Tracy is currently working toward completing her Master of Psychotherapy and Spirituality degree at St. Stephen's College in Edmonton. This program integrates both psychology and spirituality, and is broadly inclusive of all spiritual perspectives and traditions.
Tracy has an extensive background in diversity education and community development. She works with the BC provincial government, where she designs and manages programs which build the capacity of organizations and communities to be inclusive of cultural diversity and new immigrants. Prior to her work in government she delivered diversity education to schools and service providers, and facilitated peace-building projects in Sudan.
Honouring diversity and difference is central to her practice. Tracy values collaboration, transparency and compassion and sees counselling as an opportunity for clients to explore, gain insight, and access their inner strengths to make concrete change. She is also an artist and passionate about the potential of creative expression to support healing and growth.


text:
\section{Introduction}

The standard model of cosmology, in which the dominant matter component consists of cold dark matter (CDM) while the late-time accelerated expansion is driven by a positive cosmological constant, $\Lambda$, is highly successful at describing a number of independent observations, which constrain these parameters with a large degree of concordance. Notably, measurement of fluctuations in the Cosmic Microwave Background (CMB) \citep{Planck2018}, provides measurements of the present day expansion rate of the universe $H_{0}$, the matter density parameter $\Omega_{\rm m}$ and the matter fluctuation amplitude $\sigma_8$.

Another promising observational probe that is sensitive to and can be used to constrain many cosmological parameters is gravitational lensing, a phenomenon according to which the light of distant source images is distorted by the gravitational potentials of the foreground matter. In the strong lensing regime distant galaxies are visibly distorted into large arcs. In the weak lensing (WL) regime, which is the focus of this study, this effect is much smaller, and the weak lensing signal is measured through the correlations in distortions of many source galaxies \citep{Bacon2000,Kaiser2000,VanWaerbeke2000,Wittman2000}. This allows us to probe the total matter distribution of the Universe on the largest scales \citep[see][for reviews]{Bartelmann2001,Kilbinger2015}, and offers a powerful method to study the clustering of dark matter and its evolution.

Some of the most recent WL observations that supplement the parameter measurements from the CMB include the DES \citep{Troxel2018} \footnote{\url{https://www.darkenergysurvey.org/}}, HSC \citep{Hikage2019} \footnote{\url{https://hsc.mtk.nao.ac.jp/ssp/}} and KiDS \citep{Asgari2020} \footnote{\url{http://kids.strw.leidenuniv.nl/}} WL surveys. However, all of these surveys measure lower values of $\sigma_8$ compared to Planck, with a statistically significant disagreement arising in the comparison between the Planck and KiDS constraints. This is one example of the parameter tensions that have arisen in recent years, where different observations point to slightly different values of certain cosmological parameters, implying the presence of either unaccounted for systematics or new physics which are unaccounted for. Another example is the $H_0$ tension, where multiple observations find that measurements from the early Universe are broadly inconsistent with those of the late universe \citep{Verde2019}, particularly the distance scale measurement of $H_0$ based on Cepheids by the SH0ES collaboration \citep{Riess:2019cxk}.

In order to address these parameter tensions, it is important to measure cosmological parameters as precisely as possible, by maximising the information that can be extracted from a given survey. The standard approach for weak lensing surveys is to measure $\Lambda\rm{CDM}$ parameters with two-point statistics such as the shear-shear correlation function or the convergence power spectrum  \citep{Schneider2002,Semboloni2006,Hoekstra2006,Fu2008,Heymans2012,Kilbinger2013,Hildebrandt2017,Troxel2018,Hikage2019,Aihara2019,Asgari2020}. However, two-point statistics do not capture non-Gaussian information, and weak lensing data are highly non-Gaussian due to the non-linear evolution of the Universe. To address this loss, many complimentary statistics have been developed, which encapsulate information beyond two-point statistics. A common and popular example is the abundance of WL peaks, which has been shown to be complimentary to the two-point function and helps break the $\Omega_m$-$\sigma_8$ parameter degeneracy \citep{Jain2000,Pen2003,Dietrich2010}. Peaks are also shown to outperform the standard methods for constraining the sum of neutrino mass \citep{Li2018} and $w_0$ \cite{Martinet2020}. By including complimentary statistics, the measurement errors on cosmological parameters can be reduced, which will help inform the statistical significance of any parameter tensions between multiple observations.

The goal of this paper is to present parameter constraint forecasts for one such complimentary probe, WL voids. Voids are typically identified within the full 3D distribution of matter as regions of low matter density or low tracer density, for which void statistics such as their abundance, radial profiles and shapes contain useful non-Gaussian information \citep[see, e.g.,][]{White1979,Fry1986,Biswas2010,Bos2012,Lavaux2012}. Most studies use galaxy voids, which are identified as underdense regions in the galaxy distribution \citep[e.g.,][]{Paz2013,Sutter2014,Cautun2016,Nadathur2016}, where galaxy void statistics are complementary to the galaxy power spectrum and baryonic acoustic oscillations \citep[e.g.,][]{Pisani2015,Hamaus2016,Nadathur2019}. Recently, void WL profiles have also been shown to be a powerful cosmological probe \citep[see, e.g.,][]{Cai2015,Barreira2015,Gruen:2015jhr,Barreira:2016ias,Falck2018,Baker:2018mnu}.

While less explored compared with 3D voids, voids can also be identified in projection, such as in the projected galaxy distribution \citep[e.g.][]{Gruen2015,Barreira:2016ias,Sanchez2017,Cautun2018} or in a weak lensing map \citep[e.g.][]{Davies2018,Coulton2019}.
Here, we follow the latter approach and define WL voids generally as 2D regions within WL convergence maps that contain low convergence or few to no tracers. In a previous work \citep{Davies2018}, we have shown that the lensing profiles of 2D WL voids identified directly in WL convergence maps can be measured with a larger signal-to-noise ratio (SNR) than those of galaxy voids. This is because WL voids correspond to deeper underdensities projected along the line of sight than galaxy voids, and hence they have larger tangential shear profiles. This also means that WL voids are better at constraining alternative cosmological models, such as modified gravity models, than galaxy voids \citep{Davies2019b}. Additionally, compared to other WL statistics, WL voids are less affected by baryonic physics \citep{Coulton2019}.

In this paper we use the \emph{cosmo}-SLICS{} simulation suite \citep{Harnois2019} to identify a particular class of WL voids, the {\it tunnels}, for a range of cosmological parameters. We use this data to train a Gaussian process regression emulator, which, combined with Markov chain Monte Carlo, allows us to generate likelihood contours and provide forecast parameter constraints for an \textsc{lsst}-like survey.

The tunnel algorithm we use here is one possible choice of WL void finder. In fact, similar to voids identified in the galaxy distribution \citep[e.g.][]{Colberg2008,Cautun2018,Baker:2019gxo}, there are several void finding methods that have been successfully applied to WL maps. For example, \citet{Davies2020} have carried out a detailed analysis on the impact that varying the WL void definition might have on the resulting WL void statistics.
They have found that the `tunnel' void finding algorithm offers a great trade off between maximising the observable tangential shear profile SNR and minimising the impact of observational noise on the void statistics. Therefore, we limit our analysis to only tunnels, and we defer a more detailed study comparing the parameter constraining powers of different void finders to a future work. For galaxy voids, studies have shown that combining different void definitions can lead to improved cosmological constraints \citep[e.g.][]{Paillas2019}.

The layout of the paper is as follows. In Section \ref{sec:theory} we outline the relevant theory for WL observations. In Section \ref{sec:Mock data etc} we describe our mock observational data, emulation and likelihood analysis pipeline and void finding algorithm. In Section \ref{sec:statistics} we present the WL void statistics used in our analysis and in Section \ref{sec:parameter constraints forecast} we present our parameter constraint forecasts. Finally we conclude in Section \ref{sec:conclusion}. For completeness, we also have two appendices where we study respectively the accuracy of our emulator and the impact of varying the smoothing scale of WL maps.

\section{Theory}\label{sec:theory}

The lens equation for a gravitationally lensed image is
\begin{equation}
    \pmb{\alpha} = \pmb{\beta} - \pmb{\theta} \, ,
\end{equation}
where $\pmb{\alpha}$ is the deflection angle between $\pmb{\beta}$, the true position of the source on the sky, and $\pmb{\theta}$, the observed position of the lensed image.
The corresponding Jacobian matrix of the (linear) lens mapping is the deformation matrix \textbf{A},
\begin{equation}
    A_{ij} = \frac{\partial \beta_{i}}{\partial \theta_{j}} = \delta_{ij} - \frac{\partial \alpha_{i}}{\partial \theta_{j}} \, .
    \label{eq:amp mat}
\end{equation}
Under the Born approximation and neglecting lens-lens coupling and other second-order effects, the deflection angle can be expressed as the gradient of a 2D lensing potential $\psi$,
\begin{equation}
    \pmb{\alpha} = \pmb{\nabla}\psi \, ,
    \label{eq:alpha}
\end{equation}
where $\psi$ is given by
\begin{equation}
    \psi(\pmb{\theta},\chi) = \frac{2}{c^2} \int_0^{\chi} \frac{\chi - \chi'}{\chi \chi'} \Phi(\chi' \pmb{\theta},\pmb{\theta}) d\chi' \, .
    \label{eq:lensing potential}
\end{equation}
Here, $\chi$ is the comoving distance from the observer to the source and $\chi'$ is the comoving distance from the observer to the continuously-distributed lenses, which is also the integration variable. $\Phi$ is the 3D lensing potential of the lens, and $c$ the speed of light. $\Phi$ is related to the non-relativistic matter density contrast, $\delta= \rho/\bar{\rho} - 1$, through the Poisson equation
\begin{equation}
    \nabla^2\Phi = 4 \pi G a^2 \bar{\rho} \delta \, ,
    \label{eq:Poisson equation}
\end{equation}
where $\rho$ is the matter density of the Universe (with a bar denoting the mean),
$G$ is the gravitational constant and $a$ is the scale factor.

Eq.~\eqref{eq:lensing potential} shows that the lensing potential is a line-of-sight integral of the matter distribution from the source to the observer. The contribution that matter at distance $\chi'$ along the line of sight makes to the total lensing potential is weighted by $(\chi - \chi') / \chi \chi'$ and so depends on its distances from the source and observer.

Eq.\eqref{eq:alpha} allows Eq. \eqref{eq:amp mat} to be expressed in terms of $\psi$
\begin{equation}
    A_{ij} = \delta_{ij} - \partial_{i} \partial_{j} \psi \, ,
\end{equation}
where partial derivatives are taken with respect to $\pmb{\theta}$. The  matrix $\pmb{A}$ can be parameterised through the more physically instructive terms convergence, $\kappa$, and shear, $\gamma = \gamma_1 + i\gamma_2$, as
\begin{equation}
    \pmb{A} =
    \begin{pmatrix}
    1 - \kappa -\gamma_1 & -\gamma_2\\
    -\gamma_2 & 1-\kappa+\gamma_1
    \end{pmatrix}
    \, .
\end{equation}
This parameterisation allows the convergence and shear to be related to the lensing potential via
\begin{equation}
    \kappa \equiv \frac{1}{2} \nabla^2_{\pmb{\theta}} \psi \, ,
    \label{eq:convergence}
\end{equation}
and
\begin{equation}
    \gamma_1 \equiv \frac{1}{2}\left(\nabla_{\pmb{\theta}_1}\nabla_{\pmb{\theta}_1}-\nabla_{\pmb{\theta}_2}\nabla_{\pmb{\theta}_2}\right)\psi,
    \quad\quad\quad
    \gamma_2 \equiv \nabla_{\pmb{\theta}_1}\nabla_{\pmb{\theta}_2}\psi,
    \label{eq:shear}
\end{equation}
where $\nabla_{\pmb{\theta}} \equiv (\chi')^{-1}\nabla$. Eq.~\eqref{eq:convergence} can be interpreted as a 2D Poisson equation, and so by substituting Eq.~\eqref{eq:Poisson equation} and Eq.~\eqref{eq:lensing potential} into Eq.~\eqref{eq:convergence}, the convergence can be expressed in terms of the matter overdensity
\begin{equation}
    \kappa(\pmb{\theta},\chi) = \frac{3H_0^2\Omega_{\rm{m}}}{2c^2}\int_0^{\chi}\frac{\chi - \chi'}{\chi} \chi' \frac{\delta(\chi'\pmb{\theta},\chi')}{a(\chi')} d\chi' \, .
    \label{eq:conv source}
\end{equation}
This shows that the observed WL convergence can be interpreted as the projected density perturbation along the line of sight, weighted by the lensing efficiency factor $(\chi-\chi')\chi'/\chi$. Here, the lensing efficiency is greatest at $\chi' = \chi / 2$, when the lens is halfway between the source and the observer.

The above derivation assumes a fixed source plane. However, in real WL observations, the source galaxies do not occupy a single plane at a fixed distance from the observer. The observed catalogue of source galaxies has a probability distribution $n(\chi)$ that spans over a range of $\chi$ values, and Eq. \eqref{eq:conv source} must be weighted by this source galaxy distribution in order to obtain $\kappa(\pmb{\theta})$ \citep[see, e.g.,][for details]{Kilbinger2015}
\begin{equation}
    \kappa(\pmb{\theta}) = \int_0^{\chi} n(\chi') \kappa(\pmb{\theta},\chi') d\chi' \, .
\end{equation}

In this work, we measure the $\kappa$ profile, $\kappa(r)$, in and around WL voids. However, as $\kappa(r)$ is not directly observable, it is also useful to relate it to the radial tangential shear profile, $\gamma_t(r)$, through
\begin{equation}
    \gamma_{\rm{t}}(r) = \bar{\kappa}(< r) - \kappa (r)
    \label{eq:gamma_t} \;,
\end{equation}
where
\begin{equation}
    \bar{\kappa}(< r) = \frac{1}{\pi r^2}\int_0^{r} 2 \pi r' \kappa(r') dr'
    \;
\end{equation}
is the mean enclosed convergence within radius $r$. Notice that here and throughout this paper we use $r$ rather than $\theta$ to
represent the 2D
distance from the void centre.

WL observations rely on accurately measuring the shapes of galaxies, and cross correlating the shapes of neighbouring galaxies. However, any correlation in shape due to lensing is dominated by the random shapes and orientations of galaxies, which is the leading source of noises in WL observations, referred to as galaxy shape noise (GSN).
Since the lensing signal is weak by definition, when identifying WL peaks (local maxima in the convergence field $\kappa(\pmb{\theta})$) it is convenient to express the convergence relative to the standard deviation of the corresponding GSN component of the field. This is given by
\begin{equation}
    \nu = \frac{\kappa}{\sigma_{\rm{GSN}}}
    \label{eq:nu}
\end{equation}
where $\sigma_{\rm{GSN}}$ is the standard deviation of the contributions to the signal from galaxy shape noise. $\sigma_{\rm{GSN}}$ can be calculated by generating mock GSN maps and applying any transformations also applied to the convergence maps, such as smoothing. Mock GSN maps are generated by assigning to pixels random convergence values from a Gaussian distribution with standard deviation
\begin{equation}
        \sigma_{\rm{pix}}^2 = \frac{\sigma_{\rm{int}}^2}{2 \theta_{\rm{pix}} n_{\rm{gal}} }
    \;,
    \label{eq: GSN gaussian}
\end{equation}
where $\theta_{\rm{pix}}$ is the width of each pixel, $\sigma_{\rm{int}}$ is the intrinsic ellipticity dispersion of the source galaxies, and $n_{\rm{gal}}$ is the measured source galaxy number density. In this work we use $\sigma_{\rm{int}} = 0.28$ and $n_{\rm{gal}} = 20 $ arcmin$^{-2}$ as will be discussed in Section \ref{sec:Numerical simulations} .

It will also be useful to compare constraints from Eq. \ref{eq:gamma_t} to the standard shear two-point correlation function, which is given by

\begin{equation}
    \xi_{\pm}(\pmb{\theta}) = \langle \gamma_t \gamma_t \rangle \pm \langle \gamma_{\times} \gamma_{\times} \rangle = \frac{1}{2\pi} \int_0^{\infty} dl l P_{\kappa}(l) J_{0,4}(l\pmb{\theta}) \, ,
\end{equation}

where $\gamma_t = -\mathbb{R}(\gamma e^{-2i\phi})$ (equivalent to Eq. \eqref{eq:gamma_t}, but presented for completeness), $\gamma_{\times} = -\mathbb{I}(\gamma e^{-2i\phi})$, $\phi$ is the polar angle of the separation vector $\pmb{\theta}$ , $J_0$ and $J_4$ are the Bessel functions for $\xi_{+}$ and $\xi_{-}$ respectively, and $l$ is the Fourier mode.

\section{Methodology}
\label{sec:Mock data etc}

In this section we describe the methodology followed in this work, including the simulations and mock lensing data, likelihood analysis and the weak lensing void (tunnels) finding algorithm.

\subsection{Mock Data}
\label{sec:Numerical simulations}

In this work we use mock WL convergence maps generated from the \emph{cosmo}-SLICS{} simulation suite \citep{Harnois2019}, which we briefly outline in this subsection.

\begin{figure}
    \centering
    \includegraphics[width=\columnwidth]{Figures/Cosmolslics_nodes.pdf}
    \vskip -.1cm
    \caption{The four dimensional parameter space ($[\Omega_m,S_8,h,w_0]$) sampled by the \emph{cosmo}-SLICS{} simulation suite. The fiducial cosmology is indicated by a star with parameter values [$0.29$, $0.82$, $0.69$, $-1.00$]. We have highlighted two additional nodes with blue ([$0.48$, $0.68$, $0.64$, $-0.77$]) and red ([$0.17$, $0.86$, $0.79$, $-1.69$]) stars, which are selected as nodes in separate regions of the parameter space, used to exemplify the behavior of WL voids as a function of cosmological parameters.
    }
    \label{fig:cosmo nodes}
\end{figure}

The \emph{cosmo}-SLICS{} suite is a set of N-body dark-matter-only simulations run for 26 cosmology nodes in the $[\Omega_m,S_8,h,w_0]$ parameter space. Here $\Omega_m$ is the matter density parameter today, $S_8 = \sigma_8 (\Omega_m/0.3)^{0.5}$, $h = H_0 / 100 {\rm kms}^{-1} \ensuremath{~\mathrm{Mpc}}^{-1}$ is the reduced Hubble constant, and $w_0$ is the dark energy equation of state parameter, which is assumed to be a constant. The $\sigma_8$ parameter is the present-day root-mean-squared matter density perturbation smoothed on $8h^{-1}\ensuremath{~\mathrm{Mpc}}$ scales.

The four dimensional parameter space is sampled using a Latin hypercube, which is a sampling algorithm designed to give a high interpolation accuracy for a low node count. The exact cosmological parameter space that is modelled by each simulation node is shown in Fig.~\ref{fig:cosmo nodes}. At each node, a carefully-designed pair of simulations are run, for which sampling variance is highly suppressed. This is achieved by selecting a pair of initial conditions out of a large number of random realisations, such that the mean matter power spectrum closely matches the ensemble average. The random phases of this pair of initial conditions are used for all cosmology nodes. The simulation volume is a cube with length $L = 505 \ensuremath{~h^{-1}\mathrm{Mpc}}$, with $N = 1536^{3}$ dark matter particles.

For each node, 50 pseudo-independent light-cones are constructed by resampling projected mass sheets, which are then ray-traced under Born approximation to construct lensing maps and catalogues \citep[see][for full details about the light-cone and catalogue construction]{Harnois2019}.

We use the \emph{cosmo}-SLICS{} source catalogue down-sampled to match $\textsc{lsst}$ specifications with a source redshift distribution of $z_s = [0.6,1.4]$, which gives a conservative source galaxy number density of $20 \arcmin^{-2}$. From this we generate 50 WL convergence maps for each of the 26 cosmology nodes, with a sky coverage of \map{10} each and pixel grid of dimensions $3600^{2}$ \citep{Giblin2018}. These maps are smoothed with a Gaussian filter with smoothing scale $\theta_s = 1 \arcmin$.

\subsection{Emulation and likelihood analysis}
\label{sec:Emulation}

In this subsection, we outline the procedure used to test the sensitivity of WL void statistics to the cosmological parameters $\Omega_m$, $S_8$, $h$ and $w_0$.

The first step is to measure the WL void statistics from the 50 convergence maps for each of the 26 cosmo-SLICS cosmologies shown in Fig. \ref{fig:cosmo nodes}. Then, in order to make predictions of the WL void statistics at arbitrary points in the 4D parameter space shown in Fig.~\ref{fig:cosmo nodes}, we use a Gaussian process (GP) regression emulator from scikit-learn \citep{Pedregosa2011} to interpolate the void statistics between nodes. GP regression is a non-parametric Bayesian machine learning algorithm used to make probabilistic predictions that are consistent with the training data \citep[see, e.g.,][for some of its early applications in cosmology]{Habib2007,Schneider2008}. The emulator requires the training data to sample the parameter space sufficiently, and generally the accuracy of the emulator is limited by the availability of training data. The accuracy of the GP emulator trained on \emph{cosmo}-SLICS{} was tested extensively and found to yield few per cent accuracy in its predictions of weak lensing two-point correlation functions \citep{Harnois2019}, density split statistics \citep{Burger2020} persistent homology statistics \cite{Heydenreich2020} and aperture mass statistics \citep{Martinet2020}. In this work the average void statistics and their standard errors at each node are used as the training data for the emulator. We present results showing the accuracy of the emulator in Appendix \ref{app:cross_validation}.

Finally, once the emulator has been trained we use Monte Carlo Markov Chain (MCMC) to estimate the posteriors of the parameters for the entire parameter space and produce likelihood contours. We use the emcee python package \citep{Foreman2013} to conduct the MCMC analysis in this work sampling the 4D parameter space as follows.

We employ a Bayesian formalism, in which the likelihood, $P(\pmb{p}|\pmb{d})$, of the set of cosmological parameters $\pmb{p} = [\Omega_m,S_8,h,w_0]$ given a data set $\pmb{d}$, is given, according to Bayes' theorem, by
\begin{equation}
    P(\pmb{p}|\pmb{d}) = \frac{P(\pmb{p}) P(\pmb{d}|\pmb{p})}{P(\pmb{d})} \, ,
\end{equation}
where $P(\pmb{p})$ is the prior, $P(\pmb{d}|\pmb{p})$ is the likelihood of the data conditional on the parameters, and $P(\pmb{d})$ is the normalisation. In our analysis we use flat priors with upper and lower limits respectively for $\Omega_{rm m}$: [0.10, 0.55], $S_8$: [0.61, 0.99], $h$: [0.45, 0.90],
$w_0$: [-1.99, -0.52]. We note that these priors slightly extend past the parameter space in Fig. \ref{fig:cosmo nodes}, which we have expanded to ensure the likelihood contours presented in this work to not cross the prior boundary.

The log likelihood can be expressed as
\begin{equation}
    \log(P(\pmb{d}|\pmb{p})) = -\frac{1}{2} \left[\pmb{d} - \mu(\pmb{p})\right]C^{-1}\left[\pmb{d} - \mu(\pmb{p})\right] \, ,
    \label{eq:log likelihood}
\end{equation}
where $\mu(\pmb{p})$ is the prediction generated by the emulator for a set of parameters $\pmb{p}$, and $C^{-1}$ is the inverse covariance matrix. In practice we use the emulator's prediction of the void statistics at the fiducial cosmology as the data $\pmb{d}$. The likelihood returns a 4D probability distribution that indicates how well different regions of the parameter space describe the input data $\pmb{d}$. Note that in Eq.~\eqref{eq:log likelihood} we have assumed that the covariance matrix does not vary with a change in the cosmological parameters.

We calculate the covariance matrices from the $50$ WL map realisations from the fiducial cosmology, and divide it by a factor of $180$ in order to rescale the covariance matrix from a $100$ deg$^{2}$ area to the \textsc{lsst}{} survey area, which we take as $18,000$ deg$^{2}$. We also multiply the inverse covariance matrix by a factor $\alpha$, which accounts for the bias that is present when inverting a noisy covariance matrix \citep{Anderson2003,Hartlap2007}, given by:
\begin{equation}
    \alpha = \frac{N - N_{\rm{bin}} - 2}{N - 1} \, ,
\end{equation}
where $N = 50$ is the number of weak lensing maps that have been used to calculate the covariance matrix and $N_{\rm{bin}}$ is the number of bins for which the statistic is computed. We note however that \cite{Sellentin2016} present an alternative approach to robustly account for the uncertainty in the estimated covariance, via a student-$t$ likelihood distribution.

\subsection{The tunnel algorithm}

\begin{figure*}
    \centering
    \includegraphics[width=2\columnwidth]{Figures/visualisation.pdf}
    \caption{(Colour Online) A visualisation of WL peaks (green points) used to identify the tunnels (white circles) in the WL convergence maps (colour map). The left panel shows tunnels for the fiducial cosmology, while the middle and right panels show tunnels for the blue and red cosmologies highlighted in Fig.~\ref{fig:cosmo nodes} respectively. The colour bar on the right indicates the convergence field in units of $\nu=\kappa/\sigma_{\rm GSN}$.
    }
    \label{fig:visualisation}
\end{figure*}

To identify WL voids, we use the tunnel algorithm initially proposed in \cite{Cautun2018}, which identifies the largest circles in a 2D tracer catalogue that are empty of tracers. We choose to use this void finding algorithm since, compared with several other common 2D void finders, it gives void lensing profiles with high SNR, whilst also being least affected by the observational noises associated with weak lensing measurements, such as galaxy shape noise \citep{Davies2020}. The tunnel algorithm requires an input tracer catalogue to identify voids. For the identification of WL voids, we use WL peaks as tracers of the underlying convergence field (this avoids the necessity to have a synthetic 2D galaxy map for this analysis). Here we define WL peaks as local maxima in the WL convergence map as in Eq. \eqref{eq:nu}.

To identify tunnels, we first construct a Delaunay triangulation of the tracers (WL peaks). This produces a unique tessellation of the map with triangles, where each vertex is a tracer and the tessellated triangles enclose no tracers. From each triangle, a corresponding circumcircle can be defined, which is a circle that is directly on top of its Delaunay triangle with all vertices of the latter residing on the circumcircle's circumference. This tessellation is unique, and by definition gives circles that do not enclose any tracers. To avoid identifying the same regions as voids multiple times, we discard any circumcircles whose centers reside inside a larger circumcircle. The resulting list represents our tunnel catalogue, where each tunnel is characterized in terms of the centre and radius of its corresponding circumcircle.

The WL peak catalogues that may be used to identify tunnels contain peaks with a range of amplitudes (or heights) $\nu$. WL peaks of different amplitudes trace different components of the WL map, where the peaks with low or negative amplitudes trace underdense regions of the map, and those with high amplitudes trace overdense regions. Furthermore, peaks with low amplitudes are more susceptible to either being created or contaminated by GSN. It is therefore convenient to generate multiple sub-catalogues of a given WL peak catalogue, by retaining only the peaks with amplitudes larger than a given $\nu$ value. Varying the $\nu$ thresholds allows us to study how the tunnels respond to tracer catalogues with different properties. In this work we use WL peak catalogues with amplitudes of $\nu > 1,2$ and $3$ to identify tunnels, and will also use these $\nu$ values to denote the corresponding tunnel catalogues.

In Fig. \ref{fig:visualisation} we show a visualisation of tunnels identified from catalogues of WL peaks with amplitudes $\nu > 2$. The figure shows WL maps, WL peaks and tunnels for the fiducial cosmology (left), and two sample cosmologies, blue (middle) and red (right) to exemplify the impact of changing cosmological parameters. Here it can be seen in the bottom left part of the panels that the red cosmology, which has the highest $S_8$ value of the three reference cosmologies, contains more overdense (orange) regions than the other two cosmologies. The increased overdensity in the bottom left of the red-cosmology panel leads to more WL peaks, which generates more small tunnels and fewer large tunnels relative to the other two cosmologies. This highlights how changing the cosmological parameters changes the structure observed in WL maps and the corresponding WL void properties.

\section{Weak lensing void statistics}
\label{sec:statistics}

In this section we present the weak lensing void statistics used in this analysis, showing their abundance in Section \ref{sec:void abundance}, and the tangential shear profiles in Section \ref{sec:lensing profiles}.

\subsection{Void abundance}
\label{sec:void abundance}

\begin{figure*}
    \centering
    \includegraphics[width=2\columnwidth]{Figures/Void_abundances.pdf}
    \caption{(Colour Online) The differential void abundance as a function of void radius $R_V$. The three panels correspond to voids identified in different WL peak catalogues, with peak heights $\nu > 1,2$ and $3$ (from left to right). The void abundances for all cosmologies in Fig. \ref{fig:cosmo nodes} are plotted in grey. Results for the fiducial (black), red (red) and blue (blue) cosmologies are over-plotted in colour.}
    \label{fig:void abundance}
\end{figure*}

\begin{figure*}
    \centering
    \includegraphics[width=2\columnwidth]{Figures/tangential_shear_profiles.pdf}
    \caption{(Colour Online) The tangential shear profiles as a function of re-scaled distance to void centre, $r/R_V$. The three panels correspond to voids identified in three WL peak catalogues with peak heights $\nu > 1,2$ and $3$ (from left to right). The tangential shear profiles for all cosmologies in Fig. \ref{fig:cosmo nodes} are plotted in grey. Results for the fiducial cosmology (black), red (red) and blue (blue) cosmologies are over-plotted in colour. }
    \label{fig:tangential shear}
\end{figure*}

Fig.~\ref{fig:void abundance} shows the differential void abundance per unit area as a function of void radius. The three panels correspond to voids identified in three WL peak catalogues, with peak heights $\nu > 1,2$ and $3$. Void abundances for each of the nodes in Fig. \ref{fig:cosmo nodes} are plotted in grey, the fiducial cosmology in black and two sample cosmologies in colour (blue and red -- corresponding to the two cosmologies in the middle and right panels of Fig.~\ref{fig:visualisation}).

The figure shows that as the $\nu$ threshold increases, the total number of WL voids decreases (given by the area under the curves), and the average size of the voids increases. The spread in the void abundances over all cosmologies is largest for the $\nu > 3$ catalogue. However, the data is also noisier in this catalogue, because there are fewer peaks with $\nu>3$ and subsequently fewer tunnels.

The red cosmology produces more large voids for the $\nu > 1$ and $2$ catalogues than the fiducial and blue cosmologies. However, the same behaviour is not seen for the $\nu > 3$ catalogue, which may indicate that the sensitivity of the void abundance to specific cosmological parameters changes as fewer tracers are used to identify WL voids. The red cosmology has the largest $S_8$ and smallest $\Omega_m$ compared to the fiducial and blue cosmologies. Increasing $S_8$ or $\Omega_m$ increases the clustering of matter which leads to a wider range of WL void sizes, as we have seen in Fig.~\ref{fig:visualisation}: this is because the enhanced clustering creates more peaks with $\nu>1$ or $2$ in dense regions, reducing the void sizes there, and at the same time reduces the amplitudes of some low peaks in underdense regions, increasing void sizes there. On the other hand, for the $\nu>3$ catalogue, the peaks are sparser in all three cosmologies (hence voids are larger), and the fact that the red cosmology has more peaks at $\nu>3$ again restricts the sizes of its voids, this time affecting the largest ones.

For $\nu > 1$ the fiducial cosmology produces the fewest large voids compared to the red and blue cosmologies, however for $\nu > 3$ it produces the most large voids. The change in relative behaviour between the fiducial, red and blue cosmologies as the $\nu$ threshold increases, indicates that void abundances measured from different WL peak catalogues contain complementary information to each other. We will see this point more clearly later when looking at the constraints from void abundances.

\begin{table}
    \centering
    \caption{Forecast of percentage uncertainties obtained from various WL void statistics for an \textsc{lsst}-like survey. The first block of 4 rows show 68\% CL while the bottom 4 rows show
    95\% CL. In each block, the results shown in the first three lines are quoted from the tightest contours in each figure in Section \ref{sec:parameter constraints forecast} (see first column for more details). In the last line of each block, `$\gamma$-2PCF' stands for the parameter constraints using the cosmic shear two-point correlation functions for the same maps as used for the cosmic void statistics.
    }
    \begin{tabular}{ l cccc }
        \hline\hline
        Statistic & $\Omega_m$ & $S_8$ & $h$ & $w_0$ \\
        \hline \\[-.25cm]
        \multicolumn{5}{c}{\bf 68\% confidence limits}
        \\[.1cm]

        $dn/dR_v$ (combined) & $4.3\%$ & $0.7\%$ & $4.7\%$ & $7.0\%$ \\
        $\gamma_t$ (combined) & $6.8\%$ & $1.3\%$ & $6.3\%$ & $12.5\%$  \\
        $dn/dR_v$ and $\gamma_t$ ($\nu > 1$) & $4.9\%$ & $0.8\%$& $5.0\%$ & $6.9\%$ \\
        $\gamma$-2PCF & $4.5\%$ & $0.8\%$ & $4.1\%$ & $7.8\%$ \\

        \hline\\[-.25cm]
        \multicolumn{5}{c}{\bf 95\% confidence limits}
        \\[.1cm]

        $dn/dR_v$ (combined) & $8.3\%$ & $1.4\%$ & $8.9\%$ & $13.5\%$ \\
        $\gamma_t$ (combined) & $13.3\%$ & $2.6\%$ & $12.3\%$ & $25.1\%$ \\
        $dn/dR_v$ and $\gamma_t$ ($\nu > 1$) & $9.7\%$ & $1.6\%$ & $10.0\%$ & $13.9\%$ \\
        $\gamma$-2PCF & $12.6\%$ & $2.2\%$ & $11.5\%$ & $23.0\%$ \\

        \hline\hline
    \end{tabular}
    \label{tab:percentage errors}
\end{table}

\begin{figure*}
    \centering
    \includegraphics[width=2\columnwidth]{Figures/void_abundance_contours_2.pdf}
    \caption{(Colour Online) Constraint forecasts on cosmological parameters measured from void abundances. Contours are shown for WL voids identified in WL peak catalogues with $\nu > 1$ (blue), $\nu > 2$ (orange), $\nu > 3$ (green) and the combination of all three catalogues (red). The true cosmological parameter values used to generate the data are indicated by the black point. The diagonal panels show the 1D marginalised probability distribution, and remaining panels show the marginalised 2D probability contours enclosing the 68\% and 95\% confidence intervals. The table in the top right shows true parameter values (top) and the inferred parameter values for the different peak catalogues with 68\% (upper section) and 95\% (lower section) confidence limits.
    }
    \label{fig:VA contours}
\end{figure*}

\begin{figure*}
    \centering
    \includegraphics[width=2\columnwidth]{Figures/gamma_profiles_contours_2.pdf}
    \caption{(Colour Online) The same as Fig.~\ref{fig:VA contours} but for the tangential shear profiles. See the caption in Fig.\ref{fig:VA contours} for more details}
    \label{fig:gamma contours}
\end{figure*}

\begin{figure*}
    \centering
    \includegraphics[width=2\columnwidth]{Figures/combined_contours_wshear2pcf.pdf}
    \caption{(Colour Online) The same as Fig.~\ref{fig:VA contours} but for the tangential shear profiles (blue), void abundance (orange), and combination of the two (green). Results are shown for $\nu > 1$ WL peaks. See the caption in Fig.~\ref{fig:VA contours} for more details. Note that, for comparison, we have added the contours from the shear-shear two-point correlation function (without tomography) extracted from the same maps in grey colour, and the corresponding constraints on the parameters are also listed in the table in grey.}
    \label{fig:VA and gamma contours}
\end{figure*}

\begin{figure*}
    \centering
    \includegraphics[width=2\columnwidth]{Figures/combined_2.pdf}
    \caption{(Colour Online) The same as Fig.~\ref{fig:VA contours} but for the combination of the tangential shear profiles and the
    void abundance. Results are shown for the three WL Peak catalogues with $\nu > 1$ (blue), $\nu > 2$ (orange), $\nu > 3$ (green). See the caption in Fig.~\ref{fig:VA contours} for more details.}
    \label{fig:combined all nu}
\end{figure*}

\subsection{Lensing tangential shear profiles}
\label{sec:lensing profiles}

Fig.~\ref{fig:tangential shear} shows the tangential shear profiles for WL voids, where the panels (from left to right) show WL voids identified in the $\nu > 1,2$ and $3$ catalogues. Tangential shear profiles for all cosmologies are plotted in grey, with the fiducial and two highlighted cosmologies plotted in colour as in Fig.~\ref{fig:void abundance}. The tangential shear profiles are calculated by first measuring the convergence profiles in annuli centered on the void center (pixels are interpolated for small annuli), where the number of annuli used is the lensing profile bin number. The annuli are then stacked as a function of relative angular size ($r/R_V$), weighted by their corresponding void area. Using Eq. \ref{eq:gamma_t}, this is then converted to the tangential shear profiles.

The tangential shear profiles plotted here are negative, which indicates that the WL voids behave like concave lenses and their interiors correspond to underdense regions. The figure shows that as the $\nu$ threshold increases, the depth of the tangential shear profiles at $r/R_V = 1$ decreases, but meanwhile the spread in the amplitude of the tangential shear profiles, as well as the spread in the width of the peak around $r/R_V=1$, amongst all cosmologies, increases.
Note that the peaks of the tangential shear profiles appear to be narrower as the $\nu$ threshold increases, but this is an artificial consequence of the fact that these plots are made against $r/R_V$ with the void radius $R_V$ larger for larger $\nu$ thresholds.

For the $\nu > 1$ catalogue, the fiducial, red and blue cosmologies all lie on top of each other. For $\nu > 2$, the red cosmology (with the largest $S_8$ value) has a deeper tangential shear profile compared to the other two cosmologies. For $\nu > 3$ the difference in amplitude increases further between the three reference cosmologies, with the fiducial cosmology having the lowest (absolute) amplitude, however the general trend between the three reference cosmologies is the same for all $\nu$ thresholds. Part of this can again be attributed to the high $S_8$ value in the red cosmology, which enhances the clustering of matter, resulting in low-density regions becoming more underdense. However, the three reference cosmologies have very distinct values for the other three parameters, in particular $\Omega_m$, which means that an intuitive and yet complete explanation of their relative behaviours is difficult to gauge by eye.

The observation that, although the spread in shapes amongst all cosmologies increases with the $\nu$ threshold, the general order in which they appear does not change, indicates that there may not be much complementary information between tangential shear profiles measured from different $\nu$ thresholds. Also, while the differences between the different cosmologies are larger for the $\nu>3$ catalogue, the fact that there are relatively fewer voids in this catalogue means that its constraining power is not necessarily stronger than the other two catalogues, as we will see shortly.

\section{Parameter constraints forecast}
\label{sec:parameter constraints forecast}

In this section we present parameter constraint forecasts for an \textsc{lsst}-like survey from the void abundances and tangential shear profiles of WL voids, as well as their combinations.

\subsection{Void abundance constrains}
\label{sec:VA constraints}

Fig.~\ref{fig:VA contours} shows the likelihood contours for measuring the four cosmological parameters with WL void abundance. The diagonal panels of the figure show the 1D marginalised likelihood distribution and remaining panels show the 2D marginalised likelihood contours.
For each inference case, the inner and outer contours indicate respectively the 68\% and 95\% confidence limits (CL). As mentioned above, we use the fiducial cosmology as our `observed' data set, which is indicated by the black point. The figure shows results for three $\nu$ thresholds with $\nu > 1$ (blue), $\nu > 2$ (orange) and $\nu > 3$ (green). We also show results for the combination of all three catalogues (red). The table in the top right of the figure indicates the estimated cosmological parameters with their corresponding 68\% (top) and 95\% (bottom) CL, for each of the contours.

All of the contours presented in this figure are centered on the true (fiducial) cosmology, indicating that the WL void abundances are able to measure cosmological parameters without any bias. The tightest contours are for the $\Omega_m - S_8$ plane, which is expected since these are the cosmological parameters to which WL analysis is the most sensitive.

For nearly every combination of parameters, the three contours for the $\nu > 1$, $2$ and $3$ void catalogues occupy different parts of the marginalised 2D parameter space, or have different degeneracy directions, where most of the overlapping occurs around the true values. As suggested by the behaviour of the three reference cosmologies discussed in Section \ref{sec:void abundance}, this indicates that void abundances measured from different catalogues contain complementary information to each other. We therefore also show parameter likelihood contours for the combination of the WL void abundances from the three catalogues in red.

The `Combined' contours are smaller than any of the individual contours, for all combinations of parameters. This shows that parameter measurements from the WL void abundances are significantly improved when multiple catalogues are used. The $68\%$ and $95\%$ CL percentage accuracy that the combined WL void abundance is able to measure the parameters with is shown in \autoref{tab:percentage errors}. We note that when combining multiple catalogues, we had to reduce the number of bins used for each catalogue to avoid having a data vector with more bins than the number of WL map realisations that we had available for calculating the covariance matrix. For the individual catalogues we use 20 bins, and when combining catalogues we use 10 bins per catalogue giving 30 in total. This coarser binning per catalogue means that the red contours presented here may be larger than the `true' contours. We plan to run further simulations and make more mock WL maps in the future, to improve the constraint here.

Theoretically, the abundance of tunnels identified from a WL peak catalogue depends not only on the number of peaks, but also on their clustering pattern. We therefore expect that the information contained in WL void abundance and peak correlation functions may have substantial overlapping. The latter probe has been studied in details by \cite{Davies2019}, with certain scaling properties observed. While it is beyond the scope of the current work, we will conduct a similar analysis by forecasting the parameter constraining power by WL peak two-point correlation functions in a followup study.

\subsection{Tangential shear constraints}

Fig.~\ref{fig:gamma contours} shows likelihood contours for the four cosmological parameters from tangential shear profiles. The colours of the contours correspond to the same void catalogues as in Fig.~\ref{fig:VA contours}. Again, the contours are smallest in the $\Omega_m$-$S_8$ plane, and each contour is roughly centered on the true parameter values indicated by the black point, showing that the tangential shear profiles are able to measure these parameters without any bias.
The figure shows that the contours from $\nu > 1$ and $\nu > 2$ are similar in size, and the $\nu > 3$ contours are significantly larger and in most cases entirely enclose the other contours. All of the contours in this figure, unlike in the case of void abundances, occupy similar regions of the parameter space, or have similar degeneracy directions. This confirms our conclusion based on observation of Fig.~\ref{fig:tangential shear}, namely the tangential shear profiles from different peak catalogues do not offer much complementarity.

As in Section \ref{sec:VA constraints}, we combine the tangential shear profiles from all three catalogues to generate `Combined' likelihood contours. Note that for individual catalogues the tangential shear profiles are calculated with 30 bins each, and when combined, individual catalogues are plotted with 10 bins each, giving 30 in total. Again, this is to avoid having more data bins than WL map realisations used to calculate the covaraince matrix. We expect this approach might give contours that are slightly larger than the truth, due to the coarser binning. By combining catalogues, we find only a small improvement in contour size relative the the $\nu > 1$ catalogue, which again suggests that there is very little complementary information between the different $\nu$ catalogues for the tangential shear profiles.

The strongest constraints from the tangential shear profiles are for the combined contour. We summarise the 68\% and 95\% CL for the $\gamma_t$ combined case in \autoref{tab:percentage errors}.

\subsection{Constraints by combining void abundance and tangential shear}

In this section we present parameter constraint forecasts for the combination of the WL void abundance and tangential shear profiles.

Fig.~\ref{fig:VA and gamma contours} shows contours for the tangential shear profiles (blue) WL void abundance (orange) and the combination of the two (green) for the $\nu>1$ catalogue. We also include the shear-shear two-point correlation function constraints as a comparison. We follow \cite{Asgari2020} and sample the 2PCF using 9 logarithmically-spaced angular separation bins from 0.5 to 300 arcmin, and use only the $\xi_+$ correlation function. Note that the WL void abundance and tangential shear profile contours shown in this figure are also presented as the blue contours in Fig.~\ref{fig:VA contours} and Fig.~\ref{fig:gamma contours} respectively. We include these contours in this figure to illustrate how the contours behave when combining the two probes.

For all combinations of parameters, the WL void abundance contours and the tangential shear contours occupy similar regions of the parameter space and have similar degeneracy directions. However there is a large difference in size between the two for all panels that include $h$.

The combined contours are smaller than the contours from the tangential shear profiles alone, which shows that there is a benefit of combining these two probes. When comparing the tangential shear contours and the combined contours, the parameter with the greatest relative increase in accuracy is $S_8$. This is shown in the table in the top right of the figure, where it can be seen that the combined errors are roughly half the tangential shear errors, although this seems to be mainly due to the fact that void abundance alone can give much stronger constraint on $S_8$. Finally, the parameter with the greatest absolute increase in accuracy enabled by this combination is $w_0$, which suggests that using different probes to strengthen constraints could be particularly important for dark energy properties.

Note that, as in the previous subsections, we have had to reduce the number of bins with which the two statistics are calculated when creating the combined contours, to avoid having a data vector with more bins than the number of realisations used for the covariance matrix calculation. For the combined contours in this figure, we use the void abundance calculated with 10 bins, and the tangential shear profiles calculated with 20 bins.

We show the percentage errors (at $68\%$ and $95\%$ CL) for the combination of the void abundance and tangential shear profiles, as well as those of the shear two-point correlation function ($\gamma$-2PCF), in \autoref{tab:percentage errors}. The grey contours in Fig.~\ref{fig:VA and gamma contours} show that the shear two-point correlation function provides similar, constraints to those from the combination of the void abundance and tangential shear profiles, for all parameters at the $68\%$ CL. Where constraints on $\Omega_m$ and $h$ are slightly better for the shear two-point correlation function. And constraints on $w_0$ are tighter for the combined void abundance and tangential shear. However, for the $95\%$ CL, constraints are better for the combined void abundance and tangential shear profiles for all parameters, relative to the shear two-point correlation function.

As shown by \autoref{tab:percentage errors}, the $68\%$ CL constraints for the combined void abundance with three $\nu$ thresholds are tighter than those of the shear two-point correlation function for $\Omega_m$, $S_8$ and $w_0$. For the $95\%$ CL constraints, the difference is even larger, where better constraints come form the combined void abundance for all parameters. The difference is most significant for $w_0$, where the cause of this behaviour is due to the tails in the contours for the shear two-point correlation function in the 2D planes which are most prominent for $w_0$. We expect that combining void abundance and $\gamma$-2PCF will further tighten the constraints, though this will be left for a future study.

Fig. \ref{fig:combined all nu} shows contours for the WL void abundance and tangential shear profiles combined, for the three catalogues $\nu > 1$ (blue), $2$ (orange) and $3$ (green) respectively. The smallest contours are for the $\nu > 1$ catalogue (which has been shown in green colour in Fig.~\ref{fig:VA and gamma contours}) and increasing the $\nu$ threshold increases the contour size, where the larger contours almost entirely enclose the smaller contours. This is because the number of voids decreases with $\nu$ increasing, meaning that the statistical uncertainties increase and the constraining power is weakened.

Nevertheless, it is interesting to note that in Fig. \ref{fig:gamma contours} the tangential shear contours for the $\nu > 3$ catalogue are large. The same is also true in Fig. \ref{fig:VA contours} with the WL void abundance for the same catalogue. The resulting contour when the two statistics are combined however is significantly smaller, as shown by the green contour in Fig. \ref{fig:combined all nu}. So even for this catalogue where individual constraints are poor, their combination is highly beneficial.

\section{Discussion and conclusions}
\label{sec:conclusion}

In this paper we have tested the sensitivity of the WL void abundances and tangential shear profiles to four cosmological parameters: $\Omega_m$, $S_8$, $h$ and $w_0$. To this end, we have trained a Gaussian Process emulator with 26 cosmologies sampled in this 4D parameter space using Latin hypercube, which can be used to predict these two void statistics for arbitrary cosmologies (within the range spanned by the training cosmologies). We have investigated the impact of changing the number of WL peaks used as tracers to identify voids, and ran Markov Chain Monte Carlo samplings from our mock weak lensing data to forecast the accuracies at which these four parameters can be constrained by a future, \textsc{lsst}-like, lensing survey, using different combinations of the above WL void statistics.

The results from Fig.~\ref{sec:parameter constraints forecast} show that the WL void abundance combined over all catalogues gives the tightest parameter constraints, where the greatest sensitivity is to the $S_8$ parameter. This is because the abundances of WL voids identified from WL peak catalogues at different $\nu$ thresholds have different dependencies and degeneracy directions in the studied parameter space. We suspect that there is a close interlink between the void abundance and the peak two-point correlation function, but will defer a detailed study of the latter to a follow-up work. For now, we conclude that complementary information is contained in the abundances of voids from different WL peak catalogues, a fact that should be utilised in order to maximise the use and scientific return of future lensing data.

WL void tangential shear profiles, in contrast, provide less tight constraints on the same cosmological parameters, and the results from different peak catalogues do not seem to be complementary to each other. In particular, for low-$\nu$ peak catalogues such as $\nu>1$ (Fig.~\ref{fig:gamma contours}), there is little degeneracy between $\Omega_m$ and $S_8$; this is because $S_8$ is designed to break the degeneracy between $\Omega_m$ and $\sigma_8$ for standard WL analysis, e.g., shear two-point correlation function, and the low-$\nu$ peaks have little bias with respect to the underlying convergence field, so that their tangential shear profiles follow more closely the parameter dependency of the shear two-point correlation functions. WL void abundances, on the other hand, can have further degeneracies between $\Omega_m$ and $S_8$ (as seen in Fig.~\ref{fig:VA contours}), indicating that they have different degeneracy directions between $\Omega_m$ and $\sigma_8$ compared with the shear two-point function, and therefore can lead to additional constraints to the latter.

Nevertheless, we highlight that the above conclusions only apply to the 4D parameters space that we have focused on in this work, and may change if additional physics is introduced into the cosmological model. In \cite{Davies2019b} we found that the tangential shear profiles are able to distinguish between modified gravity models with a larger signal-to-noise ratio than the void abundance. This suggests that there may be other cosmological parameters not studied here, such as those governing modified gravity laws, to which the tangential shear profile is more sensitive than the WL void abundance. We leave an exploration of this possibility to future works.

Finally, we have found that combining void abundance and tangential shear is another way to obtain tighter parameter constraints. Even for the $\nu>3$ catalogues, for which these two void statistics give poor individual constraints, significant improvement has been found with their synergy.

Overall, we find that weak lensing voids can be a promising cosmological probe to constrain models. The cosmological parameter to which the WL void statistics are most sensitive is $S_8$, which can be measured at the sub percent level (68\% CL). We also find that $\Omega_m$ can be measured to within $\simeq 4\%$, $h$ to within $\simeq 5\%$ and $w_0$ within $\simeq 7\%$ (all 68\% CL).

As a comparison, we find that parameter constraints from the combined void abundances are comparable to those from the shear two-point correlation function (which were obtained from the same WL maps) at the $68\%$ CL, however the constraints are tighter for the combined void abundance at the $95\%$ CL. We note that we are unable to combine multiple catalogues without reducing the number of bins so the results presented here are upper limits. Further combinations such as combining the void abundance and tangential shear profiles over all catalogues, will show the maximum constraining power of WL voids, where it would also be possible to include further peak catalogues with higher or lower $\nu$ thresholds. However, combining so many statistics into a single data vector requires a larger number of simulations than those currently available to us, so we leave such a study to a future project. We also note that constraints from the shear two-point correlation function can be improved by using tomography \citep{Martinet2020}, and it is therefore also important to test how tomography can improve the constraints from WL void statistics in the future.

Throughout this study, we have adopted a Gaussian smoothing of $\theta_s=1$ arcmin. It may also be interesting to study how the parameter constraints depend on the smoothing scale used to smooth the WL convergence maps. We know that using larger smoothing scales increases the size of the WL voids and reduces their total number \citep{Davies2020}. A larger number of WL map realisations will then be required in order to accurately measure WL void statistics for larger smoothing scales, so we leave such a study to future work. Nevertheless, we have performed a test by using a larger smoothing scale, $\theta_s=2$ arcmin, and in Appendix \ref{app:2 arcmin smoothing} we give a brief summary of the resulting parameter constraints. We can see that the results are similar to what we have found for a 1 arcmin smoothing, cf.~Fig.~\ref{fig:VA and gamma contours}.

We have mentioned several potential follow-up works above, including a more in-depth study based on a larger number of simulations. Having a greater number of WL map realisations will also allow us to use larger data vectors in our likelihood analysis. This would allow us to calculate likelihood contours for the tangential shear profiles combined over all catalogues with a greater number of bins, which may decreases the errors on the parameter estimates. This is because coarse binning of the statistics increases the contour size. It will further enable us to combine both the void abundances and their tangential shear profiles for all peak catalogues in a single analysis, which again we have not been able to do in this work.

It will also be important to develop an understanding of how the void function is affected by systematics including intrinsic alignments, baryonic feedback, and masking (which can bias statistics measured from convergence maps, e.g. see \cite{Giblin2018} ), which we leave to future study.

In \cite{Davies2020} we studied the differences in WL void statistics between WL voids identified from different void finders. We found that the tunnel algorithm offered one of the best compromises between high signal-to-noise ratio and small impact from galaxy shape noise in the tangential shear profiles. However, it will also be interesting to assess the constraining power of WL voids identified using other void finders such as the watershed algorithm. The aim is to have a fully comprehensive study of the many different and unexplored ways to use future high-quality weak lensing data to maximise our ability to test cosmological models and constrain cosmological parameters.

\section*{Acknowledgements}
We thank Ludovic Van Waerbeke for the code used in the weak lensing convergence map construction.

CTD is funded by a UK Science and Technology Facilities Council (STFC) PhD studentship through grant ST/R504725/1. MC is supported by the EU Horizon 2020 research and innovation programme under a Marie Sk{\l}odowska-Curie grant agreement 794474 (DancingGalaxies). JHD acknowledges support from an STFC Ernest Rutherford Fellowship (project reference ST/S004858/1). BG acknowledges the support of the Royal Society through an Enhancement Award (RGF/EA/181006). YC acknowledges the support of the Royal Society through a University Research Fellowship and an Enhancement Award. BL is supported by an ERC Starting Grant, ERC-StG-PUNCA-716532, and additionally supported by the STFC Consolidated Grants [ST/P000541/1, ST/T000244/1].

Computations for the N-body simulations were enabled by Compute Ontario (www.computeontario.ca), Westgrid (www.westgrid.ca) and Compute Canada (www.computecanada.ca). The SLICS numerical simulations can be found at http://slics.roe.ac.uk/, while the cosmo-SLICS can be made available upon request. This work used the DiRAC@Durham facility managed by the Institute for Computational Cosmology on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). The equipment was funded by BEIS capital funding via STFC capital grants ST/K00042X/1, ST/P002293/1, ST/R002371/1 and ST/S002502/1, Durham University and STFC operations grant ST/R000832/1. DiRAC is part of the National e-Infrastructure.

\bibliographystyle{mnras}

redpajama_set_name: RedPajamaArXiv


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Do I need a domain name to set up a website?
No, you don't need a domain name to set up your website with MYOB Atlas. When you sign up, you'll be provided with a temporary domain name. And during the setup wizard (or at any time after), you can register a domain name. For more information, see Registering domain names.
However, this temporary domain name is just to get you started with your website. If you don't want to use it to access your website, you can register a domain name, and then make sure it's set as your primary domain name. Or, if you've registered domain names with a registrar other than MYOB, link these domain names to your website, and then set one of these as your primary domain name.
When can I claim my free domain name?
After you've completed the setup wizard and gone live with your site, you'll be able to get your free domain name by clicking Get free domain name in the Things to do toolbar. For more information, see Registering domain names.
How do I know if I'm logged in to my website?
If you're logged in to your website, you'll see the MYOB Atlas menu along the top of your website.
I'm having trouble logging in using my Facebook account.
If you're having trouble logging into your website using your Facebook account, try removing the MYOB Atlas app from your Facebook account. For more information, see Facebook's help article How do I remove or delete an app from my account?
To see the Tell the world link, you first need to register a domain name. After this has been processed (which can take up to a day), the Tell the world link will appear.
Your domain name registrar is the company who you registered your domain name through. They're responsible for updating your registrant details and renewing your domain name when it expires.
It may take up to 24 hours for a domain name to be registered. After this time, you'll be able to use it to access your MYOB Atlas website.
To check the status of your domain name registration, click My domain names in the Settings toolbar, then check the status next to the domain name in the Domain names purchased through MYOB section. If it shows 'Pending registration', the registration is still being processed.
What if I don't receive my verification code?
If you don't receive your verification code by SMS within a few minutes, you can either click the Use a different phone number button to try a different phone number, or you can contact MYOB Atlas by submitting a Domain name enquiry form.
Do I own my free .co.nz domain name?
As the owner of your business and your MYOB Atlas website, you own the license to your free .co.nz domain name (and all other domain names that you register through MYOB Atlas) for the period of time specified in your license. MYOB is your domain name registrar, and is responsible for updating your registrant details and renewing your domain name when it expires.
How many domain names can I claim?
MYOB Atlas offers you one free .co.nz domain name for one year per business. However, you can purchase an unlimited number of additional domain names through MYOB Atlas.
Where can I find my domain name's expiry date?
Click My domain names in the Settings toolbar, then check the expiry date next to your domain names in the Domain names purchased through MYOB section.
How long can I register domain names for?
Top-level domain names (such as .com, .org and .net) and second-level domain names (such as .co.nz and .net.nz) can be registered for a period of 1, 2, 3, 5 or 10 years.
You can renew your domain names at the end of each registration period.
How much will it cost me to renew my domain names?
It will cost you $30 a year (inc GST)* to renew each domain name.
* This price is accurate at the time of writing.
How do I know if there's a problem registering or paying for my domain names?
If a problem occurs while registering or paying for your domain names, you'll receive an email with the relevant details.
How do I change the registrant email address for my domain name?
The registrant contact information for your domain name is the information that your domain name registrar (the company who you registered your domain name through) uses to contact you, the registrant. If your domain name is registered with MYOB Atlas, your registrant contact information-including your registrant email address-will be the same as the contact information for your MYOB Atlas account (that is, the contact information MYOB Atlas uses for all communications with you).
To update this contact information, click My account on the Settings menu, then click Edit account information in the Account information section and make your changes.
This will update both your registrant contact information (for any domain names registered with MYOB Atlas) and the contact information for your MYOB Atlas account.
How does the 'Get vCard' link work?
By clicking the Get vCard link in the Contact information section of your site, your visitors can download a vCard (.vcf) file. This file saves your contact details in a format that can be imported into visitors' email clients, such as Microsoft Outlook.
Can I hide my business and contact information from visitors?
Yes, you can set the Content visibility settings for these features so that your business and contact information is hidden from your website. For more information, see Setting visibility options for your website content.
The maximum size for banner images depends on your chosen theme. For more information, see How will my logo and banner look with my chosen theme? in this article.
This depends on your chosen theme. For more information, see How will my logo and banner look with my chosen theme? in this article.
MYOB Atlas is free for the first 12 months. After your free trial ends, the number of layouts and colour palettes you can select from depends on which Atlas plan you choose. For information on MYOB Atlas pricing, log in to your website and click My account on the Settings menu.
The number of layouts and colour palettes you can select from depends on which Atlas plan you're on. If you can't access the one you want, it means that your current plan doesn't include access to it, and you'll need to upgrade your plan to use it.
We suggest that you don't enter a PO Box address on the Find us page. This is because they can't be verified by Google Maps and may not appear correctly. However, you can enter your PO Box address in the Contact information section. For more information, see Publishing your business and contact information.
How do I change the default Sort by option?
Click the Edit settings link on the Gallery page, and then select a different option from the Sort gallery items by drop-down list.
How do I change the default display order of my gallery items?
Click the Edit display order link on the Gallery page, and reorder your gallery items by clicking and dragging them with your mouse. Then, when you sort your items by Default, they'll appear in the order you've chosen.
How many item categories can I create?
You can create as many categories for gallery items as you want. However, you can only apply one category to each gallery item.
What size will my item image be?
Detailed view: the width is 470 pixels and the height varies depending on the original image size.
If the width of your original image is not 470 pixels, it will be resized to 470 pixels, and the height will be resized so it's proportional to this. When visitors click an item image while in the detailed view, the image will appear in a pop-up window in its original size.
Why can't I upload my item image?
Note that if you have a slow internet connection, it might take some time to upload your image. You'll need to wait for the upload to finish before you click Add.
What happens if a visitor asks a question about one of my gallery items?
If a visitor uses the Ask a question form to ask a question about a gallery item, you'll receive an email with details of the visitor and the question they submitted. This will be send to the email address you've listed as your account email address. You can find this address by clicking My account in the Settings menu, and checking the Account email field in the Account information section.
If you've created a Google AdWords ad campaign, you can use Conversion Tracking to track visitors who click on your ad, then complete the Ask a question form for a gallery item. For more information, see Integrating Google tools with your website.
Can I hide my gallery from visitors?
Yes, you can set the Content visibility settings for this feature so that your gallery is hidden from your website. For more information, see Setting visibility options for your website content.
What if I don't want to sell an item any more (say, if I run out of stock)?
If you don't want to sell an item any more, click Gallery on the public menu, and then click the Edit link at the bottom of the gallery item. Deselect the Make item available for purchase option to stop selling the item.
If you've only temporarily run out of stock of an item, you can reselect this option once you get more in stock.
The large cog ( ) contains editing options for the Links feature as a whole, whereas each small cog ( )contains editing options for the individual link it appears beneath. For more information, see Finding your way around your website.
Google Analytics account, you can track visitors and get a overview of how they're interacting with your website. This means you'll be able to see how many visitors have accessed your subscription form. For more information, see Integrating your Google Analytics account. Note: Google Analytics won't be able to show you the details your visitors enter in the subscription form. To see these details, you'll need to download and view the CSV file. For more information, see Sending newsletters to your customers in this article.
Do I have to link my existing domain names to my MYOB Atlas website?
No, you don't. But if you don't have a website for your existing domain names, we suggest that you link them to your MYOB Atlas website. This means that visitors who go to your existing web addresses will see your MYOB Atlas website.
Can I link a domain name that I don't own?
No, you can't. If you don't own a domain name, you can't change its DNS settings, and therefore can't link it to your MYOB Atlas website.
How do I find out which company is managing my domain names?
To find out who is managing your domain names, you can do a whois search.
To do this, do a Google search for the term 'whois search'. You'll find numerous websites that allow you to search the whois records for your domain names. The whois records will include details about the company who manages your domain names.
Does my domain name management company need to add both values for the A record?
If possible, we suggest that your domain name management company adds both values (54.252.114.40 and 54.252.114.46) as your domain name's A record. If they can only add one value, they can use either one.
How do I unlink a domain name?
To unlink a domain name that you've linked to your website, click My account in the Settings toolbar, and then click Manage domain names in the Domain names section. In the Linked domain names section of the My domain names page, click Unlink next to the domain name that you want to unlink. The domain name will be unlinked from your website immediately.
Note that you should contact the company who manages your domain name to update the DNS (Domain Name Server) settings for your domain name.
What's the difference between my primary domain name and secondary domain names?
Your primary domain name is the domain name that all of your other domain names redirect to.
For example, let's say your primary domain name is bluebowlhomewares.co.nz, and your secondary domain name is bluebowlhomewares.myob.net. If visitors enter bluebowlhomewares.myob.net in the address bar of their web browser, they'll be taken to your MYOB Atlas website. Once your website loads, the domain name they'll see in the address bar will be bluebowlhomewares.co.nz.
How do I know which domain name is my primary domain name?
It's been 24 hours and my new primary domain name still isn't active.
If you've set a linked domain name as your primary domain name, you may need to check whether this domain name is linked properly. For more information about how to link a domain name, see Linking domain names to your website.
What does it cost to use MYOB Atlas' email forwarding feature?
MYOB Atlas is free for the first 12 months. In the second year, you have choose a plan. The number of email addresses you can create depends on which plan you choose. For information on MYOB Atlas pricing, log in to your website and click My account on the Settings menu.
How do I edit or delete an email address?
To edit or delete an email address you've set up to use email forwarding with, click Email forwarding in the Settings menu. In the Email addresses section at the bottom of the Email forwarding page, click Edit or Delete next to the email address you want to edit or delete.
How and why would I make an email address inactive?
You may need to make one or more of your email addresses inactive if you choose to downgrade your MYOB Atlas plan. In this case, the number of active email addresses you're allowed will decrease. Before you downgrade you'll be asked to make sure that the addresses you want to keep are active, and all others are set as inactive.
To make an email address inactive, click Email forwarding in the Settings menu, and click the Edit link next to the email address you want make inactive. On the Edit email address page, deselect the Make this email address active option. Once you're finished, click Update.
How do I know if my new email address is working?
In the Email addresses list at the bottom of the Email forwarding page, you can see the time and date of the last email forwarded for each of your email addresses.
If you think your email address might not be working, try sending a test email to it and checking whether this time and date is updated. If you don't receive the test email, check that you've entered the correct email address to forward to, and that the email hasn't been sent to your Spam folder.
Note: Make sure you send your test email from a different address to the one that will be receiving forwards. This is because some email providers aren't able to forward emails to the same address that they've been sent from.
You need to use your MYOB Atlas website's primary domain name as your web address.
There are three options for verifying that your conversion tracking code was implemented correctly. For more information, see What can I do to verify that the conversion was implemented correctly?
What does it cost me to sell online through my MYOB Atlas website?
MYOB Atlas is free for the first 12 months. For detailed information on what it will cost after this time, log in to your website and click My account on the Settings menu.
Detailed view: the width is 470 pixels and the height varies depending on the original image size. If the width of your original image is not 470 pixels, it will be resized to 470 pixels, and the height will be resized so it's proportional to this.
When visitors click an item image while in the detailed view, the image will appear in a pop-up window in its original size.
If a visitor uses the Ask a question form to ask a question about a gallery item, you'll receive an email with details of the visitor and the question they submitted. This will be send to the email address you've listed as your account email address.You can find this address by clicking My account in the Settings menu, and checking the Account email field in the Account information section.
Yes, you can set the Content visibility settings for this feature so that your gallery is hidden from your website. For more information, see Editing content visibility.
MYOB Atlas is free for the first 12 months with GetOnline.co.nz.
For detailed information on what it will cost after this time, log in to your website and click My account on the Settings menu.


There are lots of ways in which your organisation can support Halton Haven Hospice and help in your local community!
We can help you with lots of ideas and tips for raising funds, provide support where it's needed and assist you with events.
builds your brand in the local community.
helps you meet your corporate social responsibility goals.
provides unique opportunities to network.
raising funds for a specific hospice project.
encouraging employees to take part in our fundraising events.
encouraging a payroll giving scheme.
matching employees' fundraising efforts with £ for £ sponsorship.
There's no better way to build morale than to create an in-house challenge . A race, football match, walk or other event that gets people working together in teams and competing against each other to raise money!
You don't have to organise your own event you can encourage your employees to take part in one of our annual events such as Santa Dash. We have a strong calendar of fundraising events.
We organise a wide range of well-attended fundraising events which can provide a great place for you to advertise and show your support for the hospice at the same time.
Giving to a charity via your payroll is both simple and tax-effective. Around five million people in the UK do this each year because it's a more tax-efficient way to give to charity, since deductions are made before tax. You can get more information about payroll giving from the HM Revenue & Customs website. You can also use a service such as Pennies for Halton Heaven to make things even simpler, nominating us as your charity.
Companies can claim tax relief for donations to UK charities as a charge against income in their corporation tax. If you're a partner or sole trader, then gifts to charities can be claimed on your tax self-assessment form. When gifts are made from partnerships, they are treated as donations of an equal amount from each partner, unless the partners specify otherwise. Please talk to your accountant or tax advisor for more information or check the information about gifts made to charity from companies on the HM Revenue & Customs website.
Giving time to the hospice is inspirational and really does help us to provide our services. This can be a secondment, for a specific activity or role (which is a legitimate business expense), or perhaps they can volunteer to help out at one of our events.