Gunn 2010). A significant restructuring
of template fitting codes would be required. One advantage in our
analysis is that the expectation of minimal dust at high redshifts
substantially mitigates the degeneracy between reddening and the
age of the stellar population that exists at lower redshifts.
Although existing codes do not support the required templates, it
is still useful to examine the spectra they predict for a single stellar
population. Simulated spectra (Fig. 5) were produced for a single
stellar population with an age of 109 yr with the IMF of equation (10)
for a variety of redshifts and dust content, with the PYTHON-FSPS code
(Conroy et al. 2009; Conroy & Gunn 2010). This is what would be
seen if a galaxy formed with a single starburst at the specified red-
shift and were then observed 109 yr later. The effect of the modified
IMF strongly depends on redshift and is distinguishable from that
of dust content in the (rest-frame) near-infrared (IR) but may be
masked by the effects of dust at longer wavelengths. As a result
photometry including mid-IR observations should contain enough
information to test these predictions if template-fitting codes can be
appropriately modified.
3In particular Guszejnov et al. (2017) find that the protostellar heating model
with f(T)∝T−1/18 produces less than the observed variation, while the other
models we have discussed produce somewhat more than is observed. This
favours the former but does mean that nearly all of the observed variation in
the knee of the IMF among different clusters must be explained as a result
of observational uncertainties rather than intrinsic scatter. It could also be
that there are as-yet unknown sources of variation or that the variation other
models predict is suppressed by selection effects acting in the environments
conducive to star formation.
4This point is discussed in some detail by Guszejnov et al. (2017) along
with other caveats in their section 3.1.
Figure 5. The relative flux difference 
F/F between the standard Kroupa
(2001) IMF and several modified IMFs is shown as a function of wavelength
λ for a population of age 1 Gyr. All differences were computed between rest-
frame spectra. The reference spectrum was computed with low dust content,
while the modified ones were computed with two different dust models (low
and high). The effect of the modified IMF strongly depends on temperature
and largely amounts to a rescaling. This makes it distinguishable from that
of dust content in the rest-frame near-IR.
In summary, it should be possible to conclusively determine
whether this modified IMF is responsible for the spectra of high-
redshift galaxies that will be observed by JWST/NIRSpec. Sub-
stantial modifications are required to investigate this with existing
codes but, once performed, photometry of high-redshift galaxies
will likely be sufficient for this purpose, particularly at high enough
redshift that dust-driven extinction is minimal.
6 C O M P L I C AT I O N S
There are three potentially significant complications to this picture
but we conclude that none are likely to fundamentally alter it. First,
if the typical age of stars increases at higher redshift, then, even
though the IMF would be bottom-heavy, after convolving it with
the star formation history of a galaxy, the net result could be to leave
the mass-to-light ratio unchanged. However, this is counter to what
is both expected theoretically and observed out to z ≈ 6 (Steinhardt
& Speagle 2014) and so, if anything, likely operates in the opposite
direction.
Secondly, if the typical density of molecular clouds at high red-
shift were once greater than today that could counteract the increase
in the Jeans mass. This is unlikely for the same reason that a de-
creasing cloud size is unlikely: the early molecular clouds were
likely larger and more diffuse than those of our galaxy simply be-
cause there was less time available for them to collapse and develop
density gradients. However, other parameters including the cloud
velocity dispersion and metallicity likely also change with time and
could conceivably counteract the effects of temperature variation.
Without a much more detailed analysis we cannot eliminate this
possibility and so it is important to bear in mind.
Finally, while we have characterized the physics of star formation
by the parameters κ , ρ, and T, in practice these quantities are really
drawn from a joint probability distribution. That is, each galaxy
has many molecular clouds with a variety of masses, temperatures,
densities, and so on, so it is an approximation to replace these
distributions by typical quantities as we have done. At z < zc, this
implies that there should be clouds that would have T < TCMB(z)
were it not for the CMB heating them, meaning that the effect
ought to be observable at lower redshift than our treatment would
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The CMB and the IMF 4271
suggest. Likewise at z > zc, there ought to be some clouds that
would be hotter than TCMB(z) even without CMB heating and that
would therefore be unaffected either way. This means that the effect
is not quite as strong at moderate z > zc as we would predict. The
practical effect of us considering distributions rather than single
representative quantities is one of smoothing the z dependence of
the mass-to-light ratio. The width of this smoothing is

z ≈ zc 
T
TISM
= 
T
TCMB
, (21)
where 
T ≈ 11 K is the spread in molecular cloud
temperatures.5This suggests 
z ≈ 4, so this approximation
likely does mean that the effect should be visible as a small
correction that becomes larger as we look to increasing redshift.
7 D ISCUSSION
Both theory and observation strongly suggest that gas temperatures
in star-forming regions of many galaxies should be higher than
in star-forming regions of the Milky Way. We have shown that
failure to account for the resultant bottom-light IMF leads to an
overestimation of both SFR and stellar mass for these galaxies.
This result has broad implications for both our observational and
theoretical understanding of galaxy evolution. The most important
effects on current observations of galaxy evolution are as follows.
(i) Every z  6 photometric stellar mass and SFR is overesti-
mated by current fitting techniques, because the CMB increases the
temperature in star-forming regions. This may also help to explain
the possible tension between the inferred stellar masses of high-
redshift galaxies and theoretical halo mass functions (Steinhardt
et al. 2016). However, several other explanations in which the UV
luminosity to halo mass ratio would evolve sharply have also been
proposed (Finkelstein et al. 2015; Trac et al. 2015; Mashian et al.
2016).
(ii) The stellar mass and SFR of nearly every z > 1 star-forming
galaxy are overestimated by current fitting techniques, because CR
heating increases the temperature in star-forming regions. Because
this effect varies with SFR and therefore also with stellar mass for
galaxies on the star-forming main sequence, a correction for it also
alters the shape of inferred mass functions. Tentative evidence of
this has been seen in the centre of the Milky Way, which has both
a top-heavy IMF (Lu et al. 2013) and enhanced CR density (Goto
2013), though the latter occurs over a somewhat larger region than
the former. CR heating may also contribute to an explanation for the
apparent difference in the shapes of high-redshift stellar mass and
halo mass functions (Leauthaud et al. 2010, 2012; Gonzalez et al.
2013; Behroozi & Silk 2015; Steinhardt et al. 2016; Davidzon et al.
2017).
(iii) The star-forming main sequence is narrower than previously
believed. At fixed mass and redshift, SFR overestimation is larger
in galaxies with higher SFRs. Thus correcting for this effect reduces
the spread of the star-forming main sequence.
(iv) Accounting for these effects properly is difficult with current
template fitting codes. There are several codes (e.g. those of Arnouts
et al. 1999; Conroy et al. 2009; Kriek et al. 2009) that allow different
choices of IMF. However, we find here that the correct choice of
IMF is linked to other fit parameters, including redshift and SFR,
5More formally 
T is the standard deviation in temperature among star-
forming clouds measured by Svoboda et al. (2016). This was computed by
fitting a normal distribution to the 25th and 75th percentiles.
which are currently treated independently. In addition, modelling
the existing stellar population requires not merely a star formation
history but also a proper treatment of the linked variation in the IMF
over the course of that history.
Although these observational and analytical challenges are diffi-
cult, these same effects provide several tantalizing possibilities for
the development of new models of feedback in galaxy evolution. In
particular, there should be strong feedback between star formation
and temperature-driven changes to the IMF in star-forming galaxies.
An increased SFR produces more CRs, resulting in higher tempera-
ture gas and dust. An increased temperature in star-forming regions
produces a more bottom-light IMF. This increase in the fraction of
massive stars in turn increases the CR density generated at fixed
SFR. However, it also decreases the ability of molecular clouds to
collapse so reducing the overall SFR.
Depending upon the magnitude of these two effects, two different
behaviours are possible. If the reduction in SFR is sharper than
the increase in CR production per unit SFR, it provides negative
feedback, leading to an equilibrium between SFR, the IMF, and
gas temperature. It may be possible to develop a model in which
these effects lead to an explanation for the observed star-forming
main sequence as an equilibrium solution. If so, there is copious
data available to test such a model. It should also be noted that
the natural time-scale for such feedback would be the average gap
between star formation and CR production, which, depending upon
the IMF, should be of order 1–3 Gyr because this is the typical age
of stars with sufficient mass to produce CRs. This is longer than the
dynamical time-scale for a typical star-forming galaxy but similar
to the feedback time-scale estimated from the scatter in redshift of
the star-forming main sequence (Steinhardt & Speagle 2014).
However, if the increase in CR production outweighs the reduc-
tion in SFR, it instead produces positive feedback. This runaway
process produces a galaxy with very high gas and dust temperatures,
an IMF consisting only of massive stars with short lifetimes, and
rapid production of dust and metals. Although the true SFR in such
a galaxy would be relatively low, the abundance of massive stars and
their high luminosity would indicate a very young stellar popula-
tion with a very high SFR if analysed with current techniques with a
Milky Way IMF. This solution could be a reasonable description of
the origin of starburst galaxies, which have many of these properties
(Weedman et al. 1981). To make a specific model for either case
requires a treatment of the various cooling mechanisms in order to
fully describe the feedback between changes in CR production and
temperature. At present, such models are very poorly constrained
by extragalactic observations, and even less so at high redshift.
We shall consider a broader set of models in the future. However,
we note here that one generic prediction is that if all parameters
except the SFR are fixed in a galaxy, then at lower SFRs there
would be an equilibrium solution and at sufficiently high SFRs, the
galaxy would instead enter the runaway regime. A galaxy following
this path would first grow along an equilibrium track, potentially
one that would be well fitted by the observed star-forming main
sequence. Galaxies growing along the main sequence increase their
SFR over time, and therefore such a galaxy would eventually hit
the runaway regime and become a starburst galaxy. Finally, the
rapid heating of gas would prevent the formation of new stars and
the galaxy would become quiescent. Furthermore, these transitions
would be entirely secular, being driven by the long-term evolution
of the galaxy rather than by external triggers. Empirical models
with some of these properties have been described in recent years
(Conroy & Wechsler 2009; Peng et al. 2010; Steinhardt & Speagle
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4272 A. S. Jermyn, C. L. Steinhardt and C. A. Tout
2014; Toft et al. 2014; Peng, Maiolino & Cochrane 2015), so this
is a promising theoretical avenue to explore. It is possible that the
temperature dependence of the IMF may bring the star-forming
main sequence under the same theoretical umbrella as the starburst
and quiescent regimes and it is certainly the case that continuing to
neglect this phenomenon will yield misleading results.
AC K N OW L E D G E M E N T S
The authors thank Max Pettini for productive conversations on star
formation and Rob Izzard for suggested reading on chemical evolu-
tion. ASJ thanks the UK Marshall Commission for financial support.
CLS thanks the ERC Consolidator Grant funding scheme (project
ConTExt, grant number 648179) and the Carlsberg Foundation for
support. CAT thanks Churchill College for his fellowship.
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