Massive Neutrinos and the Non-linear Matter Power Spectrum

1.2 MB·English

by  Simeon Bird

Mon.Not.R.Astron.Soc.000,1–12(2011) Printed16January2012 (MNLATEXstylefilev2.2)
Massive Neutrinos and the Non-linear Matter Power
Spectrum
Simeon Bird 1(cid:63), Matteo Viel 2,3† and Martin G. Haehnelt 1‡
2
1Institute of Astronomy and Kavli Institute for Cosmology, Madingley Road, Cambridge CB3 0HA, U.K.
1
2INAF - Osservatorio Astronomico di Trieste, Via G.B. Tiepolo 11, I-34131 Trieste, Italy
0
3INFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, Italy
2

n
a
J

3
ABSTRACT
1
WeperformanextensivesuiteofN-bodysimulationsofthematterpowerspectrum,in-

corporatingmassiveneutrinosintherangeM =0.15−0.6eV,probingthenon-linear
] ν
O regime at scales k < 10hMpc−1 at z < 3. We extend the widely used HALOFIT ap-
C proximation to account for the effect of massive neutrinos on the power spectrum.
In the strongly non-linear regime HALOFIT systematically over-predicts the suppres-
.
h sion due to the free-streaming of the neutrinos. The maximal discrepancy occurs at
p k ∼ 1hMpc−1, and is at the level of 10% of the total suppression. Most published
-
o constraints on neutrino masses based on HALOFITare not affected, as they rely on
r data probing the matter power spectrum in the linear or mildly non-linear regime.
t However, predictions for future galaxy, Lyman-α forest and weak lensing surveys ex-
s
a tendingtomorenon-linearscaleswillbenefitfromtheimprovedapproximationtothe
[ non-linear matter power spectrum we provide. Our approximation reproduces the in-

duced neutrino suppression over the targeted scales and redshifts significantly better.
2
We test its robustness with regard to changing cosmological parameters and a variety
v
6 of modelling effects.
1
Key words: neutrinos - cosmology: large-scale structure of Universe - cosmology:
4
dark matter
4
.
9
0
1
1 1 INTRODUCTION fixing the Hubble constant implies Mν (cid:46) 15 eV (Gershtein
: & Zel’Dovich 1966; Lesgourgues & Pastor 2006). The cos-
v Avarietyofsolar,atmospheric,reactorandacceleratorneu-
mic microwave background (CMB; e.g. Komatsu & et al.
i trino experiments have firmly established the existence of
X (2011)) combined with measurements of the matter power
neutrino flavour oscillations, implying that neutrinos have
spectrum from galaxy surveys provide more stringent con-
r atleasttwonon-zeromasseigenstates(Becker-Szendy&et
a straints,presentlyaroundM (cid:46)0.6eV(Elgaroyetal.2002;
al.1992;Fukuda&etal.1998;Ahmed&etal.2004).These ν
Seljaketal.2006;Tegmarketal.2006;Grattonetal.2008;
experiments yield a lower limit on the sum of the neutrino
Mantzetal.2010;Thomasetal.2010;Reidetal.2010).Sel-
masses; M > 0.05eV. Classical β-decay experiments pro-
ν jak et al. (2006) combined the Lyman-α forest flux power
vide an upper limit on the mass of the electron neutrino of
spectrum with the CMB to claim the tightest constraint
2.2eV(Lobashev2003),butmayinfuturereachconstraints
so far; M < 0.17 eV. This strong limit may, however, be
as low as 0.2 eV (Wolf & KATRIN collaboration 2010). ν
partly caused by a slight tension between the power spec-
Sensitivity to well below 1 eV is necessary to distin-
trum amplitude preferred by the Lyman-α forest data and
guish the neutrino mass hierarchy (Jimenez et al. 2010),
thatpreferredbytheCMB.Inanycase,itshouldbetested
which is of particular interest when formulating theories of
andimprovedbyupcomingdata(Slosar&etal.2011).The
particle physics beyond the standard model. Fortunately,
Lyman-α data alone provides a limit of M < 1 eV (Viel
such strong constraints can be obtained from the cosmic ν
2005; Viel et al. 2010), slightly stronger than that provided
neutrino background. Simply requiring Ω (cid:54)Ω ≈0.3 and
ν M by the CMB data alone (Larson & et al. 2011).
The sensitivity of the matter power spectrum to neu-
trino mass is due to the free-streaming of the neutrinos
(cid:63) E-mail:[email protected]
† E-mail:[email protected] which suppresses the growth of structure. On large scales,
‡ E-mail:[email protected] where matter perturbations can be described by linear the-
(cid:13)c 2011RAS
2 S. Bird et al.
ory,thiseffectisrelativelystraightforwardtoquantifywith
Matter Power Spectra at z=0,1
numerical simulations or approximate schemes (e.g. Bond 102
et al. (1980); Zeldovich et al. (1980); Ma & Bertschinger
(1994); Valdarnini et al. (1998)) using standard Boltzmann
codes (Lewis et al. 2000; Lesgourgues & Tram 2011). Prob- 1)
ingtheeffectofneutrinosonsmaller,morenon-linearscales h-
c
ismorechallenging,buthasbeenattemptedby,forexample, Mp101
Setaiatol.e(t2a0l0.9()2,0w08it,h20e0x9t,e2n0s1io1n),sWtoonpger(t2u0r0b8a)taionndtLheesogroyu.rAgulseos k)/k /(
of interest here is the approach of Abazajian et al. (2005); (
Hannestad et al. (2005), who described the fully non-linear 2∆
regime by perturbatively evaluating the effect of massive 100
neutrinos on halo profiles.
However, accurately probing the effect of massive neu- 10-1 100 101
trinosonthegrowthofstructurearbitrarilyfarintothenon- k /(h Mpc-1)
linear regime requires fitting formulae precisely calibrated
from high dynamic range N-body simulations. The formula Figure 1. Matter power spectra from a simulation with mas-
wepresentherebuildsonthewidely-usedHALOFIT (Smith sive neutrinos (dashed lines), and Mν = 0.6 eV, compared to
et al. 2003). HALOFIT models the non-linear matter power thecorrespondingpowerspectrafromaΛCDMsimulation(solid
spectrum by assuming that the growth of structure is due lines). Dotted lines show the linear theory prediction for mas-
to a combination of the density structure of individual DM siveneutrinos,whiledot-dashedlinesshowthelineartheorypre-
diction for ΛCDM. Upper (green) lines show spectra at z = 0,
haloes and the clustering of these DM haloes relative to
while lower (black) lines show them at z = 1. The simulations
each other. It has been calibrated against a wide range of
shown are S60 (small scales) and L60 (large scales), whose pa-
CDM simulations, superseding earlier prescriptions based
rameters are described in Table 1. S60 shows only modes where
on the stable clustering hypothesis (Hamilton et al. 1991;
k > 4×2π/ Box-size, so sample variance is small, and L60 is
Peacock & Dodds 1996). At low and moderate redshifts cut off at k = 0.4hMpc−1, for clarity. Note that the absolute
HALOFIT models the non-linear growth of structure in a powerspectrumisslightlylessconvergedwithrespecttoboxsize
ΛCDM cosmology with an accuracy of (cid:46) 5−10% (Heit- thantheratioofthemassiveneutrinopowerspectrumwithand
mann et al. 2010) for k < 1hMpc−1, but it has not been withoutmassiveneutrinos.
calibrated for non-zero neutrino masses.
Fortunately, the quality of the numerical simulations
simulatingtheeffectofmassiveneutrinos,describingmeth-
has advanced to the point where it is possible to model the
odsbasedonbothparticleandfourier-spacerepresentations
impact of neutrinos to the level of a few percent, despite
the difficulties presented by the large thermal dispersion of of neutrinos. In Section 2.4, we summarise HALOFIT. Fi-
nally,Section2.3givesdetailsofournumericalsimulations.
light neutrinos (Brandbyge et al. 2008; Viel et al. 2010). A
quantitativenumericalstudyofthesuppressionofthemat-
ter power spectrum due to free-streaming of neutrinos ap-
pears timely, as future matter power spectrum surveys aim 2.1 Massive neutrinos in linear theory
to make measurements accurate to a few percent. Realising
Neutrinos in the very early Universe are relativistic and
the potential of these measurements to constrain the neu-
tightly coupled to the primordial plasma. They decouple
trino mass requires tight control of systematic error from
while still relativistic and then redshift adiabatically. The
theoretical predictions.
transition to non-relativistic behaviour takes place when
We present results from simulations with the re- M =3T ,ataredshiftof(1+z)=2×103(M /1eV).How-
ν ν ν
cently developed version of the Tree-SPH code GADGET-3 ever, it is some time later before relativistic corrections are
(Springel 2005; Viel et al. 2010). This suite of simulations
negligible, especially in the high-velocity tail of the Fermi-
wasdesignedtoevaluatetheperformanceofHALOFIT pre- Dirac distribution. Once non-relativistic, massive, weakly-
dictions when compared to full numerical simulations. We
interacting neutrinos behave qualitatively as a species of
useoursimulationstocalibrateanimprovedfittingformula,
warm/hot dark matter, suppressing fluctuations on scales
capableofaccuratelyreproducingtheeffectofmassiveneu-
smaller than their thermal free-streaming length.
trinosonthenon-linearmatterpowerspectrum.Wediscuss
This suppression has a distinctive shape in the linear
our methods in Section 2, present our results in Section 3,
regime. Neutrinos of mass M have energy density Ω , and
ν ν
and conclude in Section 4.
make up a fraction, f , of the dark matter, where
ν
M Ω
Ω = ν , f = ν . (1)
ν 93.14h2eV ν Ω
M
2 MODELLING THE NON-LINEAR MATTER
Ω is the present day matter density and h is the Hubble
POWER SPECTRUM M
constant. The residual comoving peculiar thermal velocity
Here we briefly describe the numerical methods we use to of the neutrinos at redshift z is
simulatestructuregrowthinthepresenceofmassiveneutri- (cid:20) (cid:21)
1eV
nos,aswellastheapproximationsunderlyingHALOFIT.In vth ≈150(1+z) M km/s. (2)
Section2.1wereviewthepredictedeffectofneutrinosinlin- ν
ear theory. In Section 2.2 we discuss the strategies used for ByanalogywiththeJeanslength,wecandefineaco-moving
(cid:13)c 2011RAS,MNRAS000,1–12
The Matter Power with Neutrinos 3
free-streaming scale 2.2.1 Particle-based Neutrinos
(cid:114)
2H(t)a(t) Massive neutrinos can be incorporated into cold dark mat-
k = . (3)
FS 3 v (t) ter simulations as a separate low-mass collisionless particle
th
Neutrinoscannotclusteronscalessmallerthank ,andso species. Structure growth is primarily driven by the cold
FS
the matter power spectrum, defined as dark matter. The neutrino species has significant thermal
dispersion, preventing it from clustering on small scales.
P(k)=|δ˜(k)|2, (4)
This allows us to avoid following the small-scale evolu-
whereatildedenotesaFouriertransformedquantity,issup- tion of the neutrino component, first by ignoring the short-
pressed by a constant factor proportional to the neutrino rangetreeforcefortheneutrinosandsecondbysettingthe
energy density, f . The Boltzmann equation can be solved timesteppurelyfromthedarkmatterspecies,effectivelyre-
ν
numericallytoshowthatthefractionalchangeinthepower laxing the Courant conditions for the neutrino component.
spectrum for f <0.07 is Brandbyge et al. (2008) and Viel et al. (2010) found that
ν
δP these approximations significantly speed up computation
P ≈−8fν. (5) while having a negligible impact on results.
One potential problem with particle based simulations
Onlargerscales,k<k ,neutrinoshavefree-streamedonly
FS
is thermal shot noise from the finite sampling of the neu-
foraportionofthecosmichistory,inducinglesssuppression.
trino distribution (Wang & White 2007), especially for the
Thelargestscaleonwhichneutrinosalterthematterpower
higherthermalvelocitiesencounteredathighredshiftorfor
spectrum is given by the wavenumber for which they free-
particularly low mass neutrinos. We deal with this by ex-
streamed only immediately after becoming non-relativistic
plicitly checking the effect of shot noise, and increasing the
(Lesgourgues & Pastor 2006)
neutrino particle number should it be non-negligible.
(cid:115)
(cid:20) (cid:21)
M
k ∼0.018 Ω ν hMpc−1. (6)
nr M 1eV
2.2.2 Fourier-space Neutrinos
Today, neutrinos in the observationally acceptable range
have k ∼ 10−3, placing the free-streaming effect at the The effect of neutrinos on the dark matter can also be ap-
nr
scales probed by galaxy and Lyman-αmeasurements of the proximated by adding their gravitational force to that of
matter power spectrum, as illustrated by Figure 1, which the dark matter in Fourier space (Brandbyge & Hannestad
shows non-linear matter power spectra estimated from our 2009),aspartoftheparticlemeshcomputation.Themassive
simulations. For further information on the effect of neutri- neutrinocomponentisassumedtohavethepowerspectrum
nos in linear theory see, e.g. Kolb & Turner (1990); Bond predicted by linear theory, in our case computed using the
etal.(1980);Eisenstein&Hu(1999)andreferencestherein. BoltzmanncodeCAMB1 (Lewisetal.2000).Sincetheneu-
trinosarenotfolloweddirectly,wecanavoidshotnoiseand
the need to deal with large thermal velocities when com-
2.2 Incorporating massive neutrinos into N-body
puting their effect on the overall growth of structure. How-
simulations
ever, because the neutrinos are evolved using linear theory,
The first N-body simulations incorporating massive neu- Fourier-spacesimulationsneglectanynon-linearevolutionin
trinos were aimed at evaluating their potential as a dark theneutrinocomponentitself,includingback-reactionfrom
matter candidate (White et al. 1983; Klypin et al. 1993; non-linearitiesinthedarkmatter.AswewilldiscussinSec-
Ma&Bertschinger1994).Recentlyattentionhasshiftedto tion 3 these effects are important, and so we shall focus
their perturbative effect as a sub-dominant component in a mainly on simulations involving particles.
mainlyΛCDMuniverse.WebuildontheworkofVieletal.
(2010),whoincorporatedmassiveneutrinosintoGADGET-
2.3 The numerical simulations
3(Springel2005)intwoways;asanewparticlespecies,and
usingaFourier-spacerepresentation(Brandbyge&Hannes- We primarily used particle-based simulations, summarised
tad2009).Vieletal.(2010)havedescribedtheimplementa- inTable1.WealsoranseveralsimulationsusingtheFourier-
tionindetailandperformedextensivevalidationtests.Here space implementation of neutrinos using a grid with 5123
webrieflyreviewthetwoimplementationsofneutrinos.We elements, but as these were not used in the final results we
refer the reader to Brandbyge et al. (2008); Brandbyge & omit them from the table for clarity.
Hannestad (2009) and Viel et al. (2010) for further details. Our fiducial simulations have 5123 dark matter parti-
Both the particle and the Fourier space implementations cles in a box of 150Mpch−1. We tested M = 0.15 eV,
ν
incorporate massive neutrinos directly into the simulations 0.3 eV and 0.6 eV, running 6 simulations for each value of
and so are significantly more accurate than simulations al- M with different realisations of the initial conditions, to
ν
tering only the initial conditions of a ΛCDM simulation, as suppress cosmic variance. To check we had correctly sam-
e.g. in Agarwal & Feldman (2011). pledthelargescalemodes,toexplorethelinearregimeand
All our simulations are inherently Newtonian, ignoring tocomparedirectlytoVieletal.(2010),weranalsosimula-
relativistic corrections. It is therefore important to set up tionswithboxsizesof512Mpch−1.Tofacilitatecomparison
theinitialconditionsofthesimulationatsufficientlylowred- withBrandbygeetal.(2008)andVieletal.(2010),ourfidu-
shiftthatthevastmajorityofneutrinosarenon-relativistic. cialcosmologyhadΩ =0.3,aspectralindexn =1.0,the
M s
Otherwise,neutrinoswillbetreatedasmassivewhilestillef-
fectivelymassless,over-estimatingtheireffectbyaconstant
factor in the linear regime, at the highest redshifts. 1 http://www.camb.info
(cid:13)c 2011RAS,MNRAS000,1–12
4 S. Bird et al.
Name Mν (eV) Box(Mpch−1) ND1/M3 Nb1a/r3 Nν1/3 No.Seeds zi ΩM As/10−9 ns h σ8 (derived)
S15 0.15 150 512 0 512 6 24 0.3 2.27 1.0 0.7 0.84
S30 0.3 150 512 0 512 6 49 0.3 2.27 1.0 0.7 0.80
S60 0.6 150 512 0 512 6 99 0.3 2.27 1.0 0.7 0.73
L15 0.15 512 512 0 512 1 24 0.3 2.27 1.0 0.7 0.84
L30 0.3 512 512 0 512 1 49 0.3 2.27 1.0 0.7 0.80
L60 0.6 512 512 0 512 1 99 0.3 2.27 1.0 0.7 0.73
S15NU 0.15 150 512 0 1024 1 24 0.3 2.27 1.0 0.7 0.84
S60NU 0.6 150 512 0 1024 1 99 0.3 2.27 1.0 0.7 0.73
S60AS 0.6 150 512 0 512 1 99 0.3 2.0 1.0 0.7 0.73
S60NS 0.6 150 512 0 512 1 99 0.3 2.27 0.9 0.7 0.73
S60ND 0.6 150 640 0 640 1 99 0.3 2.27 1.0 0.7 0.73
S60H 0.6 150 512 0 512 1 99 0.3 2.27 1.0 0.75 0.79
S30OM 0.3 150 512 0 512 2 49 0.25 2.27 1.0 0.7 0.68
S30OL 0.3 150 512 0 512 1 49 0.4 2.27 1.0 0.7 1.07
S60BA 0.6 150 512 512 512 1 99 0.3 2.27 1.0 0.7 0.73
Table1.Summaryofsimulationparameters.σ8hasbeenderivedfromtheotherparameters.Foreachsimulationwithmassiveneutrinos
weranaΛCDMsimulationwithidenticalparameters,butwithoutmassiveneutrinos,inordertocomputetheirratio.Thevalueofσ8
givenisthatforthesimulationwithmassiveneutrinos.
Hubble parameter h = 0.7, and amplitude of the primor- ters, suggesting that it is also independent of the neutrino
dial power spectrum A =2.27×10−9. This implies that a mass, and thus should not affect our results.
s
simulation without neutrinos has σ =0.87 at z =0, but a
8
simulation with M = 0.3 eV has σ = 0.8. We tested our
ν 8
2.4 The HALOFIT formula
formulausingavarietyofsimulationswithdifferentcosmo-
logical parameters, as summarised in Table 1. HALOFIT (Smith et al. 2003) estimates non-linear growth
from a given linear theory power spectrum, based on the
Theinitialredshiftsofoursimulationswerez =99for
i principles of the halo model. These assert that the growth
M = 0.6 eV, z = 49 for M = 0.3 eV and z = 24 for
ν i ν i ofhalosshoulddependonlyonthelocalphysicsatthescale
M = 0.15 eV, which we found to be the earliest redshift
ν of the halo, and not the details of the pre-collapse mate-
where the neutrinos were fully non-relativistic. Our initial
rial,northelarger-scaledistributionofmatter.Thusgrowth
conditions were generated using the Zel’dovich approxima-
deep in the non-linear regime becomes independent of the
tion, and the linear theory power spectrum was calculated
detailsofthelinearpowerspectrum,dependingonlyonthe
by CAMB. We acknowledge that second order Lagrangian
non-linear scale and the slope and curvature of the power
perturbation theory is in principle more accurate, but our
spectrum. In the mildly non-linear regime, the non-linear
simulations are started at sufficiently high redshift that the
correction is assumed to be a function of the instantaneous
effect should be small. The transfer function used for our
linear power spectrum, without explicit dependence on the
cold dark matter (CDM) particles was a weighted average
historical growth rate. HALOFIT is accurate to 5−10% at
ofthelineartheorytransferfunctionsforCDMandbaryons,
z<3(Heitmannetal.2010),suggestingthattheseapprox-
to account for the slight difference between them (Somogyi
imations are reasonably accurate for cold dark matter.
& Smith 2010). Neutrinos and CDM had identical initial
HALOFIT realisestheabovebysplittingthenon-linear
randomphaseinformation,tosimulateadiabaticinitialcon-
power spectrum into two components. A quasilinear term
ditions.
∆2, and a non-linear term ∆2. This means the dimension-
Q H
The numerical convergence of our results and the im- less power spectrum can be written as,
pact of neutrino shot noise were checked with simulations
k3P(k)
S60ND, S15NU and S60NU. We further checked the effect ∆2 ≡ =∆2 +∆2 . (7)
2π2 Q H
of baryonic physics with simulation S60BA, which included
5123baryonparticles.S60BAmodelledradiativecoolingand Both terms depend on the non-linear scale, kσ, defined as
thewave-numberwheretheperturbationamplitude,filtered
star formation using the default multiphase model of GAD-
byaGaussian,becomesgreaterthanunityandlineartheory
GET-3(Springel&Hernquist2003).Differencesmuchlarger
than10%havebeenfoundfork>20hMpc−1betweensimu- breaks down.
The quasilinear term, ∆2, is given as an enhancement
lationsimplementingdifferentradiativeprocesses(e.g.metal Q
tothelinearpowerspectrum,andincorporatesanexponen-
cooling)orfeedbackrecipesinΛCDMmodels(seee.g.Rudd
tialcut-offensuringitisnegligibleonsmallscales.Theform
etal.(2008);Guilletetal.(2010);vanDaalenetal.(2011)).
of the quasilinear term is,
Furthermore,themodelofAGNfeedbacksuggestedbyvan
Daalen et al. (2011) in order to reproduce observations of (cid:20)(1+∆2(k))β(n)(cid:21)
∆2(k)=∆2(k) L exp[−f(y)]. (8)
groupsofgalaxiesproducesapowerspectrumsuppressionof Q L 1+α(n)∆2(k)
roughly 10% at k = 1hMpc−1 and 1% at k = 0.3hMpc−1. dlnσ2(cid:12)(cid:12) L
We did not include this effect; however, van Daalen et al. Here 3+n= (cid:12) , (9)
dlnR(cid:12)
(2011) found it to be independent of cosmological parame- k=kσ
(cid:13)c 2011RAS,MNRAS000,1–12
The Matter Power with Neutrinos 5
so that n is the effective spectral index of the power spec- As mentioned in Section 2.3, we performed checks for
trum.β andαarefunctionsofnandf(y)=y/4+y2/8.We numerical convergence. A simulation with increased reso-
referreadersinterestedintheexactvaluesofthesefunctions lution, S60ND (see Table 1), had a relative matter power
toAppendixCofSmithetal.(2003).∆2 isthedimension- spectrum which differed from the fiducal simulation, S60,
L
less linear theory power spectrum. by1−2%fork<7hMpc−1.Furthermore,therelativemat-
Non-lineargrowth,∆2,dominateswheny≡k/k (cid:29)1. ter power spectra for our two different box sizes, 512 and
H σ
The form of ∆H used by HALOFIT is, 150Mpch−1, were always in good agreement on the largest
scales probed by the smaller box, showing that our smaller
∆2(cid:48)(k)
∆2(k)= H , (10) box size was sufficient for an accurate estimate of the sup-
H 1+µ(n)/y+ν(n)/y2
pressionofmatterpowerspectrumduetothefree-streaming
a(n,C)y3f1(Ω) oftheneutrinos.Thelargerboxshowedaslightlyincreased
∆2(cid:48)(k)= .
H 1+b(n,C)yf2(Ω)+[c(n,C)f3(Ω)y]3−γ(n,C) suppression on small scales, probably due to limited reso-
(11) lution. We found that the effect of baryons was, at all red-
d2lnσ2(R)(cid:12)(cid:12) shifts, less than 1% at k < 8hMpc−1, gradually increasing
Where C = (cid:12) , (12) on smaller scales, in agreement with the results of e.g. Jing
dlnR2 (cid:12)
k=kσ et al. (2006); van Daalen et al. (2011).
thespectralcurvatureofthepowerspectrum.a,b,c,µandν SimulationsS15NUandS60NUwererunwithahigher
arefunctionsofnandC,andν(n)isnotrelatedtoneutrinos. number of neutrino particles, to check explicitly the effect
f arefunctionsofΩ ,thetotalmatterdensity.Forthe of shot noise. We found that, although the neutrino power
1,2,3 M
values of these functions, we again refer to Appendix C of spectrum is indeed shot noise dominated on small scales,
Smithetal.(2003).Notethatthenon-lineargrowthdepends the power on these scales is so small that their impact on
only on y, n, C and the matter fraction. thegrowthofthedarkmatterpowerspectrumisnegligible.
Massive neutrinos affect this prescription because they Hence shot noise has only a minor impact for simulations
suppress the linear power spectrum, increasing the non- with a box size of 150Mpch−1, at the level of 1% at the
linear scale and retarding growth. HALOFIT estimates the relevant scales and redshifts.
effect of this suppression on non-linear power, but neglects
any effect of the neutrinos on the process of halo collapse,
as well as the effects of non-linear growth on the neutrinos 3.1 Particle and Fourier-space implementation of
themselves. neutrinos
Figure2showsthatparticleandFourier-spaceimplementa-
tions of neutrinos give similar results for the matter power
3 RESULTS spectrum,andbothagreewithlineartheoryonlargescales.
Theparticleimplementation,however,showsasmallersup-
Figure 2 shows ratios between the matter power spectrum
pression,whichscalesroughlylinearlywithM ,particularly
ν
for massive neutrinos, and ΛCDM. Plotted are the predic-
near the trough of the dip.
tions of linear theory, of simulations S60 and L60, and of
This discrepancy has again a simple physical explana-
grid-basedsimulationswithidenticalparameters.Themax-
tion; structure growth in the dark matter induces structure
imalpowerspectrumsuppressioninlineartheoryispropor-
growthintheneutrinocomponent,asthelessenergeticneu-
tional to −8f , compared to a maximal non-linear suppres-
ν trinos fall into the gravitational wells created by the dark
sion of approximately −10f for z (cid:54) 1, and −10.5f for
ν ν matter. This increases the power in the neutrinos, drawing
z=2.AsdiscussedinSection2.1,lineartheorypredictsan
them closer to the non-linear dark matter power spectrum
increasingsuppressionapproachinganalmostflatplateauon
and leading to a reduced neutrino suppression. Essentially
small scales. In the numerical simulations, the flat plateau
this is a backreaction effect, where the dark matter drags
is replaced by a minimum. The suppression increases up
the neutrinos with it; since it results from the non-linear
to k ∼ 1hMpc−1, after which it gradually decreases. This
growth, it is not being fully accounted for by linear theory
distinctive spoon shaped pre-virialization feature was also
neutrinos. Note that structure growth in the neutrino com-
found by Brandbyge et al. (2008) and Viel et al. (2010),
ponent itself remains linear; the effect is due to non-linear
andinCDMsimulationsbyLokasetal.(1996);Smithetal.
growth in the dark matter. Because this effect is important
(2007).
at the accuracy we wish to achieve, our main results (Fig-
Wecanunderstandthisshapebyconsideringtheeffect
ures 3, 4 and 5) are based on simulations with the particle
of neutrinos on non-linear growth. The suppression caused
implementation of neutrinos.
by massive neutrinos delays the onset of non-linear growth
and increases the wave-number of the non-linear scale. Be-
tweenthenon-linearscaleforthesimulationwithoutneutri-
3.2 Comparison to the HALOFIT model
nosandthenon-linearscaleforthesimulationincorporating
neutrinos, the suppression will be enhanced by the absence Figures 3, 4 and 5 show the main results of this paper.
of non-linear growth. On the smallest scales, the non-linear EachFigureshowsthesuppressionofthematterpowerspec-
growth lost as a consequence of the massive neutrinos be- trumcausedbymassiveneutrinosatfourredshiftsnapshots
comes an ever-smaller fraction of the total, and the extra (z =0,0.5,1 and 2), compared to the predicted effect from
suppression decreases. Eventually non-linear growth comes HALOFITand linear theory. Figure 3 shows Mν = 0.6 eV,
todominatethelineareffect,andthesuppressionislessthan Figure 4 has M = 0.3 eV and finally Figure 5 has M =
ν ν
predicted by linear theory. 0.15 eV.
(cid:13)c 2011RAS,MNRAS000,1–12
6 S. Bird et al.
Change in ∆2(k) for M =0.6,z=0 Change in ∆2(k) for M =0.6,z=1
1.0 ν 1.0 ν
0.9 0.9
M0.8 M0.8
D D
2∆C 2∆C
/ /
2∆ν0.7 2∆ν0.7
0.6 0.6
0.5 0.5
10-2 10-1 100 101 10-2 10-1 100 101
k /(h Mpc−1) k /(h Mpc−1)
Figure 2. The ratio between the matter power spectrum at z =0 (Left) and z =1 (Right) for simulations with and without massive
neutrinos(Mν =0.6eV).Solidgreenlinesshowresultsforsimulationsusingparticles,whiledashedredlinesshowresultsfromFourier-
spacesimulations.Theblackdot-dashedlinesshowthepredictedeffectfromlineartheory.ThesimulationsusedwereL60andS60;box
sizes were 512Mpch−1 and 150Mpch−1, with parameters shown in Table 1. For the smaller box sizes, more than one realisation of
structureisavailable,andhenceweshow(dottedgreylines)theoneσ errorduetosamplevariance,asestimatednumerically.Particle
and Fourier-space methods use the same initial structure realisations. The Fourier-space method has more power in the largest scale
mode,probablybecausethephasestructureoftheneutrinocomponenthasbeenpreventedfromevolving,enhancingtheeffectofsample
variance.
Change in ∆2(k) for M =0.6,z=0 Change in ∆2(k) for M =0.6,z=0.5
1.0 ν 1.0 ν
0.9 0.9
M0.8 M0.8
D D
2∆C 2∆C
/ /
2∆ν0.7 2∆ν0.7
0.6 0.6
0.5 0.5
10-2 10-1 100 101 10-2 10-1 100 101
k /(h Mpc−1) k /(h Mpc−1)
Change in ∆2(k) for M =0.6,z=1 Change in ∆2(k) for M =0.6,z=2
1.0 ν 1.0 ν
0.9
0.9
0.8
M0.8 M
D D
2∆C 2∆C0.7
/ /
2∆ν0.7 2∆ν
0.6
0.6
0.5
0.5 0.4
10-2 10-1 100 101 10-2 10-1 100 101
k /(h Mpc−1) k /(h Mpc−1)
Figure 3. The effect of massive neutrinos on the matter power spectrum for a neutrino mass of Mν = 0.6 eV. Solid lines show the
ratio between simulations with and without massive neutrinos, for both L60 (red), with a 512Mpch−1 box and S60 (orange), with a
150Mpch−1 box.Initialredshiftwas99.ThebluedashedlineshowstheestimatedratiousingHALOFIT,whiletheblackdashedline
showsthepredictionfromlineartheory.
(cid:13)c 2011RAS,MNRAS000,1–12
The Matter Power with Neutrinos 7
Change in ∆2(k) for M =0.3,z=0 Change in ∆2(k) for M =0.3,z=0.5
1.00 ν 1.00 ν
0.95 0.95
M0.90 M0.90
D D
2∆C 2∆C
/ /
2∆ν0.85 2∆ν0.85
0.80 0.80
0.75 0.75
10-2 10-1 100 101 10-2 10-1 100 101
k /(h Mpc−1) k /(h Mpc−1)
Change in ∆2(k) for M =0.3,z=1 Change in ∆2(k) for M =0.3,z=2
1.00 ν 1.00 ν
0.95
0.95
0.90
M0.90 M
D D
2∆C 2∆C0.85
/ /
2∆ν0.85 2∆ν
0.80
0.80
0.75
0.75 0.70
10-2 10-1 100 101 10-2 10-1 100 101
k /(h Mpc−1) k /(h Mpc−1)
Figure 4. The effect of massive neutrinos on the matter power spectrum for a neutrino mass of Mν = 0.3 eV. Solid lines show the
ratio between simulations with and without massive neutrinos, for both L30 (red), with a 512Mpch−1 box and S30 (orange), with a
150Mpch−1 box.Initialredshiftwas49.ThebluedashedlineshowstheestimatedratiousingHALOFIT,whiletheblackdashedline
showsthepredictionfromlineartheory.
HALOFIT clearly over-predicts the suppression of the for a given redshift being proportional to fν, although the
matterpowerspectrumduetomassiveneutrinosinthenon- redshift dependence is more complicated.
linear regime. The cause is similar to that discussed in Sec-
tion3.1;HALOFIT includesmassiveneutrinosonlythrough HALOFIT over-predicts the effect of neutrinos on the
the linear theory neutrinos suppression on the non-linear smallest scales. This is not due to neutrino physics, but is
scale, and thus neglects any back-reaction from the dark a discrepancy induced because, as found by Hilbert et al.
matter.Thisinterpretationissupportedbythegoodagree- (2009), HALOFIT under-predicts the growth of non-linear
ment of HALOFITwith the Fourier-space simulations. powerfork>2hMpc−1inaΛCDMuniversebyuptoafac-
toroftwo.Wecorrectedthisbyre-fittingtheHALOFIT pa-
The largest discrepancy, around 10% of the total sup- rameterthatcontrolsthesmall-scalepowerusingourΛCDM
pression,occursatk∼1hMpc−1.Thelocationofthemaxi- simulations,afterwhichwecouldreproducetheasymptotic
malsuppressioninthenumericalsimulationmovestolarger neutrinoeffect.WedetailourmodificationsinAppendixA.
scales at lower redshifts, an effect which is again not cap- NotethatHALOFIT alsohasreducedaccuracyatz=3−4,
turedbyHALOFIT,althoughitwaspresenttosomeextent witherrorsof15−20%,comparedto5−10%atz=0−2.
in our Fourier-space simulations. Furthermore, the ampli- Becauseofthis,itfailstoaccuratelypredictthelocationof
tude of the suppression decreases with redshift, which is to thepeaknon-linearsuppressionatz>2.Wedidnotcorrect
beexpectedfromourdiscussionin3.1;asnon-lineargrowth thiseffect,asourattemptswerefoundtonegativelyimpact
occursinthedarkmattermoreneutrinosfallintothegrav- accuracy at lower redshifts.
ity wells. A cross-over redshift occurs at z = 1; on very
small scales the suppression here is equal to that of linear We have also performed some simulations varying the
theory, while at lower redshifts it is less. Note also that in cosmological parameters from our fiducial values, as de-
the quasilinear regime, 0.05<k <0.2hMpc−1, the simula- scribedinSection2.3.Theresultsofthesesimulationswere
tions clearly agree much better with HALOFIT rather than similar to those for our fiducial cosmology, and agree with
linear theory The dependence of our results on M is well theresultsdiscussedinVieletal.(2010).Thedependenceof
ν
describedbyalinearrelation,withthemaximalsuppression thenon-linearsuppressiononcosmologyislargelycaptured
(cid:13)c 2011RAS,MNRAS000,1–12
8 S. Bird et al.
Change in ∆2(k) for M =0.15,z=0 Change in ∆2(k) for M =0.15,z=0.5
1.00 ν 1.00 ν
0.98 0.98
0.96 0.96
M M
D D
2∆C0.94 2∆C0.94
/ /
2∆ν 2∆ν
0.92 0.92
0.90 0.90
0.88 0.88
10-2 10-1 100 101 10-2 10-1 100 101
k /(h Mpc−1) k /(h Mpc−1)
Change in ∆2(k) for M =0.15,z=1 Change in ∆2(k) for M =0.15,z=2
1.00 ν 1.00 ν
0.98 0.98
0.96 0.96
M M
D0.94 D0.94
2∆C 2∆C
/ /
2∆ν0.92 2∆ν0.92
0.90 0.90
0.88 0.88
0.86 0.86
10-2 10-1 100 101 10-2 10-1 100 101
k /(h Mpc−1) k /(h Mpc−1)
Figure 5. The effect of massive neutrinos on the matter power spectrum for a neutrino mass of Mν = 0.15 eV. Solid lines show the
ratio between simulations with and without massive neutrinos, for both L15 (red), with a 512Mpch−1 box and S15 (orange), with a
150Mpch−1 box.Initialredshiftwas24.ThebluedashedlineshowstheestimatedratiousingHALOFIT,whiletheblackdashedline
showsthepredictionfromlineartheory.
by HALOFIT, with the exception of a weak dependence on the only current probe of the matter power spectrum at
Ω , which we include in our fitting formula. these redshifts is the Lyman-α forest, which should be ex-
M
aminedbymodellingthefluxpowerdirectlywithfullhydro
simulations, as in Viel et al. (2010). Hence we do not at-
3.3 Towards an improved fitting formula tempt to provide a good fit for z >3. We checked that our
In the previous Section, we compared our simulations to fit was not significantly altered by excluding the simulation
HALOFIT. While there was some discrepancy, it seemed to results at z=3.
capturemanyoftheimportantphysicalfeatures.Inlightof Beforeaddressingtheeffectofneutrinos,weattempted
this,wehavedesignedanimprovedfittingformulabymodi- tocorrectthelackofpowershownbyHALOFITonthesmall-
fyingHALOFITtoexplicitlyaccountfortheeffectofmassive estscales.Eq.12showsthattheasymptoticbehaviourofthe
neutrinos. Our fitting formula was derived by performing a non-linear term in HALOFITis given by
swehrieerseo∆f2sliemasits-stqhueamreasttmerinpimowiseartisopnesctorfuLm=fro∆m2simth/e∆s2fiimt−ula1-, ∆2H → c(na)(fn3)(Ω)y3(f1(Ω)−1)+γ(n) ∼yγ(n). (13)
tionand∆2 istheequivalentfromthefittingformula.Both
fit Thus, to correct the behaviour on small scales we altered
spectrahavebeenrebinnedtobeevenlyspacedinlogk.To
γ(n),fittingittotheΛCDMsimulationslabelledinTable1
findaminimum,weusedPowell’smethodasimplementedin
SciPy2. To avoid any contamination from sample variance, asS60,S60AS,S60NS,S60HandS30OL.Wewereinterested
weomittedscaleswithk<6 2π ,whereBoxistheboxsize. onlyintheasymptoticbehaviour,soweomittedtheresults
Box from the large simulation boxes from the fit. We thereby
The redshift bins used in our fits were z=0,0.2,0.5,1,2,3.
included extremely small scales into the fit; down to k =
HALOFIT models the effect of non-linear growth at z = 4
30hMpc−1. The numerical minimisation algorithm proved
rather poorly, and there is some residual shot noise present
unstable when constrained to k < 6hMpc−1. We checked
attheseredshiftsinourneutrinosimulations.Furthermore,
that the resulting fit is a good match to larger scales.
The largest remaining contribution to the discrepancy
2 http://www.scipy.org betweenHALOFITandoursimulationsisthedecreaseinthe
(cid:13)c 2011RAS,MNRAS000,1–12
The Matter Power with Neutrinos 9
maximal neutrino suppression due to neutrinos falling into when compared to the fiducial cosmology, although there
dark matter halos. This is proportional to f and roughly wereoneortwooutlierscausedbyrealisationnoise.Check-
ν
constant on large scales. To account for it we modified the ing against S30OM, which has Ω =0.25, produced some-
M
nonlinear term with the ansatz what larger errors at the highest redshift bin, with z = 3.
Q = lfν , (14) The lower value of ΩM means that z =3 is deeper into the
ν 1+mk3 quasilinear regime, where HALOFIT is less accurate.
(∆ν )2 =∆2 (1+Q ), (15) Figure6showstheperformanceofourimprovedfitting
NL NL ν formula, in a similar format to that of Figure 3. It is clear
with l and m found by minimisation. We considered other that our improved formula captures the suppression due to
powers of k in the denominator of Qν, as well as additional neutrinossignificantlymoreaccuratelythanHALOFIT,cor-
terms and powers of the non-linear scale kσ, but none of rectlypredictingthelocationanddepthofthemaximalsup-
these significantly enhanced the fit. pression to within the quoted errors.
To correctly match the location of the maximal sup-
pression at higher redshift, z (cid:62) 2, we needed to alter the
quasilinear term, reflecting a leakage of non-linear power
4 DISCUSSION AND CONCLUSIONS
from small scales to larger scales caused by the high ther-
maldispersionoftheneutrinos.Therefore,wemodifiedEq. We have performed a suite of detailed simulations of the
9 as follows, matter power spectrum, incorporating the effects of mas-
(cid:34)(1+∆˜2(k))β˜(n)(cid:35) sive neutrinos using both particle and Fourier-space meth-
∆2(k)=∆2(k) L exp[−f(y)], (16) ods, but focussing on simulations with neutrino particles.
Q L 1+α(n)∆˜2(k)
L We have compared our results to those predicted by the
∆˜2 =∆2(cid:18)1+ pfνk2 (cid:19), (17) widelyusedHALOFIT approximationtothenon-linearmat-
L L 1+1.5k2 ter power spectrum, and presented an improved fitting for-
β˜(n)=β(n)+f (r+sn2), (18) mula.
ν Observationalconstraintsontheneutrinomasspriorto
where p, r and s were found again by minimisation. These thisworkhaveeitherusedlineartheory(Reidetal.2010)or
coefficients, as well as l and m were found by simultane- HALOFIT toestimatetheeffectofmassiveneutrinosonthe
ously fitting to the simulations S15, S30, S60, L15, L30 matterpowerspectrum.Thomasetal.(2010),forexample,
and L60. For the larger box sizes, we used only scales used photometric redshifts at z = 0.45 − 0.65, on scales
with k < 0.5hMpc−1, and for the smaller boxes, we used k (cid:46) 0.2hMpc−1. This is in the weakly non-linear regime,
k<6hMpc−1. where we find HALOFIT to be accurate. Hence our results
Finally,therewassomeresidualdependenceonΩM.To should not significantly change their published constraints
accountforthis,wemodifiedlasl→l−t(ΩM−0.3),andfit and that of studies probing similar scales.
tseparately,basedonaminimisationwithsimulationsS30, However,whileHALOFIT reproducesthesuppresionof
S30OMandS30OL.Thenumericalvaluesofthecoefficients thematterpowerspectrumduetothefree-streamingofneu-
are shown in Appendix A, together with a summary of the trinos well on some scales, in the fully non-linear regime it
fitting formula. The reduced error in the fit can be defined systematically over-predicts the level of the effect. Further-
as more, HALOFIT fails to capture the redshift dependence of
(cid:114) theamplitudeandscaleofthepeaksuppression.Ourquan-
L2
E = , (19) titative numerical study of the non-linear regime will thus
N
d certainly be of importance for future galaxy (Wang et al.
where Nd is the number of data points used in the fit. We 2005; Schlegel et al. 2009; Carbone et al. 2011), CMB lens-
find E = 0.007 with our coefficients, compared to E = 0.04 ing (Cooray 1999; Kaplinghat et al. 2003), Lyman-α forest
with the unmodifed HALOFIT. (Gratton et al. 2008) and weak lensing (Ichiki et al. 2009;
A rough estimate of the error on our fitting formula, Beaulieu & et al. 2010) surveys. Abazajian & et al. (2011)
including realisation noise at the 1−σ level, is given by have published a review of the potential constraints on the
(cid:15)=E(k,z)P(k), (20) neutrinomassfromavarietyofobservationalmeasurements,
manyofwhichreachtheM ∼0.1eVrange.Realisingthese
log[1+k/k (z)] ν
E(k,z)= σ f , (21) limitswillrequiretheoreticalpredictionsofthepowerspec-
1+log[1+k/k (z)] ν
σ trum accurate to the percent level, significantly exceeding
where k (z) is again the non-linear scale. Differences be- the accuracy of current prescriptions.
σ
tween the fitting formula and the simulations change sign We have proposed an improved fitting formula which
betweenz=0andz=0.2,suggestingthedominantcontri- modifiesHALOFIT totakeaccountoftheeffectsofmassive
bution is realisation noise. The quoted errors are valid for neutrinos for k < 7 hMpc−1 at z (cid:54) 3. Errors are propor-
k < 7hMpc−1, and z (cid:54) 3. At z > 3 our fitting formula is tional to the amplitude of the suppression, and thus to f ;
ν
somewhatlessaccurate.Themaximalerroroverallscalesis forM =0.3eV,thelargesterrorisroughly2%ofthemat-
ν
E =f ; for M =0.3, E ∼2%. terpowerspectrum.Themaximalnon-linearsuppressionwe
max ν ν max
Although they were not included in the fit, we checked findisaround−10f ,significantlygreaterthanthe−8f ex-
ν ν
that our formula accurately reproduced simulations with pectedfromlineartheory.Henceconstraintsobtainedusing
different cosmologies, chosen to be at the bounds of the our formula should be tighter than predicted by forecasts
rangeallowedbycurrentdata.Thefittingformulaerrorsfor using linear theory.
S60AS, S60NS, and S60H were very similar to those found Saito et al. (2008, 2009, 2011) and Wong (2008) have
(cid:13)c 2011RAS,MNRAS000,1–12
10 S. Bird et al.
Change in ∆2(k) for M =0.6,z=0 Change in ∆2(k) for M =0.6,z=0.5
1.0 ν 1.0 ν
0.9 0.9
M0.8 M0.8
D D
2∆C 2∆C
/ /
2∆ν0.7 2∆ν0.7
0.6 0.6
0.5 0.5
10-2 10-1 100 101 10-2 10-1 100 101
k /(h Mpc−1) k /(h Mpc−1)
Change in ∆2(k) for M =0.6,z=1 Change in ∆2(k) for M =0.6,z=2
1.0 ν 1.0 ν
0.9 0.9
M0.8 M0.8
D D
2∆C 2∆C
/ /
2∆ν0.7 2∆ν0.7
0.6 0.6
0.5 0.5
10-2 10-1 100 101 10-2 10-1 100 101
k /(h Mpc−1) k /(h Mpc−1)
Change in ∆2(k) for M =0.15,z=0 Change in ∆2(k) for M =0.15,z=1
1.00 ν 1.00 ν
0.98 0.98
0.96 0.96
M M
D D
2∆C0.94 2∆C0.94
/ /
2∆ν 2∆ν
0.92 0.92
0.90 0.90
0.88 0.88
10-2 10-1 100 101 10-2 10-1 100 101
k /(h Mpc−1) k /(h Mpc−1)
Figure 6. The effect of massive neutrinos on the matter power spectrum for a variety of neutrino masses. Solid lines show the ratio
betweensimulationswithandwithoutmassiveneutrinos,forboth512Mpch−1 (red)and150Mpch−1 (orange)boxes.Thebluedashed
lineshowstheestimatedratiousingourimprovedfittingformula,whiletheblackdashedlineshowsthepredictionfromlineartheory.
proposed an approximation to the non-linear matter power make forecasts for the accuracy of neutrino mass measure-
spectrumbasedonone-loopperturbationtheory,whileLes- ments of future surveys. We should stress, however that we
gourguesetal.(2009)havemodelledtheeffectsofneutrinos have concentrated here on the relative impact of the neu-
with an approach based on a time renormalisation group trinos; HALOFITitself only predicts the non-linear power
flow.Theseapproachesaretosomeextentcomplementaryto spectrumtoanaccuracyof5−10%(Heitmannetal.2010)
our efforts; while these formulae are designed for the quasi- with larger discrepancies at very small scales and high red-
linear regime, for k < 0.3hMpc−1 or z > 1, our formula shifts (z >3). Detailed quantitative analysis of actual data
should be useful even in the fully non-linear regime, at the reachingintothenon-linearregimewillthusstillneedcare-
lowest redshifts. ful calibration with accurate numerical simulations. These
will have to be tailored to the redshift and scales relevant
Our fitting formula should be sufficiently accurate to
(cid:13)c 2011RAS,MNRAS000,1–12