Mapping LSS — Mike Hudson

How it works

Galaxies trace the underlying dark-matter density field with a (scale- and population-dependent) bias b. A whole-sky redshift survey therefore gives us a noisy, biased, redshift-space-distorted estimate of δ_m(x) at z ≈ 0. The job of an LSS reconstruction is to invert that: clean up the shot noise, undo the redshift-space distortions, and produce a best-estimate of the underlying matter density on a 3D grid — along with its uncertainty.

Modern reconstructions use Wiener filtering, iterative linear-theory schemes, or fully Bayesian forward modelling (such as Hamiltonian Monte Carlo over an initial-conditions prior). The reconstructed density field is what we then use to predict peculiar velocities, identify filaments and voids, and provide a template for cross-correlation analyses.

Because the reconstruction depends on the assumed bias and growth rate, it must be calibrated externally. We do that by comparing the predicted peculiar velocity field to direct distance-indicator measurements — see the Cosmic flows page.

Surveys and reconstructions

2M++

The 2M++ galaxy redshift catalogue (Lavaux & Hudson 2011) is an all-sky compilation of ~70,000 redshifts to K < 12.5, built by combining 2MASS photometry with redshifts from 2MRS, 6dFGRS, and SDSS. Its near-IR selection minimises Galactic extinction effects and gives uniform completeness over the celestial sphere — properties that make it the natural input for local density-field reconstructions. Carrick et al. 2015 and Boruah et al. 2020 used 2M++ as the input to the linear-theory reconstruction whose predicted velocity field was then compared against Tully–Fisher and SNe Ia distances respectively.

Aquila Consortium & Manticore

The Aquila Consortium develops Bayesian forward-modelling reconstructions of cosmological large-scale structure. I am involved in Manticore, a successor to the linear-theory 2M++ reconstruction used by Carrick et al. 2015 and Boruah et al. 2020. Manticore samples the posterior over the initial conditions via Hamiltonian Monte Carlo and evolves each sample forward through non-linear gravitational structure formation, giving a fully Bayesian posterior on the present-day density and velocity fields — richer in uncertainty quantification than the Wiener-filter approach that preceded it, and directly usable for joint analyses with peculiar-velocity samples.

Calibrating the method (Amber Hollinger)

The accuracy of fσ₈ from comparing observed peculiar velocities to density-field predictions depends sensitively on reconstruction choices and the selection function of the input catalogue. Amber Hollinger's PhD work systematically quantified these systematics.

In Hollinger & Hudson 2021 we used N-body simulations and semi-analytic galaxy models to test the optimal smoothing length, the choice of tracer (dark-matter halos vs galaxies), noise in halo-mass estimates and in the stellar-to-halo-mass relation, and finite-volume effects. With a 4 h⁻¹ Mpc Gaussian smoothing the method is unbiased at the ~5% level; cosmic variance contributes a further ~5% for a 2M++-sized volume.

In Hollinger & Hudson 2024 we extended the analysis to the specific selection effects of 2M++: flux-limited sampling, survey edges, and Galactic-plane obscuration. The reconstruction is unbiased near the origin but biased high by a factor 1.04 ± 0.01 in the 100–180 h⁻¹ Mpc range. Correcting for this and applying the method to recent peculiar-velocity samples gives fσ₈^lin = 0.362 ± 0.023 — the most recent low-redshift growth-rate measurement from my group, in tension with Planck CMB extrapolations and consistent with the broader late-time S₈ tension. The residual scatter in predicted 2M++ velocities is ~170 km s⁻¹ per galaxy, growing to ~200 km s⁻¹ beyond 180 h⁻¹ Mpc. The recovered bulk flows are consistent with ΛCDM expectations — not in tension with the standard model.

Looking ahead