Field-level inference of primordial non-Gaussianity from DESI:
Validation on simulations

Hugo SIMON, 
PhD student supervised by
Arnaud DE MATTIA and François LANUSSE

2025/12/09

Field-level inference of primordial non-Gaussianity from galaxy redshift surveys

Hugo SIMON, 
PhD student supervised by
Arnaud DE MATTIA and François LANUSSE

2025/11/27

Field-level inference of primordial non-Gaussianity from galaxy redshift surveys

Hugo SIMON, 
PhD student supervised by
Arnaud DE MATTIA and François LANUSSE

CoBALt, 2025/06/30

Field-level inference of primordial non-Gaussianity from galaxy redshift surveys

Hugo SIMON, 
PhD student supervised by
Arnaud DE MATTIA and François LANUSSE

CoBALt, 2025/06/30

Optimal extraction of primordial non-Gaussian signal from galaxy redshift survey

Hugo SIMON, 
PhD student supervised by
Arnaud DE MATTIA and François LANUSSE

Sesto, 2025/07/17

Field-level analysis of primordial non-Gaussianity with DESI tracers

Hugo SIMON, 
PhD student supervised by
Arnaud DE MATTIA and François LANUSSE

PNG Meeting, 2025/06/18

The universe recipe (so far)

$$\frac{H}{H_0} = \sqrt{\Omega_r + \Omega_b + \Omega_c+ \Omega_\kappa + \Omega_\Lambda}$$

instantaneous expansion rate

energy content

Cosmological principle + Einstein equation

+ Inflation

\(\delta_L \sim \mathcal G(0, \mathcal P)\)

\(\sigma_8:= \sigma[\delta_L * \boldsymbol 1_{r \leq 8}]\)

initial field

primordial power spectrum

std. of fluctuations smoothed at \(8 \text{ Mpc}/h\)

Cosmological inference

DESI+2024

\(\Omega\)

\(P\)

\(\Omega := \{ \Omega_m, \Omega_\Lambda, H_0, \sigma_8, f_\mathrm{NL},...\}\)

inference

\(P\)

\(\Omega\)

\(\delta_L\)

\(\delta_g\)

Cosmological inference

\(\Omega := \{ \Omega_m, \Omega_\Lambda, H_0, \sigma_8, f_\mathrm{NL},...\}\)

\(\Omega\)

\(\delta_L\)

\(\delta_g\)

inference

\(\delta_L\)

\(\delta_g\)

\(\Omega\)

Cosmological inference

DESI+2024

\(\Omega\)

\(P\)

inference

\(\Omega := \{ \Omega_m, \Omega_\Lambda, H_0, \sigma_8, f_\mathrm{NL},...\}\)

\(\Omega\)

\(\delta_L\)

\(\delta_g\)

\(P\)

Cosmological inference

\(\Omega := \{ \Omega_m, \Omega_\Lambda, H_0, \sigma_8, f_\mathrm{NL},...\}\)

\(\Omega\)

\(\delta_L\)

\(\delta_g\)

inference

\(\delta_L\)

\(\delta_g\)

\(\Omega\)

\(128^3\) PM on 8GPU: 
4h MCLMC vs. \(\geq\) 80h HMC

Fast & differentiable model with

Simon+2025

Field-level inference

Summary stat inference

\(\Omega\)

\(s\)

\(\delta_g\)

\(\Omega\)

\(\delta_L\)

\(s\)

marginalize

condition

marginalize

\(\Omega\)

\(s\)

\(\delta_g\)

\(\Omega\)

\(\delta_L\)

condition

Two approaches to cosmological inference

Cosmo model

\(\mathrm{p}(\Omega,s)\)

\(\mathrm{p}(\Omega \mid s)\)

\(\Omega\)

\(\delta_g\)

\(\mathrm{p}(\Omega,\delta_L,\delta_g, s)= \mathrm{p}(s \mid \delta_g) \, \mathrm{p}(\delta_g \mid \Omega,\delta_L)\, \mathrm{p}(\delta_L \mid \Omega)\, \mathrm{p}(\Omega)\)

\(\mathrm{p}(\Omega,\delta_L \mid \delta_g)\)

\(\mathrm{p}(\Omega \mid \delta_g)\)

\(\delta_g\)

\(\Omega\)

\(\delta_L\)

\(s\)

Two approaches to cosmological inference

Cosmo model


Problem:

  • \(s\) is too simple \(\implies\) lossy compression
  • \(s\) is too complex \(\implies\) intractable marginalization



The Problem:

  • high-dimensional integral $$\mathrm{p}(\Omega \mid \delta_g) = \int \mathrm{p}(\Omega, \delta_L \mid \delta_g) \;\mathrm d \delta_L$$
  • To probe scales of \(15\ \mathrm{Mpc}/h\) in DESI volume, \(\operatorname{dim}(\delta_L) \simeq 1024^3\)
     

The Promise:

  • "lossless" explicit inference

Nguyen+2024

Field-level inference

Summary stat inference

  1. Prior on
    • Cosmology \(\Omega\)
    • Initial field \(\delta_L\)
    • EFT parameters
      (Dark matter-galaxy connection)
  2. LSS formation: 2LPT or PM
    (BullFrog or FastPM)
  3. Apply galaxy bias
  4. Redshift-Space Distortions
  5. Observational noise

Field-Level Modeling

Fast and differentiable model thanks to                        (\(\texttt{NumPyro}\) and \(\texttt{JaxPM}\))

How to N-body-differentiate?

\((\boldsymbol q, \boldsymbol p)\)

\(\delta(\boldsymbol x)\)

\(\delta(\boldsymbol k)\)

paint*

read*

fft*

ifft*

fft*

*: differentiable, e.g. with                    via \(\texttt{JaxPM}\), in \(\mathcal O(n \log n)\)

apply forces
to move particles

solve Vlasov-Poisson
to compute forces

\(\begin{cases}\dot {\boldsymbol q} \propto \boldsymbol p\\ \dot{\boldsymbol p} = \boldsymbol f \end{cases}\)

\(\begin{cases}\nabla^2 \phi \propto \delta\\ \boldsymbol f = -\nabla \phi \end{cases} \implies \boldsymbol f \propto \frac{i\boldsymbol k}{k^2} \delta\)

MCMC sampling

High-dimensional sampling is hard

𝓐 𝓭𝓻𝓾𝓷𝓴 𝓶𝓪𝓷 𝔀𝓲𝓵𝓵 𝓯𝓲𝓷𝓭 𝓱𝓲𝓼 𝔀𝓪𝔂 𝓱𝓸𝓶𝓮, 𝓫𝓾𝓽 𝓪 𝓭𝓻𝓾𝓷𝓴 𝓫𝓲𝓻𝓭 𝓶𝓪𝔂 𝓰𝓮𝓽 𝓵𝓸𝓼𝓽 𝓯𝓸𝓻𝓮𝓿𝓮𝓻 (\(\mathrm p \approx 0.66\))
                                         🌸 𝓢𝓱𝓲𝔃𝓾𝓸 𝓚𝓪𝓴𝓾𝓽𝓪𝓷𝓲

\(-\nabla\)

\(d \approx 1\)

🏠

🚶‍♀️

To maintain constant move-away probability, step-size \(\simeq d^{-1/2}\)

\(d \gg 1\)

🪺

🐦

Canonical MCMC samplers

Recipe😋 to sample from \(\mathrm p \propto e^{-U}\)

  • take particle with position \(\boldsymbol q\), momentum \(\boldsymbol p\), mass matrix \(M\), and Hamiltonian $$\mathcal H(\boldsymbol q, \boldsymbol p) = U(\boldsymbol q) + \frac 1 2 \boldsymbol p^\top M^{-1} \boldsymbol p$$
     
  • follow Hamiltonian dynamics during time \(L\)
    $$\begin{cases} \dot {{\boldsymbol q}} = \partial_{\boldsymbol p}\mathcal H = M^{-1}{{\boldsymbol p}}\\ \dot {{\boldsymbol p}} = -\partial_{\boldsymbol q}\mathcal H = - \nabla U(\boldsymbol q)  \end{cases}$$and refresh momentum \(\boldsymbol p \sim \mathcal N(\boldsymbol 0,M)\)
     
  • usually, perform Metropolis adjustment
     
  • this samples canonical ensemble $$\mathrm p_\text{C}(\boldsymbol q, \boldsymbol p) \propto e^{-\mathcal H(\boldsymbol q, \boldsymbol p)} \propto \mathrm p(\boldsymbol q)\,\mathcal N(\boldsymbol 0, M)$$
     

gradient guides particle toward high density sets

scales poorly with dimension

must average over all energy levels

Hamiltonian Monte Carlo (e.g. Neal2011)

Microcanonical MCMC samplers

Recipe😋 to sample from \(\mathrm p \propto e^{-U}\)

  • take particle with position \(\boldsymbol q\), momentum \(\boldsymbol p\), mass matrix \(M\), and Hamiltonian $$\mathcal H(\boldsymbol q, \boldsymbol p) = \frac {\boldsymbol p^\top M^{-1} \boldsymbol p} {2 m(\boldsymbol q)} - \frac{m(\boldsymbol q)}{2} \quad ; \quad m=e^{-U/(d-1)}$$
     
  • follow Hamiltonian dynamics during time \(L\)
    $$\begin{cases} \dot{\boldsymbol q} = M^{-1/2} \boldsymbol u\\ \dot{\boldsymbol u} = -(I - \boldsymbol u \boldsymbol u^\top) M^{-1/2} \nabla U(\boldsymbol q) / (d-1) \end{cases}$$ and refresh \(\boldsymbol u \leftarrow \boldsymbol z/ \lvert \boldsymbol z \rvert \quad ; \quad \boldsymbol z \sim \mathcal N(\boldsymbol 0,I)\)
     
  • usually, perform Metropolis adjustment
     
  • this samples microcanonical/isokinetic ensemble $$\mathrm p_\text{MC}(\boldsymbol q, \boldsymbol u) \propto \delta(H(\boldsymbol q, \boldsymbol u)) \propto  \mathrm p (\boldsymbol q) \delta(|\boldsymbol u|^2 - 1)$$

single energy/speed level

let's try avoiding that

gradient guides particle toward high density sets

MicroCanonical HMC (Robnik+2022)

Canonical/Microcanonical MCMC samplers

 

  • \(\mathcal H(\boldsymbol q, \boldsymbol p) = \frac {\boldsymbol p^\top M^{-1} \boldsymbol p} {2 m(\boldsymbol q)} - \frac{m(\boldsymbol q)}{2} \quad ; \quad m:=e^{-U/(d-1)}\)













     
  • samples microcanonical/isokinetic ensemble $$\mathrm p_\text{MC}(\boldsymbol q, \boldsymbol p) \propto \delta(\mathcal H(\boldsymbol q, \boldsymbol p)) \propto  \mathrm p (\boldsymbol q) \delta(\dot{\boldsymbol q}^\top M \dot{\boldsymbol q} - 1)$$

Hamiltonian Monte Carlo (e.g. Neal2011)

MicroCanonical HMC (Robnik+2022)

  • \(\mathcal H(\boldsymbol q, \boldsymbol p) = U(\boldsymbol q) + \frac 1 2 \boldsymbol p^\top M^{-1} \boldsymbol p\)













     
  • samples canonical ensemble $$\mathrm p_\text{C}(\boldsymbol q, \boldsymbol p) \propto e^{-\mathcal H(\boldsymbol q, \boldsymbol p)} \propto \mathrm p(\boldsymbol q)\,\mathcal N(\boldsymbol 0, M)$$

What it looks like

Inferring jointly cosmology, bias parameters, and initial matter field allows full universe history reconstruction

Simon+2025

million-dimensional inference:
4h on a single GPU node

PRELIMINARY

Previous results
on LRG SGC footprint (self-specified)

\(k_\mathrm{max} \approx 0.04\ h/\mathrm{Mpc}\)

\(\sigma[f_\mathrm{NL}] \approx 20\), consistent with power spectrum analysis (Chaussidon+2024)

Current status:









 

  • model validation (w/o PNG) on AbacusSummit
  • model validation (w/ PNG) on PNG-Unitsims
  • Systematics model validation on contaminated mocks
  • Application to DESI DR1 LRG and QSO
Part Implementation Validation
MCMC ✅️ ✅️
LSS formation ✅️ ✅️
Galaxy bias ✅️ ✅️
Galaxy stochasticity ✅️ 🗘
Selection ✅️ 🗘
Lightcone ✅️
Integral Constraint ✅️
Imaging
  • Different samplers and strategies used for field-level (e.g. Lavaux+2018, Bayer+2023). Additional comparisons required.
     
  • We provide a consistent benchmark for field-level from galaxy surveys. Build upon \(\texttt{NumPyro}\) and \(\texttt{BlackJAX}\).

Samplers comparison

= NUTS within Gibbs
= auto-tuned HMC

= adjusted MCHMC
= unadjusted Langevin MCHMC

10 times less evaluations required

Unadjusted microcanonical sampler outperforms any adjusted sampler 

Model preconditioning

  • Sampling is easier when target density is isotropic Gaussian












     
  • The model is reparametrized assuming a tractable Kaiser model:
    linear growth + linear Eulerian bias + flat sky RSD + Gaussian noise

10 times less evaluations required

Benchmark results

\(128^3\) PM on 8GPU: 
4h MCLMC vs. \(\geq\)80h NUTS

Mildly dependent with respect to formation model and volume

Probing smaller scales could be harder

Handling bias in unadjusted MCLMC

  1. Microcanonical dynamics \(\implies\) energy should not vary
  2. Numerical integration yields quantifiable errors that can be linked to bias
  3. Stepsize can be tuned to ensure controlled bias, see Robnik+2024

reducing stepsize rapidly brings bias under Monte Carlo error

  • Effective Sample Size (ESS)
    • number of i.i.d. samples that yield same statistical power.
    • For sample sequence of size \(N\) and autocorrelation \(\rho\) $$N_\textrm{eff} = \frac{N}{1+2 \sum_{t=1}^{+\infty}\rho_t}$$so aim for as less correlated sample as possible.








       
  • Main limiting computational factor is model evaluation (e.g. N-body), so characterize MCMC efficiency by \(N_\text{eval} / N_\text{eff}\)

Andy Jones

How to compare samplers?

EFT-based Field-Level modeling

  1. Sample initial conditions and add PNG
    $$\phi_{\mathrm{NL}}=\phi+{\color{purple}f_{\mathrm{NL}}}\phi^{2}$$
  2. Compute Lagrangian bias of particles at \(\boldsymbol q^\mathrm{in}\) $$\mathcal O_{\rm L}=1+{\color{purple}b_{1}}\,\delta_{\rm L}+{\color{purple}b_{2}}\delta_{\rm L}^{2}+{\color{purple}b_{s^2}}s^{2}+ {\color{purple}b_{\nabla^2}} \nabla^2 \delta _{\rm L}\\\!\!\!\!\!\!\! + {\color{purple}f_{\rm NL} b_\phi}\phi + {\color{purple} f_{\rm NL} b_{\phi\delta}}  \phi \delta_{\rm L}$$
  3. Displace particles to \(\boldsymbol q^\mathrm{fin}\)
    \(\boldsymbol q^\mathrm{fin} = \boldsymbol q^\mathrm{2LPT} + H^{-1}\dot {\boldsymbol q}^\mathrm{2LPT}_\parallel + {\color{purple}b_{\nabla_\parallel}} \nabla_\parallel \delta_\mathrm{L}(\boldsymbol q^\mathrm{in})\)
  4. Paint particles on grid with kernel \(K\) and weight \(\mathcal O_{\rm L}\)
    $$(1+\delta_g)(\boldsymbol x) = \int K(\boldsymbol x - \boldsymbol q^\mathrm{fin}) \mathcal O_L(\boldsymbol q^\mathrm{in})\, \mathrm d \boldsymbol q^\mathrm{in}$$
  5. Noise via galaxy stochasticity
    $$n_g \sim \mathcal N({\color{purple} \bar n_g} (1+\delta_g),\, {\color{purple}\bar n_g \sigma_0}(1+{\color{purple}\sigma_\delta}\delta_g))$$

3 PNG parameters, 2 options:

  • infer the 3 as independent
  • assume "universality" relations
    $$\begin{align*}b_\phi &=2\delta_c({\color{purple} b_1}+1-p)\\b_{\phi \delta} &=2 (\delta_c {\color{purple} b_2}+ {\color{purple} b_1})\end{align*}$$(Lagrangian form)

Fast and differentiable model with                        

Fitting AbacusSummit+HOD

  • CDM: fix initial conditions. Match within \(0.5\%\) at field-level for \(k_\mathrm{nyq} < 0.1 h/\mathrm{Mpc} \)                                                                                                                              







 

  • Tracer (LRG, \(z=0.8\)): fix initial conditions and optimize on EFT parameters                                                                                                                

$$\sqrt{P_{\delta} / P_{\delta^\mathrm{true}}}$$  = amplitude info

$$P_{\delta,\delta^\mathrm{true}} / \sqrt{P_{\delta}P_{\delta^\mathrm{true}}}$$ = phase info

Galaxy stochasticity characterization

EFT says \({\color{purple}\sigma_0}(1+{\color{purple}\sigma_\delta}\delta_g^\mathrm{det}))\)
Poisson \(\simeq \sigma_0=\sigma_\delta=1\), but fit shows sub-Poisson

For positivity, we take:
\({\color{purple}\sigma_0}\ln(1+e^{1+{\color{purple}\sigma_\delta}\delta_g^\mathrm{det}})\)

Quite Gaussian 👍
So \(\delta_g \sim \mathcal N(\delta_g^\mathrm{det},\, {\color{purple}❓})\)

EFT says \({\color{purple}\sigma_0}(1+{\color{purple}\sigma_{2}}k^2 + {\color{purple}\sigma_{\mu,2}}(\mu k)^2)\)
Negligible for currently probed scales.
If not, how to implement efficiently in real domain?

Galaxy stochasticity = \(\delta_g^\mathrm{true} -\delta_g^\mathrm{det}\), and we take \(\delta_g^\mathrm{det}\) to be EFT best fit.

Preliminary FLI results

  • CDM:





     
  • Tracer (LRG, \(z=0.8\)):

\((2\ \mathrm{Gpc}/h)^3,\, \operatorname{dim}(\delta_L) = 96^3,\,k_\mathrm{nyq} = 0.1 h / \mathrm{Mpc}\)

\((2\ \mathrm{Gpc}/h)^3,\, \operatorname{dim}(\delta_L) = 48^3\), \(k_\mathrm{nyq} = 0.05 h / \mathrm{Mpc}\)

PRELIMINARY

Infer the initial conditions and \(\sigma_8\)

Preliminary FLI results on PNG

On AbacusSummit + HOD mock (\(f_\mathrm{NL}= 0\))

For \(k_\mathrm{nyq} = 0.05\ h/\mathrm{Mpc}\), inference compatible with \(f_\mathrm{NL}= 0\).
TBD: posterior calibration tests

\((2\ \mathrm{Gpc}/h)^3,\, \mathrm{LRG}\, z=0.8\)

PRELIMINARY

Preliminary FLI results on PNG

On AbacusSummit + HOD mock (\(f_\mathrm{NL}= 0\))

For \(k_\mathrm{nyq} = 0.1\ h/\mathrm{Mpc}\),
\(f_\mathrm{NL} b_\phi\) compatible, but not \(f_\mathrm{NL} b_{\phi \delta}\) nor \(f_\mathrm{NL}\)

For \(k_\mathrm{nyq} = 0.05\ h/\mathrm{Mpc}\), inference compatible with \(f_\mathrm{NL}= 0\).
TBD: posterior calibration tests

\((2\ \mathrm{Gpc}/h)^3,\, \mathrm{LRG}\, z=0.8\)

PRELIMINARY

Preliminary FLI results on PNG

On \(f_\mathrm{NL}\neq 0\) FastPM + HOD mocks (courtesy of Edmond)

Next steps:

  • confirm the calibration at relevant scales on PNG-Unitsims
  • light-cone, survey selection, imaging...
    validation on contaminated mocks
PRELIMINARY

\((2.76\ \mathrm{Gpc}/h)^3,\,k_\mathrm{nyq} = 0.036 h/ \mathrm{Mpc},\\\, \mathrm{QSO}\, z=1,\,\operatorname{dim}(\delta_L) = 48^3\)

Preliminary FLI results on PNG

On \(f_\mathrm{NL}\neq 0\) FastPM + HOD mocks (courtesy of Edmond)

PRELIMINARY

\((2.76\ \mathrm{Gpc}/h)^3,\,k_\mathrm{nyq} = 0.073 h/ \mathrm{Mpc},\\ \mathrm{QSO}\, z=1,\, \operatorname{dim}(\delta_L) = 96^3\)

Next steps:

  • confirm the calibration at relevant scales on PNG-Unitsims
  • light-cone, survey selection, imaging...
    validation on contaminated mocks

Preliminary FLI results on PNG

On \(f_\mathrm{NL}\neq 0\) FastPM + HOD mocks (courtesy of Edmond)

PRELIMINARY

\((2.76\ \mathrm{Gpc}/h)^3,\,k_\mathrm{nyq} = 0.073 h/ \mathrm{Mpc},\\ \mathrm{QSO}\, z=1,\, \operatorname{dim}(\delta_L) = 96^3\)

PRELIMINARY

\((2.76\ \mathrm{Gpc}/h)^3,\,k_\mathrm{nyq} = 0.036 h/ \mathrm{Mpc},\\\, \mathrm{QSO}\, z=1,\,\operatorname{dim}(\delta_L) = 48^3\)

Next steps:

  • confirm the calibration at relevant scales on PNG-Unitsims
  • light-cone, survey selection, imaging...

Thank you!

 BaoBan comics

Primordial Non-Gaussianity from galaxies

Local-type PNG is constrained by the induced scale-dependent bias 

\(\phi_{\mathrm{NL}}=\phi+{\color{purple}f_{\mathrm{NL}}}\phi^{2}\)

\(\delta(\boldsymbol k)\simeq\left(b_{1}+ b_\phi {\color{purple}f_\mathrm{NL}}k^{-2} \right) \delta_L(\boldsymbol k)\)

\(f_{\mathrm{NL}}=-3.6_{-9.1}^{+9.0}\)

Chaussidon+2024

Field-level modeling of PNG

$$\begin{align*}w_g&=1+{\color{purple}b_{1}}\,\delta_{\rm L}+{\color{purple}b_{2}}\delta_{\rm L}^{2}+{\color{purple}b_{s^2}}s^{2}+ {\color{purple}b_{\nabla^2}} \nabla^2 \delta _{\rm L}\\&\quad\quad\! + {\color{purple}b_\phi f_{\rm NL}} \phi + {\color{purple} b_{\phi\delta} f_{\rm NL}} \phi \delta_{\rm L}\\\Delta \boldsymbol q_\parallel &= H^{-1} \dot{\boldsymbol q}_\parallel  +  {\color{purple}b_{\nabla_\parallel}} \nabla_\parallel \delta_\mathrm{L}\end{align*}$$

\(\phi_{\mathrm{NL}}=\phi+{\color{purple}f_{\mathrm{NL}}}\phi^{2}\)

Primordial to Linear
Transfer function

modified from Andrews+2024

Galaxy + velocity
bias model

\(\boldsymbol q_\mathrm{LPT} \simeq \boldsymbol q_\mathrm{in} + \Psi_\mathrm{LPT}(\boldsymbol q_\mathrm{in}, z(\boldsymbol q_\mathrm{in}))\)
one-shot 2LPT light-cone

\(n_g^\mathrm{obs}(\boldsymbol q) = (1+\delta_g(\boldsymbol q))\, {\color{purple}\bar n_g(\,r)}\, {\color{blue}W(\boldsymbol q)}\, {\color{purple}\beta_i} {\color{green}T^i(\theta)}\)
RIC relax + selection + imag. templates

\(\delta_g \sim \mathcal N(\delta_g^\mathrm{det}, \sigma^2)\) with
\(\sigma(k) = {\color{purple}\sigma_0}(1+{\color{purple}\sigma_2} k^2 + {\color{purple}\sigma_{\mu2}}(k\mu)^2)\)

EFT-based modeling, many scale cuts alleviating discretization effects (see Stadler+2024)

EFT-based Field-Level modeling

  1. Sample initial conditions and add PNG
    $$\phi_{\mathrm{NL}}=\phi+{\color{purple}f_{\mathrm{NL}}}\phi^{2}$$
  2. Initialize particles at \(\boldsymbol q^\mathrm{in}\)
  3. Compute their Lagrangian bias expansion
    $$\mathcal O_{\rm L}=1+{\color{purple}b_{1}}\,\delta_{\rm L}+{\color{purple}b_{2}}\delta_{\rm L}^{2}+{\color{purple}b_{s^2}}s^{2}+ {\color{purple}b_{\nabla^2}} \nabla^2 \delta _{\rm L}\\\!\!\!\!\!\!\! + {\color{purple}f_{\rm NL} b_\phi \phi} + {\color{purple} f_{\rm NL} b_{\phi\delta}}  \phi \delta_{\rm L}$$
  4. Displace particles to \(\boldsymbol q^\mathrm{fin}\)
    \(\boldsymbol q^\mathrm{fin} = \boldsymbol q^\mathrm{LPT} + H^{-1}\dot {\boldsymbol q}^\mathrm{LPT}_\parallel + {\color{purple}b_{\nabla_\parallel}} \nabla_\parallel \delta_\mathrm{L}(\boldsymbol q^\mathrm{in})\),
    with \(\boldsymbol q^\mathrm{LPT} = \boldsymbol q^\mathrm{in} + \Psi^\mathrm{LPT}(\boldsymbol q^\mathrm{in}, \delta_{\rm NL})\)
  5. Paint particles on grid with kernel \(K\) and weight \(\mathcal O_{\rm L}\)
    $$(1+\delta_g)(\boldsymbol x) = \int K(\boldsymbol x - \boldsymbol q^\mathrm{fin}) \mathcal O_L(\boldsymbol q^\mathrm{in})\, \mathrm d \boldsymbol q^\mathrm{in}$$
  6. Noise via galaxy stochasticity
    $$n_g \sim \mathcal N({\color{purple} \bar n_g} (1+\delta_g),\, {\color{purple}\bar n_g \sigma_0}(1+{\color{purple}\sigma_\delta}\delta_g))$$

\(k_\mathrm{evolve}\)
(LPT, bias)

\(k_\mathrm{paint}\)

\(k_\mathrm{final}\)

\(k_\mathrm{init}\)

In practice, discreteness reduction: oversampling, deconvolution, interlacing, kernel choice (NUFFT-like, see e.g. Stadler+2024)

3 PNG parameters, 2 options:

  • infer the 3 as independent
  • assume "universality" relations
    $$\begin{align*}b_\phi &=2\delta_c({\color{purple} b_1}+1-p)\\b_{\phi \delta} &=2 (\delta_c {\color{purple} b_2}- {\color{purple} b_1})\end{align*}$$(Lagrangian form)

A word on integral constraints

de Mattia+2019

Radial Integral Constraint
\(\delta_g \propto n_g - \braket{n_g}\approx n_g - \bar n_g(r)\)
i.e. impose \(\bar \delta_g(r) = 0\)

Global Integral Constraint
\(\delta_g \propto n_g - \braket{n_g} \approx n_g - \bar n_g\)
i.e. impose \(\bar \delta_g = 0\)

To be answered

  • 3 \(f_\mathrm{NL}\) "values" to infer:
    • \(f_\mathrm{NL}\) in init field, \(b_\phi f_\mathrm{NL}\) and \(b_{\phi\delta} f_\mathrm{NL}\) in galaxy bias
    • "universality" relations robust at field-level?
      \(b_\phi = 2 \delta_c (1 + b_1 - p)\) and \(b_{\phi \delta} = b_\phi - b_1 + \delta_c b_2\)
  • Redshift varying biases? templates?
    • \(b_1(z) = a_1 (1+z)^2 + c_1\)?    \(b_2(z)\), \(b_s^2(z)\),...?
    • Redshift bins? Interpolation?
  • Max resolution we can robustly + computationally push to
    • \(k_\mathrm{max} \leq 0.14\, h/\mathrm{Mpc}\)?

Where we are

Tally

  • Currently visiting Montréal in Laurence Perreault-Levasseur team
  • Talks:
    • Optimal cosmo information extraction at Sesto (for Euclid people)
    • DESI meeting at Berkeley (for colab)
    • CoBALt at Institut Pascal (for inflation theorists)
    • Bayesian Deep Learning 3 at APC (for deep learners)
    • ED Festival (for particle physicists)
    • GDR Cophy 2h tutorial
  • Papers:
    • stat paper at NeurIPS2024 (from master internship)
    • Benchmarking Field-Level in review on JCAP
    • PNG measurement at the field level in prep
  • Teaching: Bachelor 2 Biostats (20h) and Master 1 Maths (15) courses at UPsaclay
  • Formation:
    • VSS, Science ethics, Sustainable dev. (Open Science left)
    • Euclid summer school

Next steps

  • Scientific:
    • Validation for PNG inference and application to DESI
    • Alternative sampling method for field-level inference
    • Going Multi-GPU
  • Manuscript:
    • Detailed plan at the end of December, first chapter in January
  • Looking and applying for postdocs this Autumn,
    hence the meetings and visits...

Hugo SIMON
PhD student supervised by
Arnaud DE MATTIA and François LANUSSE

  • Field-level Bayesian inference
  • High-dimensional sampling
  • Differentiable N-body simulations
  • Primordial non-Gaussianity from DESI

Hugo SIMON
PhD student at CEA Paris-Saclay, supervised by
Arnaud DE MATTIA and François LANUSSE

  • Field-level Bayesian inference
  • High-dimensional sampling
  • Differentiable N-body simulations
  • Primordial non-Gaussianity field-level inference from DESI