$(\Omega_{\mathrm{m}},\sigma_{8},w_{0}^{\mathrm{de}})=(0.28,0.82,-0.96)$. The
number counts are the mean over 400 realizations of 36 deg2. Focusing on the
range $2.5\leq\nu\leq 4.5$, we find that the average number counts of Gaussian
filtering with $\theta_{\ker}=$ 1.2, 2.4, and 4.8 arcmin correspond
respectively to starlet filtering with $\theta_{\ker}=$ 4, 8, and 16 arcmin.

Comparing different filtering techniques for peak counts is the subject of
this analysis. This comparison contains (1) the Gaussian kernel, (2) the
starlet function, (3) the aperture mass with the hyperbolic tangent function,
and (4) the nonlinear filtering technique MRLens.

The linear filters are parametrized with a single parameter, which is the size
of the kernel $\theta_{\ker}$. In order to compare in a pertinent way bewteen
multiscale information captured by different filter shapes, two rules are
proposed. The first is to choose $\theta_{\ker}$ such that the 2-norms have
the same value if kernels are normalized (by their respective 1-norms). The
reason for this is that if the ratio of the 2-norm to the 1-norm is identical,
then the comparison is based on the same global noise level (Eq. 8.2). Table
8.1 shows various values of $\theta_{\ker}$ that are used in this study and
the corresponding $\sigma_{\mathrm{noise}}$ for different linear filters. For
the Gaussian filter with $\theta_{\ker}=$ 1.2, 2.4, and 4.8 arcmin, the
corresponding scales are $\theta_{\ker}=$ 2, 4, and 8 arcmin for the starlet
function.

The second way is to calculate peak-count histograms, and set $\theta_{\ker}$
such that peak abundance is similar. Figure 8.2 shows an example for the
Gaussian and starlet kernels with $\theta_{\ker}$ taken from Table 8.1. We can
observe that, for Gaussian filtering with $\theta_{\ker}=$ 1.2, 2.4, and 4.8
arcmin, the correspondence for starlet filtering based on peak abundance is
$\theta_{\ker}=$ 4, 8, and 16 arcmin, if we focus on peaks with
$2.5\leq\nu\leq 4.5$.

The MRLens filter handles different scales by the wavelet transform mentioned
in Sect. 8.3.2. The most important parameter is the FDR $\alpha$. It is set to
$\alpha=0.05$ for this analysis.

#### 8.4.2 Settings for the pipeline: from the mass function to peak
catalogues

Figure 8.3: Diagram illustrating the processing pipeline for the filtering comparison. Effects which are not considered are made faint. A more detailed description for all variations can be found in Sect. 5.4. Parameter | Symbol | Value
---|---|---
Lower sampling limit | $M_{\mathrm{min}}$ | $5\times 10^{12}\leavevmode\nobreak\ M_{\odot}/h$
Upper sampling limit | $M_{\mathrm{max}}$ | $10^{17}\leavevmode\nobreak\ M_{\odot}/h$
Maximum halo redshift | - | 3
Number of halo redshift bins | - | 60
NFW inner slope | $\alpha$ | 1
$M$-$c$ relation amplitude | $c_{0}$ | 11
$M$-$c$ relation power law index | $\beta_{\mathrm{NFW}}$ | 0.13
Intrinsic ellipticity dispersion | $\sigma_{\epsilon}$ | 0.4
Galaxy number density | $n_{\mathrm{gal}}$ | 12 arcmin-2
Pixel size | $\theta_{\mathrm{pix}}$ | 0.8 arcmin
Effective field area | - | 36 deg2
Threshold for filling factor | $\lambda$ | 0.5
Table 8.2: List of parameter values adopted in the study of this chapter.

Compared to previous analyses, more realistic observational features are
accounted for now. We apply a redshift distribution for source galaxies,
include masks, construct the convergence from the reduced shear instead of
computing it directly, test different filters, determine the noise level
locally, and include the equation of state of dark energy for the constraints.
A summary is illustrated by Fig. 8.3. Details are described below.

##### Fast simulations

For fast simulations, the settings remain similar to previous chapters. Halos
are sampled from the model of Jenkins et al. (2001, Eq. 3.81). The sampling
range is set to $M=[5\times 10^{12},10^{17}]\ M_{\odot}/h$. This is done for
60 equal redshift bins from 0 to 3, on a field adequately larger than 36 deg2
so that border effects are properly eliminated. Halo correlation is neglected.
The truncated NFW profiles (Eq. 3.92) are assigned to each halo. The $M$-$c$
relation is given by Eq. (3.93).

##### Source catalogues

Instead of sources at a fixed plane, a realistic redshift distribution is
considered here. This is assumed to be a gamma distribution following
Efstathiou et al. (1991):

$\displaystyle
p(z)=\frac{z^{2}}{2z^{3}_{0}}\exp\left(-\frac{z}{z_{0}}\right),$ (8.11)

where $z_{0}=0.5$ is the pivot redshift value. The positions of sources are
random, but masking has been taken into account. The masked area of the W1
field of CFHTLenS has been taken as the characteristic mask and is applied to
all simulations. The source number density is set to $n_{\mathrm{gal}}=12\
\mathrm{\SIUnitSymbolArcminute}^{-2}$, which corresponds to a CFHTLenS-like
survey (Heymans et al. 2012). The intrinsic ellipticity dispersion is
$\sigma_{\epsilon}=0.4$, which is also close to the CFHTLenS survey (Kilbinger
et al. 2013).

##### Ray-tracing

For each galaxy, we compute $\kappa_{\mathrm{proj}}$ and
$\gamma_{\mathrm{proj}}$ following

$\displaystyle\kappa_{\mathrm{proj}}(\boldsymbol{\theta})=\sum\kappa_{\mathrm{halo}}(\boldsymbol{\theta},w_{\ell},w_{\mathrm{s}})\
\ \ \text{and}\ \ \
\gamma_{\mathrm{proj}}(\boldsymbol{\theta})=\sum\gamma_{\mathrm{halo}}(\boldsymbol{\theta},w_{\ell},w_{\mathrm{s}}),$
(8.12)

where the sums run over all lens-source pairs and $\kappa_{\mathrm{halo}}$ and
$\gamma_{\mathrm{halo}}$ are given respectively by Eqs. (4.82) and (4.83). The
observed ellipticity is computed as
$\epsilon^{(\mathrm{o})}=(\epsilon^{(\mathrm{s})}+g_{\mathrm{proj}})/(1+g_{\mathrm{proj}}^{*}\epsilon^{(\mathrm{s})})$,
where
$g_{\mathrm{proj}}\equiv\gamma_{\mathrm{proj}}/(1-(\kappa_{\mathrm{proj}}-\overline{\kappa}_{\mathrm{proj}}))$
is the reduced shear and $\epsilon^{(\mathrm{s})}$ is the intrinsic
ellipticity, taken from a random generator. The subtraction from
$\kappa_{\mathrm{proj}}$ of its mean over the field is required by the fact
that it is always positive and can not be considered as the true convergence,
as already evoked earlier.

##### Map making

Except for the aperture mass, galaxies are first binned into map pixels for
the reason of efficiency. The mean of $\epsilon^{(\mathrm{o})}$ is taken as
the pixel’s value. The pixel size is set to 0.8 arcmin. This results in
regularly spaced data so that the algorithm can be accelerated. Then, the
linear KS inversion is applied before filtering. The iterative correction for
the reduced shear is not used. By applying exactly the same processing to both
observation and model prediction, we expect the systematics related to
inversion (e.g. boundary effects, missing data, and negative mass density) to
be similar so that the comparison is unbiased.

For the aperture mass, the pixel’s value is evaluated by convolving directly
the lensing catalogue with the $Q$ filter (Eq. 8.8), successively placed at
the center of each pixel (see also Marian et al. 2012; Martinet et al. 2015).
The choice of filter sizes is detailed in Sect. 8.4.1.

##### Peak selection

For linear filters (the Gaussian, the starlet, the aperture mass), peaks are
now selected based on their local noise level, determined by Eq. (8.3). In
this case, the S/N of a peak is a local maximum with

$\displaystyle\nu(\boldsymbol{\theta})\equiv\left\\{\begin{array}[]{ll}(\kappa\ast
W)(\boldsymbol{\theta})/\sigma_{\mathrm{noise}}(\boldsymbol{\theta})&\text{if
Gaussian or starlet,}\\\
M_{\mathrm{ap}}(\boldsymbol{\theta})/\sigma_{\mathrm{noise}}(\boldsymbol{\theta})&\text{if
aperture mass.}\end{array}\right.$ (8.15)

For the nonlinear filter, the notion of noise level does not easily apply.
Actually, to determine the significance of a rare event from any distribution,
instead of using the empirical standard deviation, it is more rigorous to
obtain first the $p$-value and find how much $\sigma$ this value is associated
with if the distribution was Gaussian. However, even if we compute the
standard deviation instead, this process is still too expensive
computationally for our purpose. Therefore, we simply select peaks on
$\kappa$.

Because of masks, only pixels satisfying a specific condition are examined.
This criterion is based on the filling factor $f(\boldsymbol{\theta})$ (Van
Waerbeke et al. 2013; Liu X. et al. 2015b). A local maximum is selected as a
peak only if $f(\boldsymbol{\theta})\geq\lambda\mathrm{bar}{f}$, where
$\mathrm{bar}{f}$ is the mean of $f$ over the map. We set $\lambda=0.5$. For
analyses using binning, the filling factor is simply defined as the number of
galaxies $N(\boldsymbol{\theta})$ inside the pixel at $\boldsymbol{\theta}$.
For the aperture mass, it is the $Q$-weighted sum of the number counts. In
other words,

$\displaystyle
f(\boldsymbol{\theta})\equiv\left\\{\begin{array}[]{ll}N(\boldsymbol{\theta})&\text{if
galaxies are binned,}\\\ \sum_{i}Q(\boldsymbol{\theta}_{i})&\text{for the
aperture mass,}\end{array}\right.$ (8.18)

where $\boldsymbol{\theta}_{i}$ is the position of the $i$-th galaxy. Also, we
only count pixels in the inner area to avoid border effects. The size of this
effective field is 36 deg2.

#### 8.4.3 Data vector definitions

The data vector $\boldsymbol{x}$, for linear filters, is defined as the
concatenation of S/N histograms from various scales. In Chap. 6, we have seen
that the number counts from histograms are the most appropriate form to derive
cosmological information from peak counts. After testing several values of
$\nu_{\mathrm{min}}$, we only keep peaks above $\nu_{\mathrm{min}}=1$ for each
kernel size. This choice maximizes the figure of merit of parameter
constraints. Thus, the statement from Yang et al. (2013) that ignoring peaks
with $\nu\leq 3$ corresponds to a loss of cosmological information is
reconfirmed. For each scale, peaks are binned with width of $\Delta\nu=0.5$ up
to $\nu=5$, and the last bin is $[5,+\infty[$.

For the nonlinear filter, peaks are binned directly by their $\kappa$ values
into [0.02, 0.03, 0.04, 0.06, 0.10, 0.16, $+\infty$[. This configuration is
chosen such that the average count per bin is large enough to assume a
Gaussian fluctuation.

#### 8.4.4 Sampling in the parameter space

The free parameters to constrain in this study are
$(\Omega_{\mathrm{m}},\sigma_{8},w_{0}^{\mathrm{de}})$. The values of other
cosmological parameters are $h=0.78$, $\Omega_{\mathrm{b}}=0.047$, and
$n_{\mathrm{s}}=0.95$. The Universe is assumed to be flat.

The mock observation is generated by a realization of our model, using a
particular set
$(\Omega_{\mathrm{m}},\sigma_{8},w_{0}^{\mathrm{de}})=(0.28,0.82,-0.96)$ as
input parameters. In this way, we only focus on the precision of our model.
Simulations runs are proceeded in two different ways. The first one consists
of interpolating the likelihood, from which we draw credible regions from
Bayesian inference, and the second is ABC.

Figure 8.4: Distribution of evaluated parameter points on the
$\Omega_{\mathrm{m}}$-$\sigma_{8}$ plane. This figure can be considered as a
slice of points with the same $w_{0}^{\mathrm{de}}$. There are in total 46
slices of 816 points. Figure 8.5: Maps taken from one of the simulations. The
truth map is made by calculating $\kappa_{\mathrm{proj}}$ without noise. The
panels of the rest are different filtering techniques applied on the map
obtained from a KS inversion after calculating
$\epsilon^{(\mathrm{o})}=(\epsilon^{(\mathrm{s})}+g_{\mathrm{proj}})/(1+g_{\mathrm{proj}}^{*}\epsilon^{(\mathrm{s})})$.
The black areas are masks. The unit of kernel sizes is arcmin.

In the likelihood analysis, the copula likelihood (Eq. 6.30) is evaluated on a
grid. The range of $w_{0}^{\mathrm{de}}$ is [-1.8, 0], with $\Delta
w_{0}^{\mathrm{de}}=0.04$. Concerning $\Omega_{\mathrm{m}}$ and $\sigma_{8}$,
only some particular values are chosen for evaluation in order to reduce the
computing costs. This results in 816 points in the
$\Omega_{\mathrm{m}}$-$\sigma_{8}$ plane, as displayed in Fig. 8.4, and the
total number of parameter sets is 37536. For each parameter set, $N=400$
realizations of our model are performed. Each realization produces data
vectors for the Gaussian kernel, the starlet kernel, and MRLens, so that the
comparisons between cases are based on the same stochasticity. The aperture
mass is not included here because of the time consuming convolution of the
unbinned shear catalogue with the filter $Q$. A map example is displayed in
Fig. 8.5 for the three cases and the input simulated $\kappa$ field.

Filter |  $\theta_{\ker}$ [arcmin] or $\alpha$ | Number of bins | $d$
---|---|---|---
Gaussian |  $\theta_{\ker}=$ 1.2, 2.4, 4.8 | 9 $\nu$ bins | 27
Starlet |  $\theta_{\ker}=$ 2, 4, 8 | 9 $\nu$ bins | 27
$M_{\mathrm{ap}}$ $\tanh$ |  $\theta_{\ker}=$ 2.125, 4.25, 8.5 | 9 $\nu$ bins | 27
MRLens | $\alpha=0.05$ | 6 $\kappa$ bins | 6
Table 8.3: Definition of the data vector $\boldsymbol{x}$ for PMC ABC runs.
The 9 bins of $\nu$ are [1, 1.5, 2, $\ldots$, 4, 4.5, 5, $+\infty$[, and the 6
bins of $\kappa$ are [0.02, 0.03, 0.04, 0.06, 0.10, 0.16, $+\infty$[. The
symbol $d$ is the total dimension of $\boldsymbol{x}$, and $\alpha$ stands for
the input value of FDR for MRLens.

In the ABC analysis, all four filters are used. For the three first linear
cases, the data vector $\boldsymbol{x}$ is composed of three scales. Each
scale has the same nine S/N bins as in the likelihood analysis, which results
in 27 bins in total. For MRLens, $\boldsymbol{x}$ is a 6-bin $\kappa$
histogram, also identical to the likelihood analysis (Table 8.3).

Concerning the ABC parameters, we use 1500 particles in the PMC process. The
iteration stops when the success ratio of accept-reject processes falls below
1%. Finally, we test two distances. Between the sampled data vector
$\boldsymbol{x}$ and the observed one, $\boldsymbol{x}^{\mathrm{obs}}$, we
consider a simplified distance $D_{1}$ and a fully correlated one $D_{2}$,
which are respectively defined as

$\displaystyle D_{1}\left(\boldsymbol{x},\boldsymbol{y}\right)$
$\displaystyle\equiv\sqrt{\sum_{i}\frac{\left(x_{i}-y_{i}\right)^{2}}{C_{ii}}},$
(8.19) $\displaystyle D_{2}\left(\boldsymbol{x},\boldsymbol{y}\right)$
$\displaystyle\equiv\sqrt{\left(\boldsymbol{x}-\boldsymbol{y}\right)^{T}\boldsymbol{C}^{-1}\left(\boldsymbol{x}-\boldsymbol{y}\right)},$
(8.20)

where $C_{ii}$ and $\boldsymbol{C}^{-1}$ are now independent from cosmology,
estimated under
$(\Omega_{\mathrm{m}},\sigma_{8},w_{0}^{\mathrm{de}})=(0.28,0.82,-0.96)$.

### 8.5 Results

We again use the uncertainty on $\Sigma_{8}$ and the FoM on the
$\Omega_{\mathrm{m}}$-$\sigma_{8}$ plane to qualify constraints. Here,
$\Sigma_{8}$ is defined differently:

$\displaystyle\Sigma_{8}\equiv\left(\frac{\Omega_{\mathrm{m}}+\beta}{1-\alpha}\right)^{1-\alpha}\left(\frac{\sigma_{8}}{\alpha}\right)^{\alpha}.$
(8.21)

The motivation for this definition is to measure the “contour width”
independently from $\alpha$. With the common definition
$\Sigma_{8}\equiv\sigma_{8}(\Omega_{\mathrm{m}}/\text{pivot})^{\alpha}$, the
uncertainty on $\Sigma_{8}$ under different $\alpha$ does not correspond to
the same measure. The 1-$\sigma$ error bar on $\Sigma_{8}$,
$\Delta\Sigma_{8}$, is obtained using the same method as in Chap. 6.

#### 8.5.1 With the copula likelihood

First, we test the maximum information that Gaussian kernels can extract.
Table 8.4 shows the FoM from the marginalized likelihood. We can see that
adding $\theta_{\ker}=$ 2.4 and 4.8 arcmin to the filter with 1.2 arcmin has
no significant effect on constraints. The constraints from the smallest filter
are the most dominant ones among all.

Figure 8.6: $\Omega_{\mathrm{m}}$-$\sigma_{8}$ constraints from four different
cases. Left panel: the Gaussian case (colored regions) and the starlet case
with three corresponding scales based on the noise level (solid and dashed
contours, $\theta_{\ker}=$ 2, 4, and 8 arcmin). Right panel: the starlet case
with three corresponding scales based on number counts (colored regions,
$\theta_{\ker}=$ 4, 8, and 16 arcmin) and the MRLens case (solid and dashed
contours). The Gaussian and count-based starlet cases yield almost identical
constraints. Between four cases, the best result is given by the noise-based
starlet case. Black stars represent the input cosmology. Grey zones are
excluded in this analysis.

Next, we use all three Gaussian scales as the reference for the comparisons
with the starlet function. As mentioned in Sect. 8.4.1, for the Gaussian
filter scales of 1.2, 2.4, and 4.8 arcmin, we chose scales for the starlet
based on two criteria: for an equal noise level, these are 2, 4, and 8 arcmin,
and for equal number counts the corresponding scales are 4, 8, and 16 arcmin.
The results are shown in Fig. 8.6. For the equal-number-count criterion, we
see that if each scale gives approximately the same number of peaks, the
$\Omega_{\mathrm{m}}$-$\sigma_{8}$ constraints obtained from the Gaussian and
the starlet are similar (colored regions in the left and right panels).
However, the starlet kernel leads to tighter constraints than the Gaussian
when we match the same noise levels (lines and colored regions in the left
panel). This results suggests that compensated kernels could be more powerful
to extract cosmological information than non-compensated filters.

Figure 8.7: $\Omega_{\mathrm{m}}$-$\sigma_{8}$ 1-$\sigma$ region from
individual scales of the starlet kernel. The plotted scales are 2 (blue), 4
(red), and 8 arcmin (green). The left panel is the result from the observation
vector used in the analysis; the right panel represents the constraints from
another realization of the observation. We see that the behaviors between
different scales is more likely due to stochasticity. Grey zones are excluded
in this analysis.

We also draw constraints from individual scales of the starlet filter in (Fig.
8.7, left panel). It shows a very different behavior and seems to suggest that
different scales could be sensitive to different cosmologies. However, this is
actually a stochastic effect. We verify this statement by redoing the
constraints with other observation vectors. It turns out that the scale-
dependent tendency disappears (Fig. 8.7, right panel). Nevertheless, when
different cosmologies are preferred by different scales, the effect is less
pronounced for the Gaussian filter. This is likely due to the fact that the
starlet is a compensated filter, which is a band-pass function in the Fourier
space. Since different filtering scales could be sensitive to different mass
ranges of the mass function, band-pass filters weaken the correlations between
scales (Fig. 8.9) and separate better the multiscale information. The
stochasticity of the observation vector suggests that the simulated field of
view is rather small. While this should not affect the filter comparison nor
contour sizes, actual cosmological constraints seem to require substantially
larger data sets.

The right panel of Fig. 8.6 shows the constraints from nonlinear filtering
using MRLens (solid and dashed lines). We observe that MRLens conserves a
strong degeneracy between $\Omega_{\mathrm{m}}$ and $\sigma_{8}$. The reasons
for this result are various. First, large and small scales tend to be
sensitive to different halo masses which could help break the degeneracy.
Using the combined strategy loses this advantage. Second, a strict FDR has
been chosen. This rules out most of the spurious peaks, but also a lot of the
signal. Third, as mentioned before, it is inappropriate to define signal-to-
noise ratio when the filter is not linear. As a consequence, it is hardly
possible to find bins for $\kappa$ peaks which are equivalent to $\nu$ bins in
linear filtering. This is supported by Fig. 8.5, where we observe less peaks
in the MRLens map than in the other maps. Last, because of a low number of
peaks, the binwidths need to be enlarged to contain larger number counts and
to get closer to a Gaussian distribution, and large binwidths also weaken the
signal.

A possible solution for exploring the MRLens technique is to enhance the FDR
and to redesign the binning. By increasing the number of peaks, thinner bins
would be allowed. Another solution to better account for rare events in the
current configuration is to use the Poisson likelihood. Finally, one could
adopt the separated strategy, i.e. turning back to the methodology used by
Pires et al. (2009a) that consists in estimating the peak abundance in the
different scales before final reconstruction. In this study, the comparison
between “linear and nonlinear techniques” is basically the one between the
“separated and combined strategies”.

Filter |  $\theta_{\ker}$ [arcmin] or $\alpha$ | $\Delta\Sigma_{8}$ | FoM
---|---|---|---
Gaussian |  $\theta_{\ker}=$ 1.2 | 0.045 | 19.1
Gaussian |  $\theta_{\ker}=$ 1.2, 2.4, 4.8 | 0.046 | 20.7
Starlet |  $\theta_{\ker}=$ 2, 4, 8 | 0.046 | 23.4
Starlet |  $\theta_{\ker}=$ 4, 8, 16 | 0.044 | 21.2
Starlet |  $\theta_{\ker}=$ 2, 4, 8, 12, 16 | 0.045 | 24.8
MRLens |  $\alpha=$ 0.05 | 0.046 | 16.2
Table 8.4: Quality indicators for $\Omega_{\mathrm{m}}$-$\sigma_{8}$
constraints with likelihood. All cases figured below use number counts on $g$
peaks. The quantity $\Delta\Sigma_{8}$ stands for the width of the contour,
while the FoM is related to the area. In our study, combining five scales of
starlet yield the best result in terms of FoM.

Table 8.4 measures numerical qualities for constraints with different
filtering techniques. It indicates that the width of contours does not vary
significantly. The tightest constraint that we obtain is derived from a
compensated filter.

Figure 8.8: $\Omega_{\mathrm{m}}$-$\sigma_{8}$-$w_{0}^{\mathrm{de}}$
constraints using starlet with three scales. Left panel: constraints with the
likelihood. Fluctuations on both lower panels are due to usage of the copula
likelihood. Right panel: constraints with ABC using the distance $D_{2}$. Each
panel represents the marginalization over one of the three parameters. Black
stars are the input cosmology. As far as $w_{0}^{\mathrm{de}}$ is concerned,
the constraints are weak, but the degeneracies are clear.

Regarding results for $w_{0}^{\mathrm{de}}$, we show a representative case of
starlet with $\theta_{\ker}=$ 2, 4, and 8 arcmin. The left panel of Fig. 8.8
presents the marginalized constraints of each doublet of parameters that we
study. Those containing $w_{0}^{\mathrm{de}}$ are noisy because of the usage
of the copula likelihood. We may see that the current configuration of our
model does not allow to impose constraints on $w_{0}^{\mathrm{de}}$. To
measure this parameter, it could be useful to perform a tomography analysis to
separate information of different stages of the late-time Universe.
Nevertheless, our results successfully highlight the degeneracies of
$w_{0}^{\mathrm{de}}$ with two other parameters. We fit the posterior density
with:

$\displaystyle I_{1}$
$\displaystyle=\Omega_{\mathrm{m}}-a_{1}w_{0}^{\mathrm{de}},$ (8.22)
$\displaystyle I_{2}$ $\displaystyle=\sigma_{8}+a_{2}w_{0}^{\mathrm{de}}.$
(8.23)

We obtain for the slopes $a_{1}=0.108$ and $a_{2}=0.128$ for the left panel of
Fig. 8.8. The results for the other filter functions are similar.

#### 8.5.2 With PMC ABC

We perform parameter constraints using the PMC ABC algorithm for our four
cases. In the right panel of Fig. 8.8, we show the results derived from the
starlet case using the fully correlated distance $D_{2}$. The contours are
marginalized posteriors for all three pairs of parameters. They show the same
degeneracy as we have found with the likelihood. We measure $a_{1}$ and
$a_{2}$ from the ABC posteriors and obtain $a_{1}=0.083$ and $a_{2}=0.084$.

Filter | Constraints | $\Delta\Sigma_{8}$ | FoM
---|---|---|---
Gaussian | Likelihood | 0.046 | 20.7
Gaussian | ABC, $D_{1}$ | 0.043 | 16.3
Gaussian | ABC, $D_{2}$ | 0.059 | 11.7
Starlet | Likelihood | 0.054 | 23.4
Starlet | ABC, $D_{1}$ | 0.050 | 15.5
Starlet | ABC, $D_{2}$ | 0.054 | 15.7
$M_{\mathrm{ap}}$ tanh | ABC, $D_{1}$ | 0.037 | 19.4
$M_{\mathrm{ap}}$ tanh | ABC, $D_{2}$ | 0.043 | 15.5
MRLens | Likelihood | 0.046 | 16.2
MRLens | ABC, $D_{1}$ | 0.045 | 11.5
MRLens | ABC, $D_{2}$ | 0.045 | 12.5
Table 8.5: Quality indicators for $\Omega_{\mathrm{m}}$-$\sigma_{8}$
constraints with PMC ABC. The quantity $\Delta\Sigma_{8}$ stands for the width
of the contour, while the FoM is related to the area. ABC is used with two
different distances $D_{1}$ and $D_{2}$ respectively given by Eqs. (8.19) and
(8.20). Here, we also put values from likelihood constraints using the same
scales in this table for comparison. The kernel sizes for linear methods are
defined in Table 8.3. Figure 8.9: Correlation coefficient matrices under the
input cosmology. Left panel: the Gaussian case with $\theta_{\ker}=1.2$, 2.4,
and 4.8 arcmin. Right panel: the starlet case with $\theta_{\ker}=2$, 4, and 8
arcmin. Each of the 3$\times$3 blocks corresponds to the correlations between
two filter scales. With each block, the S/N bins are [1, 1.5, 2, $\ldots$, 5,
+$\infty$[. The data vector by starlet is less correlated.

Using the same starlet filters, we compare two distances used for PMC ABC
runs. When $D_{1}$ is used with the starlet, i.e. data are treated as if
uncorrelated, we find that the contour sizes do not change (see Table 8.5)
compared to $D_{2}$. For the Gaussian case, however, constraints from $D_{1}$
are tighter than those from $D_{2}$. This phenomenon is due to the off-
diagonal elements of the covariance matrix. For non-compensated filters, the
cross-correlations between bins are much stronger, as shown in Fig. 8.9. If
these cross-correlations are ignored, the repeated peak counts in different
bins are not properly accounted for. This overestimates the additional
sensitivity to massive structures, and therefore produces overly tight
constraints. As shown in Fig. 8.9, in the Gaussian case, adjacent filter
scales show a 20–30% correlation. The blurring of the off-diagonal stripes
indicate a leakage to neighboring S/N bins due to noise, and the fact that
clusters produce WL peaks with different S/N for different scales. On the
contrary, in the case of the starlet, except for the highest S/N bin there are
negligible correlations between different scales.

Figure 8.10: ABC $\Omega_{\mathrm{m}}$-$\sigma_{8}$ constraints from the
starlet and the aperture mass. The distance $D_{2}$ is used in both cases. The
black star is the input cosmology. The difference between two cases is that
another observation data vector is created for the aperture mass and the
direct comparison is not valid anymore.

Table 8.5 shows the ABC constraints from both the aperture mass and the
starlet. We find that the FoM are close. However, in Fig. 8.10, we see that
the contours from the aperture mass is shifted toward
high-$\Omega_{\mathrm{m}}$ regions. The explanation for this shift is once
again the stochasticity. We simulated another observation data vector for
$M_{\mathrm{ap}}$, and the maximum-likelihood point for different methods do
not coincide.

From Table 8.5, one can see that the difference between MRLens and linear
filters using ABC is similar to using the likelihood. This suggests once again
that the combined strategy leads to less tight constraints than the separated
strategy. Note that we also try to adjust $\alpha$ and run PMC ABC. However,
without modifying the $\kappa$ bin choice, the resulting constraints do not
differ substantially from $\alpha=0.05$.

Figure 8.11: Comparison of
$\Omega_{\mathrm{m}}$-$\sigma_{8}$-$w_{0}^{\mathrm{de}}$ constraints between
likelihood and ABC. Left panel: constraints with Gaussian smoothing. Right
panel: constraints with starlet filtering. Although ABC tolerates higher
$\Omega_{\mathrm{m}}$ and lower $\sigma_{8}$ in both cases, two methods agree
with each other.

Finally, we show the likelihood and ABC constraint contours for the Gaussian
and starlet cases in Fig. 8.11. It turns out that ABC contours are
systematically larger in the high-$\Omega_{\mathrm{m}}$, low-$\sigma_{8}$
region. This phenomenon was not observed previously. We speculate that by
including a third parameter $w_{0}^{\mathrm{de}}$ the contour becomes less
precise, and ABC might be more sensitive to this effect. Note also that KDE is
a biased estimator of posteriors. It smooths the posterior and makes contours
broader. Nevertheless, the ABC and likelihood constraints agree with each
other. To be free from the bias, a possible alternative is to map the samples
to a Gaussian distribution via some nonlinear mapping techniques (Schuhmann et
al. 2016).

##### Summary

In this chapter, a new standard of comparisons between various filtering
techniques has been presented. Comparing directly the constraint contours
instead of purity and completeness of the cluster detection leads to a more
direct measurement of cosmological information extraction.

The study shows a preference of compensated filters rather than non-
compensated ones. Also, the separated strategy outperforms the combined
strategy in terms of multiscale information extraction.

Constraints from the likelihood seem to be tighter than from ABC. Since in
Sect. 6.4 we found that the copula likelihood closely approximates the true
one, ABC probably overestimate the true parameter errors.

Without tomography, $w_{0}^{\mathrm{de}}$ could not be constrained tightly.
However, we observe that it is degenerate with $\Omega_{\mathrm{m}}$ and
$\sigma_{8}$.

Our model has been improved to be adapted to more realistic observational
conditions. We will now present a data application in the next chapter.

## Chapter IX Data applications

##### Overview

In this chapter, we apply our analysis to three data sets. The cosmological
constraints are performed using our peak-count model and ABC. We will show
preliminary results, possible improvements, and the perspectives of this work
at the end.

### 9.1 Data descriptions

Three data sets are processed: the full data from the Canada-France-Hawaii
Telescope Lensing Survey (CFHTLenS), the data releases 1 and 2 from the Kilo-
Degree Survey (KiDS DR1/2), and the science verification data from the Dark
Energy Survey (DES SV).

##### CFHTLenS

The CFHTLenS data (Heymans et al. 2012; Erben et al. 2013) are composed of
four separated fields W1, W2, W3, and W4. The total area is 154 deg2. The
total unmasked area is about 126 deg2. Shape measurement was done with
Lensfit, which is a model-fitting algorithm (Miller et al. 2013). Galaxies are
selected with redshift, measurement weight, and mask flags, respectively as
$0.2<$ Z_B $<1.3$, $\mathtt{weight}>0$, and $\mathtt{MASK}<2$
\sharp1\sharp1$\sharp$1All science analyses of the CFHTLenS team are performed
with $\mathtt{MASK}<2$. These objects can be safely used for most scientific
purposes. (Erben et al. 2013). This results in 6.1 million galaxies. The raw
density is 13.53 arcmin-2.

Figure 9.1: Illustrations of all eight patches (blue frames) from CFHTLenS.
The four panels represent respectively the W1, W2, W3, and W4 fields. Red
points are the center of patches. Unmasked areas are showed in yellow.

To process peak counts, the W1 field, which is relatively large, is cut into
four patches. Also, the W4 field which has a irregular shape is separated into
two patches. Thus, the whole data set is represented by eight patches of
different sizes (Fig. 9.1). For each patch, border effects are carefully
removed. This means that some galaxies near the edges are sacrificed and only
peaks in the inner area are counted. We define it as the effective area in the
same way as in previous chapters, which is 112 deg2 in this case.

##### KiDS DR1/2

KiDS DR1/DR2 (de Jong et al. 2015) contains 109 tiles of 1 deg2 all near the
equator. Concerning the lensing catalogue, KiDS uses the same algorithm as
CFHTLenS. We select sources following the criteria suggested by Kuijken et al.
(2015): MAN_MASK $=0$, $0.005<$ Z_B $<1.2$, $\mathtt{weight}>0$, and
$\mathtt{SNratio}>0$. In addition to these, due to calibration limits, we
include also c1_best $>-50$ and c2_best $>-50$ to cut out values of -99
(Kuijken et al. 2015). The final selection includes 2.4 million galaxies
distributed on an unmasked area of 75 deg2. The raw density is 8.87 arcmin-2.

Figure 9.2: Illustrations of all four patches (blue frames) from KiDS DR1/2.
Three of them are taken from the G09 field. The rest is taken from the G12
field. Red points are the center of patches. Unmasked areas are showed in
yellow.

Regarding the fact that (1) the KiDS data are more contaminated by masks and
(2) some 1-deg2 tiles are missing from a contiguous region, we only retain a
small part of the whole data set. This is organized as four patches with a
total area of 41 deg2 (Fig. 9.2). After cutting out edges, the effective area
is 30 deg2.

##### DES SV

The DES SV data (Dark Energy Survey Collaboration et al. 2016; Jarvis et al.
2016) contains a large field named SPT-E and some other small fields. Here we
only focus on the SPT-E field, which covers an unmasked area of 138 deg2. We
use the ngmix catalogue provided by the collaboration. The flag NGMIX_FLAG
$=0$, which is a combination of several flags, validates that a source has a
good ngmix measurement \sharp2\sharp2$\sharp$2See
http://des.ncsa.illinois.edu/.. An additional criterion is the redshift
information, with $0.3<$ MEAN_PHOTOZ $<1.3$, suggested by Kacprzak et al.
(2016). The total number of galaxies is 3.3 millions, with raw density 6.63
arcmin2.

Figure 9.3: Illustrations of all five patches (blue frames) from DES SV. Red
points are the center of patches. Unmasked areas are showed in yellow.

In this study, the SPT-E field is divided into five patches (Fig. 9.3). The
resulting effective area is 115 deg2.

### 9.2 Methodology

Parameter | Symbol | CFHTLenS | KiDS DR1/2 | DES SV
---|---|---|---|---
Source distribution parameter | $z_{\mathrm{min}}$ | 0.2 | 0.005 | 0.3
Source distribution parameter | $z_{\mathrm{max}}$ | 1.3 | 1.2 | 1.3
Source distribution parameter | $\alpha_{\mathrm{gal}}$ | 0.474 | 1.255 | 2.958
Source distribution parameter | $\beta_{\mathrm{gal}}$ | 5.990 | 0.698 | 1.048
Source distribution parameter | $z_{0}$ | 1.157 | 0.128 | 0.217
Galaxy number density [arcmin-2] | $n_{\mathrm{gal}}$ | 13.53 | 8.87 | 6.63
Ellipticity dispersion | $\sigma_{\epsilon}$ | 0.449 | 0.516 | 0.400
Pixel size [arcmin] | $\theta_{\mathrm{pix}}$ | 0.72 | 0.9 | 1.05
Number of filtering scales | - | 4 | 3 | 3
Table 9.1: List of parameter values adopted in the study of this chapter.

The analysis is processed with the same method as described in Sect. 8.4.2,
with $M_{\mathrm{min}}=2\times 10^{12}\ M_{\odot}$. Each survey is modelled
separately with the corresponding parameters such as $n_{\mathrm{gal}}$,
$\sigma_{\epsilon}$, $\theta_{\mathrm{pix}}$, filtering scales, and the source
redshift distribution (see Table 9.1). At the end, three peak histograms are
joined. The details are described in the following paragraphs.

##### Source redshift distribution

The redshift distribution of the sources is assumed to be a general gamma
distribution:

$\displaystyle
p(z)\propto\left(\frac{z}{z_{0}}\right)^{\alpha_{\mathrm{gal}}}\exp\left(-\left(\frac{z}{z_{0}}\right)^{\beta_{\mathrm{gal}}}\right),$
(9.1)

where the normalization is computed for the interval [$z_{\mathrm{min}}$,
$z_{\mathrm{max}}$]. For each survey, $z_{\mathrm{min}}$ and
$z_{\mathrm{max}}$ are given by the suggested flags. Then, we fit
$\alpha_{\mathrm{gal}}$, $\beta_{\mathrm{gal}}$, and $z_{0}$ to the redshift
distribution of the catalogues.

##### Ellipticity dispersion and galaxy density

The noise from the catalogue has two origins: intrinsic shape dispersion and
shape measurement errors. Unfortunately, it is impossible to separate these
two components from observations. For this reason, cosmologists define the
effective number density, such that when this quantity is combined with the
assumed intrinsic dispersion, the total variance will match the one in the
catalogue. Therefore, the effective density is always smaller than the raw
density. The less the galaxies, the higher the Poisson noise; and this Poisson
part will account for the measurement errors neglected in the effective
formalism.

However, this is actually a redundant step. First, the true shape dispersion
is not Gaussian, neither are the measurement errors. Separating both
components in the Gaussian formalism seems very unrealistic. Second, the
intrinsic dispersion is never known. Although Jarvis et al. (2016) claim that
it is possible via a fine-tuned combination of weights, the results obtained
by this technique are not consistent between different surveys. Finally, we
never need the value of shape dispersion alone in the model. Modelling the
combination of both components as a Gaussian noise is much simpler than
considering them with separate models and assuming further approximations.

As a result, we adopt a different approach to determine the density and the
dispersion. The idea is simple: given a catalogue with its total variance, we
look for the Gaussian dispersion which corresponds to the raw density, so that
the total variance matches. This leads to two advantages. First, only one
assumption is needed: the Gaussianity of the combination of the intrinsic
dispersion and the shape measurement. Second, the number density that will be
used in our model is larger. This reduces the bias generated by the KS
inversion when the distribution of sources is too irregular.

More explicitly, the total variance is defined as the empirical weighted
variance taken the multiplicative bias into account:

$\displaystyle\sigma_{\mathrm{tot}}^{2}=\frac{\sum_{i}w_{i}^{2}\left(\epsilon_{i,1}^{(\mathrm{c})2}+\epsilon_{i,2}^{(\mathrm{c})2}\right)}{\sum_{i}w_{i}^{2}(1+m_{i})^{2}},$
(9.2)

where
$\epsilon^{(\mathrm{c})}_{i,j}\equiv\epsilon^{(\mathrm{o})}_{i,j}-c_{i,j}$ is
the $j$-th component of the observed ellipticity $\epsilon_{i}^{(\mathrm{o})}$
of the $i$-th galaxy corrected with the additive bias $c_{i}$, $w_{i}$ is the
weight, $m_{i}$ is the multiplicative bias, and the sum runs over all galaxies
in the catalogue. The Gaussian dispersion $\sigma_{\epsilon}$ corresponding to
the raw density satisfies

$\displaystyle\sigma_{\mathrm{tot}}^{2}=\frac{\sigma_{\epsilon}^{2}}{2N_{\mathrm{gal}}},$
(9.3)

where $N_{\mathrm{gal}}$ is the total number of sources. In this case, the
derived $\sigma_{\epsilon}$ can have a relatively large number. In practice,
we truncate at $|\epsilon|=\pm 1$ to avoid unphysical values. The violation of
the Gaussian variance by the truncation is ignored.

##### Pixel size and filter scales

The pixel size $\theta_{\mathrm{pix}}$ is determined based on
$n_{\mathrm{gal}}$. The goal is to avoid having pixels with a low number of
galaxies. We expect $\theta_{\mathrm{pix}}$ to satisfy

$\displaystyle
P_{\mathrm{bar}{N}}(N=0)+P_{\mathrm{bar}{N}}(N=1)+P_{\mathrm{bar}{N}}(N=2)\leq
0.03,$ (9.4)

where $\mathrm{bar}{N}=n_{\mathrm{gal}}\theta_{\mathrm{pix}}^{2}$ is the
expected number of galaxies in a pixel, and $P_{\lambda}(N)$ is the
probability associated at number $N$ for the Poisson distribution with
parameter $\lambda$. This means that we expect the probability of having less
than three galaxies in a pixel to be smaller than 3%. The values in Table 9.1
satisfy this criterion.

Data set | CFHTLenS
---|---
Filter scales $\theta_{\ker}$ [arcmin] | 1.44, 2.88, 5.76, 11.52
S/N bins | [2.5, 2.75, 3.0, 3.25, 3.5, 3.75, 4.0, 4.25, 4.75, $+\infty$[
Data set | KiDS DR1/2
Filter scales $\theta_{\ker}$ [arcmin] | 1.8, 3.6, 7.2
S/N bins | [2.5, 2.75, 3.0, 3.25, 3.5, $+\infty$[
Data set | DES SV
Filter scales $\theta_{\ker}$ [arcmin] | 2.1, 4.2, 8.4
S/N bins | [2.5, 2.75, 3.0, 3.25, 3.5, 3.75, 4.0, 4.5, $+\infty$[
Table 9.2: List of scales and bins used in the study of this chapter. The
dimension of the data vector is $4\times 9+3\times 5+3\times 8=75$.

Concerning the filter, we only use the starlet filtering in this study. The
filter scales are chosen to be some even multiples of the pixel size. The
reason is that if $\theta_{\ker}=2N\theta_{\mathrm{pix}}$ where
$N=1,2,\ldots$, then the starlet is strictly compensated in its discrete sum.
These values are some natural choices for wavelet functions. We pick four
scales for CFHTLenS and three scales for the others. This results in 1.44,
2.88, 5.76, and 11.52 arcmin for CFHTLenS; 1.8, 3.6, and 7.2 arcmin for KiDS;
and 2.1, 4.2, and 8.4 arcmin for DES.

##### Data vector

Figure 9.4: Correlation matrix from the data vector defined in this study. The
components are very weakly correlated with each other.

The data vector is defined as the concatenation of peak histograms of all
scales from three surveys. The binwidth is defined slightly differently
depending on the survey (see Table 9.2). The reason for this is to have enough
number counts in each bin ($\gtrsim 8$) such that the Poisson distribution can
be approximated by a Gaussian one. This results in a data vector of dimension
75. The correlation matrix of this data vector is shown in Fig. 9.4.

##### Parameter sampling

We only perform the ABC analysis in this chapter. Two runs have been carried
out. The first focuses on three free parameters, $\Omega_{\mathrm{m}}$,
$\sigma_{8}$, and $w_{0}^{\mathrm{de}}$. The second one includes two
additional parameters, $c_{0}$ and $\beta$, which define the halo $M$-$c$
relation (Eq. 3.93).

Concerning the configuration of ABC, we take a flat prior in the studied
parameter space. The number of particles is 2400. The shutoff parameter is set
to 1%. The distance is the same as Eq. (8.20), accounting for the full
correlation.

### 9.3 Preliminary results

Figure 9.5: Preliminary result of $\Omega_{\mathrm{m}}$-$\sigma_{8}$
constraints with CFHTLenS-KiDS-DES joint data sets. This figure shows the
evolution of the PMC ABC. The blue solid and the red dashed lines represent
respective 1- and 2-$\sigma$ contours. Green points are accepted ABC
particles. Figure 9.6: Preliminary result of
$\Omega_{\mathrm{m}}$-$\sigma_{8}$ constraints with CFHTLenS-KiDS-DES joint
data sets. This figure shows the posterior from the final iteration. The blue
solid and the red dashed lines represent respective 1- and 2-$\sigma$
contours. Green points are accepted ABC particles. The orange curve is the
best power-law fit.

Figure 9.5 shows the primary results that we obtain from the joint data
constraints. In this figure, we only focus on $\Omega_{\mathrm{m}}$ and
$\sigma_{8}$ from the run with three free parameters. As $\epsilon$ decreases,
the contour size reduces. The constraints from the last iteration are shown in
Fig. 9.6. The orange curve shows the best power-law fit (i.e. best
$\Sigma_{8}$ fit, Eq. 8.21) between $\Omega_{\mathrm{m}}$ and $\sigma_{8}$. It
indicates a very good agreement with the Planck cosmology
($\Omega_{\mathrm{m}}=0.308\pm 0.012$, $\sigma_{8}=0.8149\pm 0.0093$). The
contour is consistent with constraints from other lensing surveys using two-
point statistics (CFHTLenS: Kilbinger et al. 2013, Joudaki et al. 2016; KiDS:
Hildebrandt et al. 2016; DES: The Dark Energy Survey Collaboration et al.
2015) or peaks (CFHTLenS: Liu J. et al. 2015a; CFHT Stripe 82: Liu X. et al.
2015b; DES: Kacprzak et al. 2016). However, since our results are only
preliminary, the comparisons are not shown. Here, we only give the
characteristics of the preliminary constraints: the error on the orange curve
is $\Delta\Sigma_{8}=0.13$, obtained by the same method as Chap. 8, and the
FoM is 5.2.

In this preliminary result, the constraints seem to be larger than expected.
If we compare the FoM from data with those from Table 8.5, then the difference
is about a factor of 3. Since the settings for the results from Table 8.5 have
been designed to be very similar to the CFHTLenS data set, we believe that
they can be compared directly. The reason of these looser contours is believed
to be the selection of S/N bins and the bias from our model. A quick
comparison (not plotted) with the Mice simulations (5200 deg2) indicates that
the peak function given by our model is tilted from the one derived from
$N$-body simulations. Typically, our model overestimates high peaks and
underestimates low peaks. If this is the case, then changing the input
cosmology of our model, which leads to a global change in all bins of the peak
functions, could result in similar differences with regard to the observation,
therefore degrades the constraint power. In this scenario, the bad accuracy
reduces the precision, and this is susceptible to be the case of our
preliminary result. For the future, the improvement will be to compare closely
our model with the Mice simulations to quantify the bias, and to adopt only
bins with weak bias for constraints. With this refinement, we expect that the
final results yield a similar FoM and $\Delta\Sigma_{8}$ to the values from
Table 8.5.

Figure 9.7: Preliminary result of
$\Omega_{\mathrm{m}}$-$\sigma_{8}$-$w_{0}^{\mathrm{de}}$ constraints with
CFHTLenS-KiDS-DES joint data sets. Figure 9.8: Preliminary result of
$\Omega_{\mathrm{m}}$-$\sigma_{8}$-$w_{0}^{\mathrm{de}}$-$c_{0}$-$\beta$
constraints with CFHTLenS-KiDS-DES joint data sets.

Figure 9.7 shows all marginalized constraints from the run with three free
parameters. The degeneracy between $w_{0}^{\mathrm{de}}$ and other parameters
is less significant because of ABC. Kernel density estimation smooths the
contours. In Fig. 9.8, we show the constraints with five free parameters. We
notice that the constraint on $c_{0}$ is very broad. However, concerning
$\beta$, the values below $-0.6$ are already been excluded by the preliminary
study. By including two additional parameters, the constraints on
$\Omega_{\mathrm{m}}$ and $\sigma_{8}$ become larger, which is expected.

### 9.4 Perspectives

In spite of the three-year investment, several questions concerning peak-count
modelling still remained unanswered. Below is a list non-exhausted of ideas
and perspectives that future studies could investigate.

###### Halo modelling

All along the thesis, we adopt the spherical NFW modelling for halo density
profiles. However, $N$-body simulations indicate well the asphericity of dark-
matter halos. Without reproducing their true shapes, triaxial halos or
substructures can be a more realistic option for lens modelling. Meanwhile,
baryonic effects can also modify the profiles. These different possible
improvements concerning halo geometry are worth taking a closer look. Next, in
Fig. 5.5, we have already seen that the $M$-$c$ relation of halos suffers from
a huge scatter. Therefore, taking this intrinsic scatter into account can lead
to a more accurate modelling than a deterministic relation. Finally, halo
correlation is another subject to investigate in detail. We have seen that
neglecting correlation leads to a decrease of peak counts which is compensated
by the replacement of $N$-body halo profiles with their NFW proxies. Why do
these two effects vary in opposite ways? Do they really compensate? These are
some questions to investigate for the future.

###### Source modelling

The improvement for source modelling includes at least three aspects:
photo-$z$, galaxy clustering, and intrinsic alignment. The actual photometric
estimation of galaxy redshifts is scattered and can be biased. For real
surveys, we need to correct the bias for peak counts, and to take the
propagation of the scatter into account. Meanwhile, instead of using
randomized sources as in this work, adopting the exact source catalogue would
reduce the stochasticity. Using the survey source catalogue will also account
for galaxy clustering, which is a potential source of bias for peak counts
(Kacprzak et al. 2016). Finally, intrinsic alignment plays an important role
in weak lensing analyses. Understanding its impacts on weak-lensing peaks will
be crucial for future large and high-quality surveys.

###### Filtering

To improve our knowledge about $w_{0}^{\mathrm{de}}$, tomography filters will
be required. This allows us to extract information from different cosmic eras.
Concerning mask effects, inpainting could be interesting as a tool to enhance
the quantity of information. Even if the inpainted information is not included
in statistics, the technique will still help to regularize mask effects and
reduce the systematics.

###### Larger data sets

Ongoing lensing surveys such as KiDS, DES, HSC, and J-PAS, and upcoming
surveys such as Euclid, LSST, WFIRST all aim for providing larger data sets
than CFHTLenS, considered as the current state of the art. Information from
deeper and wider fields will help us refine cosmological constraints with
weak-lensing peak counts.

###### Beyond $\boldsymbol{\Lambda}$CDM

Weak-lensing peak counts are sensitive to cosmological models beyond
$\Lambda\mathrm{CDM}$ which change the mass function. Probing modified gravity
is a future task of weak-lensing peaks. Constraints on parameters of modified
gravity have recently been studied using CFHTLenS data (Liu et al. 2016).
Similarly, peak counts can also be used for studying the effective number of
neutrinos and the neutrino mass. The shape of the mass function changes as
different models are considered.

###### Beyond peak counts

Beyond the abundance, peaks could provide other information potentially
interesting for extracting cosmological information. For example, Marian et
al. (2013) estimated the Fisher information derived from tangential shear
profiles of peaks and peak-peak correlations. These are some observables to
explore for future studies.

##### Summary

We provide constraints on $\Omega_{\mathrm{m}}$, $\sigma_{8}$,
$w_{0}^{\mathrm{de}}$, $c_{0}$, and $\beta$ with data sets from CFHTLenS, KiDS
DR1/2, and DES SV. This is done with our fast stochastic forward model and
ABC.

We also indicate that the result can be improved by excluding bins with large
number-count bias, which will be done in the near future.

Finally, possible improvements for peak-count modelling are listed. We look
forward to larger statistics provided by future surveys for improving our
knowledge about the Universe.

## Chapter X Conclusion

Science may be described as the art of systematic over-simplification.

— Karl Popper

This thesis studies cosmology with weak-lensing peak counts. In my work, a new
model has been proposed, peak statistics has been analyzed, and cosmological
constraints have been obtained.

The new model adopts a stochastic forward approach. This strategy permits to
include real-world effects and guarantees flexibility. In addition, it is
time-saving compared to the $N$-body process. These advantages make our model
the ideal choice for modelling peak counts.

Starting with a mass function, our model simulates uncorrelated halos with NFW
profiles. It computes the projected mass from this simulation, and adds noise
to generate galaxy shape catalogues. Then it applies a mass-mapping technique
to obtain a convergence map, and counts and regroups peaks by their S/N. The
model is implemented by the public code Camelus.

The model is based on two hypotheses. First, diffuse and unbound matter has
little impact; second, halo correlation contributes little to peak counts.
This halo approach, corresponding to the one-halo term, leads to a very good
approximation to peaks from $N$-body simulations. We examined the model in
detail and discovered two compensated effects: the replacement of irregular
halo profiles by the NFW ones enhances peak counts, and decorrelation
decreases numbers.

In this study, the Gaussian likelihood assumption has been tested in different
ways. On the one hand, the cosmological dependence of the covariance can not
be neglected. Assuming a constant covariance leads to a loss of constraint
power. On the other hand, the constraints from the Gaussian likelihood agree
well with the results from generalized techniques: the copula likelihood and
non-parametric methods such as the true likelihood and the $p$-value. The
conclusion is that the Gaussian likelihood is a good approximation to weak-
lensing peak counts.

Beyond the likelihood analysis, approximate Bayesian computation (ABC)
provides an alternative for parameter constraints. It is especially
competitive for problems with complex likelihood shapes. It has been shown
that ABC yields a consistent result compared with the likelihood in our
studied framework, and can reduce the computation time by two orders of
magnitude.

This study has also provided a framework in which filtering techniques can be
fairly compared. Regarding the complexity of how cosmology is encrypted in the
lensing observables, it is very difficult to determine directly an optimal
filter which minimizes the contour sizes from peak counts. However, ranking
filters by their performance is possible. This methodology has been followed
for filter comparisons. We concluded that compensated filters are more
competitive than the usual Gaussian functions and that the separated strategy
is well more recommended than the combined strategy when dealing with
multiscale information.

This peak-count analysis is applied to three survey data jointly: CFHTLenS,
KiDS DR1/2, and DES SV. So far, preliminary results agree well with Planck,
also with two-point-statistic results from CFHTLenS, KiDS, and DES. The
quality of the constraints is characterized by $\Delta\Sigma_{8}=0.13$ and FoM
$=5.2$.

Various improvements for the new model are possible. How to properly model the
halo correlations? How to deal with the intrinsic scatter of the halo
concentration parameters? Does baryonic physics influence peak counts? How do
peaks response to intrinsic alignment of the galaxies? What additional
information can we extract from a tomography filter? What are the numbers of
peaks in a massive neutrino cosmology and in a modified gravity model? These
are the questions worth investigating for the future.

Conceptually, two ideas have been promoted and put forward by this thesis. The
first one is the notion of fast stochastic forward modelling, which provides
an approach to predict non-Gaussian observables and to account for complex
survey effects; the other one is approximate Bayesian computation, which is a
promising likelihood-free parameter inference tool. As the need for exploring
higher-order statistics increases, it is expected that these two concepts rise
for future cosmological studies.

## Abbreviations

1D | one-dimensional
---|---
2D | two-dimensional
2PCF | two-point-correlation function
3D | three-dimensional
3PCF | three-point-correlation function
ABC | approximate Bayesian computation
AGN | active galactic nucleus
BAO | baryon acoustic oscillations
Calclens | Curved-sky grAvitational Lensing for Cosmological LightconE simulatioNS
Camelus | Counts of Amplified Mass Elevations from Lensing with Ultrafast Simulation
CCD | charge-coupled device
CDC | cosmology-dependent covariance
CDF | cumulative distribution function
CDM | cold dark matter
CFHT | Canada-France-Hawaii Telescope
CFHTLenS | Canada-France-Hawaii Telescope Lensing Survey
CFHTLS | Canada-France-Hawaii Telescope Legacy Survey
CMB | cosmic microwave background
COSEBI | complete orthogonal sets of $E$-/$B$-mode integral
COSMOS | Cosmological Evolution Survey
CPU | central processing unit
cst | constant
DES | Dark Energy Survey
DES SV | Dark Energy Survey science verification
DIM-ACAV | Domain d’Intérêt Majeur en Astrophysique et Conditions d’Apparition de la Vie
DLS | Deep Lens Survey
DM | dark matter
FASTLens | FAst STatistics for weak Lensing
FFT | fast Fourier transform
FLRW | Friedmann-Lemaître-Robertson-Walker
FOF | friends of friends
FoM | figure of merit
FORS | FOcal Reducer Spectrograph
FSF | fast stochastic forward
FSL | Fan-Shan-Liu
GaBoDS | Garching-Bonn Deep Survey
Gadget | GAlaxies with Dark matter and Gas intEracT
Glimpse | Gravitational Lensing Inversion and MaPping with Sparse Estimators
GPU | graphics processing unit
HEALPix | Hierarchical Equal Area isoLatitude Pixelization
HOD | halo occupation distribution
HSC | Subaru Hyper Suprime-Cam
HST | Hubble Space Telescope
IA | intrinsic alignment
iid | independent and identically distributed
J-PAS | Javalambre Physics of the Accelerating Universe Astrophysical Survey
KDE | kernel density estimation
KiDS | Kilo-Degree Survey
KiDS DR1/2 | Kilo-Degree Survey data releases 1 and 2
KS | Kaiser-Squires
LPT | Lagrangian perturbation theory
LSS | large-scale structure
LSST | Large Synoptic Survey Telescope
$\mathsf{M}$-$\mathsf{c}$ | mass-concentration
MCMC | Markov Chain Monte Carlo
MF | Minkowski functional
Mice | Marenostrum Institut de Ciencias de l’Espai
MPI | message passing interface
MRLens | Multi-Resolution methods for gravitational LENSing
NFW | Navarro-Frenk-White
Nicaea | NumerIcal Cosmology And lEnsing cAlculations
PDF | probability density function
photo-$\mathsf{z}$ | photometric redshift
Pinocchio | PINpointing Orbit-Crossing Collapsed HIerarchical Objects
PMC | population Monte Carlo
PNCG | Programme National de Cosmologie et Galaxies
PSF | point spread function
Rockstar | Robust Overdensity Calculator using $K$-Space Topologically Adaptive Refinement
SCDM | standard cold dark matter
SDSS | Sloan Digital Sky Survey
SIS | singular isothermal sphere
SMC | sequential Monte Carlo
S/N | signal-to-noise ratio
SNIa | supernova of type Ia
SO | spherical overdensity
SZ | Sunyaev–Zel’dovich
VLT | Very Large Telescope
WFIRST | Wide Field Infrared Survey Telescope
WL | weak lensing
WMAP | Wilkinson Microwave Anisotropy Probe
$\mathsf{\Lambda}$CDM | cosmological constant and cold dark matter

## Notation

boldface | vector or matrix
---|---
lower case | physical coordinates and radii
upper case |  comoving coordinates and radii, except for $w$, $f_{K}(w)$
$\dot{\mbox{}}$ | time derivative
$\widehat{\mbox{}}$ | estimator
$\widetilde{\mbox{}}$ | Fourier transform
∗ | complex conjugate
$\ast$ | convolution operator
$\nabla$ | gradient operator
$\nabla_{\perp}$ | transverse derivative operator
$\mathbbm{1}$ | indicator function
$a$ | scale factor
$A$ | amplitude of the power spectrum
$A_{\mathrm{pix}}$ | pixel area
$A_{\epsilon}$ | probability of passing the one-sample test
$\mathcal{A}$ | distortion matrix
$B$ | brightness (flux density)
$\mathrm{c}$ | light speed
$c$ | additive correction for shape measurement
$c$ | copula density
$c$ | halo concentration
$c_{0}$ |  amplitude parameter for the halo $M$-$c$ relation
$C$ | copula
$C$ | two-point-correlation function
$\boldsymbol{C}$ | covariance matrix
$d$ | dimension of a data vector
$D$ | differential operator
$D$ | distance function
$D_{+}$ | growth factor
$D_{\mathrm{A}}$ | angular diameter distance
$D_{\mathrm{H}}$ | Hubble distance at the current time
$D_{\ell}$ | angular diameter distance between the lens and the observer
$D_{\ell\mathrm{s}}$ | angular diameter distance between the lens and the source
$D_{\mathrm{L}}$ | luminosity distance
$D_{\mathrm{p}}$ | proper distance
$D_{\mathrm{s}}$ | angular diameter distance between the source and the observer
$e_{N}$ |  change of variables used in Chap. 5
$E$ | energy
$E$ | ratio of Hubble parameters
$f$ | dimension of a parameter vector
$f$ | filling factor
$f$ | mass function, multiplicity function
$f$ |  quantity for NFW profiles
$f_{K}$ | comoving transverse distance
$f_{\mathrm{NL}}$ | amplitude of the primordial non-Gaussianity
$F$ | cumulative distribution function
$g$ | lens efficiency
$g$ | lensing reduced shear
$\boldsymbol{g}$ | gravitational field
$g_{\mu\nu}$ | metric
$\mathrm{G}$ | gravitational constant
$G$ | dimensionless projected mass
$G$ | Green function
$G_{\mu\nu}$ | Einstein tensor
$h$ | dimensionless Hubble parameter at the current time
$H$ | Hubble parameter
$\boldsymbol{H}$ | bandwidth matrix for kernel density estimation
$H_{0}$ | Hubble parameter at the current time
$I$ | integral
$I$ | surface brightness (brightness density)
$k$ |  norm of the vector $\boldsymbol{k}$
$\boldsymbol{k}$ |  wave vector in 3D Fourier space
$K$ | smoothed noiseless convergence
$K$ | spacetime curvature
$K_{N}$ | smoothed noisy convergence
$\boldsymbol{\ell}$ |  wave vector in 2D Fourier space
$L$ | linear differential operator
$L$ | log-likelihood function
$L$ | luminosity
$\mathcal{L}$ | likelihood function
$\ln$ | natural logarithm, to base e
$\log$ | common logarithm, to base 10
$m$ | multiplicative correction for shape measurement
$M$ | mass
$M_{\star}$ | pivot mass
$M_{\odot}$ | solar mass
$M_{\mathrm{ap}}$ | aperture mass
$M_{\mathrm{max}}$ | upper limit of mass sampling
$M_{\mathrm{min}}$ | lower limit of mass sampling
$n$ | mass function, halo number density
$n$ | noise field
$n_{\mathrm{gal}}$ | galaxy number density
$n_{\mathrm{peak}}$ |  peak function, PDF of having a peak with a specific S/N
$n_{\mathrm{s}}$ | scalar index, power law index of the scalar field power spectrum
$N$ | number of realizations
$N$ | smoothed noise field
$N_{\mathrm{gal}}$ | number of galaxies
$\mathcal{N}(\boldsymbol{\mu},\boldsymbol{C})$ |  normal law with the mean $\boldsymbol{\mu}$ and the covariance $\boldsymbol{C}$
$p$ |  $p$-value
$p$ | pressure
$p$ | source redshift distribution
$p_{i}$ |  $p$-value corresponding to the $i$-$\sigma$ significance
$P$ | distribution in the data space
$P$ | power spectrum
$P(\cdot|\boldsymbol{\pi})$ |  probability density of a stochastic model under a parameter set $\boldsymbol{\pi}$
$\mathcal{P}$ | distribution in the parameter space
$\mathcal{P}(\cdot)$ | prior distribution
$\mathcal{P}(\cdot|\boldsymbol{x}^{\mathrm{obs}})$ | posterior distribution
$P_{\kappa}$ |  2D convergence power spectrum
$Q$ | aperture mass filter in shear space
$Q$ | number of particles
$Q$ | second-order moment tensor
$r$ | success ratio
$r_{\mathrm{s}}$ | scale radius
$r_{\mathrm{stop}}$ | shutoff parameter
$r_{\mathrm{vir}}$ | physical virial radius
$\boldsymbol{r}$ | physical coordinates
$R$ | Ricci scalar
$R_{\mathrm{vir}}$ | comoving quantity corresponding to the physical virial radius
$R_{\mu\nu}$ | Ricci tensor
$\boldsymbol{R}$ | comoving coordinates
$s$ | summary statistic
$S$ | source function
$S$ | surface
$t$ | cosmic time
$T$ | number of iterations
$T_{\mu\nu}$ | stress-energy tensor
$\mathcal{T}$ | optical tidal matrix
$u$ | percentile
$U$ | aperture mass filter in convergence space
$\boldsymbol{u}$ | velocity
$\boldsymbol{v}$ | peculiar velocity
$V$ | volume
$w$ | comoving radial distance
$w$ | equation of state
$w_{0}^{\mathrm{de}}$ | equation of state of dark energy
$W$ | filter function
$W_{\boldsymbol{C}}$ |  Gaussian smoothing kernel with covariance $\boldsymbol{C}$
$W_{R}$ |  filter function with characteristic size $R$
$x_{N}$ |  change of variables used in Chap. 5
$\boldsymbol{x}$ | data vector, vector of observables
$\boldsymbol{x}^{\mathrm{mod}}$ | data vector from the model prediction
$\boldsymbol{x}^{\mathrm{obs}}$ | data vector from the observation
$z$ | complex number
$z$ | redshift
$z_{\mathrm{s}}$ | source redshift
$\alpha$ |  FDR parameter for MRLens
$\alpha$ |  inner slope parameter for NFW profiles
$\alpha$ | slope parameter for brightness power law
$\alpha$ |  slope parameter for $\Sigma_{8}$
$\alpha_{\mathrm{gal}}$ | source distribution parameter
$\beta$ |  dragging value for $\Sigma_{8}$
$\beta$ | mass power law index for the halo mass-concentration relation
$\beta_{\mathrm{gal}}$ | source distribution parameter
$\boldsymbol{\beta}$ | unlensed angular position
$\gamma$ | lensing shear
$\gamma_{\ast}$ |  quantity used in Chap. 5
$\gamma_{+}$ | tangential shear
$\gamma_{\times}$ | cross shear
$\delta$ | density contrast
$\delta$ | difference between summary statistics
$\delta$ | Dirac delta function
$\delta$ | Kronecker delta
$\delta_{\mathrm{c}}$ | linear threshold for density contrast
$\Delta$ | overdensity
$\Delta^{2}$ | dimensionless power spectrum
$\epsilon$ | observed ellipticity
$\epsilon$ | tolerance level
$\epsilon^{(\mathrm{s})}$ | intrinsic ellipticity
$\theta$ |  change of variables used in Chap. 3
$\theta$ |  distance in the angular space, norm of the vector $\boldsymbol{\theta}$
$\boldsymbol{\theta}$ |  2D angular position coordinates
$\theta_{\ast}$ |  quantity used in Chap. 5
$\theta_{1}$ | angular coordinate component, aligned with right ascension
$\theta_{2}$ | angular coordinate component, aligned with declination
$\theta_{\mathrm{G}}$ | Gaussian kernel size
$\theta_{\ker}$ | kernel size
$\theta_{N}$ |  change of variables used in Chap. 5
$\theta_{\mathrm{pix}}$ | pixel width
$\theta_{\mathrm{s}}$ | angular scale radius
$\theta_{\mathrm{vir}}$ | angular virial radius
$\Theta$ | Heaviside step function
$\kappa$ | lensing convergence
$\Delta\kappa$ |  difference between the true convergence and $\kappa_{\mathrm{halo}}$
$\kappa_{0}$ |  free constant from the KS inversion
$\kappa_{1}$ |  correction for $\kappa_{\mathrm{proj}}$
$\kappa_{\mathrm{halo}}$ | lensing convergence from the projected mass of a halo
$\kappa_{\mathrm{proj}}$ | lensing convergence from the projected mass along a line of sight
$\kappa_{n}$ | convergence with noise
$\lambda$ | affine parameter
$\lambda$ | eigenvalue
$\lambda$ | mass proportion within a specified range
$\lambda$ | mass-sheet degeneracy
$\lambda$ | threshold for filling factor
$\lambda$ | wavelength
$\Lambda$ | cosmological constant
$\mu$ | lensing magnification
$\nu$ |  S/N of peaks
$\boldsymbol{\xi}$ | physical separation vector between two geodesics
$\boldsymbol{\Xi}$ | comoving separation vector between two geodesics
$\boldsymbol{\pi}$ | parameter vector
$\boldsymbol{\pi}^{\mathrm{in}}$ | input parameter vector
$\rho$ | matter density
$\mathrm{bar}{\rho}$ | background matter density
$\mathrm{bar}{\rho}_{0}$ | background matter density at the current time
$\rho_{\mathrm{crit}}$ | critical density at the current time
$\rho_{\mathrm{s}}$ | central mass density
$\sigma$ | density dispersion
$\sigma_{8}$ |  matter fluctuation under a spherical smoothing with radius 8 $\text{Mpc}/h$
$\sigma_{i}$ |  $i$-th order moment of a smoothed noise field
$\sigma_{\mathrm{noise}}^{2}$ | variance of a smoothed noise field, squared zero-order moment
$\sigma_{\mathrm{pix}}^{2}$ | variance of noise in a pixel
$\sigma_{\epsilon}^{2}$ | variance of the intrinsic ellipticity dispersion, sum of both components
$\Sigma$ | surface mass density
$\Sigma_{8}$ |  tilted matter fluctuation defined by Eq. (6.4) or Eq. (8.21)
$\Delta\Sigma_{8}$ |  width of the banana-shaped contour on the $\Omega_{\mathrm{m}}$-$\sigma_{8}$ plane
$\Sigma_{\mathrm{crit}}$ | critical surface mass density
$\tau$ | conformal time
$\phi$ | probability density of a normal distribution
$\phi$ | reduced Newtonian potential
$\varphi$ | rotation angle
$\Phi$ | Newtonian potential
$\chi_{f}^{2}$ |  probability density of the chi-squared distribution with $f$ degrees of freedom
$\psi$ | lensing potential
$\omega$ | weight, weight function
$\Omega$ | solid angle
$\Omega_{\mathrm{b}}$ | dimensionless baryon density at the current time
$\Omega_{K}$ | dimensionless density corresponding to the space-time curvature
$\Omega_{\mathrm{m}}$ | dimensionless matter density at the current time
$\Omega_{\mathrm{r}}$ | dimensionless radiation density at the current time
$\Omega_{\Lambda}$ |  dimensionless density corresponding to the cosmological constant for a $\Lambda\mathrm{CDM}$ model

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