Enrique Paillas, Pauline Zarrouk, Yan-Chuan Cai, Will Percival, Sesh Nadathur, Mathilde Pinon, Arnaud de Mattia, Florian Beuler

Constraining vΛCDM:

 beyond two-point functions

 

Carolina Cuesta-Lazaro

IAIFI fellow - MIT/CfA

arXiv:2209.04310

(\vec{\theta}_i, z_i)
P(\mathcal{C}|\,\,\,\,\,\,\,\,\,\,)

?

The golden days of Cosmology:

A five parameter Universe

\Omega_m
\Omega_b
\Omega_\Lambda
A_s
n_s

Initial Conditions

Dynamics

Dark energy

Dark matter 

Ordinary matter 

Amplitude initial density field

Scale dependence

z \approx 1100

Neutrino mass first evidence beyond SM physics + LSS most accurate measurement

Primordial Non-gaussianity to probe the physics of inflation (single/multi field, particle content)

Galaxy formation not only a nuisance to margnilize over!

Large scale modifications of gravity using growth to detect the existence of fifth forces 

What we expect to learn from the LSS

\delta = \frac{\rho - \bar{\rho}}{\bar{\rho}}
t = 380,000 \: \mathrm{years}
\delta = \red{F}(\delta_i)

Linear

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
\bar{\xi}(R_s)
R_s
1
1
1
2
2
4
5
5
5
3

arxiv:1911.11158

1
1
1
2
2
4
5
5
5
3

Autocorrelation

10
50
90
130

Cross-correlation with haloes

Monopole

Quadrupole

Voids

Clusters

10
50
90
130
s^2
s^2
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

Estimating sensitivity to cosmology: Fisher information

The Quijote simulations

arXiv:1909.05273

Covariance

Derivatives

\frac{\partial \bm{S}}{\partial \bm{\theta}} \simeq \frac{{\bm S}(\bm{\theta} + {\rm d}\bm{\theta}) - {\bm S}(\bm{\theta} - {\rm d}\bm{\theta})}{2 \rm{d}\bm{\theta}}

Finite differences

F_{\alpha \beta} = \frac{\partial S}{\partial \theta_\alpha} C^{-1} \frac{\partial S}{\partial \theta_\beta}
0
C = \frac{1}{n_{\rm sim} - 1} \sum_{k=1}^{n_{\rm sim}} \left({\bm{S}_{k}} - \overline{\bm{S}}\right)\left({\bm{S}_{k}} - \overline{\bm{S}}\right)
F_{\alpha \beta} = \frac{\partial S}{\partial \theta_\alpha} C^{-1} \frac{\partial S}{\partial \theta_\beta}

Where does the information come from?

?

Density split in a Gaussian Random Field

But, can we estimate densities realistically?

?

Real

Redshift

Cross-correlation between quintiles and haloes

Monopole

Quadrupole

s^2
s^2

Monopole

Quadrupole

Autocorrelation of quintiles

s^2
s^2

Comparison against other works that have used the same set of tracers

\theta

Simulated Data

Data

Prior

Posterior

x_\mathrm{obs}
x
P(\theta|x_\mathrm{obs})

Inverse problem

Forwards model

+ Galaxy-Halo connection

+ Cut-sky

+ Lightcone

+ Alcock-Paczinsky

+ Fiber collisions

Forward Model

N-body simulations

Observations

Density split in a SDSS BOSS

CMASS 0.46 < z < 0.6

Galaxy-Halo connection

Main Assumptions

  1. Galaxies don't impact dark matter clustering
  2. Number of galaxies depends on halo mass only

Extensions: Get creative!

But these extensions might limit our constraining power

a) Use hydro simulations to limit options

b) Mask the data to optimise robustness?

Velocity bias

Assembly bias

Environment

Concentration

Formation time

...

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

S(r)
\Omega_m
\Omega_\Lambda
...
\log(\mathcal{L}) \propto (S - S_\mathrm{obs}) C^{-1}(S - S_\mathrm{obs})

Neural Network emulator

\mathcal{O}(100) \,\,\, \mathrm{Nbody \, simulations}

Implicit likelihood inference with normalising flows

x = f(z), \, z = f^{-1}(x)
p(\mathbf{x}) = p_z(f^{-1}(\mathbf{x})) \left\vert \det J(f^{-1}) \right\vert

No assumptions on the likelihood (likelihoods rarely Gaussian!)

 

No expensive MCMC chains needed to estimate posterior

\Omega_M
\Omega_\Lambda
\sigma_8

Input

x

 

Neural network

f

Representation

(Summary statistic)

r = f(x)

Output

 

Increased interpretability through structured inputs

Modelling cross-correlations

P(\theta|x_\mathrm{obs})

What ML can do for cosmology

  • ML to accelerate non-linear predictions and density estimation

 

  • 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?

cuestalz@mit.edu