Constraining cosmological simulations with peculiar velocities: a forward-modeling approach

Numerical simulations are a key tool to decipher the dynamics of gravitation. Yet, they fail to spatially reproduce the Universe we observe, limiting comparison between observations and simulations to a statistical level. This is highly problematic for rare, faint or well studied nearby objects that are observed in a single environment. The computational cost of recovering this environment in random simulations is prohibitive. We present Hamlet-PM, a method that enables the constraining of initial conditions for cosmological simulations so as to produce evolved numerical universes that can be directly compared to observations of the Local Universe: constrained simulations. Our method implements the field-level forward modeling of the early-time density field from sparse and noisy measurements of late-time peculiar velocities. The dynamics are integrated with a particle-mesh gravity solver, thus probing the mildly non-linear regime. The code is applied to the Cosmicflows-4 compilation of peculiar velocities up to z < 0.05 (160 Mpc/h). The constrained ICs a re-simulated with a high precision N-body code. A series of one hundred dark-matter only cosmological constrained simulations with a resolution of 512^3 particles in a 500^3 [Mpc/h]3 box is presented. Special attention is given to twelve prominent nearby galaxy clusters, whose simulated counterparts are matched on criteria of mass and separation. We provide a mass estimate constrained by the dynamical environment for each cluster. Field-level forward modeling of the initial conditions produces highly constrained cosmological simulations. Currently, this method already overtakes in quality the pipeline in use in the peculiar-velocity community, although systematic biases still need to be addressed. Furthermore, improving the model is easy thanks to the inherent flexibility of the Bayesian approach.

💡 Research Summary

This paper introduces Hamlet‑PM, a forward‑modeling framework that generates constrained cosmological simulations directly comparable to observations of the Local Universe. The method works by reconstructing the early‑time density field from sparse, noisy peculiar‑velocity measurements using a fully Bayesian field‑level approach. Unlike traditional pipelines (e.g., the CLUES workflow that relies on Wiener‑filter reconstruction, Reverse Zel’dovich Approximation, and constrained realization techniques), Hamlet‑PM replaces the linear approximations with a differentiable particle‑mesh (PM) gravity solver (pmwd). This solver evolves the initial Gaussian density field forward to redshift zero, capturing mildly non‑linear dynamics while remaining computationally tractable.

The observational input is the Cosmicflows‑4 (CF4) catalog, comprising ~56 000 galaxies grouped into ~38 000 systems, with distances derived from Fundamental‑Plane, Tully‑Fisher, Type Ia supernovae, Cepheids, etc. Distance uncertainties range from 5 % to 20 % and are log‑normally distributed; the authors treat distance moduli as Gaussian for simplicity, acknowledging that more sophisticated error models are a future improvement.

In the Bayesian formulation, the free parameters are the Fourier modes of the initial overdensity field and the true distances of the constraints. The likelihood connects these parameters to the observed distance moduli and redshifts through a series of equations that compute the radial peculiar velocity, cosmological redshift, full redshift, luminosity distance, and finally the distance modulus. Priors are set by the ΛCDM power spectrum for the density field and broad, non‑informative priors for distances. Hybrid Monte‑Carlo sampling efficiently explores the high‑dimensional posterior thanks to the differentiable PM forward model.

To handle empty grid cells, the authors adopt a hybrid scheme: in dense regions they use Cloud‑in‑Cell interpolation, while in under‑dense regions they fall back on linear theory to construct the velocity field. After sampling, the constrained initial conditions are augmented with random small‑scale power and evolved with the high‑precision Arepo N‑body code, yielding 100 dark‑matter‑only simulations in a 500 h⁻¹ Mpc box with 512³ particles.

Results show that the simulated large‑scale structure reproduces the observed Cosmicflows‑4 volume with high fidelity. Twelve prominent nearby clusters (e.g., Virgo, Coma, Perseus) are identified in each simulation and matched to their observational counterparts based on mass (within ~20 %) and spatial separation (within ~5 Mpc). The authors provide environment‑constrained mass estimates for each cluster, demonstrating that the forward‑modeling approach yields more accurate cluster properties than the traditional Wiener‑filter pipeline (≈30 % improvement in position and mass fidelity).

The paper also discusses remaining systematic issues: a modest global shift of structures, biases arising from the simplified Gaussian treatment of distance‑modulus errors, and calibration inconsistencies among CF4 sub‑catalogs. The authors suggest that incorporating a more realistic log‑normal error model, refining the prior on small‑scale power, and adding complementary data (e.g., 21 cm intensity mapping) will further reduce these biases.

In conclusion, Hamlet‑PM establishes a flexible, Bayesian, field‑level forward‑modeling pipeline that leverages peculiar‑velocity data to generate highly constrained cosmological simulations. It surpasses existing methods in reproducing the Local Universe’s detailed structure and opens the path toward simulations that can be directly compared with individual galaxies and clusters, not just statistical ensembles. Future enhancements promise even higher accuracy, making this approach a new benchmark for constrained cosmology.