Consistency relations for the Lagrangian halo bias and their implications

2.8 MB·

by  Kwan Chuen Chan

MNRAS000,1–17(2016) Preprint4April2017 CompiledusingMNRASLATEXstylefilev3.0
Consistency relations for the Lagrangian halo bias and
their implications
Kwan Chuen Chan,1(cid:63) Ravi K. Sheth,2 Roma´n Scoccimarro3
1 Institute of Space Sciences, IEEC-CSIC, Campus UAB, Carrer de Can Magrans, s/n, 08193 Bellaterra, Barcelona, Spain
2 Center for Particle Cosmology, University of Pennsylvania, 209 S. 33rd St., PA 19104, Philadelphia, USA
3 Center for Cosmology and Particle Physics, Department of Physics, New York University, NY 10003, New York, USA
7
1 AcceptedXXX.ReceivedYYY;inoriginalformZZZ
0
2
ABSTRACT
r
p The protohalo patches from which halos form are defined by a number of constraints
A imposed on the Lagrangian dark matter density field. Each of these constraints con-
tributestobiasingthespatialdistributionoftheprotohalosrelativetothematter.We

3 showhowmeasurementsofthisspatialdistribution–linearcombinationsofprotohalo
bias factors – can be used to make inferences about the physics of halo formation.
]
Ouranalysisexploitsthefactthathalobiasfactorssatisfyconsistencyrelationswhich
O
encode this physics, and that these relations are the same even for sub-populations
C
in which assembly bias has played a role. We illustrate our methods using a model
. in which three parameters matter: a density threshold, the local slope and the curva-
h
p tureofthesmootheddensityfield.Thelattertwoarenearlydegenerate;ourapproach
- naturally allows one to build an accurate effective two-parameter model for which the
o consistency relations still apply. This, with an accurate description of the smooth-
r
t ing window, allows one to describe the protohalo-matter cross-correlation very well,
s both in Fourier and configuration space. We then use our determination of the large
a
scalebiasparameterstogetherwiththeconsistencyrelations,toestimatetheenclosed
[
density and mean slope on the Lagrangian radius scale of the protohalos. Direct mea-

2 surements of these quantities, made on smaller scales than those on which the bias
v parameters are typically measured, are in good agreement.
1
0 Key words: large-scale structure of Universe
7
1
0
.
1
1 INTRODUCTION paper is to demonstrate that we can indeed extract infor-
0
mationaboutthesmallscalephysicsofhaloformationfrom
7 Halos and the galaxies they host are biased tracers of the
measurements of the large scale clustering of halos. In par-
1 darkmatterdensityfield(Kaiser1984;Bardeenetal.1986;
: ticular, our methodology allows one to estimate if assembly
v Mo & White 1996). To extract cosmological information bias effects, of the sort first identified by Sheth & Tormen
i from galaxy surveys, this bias must be understood. In the
X (2004), are present in the halo population. In this respect,
best studied models of halo formation, the bias is a conse-
our work complements that of Castorina et al. (2016a,b);
r quence of the fact that the protohalo patches from which
a whereas they used configuration space methods to address
halos form are defined by a number of constraints imposed
similar issues, we, like Modi et al. (2016), use Fourier space
on the initial Lagrangian dark matter density field. Each
measurements.
one of these constraints contributes to biasing the spatial
distribution of the protohalos relative to the matter. Thus, This paper is organized as follows. We provide a new
inprinciple,thereisvaluablehaloformationphysicshidden derivation of the consistency relations for the linear La-
in the bias parameters. In fact, the bias parameters sat- grangian bias parameters using a straightforward linear al-
isfy a hierarchy of consistency relations (Musso et al. 2012; gebramethodinSec.2.1.Sinceprotohalosareextendedob-
Paranjapeetal.2013).Theseconsistencyrelationsnotonly jects,wediscusstheimportanceofsmoothinginSec.2.2.In
allow us to check the self-consistency of the bias prescrip- Sec. 2.3 we illustrate our arguments with some specific ex-
tion, but they also potentially open the road to learning amples,whichweusetoaddressthequestionofdegeneracies
about the physics of halo formation. The main goal of this betweenparametersandeffectiveversusexactmodelsofthe
physics.Aftershowingthemeasurementsofthecorrelations
inconfigurationandFourierspaceinSec.3.1,inSec.3.2we
(cid:63) E-mail:[email protected](KCC) discussourdirectestimatesofquantitieswhicharethought
(cid:13)c 2016TheAuthors
2 K. C. Chan, R. K. Sheth, and R. Scoccimarro
tomatterforthesmallscalephysicsofhaloformation,some where p(∆|C = C) is the distribution of ∆ conditioned on
ofwhicharenovel.InSec.3.3,weuseatwo-biasparameter C = C. In general, we must integrate over the distribution
model to fit measurements of the Fourier space bias signal. of the constraint Π(C) as well:
The halo collapse threshold inferred from the consistency (cid:82) (cid:82)
dCΠ(C) d∆p(∆|C)∆
relation is compared with the direct measurements of the (cid:104)∆|C(cid:105)= (cid:82) (cid:82) . (2)
dCΠ(C) d∆p(∆|C)
overdensity within Lagrangian protohalos in Sec. 3.4, and
a similar test of the profile slope is in Sec. 3.5. We revisit We will often assume that ∆ is the density field
thephysicalmeaningoftheconsistencyrelationsinSec.3.6. smoothed on a large scale at the same spatial position at
We summarize our findings and conclude in Sec. 4. An Ap- which the constraints were specified, but we can take it to
pendixisdevotedtothestudyofthecorrelationfunctionin beotherfieldsatotherpositionsifwewish.If∆istakento
realspace,andshowsthatalthoughbiasparametersmayde- bethedensity,(cid:104)∆|C(cid:105)wouldbetheprofileoftheLagrangian
pendonsmoothingwindow,thecombinationwhichmatters halos. The large-scale field only serves as a surrogate for
for the consistency relation does not. extracting the smaller scale halo formation physics. This
physicsisencodedintheconstraintC =(C ,...,C ),which
1 n
weexpressintermsofnormalized(zero-meanunitvariance)
2 LAGRANGIAN CONSTRAINTS, BIAS AND random variables. For example, it can be (ν,x,...), where
CONSISTENCY RELATIONS ν isthepeakheightandxitscurvature.TheFouriertrans-
forms of these quantities are defined as
Although ultimately we are interested in halos in Eulerian
space, as they are potentially observable, the modelling of W(kR)
ν(k)= √ δ (k), (3)
halopropertiesoftenstartsinLagrangianspace.Thisispri- s m
0
marilybecausethestatisticsoftheLagrangianfieldareeas- k2W(kR)
ier to describe, particularly if the initial conditions were x(k)= √s δm(k), (4)
Gaussian. The best-studied models are the peak (Kaiser 2
1984;Bardeenetal.1986;Desjacques&Sheth2010),excur- where δ (k) is (a Fourier mode of) the dark matter den-
m
sion set (Press & Schechter 1974; Bond et al. 1991; Musso sitycontrast,andW isthesmoothingwindow.Thespectral
et al. 2012) and excursion set peak (Appel & Jones 1990; moment s is defined as
j
Paranjape & Sheth 2012; Paranjape et al. 2013; Biagetti (cid:90) dk
et al. 2014; Dizgah et al. 2016) approaches. In all three ap- s (R)≡ k2(1+j)P (k)W2(kR), (5)
j 2π2 m
proaches, describing how the initial Lagrangian protohalos
evolvetoformthefinalEulerianhalosisaseparatestep.In where P (k) is the linear power spectrum evaluated at z=
m
whatfollows,wedonotconsiderthissecondstep,exceptto 0. In what follows, we will also be interested in the ‘slope’
point out that the way forward is described in Desjacques variable
etal.(2010).Forarecentreviewonhalobias,seeDesjacques
1 dW(kR)
et al. (2016). u(k)= √s ds (R) δm(k), (6)
u 0
where
2.1 General formalism
(cid:90) dk (cid:18)dW(kR)(cid:19)2
s (R)= k2P (k) . (7)
Inoneofthesimplestmodels,aprotohaloisidentifiedwith u 2π2 m ds (R)
0
anypositionwherethesmoothedfieldexceedsa(physically
motivated) threshold. The peak model adds the additional In real space, these variables are
constraints that the spatial gradient of the smoothed field 1 (cid:90) δ(r)
should vanish and that the curvature should be negative. ν(r)= √s d3r(cid:48)W(r(cid:48);R)δm(r−r(cid:48))= √s , (8)
0 0
Theseconstraintsonthescaleoftheprotohalopatchimpact
1 (cid:90) ∇2δ(r)
correlations between the protohalo centers and the large- x(r)=−√ d3r(cid:48)∇2 W(r(cid:48);R)δ (r−r(cid:48))=− √r ,
s r(cid:48) m s
scale matter distribution. Our goal is to extract these con- 2 2
(9)
straints from large-scale cross correlations.
However, to do so, we must first address the fact that 1 (cid:90) dW(r(cid:48);R) 1 dδ(r)
u(r)= √ d3r(cid:48) δ (r−r(cid:48))= √ ,
‘smoothing’ is common to all halo formation models. I.e., s ds (R) m s ds (R)
u 0 u 0
it is the average properties of the field centered on a patch (10)
in the initial conditions which determine whether or not it
where we have denoted the smoothed dark matter density
will become a halo. Therefore, the shape of the smoothing
field by δ. These variables illustrate a few of the ways in
window W is expected to play an important role. For this
whichthesmoothingwindowW appearsintheconstrained
reason,itisimportanttonotethattheanalysiswhichfollows
variables. We are now ready to consider generic relations
is generic for all window choices; we will only specify our
between the constraints.
choice of window in the next subsection.
TheconstrainedGaussianfieldisstillGaussianandits
Wesupposethattheconstraintsaregivenbythevector
mean and variance are well-known (see e.g. Appendix D in
C. If the constraint variables are set to be C = C, then
Bardeenetal.1986).InparticularwhenC isconstrainedto
the expectation value of the large-scale field ∆ given the
havesomespecificvaluesC,thentheconditionalmeanof∆
constraints is
(cid:82) is
d∆p(∆|C =C)∆
(cid:104)∆|C =C(cid:105)= (cid:82) d∆p(∆|C =C) , (1) (cid:104)∆|C =C(cid:105)=(cid:104)∆C(cid:105)j(cid:104)CC(cid:105)−1jkCk, (11)
MNRAS000,1–17(2016)
Consistency relations and implications 3
wherethevector(cid:104)∆C(cid:105)denotesthecrosscorrelationbetween particular that if the constrained variables must be aver-
∆ and the constraint variables, and (cid:104)CC(cid:105) is the covariance aged over, then this will yield averaged values of b(j) whose
1
matrix between the constraint variables. valueswilldependontherangeoverwhichtheaveragewas
Clearly, the right hand side of Eq. 11 is a sum of many taken. Inserting these averaged values in Eq. 15 yields the
terms.Ifwemultiply(cid:104)∆C(cid:105)(cid:104)CC(cid:105)−1 first,thenthetermswill corresponding averaged value of C . Since the same proce-
1
be grouped according to C . For bias, we instead wish to dure works whatever the range over which the average was
k
group terms by their scale dependence, which means we taken,Eq.15showsthatthebiasfactorsofan‘assemblybi-
group by the elements of (cid:104)∆C(cid:105). We define the linear bias ased’ population can be used to infer what was the physics
coefficients as which led to the assembly bias. While one might have wor-
(cid:104)∆C(cid:105) √ ried that the procedure for recovering the physics of forma-
(cid:104)∆|C =C(cid:105)= (cid:104)CC(cid:105)j s0b(1j), (12) tion might have been different for each sub-population, the
1j
analysis above shows that this is not the case.
where b(j) is the jth linear bias coefficient Eq. 15 has been derived previously, in the context of a
1
1 specific model in which n = 2, following a rather different
b(1j) = √s (cid:104)CC(cid:105)1j(cid:104)CC(cid:105)−1jkCk. (13) approach(Mussoetal.2012;Paranjapeetal.2013),whereit
0
was called a ‘consistency’ relation. Our analysis shows how
(Note that j is not summed over in Eq. 1√3.) In Eq. 12 we to extend the consistency relation to arbitrary n and arbi-
have divided by (cid:104)CC(cid:105)1j and introduced s0 so that the trary constraints. On the other hand, the analysis above is
dimension of the resultant bias parameter b(j) agrees with forthefirstorder(sometimescalled‘linear’)biasparameter.
1 √
the usual expansion in the density contrast. Note that s0 Thereareconsistencyrelationsassociatedwithhigherorder
appears quite differently from other variables, so we shall bias parameters (Musso et al. 2012; Paranjape et al. 2013).
treat it differently (e.g., when we consider bias at differ- Based on the results of Musso et al. (2012), we write down
enttimes).Sometimesitisadvantageoustousethevariable thehighorderconsistencyrelationsinSec.3.6.Castorina&
√
s b(j) because it can be completely expressed in terms of Sheth (2017) show how to extend these to n arbitrary vari-
0 1
the normalized dimensionless variables. We also stress that ablesandconstraintsusingageneratingfunctionapproach.
the b(j) defined here have not (yet) been averaged over the So it would be interesting to extend the simple linear al-
1
constraint, i.e. they explicitly depend on C. gebra method developed here to describe higher order bias.
Eq. 13 is interesting because it enables us to express However, doing so is beyond the scope of this work.
the bias parameters directly in terms of the constraints C,
making the bias problem simply a linear algebra problem.
Inparticular,wecaninvertEq.13toexpressCk intermsof 2.2 Smoothing windows
the b(j):
1
TheanalysisaboveisvalidregardlessoftheformofW.We
√
C =(cid:104)CC(cid:105) s0b(1j). (14) now turn to the question of what one should use for W.
k kj (cid:104)CC(cid:105) We expect it to have a characteristic scale R, so its Fourier
1j
transformwillbeafunctionofthecombinationkR.Inwhat
Therearenrelationsbetweenthebiasparameters,wheren
follows, we will use the effective window function proposed
isthedimensionof(cid:104)CC(cid:105),andtheysimplyreflecttheunder-
in Chan et al. (2015), the Fourier transform of which is
lying constraints on halo formation. Note that the bias in
Eq. 13 and the consistency relations in Eq. 14 only depend
W (kR)=W (kR)W (kR/5), (16)
Eff TH G
onC,butnot∆.Thisechoesourpreviousassertionthatthe
large-scale field ∆ is only used to extract the constraint C. where W and W denote the top hat and Gaussian win-
TH G
Inaddition, recall that we have yet to average over the dow function respectively:
constraint variables. Since averaging is a linear operation,
W (x)=(3/x3)(sinx−xcosx), (17)
ouranalysisalsoshowsthatpartialaverageswillalsosatisfy TH
similarconsistencyrelations.Aswediscusslater,thismeans W (x)=e−x2/2. (18)
G
that our analysis is immune to what is known as ‘assembly
bias’. I.e., although the numerical values of ‘assembly bi- There are a number of reasons why we will work with
ased’ bias factors may be different from those of the parent W ,ratherthanW orW .Butbeforewelistthem,itis
Eff TH G
population, the algebraic relations between the bias factors worth noting that although all three filters tend to unity at
– Eq. 14 – will be the same as for the parent population. x(cid:28)1,theyarenotallequallycompact.Acrudemeasureof
The full consistency relations are already given by theextentofthesefiltersisgivenbyexpandingW tolowest
Eq. 14. However, the first of these consistency relations for orderink.ThenW (x)≈1−3x2/25,W (x)≈1−x2/10,
Eff TH
C isparticularlysimple.Byputtingk=1inEq.14,weget andW (x)≈1−x2/2,whichsuggeststhat,ifwewantedto
1 G
replace W by a tophat or a Gaussian, then we should set
Eff
(cid:88)b(j) = √C1 . (15) RG2 = (6/25)RE2ff and RT2H = (6/5)RE2ff. Thus, the tophat
1 s is the most compact of the three, and the Gaussian is the
0
j most extended. This difference will be important below.
Eq. 15 is interesting because it shows that the value The first reason we like W is purely technical. Al-
Eff
oftheconstrainedvariableC canbedeterminedifonehas though many of the expressions above are formally well-
1
measuredthebiasparametersb(j).Thisshowsexplicitlywhy definedforanarbitrarysmoothingwindow,inpractice,some
1
measurements of the large scale bias can be used to learn lead to divergences. For example, s defined in Eq. 5 plays
j
about the small-scale physics of halo formation. Notice in an important role in peak theory. However, for W , the
TH
MNRAS000,1–17(2016)
4 K. C. Chan, R. K. Sheth, and R. Scoccimarro
2.3 Specific examples
0.90
It is common to set C = ν, although this is not neces-
1
0.85 sary. In the simplest peak model, a proto-halo patch sat-
eff.0.80 isfies three constraints: the value of the enclosed overden-
o sity (the height), the first derivative with respect to spatial
c
on 0.75 νx position (which must be zero to be a local peak), and the
ati second derivative with respect to spatial position (the peak
el0.70 ›νufi curvature) (Bardeen et al. 1986). In the simplest excursion
corr0.65 ›xufi setpeakmodel,thereisanadditionalconstraintonthefirst
ss › fi derivativewithrespecttosmoothingscale(theexcursionset
cro0.60 slope)(Appel&Jones1990;Musso&Sheth2012;Paranjape
&Sheth2012).Moresophisticatedmodelsincludetheshear
0.55
and the shape (e.g. Sheth et al. 2013; Biagetti et al. 2014;
0.510011 1012 1013 1014 1015 1016 Castorina et al. 2016b); we will ignore these complications.
M/M h−1 The first spatial derivative is independent of the other
fl
threevariables,anditisconstrainedtobezeroanyway.So,
in practice, n=2 for the peak model and n=3 for excur-
Figure 1. Cross correlation coefficients (cid:104)νx(cid:105) (solid blue), (cid:104)νu(cid:105)
sion set peak. It is typical to order these as C = (ν,x,u),
(dashedgreen)and(cid:104)xu(cid:105)(dotted-dashedred)asafunctionofhalo
where the three variables have zero mean, and they have
mass,whenthesmoothingwindowisgivenbyEq.16.
been normalized by the rms values of the height, curvature
and slope, respectively. Therefore, (cid:104)CC(cid:105) has unity along its
diagonal. In addition, since x≈u, we will use this example
to show how our formulation of consistency relations works
integral above diverges if j ≥ 2, whereas it is well-behaved
when n≥2, as well as to address the question of what one
for W .
Eff shoulddowhendifferentphysicalparametersarenearlyde-
The second is more physical. With WEff, Chan et al. generate.
(2015) were able to provide an accurate description of pro-
E.g., if x and u are nearly degenerate, shouldn’t the
tohalosinsimulationsusingjusttwobiasparameters:those
bias factors for each be similar? If so, how is it that the
associated with ν and u. In contrast, with W , just ν and
G n=3consistencyrelationdoesnotdoublecountthecontri-
u arenotenoughtoaccuratelydescribe protohalos. Thatis
bution from the two terms? Alternatively, what form does
tosay,W allowsustousethen=2consistencyrelations
Eff the consistency relation take if fundamentally both x and
above to constrain halo formation. I.e., of the potentially
u matter, but we suppose that only one of them matters?
vast number of physical parameters which may matter for Can we effectively group the b(2) and b(3) terms together
1 1
haloformation,W allowsustoworkwithjustn=2.This
Eff in the consistency relations? Such questions are relevant to
is a very useful simplification.
the issue of efficiency in describing the halo distribution:
Finally, WEff allows us to illustrate some subtle issues How many variables are sufficient to capture the physics?
associated with our approach. For example, x = u for a In what follows, we will consider both the 2-parameter and
Gaussian smoothing window because 3-parameter models.
Theprevioussectionshowedthatlinearcombinationsof
dW (kR) s (R)
G = 0 k2R2W (kR). (19) the bias factors allow one to isolate the dependence of halo
dlns (R) 2R2s (R) G
0 1 formation on ν, x and u. To see this in practice, suppose
that n=1 and C =ν. Then
Therefore, for general smoothing windows, such as our 1
√
Eq. 16, we expect to find that although x and u are for- b(1) =ν / s , (23)
1 c 0
mally different, in practice, constraints on x may be rather
degenerate with those on u. where νc would typically be given by the spherical collapse
To illustrate this point, Fig. 1 shows the cross correla- (SC) value νsc
tion coefficients √
ν =δ (z)/ s , (24)
sc sc 0
s
(cid:104)νx(cid:105)= √s1s ≡γνx, (20) with
0 2
D(z=0)
1/2 δ (z)= δ , (25)
(cid:104)νu(cid:105)= √s s ≡γνu, (21) sc D(z) sc
0 u
1/2 ds where δ = 1.68 is the SC threshold and D is the linear
(cid:104)xu(cid:105)= √ 1 ≡γ . (22) sc
s s ds xu growth factor. Eq. 23 is the expression for biased tracers
2 u 0
which appears in Kaiser (1984). If the biased tracers were
as a function of smoothing scale R (recall halo mass M ∝ constrained to have δ ≥ δ rather than δ = δ , then,
sc sc
R3). Notice that(cid:104)xu(cid:105)≈1, and(cid:104)νx(cid:105) and(cid:104)νu(cid:105) are similar in because the constraint C would include an integral over
magnitude, as expected (recall that (cid:104)xu(cid:105) = 1 for Gaussian δ≥δ ,thebiasfactorb(1) wouldalsobegivenbyintegrat-
sc 1
smoothing).Inwhatfollows,wewillusethisfacttoaddress ing over the allowed range, and ν would also be replaced
sc
a number of issues which arise when different physical pa- by its average over the range. Note that for notational sim-
rameters are nearly degenerate. plicity we have denoted the smoothed dark matter density
MNRAS000,1–17(2016)
Consistency relations and implications 5
W ∗δ simply as δ, and the window function is implicitly expansion reads
m
included in δ and the constraint variables. s
(cid:104)∆|C =(ν ,x ,u )(cid:105)=(cid:104)∆δ(cid:105)b(1)−(cid:104)∆∇2δ(cid:105) 0b(2)
Thenextnon-trivialcaseisn=2,withC =(ν,x)inthe c c c 1 s 1
1
peakmodel,and(ν,u)intheexcursionsetapproach,where (cid:68) dδ (cid:69)
ν is the overdensity as before, u its slope with respect to + 2 ∆ds s0b(13), (31)
0
smoothingscale,andxitscurvaturewithrespecttospatial
and Eq. 14 becomes
position. E.g., in the peak model,
(cid:104)CC(cid:105)=(cid:18) γ1νx γ1νx (cid:19), (26)  xνcc =√s0 γνxb(11)+b(1b1(1)2+)/γb(1ν2x)++bb(1(133))γxu/γνu .
uc γνub(11)+b(12)γxu/γνx+b(13)/γνu
and the bias expansion is given by
(32)
s
(cid:104)∆|C =(ν ,x )(cid:105)=(cid:104)∆δ(cid:105)b(1)−(cid:104)∆∇2δ(cid:105) 0b(2), (27) Inthisformitiseasytoseethattheconsistencyrelationis
c c 1 s 1
1 satisfied.Clearly,ourpreviousremarksaboutassemblybias
applyheretoo.Ourmaingoalinthisexampleistoillustrate
with the bias parameters being
what happens when some of the parameters, in this case x
ν −γ x and u, are nearly degenerate. E.g., for Gaussian smoothing
b(1) = √c νx c (28)
1 s (1−γ2 ) (cid:104)xu(cid:105)=1, and (cid:104)νu(cid:105)=(cid:104)νx(cid:105) so the expressions above clearly
0 νx
b(12) = γν√xs(x(c1−−γγν2xν)c), (29) dxecpaenndduocnlayreotnheb(11s)amane.d (b(12)+b(13)), and the relations for
0 νx
The bias factors, written explicitly in terms of the cor-
whereν ,andx denotetheconstrainedvalues.Theconsis- relation coefficients of the constrained variables, are
c c
tency relations read
ν −γ x γ −γ γ b(3)
b(1) = √c νx c − νu ux xν 1 , (33)
(cid:18) ν (cid:19) √ (cid:32) b(1)+b(2) (cid:33) 1 s0(1−γν2x) 1−γν2x γνu
xcc = s0 γνxb(111)+b(112)/γνx . (30) b(12) = γν√xs(x(c1−−γγνx2ν)c) −γνxγux1−−γγu2νγνxbγ(13), (34)
0 νx νx νu
aFnodr t−h(cid:104)e∆e∇xc2uδr(cid:105)s(iso0n/ss1e)t→ap2p(cid:104)r∆oadcδh/,dslinm√sp0l(cid:105)y. replace γνx → γνu b(13) = √γνsu0 uvca−r(u(cid:104)u|ν|νc,c,xxc)c(cid:105), (35)
Notice that b(1) +b(2) = ν / s , just as when n =
1 1 c 0 where
1. However, b(1) here will be different from when n = 1 if
1 (γ −γ γ )ν +(γ −γ γ )x
b(12) (cid:54)=0.Again,allowingarangeofδcvaluesmeansonem√ust (cid:104)u|νc,xc(cid:105)= νu xu νx 1c−γ2 xu νx νu c (36)
replace b(1) and b(2) by their averaged values, and ν / s νx
1 1 c 0
should also be replaced by its average. and
E.g., a simple model of assembly bias would be to as- 1−γ2 −γ2 −γ2 +2γ γ γ
var(u|ν ,x )= νx ux uν νx νu ux. (37)
sume that all objects have the same height δc, but different c c 1−γ2
νx
curvaturesorslopes(Dalaletal.2008;Musso&Sheth2014;
Lazeyras et al. 2016). Eq. 28 and 29 show that populations Thefirsttermsontherhsoftheexpressionsforb(1) andb(2)
1 1
withdifferentx willhavedifferentbiasparametersb(1) and are the same as when n = 2 (compare Eq. 28 and 29), so
c 1
b(2).However,becausewehaveassumedtheyhavethesame it appears to be straightforward to address the question of
1 the impact of the third variable. However, when γ → 1,
heightν ,thesumb(1)+b(2) (thefirstconsistencyrelation) xu
c 1 1 both(u −(cid:104)u|ν ,x (cid:105))andvar(u|ν ,x )vanish,thusb(3) and
willbe the same for the differentpopulations. Onthe other c c c c c 1
hence b(1) and b(2) are indeterminate. As we discussed, in
hand, because the left hand side of the second consistency 1 1
this limit, the problem indeed reduces to the n = 2 case,
relationequalsx ,thisonewillbedifferentforthedifferent
c
populations.Thus,measuringthelargescalebiasfactorsb(1) and we can choose the second variable to be either x or u.
1 We will return to this shortly.
and b(2) for the populations and then using the two consis-
1 Wenowturntothequestionofestimatingthebiascoef-
tency relations in Eq. 30 allows one to discover that the
ficientsfromcross-correlationswiththedarkmatterdensity
populationsonlydifferincurvatureandnotinheight.Since
field ∆. The scale dependence of bias comes from the inter-
this argument obviously holds if the different populations
playofthreeterms:(cid:104)∆ν(cid:105),(cid:104)∆x(cid:105)and(cid:104)∆u(cid:105),whicharedefined
are constrained to have different ranges in δ and x, this ex-
by
ample shows explicitly how the same methodology can be
used to learn about the physics of assembly bias, without (cid:104)∆ν(cid:105)= √1 (cid:90) dk k2P (k)W(kR), (38)
having a priori knowledge that it was even present in the s0 2π2 m
firstplace.Conversely,ifwehavesomewaytodirectlymea- 1 (cid:90) dk
(cid:104)∆x(cid:105)= √ k4P (k)W(kR), (39)
sure δc and xc, e.g. using the direct measurements outlined s2 2π2 m
in Sec. 3, then the consistency relations provide a way to 1 (cid:90) dk dW(kR)
detect assembly bias. (cid:104)∆u(cid:105)= √ k2P (k) . (40)
s 2π2 m ds (R)
Finally,weconsidern=3withC =(ν,x,u);recallthat u 0
u is associated with the excursion set upcrossing constraint (We note in passing that in these expressions there is a
ontheslopeofthesmootheddensityfield(Eq.6).Thebias window function only for the protohalo/peak; there is no
MNRAS000,1–17(2016)
6 K. C. Chan, R. K. Sheth, and R. Scoccimarro
1.0
1.5 W γν+T+
(s0/s1)k2W γ T
ν
2dW/dlns0 0.5 − −
1.0

±
0.5 T 0.0
±
γν
0.0
0.5
0.5
10-2 10-1 100 101 102 103 1.100-2 10-1 100 101 102 103
kR kR
Figure 2. The three terms contributing to the effective cross Figure 3. The terms γν+T+ (solid blue) and γν−T− (dashed
bias in Eq. 41, when M = 5×1013M(cid:12)h−1 and the smoothing red) when M = 5×1013M(cid:12)h−1, shown as a function of kR.
window is given by Eq. 16. The dashed and dot-dashed terms, Notethatγν−T− remainsclosetozerocomparedtoγν+T+ until
whichareduetothecurvatureandslopeconstraints,haverather kR∼3.
similarshapes.
and
window on ∆. Also, for a Gaussian window, (cid:104)∆u(cid:105) = (cid:104)∆x(cid:105) 1−γ2 −γ2
var(x |ν ,x )= ν+ ν−. (47)
so we expect this to remain approximately true for other − c +c 1−γ2
ν+
windows.)FouriertransformingEq.31,anddividingbythe
It is again easy to see that the consistency relation is
cross correlation between ∆ and the dark matter density
the same as before. However, now var(x |ν ,x ) is well-
contrast δ , yields − c +c
m
behaved, so it is clear that, when γ (cid:28) 1 (when x = u,
(cid:104)∆|C =C(cid:105) ν−
b(3)(k)≡ γ is still indeterminate), then b(−) (cid:28) 1 and the system
eff (cid:104)∆δ (cid:105) ν− 1
m reduces to one involving just two constrained variables, ν
c
s dW
=Wb(1)+ 0 (kR)2Wb(2)+2 b(3). (41) and x+c =(xc+uc)/2.
1 R2s1 1 dlns0 1 UsingEq.12,thecorrespondingeffectivebiasexpansion
Fig. 2 compares the scale dependence of the three terms on reads
the rhs when W is given by Eq. 16. For kR(cid:46)0.5, only the b3(cid:48) =b(1)W +b(+)T +b(−)T , (48)
first term is non-vanishing: hence, b(1) can be determined eff 1 1 + 1 −
1
from the bias signal at kR (cid:46) 0.5. The other two terms are where
rather similar. Therefore, when Eq. 41 is used to fit the s k2 γ dW γ
T = 0 W νx ±2 νu . (49)
k-dependence of the measured bias signal, the derived con- ± s γ ±γ dlns γ ±γ
1 νx νu 0 νx νu
straints on b(2) and b(3) are likely to be degenerate, along a
1 1 (In the ± sign, + and − are for T and T respectively.)
locus of approximately constant b(2)+b(3). + −
1 1 The quantities which actually matter for the bias ex-
Having shown why b(12) and b(13) are approximately pansionareγν±T±.Fig.3showsthatforkR(cid:46)3,γν−T− (cid:28)
degenerate, we are ready to reconsider the issue of γ T and so the b(−)T term can be neglected. There-
var(u|ν,x)→0.Itisparticularlyinstructivetoconsiderthe foνr+e,+when fitting to1data−at kR (cid:46) 3, it should be a good
following variables
approximation to neglect the x variable and use only the
−
x ≡ x±u for which γ ≡ γνx±γνu . (42) effective two-parameter model. This is important, as when
± (cid:112) ν± (cid:112)
2(1±γ ) 2(1±γ ) fittingdata,onedoesnotknowapriorihowmanyvariables
xu xu
These variables are uncorrelated with each other, and this are important for the physics. The fact that b1(−)T− (cid:28) 1
simplifies the analysis. By analogy with Eq. 33-37, the bias meansthatworkingwiththeeffectivetwo-variableproblem
factors are will not yield biased results.
ν −γ x b(−)
b(1) = √c ν+ +c − 1 , (43)
1 s (1−γ2 ) 1−γ2
0 ν+ ν+ 3 CROSS CORRELATIONS AND
γ (x −γ ν ) γ2 b(−) LAGRANGIAN BIAS CONSISTENCY
b(+) = ν+√ +c ν+ c + ν+ 1 , (44)
1 s (1−γ2 ) 1−γ2 RELATIONS
0 ν+ ν+
b(−) = √γν− x−c−(cid:104)x−|νc,x+c(cid:105) (45) As discussed in the previous section, the cross correlation
1 s0 var(x−|νc,x+c) between the dark matter field and the Lagrangian protoha-
where los encodes the constraints used to define the Lagrangian
γ protohalos. We shall check this using Lagrangian protoha-
(cid:104)x |ν ,x (cid:105)= ν− (ν −γ x ) (46)
− c +c 1−γ2 c ν+ +c losobtainedinnumericalsimulations.Todoso,weidentify
ν+
MNRAS000,1–17(2016)
Consistency relations and implications 7
z=0.97 z=0
6 6
1.1e+13 1.1e+13
2.2e+13 2.2e+13
5 5
4.6e+13 4.6e+13
4.8e+13 9.4e+13
9.9e+13 4.8e+13
4 4
2.0e+14 9.9e+13
2.0e+14
ξc3 3 4.2e+14
2 2
1 1
0 0
10-2 10-1 100 101 10-2 10-1 100 101
r/R r/R
Lag Lag
Figure 4. Cross correlation between Lagrangian protohalo centers and the dark matter measured in the Oriana (circles) and Carmen
(triangles) simulations and extrapolated to z = 0 using the linear growth factor. The corresponding Eulerian halos were identified at
z=0.97(left)and0(right).Insetlistsmassesinunitsof M(cid:12)h−1.Lessmassiveprotohaloshavesteeperprofiles,ingoodagreementwith
agenericpredictionofpeaktheory.
Eulerianhalosatsomeredshiftz,andtracetheconstituent scaledbyRLag,whichisdefinedas
particles in each halo back to the initial redshift z . The
∗
position of the Lagrangian protohalo is estimated using the (cid:18) M (cid:19)1/3
center of mass of the particles in the Lagrangian space. RLag≡ 4πρ¯m/3 (50)
In this work, we use two sets of simulations from the
LasDamas project, denoted by Oriana and Carmen respec- (ρ¯m isthecomovingdensityofthedarkmatter).However,there
tively.BothsetsassumeaflatΛCDMmodelwiththecosmo- issomeresidualmassdependence:ξc dropsslightlymorerapidly
logicalparameters,Ωm =0.25,ΩΛ =0.75andσ8 =0.8.The for lower mass halos. The plot shows that, within RLag, ξc is
initial conditions are Gaussian with spectral index ns = 1, largerforlessmassivehalos,butthemasstrendreversesbeyond
and transfer function output from CMBFAST (Seljak & Zal- RLag: i.e., less massive protohalos have steeper profiles. This is
darriaga1996).Theinitialdisplacementfieldsaresetusing2LPT inqualitativeagreementwithagenericpredictionofpeaktheory
(Crocce et al. 2006) at z∗ =49. The simulations are evolved us- (Sheth1999).Inthepresentcontext,thisconfirmationofthepeak
ingthepubliccodeGadget2(Springel2005).IntheOrianasim- predictionisinteresting,butitisonlyameanstoanend.
ulations, there are 12803 particles in a cubic box of size 2400 The cross power spectrum Pc(k) between the initial La-
Mpch−1. For the Carmen simulations, there are 11203 particles grangian protohalo overdensity field δ , and that of the dark
halo
inaboxofsize1000 Mpch−1.ThustheparticlemassinOriana matterattheinitialtime,δm,isdefinedas
and Carmen is 4.57×1011 and 4.94×1010M(cid:12)h−1 respectively.
We use 5 realizations from Oriana and 7 from Carmen. In each, (cid:104)δhalo(k1)δm(k2)(cid:105)=(2π)3Pc(k1)δD(k1+k2), (51)
theEulerianhalosareidentifiedusingaFriends-Of-Friendsalgo-
rithm(Davisetal.1985)withlinkinglengthb=0.156atz=0.97 where δD is the Dirac delta function. Note that Pc(k) is the
and 0. Weonlyconsiderhaloswithatleast65particles.Webin Fourier transform of ξc, and that δhalo here should not be con-
the halos into thin narrow mass bins of width ∆lnM = 0.157. fused with δc, the overdensity within a protohalo patch, which
Toavoidcontaminationfromevolution,byLagrangianspacewe playedamajorroleintheprevioussection.
meantheinitialGaussiandensityfieldevaluatedatthegridpo- SincetheLagrangianmatterpowerspectrumis
sitionsinsteadofthedisplaced2LPTfield.
(cid:104)δm(k1)δm(k2)(cid:105)=(2π)3Pm(k1)δD(k1+k2), (52)
3.1 Lagrangian cross-correlations in configuration wedefinetheLagrangiancrossbiasparameter
and Fourier space
Wemeasurecross-correlationsbetweentheinitialLagrangianpro- bc(k,z)≡ DD((z0∗))(cid:16)PPcm(k(,kz,∗z;∗z)) −1(cid:17), (53)
tohalo positions and the dark matter at the initial time using
configurationspaceandFouriermethods. where D is the linear growth factor. Note that we extrapolate
Theconfigurationspacesignal,ξcissimplythemeandensity theLagrangianbiasparametertoz=0usinglinearevolutionof
profile around the Lagrangian halo centre (e.g. Peebles 1980). bias. The presence of unity in Eq. 53 is due to the finite initial
Fig. 4 shows this cross-correlation for a range of narrow mass redshiftinsimulation(see,e.g.,Chanetal.2012).Theargument
bins;muchofthedependenceonmassisremovedifdistancesare zdenotestheredshiftatwhichtheEulerianhaloswereidentified.
MNRAS000,1–17(2016)
8 K. C. Chan, R. K. Sheth, and R. Scoccimarro
z=0.97 z=0
7 7
1.1e+13 1.1e+13
6 2.2e+13 6 2.2e+13
4.6e+13 4.6e+13
) 5 4.8e+13 5 9.4e+13
R 9.9e+13 4.8e+13
(
0 2.0e+14 9.9e+13
s 4 4
n 2.0e+14
dl 4.2e+14
/ 3 3
)
R
(
W 2 2
c
ξ
d
2 1 1
0 0
1 1
10-1 100 101 10-1 100 101
R/R R/R
Lag Lag
Figure 5. The slope variable 2dξcW/dlns0, shown as a function of the smoothing scale R expressed in units of RLag. The slope is
maximalaroundRLag,withlessmassivehaloshavingsteeperslopes.
2.5 7
2.4
)
2.3 Lag6
R
2.2 (0
s
n5
2.1fi dl
h /
δ )
2.0› g
La4
R
1.9 (
W
1.8 dξc3
2
1.7
1.60.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 20.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
νsc νsc
Figure 6. Direct estimates of the mean density enclosed within Figure 7. Same as Fig. 6, but now for the slope variable
RLag√, shown as a function of the scaled mass variable νsc ≡ 2dξcW/dlns0.Again,theconfiguration-andFourier-basedmeth-
δsc/ s0. Green symbols show measurements based on Eq. 57 odsareingoodagreement.
and red ones show the traditional estimator based on Eq. 55.
Data from the Carmen and Oriana simulation sets at z = 0.97
and 0 are used. The triangles (upper set) denote the estimates sityprofile,soitisjust
usingWTH;thecircles(lowerset)usedWEff;inbothcases,lower (cid:90)
massprotohalosaremoreoverdense. ξcW(R)= dr4πr2W(r;R)ξc(r). (55)
HereW istheinverseFouriertransformofthesmoothingwindow
definedearlier,soithasunitsofinversevolume.E.g.,foratophat,
3.2 Direct estimates of the mean enclosed W =Θ(R/r−1)/(4πR3/3),andforW ,seeChanetal.(2015).
Eff
overdensity and slope I.e.,ξcW(R)isthecrosscorrelationξc(r)showninFig.4,smoothed
bytheeffectivewindowfunction.
The consistency relations for the enclosed overdensity and its
Similarly,theslopevariableis
slope,bothevaluatedonthescaleRLag,statethat
(cid:18) uνcc (cid:19)=√s0(cid:32) γνubb(1(111))++bb(1(122))/γνu (cid:33) (54) (cid:104)wsUmhhoe(cid:105)roet≡hδe(cid:104)hddδohins(Rstc)ha/eldesRo0v((eRirt)di(cid:105)esn=nsoidttyξδcWw(Rith)d/iendfisn0ae(dRpi)rn,oEtoqh.a5l1o!).pFaotcrhreaws(ho5en6ns)
halo
(compareEq.30).Wetesttheserelationsbymeasuringthequan- which will become clear shortly, Fig. 5 shows a related measure
titiesontheleft-andright-handsides. of the slope: 2s0(cid:104)Uh(cid:105)≡2dξcW(R)/dlns0(R). As for ξc, normal-
Theenclosedoverdensityisobtainedbysmoothingtheden- izing distances by RLag removes most of the mass dependence.
MNRAS000,1–17(2016)
Consistency relations and implications 9
z=0.97 z=0
4.8e+13 4.8e+13
4 1.4e+14 4 1.4e+14
5.2e+12 4.2e+14
1.5e+13 5.2e+12
4.6e+13 1.5e+13
3 3
4.6e+13
2 2

c
b
1 1
0 0
1 1
10-2 10-1 100 101 102 10-2 10-1 100 101 102
kR kR
Lag Lag
Figure8. Thebiasparameterbc,obtainedfromtheFourierspacecross-correlationbetweenthedarkmatterandLagrangianprotohalos
patches of halos identified at z =0.97 (left panel) and z =0 (right panel), in the Oriana (circles) and Carmen (triangles) simulations.
Halosspanawiderangeofmasses(legendisinunitsof M(cid:12)h−1).SmoothcurvesshowtheresultoffittingEq.59tothesemeasurements.
For all masses, we find that the slope has an obvious maximum we plotted bc against kRLag rather than k. This plot is similar
onscalesthatareveryclosetoRLag,andthismaximumislarger toFig.5inChanetal.(2015).Thereweusedittohighlightthe
for the less massive halos. Also note that there is an additional factthatEq.16iscrucialforestimatingtheLagrangianbiaspa-
smalldependenceonz.ThemaximumfeatureinFig.5arisesbe- rametersreliably,butwedidnotgointothedetailsofthebiasing
causethereisanintrinsicwindowfunctioninξc,andξcW changes model.Hereweinsteadfocusonthebestfitbiasparameters,and
rapidlywhentheexternalwindowfunctionhaslargeoverlapwith exploretheconsistencyrelationsamongthem.
theintrinsicone.ThishappenswhenR∼RLag,andhenceapeak Ourgoalhereistomodelthemeasurementsofbc shownin
results. Fig. 8. We start with the simple two-parameter model for the
Thequantitieswhicharemostdirectlyrelatedtotheconsis- Lagrangiancrossbiasbc:
tency relations are the mean enclosed overdensity and slope on
dW(kR)
sthcaelmeRfrLoamg.ξWc,hthileeytchaenparlesvoiobueswerxipttreenssaiosnssusmhsowovherowFotuorieesrtismpaactee beff(k)=b10W(kR)+b012dlns0(R). (59)
quantities:
(cid:104)δh(cid:105)=ξcW(RLag)=(cid:90) d2kπk22 Pc(k)W(kRLag) (57) Atehxsecewdpeetnmtsheitanyttitoohunrreedsb,h0t1ohldiestawwnhodabtsilatohspepeyacrcoaanmllsetbrt1ae1ri)ns.tbsW10(eMahnuadsvsebo0c1ehtaarnailgs.eed2f0rno1mo2-,
and tation from b(11) in Sec. 2 to b10, etc., to highlight the fact that
(cid:104)Uh(cid:105)= ddξscW0((RRLLaagg)) =(cid:90) d2kπk22 Pc(k)ddWs0((kRRLLaagg)). (58) tshimeubliaatsieosnsinaSreeca.v2eraargeeduno-vaevrertahgeecdo,nwshtrialeintthsoasesimneEasqu.r2e.dinthe
ThesmoothblackcurvesinFig.8showtheresultoftreating
Figs. 6 and 7 show that our direct estimates of (cid:104)δ (cid:105) and (cid:104)U (cid:105)
usingPc(k)areinexcellentagreementwiththemorhetraditionhal bm1e0a,sbu0r1emaenndtsR. Inasprfraeceticpea,rtaomeentesursrewthheant tfihtetinlogw-Ekq.pa5r9t,twohtihche
estimators which are based on ξc(r). Both estimators in Fig. 6 haslargeerrorbarsrelativetothehighk-region,isproperlyfitted,
show that the enclosed overdensity is larger for smaller masses
(vsitoruicstdlyirsepcteamkeinagsu,rfoermsemntaslloerftνhsce)m.Teahnisoisveirndaegnrseiteymwenitthwinitphrpotroe-- wkReLfiargst<de0te.1r5m.inWeeb1c0anbydfiotttihnigs tboectahuesleowWk→con1staanndtptahretsucpalteo-
halopatches(Shethetal.2001;Robertsonetal.2009;Eliaetal. dependenttermvanishesatk(cid:28)RLag (c.f.Fig.2).Wethenkeep
2012;Despalietal.2013). thebest-fitb10fixedandfittheremainingparametersb01andR.
Evidently,thissimplemodelforb isabletoprovideagoodfit
With these direct estimates of the overdensity and slope in eff
overtheentirerangeofk.
hand, we are now ready to estimate the other quantities which
appearintheconsistencyrelations:large-scalebiasfactors. Fig. 9 shows the best-fit parameters, b10, b01, and(cid:112)R as a
functionofhalomass,expressedintermsofνsc=δsc(z)/ s0(M)
where large M has large νsc). Note that s0 uses Eq. 16 for the
3.3 Lagrangian cross bias parameters in Fourier smoothingwindow,andthezdependenceofδscallowsustoeasily
compare measurements for halos identified at different redshifts
space
(inthiscasez=0.97and0).Thebiasparametersfromdifferent
The symbols in Fig. 8 show the Lagrangian protohalo bias pa- zcoincidewitheachotherratherwell,inagreementwithprevious
rameters bc for halos identified at z = 0.97 and 0 for a range work(Sheth&Tormen1999).Wefindthat,althoughthebestfit
of halo masses (as indicated) obtained using Eq. 53 from the Pc RisclosetoRLag,thebest-fitR/RLagdecreasesasνscincreases.
measurements.Someofthemassdependenceisremovedbecause For low νsc, especially νsc (cid:46) 1.5, the model does not work very
MNRAS000,1–17(2016)
10 K. C. Chan, R. K. Sheth, and R. Scoccimarro
3.0 1.30 1.14
2.5 1.25 1.12
1.20 1.10
2.0
1.15 1.08
s0 1.5 s0 Lag
p 1.p10 R1.06
0 1 /
b1 1.0 b0 R
1.05 1.04
z=0.97, O
0.5 z=0, O
1.00 1.02
z=0.97, C
z=0, C
0.0 SC 0.95 1.00
EC
0.5 0.90 0.98
1.0 1.5 2.0 2.5 3.0 3.5 4.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0
ν ν ν
sc sc sc
Figure 9. Best-fit bias parameters b10 and b01, and window scale R as a function of protohalo mass, shown as the scaled variable
(cid:112)
νsc=δsc(z)/ s0(M).SymbolsshowmeasurementsintheOriana(bluecircles)andCarmen(redtriangles)simulationsrespectively,for
halosidentifiedatz=0.97(dashed)and0(solid).Thepanelsforb10andb01alsoshowpredictionsfromtheexcursionsetpeakmodelin
whichhalosareassumedtoformfromadeterministicsphericalcollapse[(δc=δsc)solidyellowcurve]orastochasticellipsoidalcollapse
in(Paranjapeetal.2013)(cyanbandbrackets0.1≤β≤0.5).
well. There are additional small differences between the results Section3.6.Fornow,wesimplynotethattheconsistencyrelation
fromCarmenandOrianasimulations. has used just n = 2 large scale bias factors to correctly predict
In principle, we should be able to use our Fourier space es- the enclosed Lagrangian overdensity of the protohalos measured
timates of Pc (essentially the solid curves in Fig. 8) to predict onmuchsmallerscales.
themeasuredξc ofFig.4.Sincethisisnotthemainfocusofour It is possible that the slight discrepancy which is apparent
study,weonlyshowtheresultinFig.A3. at lower νsc is indicating that our model, which assumes there
Inaddition,ithappensthatourmeasurementsofb10andb01 are only two important parameters, is overly simplistic. We ar-
arereasonablywelldescribedbytheexcursionsetpeakformulae guedthat,inprinciple,ouranalysispermitsonetoaddasmany
ofParanjapeetal.(2013)providedoneassumeshalosformedfrom parameters as desired to the bias prescription, so that the effec-
an ellipsoidal collapse with some stochasticity. In that model, a tivebiasagreeswiththesimulationresultswell.Thatoursimple
√
stochastic term proportional to s0 is added to the spherical modelreturnsanestimatewhichiswithinafewpercentofthedi-
collapsebarriersothatthebarrierincreasesasνscdecreases.The rectmeasurementisanon-trivialself-consistencycheckthatthe
amountofstochasticityiscontrolledbythefreeparameterβ.The simplest requirements (recall we have just two bias parameters)
cyan band shows the range spanned by their models with 0.1≤ alreadycapturemostoftheeffectsofbias.Theagreementshown
β ≤ 0.5. The yellow curve shows β = 0 (spherical collapse and inFig.10meansthatconsistencyrelationshaveopenedthedoor
no stochasticity). While the general agreement is encouraging, tousingthescaledependenceofbiastoconstrainhaloformation
constrainingspecificcollapsemodelsisnotthemainfocusofour physics.Encouragedbythisresult,wenowstudytheconsistency
study(formorediscussionofmodels,seeCastorinaetal.2016b). relationassociatedwiththesecondvariable:theslope.
Rather,ourgoalistotesttheaccuracyoftheconsistencyrelation,
Eq.54.
3.5 Consistency relation for the slope
Wewillnowstudytheconsistencyrelationfortheslopeurather
3.4 Consistency relation for the smoothed thantheenclosedoverdensityδ .Alittlealgebrashowsthatthe
h
enclosed overdensity secondofEqs.54canbere-writtenas
The first consistency relation of Eq. 54 equates the sum of the γνub10+ b01 = √(cid:104)u(cid:105) =2γνu dξcW(R), (60)
large scale bias factors to (cid:104)δh(cid:105)/s0, where (cid:104)δh(cid:105) is the mean over- γνu s0 ds0(R)
density within the protohalo patch. The red and green symbols whereuistheslopevariablenormalizedbyitsrmsvalue,defined
in the upper panel of Fig. 10 show our direct estimates of (cid:104)δ (cid:105) inEq.6,andξW(R)istheenclosedoverdensitydefinedinEq.55
h c
usingEq.55andEq.57withWEff (i.e.,theyarethelowersetof evaluated on scale RLag (also see Eq. 57), so dξcW/ds0 is the
symbols in Fig. 6). The blue symbols and curves show our esti- originalunnormalizedslope(Eq.58).
mateof(cid:104)δh(cid:105)fromthelargescalebiasfactors:(b10+b01)s0(M). The first consistency relation states that s0 times the sum
The lower panel shows that the consistency relation estimate of of the bias factors is an estimator of δ . For similar reasons, it
h
(cid:104)δ (cid:105) is within 4% of the direct estimate (we used the one shown isinterestingtoviewthesecondconsistencyrelationasequating
h
bythegreensymbols).Thisagreementisremarkable,giventhat s0(b10+b01/γν2u) to 2dξcW/dlns0 on scale RLag. We take this
the two estimates have very different systematics: that based on latter quantity from the direct measurements shown in Fig. 5;
b10 andb01 isderivedfromtwo-pointmeasurementsonrelatively these are shown as the red curves in the upper panel of Fig. 11.
large scales, whereas the traditional view of the direct measure ThegreensymbolsandcurvesshowtheFourier-basedestimator
is that it is more like a one-point measurement on substantially ofEq.58.Theseclearlyshowsteeperslopesforlowermassproto-
smallerscales.Wewillhavemoretosayaboutthisagreementin halos,whichweassertedearlierwasagenericpredictionofpeak
MNRAS000,1–17(2016)