Table Of ContentPrimordial Bispectrum and Trispectrum Contributions to the Non-Gaussian
Excursion Set Halo Mass Function with Diffusive Drifting Barrier
Ixandra E. Achitouv⋆, and Pier Stefano Corasaniti⋆
†
⋆
Laboratoire Univers et Th´eories (LUTh), UMR 8102 CNRS,
Observatoire de Paris, Universit´e Paris Diderot,
5 Place Jules Janssen, 92190 Meudon, France and
† Laboratoire Astroparticule Particule Cosmologie, 10,
rue Alice Domon et Lonie Duquet 75205 Paris, France
The high-mass end of the halo mass function is a sensitive probe of primordial non-Gaussianity
(NG). In a recent study [9] we have computed the NG halo mass function in the context of the
2
Excursion Set theory and shown that the primordial NG imprint is coupled to that induced by
1
the non-linear collapse of dark matter halos. We also found an excellent agreement with N-body
0
simulation results. Here, we perform a more accurate computation which accounts for the interval
2
validityofthebispectrumexpansiontonext-to-leadingorderandextendthecalculation tothecase
l of a non-vanishingprimordial trispectrum.
u
J
9
I. INTRODUCTION path-integral formulation of the Excursion Set Theory
1
the authors have shown that the first-crossing distribu-
tion of correlated random walks can be computed as a
] TheExcursionSetTheoryinitiallyintroducedbyBond
O perturbative expansion about the Markovian solution.
et al. [1] provides a self-consistent mathematical frame-
C Using this methodology it has been possible to derive
worktoinferthepropertiesofthehalomassdistribution
analyticalformulae for the halo massfunction under dif-
h. from the statistics of the initial density field. The for- ferent halo collapse model assumptions as well as Gaus-
p malism generalizes the original Press-Schechter idea [2]
sian and non-Gaussian (NG) initial conditions [4–6].
- byformulatingthe halomasscountingproblemasoneof
o In a series of papers [7–9] we haveused this formalism
stochastic calculus. The starting point is the realization
r to evaluate the imprint of the non-spherical collapse of
t that at any location in space the linear matter density
s halos on the mass function. To this end we have intro-
a fluctuationfieldperformsarandomwalkasfunctionofa
duced an effective stochastic Diffusive Drifting Barrier
[ filtering scale R. In average, this scale naturally defines
(DDB) model which parametrizes the main features of
a mass scale M = ρ¯V(R), where ρ¯ is the mean mat-
1 the ellipsoidal collapse of halos. Accounting for such ef-
v ter density and V(R) the enclosed spatial volume. By
fects can reproduce the halo mass function from N-body
6 counting the number of trajectories which first-cross a
simulations with remarkable accuracy both for Gaussian
9 collapsethresholditisthenpossibletocomputethefrac-
7 tion of mass elements in halos F(M) and consequently and non-Gaussian initial conditions.
4 derive the halo mass function dn/dM = (1/V)dF/dM. Here,weextendtheworkpresented[9]toderiveamore
7. The requirement of first-crossing is key to solving the accurate expression of the contribution to the halo mass
0 so called “cloud-in-cloud” problem affecting the original function of the primordial bispectrum expanded in the
2 Press-Schechter result. In fact, the first-crossing condi- largescalelimittonext-to-leadingorderandcomputethe
1 tionguaranteesthatinthesmallscalelimit(R 0)and leadingordercontributionoftheprimordialtrispectrum.
: →
v independently of the properties of the randomwalks the The paper is organizedas follows. In Section II we re-
i fraction of mass into collapsed objects always tends to view the path-integral formulation of the Excursion Set
X
unity. andits applicationto Gaussianand non-Gaussianinitial
r
a The Excursion Set is analytically solvable in the case conditions in the case of the DDB model. In Section III
we evaluate a lower limit on the interval validity of the
of uncorrelated (Markov) random walks for which the
bispectrum expansion at next-to-leading order. In Sec-
evaluation of the first-crossing distribution is reduced to
tion IV and V compute the bispectrum and trispectrum
solving a standard diffusion problem. However,uncorre-
contribution to the mass function respectively. Finally,
lated random walks are generated by a special filtering
we present our conclusion in Section VI.
of the linear density field which corresponds to a non-
physical halo mass definition. In contrast any filtering
which specifies a physically meaningful mass generates
correlatedrandomwalksforwhichF(M)canbeinferred
II. EXCURSION SET MASS FUNCTION AND
only through a numerical computation. This has repre-
DIFFUSIVE DRIFTING BARRIER
sented a major limitation since Monte Carlo simulations
arecomputationally expensiveand moreoverdo not pro-
vide the same level of physical insight of analytic solu- Here, we briefly review the main features of the path-
tions. The seminal work by Maggiore & Riotto [3] has integral formulation of the Excursion Set Theory. First,
made a major step forward in this direction. Using the let us introduce the variance of the linear density field δ
2
filtered on a scale R: the connected correlators. Once these are specified then
Eq.(2)canbeintegratedinthecontinuouslimittofinally
1
σ2(R) S(R)= dkk2P(k)W˜2(k,R), (1) obtain the first-crossing distribution
≡ 2π2
Z
where P(k) is the linear matter power spectrum and dF (S)= ∂ ∞dY Π(Y ,Y,S) , (5)
0
W˜2(k,R) is the Fourier transform of the filter function dS ≡F −∂S (cid:20)ZYc (cid:21)
in realspace, W(x,R). As already mentioned, by select-
ing a volume V(R) = d3xW(x,R) the filter naturally and the halo mass function is given by
associates a mass to the enclosed region, M = ρ¯V(R).
Thus we have a one-toR-one relation between R (or M) dn ρ¯ dlogσ−1
=f(σ) , (6)
and S. dM M2 dlogM
In the Excursion Set Theory the filtered density field
δ(x,R)atanyrandompointinspaceperformsarandom wheref(σ)=2S (S)isthesocalled“multiplicity”func-
F
walk as function of R, M or equivalently S which plays tion.
the role of a pseudo-time variable. The random walks
start at S = 0 with δ = 0, since in the large scale limit
(R ) we have S 0 and the matter density dis-
→ ∞ → A. Gaussian Initial Conditions
tribution tends toward homogeneity, i.e. δ 0. We are
→
interested in counting trajectories that at a given value
of S cross for the first time a collapse density threshold LetusconsideraGaussiandensityfieldsmoothedwith
B such that δ =B. This threshold encodes all informa- a top-hat filter in real space. On average the density
tions on the gravitational collapse of halos. In order to field is homogeneous, thus implying that δ(S) c = 0.
h i
model features of the non-sphericalcollapse, the absorb- Therefore, due to the Gaussian nature of the field the
ing barrier B is promoted to a stochastic variable (see only non-vanishing connected correlator is the 2-point
e.g. [4, 10, 11]) also performing a random walk as func- function δ(S)δ(S′) c, while all higher-order connected
h i
tion of S (i.e. R or M). In such a case it is convenient correlators identically vanish. Maggiore & Riotto [3]
to introduce Y =B δ, which performs a random walk have shown that for standard cosmological scenarios
starting at Y(0)=Y−with barrier crossing at Y =0. with Cold Dark Matter power spectra, the 2-point func-
0 c
Our goal is to compute the probability distribution tion smoothed with a sharp-x is well approximated by
Π(Y0,Y,S) of trajectories starting at Y0 which reachthe hδ(S)δ(S′)ic = min(S,S′) + ∆(S,S′), where the first
value Y at S without ever touching the barrier Y = 0. termcorrespondsto Markovrandomwalks generatedby
c
This can be computed as a path-integral over the en- asharp-kfilterandthesecondtermiswellapproximated
semble of the random trajectories (see [3] for a detailed by ∆(S,S′)=κS/S′(S′ S) with a nearly constant am-
−
derivation). Let us discretize the pseudo-time variable S plitude κ < 1. Thus, the pseudo-time correlations in-
in equally spaced steps, ∆S =ǫ, such that S =kǫ with duced by the filter function can be treated as small cor-
k
k = 1,..,n. The probability distribution of trajectories rection about the Markoviancase and the mass function
startingatY andendinginY atS andthathavenever obtained using a perturbative expansionof the partition
0 n n
crossed the barrier before is given by function in the path-integral in powers of κ.
Concerningthebarrierrandomwalks,in[7,8]wehave
∞ ∞ introducedastochasticmodelwithlineardriftandGaus-
Π (Y ,Y ,S )= dY .. dY W(Y ,..,Y ,S ),
ǫ 0 n n 1 n 1 0 n n
ZYc ZYc − sian diffusion characterized by hB(S)i = δc + βS and
(2) B(S)B(S ) = D min(S,S ), where δ is the linearly
′ c B ′ c
h i
where extrapolated critical spherical collapse density, β is the
average linear rate of deviation from the spherical col-
n
W(Y0,..,Yn,Sn)= DλeiPni=1λiYihe−iiP=1λiY(Si)i, laabposuetptrheedicatvieornagaend1.DIBn issutchheaamcapsleittuhdeenoofnt-hveanscisahtitnegr
Z
(3)
where the brackets ... refer to an ensemble average of
h i
the random walks and the averaged quantity is the ex-
plonential of 1 IntheExcursionSetthebarrierdiffusioncoefficientparametrizes
the stochasticity inherent to the ellipsoidal collapse of halos.
∞ ( i)p n n However, itis important to keep inmindthat inthe Excursion
Z = −p! ... λi1...λiphYi1..Yipic (4) Set halos can form out of any random position. On the other
hand,numericalsimulationsshowthathalosformpreferentially
Xp=1 iX1=1 iXp=1 out of peaks of the linear density field as suggested by the hi-
which is the partition function of the system written erarchical model of structure formation. Thus when comparing
in terms of the p-point connected correlation functions with N-body results the value of DB can be biased by the un-
derlyingassumption ofthe ExcursionSetapproach (see[12] for
Y ..Y of the random walks. Thus, the the proper-
h i1 ipic an extension of the formalism to random walks around density
ties of the stochastic system are entirely determined by peaks).
3
connected correlators of the Y variable are δ(S )δ(S )δ(S ) with
i j k c
−h i
d3k d3k d3k
hY(S)ic = δc+βS (7) hδ(Si)δ(Sj)δ(Sk)ic = (2π)i3(2π)j3(2π)k3W˜(ki,Ri[Si])
Y(S)Y(S′) c = (1+DB)min(S,S′)+∆(S,S′).(8) Z
h i W˜(k ,R [S ])W˜(k ,R [S ]) (k ) (k ) (k )
j j j k k k i j j
× M M M ×
Substituting these expressions in Eq. (4) and perform- ζ(k )ζ(k )ζ(k ) ,
i j k c
ing a double expansion in κ and β we have derived the ×h i
(15)
Gaussian multiplicity function
where W˜ is the Fourier transform of the sharp-x filter,
fG(σ)=f0(σ)+fκ=1(σ) (9) M(k) = 2/(5H02Ωm)T(k)k2, H0 is the Hubble constant,
Ω the matter density, T(k) the transfer function and
m
where f0(σ) is the Markovian contribution and fκ=1(σ) ζ(k) is the curvature perturbation with
isthefiltercorrectiontofirstorderinκanduptosecond
ζ(k )ζ(k )ζ(k ) =(2π)3δ (k +k +k )B(k ,k ,k ),
order in β which read as i j k c D i j k i j k
h i
(16)
whereB(k ,k ,k )isthesocalled“reduced”bispectrum.
f0(σ)= δc 2ae−2σa2(δc+βσ2)2 (10) By expaindjingkEq. (3) in powers of the amplitude of
σ π
r thereducedbispectrum(usuallyparametrizedbytheco-
efficient f ), we obtain to first-order in f the non-
and NL NL
Gaussian part of the first-crossing distribution
fκ=1(σ)=f1m,β−=m0(σ)+f1m,β−(1m)(σ)+f1m,β−(2m)(σ) (11) (S)= ∂ F (S) (17)
NG NG
F −∂S
with
where F (S) is the continuous limit of
NG
f1m,β−=m0(σ)=−κ˜δσcr2πa(cid:20)e−2aσδc22 − 12Γ(cid:18)0,a2δσC22(cid:19)(cid:21), (12) FNG(S)= 61 n hδ(Si)δ(Sj)δ(Sk)ic×
i,jX,k=0 (18)
∞ ∞
dY dY ...dY ∂ ∂ ∂ W (Y ,...,Y,S),
1 n 1 i j k 0 0
f1m,β−(1m)(σ)=−aδcβ κ˜Erfc δc 2σa2 +f1m,β−=m0(σ) , ×ZYc ZYc −
(cid:20) (cid:18) r (cid:19) ((cid:21)13) whereW0(...)istheGaussianMarkovianprobabilityden-
sity distribution. Eq. (18) can be evaluated provided we
have an analytical expression for the primordial bispec-
β
f1m,β−(2m)(σ)=−aβ 2σ2f1m,β−=m0(σ)+δcf1m,β−(1m)(σ) , (14) trum. In [9] we have used the standardapproachof con-
(cid:20) (cid:21) sidering a triple Taylor series of the primordial bispec-
trum in the largescale limit. In the next Section we will
where κ˜ = aκ and a = 1/(1 + D ). Equation (14)
B study indetailthe rangeofvalidityofsuchanexpansion
includes a term (β2) which was missing in the orig-
and infer the relevant contribution to the non-Gaussian
O
inal derivation presented in [7, 8]. As we explain in
halo mass function.
Appendix A this is due to having neglected a fac-
tor exp( β2S/2) in the computation of the probabil-
ity Πǫ(Yc−,Yc,S)2 and which enters the calculationof the III. INTERVAL VALIDITY OF PRIMORDIAL
memory-of-memory term to first order in κ. BISPECTRUM EXPANSION
Letusexpandthe bispectrumEq.(15)inatripleTay-
B. Non-Gaussian Initial Conditions lor series around S S S S (for convenience we
i j k
∼ ∼ ∼
set S =S ):
n
In the case of non-Gaussian initial conditions the
∞ ( 1)p+q+r
higher-order connect correlators of the linear density δ(S )δ(S )δ(S ) = − (S S )p
i j k c i
h i p!q!r! − ×
field are non-vanishing. Let us consider the case of pri-
p,q,r=0
X
mordial non-Gaussianity sourced by a bispectrum term,
(S S )q(S S )rG(p,q,r)(S)
hence in addition to Eq. (7) and (8), the partition func- × − i − k 3
(19)
tioncontainsthe contributionofanon-vanishing3-point
connected correlation function Y(S )Y(S )Y(S ) =
i j k c where
h i
dp dq dr
G(p,q,r)(S) δ(S )δ(S )δ(S ) .
3 ≡ dSpdSq dSrh i j k ic
i j k (cid:12)i,j,k=n
2 WethankRubenvanDrongelenforpointingthistous. (cid:12) (20)
(cid:12)
(cid:12)
4
Weexpectthesignatureofprimordialnon-Gaussianity We can now derive a lower limit, S , on the value
min
tobestrongeratlargescales(S 0)wheretheevolution of S ,S ,S for which such an expansion remains valid.
i j k
→
of the density field remains linear. Hence, the leading This is obtained by imposing the next-to-leading order
order contribution to the halo mass function is given by term to be smaller than the leading one. We find
the lowestorder termofthe bispectrum expansion. This
correspondstohavingp+q+r =0whichgivestheleading (S)
3
order term δ3(S) that can be computed numerically Smin =S 1− S (S) , (25)
using Eq. (1h5) foria given type of primordial NG. It is (cid:20) U3 (cid:21)
convenienttointroducethenormalizedskewness (S)=
3 usingthefitting formulaefor (S)and (S)derivedin
δ3(S) /S2. In [9] we have provided fitting formSula for S3 U3
[9] we find that to good approximationS =S/α with
h i min
Slo3c(aSl)anadndeqitusiladteerriavaltnivoens-Gaacucussriaatneittyo. a few percent for α−lo1c = 0.373 and α−eq1ui = 0.382 for local and equilateral
NG respectively.
The next-to-leading order contribution is given by
three terms corresponding to the case p + q + r = 1.
Hence, up to next-to-leading order the primordial bis-
pectrum reads as IV. NON-GAUSSIAN HALO MASS FUNCTION
AND BISPECTRUM EXPANSION ACCURACY
δ(S )δ(S )δ(S ) = δ3(S) (S S )G(1,0,0)(S)+
h i j k ic h i− − i 3
(S S )G(0,1,0)(S) (S S )G(0,0,1)(S), Having inferred a lower limit on the interval validity
− − j 3 − − k 3 of the bispectrum expansion up to next-to-leading order
(21)
we can infer a more accurate estimate of its contribu-
tion to the multiplicity function. Given the bispectrum
since S S S we can collect the terms in (S S ),
i ∼ j ∼ k − i expansion Eq. (24) we can split Eq. (18) as
moreover by computing G(1,0,0), G(0,1,0) and G(0,0,1) as
3 3 3
derivatives of Eq. (15) one can notice that
F (S)=FL (S)+FNL(S). (26)
NG NG NG
As shown in [9] the leading order term is given by
dR d
G1,0,0(S)+G0,1,0(S)+G0,0,1(S)= δ3(R) (22)
3 3 3 dS dRh i
FL (S)= 1S2S (S) ∞dY ∂3 Π (Y ,Y,S) (27)
and introducing NG 6 3 ∂Y3 0 0
ZYc c
1 dR d
U3(S)= S dS dRhδ3(R(S))i, (23) where Π0(Y0,Y,S) is the probability distribution of the
GaussianrandomwalksinthecaseoftheDiffusingDrift-
we can rewrite Eq. (21) as ing Barrier model given by Eq. (8) in [7]. The integral
canbecomputedanalyticallytofinallyobtaintheleading
δ(S )δ(S )δ(S ) S2 (S) (S S )S (S). (24) order contribution to the multiplicity function [9]:
i j k c 3 i 3
h i ≈ S − − U
a(δ +βσ2)2
a 2a c a2 a a2
fL (σ)= σe− 2σ2 S (σ) δ4 2 δ2 1+3 βδ3+3aβδ +a2β3σ2δ +3a2β2δ2+13aβ2σ2 +
NG 6 π 3 σ4 c − σ2 c − σ2 c c c c
r (cid:26) (cid:20) (cid:21)
dS (σ) a 2 a dS (σ)
+dlo3gσ σ2δc2−1+3aβδc+4aβ2σ2 + 3a3β3σ4e−2aβδcErfc 2σ2(δc−βσ2) 4S3(σ)+ dlo3gσ
(cid:20) (cid:21)(cid:27) (cid:20)r (cid:21)(cid:26) (cid:27)
(28)
On the other hand let us detail more the derivationof [9]. In such a case we have from Eq. (18) that
the next-to-leadingorderterm whichdiffers fromthatof
1 imax
FNL(S)= SU (S) (S S )
NG −6 3 − i ×
iXmin
∞ ∞ ∞
dY dY ... dY ∂ ∂ ∂ W ,
1 n 1 i j k 0
×ZYc j,k ZYc ZYc −
X
(29)
5
wherewehavedecomposedthesuminEq.(18)andkept and
only the terms up to n 1. In fact, as shown in [9]
the integral in dYn of the−terms with i,j,k =n vanishes ∂ ∂2INαL = a S dSi e−2Sai(Y0+βSi)2
sisinecaesythteoinshteogwratnhdastaretotal∂d2e/ri∂vYat2i,vetsh.uFsurthermoreit ∂S ∂Yc2 −π ZαS Si3/2 ×
j,k → c 2aβ+a2β2δ +aδc 3+2aβδ +aδc2
c c
P × − S − S ×
(cid:26) i (cid:20) i(cid:21)(cid:27)
FNNGL(S)=−61SU3(S)i=nXi−m1in(S−Si)× ×(e−2√a2βS2(−S−SSii) (cid:20)1− α1 + α1 S−SiSi + α1aβ2√SS+−SSii(cid:21)+
∞ ∂2 ∞ ∞ aπ 1 a
×ZYc dY ∂Yc2(cid:20)ZYc dY1...ZYc dYn−1∂iW0(cid:21), +r 2 β(cid:18)1− α(cid:19)Erfc(cid:20)−βr2(S−Si)(cid:21)(cid:27).
(30) (35)
where the sum is bounded from below due to the fact
that S < S < S. The multiple integral in the above
min i
expression can be computed by part and using the fact
that W obeys the Chapman-Kolmogorovequation
0
W (Y ,..,Yˆ,..,Y,S)=W (Y ,..,Y ,S S )
0 0 i 0 0 c i− imin × (31)
W (Y ,..,Y,S S +S )
× 0 c − i imin
we have
n 1
1 −
FNL = SU (S) (S S )
NG −6 3 − i ×
i=Ximin
∞ ∂2 S S
dY Π Y ,Y ,S Π Y ,Y,S S + .
× ∂Y2 0 c i− α c − i α
ZYc c (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
(32)
where Π(Y ,Y ,S S/α)andΠ(Y ,Y,S S +S/α)are
0 c i c i
− −
given by Eq. (3.20) and (3.21) in [9] respectively. In
the continuous limit and taking into account that the
bispectrumexpansionuptonext-to-leadingorderisvalid
in the range Smin ≈ S/α < Si < S we have in=−i1min →
S
dS .
S/α i P
R Finally, we obtain the first-crossing distribution to
next-to-leading order which reads as
1 ∂2Iα dU ∂ ∂2Iα
NL(S)= NL U +S 3 +SU NL
FNG −6 dY2 3 dS 3∂S dY2
(cid:26) c (cid:20) (cid:21) c (cid:27)
(33)
with
∂∂2IYNαc2L =−πa ZαSS Sdi3S/i2e−2Sai(δc+βSi)2(cid:20)S(cid:18)1− α1(cid:19)−Si(cid:21)× FIG. 1: Leading order (red dash line), next-to-leading order
δ δ2 (blue dot line) contribution to the non-Gaussian multiplicity
2aβ+a2β2δ +a c 3+2aβδ +a c function(blacksolidline)forlocal(toppanel)andequilateral
c c
× − S − S ×
(cid:26) i (cid:20) i(cid:21)(cid:27) (bottom panel) non-Gaussianities respectively.
β a
× e−a2β2(S−Si)+ 2 πa(S−Si)Erfc −β 2(S−Si) The integrals in Eq. (34) and (35) can be computed
(cid:26) (cid:20) r (cid:21)(cid:27)
p (34) numerically to finally evaluate the next-to-leading order
6
contribution to the multiplicity function, V. TRISPECTRUM CONTRIBUTION TO THE
NON-GAUSSIAN HALO MASS FUNCTION
fNNGL(σ)=2σ2FNNGL(σ2). (36) A number of scenarios of primordial inflation predict
deviationfromGaussianityofthe form(see e.g. [14,15])
unIintsFofigf. (1fworelopcloatl(ftNLopG(pσa)n,efl)NNaGLn(dσe)qaunildattehraeilr(bsuotmtomin ζ =ζG+ 53fNL(ζG2 −hζG2i)+ 295gNLζG3 +O(ζG4), (37)
NL
panel)non-Gaussianity. WehavesettheDDBmodelpa-
where ζ is the primordial curvature perturbation and
rameterstothevaluesbestfittingtheGaussianhalomass G
g is the amplitude of the cubic term which give rises
function inferred from Gaussian N-body simulations in NL
toanon-vanishing4-pointconnectedcorrelationfunction
[13]. The mass interval shown here corresponds to that
of the linear density field given by
probed by these numerical simulations. We may notice
that in the local and equilateral cases the leading or-
d3k d3k d3k d3k
der term is larger than the next-to-leading order one at i j k l
δ(S )δ(S )δ(S )δ(S ) =
high-masses (log σ 1 >0.1), while at lower masses the h i j k l ic (2π)3(2π)3(2π)3(2π)3
10 − Z
next-to-leading order is larger. This is expected since W˜(k ,R )W˜(k ,R )W˜(k ,R )W˜(k ,R )
i i j j k k l l
as already mentioned the signature of primordial non- ×
(k ) (k ) (k ) (k ) ζ(k )ζ(k )ζ(k )ζ(k )
Gaussianityatlargemassesresultsofthe lowestorderin ×M i M j M k M l h i j k l ic
(38)
the bispectrum expansion.
We can now evaluate the overall contribution to the where ζ(k )ζ(k )ζ(k )ζ(k ) =(2π)3δ (k +k +k +
i j k l c D i j k
halo multiplicity function, f(σ) = fG(σ)+fNG(σ). In kl)T(khi,kj,kk,kl)andT(ki,kij,kk,kl)isthetrispectrum.
Fig.2weplottherelativedifferenceoftheNGhalomass The amplitude of the trispectrum is affected not only by
functionwithandwithoutnext-to-leadingordertermfor the cubic term in Eq. (37) and parametrized in terms
local (top panel) and (bottom panel) equilateral non- of g , but also from the skewness as parametrized by
NL
Gaussianity respectively in the case of fNL = 150. As fNL. In particular, For models in which curvature per-
wecanseethe differencesisnolargerthan2%inthelow turbations are sourced by a single scalar field, as in the
massrange,hence the next-to-leadingtermremainsneg- case of the curvaton model (see e.g. [16]) the skewness
ligibleevenforlargenon-Gaussianitiesinthe massrange contribution to the kurtois is given by τ2 =36/25f2 .
corresponding to halos with M > 1013M and can be NL NL
We can compute the trispectrum contribution to the
neglected for practical purposes. ⊙
multiplicity function by including the 4-point connect
correlatorinthepartitionfunctionandexpandthepath-
integralforsmallvaluesofthetrispectrumamplitude. As
in the case of the bispectrum, to leading order in a large
scale expansion the trispectrum can be approximated as
δ(S )δ(S )δ(S )δ(S ) δ4(S) (39)
i j k l c c
h i ≃h i
To first-order in the trispectrum amplitude we have
∂
Tri,L(S)= FTri,L(S), (40)
FNG −∂S NG
where FTri,L(S) is the continuous limit of
NG
1
FTri,L(S)= δ4(S)
NG −4! h ic×
i,j,k,l
X (41)
∞ ∞
dY dY ...dY ∂ ∂ ∂ ∂ W .
1 n 1 i j k l 0
×ZYc ZYc −
FIG.2: Relativedifferencebewteenofthenon-Gaussianhalo Using the fact that ∂4/∂Y4 we obtain
mass function with and without next-leading order contri- i,j,k,l → c
bution in the case of local (panel a) and equilateral non- P
Gaussianity (panel b). FTri,L(S)= 1 δ4(S) ∂4 ∞Π (Y ,Y,S)dY,
NG −4!h ic∂Y4 0 0
c ZYc
(42)
7
contribution to the multiplicity function:
fNTrGi,L(σ)=2S4(σ)σ6(aβ)4e−2aβδc×
a
×Erfc 2σ2(δc−βσ2) +S4(σ)e−2σa2(δc+βσ2)2×
(cid:20)r (cid:21)
2a 1 11 1
(aσ)2 βσ2+ aβ3σ4 δ +a(βσ)2δ +
c c
× πS −2 6 − 8
r (cid:20)
1 1 1 1 δ3
+ a2(βσ)4δ + (aσδ )2β3+ (aβ)2δ3 a c +
24 c 6 c 4 c − 6 σ2
1a2 1 δ5 1d (σ)
+ 6σ2βδc4+ 24a2σc4 + 3 dSlo4gσσ6(aβ)4e−2aβδc×
(cid:21)
×Erfc 2σa2(δc−βσ2) + ddSlo4(gσσ)e−2σa2(δc+βσ2)2×
(cid:20)r (cid:21)
2a 1 1 1
(aσ)2 βσ2+ aβ3σ4 δ +
c
× πS −6 3 − 8
r (cid:20)
7 1 1 δ3
+ a(βσ)2δ + aβδ2+ a c
24 c 6 c 24 σ2
(cid:21)
(45)
It can be noticed that by setting the barrier model
parameters to the spherical collapse values a = 1 and
β = 0 we recover the formula derived in [17]. It is also
worth noticing that in the spherical collapse limit and
neglectingthe filtercorrectiontofirstorderinκ,the NG
multiplicity function given by the sum of the Markovian
term,the bispectrumandtrispectrumleadingordercon-
tributions has the same functional form as that derived
in [18] using the Edgeworth expansion to describe the
non-Gaussian probability distribution of the initial den-
sity perturbations.
In Fig. 3 we plot the multiplicity function for the bis-
pectrumandtrispectrumrespectivelyfordifferentvalues
of f and g within Cosmic Microwave Background
NL NL
FIG. 3: Leading order contribution of the bispectrum (red limits [20], having used the the fitting formulae of (σ)
4
S
dash line), the trispectrum (blue dot line) and their sum for derived in [19] and set the DDB model parameters to
local typeof primordial non-Gaussianity. their Gaussian value, which is a good approximation for
f < 150. We can see that even for g = 106 and
NL NL
τ = 104 the trispectrum exceeds the bispectrum sig-
the integral can be computed analytically, NL
nal only in a very limited low mass range. In contrast
∂∂Y44 ∞Π(Y0,Y,S)dY = 2πaSe2aS(δc+βS)2× fGoarugsNsiaLn=mu0ltiapnlidcitτyNfLun=ctio1n04retmheaineffsevcetryonsmtahlel cnoomn--
c ZYc r pared to that of the bispectrum.
a2 a2 a3 a3
16(aβ)3+8 β+6 Y 14 β2δ 8 βδ2+
× − S S2 0− S c− S2 c
(cid:20)
a3 a VI. CONCLUSION
−2S3δc3 −8(aβ)4e−2aβY0Erfc 2S(Y0−βS) .
(cid:21) (cid:20)r (cid:21) (43) Thehalomassfunctioncarriesanimprintofthestatis-
tics of the primordial density field as well as the proper-
As in the case of the bispectrum, it is convenient to ties of the halo collapse process. The path-integral for-
introduce the reduced cumulant of order 4 mulation of the Excursion Set theory provides a pow-
erful and self-consistent mathematical framework to ac-
δ4(R)
(R) h i, (44) count for these effects on the halo mass function. Here,
S4 ≡ S3
we have extended a previous analysis [9] and performed
substituting in Eq. (42) and evaluating the first-crossing a more accurate derivation of the contribution of the
distribution Eq. (40) we finally obtain the trispectrum primordial bispectrum expanded in the large scale limit
8
to next-to-leading order for the Diffusive Drifting Bar- ing at the barrier, Π (Y ,Y ,S ) which enters the calcu-
ε c c n
rier model introduced in [7, 8]. We have shown that lation of the memory-of-memory term due to the non-
the next-to-leading order term of the primordial bispec- Markovian filter corrections (see Eq. (11) in [7]). Fol-
trum decomposition contributes to no more than 2% lowing the derivation presented in [8] let us consider the
∼
of the non-Gaussian mass function. Thus, for all practi- Chapman-Kolmogorovequation:
cal purposes it can be neglected. We have also derived
ananalytic formulafor the trispectrum contribution. As
∞
Π (Y ,Y ,S )= dY ψ (∆Y βε)
idneptehnedcsasoenotfertmhesbwishpicehctrcuomup,lethtehemuplatripamliceitteyrsfuenncctoiodn- ε 0 n n ZYc n−1 ε − × (A1)
Π (Y ,Y ,S ),
ing the ellipsoidal collapse of halos with the primordial × ε 0 n−1 n−1
four-point correlationfunction. Also in this case we find
with
that in the spherical collapse limit the trispectrum con-
tribution reduces to the functional form derived in the
Press-Schechter formalism using the Edgeworth expan- ψε(∆Y βε)= a e−a(∆Y2−εβε)2, (A2)
sion. However, in order to reproduce N-body simulation − 2πε
r
resultsthelatterrequirestwoad-hocprescriptions. First,
the non-Gaussian prediction is rescaled by a Gaussian expanding in Taylor seriesthe left-hand-side of Eq. (A1)
simulationcalibratedmultiplicityfunction,suchastoac- in powers of ε and the right-hand-side in powers of ∆Y
count for the imprints of the ellipsoidal collapse. Thus, we obtain
implicitly assuming that the effect of the non-spherical
collapse of halos on the mass function is independent of
the amplitude of primordial non-Gaussianity. However, ∂Πε ε2∂2Πε
Π +ε + +...=
this is not the case since our derivation explicitly show ε ∂S 2 ∂S2
n n
anocno-uGpaliunsgsibaentwcoenentrtibhuethioanlo. cTolhlaepns,eapnareammpeitreicrasla“nfdudthgee = 1 Y√−ε2βε−/aYc dx 2ε n2 x+ εβ ne−x2.
factor” is introduced to correctly reproduce the shape √π Z−∞ (cid:18) a (cid:19) 2ε/a!
ofnon-GaussianmassfunctionfromN-bodysimulations. (A3)
p
Hence, it is not surprising that such analyses find that
for large primordial non-Gaussianities the fudge factor
ItisworthnoticingthattheMarkovianprobabilitydis-
depends on f . The ExcursionSet derivationprovided
NL tribution starting at the barrier value Y = Y vanishes
0 C
hereshowsthatthesetwoempiricalprescriptionsarenot
in √ε while for Y = Y = Y the first finite corrections
0 c
necessary and provides a better physical insight on how
to the probability distribution function is in ε. At order
the stastics of the initial density field and the non-linear
one in ε for Y =Y , this equation gives
c
collapse of halos influence the mass function over the
range of mass which will be probed by the upcoming
∂Π a∂2Π β∂Π
generation of cluster surveys. ε = ε ε, (A4)
∂S 2 ∂Y2 − 2 ∂Y
n
whichhasthesameformofthantheFokker-Planckequa-
Acknowledgments
tionassociatedwiththeMarkoviansolutionfortheDDB
modelinthecontinuouslimitwhosesolutionforabarrier
We thank James G. Bartlett for his support and
atagenericpointY isgivenbyEq.(3.11)in[9]. Insuch
his advices. We also thank Ruben van Drongelen c
case for Y = Y = Y the drifting term stay in factor of
and Koenraad Schalm for useful discussions. I. Achi- 0 c
Gaussianminusanti-Gaussian. Thereforewecanassume
touv is supported by a scholarship of the ‘Minist`ere de
that
l’Education Nationale, de la Recherche et de la Tech-
nologie’(MENRT). The researchleading to these results
has receivedfunding fromthe EuropeanResearchCoun-
cil under the European Community’s Seventh Frame- e−aβ2Sn/2
Π (Y ,Y ,S )=Cε . (A5)
work Programme (FP7/2007-2013Grant Agreement no. ε c c n S3/2
n
279954).
The value of the constant C can be evaluating using
thepath-integralEq.(2)forn=2withY =Y =Y and
0 2 c
Appendix A theexplicitformofW (..)givenbyEq.(A3)in[8]. Thus,
0
equating Eq. (A5) on the left-hand-side to the result of
Here, we derive the finite correction to the probabil- theintegraloverW (...)ontheright-hand-sideweobtain:
0
ity distribution of the random walks starting and end- C = a .
2π
p
9
[1] J. R. Bond, S. Cole, G. Efstathiou & G. Kaiser, Astro- Astron. Soc. 323, 1 (2001)
phys.J. 379, 440 (1991) [12] A. Paranjape, R.K. Sheth,arXiv:1206.3506
[2] W. H. Press & P. Schechter, Astrophys. J. 187, 425 [13] A. Pillepich, C. Porciani, O. Hahn,Mont. Not. Roy.As-
(1974) tron. Soc. 402, 191 (2008)
[3] M.Maggiore&A.Riotto,Astrophys.J.711,907(2010) [14] M. Sasali, J. Valiviita, D. Wands, Phys. Rev. D 74,
[4] M.Maggiore&A.Riotto,Astrophys.J.717,515(2010) 103003 (2006)
[5] M.Maggiore&A.Riotto,Astrophys.J.717,526(2010) [15] K. Enqvist,T. Takahashi, JCAP 09, 012 (2008)
[6] A. De Simone, M. Maggiore & A. Riotto, Mont. Not. [16] D.H. Lyth,D. Wands,Phys. Lett.B 524, 5 (2002)
Roy.Astron.Soc. 412, 2587 (2011) [17] M. Maggiore, A. Riotto, Mont. Not. Roy. Astron. Soc.,
[7] P.S.CorasanitiandI.E.Achitouv,Phys.Rev.Lett.106, 405, 1244 (2010)
241302 (2011) [18] M., Lo Verde, A. Miller, S. Shandera, L. Verde, JCAP
[8] P.S. Corasaniti and I.E. Achitouv, Phys. Rev. D 84, 04, 014 (2008)
023009 (2011) [19] S. Yokoyama, N. Sugiyama, S. Zaroubi, J. Silk, Mont.
[9] I.E.Achitouvand P.S.Corasaniti, JCAP02,002 (2012) Not. Roy.Astron. Soc., 417, 1074 (2011)
[10] E. Audit, R. Teyssier & J.-M. Alimi, Astron. & Astro- [20] J. Smidt et al., Phys.Rev.D 81, 123007 (2010)
phys.325, 439 (1997)
[11] R.K. Sheth, H. J. Mo & G. Tormen, Mont. Not. Roy.
