Optimal Constraints on Local Primordial Non-Gaussianity from the Two-Point Statistics of Large-Scale Structure

1.1 MB·English

by  Nico Hamaus

Optimal Constraints on Local Primordial Non-Gaussianity from the Two-Point
Statistics of Large-Scale Structure
Nico Hamaus,1,∗ Uroˇs Seljak,1,2,3 and Vincent Desjacques1
1Institute for Theoretical Physics, University of Zurich, 8057 Zurich, Switzerland
2Physics Department, Astronomy Department and Lawrence Berkeley National Laboratory,
University of California, Berkeley, California 94720, USA
3Ewha University, Seoul 120-750, S. Korea
(Dated: January 19, 2013)
2 One of the main signatures of primordial non-Gaussianity of the local type is a scale-dependent
1 correction to the bias of large-scale structure tracers such as galaxies or clusters, whose amplitude
0 depends on the bias of the tracers itself. The dominant source of noise in the power spectrum
2 of the tracers is caused by sampling variance on large scales (where the non-Gaussian signal is
strongest) and shot noise arising from their discrete nature. Recent work has argued that one can
n
avoid sampling variancebycomparing multipletracersof differentbias, and suppressshotnoise by
a
optimally weighting halos of different mass. Here we combine these ideas and investigate how well
J
the signatures of non-Gaussian fluctuations in the primordial potential can be extracted from the
9
two-point correlations of halos and dark matter. On the basis of large N-body simulations with

local non-Gaussianinitialconditionsandtheirhalocatalogs weperform aFishermatrixanalysisof
]
O thetwo-pointstatistics. Compared tothestandard analysis, optimal weightingand multiple-tracer
C techniques applied to halos can yield up to 1 order of magnitude improvements in fNL-constraints,
even if the underlying dark matter density field is not known. In this case one needs to resolve all
h. halos down to 1010h−1M⊙ at z =0, while with the dark matter this is already achieved at a mass
p thresholdof1012h−1M⊙. Wecompareournumericalresultstothehalomodelandfindsatisfactory
- agreement. Forecasting the optimal fNL-constraints that can be achieved with our methods when
o applied to existing and future survey data, we find that a survey of 50h−3Gpc3 volume resolving
tr all halos down to 1011h−1M⊙ at z = 1 will be able to obtain σfNL ∼ 1 (68% cl), a factor of ∼ 20
s improvement over the current limits. Decreasing the minimum mass of resolved halos, increasing
a
the surveyvolume or obtaining the dark matter maps can furtherimprove theselimits, potentially
[
reaching the level of σfNL ∼ 0.1. This precision opens up the possibility to distinguish different
2 typesof primordial non-Gaussianity and to probeinflationary physics of thevery early Universe.
v
1 PACSnumbers: 98.80.-k,98.62.-g,98.65.-r
2
3
2 I. INTRODUCTION a characteristic scale dependence on large scales in the
. presence of primordial non-Gaussianity of the local type
4
0 Adetection of primordialnon-Gaussianityhas the po- [8]. The power spectrum picks up an additional term
1 tential to test today’s standard inflationary paradigm proportional to fNL(bG 1), where bG is the Gaussian
−
1 andits alternativesfor the physics ofthe early Universe. bias of the tracer and fNL is a parameter describing the
v: Measurements of the CMB bispectrum furnish a direct strength of the non-Gaussian signal. However, the pre-
i probe of the nature of the initial conditions (see, e.g., cision to which we can constrain fNL is limited by sam-
X
[1–5]andreferencestherein), butarelimited by the two- pling variance on large scales: each Fourier mode is an
r dimensionalnatureoftheCMBanditsdampingonsmall independent realization of a (nearly) Gaussian random
a
scales. However, the non-Gaussian signatures imprinted field, so the ability to determine its rms-amplitude from
in the initial fluctuations of the potential gravitation- a finite number of modes is limited. Recent work has
ally evolve into the large-scale structure (LSS) of the demonstrated that it is possible to circumvent sampling
Universe, which can be observed in all three dimensions variance by comparing two different tracers of the same
and whose statistical properties can be constrained with underlyingdensityfield[9–12]. Theideaistotakethera-
galaxy clustering data (for recent reviews, see [6, 7]). tio of power spectra fromtwo tracersto (at leastpartly)
One of the cleanest probes is the galaxy (or, more cancel out the random fluctuations, leaving just the sig-
generally, any tracer of LSS including clusters, etc.) nature of primordial non-Gaussianity itself.
two-point correlation function (in configuration space) Anotherimportantlimitationarisesfromthefactthat
or power spectrum (in Fourier space), which develops galaxiesarediscretetracersoftheunderlyingdarkmatter
distribution. Therefore, with a finite number of observ-
able objects, the measurement of their power spectrum
is affected by shot noise. Assuming galaxiesare sampled
∗ [email protected] fromaPoissonprocess,thisaddsaconstantcontribution
2
to their powerspectrum, it is given by the inversetracer peak-backgroundsplitargumenttotheGaussianpieceof
number density 1/n¯. This is particularly important for Bardeen’s potential, one finds a scale-dependent correc-
massive tracerssuch as clusters, since their number den- tion to the linear halo bias [8, 21, 22]:
sity is very low. Yet they are strongly biased and there-
b(k,f )=b +f (b 1)u(k,z), (3)
fore very sensitive to a potential non-Gaussian signal. NL G NL G
−
Recent work has demonstrated the Poisson shot noise
where b is the scale-independent linear bias parameter
model to be inadequate [13–15]. In particular, [13, 14] G
of the corresponding Gaussian field (f =0) and
haveshownthatamass-dependentweightingcanconsid- NL
erably suppress the stochasticity between halos and the 3δ Ω H2
darkmatterandthusreducetheshotnoisecontribution. u(k,z) c m 0 . (4)
≡ k2T(k)D(z)c2
In view of constraining primordialnon-Gaussianityfrom
LSS,this canbe a veryhelpfultooltofurther reducethe Here, δ 1.686 is the linear critical overdensity for
c
≃
error on f . spherical collapse. Corrections to Eq. (3) beyond linear
NL
Bothofthesemethods(samplingvariancecancellation theory have already been worked out and agree reason-
and shot noise suppression) have so far been discussed ably well with numerical simulations [23–26]. Also, the
separately in the literature. In this paper we combine dependence of the halo bias on merger history and halo
the two to derive optimal constraints on f that can formationtimeaffectstheamplitudeofthenon-Gaussian
NL
beachievedfromtwo-pointcorrelationsofLSS.Weshow corrections in Eq. (3) [22, 27–29], which we will neglect
that dramatic improvements are feasible, but we do not here.
imply that two-point correlations achieve optimal con-
straints in general: further gains may be possible when
considering higher-order correlations, starting with the III. FISHER INFORMATION FROM THE
bispectrum analysis [16] (three-point correlations). TWO-POINT STATISTICS OF LSS
This paper is organized as follows: Sec. II briefly re-
views the impact of local primordial non-Gaussianity on It is believed that all discrete tracers of LSS, such as
the halo bias, and the calculation of the Fisher informa- galaxies and clusters, reside within dark matter halos,
tion content on fNL from two-point statistics in Fourier collapsednonlinearstructuresthatsatisfythe conditions
space is presented in Sec. III. In Sec. IV we apply our for galaxyformation. The analysisofthe fullcomplexity
weighting and multitracer methods to dark matter halos of LSS is therefore reduced to the information content
extractedfromaseriesoflargecosmologicalN-bodysim- in dark matter halos. In this section we introduce our
ulations and demonstrate how we can improve the fNL- model for the halo covariance matrix and utilize it to
constraints. These results are confronted with the halo compute the Fisherinformationcontentonf fromthe
NL
model predictions in Sec. V before we finally summarize two-point statistics of halos and dark matter in Fourier
our findings in Sec. VI. space. We separatelyconsider two cases: first halos only
and secondhalos combinedwith dark matter. While the
observation of halos is relatively easy with present-day
II. NON-GAUSSIAN HALO BIAS galaxy redshift surveys, observing the underlying dark
matter is hard, but not impossible: weak-lensing tomog-
Primordialnon-Gaussianityofthelocaltypeisusually raphy is the leading candidate to achieve that.
characterized by expanding Bardeen’s gauge-invariant
potential Φ about the fiducial Gaussian case. Up to sec-
ondorder,itcanbeparametrizedbythemapping[17–20] A. Covariance of Halos
Φ(x)=ΦG(x)+fNLΦ2G(x), (1) 1. Definitions
where Φ (x) is an isotropic Gaussian random field and
G We write the halo overdensity in Fourier space as a
f adimensionlessphenomenologicalparameter. Ignor-
NL vector whose elements correspond to N successive bins
ing smoothing (we will consider scales much larger than
the Lagrangiansize of a halo), the linear density pertur- δ (δ ,δ ,...,δ )⊺ . (5)
bationδ is relatedto Φthroughthe Poissonequationin h ≡ h1 h2 hN
0
Fourier space, Inthispaperwewillonlyconsiderabinninginhalomass,
but the following equations remain valid for any quan-
δ (k,z)= 2k2T(k)D(z)c2Φ(k), (2) tity that the halo density field depends on (e.g., galaxy-
0 3 ΩmH02 luminosity, etc.). The covariance matrix of halos is de-
fined as
where T(k) is the matter transfer function and D(z) is
the linear growthrate normalizedto 1+z. Applying the C δ δ⊺ , (6)
h ≡h h hi
3
i.e.,theouterproductofthevectorofhalofieldsaveraged Insteadwewillinvestigatetheshotnoisematrixwiththe
within a k-shell in Fourier space. Assuming the halos to help of N-body simulations.
belocallybiasedandstochastictracersofthedarkmatter The Gaussian case has already been studied in [14].
density field δ, we can write Simulations revealed a very simple eigenstructure of the
shot noise matrix: for N >2 mass bins of equal number
δh =bδ+ǫ, (7) density n¯ it exhibits a (N 2)-dimensional degenerate
−
(N−2)
subspace with eigenvalue λ = 1/n¯, which is the
and we define P
expected result from Poisson sampling. Of the two re-
b hδhδi (8) msuapipnrinesgseedig(eλnva)lwueitshλr±es,poencettisoethnehavnacluede 1(/λn¯+.)Tahnedsohnoet
≡ δ2 −
h i noise matrix can thus be written as
as the effective bias, which is generally scale-dependent
E =n¯−1I+(λ n¯−1)V V⊺+(λ n¯−1)V V⊺ , (13)
and non-Gaussian. ǫ is a residual noise-field with zero +− + + −− − −
meanandwe assumeit to be uncorrelatedwiththe dark where I is the N N identity matrix and V are the
±
matter, i.e., ǫδ =0 [30]. ×
h i normalizedeigenvectorscorrespondingtoλ±. Itsinverse
In each mass bin, the effective bias b shows a distinct
takes a very similar form
dependenceonf . Inwhatfollows,wewillassumethat
NL
b is linear in fNL, as suggested by Eq. (3): E−1 =n¯I+(λ−+1−n¯)V+V+⊺+(λ−−1−n¯)V−V−⊺ . (14)
b(k,f )=b +f b′(k). (9) The halo model [38] can be applied to predict the func-
NL G NL
tional form of λ and V (see [14] and Sec. V). This
± ±
Here,bGistheGaussianeffectivebiasandb′ ∂b/∂fNL. approachis howevernot expected to be exact, as it does
Finally, we write P δ2 for the nonlinear d≡ark matter notensuremass-andmomentumconservationofthedark
≡h i
power spectrum and assume ∂P/∂fNL = 0. This is a matter density field and leads to white-noise-like contri-
good approximation on large scales [31–33]. Thus, the butions in both the halo-matter cross and the matter
model from Eq. (7) yields the following halo covariance auto power spectra which are not observed in simula-
matrix: tions [39]. Yet, the halo model is able to reproduce the
eigenstructure of E fairly well [14] and we will use it for
Ch =bb⊺P +E , (10) making predictions beyond our N-body resolution limit.
In the Gaussian case one can also relate the dominant
where the shot noise matrix E was defined as
eigenmode V with corresponding eigenvalue λ to the
+ +
second-ordertermarisinginalocalbias-expansionmodel
E ǫǫ⊺ . (11)
≡h i [34, 35], where the coefficients bi are determined analyt-
In principle, E can contain other components than pure ically from the peak-background split formalism given a
halo mass function [40, 41]. In non-Gaussian scenarios
Poisson noise, for instance higher-order terms from the
this canbe extendedto a multivariate expansionin dark
bias expansion[34–36]. Here and henceforth, we will de-
fineE astheresidualfromtheeffectivebiastermbb⊺P in matter density δ and primordial potential Φ including
C , and allow it to depend on f . Thus, with Eqs. (8) bias coefficients for both fields [16, 23]. For the calcula-
h NL tion of E we will however restrict ourselves to the Gaus-
and (10) the shot noise matrix can be written as
sian case and later compare with the numerical results
δ δ δ⊺δ of non-Gaussian initial conditions to see the effects of
E = δ δ⊺ h h ih h i . (12) f onE anditseigenvalues. Thesuppressedeigenmode
h h hi− δ2 NL
h i V− with eigenvalue λ− can also be explained by a halo-
exclusion correction to the Poisson-sampling model for
This agrees precisely with the definition given in [14]
halos, as studied in [33].
for the Gaussian case, however it also takes into ac-
In what follows, we will truncate the local bias expan-
countthepossibilityofascale-dependenteffectivebiasin
sion at second order. Therefore, we shall assume the
non-Gaussianscenarios,suchthattheeffectivebiasterm
bb⊺P always cancels in this expression [37]. following model for the halo overdensityinconfiguration
space
Reference[10]alreadyinvestigatedtheFisherinforma-
tioncontentonprimordialnon-Gaussianityfortheideal-
δ (x)=b δ(x)+b δ2(x)+n (x)+n (x). (15)
ized caseof a purely Poissonianshot noise component in h 1 2 P c
the halo covariance matrix. In [15], the halo covariance Here,n istheusualPoissonnoiseandn acorrectionto
P c
wassuggestedtobeofasimilarsimpleform,albeitwitha accountfordeviationsfromthe Poisson-samplingmodel.
modifieddefinitionofhalobiasandadiagonalshotnoise In Fourier space, this yields
matrix. In this work we will consider the more general
model of Eq. (10) without assuming anything about E. δ (k)=b δ(k)+b (δ δ)(k)+n (k)+n (k), (16)
h 1 2 P c
∗
4
where the asterisk-symbol denotes a convolution. The 106
Poisson noise n arises from a discrete sampling of the
P
field δ with a finite number of halos, it is uncorrelated
h
wanitdhittshpeouwnedresrplyecintrgumdarisk nmantt⊺er=de1n/sn¯ity(,PohnisPsoδni w=hit0e, <(δ∗δ)2>
nleoaidses)t.oWe further assumehhnPPnP⊺cii = hncδi = 0, which 3pc] 105 εδ2
M
(δ δ)δ h-3 <(δ∗δ)δ>
b=b1+b2h ∗δ2 i , (17) P [ 104
h i
<δ2>
C =b b⊺ δ2 +(b b⊺+b b⊺) (δ δ)δ
h 1 1h i 1 2 2 1 h ∗ i
+b b⊺ (δ δ)2 + n n⊺ + n n⊺ , (18)
2 2h ∗ i h P Pi h c ci 103
0.01 0.10
(δ δ)δ 2 k [hMpc-1]
E =n¯−1I+b b⊺ (δ δ)2 h ∗ i + n n⊺ .
2 2 h ∗ i− δ2 h c ci
(cid:20) h i (cid:21) (19) FIG.1. ShotnoiseEδ2 ofthesquareddarkmatterdensityfield
δ2asdefinedinEq.(21)withbothGaussian(solidgreen)and
Hence,wecanidentifythenormalizedvectorb /b with
theeigenvectorV ofEq.(13)withcorrespond2in|g2e|igen- non-Gaussian initial conditions with fNL = +100 (solid red)
+ and fNL = −100 (solid yellow) from N-body simulations at
value
z = 0. Clearly, Eδ2 is close to white-noise like in all three
λ+ =b⊺2b2Eδ2 +n¯−1 , (20) scpasaecse.(dTahsheeadu),toitspocwroesrssppoewcterrumspehc(tδr∗uδm)2ih(δo∗fδδ)2δiinwFitohurtiheer
ordinarydarkmatterfieldδ (dot-dashed),aswellastheordi-
where we define
nary darkmatterpowerspectrumhδ2i(dotted)areoverplot-
(δ δ)δ 2 ted for the corresponding values of fNL. The squared dark
Eδ2 ≡h(δ∗δ)2i− h ∗δ2 i . (21) matter field δ2 can beinterpreted as a biased tracer of δ and
h i therefore shows the characteristic fNL-dependence of biased
fields (like halos) on large scales.
In[36]thistermisabsorbedintoaneffectiveshotnoise
power,sinceitbehaveslikewhitenoiseonlargescalesand
arisesfromthepeaksandtroughsinthedarkmatterden-
2. Likelihood and Fisher information
sityfieldbeingnonlinearlybiasedbytheb -term[42]. We
2
evaluated δ2 along with the expressions that appear in
Eq.(21) wEith the help of our dark matter N-body simu- In order to find the best unbiased estimator for fNL,
lations for Gaussian and non-Gaussian initial conditions we have to maximize the likelihood function. Although
(for details about the simulations, see Sec. IV). wearedealingwithnon-Gaussianstatisticsofthedensity
TheresultsaredepictedinFig.1. δ2 obviouslyshows field,deviationsfromtheGaussiancaseareusuallysmall
a slight dependence on f , but it reEmains white-noise- in practical applications (e.g., [22, 43, 44]), so we will
NL
likeeveninthenon-Gaussiancases. Thef -dependence consider a multivariate Gaussian likelihood
NL
of this term has not been discussed in the literature yet,
1 1
but it can have a significant impact on the power spec- L = exp δ⊺C−1δ . (22)
(2π)N/2√detC −2 h h h
trum of high-mass halos which have a large b -term; see h (cid:18) (cid:19)
2
Eq. (20). A discussion of the numerical results for ha- Maximizing this likelihoodfunctionis equivalentto min-
los, specifically the fNL-dependence of λ+, is conducted imizing the following chi-square,
later in this paper. It is also worth noticing the f -
NL
dependence of (δ δ)2 and (δ δ)δ . The properties of χ2 =δ⊺C−1δ +ln(1+α)+ln(detE), (23)
the squared dahrk∗matiter fiehld ∗δ2(x)i are similar to the h h h
ones of halos, namely, the k−2-correction of the effective where we dropped the irrelevant constant Nln(2π) and
bias in Fourier space, which in this case is defined as used
bδ2 ≡h(δ∗δ)δi/hδ2i and appears in Eq. (17). detC =det(bb⊺P +E)=(1+α)detE , (24)
ThelastterminEq.(19)correspondstothesuppressed h
eigenmodeofthe shotnoisematrix. Bothits eigenvector with α b⊺E−1bP. For a single mass bin, Eq. (23)
and eigenvalue can be described reasonably well by the ≡
simplifies to
halo model [14]. The argument of [33] based on halo
exclusion yields a similar result while providing a more δ2
χ2 = h +ln b2P + . (25)
intuitive explanation for the occurrence of such a term. b2P + E
E
(cid:0) (cid:1)
5
The Fisher informationmatrix [45] for the parameters combining galaxyredshift surveyswith lensing tomogra-
θ andθ andtherandomvariableδ withcovarianceC , phy [50], but the prospects are somewhat uncertain. We
i j h h
as derived from a multivariate Gaussian likelihood [46, will simply add the dark matter overdensity mode δ to
47], reads the halo overdensity vector δ , defining a new vector
h
Fij ≡ 12Tr ∂∂CθhC−h1∂∂CθhC−h1 . (26) δ ≡(δ,δh1,δh2,...,δhN)⊺ . (31)
(cid:18) i j (cid:19) In analogy with the previous section, we define the co-
With the above assumptions, the derivative of the halo variance matrix as C δδ⊺ and write
covariance matrix with respect to the parameter f is ≡h i
NL
∂Ch = bb′⊺+b′b⊺ P +E′ , (27) C= hδδ2δi hδδ⊺hδδ⊺i = bPP bC⊺P . (32)
∂fNL (cid:18)h h i h h hi(cid:19) (cid:18) h (cid:19)
with E′ ∂E/∂f . (cid:0)The inverse(cid:1)of the covariance ma-
NL
≡
trix can be obtained by applying the Sherman-Morrison 2. Likelihood and Fisher information
formula [48, 49]
E−1bb⊺E−1P Upon inserting the new covariance matrix into the
C−1 =E−1 , (28) Gaussian likelihood as defined in Eq. (22), we find the
h − 1+α
chi-square to be
where again α b⊺E−1bP. On inserting the two previ-
≡ χ2 =δ⊺C−1δ +ln(detE), (33)
ous relations into Eq. (26), we eventually obtain the full
expressionforF intermsofb, b′, E, E′ andP (see
fNLfNL where we used
Appendix A forthe derivationofEq.(A11)). Neglecting
tinhfeorfmNLa-tdioenpeonndfenceboefcEom, ie.se., setting E′ ≡0, the Fisher detC=detChdet P −b⊺C−h1bP2 =P detE , (34)
NL
andwestillassumeP(cid:0)tobeindependen(cid:1)toff andthere-
αγ+β2+α αγ β2 NL
F = − , (29) foredropthetermln(P)inEq.(33). Intermsofthehalo
fNLfNL (1+α(cid:0))2 (cid:1) anddarkmatteroverdensities,thechi-squarecanalsobe
expressed as
with α b⊺E−1bP, β b⊺E−1b′P and γ b′⊺E−1b′P.
≡ ≡ ≡
For a single mass bin, Eq. (A11) simplifies to Eq. (A12), χ2 =(δ bδ)⊺E−1(δ bδ)+ln(detE), (35)
h h
− −
bb′P + ′/2 2 which is equivalent to the definition in [14] (where the
F =2 E . (30)
fNLfNL b2P + last term was neglected). The corresponding expression
(cid:18) E (cid:19)
for a single halo mass bin reads
Thisimpliesthateveninthelimitofaverywell-sampled
phaolwoerdens(iatnydfineeldgle(cn¯tin→g ∞′))thweitFhisnheegrliignifbolremsahtoiotnncooinse- χ2 = (δh−bδ)2 +ln( ). (36)
E E E
tentonfNL thatcanbeextractedpermodefromasingle E
halomassbinislimitedtothevalue2(b′/b)2. Thisisdue For the derivative of C with respect to fNL we get
tothefactthatwecanonlyconstrainf fromachange
NL ∂C 0 b′⊺P
inthehalobiasrelativetotheGaussianexpectation,not = . (37)
from a measurement of the effective bias itself. The lat- ∂fNL (cid:18)b′P bb′⊺P +b′b⊺P +E′ (cid:19)
ter can only be measured directly if one knows the dark
Performing a block inversion, we readily obtain the in-
matter distribution, as will be shown in the subsequent
verse covariance matrix,
paragraph. However, the situation changes for several
halo mass bins (multiple tracers as in [9]). In this case, (1+α)P−1 b⊺E−1
theFisherinformationcontentfromEqs.(A11)and(29) C−1 = E−1b −E−1 . (38)
can exceed the value 2(b′/b)2 (see Sec. IV and V). (cid:18) − (cid:19)
AsshowninAppendixB,theFisherinformationcontent
on f now becomes
NL
B. Covariance of Halos and Dark Matter
F =γ+τ , (39)
fNLfNL
1. Definitions with γ b′⊺E−1b′P and τ 1Tr E′E−1E′E−1 . For a
≡ ≡ 2
single halo mass bin this further simplifies to
We will now assume that we possess knowledge about (cid:0) (cid:1)
the darkmatter distributionin additionto the haloden- b′2P 1 ′ 2
F = + E . (40)
sity field. In practice one may be able to achieve this by fNLfNL 2
E (cid:18)E (cid:19)
6
Itisworthnotingthat,incontrasttoEq.(30),theFisher putedinconfigurationspaceviainterpolationofthepar-
information from one halo mass bin with knowledge of ticles onto a cubical mesh with 5123 grid points using a
the dark matter becomes infinite in the limit of vanish- cloud-in-cell mesh assignment algorithm [57]. We then
ing . Inthislimittheeffectivebiascanindeedbedeter- perform a fast fourier transform to compute the modes
E
minedexactly,allowinganexactmeasurementoff [9]. of the fields in k-space.
NL
For each of our Gaussian and non-Gaussian realiza-
tions,wematchthetotalnumberofhalostotheonereal-
IV. APPLICATION TO N-BODY SIMULATIONS izationwiththeleastamountofthembydiscardinghalos
from the low-mass end. This abundance matching tech-
niqueensuresthatweeliminateanypossiblesignatureof
We employ numerical N-body simulations with both
primordialnon-Gaussianityinducedbythe unobservable
Gaussianandnon-Gaussianinitialconditionstofindsig-
f -dependence ofthe halomassfunction. Itguarantees
natures of primordial non-Gaussianity in the two-point NL
a constantvalue 1/n¯ of the Poissonnoise for both Gaus-
statisticsofthefinaldensityfieldsinFourierspace. More
sianandnon-Gaussianrealizations. Adependence ofthe
precisely, we consider an ensemble of 12 realizations of
Poisson noise on f would complicate the interpreta-
box-size1.6h−1Gpc(thisyieldsatotaleffectivevolumeof NL
V 50h−3Gpc3). Eachrealizationis seeded with both tion of the Fisher information content. Note also that,
eff ≃ in order to calculate the derivative of a function with
Gaussian(fNL =0)andnon-Gaussian(fNL =±100)ini- respect to f , we apply the linear approximationF
tial conditions of the local type [31], and evolves 10243 NL
ppaarratimcleetserosf amreasΩs 3.0=×0.1207191,h−Ω1M=⊙.0.7T2h1e, cΩosm=ol0o.g0i4c6a,l ∂F F(fNL =+100)−F(fNL =−100) , (41)
m Λ b ∂f ≃ 2 100
NL
σ = 0.81, n = 0.96, and h = 0.7, consistent with the ×
8 s
wmap5 [51] best-fit constraint. Additionally, we con- whichexploitsthestatisticsofallournon-Gaussianruns.
sider one realization with each f = 0, 50 of box-size Alltheerrorbarsquotedinthispaperarecomputedfrom
NL
1.3h−1Gpcwith15363particlesofmass4.±7 1010h−1M the variance amongst our 12 realizations.
⊙
×
to assess a higher-resolution regime. The simulations
wereperformedonthesupercomputerzbox3 attheUni-
versity of Zu¨rich with the gadget ii code [52]. The ini- A. Effective bias and shot noise
tialconditionswerelaiddownatredshiftz =100byper-
turbing a uniform mesh of particles with the Zel’dovich At the two-point level and in Fourier space, the clus-
approximation. tering of halos as described by Eq. (10) is determined
To generate halo catalogs, we employ a friends-of- by two basic components: effective bias and shot noise.
friends (FOF) algorithm [53] with a linking length equal Since the impact of primordial non-Gaussianity on the
to 20% of the mean interparticle distance. For compar- nonlineardarkmatterpowerspectrumP isnegligibleon
ison, we also generate halo catalogs using the ahf halo large scales (see Fig. 1), the dependence of both b and
finder developed by [54], which is based on the spherical E on f must be known if one wishes to constrain the
NL
overdensity(SO)method[55]. Inthiscase,weassumean latter. In the following sections, we will examine this
overdensity threshold ∆ (z) being a decreasing function dependence in our series of N-body simulations.
c
of redshift, as dictated by the solution to the spherical
collapse of a tophat perturbation in a ΛCDM Universe
[56]. In both cases, we require a minimum of 20 parti- 1. Effective bias
clesperhalo,whichcorrespondstoaminimumhalomass
Mmin 5.9 1012h−1M⊙ forthe simulations with 10243 In the top left panel of Fig. 2, the effective bias b in
≃ ×
particles. ForGaussianinitialconditionstheresultingto- the fiducial Gaussian case (f = 0) is shown for 30
NL
talnumberdensityofhalosisn¯ 7.0 10−4h3Mpc−3and consecutive FOF halo mass bins as a function of wave
4.2 10−4h3Mpc−3 fortheFOF≃and×SOcatalogs,respec- number. In the large-scale limit k 0, the measure-
× →
tively. Note that the FOF mass estimate is on average mentsareconsistentwithbeingscale-independent,asin-
20% higher than the SO mass estimate. For our 15363- dicated by the dotted lines which show the average of
apnadrtin¯cles4s.i0mul1a0t−io3nh3wMepocb−t3airnesMulmtiinng≃fr9o.m4×th1e0F11OhF−1hMal⊙o bd(ekn,oftNedL b= .0)Aovtelraragllermwoadvees wnuitmhbkers≤, t0h.e03d2ehvMiaptcio−n1s,
≃ × G
finder. can be attributed to higher-order bias terms, which are
ThebinningofthehalodensityfieldintoN consecutive most important at high mass. Relative to the low-k av-
mass bins is done by sortingall halosby increasingmass eraged, scale-independent Gaussian bias b , these cor-
G
anddividingthisorderedarrayintoN binswithanequal rections tend to suppress the effective bias at low mass,
number of halos. The halos of each bin i [1...N] are whereas they increase it at the very high-mass end (see
∈
selectedseparatelytoconstructthehalodensityfieldδ . Eq.(17)). TherightpanelofFig.2showsthelarge-scale
hi
Thedensityfieldsofdarkmatterandhalosarefirstcom- average b as a function of halo mass, as determined
G
7
3.5 3.0
3.0
0) 2.5
b(f=NL 12..50 2.5
1.0
G 2.0
b
0.015
∂/fNL 0.010 1.5
b
∂ 0.005
0.000
1.0
0.01 0.10 1013 1014
k [hMpc-1] M [h-1M ]
O •
FIG.2. LEFT: Gaussian effectivebias(top) and itsderivativewith respect tofNL (bottom) forthecase of 30mass bins. The
scale-independentpartbG isplottedindottedlinesforeachbin;itwasobtainedbyaveragingallmodeswithk≤0.032hMpc−1.
RIGHT: Large-scale averaged Gaussian effective bias bG from the left panel (dotted lines) plotted against mean halo mass.
Thesolidlinedepictsthelinear-orderbiasderivedfromthepeak-backgroundsplitformalism. Allerrorbarsareobtainedfrom
thevariance of our 12 boxes to their mean. Results are shown for FOFhalos at z=0.
from 30 halo mass bins, each with a number density of the rightpanelshowsthe twoimportanteigenvectorsV
+
n¯ 2.3 10−5h3Mpc−3. Thesolidlineisthelinear-order andV (top)alongwiththeirderivatives(bottom). The
−
bia≃s as×derived from the peak-background split formal- eigenstructure of E is accurately described by Eq. (13),
ism[40,41]. WefindagoodagreementwithourN-body even in the non-Gaussian case. Namely, we still find one
data, only at masses below 8 1012h−1M deviations enhanced eigenvalue λ and one suppressed eigenvalue
⊙ +
∼ ×
appear for halos with less than 30 particles [58]. λ . The remaining N 2 eigenvalues λ(N−2) are degen-
∼ − − P
The bottom left panel of Fig. 2 depicts the derivative erate with the value 1/n¯, the Poisson noise expectation.
of b with respect to fNL for each of the 30 mass bins. ThismeansthatourGaussianbias-expansionmodelfrom
The behavior is well described by the linear theory pre- Eq. (16) still works to describe E in the weakly non-
diction of Eq. (3), leading to a k−2-dependence on large Gaussian regime.
scales which is more pronounced for more massive ha- Notehoweverthat,owingtosamplingvariance,thede-
los (for quantitative comparisons with simulations, see composition into eigenmodes becomes increasingly noisy
[31, 32, 59]). Thus, the amplitude of this effect grad- towardslargerscales. This leadsto anartificialbreaking
ually diminishes towards smaller scales and even disap- of the eigenvalue degeneracy which manifests itself as a
pears around k 0.1hMpc−1. Note that [31] argued for scatter around the mean value 1/n¯. This scatter is the
∼
anadditionalnon-Gaussianbiascorrectionwhichfollows major contribution of sampling variance in the halo co-
from the fNL-dependence of the mass function. This k- variance matrix Ch. Although we can eliminate most of
independentcontributionshouldinprinciplebe included itbysettingλ(N−2) 1/n¯,aresidualdegreeofsampling
inEq.(3). However,ascanbeseeninthelowerleftplot, P ≡
variancewillremaininλ andλ ,aswellasinbandP.
+ −
it is negligible in our approach (i.e., all curves approach
As is apparentfrom the left panel in Fig. 3, the domi-
zeroathighk)owingtothematchingofhaloabundances
nant eigenvalue λ exhibits a small, but noticeable f -
between our Gaussian and non-Gaussian realizations. + NL
dependencesimilartothatof δ2 inFig.1,whichisabout
E
2% in this case. Its derivative, ∂λ /∂f , clearly domi-
+ NL
natesthederivativeofallothereigenvalues(whichareall
2. Shot noise matrix consistent with zero due to matched abundances). Only
the derivative of the suppressed eigenvalue λ shows a
−
The shot noise matrix E has been studied using simu- similar fNL-dependence of 2%, albeit at a much lower
∼
lations with Gaussian initial conditions in [14]. Figure 3 absolute amplitude. To check the convergence of our re-
displays the eigenstructure of this matrix for f = 0 sults, we repeated the analysis with 100 and 200 bins
NL
(solid curves) and fNL = 100 (dashed and dotted and found both derivatives of λ+ and λ− to increase,
±
curves). The left panel depicts all the eigenvalues (top) supporting an fNL-dependence of these eigenvalues.
andtheirderivativeswithrespecttof (bottom),while By contrast, the eigenvectors V and V shown in
NL + −
8
1.0
1•105
3c] 8•104 0.5
p
M 6•104 ±
3 V
λh- [ 4•104 0.0
2•104
0 -0.5
3c] 30 5•10-5
p 20
h-3M 10 ∂/fNL 0
∂/f [NL -100 ∂V± -5•10-5
λ
∂ -20 -1•10-4
0.01 0.10 1013 1014
k [hMpc-1] M [h-1M ]
O •
FIG.3. Eigenvalues(leftpanel)andeigenvectors(rightpanel)oftheshotnoisematrixE forfNL =0(solid),+100(dashed)and
−100 (dotted) in thecase of 30 mass bins. Their derivatives with respect to fNL are plotted underneath. For clarity, only the
two eigenvectors V± along with their derivativesare shown in theright panel. The straight dotted line in the upperleft panel
depictsthevalue1/n¯ andthered (dot-dashed)curveinthetoprightpanelshows b2(M)computedfrom thepeak-background
split formalism, scaled to the valueof V+ at M ≃3×1013h−1M⊙. Resultsare shown for FOFhalos at z=0.
the right panel of Fig. 3 exhibit very little dependence ing weighted field will show the same f -dependence.
NL
on f (the different lines are all on top of each other). However,thisf -dependencecannotimmediatelybeex-
NL NL
ThederivativesofV andV withrespecttof shown ploitedto constrainprimordialnon-Gaussianity,because
+ − NL
in the lower panel reveal a very weak sensitivity to fNL the Fourier modes of δ2 are heavily correlated due to
E
which is less than 0.5% for most of the mass bins (for the convolution of δ with itself in Eq. (21), and thus do
the most massive bin it reaches up to 1%). We repeated not contribute to the Fisher information independently.
the same analysiswith 100and200mass bins andfound Thebottomlineisthatforincreasinglymassivehalobins
thattherelativedifferencesbetweenthemeasurementsin withlargeb2,theterm δ2 makesanimportantcontribu-
E
Gaussianandnon-Gaussiansimulationsfurtherdecrease. tion to the halo power spectrum and shows a significant
We thus conclude that the eigenvectors V and V can dependence on f . It is important to take into account
+ − NL
be assumed independent of f to a very high accuracy. this dependence when attempting to extract the best-fit
NL
value of f from high-mass clusters, so as to avoid a
Our findings demonstrate that the two-point statis- NL
possible measurement bias. Although it provides some
tics of halos are sensitive to primordial non-Gaussianity
additional information on f , we will ignore it in the
beyond the linear-order effect of Eq. (3) derived in NL
followingandquoteonlylowerlimitsontheFisherinfor-
[8, 21, 22]. However, the corrections are tiny if one con-
mation content.
sidersasinglebincontainingmanyhalosofverydifferent
mass(see[37])duetomutualcancellationsfromb -terms
2
ofoppositesign. Onlytwospecificeigenmodesoftheshot
noisematrix(correspondingtotwodifferentweightingsof B. Constraints from Halos and Dark Matter
thehalodensityfield)inheritasignificantdependenceon
f . Thisismostprominentlythecasefortheeigenmode
NL Letusfirstassumetheunderlyingdarkmatterdensity
correspondingtothehighesteigenvalueλ . Itseigenvec-
+ field δ is available in addition to the galaxy distribution.
tor,V ,isshowntobecloselyrelatedtothesecond-order
+ Although this can in principle be achieved with weak-
bias b in Eq. (19). As can be seen in the upper right
2 lensing surveys using tomography, the spatial resolution
panel of Fig. 3, V measured from the simulations, and
+ will not be comparable to that of galaxy surveys. To
thefunctionb (M)calculatedfromthepeak-background
2 mimic the observed galaxy distribution we will assume
split formalism [40, 41], agree closely (note that b (M)
2 that each dark matter halo (identified in the numerical
has been rescaled to match the normalized vector V ).
+ simulations) hosts exactly one galaxy. A further refine-
In the continuous limit this implies that weighting the ment in the description of galaxies can be accomplished
halodensity field withb (M)selects the eigenmode with with the specification of a halo occupation distribution
2
eigenvalueλ giveninEq.(20). Sinceλ dependsonf for galaxies [15, 60], but we will not pursue this here.
+ + NL
through the quantity δ2 defined in Eq. (21), the result- Instead, we can think of the halo catalogs as a sample
E
9
of central halo galaxies from which satellites have been 104.0 8.5. In fact, the closest match to the input
− ±
removed. We also neglect the effects of baryons on the f -values is obtained for a slightly larger q of 0.8.
NL
≃
evolution of structure formation, which are shown to be Notethat[59]attributedthissuppressiontoellipsoidal
marginally influenced by primordial non-Gaussianity at collapse. However, this conclusion seems rather unlikely
late times [61]. since ellipsoidal collapse increases the collapse threshold
or, equivalently, implies q > 1 [62]. A more sensible ex-
planationarisesfromthe factthatalinkinglengthof0.2
1. Single tracer: uniform weighting times the mean interparticle distance can select regions
with an overdensity as low as ∆ 1/0.23 = 125 (with
∼
respect to the mean background density ρ¯ ), which is
In the simplest scenario we only consider one sin- m
much less than the virial overdensity ∆ (z = 0) 340
gle halo mass bin. In this case, all the observed halos c
≃
associated with a linear overdensity δ (see [56, 63, 64]).
(galaxies) of a survey are correlated with the underly- c
Therefore, we may reasonably expect that, on average,
ing dark matter density field in Fourier space to deter-
FOF halos with this linking length trace linear overden-
minetheirscale-dependenteffectivebias,whichcanthen
sities of height less than δ .
be compared to theoretical predictions. In practice, this c
InthecaseofSOhalos,however,weobservetheoppo-
translatesintofitting ourtheoreticalmodelforthescale-
sitetrend. AsisapparentintherightpanelofFig.4,the
dependenteffectivebias,Eq.(3),totheFouriermodesof
model from Eq. (3) overestimates the amplitude of pri-
the density fields and extracting the best fitting value of
mordial non-Gaussianity by roughly 40%. This is some-
f togetherwithits uncertainty. Fora singlehalomass
NL
whatsurprisingsincetheoverdensitythreshold∆ 340
bin,wecanemployEq.(36)andsumoveralltheFourier c
≃
used to identify the SO halos at z = 0 is precisely the
modes.
virialoverdensitypredictedby the sphericalcollapseofa
In the Gaussian simulations, we measure the scale-
independenteffectivebiasb viatheestimator δ δ / δ2 linear perturbation of height δc. As we will see shortly,
G h
and the shot noise via (δ b δ)2 , and avehragie ohveir however, an optimal weighting of halos can remove this
all modes with k E0.03h2hMh−pc−G1. Iin practice, b and overshoot and therefore noticeably improve the agree-
≤ G ment between model and simulations.
are not directly observable, but a theoretical predic-
E
tion basedonthe peak-backgroundsplit [40, 41] and the
halo model [14] provides a reasonable approximation to
2. Single tracer: optimal weighting
the measured b and , respectively, (see Sec. V). Note
G
E
that for bins covering a wide range of halo masses, the
As demonstrated in [14], the shot noise matrix E ex-
f -dependence of the shotnoise is negligible [37] and it
NL
hibitsnonzerooff-diagonalelementsfromcorrelationsbe-
is well approximated by its Gaussian expectation.
tween halos of different mass. Thus, in order to extract
Figure 4 shows the best fits of Eq. (3) to the simu-
the full information on halo statistics, it is necessary to
lations with f = 0, 100 using all the halos of our
NL
± includethesecorrelationsintoouranalysis. Forthispur-
FOF(left panel) andSO catalogs(rightpanel). Inorder
pose, we must employ the more general chi-square of
to highlight the relative influence of f on the effec-
NL
Eq. (35). The halo density field is split up into N con-
tive bias, we normalize the measurements by the large-
secutive mass bins in order to construct the vector δ ,
scale Gaussian average b and subtract unity. The re- h
G and the full shot noise matrix E must be considered.
sulting best-fit values of f along with their one-sigma
NL
However, this approach can be simplified, since we
errorsarequotedinthelowerrightforeachcaseofinitial
knowthatE exhibitsoneparticularlyloweigenvalueλ .
conditions. The 68%-confidence region is determined by −
Because the Fisher information content on f from
the condition ∆χ2(f ) = 1. Note that we include only NL
FouriermodesuptokNL 0.032hMpc−1inthefit,aslinear Eq. (39) is proportional to the inverse of E (this is true
≃ at least for the dominant part γ), it is governed by the
theory begins to break down at higher wave numbers.
eigenmodecorrespondingtothiseigenvalue. In[14]ithas
Obviously, the best-fit values for f measured from
NL beenshownthatthiseigenmodedominatestheclustering
the FOF halo catalogs are about 20% below the input
signal-to-noise ratio. In the continuous limit (infinitely
values. A suppression of the non-Gaussian correction to
many bins), it can be projected out by performing an
the bias of FOF halos has already been reported by [32,
appropriate weighting of the halo density field. The cor-
59]. Theseauthorsshowedthatthereplacementδ qδ
c → c respondingweightingfunction,denotedasmodified mass
with q = 0.75 in Eq. (3) yields a good agreement with
weighting with functional form
their simulation data. In our framework, including this
“q-factor” is equivalent to exchanging f f /q and
NL NL w(M)=M +M , (42)
→ 0
σ σ /q, owing to the linear scaling of Eq. (3)
fNL → fNL
withδ . Repeatingthechi-squareminimizationwithq = was found to minimize the stochasticity of halos with
c
0.75 yields best-fit values that are consistent with our respect to the dark matter. Here, M is the individual
inputvalues,namely f =+107.0 8.3,+1.8 8.7and halo mass and M a constant whose value depends on
NL 0
± ±
10
uniform FOF-halos uniform SO-halos
0.5 b = 1.31, ε = 1939.9 h-3Mpc3 0.5 b = 1.32, ε = 2871.5 h-3Mpc3
G G
1 1
- -
G G
b b
)/L 0.0 )/L 0.0
N N
k,f k,f
b( b(
f = +79.9 ± 6.6 f = +137.6 ± 7.8
NL NL
-0.5 f = +3.1 ± 6.3 -0.5 f = +2.1 ± 7.0
NL NL
f = -78.2 ± 6.3 f = -143.8 ± 7.0
NL NL
0.01 0.10 0.01 0.10
k [hMpc-1] k [hMpc-1]
FIG. 4. Relative scale dependence of the effective bias from all FOF (left panel) and SO halos (right panel) resolved in our
N-body simulations (Mmin ≃5.9×1012h−1M⊙), which are seeded with non-Gaussian initial conditions of the local type with
fNL = +100,0,−100 (solid lines and data points from top to bottom). The solid lines show the best fit to the linear theory
modelofEq.(3),takingintoaccountallthemodestotheleftofthearrow. Thecorrespondingbest-fitvaluesarequotedinthe
bottom right of each panel. The dotted lines show the model evaluated at the input values fNL =+100,0,−100. The results
assume knowledge of thedark matter density field and an effectivevolume of Veff ≃50h−3Gpc3 at z =0.
weighted FOF-halos weighted SO-halos
0.5 b = 1.73, ε = 307.2 h-3Mpc3 0.5 b = 1.83, ε = 863.9 h-3Mpc3
G G
1 1
- -
G G
b b
)/L 0.0 )/L 0.0
N N
k,f k,f
b( b(
f = +80.0 ± 1.0 f = +100.0 ± 1.5
NL NL
-0.5 f = -1.3 ± 1.1 -0.5 f = -1.6 ± 1.6
NL NL
f = -85.4 ± 1.1 f = -108.1 ± 1.6
NL NL
0.01 0.10 0.01 0.10
k [hMpc-1] k [hMpc-1]
FIG. 5. Same as Fig. 4, but for weighted halos that have minimum stochasticity relative to the dark matter. Note that the
one-sigmaerrorsonfNL arereducedbyafactorof∼5comparedtouniformweighting. InthecaseofSOhalostheinputvalues
for fNL are well recovered by thebest-fit,while FOFhalos still show a suppression of ∼20% (q≃0.8) in thebest-fit fNL.
the resolution of the simulation. It is approximately 3 eigenmode, it simplifies to the form of Eq. (36) with the
times the minimum resolved halo mass M , so in this halofieldδ beingreplacedbytheweightedhalofieldδ .
min h w
case M 1.8 1013h−1M . The weighted halo density Note also that b and have to be replaced by the cor-
0 ⊙ G
≃ × E
field is computed as responding weighted quantities (see [14]).
w(M )δ w⊺δ The results are shown in Fig. 5 for both FOF and SO
δw = Pi iw(Mi i)hi ≡ w⊺11h , (43) honalofs. Wbye aobfsaecrtvoeraofrema4rkab6le(dreepdeuncdtiionng ionntthheeehrraolor
NL
∼ −
where we have combPined the weights of the individual finder)whenreplacingthe uniformsampleusedinFig.4
mass bins into a vector w in the last expression. Be- by the optimally weighted one. While for the FOF halos
causethechi-squareinEq.(35)isdominatedbyonlyone the predicted amplitude of the non-Gaussian correction