arXiv:1112.4940v1 [astro-ph.CO] 21 Dec 2011
The WiggleZ Dark Energy Survey:
Cosmological neutrino mass constraint from blue high-redshift galaxies
Signe Riemer–Sørensen1, Chris Blake2, David Parkinson1, Tamara M. Davis1, Sarah Brough3, Matthew
Colless3, Carlos Contreras2, Warrick Couch2, Scott Croom4, Darren Croton2, Michael J. Drinkwater1,
Karl Forster5, David Gilbank6, Mike Gladders7, Karl Glazebrook2, Ben Jelliffe4, Russell J. Jurek8,
I-hui Li2, Barry Madore9, D. Christopher Martin5, Kevin Pimbblet10, Gregory B. Poole2, Michael
Pracy4, Rob Sharp3,11, Emily Wisnioski2, David Woods12, Ted K. Wyder5and H.K.C. Yee13
1School of Mathematics and Physics, University of Queensland, QLD 4072, Australia∗
2Centre for Astrophysics & Supercomputing, Swinburne University
of Technology, P.O. Box 218, Hawthorn, VIC 3122, Australia
3Australian Astronomical Observatory, P.O. Box 296, Epping, NSW 1710, Australia
4Sydney Institute for Astronomy, School of Physics, University of Sydney, NSW 2006, Australia
5California Institute of Technology, MC 278-17,
1200 East California Boulevard, Pasadena, CA 91125, United States
6Astrophysics and Gravitation Group, Department of Physics and Astronomy,
University of Waterloo, Waterloo, ON N2L 3G1, Canada
7Department of Astronomy and Astrophysics, University of Chicago,
5640 South Ellis Avenue, Chicago, IL 60637, United States
8CSIRO Astronomy & Space Sciences, Australia Telescope National Facility, Epping, NSW 1710, Australia
9Observatories of the Carnegie Institute of Washington,
813 Santa Barbara St., Pasadena, CA 91101, United States
10School of Physics, Monash University, Clayton, VIC 3800, Australia
11Research School of Astronomy & Astrophysics,
Australian National University, Weston Creek, ACT 2611, Australia
12Department of Physics & Astronomy, University of British Columbia,
6224 Agricultural Road, Vancouver, BC V6T 1Z1, Canada and
13Department of Astronomy and Astrophysics, University of Toronto,
50 St. George Street, Toronto, ON M5S 3H4, Canada
(Dated: December 22, 2011)
The absolute neutrino mass scale is currently unknown, but can be constrained from cosmology.
The WiggleZ high redshift star-forming blue galaxy sample is less sensitive to systematics from non-
linear structure formation, redshift-space distortions and galaxy bias than previous surveys. We
obtain a limit of?mν < 0.60eV (95% confidence) for WiggleZ+Wilkinson Microwave Anisotropy
Probe. Combining with priors on the Hubble Parameter and the baryon acoustic oscillation scale
gives?mν < 0.29eV, which is the strongest neutrino mass constraint derived from spectroscopic
galaxy redshift surveys.
Neutrinos are the lightest massive known particles, yet
are treated as exactly massless by the Standard Model
of particle physics.However, neutrino oscillation ex-
periments using solar, atmospheric, and reactor neutri-
nos have measured mass differences between the three
species. The Heidelberg-Moscow experiment has limited
the mass of the electron neutrino to be less than 0.35eV
using β−spectroscopy , but no current experiment can
measure the absolute neutrino mass. Massive neutrinos
affect the way large-scale cosmological structures form,
and consequently we can infer a limit on the sum of neu-
trino masses from the matter distribution of the Universe
The cosmic microwave background (CMB) provides an
upper limit on the sum of neutrino masses of?mν <
1.3eV [3, all limits are 95% confidence]. Combining with
large-scale structure measurements such as the galaxy
power spectrum [4–6], cluster mass function , or the
scale of baryon acoustic oscillations [BAO, 3, 4] tightens
the constraints to?mν? 0.3eV by breaking degenera-
cies with other parameters. Neutrino mass constraints
are important goals of current and future galaxy surveys
e.g. Baryon Oscillation Spectroscopic Survey , Dark
Energy Survey  and Euclid . In this letter we use
the galaxy power spectrum from the WiggleZ Dark En-
ergy Survey to constrain the sum of neutrino masses.
The WiggleZ galaxy survey has advantages over previ-
ous surveys allowing the measured clustering pattern to
be fitted at smaller scales: 1) At the scales affected by
massive neutrinos, structure formation is non-linear to-
day and consequently difficult to model. At higher red-
shifts, the non-linear effects are smaller; WiggleZ is the
largest high-redshift galaxy survey. 2) The translation
(bias) between the observed galaxy distribution and the
dark matter distribution influenced by massive neutrinos
depends on the observed galaxy type. Previous studies
[e.g. 4, 11] measured red galaxies which tend to cluster
in the centers of dark matter halos, whereas the star-
forming blue WiggleZ galaxies avoid the densest regions
leading to a less scale-dependent bias relation, reducing
Galaxy redshifts are not entirely determined by the
Hubble flow, but also by their bulk flows and move-
ment (redshift-space distortions) providing a challenge
when comparing observations (redshift-space) with the-
ory (real-space). Through exhaustive tests using numer-
ical dark-matter simulations of the WiggleZ survey, we
demonstrate the breakdown of common models at small
scales, and calibrates a new non-linear fitting formula.
THE WIGGLEZ DARK ENERGY SURVEY
The WiggleZ Dark Energy Survey was designed to de-
tect the BAO scale at higher redshifts than was possi-
ble with previous datasets. The 238,000 galaxies are se-
lected from optical galaxy surveys and ultraviolet imag-
ing by the Galaxy Evolution Explorer to map seven re-
gions of the sky with a total volume of 1Gpc3in the
redshift range z < 1 . We split the data into four
redshift bins of width ∆z = 0.2 with effective redshifts of
zeff= [0.22,0.41,0.6,0.78]. The power spectra, Pobs, and
covariance matrices, C, are measured in ∆k = 0.01h
Mpc−1bins using the optimal-weighting scheme pro-
posed by Feldman et al.  for a fiducial cosmological
model . The Gigaparsec WiggleZ Survey (GiggleZ)
simulations  provide a means for testing and calibrat-
ing our modeling algorithms. The GiggleZ simulations
show that over the range of scales and halo masses rele-
vant for this Letter, the galaxy bias is scale-independent
to within 1% .
Large-scale structure alone cannot determine all cos-
mological parameters, so we include data from the CMB
as measured by the Wilkinson Microwave Anisotropy
Probe (WMAP). To compute the parameter likelihoods,
we use importance sampling [16, 17] of the WMAP7
Markov Chain Monte Carlo (MCMC) chains from fitting
to WMAP alone and the chains combining WMAP with
the BAO scale from SDSS LRGs  and a prior on the
Hubble parameter  (BAO+H0, available online ).
Matter power spectra: We test six different ap-
proaches for generating a model redshift-space galaxy
power spectrum. This section explains the aspects com-
mon to all models. First we calculate the matter power
spectrum, Pm, for each redshift bin for the set of cosmo-
logical parameters θ = [Ωc (cold dark matter density),
Ωb(baryon density), ΩΛ(dark energy density), Ων(neu-
trino density), h (Hubble parameter), ns(spectral index),
R(amplitude of primordial density fluctuations)]. We
assume a standard flat ΛCDM cosmology with no time
variation of w in agreement with observational data .
The effective number of neutrinos is fixed to Neff= 3.04
assuming no sterile neutrinos or other relativistic degrees
of freedom. The Pmis converted to a galaxy power spec-
trum, Pgal, using one of the six approaches described
Scaling and convolution:
compared to the observed power spectrum, Pobs, the
survey geometry and the fiducial cosmological model
used when measuring the power spectra must be
accounted for by Alcock-Paczynski scaling, a3
A,fidH(z)), and convolution with the sur-
vey window function, Wij [17, 20, 21]:
Before Pgal can be
where DAis the angular diameter distance and H(z) is
the Hubble parameter. Details of the window function
can be found in . For all models, we marginalise an-
alytically  or numerically over a linear galaxy bias
Likelihood: We assume the power spectra to be dis-
tributed as a multivariate Gaussian so the likelihood can
be determined as:
− 2ln(L(θ)) = χ2=
where θ is the set of cosmological parameters (includ-
ing galaxy bias), ∆j ≡ [Pobs(kj,θ) − Pcon(kj,θ)] with
Pcon(kj) being the convolved power spectrum in the j’th
bin, and Cjkthe covariance matrix.
The power spectrum measurements in the seven sur-
vey regions are treated as independent observations, and
their χ2’s combined by addition. We require the bias to
be the same for all regions at a given redshift, but allow
the bias to vary between redshifts since the galaxy lumi-
nosities evolve with redshift due to the survey magnitude
and colour cuts.
Importance sampling: We use importance sampling
to re-weight the WMAP MCMC likelihood chains .
For a chain of parameter values θ drawn from a likeli-
hood, L, it is possible to re-weight the likelihoods with
an independent sample from the same underlying dis-
tribution. The WMAP and WiggleZ power spectra are
independent measurements, so their likelihoods can be
combined by multiplication thus the weight, ωi
each element in the MCMC chain, i, can be re-weighted
Using the CosmoMC software  for a subsample of
the data, we have checked that the preferred regions of
parameter space for WMAP and WiggleZ overlap, and
consequently importance sampling is a valid method. A
CosmoMC module for the WiggleZ power spectra is un-
der development . We have also fitted the WiggleZ
power spectra alone over the range k = 0.02−0.2hMpc−1
varying only Ωmand fb, where fb= Ωb/Ωm, keeping all
other parameters fixed at the WMAP7 best fit values
(WiggleZ alone can not constrain all the cosmological
parameters). The results are consistent with the mea-
surements of these parameters using WMAP data alone.
Neutrino mass constraint: The neutrino mass limit
is calculated from the histogram of the WMAP MCMC
chain likelihoods re-weighted by WiggleZ. The X% con-
fidence upper limit on?mν is the value of Mlim
Using the WMAP chains alone gives a 95% confidence
limit on the neutrino mass of?mν < 1.3eV  and
?mν< 0.55eV when combining with BAO+H0.
= X/100. (3)
APPROACHES TO MODELING
At low redshift structure formation is no longer linear
for the small scales affected by massive neutrinos k ?
0.1hMpc−1[2, 26–28]. The standard way of determining
the matter power spectrum of non-linear structure for-
mation is the phenomenological Halofit calculation 
distributed with CAMB . Halofit has been derived
for massless neutrinos, but using hydro-dynamical SPH
simulations Bird et al.  demonstrated that including
realistic neutrinos in the simulations affected the power
spectrum less than 1% for k < 0.3hMpc−1. Simulations
show that redshift-space distortions become k-dependent
at low redshift and consequently are degenerate with neu-
trino mass .
With the aim of constraining neutrino mass, Swan-
son et al.  investigated 12 different models for non-
linear structure formation, galaxy bias, and redshift-
space distortions. They concluded that models with only
one free parameter are unable to provide a good fit for
kmax? 0.1−0.2hMpc−1for the SDSS red and blue galax-
ies. Blake et al.  fitted 18 different models to the
2D redshift-space WiggleZ power spectrum for a fidu-
cial cosmology, and concluded that the best fitting mod-
els are those of Jennings et al.  and Saito et al. 
but the latter is very computationally expensive. With
these conclusions in mind, we have selected six different
approaches for our analysis. The models are shown in
Fig. 1 for a fixed cosmology and described below. At low
values of k, the large-scale clustering can be treated as
linear, and the theory is quite robust, so we expect little
difference between the models. Throughout the analysis
we have fixed the lower limit kmin= 0.02hMpc−1which
corresponds to the largest modes observed in each of the
WiggleZ regions. All results are presented as a function
A) Linear: We use CAMB to calculate the linear
matter power spectrum, to which we add a linear bias
0.10.1 0.20.2 0.30.3 0.4 0.40.50.5
k [h Mpc−1]k [h Mpc−1]
FIG. 1. A weighted average of the WiggleZ power spectra in
the survey regions and redshifts, and the six models for the
best fit cosmology of F). The models are: A) blue, B) green,
C) magenta, D) cyan, E) red, F) thick orange.
model with redshift-space distortions in the Kaiser limit
. The model is valid within a few percent for k ≤
B) Non-linear structure formation:
Halofit to calculate the non-linear matter power spec-
trum, Phf. The bias and redshift-space distortions are
treated as for model A).
C) Non-linear with fitting formula for redshift-
space distortions and pairwise velocities: Combin-
ing the ansatz of Scoccimarro  with fitting formulae
derived from simulations, the model of Jennings et al. 
describes the time evolution of the power spectra of dark
matter density fluctuations and their velocity divergence
field. The details of our implementations of this model
are given in Parkinson et al. .
D) Non-linear with fitting formula for redshift-
damping: The fitting formulae used in model C) were
derived for dark matter particles and not for halos. Set-
ting all galaxy velocity dispersions to zero in model C)
provides a better fit to the GiggleZ halo catalogues, and
we have treated this special case as a separate model.
E) Non-linear with pairwise galaxy velocity
damping: Non-linear structure formation leads to in-
creased peculiar galaxy velocities at low redshift, which
damps the observed power spectrum. The effect can be
described by the empirical model :
Pgal(k) = b2
1 + (kfσvµ)2dµ
where f is the cosmic growth rate, µ =ˆk · ˆ z is the cosine
of the angle between the wave vector,ˆk, and the direction
of the line of sight, ˆ z, and the one-dimensional velocity
dispersion, σv(in units of h−1Mpc), is given by :
Setting σv= 0h−1Mpc we recover model B).
F) N-body simulation calibrated approach: All
the non-linear effects are present in an N-body simula-
tion for a fiducial cosmology and can be implemented
following the approach of Reid et al. . For each trial
gal(k) = b2Ptrial
damped(k) = Ptrial
and fdamp(k) = exp(−(kσv)2) with σv given by Eqn. 5.
the power spectrum of a set of halos in GiggleZ simula-
tion chosen to match the clustering amplitude of WiggleZ
galaxies. Pnwand Phf,nware the power spectra without
the acoustic peaks, for the linear and Halofit power spec-
tra respectively. They are calculated from a spline fit
to the CAMB power spectra following the approach of
Jennings et al.  and Swanson et al. . b2is related
to galaxy bias. The second factor in Eqn. 6 represents
the smooth power spectrum of the trial cosmology. The
third factor defines the acoustic peaks and their broaden-
ing caused by the bulk-flow motion of galaxies from their
initial positions in the density field, and the fourth factor
describes all additional non-linear effects in the N-body
Performance of the approaches: We tested the
different approaches by fitting to the z = 0.6 power
spectrum of a GiggleZ halo catalogue matching the clus-
tering amplitude of WiggleZ galaxies to two sets of 2D
parameter grids: Ωm− fb, and Ωm− ns, with the re-
maining parameters fixed at the GiggleZ fiducial cos-
mology values.We chose these grids because the pa-
rameters are susceptible to degeneracies with neutrino
mass. In both cases we obtain very similar conclusions,
so here we only present the results of Ωm− fb.
kmax< 0.2hMpc−1most of the models produce a good
fit, whereas for kmax= 0.3hMpc−1model B), C) and E)
give reduced χ2values above 1.5. The upper panel of
Fig. 2 shows the χ2for the fiducial GiggleZ cosmological
parameters, which is a measure of how well the models
recover the input parameters. The lower panel shows
the difference between χ2of the GiggleZ values and the
best fit, indicating how far the best fit is from the input
values. We assume that the N-body simulation, which
provides a complete census of the relevant non-linear ef-
fects, yields the most accurate clustering model. In this
sense the good performance of model F) (Fig. 2) is a con-
sistency check, and the variations of results produced by
the other models are due to the breakdown in their per-
formances compared to the simulation. We chose to fit
with kmax= 0.3hMpc−1, above which the effects of mas-
GiggleZ(k) is found from a 5thorder polynomial fit to
kmax [h Mpc−1]
FIG. 2. Upper: Reduced χ2of models fitted to the N-body
simulation halo catalogue for the GiggleZ fiducial cosmology
values. In absence of systematic errors the models should
recover the input cosmology with χ2/dof = 1. Lower: Dif-
ference in reduced χ2values when using the GiggleZ fiducial
cosmological parameters and the best fit values. The models
are: A) blue, B) green, C) magenta, D) cyan, E) red, F) thick
sive neutrinos are not properly implemented in CAMB
RESULTS AND DISCUSSION
In Fig. 3 the upper panel shows the χ2as a function
of kmax for the best fitting parameter values for each
of the six approaches, and the lower panel shows the
corresponding neutrino mass constraints. Although all
models produce similar χ2values, our comparison with
the full N-body simulation catalogue (Fig. 2) revealed
that systematic errors arise when models A) to E) are
fit across the range of scales kmax< 0.3hMpc−1. Using
the fully-calibrated model F), we obtain?mν< 0.60eV
for WMAP+WiggleZ with kmax = 0.3hMpc−1. Com-
bining with BAO+H0 reduces the uncertainty in Ωm
and H0, leading to stronger neutrino mass constraints.
Without WiggleZ, the WMAP+H0+BAO dataset gives
?mν < 0.55eV whereas combining with WiggleZ adds
information about the power spectrum tilt (ns).
resulting neutrino mass constraint is?mν < 0.29eV
for model F) and kmax = 0.3hMpc−1.
probability distributions of
kmax= 0.3hMpc−1are shown in Fig. 4. It is clear how
adding WiggleZ data to the fit narrows the distributions
(dotted to solid) both with and without the inclusion
of BAO+H0. This is the strongest neutrino mass limit
so far derived from spectroscopic redshift galaxy surveys.
The advantages of WiggleZ are a higher redshift for which
?mν for model F) with
kmax [h Mpc−1]
FIG. 3. Upper: Reduced χ2as a function of kmax for each
of the six approaches. Lower: Upper limits on?mν as a
function of kmax. The models are: A) blue, B) green, C)
magenta, D) cyan, E) red, F) thick orange. The dashed grey
line is the lower limit from oscillation experiments, and the
black lines are upper limits from WMAP+BAO+H0 (dotted)
and WiggleZ+WMAP+BAO+H0 (solid).
0.0 0.2 0.4 0.60.81.0 1.2 1.4
from fitting model F) with kmax
WiggleZ+WMAP+BAO+H0 (solid black).
grey line is the lower limit from oscillation experiments, and
the vertical lines are 95% confidence upper limits.
The relative probability distribution of
the structure formation is linear to smaller scales, and
a simple galaxy bias for the strongly star-forming blue
emission line galaxies.
Our result is comparable to that obtained using pho-
tometric redshift galaxy surveys [?mν < 0.28eV,
but the systematics in the two data set are completely
different. For example, imaging surveys are potentially
susceptible to systematic errors from the imprint of stars
on the selection function  and the shape of the redshift
distribution. The high redshift and blue galaxies of Wig-
gleZ allow us to fit the power spectrum to smaller scales
than previous surveys (both spectroscopic and photomet-
ric), where the effect of the neutrinos is larger, and get a
similar neutrino mass constraint from a smaller, but well
understood galaxy sample. Also the result from galaxy
clusters [?mν< 0.33eV, 7] is similar, but with differ-
ent systematics. Since the data sets are all independent,
they can potentially be combined in the future to provide
even stronger constraints.
Acknowledgements: SRS acknowledges financial
support from The Danish Council for Independent Re-
support from the Australian Research Council through
Discovery Project grants DP0772084 and DP1093738.
SC and DC acknowledge the support of the Australian
Research Council through QEII Fellowships.
search was supported by CAASTRO: http://caastro.org.
GALEX (the Galaxy Evolution Explorer) is a NASA
Small Explorer, launched in April 2003. We gratefully
acknowledge NASA support for construction, operation
and science analysis for the GALEX mission, developed
in co-operation with the Centre National d’Etudes Spa-
tiales de France and the Korean Ministry of Science and
Technology. We thank the Anglo-Australian Telescope
Allocation Committee for supporting the WiggleZ sur-
vey over 9 semesters, and we are very grateful for the
dedicated work of the staff of the Australian Astronom-
ical Observatory in the development and support of the
AAOmega spectrograph, and the running of the AAT.
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