avalable Euclid France 2021 talk

Forecasting the power of Higher Order Weak Lensing Statistics with automatically differentiable simulations

Denise Lanzieri

slides at github.com/dlanzieri/talks/Euclid_France_2021

the $\Lambda$CDM view of the Universe

Weak Gravitational lensing

Traditional cosmological inference

How do we make the most of the available data?

HSC cosmic shear power spectrum
HSC Y1 constraints on $(S_8, \Omega_m)$
(Hikage,..., Lanusse, et al. 2018)
  • Data reduction

  • Compute summary statistics based on 2pt functions,
    e.g. the power spectrum

  • Run an MCMC to recover a posterior on model parameters, using an analytic likelihood $$ p(\theta | x ) \propto \underbrace{p(x | \theta)}_{\mathrm{likelihood}} \ \underbrace{p(\theta)}_{\mathrm{prior}}$$
Main limitation: The lensing convergence field is inherently and significantly non-Gaussian
The two-point statistics do not fully capture the non-Gaussian information encoded in the peaks of the matter distribution

2 Point Statistics: A Suboptimal Measure

How to maximize the information gain?

(Ajani, et al. 2020)
  • Approaches based on measuring high-order correlations to access the non-Gaussian information
    e.g. the Peaks count

  • Simulation-based approaches

Main limitation: Gradient-based
  • Costly as they require a large number of simulations
  • Intractable for more than 3 or 4 cosmological parameters.

  • Numerical Differentiation

    \[\begin{equation} \left.\frac{df(x)}{dx}\right|_{x_1} \approx \frac{f(x_1+h)-f(x_1)}{h} \end{equation} \]
      Flaws :
    • It’s numerically very unstable

    • It’s very expensive in term of simulation time

    • Computes an approximation

    Different approach :

    Automatic Differentiation

    Automatic Differentiation and Gradients in TensorFlow

    • Automatic differentiation allows you to compute analytic derivatives of arbitraty expressions:
      If I form the expression $y = a * x + b$, it is separated in fundamental ops: $$ y = u + b \qquad u = a * x $$ then gradients can be obtained by the chain rule: $$\frac{\partial y}{\partial x} = \frac{\partial y}{\partial u} \frac{ \partial u}{\partial x} = 1 \times a = a$$

    • To differentiate automatically, TensorFlow remember what operations happen in what order during the forward pass and traverses this list of operations in reverse order to compute gradients.

      Advantages :
    • Derivative as functions

    • No numerical approximation

    • High speed

    Cosmological N-Body Simulations

    How do we simulate the Universe in a fast and differentiable way?

    The Particle-Mesh scheme for N-body simulations

    The idea: approximate gravitational forces by estimating densities on a grid.
    • The numerical scheme:

      • Estimate the density of particles on a mesh
        $\Longrightarrow$ compute gravitational forces by FFT

      • Interpolate forces at particle positions

      • Update particle velocity and positions, and iterate

    • Fast and simple, at the cost of approximating short range interactions.
    $\Longrightarrow$ Only a series of FFTs and interpolations.

    Introducing FlowPM: Particle-Mesh Simulations in TensorFlow

    											import tensorflow as tf
    											import flowpm
    											# Defines integration steps
    											stages = np.linspace(0.1, 1.0, 10, endpoint=True)
    											initial_conds = flowpm.linear_field(32,        # size of the cube
    											100,       # Physical size
    											ipklin,    # Initial powerspectrum
    											# Sample particles and displace them by LPT
    											state = flowpm.lpt_init(initial_conds, a0=0.1)
    											# Evolve particles down to z=0
    											final_state = flowpm.nbody(state, stages, 32)
    											# Retrieve final density field
    											final_field = flowpm.cic_paint(tf.zeros_like(initial_conditions),
    											with tf.Session() as sess:
    											sim = sess.run(final_field)
    • Seamless interfacing with deep learning components

    Potential Gradient Descent (PGD)

      Flow N-body PM simulation:

      • Fast (limited number of time steps while enforcing the correct linear growth)

      • Cannot give accurate halo matter profiles or matter power spectrum

      The PGD idea : mimics the physics that is missing
      $\Longrightarrow$ Halo virialization
    (Biwei Dai et al. 2018)

    Mocking the weak lensing universe: The Born Approximation

    • Numerical simulation of WL features rely on ray-tracing through the output of N-body simulations​

    • Knowledge of the Gravitational potential and accurate solvers for light ray trajectories is computationally expensive

    • Born approximation , only requiring knowledge of the density field, can be implemented more efficiently and at a lower computational cost

    \[\begin{equation} \kappa_{born}(\boldsymbol{\theta},\chi_s)= \frac{3H_0^2 \Omega_m}{2c^2} \int_0^{\chi_s} d\chi \frac{\chi}{a(\chi)} W(\chi,\chi_s) \delta(\chi \boldsymbol{\theta},\chi). \end{equation} \]

    Proof of Concept

    • Convergence map at z=1.0, based on a 3D simulation of $128^3$ particles for side. The 2D lensing map has an angular extent of $5^{\circ}.$
    • Angular Power Spectrum $C_\ell$

    Proof of Concept: Peak counts

  • Local maxima in the map

  • Non-Gaussian information

  • Trace overdense regions
  • Proof of Concept: $l_1$norm

    • We take the sum of all wavelet coefficients of the original image $\kappa$ map in a given bin $i$ defined by two values, $B_i$ and $B_{i+1}$,

    • Non-Gaussian information

    • Information encoded in all pixels

    • Avoids the problem of defining peaks and voids

    A first application: Fisher forecasts

    We are testing the simulations to reproduce a LSST Y1-like setting
    (Preliminary results, presented on behalf of LSST DESC)

    \[\begin{equation} F_{\alpha, \beta} =\sum_{i,j} \frac{d\mu_i}{d\theta_{\alpha}} C_{i,j}^{-1} \frac{d\mu_j}{d\theta_{\beta}} \end{equation} \]

    • Use Fisher matrix to estimate the information content extracted with a given statistic

    • Derivative of summary statistics respect to the cosmological parameters.
      e.g. the $\Omega_c$, $\sigma_8$

    • Fisher matrices are notoriously unstable
      $\Longrightarrow$ they rely on evaluating gradients by finite differences.

    • They do not scale well to large number of parameters.

    A first application: Fisher forecasts

  • Use Fisher matrix to investigate the constraining power of different statistics
  • A first application: Fisher forecasts

  • Use Fisher matrix to estimate the information content extracted with a given statistic for more cosmological parameters simultaneously
  • Fisher constraints from the full peak count statistics



    What can be gained by simulating the Universe in a fast and differentiable way?

    • Investigate the constraining power of various map-based higher order weak lensing statistics and control the systematics

    • Differentiable physical models for fast inference ( made efficient by very fast lensing lightcone and having access to gradient)

    • Even analytic cosmological computations can benefit from differentiability.

    Thank you !