Evidence for Quadratic Tidal Tensor Bias from the Halo Bispectrum

0.91 MB·English

by  Tobias Baldauf

Evidence for Quadratic Tidal Tensor Bias from the Halo Bispectrum
Tobias Baldauf,1,∗ Uroˇs Seljak,1,2,3,4 Vincent Desjacques,5 and Patrick McDonald2,6
1Institute for Theoretical Physics, University of Zurich, Zurich, Switzerland
2Lawrence Berkeley National Laboratory, Berkeley, CA, USA
3Physics Dept. and Astronomy Dept., University of California, Berkeley, CA, USA
4Institute for the Early Universe, EWHA Womans University, Seoul, South Korea
5Department of Theoretical Physics and Center for Astroparticle Physics, University of Geneva, Geneva, Switzerland
6Physics Dept., Brookhaven National Laboratory, Upton, NY, USA
(Dated: January 24, 2012)
Therelationbetweentheclusteringpropertiesofluminousmatterintheformofgalaxiesandthe
underlying dark matter distribution is offundamental importance for the interpretation ofongoing
and upcoming galaxy surveys. The so called local bias model, where galaxy density is a function
of local matter density, is frequently discussed as a means to infer the matter power spectrum
or correlation function from the measured galaxy correlation. However, gravitational evolution
generatesatermquadraticinthetidaltensorandthusnon-localintheEuleriandensityfield,even
if this term is absent in the initial conditions (Lagrangian space). Because the term is quadratic,
it contributes as a loop correction to the power spectrum, so the standard linear bias picture still
2
appliesonverylargescales,however,itcontributesatleadingordertothebispectrumforwhichitis
1
significantonallscales. SuchatermcouldalsobepresentinLagrangianspaceifhaloformationwere
0
influenced by the tidal field. We measure the corresponding coupling strengths from the matter-
2
matter-halo bispectrum in numerical simulations and find a non-vanishing coefficient for the tidal
n tensor term. We find no scale dependence of the inferred bias parameters up to k ∼ 0.1 hMpc−1
a
and that the tidal effect is increasing with halo mass. While the local Lagrangian bias picture is a
J
better description of our results than the local Eulerian bias picture, our results suggest that there
3 mightbeatidaltensorbiasalreadyintheinitialconditions. Wealsofindthatthecoefficientsofthe
2
quadraticdensitytermdeviatequitestronglyfromthetheoreticalpredictionsbasedonthespherical

collapse model and a universal mass function. Both quadratic density and tidal tensor bias terms
]
O mustbeincludedinthemodelingofgalaxyclusteringofcurrentandfuturesurveysifonewantsto
achieve the high precision cosmology promise of these datasets.
C
.
h
p
-
o I. INTRODUCTION
r
t
s LargeScaleStructure(LSS),thelarge-scaledistributionofmatterintheUniverse,containsawealthofinformation
a
aboutthehistoryandcompositionoftheUniverseaswellasfundamentalphysics. Forinstance,LSShasthepotential
[
to constrain neutrino masses or modifications of gravity, which however requires percent level accuracy for the theory

1 and observations. Besides gravitational lensing, which is sensitive to the total matter distribution, the positions of
v
galaxies are the main observable and tool to infer the underlying matter distribution. They have the advantage of
7
higherstatisticalpowerrelativetoweaklensingsurveys. OngoingandupcomingLSSsurveyssuchasBOSS,BigBOSS,
2
8 EUCLID, DES will provide an unprecedented quality of galaxy clustering data, which needs to be properly analyzed.
4 A crucial step in the analysis of galaxy surveys is to connect the distribution of the tracer to the underlying
. distribution of matter. The first step in this logical chain is the realization that galaxies form preferentially in the
1
potential wells of collapsed dark matter halos, where the hot gas can cool sufficiently fast [1, 2]. This leads to the
0
2 questionofhowtheclusteringpropertiesofdarkmatterhalosrelatetotheclusteringofmatteringeneral. Theanswer
1 to this question is usually phrased in terms of a relation between the overdensities in these two fields and is dubbed
: a halo biasing scheme. With the ever increasing computing power it is in principle possible to generate templates for
v
the survey analysis for standard ΛCDM and even modified gravity models using N-body simulations. This approach
i
X becomesveryexpensivewhenitistobeusedinMarkov-Chain-Monte-Carloparameterinferencemethodsanddoesnot
r provide insight into the underlying clustering properties. We thus consider it important to understand the properties
a
ofhaloclusteringbytestingtheoreticalprescriptionsonsimulationswiththefinalgoalofdevisinganalytical andthus
easily evaluable models for the survey analysis.
The so called local biasing model [3, 4], where galaxy and halo density is a function of local matter density, has
been the most popular model used in previous work. In the simplest version one adds another contribution that
∗Electronicaddress: [email protected]
2
scales quadratically with density, the quadratic density bias term. Recent work has argued, based on symmetry and
analyticity arguments, that there are additional terms not included in the local bias model that appear in the power
spectrum and are formally at the same order as that of quadratic bias [5]. One of these terms is quadratic in density
butnon-localandcanbewrittenasthesquareofthetidaltensor. Thefirstgoalofthispaperistoprovideadditional
theoretical motivation for inclusion of this term in the analysis of galaxy clustering.
The working horses in LSS analysis are the two-point functions, the correlation function and the power spectrum.
Purely Gaussian, linear fields are completely characterized by their two point function. However, non-linear phe-
nomena in galaxy and halo formation as well as non-linear gravitational clustering can generate the full hierarchy of
n-point functions. These higher order statistics might be difficult to measure in the sky due to non-trivial survey
windows and redshift space distortions, but they can be easily extracted from N-body simulations. The simplest
statistic beyond the power spectrum is the bispectrum
δ(k )δ(k )δ(k ) =B(k ,k ,k ) (2π)3δ(D)(k +k +k ), (1)
1 2 3 1 2 3 1 2 3
(cid:104) (cid:105)
whichiswellsuitedforthestudyofnext-to-leadingordereffectsincosmicdensityfields[6–10]. Whilethesecontribute
onlylooptermstothepowerspectrumtheyaretheleadingordertermsforthebispectrum, whichvanishesforpurely
linear Gaussian fields.
The second aim of this study is to probe the halo bispectrum for Gaussian initial conditions in the low-k regime, in
order to extract the two terms that lead to a quadratic coupling between the number density of collapsed objects and
the long wavelength matter fluctuations, quadratic density bias and quadratic tidal tensor bias. We present a study
oftheirscaleandmassdependenceusingN-bodysimulationsandwecomparethenumericalresultstothetheoretical
expectations based on simple halo bias models.
Thispaperisorganizedasfollows. In IIwereviewthestandardformulationofthebiasmodelanddiscusspossible
§
extensions. III describes the simulations, the bispectrum measurement and data reduction as well as the parameter
§
estimation. The results are presented in IV. Finally, in V we discuss our findings and their implications as well as
§ §
possible directions for future investigation.
II. THE BIAS MODEL: LOCAL AND NON-LOCAL FORMS
A. Standard Formulation: Local Bias
The formation of galaxies and their host dark matter halos is a complicated highly non-perturbative process. It
is, however, reasonable to assume that certain properties of collapsed objects, for instance their number density, are
related to the coarse grained underlying matter density field in the same region of space. Neglecting complications
arising from gas physics, the number density of collapsed objects can be written as a functional of the underlying
matter density perturbation [4]
δ (x,η)= [δ(x(cid:48),η)]. (2)
h
F
Onlargescales,thisfunctionaliscommonlyapproximatedbyalocal andlinear biasmodelδ (x,η)=b δ(x,η)(plus
h 1
generallyanoisetermwhichwewillavoidinthispaperbyonlylookingatcross-correlationswithmass). Thenextstep
is to give up on linearity and to introduce the second order local bias model δ (x,η)= b δ(x,η)+b δ2(x,η). This
h 1 2
model has been well studied in the literature in combination with Standard Perturbation Theory (SPT) and leads to
non-trivial renormalizations of the leading order bias parameter [11]. Measurements of the quadratic bias parameters
of this model in the bispectrum have lead to contradictory results, which raises doubts about the completeness of the
model [12] (see also [13]).
B. Tidal Terms
The power series expansion of the functional presented above is certainly overly simplified and one should consider
whether other terms could influence the number density of collapsed objects. As proposed in [5], the environmental
dependence of halo formation could lead to a dependence on the tidal field, as quantified by the tidal tensor
1
s (x,η)=∂ ∂ Φ(x,η) δ(K)δ(x,η). (3)
ij i j − 3 ij
Note that we absorbed the constants in the Poisson equation into the gravitational potential ∇2Φ(x,η) = δ(x,η)
and subtract out the trace from the tidal tensor because it is degenerate with the density field. The corresponding
3
expression in Fourier space is given by
k k 1
s (k,η)= i j δ(K) δ(k,η). (4)
ij k2 − 3 ij
(cid:18) (cid:19)
The halo overdensity is a scalar quantity and can thus only depend on scalars. The simplest scalar that can be
constructedfromthetidaltensorisgivenbys2(x)=s (x)s (x)whichinFourierspaceisexpressedbytheconvolution
ij ij
d3k(cid:48)
s2(k,η)= S (k(cid:48),k k(cid:48))δ(k(cid:48),η)δ(k k(cid:48),η) , (5)
(2π)3 2 − −
(cid:90)
where we have implicitly defined the kernel
(q q )2 1
S (q ,q )= 1· 2 . (6)
2 1 2 q2q2 − 3
1 2
Following [5], the halo density field up to second order can be written as
δ (x,η)=b δ(x,η)+b δ2(x,η) δ2(x,η) +b s2(x,η) s2(x,η) , (7)
h 1 2 s2
− −
where we absorbed prefactors of 1/2 into the(cid:2)bias param(cid:10)eters. W(cid:11)e(cid:3)trunca(cid:2)ted the ser(cid:10)ies at sec(cid:11)o(cid:3)nd order, since higher
order terms influence the bispectrum only through loop corrections, which are believed to be subdominant on large
scales. As can be easily verified from the above definitions one has s2(x,η) =2/3 (1)δ2(x,η) .
(cid:10) (cid:11) (cid:10) (cid:11)
C. Lagrangian Bias
IntheusualLagrangianbiaspicture, thegalaxyformationsitesareidentifiedintheprimordialdensityfield, andit
is assumed that the primordial halo density field at initial time η can be written as a power series in the primordial
i
matterfluctuations. Forcalculationalconvenience,theexpansioncanberewrittenintermsofthelinearlyextrapolated
density field δ(q,η)=D(η)/D(η )δ(q)
i
(L) (L)
b (η) b (η)
δ (q)= l i δl(q) δl(q) = l δl(q,η) δl(q,η) , (8)
h
l! − l! −
l l
(cid:88) (cid:2) (cid:10) (cid:11)(cid:3) (cid:88) (cid:2) (cid:10) (cid:11)(cid:3)
where b(L)(η) = D(η )/D(η) lb(L)(η ) and q is the Lagrangian position. Here, we introduced the conformal time
l i l i
adη = dt and the linear growth factor D(η), normalized to unity at present time. We start the sum from l = 1
(cid:0) (cid:1)
becausethel=0termvanishesbyrequiringthatthehalofieldhasavanishingmean. Incontrasttothis,theEulerian
bias model expands the halo density field at a certain point in time in the non-linear matter density field at the same
time. Motivation for a Lagrangian nature of halo bias comes from the peak model [14–17], where the peaks of the
primordial density field are associated with the formation sites of protohaloes.
It now remains to connect the Lagrangian density fields to the observable Eulerian ones. The continuity equation
for halos requires
[1+δ (x,η)] d3x=[1+δ (q)] d3q, (9)
h h
where
[1+δ(x,η)] d3x= d3q (10)
is the continuity equation for the underlying dark matter field. Note that the Lagrangian density field we work with
is always the linear density field, because this was the definition of the bias expansion in Eq. (8) – more general
expansions are possible in principle.
The Lagrangian and Eulerian positions are related by x(q,η)=q+Ψ(q,η), thus up to third order in the density
field we have
1
δ(q)=δ(x,η) Ψ (q,η)∂ δ(q)+ Ψ (q,η)Ψ (q,η)∂ ∂ δ(q). (11)
i i i j i j
− 2
4
In contrast to Eulerian perturbation theory, where the density is the central quantity, Lagrangian Perturbation
Theory (LPT) has the displacement field Ψ as the central dynamic quantity and the density fields are only derived
quantities(forthebasicsofLPTanditsrelationtoSPTseeAppendixA).Forsimplicity,wewillfocusonthematter-
only Einstein-de Sitter Universe for the theoretical calculations throughout this paper. Using the Lagrangian bias
expansion (8) in (9) and expressing the Lagrangian position field in terms of the Eulerian position, we have (see also
[18])
δ (x,η)= 1+b(L)(η) (1)δ(x,η)
h 1
(cid:16) (cid:17) 1
+b(L)(η)δ2(x,η)+ b(L)(η) (1)δ2(x,η) (1)δ2(x,η) b(L)(η)Ψ(q,η) ∇δ(x,η)+(2)δ(x,η). (12)
1 2 2 − − 1 ·
(cid:104) (cid:10) (cid:11)(cid:105)
Reorganizing the terms in order to have the first order bias multiply the full second order matter density field, we
obtain
δ (x,η)= 1+b(L)(η) (1)δ(x,η)+(2)δ(x,η)
h 1
(cid:16) 4 1(cid:17)(cid:16) (cid:17) 2
+ b(L)+ b(L)(η) (1)δ2(x,η) (1)δ2(x,η) b(L)(η) s2(x,η) s2(x,η) , (13)
21 1 2 2 − − 7 1 −
(cid:18) (cid:19)(cid:104) (cid:10) (cid:11)(cid:105) (cid:104) (cid:10) (cid:11)(cid:105)
where we used that the second order mass density (in SPT and LPT) can be written in configuration space as (see
e.g. [19])
17 2
(2)δ(x,η)= (1)δ2(x,η) Ψ(x,η) ∇δ(x,η)+ s2(x,η). (14)
21 − · 7
We see that the functional form of the above result agrees with Eq. (7), but complements it with a dynamical
perspective. In particular, we see that even in the absence of a tidal tensor bias in the initial conditions, such a term
will be generated by subsequent gravitational evolution. While our result in Eq. (13) is consistent with the relations
presented in [18], we have reorganized terms in order to have the first order bias multiplying the second order density
field Eq. (14) and to make the formal equivalence with the phenomenological picture presented above more obvious.
Furthermore, there are no b(L) factors, because we fixed the original Taylor series for δ (q) to have vanishing mean.
0 h
D. Coevolution of Halos and Dark Matter
In this subsection we will consider an (at first glance) Eulerian approach to halo clustering and consider the
coevolutionofthecoupledhalo-darkmatterfluid(Forasimilarapproachincombinationwithresummedperturbation
theory see [20]). Assuming vanishing velocity bias v = v and conservation of halo number, we can write down a
h
coupled system of differential equations for the matter and halo fluid [38], namely the continuity equation for halos,
the continuity equation for the matter and the combined Euler and Poisson equations for matter
d3k(cid:48)
δ(cid:48)(k,η)+θ(k,η)= α(k(cid:48),k k(cid:48))θ(k(cid:48),η)δ (k k(cid:48),η) (15)
h − (2π)3 − h −
(cid:90)
d3k(cid:48)
δ(cid:48)(k,η)+θ(k,η)= α(k(cid:48),k k(cid:48))θ(k(cid:48),η)δ(k k(cid:48),η) (16)
− (2π)3 − −
(cid:90)
3 d3k(cid:48)
θ(cid:48)(k,η)+ (η)θ(k,η)+ Ω (η) 2(η)δ(k,η)= β(k(cid:48),k k(cid:48))θ(k(cid:48),η)θ(k k(cid:48),η) (17)
H 2 m H − (2π)3 − −
(cid:90)
Here we introduced the velocity divergence θ(x) = ∇ v(x). The matter equations at first order are solved by
(1)θ(k,η) = (η)(1)δ(k,η) and (1)δ(k,η) = D(η)(1)δ (k·), where (1)δ (k) is the present day linear overdenisty. The
0 0
−H
second order solution for the matter yields
d3k(cid:48)
(2)δ(k,η)= F (k(cid:48),k k(cid:48))(1)δ(k(cid:48),η)(1)δ(k k(cid:48),η) (18)
(2π)3 2 − −
(cid:90)
d3k(cid:48)
(2)θ(k,η)= (η) G (k(cid:48),k k(cid:48))(1)δ(k(cid:48),η)(1)δ(k k(cid:48),η) (19)
−H (2π)3 2 − −
(cid:90)
5
where the second order SPT mode coupling kernels are defined as [21]
5 2
F (q ,q )= α(q ,q )+ β(q ,q ) (20)
2 1 2 1 2 1 2
7 7
3 4
G (q ,q )= α(q ,q )+ β(q ,q ) (21)
2 1 2 1 2 1 2
7 7
with
(q +q ) q 1 q q
α(q ,q )= 1 2 · 1 β(q ,q )= (q +q )2 1· 2 (22)
1 2 q2 1 2 2 1 2 q2q2
1 1 2
The first order equation for the halos can be solved using the local bias ansatz (1)δ (k,η)=b(E)(η)(1)δ(k,η), which
h 1
then gives the time evolution of the first order bias as
(E)
b (η) 1 D(η)
1 − = i . (23)
b(E)(η) 1 D(η)
1 i −
This relation known as debiasing, i.e., at late times the bias converges to unity and halo and matter density field
agree [22].
Solving Eq. (15) at second order using the second order matter solutions, we obtain
d3k(cid:48)
(2)δ (k,η)=(2)δ (k,η)+b(E)(η) F (k(cid:48),k k(cid:48))δ(k(cid:48),η)δ(k k(cid:48),η)
h h i 1 (2π)3 2 − −
(cid:90)
4 d3k(cid:48)
+ b(E)(η) 1 δ(k(cid:48),η)δ(k k(cid:48),η) (24)
21 1 − (2π)3 −
(cid:16) (cid:17)(cid:90)
2 d3k(cid:48)
b(E)(η) 1 S (k(cid:48),k k(cid:48))δ(k(cid:48),η)δ(k k(cid:48),η),
− 7 1 − (2π)3 2 − −
(cid:16) (cid:17)(cid:90)
where we assumed D(η ) D(η). Here, we have isolated the part proportional to the second order matter field in
i
the first line. Thus, the d(cid:28)ynamical evolution naturally introduces a δ2(x) and a s2(x) term, even in the absence
thereof at some initial time η. Translating back to real space, we see that this has the same functional form as
i
Eq. (7) and agrees with the Lagrangian bias picture. The equivalence is even more obvious if we have (2)δ (x,η) =
h i
b(L)(η)(1)δ2(x,η)/2=b(L)(η)(1)δ2(x,η)/2incorrespondencetotheLagrangianbiaspicturediscussedabove(seenext
2 i i 2
subsection for a relation between the parameters of the models discussed here). We note in retrospect that, while the
calculation took an Eulerian form, the specification that galaxies formed at an early time tracing the initial density
field (or at least the location where they will form is determined this way), and then just follow gravity is actually
the same one made in the Lagrangian calculation.
E. Relation between Eulerian and Lagrangian bias parameters
As we have seen above, the Lagrangian bias model, the coevolution model and the educated guess of [5] all lead to
the same functional form for the halo density field if one identifies the parameters of the model as
(E) (L)
b =b (η)=1+b (η) (25)
1 1 1
1 4 1
(E) (L) (L)
b = b (η)= b (η)+ b (η) . (26)
2 2 2 21 1 2 2
This identification agrees with the one of the spherical collapse picture [23]. Note however that our results are not
relying on the spherical collapse dynamics. For the prefactor of the tidal field scalar we have
2 2
(E) (L)
b = b (η) 1 = b (η). (27)
s2 −7 1 − −7 1
(cid:16) (cid:17)
In this model there is no tidal field bias in Lagrangian space, consistent with the spherical collapse model, although
in the ellipsoidal collapse model [24–27] such a term would be allowed.
Whencomparingourmeasurementstotheoreticalbiasmodels,weconsiderthebiasderivedfromtheSheth-Tormen
[28] mass function with parameters p=0.15 and q =0.75 that were shown to be in good agreement with first order
6
bias in N-body simulations [29], although we checked that the predictions do not differ much from those using the
original values in [28]. The Lagrangian bias parameters are then given by the first and second derivative of the mass
function n(M) with respect to a long wavelength background fluctuation δ. For universal mass functions these can
l
be rewritten as derivatives with respect to the peak height ν(M,η)=δ2/σ2(M,η)
c
1 ∂n 12ν ∂n
(L)
b = = (28)
1 n¯∂δ −n¯ δ ∂ν
l c
1∂2n 4ν2∂2n 2 ν ∂n
(L)
b = = + , (29)
2 n¯ ∂δ2 n¯ δ2 ∂ν2 n¯δ2 ∂ν
l c c
whereδ =1.686isthecriticaldensityforsphericalcollapse. ThederivativesoftheSheth-Tormenmassfunctionread
c
1∂n qν 1 p
= − (30)
n∂ν − 2ν − ν(1+(qν)p)
1∂2n p2+νpq (qν)2 2qν 1
= + − − . (31)
n∂ν2 ν2(1+(qν)p) 4ν2
The mass dependence of the bias function is presented in Figure 4 and discussed in IV.
§
F. Bispectra
The leading order contribution to the bispectrum arises from quadratic terms in the fields. Higher order couplings
enteronlythroughloopcorrections,whichgainimportanceforhighk. Thisisequivalenttothesituationinthepower
spectrum, wherelineartermsinthefieldaredominantonlargescalesandloopcorrectionsfromquadraticandhigher
order terms gain importance for high k.
The tree-level matter bispectrum in SPT is given by
B (k ,k ,k )=2P(k )P(k )F (k ,k )+2cyc. , (32)
mmm 1 2 3 1 2 2 1 2
where cyc. symbolizes the two cyclic permutations of the k-vectors in the power spectrum and mode coupling kernel.
From an observational point of view the halo auto bispectrum B is probably the most appealing statistic. Unfor-
hhh
tunately it is suffering from shotnoise, which might deviate from its fiducial Poisson form 1/n¯ [30]. Besides the halo
auto power spectrum, there are two halo-matter cross bispectra B and B , where either one or two matter
mhh mmh
fields are correlated with two or one halo fields, respectively. One further needs to state whether the cross bispectra
are symmetrized over the k modes or not.
Our focus is not on observability but on understanding the clustering properties of dark matter halos in N-body
simulations and devising a theoretical framework that can later be used to analyze real data. Thus, we will consider
the un-symmetrized matter-matter-halo cross bispectrum defined as
B(unsym)(k ,k ,k ) (2π)3δ(D)(k +k +k )= δ(k )δ(k )δ (k ) , (33)
mmh 1 2 3 1 2 3 (cid:104) 1 2 h 3 (cid:105)
where the halo density field is always on the k -mode. This particular configuration might not be the one with the
3
highest signal to noise ratio but it has a couple of quite useful properties for our study: a) The cross-bispectrum
does not suffer from a spurious shotnoise contamination and is thus a clean probe of the clustering of halos b) the
functional simplicity of the second order bias contributions in terms of k , k and the enclosed angle µ where the b
1 2 2
and b contributions are basically orthogonal (see below). The latter should allow for a clear distinction between the
s2
standard second order bias picture or its possible extensions discussed above.
As noted above, all of the models under consideration share the same functional form (7) for the second order halo
density field. Only the standard quadratic bias model has b =0. Therefore, we can write down the bispectrum as
s2
1
B(unsym)(k ,k ,k ) b B (k ,k ,k )=2P(k )P(k ) b +b µ2 . (34)
mmh 1 2 3 − 1 mmm 1 2 3 1 2 2 s2 − 3
(cid:20) (cid:18) (cid:19)(cid:21)
Note that we use a parametrization where the factors of 1/2 are absorbed into the bias parameters for simplicity. We
will restore these prefactors only at the very end, when we are comparing our bias measurements to the theoretical
bias functions. A non-vanishing b in the above equation would be a clear evidence for a dynamical biasing picture.
s2
After dividing by the two matter power spectra, the remaining quantity is a function of the angle µ only, which
simplifies the combination of information from several scales k and k .
1 2
7
1010 1010
bin I bin II
3]− 3]−
pc pc
M 109 M 109
3h 3h
[ [
B B
108 108
1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0
− − µ − − µ
1010 1010
bin III bin IV Bm(tmhemo) Bm(tmheho)b1,b2,bs2
Bm(tmheho)b1 Bm(smimm)
Bm(tmheho)b1,b2 Bm(smimh)
3]− 3]−
pc pc
M 109 M 109
3h 3h
[ [
B B
108 108
1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0
− − µ − − µ
FIG. 1: Matter (black points) and halo-matter-matter (red points) bispectra as a function of triangle shape for configuration
k1 = 0.052 hMpc−1 k2 = 0.06 hMpc−1. The black solid line shows the tree level prediction for the matter bispectrum, the
red solid line has b1 only and dashed and dashed-dotted lines are adding b2 and bs2. The list of bias parameters behind the
theoretical cross bispectra in the legend indicates the parameters that were considered for the corresponding curve. The error
barsareestimatedfromthebox-to-boxvarianceofthebispectrummeasurement. Notethatthisisonlyasmallfractionofthe
total bispectrum information that our simulations contain, and the log scale also de-emphasises what are actually significant
differences between the fit with and without s2 (these will be highlighted later).
b1 ∆b1 b2 ∆b2 bs2 ∆bs2 M[h−1M(cid:12)]
I 1.142 0.002 −0.37 0.01 −0.07 0.03 9.68×1012
II 1.409 0.002 −0.42 0.01 −0.21 0.04 2.90×1013
III 1.954 0.004 −0.12 0.02 −0.38 0.06 8.58×1013
IV 2.889 0.010 1.25 0.03 −0.63 0.11 2.48×1014
TABLE I: Best fit bias parameters, their errors and mean mass for our four mass bins. The bias parameters are compared to
the theoretical bias functions in Figure 4. The first order bias b1 is extracted from the halo-matter cross power spectrum and
the second order bias parameters are inferred from the cross bispectrum.
8
0.0
0.0 bin I bin II
0.2
0.2 −
−
b2 0.4 b2−0.4
−
0.6
0.6 −
−
0.8
0.8 −
−
0.4
0.2
0.2
0.0
0.0
b2s b2s−0.2
0.2
− 0.4
−
0.4
− 0.6
−
0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
kmax[hMpc−1] kmax[hMpc−1]
0.2 bin III 1.6 bin IV
0.0 1.4
b2 0.2 b21.2
−
0.4 1.0
−
0.6 0.8
− 0.2
0.0 −
0.4
0.2 −
−
b2s−0.4 b2s−0.6
0.8
0.6 −
−
1.0
0.8 −
−
0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
kmax[hMpc−1] kmax[hMpc−1]
FIG. 2: Convergence of the measured b2 (upper panels) and bs2 (lower panels) parameters with increasing maximum k-mode
for the four mass bins. The horizontal red and blue lines show the constraints obtained for our fiducial kmax =0.07 hMpc−1.
The pivot data point is highlighted by the gray shaded region.
III. SIMULATIONS & BISPECTRUM ESTIMATION
A. The Simulations
We are studying the cosmic density field in a suite of 11 dark matter only simulations with box size of L =
1600 h−1Mpc, which are an extension of the simulations described in [31]. The ΛCDM model is based on best-fit
parameters inferred from the WMAP 5-year data release [32]. Thus, we adopt a mass density parameter Ω =0.279,
m
a baryon density parameter Ω =0.0462, a Hubble constant h=0.7, a spectral index n =0.96, and a normalization
b s
ofthecurvatureperturbationsof∆2 =2.21 10−9 atthepivotpointk =0.02Mpc−1. Thisnormalizationleadstoa
R ×
present day fluctuation amplitude of σ 0.81. The initial conditions are set up a redshift z =99. The gravitational
8 i
evolution of the N = 10243 particles≈is integrated using the publicly available Gadget2 code [33]. Simulations
p
size, particle number and matter density parameter yield a particle mass of 3 1011 h−1M . Dark matter halos are
(cid:12)
×
identifiedusingaFriends-of-Friendshalofinderwithalinkinglengthof0.2timesthemeaninterparticlespacing. Only
halos exceeding 20 particles are considered for our analysis, corresponding to a minimum halo mass of approximately
6 1012 h−1M . We consider four mass bins, each spanning a factor of three in mass. For the estimation of the
(cid:12)
×
statistics, particles are interpolated on a N = 512 grid using the Cloud-in-Cell (CIC) algorithm and the gridded
c
density field is corrected for the window of the grid. The matter-matter-halo and matter bispectrum are measured
for low k-modes k <0.12 hMpc−1 in the simulation output at redshift z =0. The bispectrum measurement scales as
the sixth power of the number of grid cells per dimension, which makes it computationally very expensive to extract
the full bispectrum information at higher k. It would still be interesting to extent the measurement to higher k in
the future to determine the scale of breakdown of the tree level calculation.
9
2.0 2.0
bin I bin II
1.5 1.5
1.0 1.0
M 0.5 M 0.5
0.0 0.0
0.5 0.5
− −
1.0 1.0
− 1.0 0.5 0.0 0.5 1.0 − 1.0 0.5 0.0 0.5 1.0
− − µ − − µ
2.0 2.0
bin III bin IV
1.5 1.5
1.0 1.0
M 0.5 M 0.5
0.0 0.0
b2L0(µ)
−0.5 −0.5 b2L0(µ)+bs2L2(µ)
combinedkmax=0.07hMpc−1
1.0 1.0
− 1.0 0.5 0.0 0.5 1.0 − 1.0 0.5 0.0 0.5 1.0
− − µ − − µ
FIG. 3: Residual shape dependence of the halo bispectrum for our reduced bispectrum defined in Eq. (35). The blue data
points with error bars show the result of the combined reduced bispectrum defined in Eq. (37) including all the configurations
uptokmax =0.07hMpc−1. Thehorizontaldashedlineshowsthemodelincludingb2 only, thesolidbluelineshowsthemodel
including both b2 and bs2.
B. Bispectrum estimation and data reduction
The bispectrum modes must satisfy the triangle condition k +k +k = 0, thus the shape of the bispectrum is
1 2 3
fully specified by two lengths and one angle, which we choose as k ,k and µ = k k /k k . Since δ(x) is a real
1 2 1 2 1 2
valued field, the Fourier modes have to satisfy δ∗(k) = δ( k). Consequently, the ima·ginary part of the bispectrum
−
cancels when we add δ(k )δ(k )δ(k ) and δ( k )δ( k )δ( k ). We consider bins that are logarithmically spaced in
1 2 3 1 2 3
− − −
k and k and linearly in µ. We add all the bispectrum amplitudes that fall into the bin centered at (k ,k ,µ).
1 2 1 2
Asafirststepinouranalysiswesubtractouttheb contributionproportionaltothematterbispectrumanddivide
1
by the power spectrum measured in the same simulation and k-bins to cancel part of the cosmic variance
Bˆ(unsym)(k ,k ,µ) ˆb Bˆ (k ,k ,µ)
Mˆ(k ,k ,µ)= mmh 1 2 − 1 mmm 1 2 . (35)
1 2 2Pˆ (k )Pˆ (k )
mm 1 mm 2
Comparing to Eq. (34), the resulting statistic should be a function of µ only. The hat is used to mark quantities
estimatedfromthesimulations. Thisquantityisdistinctfromthetheusualdefinitionofthereducedbispectrumsince
the power spectra in the denominator do not depend on k and can thus be controlled by limiting k and k . The
3 1 2
first order bias parameterˆb is estimated from the halo-matter cross power spectrum on large scales 0.015 hMpc−1 <
1
k < 0.03 hMpc−1, where the linear bias model P = b P is believed to be accurate. We show the k-dependence
hm 1 mm
of Pˆ (k)/Pˆ (k) and the inferred bias parameters in Fig. 4. Note that the resulting statistic depends only on the
hm mm
magnitude of the k and k modes, so that we can ensure the validity of the perturbative expansion by limiting these
1 2
modes accordingly.
10
4
ˆb1fromPmh b(1E) 23..9044
3 2ˆb2fromBmmh b(2E) b12.84
ˆbs2fromBmmh 2/7 b(E) 1 2.74 binIV
− 1 − 2.10
(cid:16) (cid:17) 2.00
2 b11.90
1.80 binIII
bi
1.56
1 1.46
b11.36
1.26 binII
0 1.29
1.19
b11.09
0.99 binI
1
− 1013 1014 0.01 0.02 0.03 0.04 0.06 0.08 0.10
M[h−1M ] k[hMpc−1]
(cid:12)
FIG. 4: Left panel: Mass dependence of the bias parameters and theoretical predictions. The points with error bars are our
best fit parameters for ˆb1, 2ˆb2 and ˆbs2. The numerical values of the data points are given in Table I. The curves show the
corresponding theoretical bias functions as calculated using the relations in §IIE. The measurements for ˆb1 are in a good
agreementwiththetheory,thereisacleardeviationfortheˆbs2 andˆb2 measurementforthetwocentralmassbins. Rightpanel:
Ratio of the simulation halo matter and matter power spectra Pˆhm(k)/Pˆmm(k) and first order bias parameters inferred using
the data points highlighted by the shaded region.
When showing the reduced data as a function of µ only, we reduce them as follows
2
Mˆ(k ,k ,µ) M(µ)
χ2 = 1 2 − (36)
M ∆M(k ,k ,µ)
(cid:32) 1 2 (cid:33)
k(cid:88)1,k2
−1
Mˆ(k ,k ,µ) 1
1 2
M(µ)= (37)
∆M2(k ,k ,µ) ∆M2(k ,k ,µ)
1 2 1 2
k(cid:88)1,k2 k(cid:88)1,k2
 
−1/2
1
∆M(µ)= (38)
 ∆M2(k ,k ,µ)
1 2
k(cid:88)1,k2
 
for each µ.
Thecosmicvarianceofthebispectrumestimatescouldinprinciplebemeasuredfromthestandarddeviationbetween
oursimulationboxes. Giventhesmallnumberofboxesthisapproachisboundtogiveaverynoisyestimate. Sincewe
are using the error for the weighting of the modes, we would like to avoid a spurious upweighting of modes which by
chance have a low simulation to simulation variance. For this purpose we prefer a smooth error estimate. As shown
in [34] the variance of the bispectrum is given by
V 1 1
∆B2 (k ,k ,µ)=s f P (k )P (k ) P (k )+ , (39)
mmh 1 2 123V (2π)3 mm 1 mm 2 hh 3 n¯
123 (cid:18) h(cid:19)
where V =(2π)3/L3 is the volume of the fundamental cell, n¯ is the number density of the tracer and s takes on
f h 123
values of 1, 2 and 6 for general, isosceles and equilateral triangles, respectively. The norm volume is given by
8π2
V = k3k3(dlnk)2 dµ (40)
123 (2π)6 1 2
for our bins, which are logarithmically spaced in k , k and linearly spaced in µ. The above just quantifies the
1 2
diagonals of the covariance matrix between the different triangle shapes and scales, but the correlations between