Theoretical higher order predictions for wavelet l1-norm
Vilasini Tinnaneri Sreekanth, Sandrine Codis, Alexander Barthlemey, Jean-Luc Starck
Colloque national Action Dark Energy- 07 Nov 2023
Weak Lensing is one of the crucial probe in modern cosmology.
- Direct observations of the mass distribution
- Can help us in better understanding the universe
The usual method of analysing the WL map is by using the 2 point statistics.
Data
2 point statistics
Theory
2 point statistics
Likelihood/MCMC
2 point statistics is not sufficient. Why?
$\Longrightarrow$ The power spectrum shows power as the mean squared amplitude at each frequency line but includes no phase information.
Higher-order statistics
Higher-order statistics
Data
Higher order statistics
Simulation
Higher order statistics
Likelihood/MCMC
Data
Higher order statistics
Simulation
Multiple realisations
Higher order statistics
Likelihood/MCMC
- Need huge amount of storage
- Huge amount of computational resourses
This work
Data
Higher order statistics
Simulation
Multiple realisations
Higher order summary statistics
Likelihood/MCMC
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Theory for HOS
wavelet $\ell_1$-norm PDF
One-point Probability Distribution Functions
- Contain a significant amount of cosmological information
- potential to break cosmological parameter degeneracies
- Has the advantage of being straightforward to measure.
wavelet $\ell_1$-norm
- $\ell_1 = \sum_k|x_k|$
- the $\ell_1$-norm carries the information encoded in all pixels of the map.
- It is shown in Ajani et al. (2021) that it remarkably outperforms commonly used summary statistics, such as the power spectrum or the combination of peak and void counts.
Wavelet decomposition of a convergence map: example
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My work: Theoretical prediction of wavelet $\ell_1$-norm based on Large Deviation Theory
- Part 1: What is LDT?
- Part 2: Prediction of one-Point PDF
- Part 3: Derivation of wavelet $\ell_1$-norm
Part 1: Intuition to LDT
- Central Limit Theorem gives accurate information around the mean.
- But what about the tail? $\hookrightarrow$Large Deviations Theory concerns the asymptotic behavior of remote tails of sequences of probability distributions.
- If we know the behaviour of the tail $\Longrightarrow$ we can get the entire probability distribution.
LDT in Cosmology and prediction of the one-point PDF
Rate Function
Cumulant Generating Function
PDF
Rate Function
From the first principles of Cosmology $\downarrow$ Spherical Collapse Model (one-to-one mapping between initial and final densities)
$\zeta(\bar{\tau_k}) = \rho_k = \Bigl( 1 - \frac{\bar{\tau_k}}{\nu}\Bigr)^{-\nu}$
$\psi (\{ \rho_i\}) = \frac{\sigma^2_{R1}}{2} \sum_{k,j}\Xi_{kj}(\{\tau_i \}) \bar{\tau_k} \bar{\tau_j}$
Cumulant Generating Function
PDF
Rate Function
From the first principles of Cosmology $\downarrow$ Spherical Collapse Model
$\zeta(\bar{\tau_k}) = \rho_k = \Bigl( 1 - \frac{\bar{\tau_k}}{\nu}\Bigr)^{-\nu}$
$\psi (\{ \rho_i\}) = \frac{\sigma^2_{R1}}{2} \sum_{k,j}\Xi_{kj}(\{\tau_i \}) \bar{\tau_k} \bar{\tau_j}$
Cumulant Generating Function
Legendre-Fenschel transformation
$\varphi_{\{ \rho_i\}}(\{ \lambda_i\}) = \sup_{\{ \rho_i\}} \Bigl[ \sum_{i}\lambda_i\rho_i - \psi_{\{ \rho_i\}} (\{ \rho_i\})\Bigr]$
PDF
Rate Function
From the first principles of Cosmology $\downarrow$ Spherical Collapse Model
$\zeta(\bar{\tau_k}) = \rho_k = \Bigl( 1 - \frac{\bar{\tau_k}}{\nu}\Bigr)^{-\nu}$
$\psi (\{ \rho_i\}) = \frac{\sigma^2_{R1}}{2} \sum_{k,j}\Xi_{kj}(\{\tau_i \}) \bar{\tau_k} \bar{\tau_j}$
Cumulant Generating Function
Legendre-Fenschel transformation
$\varphi_{\{ \rho_i\}}(\{ \lambda_i\}) = \sup_{\{ \rho_i\}} \Bigl[ \sum_{i}\lambda_i\rho_i - \psi_{\{ \rho_i\}} (\{ \rho_i\})\Bigr]$
$\varphi_{\{ \rho_i\}}(\{ \lambda_i\}) = \sup_{\{ \rho_i\}} \Bigl[ \sum_{i}\lambda_i\rho_i - \psi_{\{ \rho_i\}} (\{ \rho_i\})\Bigr]$
PDF
Inverse laplace transformation
$P(\kappa) = \int_{-i\infty}^{+i\infty} \frac{dy}{2\pi i} exp(-\lambda \kappa + \phi_{\kappa,\theta}(\lambda))$
$P(\kappa) = \int_{-i\infty}^{+i\infty} \frac{dy}{2\pi i} exp(-\lambda \kappa + \phi_{\kappa,\theta}(\lambda))$
Filter Choice
- Mass sheet degeneracy $\rightarrow$ Use aperture mass statistic $\hookrightarrow$ Equivalent to using a compensated filter on the convergence map
- Wavelet filters are formally inidentical to aperture mass A. Leonard et al, "Fast Calculation of the Weak Lensing Aperture Mass Statictic"
In this work: : use a function of concentric disks
$\hookrightarrow$ $M_{ap}(\nu) = \kappa_{<\theta_2}(\nu) - \kappa_{<\theta_1}(\nu)$
$\hookrightarrow$ $M_{ap}(\nu) = \kappa_{<\theta_2}(\nu) - \kappa_{<\theta_1}(\nu)$
Deriving wavelet $\ell_1$-norm from PDF
$l_1^{j,i} = \sum_{u=1}^{coef(Sj,i)} |S_{j,i}[u]|$
$S_{j,i} = {w_{j,k}/B_i < w_{j,k} < B_{i+1}}$ $w_{j,k}$ is a wavelet coefficient
- information encoded in all pixels
- automatically includes peaks and voids
- multi-scale approach
- avoids the problem of defining peaks and voids
This is equivalent to obtaining the $\ell_1$-norm from the PDF using counts x bin method
Results
Preliminary
Preliminary
Preliminary
Conclusion
- We need different analytical methods to extract non-Gaussianities $\hookrightarrow$ Using Higher-Order statistics
- Current methods use simulations based approach $\rightarrow$ Highly resource intensive $\hookrightarrow$ Need theoretical modelling
- Use LDT based approach to obtain the PDF for aperture mass maps $\hookrightarrow$ Derieved wavelet $\ell_1$-norm from PDF
- Future work: Extending to any wavelet filter
$\hookrightarrow$ Wavelet $\ell_1$-norm is shown to be a better estimator in comparision to powerspectrum
Thank you!
Questions?
Extra slides
Wavelets
A wavelet is a waveform of effectively limited duration that has an average value of zero and nonzero norm.
Especially useful in cases where we have discontinuties, trends etc that other techniques often tend to miss
Wavelet transform vs fourier transform
