FEATURE

Kavli IPMU Professor

Masahiro Takada
Research Area： Cosmology

Cosmology and Statistics

1. Introduction
Observational cosmology is an exciting research
field. High-precision cosmology data sets, as represented by the cosmic microwave background(CMB) experiments, have allowed us to address
the most fundamental questions of the Universe,such as dark matter, dark energy, and the age of
the Universe. 21st century cosmology requires boththeory and observation and has now grown to be
an experimental science.
Since some articles in the previous IPMU News
touched on these advances in cosmology, we would
like to discuss a different topic in this article, namely,
statistical aspects of cosmology. Cosmologicaltheory and methodology involve various statisticalconcepts. Cosmological data sets may lookcomplicated at first glance. For instance, thefundamental observational quantity for CMB is theblack-body radiation temperature of CMB photons
observed in each direction in the celestial sky. In thecase of a galaxy survey, the spatial distribution ofgalaxies is the fundamental observational quantity.
How can these cosmological data sets be quantified
and compared with theory? What are the premises
and assumptions made in cosmological analysis andwhat are the limitations? The goal of this article isto discuss these questions.

2. Cosmological Principle, Ergodic Hypothesis and Two-Point Correlation
Suppose that F (θ) is a fluctuation field obtained

4from a cosmological data set. In the case of CMB,
it is the CMB temperature fluctuation field, definedas F(θ) ≡ [T (θ) − T̄]/T̄, where T (θ) is the CMB
temperature in the direction θ and T̄ is the averagetemperature over the sky (see the upper-left plotof Fig. 1). Although a similar field can be definedin three-dimensional space, letʼs consider the twodimensional field in the following. The field F(θ) is
observed with a finite angular resolution in actual
observation, and would therefore be given in
discretized pixels. The Planck satellite observed the
CMB sky with angular resolution of about 5 arcmin,providing the CMB temperature fluctuation field in
about 5 millions pixels. This is an enormous data set.
However, cosmology theory cannot reproduce theF(θ) field as is observed. More precisely, to build amodel to reproduce the observed F(θ), one would
need to introduce too many model parametersthat can be as many as the degrees of freedom of
the data. This is obviously not interesting, and we
do not want to make such a fruitless effort. For
this, a cosmological analysis usually employs thecosmological principle:• the Universe is homogeneous and isotropic in astatistical sense.
In simpler terms, it says that there is no spatial
direction or position in the Universe, or we on the
Earth are not located at a spatial position in the
Universe. Thus, this principle would be regarded as ademocratic concept.
If we accept the cosmological principles, we canKavli IPMU News No. 26 June 2014

Figure 1. A flowchart of cosmological analysis. Upper-left plot : A cosmological fluctuation field F (θ). In this example, the temperaturefluctuation field of the cosmic microwave background (CMB), taken from the Planck data. Upper-right plot : A Fourier decomposition of the
field. Lower-right plot : A power spectrum estimation from the Fourier coefficients of the field, F̃l. The gray points (in the upper panel) show
the measured value at each wavenumber bin (more exactly multipole bin in the spherical harmonics expansion). The blue points are theaverage among the several wavenumber bins. The error bars account for statistical uncertainties in each bin arising from both the sample
variance due to a finite number of the Fourier modes and observational effects such as the instrumental noise. The red curve is the best-fit
theoretical model. Lower- left plot : An example of parameter estimation, obtained by comparing the measurement with the model prediction.
The confidence region of each parameter is obtained by properly propagating the measurement errors into parameters.

consider that the observed field F (θ) is a representative sample of the parent population of {F(θ)}
that exists in a vast universe. Again in the CMB case,
we assume that even if an independent observer
who is located somewhere far away from us in
the Universe observed the CMB, he would see asimilar field of the CMB fluctuations as what we
observed. We need to statistically quantify theextent of similarity, however; i.e., how typical or
representative the field we observed is compared tothe expectation, as we will discuss below.
A Fourier decomposition is useful to quantify theobserved field F (θ). That is, we decompose the field
into orthogonal functions of different wavenumbers:

F (θ) =1Ωsl

F̃ l ei l ·θ(1)Here we ignored the curvature of the celestial
sphere for simplicity and assumed that the sky is a
two-dimensional flat space. If Ωs is an area of theobserved region, the fundamental wavenumber of
1/2
1/2
the Fourier decomposition lf ≡ 2π/Ωs (Ωs is a oneside length of the observed area). Then the Fouriermodes are given as l = lf (nx, ny) (nx, ny = ±1, ±2, ...).
The Fourier coefficient F̃l is a quantity to describe
how much the Fourier mode of wavenumber l

Featurecontributes to the observed field.
The Fourier coefficient is generally expressed asF̃l = |F̃l| eiφl, carrying two degrees of freedom: theamplitude and the phase. Due to the statisticalisotropy of the cosmological principle, the phase
unlikely carries useful information.1 Hence, as
1

If there is any correlation between the phases of different Fourier modes, iteasily causes an anisotropic or direction-aligned pattern of the field.

5

a useful quantity to characterize the amplitude
information of the Fourier mode, we can define thepower spectrum estimator asP̂ F (l) ≡1
N mode (l)Ω s| l |∈ l

F̃ l

2

(2)

where Σ|l'|∈l denotes a summation over all the Fouriermodes satisfying |l'| ∈ l to within the bin width.
Nmode(l) is the number of the Fourier modes in thesummation, and is approximated for the mode l >> lf as

N mode (l) ≡| l |∈l

2πl ∆ l(2π ) 2 / Ωs(3)where ∆l is the bin width around the mode l, which
an observer has to choose. Here 2πl∆l is the areaof Fourier space used for the power spectrum
22
estimation, while (2π) /Ωs = l f is the area of thefundamental mode element. Thus, the power
spectrum estimation allows for a significant datacompression, from the vast information content of
two-dimensional data to the one-dimensional scalarquantity, P̂F (l). By ignoring the phase informationof the Fourier coefficients and assuming that theFourier modes with the same length l = |l| areequivalent to each other due to the cosmologicalprinciple, we estimate the power spectrum from the2average of the Fourier coefficients |F̃l| . This would
be the simplest statistical quantity to characterize
the amplitude of the Fourier coefficients.
What is the main difference between laboratory
physics and cosmological experiments? In mostcases a physics experiment can be repeatedly done
at a laboratory. Then, the expectation value and thestatistical error can be estimated from the averageand variance of the independent experiment events(realizations). On the other hand, we cannot repeat
a cosmological experiment: there is one observation
region or there is only one universe even if the all-sky
survey is done. Thus, in a cosmological analysis weapproximate the ensemble average of independent
realizations by the sample average ; for the above
example, the power spectrum is estimated from theaverage over the Fourier modes of the same length,6|F̃l|2, which are all drawn from the same realization(one survey region). This is sometimes called theergodic hypothesis in cosmology. A cosmologist
has to take into account statistical uncertainty andlimitation arising from this hypothesis.

3. Primordial Gaussian Random
Fluctuations and Sample VarianceIn the preceding section we introduced the power
spectrum that is a standard statistical quantity incosmology. In fact, there is a case that the power
spectrum carries all the statistical information ofdata. You may wonder, is there such a convenientcase for us? , but the answer is indeed yes! As wewill see below, our Universe appears to be simple
and beautiful in a statistical sense.
The inflationary scenario provides a plausible
mechanism to explain the origin of cosmologicalfluctuation fields such as the CMB temperaturefluctuations and large-scale structure. It is a scenariothat the universe underwent an exponential
expansion at the beginning of hot Big Bang universe.
Since the universe itself is tiny during the inflationera, one needs to consider a quantization of a fieldthat causes an inflationary expansion, which is often
called inflaton. Due to the uncertainty principleof quantum mechanics, the inflaton inevitably
has quantum fluctuations. Then the inflationary
scenario predicts that the quantum fluctuations
are stretched out by the exponential expansion tomacroscopic scales, leaving classical fluctuations on
horizon scales (the scales beyond causal contact).
In quantum field theory, different wavenumber
modes (k) correspond to different quantum states.
Furthermore, it is assumed in a standard inflation
model that an interaction involved in the inflation
field is small, and different quantum states are
independent from each other. That is, if we denotethe primordial classical fluctuation field by ζ̃ k (often
called the primordial curvature perturbation), thefollowing condition is satisfied:3 3〈ζ̃ k ζ̃ k'〉 ≡ Pζ (k)(2π) δD(k + k')(4)

Kavli IPMU News No. 26 June 2014

where Pζ (k) is the primordial power spectrum.
δ3D(k + k') is the three-dimensional Dirac delta
function imposing that the fields of different
wavenumbers are independent. In addition, we
assumed that the primordial field is statisticallyisotropic, leading the power spectrum Pζ to depend
only on the length k = |k|, which is valid as long as
an inflation expansion is isotropic.
Thus, the inflationary scenario naturally predicts
the generation of isotropic classical perturbations
across the whole universe originating from random,
free quantum fluctuations. More precisely, we
expect that the phase of the primordial fluctuation
field, ζ̃ k, is a random zero-mean variable, and only
the amplitude contains the physical information,
where the typical amplitude is given by theprimordial power spectrum Pζ (k). This means thatthe primordial field ζ(x) is a Gaussian random field.
The Gaussian field has simple statistical properties.
The even order correlation functions2 are given bya product of the two-point correlation function,
which is the inverse Fourier transform of the power
spectrum. The odd order correlation functions3 areall vanishing. That is, the statistical properties ofa Gaussian random field are fully described by its
power spectrum.
One problem of cosmic structure formation issolving how the perturbations of each componentof radiation, baryon and dark matter have evolvedin an expanding universe that has undergone fromthe radiation-, matter-, and dark-energy dominatedera, given the initial conditions such as those set by
the inflation models. As long as the amplitude of
the perturbations is much smaller than unity, we
can use the linearized perturbation theory to solve
the dynamical evolution of multi-component system
based on the linearized Einstein equations and thelinearized Boltzmann equations. The nice thing isthat different Fourier modes evolve independently inthe linear regime. This means that the perturbationsin the linear regime preserve statistical properties ofthe primordial perturbations. In fact the observed
CMB temperature fluctuation field, which is wellin the linear regime, is found to be in remarkableagreement with a Gaussian random field, which isconsidered as one of the pieces of strong evidenceof the inflationary scenario.
Hence, a power spectrum estimation fromthe cosmological data can be regarded as an
appropriate approach, motivated by the inflationary
scenario. As we stated in the previous section, we
need to take into account statistical uncertainty inthe power spectrum estimation arising due to a
finite number of the Fourier modes. This uncertainty
is called the sample variance. The statisticaluncertainty in the power spectrum estimation
is given by the covariance matrix, and can beanalytically computed for a Gaussian field:

Cov[P̂F (l), P̂F (l )] ≡ P̂F (l) P̂F (l ) − P̂F (l) P̂F (l )=2
δ K PF (l) 2
Nmode (l) ll(5)Here δ ll' is a Kronecker-type delta function, definedKin that δ ll' =1 if l =l' to within the bin width,K
otherwise δ ll' =0. Thus the covariance matrix fora Gaussian field has only diagonal elements.4 In
other words, the power spectra of different bins
are independent from each other. For a nonGaussian field, the covariance matrix has additional
contribution that arises from the 4-point correlation
function that cannot be expressed by a combinationof the power spectra, causing correlations
between the power spectra of different bins. In
actual observation, other statistical noise such as
instrument noise also contributes to the covariance,
but we ignore the contribution for simplicity in thefollowing.
The covariance elements at each l bin give a
1σ-distribution of the measurement values aroundthe expectation value, when the power spectrumat the l bin is estimated from each of the survey
realizations for the fixed area Ωs. Hence an expected
statistical significance of the power spectrum
K

For instance, the 4-point correlation function is given by〈ζ(x1)ζ(x2)ζ(x3)ζ(x4)〉.
For instance, the 3-point correlation function is given by〈ζ(x1)ζ(x2)ζ(x3)〉.
4
The prefactor 2 on the r.h.s arises from the fact F̃l = F̃*−l due to real conditionof the fluctuation field F (θ).

2
3

7

Featureestimation at each l bin is

P F (l)σ (P F (l))

2=

N mode (l)2(6)

where σ(PF(l)) = [Cov(P̂F (l), P̂F (l))] . For a Gaussian
case the statistical significance does not dependon the power spectrum, but rather depends only
on the number of Fourier modes around the binl, Nmode(l). Because of Nmode(l) ∝ Ωsl∆l, the statisticalsignificance is greater for a larger area, a higher
wavenumber bin l, and a wider bin width ∆l.
l/2

The lower-right panel of Fig. 1 shows the CMB
temperature power spectrum measured fromthe Planck data. The plot clearly shows that themeasurement accuracy is increasingly higher athigher wavenumber bin l (more exactly the higher
multipoles that are the order in spherical harmonics
expansion). The error bars around blue data points
denote the statistical uncertainties due to thesample variance as well as the instrumental noise.
Again we would like to emphasize that an observer
has to model the sample variance.
Once the statistical uncertainties in the power
spectrum measurement are given, one can compare
the measurement with theory:P̂F (l) ←→ P model[l; Pζ (k), Ωm0h2, Ωb0h2, Ωde, ...]

(7)

Here we assumed on the r.h.s that the model powermodel
spectrum, PF (l), is given as a function of theprimordial power spectrum and other cosmologicalparameters that specify the cosmic expansion
history. This assumption is valid as long as thefluctuations are in the linear regime. First, we canstudy whether a model of interest can give a good
fit to the measurement to within the error bars.
Next, by propagating the measurement errors into
parameter estimation, we can estimate a confidence
region for each parameter. The lower-left panel
shows such an example. Strikingly, the Planck
team successfully attained the ultimate statisticalprecision allowed by the sample variance up to l ~2000, as it uses the all-sky data and the instrument
noise is negligible up to the multipoles. Namely,

8

since the power spectrum contains all the statisticalinformation for a Gaussian field, they were able
to use all the CMB temperature information in
the Universe in order to extract the cosmologicalinformation.
What we have so far described is summarized
by the following three remarks. (1) We measure
the power spectrum from a cosmological data
set assuming the cosmological principle that the
Universe is statistically isotropic and homogeneous.
(2) Assuming the ergodic hypothesis that an
observed region is a representative sample randomly
drawn from the parent populations, an observerneeds to model sample variance effects in the
power spectrum estimation. (3) If the cosmologicalfluctuation field is a Gaussian random field aspredicted by the inflationary scenario, the power
spectrum contains all the statistical informationof the field. If any of these three breaks down, thepower spectrum is no longer the optimal quantity
to extract all the statistical information.

4. Large-Scale Structure Formation:
Nonlinearity of Gravity
While we have so far mainly discussed thelinear Gaussian field such as the CMB field, in thissection we consider the cosmological data setsavailable from galaxy surveys that are aimed at
exploring the accelerating universe. The dynamical
evolution of fluctuations after the CMB epoch
is driven mainly by the spatial inhomogeneities
of dark matter distribution. The current standard
scenario is that the seed perturbations have grown
due to gravitational instability and then formed
the present-day, cosmic hierarchical structurescontaining nonlinear structures such as galaxies,
clusters of galaxies and large-scale structure seen
through the distribution of galaxies. This scenario isthe cold dark matter dominated structure formation
model (hereafter simply CDM model). The CDM
model predicts a bottom-up structure formation:
small-scale nonlinear structures are first formed

Kavli IPMU News No. 26 June 2014

Figure 2. An example of N-body simulations for the cold dark
matter (CDM) dominated the structure formation model. Even if
starting from the initial Gaussian condition, the spatial distributionof dark matter, as represented by N-body particles, display rich
non-Gaussian features as a consequence of complex nonlinear
gravitational instability. Dark matter halos appear at the spatiallysmall places where dark matter particles particularly cluster. Very
> 1015M○• , tend to
massive halos of cluster scales, with masses ~appear at the intersection of cosmic webs/filaments (the obvious
example is the halo at the center of the plot). The amount of dark
matter contained in halos with masses greater than galactic scales
1012M○• is a few 10 percent of the total matter in the universe. On
the other hand, about 70 percent of the spatial volume is occupiedby the spatially big, under-density regions – the so-called voids.
Reflecting such asymmetry, the dark matter distribution becomes tohave non-vanishing values for all the n-point correlation functions.

and then merge in a continuous hierarchy to form
larger-scale structures.
We assume that CDM particles have negligible
thermal velocity (cold) and interact with otherparticles only via gravity. We can show that, on
scales well within horizon scale and from a spatiallycoarse-grained viewpoint, the time evolution ofdark matter clustering follows the pressureless andirrotational fluid equations in an expanding universe:∂δm 1+ ∇ [(1 + δm) vm ] = 0∂ta
11∂vm ȧ
+ vm + (vm · ∇ ) vm = − ∇ φ
∂t
a
a
a

(8)

∇ 2 φ = 4πG ρ̄ m a2 δmHere a(t) is the scale factor that is an increasing
function with cosmic expansion, δm(x) ≡ [ρm(x) −ρ̄m]/ρmis the mass density fluctuation field, vm(x) is thepeculiar velocity field, and ϕ(x) is the gravitational
potential field. In a completely isotropic andhomogeneous universe, δm=|vm|= 0 everywhere.
We can study the dynamical evolution of the dark
matter fields by solving these equations, startingfrom the initial conditions constrained by the
CMB observations. As is obvious from the above
equations, the dark matter fields linearly evolve as
long as the amplitudes of the perturbations aremuch smaller than unity, i.e. |δm|=|vm|<< 1 in units ofc = 1 for speed of light. Once the nonlinear terms
δmvm and (vm · ∇)vm become non-negligible comparedto the linear terms as time goes by, however, thedark matter fields enter into the nonlinear regime.
The nonlinear evolution induces a mode coupling
between different Fourier modes, leading to acomplex evolution of the dark matter fields. Thus,
even if the initial fields are Gaussian, the nonlinearityof gravity induces non-Gaussian features in thespatial distribution of dark matter. The degrees of
non-Gaussianity become greater at smaller scales
and at lower redshift.
Thus, the power spectrum no longer describes thefull information of dark matter distribution in thelate-time universe . In fact, N-body simulation based
studies for the CDM model, as illustrated in Fig. 2,
have shown that the all n-point correlation functionsof the dark matter distribution generally become
non-vanishing .
Having described the non-Gaussian featuresof large-scale structure, one question arises. As
described in the previous sections, the power
spectrum describes the full information containedin the linear, Gaussian fluctuation field in theearly universe, which gives the initial conditions ofstructure formation. On the other hand, the dark
matter fluctuation field displays non-Gaussianfeatures, and has non-vanishing values for all
higher-order correlation functions beyond thepower spectrum. Since we cannot extract any
extra information beyond the initial conditions inanalogy to the second law of thermodynamics,
we can consider that some of the initial Gaussian

9

Feature

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Cumulative information content: I(<lmax )Figure 3. The cumulative information content contained in
the weak lensing power spectrum. The weak lensing field isthe projected field of the mass density fluctuation field along
the line-of-sight, between an observer and source galaxies at
redshift zs = 1. Here we assume Ωs = 1400 sq. degrees for thesurvey area. The cumulative information content is obtainedby summing the signal-to-noise ratio of the power spectrum
measurement (e.g., see Eq. 6) from the minimum wavenumber
lmin = 72 up to a certain maximum wavenumber lmax as denotedby the x-axis. The dotted curve shows the information content
if the weak lensing field is Gaussian as expected from the initial
linear fluctuation field – the maximum information content.
In this case, the information content follows a simple scaling
given by I(< lmax) ∝ lmaxΩs1/2. The circle points show the result
obtained by using mock catalogs of the weak lensing fieldthat are constructed by using N-body simulations of nonlinear
large-scale structure for the CDM model as in Fig. 2. In this case
we fully take into account correlations between the power
spectra of different bins, due to the non-Gaussianity of the
weak lensing field. The solid curve shows the analytical model,
showing remarkably nice agreement with the simulations. The
dashed curve is the analytical result, if we ignore the effect ofsuper-survey modes on the covariance (see text for details).

Simulation

maximum multipole: l max

information leak into the higher-order correlation
functions as a consequence of the nonlinear
structure formation. Namely, the question is• Can we recover the initial Gaussian information
content from the present-day dark matter
distribution by combining the statisticalquantities?
This is indeed an unresolved problem in the fieldof cosmology. If the dynamical system is timereversible, we can back to the initial conditions,and therefore should be able to recover the initial
information content from observables of the current
data (the final state). In the bottom-up CDM modelthe density fluctuations at sufficiently large scales
are still in the linear regime, preserve the Gaussianityof the initial field, and therefore we can recover the
initial information. On the other hand, the density
fluctuations at very small scales are in the strongly
nonlinear regime (|δm| >> 1) and the scales may
have already lost the initial memory. For example,
consider dark matter particles that are bound ina dark matter halo. Such particles may have acomplex orbit scattering or oscillating around thehalo center during their evolution history, so the10

snapshot information alone of the final state would
not allow us to perfectly back their distribution totheir initial positions in a time-reversible way. At
the intermediate scales, which are in the weakly
nonlinear regime, we may be able to recover mostof the initial information. It is still not clear, however,to what extent we can recover the Gaussian
information from the available observables fromthe scales. In fact, upcoming galaxies surveys aim atachieving a high-precision measurement of galaxy
clustering or weak lensing observables in the weakly
nonlinear regime, in order to use the measurements
to do cosmology. Hence this question is very
important for cosmologists.
In recent years we have been addressing theabove question based on both N-body simulations
and analytical model of structure formation. Fig.
3 shows one of our results. Here, we studied
how much information the weak lensing power
spectrum in the late-time universe carries, by using
mock catalogs of weak lensing observables thatwere constructed using N-body simulation outputsof CDM structure formation. At small wavenumberl ~ 100, the weak lensing power spectrum recovers
almost all the Gaussian information. However, at thelarger wavenumber l >~ a few 100, the information

Kavli IPMU News No. 26 June 2014

content of the power spectrum is significantly
reduced compared to the Gaussian information,
more than a factor of 2 at l ~ 1000, from aroundwhich multipoles we expect to extract most usefulcosmological simulation for upcoming surveys suchas HSC. Thus the plot clearly shows that the power
spectrum alone cannot extract all the information
contained in the weak lensing field, and morethan half of the Gaussian information has gonesomewhere! We have indeed found in other workthat the 3-point correlation function does add theinformation. However, the combined informationof the 2- and 3-point correlation functions is notstill as sufficient as the initial Gaussian information.
This implies that the 4-point correlation function
is important, which is even more complicated tomeasure.
Through these studies, we are also finding
something very interesting we did not expect. We
found that most of the information lost is caused by
Fourier modes that are comparable with or beyond
a survey region – super survey modes. Obviously
super survey modes are not observable. This supersurvey mode is regarded as a constant offset in themean density inside the survey region compared tothe cosmic mean (the ensemble averaged mean).
However, we cannot know whether a modulationin the mean density is positive or negative. Supposethat a survey region is embedded into a slightly
coherent, over-density region. In this case, the survey
region is considered as a slightly positive curvature
universe. Due to the nonlinear nature of gravity, thepositive super-survey mode becomes coupled withall the Fourier modes inside the survey volume once
the nonlinear structure formation evolves. That is,
the time evolution of all sub-volume Fourier modes
is enhanced compared to what we naively expect.
We found that the mode-coupling between the
super-survey mode and the Fourier modes inside
the survey region causes a significant contribution
to the sample variance in the power spectrum
measurement. We then succeeded in formulating
this effect in a unified form that can be applied to

any large-scale structure probes. The solid curvein Fig. 3 shows the analytical model prediction for
the information content including the effect of thesuper-sample mode, showing a remarkably niceagreement with the simulation results. Our finding
opens up a new and interesting possibility: if we
can properly take into account the effect of supersurvey mode when comparing the measured power
spectrum with theory, we may be able to infer the
existence of the super survey mode, which lies invery, very large length scales, for upcoming widearea surveys. This is potentially very interesting, andwe are planning to further explore this effect.

5. Future Prospects
Upcoming galaxy surveys such as the SuMIRegalaxy survey are increasingly expensive, both interms of time and cost. We want to attain the fullpotential of these surveys in order to tackle themost fundamental, yet profound problems suchas the nature of dark energy. In this article wehave described the hypotheses, assumptions andprocedures that are often used when measuring
statistical quantities from a cosmological data set
and then using those to estimate cosmologicalparameters. The CMB experiment attained the
maximum success in terms of the extracted
information content, largely because the CMB field isGaussian, the simplest field. For galaxy surveys, on theother hand, it is not yet clear which observables aremost optimal to extract the maximum information.
Given these facts, an interdisciplinary field between
cosmology and statistics will be more and morestrengthened. If you have any new idea, you should
work on this new subject!
References

M. Takada and S. Bridle, New J. Phys. 9, 446 (2007)
M. Takada and B. Jain, MNRAS 395, 2065 (2009)
M. Sato et al., Astrophys. J. 701, 945 (2009)
I. Kayo, M. Takada, and B. Jain, MNRAS 429, 344 (2013)
M. Takada and W. Hu, Phys. Rev. D 87, 123504 (2013)
Y. Li, W. Hu, and M. Takada, Phys. Rev. D 89, 083519 (2014)
M. Takada and D. N. Spergel, MNRAS 441, 2456 (2014)11

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