# Learning summary statistics with ML

Carolina Cuesta-Lazaro

Newcastle Astro Journal Club

Collaborators: Cheng-Zong Ruan, Yosuke Kobayashi, Enrique Paillas, Alexander Eggemeier, Pauline Zarrouk, Sownak Bose, Takahiro Nishimichi, Baojiu Li, Carlton Baugh

Medical Imaging

Epidemiology: Agent Based simulations

OBSERVED

SIMULATED

Cosmology

Simulations

HPC

Science question

Statistics ML

## Fifth forces modify structure growth

GROWTH

- GRAVITY

- FIFTH FORCE

+ EXPANSION

Credit: Cartoon depicting Willem de Sitter as Lambda from Algemeen Handelsblad  (1930).

S_8 = \sigma_8 \sqrt{\Omega_m / 0.3}

## Resolving tensions

(\vec{\theta}_i, z_i)
z_i = z_{\mathrm{Cosmological} }
+ z_{\mathrm{Doppler}}
\chi(z) = \int_0^z \frac{dz'}{H(z')}
+ \frac{v_{\mathrm{pec}}}{aH(a)}
\chi_i

Early Universe

~linear

Gravity

Late Universe

Non-linear

Credit: S. Codis+16

\delta = \frac{\rho - \bar{\rho}}{\bar{\rho}} << 1
\delta >> 1

Non-linearity = PT predictions  inaccurate

Credit: S. Codis+16

Early Universe

~linear

Gravity

Late Universe

Non-linear

Credit: S. Codis+16

Non-Guassianity

Second moment not optimal

\delta = \frac{\rho - \bar{\rho}}{\bar{\rho}} << 1
\delta >> 1

## Machine Learning as a solution to

• Non-linearities Produce accurate predictions based on N-body simulations
• Non-Gaussianity Extract cosmological information at the field level

Cosmology =

\{\vec{c}\}

Main Assumptions

1. Galaxies don't impact dark matter clustering
2. Number of galaxies depends on halo mass only
1. We don't know the Initial Conditions
2. Data is very high dimensional
3. Large number of parameters to constrain
4. N-body sims extremely slow to run! (Sampling parameter space > O(10^6) calls)

Cosmology =

Galaxy =

\{\vec{c}\}
\{\vec{g}\}
\Omega_M
P(\vec{c}|\vec{D})

?

## Summarise the data

\mathcal{O}(100)

N-body simulations

\xi_{gg} = f(\vec{c}, \vec{g}, z)
\mathcal{O}(10^5)

Likelihood evaluations

Credit: https://cs231n.github.io/convolutional-networks/

\hat{\xi}(r)
\mathcal{L} = \frac{1}{n} \sum |\hat{\xi}(r) - \xi(r)|
\Omega_m
r
\Omega_\Lambda
...

## What to emulate?

• Flexibility: Vary galaxy tracers, and their cross-correlations. Marginalising over g requires flexible g!
• 1% accuracy 1-sigma accuracy:
• Emulator only as good as data used for training
• Model clustering and mapping between real and redshift space separately
\xi_{hh}^S = \red{F}(\blue{\xi_{hh}^R(r|\vec{c})}, \blue{v^{i}_{hh}(r|\vec{c})})
\xi_{gg}^S(\vec{s}|\vec{c},\vec{g},z) = \red{\mathcal{G}}(\blue{\xi_{hh}^S}(\vec{s}|\vec{c},z), \vec{g})

Neural Net

Analytical

1+\xi^S(s_\perp, s_\parallel) = \int dr_\parallel \left(1 + \blue{\xi^R(r)}\right) \red{\mathcal{P}(v_\parallel=s_\parallel-r_\parallel|r_\perp, r_\parallel)}
\blue{\xi^R(r)}
\xi^S(s_\perp, s_\parallel)

Cosmology =

\{\vec{c}\}
\vec{c}_i

Neural Network Emulator

\vec{c}, \mathrm{redshift}, M_h
\xi^R_{hh}(r|M_h)
v_{hh}(r|M_h)

1) Very fast -> MCMC

2) Halo-Galaxy mapping modelled very accurately

3) Allows for flexible implementations of Halo-Galaxy connection

4) Modelling RSD through the Streaming Model simplifies the functions the emulator needs to learn

\xi^R_{hh}(r|c_i, M_h)
v_{hh}(r|c_i, M_h)

Galaxy =

\{\vec{g}\}
\xi_{gg} \propto \int d M_h W(\vec{g}_j, M_h) \xi_{hh}(M_h)
r_\mathrm{min} = 0.1 \, h^{-1} \mathrm{Mpc}
r_\mathrm{min} = 3 \, h^{-1} \mathrm{Mpc}
r_\mathrm{min} = 20 \, h^{-1} \mathrm{Mpc}

Cosmology

Centrals

Satellites

How much information are we throwing away by summarising in two piont functions?

How much information are we throwing away by summarising the data?

\bar{\xi}(R_s)
R_s

## Density-dependent clustering

1
1
1
2
2
4
5
5
5
3

Clusters

r [h^{-1} \mathrm{Mpc}]

Voids

F_{\alpha \beta} = \mathbb{E} \left[\frac{\partial^2 \ln \mathcal{L}(x|\theta)}{\partial \theta_i \partial \theta_j} \right] = \frac{\partial S}{\partial \theta_\alpha} C^{-1} \frac{\partial S}{\partial \theta_\beta}
\delta \theta_\alpha \geq \left( F^{-1} \right)_{\alpha \alpha}
\frac{\partial \log \mathcal{L}(x|\theta)} {\partial \theta} = 0
\Omega_b
h
\sigma_8
n_s
M_\mathrm{min}

0.08

0.05

0.02

0.7

0.4

PRELIMINARY

0.85

0.80

1.1

1.0

0.9

3.5

0.9

3.0

\Omega_m

0.33

0.08

0.28

\Omega_b
h
\sigma_8
n_s
M_\mathrm{min}

0.03

0.07

0.4

0.7

0.8

0.86

0.87

1.06

0.87

3.0

3.5

\mathrm{2PCF}
\mathrm{DS}_{1+2+3+4+5} \, \, \mathrm{(z \, space)}
\mathrm{DS}_{1+2+3+4+5} \, \, \mathrm{(r \, space)}
\Omega_M
\Omega_\Lambda
\sigma_8

Input

x

Neural network

f

Representation

(Summary statistic)

r = f(x)

Output

o = g(r)

Increased interpretability through structured inputs

Modelling cross-correlations

## ML and cosmology

• ML to accelerate non-linear predictions:  allow MCMC sampling of non-linear scales
• Precision of future surveys: what and how we emulate will have an impact on cosmological constraints

• Can ML extract **all** the information that there is at the field-level in the non-linear regime?
• Compare data and simulations, point us to the missing pieces?

By carol cuesta

• 310