Testing cosmology with galaxy clusters, the CMB and galaxy clustering
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This document summarizes a presentation on testing cosmology using galaxy clusters, the cosmic microwave background, and galaxy clustering. It discusses combining measurements of cosmic growth and expansion from these sources to constrain departures from general relativity. Models are presented for linear, time-dependent departures from GR. Constraints on parameters like the growth index γ are shown from combinations of clusters, CMB, and galaxy data. Tightening constraints are achieved by adding baryon acoustic oscillation, supernova, and Hubble constant data. The document also briefly discusses using cluster counts to constrain primordial non-Gaussianity.
![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](https://image.slidesharecdn.com/drapettisuperjedimauritius13july4-130805090921-phpapp02/85/Testing-cosmology-with-galaxy-clusters-the-CMB-and-galaxy-clustering-4-320.jpg)
![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](https://image.slidesharecdn.com/drapettisuperjedimauritius13july4-130805090921-phpapp02/85/Testing-cosmology-with-galaxy-clusters-the-CMB-and-galaxy-clustering-16-320.jpg)
![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](https://image.slidesharecdn.com/drapettisuperjedimauritius13july4-130805090921-phpapp02/85/Testing-cosmology-with-galaxy-clusters-the-CMB-and-galaxy-clustering-17-320.jpg)
![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](https://image.slidesharecdn.com/drapettisuperjedimauritius13july4-130805090921-phpapp02/85/Testing-cosmology-with-galaxy-clusters-the-CMB-and-galaxy-clustering-18-320.jpg)
![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](https://image.slidesharecdn.com/drapettisuperjedimauritius13july4-130805090921-phpapp02/85/Testing-cosmology-with-galaxy-clusters-the-CMB-and-galaxy-clustering-25-320.jpg)
Testing cosmology with galaxy clusters, the CMB and galaxy clustering
- 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
- 5. 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 Gold: clusters+CMB+galaxies (+BAO+SNIa+SH0ES) ! = 0.616 ± 0.061 "8 = 0.791± 0.019 !m = 0.277± 0.011 H0 = 70.2 ±1.0
- 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