1. July 4, 2013 SuperJEDI Mauritius
Testing cosmology with galaxy clusters,
the CMB and galaxy clustering
David Rapetti
DARK Fellow
Dark Cosmology Centre, Niels Bohr Institute
University of Copenhagen
In collaboration with
Steve Allen (KIPAC), Adam Mantz (KICP), Chris Blake
(Swinburne), David Parkinson (Queensland), Florian Beutler
(LBNL), Sarah Shandera (Pennsylvania)
2. July 4, 2013 SuperJEDI Mauritius
Combined constraints on growth and
expansion: breaking degeneracies
A combined measurement of cosmic growth and expansion
from clusters of galaxies, the CMB and galaxy clustering ,
MNRAS 2013 (arXiv:1205.4679)
David Rapetti, Chris Blake, Steven Allen, Adam Mantz, David Parkinson, Florian Beutler
3. July 4, 2013 SuperJEDI Mauritius
GR γ~0.55
Modeling linear, time-dependent
departures from GR
Linear power spectrum
Variance of the
density fluctuations
General Relativity Phenomenological parameterization
Growth rate
Scale independent in the
synchronous gauge
Number density of
galaxy clusters
4. July 4, 2013 SuperJEDI Mauritius
Modeling linear, time-dependent
departures from GR
Having measurements of σ8(z)
allows us to obtain f(z)
To measure g we need growth, f(z),
and expansion, Ωm(z), measurements
From measurements of the shape of the galaxy power spec-
trum and correlation function, we use constraints on the
product f(z) σ8(z) and on the quantity F(z), where the lat-
ter are purely expansion history constraints, i.e. on Ωm(z).
For this data set, both of these constraints are crucial to
measure γ = ln f(z)/ ln Ωm(z).10
However, having a con-
straint on f(z) σ8(z), instead of on f(z), yields a positive cor-
relation between γ and σ8 (see Figure 1) as long as Ωm < 1.
The faster the perturbations grow (small γ), the smaller
the perturbation amplitude, σ8, needs to be to provide the
same amount of anisotropy in the distribution of galaxies,
f(z) σ8(z). Note that the current uncertainty on the bias of
baryonic matter limits the ability of using the normaliza-
tion of the galaxy power spectrum to measure σ8, and thus
to break the degeneracy with γ.
3.3 Cluster abundance and masses
For clusters, we have direct constraints on σ8(z) and Ωm(z)
from abundance, mass calibration and gas mass fraction
data (see Sections 2.2 and 4.1). σ8(z) measurements pro-
vide us with constraints not only on σ8(z = 0), from
the local cluster mass function, but also on the growth
rate f(z) = −(1 + z)d ln σ8(z)/dz, from which together
with those on Ωm(z), we can constrain γ. The evolution of
σ8(z) = σ8e−g(z)
depends on γ, Ωm and w as follows
10 Note that without the AP eﬀect constraints on Ωm(z), no rel-
evant constraints on γ can be obtained from RSD measurements
alone. For the same reason, the addition of the BAO constraints
on Ωm(z) improves signiﬁcantly the measurement of γ for the
combination gal+BAO (see the right panel of Figure 1).
ation function
ameters in the
ell as the non-
lie within the
ies in F(z) and
(2011) used in
sses a range of
3.3 Cluster abundance and masses
For clusters, we have direct constraints on σ
from abundance, mass calibration and ga
data (see Sections 2.2 and 4.1). σ8(z) mea
vide us with constraints not only on σ8
the local cluster mass function, but also
rate f(z) = −(1 + z)d ln σ8(z)/dz, from
with those on Ωm(z), we can constrain γ. T
σ8(z) = σ8e−g(z)
depends on γ, Ωm and w a
10 Note that without the AP eﬀect constraints
evant constraints on γ can be obtained from RS
alone. For the same reason, the addition of the
on Ωm(z) improves signiﬁcantly the measurem
combination gal+BAO (see the right panel of F
Growth and e
g(z) =
z
0
(1 + z )−1
p(z ) − 1
−γ
p(z )γ
dz (8)
= (3wγ)−1
[λ(z) − λ(0)] , (9)
where λ(z) = [p(z) − 1]1−γ
p(z)γ
2F1 [1, 1; 1 + γ; p(z)], 2F1
is a hypergeometric function, p(z) = p0(1 + z)−3w
and
p0 = Ωm/(Ωm − 1). In practice, a negative degeneracy be-
tween σ8 and γ exists due to the limited precision of clus-
ter mass estimates, but it is notably smaller than those de-
scribed above (see Figure 1). Within the precision of the
data, indistinguishable cluster mass functions can be pro-
Growth
g(z) =
z
0
(1 + z )−1
p(z ) − 1
−γ
p(z )γ
dz
= (3wγ)−1
[λ(z) − λ(0)] ,
where λ(z) = [p(z) − 1]1−γ
p(z)γ
2F1 [1,1;1 + γ;p(z
is a hypergeometric function, p(z) = p0(1 + z)−3
p0 = Ωm/(Ωm − 1). In practice, a negative degenera
tween σ8 and γ exists due to the limited precision o
Growth and ex
g(z) =
z
0
(1 + z )−1
p(z ) − 1
−γ
p(z )γ
dz (8)
= (3wγ)−1
[λ(z) − λ(0)] , (9)
where λ(z) = [p(z) − 1]1−γ
p(z)γ
2F1 [1, 1; 1 + γ; p(z)], 2F1
is a hypergeometric function, p(z) = p0(1 + z)−3w
and
p0 = Ωm/(Ωm − 1). In practice, a negative degeneracy be-
tween σ8 and γ exists due to the limited precision of clus-
ter mass estimates, but it is notably smaller than those de-
scribed above (see Figure 1). Within the precision of the
Growth and expansion
g(z) =
z
0
(1 + z )−1
p(z ) − 1
−γ
p(z )γ
dz (8)
= (3wγ)−1
[λ(z) − λ(0)] , (9)
where λ(z) = [p(z) − 1]1−γ
p(z)γ
2F1 [1, 1; 1 + γ; p(z)], 2F1
is a hypergeometric function, p(z) = p0(1 + z)−3w
and
p0 = Ωm/(Ωm − 1). In practice, a negative degeneracy be-
tween σ8 and γ exists due to the limited precision of clus-
ter mass estimates, but it is notably smaller than those de-
scribed above (see Figure 1). Within the precision of the
data, indistinguishable cluster mass functions can be pro-
4.1.1
We mo
with a
where
log10[E
Growth and expansion from clu
z
0
(1 + z )−1
p(z ) − 1
−γ
p(z )γ
dz (8)
(3wγ)−1
[λ(z) − λ(0)] , (9)
= [p(z) − 1]1−γ
p(z)γ
2F1 [1, 1; 1 + γ; p(z)], 2F1
geometric function, p(z) = p0(1 + z)−3w
and
Ωm − 1). In practice, a negative degeneracy be-
nd γ exists due to the limited precision of clus-
timates, but it is notably smaller than those de-
ove (see Figure 1). Within the precision of the
4.1.1 Scaling re
We model the L
(m) =
with a log-norma
where ≡ lo
Growth an
g(z) =
z
0
(1 + z )−1
p(z ) − 1
−γ
p(z )γ
dz
= (3wγ)−1
[λ(z) − λ(0)] ,
where λ(z) = [p(z) − 1]1−γ
p(z)γ
2F1 [1, 1; 1 + γ; p(z)],
is a hypergeometric function, p(z) = p0(1 + z)−3w
p0 = Ωm/(Ωm − 1). In practice, a negative degeneracy
tween σ8 and γ exists due to the limited precision of c
ter mass estimates, but it is notably smaller than those
scribed above (see Figure 1). Within the precision of
Rapetti et al 13
6. July 4, 2013 SuperJEDI Mauritius
Flat ΛCDM + γ: full pdf’s
Gold, solid line:
clusters+CMB (ISW)+galaxies
Red, dashed line:
clusters
Blue, dotted line:
CMB (ISW)
Green, long-dashed line:
galaxies
Rapetti et al 13
7. July 4, 2013 SuperJEDI Mauritius
Gold, solid line:
clusters+CMB (ISW)+galaxies
Red, dashed line:
clusters
Blue, dotted line:
CMB (ISW)
Green, long-dashed line:
galaxies
Flat ΛCDM + γ: full pdf’s
Rapetti et al 13
8. July 4, 2013 SuperJEDI Mauritius
Flat ΛCDM + growth index γ
Rapetti et al 13
clusters (XLF+fgas): BCS+REFLEX
+MACS
CMB (ISW): WMAP
galaxies (RSD+AP): WiggleZ
+6dFGS+BOSS
For General Relativity γ~0.55
Magenta: clusters+galaxies
Purple: clusters+CMB
Turquoise: CMB+galaxies
Gold: clusters+CMB+galaxies
9. July 4, 2013 SuperJEDI Mauritius
Rapetti et al 13
For General Relativity γ~0.55
Magenta: clusters+galaxies
Purple: clusters+CMB
Turquoise: CMB+galaxies
Gold: clusters+CMB+galaxies
Platinum: clusters+CMB+galaxies
+BAO (Reid et al 12; Percival et al
10)+SNIa (Suzuki et al 12)
+SH0ES (Riess et al 11)
Flat wCDM + growth index γ: growth plane
10. July 4, 2013 SuperJEDI Mauritius
Rapetti et al 13
Flat wCDM + growth index γ: expansion planes
Platinum: clusters + CMB + galaxies + BAO (Reid et al 12; Percival et al 10)
+ SNIa (Suzuki et al 12) + SH0ES (Riess et al 11)
11. July 4, 2013 SuperJEDI Mauritius
Rapetti et al 13
For General Relativity γ~0.55
Magenta: clusters+galaxies
Purple: clusters+CMB
Turquoise: CMB+galaxies
Gold: clusters+CMB+galaxies
Flat wCDM + growth index γ: expansion+growth
12. July 4, 2013 SuperJEDI Mauritius
Rapetti et al 13
Flat wCDM + growth index γ: expansion+growth
For General Relativity γ~0.55
For ΛCDM w=-1
Gold: clusters+CMB+galaxies
Platinum: clusters+CMB+galaxies
+BAO+SNIa+SH0ES
! = 0.604± 0.078
"8 = 0.789 ± 0.019
w = !0.967!0.053
+0.054
"m = 0.278!0.011
+0.012
H0 = 70.0 ±1.3
13. July 4, 2013 SuperJEDI Mauritius
Flat wCDM + γ: full pdf’s
Red, dashed line: clusters; Purple, dotted line: clusters+CMB; Gold, solid line:
clusters+CMB+galaxies; Platinum, long-dashed line: all
Rapetti et al 13
14. July 4, 2013 SuperJEDI Mauritius
Beyond ΛCDM: Primordial non-Gaussianity
X-ray cluster constraints on non-Gaussianity ,
arXiv:1304.1216
Sarah Shandera, Adam Mantz, David Rapetti, Steven Allen
15. July 4, 2013 SuperJEDI Mauritius
Testing the Gaussianity of the
primordial fluctuations
1. When cumulants beyond skewness (correlations beyond the
bispectrum) are important, we can only properly describe the non-
Gaussianity with a one-parameter model if we can use this
parameter to specify the amplitude of all the correlations.
2. We assume two different ways to scale higher moments with the
skewness based on particle physics models of inflation.
3. Cluster counts probe smaller scales (0.1-0.5h/Mpc) than the CMB
and the galaxy bias.
4. Cluster counts are sensitive to any non-Gaussianity and to higher
order moments of the probability distribution of primordial
fluctuations.
16. July 4, 2013 SuperJEDI Mauritius
Testing the Gaussianity of the
primordial fluctuations
D D
spectrum. The three-point correlation is then just a function of two independent momenta
and is called the bispectrum.
The local, equilateral and orthogonal bispectra are shown in Eq.(2.3) below. Inter-
estingly, though, object number counts are not sensitive to the details of the bispectrum’s
momentum dependence. Instead, only the integrated moments of the smoothed density ﬂuc-
tuations δR are relevant. For example,
δ3
R =
d3k1
(2π)3
d3k2
(2π)3
d3k3
(2π)3
M(k1, R, z)M(k2, R, z)M(k3, R, z)Φ(k1)Φ(k2)Φ(k3)c
(1.1)
where the terms M(ki, R, z) contain a window function, the corresponding factors from the
Poisson equation, the transfer function and the growth factor converting the linear perturba-
tion in the gravitational potential to the smoothed density perturbation. (The full expressions
for these quantities can be found in Appendix A.) We characterize the non-Gaussianity by
the dimensionless ratios of the cumulants of the density ﬁeld
Mn,R =
δn
Rc
δ2
Rn/2
(1.2)
which are by construction redshift independent and nearly independent of the smoothing
scale, R, if the primordial bispectrum is scale independent.1
When cumulants beyond the skewness (correlations beyond the bispectrum) are relevant,
a one-parameter model is only useful if we can use it to specify the amplitude of all the
correlations. In this paper we use M3 and a choice for how higher moments scale with M3
to describe non-Gaussian ﬂuctuations. The scalings we consider are motivated by particle
1
Scale independence means that the bispectrum (e.g., those in Eq.(2.3)) contains no length scale other
than the factors k−1
i in the P(ki) terms.
trum.
eral and orthogonal bispectra are shown in Eq.(2.3) below. Inter-
t number counts are not sensitive to the details of the bispectrum’s
. Instead, only the integrated moments of the smoothed density ﬂuc-
t. For example,
3k2
π)3
d3k3
(2π)3
M(k1, R, z)M(k2, R, z)M(k3, R, z)Φ(k1)Φ(k2)Φ(k3)c
(1.1)
R, z) contain a window function, the corresponding factors from the
ansfer function and the growth factor converting the linear perturba-
potential to the smoothed density perturbation. (The full expressions
be found in Appendix A.) We characterize the non-Gaussianity by
s of the cumulants of the density ﬁeld
Mn,R =
δn
Rc
δ2
Rn/2
(1.2)
ion redshift independent and nearly independent of the smoothing
ial bispectrum is scale independent.1
eyond the skewness (correlations beyond the bispectrum) are relevant,
is only useful if we can use it to specify the amplitude of all the
per we use M3 and a choice for how higher moments scale with M3
an ﬂuctuations. The scalings we consider are motivated by particle
ans that the bispectrum (e.g., those in Eq.(2.3)) contains no length scale other
P(ki) terms.
eﬀect of Primordial non-Gaussianity on object number counts
tool is a series expansion for the ratio of the non-Gaussian mass function to
sian one. The expansion we use is based on a Press-Schechter model for halo
applied to non-Gaussian probability distributions for the primordial ﬂuctuations.
d derivation of the non-Gaussian mass function we use is given in Appendix A
developed in [30–32]. The weakly non-Gaussian probability distributions that the
tion is based on are asymptotic expansions that deviate substantially from the
bability density function (PDF) for suﬃciently rare ﬂuctuations. Fortunately, our
is already suﬃciently constrained to determine that the clusters in our sample
in that regime. However, the clusters are suﬃciently rare that truncating the
below at a single term (the skewness) is not suﬃcient to test the full range of
at are only as skewed as current CMB constraints allow.
dd non-Gaussianity to the cosmology by considering a mass function of the form
dn
dM
NG
=
dn
dM
T,M300
nNG
nG
Edgeworth
(2.1)
ﬁrst term on the right hand side is the Gaussian mass function of Tinker et al.
usters identiﬁed as spheres containing a mean density 300 times that of the mean
nsity of the Universe, 300 ¯ρm(z). The ratio of the non-Gaussian mass function to
ian one will be given as a series expansion, deﬁned below. This factor will be
n of mass, redshift, and parameters that characterize the amplitude of the non-
ty, which we deﬁne next.
ametrizing the level of non-Gaussianity
ect number counts are not sensitive to the details of the momentum space corre-
Hierarchical scaling (local)
Feeder scaling (two field model)
Non-Gaussian mass function
Dimensionless ratios of the cumulants of the density field
Since object number counts are not sensitive to the details of the momentum space corre-
lations, we consider the dimensionless, connected moments (the cumulants, divided by the
appropriate power of the amplitude of ﬂuctuations) of the density ﬂuctuations smoothed on
a given scale R, as deﬁned in Eq.(1.2). Most constraints on non-Gaussianity have so far been
reported for a parameter that measures the size of the three-point correlation in momentum
space, or bispectrum. This is an extremely useful ﬁrst statistic because this correlation should
be exactly zero if the ﬂuctuations were exactly Gaussian. However, because the bispectrum
is a function of two momenta, the non-Gaussian parameters most often quoted assume a
shape for the bispectrum.
A generic homogeneous and isotropic bispectrum for the potential Φ can be written as
Φ(k1)Φ(k2)Φ(k3)
c
= (2π)3
δ3
D(k1 + k2 + k3) B(k1, k2, k3) (2.2)
where the function B(k1, k2, k3) determines the shape. Bispectra are colloquially named by
the (triangle) conﬁguration of the three momentum vectors that are most strongly correlated.
To interpret our constraints on M3 in terms of familiar bispectra, we consider the templates
for ‘local’, ‘equilateral’ and ‘orthogonal’ bispectra:
Blocal = 2flocal
NL (P(k1)P(k2) + P(k1)P(k3) + P(k2)P(k3)) (2.3)
Bequil = 6fequil
NL [−P(k1)P(k2) + 2 perm. − 2(P(k1)P(k2)P(k3))2/3
+P(k1)1/3
P(k2)2/3
P(k3) + 5 perm.]
Borth = 6forth
NL [−3P(k1)P(k2) + 2 perm. − 8(P(k1)P(k2)P(k3))2/3
+3P(k1)1/3
P(k2)2/3
P(k3) + 5 perm.]
– 4 –
Integrated moments of the smoothed density fluctuations
Generic homogeneous and isotropic bispectrum of the potential
r counts are sensitive to the value of the total skewness and to the scaling of
ts, rather than any details of the momentum space correlations.
on to the dependence on a parameter like fNL, the cumulants also have nu-
ients that typically have to do with combinatorics. For example, beginning
the bispectrum contains three terms linear in flocal
NL , each with two equivalent
the expectation value of pairs of ﬁelds ΦG. We will the choose the constants
ality equal to combinatoric factors for the moments that are generated in the
nd a simple two-ﬁeld extension that gives feeder scaling:4
Hierarchical Mh
n = n! 2n−3
Mh
3
6
n−2
(2.7)
Feeder Mf
n = (n − 1)! 2n−1
Mf
3
8
n/3
. (2.8)
aling of the moments, we can determine a series expansion for the probability
nd for the mass function that can be consistently truncated at some order in
single parameter scenarios, we report constraints in terms of the scaling as-
umber counts are sensitive to the value of the total skewness and to the scaling of
oments, rather than any details of the momentum space correlations.
ddition to the dependence on a parameter like fNL, the cumulants also have nu-
oeﬃcients that typically have to do with combinatorics. For example, beginning
1.3), the bispectrum contains three terms linear in flocal
NL , each with two equivalent
ake the expectation value of pairs of ﬁelds ΦG. We will the choose the constants
tionality equal to combinatoric factors for the moments that are generated in the
atz and a simple two-ﬁeld extension that gives feeder scaling:4
Hierarchical Mh
n = n! 2n−3
Mh
3
6
n−2
(2.7)
Feeder Mf
n = (n − 1)! 2n−1
Mf
3
8
n/3
. (2.8)
en scaling of the moments, we can determine a series expansion for the probability
on and for the mass function that can be consistently truncated at some order in
ents.
the single parameter scenarios, we report constraints in terms of the scaling as-
physics models of inﬂation, and our constraints on the total dimensionless skewness can
always be re-written in terms of a particular bispectrum using Eq. (1.1) and Eq. (1.2).
Most previous work on the utility of cluster counts to constrain non-Gaussianity has
focused on the local ansatz [18, 19], where one assumes that the non-Gaussian ﬁeld Φ(x) is
a simple, local transformation of a Gaussian ﬁeld ΦG(x):
Φ(x) = ΦG(x) + flocal
NL [ΦG(x)2
− ΦG(x)2
]. (1.3)
In this useful model, flocal
NL is the single parameter that all correlation functions depend on,
and the cumulants scale2 as (flocal
NL )n−2. Non-Gaussianity of the local type has a bispectrum
that most strongly correlates Fourier modes of very diﬀerent wavelengths. This particular
mode coupling generates strong signals in other large scale structure observables – most no-
tably introducing a scale dependence in the bias of any biased tracer of the underlying dark
matter distribution [11, 14, 20]. For this reason, papers that have analyzed the potential for
17. July 4, 2013 SuperJEDI Mauritius
Testing the Gaussianity of the
primordial fluctuations
Hierarchical scaling (local)
Feeder scaling (two field model)
reported for a parameter that measures the size of the three-point correlation in momentum
space, or bispectrum. This is an extremely useful ﬁrst statistic because this correlation should
be exactly zero if the ﬂuctuations were exactly Gaussian. However, because the bispectrum
is a function of two momenta, the non-Gaussian parameters most often quoted assume a
shape for the bispectrum.
A generic homogeneous and isotropic bispectrum for the potential Φ can be written as
Φ(k1)Φ(k2)Φ(k3)
c
= (2π)3
δ3
D(k1 + k2 + k3) B(k1, k2, k3) (2.2)
where the function B(k1, k2, k3) determines the shape. Bispectra are colloquially named by
the (triangle) conﬁguration of the three momentum vectors that are most strongly correlated.
To interpret our constraints on M3 in terms of familiar bispectra, we consider the templates
for ‘local’, ‘equilateral’ and ‘orthogonal’ bispectra:
Blocal = 2flocal
NL (P(k1)P(k2) + P(k1)P(k3) + P(k2)P(k3)) (2.3)
Bequil = 6fequil
NL [−P(k1)P(k2) + 2 perm. − 2(P(k1)P(k2)P(k3))2/3
+P(k1)1/3
P(k2)2/3
P(k3) + 5 perm.]
Borth = 6forth
NL [−3P(k1)P(k2) + 2 perm. − 8(P(k1)P(k2)P(k3))2/3
+3P(k1)1/3
P(k2)2/3
P(k3) + 5 perm.]
– 4 –
Generic homogeneous and isotropic bispectrum of the potential
r counts are sensitive to the value of the total skewness and to the scaling of
ts, rather than any details of the momentum space correlations.
on to the dependence on a parameter like fNL, the cumulants also have nu-
ients that typically have to do with combinatorics. For example, beginning
the bispectrum contains three terms linear in flocal
NL , each with two equivalent
the expectation value of pairs of ﬁelds ΦG. We will the choose the constants
ality equal to combinatoric factors for the moments that are generated in the
nd a simple two-ﬁeld extension that gives feeder scaling:4
Hierarchical Mh
n = n! 2n−3
Mh
3
6
n−2
(2.7)
Feeder Mf
n = (n − 1)! 2n−1
Mf
3
8
n/3
. (2.8)
aling of the moments, we can determine a series expansion for the probability
nd for the mass function that can be consistently truncated at some order in
single parameter scenarios, we report constraints in terms of the scaling as-
umber counts are sensitive to the value of the total skewness and to the scaling of
oments, rather than any details of the momentum space correlations.
ddition to the dependence on a parameter like fNL, the cumulants also have nu-
oeﬃcients that typically have to do with combinatorics. For example, beginning
1.3), the bispectrum contains three terms linear in flocal
NL , each with two equivalent
ake the expectation value of pairs of ﬁelds ΦG. We will the choose the constants
tionality equal to combinatoric factors for the moments that are generated in the
atz and a simple two-ﬁeld extension that gives feeder scaling:4
Hierarchical Mh
n = n! 2n−3
Mh
3
6
n−2
(2.7)
Feeder Mf
n = (n − 1)! 2n−1
Mf
3
8
n/3
. (2.8)
en scaling of the moments, we can determine a series expansion for the probability
on and for the mass function that can be consistently truncated at some order in
ents.
the single parameter scenarios, we report constraints in terms of the scaling as-
physics models of inﬂation, and our constraints on the total dimensionless skewness can
always be re-written in terms of a particular bispectrum using Eq. (1.1) and Eq. (1.2).
Most previous work on the utility of cluster counts to constrain non-Gaussianity has
focused on the local ansatz [18, 19], where one assumes that the non-Gaussian ﬁeld Φ(x) is
a simple, local transformation of a Gaussian ﬁeld ΦG(x):
Φ(x) = ΦG(x) + flocal
NL [ΦG(x)2
− ΦG(x)2
]. (1.3)
In this useful model, flocal
NL is the single parameter that all correlation functions depend on,
and the cumulants scale2 as (flocal
NL )n−2. Non-Gaussianity of the local type has a bispectrum
that most strongly correlates Fourier modes of very diﬀerent wavelengths. This particular
mode coupling generates strong signals in other large scale structure observables – most no-
tably introducing a scale dependence in the bias of any biased tracer of the underlying dark
matter distribution [11, 14, 20]. For this reason, papers that have analyzed the potential for
A generic homogeneous and isotropic bispectrum for the potential Φ can be written as
Φ(k1)Φ(k2)Φ(k3)
c
= (2π)3
δ3
D(k1 + k2 + k3) B(k1, k2, k3) (2.2)
where the function B(k1, k2, k3) determines the shape. Bispectra are colloquially named by
the (triangle) conﬁguration of the three momentum vectors that are most strongly correlated.
To interpret our constraints on M3 in terms of familiar bispectra, we consider the templates
for ‘local’, ‘equilateral’ and ‘orthogonal’ bispectra:
Blocal = 2flocal
NL (P(k1)P(k2) + P(k1)P(k3) + P(k2)P(k3)) (2.3)
Bequil = 6fequil
NL [−P(k1)P(k2) + 2 perm. − 2(P(k1)P(k2)P(k3))2/3
+P(k1)1/3
P(k2)2/3
P(k3) + 5 perm.]
Borth = 6forth
NL [−3P(k1)P(k2) + 2 perm. − 8(P(k1)P(k2)P(k3))2/3
+3P(k1)1/3
P(k2)2/3
P(k3) + 5 perm.]
where the power spectrum, P(k), is deﬁned from the two-point correlation function by
Φ(k1)Φ(k2)
= (2π)3
δ3
D(k1 + k2)P(k1) ≡ (2π)3
δ3
D(k1 + k2)2π2 ∆2
Φ(k0)
k3
1
k1
k0
ns−1
(2.4)
where ∆Φ(k0) is the RMS amplitude of ﬂuctuations at a pivot point k0 and any running
of that amplitude with scale is parametrized with the spectral index ns. In the best ﬁt
cosmology from the seven-year WMAP data, baryon acoustic oscillations and Hubble pa-
rameter measurements, the pivot point is k0 = 0.002 Mpc−1
, the spectral index is a constant
18. July 4, 2013 SuperJEDI Mauritius
Testing the Gaussianity of the
primordial fluctuations
oments.
or the single parameter scenarios, we report constraints in terms of the scaling as-
and the parameter M3, which can be compared with other constraints on particular
trum shapes using Table 1.
The mass function in terms of M3 and the scaling of higher moments
l assume the non-Gaussian factor in the mass function of Eq.(2.1) takes the following
nNG
nG
Edgeworth
≈ 1 +
Fh,f
1 (M)
F
0(M)
+
Fh,f
2 (M)
F
0(M)
+ . . . (2.9)
term in the series is normalized by the Press-Schechter Gaussian term, F
0(M) =
2/
√
2π)(dσ/dM)(νc/σ), where νc = δc/σ, δc = 1.686 is the collapse threshold, and σ =
is the variance in density ﬂuctuations smoothed on the appropriate scale (Eq.(A.4)).
ugh the ﬁrst term, Fh
1 (M) or Ff
1 (M), is proportional to M3 regardless of how the
moments scale, the exact form of all higher order terms depends on the choice of
. For the hierarchical and feeder scaling, Fh
n (M) and Ff
n (M) are given in Eq.(A.14)
Appendix. Truncating this series after the ﬁrst term is clearly unphysical since no
bility distribution with only a non-zero skewness can be positive everywhere. Although
me objects (low mass, low redshift) this truncation does not cause a signiﬁcant error,
er ﬂuctuations it does. Keeping higher terms in the series is therefore important. How
cant these terms are in the context of cluster constraints depends on the mass and red-
f the objects as well as the amplitude and scaling of the non-Gaussianity considered.
tion 5, we show several examples to illustrate how relevant the higher terms are as a
on of mass, redshift, skewness and scaling. Although this mass function has been shown
ee reasonably well with simulations, it does not come from a ﬁrst principles derivation.
tion 5 we also contrast it to the Dalal et al mass function from simulations of the local
[20].
For the single parameter scenarios, we report constraints in terms of
sumed and the parameter M3, which can be compared with other constrain
bispectrum shapes using Table 1.
2.2 The mass function in terms of M3 and the scaling of higher m
We will assume the non-Gaussian factor in the mass function of Eq.(2.1) tak
form:
nNG
nG
Edgeworth
≈ 1 +
Fh,f
1 (M)
F
0(M)
+
Fh,f
2 (M)
F
0(M)
+ . . .
Each term in the series is normalized by the Press-Schechter Gaussian t
(e−ν2
c /2/
√
2π)(dσ/dM)(νc/σ), where νc = δc/σ, δc = 1.686 is the collapse thr
σ(M) is the variance in density ﬂuctuations smoothed on the appropriate s
Although the ﬁrst term, Fh
1 (M) or Ff
1 (M), is proportional to M3 regard
higher moments scale, the exact form of all higher order terms depends o
scaling. For the hierarchical and feeder scaling, Fh
n (M) and Ff
n (M) are giv
of the Appendix. Truncating this series after the ﬁrst term is clearly unph
probability distribution with only a non-zero skewness can be positive everyw
for some objects (low mass, low redshift) this truncation does not cause a s
for rarer ﬂuctuations it does. Keeping higher terms in the series is therefore i
signiﬁcant these terms are in the context of cluster constraints depends on th
shift of the objects as well as the amplitude and scaling of the non-Gaussia
In Section 5, we show several examples to illustrate how relevant the higher
function of mass, redshift, skewness and scaling. Although this mass function
to agree reasonably well with simulations, it does not come from a ﬁrst princ
In Section 5 we also contrast it to the Dalal et al mass function from simulat
terms of the scaling as-
constraints on particular
higher moments
.(2.1) takes the following
. . . (2.9)
aussian term, F
0(M) =
lapse threshold, and σ =
opriate scale (Eq.(A.4)).
M3 regardless of how the
epends on the choice of
M) are given in Eq.(A.14)
arly unphysical since no
ve everywhere. Although
cause a signiﬁcant error,
herefore important. How
nds on the mass and red-
-Gaussianity considered.
he higher terms are as a
function has been shown
ﬁrst principles derivation.
Now for either scaling, truncating the series at some ﬁnite s in the sums above keeps all terms
up to the same order in M3: Ms
3 for hierarchical scalings and M
s/3
3 for feeder scalings.
To write the mass function we will need derivatives of all the terms in the expansion
with respect to mass (or smoothing scale). In general, the derivatives can be found using the
relationship for the Hermite polynomials:
νHn(ν) −
dHn(ν)
dν
= Hn+1(ν) . (A.12)
The ratio of the non-Gaussian Edgeworth mass function to the Gaussian has the same struc-
tural form for either scaling:
nNG
nG
Edgeworth
≈ 1 +
Fh,f
1 (M)
F
0(M)
+
Fh,f
2 (M)
F
0(M)
+ . . . (A.13)
with the derivatives of each term F
s = dFs/dM for s ≥ 1:
Fh
s (ν) = F
0
{km}h
Hs+2r
s
m=1
1
km!
Mm+2,R
(m + 2)!
km
(A.14)
+Hs+2r−1
σ
ν
d
dσ
s
m=1
1
km!
Mm+2,R
(m + 2)!
km
Ff
s (ν) = F
0
{km}f
Hs+2
s
m=1
1
km!
Mm+2,R
(m + 2)!
km
+Hs+1
σ
ν
d
dσ
s
m=1
1
km!
Mm+2,R
(m + 2)!
km
where the {km} again satisfy the relationships given below Eq.(A.11) and we have used
F
0 =
e−
ν2
c
2
√
dσ νc
. (A.15)
Press-Schechter normalization
Edgeworth expansion
Hierarchical scaling
Feeder scaling
19. July 4, 2013 SuperJEDI Mauritius
Shandera et al 13
Gaussian distribution Μ3=0
Purple: clusters
Gold: clusters+CMB
Flat ΛCDM + beyond skewness: hierarchical
−0.2 −0.1 0.0 0.1 0.2
0.70.80.91.01.1
M3
σ8
●
●
clusters
clusters+CMB
●
●
●
●
●
●
Hierarchical model
−600 −300 0 300 600
fNL
local
103
M3 = !1!28
+24
!8 = 0.81!0.03
+0.02
fNL
local
= !3!91
+78
20. July 4, 2013 SuperJEDI Mauritius
Shandera et al 13
Gaussian distribution Μ3=0
Purple: clusters
Gold: clusters+CMB
Flat ΛCDM + beyond skewness: feeder
−0.04 −0.02 0.00 0.02 0.04
0.70.80.91.01.1
M3
σ8
●
●
clusters
clusters+CMB
●
●
●
●
●
●
Feeder model
−100 −50 0 50 100
fNL
local
103
M3 = !1!28
+24
!8 = 0.81!0.03
+0.02
fNL
local
= !14!21
+22
21. July 4, 2013 SuperJEDI Mauritius
Shandera et al 13
Gaussian distribution Μ3=0
Purple: clusters
Gold: clusters+CMB
Flat ΛCDM + beyond skewness: hierarchical
−0.2 −0.1 0.0 0.1 0.2
1.11.21.31.41.51.6
M3
βlm
●
●
clusters
clusters+CMB
●
●
●
●
●
●
Hierarchical model
−600 −300 0 300 600
fNL
local
103
M3 = !1!28
+24
!lm =1.33!0.08
+0.07
fNL
local
= !3!91
+78
22. July 4, 2013 SuperJEDI Mauritius
Shandera et al 13
Gaussian distribution Μ3=0
Purple: clusters
Gold: clusters+CMB
Flat ΛCDM + beyond skewness: feeder
−0.04 −0.02 0.00 0.02 0.04
1.11.21.31.41.51.6
M3
βlm
●
●
clusters
clusters+CMB
●
●
●
●
●
●
Feeder model
−100 −50 0 50 100
fNL
local
103
M3 = !1!28
+24
!lm =1.32!0.05
+0.06
fNL
local
= !14!21
+22
23. July 4, 2013 SuperJEDI Mauritius
Flat ΛCDM + beyond skewness: redshift
−0.2 −0.1 0.0 0.1 0.2 0.3
0.60.70.80.91.01.1
M3
σ8
●
●
clusters (all)
clusters (z 0.3)
●
●
●
●
●
●
Hierarchical model
−600 −300 0 300 600 900
fNL
local
Shandera et al 13
−0.04 −0.02 0.00 0.02 0.04
0.60.70.80.91.01.1 M3
σ8
●
●
clusters (all)
clusters (z 0.3)
●
●
●
●
●
●
Feeder model
−150 −100 −50 0 50 100 150
fNL
local
24. July 4, 2013 SuperJEDI Mauritius
Shandera et al 13
13.5 13.75 14. 14.25 14.5 14.75 15. 15.25 15.5
1.0
1.1
1.2
1.3
1.4
z0, fNL30
dn
dM
NG dn
dM
G
z = 0 , flocal
NL = 30
LMSV, skew only
LMSV, hierarchical
LMSV, feeder
Dalal et al.
Log10(M/M⊙ h−1
)
13.5 13.75 14. 14.25 14.5 14.75 15. 15.25 15.5
1.0
1.1
1.2
1.3
1.4
z1, fNL30
dn
dM
NG dn
dM
G
z = 1.0 , flocal
NL = 30
Log10(M/M⊙ h−1
)
13.5 13.75 14. 14.25 14.5 14.75 15. 15.25 15.5
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
z0, fNL100
dn
dM
NG dn
dM
G
z = 0 , flocal
NL = 100
Log10(M/M⊙ h−1
)
13.5 13.75 14. 14.25 14.5 14.75 15. 15.25 15.5
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
z1, fNL100
dn
dM
NG dn
dM
G
z = 1.0 , flocal
NL = 100
Log10(M/M⊙ h−1
)
25. July 4, 2013 SuperJEDI Mauritius
Shandera et al 13
Table 3. The constraints on the skewness can be converted to constraints on the amplitude of any
bispectrum. The shape of the bispectrum is independent of the scaling, although the usual local
ansatz corresponds to a local-shape bispectrum with hierarchical moments.
Scaling Data Local Bispectrum Equil. Bispectrum Orthog. Bispectrum
h CL −73+129
−113 −271+482
−422 346+538
−615
h CL+CMB −3+78
−91 −12+289
−338 15+430
−369
f CL −28+35
−13 −106+134
−48 130+60
−164
f CL+CMB −14+22
−21 −52+85
−79 63+97
−104
skew-only CL −29+532
−78 −105+1916
−280 146+389
−2658
skew-only CL+CMB −9+234
−65 −35+841
−234 48+324
−1167
ensity
1.01.5
Hierarchical
Feeder
tes from follow-up data, or the mass/redshift ranges
on the full set of cosmological and scaling relation
urselves to a single, limited, but informative compar-
ge when data at z ≥ 0.3 are excluded. In detail, this
s, of which 61 have follow-up data, compared to 237
Figure 5 for single-parameter non-Gaussian models
ings, the constraining power of this low-redshift data
he low-redshift clusters only are shown in Table 2.
re
-Gaussianity from cluster counts have been done by
y mission, Sartoris et al. [43] for future X-ray surveys
pe concept, Oguri [44] for a variety of future optical
elected clusters in the Dark Energy Survey (DES),
veys using the Sunyaev-Zel’dovich eﬀect. There have
on non-Gaussianity: two based on clusters detected
who ﬁnd flocal
NL = −192±310, and Williamson et al.
one based on the SDSS maxBCG cluster catalogue,
82 ± 317.8
ts use a variety of prescriptions for the non-Gaussian
e mass functions are given in Eq.(A.19) in the Ap-
e levels of non-Gaussianity shown are M3 = 0.009
nly cluster number counts, without including either the cluster
Benson et al 13 (SPT+CMB)
mber of clusters with mass estimates from follow-up data, or the mass/redshift ranges
, necessarily impacts constraints on the full set of cosmological and scaling relation
ters. Consequently, we conﬁne ourselves to a single, limited, but informative compar-
asking how our constraints change when data at z ≥ 0.3 are excluded. In detail, this
shift sample contains 203 clusters, of which 61 have follow-up data, compared to 237
for the full data set. As shown in Figure 5 for single-parameter non-Gaussian models
he full hierarchical and feeder scalings, the constraining power of this low-redshift data
gniﬁcantly reduced. Results for the low-redshift clusters only are shown in Table 2.
mparison with the literature
s forecasts for constraints on non-Gaussianity from cluster counts have been done by
h et al. [7] for the eROSITA X-ray mission, Sartoris et al. [43] for future X-ray surveys
ing the Wide Field X-ray Telescope concept, Oguri [44] for a variety of future optical
, Cunha et al. [6] for optically selected clusters in the Dark Energy Survey (DES),
k and Pierpaoli [8] for future surveys using the Sunyaev-Zel’dovich eﬀect. There have
ree previous cluster constraints on non-Gaussianity: two based on clusters detected
PT survey, by Benson et al. [15], who ﬁnd flocal
NL = −192±310, and Williamson et al.
ho report flocal
NL = 20 ± 450; and one based on the SDSS maxBCG cluster catalogue,
a et al. [17], who have flocal
NL = 282 ± 317.8
he existing forecasts and constraints use a variety of prescriptions for the non-Gaussian
nction (listed in Table 4). These mass functions are given in Eq.(A.19) in the Ap-
and are plotted in Figure 6. The levels of non-Gaussianity shown are M3 = 0.009
result corresponds to their analysis of only cluster number counts, without including either the cluster
Williamson et al 11 (SPT+CMB)
n, however, since any attempt to reduce the overall size of the sample,
s with mass estimates from follow-up data, or the mass/redshift ranges
mpacts constraints on the full set of cosmological and scaling relation
ently, we conﬁne ourselves to a single, limited, but informative compar-
r constraints change when data at z ≥ 0.3 are excluded. In detail, this
ontains 203 clusters, of which 61 have follow-up data, compared to 237
a set. As shown in Figure 5 for single-parameter non-Gaussian models
ical and feeder scalings, the constraining power of this low-redshift data
uced. Results for the low-redshift clusters only are shown in Table 2.
ith the literature
constraints on non-Gaussianity from cluster counts have been done by
the eROSITA X-ray mission, Sartoris et al. [43] for future X-ray surveys
Field X-ray Telescope concept, Oguri [44] for a variety of future optical
[6] for optically selected clusters in the Dark Energy Survey (DES),
li [8] for future surveys using the Sunyaev-Zel’dovich eﬀect. There have
luster constraints on non-Gaussianity: two based on clusters detected
Benson et al. [15], who ﬁnd flocal
NL = −192±310, and Williamson et al.
= 20 ± 450; and one based on the SDSS maxBCG cluster catalogue,
who have flocal
NL = 282 ± 317.8
casts and constraints use a variety of prescriptions for the non-Gaussian
in Table 4). These mass functions are given in Eq.(A.19) in the Ap-
d in Figure 6. The levels of non-Gaussianity shown are M3 = 0.009
Mana et al 13 (MaXBCG)
and only for non-Gaussianity of the local type. A more precisely calibrated, more general
non-Gaussian mass function will be important for any future analysis of non-Gaussianity
with clusters.
Table 4. Gaussian mass functions and non-Gaussian extensions used in the literature. The non-
Gaussian mass functions are either the ﬁrst order semi-analytic expression from LoVerde et al. [30]
(LMSV) or the mass function calibrated on N-body simulations of the local ansatz by Dalal et al.
[20]. All non-Gaussian mass functions also make use of a Gaussian mass function such as those ﬁt by
Sheth and Tormen [46], Warren et al. [47], Jenkins et al. [48] or Tinker et al. [33, 49].
Author Mass Function used
Benson [15] Jenkins + fDalal
Cunha [6] Jenkins + fDalal
Mak [8] Tinker + fLMSV,skew only
Mana [17] Tinker + fLMSV,skew only
Oguri [44] Warren +fLMSV,skew only
Pillepich [7] Tinker + fLMSV,skew only
Sartoris [43] Sheth-Tormen + fLMSV,skew only
Williamson [16] Jenkins + fDalal
This work Tinker + fLMSV,many terms
Apart from the non-Gaussian mass function, these forecasts and analyses diﬀer from
one another and from ours in two principal ways: the form and complexity assumed for the
mass–observable relation and its intrinsic scatter, and priors on the associated parameters.
The most pessimistic forecasts in the literature ﬁnd marginalized one sigma errors on flocal
NL
calibrated on simulations of the local ansatz, in principle it should include information about
higher moments. This technique, though, has only been tried against one set of simulations
and only for non-Gaussianity of the local type. A more precisely calibrated, more genera
non-Gaussian mass function will be important for any future analysis of non-Gaussianity
with clusters.
Table 4. Gaussian mass functions and non-Gaussian extensions used in the literature. The non
Gaussian mass functions are either the ﬁrst order semi-analytic expression from LoVerde et al. [30
(LMSV) or the mass function calibrated on N-body simulations of the local ansatz by Dalal et al
[20]. All non-Gaussian mass functions also make use of a Gaussian mass function such as those ﬁt by
Sheth and Tormen [46], Warren et al. [47], Jenkins et al. [48] or Tinker et al. [33, 49].
Author Mass Function used
Benson [15] Jenkins + fDalal
Cunha [6] Jenkins + fDalal
Mak [8] Tinker + fLMSV,skew only
Mana [17] Tinker + fLMSV,skew only
Oguri [44] Warren +fLMSV,skew only
Pillepich [7] Tinker + fLMSV,skew only
Sartoris [43] Sheth-Tormen + fLMSV,skew only
Williamson [16] Jenkins + fDalal
This work Tinker + fLMSV,many terms
Apart from the non-Gaussian mass function, these forecasts and analyses diﬀer from
one another and from ours in two principal ways: the form and complexity assumed for the
mass–observable relation and its intrinsic scatter, and priors on the associated parameters
better with simulation results if a reduced collapse threshold, δc ∼ 1.5, is used. If that ad
justment is made, the Dalal et al mass function would deviate more from the Gaussian tha
LMSV; see [45] for a comparison of all these cases. Since the Dalal et al. mass function wa
calibrated on simulations of the local ansatz, in principle it should include information abou
higher moments. This technique, though, has only been tried against one set of simulation
and only for non-Gaussianity of the local type. A more precisely calibrated, more genera
non-Gaussian mass function will be important for any future analysis of non-Gaussianit
with clusters.
Table 4. Gaussian mass functions and non-Gaussian extensions used in the literature. The non
Gaussian mass functions are either the ﬁrst order semi-analytic expression from LoVerde et al. [30
(LMSV) or the mass function calibrated on N-body simulations of the local ansatz by Dalal et a
[20]. All non-Gaussian mass functions also make use of a Gaussian mass function such as those ﬁt b
Sheth and Tormen [46], Warren et al. [47], Jenkins et al. [48] or Tinker et al. [33, 49].
Author Mass Function used
Benson [15] Jenkins + fDalal
Cunha [6] Jenkins + fDalal
Mak [8] Tinker + fLMSV,skew only
Mana [17] Tinker + fLMSV,skew only
Oguri [44] Warren +fLMSV,skew only
Pillepich [7] Tinker + fLMSV,skew only
Sartoris [43] Sheth-Tormen + fLMSV,skew only
Williamson [16] Jenkins + f
Mana [17] Tinker + fLMSV,skew only
Oguri [44] Warren +fLMSV,skew only
Pillepich [7] Tinker + fLMSV,skew only
Sartoris [43] Sheth-Tormen + fLMSV,skew only
Williamson [16] Jenkins + fDalal
This work Tinker + fLMSV,many terms
he non-Gaussian mass function, these forecasts and analyses diﬀer from
om ours in two principal ways: the form and complexity assumed for the
lation and its intrinsic scatter, and priors on the associated parameters.
tic forecasts in the literature ﬁnd marginalized one sigma errors on flocal
NL
g, some cases analyzed in [6, 7]). Those results assume that the scaling
onstrained solely through self-calibration [50] rather than with estimates
which can signiﬁcantly boost the constraining power [51]. In addition,
ume signiﬁcant photometric redshift errors [7]. As outlined in Section 4,
n our sample have spectroscopic redshifts and for nearly half we also have
ta that signiﬁcantly improve the mass determinations.
PT results, Benson et al. use a smaller area of the survey than Williamson
improved mass calibration and extend their sample to lower SZ detection
wer mass). In comparison, our cluster data set is signiﬁcantly larger than
cluster samples, contains more massive clusters (although at lower red-
intrinsic scatter in the mass–observable relation (although the parameters
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