Will ML shape the future of cosmology?

 

A journey through obstacles and potential approaches to overcoming them

 

Carolina Cuesta-Lazaro

IAIFI Fellow (MIT/CfA)

The golden days of Cosmology:

A five parameter Universe

\Omega_m
\Omega_b
\Omega_\Lambda
A_s
n_s

Initial Conditions

(Inflation)

Dynamics

Dark energy

Dark matter 

Ordinary matter 

Amplitude initial density field

Scale dependence

t = 400,000 years

DESI: Dark Energy Spectroscopic Instrument

~35 Million spectra!

(Image Credit: Jinyi Yang, Steward Observatory/University of Arizona)

(Image Credit: D. Schlegel/Berkeley Lab using data from DESI)

Neutrino mass hierarchy 

Primordial Non-gaussianity 

Galaxy formation 

Large scale modifications of gravity 

using growth to detect the existence of fifth forces

LSS might provide the most accurate measurement

to probe the physics of inflation (single/multi field, particle content)

not only a nuisance to margnilize over!

1

Observe galaxies

4

Pick your favourite analytical likelihood (Gaussian!)

5

Compute ~1Million times to get posterior

Constrain cosmology 101

Missing Information!

Perturbation Theory inaccurate / hard to compute

Is it always true?

3

Work on your analytical theory

2

Count pairs as a function of distance

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

Enrique Paillas

Waterloo

arXiv:2209.04310

How much information do we lose?

Bispectrum

Wavelet Scattering Transform

arXiv:1909.11107

arXiv:2204.13717

Marked Power Spectrum

arXiv:2206.01709

ML for Large Scale Structure:

A wish list

Generative models

Learn p(x)

Sample simulations with different parameter values quickly

1

Evaluate their likelihood the field level

2

Do not make assumptions on the likelihood's form

3

Latent Generative Models: 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

(Image Credit: Phillip Lippe)

p(x) = \int dz \, p(x|z)
z_T
z_{0}
z_{1}

Diffusion Models

Reverse diffusion: Denoise previous step

Forward diffusion: Add Gaussian noise (fixed)

z_{2}
p_\theta(z_{t-1}|z_t)
q(z_t|z_{t-1})

Siddharth Mishra-Sharma

Should we be using CNNs?

Galaxy positions

+ Magnitudes, velocities ...

Dark matter density field

(Image Credit: SIMBIG)

z_T
z_{0}
z_{1}

Diffusion on sets

z_{2}
p_\theta(z_{t-1}|z_t)
q(z_t|z_{t-1})

Reverse diffusion: Denoise previous step

Forward diffusion: Add Gaussian noise (fixed)

p_\theta(z_{t-1}|z_t) = \mathcal{N}(z_{t-1}|\mu_\theta(z_t, t), \sigma_t^2 \mathcal{I})

Set

Set

\mu_\theta

Neural network

z_{t-1}
z_{t}

Credit: Siddharth Mishra-Sharma

+ Galaxy formation

+ Observational systematics (Cut-sky, Fiber collisions)

+ Lightcone, Redshift Space Distortions....

Forward Model

N-body simulations

Observations

 SIMBIG arXiv:2211.00723

We can simulate the observable Universe, we just need hydrodynamical simulations

25 \, h^{-1}\mathrm{Mpc}

What are subresolution models?

10^{10} - 10^{11} M_\odot

Super massive black hole seeding in dark matter halos

BH feedback impacts galactic scales

Black holes can also growth through mergers

Effective models of astrophysical processes needed due to limited numerical resolutions or limited physical models

they can even teleport!

We can't model galaxy formation, how do we make our models robust?

Robust Summarisation

\mathcal{L} = I(S(x_\mathrm{sims})|\theta_\mathrm{sims}) + \blue{\lambda p_\mathrm{sims}(S(x_\mathrm{obs}))}
S:

Summariser (neural net)

(\theta_\mathrm{sims}, x_\mathrm{sims})
x_\mathrm{obs}
\mathcal{L} = I(S(x_\mathrm{sims})|\theta_\mathrm{sims})

Increase the evidence of the observations

Sims misspecified

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

Graph Neural Networks in a nutshell

\mathcal{G} = h^{L}_i, e^{L}_{ij} \rightarrow h^{L+1}_i, e^{L+1}_{ij}
e^{L+1}_{ij} = \phi_e(e^L_{ij}, h^L_i, h^L_j)
h^{L+1}_{i} = \phi_h( h^L_i, \mathcal{A}_j e^{L+1}_{ij})

edge embedding

node embedding

e_{ij}
h_i = \{\mathrm{positions}, \mathrm{velocities}...\}
h_j

Introducing homogeneity and isotropy

Redshift Space Distortions break isotropy

(Credit: E(3)-equivariant graph neural networks for data-efficient and accurate interatomic potentials)

 

arXiv:2202.05282

 

Normalising flows as generative models

arXiv:2202.05282

 

  • Fast theory models
  • Do all these summaries combined get all info?
    • Optimal information extraction
  • Density estimation
  • Improving current simulations?
  • How can we deal with model mispecification?

 

ML to the rescue for

arXiv:2107.00630

- \log p(x) \leq -\mathrm{VLB}(x) =
\purple{\mathrm{Prior Loss}} + \gray{\mathrm{Recon Loss}} + \blue{\mathrm{Diffusion Loss}}
\purple{\mathrm{Prior Loss} = D_\mathrm{KL} (q(z_T|x)|p(z_T))}
\gray{\mathrm{Recon Loss} = \mathbb{E}_{q(z_0|x)} [ -\log p(x|z_0)]}
\blue{\mathrm{Diffusion Loss} = }
\blue{\sum_{t} D_\mathrm{KL} \left[ q(z_{t-1}|z_{t},x) || p_\theta (z_{t-1}|z_{t}) \right]}

Gaussian,

fully known

Also Gaussian,

but learned mean

Invariance vs Equivariance

r
\theta
\phi

Invariance

Scalar interactions

r

Equivariance

What can we do with vectors? 

Tensor products

v_i
v_j

Mario Geiger

Siddharth Mishra-Sharma

Making homogeneous and isotropic universes

p(x)
\mu_\theta(z_t,t)
p(z_T)

Invariant

to rotations and translations

Equivariant

Invariant

=
p(
)
p(
p(
)

Invariant

=
p(
)
p(
=
p(
)
p(
(\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

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By carol cuesta

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