AstroAI lunch talk | Harvard CfA | 22. September 2025

Probabilistic reconstruction of peculiar velocity fields in 3D Constança Tropa! Supervisors: Carol Cuesta-Lazaro, Daniel Eisenstein" AstroAI and ITC! Center for Astrophysics | Harvard & Smithsonian! Institute of Particle Physics and Astrophysics |#ETH Zürich My Project - Motivation GOAL: Study large scale structure Dark Matter distribution Galaxy distribution Galaxy distribution in redshift space Infer and ζ= (Ωm,σ8) b=δgal /δDM + peculiar velocities H0 No direct observation + bias Simultaneous constraint PRSD k Pk Remove peculiar velocities (vpec) Methods Remove peculiar velocities (vpec) Overview Simultaneous constraint Probabilistic! Simulations -More universes!" -Nonlinear evolution, analytic models fail" -Galaxies = biased tracers" -Quijote N-body (z = 0) -Sampling problem! High-dim: " -No analytical likelihood" -Generative models capture uncertainty 1283∼2×106 Machine Learning Infer and ζ= (Ωm,σ8) b=δgal /δDM PARAMETER + BIAS ! INFERENCE VELOCITY FIELD RECONSTRUCTION 1 2 p(ζ,b|δRSD g)p(vnl |δRSD g,vRSD lin ,ζ,b) p(vRSD nl ,ζ,b|δRSD g,vRSD lin ) 1 Parameter estimation p(ζ,b|tracer) ζ= (Ωm,σ8) Six tracers 1 Power spectrum: p(ζ,b|PRSD k) ζ= (Ωm,σ8) Six tracers Power spectrum ! in redshift space Power spectrum 1 Density fields: p(ζ,b|δRSD) Six tracers Power spectrum ! in redshift space Power spectrum Galaxy overdensity fields Matter overdensity fields Galaxy overdensity fields ! in redshift space ζ= (Ωm,σ8) Six tracers Power spectrum ! in redshift space Power spectrum Galaxy overdensity fields Matter overdensity fields Galaxy overdensity fields ! in redshift space Galaxy overdensity RSD +! mean velocity field RSD Adding velocities: p(ζ,b|δRSD,vRSD pec ) 1 ζ= (Ωm,σ8) Six tracers Power spectrum ! in redshift space Power spectrum Galaxy overdensity fields Matter overdensity fields Galaxy overdensity fields ! in redshift space 1 ζ= (Ωm,σ8) Parameter estimation p(ζ,b|tracer) Galaxy overdensity RSD +! mean velocity field RSD 1 Model architecture and loss ℒ(θ) = −1 N N ∑ i=1 log pθ(ζ(i)∣z(i)) Negative log-likelihood loss Linear Theory 2 Gaussian smoothing ( Mpc) R= 10 δ=−  ∇ ⋅ v aHf f=Ω0.55 m δgal b=δDM Assumes some cosmology Linear Theory 2 Velocity field reconstruction Gaussian smoothing ( Mpc) R= 10 Assumes some cosmology δ=−  ∇ ⋅ v aHf f=Ω0.55 m δgal b=δDM vnl vlin Flow Matching Model Linear velocity field Nonlinear velocity field CONDITION ON Galaxy overdensity field in redshift space Learn flow! modelled with ResUNet p(vnl |δRSD g,vlin) 2 dx dt =uθ(x(t), t)“flow” 1. Sample a pair, " 2. Compute the true instantaneous flow ! 3. Predict flow velocity with ResUNet " 4. Evaluate loss " 5. Back-propagate through the ResUNet’s parameters θ xt= (1 −t)x0+t x1 utrue =x1−x0 upred(xt,t) ℒ=𝔼[∥upred −utrue∥2] Learn flow! modelled with ResUNet Flow Matching Model 2 x0=vlin x1=vnl Flow Matching Model 2 x0=vlin 1. Sample a pair, " 2. Compute the true instantaneous flow ! 3. Predict flow velocity with ResUNet " 4. Evaluate loss " 5. Back-propagate through the ResUNet’s parameters θ xt= (1 −t)x0+t x1 utrue =x1−x0 upred(xt,t) ℒ=𝔼[∥upred −utrue∥2] Learn flow! modelled with ResUNet x1=vnl Learn flow! modelled with ResUNet 2 - interpolated physical velocity grid (+ density)" -Time embedding “trick”" 1283 xt uθ(x(t), t) Output: Learn flow! modelled with ResUNet 1. Sample a pair, " 2. Compute the true instantaneous velocity ! 3. Predict flow velocity with ResUNet " 4. Evaluate loss " 5. Back-propagate through the ResUNet’s parameters θ! 6. At inference: " 1. Sample " 2. Integrate Neural ODE ! ! xt= (1 −t)x0+t x1 utrue =x1−x0 uθ(x(t), t) ℒ=𝔼[∥uθ−utrue∥2] xt dx dt =uθ(x(t), t) x(1) = x(0) + ∫1 0 uθ(x(t), t)dt Flow Matching Model 2 x0=vlin x1=vnl 2 3D Velocity Field Reconstruction Linear ! velocity field Nonlinear velocity field Predicted velocity field 2 Linear velocity field Nonlinear velocity field Predicted velocity field Velocity Field Reconstruction Mean prediction Std of samples 2 Statistics for vx Power spectrum Cross correlation Putting everything together… θ1= {Ωm,1,σ8,1,b1} θ2= {Ωm,2,σ8,2,b2} θ3= {Ωm,3,σ8,3,b3} p(ζ,b|δRSD g) 1 δ=−  ∇ ⋅ v aHf f=Ω0.55 m δgal b=δDM Putting everything together… θ1= {Ωm,1,σ8,1,b1} θ2= {Ωm,2,σ8,2,b2} θ3= {Ωm,3,σ8,3,b3} p(ζ,b|δRSD g) p(vnl |δRSD g,vlin,ζ,b) 1 2 θ3= {Ωm,1,σ8,1,b1} Putting everything together… p(vnl,ζ,b|δRSD g,vlin) θ1= {Ωm,1,σ8,1,b1} θ2= {Ωm,2,σ8,2,b2} θ3= {Ωm,3,σ8,3,b3} p(ζ,b|δRSD g) p(vnl |δRSD g,vlin,ζ,b) 1 2 θ1= {Ωm,1,σ8,1,b1} θ2= {Ωm,2,σ8,2,b2} Putting everything together… p(vnl,ζ,b|δRSD g,vlin) θ1= {Ωm,1,σ8,1,b1} θ2= {Ωm,2,σ8,2,b2} θ3= {Ωm,3,σ8,3,b3} p(ζ,b|δRSD g) p(vnl |δRSD g,vlin,ζ,b) 1 2 θ= {Ωm,σ8,b} ± •Application target: DESI" •Need to make more realistic mock, add survey effects " •Populate halos galaxies using an HOD model" •We have a periodic box handle survey mask as to account for unobserved regions (e.g. bright stars / Milky Way obscuration)" •We assume we know all galaxies not true (e.g. fiber collisions) → → How can we use this? [1] C. F. Park, N. Mudur, C. Cuesta-Lazaro, Y. Ni, V. Ono, and D. P. Finkbeiner, “3D Reconstruction of Dark Matter Fields with Diffusion Models: Towards Application to Galaxy Surveys”. [2] V. Ono, C. F. Park, N. Mudur, Y. Ni, C. Cuesta-Lazaro, and F. Villaescusa-Navarro, “Debiasing with Diffusion: Probabilistic reconstruction of Dark Matter fields from galaxies with CAMELS,” Mar. 15, 2024, arXiv: arXiv:2403.10648. doi: 10.48550/arXiv.2403.10648. [3] Y. Lipman, R. T. Q. Chen, H. Ben-Hamu, M. Nickel, and M. Le, “Flow Matching for Generative Modeling,” Feb. 08, 2023, arXiv: arXiv:2210.02747. doi: 10.48550/arXiv.2210.02747. [4] C. Cuesta-Lazaro, A. E. Bayer, M. S. Albergo, S. Mishra-Sharma, C. Modi, and D. J. Eisenstein, “Joint cosmological parameter inference and initial condition reconstruction with Stochastic Interpolants”. [5] A. E. Bayer, C. Modi, and S. Ferraro, “Joint velocity and density reconstruction of the Universe with nonlinear differentiable forward modeling,” J. Cosmol. 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