Table Of ContentConstraining the Evolution of Dark Energy with a Combination of
Galaxy Cluster Observables
Sheng Wang1,2, Justin Khoury3, Zolt´an Haiman4 and Morgan May1
1Brookhaven National Laboratory, Upton, NY 11973–5000, USA
2Department of Physics, Columbia University, New York, NY 10027, USA
3Institute for Strings, Cosmology and Astroparticle Physics,
Columbia University, New York, NY 10027, USA
5 4Department of Astronomy, Columbia University, New York, NY 10027, USA
0
0 Weshowthattheabundanceandredshift distribution(dN/dz)ofgalaxyclustersinfuturehigh–
2 yield cluster surveys, combined with the spatial power spectrum (Pc(k)) of the same clusters, can
place significant constraints on the evolution of the dark energy equation of state, w = w(a).
an We evaluate the expected errors on wa = −dw/da and other cosmological parameters using a
Fishermatrixapproach,andsimultaneouslyincludingclusterstructureevolutionparametersinour
J
analysis. We study three different types of forthcoming surveys that will identify clusters based
3
on their X–ray emission (such as DUO,the Dark Universe Observatory), their Sunyaev–Zel’dovich
1
(SZ) decrement (such as SPT, the South Pole Telescope), or their weak lensing (WL) shear (such
as LSST, the Large Synoptic Survey Telescope). We find that combining the cluster abundance
2
andpowerspectrumsignificantlyenhancesconstraintsfromeithermethodalone. Weshowthatthe
v
1 weak-lensing survey can deliver a constraint as tight as ∆wa ∼ 0.1 on the evolution of the dark
3 energy equation of state, and that the X–ray and SZ surveys each yield ∆wa ∼ 0.4 separately, or
3 ∆wa ∼ 0.2 when these two surveys are combined. For the X–ray and SZ surveys, constraints on
6 darkenergy parametersare improvedbyafactor oftwo bycombining theclusterdatawith cosmic
0 microwavebackground(CMB) anisotropy measurementsbyPlanck, butdegrade byafactor of two
4 if the survey is required to solve simultaneously for cosmological and cluster structure evolution
0 parameters. The constraint on wa from the weak lensing survey is improved by ∼ 25% with the
/ addition of Planck data.
h
p
o- I. INTRODUCTION tistical constraints available on curvature Ωk [8]; assess-
r ingtheimpactofsamplevariance[9]andotheruncertain-
t ties[10]onparameterestimates;andcontrollingsuchun-
s
It has long been realized that clusters of galaxies pro-
a certaintiesbyutilizinginformationfromthe shapeofthe
v: vide a useful probe of fundamental cosmological param- cluster mass function dN/dM [11]. Closest to the sub-
eters. The formation of the massive dark matter po-
i ject of the present paper, Weller et al. [12] and Weller &
X tential wells is dictated by simple gravitational physics,
Battye [13] considered constraints on the time evolution
r and the abundance and redshift distribution of clusters of the dark energy equation of state in forthcoming SZE
a (dN/dz) should be determined by the geometry of the
cluster surveys.
universe and the power spectrum of initial density fluc-
Recent studies have elucidated the additional cosmo-
tuations. Early studies of nearby clusters used this re-
logical information available from the spatial distribu-
lation to constrain the amplitude σ of the power spec-
8 tion of galaxy clusters through a measurement of their
trum(e.g., [1,2]). Subsequentworks(e.g., [3,4,5])have
three–dimensional power spectrum P (k) [14] (see [15]
c
shownthattheredshift–evolutionoftheobservedcluster
for a more general treatment of extracting cosmologi-
abundanceplacesusefulconstrainsonthematterdensity
cal information from redshift surveys). The power spec-
parameter Ω .
m trumcontainscosmologicalinformationfromtheintrinsic
Thenextgenerationofsurveys,utilizingthe Sunyaev– shape of the transfer function [16] and from baryon fea-
Zel’dovich effect (SZE), X–ray flux, or weak lensing sig- tures [17, 18, 19]. The existing sample of 400 nearby
∼
naturestoidentify galaxyclusters,willbe abletodeliver clustersintheREFLEXsurveyhasalreadybeenusedto
large catalogs, containing many thousands of clusters, derive their power spectrum; combined with the number
withcomplementaryselectioncriteria. Suchforthcoming counts, this has yielded constraints on σ and Ω that
8 m
datasets have rekindled a strong theoretical and exper- are consistent with other recent determinations [20].
imental interest in galaxy clusters. In particular, Wang Most importantly, the cluster abundance and power
& Steinhardt [6] first argued that the cluster abundance spectrum can provide two independent powerful probes
can be used to probe the properties of dark energy, and of cosmological parameters from a single dataset. The
Haiman, Mohr & Holder [7] showed that a survey with dependence of dN/dz and P (k) on the cosmologicalpa-
c
severalthousandclusterscanyieldprecisestatisticalcon- rametersaredifferent. Combiningthetwopiecesofinfor-
straints on both its density (Ω ) and its equation of mationcanthereforebreakdegeneraciespresentineither
DE
state (w P/ρ). Several subsequent recent works have method alone, and yield tighter statistical constraints.
≡
focusedonvariousaspectsofextractingcosmologicalpa- Furthermore, this can be used to significantly reduce
rametersfromhigh–yield,futuresurveys,suchasthesta- systematic errors arising from the mass–observable rela-
2
tion,makingclustersurveys“self–calibrating”[21]. This where σ2(M,z) is the variance of the linear density field
self–calibration is especially strong when the abundance at redshift z, smoothed with a spherical top–hat fil-
and power spectrum information is combined with even ter which would enclose mass M at the mean present–
a modest follow–up mass calibration program[22]. day matter density ρ [56]. The Jenkins et al. mass
m
Inthispaper,wereturntothequestionofconstraining function was derived from numerical simulations, and
the time–evolution of the dark energy equation of state. its self–similar form is demonstrated to be accurate to
Specifically, we ask the question: can improved con- within 15% in three widely separated cosmologies (al-
∼
straints be obtained on the time–evolution of w = w(z) though see [27] who find a more significant cosmology–
when the cluster counts and power spectrum are com- dependenceofthemassfunction). Jenkinset al. identify
bined? We quantify the statistical constraints expected simulated clusters using M , the mass enclosed within
180
to be available from future samples of > 10,000 galaxy a spherical overdensity of 180 with respect to the mean
clusters. ∼ matter density. However, it is customary to define the
We study constraints from three different types of relationbetweenX–rayorSZEfluxandhalomassM ,
200
forthcoming cluster surveys. The proposed DUO (Dark definedastheclustermassenclosedwithinaspherewith
Universe Observatory) [23] X–ray survey will be per- mean interior overdensity of 200 relative to the critical
formedby anEarth–orbitingsatellite consisting ofseven density. To combine this relationwiththe mass function
telescopes that take a wide survey of the sky in soft X– in Eq. (1), we convert M to M assuming that the
200 180
ray bands. The SPT–like (South Pole Telescope) [24] halodensity profileis describedby the NFW modelwith
SZE survey will be performed by an 8–meter precision a concentration parameter of c =5 [28].
nfw
submillimeter–wave telescope detecting distant galaxy Thespatialdistributionofclustersisassumedtofollow
clusters by their Sunyaev–Zel’dovich decrement. The the spatial distribution of the dark matter halos and is
LSST–like(Large Synoptic Survey Telescope) [25]survey specifiedbytheclusterpowerspectrumP (k). Wefollow
c
willbe performedbyaground–basedtelescopedetecting Hu& Haiman[14]andobtainP (k) fromthe underlying
c
clusters by their weak lensing shear signature on back- mass power spectrum, P(k), modified by redshift–space
ground galaxies. distortions [29]
The most important differences between the present
paper and earlier works that have addressed the time– k 2 2
P (k ,k )= 1+β k b2 P(k), (2)
evolution of the equation of state [12, 13] are that here c ⊥ k k
(i) we simultaneously include the abundance and power " (cid:18) (cid:19) #
spectrum in our analysis; (ii) in addition to the cosmo-
logicalparameters,wesimultaneouslyincludeseveralpa- k2 =k2 +k2, (3)
⊥ k
rameters that describe cluster structure and evolution;
and (iii) we study three different types of forthcoming where k⊥ and kk are the wavenumbers of the sinusoidal
surveys. We also chose a different fiducial model for fluctuation modes transverse and parallel to the line of
our analysis (one close to the standard “concordance” sight, respectively. The redshift–distortion parameter β
cosmology). Our calculational method, based on Fisher is defined by [29]
matrices, is, on the other hand, only a simple approxi-
1dlnD
grow
mationtotheMonte–Carlolikelihoodanalysisperformed β = , (4)
b dlna
in [12, 13].
This paper is organized as follows. In II, we describe where Dgrow is the linear growth rate, and a is the ex-
our basic calculational methods. In III,§we present our pansionfactornormalizedtounitytoday. Theparameter
results for different future cluster su§rveys. In IV, we b in Eqs. (2) and (4) represents the linear bias averaged
critically discuss our results, including their un§certain- over all halos at redshift z:
ties, and summarize the implications of this work. −1
∞ dn(M,z) ∞ dn
b(z)= b(M)dM dM ,
dM dM
ZMmin(z) "ZMmin(z) #
II. CALCULATIONAL METHOD (5)
whereM (z)isthemassofthesmallestdetectableclus-
min
A. Simulating Cluster Surveys ter, which depends on the type of survey as discussed in
II.D. The bias parameter of halos of a fixed mass M,
§
We follow the standard approach, and identify galaxy b(M), is assumed to be scale independent and given by
clusters with dark matter halos. The differential comov-
aδ2/σ2 1 2p/δ
ing number density of clusters is given by Jenkins et b(M)=1+ c − + c , (6)
δ 1+(aδ2/σ2)p
al. [26] c c
with a = 0.75 and p = 0.3 providing the best fits to the
dn ρ dlnσ−1(M,z)
m clusteringmeasuredincosmologicalsimulations[30]. Fi-
(z,M) = 0.301
dM M dM nally,δ representsthethresholdlinearoverdensitycorre-
c
exp[ lnσ−1(M,z)+0.643.82],(1) sponding to spherical collapse, whose value is δ =1.686
c
× −| |
3
in an Einstein–de Sitter universe. We keep it fixed
TABLE I:Parameters for thePlanck Survey.
throughout the calculation, as it was shown to be only
weakly dependent on cosmology and redshift in other Frequency(GHz) 100 143 217
models [6]. θc (arcmin) 10.7 8.0 5.5
σc,T (µK) 5.4 6.0 13.1
σc,E (µK) −− 11.4 26.7
B. Fisher Matrix Technique ℓc 757 1012 1472
The Fisher matrix formalism allows a forecasting of
We construct the Fisher matrix for the redshift–space
the ability of a given survey to constrain cosmological
power spectrum as [14]
parameters [31]. It gives a lower bound to the statistical
uncertainty of each model parameter that is to be fit by
future data. The well–known advantages of the Fisher Fpower = ∂ln(k⊥2kkPc)ij ∂ln(k⊥2kkPc)ij (VkVeff)ij,
matrix technique are that (i) it allows a quick estimate µν ∂pµ ∂pν 2
i,j
X
of errors in a multi–dimensional parameter space, since (10)
the likelihood functions do not have to be evaluated at where P (k) is the cluster power spectrum. The two–
c
each point of the multi–dimensional grid, and (ii) con- dimensional k–space cells and the set of redshift bins
straints from independent data sets or methods can be are labeled by i and j, respectively. The factor of
easilycombinedbysimplysummingtheindividualFisher (V V /2)−1 estimates the uncertainty (∆P /P )2 in the
k eff c c
matrices. measured power spectrum, including the effects of shot
The Fisher matrix is defined as noise and cosmic variance [32]. Here V is the effective
eff
volume probed by the survey
∂2
F = L , (7)
ij ∂p ∂p n¯(z )P (k) 2
(cid:28) i j(cid:29) V (k)= dV j c , (11)
eff s
1+n¯(z )P (k)
where = lnL is the log–likelihood function, and Z (cid:20) j c (cid:21)
L −
where pi’s are the various model parameters which, in where n¯ is the expected average number density, and Vk
our case, include both cosmological parameters and pa- is the cylindrical volume factor in k–space:
rametersdescribingcluster structureandevolution. The
inverse (F−1) gives the best attainable covariance ma- 2π∆(k2)∆k
ij V = ⊥ k . (12)
trix, regardless of any specific method used to estimate k (2π)3
the parameters from the data [31]. In particular, the
best statistical uncertainty attainable on any individual We sum over 292 linearly spaced k cells from k =
⊥,k
parameterpi,aftermarginalizationoverallotherparam- 0.005 to 0.15 Mpc−1, thus defining a cylinder in three–
eters, is (F−1)1/2. dimensional k–space. We sum over redshift bins of size
ii
We construct the Fisher matrix for the redshift dis- ∆z = 0.2, between z = 0 and z = 2.0 for the
min max
tribution of the number density of galaxy clusters as [8] DUO–like survey and the SPT–like survey, and between
z =0.1andz =1.4forthe LSST–likesurvey. The
min max
choice of this relatively wide redshift bin size is dictated
Fcounts = ∂Ni∂Ni 1 , (8) bytheneedtohaveasufficientnumberofclustersineach
µν ∂pµ ∂pν Ni binforanaccuratedeterminationofthe powerspectrum
Xi (N >1,000), as well as a wide enough ∆z that includes
where radi∼al modes with k⊥,k 0.005 Mpc−1.
≈
Finally, in addition to the constraints from clusters
d2V ∞ dn(M,z ) consideredhere,weconstructthe Fishermatrixthatcan
i
N =∆Ω∆z (z ) dM (9)
i dzdΩ i dM be used to forecast cosmological parameter errors from
ZMmin(zi) the temperature and polarization anisotropy of the cos-
mic microwave background (CMB). We have in mind a
is the number of clusters above the detection threshold
near–futuresurvey suchas Planck[33] thatwill measure
in each redshift bin i centered at z . In Eq. (9), ∆Ω
i
is the solid angle covered by a survey, d2V/dzdΩ is the temperature and E–mode polarization auto–correlation
(respectively, TT and EE), as well as temperature–
comovingvolumeelement,anddn/dM istheclustermass
polarizationcross–correlation(TE). We neglect B–mode
function (see Eq. (1)). We sum over redshift bins of size
polarization. The full CMB Fisher matrix is then given
∆z = 0.05, between z = 0 and z = 2.0 for the
min max
by [34, 35]
DUO X–ray survey and the SPT–like SZE survey, and
between z = 0.1 and z = 1.4 for the LSST–like
min max ∂C ∂C
survey, although accurate redshifts are not required for Fcmb = X,ℓCov−1(C ,C ) Y,ℓ , (13)
µν ∂p X,ℓ Y,ℓ ∂p
the dN/dz test. µ ν
ℓ X,Y
XX
4
where X,Y run over TT, EE and TE correlations. The matter (CDM) cosmologicalmodel, dominated by a cos-
covariance matrix, Cov(C ,C ), has elements mological constant (Λ). The sensitivity of our results to
X,ℓ Y,ℓ
the choice of the fiducial parameters is discussed in IV
2 §
Cov(C ,C ) = (C +B−2)2 below. The parameters are adopted from recent mea-
TT,ℓ TT,ℓ (2ℓ+1)fsky TT,ℓ T,ℓ surementsbytheWilkinson Microwave AnisotropyProbe
2 (WMAP),assummarizedinTable1of[39]: baryonden-
Cov(CEE,ℓ,CEE,ℓ) = (2ℓ+1)fsky(CEE,ℓ+BE−,2ℓ)2 sity Ωbh2 = 0.024, matter density Ωmh2 = 0.14, dark
energydensityinunitsofthecriticaldensityΩ =0.73
1 DE
Cov(C ,C ) = [C2 + (or Hubble constant H = 100h km s−1 Mpc−1 with
TE,ℓ TE,ℓ (2ℓ+1)f TE,ℓ 0
sky h = 0.72), with present–day normalization σ = 0.9 and
8
(CTT,ℓ+BT−,2ℓ)(CEE,ℓ+BE−,2ℓ)] scalar power–law slope ns = 1 of the primordial power
2 spectrum. Following Linder [19], we parameterize the
Cov(CEE,ℓ,CTE,ℓ) = (2ℓ+1)f CTE,ℓ(CEE,ℓ+BE−,2ℓ) evolving dark energy equation of state as
sky
2 z
Cov(C ,C ) = C (C +B−2) w(z)=w +w (1 a)=w +w , (16)
TT,ℓ TE,ℓ (2ℓ+1)f TE,ℓ TT,ℓ T,ℓ 0 a − 0 a1+z
sky
2
Cov(CTT,ℓ,CEE,ℓ) = (2ℓ+1)f CT2E,ℓ, (14) with values in our fiducial model chosen to be w0 = −1
sky and w =0. An alternative parameterization sometimes
a
used in the literature is w(z)=w +w z. The errors we
where f is the fraction of the sky covered. The B ’s 0 z
sky T,ℓ obtain here on w should be divided by approximately
and B ’s account for experimental noise for temper- a
E,ℓ
a factor of two to obtain the corresponding errors on
ature and polarization measurements, respectively, and
w . This follows from Taylor–expanding Eq. (16) about
are given by [36] z
z =1/2, which is approximately where the sensitivity of
B2 = (σ θ )−2e−ℓ(ℓ+1)/ℓ2c, (15) cluster surveys peak.
ℓ c c
c
X
where the sum is over the different frequency channels D. Survey Parameters
denoted by c, σ is the sensitivity, θ is the beam width,
c c
and lc ≡2√2ln2/θc is the corresponding“cut–off” mul- To determine the detection mass limit Mmin(z) in
tipole. Equation 15 assumes that different channels pro- Eq. (1), a mass–observable relation is needed. We con-
vide independent constraints. We follow previous the-
sider three surveys for our analysis, a flux limited X–ray
oretical “error forecast” work in adopting this assump- survey,suchasDUO,anSZEsurveythatissimilartothe
tion; however, we note that this implicitly assumes that
onetobe carriedoutwiththeSouthPoleTelescope,and
allforegroundswereperfectlyremovedfromthetempera- a weak lensing survey that is similar to that planned for
tureandpolarizationmaps. Inreality,imperfectremoval
the LSST. For the X–ray and SZE surveys, we impose a
of foregrounds will induce correlations among the chan- minimum mass of 1014h−1M (if M (z) as computed
⊙ min
nels,whichwillhavetobetakenintoaccountinarefined below turns out to be less than 1014h−1M ) since less
⊙
analysis. Modeling foregrounds and the expected preci- massivehaloscorrespondto smallclustersorgroupsand
sionwithwhichtheycanberemovedisbeyondthescope
are likely to depart from the scaling relations adopted
of the present paper. here. We do not impose this lower bound for the weak
In this paper we focus on the Planck survey for con- lensing survey since, in principle, the dark matter halos
creteness. Thissurveywillmeasuretemperatureandpo- oflow–massclusters orgroupsshouldstillproduceshear
larization anisotropy in three frequency bands, namely signals with a well–defined mass–shear relation.
100, 143 and 217 GHz, with fractional sky coverage of
FortheDUO–likeX–raysurvey,weadoptabolometric
f 0.8. The parameters for this experiment are
sky flux–mass relation of the form:
≈
listed in Table I (taken from [36]). The various C ’s
ℓ
are calculated up to ℓmax = 2,000, as appropriate for fx(z)4πd2L =AxM2β0x0E2(z)(1+z)γx, (17)
Planck, using KINKFAST [37], a modified version of
CMBFAST [38] tailored for time-varying w. where f in units of erg s−1 cm−2 is the bolometric flux
x
limit,d inunitsofMpcistheluminositydistance,M
L 200
is the mass of the cluster, and H(z) = H E(z) is the
0
C. Fiducial Cosmology Hubble parameter at redshift z. Following Majumdar &
Mohr [21], we adopt log (A ) = 4.159, β = 1.807,
10 x − x
TheFishermatrixformalismestimateshowwellasur- and γ = 0 as fiducial values. We model the DUO ob-
x
vey candistinguish a fiducialmodel of the universe from servations as a combination of a “wide” survey, covering
other models. The results depend on the fiducial model a sky area of ∆Ω = 6,000 deg2 with a bolometric flux
itself. Throughout the paper, we take a 7–dimensional limitoff >1.75 10−14erg s−1 cm−2(correspondingto
x
parameterization of a spatially–flat (Ω = 0), cold dark f > 7 10−14 er×g s−1 cm−2 in the 0.5 : 10 keV band);
k x
×
5
and a “deep” survey, spanning ∆Ω = 150 deg2 with a 1e+06
DUO wide
bolometric flux limit of f > 2.25 10−14 erg s−1 cm−2 DUO deep
(f > 9 10−15 erg s−x1 cm−2 ×in the 0.5 : 10 keV SLPSTS-Tli-klieke
x
×
band). Withtheseparameters,forourfiducialcosmolog- 10000
icalmodel,thewidesurveyyields 10,000clusterswhile
∼ Ω
the deep survey yields 1,500 clusters. These numbers d
∼ z
areconsistentwiththeexistingdataonthelogN logS d100
relation for clusters in soft X–ray bands [40] an−d also N/
2
with independent estimates for the total number of clus- d
ters expected to be detected by DUO [21].
1
For an SPT–like SZE survey, we adopt an SZE flux–
mass relation:
fsz(z)d2A =f(ν)fICMAszM2β0s0zE2/3(z)(1+z)γsz, (18) 0.010 0.5 1 1.5 2
z
wheref inunitsofmJyistheobservedfluxdecrement,
sz
d inunitsofMpcistheangulardiameterdistance,f(ν) FIG. 1: Expected number of clusters per redshift per unit
A
is the frequency dependence of the SZE distortion, and solid angle for the fiducial cosmology.
f is the mass fraction of the intra–cluster medium.
ICM
We model the SPT–like SZE survey as a flux–limited
survey with f > 5 mJy at 150 GHz. While this is We model the LSST-like survey to have a constant
sz
an oversimplification, the threshold value approximately detection threshold of κG = 4.5σnoise [42]. The noise
representsthetotalfluxdecrementofthesmallestcluster is given by the ratio of the mean ellipticity dispersion
that can be detected at 5σ significance with SPT. [57] (σǫ) of galaxies and the total number of background
βWsze =als1o.6a8d,oγpstzt=he0,fidanudciaclovpearrianmge∆teΩrs=lo4g,1000(A0sdze)g=2.8W.9e, σgan2loaisxeie=sσcoǫ2n/(t4aπinθeG2dnwg)it.hWineaadsmopototσhǫin=g0a.3pearntudrae θnGum[4b3e]r,
alsoassumefICM =0.12. Withtheseparameters,forour density of background galaxies ng = 65 arcmin−2 [44],
fiducial cosmological model, this survey yields 20,000 and the angular smoothing scale θG = 1 arcmin, which
clusters. ∼ corresponds to optimal S/N for the range of cluster
Finally, for the LSST–like WL survey, we follow masses and redshifts we considered. We take the survey
Hamana et al. [41] to find a relation between the di- to cover ∆Ω=18,000 deg2 and to extend over the clus-
mensionless shear and halo mass, given by ter redshift range 0.1 z 1.4. This yields 200,000
≤ ≤ ∼
clusters for our fiducial cosmology.
M /(πr2) Evidently,thevarioussurveysconsideredherespandif-
κ =α(θ ) vir s . (19)
G G Σ ferent redshift range, sky coverage and sensitivity. It is
(cid:20) cr (cid:21) thereforeusefultocomparetheexpectednumberofclus-
Here κ is the average shear within a Gaussian filter of ters per redshift and unit solid angle for each survey for
G
angularsizeθ ;M andr arethemassandscaleradius our fiducial cosmology. This is shown in Fig. 1. We see
G vir s
ofaclusterwithanNFWdensityprofile(r =r /c ); that the SPT–like survey is the most sensitive probe at
s vir nfw
z is the redshift of the cluster; and d is the angular high redshift, a consequence of the fact that the mass
l A
diameter distance to the cluster. The coefficient α is limit for SZE surveys is nearly redshift–independent. In
given by comparison,theDUOX–rayandLSST–likesurveysdrop
more sharply with redshift. Note that the fixed “mass
α= 0∞dx (x/x2G) exp −x2/x2G f(x), (20) floor” of 1014h−1M⊙ determines the number of clus-
log(1+c ) c /(1+c ) ters at z < 0.2,0.25,0.5 for DUO wide, SPT–like and
R nfw − n(cid:0)fw (cid:1)nfw
DUO deep, respectively. In comparison, the LSST–like
where x=φ/θs a dimensionless angular coordinate, and counts are dominated by small clusters or groups with
θxG=≡r θ/Gd/θ(sz )codrerneosptionngdtshetoantghuelasrmsocoatlehirnagdisucsa.leT,hweidtih- M ≤1014h−1M⊙.
s s A l
mensionlesssurfacedensityprofilef(x)isgivenbyequa-
tion (7) in Hamana et al. [41]. Finally, the mean inverse III. RESULTS
critical surface mass density is given by
Before discussing our results for the cluster surveys,
∞
Σ−1 = 4πGa(z )χ(z ) zl dz dn/dz (1−χ(zl)/χ(z)), we first summarize constraints from the CMB alone.
cr c2 l l R 0∞dz dn/dz Ourprojectionsforthe Plancksatellite,usingtheseven–
(21) parameter Fisher matrix, are listed in Table II and are
R
where a is the scale factor and χ denotes the comoving consistent with well–known previous forecasts [35, 45,
radial distance (valid for the flat universe with Ω = 0 46]. This table shows the power of the CMB in con-
k
we are assuming). strainingthematterandbaryondensity,Ω h2andΩ h2,
m b
6
powerspectrum(P (k))[58]. Thistableaddressestheis-
c
TABLE II: Estimated Cosmological Parameter Constraints
sue of the relative merits of survey size versus depth.
from Planck.
Starting with the counts, we see that the constraints
for the wide and the deep surveys are of the same or-
Planck Survey
der, even though the latter yields about 7 times fewer
∆ΩDE 0.035 clusters. This is because the deep survey, despite its
∆Ωmh2 0.0012 limited angular coverage, measures a higher fraction of
∆σ8 0.041 high–redshiftclusters. Forthepowerspectrum,however,
∆w0 0.32 the constraints are most sensitive to the total number
∆wa 1.0 of clusters. Indeed, the errors on most parameters dif-
∆Ωbh2 0.00014 fer by roughly a factor of Nwide/Ndeep ≈ 2.6, where
N and N are the total number of clusters for the
∆ns 0.0035 wide deep p
respective surveys. The power spectrum P (k) delivers
c
good constraints on the densities (Ω’s) and on σ in the
8
TABLE III: Estimated Cosmological Parameter Constraints wide survey, but has little sensitivity to w0 and wa.
from DUO.ThedN/dz column includespriors from WMAP: Table IV addresses the issue of the relative merits of
∆Ωbh2 =0.0010, and ∆ns =0.040. dN/dz, Pc(k), their combination, and their combination
Surveyand Parameter Constraints dN/dz Pc(k) withCMBanisotropydatafromPlanck,forallthreesur-
DUO Wide (6,000 deg2) veys. The top third of Table IV lists the results of com-
bining DUO wide and deep. Of particular interest is the
∆ΩDE 0.14 0.037
∆Ωmh2 0.25 0.096 third column from the left, which shows the constraints
obtained by adding the Fisher matrices for dN/dz and
∆σ8 0.16 0.10
P (k) for the combined survey. These are, in short, the
c
∆w0 0.16 0.59 most optimistic error bars from DUO alone. The ta-
∆wa 0.92 3.2 ble alsoillustrates the powerofcombiningcluster counts
∆Ωbh2 0.0010 0.023 withtwo–pointfunctionstatistics. Indeed,the combined
∆ns 0.040 0.18 error bars (column 3) for ΩDE and Ωmh2 are about two
DUO Deep (150 deg2) timessmallerthanthosederivedfromeitherdN/dz (col-
umn 1) or P (k) (column 2) alone. Finally, we see in the
∆ΩDE 0.097 0.11 c
∆Ωmh2 0.33 0.25 lastcolumn that combining DUO and Planckfurther re-
duces the uncertainty on w and w by approximately a
∆σ8 0.040 0.25 0 a
factor of two. This underscores the complementarity of
∆w0 0.29 0.78 cluster and CMB data in uncovering the nature of the
∆wa 2.5 3.7 dark energy.
∆Ωbh2 0.0010 0.059 The middle third of Table IV shows our results for
∆ns 0.040 0.49 the SPT–like SZE survey. Overall, the constraints on
thecosmologicalparametersaresimilartothoseavailable
from the DUO–like survey.
respectively, as well as the spectral tilt, n . However, as
s The bottom third of Table IV shows our estimated
is well known, the equation of state of the dark energy,
parameter uncertainties for the LSST–like cluster sur-
parameterized by w and w , is poorly constrained by
0 a vey. Comparing the third column with the previous two
CMB observations alone [47, 48]. This is because the
againconfirmsthepowerofcombiningcountswithpower
dependence of the CMB power spectrum on these two
spectrum. The constraints on w and w are of the or-
parameters comes mainly from the distance to last scat- 0 a
der of a few percent. These remarkably tight bounds
tering, d , which involves a double integral of w(z):
LS (comparing favorablywith those fromthe Planck survey
dLS zrec dz [Table II] for all cosmological parameters except Ωmh2,
, Ω h2 andn ) arethe resultofthe veryhighclusteryield
3000Mpc ≈ Ω h2(1+z)3+(1 Ω )h2g(z) b s
Z0 m − m ofthissurvey. Toexaminethesensitivityoftheseresults
(22)
p to the inclusion of the lowest mass clusters, we follow
where z is the redshift of recombination, and where
rec a more conservative approach by imposing a minimum
z (1+w(z′))dz′ massof2 1014h−1M⊙. Thisreducesthenumberofclus-
g(z) exp 3 . (23) ters to ×50,000. The constraints on w(a) from LSST
≡ (cid:26) Z0 1+z′ (cid:27) alone de∼grade by a factor of about two (consistent with
Since w and w only appear inside this double integral, √N scaling of statistical errors), to ∆w = 0.030 and
0 a 0
there is a severe degeneracy that can leave d nearly ∆w = 0.23. When combined with Planck, the errors
LS a
invariant under changes in these parameters. are nearly unaffected: ∆w = 0.024 and ∆w = 0.070.
0 a
Table III summarizes the results for the DUO wide Thereforeweareconfidentthatwithenoughclustersand
anddeep surveys,both fromcluster counts (dN/dz) and combining with Planck, LSST can constrainw to a few
a
7
TABLE IV: Estimated Cosmological Parameter Constraints from Clusters and CMB Combined. The dN/dz column includes
priors from WMAP: ∆Ωbh2 =0.0010, and ∆ns =0.040.
Survey and Parameter Constraints dN/dz Pc(k) dN/dz + Pc(k) dN/dz + Pc(k) + Planck
DUO Combined
∆ΩDE 0.011 0.032 0.0074 0.0064
∆Ωmh2 0.022 0.084 0.0098 0.00041
∆σ8 0.016 0.088 0.012 0.011
∆w0 0.10 0.45 0.096 0.061
∆wa 0.48 2.3 0.45 0.19
∆Ωbh2 0.0010 0.021 0.0010 0.00011
∆ns 0.040 0.15 0.033 0.0024
SPT-like Survey (4,000 deg2)
∆ΩDE 0.036 0.033 0.014 0.0097
∆Ωmh2 0.049 0.056 0.0083 0.00027
∆σ8 0.031 0.064 0.018 0.012
∆w0 0.22 0.41 0.15 0.082
∆wa 0.59 1.8 0.46 0.18
∆Ωbh2 0.0010 0.014 0.00099 0.00011
∆ns 0.040 0.094 0.029 0.0023
LSST-like Survey (18,000 deg2)
∆ΩDE 0.0053 0.0080 0.0024 0.0023
∆Ωmh2 0.026 0.021 0.0048 0.00024
∆σ8 0.0035 0.022 0.0025 0.0024
∆w0 0.051 0.10 0.024 0.023
∆wa 0.086 0.47 0.077 0.061
∆Ωbh2 0.0010 0.0050 0.00097 0.00010
∆ns 0.040 0.040 0.015 0.0022
percent. Finally,wefindthatitisessentialtoincludethe P (k)), we note that, for DUO, adding the power spec-
c
cosmology–dependenceofthelimitingmassfortheLSST trumdoesnotsignificantlytightenthe constraintsonw
0
survey. Repeating our analysis adopting the redshift– and w in this case. However, for the “self–calibration”
a
dependent mass limit from the fiducial cosmology, and case(TableV),combiningthetwomethodshelpsgreatly
not allowing it to vary with cosmology, results in an in- on all the constraints. For SPT–like and LSST–like sur-
crease by a factor of 3–4 in the uncertainties. veys, combining two methods always gives stronger con-
straints. This is clearly illustrated in Fig. 2.
In Table V, we repeat the analysis for the DUO and
SPT–like surveys, but this time taking into account the An obvious method to cross-check for systematic ef-
uncertaintyinthe structureandevolutionofclusters. In fects due to cluster structure and evolution, and to im-
otherwords,werequirethattheclustersurveysconstrain proveconstraints,istocombinetheX-rayandSZEdata.
not only the cosmology, but also the parameters of the Wehavefoundthattheuncertaintiesinw andw reduce
0 a
mass–observable relation. For DUO, this is modeled by to0.083and0.35whentheself-calibratedSPTandDUO
includingtheparametersA ,β andγ ofEq.(17)inthe samplesareconsideredincombination,andto 0.060and
x x x
Fisher matrix analysis. Similarly, for the SPT–like sur- 0.22 when Planck is added to this combined sample.
vey,weinclude A ,β andγ fromEq.(18). Forboth
sz sz sz Uncertaintiesinclusterstructureandevolutionshould
surveys,weseethataddingself–calibrationincreasesthe
be less severe for WL signatures, which probe the dark
error on w and w by a factor of < 2 in comparison
0 a matter potential directly. However, to compare the WL
with the corresponding results in Tab∼le IV. Overall, we
error forecasts more fairly with the SZ and X–ray pre-
see that including self–calibrationparameters still yields
dictions, in Table VI we show LSST predictions that in-
verygoodconstraintsonthe cosmology. The constraints
corporate two additional uncertainties inherent to WL
fromdN/dz+P (k)anddN/dz+P (k)+Planckareshown
c c selection: false detections andincompleteness. We relied
graphicallyinFigs.2and3,includingself–calibrationfor
on published results of numerical simulations [41, 42] to
the DUO–like and SPT–like surveys.
calibrate our results: we assumed 25% of detections are
Coming back to Table IV and comparing the first col- false, and 30% of real clusters are undetected. These ef-
umn from the left (dN/dz) with the third (dN/dz + fects increase the parameter errors, relative to those in
8
TABLE V: Parameter Constraints including Self–Calibration. The dN/dz column includes priors from WMAP: ∆Ωbh2 =
0.0010, and ∆ns=0.040.
Surveyand Parameter Constraints dN/dz Pc(k) dN/dz + Pc(k) dN/dz + Pc(k) + Planck
DUO Combined
∆ΩDE 0.030 0.043 0.015 0.012
∆Ωmh2 0.14 0.091 0.0098 0.00067
∆σ8 0.058 0.12 0.016 0.013
∆w0 0.44 0.53 0.20 0.15
∆wa 1.2 2.5 0.82 0.46
∆Ωbh2 0.0010 0.022 0.0010 0.00011
∆ns 0.040 0.18 0.034 0.0027
∆logAx 0.27 0.18 0.050 0.037
∆bx 0.29 0.18 0.030 0.027
∆γx 0.67 0.63 0.17 0.081
SPT-like survey
∆ΩDE 0.13 0.037 0.017 0.014
∆Ωmh2 0.65 0.077 0.0086 0.00063
∆σ8 0.14 0.099 0.019 0.017
∆w0 0.42 0.46 0.19 0.14
∆wa 2.4 1.8 0.81 0.40
∆Ωbh2 0.0010 0.018 0.0010 0.00011
∆ns 0.040 0.15 0.032 0.0026
∆logAsz 0.59 0.35 0.12 0.056
∆bsz 0.79 0.35 0.12 0.063
∆γsz 1.7 0.62 0.19 0.057
TABLE VI: Calibrated Cosmological Parameter Constraints from LSST and CMB Combined. The dN/dz column includes
priors from WMAP: ∆Ωbh2 =0.0010, and ∆ns =0.040.
Parameter Constraints dN/dz Pc(k) dN/dz + Pc(k) dN/dz + Pc(k) + Planck
LSST-like Survey
∆ΩDE 0.0081 0.012 0.0037 0.0033
∆Ωmh2 0.038 0.032 0.0059 0.00024
∆σ8 0.0054 0.034 0.0038 0.0037
∆w0 0.079 0.16 0.037 0.036
∆wa 0.13 0.72 0.12 0.093
∆Ωbh2 0.0010 0.0077 0.00098 0.00010
∆ns 0.040 0.062 0.021 0.0022
the bottom thirdofTable IVby 50%(exceptfor Ω h2 from simple models of cluster structure and evolution.
b
≈
andn ,whicharesignificantlydeterminedbytheWMAP WerequiredtheSZandX–raysurveysto“self–calibrate”
s
priors and are less affected). See the next section for a and constrain structure and evolution parameters simul-
more detailed discussion. taneouslywithcosmology. However,thepower–lawform
of the relations we adopted will have to be further con-
strained. This should be feasible by combining the three
IV. DISCUSSION observables in the different wave bands for a subset of
the samples,andbyaddingnew observables(suchasthe
In the previous section, we derived constraints on cos- shapeofthemassfunction,theangularsize,velocitydis-
mological parameters from future SZE, X–ray, and WL persion) which we have not considered here. In the case
surveys. Our Fisher matrix approach should be inter- of the WL sample, we relied on results from numerical
preted as yielding lower limits on the achievable statisti- simulationstocalibratethe mass–observablerelation,an
cal errors. In our analysis, we adopted unique relations approachthat can be refined with a largersuite of simu-
between the observables and cluster mass, which come lations in the future.
9
DUO 0.84 DUO
2
0.80
1
wa DE0.76
0 W
0.72
-1 0.68
dN/dz
0.64
-2 Pc(k)
dN/dz + Pc(k) 0.60
SPT 0.84 SPT
2
0.80
1
0.76
wa DE
0 W
0.72
-1 0.68
0.64
-2
0.60
LSST 0.84 LSST
2
0.80
1
wa DE0.76
0 W
0.72
-1 0.68
0.64
-2
0.60
-1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4
FIG. 2: Constraints on dark energy pawr0ameters: w0 and wa (left), w0 and ΩDE (right)wf0or a DUO–like X–ray survey (top),
an SPT–like SZE survey (middle), and an LSST–like weak lensing survey (bottom). The three curves in each figure show the
constraintsavailablefromdN/dz (dashed),Pc(k)(dotted),andfromtheircombination(solid). Thestar–shapedsymbolatthe
center of each figure indicates our fiducial cosmology. The constraints for X–ray survey and SZE survey are calculated for the
self–calibration case. The constraints are marginalized over all other cosmological and relevant structure parameters. In all
cases, the constraints from the combination of dN/dz and Pc(k) are at least a factor of two stronger than from either method
alone.
The Fisher matrix technique, especially the way of parameters involves a nonlinear inversion of the Fisher
combining independent constraints by summing individ- matrix, which makes isolating the various sources of in-
ualFishermatrices,allowsustoexplorethephysicalori- formationdifficult. Asanexample,werepeatedtheanal-
ginofthecosmologicalinformation. AglanceatEqs.(8) ysis above, but keeping either the volume factor, k2k ,
⊥ k
and (10) reveals that cosmology enters through several orthebiasparameterattheirvaluesinthefiducialmodel
physical quantities into the Fisher matrices, such as the (i.e., excluding their derivatives from computing power
cosmic volume, growth function, transfer function, bias, spectrum Fisher matrices). We find that as a result, the
etc. Unfortunately, the marginalization over all other constraintonsomeoftheparametersimprove,whileoth-
10
DUO DUO
0.8 0.76
0.4
0.74
wa DE
0.0 W
0.72
-0.4
CMB
-0.8 dN/dz + Pc(k) 0.70
dN/dz + Pc(k) +CMB
SPT SPT
0.8 0.76
0.4
0.74
wa DE
0.0 W
0.72
-0.4
-0.8 0.70
LSST LSST
0.8 0.76
0.4
0.74
wa DE
0.0 W
0.72
-0.4
-0.8 0.70
-1.4 -1.2 -1.0 -0.8 -0.6 -1.4 -1.2 -1.0 -0.8 -0.6
FIG.3: Constraintsondarkenergyparawm0eters: w0 andwa (left),w0 andΩDE (right),bywc0ombiningaDUO–likeX–raysurvey
(top), an SPT–like SZE survey (middle), or an LSST–like weak lensing survey (bottom) with Planck–like CMB observations.
Thethreecurvesin each figureshow theconstraints available from thecombination of dN/dz and Pc(k) (dotted),CMB alone
(dot–dashed), and from the combination of all of the above (solid). As in Fig. 2, the constraints for X–ray survey and SZE
surveys are calculated for the self–calibration case, and constraints are marginalized over all other relevant parameters. Note
thedifferent scales of thehorizontal and vertical axes compared to Fig. 2.
ers degrade, which does not offer a useful description of factor of normalization would be degenerate with σ . In
8
the amount of information the volume factor or the bias thefiducialmodel,b (z)followsfromEq.(5),andγ =0.
0 b
parameter provides. The constraints from cluster power spectrum including
the “non–standard evolution parameter” γ causes only
Inaddition,weexploredtheimplicitassumptionmade b
aminordegradation(under10%)oftheconstraintsonw
abovethatthebiasparameterbcanbepreciselymodeled. 0
andw (relativetocolumn2inTableIV),suggestingthat
We take a similar approach to that used in addressing a
thebiasfactordidnotdrivethe cosmologicalconstraints
“self–calibration”,by modeling the bias as b=b (z)(1+
0
z)γb. Weeffectivelyincludeanadditionalnewparameter, we derived from the power spectrum.
γ , in the Fisher matrix analysis. Note that a constant The results from the power spectrum also depend on
b
