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ACCEPTED FOR PUBLICATION APJ, 2006 JUN 13
Preprint typeset using LATEX style emulateapj v. 11/12/01
ON THE NORMALISATION OF THE COSMIC STAR FORMATION HISTORY
ANDREW M. HOPKINS1 , JOHN F. BEACOM2,3
1. School of Physics, University of Sydney, NSW 2006, Australia; email: ahopkins@physics.usyd.edu.au
2. Dept. of Physics, The Ohio State University, 191 W. Woodruff Ave., Columbus, OH 43210; email: beacom@mps.ohio-
state.edu
3. Dept. of Astronomy, The Ohio State University, 140 W. 18th Ave., Columbus, OH 43210
Accepted for publication ApJ, 2006 Jun 13
ABSTRACT
Strong constraints on the cosmic star formation history (SFH) have recently been established using ultravio-
let and far-infrared measurements, refining the results of numerous measurements over the past decade. Taken
together, the most recent and robust data indicate a compellingly consistent picture of the SFH out to redshift
z ≈ 6, with especially tight constraints for z . 1. We fit these data with simple analytical forms, and derive
conservative bands to indicate possible variations from the best fits. Since the z . 1 SFH data are quite pre-
cise, we investigate the sequence of assumptions and corrections that together affect the SFH normalisation, to
test their accuracy, both in this redshift range and beyond. As lower limits on this normalisation, we consider
the evolution in stellar mass density, metal mass density, and supernova rate density, finding it unlikely that the
SFH normalisation is much lower than indicated by our direct fit. Additionally, predictions from the SFH for
supernova type Ia rate densities tentatively suggests delay times of ∼ 3 Gyr. As a corresponding upper limit on
the SFH normalisation, we consider the Super-Kamiokande (SK) limit on the electron antineutrino (νe) flux from
past core-collapse supernovae, which applies primarily to z . 1. We find consistency with the SFH only if the
neutrino temperatures from SN events are relatively modest. Constraints on the assumed initial mass function
(IMF) also become apparent. The traditional Salpeter IMF, assumed for convenience by many authors, is known
to be a poor representation at low stellar masses (. 1M⊙), and we show that recently favoured IMFs are also
constrained. In particular, somewhat shallow, or top-heavy, IMFs may be preferred, although they cannot be too
top-heavy. To resolve the outstanding issues, improved data are called for on the supernova rate density evolution,
the ranges of stellar masses leading to core-collapse and type Ia supernovae, and the antineutrino and neutrino
backgrounds from core-collapse supernovae.
Subject headings: galaxies: evolution — galaxies: formation — galaxies: starburst — neutrinos — supernovae:
general
1. INTRODUCTION
In the past few years measurement of the evolution of
galaxy luminosity functions at a broad range of wavelengths
has rapidly matured. One consequence of this has been the
refinement in our understanding of how the space density of
the galaxy star formation rate (SFR) evolves (Lilly et al. 1996;
Madau et al. 1996). In particular the recent results from the
Sloan Digital Sky Survey (SDSS), the Galaxy Evolution Ex-
plorer (GALEX), and Classifying Objects by Medium-Band
Observations (COMBO17) at ultraviolet (UV) wavelengths,
and from the Spitzer Space Telescope at far-infrared (FIR)
wavelengths, now allows this cosmic star formation history
(SFH) to be quite tightly constrained (to within ≈ 30 − 50%)
up to redshifts of z ≈ 1. Combined with measurements of
the SFH at higher redshifts from FIR, sub-millimeter, Balmer
line and UV emission, the SFH is reasonably well determined
(within a factor of about 3 at z & 1) up to z ≈ 6 (e.g., Hopkins
2004).
Additional results from the Super-Kamiokande (SK) parti-
cle detector provide a strong limit on the electron antineutrino
(νe) flux, 1.2 cm−2 s−1 (forEν > 19.3 MeV), originating from
supernova type II (SNII; here and throughout, we assume the
inclusion of all core-collapse supernovae: types II, Ib, and
Ic) events associated with the SFH (Malek et al. 2003). This
limit on the diffuse supernova neutrino background (DSNB)
acts to constrain the normalisation of the SFH (Fukugita &
Kawasaki 2003; Ando 2004; Strigari et al. 2004, 2005). An ex-
ploration of quantities predicted from the SFH, the stellar and
metal mass density evolution, and supernova (SN) rate evolu-
tion, provides further insight into the allowable normalisation
of the SFH (Strigari et al. 2005). This series of interconnected
physical properties of galaxies and SNe provides an emerging
opportunity for determining the level of the SFH normalisation,
and the SFH measurements particularly for z . 1 now have
the precision to allow this exploration of their accuracy. Con-
straining the normalisation of the SFH will support a range of
quantitative analyses of galaxy evolution, including the mass-
dependence of the SFH (e.g., Papovich et al. 2006; Juneau et
al. 2005; Heavens et al. 2004), and the reasons underlying the
decline in the SFH to low redshifts (e.g., Bell et al. 2005).
The sequence of assumptions explored here starts from ob-
served luminosity density measurements. (1) To these, dust cor-
rections (where necessary), SFR calibrations and IMF assump-
tions are applied, to produce corrected SFH measurements.
(2) From the SFH, assumptions about the high-mass IMF frac-
tion that produce SNII lead to predictions for the SNII rate den-
sity. This can be compared directly with measurements of this
quantity. (3) Assumptions about the neutrino emission per SNII
then give a prediction of the DSNB, for comparison with the SK
limit. Since optical SNII can be hidden from observations by
dust obscuration, the present SNII rate density measurements
may merely be lower limits. In contrast, since neutrinos are un-
affected by dust, the DSNB provides an absolute upper limit on
the true SNII rate. The consistency that we find between the
1

2 A. M. Hopkins, J. F. Beacom
SFH data and the SK upper limit on the DSNB indicates that
there is little room to increase the overall normalisation of the
SFH. Thus, this constrains each of the dust corrections, the as-
sumed IMF, and the neutrino emission per supernova. None can
be increased significantly without requiring at least one of the
others to be reduced below its expected range. Below, we focus
especially on the assumed IMF, noting the more detailed treat-
ment of the neutrino emission per supernova by Yu¨ksel et al.
(2005). A parallel set of assumptions to step (2) regarding the
generation of SNIa lead to predictions for the SNIa rate density,
and this is explored in some detail with tantalising implications
regarding the extent of the SNIa delay time.
In § 2 we update the SFH data compilation of Hopkins (2004)
and address some of the assumptions that affect the normali-
sation. We identify the best parametric fit to the most robust
subset of this data in § 3, consistent with the νe limits from SK.
In § 4 we present the results of this fitting in terms of the stel-
lar and metal mass density evolution and the SNII and SNIa
rate evolution. The implications for the assumed IMF and SNIa
properties are discussed further in § 5.
The 7371 cosmology is assumed throughout with H0 =
70 km s−1 Mpc−1, ΩM = 0.3, ΩΛ = 0.7 (e.g., Spergel et al.
2003).
2. THE DATA
The compilation of Hopkins (2004) was taken as the start-
ing point for this analysis, shown in Figure 1 as grey points.
These data are reproduced from Figure 1 of Hopkins (2004),
and use their “common” obscuration correction where neces-
sary. Additional measurements are indicated in colour in Fig-
ure 1. For z . 3 these consist of FIR (24µm) photometry
from the Spitzer Space Telescope (Pe´rez-Gonza´lez et al. 2005;
Le Floc’h et al. 2005), and UV measurements from the SDSS
(Baldry et al. 2005), GALEX (Arnouts et al. 2005; Schimi-
novich 2005) and the COMBO17 project (Wolf et al. 2003). At
z = 0.05 a new radio (1.4 GHz) measurement is shown (Mauch
2005), which is highly consistent with the FIR results, as ex-
pected from the radio-FIR correlation (Bell 2003a). Also at low
redshift (z = 0.01) is a new Hα derived measurement (Hanish
et al. 2006). At higher redshifts additional SFH measurements
come from the Hubble Ultra Deep Field (UDF, Thompson et al.
2006), and from various photometric dropout analyses, probing
rest-frame UV luminosities (Bouwens et al. 2003a,b; Ouchi et
al. 2004; Bunker et al. 2004; Bouwens et al. 2005a). The UDF
measurements of Thompson et al. (2006) are derived through
fitting spectral energy distributions to the UDF photometry us-
ing a variety of templates with a range of underlying assump-
tions. In particular this includes different IMF assumptions for
different templates. Although we show these measurements in
Figure 1 for illustrative purposes (having scaled them assuming
they were uniformly estimated using a Salpeter 1955 IMF), we
do not include them in subsequent analyses as there is no clear
process for scaling these measurements to our assumed IMFs
in the absence of a common original IMF assumption.
2.1. SFR calibrations
Throughout we assume the same SFR calibrations as Hop-
kins (2004). Uncertainties in the calibrations for different SFR
indicators will correspond to uncertainties in the resulting SFH
normalisation for that indicator. Issues regarding SFR calibra-
tions are detailed in Moustakas et al. (2006), Kennicutt (1998)
and Condon (1992). Perhaps the most uncertain calibrations
are the radio (1.4 GHz) and FIR indicators (although the [OII]
indicator has a similar level of uncertainty). For FIR SFRs,
Kennicutt (1998) indicates a variation of about 30% between
calibrations in the literature. Bell (2003a) refines the 1.4 GHz
calibration of Condon (1992) following an exploration of the
origins of the radio-FIR correlation, and the implication is that
the radio SFR calibration has about the same uncertainty as the
FIR, assuming no contamination by emission from an active
galactic nucleus (AGN). More significantly, though, for indi-
vidual galaxies there can be large differences, up to an order
of magnitude, in the SFRs inferred through different indicators
(e.g. Hopkins et al. 2003), although on average for large sam-
ples there is a high level of consistency. This is reflected in the
overall consistency between SFR densities, ρ˙∗, estimated from
different indicators, with at most about a factor of two or three
variation (which also includes the uncertainty in dust obscura-
tion corrections, where relevant). This scatter is still notably
larger than the uncertainties in individual SFR calibrations, and
is suggestive of the overall level of systematic uncertainty in the
individual calibrations. It is likely that this reflects subtleties
such as low-level AGN contamination in various samples, the
difficulties with aperture corrections where necessary, dust ob-
scuration uncertainties (discussed further below), and other is-
sues. It is for these reasons that we neglect the details of the
underlying SFR calibrations, as their small formal uncertainties
are dominated by these larger systematics. Further, the effect of
these systematics between different SFR indicators acts to in-
crease the scatter in the overall SFH compilation, rather than
to systematically shift all measurements in a common direc-
tion. So even the factor of two to three variation here cannot be
viewed precisely as an uncertainty on the SFH normalisation.
In this sense, the level of consistency between ρ˙∗ measurements
using dramatically different samples and SFR indicators, over
the whole redshift range up to z ≈ 6, is actually quite encour-
aging.
2.2. Dust obscuration corrections
The issue of dust corrections is complex and has been ad-
dressed by many authors (e.g., Buat et al. 2005; Bell 2003b;
Buat et al. 2002; Calzetti 2001). Hopkins (2004) com-
pared assumptions of a “common” obscuration correction to a
luminosity-dependent correction on the SFH. The latter leads,
for UV and emission line estimators, to somewhat higher values
for ρ˙∗ at higher redshift, although both methods give results es-
sentially consistent with SFH estimators unaffected by obscura-
tion. Bell (2003a) shows that SFRs derived from summing total
IR (8−1000µm) and UV indicators using the SFR calibrations
of Kennicutt (1998) adopted here, are consistent within a factor
of 2 of obscuration corrected Hα SFRs. This technique, sug-
gested as the preferred method by Iglesias-Paramo et al. (2006),
has the strong advantage that it avoids assumptions about the
extent or form of the obscuration, and variations due to possi-
ble luminosity bias in the UV selected sample (e.g., Hopkins et
al. 2001; Sullivan et al. 2001; Afonso et al. 2003).
To implement effective obscuration corrections for the UV
measurements at z . 1 (Baldry et al. 2005; Wolf et al. 2003;
Arnouts et al. 2005), we thus take advantage of the well-
established FIR SFR densities up to z = 1 from Le Floc’h et
al. (2005). The UV data at z ≤ 1 are “obscuration corrected”
1 Thanks to Sandhya Rao (Rao et al. 2006) for this terminology.

On the normalisation of the Cosmic SFH 3
by adding the FIR SFR density from Le Floc’h et al. (2005) to
each point. As shown by Bell (2003a) for individual systems,
this technique results in ρ˙∗ estimates consistent with the obscu-
ration corrected Hα measurements, in particular the recent esti-
mates from Hanish et al. (2006) and Doherty et al. (2006), (also
compare the current Figure 1 to Figure 1 of Hopkins 2004).
This result is consistent with the interpretation of Takeuchi et
al. (2005) that about half the SFR density in the local universe
is obscured by dust (see also Martin et al. 2005), increasing
to about 80% by z ≈ 1, a trend that can be seen in the differ-
ent slopes of the (obscuration corrected) UV measurements and
FIR measurements in Figure 1. For obscuration corrections to
the UV data between 1 < z < 3 we rely on the fact that the FIR
measurements of Pe´rez-Gonza´lez et al. (2005) are quite flat in
this domain, as well as being highly consistent with those of Le
Floc’h et al. (2005) at z < 1, and add the constant SFR density
corresponding to that of Le Floc’h et al. (2005) at z = 1. This
is also consistent with the recent measurements of obscuration
corrections for UV luminosities at z ≈ 2 by Erb et al. (2006),
who find a typical correction factor of ≈ 4.5. At higher red-
shifts we apply a “common” obscuration correction to the UV
data as detailed in Hopkins (2004). The reliability of this as-
sumption is open to question, but may not be unreasonable, as
a comparison with the the measured colour excesses of Ouchi et
al. (2004) illustrates. The obscuration corrected SFR densities
of Ouchi et al. (2004) at z = 4− 5 using their measured colour
excesses are actually marginally higher than we derive with the
“common” obscuration correction.
2.3. UV data at high-z
At z & 2, some clarification is necessary regarding the UV
derived ρ˙∗ measurements. The two UV measurements at z ≈ 2
and z ≈ 3 are taken from the Hubble Deep Field (HDF) sample
of Arnouts et al. (2005), and have been corrected for obscura-
tion by adding to them a constant FIR SFR density equal to that
at z = 1 from Le Floc’h et al. (2005). The three UV points
at 2 . z . 5 with comparatively low ρ˙∗ come from Bouwens
et al. (2003a), and are based on photometric dropouts using the
cloning technique detailed by those authors. These points are
low as they correspond to only the high-L end of the luminos-
ity function, and are not used here in any subsequent analysis.
The two UV measurements at z ≈ 4 and z ≈ 5 are from Ouchi
et al. (2004), corrected using a “common” obscuration correc-
tion, and are marginally lower than the ρ˙∗ derived by those au-
thors using their measured E(B − V ) colour excesses. This
slight difference has a negligible impact on the fitted paramet-
ric forms detailed below, and does not affect any subsequent
analysis. At z ≈ 6 there are now a large number of esti-
mates in the literature, primarily using the photometric dropout
technique and hence relying on accurate photometry and colour
measurements, all of which are based on small, deep fields (in-
cluding the HDF, the Great Observatories Origins Deep Survey,
GOODS, and the UDF). The two highest measurements of ρ˙∗
at z ≈ 6 come from GOODS (Giavalisco et al. 2004) and i-
dropouts in two deep ACS fields (Bouwens et al. 2003b). These
estimates appear to be high compared to subsequent photomet-
ric dropout analyses, and this seems to be a result of sample
contamination due to large colour uncertainties from low S/N
photometric measurements (A. Bunker 2005, private communi-
cation). These two points are not used in subsequent analysis.
The lowest measurement of ρ˙∗ at z ≈ 6 is from Bunker et al.
(2004), based on UDF dropouts, and probes to 0.1L∗. Contri-
butions from fainter sources are unlikely to increase this mea-
surement by more than a factor of two. The measurement of ρ˙∗
between these extremes comes from Bouwens et al. (2005a),
incorporating the largest current sample of i-dropouts from the
UDF, UDF parallels, and GOODS. This measurement super-
sedes an earlier measurement (Bouwens et al. 2004a) using only
the UDF parallel fields, which is not shown.
Although the issue of dust obscuration at high redshift is still
highly uncertain, some data is beginning to be obtained. In ad-
dition to the E(B − V ) estimates from Ouchi et al. (2004), in-
triguing evidence for significant obscuration (AV ≈ 1 mag) at
z = 6.56 has recently been established through Spitzer obser-
vations of a lensed Lymanα (Lyα) emitting source (Chary et al.
2005). This implies that the first epoch of star formation in this
source must have occurred around z ≈ 20, and moreover that
the “common” obscuration corrections applied to UV luminos-
ity based SFR densities at 3 . z . 6 may not be unreasonable.
This is also supported by spectroscopic Lyα emission measure-
ments of Lyman Break Galaxies (LBGs) at z ≈ 5 (Ando et al.
2005b), suggesting that the bright LBGs (at least) lie in dusty,
chemically evolved systems at this redshift.
At redshifts 6 < z < 10 there have also been recent exciting
estimates of the SFH (Bouwens et al. 2004b, 2005b,c) based on
UV luminosities inferred using the photometric dropout tech-
nique. These measurements strongly suggest that the decline in
the SFH seen around z ≈ 6 continues to higher redshifts. Given
the very small samples involved, the complete uncertainty re-
garding the level of obscuration at these redshifts, and more
importantly the minimal impact that these data have on the in-
tegrated properties of the SFH being explored here, we do not
include these points in any of our subsequent analysis. It is in-
teresting to note, though, that they do appear in general to be
consistent with all our results based on the SFH at z . 6.
To summarise the impact of dust obscuration corrections on
the normalisation of the SFH, first the corrections can obviously
act only to increase, not decrease, the normalisation. The tech-
nique of UV+FIR measurements gives an effective obscuration
correction to the UV data increasing from a factor of two at
z ≈ 0 up to a factor of five at z & 1. These results are consis-
tent with obscuration corrected Hα estimates for ρ˙∗ spanning
0 < z . 2 (the range for which Hα estimates are available),
suggesting that the extent of the obscuration correction is un-
likely to be much smaller. For 0.5 . z . 2.5 this becomes less
of a concern as the SFH is dominated by contributions from
FIR measurements, unaffected by obscuration, which serve as
a lower limit to the SFH normalisation. At higher redshifts still,
the issue is less clear, but the recent results indicated above
suggest that even up to z ≈ 6 dust obscuration may be sig-
nificant. For the following analysis, the νe flux is dominated
by the z . 1 regime, where the obscuration correction seems
fairly well constrained through the UV+FIR technique.
2.4. IMF assumptions
While uncertainties in SFR calibration act to increase the
scatter in the SFH, and uncertainties in dust obscuration can
raise it to greater or lesser degrees, the choice of IMF is the only
assumption that can systematically decrease the SFH normali-
sation. While most authors over the past decade have assumed
the traditional Salpeter (1955) IMF for convenience, observa-
tions within recent years have strongly ruled it out as a robust
model for a universal IMF. A modified Salpeter form with a
turnover below 1M⊙, though, is still a reasonable model (e.g.,

4 A. M. Hopkins, J. F. Beacom
Baldry & Glazebrook 2003), and other currently favoured IMFs
include those of Kroupa (2001), Baldry & Glazebrook (2003),
and Weidner & Kroupa (2005, 2006).
The SFR calibrations for all indicators used here for deriving
the SFH estimates, (which are the same as in Hopkins 2004),
are defined assuming the Salpeter (1955) IMF. The compila-
tion of data by Hopkins (2004) converts SFH estimates from
the literature to these calibrations, where necessary, to ensure
consistent assumptions throughout. To convert SFH estimates
to an alternative IMF assumption corresponds to simple scale
factor. This scale factor is typically established through popu-
lation synthesis modelling, using codes such as PEGASE (Fioc
& Rocca-Volmerange 1997), GALAXEV (Bruzual & Charlot
2003), or STARBURST99 (Leitherer et al. 1999). These codes,
given an input IMF and SFR, can be used to infer the luminos-
ity at SFR-sensitive wavelengths (typically UV and Hα). For
different IMFs, the ratio between the resulting luminosities for
a fixed SFR is the required scale factor. To explore these scale
factors we used PEGASE.2 to infer the UV (2000 A˚) and Hα lu-
minosities, given as input a fixed, constant SFR, for those IMFs
not excluded by Baldry & Glazebrook (2003). We use the de-
fault PEGASE.2 values for most input parameters, including a
close binary mass fraction of 0.05, evolutionary tracks with stel-
lar winds, and the SNII model B of Woosley & Weaver (1995),
but we specify an initial metallicity of Z = 0.02, evolving self-
consistently. We refer the reader to Bruzual & Charlot (2003)
for a comparison between GALAXEV and PEGASE.2 (and
also Fardal et al. 2006, who explore the effect of different input
metallicities). For IMFs with progressively shallower slopes, it
can be seen that the Hα and UV luminosities scale differently,
compared to those derived assuming the Salpeter (1955) IMF,
with Hα being enhanced more quickly than the UV. We choose
to follow earlier work, and rely on the UV luminosity for our
IMF scale factors (e.g., Cole et al. 2001; Madau et al. 1998), but
note that our results would change only marginally if we used
the scale factors from the Hα luminosities.
The scale factors derived for shallower (flatter, or more top-
heavy) IMFs are, in general, smaller than those for steeper
IMFs. This results from the top-heavy IMFs producing more
high mass stars, and consequently more UV or Hα luminosity,
for a fixed total mass or SFR. Conversely, for a fixed UV or
Hα luminosity, a top-heavy IMF requires a lower SFR to re-
produce that luminosity. We emphasize that here the observed
quantity is the UV or Hα luminosity, and the derived quantity
is the SFR. We investigate here two extreme, but still realis-
tic, IMF possibilities. The factor to convert SFH measurements
from a traditional Salpeter (1955) IMF (with a power law slope
of −1.35) to the IMF of Baldry & Glazebrook (2003, hereafter
BG IMF, having a high mass power law slope of −1.15) is 0.50
(−0.305 dex). (Note that Baldry & Glazebrook 2003 quote a
high mass slope of −1.2 in their Table 2, from the local Hα
luminosity density, but also quote −1.15 as the best-fit from
the cosmic SFH. Here, as we are interested in the extremes, we
choose to use the latter.) If we used the scaling from the PE-
GASE.2 Hα luminosity, we would have a factor of 0.41 rather
than 0.50 for the BG IMF. All other IMFs explored here vary
much less in the relative scaling for the Hα and UV luminosi-
ties. To convert to the modified Salpeter A IMF (hereafter SalA
IMF, Baldry & Glazebrook 2003, with high mass power law
slope of −1.35) is a factor of 0.77 (−0.114 dex). The Kroupa
(2001) IMF (high mass power law slope of −1.3) and the modi-
fied Salpeter B IMF (hereafter SalB IMF, Baldry & Glazebrook
2003, high mass power law slope of −1.35) have scale factors
intermediate between these choices. By exploring the impact
of assuming the two extreme IMF choices we expect to provide
bounds encompassing the result from choosing any reasonable
IMF in our subsequent analysis. We refer the reader to Figure 1
of Baldry & Glazebrook (2003) for an illustration of these and
other various IMFs from the literature, and also to Table 1 of
Fardal et al. (2006).
The application of an IMF in this manner neglects, of course,
the possibility that the IMF is not universal and indeed is even
likely to be evolving itself (e.g., Kroupa 2001). Different as-
sumed star formation histories for our own galaxy may also af-
fect estimates for the local IMF (see discussion by Elmegreen
& Scalo 2006). At any epoch, though, the universe has some
average IMF, which may have a large scatter around it for indi-
vidual objects, and this average may vary with epoch. It is this
average IMF that the SFH is sensitive to, and this may be differ-
ent from what is inferred locally in the Milky Way. The current
measurements do not yet support any detailed exploration of
these issues and we ignore them for the current analysis.
3. SFH FITTING
In order to derive a νe flux from the DSNB for compari-
son with the limits from SK, it is necessary to fit some func-
tional form to the SFH in order to facilitate integration over
redshift. We choose to use the parametric form of Cole et
al. (2001) as is now commonly used by many authors: ρ˙∗ =
(a + bz)h/(1 + (z/c)d), here with h = 0.7. The individual
ρ˙∗ measurements chosen to constrain this fit are also important
since the resulting fit will obviously vary depending on the data
used. For z ≤ 1, the SFH now appears to be very tightly con-
strained by the combination of UV data from SDSS (Baldry et
al. 2005), COMBO17 (Wolf et al. 2003), and GALEX (Arnouts
et al. 2005), corrected for obscuration using the Spitzer FIR
measurements of Le Floc’h et al. (2005). This use of the FIR
measurements of Le Floc’h et al. (2005) is further supported by
their high level of consistency with those from Pe´rez-Gonza´lez
et al. (2005) and the robust local 1.4 GHz estimate from Mauch
(2005). As a consequence we use only this set of corrected
UV+FIR measurements, along with the z = 0.01 Hα estimate
of Hanish et al. (2006), to constrain the parametric fit for z < 1.
For z > 1 we use all the data available in the compilation with
exceptions as noted above (the Thompson et al. 2006 UDF esti-
mates, the two highest estimates at z ≈ 6, and the three high-L
only estimates from Bouwens et al. 2003a) and we further ex-
clude the six lowest measurements between 1 < z < 2. The lat-
ter include the three highest redshift [OII] estimates from Hogg
et al. (1998), and three UV estimates, where the “common”
obscuration correction assumed is likely to significantly under-
estimate the true level of obscuration (compare the UV points
between Figures 1 and 2 of Hopkins 2004, for example). This
is not unexpected as the UV luminosity density, in particular,
at this redshift probes a very small fraction of the total ρ˙∗ (see
also Takeuchi et al. 2005).
The parametric fitting is a simple χ2 fit to the 58 selected ρ˙∗
measurements spanning 0 ≤ z . 6. The corresponding DSNB
is calculated following the description in Beacom & Strigari
(2006, their equations 1 and 2). The ρ˙∗(z) is first converted to
a type II supernova rate history, ρ˙SNII(z), scaling by the appro-
priate integral over the IMF
ρ˙SNII(z) = ρ˙∗(z)
∫ 50
8 ψ(M)dM
∫ 100
0.1 Mψ(M)dM
(1)

On the normalisation of the Cosmic SFH 5
(see, e.g., Dahlen et al. 2004; Madau et al. 1998), where we
have neglected the small delays due to the short lifetimes
of SNII progenitors. For the two IMFs explored here we
have ρ˙SNII(z) = (0.0132/M⊙) ρ˙∗(z) for the BG IMF and
ρ˙SNII(z) = (0.00915/M⊙) ρ˙∗(z) for the SalA IMF. This il-
lustrates that the choice of IMF will affect the derived neu-
trino production through two separate but related normalisa-
tions. The first comes from how the IMF affects the nor-
malisation of the derived SFH from the observed UV, Hα, or
other SFR-sensitive luminosity, the second through this con-
version of the SFH into a SNII rate history. The combina-
tion of these two factors to some degree converge on similar
results, with steeper IMFs having higher SFH normalisations,
but lower SNII rate scalings, and vice-versa. As we show
below the BG IMF produces SNII rates that are a factor of
0.94 = (0.50/0.77)×(0.0132/0.00915)of those from the SalA
IMF (see also additional discussion below in § 5.2). We empha-
sise that, for a fixed UV, Hα, or other SFR-sensitive luminos-
ity, a shallower high-mass IMF produces fewer SNII, while a
steeper IMF produces more. If there are fewer SNII, this will
respect the upper limit on the DSNB flux, but may start to con-
flict with direct measurements of ρ˙SNII (which we take to be a
lower limit) if too few SNII are predicted. The converse applies
for the steeper IMFs, with more SNII predicted.
It should be noted, too, that the SNII rate could be de-
rived more directly from (say) the UV luminosity, rather than
through the calibration to an SFR (representative of the whole
mass range of an IMF), and then back to a SNII rate in this
way. We use this method as it conveniently allows the con-
fidence region fit to the SFH data to be used directly rather
than calculating different SNII rate conversions for each SFH
measurement, depending on the SFR-sensitive wavelength. We
have assumed throughout that all IMFs span the mass range
0.1 < M < 100M⊙. Allowing a mass range up to 125M⊙
alters most quantities by less than 1% and all quantites by less
than 2%, significantly less than the variation between different
IMF choices, the measurement uncertainties, or other uncer-
tainties affecting the SFH normalisation. The choice of stellar
mass range that gives rise to SNII is the largest assumption in
this step. Restricting the upper mass limit to 30M⊙ reduces
the scale factor by about 10% in both cases. A much greater
change is introduced by raising the lower mass limit. With a
mass range of only 10 < M < 30M⊙ in the numerator of
Equation 1, the resulting scale factors are reduced by ≈ 40%.
The predicted differential neutrino flux (per unit energy)
is then calculated by integrating ρ˙SNII(z) multiplied by the
νe emission per supernova, dN/dE′, appropriately redshifted,
over cosmic time (Fukugita & Kawasaki 2003; Ando & Sato
2004; Strigari et al. 2004, 2005; Daigne et al. 2005; Lunardini
2005):
dφ(E)
dE
= c
∫ 6
z=0
ρ˙SNII(z)
dN(E(1 + z))
dE′
(1 + z)
dt
dz
dz, (2)
where dt/dz = (H0(1 + z)
√
ΩM (1 + z)3 + ΩΛ)−1, c is the
speed of light, and we are evaluating the thermal emission spec-
trum
dN
dE′
(E′) =
Etotν
6
120
7pi4
E′2
T 4
[
eE
′/T + 1
]−1
, (3)
at the appropriately redshifted energy E′ = E(1 + z) (note
the choice of units where k = 1 so that T has units of en-
ergy). Finally, ∫ ∞19.3MeV(dφ/dE)dE is calculated to establish
the νe flux for comparison with the SK limit of 1.2 cm−2 s−1
(Malek et al. 2003). The assumption of a thermal emission
spectrum, and the associated choices of temperature, are de-
termined by the eventual decoupling (after diffusion) of neu-
trinos from the hot and dense proto-neutron star, at a radius
called the “neutrinosphere” (Raffelt 1996). We explore the im-
plication of assuming a temperature of T ≈ 4 MeV, 6 MeV,
or 8 MeV, and as in Beacom & Strigari (2006), we assume
Etotν = 3 × 1046 J= 3 × 1053 erg for the total energy carried
by all six neutrino flavours. While supernovae emit all flavours
of neutrinos and antineutrinos, and each is assumed to carry
an approximately equal fraction of the total energy, at present
the νe flavour is the most detectable. Because the temperatures
of the muon (νµ) and tauon (ντ ) antineutrinos are expected to
be higher than for νe, the effect of neutrino mixing may be
to increase the observable νe flux for a given SFH (Fukugita &
Kawasaki 2003; Ando & Sato 2004; Strigari et al. 2004; Daigne
et al. 2005); as in Beacom & Strigari (2006) and Yu¨ksel et al.
(2005) our results apply to an effective temperature after mix-
ing, making them more constraining.
Given the νe flux for each temperature, we simply scale the
best fitting SFH to ensure the SK limit is not violated. This
approach has the advantage that the shape of the SFH is de-
termined only by measured data that is differential in redshift,
while the normalisation comes directly by imposing the SK νe
limit. We also explored an alternative approach that did not
restrict the fitted shape of the SFH. In this approach, while iter-
ating over the four fitting parameters, each new potential min-
imum χ2 SFH is subjected to the DSNB limit, and the fit is
rejected if the limit is violated. This technique resulted, for the
higher νe temperatures, in fitted SFHs that were significantly
low (compared to the measurements) in the mid-to-high red-
shift range, while still matching the z = 0 and z & 4 SFH. For
the low νe temperatures, this method gave identical results to
the original approach. Our preferred approach emphasises the
direct impact of the νe data appropriately on the z . 1 mea-
surements, as there are very few detectable neutrinos received
from higher redshift. The best fit SFHs independent of the νe
limit are identical to the fits constrained by a νe temperature of
T = 4 MeV. As found and discussed in Yu¨ksel et al. (2005),
our results favour effective temperatures at the lower end of
the predicted range (Keil et al. 2003; Thompson et al. 2003;
Liebendoerfer et al. 2005; Sumiyoshi et al. 2005).
In addition to the Cole et al. (2001) parameterisation, we also
explored a piecewise linear SFH model in log(1+ z)− log(ρ˙∗)
space, in order to test the possibility that the Cole et al. (2001)
parametric model could be biasing the shape of the resulting
SFH fit in some way. In this model we allow the following six
parameters to vary: The z = 0 intercept, the slopes of three lin-
ear segments and the two redshift values at which the slopes
change. With this model we similarly explore the range of
νe temperatures, as for the Cole et al. (2001) parameterisation
above.
4. RESULTS
Figure 1 shows the current SFH data compilation (assum-
ing the SalA IMF) emphasising the additional data used in this
analysis compared to the compilation of Hopkins (2004)2. The
2 The ρ˙∗ data from Hogg et al. (1998) as given by Hopkins (2004) are incorrect, a result of an error in the cosmology conversion parameters used in that analysis.
The correctly converted data are shown here, and are smaller than those given by Hopkins (2004) by values decreasing monotonically from ≈ 43% for the z = 0.2
bin to ≈ 30% for the z = 1.2 bin. Sincere thanks go to Chun Ly for bringing this error to our attention.

6 A. M. Hopkins, J. F. Beacom
best fitting Cole et al. (2001) form for this IMF is also shown
(a = 0.0170, b = 0.13, c = 3.3, d = 5.3), as is the best-
fitting piecewise linear fit. Figure 2 shows the data used in
the fitting and the best fits assuming three temperature values
for the νe population for each IMF assumed. The Cole et al.
(2001) parameters for each case are given in Table 1. With 58
data points and 4 free parameters there are 54 degrees of free-
dom, and the best-fitting χ2 values (for the T = 4 MeV cases)
are 37.5 for both the SalA IMF and the BG IMF. This value
is perhaps somewhat lower than expected, and reflects the na-
ture of the uncertainties for the heterogeneous data used in the
fit. These come primarily from the Poisson counting statistics
of the numbers of observed galaxies contributing to each mea-
surement. No attempt has been made to resample or reanalyse
independent data to appropriately combine their uncertainties,
and the effect of multiple independent measurements at sim-
ilar redshift, each with independently calculated uncertainties
(since the data are heterogeneous), is to mimic conservative un-
certainty estimates in a homogeneous dataset. Correlations may
also exist between assumed independent SFH measurements at
similar redshift, which may also contribute to the low χ2 value.
This may arise as a result, for example, of different teams inde-
pendently analysing the same dataset, or of independent analy-
ses of common or overlapping survey areas, either at a range of
wavelengths within the same survey, or from surveys at differ-
ent wavelengths of the same region of sky.
For both assumed IMFs it can clearly be seen that the as-
sumption of T = 8 MeV, when the SFH is required to be con-
sistent with the νe flux limit, is inconsistent with the SFH mea-
surements. Also for both IMFs, the best fitting SFH assuming
T = 6 MeV is identical to that assuming T = 4 MeV. This
can be understood by considering the SalA IMF, for example,
with the higher SFH normalisation, but which also has a lower
conversion factor between ρ˙∗ and ρ˙SNII, causing the predicted
νe flux to be within the SK limit even with the assumption of
the slightly higher neutrino temperature. For both IMF assump-
tions we determine the 1 σ (grey-shaded) and 3 σ (hatched) con-
fidence regions around the best fitting SFH (corresponding to
T = 4 or 6 MeV). These are derived from the regions of pa-
rameter space with χ2 < χ2min + ∆χ2 where ∆χ2 = 4.7 and
16.0 respectively for 1 and 3 σ (see Avni 1976, who shows that
∆χ2 for q “interesting” parameters itself follows a χ2 distri-
bution with q degrees of freedom; here q = 4 for the Cole et
al. 2001 parameterisation). These confidence regions are deter-
mined independently at each redshift, intended to encompass
the envelope of all fits. They are thus likely to be somewhat
conservative, an effect that is compounded by the result of the
χ2 analysis being affected by what are effectively conserva-
tive measurement uncertainties. Subsequent Figures reproduce
these confidence regions in the predictions for stellar and metal
mass density evolution (ρ∗(z) and ρZ(z), respectively) and SN
rate evolution (ρ˙SN(z)). The shape of the νe energy spectra
corresponding to the various fits for the different IMF and tem-
perature assumptions are shown in Figure 3. These results are
very similar to the spectral shapes derived in Figure 1 of Bea-
com & Strigari (2006).
The results of the piecewise linear fitting, seen in Figure 4
and detailed in Table 2, are remarkably similar in general prop-
erties to the results in Figure 2. The grey shaded and hatched
regions here again show, respectively, the 1 σ and 3 σ confi-
dence regions, here corresponding to ∆χ2 = 7.0 and 19.8
(with q = 6). In this case the perimeters of the confidence
regions are not smooth, an artifact arising from a combination
of both the finite sampling of parameter space for which χ2
values are calculated, and the linear nature of the parameteri-
sation. With sufficiently high resolution sampling of parameter
space the confidence region perimeters would be expected to
curve more smoothly around the upper left and right corners.
This artifact does not impact upon any of our analysis, and as
the confidence regions are mainly shown to illustrate the range
of uncertainty in the fits, we do not pursue this issue further.
The piecewise linear fits again show low values of χ2 (see Ta-
ble 2) for the same reasons as given above. The same prefer-
ence for lower νe temperatures is seen. The similarities here
with the Cole et al. (2001) parameterisation are encouraging
and suggest that the the Cole et al. (2001) parameterisation has
not introduced any significant bias against specific SFH shapes.
For subsequent analysis we retain the T = 4 MeV fits using
the Cole et al. (2001) parameterisation. This does not affect our
conclusions, which remain unchanged regardless of which SFH
parameterisation we choose.
Figure 5 shows the evolution of the stellar mass density,
ρ∗(z), along with the predictions from the best fitting SFH for
the two extreme IMF assumptions (compare with the extensive
compilation of data in Figure 4 of Fardal et al. 2006). To con-
struct this diagram we need to know the fraction of the stellar
mass recycled into the interstellar medium as stellar winds or
SN ejecta, R, corresponding to each IMF (Cole et al. 2001;
Madau et al. 1998; Kennicutt et al. 1994). We follow the pre-
scription suggested by Cole et al. (2001), using the models of
Renzini & Voli (1981) and Woosley & Weaver (1995) for mass
loss due to stellar winds and supernovae respectively, and cal-
culate R = 0.40 for the SalA IMF, and R = 0.56 for the
BG IMF. The stellar mass inferred is then a fraction (1 − R)
of the time integral of the SFH (Cole et al. 2001). Convert-
ing the observed stellar mass density measurements (where a
Salpeter IMF is most commonly used) to our assumed IMFs is
achieved by scaling by the product of the SFR conversion fac-
tor and the ratio of the 1 − R factor for the chosen IMF to that
of the Salpeter IMF (where 1 − R = 0.72). The reliability of
this method was confirmed by reproducing the stellar mass esti-
mate assuming the Kennicutt (1983) IMF by Cole et al. (2001)
compared to their Salpeter IMF value. As a point of reference,
using the Salpeter (1955) IMF where these scalings are not re-
quired, a diagram very similar to the current Figure is shown in
Figure 3 of Hopkins et al. (2005).
Both plots in Figure 5 show the measurements lying system-
atically below the predictions from the SFH, although the dif-
ference becomes more significant at higher redshift (z & 1.5).
Causes for the apparent inconsistency at high redshift have been
discussed by other authors (e.g., Nagamine et al. 2004; Hopkins
et al. 2005), who suggest that, in this regime, the observations
might be missing up to half the stellar mass density. We discuss
this, and the low redshift discrepancy, further in § 5.
Figure 6 shows how the metal mass density evolves, ρZ(z),
as inferred from the SFH (Pei & Fall 1995; Madau et al. 1996).
(Compare with Hopkins et al. 2005, and for a more detailed
treatment of the evolution of separate metals, see Daigne et al.
2004.) To determine this relation from the SFH, we assume that
ρ˙∗ = 63.7 ρ˙Z (e.g., Conti et al. 2003). At z = 0 the compilation
of data from Calura & Matteucci (2004) is shown, and these au-
thors favour a value of 1.31 × 107M⊙Mpc−3, toward the low
end of the range. Values at z = 0 and z = 2.5 from Dunne et
al. (2003) are also shown, suggesting that the evolution in ρZ

On the normalisation of the Cosmic SFH 7
from the SFH may be consistent with that estimated from the
dusty submillimeter galaxy (SMG) population, although recent
results from Bouche´ et al. (2005) indicate that the SMGs may
contribute much less to the metal mass density at high redshift.
Figure 7 shows the evolution in the SN rate for both types Ia
and II SNe. Although the data to date are not yet precise, the
SN rate data have a significant advantage over the stellar and
metal mass density in that they are differential in redshift, and
are in principle more straightforward to measure. The SNII
rate density, ρ˙SNII, is calculated from the SFH as described
above in Equation 1. The SNIa rate density, ρ˙SNIa, is simi-
larly estimated, although it involves more assumptions about
the properties of SNIa events than in the case of SNII. In par-
ticular the delay time tIa between star formation and the SNIa
event and the efficiency η of producing an SNIa event from ob-
jects in the stellar mass range 3 < M < 8M⊙ are not well
constrained (and even this mass range is somewhat uncertain).
Current estimates put tIa roughly in the range of 1− 3Gyr and
η of order 1 − 5% (see discussion in Strigari et al. 2005, and
references therein). We follow Strigari et al. (2005) in assum-
ing a constant tIa = 3 Gyr, and fIa = 1/700M−1⊙ , where
fIa = η
∫ 8
3 ψ(M)dM/
∫ 100
0.1 Mψ(M)dM , to determine ρ˙SNIa
from
ρ˙SNIa(t) = η
∫ 8
3 ψ(M)dM
∫ 100
0.1 Mψ(M)dM
ρ˙∗(t− tIa). (4)
For the SalA and BG IMFs, fIa = 0.028η and fIa = 0.032η,
respectively. With our assumed value of fIa above, this corre-
sponds to assuming η ≈ 5% for both IMFs. Figure 7 also shows
the effect of assuming fIa = 1/1000M−1⊙ in the lower limit of
the 1 σ and 3 σ envelopes for the predicted ρ˙SNIa. Figure 8a
reproduces Figure 7b with the assumption of tIa = 1 Gyr, il-
lustrating the effect of the different delay times. This produces a
somewhat reduced ρ˙SNIa envelope at lower redshifts and moves
the turnover to higher redshifts.
5. DISCUSSION
5.1. Stellar mass and metal mass densities
The predictions from the SFH for both ρ∗(z) and ρZ(z) are
difficult to analyse, for different reasons. Most predictions of
ρ∗(z) based on SFH measurements seem to be larger than the
observed stellar mass density at high redshift, and numerous
simulations imply that the measurements might be underesti-
mating the total ρ∗(z) (e.g., Nagamine et al. 2004; Menci et al.
2004; Somerville et al. 2001; Granato et al. 2000). Indeed it
is suggested by Dickinson et al. (2003) that additional obscura-
tion added to their maximally old component used in estimating
stellar masses could cause arbitrarily large masses to be derived,
and it is perhaps not unreasonable to expect about a factor of
two larger stellar mass densities as a result of reasonable obscu-
ration levels. This would bring the measurements more into line
with the predictions from the SFH. Similar issues may affect
the other high redshift measurements of ρ∗(z), and other issues
that have also been raised include incomplete galaxy population
sampling and cosmic variance that may affect surveys probing
small fields of view (see discussion in Nagamine et al. 2004).
At low redshift, the discrepancy between the measurements
of ρ∗(z) and the SFH prediction is more of a concern. A first
attempt at resolving this might be to suggest that the measured
SFH is too high at z = 0, and the technique used (combining
ρ˙∗ from FIR and UV estimates) is not accurate. This cannot be
the whole solution as it does not address the problem at z ≈ 1,
where the SFH is dominated by the FIR contribution, and the
ρ∗(z) values inferred from the SFH are similarly higher than
the measurements. Another point against this solution is the
equivalent diagram in Figure 3 of Hopkins et al. (2005), where
(for a Salpeter IMF) a broader region, encompassing the major-
ity of SFH measurements, is used to predict ρ∗(z) rather than
the confidence regions fitted here. Even in that diagram, the
ρ∗(z) measurement at z = 0 lies at the extreme lower bound-
ary of the SFH prediction. This suggests that there is some-
thing more subtle underlying this discrepancy than simple ob-
scuration correction errors in the SFH. A partial solution might
be found through the underlying measurement techniques used
for the different quantities. The SFH measurements rely on in-
ferring SFRs from the luminosity generated by massive stars,
while the ρ∗(z) measurements come by inferring total stellar
masses based on the luminosity of low-mass stars. These are
connected in the prediction of ρ∗(z) from the SFH through the
assumed IMF shape, and it is possible that the discrepancy seen
in Figure 5 may be reflecting limitations in our understanding
of the relative shapes of the low and high mass ends of our as-
sumed IMFs. Although further exploration of this discrepancy
is beyond the scope of the present investigation, we refer the
reader to Fardal et al. (2006) who provide a detailed analysis
of this issue, incorporating limits from the total extragalactic
background radiation along with the SFH and stellar mass evo-
lution.
Regarding the evolution of the metal mass density, ρZ(z), in-
vestigation of predictions from the SFH are complicated by the
limited number of estimates for this quantity at z > 0. This is
observationally a difficult measurement to make, particularly
as much of the metals may exist in an ionised intergalactic
medium component. Measurements of the contribution from
various components (stellar, dust, gas) have been explored with
varying estimates for how much of the metal mass density bud-
get might be contained in different components at high redshift
(e.g., Dunne et al. 2003; Bouche´ et al. 2005). Additional discus-
sion of this issue, emphasising the (minimal) contribution from
damped Lyα absorbers, is presented by Hopkins et al. (2005).
5.2. Supernovae type II
Almost identical predictions for ρ˙SNII are obtained for both
the SalA and SalB IMFs detailed in Baldry & Glazebrook
(2003), a consequence of SalA having a slightly higher conver-
sion factor than SalB for transforming ρ˙∗ from the traditional
Salpeter IMF, and a slightly lower conversion factor to trans-
form between ρ˙∗ and ρ˙SNII. The BG IMF result (Figure 7b)
appears to be marginally more consistent with both the ρ˙SNII
and ρ˙SNIa measurements than the prediction assuming the SalA
IMF (Figure 7a) although both are acceptable. The uncertain-
ties affecting the ρ˙SNII measurements are treated in some detail
by Dahlen et al. (2004), and the error bars shown are represen-
tative of the combination of both statistical and systematic un-
certainties. The z = 0 measurement may be somewhat low, and
recent investigations (Mannucci et al. 2003; Ando et al. 2005a)
suggest that this point is in fact likely to be higher by a factor of
two or three, consistent with the SFH predictions. Dahlen et al.
(2004) indicate that the main concern in the measurements is
the level of obscuration, and that a change from their assumed
E(B − V ) = 0.15 of ∆E(B − V ) = ±0.06 would alter their
measurements by one standard deviation. It is certainly likely
that this is the case for the highest redshift point at z = 0.7,

8 A. M. Hopkins, J. F. Beacom
where obscuration is expected to be higher than at lower red-
shifts (e.g., Takeuchi et al. 2005; Hopkins 2004, see also Fig-
ure 1). This would have the effect of raising the z = 0.7 mea-
surement to be consistent with the predictions from the SFH.
The ρ˙SNII measurements provide a strong lower bound on
the normalisation of the SFH. Particularly given that uncer-
tainty regarding obscuration corrections is more likely to raise
than lower the ρ˙SNII measurements, the SFH normalisation can-
not realistically be much lower than that obtained from as-
suming the BG IMF (Figure 2b). The scaling factors for the
SFH fits suggest that this provides further evidence against the
T = 8 MeV assumption, for both assumed IMFs. Moreover,
the ρ˙SNII measurements are unlikely to be affected by sufficient
obscuration to support an SFH normalisation much higher than
that obtained with the SalA IMF. The level of obscuration to be
inferred in order to make the ρ˙SNII measurements lie above the
upper edge of the SalA IMF envelope requiresE(B−V ) ≈ 0.3
at z = 0.3 and E(B − V ) ≈ 0.5 at z = 0.7. These val-
ues are quite extreme, corresponding to correction factors of
approximately 3.3 and 6 respectively, and values this high are
not even inferred from the luminosity dependent obscuration
corrections of Hopkins (2004) for UV data at similar redshifts.
Higher SFH normalisations may still be possible, though, and
here we return to the assumptions regarding the mass range over
which we assume stars become core collapse SNe. If we allow
only a mass range of 10 < M < 30M⊙, the ρ˙SNII predictions
from the SalA IMF would be lowered by as much as a factor
of 0.6 (−0.2 dex). Assuming this mass range would place pre-
dictions from even the traditional Salpeter IMF at such a level
(i.e., ≈ −0.2 dex from the confidence region in Figure 7a). So,
at the expense of an increased lower limit of integration for core
collapse SN production, IMFs providing quite high normalisa-
tions for the SFH can still be made consistent with the ρ˙SNII
measurements.
Another point to be considered here is the issue of a possi-
ble neutrino flux generated from a population of massive stellar
objects not observable as SNII (Beacom et al. 2001; Heger et
al. 2003; Strigari et al. 2005; Daigne et al. 2005). For example,
if stars in the mass range 30 < M < 50M⊙ produce the same
kind of burst of neutrinos at the end of their life as SNII, but do
not become core collapse SNe, instead progressing directly to
a black hole or other exotic end, then the neutrino flux inferred
from the SFH via an assumed SNII rate will be biased, and the
observed SNII rate density could legitimately be ≈ 10% lower
than that predicted from the SFH using this technique. If even
lower mass progenitors become failed supernovae, then this dif-
ference could be even larger. Certainly more robust information
regarding the connection between the final stages of stellar evo-
lution and neutrino emission would help to refine this type of
analysis, and strengthen the implications regarding the favoured
SFH normalisation and corresponding IMF.
Our general conclusions here are (1) that the IMF cannot be
much more shallow at the high mass end than the BG IMF
without predicting values for ρ˙SNII that are too low (see also
Loewenstein 2006); and (2) in the absence of a lower integra-
tion limit as high as ≈ 10M⊙ for core collapse SN produc-
tion, the IMF cannot produce SFH normalisations much higher
than does the SalA IMF without predicting values for ρ˙SNII that
begin to require quite extreme obscuration corrections. Some
slightly less extreme IMFs, while still producing higher SFH
normalisations than the SalA IMF, may be allowed if the core
collapse SN lower mass cut-off lay between 8 and 10M⊙. This
would still seem to be hard to justify, though, as evidence is
growing for SN progenitors at these low masses. The SNII anal-
ysed by Van Dyk et al. (2006), for example, (2003gd), favours a
progenitor mass 8− 9M⊙ (see also Smartt et al. 2004). A sec-
ond SNII (2005cs) also favors a low progenitor mass, 9+3−2M⊙(Maund et al. 2005). To confirm and refine these IMF con-
straints, a larger selection of independent ρ˙SNII measurements,
spanning a broad range of redshift, would be invaluable. As
noted in Strigari et al. (2005), sufficiently precise SNII rate data
could be used to directly predict the DSNB flux, independently
of assumptions about the SFH and IMF.
5.3. Supernovae type Ia
The prediction for ρ˙SNIa from the SFH is also particularly
intriguing. The assumption of the fixed tIa = 3 Gyr has the
effect of matching the z & 3 turnover in the fitted SFH with the
apparent decline in ρ˙SNIa seen in the highest redshift measure-
ment from the GOODS sample of Dahlen et al. (2004). It is
possible, indeed probable, that this is simply a coincidence as it
is a single ρ˙SNIa measurement, with large uncertainties, that is
suggestive of the decline, and the turnover in the SFH is driven
almost entirely by the z ≈ 6 measurement of Bunker et al.
(2004). It is thus still highly possible that the decline in both the
SFH and ρ˙SNIa lie at somewhat higher redshift. In particular,
recent spectroscopic results from a complete magnitude limited
sample (Le Fe`vre et al. 2005) suggest that the SFH inferred at
z = 3− 4 is up to two or three times higher than that estimated
from colour-selected LBGs (Steidel et al. 1999). This may im-
ply that the shape of the SFH is flatter between 2 < z < 6 than
our current fits suggest. Even more tantalisingly, the higher
SFH estimates from the spectroscopic measurements compared
to the colour-selected samples at 3 < z < 4 suggest that the
photometric dropout selected samples at even higher redshift
(z ≈ 6) may also be underestimating the total SFH. This effect
would be compounded by the recent evidence suggesting non-
negligible obscuration at high-z (Chary et al. 2005; Ando et al.
2005b), and appears to provide evidence that the expected high
redshift decline in the SFH may still not be well established yet.
With this in mind, it is nonetheless interesting to note that the
robustness of the GOODS measurements, again, has been ex-
plored in some detail by Dahlen et al. (2004) who are confident
of the reliability of this feature in the ρ˙SNIa data. If the turnover
in the SFH also turns out to be reliable, the ρ˙SNIa evolution
can be used to constrain the delay time for SNIa (e.g., Fo¨rster
et al. 2006). The predictions for ρ˙SNIa are quite different for
tIa = 1 Gyr (Figure 8a) compared with tIa = 3 Gyr, and refin-
ing the measurements of ρ˙SNIa between 1 < z < 3 would be
quite revealing. Alternatively, if the SFH turnover does indeed
lie at z > 6 that would imply either that tIa > 3 Gyr, a result
that appears to lie outside the currently favoured range, or that
the true turnover in the ρ˙SNIa measurements is also at somewhat
higher redshift than the z ≈ 1.5 from Dahlen et al. (2004).
To explore a more physically motivated connection between
SNIa generation and the underlying stellar populations, Scan-
napieco & Bildsten (2005) introduced a two component model,
dependent on both SFR and stellar mass densities, to derive the
SNIa rate density. This has the effect of allowing a contribu-
tion to the SNIa rate from old stellar populations where the cur-
rent SFR may be low, while maintaining a contribution from
the currently star forming systems. Taking a logical next step,
Neill et al. (2006) allow a delay time to be incorporated into
such a two component model, and find a characteristic delay

On the normalisation of the Cosmic SFH 9
time τ = 3 Gyr (given a Gaussian distribution of delay times),
consistent with our simple result above, although their model
also includes a contribution from a component proportional to
ρ∗. Mannucci et al. (2006) find a bimodal distribution of de-
lay times, with a “prompt” component having a delay time of
≈ 0.1 Gyr. (Pinsonneault & Stanek 2006 find evidence that the
joint IMF of binary stars favors “twins” of nearly equal mass,
and suggest that this may provide a natural explanation for these
prompt SNIa.) To illustrate the power of a well constrained
SFH, we use our best fitting SFH for the BG IMF (from Ta-
ble 1) to explore the available parameters in a Scannapieco &
Bildsten (2005) model and, keeping in mind the cautions re-
garding the high redshift declines in both ρ˙SNIa and ρ˙∗, do not
try to overinterpret our results. We explored the connection be-
tween ρ˙SNIa, ρ˙∗ and ρ∗ using the relation
ρ˙SNIa(t) = Aρ˙∗(t− tIa) +Bρ∗(t) (5)
with all quantities in the physical units used in this investi-
gation. Performing a simple χ2 minimisation, allowing A,
B and tIa to vary, we find the best fitting solution favours
A = 1.15 × 10−3M−1⊙ , B = 0M−1⊙ yr−1, and tIa = 2.7 Gyr.
This result is illustrated in Figure 8b, and is clearly driven fairly
strongly (as expected intuitively from Figure 7) by the combi-
nation of the high-z ρ˙SNIa measurement and the turnover in
the SFH driven by the Bunker et al. (2004) measurement. The
value of A = 1.15 × 10−3M−1⊙ is consistent with the range
of 0.001 ≤ fIa ≤ 0.0014M−1⊙ assumed for the limits in Fig-
ure 7. The B = 0 result seems to arise from the strong similar-
ity in shape between the SFH and the ρ˙SNIa(z) evolution, while
the ρ∗(z) evolution has the opposite shape (increasing to lower
redshifts, rather than decreasing). It seems clear that allowing
negative values of B would allow some non-zero solution, al-
though what physical interpretation this would have is not clear.
It certainly would not be in the spirit of the model as intended
by Scannapieco & Bildsten (2005). As concluded above, and
since B = 0, our tentative inference appears to favour SNIa de-
lay times close to tIa ≈ 3 Gyr, although there are clearly much
more sophisticated models, including those allowing for a dis-
tribution in delay times, that we have not explored here, and
the locations of the apparent downturns in both ρ˙SNIa and ρ˙∗
clearly have a very strong influence on this result.
5.4. Detecting the DSNB
With the best fitting SFH models explored here, the predic-
tions for the DSNB appear to lie excitingly close to the mea-
sured νe flux limit (Figure 3). It is clear that directly observing
the DSNB will allow much greater insight into the properties of
star formation. Already the DSNB constraint indicates a pre-
ferred IMF range and normalisation for the SFH. It also illus-
trates that stronger constraints on the SFH have implications
for understanding the details of both SNII and SNIa produc-
tion, and the physical basis of neutrino generation by SNII is
intimately associated with all these predictions. Being able to
detect the DSNB and its energy spectrum will allow a more
sophisticated analysis of the detailed connections between all
these aspects of star formation and the cosmic SFH.
Methods for increasing the sensitivity of particle detectors
to DSNB antineutrinos and neutrinos have been detailed else-
where, and we briefly reiterate some of these conclusions, to il-
lustrate the potential for detecting the DSNB. Beacom & Vagins
(2004) propose loading SK with dissolved gadolinium trichlo-
ride to allow tagging of neutron captures, thus significantly
lowering backgrounds, in order to directly detect the DSNB
νe spectrum. This provides perhaps the best immediate pos-
sibility for measuring the DSNB spectrum shape, which, given
the minimum normalisation of the SFH assuming the BG IMF,
seems to be lying just below the current flux limit. Beacom
& Strigari (2006) describe how the Sudbury Neutrino Observa-
tory (SNO) could improve the current flux limit for DSNB νe
by about three orders of magnitude by coupling the background
analysis from SK with the sensitivity to νe at SNO. Combining
information on both DSNB νe and νe populations will allow
further exciting insight and constraints on SNII neutrino pro-
duction (the connection between the two is explored further by
Lunardini 2006).
6. SUMMARY
We have updated the SFH compilation of Hopkins (2004),
emphasising the strong constraints from recent UV and FIR
measurements, and refining the results of numerous measure-
ments over the past decade. An analysis of various uncertainties
that may contribute to the normalisation of the SFH has been
explored and the IMF assumptions play a key role, being es-
sentially the only assumption that can lower the normalisation.
We performed parametric fits to the SFH, using both the form of
Cole et al. (2001) and piecewise linear models, both constrained
by the SK νe limit. The results suggest that the preferred IMF
should produce SFH normalisations within the range of those
from the modified Salpeter A IMF (Baldry & Glazebrook 2003)
and the IMF of Baldry & Glazebrook (2003). They also sug-
gest that lower temperatures (T = 4 − 6 MeV) are preferred
for the νe population. It should be noted that here we have
assumed a simple Fermi-Dirac spectrum for the νe spectrum
after neutrino mixing. Since the current SK energy threshold
(19.3 MeV) is so high, however, the DSNB flux limit is sen-
sitive to the assumed shape of the spectrum, and a reduction
in the height of the spectral tail would allow a higher average
neutrino energy. This highlights the importance of lowering the
energy threshold (Beacom & Vagins 2004; Yu¨ksel et al. 2005).
Additionally, a more accurate treatment of future data would
be based not on the integrated flux above the energy threshold,
but rather on the detected event rate spectrum above the energy
threshold (Yu¨ksel et al. 2005), after weighting with the neutrino
interaction cross section (Vogel & Beacom 1999).
Based on the fits to the SFH we predict the evolution of
ρ∗, ρZ and ρ˙SN, and compare with observations. The compar-
isons with ρ˙SN are most revealing, providing the limitations on
the assumed IMF, and tentatively favouring longer delay times
(tIa = 3 Gyr) for the SNIa population. The ρ˙SNII measure-
ments provide a key constraint on the SFH normalisation, and
correspondingly on the favoured IMF. In particular, these data
bound the SFH from below, while the DSNB bounds the SFH
from above. (Due to the SK energy threshold, the DSNB flux
limit primarily constrains SNII with z . 1, the same range
in which we wish to test the factors that normalize the SFH.)
Together, these provide a novel technique for testing or verify-
ing measurements of a universal IMF, and emphasises the im-
portance of understanding the range of stellar masses leading
to the various stellar evolution core collapse outcomes. More
measurements of ρ˙SN for both type II and Ia SNe over a broader
redshift range (Oda & Totani 2005; Mesinger et al. 2006) would
help to more strongly constrain both the preferred universal
IMF and the properties of SNe. Observing the high redshift
turnover in the SNIa rate would also have strong implications
for the location of the expected high redshift turnover in the

10 A. M. Hopkins, J. F. Beacom
SFH. Direct observation of the DSNB will clearly allow much
greater insight into the physics and astrophysics of star forma-
tion and supernovae. Given the power of the existing SK νe flux
limit, any improvements in sensitivity will have a very strong
impact on constraining the product of the dust corrections, IMF
normalisation, and neutrino emission per supernova. In fact,
since reasonable choices for all of these already saturate the
SK limit, we expect that the significantly improved sensitivity
which would be enabled by adding gadolinium to SK (Beacom
& Vagins 2004) should lead to a first detection of DSNB νe
(Beacom & Vagins 2004; Strigari et al. 2004, 2005; Daigne et
al. 2005; Yu¨ksel et al. 2005).
The authors would like to thank the referee, Shaun Cole, for
rectifying a misconception in an early draft of this manuscript,
and for helpful suggestions leading to the current and much
improved version. We also thank Andy Bunker, Mark Fardal,
Louie Strigari and Mark Sullivan for helpful comments and in-
teresting discussions, and Don Neill for providing a copy of
Neill et al. (2006) prior to submission. AMH acknowledges
support provided by the Australian Research Council in the
form of a QEII Fellowship (DP0557850). JFB acknowledges
support from The Ohio State University and NSF CAREER
grant No. PHY-0547102.
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TABLE 1
SFH PARAMETRIC FITTING TO THE FORM OF COLE ET AL. (2001).
parameter modified Salpeter A IMFa Baldry & Glazebrook (2003)b
a 0.0170 0.0118
b 0.13 0.08
c 3.3 3.3
d 5.3 5.2
aFor this fit χ2 = 37.5. The scaling factors assuming T = (4, 6, 8)MeV
are (1.0, 1.0, 0.67).
bFor this fit χ2 = 37.5. The scaling factors assuming T = (4, 6, 8)MeV
are (1.0, 1.0, 0.74).

12 A. M. Hopkins, J. F. Beacom
TABLE 2
PIECEWISE LINEAR SFH PARAMETRIC FITTING.
parametera modified Salpeter A IMFb Baldry & Glazebrook (2003)c
a −1.82 −2.02
b 3.28 3.44
c −0.724 −0.930
d −0.26 −0.26
e 4.99 4.64
f −8.0 −7.8
z1 1.04 0.97
z2 4.48 4.48
aThe parameters here are the intercepts and slopes of the linear segments
in log(1+ z)− log(ρ˙∗) space, and the redshifts at which the slope changes.
The parameters a, b are the intercept and slope for the line segment between
0 ≤ z ≤ z1; c, d are intercept and slope between z1 ≤ z ≤ z2; e, f are
intercept and slope between z2 ≤ z ≤ 6. All eight parameters are shown
for convenience, but c and e are not independent and are not free parameters
in the fitting.
bFor this fit χ2 = 19.8. The scaling factors assuming T = (4, 6, 8)MeV
are (1.0, 1.0, 0.63).
cFor this fit χ2 = 19.6. The scaling factors assuming T = (4, 6, 8)MeV
are (1.0, 1.0, 0.67).
FIG. 1.— Evolution of SFR density with redshift. Data shown here have been scaled assuming the SalA IMF. The grey points are from the compilation of Hopkins
(2004). The hatched region is the FIR (24µm) SFH from Le Floc’h et al. (2005). The green triangles are FIR (24µm) data from Pe´rez-Gonza´lez et al. (2005). The
open red star at z = 0.05 is based on radio (1.4 GHz) data from Mauch (2005). The filled red circle at z = 0.01 is the Hα estimate from Hanish et al. (2006). Blue
squares are UV data from Baldry et al. (2005); Wolf et al. (2003); Arnouts et al. (2005); Bouwens et al. (2003b,a); Bunker et al. (2004); Bouwens et al. (2005a);
Ouchi et al. (2004). Blue crosses are the UDF estimates from Thompson et al. (2006). Note that these have been scaled to the SalA IMF assuming they were
originally estimated using a uniform Salpeter (1955) IMF. The solid lines are the best-fitting parametric forms (see text for details of which data are used in the
fitting). Although the FIR SFH of Le Floc’h et al. (2005) is not used directly in the fitting, it has been used to effectively obscuration-correct the UV data to the
values shown, which are used in the fitting. Note that the top logarithmic scale is labelled with redshift values, not (1 + z).

On the normalisation of the Cosmic SFH 13
FIG. 2.— SFR density data used in defining the best fitting parametric forms, and the resulting fits. (a) Assumes SalA IMF. (b) Assumes BG IMF. The shape
of the fits is determined from the SFH data alone, and a scaling factor is fit to ensure consistency with the SK νe limit (i.e., given the assumed temperature, this
quantifies how much lower the SFH normalisation has to be so as not to violate the SK limit). Solid lines assume a νe temperature of T = 4 MeV or T = 6 MeV,
and dotted T = 8 MeV. The grey shaded and hatched regions are the 1σ and 3σ confidence regions around the T = 4 MeV fits respectively. The scaling factors
are: (a) (1.0, 1.0, 0.67) and (b) (1.0, 1.0, 0.74) respectively for T = (4, 6, 8) MeV.
FIG. 3.— The νe energy spectra predicted for the various SFH fits and temperature assumptions. The solid and dashed curves are the SalA IMF and BG IMF
respectively. The T = 4 MeV and T = 6 MeV curves are consistent with the SK νe limit. The T = 8 MeV curves are inconsistent with the νe limit, and indicate
the shape of the νe spectrum derived by assuming the parametric form for the SFH corresponding to our best fit (T = 4 MeV), and setting the νe temperature to
the higher value. The thin vertical line marks E = 19.3 MeV, above which the νe contribute to the SK limit.

14 A. M. Hopkins, J. F. Beacom
FIG. 4.— As for Figure 2 but assuming a piecewise linear SFH model. (a) Assumes SalA IMF. (b) Assumes BG IMF. As before, solid lines assume a νe
temperature of T = 4 MeV or T = 6 MeV, and dotted T = 8 MeV.
FIG. 5.— Evolution of stellar mass density buildup inferred from the SFH. (a) Assumes SalA IMF (with R = 0.40). (b) Assumes BG IMF (with R = 0.56). The
grey shaded and hatched regions come from the 1σ and 3σ confidence regions around the SFH T = 4 MeV fits respectively. The details of scaling the data points
to our assumed IMFs are given in the text. The open circle is the local stellar density from Cole et al. (2001); the filled circle and filled squares represent the SDSS
and FIRES data, respectively, from Rudnick et al. (2003), scaled such that the SDSS measurement is consistent with that from Cole et al. (2001); the open stars are
from Brinchmann & Ellis (2000); and the open squares are from Dickinson et al. (2003).
FIG. 6.— Evolution of metal mass density buildup inferred from the SFH. (a) Assumes SalA IMF. (b) Assumes BG IMF. The grey shaded and hatched regions
come from the 1σ and 3σ confidence regions around the SFH T = 4 MeV fits respectively. The triangles at z = 0 are from Calura & Matteucci (2004), and the
open circles at z = 0 and z ≈ 2.5 are from Dunne et al. (2003).

On the normalisation of the Cosmic SFH 15
FIG. 7.— Evolution of SN rates inferred from the SFH. The upper curves correspond to the predictions for ρ˙SNII, and the lower for ρ˙SNIa, assuming a delay time
tIa = 3 Gyr. (a) Assumes SalA IMF. (b) Assumes BG IMF. The grey shaded and hatched regions again come from the 1σ and 3σ confidence regions around the
SFH T = 4 MeV fits respectively, except that for the SNIa region, the lower bound comes from assuming fIa = 1/1000M−1⊙ , while the upper bound assumes
fIa = 1/700M−1⊙ . The filled circles are ρ˙SNII measurements from Dahlen et al. (2004) and Cappellaro et al. (2005). The open circles are ρ˙SNIa measurements
reproduced from the compilation of Strolger & Reiss (2006) and the data of Barris & Tonry (2006) and Neill et al. (2006).
FIG. 8.— (a) As for Figure 7b, but assuming tIa = 1 Gyr. (b) The best fit to the SNIa rate from the SFH, with A = 1.15 × 10−3 and tIa = 2.7 Gyr (see
text). Crosses are the data compilation from Strolger & Reiss (2006), diamonds are data from Barris & Tonry (2006) and the square is from Neill et al. (2006). Note
different axes ranges from (a), and the linear redshift scale.