avalable LSST talk

## Denise Lanzieri

slides at github.com/dlanzieri/talks/LSST_talk

## 2 Point Statistics: A Suboptimal Measure

### 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)
• Standard Weak lensing analysis:

• Shear $\gamma$

• Data reduction

• Power spectrum
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

$\Longrightarrow$ We are dismissing most of the information!

## How to maximize the information gain?

Cosmological constraints from the combination of different summary statistics
(Ajani, et al. 2020)
• Approaches based on measuring high-order correlations to access the non-Gaussian information
e.g. the Peaks count

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

• How can we do that in a fast and accurate way?
$\Longrightarrow$ Fisher Information

## Fisher Matrix

$$$F_{\alpha, \beta} =\sum_{i,j} \frac{d\mu_i}{d\theta_{\alpha}} C_{i,j}^{-1} \frac{d\mu_j}{d\theta_{\beta}}$$$
• 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 $\Omega_c$, $\sigma_8$

• Estimate derivative with final differences

## Numerical Differentiation

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

• It’s very expensive in term of simulation time

• Computes an approximation

Different approach :

### Automatic Differentiation and Gradients in TensorFlow

• TensorFlow compute automatically derivatives of arbitrary order by applying the chain rule   repeatedly to elementary arithmetic operations

• 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.

• Derivative as functions

• No numerical approximation

• High speed

Chain rule   :

$$$\frac{dw}{du}=\frac{dw}{dv}\frac{dv}{du}$$$

## 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
batch_size=16)

# 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),
final_state[0])

with tf.Session() as sess:
sim = sess.run(final_field)

• Seamless interfacing with deep learning components

## Mocking the weak lensing universe:

• We can map the distribution of all matter over cosmic time.

• Convergence maps
$\Longrightarrow$ weighted projection of the 3D cosmological mass density along the line-of-sight.

• Access the signal in the convergence maps using a given statistic, e.g. Convergence power spectrum
Convergence map at z=1.0, based on a 3D simulation of side length 256 Mpc/h and $256^3$ particles. (Böhm, et al. 2020)

### 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

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

### Raytracing Validation

We simulate lensing lightcones implementing gravitational lensing ray-tracing in FlowPM framework

### Proof of Concept

• Convergence map at z=1.0, based on a 3D simulation of $64^3$ particles for side. The 2D lensing map has an angular extent of $5^{\circ}.$

• Angular Power Spectrum $C_\ell$

• The Jacobian: $$$\frac{dC_l}{d\Omega_c}, \frac{dC_l}{d\sigma_8}$$$ $$$F_{\alpha, \beta} =\sum_{i,j} \frac{d\mu_i}{d\theta_{\alpha}} C_{i,j}^{-1} \frac{d\mu_j}{d\theta_{\beta}}$$$
• Constraints on cosmological parameter $\Omega_c$ and $\sigma_8$:

### Status and Next steps

• Distributed version of FlowPM on several GPUs to generate large-scale mass-maps

• Peak counts statistic implement in TensorFlow framework
• Weak lensing peaks trace regions where the value of the convergence field is high $\Longrightarrow$ they are associated to massive Structures.
• We compute peaks as local maxima of the signal-to-noise field

• Compute summary statistic higher than second order to extract non-Gaussian cosmological information and infer cosmological parameters

# Conclusion

### Conclusion

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

• Differentiable physical models for fast inference
• Forecasts on differentiable higher-order statistics, including peak counts, etc.

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

### Conclusion

Everyone is most welcome to join! How to get in touch:

Thank you !