Impact of Baryonic Processes on Weak Lensing Cosmology: Power Spectrum, Non-Local Statistics, and Parameter Bias

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by  Ken Osato

Submitted to ApJ
PreprinttypesetusingLATEXstyleemulateapjv.05/12/14
IMPACT OF BARYONIC PROCESSES ON WEAK LENSING COSMOLOGY:
POWER SPECTRUM, NON-LOCAL STATISTICS, AND PARAMETER BIAS
Ken Osato1†, Masato Shirasaki1 and Naoki Yoshida1,2,3
1DepartmentofPhysics,SchoolofScience,TheUniversityofTokyo,7-3-1Hongo,Bunkyo,Tokyo113-0033,Japan
2KavliInstituteforthePhysicsandMathematicsoftheUniverse(WPI),TodaiInstitutesforAdvancedStudy,TheUniversityofTokyo,
Kashiwa,Chiba277-8583,Japan
3CREST,JapanScienceandTechnologyAgency,4-1-8Honcho,Kawaguchi,Saitama,332-0012,Japan
Submitted to ApJ
5
1 ABSTRACT
0 We study the impact of baryonic physics on cosmological parameter estimation with weak lensing
2
surveys. We run a set of cosmological hydrodynamics simulations with different galaxy formation

y models. We then perform ray-tracing simulations through the total matter density field to generate
a 100 independent convergence maps of 25 deg2 field-of-view, and use them to examine the ability
M of the following three lensing statistics as cosmological probes; power spectrum, peak counts, and
Minkowski functionals. For the upcoming wide-field observations such as Subaru Hyper Suprime-

9 Cam (HSC) survey with a sky coverage of 1400 deg2, these three statistics provide tight constraints
1 onthematterdensity,densityfluctuationamplitude,anddarkenergyequationofstate,butparameter
biasisinducedbythebaryonicprocessessuchasgascoolingandstellarfeedback. Whenweusepower
]
spectrum, peak counts, and Minkowski functionals, the magnitude of relative bias in the dark energy
O
equation of state parameter w is at a level of, respectively, δw ∼ 0.017, 0.061, and 0.0011. For HSC
C
survey, these values are smaller than the statistical errors estimated from Fisher analysis. The bias
. can be significant when the statistical errors become small in future observations with a much larger
h
surveyarea. Wefindthebiasisinducedindifferentdirectionsintheparameterspacedependingonthe
p
statistics employed. While the two-point statistic, i.e. power spectrum, yields robust results against
-
o baryonic effects, the overall constraining power is weak compared with peak counts and Minkowski
r functionals. On the other hand, using one of peak counts or Minkowski functionals, or combined
t
s analysis with multiple statistics, results in biased parameter estimate. The bias can be as large as
a 1σ for HSC survey, and will be more significant for upcoming wider area surveys. We suggest to use
[
an optimized combination so that the baryonic effects on parameter estimation are mitigated. Such
2 ‘calibrated’ combination can place stringent and robust constraints on cosmological parameters.
v Subject headings: gravitational lensing: weak — cosmological parameters
5 cosmology: theory — large-scale structure of the universe
5
0
2 1. INTRODUCTION Euclid4 and WFIRST5 are also promising. Gravita-
0 tionallensingisexpectedtobethemainsubjectofthese
Anarrayofrecentobservationsofthelarge-scalestruc-
. futuresurveysthatareaimedatstudyingthelarge-scale
1 ture of the universe such as cosmic microwave back-
structure of the universe at present and in the past.
0 ground (CMB) anisotropies (e.g., Hinshaw et al. 2013;
Weak gravitational lensing (WL) by large-scale struc-
5 Planck Collaboration et al. 2014) and galaxy clustering
ture in the universe is the promising probe into proper-
1 (e.g., Reid et al. 2010; Beutler et al. 2014) established
: the standard cosmological model called ΛCDM model. ties of dark matter and dark energy (for a review, see
v Bartelmann & Schneider 2001; Munshi et al. 2008; Kil-
In ΛCDM model, the energy content of the present-day
Xi universe is dominated by two mysterious components: binger 2014). WL causes small distortion of image of
distantsourcegalaxies,calledcosmicshear,whichreflect
r dark energy and dark matter. Dark energy realizes the
a cosmic acceleration at present and dark matter plays an directly the intervening matter distribution along a line
of sight. Recent cosmic shear observations have proved
important role of formation of rich structure in the uni-
WLmeasurementtobeapowerfultoolforstudyingdark
verse. However, we have not understood yet the nature
matter distribution in the universe, from which one can
of dark energy and the physical properties of dark mat-
extract information on the basic cosmological parame-
ter. In order to reveal the mysterious dark components
ters (e.g., Massey et al. (2007); Kilbinger et al. (2013)).
in the universe, several observational programs are pro-
Forthcoming weak-lensing surveys are aimed at measur-
posed and still under investigation. Such observational
programs include Subaru Hyper Suprime-Cam (HSC)1, ing cosmic shear over a wide area of more than 1000
the Dark Energy Survey (DES)2, and the Large Synop- deg2. These observations will address important ques-
tions of dark matter and dark energy at unprecedented
tic Survey Telescope (LSST)3. Space missions such as
precision.
†E-mail: [email protected] Unfortunately, major statistical methods to make the
1 http://www.naoj.org/Projects/HSC/index.html
2 http://www.darkenergysurvey.org/ 4 http://www.euclid-ec.org/
3 http://www.lsst.org/lsst/ 5 http://wfirst.gsfc.nasa.gov/
2 Osato, Shirasaki & Yoshida
best use of WL data in upcoming surveys are still un- tics of interest and the implementations to estimate the
der debate. The problem originates from the fact that statistics for a given WL data. In Section 3, we describe
cosmic shear follows non-Gaussian probability distribu- oursimulationsetthatincludescosmologicalsimulations
tion due to non-linear gravitational growth (Sato et al. with baryonic physics. In Section 4, we provide the de-
2009). In order to incorporate the non-linear features tails of analysis performed in this paper. In Section 5,
accurately into WL statistics, cosmological N-body sim- we show the impact of baryonic physics on WL statis-
ulations have been extensively used. Previous numerical tics. We also perform a Fisher analysis to present the
studies (Hilbert et al. 2009; Sato et al. 2009, 2011) have expected cosmological constraints in upcoming lensing
alreadyprovidedimportantguidesforcosmologicalstud- surveys. We show the results of χ2 analysis to quantify
ieswithWLstatistics. Therestillremainseveralpossible the baryonic effects on parameter estimation with WL
factors to be examined. One of the uncertainties in such statistics. Concluding remarks and discussions are given
studiesistheeffectsofbaryonic physics. Modelingbary- in Section 6.
onic effects on the WL statistics is difficult because of
the overall complexities in galaxy formation. Recent nu- 2. WEAKLENSINGSTATISTICS
merical simulations and semi-analytic methods success- We first summarize the basics of gravitational lensing
fullyreproducekeyobservationaldata(Duffyetal.2010; by large-scale structure. Weak gravitational lensing ef-
Martizzi et al. 2012; Schaye et al. 2014; Okamoto et al. fect is characterized by the image distortion of a source
2014; Martizzi et al. 2014; Schaller et al. 2014a,b; Vellis- object by the following 2D matrix:
cig et al. 2014; Pike et al. 2014). Some of these studies
focus on active galactic nuclei (AGN) feedback, which ∂βi (cid:18)1−κ−γ −γ (cid:19)
A = ≡ 1 2 , (1)
quenches star formation in massive halos and may solve ij ∂θj −γ2 1−κ+γ1
over cooling problem (i.e. the over production of stars
in numerical simulations). These simulations also show wheretheobservedpositionofasourceobjectisdenoted
that baryonic physics can change significantly the distri- by θ, the true position is β, κ is convergence, and γ
bution of both dark matter and baryons within a halo. is shear. In weak field, where gravitational potential is
However, since baryonic effects are expected to be weak small compared with c2, each component of A can be
ij
at large scale, most of WL studies so far are based on related to the second derivative of the gravitational po-
simulations with dark matter component only. For fu- tential Φ as
ture ‘precision cosmology’ with WL, neglecting various
A =δ −Φ , (2)
astrophysical processes may lead to undesirable bias of ij ij ij
cosmological parameter estimation, as has been pointed 2 (cid:90) χ ∂2
Φ = dχ(cid:48)f(χ,χ(cid:48)) Φ[r(χ(cid:48))θ,χ(cid:48)], (3)
out by several studies (Jing et al. 2006; Semboloni et al. ij c2 ∂x ∂x
2011; Zentner et al. 2013; Yang et al. 2013; Mohammed 0 i j
r(χ−χ(cid:48))r(χ(cid:48))
et al. 2014). Hence, it is crucial and timely to study the f(χ,χ(cid:48))= , (4)
effects of baryonic physics on WL statistics in detail. r(χ)
In this paper, we use several statistics to extract the
where r(χ) is angular diameter distance, and x = rθ
non-Gaussian information of WL maps. In addition to i i
represents physical separation (Bartelmann & Schneider
power spectrum (PS) of weak lensing convergence, we
2001; Munshi et al. 2008). By using the Poisson equa-
consider peak counts and the Minkowski Functionals
tion and Born approximation (Bartelmann & Schneider
(MFs). Kratochvil et al. (2010) show that peak counts
2001; Munshi et al. 2008), one can express weak lensing
onlensingmapcanbeindeedusefulforcosmologicalpa-
convergence field as
rameter estimation. High-σ lensing peaks are likely as-
sociated with massive dark matter halos along a line of 3(cid:18)H (cid:19)2 (cid:90) χ δ[r(χ(cid:48))θ,χ(cid:48)]
sight (Hamana et al. 2004; Yang et al. 2011; Hamana κ(θ,χ)= 0 Ω dχ(cid:48)f(χ,χ(cid:48)) .(5)
etal.2012). MFsaremorphologicalstatisticsforagiven 2 c m a(χ(cid:48))
0
multi-dimensional field. MFs are one of the good mea-
Born approximation yields sufficiently accurate two-
sure of topology in WL and contain the suitable cos-
point statistics (e.g., Schneider et al. 1998). In the
mologicalinformationbeyondtwo-pointstatistics(Mun-
present paper, we take into account the non-linearity of
shi et al. 2012; Kratochvil et al. 2012; Shirasaki et al.
convergence shown in Eq. (3) by performing ray-tracing
2012; Shirasaki & Yoshida 2014). Throughout this pa-
simulations through the matter density field obtained
per, we use the terms “local statistics” and “non-local
from cosmological simulations.
statistics”. Shear two-point correlation functions and
thecorrespondingpowerspectrumaremeasureddirectly
2.1. Observables
from shear that is obtained from local measurements,
i.e., from galaxy ellipticities. Peak counts and MFs are Here, we summarize three different statistics of weak
non-local statistics, because they can be obtained from lensing convergence field, power spectrum (PS), peak
convergence maps that are derived by integrating shear counts, and Minkowski Functionals (MFs). PS has com-
over the region of interest. In order to clarify the bary- plete cosmological information only if statistical prop-
onic effects on the WL statistics, we utilize a large set erties of matter fluctuation follows Gaussian distribu-
of hydrodynamical N-body simulations including vari- tion. However, non-linear structure formation induced
ousbaryonicprocesses. Wethenstudyhowthebaryonic by gravity inevitably makes the fluctuation deviate sig-
effects bias cosmological parameter estimation with WL nificantly from Gaussian. In order to extract cosmologi-
measurement. The rest of this paper is organized as fol- cal information, we will also use non-local statistics, i.e.
lows. InSection2,wesummarizethebasicsofWLstatis- peak counts and MFs.
Impact of baryonic processes on weak lensing cosmology 3
2.1.1. Power spectrum byincluding the statisticalpropertiesofshapenoise and
the impact of shape noise on peak position.
PS is one of the basic statistics for modern cosmology.
Inordertolocatepeaksonadiscretizedmapobtained
For a convergence field κ, PS is defined by the two point
fromnumericalsimulations,wedefinethepeakasapixel
correlation in Fourier space:
that is higher than eight neighboring pixels. We then
(cid:104)κ˜((cid:96))κ˜∗((cid:96)(cid:48))(cid:105)=(2π)2δ ((cid:96)−(cid:96)(cid:48))P ((cid:96)), (6) measure the number density of peaks as a function of
D κ
K. We exclude the region within 2θ from the bound-
G
where the multipole (cid:96) is related with angular scale ary of the map in order to avoid the effect of incomplete
through θ =π/(cid:96). By using Limber approximation (Lim- smoothing. In this paper, we divide peaks into two sub-
ber 1954; Kaiser 1992) and Eq. (5), one can derive the groups: “mediumpeaks(MPs)”and“highpeaks(HPs)”.
convergence power spectrum as We define MPs and HPs by the peak height in a simi-
(cid:90) χs W(χ)2 (cid:18) (cid:96) (cid:19) lar manner to Yang et al. (2013). The former is defined
Pκ((cid:96))= 0 dχ r(χ)2 Pδ k = r(χ),z(χ) , (7) bwyhetrheeascotvheerglaetntceer pcoearrkeswpiotnhds1.0to≤thKeppeeaak/kσwnoiitshe ≤3.03.≤0
K /σ ≤5.0. Here, σ isthermsofshapenoise
where P (k) represents the three dimensional matter peak noise noise
δ on smoothed map given by Eq. (23). It is important to
powerspectrum,χ iscomovingdistancetosourcegalax-
s notethatthesepeaksarethoughttohavedifferentphys-
ies and W(χ) is the lensing weight function defined as
ical origins. MPs are likely caused by the shape noise
3(cid:18)H (cid:19)2 r(χ −χ)r(χ) or/and several dark matter halos aligned along a line of
W(χ)= 0 Ω s (1+z(χ)). (8) sight(Yangetal.2011),whereasHPsareassociatedwith
2 c m r(χs) individualmassivedarkmatterhalos(e.g.,Hamanaetal.
2004). Throughoutthepaper, wesetthenumberofbins
We follow Sato et al. (2009) to estimate the conver-
to be 10 for parameter estimation when measuring peak
gence PS from numerical simulations. We measure the
count of each subgroup.
binned power spectrum of convergence field by averag-
ingtheproductofFouriermodes|κ˜((cid:96))|2. Weuse20bins
logarithmically spaced in the range of (cid:96) = 100 to 105.
For parameter estimation performed in section 4, we re- 2.1.3. Minkowski Functionals
compute PS using 10 bins logarithmically spaced in the
range of (cid:96)=100 to 2000. MFs are morphological descriptors for smoothed ran-
dom fields. There are three kinds of MFs for two-
2.1.2. Peak count dimensional maps. Each MFs of V , V , and V repre-
0 1 2
Peaks in convergence maps can be a probe of massive sent the area above the threshold ν, the total boundary
halos (Hamana et al. 2004, 2012) and thus contain cos- length, the integral of geodesic curvature along the con-
mological information (Yang et al. 2011). tours. They are given by
In practice, peaks on convergence map is defined by a
local maxima on the “smoothed” map. We do the map- 1 (cid:90)
smoothing because the observed lensing field is signifi- V (ν)≡ da, (11)
cantlycontaminatedbytheintrinsicellipticitiesofsource 0 A Qν
galaxies. Thecontaminantiscalledshapenoise,whichis 1 (cid:90) 1
indeed the major contribution to the measured shape of V1(ν)≡A 4d(cid:96), (12)
source galaxies. In practice, we use a Gaussian window ∂Qν
function in order to reduce the effect of shape noises on 1 (cid:90) 1
V (ν)≡ Kd(cid:96), (13)
WbeLwsrtitatteisntiacss.coTnhveolsumtiooonthweidthcaonfivletregrefnucnectKio(nθ;oθfGW) ca:n 2 A ∂Qν 2π
G
(cid:90)
K(θ;θ )= d2φ W (θ−φ;θ )κ(φ), (9) where K is the geodesic curvature of the contours, da
G G G
and d(cid:96) represent the area and length elements, and A
is the total area. Q and ∂Q are denoted to be excur-
where θ is the smoothing scale and W is a gaussian ν ν
G G
sionsetsandboundarysetsforthesmoothedfieldK(x),
filter given by
respectively. They are defined by
1 (cid:18) θ2+θ2(cid:19)
W (θ)= exp − 1 2 . (10)
G πθ2 θ2
G G Q ={x|K(x)>ν}, (14)
ν
One can evaluate the smoothed convergence due to an ∂Q ={x|K(x)=ν}. (15)
ν
isolated massive cluster at a given redshift by assum-
ing the universal matter density profile of dark matter
halos (e.g., Navarro et al. 1997b). A simple theoreti- Inparticular,V isequivalenttoakindofgenusstatistics
2
cal framework to predict the number density of the K and equal to the number of connected regions above the
peaksispresentedbyHamanaetal.(2004);Maturietal. threshold, minus ones below the threshold. Therefore,
(2011). Their calculation yields reasonable results when for high thresholds, V is essentially equivalent to the
2
the signal-to-noise ratio of K due to massive halos is number of peaks.
larger than ∼ 4 (see, Hamana et al. 2004, for details). For a two-dimensional Gaussian random field, the ex-
Fan et al. (2010) also consider more detailed calculation pectation values for MFs can be described by analytic
4 Osato, Shirasaki & Yoshida
functions as shown in (Tomita 1986): 10-2
ST99
(cid:20) (cid:18) (cid:19)(cid:21)
V0(ν)=1211−σ erf (cid:18)νσ−0(µν−µ,)2(cid:19) (16) 3Mpc)−1100--43 DBFEAM
V1(ν)=8√2σ10 exp − σ02 , (17) 1ex−10-5
d
V (ν)= ν−µ σ12 exp(cid:18)−(ν−µ)2(cid:19), (18) φ(10-6
2 2(2π)3/2σ3 σ2
0 0 10-7
0.00
where µ = (cid:104)K(cid:105), σ2 = (cid:104)K2(cid:105)−µ2, and σ2 = (cid:104)|∇K|2(cid:105). Al- 0.05
tGhaouusgshiaMnitFysocfatnheb0efieelvdaliusawteedakp(eMrtautrsbuab1taivrealy20if03th,e20n1o0n)-, Diff 00..1105
0.20
itisdifficulttoevaluateMFsofhighlynon-Gaussianfield
0.25
(Petrietal.2013). Inthispaper, wepayaspatialatten- 12.0 12.5 13.0 13.5 14.0 14.5 15.0 15.5
log(M /M )
tion to the non-Gaussian cosmological information ob- tot fl
tained from convergence MFs.
Fig. 1.— Top panel: We plot the halo mass functions at z=0
For discretized K maps, we employ following estima-
measured from our set of simulations. The horizontal axis repre-
tors, as shown in, e.g., Kratochvil et al. (2012), sents total mass of halos which include dark matter, gas and star
particles. The blue, green and red line corresponds to simulation
1 (cid:90) resultsaveragedover10realizationsofDMmodel,BAmodeland
V0(ν)=A Θ(K−ν)dxdy, (19)FEmodel,respectively. Theerrorbarsrepresentthestandardde-
viationofthetenrealizationsforeachmodel. Thecyanlineshows
1 (cid:90) (cid:113) model prediction by Sheth & Tormen (1999). Bottom panel: The
V1(ν)=4A δ(K−ν) Kx2 +Ky2dxdy, (20)fractionaldifferenceofbaryonicmodelsfromthefiducialmodel.
V (ν)= 1 (cid:90) δ(K−ν)2KxKyKxy−Kx2Kyy−Ky2Kxxdxdy, noted as “DM”) and two baryonic simulations. In or-
2 2πA K2 +K2 der to model the degeneracy between cosmological pa-
x y rameters in WL statistics, we run six CDM only sim-
(21)ulations with one cosmological parameter varied. Cos-
mological parameters for our models are summarized in
where Θ(x) is the Heaviside step function and δ(x) is
Table 1. The number of particles is set to be 5123 for
the Dirac delta function. The subscripts represent dif-
CDM only simulations and 2×5123 for baryonic sim-
ferentiation with respect to x or y. The first and sec-
ulations, which consist of both CDM and gas particles.
ond differentiation are evaluated with finite difference.
For the fiducial cosmological model, the mass of a par-
We precompute MFs for 100 equally spaced bins of
ticle is found to be m = 7.97×109M /h and m =
ν(cid:48) = (ν−(cid:104)K(cid:105))/σ between −10 to 10. For cosmological p (cid:12) cdm
0 6.65×109M /h,m = 1.32×109M /h for baryonic
parameter estimation, we recalculate values on equally (cid:12) gas (cid:12)
spaced 10 bins in the range −3.0 ≤ ν(cid:48) ≤ 3.0 from 100 simulations. Our baryonic simulations are based on two
models denoted as BA and FE. BA model contains adi-
bins.
abatic gas particles, which exert only adiabatic pressure
3. SIMULATION in a Smoothed Particle Hydrodynamics (SPH) manner.
In FE model, we employ the galaxy formation ‘recipes’
3.1. N-body simulations
of Okamoto et al. (2014). Our FE model corresponds to
We are interested in non-linear gravitational evolution SN+AGNmodelintheirpaperwithsomemodifications,
of large-scale structure. In order to follow the evolution which include star formation, radiative cooling, super-
accurately,weruncosmologicalN-bodysimulations. We nova (SN) feedback, stellar wind feedback and an ad hoc
use parallelized tree-PM code Gadget-3 (Springel 2005) activegalacticnuclei(AGN)feedback. Thisprescription
withbaryonicprocesses(discussedbelow)tofollowstruc- savescomputationtimeandcontainslessfreeparameters
ture formation from an early epoch (z = 99) to present than solving fully the evolution of black holes. In their
(z = 0). The initial conditions are generated by MUSIC model, the radiative cooling rate is exponentially sup-
code (Hahn & Abel 2011), which is based on the sec- pressed when the velocity dispersion within a dark mat-
ond order Lagrangian perturbation theory (e.g., Crocce ter halo exceeds some threshold. Our simulations em-
et al. 2006). The transfer function is generated by the ployasimplermethodthatswitchesoffradiativecooling
linearBoltzmanncodeCAMB(Lewisetal.2000). Thevol- when the local velocity dispersion reaches the threshold.
ume of each simulation box is comoving 240 Mpc/h on Okamoto et al. (2014) report that the sudden change of
a side. We adopt the fiducial cosmological parameters thecoolingfunctionleadstosomewhatartificialincrease
as follows: matter density Ωm = 0.279, baryon density of the stellar mass, but we expect this modification does
Ωb = 0.0463, dark energy density ΩΛ = 0.721, Hub- not affect the final results significantly.
ble parameter h = 0.70, spectral index ns = 0.972 and In order to check basic statistics of our simulations,
amplitude of scalar perturbation As = 2.41 × 10−9 at we measure halo mass function for each model and stel-
the pivot scale k = 0.002 Mpc−1. These parameters are lar mass function for FE model. We run a friends-of-
consistent with 9-year Wilkinson Microwave Anisotropy friends (FoF) halo finder SubFind (Springel et al. 2001;
Probe (WMAP) result (Hinshaw et al. 2013). Dolag et al. 2009) to identify halos in simulations. For
In this paper, we perform three kinds of cosmologi- our simulations with baryons, the procedure is different
calsimulations;colddarkmatter(CDM)simulation(de- from that for CDM only simulation. First, using only
Impact of baryonic processes on weak lensing cosmology 5
TABLE 1
Cosmological parameters used for simulations.
Run w 109As Ωm ΩΛ σ8 No. ofsim. No. ofmaps Explanation
DM −1.0 2.41 0.279 0.721 0.821 10 100 CDMonlyfiducialmodel
BA −1.0 2.41 0.279 0.721 0.821 10 100 CDMandadiabaticgas
FE −1.0 2.41 0.279 0.721 0.821 10 100 CDMandbaryonicprocesses
HighΩm −1.0 2.41 0.302 0.698 0.872 10 100 1σ higherΩm model
LowΩm −1.0 2.41 0.256 0.744 0.767 10 100 1σ lowerΩm model
Highw −0.8 2.41 0.279 0.721 0.766 10 100 higherw model
Loww −1.2 2.41 0.279 0.721 0.860 10 100 lowerw model
HighAs −1.0 2.51 0.279 0.721 0.838 10 100 1σ higherAs model
LowAs −1.0 2.31 0.279 0.721 0.804 10 100 1σ lowerAs model
Note. — Two parameters (Ωm, 109As) are varied by 1σ value of WMAP nine-year result (Hinshaw et al. 2013) and we also adjust
ΩΛ accordinglytokeeptheuniversespatiallyflat. Wealsoshowthenormalizationoflinearmatterpowerspectrumσ8,whichisaderived
parameter.
10-1 ies. Also, our simulation results are inaccurate at stellar
FE masses less than ∼1010M(cid:12), because such small galaxies
Baldry+'12 contain only a few star particles. These small discrep-
10-2
ancies are, however, unimportant in the present paper
) because we draw our main conclusions through compar-
3c−10-3 isons of multiple sets of simulations with and without
p
M baryonic components, rather than through comparisons
1−10-4 of different feedback models.
x
e
d
(∗10-5 3.2. Ray-tracing simulations
φ
Forray-tracingsimulationsofgravitationallensing,we
10-6 utilize multiple simulation boxes to generate light-cone
outputssimilarlytoWhite&Hu(2000),Hamana&Mel-
10-7 lier (2001), and Sato et al. (2009). Details of the config-
10.0 10.5 11.0 11.5 12.0 12.5 13.0
log(M /M ) uration are found in the last reference.
∗ fl We place the simulation outputs to fill the past light-
Fig. 2.— Galaxy stellar mass functions at z = 0. The cyan cone of a hypothetical observer with an angular extent
pointsshowobservationalestimatesfromBaldryetal.(2012). The 5◦×5◦,fromz =0to1. Theangulargridsizeofourmaps
redlineillustratestheresultfromFEsimulationaveragedover10 issettobe5◦/4096∼0.073arcmin. Werandomlyrotate
realizationsandtheerrorbarsrepresentthestandarddeviation.
andshiftthesimulationboxesinordertoavoidthesame
structure appearing multiple times along a line-of-sight.
dark matter particles, we find FoF groups which con-
In total, we generate 100 independent lensing maps for
tain more than 31 particles. Next, we find gas particles
the source redshift of z =1 2. We show an example
for BA model and gas and star particles for FE model source
of convergence maps obtained from each baryonic model
linked with a dark matter particle which belongs to a
(DM, BA, and FE) in Figure 3. Note that each realiza-
FoF group. Finally, we separate each FoF group into a
tion of three models use the same random seed when we
central halo and substructures. In FE model, the stel-
generate initial conditions and multiple planes.
lar component of a halo is called ‘galaxy’ and ‘halo’ in-
Inordertomakethemocklensingmapsmorerealistic,
cludes the baryonic components and dark matter. Fig-
we add random gaussian noises as shape noise to the
ure 1 and 2 represent halo and stellar mass functions
simulatedconvergencedata(e.g.,Kratochviletal.(2010)
measured from simulations at z = 0. We compare the
and Shirasaki et al. (2012)):
simulation result with theoretical prediction of Sheth &
Tormen (1999). Figure 1 clearly shows our simulation
σ2
results are well described by the theoretical prediction (cid:104)κ (x ,x )κ (x(cid:48),x(cid:48))(cid:105)= γ δ δ ,
over the typical cluster mass scale, while there are ap- noise 1 2 noise 1 2 ngalApix x1,x(cid:48)1 x2,x(cid:48)2
preciable discrepancies between DM and FE simulations (22)
in the halo mass of less than ∼1013.5M . This is partly
(cid:12)
induced by SN explosions and/or stellar winds. Energy whereδ istheKroneckerdeltasymbol,n isthenum-
x,y gal
releasedbySNexplosionsorstellarwindsexpelgasparti- berdensityofsourcegalaxies,σ isthermsofshapenoise
γ
clesfromthehalobutmassivehaloscanretaintheparti- and A is the solid angle of a pixel. In the following,
pix
clesbythedeepgravitationalpotentialwell. Asaresult, weadoptσ =0.4,n =30arcmin−2. Thesevaluesare
γ gal
themassfunctionforourFEmodelisslightlysmallerat expected to be typical for HSC survey.
smallmasses. Wealsocomparethestellarmassfunctions As described in Section 2.1.2 and 2.1.3, smoothing is
of our simulation results and the observation of galaxy required to measure non-local statistics of convergence
stellar mass function in Baldry et al. (2012). The stel- from noisy WL data. For a given smoothing scale θ ,
G
lar mass distribution in our FE model is in reasonable the rms of the shape noise after gaussian smoothing is
agreement with the observation at z ∼ 0. Because we
adopt a simpler model than the original implementation 2 Forzsource=1,thelensingweightfunctionW(χ)hasamax-
ofOkamotoetal.(2014)thatincludesradiationpressure imum at z ∼ 0.5. This means that our lensing statistics are not
feedback, our FE run produces less small-mass galax- sensitivetothelargescalestructureabovez=0.7−0.8.
6 Osato, Shirasaki & Yoshida
DM BA FE
Fig. 3.—WeshowsampleconvergencemapsofDM,BAandFEsimulationswithoutnoiseandsmoothing. Redorblueareacorresponds
tohighorlowconvergencerespectively. Therearemanyredpoints,whichareidentifiedaspeaks.
DM models with seven different cosmologies. Thus, we
TABLE 2
The configuration of bins for each observable obtain the theoretical model of Ni for a given cosmolog-
ical model as follows:
Observable Range No. ofbins
PMSP 1.0≤1K00pe≤ak(cid:96)/≤σn2oi0s0e0≤3.0 1100 N¯i(p)=N¯i(p0)+(cid:88)N¯i(ppααα++)−−Np¯αi−(pαα−)(pα−p0α),
HP 3.0≤Kpeak/σnoise≤5.0 10 α α α
MFs −3.0≤(ν−(cid:104)K(cid:105))/σ0≤3.0 10foreach (25)
Note. — We summarize the configurations of our statistical
analysis. BinsarelinearlyseparatedexceptforPSthatareevalu- where α runs 1 to 3 and the superscript 0 means fiducial
atedusinglogarithmicallyspacedbins. value p0 = (0.279,−1.0,2.41). pα± represents a vector
given by van Waerbeke (2000)
with one parameter with a higher or lower value; for
(cid:16)σ (cid:17)(cid:18) θ (cid:19)−1(cid:18) n (cid:19)−1/2 example p1+ = (0.302,−1.0,2.41) (the other parameter
σ =0.0291 γ G gal . values are given in Table.1).
noise 0.4 1arcmin 30arcmin−2 ThecovariancematrixofthedatavectorN on25deg2
i
(23) maps is estimated as
4. ANALYSIS C (p)= 1 (cid:88)R [N (r,p)−N¯ (p)][N (r,p)−N¯ (p)].
We quantify the effects of the baryonic processes on ij R−1 i i j j
weak lensing analyses in terms of errors and bias in cos- r=1
mological parameter estimation. We consider primarily (26)
a lensing survey with a sky coverage of 1400 deg2, i.e.,
WeignorethecosmologicaldependenceofC(p)(see,e.g.,
theongoingwide-fieldsurveybySubaruHyperSuprime-
Eifleretal.2009)andhenceevaluatethecovariancema-
Cam (HSC). In the following, we describe in detail the
trix by using the fiducial model, i.e. C(p0). For the
calculation of the ensemble averages of three statistics
fiducial HSC survey, we simply scale the covariance ma-
and the covariance matrix in order to derive statisti-
trix by survey area, by multiplying the covariance ma-
calimplicationsandtoestimatecosmologicalparameters
trix C (p0) in Eq. (26) by a factor of 25/1400. When
p=(Ω ,w,109A ). ij
m s calculating the inverse covariance, we include a debi-
asing correction, the so-called Anderson-Hartlap factor
4.1. Theoretical model and covariance of lensing
α=(R−n−2)/(R−1) (Hartlap et al. 2007), where R
statistics
is the number of realizations and n is the dimension of
First,wecalculatethetheoreticaltemplateandcovari-
the data vector.
ance of the WL statistics using our ray-tracing simula-
tions. The configuration of bins are given in Table 2. 4.2. Fisher analysis
The model template and covariance are based on our
Fisher analysis gives a simple forecast for statisti-
DM models. Later, we examine the baryonic effects on
cal confidence level of three cosmological parameters
WLcosmologicalanalysisbycomparisonwithDMmodel
(Ω ,w,109A ) with WL statistics. The Fisher matrix
and two baryonic models. m s
is given by
We represent the data vector as N and denote the
i
dimension of the data vector as n. Note that n can be 1
F = Tr[A A +C−1M ], (27)
largerthanten,whenmultipleobservablesarecombined. ij 2 i j ij
We derive theoretical prediction of lensing statistics by
averaging over R=100 realizations: where Ai = C−1∂C/∂pi, Mij = 2(∂N/∂pi)(∂N/∂pj)T.
The first term vanishes when the cosmological depen-
(cid:104)Ni(p)(cid:105)(cid:39)N¯i(p)≡ R1 (cid:88)R Ni(r,p). (24) dteernmce, itshweefiarkst(Edieflreivraettivael.i2s0e0v9a)l.uTatoedcobmyputhteetfihrestseocrodnedr
r=1 finite difference, which is given by
In order to consider the cosmological parameter depen- ∂N N¯(pi+)−N¯(pi−)
dence on N , we employ linear interpolation based on (cid:39) i i . (28)
i ∂pi pii+−pii−
Impact of baryonic processes on weak lensing cosmology 7
Then, the marginalized error over the other two param- 5.1. Baryonic effect on convergence statistics
eters is given by
We summarize the main results of the baryonic effects
(cid:112) on WL statistics in Figure 4, 5 and 6.
σ = (F−1) . (29)
i ii Figure 4 show PS from our maps, where DM, BA and
FE models are indicated by blue, green and red lines re-
4.3. Mock lensing surveys
spectively. Wealsoshowthetheoreticalpredictionbythe
In order to make a realistic forecast in upcoming HSC cyan line which is calculated by Eq. (7) with the mod-
survey,weemploythebootstrapmethodasinYangetal. eling of matter power spectrum, essentially an modified
(2013). Since our suite of WL maps consist of one hun- HaloFit,ofTakahashietal.(2012)Weplottheshotnoise
dred 25 deg2 maps, we randomly choose 1400/25 = 56 contribution in the bottom panel as the dashed magenta
realizations from one hundred 25 deg2 WL maps. Re- line. The error bars represent the standard deviations
peating this procedure one thousand times, we can get over one hundred maps. The top and bottom panel rep-
onethousand‘mock’HSCWLmaps. Theresultingmaps resentstoresultsfromthemapswithandwithoutnoises,
are based on the fiducial cosmological model but we also respectively. The lower portion in each panel shows the
have the same set from simulations with the different fractional differences of BA and FE models with respect
baryonic effects. Lensing statistics of interest (i.e. PS, to the DM model.
peak counts and MFs) in HSC surveys are evaluated by
averaging each statistics on a 25 deg2 map over 56 real- POWER SPECTRUM
izations.
Compared with DM model, gas pressure suppresses
We then utilize the HSC observables derived in this
small-scale ((cid:96) > 10000) structures in BA model. On
way to investigate the impact of baryonic effects on pa-
the other hand, in FE model feedback processes are effi-
rameter estimation. We perform χ2 minimization to
cient at the scales of 2000 < (cid:96) < 10000 but cooling pro-
N as follows:
i,HSC cessesenhancetheconvergencepoweratthesmallerscale
(cid:88) (cid:96) > 20000. These features by cooling and feedback pro-
χ2(r,p,m)= ∆N (C−1) ∆N , (30)
i ij j cesses are consistently found by Semboloni et al. (2011);
i,j Mohammed et al. (2014). However, the strength of the
where ∆N ≡ Nm (r)−N¯ (p), r is the index of re- baryonic enhancement and suppression is different, be-
i i,HSC i cause of the details of baryonic physics implementations
alizations and 1 ≤ r ≤ 1000, N¯ represents the the-
i and also partially because of the difference of cosmolog-
oretical model of N for our dark matter only (DM)
i ical parameters.
model, andmrepresentsthedifferenceofmodelofbary-
onic physics, i.e., BA and FE. Suppose we calculate
PEAK COUNT
the i-th data N from a dark matter only conver-
i,HSC
gence map (m = DM), we should then find the result- Figure5showsthepeakcountsmeasuredfromthesim-
ing best-fit points distribute around the fiducial point ulated convergence maps for the three models. We use
p0 = (0.279,−1.0,2.41). However, when one considers the same combination of line colors as in Figure 4. The
the case of m = BA or FE, the center of the distri- error bars represent standard deviations from one hun-
bution of best-fit values could be biased in parameter dred maps. The lower portion in each panel shows the
space of p if baryonic effects induce discrepancies be- fractional difference of BA and FE models with respect
tween N and N¯ (p0). That means χ2 values cal- to DM model, and the shaded region indicates the Pois-
i,HSC i
culated from Eq. (30) do not follow χ2 distribution for son error of the fiducial model, i.e. the square root of
baryonic models because the ensemble mean and the co- the number of peaks within a bin, In this figure, we em-
variance matrix are computed from the fiducial model, ploy 5 bins per unit S/N ratio ν/σnoise, the peak height
not baryonic models. of convergence ν divided by σnoise. For the analytical
Thereisanotherwaytoestimatethebiasofparameter prediction, we adopt the model of Hamana et al. (2004)
estimation. The parameter biases of the BA and FE with Sheth-Tormen (Sheth & Tormen 1999) mass func-
models from fiducial parameters can be computed in the tion and Navarro-Frenk-White dark matter density pro-
following manner (see, e.g., Huterer et al. 2006): file(Navarroetal.1996,1997a). Weassumetherelation
between the halo mass and concentration parameter as
δp =(cid:88)F−1(cid:88)[N¯m−N¯ (p0)](C−1) ∂Nj, (31) in (Hamana et al. 2012),
α αβ i i ij ∂p
β i,j β (cid:18) M (cid:19)−0.086
c(M,z)=7.26 (1+z)−0.71,(32)
where F is the fisher matrix given by Eq. (27) and N¯m 1012M /h
i (cid:12)
represents the average of N over 100 convergence maps
i
In Figure 5, medium peaks (MPs) correspond to the
for m=BA and FE.
In the following, we quantify the baryonic effects on peaks with 0.03∼< Kpeak ∼< 0.09, while high peaks (HPs)
parameter estimation with WL surveys by considering arethosewith0.09∼< Kpeak ∼< 0.15. HPsaretypicallyas-
thedistributionofχ2 inEq.(30)andδpα inEq.(31)for sociatedwithmassivehaloswiththemassof∼1014M(cid:12).
our BA and FE models. We then compare the bias of The number of HPs is less affected by baryonic effects
parameter estimations with the marginalized error given because the baryonic processes do not cause very strong
in Eq. (29). effect to change the number of such massive halos, as
seeninFigure1. MPsoftenoriginatefrommultiplehalos
5. RESULTS withmassesof1012−1013M alignedinthelineofsight
(cid:12)
8 Osato, Shirasaki & Yoshida
HALOFIT 104
DM
P(‘)/2π10-4 BFEA peakK103 HDBFEAaMmana+04
‘(‘+1) n/dpeak102
10-5 d 101
0.2
100
0.1 0.15
Diff 00..01 Diff 0000....01000055
0.2 0.10
102 103 104 105 0.15
multipole ‘ 0.05 0.00 0.05 0.10 0.15 0.20
Kpeak
100
HALOFIT 104
10-1 noise Hamana+04
π
2 DM DM
(‘)/10-2 BFEA peak103 BFEA
P K
1)10-3 /d 102
(‘+10-4 npeak
‘ d 101
10-5
0.010
100
0.005 0.15
Diff 00..000005 Diff 0000....01000055
0.010 0.10
102 103 104 105 0.15
multipole ‘ 0.05 0.00 0.05 0.10 0.15 0.20
Kpeak
Fig. 4.—Weplottheconvergencepowerspectrumfromonehun- Fig. 5.—Thenumberofpeaksperunitsquaredegree. Thetop
dredconvergencemaps. Thetopandbottompanelshowtheresult and bottom panel shows the result without and with noise. The
withoutandwithnoiseandsmoothing. Thesolidcyanlineshows solid cyan line shows the model prediction of peaks by Hamana
themodelpredictionusingHaloFitofTakahashietal.(2012). The et al. (2004), which assumes that the halo mass function is de-
measuredpowersfromsimulationsareslightlysmallerthanthere- scribedbySheth&Tormen(1999)andthedensityprofileofhalos
sult of HaloFit due to finite box effect. The dashed magenta line follow Navarro-Frenk-White profile (Navarro et al. 1996, 1997a).
in the noisy case shows the power spectrum of the noise. In the The lower portion in each panel shows the fractional difference
lowerportionofeachpanel,weplotthefractionaldifferencesfrom fromthefiducialmodel. Thebluefilledregionrepresentsexpected
thefiducialmodelandthebluefilledregioncorrespondstotheex- PoissonerrorofDMmodelfromonehundred1400deg2 maps.
pectedstatisticalerrorofDMmodelwhenonemeasuresthepower
spectrumfromonehundred1400deg2maps. Notethatthisregion useful to calculate the expectation values for a Gaussian
does not correspond to the error bar, i.e. the standard deviation, random field (Eqs. (16)-(18)) in order to understand the
shownintheupperpanelthatismeasuredfromoursampleof25 overall feature of MFs. In Eqs. (16)-(18), the spectral
deg2 maps. And in the top panel, this expected error of power
moments σ and σ can be calculated from the conver-
spectrumfromnoisefreemapsissosmallthatitdoesnotappear 0 1
clearly. gence power spectrum as
direction. Baryonic physics, such as SN and AGN feed- σ2 =(cid:90) d2(cid:96) (cid:96)2pP ((cid:96))|W˜ ((cid:96);θ )|2, (33)
back,reducethemassesofthesehalos(seeSection2.1.2). p (2π)2 κ G G
Hencethebaryoniceffectswoulddecreasethenumberof
MPs, but the difference is largely made unimportant in where W˜G represents the Fourier transform of Eq. (10),
the noisy convergence maps because MPs are affected which is given by exp(cid:0)−(cid:96)2θ2/4(cid:1). Note that W˜
G G
significantly by shape noise (Yang et al. 2011). decreases exponentially at (cid:96)∼> 20000 × (1arcmin/θG).
¿From the result shown in Figure 4, we expect that σ
0
MINKOWSKI FUNCTIONALS
of BA and FE models is smaller than that of DM model
Figure 6 show the MFs computed from our maps of butσ wouldbenearlythesame. Weconfirmthisnotion
1
DM, BA and FE models. Again we use the same color by direct measurement of σ and σ from our 100 maps.
0 1
foreachmodelasinthepreviousfigures. TheMFsV ,V It is important to test whether the differences between
0 1
and V are plotted from left to right. The upper (lower) the models are attributed to σ . In the above analy-
2 0
two rows represent results from noise-free (noisy) maps. sis, we measured MFs in terms of normalized threshold
Panels in the second and fourth row show the difference ν(cid:48) because, if the difference is largely owing to varia-
of BA and FE models from the DM model. The er- tion of σ , the normalization would absorb all or most
0
rorbarsrepresentstandarddeviationsfromonehundred of the differences. The difference in the MFs becomes
maps. Weemploy5binsperunitν(cid:48) =(ν−(cid:104)K(cid:105))/σ . Itis indeed slightly smaller with the normalization, but does
0
Impact of baryonic processes on weak lensing cosmology 9
1.0 5 8
DM DM DM
BA BA 6 BA
0.8 FE 4 FE FE
4
) )
V(ν)000..46 3V(ν)(10−132 5V(µ)(10−2 02
2
0.2 1
4
0.0 0 6
0.04 0.03 0.02 0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.04 0.03 0.02 0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.04 0.03 0.02 0.01 0.00 0.01 0.02 0.03 0.04 0.05
ν ν ν
0.004 0.05 0.4
BA BA BA
0.003 FE FE 0.3 FE
0.002 0.00
0.001 3)− 5)− 0.2
∆V(ν)0 00..000001 ∆V(ν)(101 0.05 ∆V(ν)(102 00..01
0.002 0.10
0.1
0.003
0.004 0.15 0.2
0.04 0.03 0.02 0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.04 0.03 0.02 0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.04 0.03 0.02 0.01 0.00 0.01 0.02 0.03 0.04 0.05
ν ν ν
1.0 9 20
DM DM DM
BA 8 BA 15 BA
0.8 FE 7 FE 10 FE
0.6 3)−6 5)− 5
V(ν)00.4 V(ν)(101435 V(ν)(102 05
0.2 2 10
1 15
0.0 0 20
0.15 0.10 0.05 0.00 0.05 0.10 0.15 0.15 0.10 0.05 0.00 0.05 0.10 0.15 0.15 0.10 0.05 0.00 0.05 0.10 0.15
ν ν ν
0.0002 0.006
0.0001 BFEA 0.004 BFEA 0.04 BFEA
0.002
0.0000 3)− 0.000 5)− 0.02
∆V(ν)0 00..00000012 ∆V(ν)(101 000...000000462 ∆V(ν)(102 00..0002
0.0003 0.008
0.04
0.010
0.0004
0.15 0.10 0.05 0.00 0.05 0.10 0.15 0.15 0.10 0.05 0.00 0.05 0.10 0.15 0.15 0.10 0.05 0.00 0.05 0.10 0.15
ν ν ν
Fig. 6.—ThethreeMFsaveragedoveronehundredconvergencemaps. Panelsinthefirstandthesecondrowsshowtheresultswithout
noise whereas the third and the fourth rows are for those with noise. The error bars show the standard deviation of one hundred maps.
Note that the range of threshold ν is different between the noise-free and noisy cases. The blue filled region corresponds to the expected
statisticalerrorofDMmodelfromonehundred1400deg2 maps.
notvanishcompletely. Therefore,thebaryonicprocesses biases divided by the marginalized errors for BA and
apparentlyaffecttheMFs. Itisnecessarytodevelopfur- FE models. To check the accuracy of our code, we also
theranalysisbeyondGaussiandescriptiontoexplainthe present results from the fiducial simulation. We plot re-
difference of MFs between baryonic and fiducial models. sults obtained from K maps with shape noise.
In the following, we summarize the baryonic effects on
5.2. Fisher forecast and bias of cosmological parameters parameter estimation with different convergence statis-
tics.
We show the results of our parameter estimation in
Figures 9to11. Ineachfigure,weshowtwo-dimensional
POWER SPECTRUM
error contours marginalized over the other parameter.
The blue, green and red dots indicate the best-fit values The top three panels in Figure 9 show the results of
for DM, BA and FE models, respectively. Red dashed parameter estimation for three (DM, BA, FE) models
lines represent the fiducial values. Note that in these using only PS. We find only small bias caused by the
three figures, the plotted parameter range is adjusted baryonicphysics. Thatisbecause,withrealisticground-
accordingtothesizeoftheforecastcirclefortheobserv- based surveys PS can be measured accurately at low (cid:96)
able. In order to make comparison between the figures about 2000 at most, which is the maximum multipole
easy,weshowthe1σconfidenceregionsofsixobservables used in our estimation. The difference between the fidu-
in the same scale in Figure 8. The biases and marginal- cialandbaryonicmodelsappearsatthesmallscale. The
izederrors ofparameters aresummarized inTable3 and PS amplitude and the shape at (cid:96)∼< 2000 are not signifi-
Figure 7. In Figure 7, green and red points show the cantlyaffectedbythebaryonicphysics. Previousstudies
10 Osato, Shirasaki & Yoshida
Ωm w 109As
PS PS PS
MP MP MP
HP HP HP
V0 V0 V0
V1 V1 V1
V2 V2 V2
V0+V1+V2 V0+V1+V2 V0+V1+V2
PS+MP+HP PS+MP+HP PS+MP+HP
PS+MP+V0 PS+MP+V0 PS+MP+V0
PS+HP+V0 PS+HP+V0 PS+HP+V0
3 2 1 0 1 2 3 3 2 1 0 1 2 3 3 2 1 0 1 2 3
bias/error bias/error bias/error
Fig. 7.—Theparameterbiasesdividedbymarginalizederrors. GreenandredpointscorrespondtoBAandFEmodelsrespectively. The
biasescausedwhenusingonlyoneobservablearemostlywithin1σ. However,incombinedanalyses,e.g. PS+MP+HP,theresultingbias
canexceedthemarginalized1σ error.
TABLE 3 0.32 PS
MP
The marginalized errors and biases HP
Datastatistics δΩm δw δ109As 0.30 VVV012
Fisherforecastmarginalized1σ error
PS 0.0275 0.276 1.01 Ωm0.28
MP 0.00853 0.110 0.0557
HP 0.0129 0.118 0.0971 0.26
V0 0.00678 0.0711 0.0831
V1 0.00767 0.0911 0.0931 0.24
V2 0.00912 0.115 0.0718 4.0
V0+V1+V2 0.00439 0.0513 0.0521
PS+MP+HP 0.00478 0.0520 0.0371 3.5
PPSS++MHPP++VV00 00..0000357427 00..00349516 00..00461622 3.0
ParameterbiasofBA As2.5
PS 0.00104 0.000285 −0.0342 910
MP −0.00280 −0.0394 −0.0187 2.0
HP −0.00745 −0.0648 0.0278
V0 −0.00385 −0.0471 0.0435 1.5
V1 −0.00349 0.0147 0.101 1.0
VV20+V1+V2 −−00..0000521781 −0.00.00345729 00..00146797 1.4 1.2 1.w0 0.8 0.6 0.24 0.26 0.Ω28m 0.30 0.32
PS+MP+HP −0.00115 −0.0145 −0.0134
PS+MP+V0 0.000493 0.00803 0.00428 Fig. 8.—Wecomparethe1σconfidenceregionscalculatedfrom
PS+HP+V0 0.00148 0.0261 0.00488 Fisher forecast using six observables separately. The confidence
ParameterbiasofFE region from PS (green line) is clearly the largest. The other non-
PS −0.00171 −0.0172 0.0139 localstatisticsyieldsubstantiallysmallerconfidenceregions.
MP −0.00591 −0.0616 0.00536
HP −0.00668 −0.0610 0.0347 onic processes. Interestingly, we find that both MPs
V0 −0.000951 −0.00112 0.0543 and HPs induce biased parameter estimation of w with
V1 −0.00559 0.00157 0.120 a level of ∼0.56σ and 0.52σ, respectively, for FE model.
V2 −0.00182 0.0135 0.0342
V0+V1+V2 −0.000412 0.0347 0.0481 The bias may be partly originated from the baryonic ef-
PS+MP+HP −0.00594 −0.0587 0.0156 fect on the shape of massive halos. Massive halos tend
PS+MP+V0 −0.00288 −0.00899 0.0544 be rounder when baryonic processes are included (e.g.,
PS+HP+V0 0.000589 0.0136 0.0241 Kazantzidisetal.2004). Thentheheightofapeakasso-
Note. —Themarginalizederrorsofthreeparametersbasedon
ciated with a single halo is reduced, and so may be the
FisherforecastofDMmodelandthebiasesofBAandFEmodels
compared with the DM model. Note that these values are not peak counts at K∼ 0.0−0.05.
normalizedbythefiducialerrors.
(Yangetal.2013;Mohammedetal.2014)alsoexamined MINKOWSKI FUNCTIONALS
differences due to the choice of the maximum multipole. Figure 10 shows the results using each of the three
In our result, the difference at large scale is quite small MFs, V ,V ,V . The center of the dots is shifted in the
0 1 2
whereasnoisedominatesatsmallerangularscales,where respective parameter space; significant parameter bias
PS can not be measured accurately. For this reason, we can be caused by the baryonic effects. Note that the
fixthemaximummultipoleat(cid:96)=2000. Thenthebaryon absolute shift itself is not very large, but that the bias
effects on PS is negligible in cosmological parameter es- with respect to the error circle is appreciable (see Fig.
timation using PS. 8 for the relative size of the error circles). When one
constructs the theoretical template of MFs without the
PEAK COUNT
modeling of baryonic physics, analyses using V , V and
0 1
The medium (bottom) three panels in Figure 9 show V cause the biased estimation of w with a level of ∼
2
the results using only MPs (HPs). As we have discussed 0.016σ, 0.017σ, and 0.12σ, respectively, for FE model.
above, medium height peaks are also dominated by in- Although it is difficult to explain the origin of this bias
trinsic shape noise but still have cosmological informa- completely, we expect the following two effects can be
tion. High peaks have nearly one-to-one relation with responsible for the biased parameter estimation: (i) the
massivehalos,whosemassesarelessaffectedbythebary- change of variance of K (i.e. σ ) and (ii) the change
0